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| author | pgww <pgww@lists.pglaf.org> | 2026-03-08 09:56:56 -0700 |
|---|---|---|
| committer | pgww <pgww@lists.pglaf.org> | 2026-03-08 09:56:56 -0700 |
| commit | e9297021e994efa64479a0891ef26e312ea15c2a (patch) | |
| tree | 42d7dd7e44602e0df7dc2c087a03d8337eb948b8 | |
| parent | 9c56a5299d8655c990a4102df4a8be46437db573 (diff) | |
| -rw-r--r-- | .gitattributes | 7 | ||||
| -rw-r--r-- | 78050-src/78050-src.htm | 44199 | ||||
| -rw-r--r-- | 78050-src/README-math.txt | 96 | ||||
| -rw-r--r-- | LICENSE.txt | 2 | ||||
| -rw-r--r-- | README.md | 4 |
5 files changed, 44302 insertions, 6 deletions
diff --git a/.gitattributes b/.gitattributes index 6833f05..d7b82bc 100644 --- a/.gitattributes +++ b/.gitattributes @@ -1,3 +1,4 @@ -* text=auto -*.txt text -*.md text +*.txt text eol=lf +*.htm text eol=lf +*.html text eol=lf +*.md text eol=lf diff --git a/78050-src/78050-src.htm b/78050-src/78050-src.htm new file mode 100644 index 0000000..8e28717 --- /dev/null +++ b/78050-src/78050-src.htm @@ -0,0 +1,44199 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + <meta charset="UTF-8"> + <title> + Principia Mathematica | Project Gutenberg + </title> + <link rel="icon" href="images/cover.jpg" type="image/x-cover"> + <style> + +body { + margin-left: 10%; + margin-right: 10%; +} + +/* General headers */ + +h1 { + text-align: center; + clear: both; +} + +/* General headers */ +h2, h3 { + text-align: center; + font-weight: bold; + margin-top: 1em; + margin-bottom: 1em; + } + +p { + margin-top: .51em; + text-align: justify; + margin-bottom: .49em; + text-indent: 1.5em; +} + +.nind {text-indent:0;} + +.nindc { + text-align: center; + text-indent: 0 + } + +.large { + font-size: 125% + } + +.space-above2 { margin-top: 2em; } +.space-below2 { margin-bottom: 2em; } + +.spa1 { + margin-top: 1em + } + +hr { + width: 33%; + margin-top: 2em; + margin-bottom: 2em; + margin-left: 33.5%; + margin-right: 33.5%; + clear: both; +} + +.antiqua {font-family: sans-serif;} + +hr.tb {width: 45%; margin-left: 27.5%; margin-right: 27.5%;} +hr.chap {width: 65%; margin-left: 17.5%; margin-right: 17.5%;} +@media print { hr.chap {display: none; visibility: hidden;} } + +div.chapter {page-break-before: always;} +h2.nobreak {page-break-before: avoid;} + +table { + margin-left: auto; + margin-right: auto; +} +table.autotable { border-collapse: collapse; } +table.autotable td { padding: 0.25em; } + +.tdl {text-align: left;} +.tdr {text-align: right;} +.tdc {text-align: center;} + +.tdlh { + text-align: left; + margin-left: 2em; + text-indent: 2em + } +.tdlh2 { + text-align: left; + margin-left: 4em; + text-indent: 4em + } + +.pagenum { /* uncomment the next line for invisible page numbers */ + /* visibility: hidden; */ + position: absolute; + left: 92%; + font-size: small; + text-align: right; + font-style: normal; + font-weight: normal; + font-variant: normal; + text-indent: 0; +} /* page numbers */ + +.right {text-align: right;} + +.allsmcap {font-variant: small-caps; text-transform: lowercase;} + +/* Dropcap */ + +.dropcap { + float: left; + font-size: 250%; + margin-top:-.7%; +} + +p.dropcap:first-letter +{ + color: transparent; + visibility: hidden; + margin-left: -0.9em; +} + +/* Images */ + +img {max-width: 100%; width: 100%; height: auto;} +.width500 {max-width: 500px;} +.x-ebookmaker .width500 {width: 100%;} + + +.figcenter { + margin: auto; + text-align: center; + page-break-inside: avoid; + max-width: 100%; +} + +/* Footnotes */ +.footnotes {border: 1px dashed;} + +.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;} + +.footnote .label {position: absolute; right: 84%; text-align: right;} + +.fnanchor { + vertical-align: super; + font-size: .8em; + text-decoration: + none; +} + +/* Transcriber's notes */ +.transnote {background-color: #E6E6FA; + color: black; + font-size:small; + padding:0.5em; + margin-bottom:5em; + font-family:sans-serif, serif; +} + +/* css needed in m2svg output: displayed equations and prevention of bad breaks*/ + .align-center { + display: block; + text-align: center; + text-indent: 0; + margin-top: 1em; + margin-bottom: 1em; + } + .nowrap { + white-space: nowrap; + } + + </style> +</head> +<body> + + +<figure class="figcenter width500" id="cover" style="width: 1744px;"> +<img src="images/cover.jpg" width="1744" height="2560" alt="Written +as a defense of logicism, this book was instrumental in developing and +popularizing modern mathematical logic."> +</figure> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_xi">[Pg xi]</span></p> +<p class="nindc"><span class="large"> +PRINCIPIA MATHEMATICA</span> +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="nindc space-above2 space-below2"> +CAMBRIDGE UNIVERSITY PRESS<br> +<span class="antiqua"><b>London</b></span>: FETTER LANE, E.C.<br> +C. F. CLAY, <span class="allsmcap">MANAGER</span><br> +</p> + +<figure class="figcenter width500" id="i_003" style="width: 183px;"> +<img src="images/i_003.jpg" width="183" height="150" alt="decorative"> +</figure> + + +<p class="nindc space-above2 space-below2"> +<span class="antiqua"><b>Edinburgh</b></span>: 100, PRINCES STREET<br> +<span class="antiqua"><b>Berlin</b></span>: A. ASHER AND CO.<br> +<span class="antiqua"><b>Leipzig</b></span>: F. A. BROCKHAUS<br> +<span class="antiqua"><b>New York</b></span>: G. P. PUTNAM'S SONS<br> +<span class="antiqua"><b>Bombay and Calcutta</b></span>: +MACMILLAN AND CO., <span class="allsmcap">LTD.</span></p> + +<p class="nindc space-above2 space-below2"> +<i>All rights reserved</i> +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h1>PRINCIPIA MATHEMATICA</h1> + +<p class="nindc space-above2 space-below2"> +<span class="allsmcap">BY</span></p> + +<p class="nindc"><span class="large"> +ALFRED NORTH WHITEHEAD, Sc.D., F.R.S.</span></p> + +<p class="nindc">Fellow and late Lecturer of Trinity College, Cambridge</p> + +<p class="nindc space-above2 space-below2"> +<span class="allsmcap">AND</span></p> + +<p class="nindc"><span class="large"> +BERTRAND RUSSELL, M.A., F.R.S.</span></p> + +<p class="nindc">Lecturer and late Fellow of Trinity College, Cambridge</p> + +<p class="nindc space-above2 space-below2"> +VOLUME I</p> + +<p class="nindc space-above2 space-below2"> +Cambridge<br> +at the University Press<br> +1910</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="nindc space-above2 space-below2"> +<span class="antiqua">Cambridge</span>:<br> +PRINTED BY JOHN CLAY, <span class="allsmcap">M.A.</span><br> +AT THE UNIVERSITY PRESS<br> +</p> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="CONTENTS_OF_VOLUME_I">CONTENTS OF VOLUME I</h2> +</div> + + +<table class="autotable"> +<tbody><tr> +<td class="tdl"> </td> +<td class="tdr"> <span class="allsmcap">PAGE</span></td> +</tr><tr> +<td class="tdl">PREFACE</td> +<td class="tdr"><a href="#Page_v">v</a></td> +</tr><tr> +<td class="tdl">INTRODUCTION</td> +<td class="tdr"><a href="#Page_1">1</a></td> +</tr><tr> +<td class="tdl"><span class="allsmcap">CHAPTER I. PRELIMINARY EXPLANATIONS OF IDEAS AND NOTATIONS</span></td> +<td class="tdr"><a href="#Page_4">4</a></td> +</tr><tr> +<td class="tdl"><span class="allsmcap">CHAPTER II. THE THEORY OF LOGICAL TYPES</span></td> +<td class="tdr"><a href="#Page_39">39</a></td> +</tr><tr> +<td class="tdl"><span class="allsmcap">CHAPTER III. INCOMPLETE SYMBOLS</span></td> +<td class="tdr"><a href="#Page_69">69</a></td> +</tr><tr> +<td class="tdl">PART I. MATHEMATICAL LOGIC.<br> +<span class="tdlh2">Summary of Part I</span></td> +<td class="tdr"><a href="#Page_91">91</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION A. THE THEORY OF DEDUCTION</span></td> +<td class="tdr"><a href="#Page_94">94</a></td> +</tr><tr> +<td class="tdlh">*1. Primitive Ideas and Propositions</td> +<td class="tdr"><a href="#Page_95">95</a></td> +</tr><tr> +<td class="tdlh">*2. Immediate Consequences of the Primitive Propositions</td> +<td class="tdr"><a href="#Page_102">102</a></td> +</tr><tr> +<td class="tdlh">*3. The Logical Product of two Propositions</td> +<td class="tdr"><a href="#Page_114">114</a></td> +</tr><tr> +<td class="tdlh">*4. Equivalence and Formal Rules</td> +<td class="tdr"><a href="#Page_120">120</a></td> +</tr><tr> +<td class="tdlh">*5. Miscellaneous Propositions</td> +<td class="tdr"><a href="#Page_128">128</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION B. THEORY OF APPARENT VARIABLES</span></td> +<td class="tdr"><a href="#Page_132">132</a></td> +</tr><tr> +<td class="tdlh">*9. Extension of the Theory of Deduction from Lower to Higher<br> +<span class="tdlh2">Types of Propositions</span></td> +<td class="tdr"><a href="#Page_132">132</a></td> +</tr><tr> +<td class="tdlh">*10. Theory of Propositions containing one Apparent Variable</td> +<td class="tdr"><a href="#Page_143">143</a></td> +</tr><tr> +<td class="tdlh">*11. Theory of two Apparent Variables</td> +<td class="tdr"><a href="#Page_157">157</a></td> +</tr><tr> +<td class="tdlh">*12. The Hierarchy of Types and the Axiom of Reducibility</td> +<td class="tdr"><a href="#Page_168">168</a></td> +</tr><tr> +<td class="tdlh">*13. Identity</td> +<td class="tdr"><a href="#Page_176">176</a></td> +</tr><tr> +<td class="tdlh">*14. Descriptions</td> +<td class="tdr"><a href="#Page_181">181</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION C. CLASSES AND RELATIONS</span></td> +<td class="tdr"><a href="#Page_196">196</a></td> +</tr><tr> +<td class="tdlh">*20. General Theory of Classes</td> +<td class="tdr"><a href="#Page_196">196</a></td> +</tr><tr> +<td class="tdlh">*21. General Theory of Relations</td> +<td class="tdr"><a href="#Page_211">211</a></td> +</tr><tr> +<td class="tdlh">*22. Calculus of Classes</td> +<td class="tdr"><a href="#Page_217">217</a></td> +</tr><tr> +<td class="tdlh">*23. Calculus of Relations</td> +<td class="tdr"><a href="#Page_226">226</a></td> +</tr><tr> +<td class="tdlh">*24. The Universal Class, the Null-Class, and the Existence of Classes</td> +<td class="tdr"><a href="#Page_229">229</a></td> +</tr><tr> +<td class="tdlh">*25. The Universal Relation, the Null Relation, and the Existence of<br> +<span class="tdlh2">Relations</span></td> +<td class="tdr"><a href="#Page_241">241</a><span class="pagenum" id="Page_xii">[Pg xii]</span></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION D. LOGIC OF RELATIONS</span></td> +<td class="tdr"><a href="#Page_244">244</a></td> +</tr><tr> +<td class="tdlh">*30. Descriptive Functions</td> +<td class="tdr"><a href="#Page_245">245</a></td> +</tr><tr> +<td class="tdlh">*31. Converses of Relations</td> +<td class="tdr"><a href="#Page_251">251</a></td> +</tr><tr> +<td class="tdlh">*32. Referents and Relata of a given Term with respect to a given<br> +<span class="tdlh2">Relation</span></td> +<td class="tdr"><a href="#Page_255">255</a></td> +</tr><tr> +<td class="tdlh">*33. Domains, Converse Domains, and Fields of Relations</td> +<td class="tdr"><a href="#Page_260">260</a></td> +</tr><tr> +<td class="tdlh">*34. The Relative Product of two Relations</td> +<td class="tdr"><a href="#Page_269">269</a></td> +</tr><tr> +<td class="tdlh">*35. Relations with Limited Domains and Converse Domains</td> +<td class="tdr"><a href="#Page_278">278</a></td> +</tr><tr> +<td class="tdlh">*36. Relations with Limited Fields</td> +<td class="tdr"><a href="#Page_291">291</a></td> +</tr><tr> +<td class="tdlh">*37. Plural Descriptive Functions</td> +<td class="tdr"><a href="#Page_293">293</a></td> +</tr><tr> +<td class="tdlh">*38. Relations and Classes derived from a Double Descriptive Function</td> +<td class="tdr"><a href="#Page_311">311</a></td> +</tr><tr> +<td class="tdl"> <br> +<span class="tdlh2">Note to Section D</span></td> +<td class="tdr"><a href="#Page_314">314</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION E. PRODUCTS AND SUMS OF CLASSES</span></td> +<td class="tdr"><a href="#Page_317">317</a></td> +</tr><tr> +<td class="tdlh">*40. Products and Sums of Classes of Classes</td> +<td class="tdr"><a href="#Page_319">319</a></td> +</tr><tr> +<td class="tdlh">*41. The Product and Sum of a Class of Relations</td> +<td class="tdr"><a href="#Page_331">331</a></td> +</tr><tr> +<td class="tdlh">*42. Miscellaneous Propositions</td> +<td class="tdr"><a href="#Page_336">336</a></td> +</tr><tr> +<td class="tdlh">*43. The Relations of a Relative Product to its Factors</td> +<td class="tdr"><a href="#Page_340">340</a></td> +</tr><tr> +<td class="tdl">PART II. PROLEGOMENA TO CARDINAL ARITHMETIC.<br> +<span class="tdlh2">Summary of Part II</span></td> +<td class="tdr"><a href="#Page_345">345</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION A. UNIT CLASSES AND COUPLES</span></td> +<td class="tdr"><a href="#Page_347">347</a></td> +</tr><tr> +<td class="tdlh">*50. Identity and Diversity as Relations</td> +<td class="tdr"><a href="#Page_349">349</a></td> +</tr><tr> +<td class="tdlh">*51. Unit Classes</td> +<td class="tdr"><a href="#Page_356">356</a></td> +</tr><tr> +<td class="tdlh">*52. The Cardinal Number 1</td> +<td class="tdr"><a href="#Page_363">363</a></td> +</tr><tr> +<td class="tdlh">*53. Miscellaneous Propositions involving Unit Classes</td> +<td class="tdr"><a href="#Page_368">368</a></td> +</tr><tr> +<td class="tdlh">*54. Cardinal Couples</td> +<td class="tdr"><a href="#Page_376">376</a></td> +</tr><tr> +<td class="tdlh">*55. Ordinal Couples</td> +<td class="tdr"><a href="#Page_383">383</a></td> +</tr><tr> +<td class="tdlh">*56. The Ordinal Number \(2_r\)</td> +<td class="tdr"><a href="#Page_395">395</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION B. SUB-CLASSES, SUB-RELATIONS, AND RELATIVE TYPES</span></td> +<td class="tdr"><a href="#Page_404">404</a></td> +</tr><tr> +<td class="tdlh">*60. The Sub-Classes of a given Class</td> +<td class="tdr"><a href="#Page_406">406</a></td> +</tr><tr> +<td class="tdlh">*61. The Sub-Relations of a given Relation</td> +<td class="tdr"><a href="#Page_412">412</a></td> +</tr><tr> +<td class="tdlh">*62. The Relation of Membership of a Class</td> +<td class="tdr"><a href="#Page_414">414</a></td> +</tr><tr> +<td class="tdlh">*63. Relative Types of Classes</td> +<td class="tdr"><a href="#Page_419">419</a></td> +</tr><tr> +<td class="tdlh">*64. Relative Types of Relations</td> +<td class="tdr"><a href="#Page_429">429</a></td> +</tr><tr> +<td class="tdlh">*65. On the Typical Definition of Ambiguous Symbols</td> +<td class="tdr"><a href="#Page_434">434</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION C. ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS</span></td> +<td class="tdr"><a href="#Page_437">437</a></td> +</tr><tr> +<td class="tdlh">*70. Relations whose Classes of Referents and of Relata belong to given<br> +<span class="tdlh2">Classes</span></td> +<td class="tdr"><a href="#Page_439">439</a></td> +</tr><tr> +<td class="tdlh">*71. One-Many, Many-One, and One-One Relations</td> +<td class="tdr"><a href="#Page_446">446</a></td> +</tr><tr> +<td class="tdlh">*72. Miscellaneous Propositions concerning One-Many, Many-One, and<br> +<span class="tdlh2">One-One Relations</span></td> +<td class="tdr"><a href="#Page_462">462</a></td> +</tr><tr> +<td class="tdlh">*73. Similarity of Classes</td> +<td class="tdr"><a href="#Page_476">476</a></td> +</tr><tr> +<td class="tdlh">*74. On One-Many and Many-One Relations with Limited Fields</td> +<td class="tdr"><a href="#Page_490">490</a><span class="pagenum" id="Page_xiii">[Pg xiii]</span></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION D. SELECTIONS</span></td> +<td class="tdr"><a href="#Page_500">500</a></td> +</tr><tr> +<td class="tdlh">*80. Elementary Properties of Selections</td> +<td class="tdr"><a href="#Page_505">505</a></td> +</tr><tr> +<td class="tdlh">*81. Selections from Many-One Relations</td> +<td class="tdr"><a href="#Page_519">519</a></td> +</tr><tr> +<td class="tdlh">*82. Selections from Relative Products</td> +<td class="tdr"><a href="#Page_524">524</a></td> +</tr><tr> +<td class="tdlh">*83. Selections from Classes of Classes</td> +<td class="tdr"><a href="#Page_531">531</a></td> +</tr><tr> +<td class="tdlh">*84. Classes of Mutually Exclusive Classes</td> +<td class="tdr"><a href="#Page_540">540</a></td> +</tr><tr> +<td class="tdlh">*85. Miscellaneous Propositions</td> +<td class="tdr"><a href="#Page_549">549</a></td> +</tr><tr> +<td class="tdlh">*88. Conditions for the Existence of Selections</td> +<td class="tdr"><a href="#Page_561">561</a></td> +</tr><tr> +<td class="tdlh"><span class="allsmcap">SECTION E. INDUCTIVE RELATIONS</span></td> +<td class="tdr"><a href="#Page_569">569</a></td> +</tr><tr> +<td class="tdlh">*90. On the Ancestral Relation</td> +<td class="tdr"><a href="#Page_576">576</a></td> +</tr><tr> +<td class="tdlh">*91. On Powers of a Relation</td> +<td class="tdr"><a href="#Page_585">585</a></td> +</tr><tr> +<td class="tdlh">*92. Powers of One-Many and Many-One Relations</td> +<td class="tdr"><a href="#Page_601">601</a></td> +</tr><tr> +<td class="tdlh">*93. Inductive Analysis of the Field of a Relation</td> +<td class="tdr"><a href="#Page_607">607</a></td> +</tr><tr> +<td class="tdlh">*94. On Powers of Relative Products</td> +<td class="tdr"><a href="#Page_617">617</a></td> +</tr><tr> +<td class="tdlh">*95. On the Equi-factor Relation</td> +<td class="tdr"><a href="#Page_626">626</a></td> +</tr><tr> +<td class="tdlh">*96. On the Posterity of a Term</td> +<td class="tdr"><a href="#Page_637">637</a></td> +</tr><tr> +<td class="tdlh">*97. Analysis of the Field of a Relation into Families</td> +<td class="tdr"><a href="#Page_654">654</a></td> +</tr> +</tbody> +</table> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p class="nindc"> +ALPHABETICAL LIST OF PROPOSITIONS +REFERRED TO BY NAMES.</p> + +<table class="autotable"> +<tbody><tr> +<td class="tdl">Name </td> +<td class="tdc">Number </td> +<td class="tdr"> </td> +</tr><tr> +<td class="tdl">Abs</td> +<td class="tdl"><b>*2·01.</b></td> +<td class="tdl">\(\vdash : p \supset {\sim}p . \supset .{\sim}p\)</td> +</tr><tr> +<td class="tdl">Add</td> +<td class="tdl"><b>*1·3.</b></td> +<td class="tdl">\(\vdash : q. \supset .p \lor q\)</td> +</tr><tr> +<td class="tdl">Ass</td> +<td class="tdl"><b>*3·35.</b></td> +<td class="tdl">\(\vdash : p .p \supset q. \supset .q\)</td> +</tr><tr> +<td class="tdl">Assoc</td> +<td class="tdl"><b>*1·5.</b></td> +<td class="tdl">\(\vdash : p \lor (q \lor r). \supset .q \lor (p \lor r)\)</td> +</tr><tr> +<td class="tdl">Comm</td> +<td class="tdl"><b>*2·04.</b></td> +<td class="tdl">\(\vdash\colon\ldotp p. \supset .q \supset r \colon \supset \colon q. \supset .p \supset r\)</td> +</tr><tr> +<td class="tdl">Comp</td> +<td class="tdl"><b>*3·43.</b></td> +<td class="tdl">\(\vdash\colon\ldotp p \supset q. p \supset r. \supset \colon p. \supset .q. r\)</td> +</tr><tr> +<td class="tdl">Exp</td> +<td class="tdl"><b>*3·3.</b></td> +<td class="tdl">\(\vdash\colon\ldotp p. q. \supset . r: \supset \colon p. \supset . q \supset r\)</td> +</tr><tr> +<td class="tdl">Fact</td> +<td class="tdl"><b>*3·45.</b></td> +<td class="tdl">\(\vdash\colon\ldotp p \supset q. \supset \colon p. r. \supset . q. r\)</td> +</tr><tr> +<td class="tdl">Id</td> +<td class="tdl"><b>*2·08.</b></td> +<td class="tdl">\(\vdash. p \supset p\)</td> +</tr><tr> +<td class="tdl">Imp</td> +<td class="tdl"><b>*3·31.</b></td> +<td class="tdl">\(\vdash\colon\ldotp p. \supset . q \supset r \colon \supset \colon p. q. \supset . r\)</td> +</tr><tr> +<td class="tdl">Perm</td> +<td class="tdl"><b>*1·4.</b></td> +<td class="tdl">\(\vdash : p \lor q. \supset . q \lor p\)</td> +</tr><tr> +<td class="tdl">Simp</td> +<td class="tdl"><b>*2·02.</b></td> +<td class="tdl">\(\vdash : q. \supset . p \supset q\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*3·26.</b></td> +<td class="tdl">\(\vdash : p. q. \supset . p\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*3·27.</b></td> +<td class="tdl">\(\vdash : p. q. \supset . q\)</td> +</tr><tr> +<td class="tdl">Sum</td> +<td class="tdl"><b>*1·6.</b></td> +<td class="tdl">\(\vdash\colon\ldotp q \supset r. \supset \colon p \lor q. \supset . p \lor r\)</td> +</tr><tr> +<td class="tdl">Syll</td> +<td class="tdl"><b>*2·05.</b></td> +<td class="tdl">\(\vdash\colon\ldotp q \supset r. \supset \colon p \supset q. \supset . p \supset r\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left:1.0em;">" </span></td> +<td class="tdl"><b>*2·06.</b></td> +<td class="tdl">\(\vdash\colon\ldotp p \supset q. \supset \colon q \supset r. \supset . p \supset r\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*3·33.</b></td> +<td class="tdl">\(\vdash : p \supset q. q \supset r. \supset . p \supset r\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*3·34.</b></td> +<td class="tdl">\(\vdash : q \supset r. p \supset q. \supset . p \supset r\)</td> +</tr><tr> +<td class="tdl">Taut</td> +<td class="tdl"><b>*1·2.</b></td> +<td class="tdl">\(\vdash : p \lor p. \supset . p\)</td> +</tr><tr> +<td class="tdl">Transp</td> +<td class="tdl"><b>*2·03.</b></td> +<td class="tdl">\(\vdash : p \supset {\sim}q. \supset . q \supset {\sim}p\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*2·15.</b></td> +<td class="tdl">\(\vdash : {\sim}p \supset q. \supset . {\sim}q \supset p\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*2·16.</b></td> +<td class="tdl">\(\vdash : p \supset q. \supset . {\sim}q \supset {\sim} p\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*2·17.</b></td> +<td class="tdl">\(\vdash : {\sim}q \supset{\sim}p. \supset . p \supset q\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*3·37.</b></td> +<td class="tdl">\(\vdash\colon\ldotp p. q. \supset . r \colon \supset \colon p. {\sim}r. \supset . {\sim}q\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*4·1.</b></td> +<td class="tdl">\(\vdash : p \supset q. \equiv . {\sim}q \supset {\sim}p\)</td> +</tr><tr> +<td class="tdl"><span style="margin-left: 1.0em;">" </span></td> +<td class="tdl"><b>*4·11.</b></td> +<td class="tdl">\(\vdash : p \equiv q. \equiv . {\sim}p \equiv {\sim}q\)</td> +</tr> +</tbody> +</table> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="ERRATA">ERRATA.</h2> +</div> + + +<p class="nind"> +p. 14, line 2, <i>for</i> "states" <i>read</i> "allows us to infer."<br> +p. 14, line 7, <i>after</i> "*3·03" <i>insert</i> "*1·7, *1·71, and *1·72."<br> +p. 15, last line but one, <i>for</i> "function of \(\phi\hat{x}\)" <i>read</i> "function \(\phi\hat{x}\)."<br> +p. 34, line 15, <i>for</i> "\(x\)" <i>read</i> "\(R\)."<br> +p. 68, line 20, <i>for</i> "classes" <i>read</i> "classes of classes."<br> +p. 86, line 2, <i>after</i> "must" <i>insert</i> "neither be nor."<br> +p. 91, line 8, <i>delete</i> "and in *3·03."<br> +p. 103, line 7, <i>for</i> "assumption" <i>read</i> "assertion."<br> +p. 103, line 25, at end of line, <i>for</i> "\(q\)" <i>read</i> "\(r\)."<br> +p. 218, last line but one, <i>for</i> "\(\Lambda\)" <i>read</i> "\(\dot{\Lambda}\)" [owing to brittleness of the<br> +<span style="margin-left: 3.0em;">type, the same error is liable to occur elsewhere].</span><br> +p. 382, last line but one, <i>delete</i> "in the theory of selections (*83·92) and."<br> +p. 487, line 13, <i>for</i> "*95" <i>read</i> "*94."<br> +p. 503, line 14, <i>for</i> "*88·38" <i>read</i> "*88·36."<br> +</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_v">[Pg v]</span></p> +<h2 class="nobreak" id="PREFACE">PREFACE</h2> +</div> + + +<p class="nind"> +<span class="dropcap">T</span>HE mathematical treatment of the principles of mathematics, which is +the subject of the present work, has arisen from the conjunction of +two different studies, both in the main very modern. On the one hand +we have the work of analysts and geometers, in the way of formulating +and systematising their axioms, and the work of Cantor and others +on such matters as the theory of aggregates. On the other hand we +have symbolic logic, which, after a necessary period of growth, +has now, thanks to Peano and his followers, acquired the technical +adaptability and the logical comprehensiveness that are essential to a +mathematical instrument for dealing with what have hitherto been the +beginnings of mathematics. From the combination of these two studies +two results emerge, namely (1) that what were formerly taken, tacitly +or explicitly, as axioms, are either unnecessary or demonstrable; (2) +that the same methods by which supposed axioms are demonstrated will +give valuable results in regions, such as infinite number, which had +formerly been regarded as inaccessible to human knowledge. Hence the +scope of mathematics is enlarged both by the addition of new subjects +and by a backward extension into provinces hitherto abandoned to +philosophy.</p> + +<p>The present work was originally intended by us to be comprised in a +second volume of <i>The Principles of Mathematics</i>. With that object +in view, the writing of it was begun in 1900. But as we advanced, it +became increasingly evident that the subject is a very much larger one +than we had supposed; moreover on many fundamental questions which +had been left obscure and doubtful in the former work, we have now +arrived at what we believe to be satisfactory solutions. It therefore +became necessary to make our book independent of <i>The Principles +of Mathematics</i>. We have, however, avoided both controversy and +general philosophy, and made our statements dogmatic in form. The +justification for this is that the chief reason in favour of any theory +on the principles of mathematics must always be inductive, <i>i.e.</i> +it must lie in the fact that the theory in question enables us to +deduce ordinary mathematics. In mathematics, the greatest degree of +self-evidence is usually not to be found quite at the beginning, but +at some later point; hence the early deductions, until they reach this +point, give reasons rather<span class="pagenum" id="Page_vi">[Pg vi]</span> for believing the premisses because true +consequences follow from them, than for believing the consequences +because they follow from the premisses.</p> + +<p>In constructing a deductive system such as that contained in +the present work, there are two opposite tasks which have to be +concurrently performed. On the one hand, we have to analyse existing +mathematics, with a view to discovering what premisses are employed, +whether these premisses are mutually consistent, and whether they are +capable of reduction to more fundamental premisses. On the other hand, +when we have decided upon our premisses, we have to build up again as +much as may seem necessary of the data previously analysed, and as +many other consequences of our premisses as are of sufficient general +interest to deserve statement. The preliminary labour of analysis does +not appear in the final presentation, which merely sets forth the +outcome of the analysis in certain undefined ideas and undemonstrated +propositions. It is not claimed that the analysis could not have been +carried farther: we have no reason to suppose that it is impossible to +find simpler ideas and axioms by means of which those with which we +start could be defined and demonstrated. All that is affirmed is that +the ideas and axioms with which we start are sufficient, not that they +are necessary.</p> + +<p>In making deductions from our premisses, we have considered it +essential to carry them up to the point where we have proved as much +as is true in whatever would ordinarily be taken for granted. But we +have not thought it desirable to limit ourselves too strictly to this +task. It is customary to consider only particular cases, even when, +with our apparatus, it is just as easy to deal with the general case. +For example, cardinal arithmetic is usually conceived in connection +with <i>finite</i> numbers, but its general laws hold equally for +infinite numbers, and are most easily proved without any mention of the +distinction between finite and infinite. Again, many of the properties +commonly associated with series hold of arrangements which are not +strictly serial, but have only some of the distinguishing properties of +serial arrangements. In such cases, it is a defect in logical style to +prove for a particular class of arrangements what might just as well +have been proved more generally. An analogous process of generalization +is involved, to a greater or less degree, in all our work. We have +sought always the most general reasonably simple hypothesis from which +any given conclusion could be reached. For this reason, especially in +the later parts of the book, the importance of a proposition usually +lies in its hypothesis. The conclusion will often be something which, +in a certain class of cases, is familiar, but the hypothesis will, +whenever possible, be wide enough to admit many cases besides those in +which the conclusion is familiar.</p> + +<p>We have found it necessary to give very full proofs, because otherwise +it is scarcely possible to see what hypotheses are really required, +or whether<span class="pagenum" id="Page_vii">[Pg vii]</span> our results follow from our explicit premisses. (It must +be remembered that we are not affirming merely that such and such +propositions are true, but also that the axioms stated by us are +sufficient to prove them.) At the same time, though full proofs are +necessary for the avoidance of errors, and for convincing those who may +feel doubtful as to our correctness, yet the proofs of propositions +may usually be omitted by a reader who is not specially interested +in that part of the subject concerned, and who feels no doubt of our +substantial accuracy on the matter in hand. The reader who is specially +interested in some particular portion of the book will probably find +it sufficient, as regards earlier portions, to read the summaries of +previous parts, sections, and numbers, since these give explanations of +the ideas involved and statements of the principal propositions proved. +The proofs in Part I, Section A, however, are necessary, since in the +course of them the manner of stating proofs is explained. The proofs of +the earliest propositions are given without the omission of any step, +but as the work proceeds the proofs are gradually compressed, retaining +however sufficient detail to enable the reader by the help of the +references to reconstruct proofs in which no step is omitted.</p> + +<p>The order adopted is to some extent optional. For example, we have +treated cardinal arithmetic and relation-arithmetic before series, but +we might have treated series first. To a great extent, however, the +order is determined by logical necessities.</p> + +<p>A very large part of the labour involved in writing the present work +has been expended on the contradictions and paradoxes which have +infected logic and the theory of aggregates. We have examined a great +number of hypotheses for dealing with these contradictions; many such +hypotheses have been advanced by others, and about as many have been +invented by ourselves. Sometimes it has cost us several months' work +to convince ourselves that a hypothesis was untenable. In the course +of such a prolonged study, we have been led, as was to be expected, to +modify our views from time to time; but it gradually became evident +to us that some form of the doctrine of types must be adopted if the +contradictions were to be avoided. The particular form of the doctrine +of types advocated in the present work is not logically indispensable, +and there are various other forms equally compatible with the truth of +our deductions. We have particularized, both because the form of the +doctrine which we advocate appears to us the most probable, and because +it was necessary to give at least one perfectly definite theory which +avoids the contradictions. But hardly anything in our book would be +changed by the adoption of a different form of the doctrine of types. +In fact, we may go farther, and say that, supposing some other way of +avoiding the contradictions to exist, not very much of our book, except +what explicitly deals with types, is dependent upon the adoption of +the doctrine of types in any form, so soon as it has been shown (as we +claim<span class="pagenum" id="Page_viii">[Pg viii]</span> that we have shown) that it is <i>possible</i> to construct a +mathematical logic which does not lead to contradictions. It should be +observed that the whole effect of the doctrine of types is negative: it +forbids certain inferences which would otherwise be valid, but does not +permit any which would otherwise be invalid. Hence we may reasonably +expect that the inferences which the doctrine of types permits would +remain valid even if the doctrine should be found to be invalid.</p> + +<p>Our logical system is wholly contained in the numbered propositions, +which are independent of the Introduction and the Summaries. The +Introduction and the Summaries are wholly explanatory, and form no part +of the chain of deductions. The explanation of the hierarchy of types +in the Introduction differs slightly from that given in <a href="#*12">*12</a> of the body +of the work. The later explanation is stricter and is that which is +assumed throughout the rest of the book.</p> + +<p>The symbolic form of the work has been forced upon us by necessity: +without its help we should have been unable to perform the requisite +reasoning. It has been developed as the result of actual practice, and +is not an excrescence introduced for the mere purpose of exposition. +The general method which guides our handling of logical symbols is due +to Peano. His great merit consists not so much in his definite logical +discoveries nor in the details of his notations (excellent as both +are), as in the fact that he first showed how symbolic logic was to +be freed from its undue obsession with the forms of ordinary algebra, +and thereby made it a suitable instrument for research. Guided by our +study of his methods, we have used great freedom in constructing, or +reconstructing, a symbolism which shall be adequate to deal with all +parts of the subject. No symbol has been introduced except on the +ground of its practical utility for the immediate purposes of our +reasoning.</p> + +<p>A certain number of forward references will be found in the notes and +explanations. Although we have taken every reasonable precaution to +secure the accuracy of these forward references, we cannot of course +guarantee their accuracy with the same confidence as is possible in the +case of backward references.</p> + +<p>Detailed acknowledgments of obligations to previous writers have not +very often been possible, as we have had to transform whatever we +have borrowed, in order to adapt it to our system and our notation. +Our chief obligations will be obvious to every reader who is familiar +with the literature of the subject. In the matter of notation, we +have as far as possible followed Peano, supplementing his notation, +when necessary, by that of Frege or by that of Schröder. A great deal +of the symbolism, however, has had to be new, not so much through +dissatisfaction with the symbolism of others, as through the fact +that we deal with ideas not previously symbolised. In all<span class="pagenum" id="Page_ix">[Pg ix]</span> questions +of logical analysis, our chief debt is to Frege. Where we differ from +him, it is largely because the contradictions showed that he, in common +with all other logicians ancient and modern, had allowed some error to +creep into his premisses; but apart from the contradictions, it would +have been almost impossible to detect this error. In Arithmetic and the +theory of series, our whole work is based on that of Georg Cantor. In +Geometry we have had continually before us the writings of v. Staudt, +Pasch, Peano, Pieri, and Veblen.</p> + +<p>We have derived assistance at various stages from the criticisms of +friends, notably Mr G. G. Berry of the Bodleian Library and Mr R. G. +Hawtrey.</p> + +<p>We have to thank the Council of the Royal Society for a grant towards +the expenses of printing of £200 from the Government Publication +Fund, and also the Syndics of the University Press who have liberally +undertaken the greater portion of the expense incurred in the +production of the work. The technical excellence, in all departments, +of the University Press, and the zeal and courtesy of its officials, +have materially lightened the task of proof-correction.</p> + +<p>The second volume is already in the press, and both it and the third +will appear as soon as the printing can be completed.</p> + +<p class="right"> +A. N. W.<br> +B. R.<br> +</p> + +<p class="nind"> +<span class="allsmcap">CAMBRIDGE</span>,<br> +<span style="margin-left: 1em;"><i>November</i>, 1910.</span><br> +</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_1">[Pg 1]</span></p> + +<h2 class="nobreak" id="INTRODUCTION">INTRODUCTION.</h2> +</div> + + +<p class="nind"> +THE mathematical logic which occupies Part I of the present work has +been constructed under the guidance of three different purposes. In the +first place, it aims at effecting the greatest possible analysis of the +ideas with which it deals and of the processes by which it conducts +demonstrations, and at diminishing to the utmost the number of the +undefined ideas and undemonstrated propositions (called respectively +<i>primitive</i> ideas and <i>primitive</i> propositions) from which it +starts. In the second place, it is framed with a view to the perfectly +precise expression, in its symbols, of mathematical propositions: to +secure such expression, and to secure it in the simplest and most +convenient notation possible, is the chief motive in the choice of +topics. In the third place, the system is specially framed to solve the +paradoxes which, in recent years, have troubled students of symbolic +logic and the theory of aggregates; it is believed that the theory of +types, as set forth in what follows, leads both to the avoidance of +contradictions, and to the detection of the precise fallacy which has +given rise to them.</p> + +<p>Of the above three purposes, the first and third often compel us to +adopt methods, definitions, and notations which are more complicated +or more difficult than they would be if we had the second object +alone in view. This applies especially to the theory of descriptive +expressions (<a href="#*14">*14</a> and <a href="#*30">*30</a>) and to the theory of classes and relations +(<a href="#*20">*20</a> and <a href="#*21">*21</a>). On these two points, and to a lesser degree on others, +it has been found necessary to make some sacrifice of lucidity to +correctness. The sacrifice is, however, in the main only temporary: in +each case, the notation ultimately adopted, though its real meaning is +very complicated, has an apparently simple meaning which, except at +certain crucial points, can without danger be substituted in thought +for the real meaning. It is therefore convenient, in a preliminary +explanation of the notation, to treat these apparently simple meanings +as primitive ideas, <i>i.e.</i> as ideas introduced without definition. +When the notation has grown more or less familiar, it is easier to +follow the more complicated explanations which we believe to be more +correct. In the body of the work, where it is necessary to adhere +rigidly to the strict logical order<span class="pagenum" id="Page_2">[Pg 2]</span> the easier order of development +could not be adopted; it is therefore given in the Introduction. The +explanations given in <a href="#CHAPTER_I">Chapter I</a> of the Introduction are such as place +lucidity before correctness; the full explanations are partly supplied +in succeeding Chapters of the Introduction, partly given in the body of +the work.</p> + +<p>The use of a symbolism, other than that of words, in all parts of the +book which aim at embodying strictly accurate demonstrative reasoning, +has been forced on us by the consistent pursuit of the above three +purposes. The reasons for this extension of symbolism beyond the +familiar regions of number and allied ideas are many:</p> + +<p>(1) The ideas here employed are more abstract than those familiarly +considered in language. Accordingly there are no words which are +used mainly in the exact consistent senses which are required here. +Any use of words would require unnatural limitations to their +ordinary meanings, which would be in fact more difficult to remember +consistently than are the definitions of entirely new symbols.</p> + +<p>(2) The grammatical structure of language is adapted to a wide variety +of usages. Thus it possesses no unique simplicity in representing the +few simple, though highly abstract, processes and ideas arising in the +deductive trains of reasoning employed here. In fact the very abstract +simplicity of the ideas of this work defeats language. Language can +represent complex ideas more easily. The proposition "a whale is +big" represents language at its best, giving terse expression to a +complicated fact; while the true analysis of "one is a number" leads, +in language, to an intolerable prolixity. Accordingly terseness is +gained by using a symbolism especially designed to represent the ideas +and processes of deduction which occur in this work.</p> + +<p>(3) The adaptation of the rules of the symbolism to the processes +of deduction aids the intuition in regions too abstract for the +imagination readily to present to the mind the true relation between +the ideas employed. For various collocations of symbols become familiar +as representing important collocations of ideas; and in turn the +possible relations—according to the rules of the symbolism—between +these collocations of symbols become familiar, and these further +collocations represent still more complicated relations between the +abstract ideas. And thus the mind is finally led to construct trains +of reasoning in regions of thought in which the imagination would be +entirely unable to sustain itself without symbolic help. Ordinary +language yields no such help. Its grammatical structure does not +represent uniquely the relations between the ideas involved. Thus, "a +whale is big" and "one is a number" both look alike, so that the eye +gives no help to the imagination.</p> + +<p><span class="pagenum" id="Page_3">[Pg 3]</span></p> + +<p>(4) The terseness of the symbolism enables a whole proposition to be +represented to the eyesight as one whole, or at most in two or three +parts divided where the natural breaks, represented in the symbolism, +occur. This is a humble property, but is in fact very important in +connection with the advantages enumerated under the heading (3).</p> + +<p>(5) The attainment of the first-mentioned object of this work, namely +the complete enumeration of all the ideas and steps in reasoning +employed in mathematics, necessitates both terseness and the +presentation of each proposition with the maximum of formality in a +form as characteristic of itself as possible.</p> + +<p>Further light on the methods and symbolism of this book is thrown by a +slight consideration of the limits to their useful employment:</p> + +<p>(\(\alpha)\) Most mathematical investigation is concerned not with +the analysis of the complete process of reasoning, but with the +presentation of such an abstract of the proof as is sufficient +to convince a properly instructed mind. For such investigations +the detailed presentation of the steps in reasoning is of course +unnecessary, provided that the detail is carried far enough to guard +against error. In this connection it may be remembered that the +investigations of Weierstrass and others of the same school have shown +that, even in the common topics of mathematical thought, much more +detail is necessary than previous generations of mathematicians had +anticipated.</p> + +<p>(\(\beta)\) In proportion as the imagination works easily in any region +of thought, symbolism (except for the express purpose of analysis) +becomes only necessary as a convenient shorthand writing to register +results obtained without its help. It is a subsidiary object of this +work to show that, with the aid of symbolism, deductive reasoning +can be extended to regions of thought not usually supposed amenable +to mathematical treatment. And until the ideas of such branches of +knowledge have become more familiar, the detailed type of reasoning, +which is also required for the analysis of the steps, is appropriate to +the investigation of the general truths concerning these subjects.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_4">[Pg 4]</span></p> +<h2 class="nobreak" id="CHAPTER_I">CHAPTER I.<br> +PRELIMINARY EXPLANATIONS OF IDEAS AND NOTATIONS.</h2> +</div> + + +<p>THE notation adopted in the present work is based upon that of Peano, +and the following explanations are to some extent modelled on those +which he prefixes to his <i>Formulario Mathematico</i>. His use of dots +as brackets is adopted, and so are many of his symbols.</p> + +<p><i>Variables.</i> The idea of a variable, as it occurs in the present +work, is more general than that which is explicitly used in ordinary +mathematics. In ordinary mathematics, a variable generally stands for +an undetermined number or quantity. In mathematical logic, any symbol +whose meaning is not determinate is called a <i>variable</i>, and +the various determinations of which its meaning is susceptible are +called the <i>values</i> of the variable. The values may be any set of +entities, propositions, functions, classes or relations, according to +circumstances. If a statement is made about "Mr A and Mr B," "Mr A" and +"Mr B" are variables whose values are confined to men. A variable may +either have a conventionally-assigned range of values, or may (in the +absence of any indication of the range of values) have as the range +of its values all determinations which render the statement in which +it occurs significant. Thus when a text-book of logic asserts that +"\(A\) is \(A\)," without any indication as to what \(A\) may be, what +is meant is that <i>any</i> statement of the form "\(A\) is \(A\)" is +true. We may call a variable <i>restricted</i> when its values are +confined to some only of those of which it is capable; otherwise, we +shall call it <i>unrestricted</i>. Thus when an unrestricted variable +occurs, it represents any object such that the statement concerned +can be made significantly (<i>i.e.</i> either truly or falsely) +concerning that object. For the purposes of logic, the unrestricted +variable is more convenient than the restricted variable, and we shall +always employ it. We shall find that the unrestricted variable is +still subject to limitations imposed by the manner of its occurrence, +<i>i.e.</i> things which can be said significantly concerning a +proposition cannot be said significantly concerning a class or a +relation, and so on. But the limitations to which the unrestricted +variable is subject do not need to be explicitly indicated, since they +are the limits of significance of the statement in which the variable +occurs, and are therefore intrinsically determined by this statement. +This will be more fully explained later<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a>.<span class="pagenum" id="Page_5">[Pg 5]</span> To sum up, the three +salient facts connected with the use of the variable are: (1) that a +variable is ambiguous in its denotation and accordingly undefined: +(2) that a variable preserves a recognizable identity in various +occurrences throughout the same context, so that many variables can +occur together in the same context each with its separate identity: and +(3) that either the range of possible determinations of two variables +may be the same, so that a possible determination of one variable +is also a possible determination of the other, or the ranges of two +variables may be different, so that, if a possible determination of +one variable is given to the other, the resulting complete phrase is +meaningless instead of becoming a complete unambiguous proposition +(true or false) as would be the case if all variables in it had been +given any <i>suitable</i> determinations.</p> + +<p><i>The uses of various letters.</i> Variables will be denoted by single +letters, and so will certain constants; but a letter which has once +been assigned to a constant by a definition must not afterwards be used +to denote a variable. The small letters of the ordinary alphabet will +all be used for variables, except \(p\) and \(s\) after <a href="#*40">*40</a>, in which +constant meanings are assigned to these two letters. The following +capital letters will receive constant meanings: \(B\), \(C\), \(D\), +\(E\), \(F\), \(I\) and \(J\). Among small Greek letters, we shall +give constant meanings to \(\epsilon\), \(\iota\) and (at a later +stage) to \(\eta\), \(\theta\) and \(\omega\). Certain Greek capitals +will from time to time be introduced for constants, but Greek capitals +will not be used for variables. Of the remaining letters, \(p\), +\(q\), \(r\) will be called <i>propositional letters</i>, and will +stand for variable propositions (except that, from <a href="#*40">*40</a> onwards, \(p\) +must not be used for a variable); \(f\), \(g\), \(\phi\), \(\psi\), +\(\chi\), \(\theta\) and (until <a href="#*33">*33</a>) \(F\) will be called <i>functional +letters</i>, and will be used for variable functions.</p> + +<p>The small Greek letters not already mentioned will be used for +variables whose values are classes, and will be referred to simply as +<i>Greek letters</i>. Ordinary capital letters not already mentioned +will be used for variables whose values are relations, and will be +referred to simply as <i>capital letters</i>. Ordinary small letters +other than \(p\), \(q\), \(r\), \(s\), \(f\), \(g\) will be used for +variables whose values are not known to be functions, classes, or +relations; these letters will be referred to simply as <i>small Latin +letters</i>.</p> + +<p>After the early part of the work, variable propositions and variable +functions will hardly ever occur. We shall then have three main kinds +of variables: variable classes, denoted by small Greek letters; +variable relations, denoted by capitals; and variables not given as +necessarily classes or relations, which will be denoted by small Latin +letters.</p> + +<p>In addition to this usage of small Greek letters for variable classes, +capital letters for variable relations, small Latin letters for +variables of type wholly undetermined by the context (these arise from +the possibility of<span class="pagenum" id="Page_6">[Pg 6]</span> "systematic ambiguity," explained later in the +explanations of the theory of types), the reader need only remember +that all letters represent variables, unless they have been defined as +constants in some previous place in the book. In general the structure +of the context determines the scope of the variables contained in it; +but the special indication of the nature of the variables employed, as +here proposed, saves considerable labour of thought.</p> + +<p><i>The fundamental functions of propositions.</i> An aggregation of +propositions, considered as wholes not necessarily unambiguously +determined, into a single proposition more complex than its +constituents, is a function <i>with propositions as arguments</i>. The +general idea of such an aggregation of propositions, or of variables +representing propositions, will not be employed in this work. But there +are four special cases which are of fundamental importance, since +all the aggregations of subordinate propositions into one complex +proposition which occur in the sequel are formed out of them step by +step.</p> + +<p>They are (1) the Contradictory Function, (2) the Logical Sum, or +Disjunctive Function, (3) the Logical Product, or Conjunctive +Function, (4) the Implicative Function. These functions in the sense +in which they are required in this work are not all independent; and +if two of them are taken as primitive undefined ideas, the other two +can be defined in terms of them. It is to some extent—though not +entirely—arbitrary as to which functions are taken as primitive. +Simplicity of primitive ideas and symmetry of treatment seem to be +gained by taking the first two functions as primitive ideas.</p> + +<p>The Contradictory Function with argument \(p\), where \(p\) is any +proposition, is the proposition which is the contradictory of \(p\), +that is, the proposition asserting that \(p\) is not true. This is +denoted by \({\sim}p\). Thus \({\sim}p\) is the contradictory function +with \(p\) as argument and means the negation of the proposition +\(p\). It will also be referred to as the proposition not-\(p\). Thus +\({\sim}p\) means not-\(p\), which means the negation of \(p\).</p> + +<p>The Logical Sum is a propositional function with two arguments +\(p\) and \(q\), and is the proposition asserting \(p\) or \(q\) +disjunctively, that is, asserting that at least one of the two \(p\) +and \(q\) is true. This is denoted by \(p \lor q\). Thus \(p \lor q\) +is the logical sum with \(p\) and \(q\) as arguments. It is also called +the logical sum of \(p\) and \(q\). Accordingly \(p \lor q\) means that +at least \(p\) or \(q\) is true, not excluding the case in which both +are true.</p> + +<p>The Logical Product is a propositional function with two arguments +\(p\) and \(q\), and is the proposition asserting \(p\) and \(q\) +conjunctively, that is, asserting that both \(p\) and \(q\) are true. +This is denoted by \(p \ldotp q\), or—in order to make the dots act as +brackets in a way to be explained immediately—by \(p \colon q\), or by +\(p \colon\ldotp q\), or by \(p \colon\colon q\). Thus \(p \ldotp q\) +is the logical product with<span class="pagenum" id="Page_7">[Pg 7]</span> \(p\) and \(q\) as arguments. It is also +called the logical product of \(p\) and \(q\). Accordingly \(p \ldotp +q\) means that both \(p\) and \(q\) are true. It is easily seen that +this function can be defined in terms of the two preceding functions. +For when \(p\) and \(q\) are both true it must be false that either +\(\sim p\text{ or }\sim q\) is true. Hence in this book \(p \ldotp q\) +is merely a shortened form of symbolism for +\[ +\sim (\sim p \lor \sim q). +\] +If any further idea attaches to the proposition "both \(p\) and \(q\) +are true," it is not required here.</p> + +<p>The Implicative Function is a propositional function with two arguments +\(p\) and \(q\), and is the proposition that either not-\(p\) or +\(q\) is true, that is, it is the proposition \(\sim p \lor q\). +Thus if \(p\) is true, \(\sim p\) is false, and accordingly the only +alternative left by the proposition \(\sim p \lor q\) is that \(q\) +is true. In other words if \(p\) and \(\sim p \lor q\) are both true, +then \(q\) is true. In this sense the proposition \({\sim} p \lor q\) +will be quoted as stating that \(p\) implies \(q\). The idea contained +in this propositional function is so important that it requires a +symbolism which with direct simplicity represents the proposition +as connecting \(p\) and \(q\) without the intervention of \({\sim} +p\). But "implies" as used here expresses nothing else than the +connection between \(p\) and \(q\) also expressed by the disjunction +"not-\(p\) or \(q\)." The symbol employed for "\(p\) implies \(q\)," +<i>i.e.</i> for "\({\sim} p \lor q\)" is "\(p \supset q\)." This +symbol may also be read "if \(p\), then \(q\)." The association of +implication with the use of an apparent variable produces an extension +called "formal implication." This is explained later: it is an idea +derivative from "implication" as here defined. When it is necessary +explicitly to discriminate "implication" from "formal implication," +it is called "material implication." Thus "material implication" is +simply "implication" as here defined. The process of inference, which +in common usage is often confused with implication, is explained +immediately.</p> + +<p>These four functions of propositions are the fundamental constant +(<i>i.e</i>. definite) propositional functions with <i>propositions +as arguments</i>, and all other constant propositional functions with +propositions as arguments, so far as they are required in the present +work, are formed out of them by successive steps. No <i>variable</i> +propositional functions of this kind occur in this work.</p> + +<p><i>Equivalence</i>. The simplest example of the formation of a more +complex function of propositions by the use of these four fundamental +forms is furnished by "equivalence." Two propositions \(p\) and \(q\) +are said to be "equivalent" when \(p\) implies \(q\) and \(q\) implies +\(p\). This relation between \(p\) and \(q\) is denoted by "\(p \equiv +q\)" Thus "\(p \equiv q\)" stands for "\((p \supset q) \ldotp (q +\supset p)\)." It is easily seen that two propositions are equivalent +when, and only when, they are both true or are both false. Equivalence +rises in the scale of importance when we come to "formal implication" +and thus to "formal equivalence." It must not be supposed that two +propositions which are equivalent are in<span class="pagenum" id="Page_8">[Pg 8]</span> any sense identical or even +remotely concerned with the same topic. Thus "Newton was a man" and +"the sun is hot" are equivalent as being both true, and "Newton was not +a man" and "the sun is cold" are equivalent as being both false. But +here we have anticipated deductions which follow later from our formal +reasoning. Equivalence in its origin is merely mutual implication as +stated above.</p> + +<p><i>Truth-values</i>. The "truth-value" of a proposition is truth if it is +true, and <i>falsehood</i> if it is false<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a>. It will be observed that the +truth-values of \(p \lor q\), \(p . q\), \(p \supset q\), \({\sim} +p\), \(p \equiv q\) depend only upon those of \(p\) and \(q\), namely +the truth-value of "\(p \lor q\)" is truth if the truth-value of +either \(p\) or \(q\) is truth, and is falsehood otherwise; that of +"\(p . q\)" is truth if that of both \(p\) and \(q\) is truth, and is +falsehood otherwise; that of "\(p \supset q\)" is truth if either that +of \(p\) is falsehood or that of \(q\) is truth; that of "\({\sim} +p\)" is the opposite of that of \(p\); and that of "\(p \equiv q\)" is +truth if \(p\) and \(q\) have the same truth-value, and is falsehood +otherwise. Now the only ways in which propositions will occur in the +present work are ways derived from the above by combinations and +repetitions. Hence it is easy to see (though it cannot be formally +proved except in each particular case) that if a proposition \(p\) +occurs in any proposition \(f(p)\) which we shall ever have occasion +to deal with, the truth-value of \(f(p)\) will depend, not upon +the particular proposition \(p\), but only upon its truth-value; +<i>i.e.</i> if \(p \equiv q\), we shall have \(f(p) \equiv f(q)\). +Thus whenever two propositions are known to be equivalent, either may +be substituted for the other in any formula with which we shall have +occasion to deal.</p> + +<p>We may call a function \(f(p)\) a "truth-function" when its argument +\(p\) is a proposition, and the truth-value of \(f(p)\) depends only +upon the truth-value of \(p\). Such functions are by no means the +only common functions of propositions. For example, "\(A\) believes +\(p\)" is a function of \(p\) which will vary its truth-value for +different arguments having the same truth-value: \(A\) may believe one +true proposition without believing another, and may believe one +false proposition without believing another. Such functions are not +excluded from our consideration, and are included in the scope of any +general propositions we may make about functions; but the particular +functions of propositions which we shall have occasion to construct +or to consider explicitly are all truth-functions. This fact is +closely connected with a characteristic of mathematics, namely, that +mathematics is always concerned with extensions rather than intensions. +The connection, if not now obvious, will become more so when we have +considered the theory of classes and relations.</p> + +<p><i>Assertion-sign</i>. The sign "\(\vdash\)," called the "assertion-sign," +means that what follows is asserted. It is required for distinguishing +a complete proposition, which we assert, from any subordinate +propositions contained in it but<span class="pagenum" id="Page_9">[Pg 9]</span> not asserted. In ordinary written +language a sentence contained between full stops denotes an asserted +proposition, and if it is false the book is in error. The sign +"\(\vdash\)" prefixed to a proposition serves this same purpose in our +symbolism. For example, if "\(\vdash (p \supset p)\)" occurs, it is +to be taken as a complete assertion convicting the authors of error +unless the proposition "\(p \supset p\)" is true (as it is). Also a +proposition stated in symbols without this sign "\(\vdash\)" prefixed +is not asserted, and is merely put forward for consideration, or as a +subordinate part of an asserted proposition.</p> + +<p><i>Inference.</i> The process of inference is as follows: a proposition +"\(p\)" is asserted, and a proposition "\(p\) implies \(q\)" is +asserted, and then as a sequel the proposition "\(q\)" is asserted. +The trust in inference is the belief that if the two former assertions +are not in error, the final assertion is not in error. Accordingly +whenever, in symbols, where \(p\) and \(q\) have of course special +determinations, +\[ +\unicode{x201c}\vdash p\unicode{x201d}\, \text{and}\, \unicode{x201c}\vdash (p \supset q)\unicode{x201d} +\] +have occurred, then "\(\vdash q\)" will occur if it is desired to +put it on record. The process of the inference cannot be reduced to +symbols. Its sole record is the occurrence of "\(\vdash q\)." It is of +course convenient, even at the risk of repetition, to write "\(\vdash p\)" +and "\(\vdash (p \supset q)\)" in close juxtaposition before +proceeding to "\(\vdash q\)" as the result of an inference. When this +is to be done, for the sake of drawing attention to the inference which +is being made, we shall write instead +\[ +\unicode{x201c}\vdash p \supset \vdash q,\unicode{x201d} +\] +which is to be considered as a mere abbreviation of the threefold +statement +\[ +\unicode{x201c}\vdash p\unicode{x201d}\, \text{and}\, \unicode{x201c}\vdash (p \supset q)\unicode{x201d}\, \text{and}\, \unicode{x201c}\vdash q.\unicode{x201d} +\] +Thus "\(\vdash p \supset \vdash q\)" may be read "\(p\), therefore +\(q\)," being in fact the same abbreviation, essentially, as this is; +for "\(p\), therefore \(q\)" does not explicitly state, what is part of +its meaning, that \(p\) implies \(q\). An inference is the dropping of +a true premiss; it is the dissolution of an implication.</p> + +<p><i>The use of dots.</i> Dots on the line of the symbols have two +uses, one to bracket off propositions, the other to indicate the +logical product of two propositions. Dots immediately preceded +or followed by "\(\lor\)" or "\(\supset\)" or "\(\equiv\)" or +"\(\vdash\)," or by "\((x)\)," "\((x, y)\)," "\((x, y, z)\)" ... or +"\((\exists x)\)," "\((\exists x, y)\)," "\((\exists x, y, z)\)" ... +or "\([({℩}x)({\phi}x)]\)" or "\([Rʻy]\)" or analogous expressions, +serve to bracket off a proposition; dots occurring otherwise serve +to mark a logical product. The general principle is that a larger +number of dots indicates an outside bracket, a smaller number +indicates an inside bracket. The exact rule as to the scope of the +bracket indicated by dots is arrived at by dividing the occurrences +of dots into three groups which we will name I, II, and III. Group I +consists of dots adjoining a sign of implication (\(\supset\)) or of +equivalence (\(\equiv\)) or of disjunction (\(\lor\)) or of equality +by definition (\(=\) Df). Group II consists of dots following brackets +indicative of an apparent variable, such as \((x)\) or \((x, y)\) or +\((\exists x)\) or<span class="pagenum" id="Page_10">[Pg 10]</span> \((\exists x,{\sim}y)\) or \([({℩}x)(\phi x)]\) +or analogous expressions<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a>. Group III consists of dots which stand +between propositions in order to indicate a logical product. Group +I is of greater force than Group II, and Group II than Group III. +The scope of the bracket indicated by any collection of dots extends +backwards or forwards beyond any <i>smaller</i> number of dots, or any +<i>equal</i> number from a group of less force, until we reach either +the end of the asserted proposition or a <i>greater</i> number of dots +or an <i>equal</i> number belonging to a group of equal or superior +force. Dots indicating a logical product have a scope which works both +backwards and forwards; other dots only work away from the adjacent +sign of disjunction, implication, or equivalence, or forward from the +adjacent symbol of one of the other kinds enumerated in Group II.</p> + +<p>Some examples will serve to illustrate the use of dots.</p> + +<p>"\(p \lor q \ldotp \supset \ldotp q \lor p\)" means the proposition +"'\(p\) or \(q\)' implies '\(q\) or \(p\).'" When we <i>assert</i> this +proposition, instead of merely considering it, we write +\[ +\unicode{x201c}\vdash : p \lor q \ldotp \supset \ldotp q \lor p,\unicode{x201d} +\] +where the two dots after the assertion-sign show that what is asserted +is the whole of what follows the assertion-sign, since there are not +as many as two dots anywhere else. If we had written "\(p : \lor : q +\ldotp \supset \ldotp q \lor p\)," that would mean the proposition +"either \(p\) is true, or \(q\) implies '\(q\) or \(p\).'" If we +wished to assert this, we should have to put three dots after the +assertion-sign. If we had written "\(p \lor q \ldotp \supset \ldotp q : +\lor : p\)," that would mean the proposition "either '\(p\) or \(q\)' +implies q, or p is true." The forms "\(p \ldotp \lor \ldotp q \ldotp +\supset \ldotp q \lor p\)" and "\(p \lor q \ldotp \supset \ldotp q +\ldotp \lor \ldotp p\)" have no meaning.</p> + +<p>"\(p \supset q \ldotp \supset : q \supset r \ldotp \supset \ldotp p +\supset r\)" will mean "if \(p\) implies \(q\), then if \(q\) implies +\(r,~p\) implies \(r\)." If we wish to assert this (which is true) we +write +\[ +\unicode{x201c}\vdash \colon\ldotp p \supset q \ldotp \supset : q \supset r \ldotp \supset \ldotp p \supset r.\unicode{x201d} +\] +Again "\(p \supset q \ldotp \supset \ldotp q \supset r : \supset \ldotp +p \supset r\)" will mean "if '\(p\) implies \(q\)' implies '\(q\) +implies \(r\),' then \(p\) implies \(r\)." This is in general untrue. +(Observe that "\(p \supset q\)" is sometimes most conveniently read +as "\(p\) implies \(q\)," and sometimes as "if \(p\), then \(q\).") +"\(p \supset q \ldotp q \supset r \ldotp \supset \ldotp p \supset r\)" +will mean "if \(p\) implies \(q\), and \(q\) implies \(r\), then \(p\) +implies \(r\)." In this formula, the first dot indicates a logical +product; hence the scope of the second dot extends backwards to the +beginning of the proposition. "\(p \supset q : q \supset r \ldotp +\supset \ldotp p \supset r\)" will mean "\(p\) implies \(q\); and if +\(q\) implies \(r\), then \(p\) implies \(r\)." (This is not true in +general.) Here the two dots indicate a logical product; since two +dots do not occur anywhere else, the scope of these two dots extends +backwards to the beginning of the proposition, and forwards to the end.</p> + +<p>"\(p \lor q \ldotp \supset \colon\ldotp p \ldotp \lor \ldotp q \supset +r : \supset \ldotp p \lor r\)" will mean "if either \(p\) or \(q\) is +true, then if either \(p\) or '\(q\) implies \(r\)' is true, it follows +that either \(p\) or \(r\) is true."<span class="pagenum" id="Page_11">[Pg 11]</span> If this is to be asserted, we +must put four dots after the assertion-sign, thus: +\[ +\unicode{x201c}\vdash \colon\colon p \lor q.\supset \colon\ldotp p. \lor .q \supset r \colon \supset .p \lor r.\unicode{x201d} +\] +(This proposition is proved in the body of the work; it is <a href="#*2·75">*2·75</a>.) If +we wish to assert (what is equivalent to the above) the proposition: +"if either \(p\) or \(q\) is true, and either \(p\) or '\(q\) implies +\(r\)' is true, then either \(p\) or \(r\) is true," we write +\[ +\unicode{x201c}\vdash \colon\ldotp p \lor q \colon p. \lor .q \supset r \colon \supset .p \lor r.\unicode{x201d} +\] +Here the first pair of dots indicates a logical product, while the +second pair does not. Thus the scope of the second pair of dots passes +over the first pair, and back until we reach the three dots after the +assertion-sign.</p> + +<p>Other uses of dots follow the same principles, and will be explained as +they are introduced. In reading a proposition, the dots should be noticed +first, as they show its structure. In a proposition containing several signs of +implication or equivalence, the one with the greatest number of dots before +or after it is the <i>principal</i> one: everything that goes before this one is stated +by the proposition to imply or be equivalent to everything that comes +after it.</p> + +<p><i>Definitions</i>. A definition is a declaration that a certain +newly-introduced symbol or combination of symbols is to mean the same +as a certain other combination of symbols of which the meaning is +already known. Or, if the defining combination of symbols is one which +only acquires meaning when combined in a suitable manner with other +symbols<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a>, what is meant is that any combination of symbols in which +the newly-defined symbol or combination of symbols occurs is to have +that meaning (if any) which results from substituting the defining +combination of symbols for the newly-defined symbol or combination +of symbols wherever the latter occurs. We will give the names of +<i>definiendum</i> and <i>definiens</i> respectively to what is defined +and to that which it is defined as meaning. We express a definition by +putting the <i>definiendum</i> to the left and the <i>definiens</i> to +the right, with the sign "=" between, and the letters "Df" to the right +of the <i>definiens</i>. It is to be understood that the sign "=" and +the letters "Df" are to be regarded as together forming one symbol. The +sign "=" without the letters "Df" will have a different meaning, to be +explained shortly.</p> + +<p>An example of a definition is +\[ +p \supset q \ldotp = \ldotp {\sim} p \lor q \qquad \text{Df.} +\]</p> + +<p>It is to be observed that a definition is, strictly speaking, no part +of the subject in which it occurs. For a definition is concerned +wholly with the symbols, not with what they symbolise. Moreover it +is not true or false, being the expression of a volition, not of a +proposition. (For this reason,<span class="pagenum" id="Page_12">[Pg 12]</span> definitions are not preceded by the +assertion-sign.) Theoretically, it is unnecessary ever to give a +definition: we might always use the <i>definiens</i> instead, and thus +wholly dispense with the <i>definiendum</i>. Thus although we employ +definitions and do not define "definition," yet "definition" does +not appear among our primitive ideas, because the definitions are no +part of our subject, but are, strictly speaking, mere typographical +conveniences. Practically, of course, if we introduced no definitions, +our formulae would very soon become so lengthy as to be unmanageable; +but theoretically, all definitions are superfluous.</p> + +<p>In spite of the fact that definitions are theoretically superfluous, it +is nevertheless true that they often convey more important information +than is contained in the propositions in which they are used. This +arises from two causes. First, a definition usually implies that +the <i>definiens</i> is worthy of careful consideration. Hence the +collection of definitions embodies our choice of subjects and our +judgment as to what is most important. Secondly, when what is defined +is (as often occurs) something already familiar, such as cardinal or +ordinal numbers, the definition contains an analysis of a common idea, +and may therefore express a notable advance. Cantor's definition of the +continuum illustrates this: his definition amounts to the statement +that what he is defining is the object which has the properties +commonly associated with the word "continuum," though what precisely +constitutes these properties had not before been known. In such cases, +a definition is a "making definite": it gives definiteness to an idea +which had previously been more or less vague.</p> + +<p>For these reasons, it will be found, in what follows, that the +definitions are what is most important, and what most deserves the +reader's prolonged attention.</p> + +<p>Some important remarks must be made respecting the variables occurring +in the <i>definiens</i> and the <i>definiendum</i>. But these will be +deferred till the notion of an "apparent variable" has been introduced, +when the subject can be considered as a whole.</p> + +<p><i>Summary of preceding statements.</i> There are, in the above, +three primitive ideas which are not "defined" but only descriptively +explained. Their primitiveness is only relative to our exposition +of logical connection and is not absolute; though of course such an +exposition gains in importance according to the simplicity of its +primitive ideas. These ideas are symbolised by "\({\sim} p\)" and "\(p +\lor q\)," and by "\(\vdash\)" prefixed to a proposition.</p> + +<p>Three definitions have been introduced: +\[ +\begin{aligned} +p \ldotp q \ldotp &= \ldotp {\sim} ({\sim} p \lor {\sim} q) \quad &\text{Df},\\ +p \supset q \ldotp &= \ldotp {\sim} p \lor q \quad &\text{Df},\\ +p \equiv q \ldotp &= \ldotp p \supset q \ldotp q \supset p \quad &\text{Df}. \\ +\end{aligned} +\]</p> + +<p><span class="pagenum" id="Page_13">[Pg 13]</span></p> + +<p><i>Primitive propositions.</i> Some propositions must be assumed +without proof, since all inference proceeds from propositions +previously asserted. These, as far as they concern the functions +of propositions mentioned above, will be found stated in <a href="#*1">*1</a>, where +the formal and continuous exposition of the subject commences. Such +propositions will be called "primitive propositions." These, like the +primitive ideas, are to some extent a matter of arbitrary choice; +though, as in the previous case, a logical system grows in importance +according as the primitive propositions are few and simple. It will be +found that owing to the weakness of the imagination in dealing with +simple abstract ideas no very great stress can be laid upon their +obviousness. They are obvious to the instructed mind, but then so are +many propositions which cannot be quite true, as being disproved by +their contradictory consequences. The proof of a logical system is +its adequacy and its coherence. That is: (1) the system must embrace +among its deductions all those propositions which we believe to be +true and capable of deduction from logical premisses alone, though +possibly they may require some slight limitation in the form of an +increased stringency of enunciation; and (2) the system must lead to no +contradictions, namely in pursuing our inferences we must never be led +to assert both \(p\) and not-\(p\), <i>i.e.</i> both "\(\vdash \ldotp +p\)" and "\(\vdash \ldotp {\sim} p\)" cannot legitimately appear.</p> + +<p>The following are the primitive propositions employed in the calculus +of propositions. The letters "Pp" stand for "primitive proposition."</p> + +<p>(1) Anything implied by a true premiss is true Pp.</p> + +<p>This is the rule which justifies inference.</p> + +<p>(2) \(\vdash \colon p \lor p \ldotp \supset \ldotp p \quad \text{Pp}\),</p> + +<p class="nind"> +<i>i.e.</i> if \(p\) or \(p\) is true, then \(p\) is true.</p> + +<p>(3) \(\vdash \colon q \ldotp \supset \ldotp p \lor q \quad \text{Pp}\),</p> + +<p class="nind"> +<i>i.e.</i> if \(q\) is true, then \(p\) or \(q\) is true.</p> + +<p>(4) \(\vdash \colon p \lor q \ldotp \supset \ldotp q \lor p \quad \text{Pp}\),</p> + +<p class="nind"> +<i>i.e.</i> if \(p\) or \(q\) is true, then \(q\) or \(p\) is true.</p> + +<p>(5) \(\vdash \colon p \lor (q \lor r) \ldotp \supset \ldotp q \lor (p \lor r) \quad \text{Pp}\),</p> + +<p class="nind"> +<i>i.e.</i> if either \(p\) is true or "\(q\) or \(r\)" is true, then +either \(q\) is true or "\(p\) or \(r\)" is true.</p> + +<p>(6) \(\vdash \colon \ldotp q \supset r \ldotp \supset \colon p \lor q \ldotp \supset \ldotp p \lor r \quad \text{Pp}\),</p> + +<p class="nind"> +<i>i.e.</i> if \(q\) implies \(r\), then "\(p\) or \(q\)" implies +"\(p\) or \(r\)."</p> + +<p>(7) Besides the above primitive propositions, we require a primitive +proposition called "the axiom of identification of real variables." +When we have separately asserted two different functions of \(x\), +where \(x\) is undetermined, it is often important to know whether +we can identify the \(x\) in one<span class="pagenum" id="Page_14">[Pg 14]</span> assertion with the \(x\) in the +other. This will be the case—so our axiom allow us to infer—if both +assertions present x as the argument to some one function, that is +to say, if \(\phi x\) is a constituent in both assertions (whatever +propositional function \(\phi\) may be), or, more generally, if \(\phi(x, y, z, \ldots)\) +is a constituent in one assertion, and \(\phi (x, u, v, \ldots)\) +is a constituent in the other. This axiom introduces notions which +have not yet been explained; for a fuller account, see the remarks +accompanying <a href="#*3·03">*3·03</a>, <a href="#*1·7">*1·7</a>, <a href="#*1·71">*1·71</a>, and <a href="#*1·72">*1·72</a> (which is the statement of +this axiom) in the body of the work, as well as the explanation of +propositional functions and ambiguous assertion to be given shortly.</p> + +<p><i>Some simple propositions</i>. In addition to the primitive +propositions we have already mentioned, the following are among the +most important of the elementary properties of propositions appearing +among the deductions.</p> + +<p>The law of excluded middle: +\[ +\vdash . p \lor {\sim}p. +\] +This is <a href="#*2·11">*2·11</a> below. We shall indicate in brackets the numbers given to +the following propositions in the body of the work.</p> + +<p>The law of contradiction (<a href="#*3·24">*3·24</a>): +\[ +\vdash . {\sim}(p.{\sim}p). +\]</p> + +<p>The law of double negation (<a href="#*4·13">*4·13</a>): +\[ +\vdash . p \equiv {\sim}({\sim}p). +\]</p> + +<p>The principle of <i>transposition</i>, <i>i.e.</i> "if \(p\) implies +\(q\), then not-\(q\) implies not-\(p\)," and vice versa: this +principle has various forms, namely +\[ +\begin{array}{l} +\text{(*4·1)}\qquad &\vdash \colon p \supset q. \equiv .{\sim}q \supset {\sim}p,\\ +\text{(*4·11)}\quad &\vdash \colon p \equiv q. \equiv . {\sim}p \equiv {\sim}q,\\ +\text{(*4·14)}\quad &\vdash \colon\ldotp p . q. \supset . r \equiv \colon p. {\sim}r . \supset. {\sim}q,\\ +\end{array} +\] +as well as others which are variants of these.</p> + +<p>The law of tautology, in the two forms: +\[ +\begin{array}{l} +\text{(*4·24)}\quad &\vdash \colon p. \equiv .p\, . p,\\ +\text{(*4·25)}\quad &\vdash \colon p. \equiv .p \lor p,\\ +\end{array} +\] +<i>i.e.</i> "\(p\) is true" is equivalent to "\(p\) is true and \(p\) +is true," as well as to "\(p\) is true or \(p\) is true." From a formal +point of view, it is through the law of tautology and its consequences +that the algebra of logic is chiefly distinguished from ordinary +algebra.</p> + +<p>The law of absorption : +\[ +\text{(*4·24)}\qquad \vdash \colon\ldotp p \supset q. \equiv \colon p. \equiv . p . q, +\] +<i>i.e.</i> "\(p\) implies \(q\)" is equivalent to "\(p\) is equivalent +to \(p . q\)." This is called the law of absorption because it shows +that the factor \(q\) in the product is<span class="pagenum" id="Page_15">[Pg 15]</span> absorbed by the factor +\(p\), if \(p\) implies \(q\). This principle enables us to replace +an implication (\(p \supset q\)) by an equivalence (\(p \ldotp \equiv\ldotp p \ldotp q)\) +whenever it is convenient to do so.</p> + +<p>An analogous and very important principle is the following: +\[ +\text{(*4·73)}\qquad \vdash \colon \ldotp q \ldotp \supset \colon p \ldotp \equiv \ldotp p \ldotp q. +\]</p> + +<p>Logical addition and multiplication of propositions obey the +associative and commutative laws, and the distributive law in two +forms, namely +\[ +\begin{align} +&\text{(*4·4)} &\vdash \colon\ldotp p \ldotp q \lor r \ldotp \equiv \colon p \ldotp q \ldotp \lor \ldotp p \ldotp r\text{,}\\ +&\text{(*4·41)} &\vdash \colon \ldotp p \ldotp \lor \ldotp q \ldotp r \colon \equiv \colon p \lor q \ldotp p \lor r\text{.}\\ +\end{align} +\] +The second of these distinguishes the relations of logical addition and +multiplication from those of arithmetical addition and multiplication.</p> + +<p><i>Propositional functions.</i> Let \({\phi}x\) be a statement +containing a variable \(x\) and such that it becomes a proposition +when \(x\) is given any fixed determined meaning. Then \({\phi}x\) is +called a "propositional function"; it is not a proposition, since owing +to the ambiguity of \(x\) it really makes no assertion at all. Thus +"\(x\) is hurt" really makes no assertion at all, till we have settled +who \(x\) is. Yet owing to the individuality retained by the ambiguous +variable \(x\), it is an ambiguous example from the collection of +propositions arrived at by giving all possible determinations to \(x\) +in "\(x\) is hurt" which yield a proposition, true or false. Also if +"\(x\) is hurt" and "\(y\) is hurt" occur <i>in the same context</i>, +where \(y\) is another variable, then according to the determinations +given to \(x\) and \(y\), they can be settled to be (possibly) the same +proposition or (possibly) different propositions. But apart from some +determination given to \(x\) and \(y\), they retain in that context +their ambiguous differentiation. Thus "\(x\) is hurt" is an ambiguous +"value" of a propositional function. When we wish to speak of the +propositional function corresponding to "\(x\) is hurt," we shall write +"\(\hat{x}\) is hurt." Thus "\(\hat{x}\) is hurt" is the propositional +function and "\(x\) is hurt" is an ambiguous value of that function. +Accordingly though "\(x\) is hurt" and "\(y\) is hurt" <i>occurring +in the same context</i> can be distinguished, "\(\hat{x}\) is hurt" +and "\(\hat{y}\) is hurt" convey no distinction of meaning at all. +More generally, \({\phi}x\) is an ambiguous value of the propositional +function \(\phi\hat{x}\), and when a definite signification \(a\) +is substituted for \(x\), \({\phi}a\) is an unambiguous value of +\({\phi}\hat{x}\).</p> + +<p>Propositional functions are the fundamental kind from which the more +usual kinds of function, such as "\(\sin x\)" or "\(\log x\)" or +"the father of \(x\)," are derived. These derivative functions are +considered later, and are called "descriptive functions." The functions +of propositions considered above are a particular case of propositional +functions.</p> + +<p><i>The range of values and total variation.</i> Thus corresponding +to any propositional function \(\phi\hat{x}\), there is a range, or +collection, of values, consisting of all the propositions (true or +false) which can be obtained by giving<span class="pagenum" id="Page_16">[Pg 16]</span> every possible determination to +\(x\) in \({\phi}x\). A value of \(x\) for which \({\phi}x\) is true +will be said to "satisfy" \(\phi\hat{x}\). Now in respect to the truth +or falsehood of propositions of this range three important cases must +be noted and symbolised. These cases are given by three propositions +of which one at least must be true. Either (1) all propositions of the +range are true, or (2) some propositions of the range are true, or (3) +no proposition of the range is true. The statement (1) is symbolised +by "\((x) \ldotp {\phi}x\)," and (2) is symbolised by "\((\exists +x) \ldotp {\phi}x\)." No definition is given of these two symbols, +which accordingly embody two new primitive ideas in our system. The +symbol "\((x) \ldotp {\phi}x\)" may be read "\({\phi}x\) always," or +"\({\phi}x\) is always true," or "\({\phi}x\) is true for all possible +values of \(x\)." The symbol "\((\exists x) \ldotp {\phi}x\)" may be +read "there exists an \(x\) for which \({\phi}x\) is true," or "there +exists an \(x\) satisfying \(\phi\hat{x}\)," and thus conforms to the +natural form of the expression of thought.</p> + +<p>Proposition (3) can be expressed in terms of the fundamental ideas now +on hand. In order to do this, note that "\({\sim}{\phi}x\)" stands for +the contradictory of \({\phi}x\). Accordingly \(\sim\phi\hat{x}\) is +another propositional function such that each value of \(\phi\hat{x}\) +contradicts a value of \({\sim}\phi\hat{x}\) and vice versa. Hence +"\((x) \ldotp {\sim}{\phi}x\)" symbolises the proposition that every +value of \(\phi\hat{x}\) is untrue. This is number (3) as stated above.</p> + +<p>It is an obvious error, though one easy to commit, to assume that +cases (1) and (3) are each other's contradictories. The symbolism +exposes this fallacy at once, for (1) is \((x) \ldotp {\phi}x\), and +(3) is \((x) \ldotp {\sim}{\phi}x\), while the contradictory of (1) is +\({\sim}\{(x) \ldotp {\phi}x\}\). For the sake of brevity of symbolism +a definition is made, namely +\[ +{\sim}(x) \ldotp {\phi}x \ldotp = \ldotp {\sim}\{(x) \ldotp {\phi}x\} \quad \text{Df}\text{.} +\]</p> + +<p>Definitions of which the object is to gain some trivial advantage in +brevity by a slight adjustment of symbols will be said to be of "merely +symbolic import," in contradistinction to those definitions which +invite consideration of an important idea.</p> + +<p>The proposition \((x) \ldotp {\phi}x\) is called the "total variation" +of the function \(\phi\hat{x}\).</p> + +<p>For reasons which will be explained in <a href="#CHAPTER_II">Chapter II</a>, we do not take +negation as a primitive idea when propositions of the forms \((x)\ldotp {\phi}x\) +and \((\exists x) \ldotp {\phi}x\) are concerned, but +we <i>define</i> the negation of \((x) \ldotp {\phi}x\), <i>i.e.</i> +of "\({\phi}x\) is always true," as being "\({\phi}x\) is sometimes +false," <i>i.e.</i> "\((\exists x) \ldotp {\sim}{\phi}x\)," and +similarly we <i>define</i> the negation of \((\exists x) \ldotp {\phi}x\) +as being \((x) \ldotp {\sim}{\phi}x\). Thus we put +\[ +\begin{align} +{\sim}\{(x) \ldotp {\phi}x\} \ldotp = \ldotp (\exists x) \ldotp {\sim}{\phi}x \quad &\text{Df}\text{,}\\ +{\sim}\{(\exists x) \ldotp {\phi}x\} \ldotp = \ldotp (x) \ldotp {\sim}{\phi}x \quad &\text{Df}\text{.}\\ +\end{align} +\]</p> + +<p>In like manner we define a disjunction in which one of the propositions +is of the form "\((x) \ldotp {\phi}x\)" or "(\(\exists x) \ldotp{\phi}x\)" +in terms of a disjunction of propositions not of this form, putting +\[ +(x) \ldotp {\phi}x \ldotp \lor \ldotp p \colon = \ldotp (x) \ldotp {\phi}x \lor p \quad \text{Df}\text{,} +\]<span class="pagenum" id="Page_17">[Pg 17]</span> +<i>i.e.</i> "either \(\phi x\) is always true, or \(p\) is true" is to +mean "'\(\phi x\) or \(p\)' is always true," with similar definitions +in other cases. This subject is resumed in <a href="#CHAPTER_II">Chapter II</a>, and in <a href="#*9">*9</a> in the +body of the work.</p> + +<p><i>Apparent variables</i>. The symbol "(\(x) . \phi x\)" denotes one +definite proposition, and there is no distinction in meaning between +"(\(x) . \phi x\)" and "\((y) . \phi y\)" when they occur in the same +context. Thus the "\(x\)" in "\((x) . \phi x\)" is not an ambiguous +constituent of any expression in which "(\(x) . \phi x\)" occurs; and +such an expression does not cease to convey a determinate meaning by +reason of the ambiguity of the x in the "\(\phi x\)." The symbol "(\(x). \phi x\)" +has some analogy to the symbol +\[ +\unicode{x201c}\int_a^{b} \phi(x) dx \unicode{x201d} +\] +for definite integration, since in neither case is the expression a +function of \(x\).</p> + +<p>The range of \(x\) in "(\(x) . \phi x\)" or "(\(\exists x) . \phi x\)" +extends over the complete field of the values of \(x\) for +which "\(\phi x\)" has meaning, and accordingly the meaning of "(\(x). \phi x\)" +or "(\(\exists x) . \phi x\)" involves the supposition +that such a field is determinate. The \(x\) which occurs in "(\(x) .\phi x\)" +or "(\(\exists x) . \phi x\)" is called (following Peano) +an "apparent variable." It follows from the meaning of "(\(\exists x). \phi x\)" +that the \(x\) in this expression is also an apparent +variable. A proposition in which \(x\) occurs as an apparent variable +is not a function of \(x\). Thus <i>e.g.</i> "(\(x) . x = x\)" will +mean "everything is equal to itself." This is an absolute constant, +not a function of a variable \(x\). This is why the \(x\) is called an +apparent variable in such cases.</p> + +<p>Besides the "<i>range</i>" of \(x\) in "(\(x) . \phi x\)" or +"(\(\exists x) . \phi x\)," which is the field of the values that +\(x\) may have, we shall speak of the "<i>scope</i>" of \(x\), meaning +the function of which all values or some value are being affirmed. If +we are asserting all values (or some value) of "\(\phi x\)," "\(\phi x\)" +is the scope of \(x\); if we are asserting all values (or some +value) of "\(\phi x \supset p\)," "\(\phi x \supset p\)" is the scope +of \(x\); if we are asserting all values (or some value) of "\(\phi x \supset \psi x\)," +"\(\phi x \supset \psi x\)" will be the scope of \(x\), and so on. The +scope of \(x\) is indicated by the number of dots after the "(\(x\))" +or "\(\exists x\)"; that is to say, the scope extends forwards until we +reach an equal number of dots not indicating a logical product, or a +greater number indicating a logical product, or the end of the asserted +proposition in which the "(\(x\))" or "\(\exists x\)" occurs, whichever +of these happens first<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a>. Thus <i>e.g.</i> +\[ +\unicode{x201c}(x) \colon \phi x . \supset . \psi x \unicode{x201d} +\] +will mean "\(\phi x\) always implies \(\psi x\)," but +\[ +\unicode{x201c}(x) \colon \phi x . \supset . \psi x \unicode{x201d} +\] +will mean "if \(\phi x\) is always true, then \(\psi x\) is true for +the argument \(x\)."</p> + +<p>Note that in the proposition +\[ +(x) \colon \phi x . \supset . \psi x +\]<span class="pagenum" id="Page_18">[Pg 18]</span> +the two \(x\)'s have no connection with each other. Since only one dot +follows the \(x\) in brackets, the scope of the first \(x\) is limited +to the "\({\phi}x\)" immediately following the \(x\) in brackets. It +usually conduces to clearness to write +\[ +\begin{align} +& (x) \ldotp {\phi}x \ldotp \supset \ldotp {\psi}y\\ +\text{rather than} \quad & (x) \ldotp {\psi}x \ldotp \supset \ldotp {\psi}x\text{,}\\ +\end{align} +\] +since the use of different letters emphasises the absence of connection +between the two variables; but there is no logical necessity to use +different letters, and it is <i>sometimes</i> convenient to use the +same letter.</p> + +<p><i>Ambiguous assertion and the real variable.</i> Any value +"\({\phi}x\)" of the function \(\phi\hat{x}\) can be asserted. Such an +assertion of an ambiguous member of the values of \(\phi\hat{x}\) is +symbolised by +\[ +\unicode{x201c}\vdash \ldotp {\phi}x.\unicode{x201d} +\]</p> + +<p>Ambiguous assertion of this kind is a primitive idea, which cannot +be defined in terms of the assertion of propositions. This primitive +idea is the one which embodies the use of the variable. Apart from +ambiguous assertion, the consideration of "\({\phi}x\)," which is an +ambiguous member of the values of \({\phi}\hat{x}\), would be of little +consequence. When we are considering or asserting "\({\phi}x\)," the +variable \(x\) is called a "real variable." Take, for example, the law +of excluded middle in the form which it has in traditional formal logic: +\[ +\unicode{x201c} a\, \text{is either}\, b\, \text{or not}\, b.\unicode{x201d} +\] +Here \(a\) and \(b\) are real variables: as they vary, different +propositions are expressed, though all of them are true. While \(a\) +and \(b\) are undetermined, as in the above enunciation, no one +definite proposition is asserted, but what is asserted is <i>any</i> +value of the propositional function in question. This can only be +legitimately asserted if, whatever value may be chosen, that value is +true, <i>i.e.</i> if all the values are true. Thus the above form of +the law of excluded middle is equivalent to +\[ +\unicode{x201c}(a, b).a\, \text{is either}\, b\, \text{or not}\, b,\unicode{x201d} +\] +<i>i.e.</i> to "it is always true that \(a\) is either \(b\) or not +\(b\)." But these two, though equivalent, are not identical, and we +shall find it necessary to keep them distinguished.</p> + +<p>When we assert something containing a real variable, as in <i>e.g.</i> +\[ +\unicode{x201c}\vdash \ldotp x = x,\unicode{x201d} +\] +we are asserting <i>any</i> value of a propositional function. When we +assert something containing an apparent variable, as in +\[ +\begin{align} +\unicode{x201c}\vdash \ldotp (x) \ldotp x = x\unicode{x201d}\\ +\text{or} \quad \unicode{x201c}\vdash \ldotp (\exists x) \ldotp x = x,\unicode{x201d} +\end{align} +\] +we are asserting, in the first case <i>all</i> values, in the second +case <i>some</i> value (undetermined), of the propositional function in +question. It is plain that<span class="pagenum" id="Page_19">[Pg 19]</span> we can only legitimately assert "<i>any</i> +value" if <i>all</i> values are true; for otherwise, since the value of +the variable remains to be determined, it might be so determined as to +give a false proposition. Thus in the above instance, since we have +\[ +\begin{align} +&\vdash \ldotp x = x\\ +\text{we may infer}\qquad\qquad &\vdash \ldotp (x) \ldotp x = x. +\end{align} +\] +And generally, given an assertion containing a real variable \(x\), we +may transform the real variable into an apparent one by placing the +\(x\) in brackets at the beginning, followed by as many dots as there +are after the assertion-sign.</p> + +<p>When we assert something containing a real variable, we cannot strictly +be said to be asserting a <i>proposition</i>, for we only obtain a +definite proposition by assigning a value to the variable, and then our +assertion only applies to one definite case, so that it has not at all +the same force as before. When what we assert contains a real variable, +we are asserting a wholly undetermined one of all the propositions +that result from giving various values to the variable. It will be +convenient to speak of such assertions as <i>asserting a propositional +function</i>. The ordinary formulae of mathematics contain such +assertions; for example +\[ +\unicode{x201c}\text{sin}^{2} x + \text{cos}^{2} x = 1\unicode{x201d} +\] +does not assert this or that particular case of the formula, nor does +it assert that the formula holds for <i>all</i> possible values of +\(x\), though it is equivalent to this latter assertion; it simply +asserts that the formula holds, leaving \(x\) wholly undetermined; +and it is able to do this legitimately, because, however \(x\) may be +determined, a true proposition results.</p> + +<p>Although an assertion containing a real variable does not, in +strictness, assert a proposition, yet it will be spoken of as asserting +a proposition except when the nature of the ambiguous assertion +involved is under discussion.</p> + +<p><i>Definition and real variables.</i> When the <i>definiens</i> +contains one or more real variables, the <i>definiendum</i> must +also contain them. For in this case we have a function of the real +variables, and the <i>definiendum</i> must have the same meaning as the +<i>definiens</i> for all values of these variables, which requires that +the symbol which is the <i>definiendum</i> should contain the letters +representing the real variables. This rule is not always observed by +mathematicians, and its infringement has sometimes caused important +confusions of thought, notably in geometry and the philosophy of space.</p> + +<p>In the definitions given above of "\(p \ldotp q\)" and "\(p \supset q\)" +and "\(p \equiv q\)," \(p\) and \(q\) are real variables, and +therefore appear on both sides of the definition. In the definition +of "\({\sim}\{(x) \ldotp {\phi}x\}\)" only the function considered, +namely \(\phi\hat{z}\), is a real variable; thus so far as concerns the +rule in question, \(x\) need not appear on the left. But when a real +variable is a function, it is necessary to indicate<span class="pagenum" id="Page_20">[Pg 20]</span> how the argument +is to be supplied, and therefore there are objections to omitting an +apparent variable where (as in the case before us) this is the argument +to the function which is the real variable. This appears more plainly +if, instead of a general function \(\phi \hat{x}\), we take some +particular function, say "\(\hat{x} = a\)," and consider the definition +of \({\sim}\{(x) . x = a\}\). Our definition gives +\[ +{\sim}\{(x) . x = a\}. = .(\exists x). {\sim}(x = a) \qquad \text{Df.} +\] +But if we had adopted a notation in which the ambiguous value "\(x = a\)," +containing the apparent variable \(x\), did not occur in +the <i>definiendum</i>, we should have had to construct a notation +employing the function itself, namely "\(\hat{x} = a\)." This does not +involve an apparent variable, but would be clumsy in practice. In fact +we have found it convenient and possible—except in the explanatory +portions—to keep the explicit use of symbols of the type "\(\phi\hat{x}\)," +either as constants [<i>e.g.</i> \(\hat{x} = a\)] or as real variables, +almost entirely out of this work.</p> + +<p><i>Propositions connecting real and apparent variables</i>. The most important +propositions connecting real and apparent variables are the following:</p> + +<p>(1) "When a propositional function can be asserted, so can the +proposition that all values of the function are true." More briefly, if +less exactly, "what holds of any, however chosen, holds of all." This +translates itself into the rule that when a real variable occurs in +an assertion, we may turn it into an apparent variable by putting the +letter representing it in brackets immediately after the assertion-sign.</p> + +<p>(2) "What holds of all, holds of any," <i>i.e.</i> +\[ +\vdash \colon (x). \phi x . \supset . \phi y. +\] +This states "if \(\phi x\) is always true, then \(\phi y\) is true."</p> + +<p>(3) "If \(\phi y\) is true, then \(\phi x\) is sometimes true," +<i>i.e.</i> +\[ +\vdash \colon \phi y . \supset . (\exists x). \phi x. +\] +An asserted proposition of the form "\((\exists x). \phi x\)" expresses +an "existence-theorem," namely "there exists an \(x\) for which +\(\phi x\) is true." The above proposition gives what is in practice +the only way of proving existence-theorems: we always have to find +some particular \(y\) for which \(\phi y\) holds, and thence to infer +"\((\exists x). \phi x\)." If we were to assume what is called the +multiplicative axiom, or the equivalent axiom enunciated by Zermelo, +that would, in an important class of cases, give an existence-theorem +where no particular instance of its truth can be found.</p> + +<p>In virtue of "\(\vdash \colon (x). \phi x . \supset . \phi y\)" and +"\(\vdash \colon \phi y . \supset . (\exists x). \phi x\)," we have +"\(\vdash \colon (x) . \phi x . \supset . (\exists x). \phi x\)," +<i>i.e.</i> "what is always true is sometimes true." This would not +be the case if nothing existed; thus our assumptions contain the +assumption that there is something. This is involved in the principle<span class="pagenum" id="Page_21">[Pg 21]</span> +that what holds of all, holds of any; for this would not be true if +there were no "any."</p> + +<p>(4) "If \(\phi x\) is always true, and \(\psi x\) is always true, then +'\(\phi x . \psi x\)' is always true," <i>i.e.</i> +\[ +\vdash \colon\ldotp (x) . \phi x \colon (x) . \psi x : \supset . (x) . \phi x . \psi x. +\] +(This requires that \(\phi\) and \(\psi\) should be functions which +take arguments of the same <i>type</i>. We shall explain this requirement at a +later stage.) The converse also holds; <i>i.e.</i> we have +\[ +\vdash \colon\ldotp (x) . \phi x . \psi x . \supset \colon (x) . \phi x : (x) . \psi x. +\]</p> + +<p>It is to some extent optional which of the propositions connecting +real and apparent variables are taken as primitive propositions. The +primitive propositions assumed, on this subject, in the body of the +work (<a href="#*9">*9</a>), are the following: +\[ +\begin{array}{l} +\text{(1)}\qquad \vdash \colon \phi x . \supset. (\exists z) . \phi z.\\ +\text{(2)}\qquad +\vdash \colon \phi x \lor \phi y. \supset . (\exists z) . \phi z, +\end{array} +\] +<i>i.e.</i> if either \(\phi x\) is true, or \(\phi y\) is true, then +(\(\exists z) . \phi z\) is true. (On the necessity for this primitive +proposition, see remarks on <a href="#*9·11">*9·11</a> in the body of the work.)</p> + +<p>(3) If we can assert \(\phi y\), where y is a real variable, then we +can assert \((x) . \phi x\); <i>i.e.</i> what holds of any, however +chosen, holds of all.</p> + +<p><span class="pagenum" id="Page_22">[Pg 22]</span></p> + +<p><i>Formal implication and formal equivalence</i>. When an implication, +say \(\phi x . \supset . \psi x\), is said to hold always, <i>i.e.</i> +when \((x) \colon \phi x . \supset . \psi x\), we shall say that +\(\phi x\) <i>formally implies</i> \(\psi x\); and propositions of the +form "\((x) \colon \phi x . \supset . \psi x\)" will be said to state +<i>formal implications</i>. In the usual instances of implication, +such as "'Socrates is a man' implies 'Socrates is mortal,'" we have +a proposition of the form "\(\phi x . \supset . \psi x\)" in a case +in which "\((x) \colon \phi x . \supset . \psi x\)" is true. In such +a case, we feel the implication as a particular case of a formal +implication. Thus it has come about that implications which are not +particular cases of formal implications have not been regarded as +implications at all. There is also a practical ground for the neglect +of such implications, for, speaking generally, they can only be +<i>known</i> when it is already known either that their hypothesis is +false or that their conclusion is true; and in neither of these cases +do they serve to make us know the conclusion, since in the first case +the conclusion need not be true, and in the second it is known already. +Thus such implications do not serve the purpose for which implications +are chiefly useful, namely that of making us know, by deduction, +conclusions of which we were previously ignorant. <i>Formal</i> +implications, on the contrary, do serve this purpose, owing to the +psychological fact that we often know "\((x) \colon \phi x . \supset .\psi x\)" +and \(\phi y\), in cases where \(\psi y\) (which follows from +these premisses) cannot easily be known directly.</p> + +<p>These reasons, though they do not warrant the complete neglect of +implications that are not instances of formal implications, are reasons +which make formal implication very important. A formal implication +states that, for all possible values of \(x\), if the hypothesis +\(\phi x\) is true, the conclusion \(\psi x\) is true. Since "\(\phi x . \supset . \psi x\)" +will always be true when \(\phi x\) is false, it is only the values of +\(x\) that make \(\phi x\) true that are <i>important</i> in a formal +implication; what is effectively stated is that, for all these values, +\(\psi x\) is true. Thus propositions of the form "all \(\alpha\) is +\(\beta\)," "no \(\alpha\) is \(\beta\)" state formal implications, +since the first (as appears by what has just been said) states +\[ +(x) \colon x\, \text{is an}\,\, \alpha . \supset . x\, \text{is an}\,\, \beta, +\] +while the second states +\[ +(x) \colon x\, \text{is an}\,\, \alpha . \supset . x\, \text{is not a}\,\, \beta. +\] +And any formal implication "(\(x) \colon \phi x . \supset . \psi x\)" +may be interpreted as: "All values of \(x\) which satisfy<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a> \(\phi x\) +satisfy \(\psi x\)," while the formal implication "(\(x) \colon \phi x. \supset . \sim{\psi x}\)" +may be interpreted as: "No values of \(x\) which satisfy \(\phi x\) +satisfy \(\psi x\)."</p> + +<p>We have similarly for "some \(\alpha\) is \(\beta\)" the formula +\[ +(\exists x) . x\, \text{is an}\,\, \alpha . x\, \text{is a}\,\, \beta, +\] +and for "some \(\alpha\) is not \(\beta\)" the formula +\[ +(\exists x) . x\, \text{is an}\,\, \alpha . x\, \text{is not a}\,\, \beta. +\]</p> + +<p>Two functions \(\phi x\), \(\psi x\) are called <i>formally +equivalent</i> when each always implies the other, <i>i.e.</i> when +\[ +(x) \colon \phi x . \equiv . \psi x, +\] +and a proposition of this form is called a <i>formal equivalence</i>. +In virtue of what was said about truth-values, if \(\phi x\) and +\(\psi x\) are formally equivalent, either may replace the other in +any truth-function. Hence for all the purposes of mathematics or of +the present work, \(\phi \hat{z}\) may replace \(\psi \hat{z}\) or +vice versa in any proposition with which we shall be concerned. Now +to say that \(\phi x\) and \(\psi x\) are formally equivalent is the +same thing as to say that \(\phi \hat{z}\) and \(\psi \hat{z}\) have +the same <i>extension</i>, <i>i.e.</i> that any value of \(x\) which +satisfies either satisfies the other. Thus whenever a constant function +occurs in our work, the truth-value of the proposition in which it +occurs depends only upon the extension of the function. A proposition +containing a function \(\phi \hat{z}\) and having this property +(<i>i.e.</i> that its truth-value depends only upon the extension of +\(\phi \hat{z}\) will be called an <i>extensional</i> function of +\(\phi \hat{z}\). Thus the functions of functions with which we shall +be specially concerned will all be extensional functions of functions.</p> + +<p>What has just been said explains the connection (noted above) between +the fact that the functions of propositions with which mathematics +is specially<span class="pagenum" id="Page_23">[Pg 23]</span> concerned are all truth-functions and the fact that +mathematics is concerned with extensions rather than intensions.</p> + +<p><i>Convenient abbreviation</i>. The following definitions give +alternative and often more convenient notations: +\[ +\begin{array}{l} +\phi x . \supset_{x} . \psi x \colon = \colon (x) \colon \phi x . \supset . \psi x \quad \text{Df},\\ +\phi x . \equiv_{x} . \psi x \colon = \colon (x) \colon \phi x . \equiv . \psi x \quad \text{Df}. +\end{array} +\] +This notation "\(\phi x . \supset_{x}. \psi x\)" is due to Peano, who, +however, has no notation for the general idea "\((x) . \phi x\)." It +may be noticed as an exercise in the use of dots as brackets that we +might have written +\[ +\begin{array}{l} +\phi x \supset_{x} \psi x . = .(x). \phi x \supset \psi x \quad \text{Df},\\ +\phi x \equiv_{x} \psi x . = .(x). \phi x \equiv \psi x \quad \text{Df}. +\end{array} +\] +In practice however, when \(\phi \hat{x}\) and \(\psi \hat{x}\) are +special functions, it is not possible to employ fewer dots than in the +first form, and often more are required.</p> + +<p>The following definitions give abbreviated notations for functions of +two or more variables: +\[ +(x, y). \phi(x, y). = \colon (x) \colon (y). \phi(x, y) \quad \text{Df}, +\] +and so on for any number of variables; +\[ +\phi(x, y) . \supset_{x, y} . \psi(x, y) \colon = \colon (x, y) \colon \phi(x, y) . \supset . \psi(x, y) \quad \text{Df}, +\] +and so on for any number of variables.</p> + +<p><i>Identity</i>. The propositional function "\(x\) is identical with +\(y\)" is expressed by +\[ +x = y. +\] +This will be defined (cf. <a href="#*13·01">*13·01</a>), but, owing to certain difficult +points involved in the definition, we shall here omit it (cf. <a href="#CHAPTER_II">Chapter II</a>). We have, of course, +\[ +\begin{array}{l} +\vdash . x = x\,\, \text{(the law of identity)},\\ +\vdash \colon x = y . \equiv . y = x,\\ +\vdash : x = y . y = z . \supset . x = z.\\ +\end{array} +\] +The first of these expresses the <i>reflexive</i> property of identity: +a relation is called <i>reflexive</i> when it holds between a term +and itself, either universally, or whenever it holds between that +term and some term. The second of the above propositions expresses +that identity is a <i>symmetrical</i> relation: a relation is called +<i>symmetrical</i> if, whenever it holds between \(x\) and \(y\), it +also holds between \(y\) and \(x\). The third proposition expresses +that identity is a <i>transitive</i> relation: a relation is called +<i>transitive</i> if, whenever it holds between \(x\) and \(y\) and +between \(y\) and \(z\), it holds also between \(x\) and \(z\).</p> + +<p>We shall find that no new definition of the sign of equality is +required in mathematics: all mathematical equations in which the sign +of equality is<span class="pagenum" id="Page_24">[Pg 24]</span> used in the ordinary way express some identity, and +thus use the sign of equality in the above sense.</p> + +<p>If \(x\) and \(y\) are identical, either can replace the other in any +proposition without altering the truth-value of the proposition; thus +we have +\[ +\vdash \colon x = y \ldotp \supset \ldotp {\phi}x \equiv {\phi}y. +\] +This is a fundamental property of identity, from which the remaining +properties mostly follow.</p> + +<p>It might be thought that identity would not have much importance, +since it can only hold between \(x\) and \(y\) if \(x\) and \(y\) +are different symbols for the same object. This view, however, does +not apply to what we shall call "descriptive phrases," <i>i.e.</i> +"the so-and-so." It is in regard to such phrases that identity is +important, as we shall shortly explain. A proposition such as "Scott +was the author of Waverley" expresses an identity in which there is a +descriptive phrase (namely "the author of Waverley"); this illustrates +how, in such cases, the assertion of identity may be important. It is +essentially the same case when the newspapers say "the identity of the +criminal has not transpired." In such a case, the criminal is known by +a descriptive phrase, namely "the man who did the deed," and we wish +to find an \(x\) of whom it is true that "\(x\) = the man who did the +deed." When such an \(x\) has been found, the identity of the criminal +has transpired.</p> + +<p><i>Classes and relations.</i> A <i>class</i> (which is the same as a +<i>manifold</i> or <i>aggregate</i>) is all the objects satisfying +some propositional function. If \(\alpha\) is the class composed of +the objects satisfying \({\phi}\hat{x}\), we shall say that \(\alpha\) +is the class <i>determined</i> by \(\phi\hat{x}\). Every propositional +function thus determines a class, though if the propositional +function is one which is always false, the class will be <i>null</i>, +<i>i.e.</i> will have no members. The class determined by the function +\(\phi\hat{x}\) will be represented by \(\hat{z} ({\phi}z)\)<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a>. Thus +for example if \({\phi}x\) is an equation, \(\hat{z} ({\phi}z)\) will +be the class of its roots; if \({\phi}x\) is "\(x\) has two legs and +no feathers," \(\hat{z} ({\phi}z)\) will be the class of men; if +\({\phi}x\) is "\(0 \lt x \lt 1\)," \(\hat{z}({\phi}z)\) will be the +class of proper fractions, and so on.</p> + +<p>It is obvious that the same class of objects will have many determining +functions. When it is not necessary to specify a determining function +of a class, the class may be conveniently represented by a single Greek +letter. Thus Greek letters, other than those to which some constant +meaning is assigned, will be exclusively used for classes.</p> + +<p>There are two kinds of difficulties which arise in formal logic; one +kind arises in connection with classes and relations and the other in +connection with descriptive functions. The point of the difficulty for +classes and relations, so far as it concerns classes, is that a class +cannot be an object suitable as an argument to any of its determining +functions. If \(\alpha\) represents<span class="pagenum" id="Page_25">[Pg 25]</span> a class and \(\phi\hat{x}\) one +of its determining functions [so that \(\alpha = \hat{z}({\phi}z)\)], +it is not sufficient that \({\phi}\alpha\) be a false proposition, it +must be nonsense. Thus a certain classification of what appear to be +objects into things of essentially different types seems to be rendered +necessary. This whole question is discussed in <a href="#CHAPTER_II">Chapter II</a>, on the +theory of types, and the formal treatment in the systematic exposition, +which forms the main body of this work, is guided by this discussion. +The part of the systematic exposition which is specially concerned with +the theory of classes is <a href="#*20">*20</a>, and in this Introduction it is discussed +in <a href="#CHAPTER_III">Chapter III</a>. It is sufficient to note here that, in the complete +treatment of *20, we have avoided the decision as to whether a class of +things has in any sense an existence as one object. A decision of this +question in either way is indifferent to our logic, though perhaps, if +we had regarded some solution which held classes and relations to be +in some real sense objects as both true and likely to be universally +received, we might have simplified one or two definitions and a few +preliminary propositions. Our symbols, such as "\(\hat{x}({\phi}x)\)" +and \(\alpha\) and others, which represent classes and relations, are +merely defined in their use, just as \(\nabla^{2}\), standing for +\[ +\frac{\partial^{2}}{{\partial}x^{2}} + \frac{\partial^{2}}{{\partial}y^{2}} + \frac{\partial^{2}}{{\partial}z^{2}}, +\] +has no meaning apart from a suitable function of \(x\), \(y\), \(z\) +on which to operate. The result of our definitions is that the way in +which we use classes corresponds in general to their use in ordinary +thought and speech; and whatever may be the ultimate interpretation +of the one is also the interpretation of the other. Thus in fact our +classification of types in <a href="#CHAPTER_II">Chapter II</a> really performs the single, +though essential, service of justifying us in refraining from entering +on trains of reasoning which lead to contradictory conclusions. The +justification is that what seem to be propositions are really nonsense.</p> + +<p>The definitions which occur in the theory of classes, by which the +idea of a class (at least in use) is based on the other ideas assumed +as primitive, cannot be understood without a fuller discussion than +can be given now (cf. <a href="#CHAPTER_II">Chapter II</a> of this Introduction and also <a href="#*20">*20</a>). +Accordingly, in this preliminary survey, we proceed to state the more +important simple propositions which result from those definitions, +leaving the reader to employ in his mind the ordinary unanalysed +idea of a class of things. Our symbols in their usage conform to +the ordinary usage of this idea in language. It is to be noticed +that in the systematic exposition our treatment of classes and +relations requires no new primitive ideas and only two new primitive +propositions, namely the two forms of the "Axiom of Reducibility" (cf. +next Chapter) for one and two variables respectively.</p> + +<p>The propositional function "\(x\) is a member of the class \(\alpha\)" +will be expressed, following Peano, by the notation +\[ +x \in \alpha\text{.} +\]<span class="pagenum" id="Page_26">[Pg 26]</span> +Here \(\in\) is chosen as the initial of the word ἐστί. "\(x \in\alpha\)" +may be read "\(x\) is an \(\alpha\)." Thus "\(x \in \text{man}\)" +will mean "\(x\) is a man," and so on. For typographical convenience we +shall put +\[ +\begin{align} +x {\sim}\in \alpha \ldotp &= \ldotp \sim(x \in \alpha) \quad &\text{Df},\\ +x, y \in \alpha \ldotp &= \ldotp x \in \alpha . y \in \alpha \quad &\text{Df}.\\ +\end{align} +\]</p> + +<p>For "class" we shall write "Cls"; thus "\(\alpha \in \text{Cls}\)" +means "\(\alpha\) is a class."</p> + +<p>We have +\[ +\vdash \colon x \in \hat{z}({\phi}z) \ldotp \equiv \ldotp {\phi}x\text{,} +\] +<i>i.e.</i> "'\(x\) is a member of the class determined by +\(\phi\hat{z}\)' is equivalent to '\(x\) satisfies \(\phi\hat{z}\),' or +to '\({\phi}x\) is true.'"</p> + +<p>A class is wholly determinate when its membership is known, that is, +there cannot be two different classes having the same membership. +Thus if \({\phi}x\), \({\psi}x\) are formally equivalent functions, +they determine the same class; for in that case, if \(x\) is a member +of the class determined by \(\phi\hat{x}\), and therefore satisfies +\({\phi}x\), it also satisfies \({\psi}x\), and is therefore a member +of the class determined by \(\psi\hat{x}\). Thus we have +\[ +\vdash \colon\ldotp \hat{z}({\phi}z) = \hat{z}({\psi}z) \ldotp \equiv \colon {\phi}x \ldotp \equiv_{x} \ldotp {\psi}x\text{.} +\]</p> + +<p>The following propositions are obvious and important: +\[ +\vdash \colon \ldotp \alpha = \hat{z}({\phi}z) \ldotp \equiv \colon x \in \alpha \ldotp \equiv_{x} \ldotp {\phi}x\text{,} +\] +<i>i.e.</i> \(\alpha\) is identical with the class determined by +\(\phi\hat{z}\) when, and only when, +"\(x\) is an \(\alpha\)" is formally equivalent to \({\phi}x\); +\[ +\vdash \colon \ldotp \alpha = \beta \ldotp \equiv \colon x \in \alpha \ldotp \equiv_{x} \ldotp x \in \beta\text{,} +\] +<i>i.e.</i> two classes \(\alpha\) and \(\beta\) are identical when, +and only when, they have the same membership; +\[ +\vdash \ldotp \hat{x}(x \in \alpha) = \alpha\text{,} +\] +<i>i.e.</i> the class whose determining function is "\(x\) is an +\(\alpha\)" is \(\alpha\), in other words, \(\alpha\) is the class of +objects which are members of \(\alpha\); +\[ +\vdash \ldotp \hat{z}({\phi}z) \in \text{Cls}\text{,} +\] +<i>i.e.</i> the class determined by the function \(\phi\hat{z}\) is a +class.</p> + +<p>It will be seen that, according to the above, any function of one +variable can be replaced by an equivalent function of the form "\(x +\in \alpha\)." Hence any extensional function of functions which holds +when its argument is a function of the form "\(\hat{z} \in \alpha\)," +whatever possible value \(\alpha\) may have, will hold also when its +argument is any function \(\phi\hat{z}\). Thus variation of classes can +replace variation of functions of one variable in all the propositions +of the sort with which we are concerned.</p> + +<p>In an exactly analogous manner we introduce dual or dyadic relations, +<i>i.e.</i> relations between two terms. Such relations will be called +simply "relations"; relations between more than two terms will be +distinguished<span class="pagenum" id="Page_27">[Pg 27]</span> as <i>multiple</i> relations, or (when the number of +their terms is specified) as triple, quadruple, ... relations, or as +triadic, tetradic, ... relations. Such relations will not concern us +until we come to Geometry. For the present, the only relations we are +concerned with are <i>dual</i> relations.</p> + +<p>Relations, like classes, are to be taken in <i>extension</i>, +<i>i.e.</i> if \(R\) and \(S\) are relations which hold between the +same pairs of terms, \(R\) and \(S\) are to be identical. We may regard +a relation, in the sense in which it is required for our purposes, as +a class of couples; <i>i.e.</i> the couple (\(x, y)\) is to be one +of the class of couples constituting the relation \(R\) if \(x\) has +the relation \(R\) to \(y\)<a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a>. This view of relations as classes of +couples will not, however, be introduced into our symbolic treatment, +and is only mentioned in order to show that it is possible so to +understand the meaning of the word relation that a relation shall be +determined by its extension.</p> + +<p>Any function \(\phi(x,y)\) determines a relation \(R\) between \(x\) +and \(y\). If we regard a relation as a class of couples, the relation +determined by \(\phi(x,y)\) is the class of couples \((x,y)\) for +which \(\phi(x,y)\) is true. The relation determined by the function +\(\phi(x,y)\) will be denoted by +\[ +\hat{x}\hat{y} \phi(x,y). +\] +We shall use a capital letter for a relation when it is not necessary +to specify the determining function. Thus whenever a capital letter +occurs, it is to be understood that it stands for a relation.</p> + +<p>The propositional function "\(x\) has the relation \(R\) to \(y\)" will +be expressed by the notation +\[ +xRy. +\] +This notation is designed to keep as near as possible to common +language, which, when it has to express a relation, generally mentions +it between its terms, as in "\(x\) loves \(y\)," "\(x\) equals \(y\)," +"\(x\) is greater than \(y\)," and so on. For "relation" we shall write +"Rel" thus "\(R \in \text{Rel}\)" means "\(R\) is a relation."</p> + +<p>Owing to our taking relations in extension, we shall have +\[ +\vdash \colon\ldotp \hat{x}\hat{y} \phi(x,y) = \hat{x}\hat{y} \psi(x,y).\equiv \colon \phi(x,y). \equiv_{x,y} . \psi(x,y), +\] +<i>i.e.</i> two functions of two variables determine the same relation +when, and only when, the two functions are formally equivalent. +\[ +\text{We have}\qquad\qquad \vdash. z \{\hat{x}\hat{y} \phi(x,y)\} w . \equiv . \phi(z,w), +\] +<i>i.e.</i> "\(z\) has to \(w\) the relation determined by the function +\(\phi(x,y)\)" is equivalent to \(\phi(z,w)\); +\[ +\begin{array}{l} +\vdash \colon\ldotp R = \hat{x}\hat{y} \phi(x,y). \equiv \colon xRy. \equiv_{x,y} . \phi(x,y),\\ +\vdash \colon\ldotp R = S .\equiv \colon xRy. \equiv_{x,y} . xSy,\\ +\vdash . \hat{x}\hat{y} (xRy) = R,\\ +\vdash . \{\hat{x}\hat{y} \phi(x,y)\} \in\, \text{Rel}. +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_28">[Pg 28]</span></p> + +<p>These propositions are analogous to those previously given for classes. +It results from them that any function of two variables is formally +equivalent to some function of the form \(xRy\); hence, in extensional +functions of two variables, variation of relations can replace +variation of functions of two variables.</p> + +<p>Both classes and relations have properties analogous to most of those +of propositions that result from negation and the logical sum. The +<i>logical product</i> of two classes \(\alpha\) and \(\beta\) is their +common part, <i>i.e.</i> the class of terms which are members of both. +This is represented by \(\alpha \cap \beta\). Thus we put +\[ +\alpha \cap \beta = \hat{x} (x \in \alpha . x \in \beta) \qquad \text{Df}. +\] +This gives us +\[ +\vdash \colon x \in \alpha \cap \beta . \equiv . x \in \alpha . x \in \beta, +\] +<i>i.e.</i> "\(x\) is a member of the logical product of \(\alpha\) and +\(\beta\)" is equivalent to the logical product of "\(x\) is a member +of \(\alpha\)" and "\(x\) is a member of \(\beta\)."</p> + +<p>Similarly the <i>logical sum</i> of two classes \(\alpha\) and +\(\beta\) is the class of terms which are members of either; we denote +it by \(\alpha \cup \beta\). The definition is +\[ +\alpha \cup \beta = \hat{x} (x \in \alpha . \lor . x \in \beta) \qquad \text{Df}, +\] +and the connection with the logical sum of propositions is given by +\[ +\vdash \colon\ldotp x \in \alpha \cup \beta . \equiv \colon x \in \alpha . \lor . x \in \beta. +\]</p> + +<p>The <i>negation</i> of a class \(\alpha\) consists of those terms \(x\) +for which "\(x \in \alpha\)" can be <i>significantly and truly</i> +denied. We shall find that there are terms of other types for which +"\(x \in \alpha\)" is neither true nor false, but nonsense. These terms +are not members of the negation of \(\alpha\).</p> + +<p>Thus the <i>negation</i> of a class \(\alpha\) is the class of terms +of suitable type which are not members of it, <i>i.e.</i> the class +\(\hat{x} (x {\sim \in \alpha})\). We call this class "\(-\alpha\)" +(read "not\(-\alpha\)"); thus the definition is +\[ +- \alpha = \hat{x} (x {\sim \in \alpha}) \qquad \text{Df}, +\] +and the connection with the negation of propositions is given by +\[ +\vdash \colon x \in -\alpha . \equiv . x {\sim \in \alpha}. +\]</p> + +<p>In place of implication we have the relation of <i>inclusion</i>. +A class \(\alpha\) is said to be included or contained in a class +\(\beta\) if all members of \(\alpha\) are members of \(\beta\), +<i>i.e.</i> if \(x \in \alpha . \supset_{x} . x \in \beta\). We write +"\(\alpha \subset \beta\)" for "\(\alpha\) is contained in \(\beta\)." +Thus we put +\[ +\alpha \subset \beta . = \colon x \in \alpha. \supset_{x} . x \in \beta \qquad \text{Df}. +\]</p> + +<p>Most of the formulae concerning \(p . q\), \(p \lor q\), \({\sim}p\), +\(p \supset q\) remain true if we substitute \(\alpha \cap \beta\), +\(\alpha \cup \beta\), -\(\alpha\), \(\alpha \subset \beta\). In place +of equivalence, we substitute identity; for "\(p \equiv q\)" was +defined as "\(p \supset q . q \subset p\)," but "\(\alpha \subset \beta . \beta \subset \alpha\)" +gives "\(x \in \alpha . \equiv_{x} . x \in\beta\)," +whence \(\alpha = \beta\).</p> + +<p><span class="pagenum" id="Page_29">[Pg 29]</span></p> + +<p>The following are some propositions concerning classes which are +analogues of propositions previously given concerning propositions: +\[ +\vdash \ldotp \alpha \cap \beta = -(-\alpha \cup -\beta)\text{,} +\] +<i>i.e.</i> the common part of \(\alpha\) and \(\beta\) is the negation +of "not\(-\alpha\) or not\(-\beta\)"; +\[ +\vdash \ldotp x \in (\alpha \cup -\alpha)\text{,} +\] +<i>i.e.</i>, "\(x\) is a member of \(\alpha\) or not\(-\alpha\)"; +\[ +\vdash \ldotp x \sim\in(\alpha \cap -\alpha), +\] +<i>i.e.</i> "\(x\) is not a member of both \(\alpha\) and not\(-\alpha\)"; +\[ +\begin{align} +&\vdash \ldotp \alpha = -(-\alpha)\text{,}\\ +&\vdash \colon \alpha \subset \beta \ldotp \equiv \ldotp -\beta \subset -\alpha\text{,}\\ +&\vdash \colon \alpha = \beta \ldotp \equiv \ldotp -\alpha = -\beta\text{,}\\ +&\vdash \colon \alpha = \alpha \cap \alpha\text{,}\\ +&\vdash \colon \alpha = \alpha \cup \alpha\text{.}\\ +\end{align} +\]</p> + +<p>The two last are the two forms of the law of tautology.</p> + +<p>The law of absorption holds in the form +\[ +\vdash \colon \alpha \subset \beta \ldotp \equiv \ldotp \alpha = \alpha \cap \beta\text{.} +\]</p> + +<p>Thus for example "all Cretans are liars" is equivalent to "Cretans are +identical with lying Cretans."</p> + +<p>Just as we have +\[\vdash \colon p \supset q \ldotp q \supset r \ldotp \supset \ldotp p \supset r, +\] +so we have +\[\vdash \colon \alpha \subset \beta \ldotp \beta \subset \gamma \ldotp \supset \ldotp \alpha \subset \gamma. +\]</p> + +<p>This expresses the ordinary syllogism in Barbara (with the premisses +interchanged); for "\(\alpha \subset \beta\)" means the same as +"all \(\alpha\)'s are \(\beta\)'s," so that the above proposition +states: "If all \(\alpha\)'s are \(\beta\)'s, and all \(\beta\)'s are +\(\gamma\)'s, then all \(\alpha\)'s are \(\gamma\)'s." (It should be +observed that syllogisms are traditionally expressed with "therefore," +as if they asserted both premisses and conclusion. This is, of course, +merely a slipshod way of speaking, since what is really asserted is +only the connection of premisses with conclusion.)</p> + +<p>The syllogism in Barbara when the minor premiss has an individual +subject is +\[ +\vdash \colon x \in \beta \ldotp \beta \subset \gamma \ldotp \supset \ldotp x \in \gamma\text{,} +\] +<i>e.g.</i> "if Socrates is a man, and all men are mortals, then +Socrates is a mortal." This, as was pointed out by Peano, is not a +particular case of "\(\alpha \subset \beta \ldotp \beta \subset \gamma\ldotp \supset \ldotp \alpha \subset \gamma\)," +since "\(x \in \beta\)" is not a particular case of "\(\alpha \subset\beta\)." +This point is important, since traditional logic is here mistaken. The +nature and magnitude of its mistake will become clearer at a later +stage.</p> + +<p>For relations, we have precisely analogous definitions and propositions. +We put +\[ +\begin{align} +&R \dot{\cap} S = \hat{x}\hat{y}(xRy \ldotp xSy) \quad \text{Df}\text{,}\\ +\text{which leads to}\quad &\vdash \colon x (R \dot{\cap} S) y \ldotp \equiv \ldotp xRy \ldotp xSy. +\end{align} +\]<span class="pagenum" id="Page_30">[Pg 30]</span> +\[ +\begin{align} +\text{Similarly} \quad R \unicode{x228d} S &= \hat{x}\hat{y}(xRy \ldotp \lor \ldotp xSy) \quad &\text{Df}\text{,}\\ +\dot{-}R &= \hat{x}\hat{y}\{\sim(xRy)\} \quad &\text{Df}\text{,}\\ +R \unicode{x2abd} S \ldotp &= \colon xRy \ldotp \supset_{x, y} \ldotp xSy \quad &\text{Df}\text{.}\\ +\end{align} +\]</p> + +<p>Generally, when we require analogous but different symbols for +relations and for classes, we shall choose for relations the symbol +obtained by adding a dot, in some convenient position, to the +corresponding symbol for classes. (The dot must not be put on the line, +since that would cause confusion with the use of dots as brackets.) But +such symbols require and receive a special definition in each case.</p> + +<p>A class is said to <i>exist</i> when it has at least one member: +"\(\alpha\) exists" is denoted by "\(\exists ! \alpha.\)" Thus we put +\[ +\exists ! \alpha \ldotp = \ldotp (\exists x) \ldotp x \in \alpha \quad \text{Df}\text{.} +\] +The class which has no members is called the "null-class," and is +denoted by "\(\Lambda\)." Any propositional function which is always +false determines the null-class. One such function is known to us +already, namely "\(x\) is not identical with \(x\)," which we denote by +"\(x \neq x\)." Thus we may use this function for defining \(\Lambda\), +and put +\[ +\Lambda = \hat{x}(x \neq x) \quad \text{Df}\text{.} +\]</p> + +<p>The class determined by a function which is always true is called the +<i>universal class</i>, and is represented by \(\text{V}\); thus +\[ +\text{V} = \hat{x}(x = x) \quad \text{Df}\text{.} +\]</p> + +<p>Thus \(\Lambda\) is the negation of \(\text{V}\). We have +\[ +\vdash \ldotp (x) \ldotp x \in \text{V}\text{,} +\] +<i>i.e.</i> "'\(x\) is a member of \(\text{V}\)' is always true"; and +\[ +\vdash \ldotp (x) \ldotp x {\sim}\in \Lambda, +\] +<i>i.e.</i> "'\(x\) is a member of \(\Lambda\)' is always false." Also +\[ +\vdash \colon \alpha = \Lambda \ldotp \equiv \ldotp {\sim}\exists! \alpha\text{,} +\] +<i>i.e.</i> "\(\alpha\) is the null-class" is equivalent to "\(\alpha\) +does not exist."</p> + +<p>For relations we use similar notations. We put +\[ +\dot{\exists}! R \ldotp = \ldotp (\exists x, y) \ldotp xRy\text{,} +\] +<i>i.e.</i> "\(\dot{\exists}! R\)" means that there is at least +one couple \(x\), \(y\) between which the relation \(R\) holds. +\(\dot{\Lambda}\) will be the relation which never holds, and +\(\dot{\text{V}}\) the relation which always holds. \(\dot{\text{V}}\) is +practically never required; \(\dot{\Lambda}\) will be the relation +\(\hat{x}\hat{y}(x \neq x . y \neq y)\). We have +\[ +\begin{align} +&\vdash \ldotp (x, y) \ldotp {\sim}(x \dot{\Lambda} y),\\ +\text{and} \qquad &\vdash \colon R = \dot{\Lambda} \ldotp \equiv \ldotp {\sim}\dot{\exists}! R\text{.} +\end{align} +\]</p> + +<p><span class="pagenum" id="Page_31">[Pg 31]</span></p> + +<p>There are no classes which contain objects of more than one type. +Accordingly there is a universal class and a null-class proper to each +type of object. But these symbols need not be distinguished, since +it will be found that there is no possibility of confusion. Similar +remarks apply to relations.</p> + +<p><i>Descriptions</i>. By a "description" we mean a phrase of the form +"<i>the</i> so-and-so" or of some equivalent form. For the present, +we confine our attention to <i>the</i> in the singular. We shall use +this word strictly, so as to imply uniqueness; <i>e.g</i>. we should +not say "\(A\) is <i>the</i> son of \(B\)" if \(B\) had other sons +besides \(A\). Thus a description of the form "the so-and-so" will +only have an application in the event of there being one so-and-so +and no more. Hence a description requires some propositional function +\(\phi\hat{x}\) which is satisfied by one value of \(x\) and by no +other values; then "the \(x\) which satisfies \(\phi\hat{x}\)" is a +description which definitely describes a certain object, though we +may not know what object it describes. For example, if \(y\) is a +man, "\(x\) is the father of \(y\)" must be true for one, and only +one, value of \(x\). Hence "the father of \(y\)" is a description of +a certain man, though we may not know <i>what</i> man it describes. A +phrase containing "the" always presupposes some initial propositional +function not containing "the"; thus instead of "\(x\) is the father of +\(y\)" we ought to take as our initial function "\(x\) begot \(y\)"; +then "the father of \(y\)" means the one value of \(x\) which satisfies +this propositional function.</p> + +<p>If \(\phi \hat{x}\) is a propositional function, the symbol +"\(({℩}x)(\phi x)\)" is used in our symbolism in such a way that it +can always be read as "the \(x\) which satisfies \(\phi \hat{x}\)." +But we do not define "\(({℩}x)(\phi x)\)" as standing for "the \(x\) +which satisfies \(\phi \hat{x}\)," thus treating this last phrase as +embodying a primitive idea. Every use of "\(({℩}x)(\phi x)\)," where +it apparently occurs as a constituent of a proposition in the place +of an object, is defined in terms of the primitive ideas already on +hand. An example of this definition in use is given by the proposition +"\(\text{E}!({℩}x)(\phi x)\)" which is considered immediately. The +whole subject is treated more fully in <a href="#CHAPTER_III">Chapter III</a>.</p> + +<p><span class="pagenum" id="Page_32">[Pg 32]</span></p> + +<p>The symbol should be compared and contrasted with "\(\hat{x}(\phi x)\)" +which in use can always be read as "the \(x\)'s which satisfy +\(\phi \hat{x}\)." Both symbols are incomplete symbols defined +only in use, and as such are discussed in <a href="#CHAPTER_III">Chapter III</a>. The symbol +"\(\hat{x}(\phi x)\)" always has an application, namely to the class +determined by \(\phi x\); but "\(({℩}x)(\phi x)\)" only has an +application when \(\phi \hat{x}\) is only satisfied by one value of +\(x\), neither more nor less. It should also be observed that the +meaning given to the symbol by the definition, given immediately below, +of \(\text{E}!({℩}x)(\phi x)\) does not presuppose that we know the +meaning of "one." This is also characteristic of the definition of any +other use of \(({℩}x)(\phi x)\).</p> + +<p>We now proceed to define "\(\text{E}!({℩}x)(\phi x)\)" so that it can +be read "the \(x\) satisfying \(\phi x\) exists." (It will be observed +that this is a different meaning of existence from that which we +express by "\(\exists\)"). Its definition is +\[ +\text{E}! ({℩}x) (\phi x) . =: (\exists c): \phi x . \equiv_{x} . x = c\quad \text{Df,} +\] +<i>i.e</i>. "the \(x\) satisfying \(\phi \hat{x}\) exists" is to mean +"there is an object \(c\) such that \(\phi x\) is true when \(x\) is +\(c\) but not otherwise."</p> + +<p>The following are equivalent forms: +\[ +\begin{aligned} +&\vdash \colon\ldotp \text{E}! ({℩}x) (\phi x) . \equiv : (\exists c) : \phi c : \phi x . \supset_{x} . x = c,\\ +&\vdash \colon\ldotp \text{E}! ({℩}x) (\phi x) . \equiv : (\exists c) . \phi c : \phi x . \phi y . \supset_{x, y} . x = y,\\ +&\vdash \colon\ldotp \text{E}! ({℩}x) (\phi x) . \equiv : (\exists c) : \phi c : x \neq c . \supset_{x} . \sim\phi x. +\end{aligned} +\]</p> + +<p>The last of these states that "the \(x\) satisfying \(\phi \hat{x}\) +exists" is equivalent to "there is an object \(c\) satisfying \(\phi\hat{x}\), +and every object other than \(c\) does not satisfy \(\phi\hat{x}\)."</p> + +<p>The kind of existence just defined covers a great many cases. Thus for +example "the most perfect Being exists" will mean: +\[ +(\exists c): x\,\,\text{ is most perfect}. \equiv_{x} . x = c, +\] +which, taking the last of the above equivalences, is equivalent to +\[ +(\exists c) : c\text{ is most perfect}: x \neq c . \supset_{x} . x\,\,\text{ is not most perfect}. +\]</p> + +<p>A proposition such as "Apollo exists" is really of the same logical +form, although it does not explicitly contain the word <i>the</i>. For +"Apollo" means really "the object having such-and-such properties," +say "the object having the properties enumerated in the Classical +Dictionary<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a>." If these properties make up the propositional function +\(\phi x\), then "Apollo" means "\((℩x)(\phi x)\)," and "Apollo exists" +means "\(\text{E}!(℩x)(\phi x)\)." To take another illustration, "the +author of Waverley" means "the man who (or rather, the object which) +wrote Waverley." Thus "Scott is the author of Waverley" is +\[ +\text{Scott} = ({℩}x)(x\,\text{ wrote Waverley}). +\] +Here (as we observed before) the importance of <i>identity</i> in +connection with descriptions plainly appears.</p> + +<p>The notation "\(({℩}x)(\phi x)\)," which is long and inconvenient, is +seldom used, being chiefly required to lead up to another notation, +namely "\(Rʻy\)" meaning "the object having the relation \(R\) to +\(y\)." That is, we put +\[ +\begin{aligned} +Rʻy = ({℩}x) (xRy)\quad \text{Df.} +\end{aligned} +\] +The inverted comma may be read "of." Thus "\(Rʻy\)" is read "the \(R\) +of \(y\)." Thus if \(R\) is the relation of father to son, "\(Rʻy\)" +means "the father of \(y\)"; if \(R\) is the relation of son to father, +"\(Rʻy\)" means "the son of \(y\)," which will<span class="pagenum" id="Page_33">[Pg 33]</span> only "exist" if \(y\) +has one son and no more. \(Rʻy\) is a function of \(y\), but not a +propositional function; we shall call it a <i>descriptive</i> function. +All the ordinary functions of mathematics are of this kind, as will +appear more fully in the sequel. Thus in our notation, "\(\sin y\)" +would be written "\(\sin ʻy\)," and "sin" would stand for the relation +which \(\sin ʻy\) has to \(y\). Instead of a variable descriptive +function \(fy\), we put \(Rʻy\), where the variable relation \(R\) +takes the place of the variable function \(f\). A descriptive function +will in general exist while \(y\) belongs to a certain domain, but not +outside that domain; thus if we are dealing with positive rationals, +\(\surd y\) will be significant if \(y\) is a perfect square, but not +otherwise; if we are dealing with real numbers, and agree that "\(\surd y\)" +is to mean the <i>positive</i> square root (or, is to mean the +negative square root), \(\surd y\) will be significant provided \(y\) +is positive, but not otherwise; and so on. Thus every descriptive +function has what we may call a "domain of definition" or a "domain of +existence," which may be thus defined: If the function in question is +\(Rʻy\), its domain of definition or of existence will be the class of +those arguments \(y\) for which we have \(\text{E}!Rʻy\), <i>i.e.</i> +for which \(\text{E}! (℩x) (xRy)\), <i>i.e.</i> for which there is one +\(x\), and no more, having the relation \(R\) to \(y\).</p> + +<p>If \(R\) is any relation, we will speak of \(Rʻy\) as the "associated +descriptive function." A great many of the constant relations which +we shall have occasion to introduce are only or chiefly important on +account of their associated descriptive functions. In such cases, it is +easier (though less correct) to begin by assigning the meaning of the +descriptive function, and to deduce the meaning of the relation from +that of the descriptive function. This will be done in the following +explanations of notation.</p> + +<p><i>Various descriptive functions of relations.</i> If \(R\) is any +relation, the <i>converse</i> of \(R\) is the relation which holds +between \(y\) and \(x\) whenever \(R\) holds between \(x\) and \(y\). +Thus <i>greater</i> is the converse of <i>less</i>, <i>before</i> +of <i>after</i>, <i>cause</i> of <i>effect</i>, <i>husband</i> of +<i>wife</i>, etc. The converse of \(R\) is written<a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a> \(\text{Cnv}ʻR\) +or \(\breve{R}\). The definition is +\[ +\begin{aligned} +\breve{R} &= \hat{x}\hat{y} (yRx) \quad &\text{Df},\\ +\text{Cnv}ʻR &= \breve{R} \quad &\text{Df}.\\ +\end{aligned} +\] +The second of these is not a formally correct definition, since we +ought to define "Cnv" and deduce the meaning of \(\text{Cnv}ʻR\). But +it is not worth while to adopt this plan in our present introductory +account, which aims at simplicity rather than formal correctness.</p> + +<p>A relation is called <i>symmetrical</i> if \(R = \breve{R}\), +<i>i.e.</i> if it holds between \(y\) and \(x\) whenever it holds +between \(x\) and \(y\) (and therefore vice versa). Identity,<span class="pagenum" id="Page_34">[Pg 34]</span> +diversity, agreement or disagreement in any respect, are symmetrical +relations. A relation is called <i>asymmetrical</i> when it is +incompatible with its converse, <i>i.e.</i> when +\(R \dot{\cap}\breve{R} = \dot{\Lambda}\), or, what is equivalent, +\[ +xRy . \supset_{x, y} . \sim{(yRx)}. +\]</p> + +<p>Before and after, greater and less, ancestor and descendant, are +asymmetrical, as are all other relations of the sort that lead to +<i>series</i>. But there are many asymmetrical relations which do not +lead to series, for instance, that of wife's brother<a id="FNanchor_11" href="#Footnote_11" class="fnanchor">[11]</a>. A relation +maybe neither symmetrical nor asymmetrical; for example, this holds of +the relation of inclusion between classes: \(\alpha \subset \beta\) +and \(\beta \subset \alpha\) will both be true if \(\alpha = \beta\), +but otherwise only one of them, at most, will be true. The relation +<i>brother</i> is neither symmetrical nor asymmetrical, for if \(x\) is +the brother of \(y\), \(y\) may be either the brother or the sister of +\(x\).</p> + +<p>In the propositional function \(xRy\), we call \(x\) the referent and +\(y\) the relatum. The class \(\hat{x}(xRy)\), consisting of all the +\(x\)'s which have the relation \(R\) to \(y\), is called the class of +referents of \(y\) with respect to \(R\); the class \(\hat{y}(xRy)\), +consisting of all the \(y\)'s to which \(x\) has the relation \(R\), +is called the class of relata of \(x\) with respect to \(R\). These +two classes are denoted respectively by \(\overrightarrow{R}ʻy\) and +\(\overleftarrow{R}ʻx\). Thus +\[ +\begin{array}{l} +\overrightarrow{R}ʻy \,= \hat{x}(xRy) \qquad \text{Df},\\ +\overleftarrow{R}ʻx = \hat{y}(xRy) \qquad \text{Df}. +\end{array} +\] +The arrow runs towards \(y\) in the first case, to show that we are +concerned with things having the relation \(R\) to \(y\); it runs away +from \(x\) in the second case to show that the relation \(R\) goes +<i>from</i> \(x\) to the members of \(\overleftarrow{R}ʻx\). It runs in +fact <i>from</i> a referent and <i>towards</i> a relatum.</p> + +<p>The notations \(\overrightarrow{R}ʻy\), \(\overleftarrow{R}ʻx\) are +very important, and are used constantly. If \(R\) is the relation of +parent to child, \(\overrightarrow{R}ʻy\) = the parents of \(y\), +\(\overleftarrow{R}ʻx\) = the children of \(x\). We have +\[ +\begin{array}{l} +&\vdash \colon x \in \overrightarrow{R}ʻy . \equiv . xRy\\ +\text{and}\qquad\qquad &\vdash \colon y \in \overleftarrow{R}ʻx . \equiv . xRy. +\end{array} +\] +These equivalences are often embodied in common language. For example, +we say indiscriminately "\(x\) is an inhabitant of London" or "\(x\) +inhabits London." If we put "\(R\)" for "inhabits," "\(x\) inhabits +London" is "\(xR\) London," while "\(x\) is an inhabitant of London" is +"\(x \in \overrightarrow{R}ʻ\text{London}\)."</p> + +<p><span class="pagenum" id="Page_35">[Pg 35]</span></p> + +<p>Instead of \(\overrightarrow{R}\) and \(\overleftarrow{R}\) we +sometimes use \(\text{sg}ʻR\), \(\text{gs}ʻR\), where "\(\text{sg}\)" +stands for "sagitta," and "\(\text{gs}\)" is "\(\text{sg}\)" backwards. +Thus we put +\[ +\begin{array}{l} +\text{sg}ʻR &= \overrightarrow{R} \qquad \text{Df},\\ +\text{gs}ʻR &= \overleftarrow{R} \qquad \text{Df}. +\end{array} +\] +These notations are sometimes more convenient than an arrow when the +relation concerned is represented by a combination of letters, instead +of a single letter such as \(R\). Thus <i>e.g.</i> we should write +\(\text{sg}ʻ(R \dot{\cap} S)\), rather than put an arrow over the whole +length of (\(R \dot{\cap} S\)).</p> + +<p>The class of all terms that have the relation \(R\) to something or +other is called the <i>domain</i> of \(R\). Thus if \(R\) is the +relation of parent and child, the domain of \(R\) will be the class of +parents. We represent the domain of \(R\) by "\(\text{D}ʻR\)." Thus we +put +\[ +\text{D}ʻR = \hat{x} \{(\exists y) . xRy\} \qquad \text{Df}. +\] +Similarly the class of all terms to which something or other has the +relation \(R\) is called the <i>converse domain</i> of \(R\); it is the +same as the domain of the converse of \(R\). The converse domain of +\(R\) is represented by "\(\text{ᗡ}ʻR\)"; thus +\[ +\text{ᗡ}ʻR = \hat{y} \{(\exists x) . xRy\} \qquad \text{Df}. +\] +The sum of the domain and the converse domain is called the field, and +is represented by \(CʻR\): thus +\[ +CʻR = \text{D}ʻR \cup \text{ᗡ}ʻR \qquad \text{Df}. +\]</p> + +<p>The <i>field</i> is chiefly important in connection with series. If +\(R\) is the ordering relation of a series, \(CʻR\) will be the class +of terms of the series, \(\text{D}ʻR\) will be all the terms except +the last (if any), and \(\text{ᗡ}ʻR\) will be all the terms except +the first (if any). The first term, if it exists, is the only member +of \(\text{D}ʻR \cap - \text{ᗡ}ʻR\), since it is the only term which +is a predecessor but not a follower. Similarly the last term (if any) +is the only member of \(\text{ᗡ}ʻR \cap - \text{D}ʻR\). The condition +that a series should have no end is \(\text{ᗡ}ʻR \subset \text{D}ʻR\), +<i>i.e.</i> "every follower is a predecessor"; the condition for no +beginning is \(\text{D}ʻR \subset \text{ᗡ}ʻR\). These conditions are +equivalent respectively to \(\text{D}ʻR = CʻR\) and \(\text{ᗡ}ʻR = CʻR\).</p> + +<p>The <i>relative product</i> of two relations \(R\) and \(S\) is +the relation which holds between \(x\) and \(z\) when there is an +intermediate term \(y\) such that \(x\) has the relation \(R\) to \(y\) +and \(y\) has the relation \(S\) to \(z\). The relative product of +\(R\) and \(S\) is represented by \(R \mid S\); thus we put +\[ +\begin{array}{l} +&R \mid S = \hat{x}\hat{z} \{(\exists y) . xRy. ySz\} \qquad \text{Df},\\ +\text{whence}\qquad\qquad &\vdash \colon x(R \mid S) z . \equiv . (\exists y) . xRy. ySz. +\end{array} +\] +Thus "paternal aunt" is the relative product of <i>sister</i> and +<i>father</i>; "paternal grandmother" is the relative product of +<i>mother</i> and <i>father</i>; "maternal<span class="pagenum" id="Page_36">[Pg 36]</span> grandfather" is the +relative product of <i>father</i> and <i>mother</i>. The relative +product is not commutative, but it obeys the associative law, +<i>i.e.</i> +\[ +\vdash \ldotp (P\mid Q)\mid R = P\mid (Q\mid R)\text{.} +\] +It also obeys the distributive law with regard to the logical addition +of relations, <i>i.e.</i> we have +\[ +\begin{align} +\vdash \ldotp P\mid (Q \unicode{x228d} R) &= (P\mid Q) \unicode{x228d} (P\mid R)\text{,}\\ +\vdash \ldotp (Q \unicode{x228d} R) \mid P &= (Q\mid P) \unicode{x228d} (R\mid P)\text{.}\\ +\end{align} +\]</p> + +<p>But with regard to the logical <i>product</i>, we have only +\[ +\begin{align} +\vdash \ldotp P\mid (Q\dot{\cap}R)\unicode{x2abd}(P\mid Q)\dot{\cap}(P\mid R)\text{,}\\ +\vdash \ldotp (Q\dot{\cap}R)\mid P\unicode{x2abd}(Q\mid P)\dot{\cap}(R\mid P)\text{.}\\ +\end{align} +\]</p> + +<p>The relative product does not obey the law of tautology, <i>i.e.</i> we +do not have in general \(R\mid R = R\). We put +\[ +R^{2} = R\mid R \quad \text{Df}\text{.} +\] +Thus paternal grandfather = (\(\text{father})^{2}\),</p> + +<p>maternal grandmother = (\(\text{mother})^{2}\).</p> + +<p>A relation is called <i>transitive</i> when \(R^{2}\unicode{x2abd}R\), +<i>i.e.</i> when, if \(xRy\) and \(yRz\), we always have \(xRz\), +<i>i.e.</i> when +\[ +xRy \ldotp yRz \ldotp \supset_{x, y, z} \ldotp xRz\text{.} +\] +Relations which generate series are always transitive; thus <i>e.g.</i> +\[ +x \gt y \ldotp y \gt z \ldotp \supset_{x, y, z} \ldotp x \gt z\text{.} +\] +If \(P\) is a relation which generates a series, \(P\) may conveniently +be read "precedes"; thus "\(xPy \ldotp yPz \ldotp \supset_{x, y, z}\ldotp xPz\)" +becomes "if \(x\) precedes \(y\) and \(y\) precedes \(z\), then \(x\) +always precedes \(z\)." The class of relations which generate series +are partially characterized by the fact that they are transitive and +asymmetrical, and never relate a term to itself.</p> + +<p>If \(P\) is a relation which generates a series, and if we have not +merely \(P^{2}\unicode{x2abd}P\), but \(P^{2}=P\), then \(P\) generates a +series which is <i>compact</i> (<i>überall dicht</i>), <i>i.e.</i> such +that there are terms between any two. For in this case we have +\[ +xPz \ldotp \supset \ldotp (\exists y) \ldotp xPy \ldotp yPz\text{,} +\] +<i>i.e.</i> if \(x\) precedes \(z\), there is a term \(y\) such that +\(x\) precedes \(y\) and \(y\) precedes \(z\), <i>i.e.</i> there is +a term between \(x\) and \(z\). Thus among relations which generate +series, those which generate compact series are those for which \(P^{2} = P\).</p> + +<p><span class="pagenum" id="Page_37">[Pg 37]</span></p> + +<p>Many relations which do not generate series are transitive, for +example, identity, or the relation of inclusion between classes. Such +cases arise when the relations are not asymmetrical. Relations which +are transitive and symmetrical are an important class: they may be +regarded as consisting in the possession of some common property.</p> + +<p><i>Plural descriptive functions.</i> The class of terms \(x\) which +have the relation +\(R\) to some member of a class \(\alpha\) is denoted by \(Rʻʻ\alpha\) or \(R_{\in}ʻ\alpha\). The definition is +\[ +Rʻʻ\alpha = \hat{x}\{(\exists y) \ldotp y \in \alpha \ldotp xRy\} \quad \text{Df}\text{.} +\] +Thus for example let \(R\) be the relation of <i>inhabiting</i>, and +\(\alpha\) the class of towns; then \(Rʻʻ\alpha\) = inhabitants of +towns. Let \(R\) be the relation "less than" among rationals, and +\(\alpha\) the class of those rationals which are of the form +\(1 - 2^{-n}\), for integral values of \(n\); then \(Rʻʻ\alpha\) will be +all rationals less than some member of \(\alpha\), <i>i.e.</i> all +rationals less than \(1\). If \(P\) is the generating relation of a +series, and \(\alpha\) is any class of members of \(P\), \(Pʻʻ\alpha\) +will be predecessors of \(\alpha\)'s, <i>i.e.</i> the segment defined +by \(\alpha\). If \(P\) is a relation such that \(Pʻy\) always exists +when \(y \in \alpha\), \(Pʻʻ\alpha\) will be the class of all terms of +the form \(Pʻy\) for values of \(y\) which are members of \(\alpha\); +<i>i.e.</i> +\[ +Pʻʻ\alpha = \hat{x}\{(\exists y) \ldotp y \in \alpha \ldotp x = Pʻy\}\text{.} +\] +Thus a member of the class "fathers of great men" will be the father +of \(y\), where \(y\) is some great man. In other cases, this will +not hold; for instance, let \(P\) be the relation of a number to any +number of which it is a factor; then \(Pʻʻ(\text{even numbers}) = \text{factors of even numbers}\), +but this class is not composed of terms of the form "<i>the</i> factor +of \(x\)," where \(x\) is an even number, because numbers do not have +only one factor apiece.</p> + +<p><i>Unit classes.</i> The class whose only member is \(x\) might be +thought to be identical with \(x\), but Peano and Frege have shown that +this is not the case. (The reasons why this is not the case will be +explained in a preliminary way in <a href="#CHAPTER_II">Chapter II</a> of the Introduction.) We +denote by "\(\iotaʻx\)" the class whose only member is \(x\): thus +\[ +\iotaʻx = \hat{y}(y = x) \quad \text{Df}\text{,} +\] +<i>i.e.</i> "\(\iotaʻx\)" means "the class of objects which are +identical with \(x\)."</p> + +<p>The class consisting of \(x\) and \(y\) will be \(\iotaʻx \cup +\iotaʻy\); the class got by adding \(x\) to a class \(\alpha\) will +be \(\alpha \cup \iotaʻx\); the class got by taking away \(x\) from a +class \(\alpha\) will be \(\alpha - \iotaʻx\). (We write \(\alpha - \beta\) +as an abbreviation for \(\alpha \cap -\beta\).)</p> + +<p>It will be observed that unit classes have been defined without +reference to the number \(1\); in fact, we use unit classes to define +the number \(1\). This number is defined as the class of unit classes, +<i>i.e.</i> +\[ +1 = \hat{\alpha} \{(\exists x) \ldotp \alpha = \iotaʻx\} \quad \text{Df}\text{.} +\] +This leads to +\[ +\vdash \colon\ldotp \alpha \in 1 \ldotp \equiv \colon (\exists x) \colon y \in \alpha \ldotp \equiv_{y} \ldotp y = x\text{.} +\] +From this it appears further that +\[ +\begin{align} +&\vdash \colon \alpha \in 1 \ldotp \equiv \ldotp \text{E}! ({℩}x) (x \in \alpha)\text{,}\\ +\text{whence} &\vdash \colon \hat{z} ({\phi}z) \in 1 \ldotp \equiv \ldotp \text{E}! ({℩}x) ({\phi}x)\text{,}\\ +\end{align} +\] +<i>i.e.</i> "\(\hat{z}({\phi}z)\) is a unit class" is equivalent to +"the \(x\) satisfying \(\phi\hat{x}\) exists."</p> + +<p><span class="pagenum" id="Page_38">[Pg 38]</span></p> + +<p>If \(\alpha \in 1\), \(\breve{\iota}ʻ\alpha\) is the only member +of \(\alpha\), for the only member of \(\alpha\) is the only term to +which \(\alpha\) has the relation \(\iota\). Thus "\(\iota ʻ\alpha\)" +takes the place of "(\({{℩}}x)(\phi x)\)," if \(\alpha\) stands for +\(\hat{z}(\phi z)\). In practice, "\(\iota ʻ\alpha\)" is a more +convenient notation than "(\({℩}x)(\phi x)\)," and is generally used +instead of "(\({℩}x)(\phi x)\)."</p> + +<p>The above account has explained most of the logical notation +employed in the present work. In the applications to various parts +of mathematics, other definitions are introduced; but the objects +defined by these later definitions belong, for the most part, rather +to mathematics than to logic. The reader who has mastered the symbols +explained above will find that any later formulae can be deciphered by +the help of comparatively few additional definitions.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a> +Cf. <a href="#CHAPTER_II">Chapter II</a> of the Introduction.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a> +This phrase is due to Frege.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a> +The meaning of these expressions will be explained later, +and examples of the use of dots in connection with them will be given +on <a href="#Page_17">pp. 17</a>, <a href="#Page_18">18</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a> +This case will be fully considered in <a href="#CHAPTER_III">Chapter III</a> of the +Introduction. It need not further concern us at present.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a> +This agrees with the rules for the occurrences of dots of +the type of Group II as explained above, <a href="#Page_9">pp. 9</a> and <a href="#Page_10">10</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a> +A value of \(x\) is said to <i>satisfy</i> \(\phi x\) or +\(\phi \hat{x}\) when \(\phi x\) is true for that value of \(x\).</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a> +Any other letter may be used instead of \(z\).</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a> +Such a couple has a <i>sense</i>, <i>i.e.</i> the couple +\((x,y)\) is different from the couple \((y,x)\), unless \(x = y\). We +shall call it a "couple with sense," to distinguish it from the class +consisting of \(x\) and \(y\). It may also be called an <i>ordered</i> +couple.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a> +The same principle applies to many uses of the proper +names of existent objects, <i>e.g</i>. to all uses of proper names +for objects known to the speaker only by report, and not by personal +acquaintance.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a> +The second of these notations is taken from Schröder's +<i>Algebra und Logik der Relative</i>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_11" href="#FNanchor_11" class="label">[11]</a> +This relation is not strictly asymmetrical, but is so +except when the wife's brother is also the sister's husband. In the +Greek Church the relation is strictly asymmetrical.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_39">[Pg 39]</span></p> +<h2 class="nobreak" id="CHAPTER_II">CHAPTER II.<br> +THE THEORY OF LOGICAL TYPES.</h2> +</div> + + +<p>THE theory of logical types, to be explained in the present Chapter, +recommended itself to us in the first instance by its ability to solve +certain contradictions, of which the one best known to mathematicians +is Burali-Forti's concerning the greatest ordinal. But the theory in +question is not wholly dependent upon this indirect recommendation: +it has also a certain consonance with common sense which makes it +inherently credible. In what follows, we shall therefore first set +forth the theory on its own account, and then apply it to the solution +of the contradictions.</p> + + +<p class="nindc space-above2"> +I. <i>The Vicious-Circle Principle</i>.</p> + + +<p>An analysis of the paradoxes to be avoided shows that they all result +from a certain kind of vicious circle<a id="FNanchor_12" href="#Footnote_12" class="fnanchor">[12]</a>. The vicious circles in +question arise from supposing that a collection of objects may contain +members which can only be defined by means of the collection as a +whole. Thus, for example, the collection of <i>propositions</i> will be +supposed to contain a proposition stating that "all propositions are +either true or false." It would seem, however, that such a statement +could not be legitimate unless "all propositions" referred to some +already definite collection, which it cannot do if new propositions are +created by statements about "all propositions." We shall, therefore, +have to say that statements about "all propositions" are meaningless. +More generally, given any set of objects such that, if we suppose the +set to have a total, it will contain members which presuppose this +total, then such a set cannot have a total. By saying that a set has +"no total," we mean, primarily, that no significant statement can be +made about "all its members." Propositions, as the above illustration +shows, must be a set having no total. The same is true, as we shall +shortly see, of propositional functions, even when these are restricted +to such as can significantly have as argument a given object \(a\). In +such cases, it is necessary to break up our set into smaller sets, each +of which is capable of a total. This is what the theory of types aims +at effecting.</p> + +<p><span class="pagenum" id="Page_40">[Pg 40]</span></p> + +<p>The principle which enables us to avoid illegitimate totalities may +be stated as follows: "Whatever involves all of a collection must not +be one of the collection"; or, conversely: "If, provided a certain +collection had a total, it would have members only definable in terms +of that total, then the said collection has no total." We shall call +this the "vicious-circle principle," because it enables us to avoid the +vicious circles involved in the assumption of illegitimate totalities. +Arguments which are condemned by the vicious-circle principle will +be called "vicious-circle fallacies." Such arguments, in certain +circumstances, may lead to contradictions, but it often happens that +the conclusions to which they lead are in fact true, though the +arguments are fallacious. Take, for example, the law of excluded +middle, in the form "all propositions are true or false." If from this +law we argue that, because the law of excluded middle is a proposition, +therefore the law of excluded middle is true or false, we incur a +vicious-circle fallacy. "All propositions" must be in some way limited +before it becomes a legitimate totality, and any limitation which makes +it legitimate must make any statement about the totality fall outside +the totality. Similarly, the imaginary sceptic, who asserts that he +knows nothing, and is refuted by being asked if he knows that he knows +nothing, has asserted nonsense, and has been fallaciously refuted by +an argument which involves a vicious-circle fallacy. In order that +the sceptic's assertion may become significant, it is necessary to +place some limitation upon the things of which he is asserting his +ignorance, because the things of which it is possible to be ignorant +form an illegitimate totality. But as soon as a suitable limitation has +been placed by him upon the collection of propositions of which he is +asserting his ignorance, the proposition that he is ignorant of every +member of this collection must not itself be one of the collection. +Hence any significant scepticism is not open to the above form of +refutation.</p> + +<p>The paradoxes of symbolic logic concern various sorts of objects: +propositions, classes, cardinal and ordinal numbers, etc. All these +sorts of objects, as we shall show, represent illegitimate totalities, +and are therefore capable of giving rise to vicious-circle fallacies. +But by means of the theory (to be explained in <a href="#CHAPTER_III">Chapter III</a>) which +reduces statements that are verbally concerned with classes and +relations to statements that are concerned with propositional +functions, the paradoxes are reduced to such as are concerned with +propositions and propositional functions. The paradoxes that concern +propositions are only indirectly relevant to mathematics, while those +that more nearly concern the mathematician are all concerned with +<i>propositional functions</i>. We shall therefore proceed at once to +the consideration of propositional functions.</p> + + +<p><span class="pagenum" id="Page_41">[Pg 41]</span></p> + +<p class="nindc space-above2"> +II. <i>The Nature of Propositional Functions.</i></p> + + +<p>By a "propositional function" we mean something which contains a +variable \(x\), and expresses a <i>proposition</i> as soon as a value +is assigned to \(x\). That is to say, it differs from a proposition +solely by the fact that it is ambiguous: it contains a variable of +which the value is unassigned. It agrees with the ordinary functions +of mathematics in the fact of containing an unassigned variable: +where it differs is in the fact that the values of the function are +propositions. Thus <i>e.g.</i> "\(x\) is a man" or "\(\sin x = 1\)" +is a propositional function. We shall find that it is possible to +incur a vicious-circle fallacy at the very outset, by admitting as +possible arguments to a propositional function terms which presuppose +the function. This form of the fallacy is very instructive, and its +avoidance leads, as we shall see, to the hierarchy of types.</p> + +<p>The question as to the nature of a function<a id="FNanchor_13" href="#Footnote_13" class="fnanchor">[13]</a> is by no means an easy +one. It would seem, however, that the essential characteristic of a +function is <i>ambiguity</i>. Take, for example, the law of identity +in the form "\(A\) is \(A\)," which is the form in which it is usually +enunciated. It is plain that, regarded psychologically, we have here a +single judgment. But what are we to say of the object of the judgment? +We are not judging that Socrates is Socrates, nor that Plato is Plato, +nor any other of the definite judgments that are instances of the law +of identity. Yet each of these judgments is, in a sense, within the +scope of our judgment. We are in fact judging an ambiguous instance of +the propositional function "\(A\) is \(A\)." We appear to have a single +thought which does not have a definite object, but has as its object +an undetermined one of the values of the function "\(A\) is \(A\)." It +is this kind of ambiguity that constitutes the essence of a function. +When we speak of "\(\phi x\)" where \(x\) is not specified, we mean one +value of the function, but not a definite one. We may express this by +saying that "\(\phi x\)" <i>ambiguously denotes</i> \(\phi a\), \(\phi b\), +\(\phi c\), etc., where \(\phi a\), \(\phi b\), \(\phi c\), etc., +are the various values of "\(\phi x\)."</p> + +<p>When we say that "\(\phi x\)" ambiguously denotes \(\phi a\), \(\phi +b\), \(\phi c\), etc., we mean that "\(\phi x\)" means one of the +objects \(\phi a\), \(\phi b\), \(\phi c\), etc., though not a definite +one, but an undetermined one. It follows that "\(\phi x\)" only has a +well-defined meaning (well-defined, that is to say, except in so far as +it is of its essence to be ambiguous) if the objects \(\phi a\), \(\phi b\), +\(\phi c\), etc., are well-defined. That is to say, a function +is not a well-defined function unless all its values are already +well-defined. It follows from this that no function can have among +its values anything which presupposes the function, for if it had, +we could not regard the objects ambiguously denoted by the function +as definite until the function was definite, while conversely, as we +have just seen, the function cannot be<span class="pagenum" id="Page_42">[Pg 42]</span> definite until its values are +definite. This is a particular case, but perhaps the most fundamental +case, of the vicious-circle principle. A function is what ambiguously +denotes some one of a certain totality, namely the values of the +function; hence this totality cannot contain any members which involve +the function, since, if it did, it would contain members involving the +totality, which, by the vicious-circle principle, no totality can do.</p> + +<p>It will be seen that, according to the above account, the values +of a function are presupposed by the function, not vice versa. It +is sufficiently obvious, in any particular case, that a value of +a function does not presuppose the function. Thus for example the +proposition "Socrates is human" can be perfectly apprehended without +regarding it as a value of the function "\(x\) is human." It is true +that, conversely, a function can be apprehended without its being +necessary to apprehend its values severally and individually. If this +were not the case, no function could be apprehended at all, since +the number of values (true and false) of a function is necessarily +infinite and there are necessarily possible arguments with which we are +unacquainted. What is necessary is not that the values should be given +individually and extensionally, but that the totality of the values +should be given intensionally, so that, concerning any assigned object, +it is at least theoretically determinate whether or not the said object +is a value of the function.</p> + +<p>It is necessary practically to distinguish the function itself from +an undetermined value of the function. We may regard the function +itself as that which ambiguously denotes, while an undetermined +value of the function is that which is ambiguously denoted. If the +undetermined value is written "\(\phi x\)," we will write the function +itself "\(\phi \hat{x}\)." (Any other letter may be used in place of +\(x\).) Thus we should say "\(\phi x\) is a proposition," but "\(\phi\hat{x}\) +is a propositional function." When we say "\(\phi x\) is +a proposition," we mean to state something which is true for every +possible value of \(x\), though we do not decide what value \(x\) is +to have. We are making an ambiguous statement about any value of the +function. But when we say "\(\phi \hat{x}\) is a function," we are not +making an ambiguous statement. It would be more correct to say that +we are making a statement about an ambiguity, taking the view that a +function is an ambiguity. The function itself, \(\phi \hat{x}\), is the +single thing which ambiguously denotes its many values; while \(\phi x\), +where \(x\) is not specified, is one of the denoted objects, with +the ambiguity belonging to the manner of denoting.</p> + +<p>We have seen that, in accordance with the vicious-circle principle, the +values of a function cannot contain terms only definable in terms of +the function. Now given a function \(\phi \hat{x}\), the values for the +function<a id="FNanchor_14" href="#Footnote_14" class="fnanchor">[14]</a> are all propositions<span class="pagenum" id="Page_43">[Pg 43]</span> of the form \(\phi x\). It follows +that there must be no propositions, of the form \(\phi x\), in which +\(x\) has a value which involves \(\phi \hat{x}\). (If this were the +case, the values of the function would not all be determinate until +the function was determinate, whereas we found that the function is +not determinate unless its values are previously determinate.) Hence +there must be no such thing as the value for \(\phi \hat{x}\) with the +argument \(\phi \hat{x}\), or with any argument which involves \(\phi\hat{x}\). +That is to say, the symbol "\(\phi(\phi \hat{x})\)" must not express a +proposition, as "\(\phi a\)" does if \(\phi a\) is a value for \(\phi \hat{x}\). +In fact "\(\phi(\phi \hat{x})\)" must be a symbol which does +not express anything: we may therefore say that it is not significant. +Thus given any function \(\phi \hat{x}\), there are arguments with +which the function has no value, as well as arguments with which it has +a value. We will call the arguments with which \(\phi \hat{x}\) has a +value "possible values of \(x\)." We will say that \(\phi \hat{x}\) is +"significant with the argument \(x\)" when \(\phi \hat{x}\) has a value +with the argument \(x\).</p> + +<p>When it is said that <i>e.g.</i> "\(\phi(\phi \hat{z})\)" is +meaningless, and therefore neither true nor false, it is necessary to +avoid a misunderstanding. If "\(\phi(\phi \hat{z})\)" were interpreted +as meaning "the value for \(\phi \hat{z}\) with the argument \(\phi\hat{z}\) +is true," that would be not meaningless, but false. It is +false for the same reason for which "the King of France is bald" +is false, namely because there is no such thing as "the value for +\(\phi \hat{z}\) with the argument \(\phi \hat{z}\)." But when, with +some argument \(a\), we assert \(\phi a\), we are not meaning to assert +"the value for \(\phi \hat{x}\) with the argument \(a\) is true"; we are +meaning to assert the actual proposition which is the value for \(\phi\hat{x}\) +with the argument \(a\). Thus for example if \(\phi \hat{x}\) +is "\(\hat{x}\) is a man," \(\phi\) (Socrates) will be "Socrates is a +man," <i>not</i> "the value for the function '\(\hat{x}\) is a man,' +with the argument Socrates, is true." Thus in accordance with our +principle that "\(\phi(\phi \hat{z})\)" is meaningless, we cannot +legitimately deny "the function '\(\hat{x}\) is a man' is a man," +because this is nonsense, but we can legitimately deny "the value for +the function '\(\hat{x}\) is a man' with the argument '\(\hat{x}\) is a +man' is true," not on the ground that the value in question is false, +but on the ground that there is no such value for the function.</p> + +<p>We will denote by the symbol "(\(x).\phi x\)" the proposition "\(\phi +x\) always<a id="FNanchor_15" href="#Footnote_15" class="fnanchor">[15]</a>," <i>i.e.</i> the proposition which asserts all the +values for \(\phi \hat{x}\). This proposition involves the function +\(\phi \hat{x}\), not merely an ambiguous value of the function. The +assertion of \(\phi x\), where \(x\) is unspecified, is a different +assertion from the one which asserts all values for \(\phi \hat{x}\), +for the former is an ambiguous assertion, whereas the latter is in no +sense ambiguous. It will be observed that "(\(x) . \phi x\)" does not +assert "\(\phi x\) with all values of \(x\)," because, as we have seen, +there must be values of \(x\) with which "\(\phi x\)" is meaningless. +What is asserted by "(\(x) . \phi x\)" is all propositions which are +values for \(\phi \hat{x}\); hence it is<span class="pagenum" id="Page_44">[Pg 44]</span> only with such values of \(x\) as +make "\(\phi x\)" significant, <i>i.e.</i> with all possible arguments, +that \(\phi x\) is asserted when we assert "(\(x) . \phi x\)." Thus +a convenient way to read "(\(x) . \phi x\)" is "\(\phi x\) is true +with all possible values of \(x\)." This is, however, a less accurate +reading than "\(\phi x\) always," because the notion of truth is not +part of the content of what is judged. When we judge "all men are +mortal," we judge truly, but the notion of truth is not necessarily in +our minds, any more than it need be when we judge "Socrates is mortal."</p> + + +<p class="nindc space-above2"> +III. <i>Definition and Systematic Ambiguity of Truth and Falsehood.</i></p> + + +<p>Since "(\(x) . \phi x\)" involves the function \(\phi \hat{x}\), it +must, according to our principle, be impossible as an argument to +\(\phi\). That is to say, the symbol "\(\phi \{(x) . \phi x\}\)" must +be meaningless. This principle would seem, at first sight, to have +certain exceptions. Take, for example, the function "\(\hat{p}\) is +false," and consider the proposition "(\(p) . p\) is false." This +should be a proposition asserting all propositions of the form "\(p\) +is false." Such a proposition, we should be inclined to say, must be +false, because "\(p\) is false" is not always true. Hence we should be +led to the proposition +\[ +\unicode{x201c}\{(p) . p\,\, \text{is false}\}\, \text{is false},\unicode{x201d} +\] +<i>i.e.</i> we should be led to a proposition in which "(\(p) . p\) is +false" is the argument to the function "\(\hat{p}\) is false," which +we had declared to be impossible. Now it will be seen that "(\(p) .p\) +is false," in the above, purports to be a proposition about all +propositions, and that, by the general form of the vicious-circle +principle, there must be no propositions about all propositions. +Nevertheless, it seems plain that, given any function, there is a +proposition (true or false) asserting all its values. Hence we are led +to the conclusion that "\(p\) is false" and "\(q\) is false" must not +always be the values, with the arguments \(p\) and \(q\), for a single +function "\(\hat{p}\) is false." This, however, is only possible if +the word "false" really has many different meanings, appropriate to +propositions of different kinds.</p> + +<p>That the words "true" and "false" have many different meanings, +according to the kind of proposition to which they are applied, is +not difficult to see. Let us take any function \(\hat{x}\), and let +\(\phi a\) be one of its values. Let us call the sort of truth which is +applicable to \(\phi a\) "<i>first</i> truth." (This is not to assume +that this would be first truth in another context: it is merely to +indicate that it is the first sort of truth in our context.) Consider +now the proposition (\(x) . \phi x\). If this has truth of the sort +appropriate to it, that will mean that every value \(\phi x\) has +"first truth." Thus if we call the sort of truth that is appropriate to +(\(x) . \phi x\) "<i>second</i> truth," we may define "\(\{(x) . \phi x\}\) +has second truth" as meaning "every value for \(\phi \hat{x}\) +has first truth," <i>i.e.</i> "(\(x)\).(\(\phi x\) has first truth)." +Similarly, if we denote by "(\(\exists x) . \phi x\)" the proposition +"\(\phi x\) sometimes," <i>i.e.</i> as we may less accurately +express it, "\(\phi x\) with some value of \(x\)," we find that +(\(\exists x) . \phi x\) has second truth if there is an \(x\) with<span class="pagenum" id="Page_45">[Pg 45]</span> +which \(\phi x\) has first truth; thus we may define "\(\{(\exists x).\phi x\}\) +has second truth" as meaning "some value for \(\phi \hat{x}\) has first +truth," <i>i.e.</i> "(\(\exists x) . (\phi x\,\text{has first truth})\)." +Similar remarks apply to falsehood. Thus "\(\{(x).\phi x\}\) has +second falsehood" will mean "some value for \(\phi \hat{x}\) has first +falsehood," <i>i.e.</i> "(\(\exists x) . (\phi x\, \text{has first falsehood})\)," +while "\(\{(\exists x).\phi x\}\) has second falsehood" +will mean "all values for \(\phi \hat{x}\) have first falsehood," +<i>i.e.</i> "(\(x). (\phi x\, \text{has first falsehood})\)." Thus the +sort of falsehood that can belong to a general proposition is different +from the sort that can belong to a particular proposition.</p> + +<p>Applying these considerations to the proposition "(\(p).p\) is false," +we see that the kind of falsehood in question must be specified. +If, for example, first falsehood is meant, the function "\(\hat{p}\) +has first falsehood" is only significant when \(p\) is the sort of +proposition which has first falsehood or first truth. Hence "(\(p) . p\) +is false" will be replaced by a statement which is equivalent to +"all propositions having either first truth or first falsehood have +first falsehood." This proposition has <i>second</i> falsehood, and is +not a possible argument to the function "\(\hat{p}\) has <i>first</i> +falsehood." Thus the apparent exception to the principle that +"\(\phi\{(x) . \phi x\}\)" must be meaningless disappears.</p> + +<p>Similar considerations will enable us to deal with "not-\(p\)" and with +"\(p\) or \(q\)." It might seem as if these were functions in which +<i>any</i> proposition might appear as argument. But this is due to a +systematic ambiguity in the meanings of "not" and "or," by which they +adapt themselves to propositions of any order. To explain fully how +this occurs, it will be well to begin with a definition of the simplest +kind of <i>truth</i> and <i>falsehood</i>.</p> + +<p>The universe consists of objects having various qualities and standing +in various relations. Some of the objects which occur in the universe +are complex. When an object is complex, it consists of interrelated +parts. Let us consider a complex object composed of two parts \(a\) +and \(b\) standing to each other in the relation \(R\). The complex +object "\(a\)-in-the-relation-\(R\)-to-\(b\)" may be capable of being +<i>perceived</i>; when perceived, it is perceived as one object. +Attention may show that it is complex; we then <i>judge</i> that +\(a\) and \(b\) stand in the relation \(R\). Such a judgment, being +derived from perception by mere attention, may be called a "judgment +of perception." This judgment of perception, considered as an actual +occurrence, is a relation of four terms, namely \(a\) and \(b\) and +\(R\) and the percipient. The perception, on the contrary, is a +relation of two terms, namely "\(a\)-in-the-relation-\(R\)-to-\(b\)," +and the percipient. Since an object of perception cannot be nothing, +we cannot perceive "\(a\)-in-the-relation-\(R\)-to-\(b\)" unless a +is in the relation \(R\) to \(b\). Hence a judgment of perception, +according to the above definition, must be true. This does not +mean that, in a judgment which <i>appears</i> to us to be one of +perception, we are sure of not being in error, since we may err in +thinking that our judgment has really been derived merely by analysis +of<span class="pagenum" id="Page_46">[Pg 46]</span> what was perceived. But if our judgment has been so derived, +it must be true. In fact, we may define <i>truth</i>, where such +judgments are concerned, as consisting in the fact that there is a +complex <i>corresponding</i> to the discursive thought which is the +judgment. That is, when we judge "\(a\) has the relation \(R\) to +\(b\)," our judgment is said to be <i>true</i> when there is a complex +"\(a\)-in-the-relation-\(R\)-to-\(b\)," and is said to be <i>false</i> +when this is not the case. This is a definition of truth and falsehood +in relation to judgments of this kind.</p> + +<p>It will be seen that, according to the above account, a judgment does +not have a single object, namely the proposition, but has several +interrelated objects. That is to say, the relation which constitutes +judgment is not a relation of two terms, namely the judging mind +and the proposition, but is a relation of several terms, namely the +mind and what are called the constituents of the proposition. That +is, when we judge (say) "this is red," what occurs is a relation of +three terms, the mind, and "this," and red. On the other hand, when +we <i>perceive</i> "the redness of this," there is a relation of two +terms, namely the mind and the complex object "the redness of this." +When a judgment occurs, there is a certain complex entity, composed of +the mind and the various objects of the judgment. When the judgment +is <i>true</i>, in the case of the kind of judgments we have been +considering, there is a corresponding complex of the <i>objects</i> +of the judgment alone. Falsehood, in regard to our present class +of judgments, consists in the absence of a corresponding complex +composed of the objects alone. It follows from the above theory that +a "proposition," in the sense in which a proposition is supposed to +be <i>the</i> object of a judgment, is a false abstraction, because a +judgment has several objects, not one. It is the severalness of the +objects in judgment (as opposed to perception) which has led people to +speak of thought as "discursive," though they do not appear to have +realized clearly what was meant by this epithet.</p> + +<p>Owing to the plurality of the objects of a single judgment, it follows +that what we call a "proposition" (in the sense in which this is +distinguished from the phrase expressing it) is not a single entity at +all. That is to say, the phrase which expresses a proposition is what +we call an "incomplete" symbol<a id="FNanchor_16" href="#Footnote_16" class="fnanchor">[16]</a>; it does not have meaning in itself, +but requires some supplementation in order to acquire a complete +meaning. This fact is somewhat concealed by the circumstance that +judgment in itself supplies a sufficient supplement, and that judgment +in itself makes no <i>verbal</i> addition to the proposition. Thus "the +proposition 'Socrates is human'" uses "Socrates is human" in a way +which requires a supplement of some kind before it acquires a complete +meaning; but when I judge "Socrates is human," the meaning is completed +by the act of judging, and we no longer have an incomplete symbol. +The fact that propositions are "incomplete symbols"<span class="pagenum" id="Page_47">[Pg 47]</span> is important +philosophically, and is relevant at certain points in symbolic logic.</p> + +<p>The judgments we have been dealing with hitherto are such as are of the +same form as judgments of perception, <i>i.e.</i> their subjects are +always particular and definite. But there are many judgments which are +not of this form. Such are "all men are mortal," "I met a man," "some +men are Greeks." Before dealing with such judgments, we will introduce +some technical terms.</p> + +<p>We will give the name of "a <i>complex</i>" to any such object as +"\(a\) in the relation \(R\) to \(b\)" or "\(a\) having the quality +\(q\)," or "\(a\) and \(b\) and \(c\) standing in the relation \(S\)." +Broadly speaking, a <i>complex</i> is anything which occurs in the +universe and is not simple. We will call a judgment <i>elementary</i> +when it merely asserts such things as "\(a\) has the relation \(R\) to +\(b\)," "\(a\) has the quality \(q\)" or "\(a\) and \(b\) and \(c\) +stand in the relation \(S\)." Then an <i>elementary</i> judgment is +true when there is a corresponding complex, and false when there is no +corresponding complex.</p> + +<p>But take now such a proposition as "all men are mortal." Here the +judgment does not correspond to <i>one</i> complex, but to many, +namely "Socrates is mortal," "Plato is mortal," "Aristotle is mortal," +etc. (For the moment, it is unnecessary to inquire whether each of +these does not require further treatment before we reach the ultimate +complexes involved. For purposes of illustration, "Socrates is mortal" +is here treated as an elementary judgment, though it is in fact not +one, as will be explained later. Truly elementary judgments are not +very easily found.) We do not mean to deny that there may be some +relation of the concept <i>man</i> to the concept <i>mortal</i> which +may be <i>equivalent</i> to "all men are mortal," but in any case this +relation is not the same thing as what we affirm when we say that all +men are mortal. Our judgment that all men are mortal collects together +a number of elementary judgments. It is not, however, composed of +these, since (<i>e.g.</i>) the fact that Socrates is mortal is no +part of what we assert, as may be seen by considering the fact that +our assertion can be understood by a person who has never heard of +Socrates. In order to understand the judgment "all men are mortal," it +is not necessary to know what men there are. We must admit, therefore, +as a radically new kind of judgment, such general assertions as "all +men are mortal." We assert that, given that \(x\) is human, \(x\) is +always mortal. That is, we assert "\(x\) is mortal" of <i>every</i> +\(x\) which is human. Thus we are able to judge (whether truly +or falsely) that <i>all</i> the objects which have some assigned +property also have some other assigned property. That is, given any +propositional functions \(\phi \hat{x}\) and \(\psi \hat{x}\), there +is a judgment asserting \(\psi x\) with every \(x\) for which we have +\(\phi x\). Such judgments we will call <i>general judgments</i>.</p> + +<p>It is evident (as explained above) that the definition of <i>truth</i> +is different<span class="pagenum" id="Page_48">[Pg 48]</span> in the case of general judgments from what it was in the +case of elementary judgments. Let us call the meaning of <i>truth</i> +which we gave for elementary judgments "elementary truth." Then when +we assert that it is true that all men are mortal, we shall mean that +all judgments of the form "\(x\) is mortal," where \(x\) is a man, have +elementary truth. We may define this as "truth of the second order" or +"second-order truth." Then if we express the proposition "all men are +mortal" in the form +\[ +\unicode{x201c}(x). x \, \text{is mortal, where}\, x\, \text{is a man}.\unicode{x201d} +\] +and call this judgment \(p\), then "\(p\) is true" must be taken to +mean "\(p\) has second-order truth," which in turn means +\[ +\unicode{x201c}(x). \unicode{x2018}x \, \text{is mortal}\unicode{x2019}\, \text{has elementary truth, where}\, x \, \text{is a man}.\unicode{x201d} +\]</p> + +<p>In order to avoid the necessity for stating explicitly the limitation +to which our variable is subject, it is convenient to replace the +above interpretation of "all men are mortal" by a slightly different +interpretation. The proposition "all men are mortal" is equivalent to +"'\(x\) is a man' implies '\(x\) is mortal,' with all possible values +of \(x\)." Here \(x\) is not restricted to such values as are men, +but may have any value with which "'\(x\) is a man' implies '\(x\) is +mortal'" is <i>significant</i>, <i>i.e.</i> either true or false. Such +a proposition is called a "formal implication." The advantage of this +form is that the values which the variable may take are given by the +function to which it is the argument: the values which the variable may +take are all those with which the function is significant.</p> + +<p>We use the symbol "(\(x).\phi x\)" to express the general judgment +which asserts all judgments of the form "\(\phi x\)." Then the judgment +"all men are mortal" is equivalent to +\[ +\unicode{x201c}(x). \unicode{x2018}x \, \text{is a man}\unicode{x2019} \, \text{implies}\, \unicode{x2018}x\, \text{is a mortal},\unicode{x2019}\unicode{x201d} +\] +<i>i.e.</i> (in virtue of the definition of implication) to +\[ +\unicode{x201c}(x) . x \,\text{is not a man or}\, x \, \text{is mortal}.\unicode{x201d} +\] +As we have just seen, the meaning of <i>truth</i> which is applicable +to this proposition is not the same as the meaning of <i>truth</i> +which is applicable to "\(x\) is a man" or to "\(x\) is mortal." And +generally, in any judgment (\(x) . \phi x\), the sense in which this +judgment is or may be true is not the same as that in which \(\phi x\) +is or may be true. If \(\phi x\) is an elementary judgment, it is true +when it <i>points</i> to a corresponding complex. But (\(x) . \phi x\) +does not point to a single corresponding complex: the corresponding +complexes are as numerous as the possible values of \(x\).</p> + +<p>It follows from the above that such a proposition as "all the judgments +made by Epimenides are true" will only be prima facie capable of truth +if all his judgments are of the same order. If they are of varying +orders, of which the nth is the highest, we may make n assertions of +the form "all the judgments of order \(m\) made by Epimenides are +true," where \(m\) has all values<span class="pagenum" id="Page_49">[Pg 49]</span> up to \(n\). But no such judgment +can include itself in its own scope, since such a judgment is always of +higher order than the judgments to which it refers.</p> + +<p>Let us consider next what is meant by the negation of a proposition +of the form "\((x) \ldotp {\phi}x\)." We observe, to begin with, +that "\({\phi}x\) in some cases," or "\({\phi}x\) sometimes," is +a judgment which is on a par with "\({\phi}x\) in all cases," or +"\({\phi}x\) always." The judgment "\({\phi}x\) sometimes" is true if +one or more values of \(x\) exist for which \({\phi}x\) is true. We +will express the proposition "\({\phi}x\) sometimes" by the notation +"\((\exists x) \ldotp {\phi}x\)," where "\(\exists\)" stands for "there +exists," and the whole symbol may be read "there exists an \(x\) +such that \({\phi}x\)." We take the two kinds of judgment expressed +by "\((x) \ldotp {\phi}x\)" and "\((\exists x) \ldotp {\phi}x\)" as +primitive ideas. We also take as a primitive idea the negation of an +<i>elementary</i> proposition. We can then define the negations of +\((x) \ldotp {\phi}x\) and (\(\exists x) \ldotp {\phi}x\). The negation +of any proposition \(p\) will be denoted by the symbol "\({\sim}p\)." +Then the negation of \((x) \ldotp {\phi}x\) will be <i>defined</i> as +meaning +\[ +\unicode{x201c}(\exists x) \ldotp {\phi}x,\unicode{x201d} +\] +and the negation of \((\exists x) \ldotp {\phi}x\) will be +<i>defined</i> as meaning "\((x) \ldotp {\sim}{\phi}x\)." Thus, in +the traditional language of formal logic, the negation of a universal +affirmative is to be defined as the particular negative, and the +negation of the particular affirmative is to be defined as the +universal negative. Hence the meaning of negation for such propositions +is different from the meaning of negation for elementary propositions.</p> + +<p>An analogous explanation will apply to disjunction. Consider the +statement "either \(p\), or \({\phi}x\) always." We will denote the +disjunction of two propositions \(p\), \(q\) by "\(p \lor q\)." Then +our statement is "\(p \ldotp \lor \ldotp (x) \ldotp {\phi}x\)." We will +suppose that \(p\) is an elementary proposition, and that \({\phi}x\) +is always an elementary proposition. We take the disjunction of +two elementary propositions as a primitive idea, and we wish to +<i>define</i> the disjunction +\[ +\unicode{x201c}p \ldotp \lor \ldotp (x) \ldotp {\phi}x.\unicode{x201d} +\] +This may be defined as "\((x) \ldotp p \lor {\phi}x\)," <i>i.e.</i> +"either \(p\) is true, or \({\phi}x\) is always true" is to mean +"'\(p\) or \({\phi}x\)' is always true." Similarly we will define +\[ +\unicode{x201c}p \ldotp \lor \ldotp (\exists x) \ldotp {\phi}x\unicode{x201d} +\] +as meaning "\((\exists x) \ldotp p \lor {\phi}x\)," <i>i.e.</i> we +define "either \(p\) is true or there is an \(x\) for which \({\phi}x\) +is true" as meaning "there is an \(x\) for which either \(p\) or +\({\phi}x\) is true." Similarly we can define a disjunction of two +universal propositions: "\((x) \ldotp {\phi}x \ldotp \lor \ldotp (y)\ldotp {\psi}y\)" +will be defined as meaning "\((x, y) \ldotp {\phi}x \lor {\psi}y\)," +<i>i.e.</i> "either \({\phi}x\) is always true or +\({\psi}y\) is always true" is to mean "'\({\phi}x\) or \({\psi}y\)' +is always true." By this method we obtain definitions of disjunctions +containing propositions of the form \((x) \ldotp {\phi}x\) or +(\(\exists x) \ldotp {\phi}x\) in terms of disjunctions of elementary +propositions; but the meaning of "disjunction" is not the<span class="pagenum" id="Page_50">[Pg 50]</span> same for +propositions of the forms (\(x) . {\phi}x\), (\(\exists x). {\phi}x\), +as it was for elementary propositions.</p> + +<p>Similar explanations could be given for implication and conjunction, +but this is unnecessary, since these can be defined in terms of +negation and disjunction.</p> + + +<p class="nindc space-above2"> +IV. <i>Why a Given Function requires Arguments of a Certain Type.</i></p> + + +<p>The considerations so far adduced in favour of the view that a +function cannot significantly have as argument anything defined in +terms of the function itself have been more or less indirect. But a +direct consideration of the kinds of functions which have functions +as arguments and the kinds of functions which have arguments other +than functions will show, if we are not mistaken, that not only +is it impossible for a function \(\phi\hat{z}\) to have itself or +anything derived from it as argument, but that, if \(\psi\hat{z}\) is +another function such that there are arguments \(a\) with which both +"\({\phi}a\)" and "\({\psi}a\)" are significant, then \(\psi\hat{z}\) +and anything derived from it cannot significantly be argument to +\(\phi\hat{z}\). This arises from the fact that a function is +essentially an ambiguity, and that, if it is to occur in a definite +proposition, it must occur in such a way that the ambiguity has +disappeared, and a wholly unambiguous statement has resulted. A few +illustrations will make this clear. Thus "\((x) . {\phi}x\)," which we +have already considered, is a function of \(\phi\hat{x}\); as soon as +\(\phi\hat{x}\) is assigned, we have a definite proposition, wholly +free from ambiguity. But it is obvious that we cannot substitute for +the function something which is not a function: "\((x) . {\phi}x\)" +means "\({\phi}x\) in all cases," and depends for its significance +upon the fact that there are "cases" of \({\phi}x\), <i>i.e.</i> upon +the ambiguity which is characteristic of a function. This instance +illustrates the fact that, when a function can occur significantly as +argument, something which is not a function cannot occur significantly +as argument. But conversely, when something which is not a function can +occur significantly as argument, a function cannot occur significantly. +Take, <i>e.g.</i> "\(x\) is a man," and consider "\(\phi\hat{x}\) +is a man." Here there is nothing to eliminate the ambiguity which +constitutes \(\phi\hat{x}\); there is thus nothing definite which +is said to be a man. A function, in fact, is not a definite object, +which could be or not be a man; it is a mere ambiguity awaiting +determination, and in order that it may occur significantly it must +receive the necessary determination, which it obviously does not +receive if it is merely substituted for something determinate in a +proposition<a id="FNanchor_17" href="#Footnote_17" class="fnanchor">[17]</a>. This argument does not, however, apply directly as +against such a statement as "\(\{(x) . {\phi}x\}\) is a man." Common +sense would pronounce such a statement to be meaningless, but it +cannot be condemned on the ground of ambiguity in its subject. We +need<span class="pagenum" id="Page_51">[Pg 51]</span> here a new objection, namely the following: A proposition is +not a single entity, but a relation of several; hence a statement in +which a proposition appears as subject will only be significant if +it can be reduced to a statement about the terms which appear in the +proposition. A proposition, like such phrases as "the so-and-so," +where grammatically it appears as subject, must be broken up into its +constituents if we are to find the true subject or subjects<a id="FNanchor_18" href="#Footnote_18" class="fnanchor">[18]</a>. But +in such a statement as "\(p\) is a man," where \(p\) is a proposition, +this is not possible. Hence "\(\{(x) \ldotp \phi x\}\) is a man" is +meaningless.</p> + + +<p class="nindc space-above2"> +V. <i>The Hierarchy of Functions and Propositions.</i></p> + + +<p>We are thus led to the conclusion, both from the vicious-circle +principle and from direct inspection, that the functions to which +a given object \(a\) can be an argument are incapable of being +arguments to each other, and that they have no term in common with the +functions to which they can be arguments. We are thus led to construct +a hierarchy. Beginning with \(a\) and the other terms which can be +arguments to the same functions to which \(a\) can be argument, we come +next to functions to which \(a\) is a possible argument, and then to +functions to which such functions are possible arguments, and so on. +But the hierarchy which has to be constructed is not so simple as might +at first appear. The functions which can take \(a\) as argument form an +illegitimate totality, and themselves require division into a hierarchy +of functions. This is easily seen as follows. Let \(f(\phi\hat{z}, x)\) +be a function of the two variables \(\phi\hat{z}\) and \(x\). Then if, +keeping \(x\) fixed for the moment, we assert this with all possible +values of \(\phi\), we obtain a proposition: +\[ +(\phi) \ldotp f(\phi\hat{z}, x). +\] +Here, if \(x\) is variable, we have a function of \(x\); but as this +function involves a totality of values of \(\phi\hat{z}\)<a id="FNanchor_19" href="#Footnote_19" class="fnanchor">[19]</a>, it +cannot itself be one of the values included in the totality, by the +vicious-circle principle. It follows that the totality of values of +\(\phi\hat{z}\) concerned in (\(\phi) \ldotp f(\phi\hat{z}, x)\) is not +the totality of all functions in which \(x\) can occur as argument, and +that there is no such totality as that of all functions in which \(x\) +can occur as argument.</p> + +<p>It follows from the above that a function in which \(\phi\hat{z}\) +appears as argument requires that "\(\phi\hat{z}\)" should not stand +for <i>any</i> function which is capable of a given argument, but +must be restricted in such a way that none of the functions which are +possible values of "\(\phi\hat{z}\)" should involve any reference to +the totality of such functions. Let us take as an illustration the +definition of identity. We might attempt to define "\(x\) is identical +with \(y\)" as meaning "whatever is true of \(x\) is true of \(y\)," +<i>i.e.</i> "\(\phi x\) always implies \(\phi y\)." But here,<span class="pagenum" id="Page_52">[Pg 52]</span> since we +are concerned to assert all values of "\(\phi x\) implies \(\phi y\)" +regarded as a function of \(\phi\), we shall be compelled to impose +upon \(\phi\) some limitation which will prevent us from including +among values of \(\phi\) values in which "all possible values of +\(\phi\)" are referred to. Thus for example "\(x\) is identical with +\(a\)" is a function of \(x\); hence, if it is a legitimate value of +\(\phi\) in "\(\phi x\) always implies \(\phi y\)" we shall be able to +infer, by means of the above definition, that if \(x\) is identical +with \(a\), and \(x\) is identical with \(y\), then \(y\) is identical +with \(a\). Although the conclusion is sound, the reasoning embodies +a vicious-circle fallacy, since we have taken "\((\phi)\ldotp\phi x\) +implies \(\phi a\)" as a possible value of \(\phi x\), which it cannot +be. If, however, we impose any limitation upon \(\phi\), it may happen, +so far as appears at present, that with other values of \(\phi\) we +might have \(\phi x\) true and \(\phi y\) false, so that our proposed +definition of identity would plainly be wrong. This difficulty is +avoided by the "axiom of reducibility," to be explained later. For the +present, it is only mentioned in order to illustrate the necessity and +the relevance of the hierarchy of functions of a given argument.</p> + +<p>Let us give the name "\(a\)-functions" to functions that are +significant for a given argument \(a\). Then suppose we take any +selection of \(a\)-functions, and consider the proposition "\(a\) +satisfies all the functions belonging to the selection in question." +If we here replace \(a\) by a variable, we obtain an \(a\)-function; +but by the vicious-circle principle this \(a\)-function cannot be a +member of our selection, since it refers to the whole of the selection. +Let the selection consist of all those functions which satisfy +\(f(\phi\hat{z})\). Then our new function is +\[ +(\phi) \ldotp \{f(\phi\hat{z}) \text{ implies } \phi x\}, +\] +where \(x\) is the argument. It thus appears that, whatever selection +of \(a\)-functions we may make, there will be other \(a\)-functions +that lie outside our selection. Such \(a\)-functions, as the above +instance illustrates, will always arise through taking a function of +two arguments, \(\phi\hat{z}\) and \(x\), and asserting all or some +of the values resulting from varying \(\phi\). What is necessary, +therefore, in order to avoid vicious-circle fallacies, is to divide our +\(a\)-functions into "types," each of which contains no functions which +refer to the whole of that type.</p> + +<p>When something is asserted or denied about all possible values or +about some (undetermined) possible values of a variable, that variable +is called <i>apparent</i>, after Peano. The presence of the words +<i>all</i> or <i>some</i> in a proposition indicates the presence of +an apparent variable; but often an apparent variable is really present +where language does not at once indicate its presence. Thus for example +"\(A\) is mortal" means "there is a time at which \(A\) will die." Thus +a variable time occurs as apparent variable.</p> + +<p>The clearest instances of propositions not containing apparent +variables are such as express immediate judgments of perception, such +as "this is red" or "this is painful," where "this" is something +immediately given. In other<span class="pagenum" id="Page_53">[Pg 53]</span> judgments, even where at first sight no +variable appears to be present, it often happens that there really is +one. Take (say) "Socrates is human." To Socrates himself, the word +"Socrates" no doubt stood for an object of which he was immediately +aware, and the judgment "Socrates is human" contained no apparent +variable. But to us, who only know Socrates by description, the +word "Socrates" cannot mean what it meant to him; it means rather +"the person having such-and-such properties," (say) "the Athenian +philosopher who drank the hemlock." Now in all propositions about "the +so-and-so" there is an apparent variable, as will be shown in <a href="#CHAPTER_III">Chapter III</a>. +Thus in what <i>we</i> have in mind when we say "Socrates is +human" there is an apparent variable, though there was no apparent +variable in the corresponding judgment as made by Socrates, provided we +assume that there is such a thing as immediate awareness of oneself.</p> + +<p>Whatever may be the instances of propositions not containing apparent +variables, it is obvious that propositional functions whose values do not +contain apparent variables are the source of propositions containing apparent +variables, in the sense in which the function \(\phi\hat{x}\) is the source of the proposition +\((x) \ldotp {\phi}x\). For the values for \(\phi\hat{x}\) do not contain the apparent variable \(x\), +which appears in \((x) \ldotp {\phi}x\); if they contain an apparent variable \(y\), this can be +similarly eliminated, and so on. This process must come to an end, since no +proposition which we can apprehend can contain more than a finite number +of apparent variables, on the ground that whatever we can apprehend must +be of finite complexity. Thus we must arrive at last at a function of as +many variables as there have been stages in reaching it from our original +proposition, and this function will be such that its values contain no apparent +variables. We may call this function the <i>matrix</i> of our original proposition +and of any other propositions and functions to be obtained by turning some +of the arguments to the function into apparent variables. Thus for example, +if we have a matrix-function whose values are \(\phi(x, y)\), we shall derive from it</p> + +<p>\((y) \ldotp \phi(x, y)\), which is a function of \(x\),</p> + +<p>\((x) \ldotp \phi(x, y)\), which is a function of \(y\),</p> + +<p>\((x,y) \ldotp \phi(x, y)\), meaning "\(\phi(x, y)\) is true with +all possible values of \(x\) and \(y\)." This last is a proposition +containing no <i>real</i> variable, <i>i.e.</i> no variable except +apparent variables.</p> + +<p>It is thus plain that all possible propositions and functions are +obtainable from matrices by the process of turning the arguments to the +matrices into apparent variables. In order to divide our propositions +and functions into types, we shall, therefore, start from matrices, +and consider how they are to be divided with a view to the avoidance +of vicious-circle fallacies in the definitions of the functions +concerned. For this purpose, we will use such letters as \(a\), +\(b\), \(c\), \(x\), \(y\), \(z\), \(w\), to denote objects which +are neither propositions nor functions. Such objects we shall call +<i>individuals</i>. Such objects will be<span class="pagenum" id="Page_54">[Pg 54]</span> constituents of propositions +or functions, and will be <i>genuine</i> constituents, in the sense +that they do not disappear on analysis, as (for example) classes do, or +phrases of the form "the so-and-so."</p> + +<p>The first matrices that occur are those whose values are of the forms +\[ +\phi x, \psi(x, y), \chi(x, y, z \ldots), +\] +<i>i.e.</i> where the arguments, however many there may be, are +all individuals. The functions \(\phi\), \(\psi\), \(\chi\) ..., +since (by definition) they contain no apparent variables, and have +no arguments except individuals, do not presuppose any totality of +functions. From the functions \(\psi\), \(\chi\) ... we may proceed +to form other functions of \(x\), such as (\(y) . \psi(x, y)\), +(\(\exists y) . \psi(x y)\), (\(y, z) . \chi(x, y, z)\), (\(y) \colon +(\exists z) . \chi(x, y, z)\), and so on. All these presuppose no +totality except that of individuals. We thus arrive at a certain +collection of functions of \(x\), characterized by the fact that they +involve no variables except individuals. Such functions we will call +"<i>first-order</i> functions."</p> + +<p>We may now introduce a notation to express "any first-order function." +We will denote any first-order function by "\(\phi ! \hat{x}\)" and +any value for such a function by "\(\phi ! \hat{x}\)." Thus "\(\phi! \hat{x}\)" +stands for any value for any function which involves no +variables except individuals. It will be seen that "\(\phi ! \hat{x}\)" +is itself a function of two variables, namely \(\phi ! \hat{z}\) and +\(x\). Thus \(\phi ! \hat{x}\) involves a variable which is not an +individual, namely \(\phi ! \hat{z}\). Similarly "(\(x). \phi ! x\)" +is a function of the variable \(\phi ! \hat{z}\), and thus involves +a variable other than an individual. Again, if \(a\) is a given +individual, +\[ +\unicode{x201c}\phi ! x\,\, \text{implies}\, \phi !\,\, \text{a with all possible values of}\,\, \phi.\unicode{x201d} +\] +is a function of \(x\), but it is not a function of the form \(\phi !x\), +because it involves an (apparent) variable \(\phi\) which is not +an individual. Let us give the name "predicate" to any first-order +function \(\phi ! \hat{x}\) (This use of the word "predicate" is +only proposed for the purposes of the present discussion.) Then the +statement "\(\phi ! x\) implies \(\phi ! a\) with all possible values +of \(\phi\)" may be read "all the predicates of \(x\) are predicates of +\(a\)." This makes a statement about \(x\), but does not attribute to +\(x\) a <i>predicate</i> in the special sense just defined.</p> + +<p>Owing to the introduction of the variable first-order function \(\phi! \hat{z}\), +we now have a new set of matrices. Thus "\(\phi ! x\)" is +a function which contains no apparent variables, but contains the two +real variables \(\phi ! \hat{z}\) and \(x\). (It should be observed +that when \(\phi\) is assigned, we may obtain a function whose values +do involve individuals as apparent variables, for example if \(\phi! x\) +is (\(y) . \psi(x, y)\). But so long as \(\phi\) is variable, +\(\phi ! x\) contains no apparent variables.) Again, if \(a\) is a +definite individual, \(\phi ! a\) is a function of the one variable +\(\phi ! \hat{z}\). If \(a\) and \(b\) are definite individuals, +"\(\phi ! a\) implies \(\psi ! b\)" is a function of the two variables +\(\phi ! \hat{z}\), \(\psi ! \hat{z}\), and so on. We are thus led to a +whole set of new matrices, +\[ +f(\phi ! \hat{z}),\, g(\phi ! \hat{z}, \psi ! \hat{z}),\, F(\phi ! \hat{z}, x),\, \text{and so on}. +\] +These matrices contain individuals and first-order functions as +arguments, but<span class="pagenum" id="Page_55">[Pg 55]</span> (like all matrices) they contain no apparent variables. +Any such matrix, if it contains more than one variable, gives rise to +new functions of one variable by turning all its arguments except one +into apparent variables. Thus we obtain the functions +\[ +\begin{align} +(\phi).g({\phi}!\hat{z},{\psi}!\hat{z})&\text{, which is a function of}~ {\psi}!\hat{z}\text{.}\\ +(x).F({\phi}!\hat{z}, x)&\text{, which is a function of}~ {\phi}!\hat{z}\text{.}\\ +(\phi).F({\phi}!\hat{z}, x)&\text{, which is a function of}~ x\text{.}\\ +\end{align} +\]</p> + +<p>We will give the name of <i>second-order matrices</i> to such matrices +as have first-order functions among their arguments, and have no +arguments except first-order functions and individuals. (It is not +<i>necessary</i> that they should have individuals among their +arguments.) We will give the name of <i>second-order functions</i> +to such as either are second-order matrices or are derived from such +matrices by turning some of the arguments into apparent variables. It +will be seen that either an individual or a first-order function may +appear as argument to a second-order function. Second-order functions +are such as contain variables which are first-order functions, but +contain no other variables except (possibly) individuals.</p> + +<p>We now have various new classes of functions at our command. In the +first place, we have second-order functions which have one argument +which is a first-order function. We will denote a variable function of +this kind by the notation \(f!(\hat{\phi}!\hat{z})\), and any value +of such a function by \(f!({\phi}!\hat{z})\). Like \({\phi}!x\), +\(f!({\phi}!\hat{z})\) is a function of two variables, namely +\(f!(\hat{\phi}!\hat{z})\) and \({\phi}!\hat{z}\). Among possible +values of \(f!({\phi}!\hat{z})\) will be \({\phi}!a\) (where \(a\) +is constant), \((x).{\phi}!x\), \((\exists x).{\phi}!x\), and so on. +(These result from assigning a value to \(f\), leaving \(\phi\) to +be assigned.) We will call such functions "predicative functions of +first-order functions."</p> + +<p>In the second place, we have second-order functions of two arguments, +one of which is a first-order function while the other is an +individual. Let us denote undetermined values of such functions by the +notation +\[ +f!({\phi}!\hat{z}, x). +\] +As soon as \(x\) is assigned, we shall have a predicative function of +\({\phi}!\hat{z}\). If our function contains no first-order function +as apparent variable, we shall obtain a predicative function of +\(x\) if we assign a value to \({\phi}!\hat{z}\). Thus, to take the +simplest possible case, if \(f!({\phi}!\hat{z}, x)\) is \({\phi}!x\), +the assignment of a value to \(\phi\) gives us a predicative function +of \(x\), in virtue of the definition of "\({\phi}!x\)." But if +\(f!({\phi}!\hat{z}, x)\) contains a first-order function as apparent +variable, the assignment of a value to \({\phi}!\hat{z}\) gives us a +second-order function of \(x\).</p> + +<p>In the third place, we have second-order functions of +individuals. These will all be derived from functions of the form +\(f!({\phi}!\hat{z}, x)\) by turning \(\phi\) into an apparent +variable. We do not, therefore, need a new notation for them.</p> + +<p><span class="pagenum" id="Page_56">[Pg 56]</span></p> + +<p>We have also second-order functions of two first-order functions, or of +two such functions and an individual, and so on.</p> + +<p>We may now proceed in exactly the same way to third-order matrices, +which will be functions containing second-order functions as arguments, +and containing no apparent variables, and no arguments except +individuals and first-order functions and second-order functions. +Thence we shall proceed, as before, to third-order functions; and so we +can proceed indefinitely. If the highest order of variable occurring in +a function, whether as argument or as apparent variable, is a function +of the \(n\)th order, then the function in which it occurs is of the +\(n + 1\)th order. We do not arrive at functions of an infinite order, +because the number of arguments and of apparent variables in a function +must be finite, and therefore every function must be of a finite order. +Since the orders of functions are only defined step by step, there +can be no process of "proceeding to the limit," and functions of an +infinite order cannot occur.</p> + +<p>We will define a function of one variable as <i>predicative</i> when +it is of the next order above that of its argument, <i>i.e.</i> of the +lowest order compatible with its having that argument. If a function +has several arguments, and the highest order of function occurring +among the arguments is the \(n\)th, we call the function predicative +if it is of the \(n + 1\)th order, <i>i.e.</i> again, if it is of +the lowest order compatible with its having the arguments it has. A +function of several arguments is predicative if there is one of its +arguments such that, when the other arguments have values assigned to +them, we obtain a predicative function of the one undetermined argument.</p> + +<p>It is important to observe that all possible functions in the above +hierarchy can be obtained by means of predicative functions and +apparent variables. Thus, as we saw, second-order functions of an +individual \(x\) are of the form +\[ +(\phi).f!(\phi ! \hat{z}, x)\, \text{or}\, (\exists \phi).f!(\phi ! \hat{z}, x) \text{or}\, (\phi, \psi).f!(\phi !\hat{z}, \psi !\hat{z}, x)\, \text{or etc.}, +\] +where \(f\) is a second-order predicative function. And speaking +generally, a non-predicative function of the \(n\)th order is +obtained from a predicative function of the \(n\)th order by turning +all the arguments of the \(n-1\)th order into apparent variables. +(Other arguments also may be turned into apparent variables.) Thus +we need not introduce as variables any functions except predicative +functions. Moreover, to obtain any function of one variable \(x\), +we need not go beyond predicative functions of <i>two</i> variables. +For the function (\(\psi).f!(\phi!\hat{z}, \psi !\hat{z}, x)\), where +\(f\) is given, is a function of \(\phi !\hat{z}\) and \(x\), and is +predicative. Thus it is of the form \(F!(\phi !\hat{z}, x)\), and +therefore (\(\phi, \psi).f!(\phi !\hat{z}, \psi !\hat{z}, x))\) is of +the form (\(\phi).F!(\phi !\hat{z}, x)\). Thus speaking generally, +by a succession of steps we find that, if \(\phi ! \hat{u}\) is a +predicative function of a sufficiently high order, any assigned +non-predicative function of \(x\) will be of one of the two forms +\[ +(\phi).F!(\phi !\hat{u}, x),\, (\exists \phi).F!(\phi !\hat{u}, x), +\] +where \(F\) is a predicative function of \(\phi ! \hat{u}\) and \(x\).</p> + +<p><span class="pagenum" id="Page_57">[Pg 57]</span></p> + +<p>The nature of the above hierarchy of functions may be restated as +follows. A function, as we saw at an earlier stage, presupposes as +part of its meaning the totality of its values, or, what comes to the +same thing, the totality of its possible arguments. The arguments to +a function may be functions or propositions or individuals. (It will +be remembered that individuals were defined as whatever is neither a +proposition nor a function.) For the present we neglect the case in +which the argument to a function is a proposition. Consider a function +whose argument is an individual. This function presupposes the totality +of individuals; but unless it contains functions as apparent variables, +it does not presuppose any totality of functions. If, however, it does +contain a function as apparent variable, then it cannot be defined +until some totality of functions has been defined. It follows that we +must first define the totality of those functions that have individuals +as arguments and contain no functions as apparent variables. These +are the <i>predicative</i> functions of individuals. Generally, a +predicative function of a variable argument is one which involves no +totality except that of the possible values of the argument, and those +that are presupposed by any one of the possible arguments. Thus a +predicative function of a variable argument is any function which can +be specified without introducing new kinds of variables not necessarily +presupposed by the variable which is the argument.</p> + +<p>A closely analogous treatment can be developed for propositions. +Propositions which contain no functions and no apparent variables may +be called <i>elementary propositions</i>. Propositions which are not +elementary, which contain no functions, and no apparent variables +except individuals, may be called <i>first-order propositions</i>. (It +should be observed that no variables except <i>apparent</i> variables +can occur in a proposition, since whatever contains a <i>real</i> +variable is a function, not a proposition.) Thus elementary and +first-order propositions will be values of first-order functions. (It +should be remembered that a function is not a constituent in one of its +values: thus for example the function "\(\hat{x}\) is human" is not a +constituent of the proposition "Socrates is human.") Elementary and +first-order propositions presuppose no totality except (at most) the +totality of individuals. They are of one or other of the three forms +\[ +\phi !x; (x). \phi !x; (\exists x).\phi !x, +\] +where \(\phi !x\) is a predicative function of an individual. It +follows that, if \(p\) represents a variable elementary proposition +or a variable first-order proposition, a function \(fp\) is either +\(f(\phi !x)\) or \(f\{(x).\phi !x\}\) or \(f\{(\exists x).\phi !x\}\). +Thus a function of an elementary or a first-order proposition may +always be reduced to a function of a first-order function. It follows +that a proposition involving the totality of first-order propositions +may be reduced to one involving the totality of first-order functions; +and this obviously applies equally to higher<span class="pagenum" id="Page_58">[Pg 58]</span> orders. The propositional +hierarchy can, therefore, be derived from the functional hierarchy, and +we may define a proposition of the nth order as one which involves an +apparent variable of the \(n - 1\)th order in the functional hierarchy. +The propositional hierarchy is never required in practice, and is only +relevant for the solution of paradoxes; hence it is unnecessary to go +into further detail as to the types of propositions.</p> + + +<p class="nindc space-above2"> +VI. <i>The Axiom of Reducibility.</i></p> + + +<p>It remains to consider the "axiom of reducibility." It will be seen +that, according to the above hierarchy, no statement can be made +significantly about "all \(a\)-functions," where \(a\) is some given +object. Thus such a notion as "all properties of \(a\)," meaning +"all functions which are true with the argument \(a\)," will be +illegitimate. We shall have to distinguish the order of function +concerned. We can speak of "all predicative properties of \(a\)," "all +second-order properties of \(a\)," and so on. (If \(a\) is not an +individual, but an object of order \(n\), "second-order properties of +\(a\)" will mean "functions of order \(n + 2\) satisfied by \(a\).") +But we cannot speak of "all properties of \(a\)." In some cases, we can +see that some statement will hold of "all \(n\)th-order properties of +\(a\)," whatever value \(n\) may have. In such cases, no practical harm +results from regarding the statement as being about "all properties of +\(a\)," provided we remember that it is really a number of statements, +and not a single statement which could be regarded as assigning another +property to \(a\), over and above all properties. Such cases will +always involve some systematic ambiguity, such as that involved in +the meaning of the word "truth," as explained above. Owing to this +systematic ambiguity, it will be possible, sometimes, to combine +into a single verbal statement what are really a number of different +statements, corresponding to different orders in the hierarchy. This is +illustrated in the case of the liar, where the statement "all \(A\)'s +statements are false" should be broken up into different statements +referring to his statements of various orders, and attributing to each +the appropriate kind of falsehood.</p> + +<p>The axiom of reducibility is introduced in order to legitimate a great +mass of reasoning, in which, prima facie, we are concerned with such +notions as "all properties of \(a\)" or "all \(a\)-functions," and +in which, nevertheless, it seems scarcely possible to suspect any +substantial error. In order to state the axiom, we must first define +what is meant by "formal equivalence." Two functions \(\phi \hat{x}\), +\(\psi \hat{x}\) are said to be "formally equivalent" when, with +every possible argument \(x\), \(\phi x\) is equivalent to \(\psi x\), +<i>i.e.</i> \(\phi x\) and \(\psi x\) are either both true or +both false. Thus two functions are formally equivalent when they are +satisfied by the same set of arguments. The axiom of reducibility is +the assumption that, given any function \(\phi \hat{x}\), there is a +formally equivalent <i>predicative</i> function,<span class="pagenum" id="Page_59">[Pg 59]</span> <i>i.e.</i> there is +a predicative function which is true when \({\phi}x\) is true and false +when \({\phi}x\) is false. In symbols, the axiom is: +\[ +\vdash \colon (\exists \psi) \colon {\phi}x \ldotp \equiv_{x} \ldotp {\psi}!x\text{.} +\] +For two variables, we require a similar axiom, namely: Given any +function \(\phi(\hat{x}, \hat{y})\), there is a formally equivalent +<i>predicative</i> function, <i>i.e.</i> +\[ +\vdash \colon (\exists \psi) \colon \phi(x, y) \ldotp \equiv_{x, y} \ldotp \psi!(x, y)\text{.} +\]</p> + +<p>In order to explain the purposes of the axiom of reducibility, and the +nature of the grounds for supposing it true, we shall first illustrate +it by applying it to some particular cases.</p> + +<p>If we call a <i>predicate</i> of an object a predicative function which +is true of that object, then the predicates of an object are only some +among its properties. Take for example such a proposition as "Napoleon +had all the qualities that make a great general." We may interpret this +as meaning "Napoleon had all the predicates that make a great general." +Here there is a predicate which is an apparent variable. If we put +"\(f({\phi}!\hat{z})\)" for "\({\phi}!\hat{z}\) is a predicate required +in a great general," our proposition is +\[ +(\phi) \colon f({\phi}!\hat{z}) ~\text{implies}~ {\phi}!(\text{Napoleon})\text{.} +\] +Since this refers to a totality of predicates, it is not itself a +predicate of Napoleon. It by no means follows, however, that there is +not some one predicate common and peculiar to great generals. In fact, +it is certain that there is such a predicate. For the number of great +generals is finite, and each of them certainly possessed some predicate +not possessed by any other human being—for example, the exact instant +of his birth. The disjunction of such predicates will constitute a +predicate common and peculiar to great generals<a id="FNanchor_20" href="#Footnote_20" class="fnanchor">[20]</a>. If we call this +predicate \({\psi}!\hat{z}\), the statement we made about Napoleon was +equivalent to \({\psi}!(\text{Napoleon})\). And this equivalence holds +equally if we substitute any other individual for Napoleon. Thus we +have arrived at a predicate which is always equivalent to the property +we ascribed to Napoleon, <i>i.e.</i> it belongs to those objects which +have this property, and to no others. The axiom of reducibility states +that such a predicate always exists, <i>i.e.</i> that any property +of an object belongs to the same collection of objects as those that +possess some predicate.</p> + +<p>We may next illustrate our principle by its application to +<i>identity</i>. In this connection, it has a certain affinity with +Leibniz's identity of indiscernibles. It is plain that, if \(x\) and +\(y\) are identical, and \({\phi}x\) is true, then \({\phi}y\) is +true. Here it cannot matter what sort of function \(\phi\hat{x}\) may +be: the statement must hold for <i>any</i> function. But we cannot +say, conversely: "If, with all values of \(\phi\), \({\phi}x\) implies +\({\phi}y\), then \(x\) and \(y\) are identical"; because "all values +of \(\phi\)" is inadmissible. If we wish to speak of "all values of +\(\phi\)," we must confine ourselves to functions of one order. We may +confine \(\phi\) to predicates, or to<span class="pagenum" id="Page_60">[Pg 60]</span> second-order functions, or to +functions of any order we please. But we must necessarily leave out +functions of all but one order. Thus we shall obtain, so to speak, +a hierarchy of different degrees of identity. We may say "all the +predicates of \(x\) belong to \(y\)," "all second-order properties of +\(x\) belong to \(y\)," and so on. Each of these statements implies +all its predecessors: for example, if all second-order properties of +\(x\) belong to \(y\), then all predicates of \(x\) belong to \(y\), +for to have all the predicates of \(x\) is a second-order property, +and this property belongs to \(x\). But we cannot, without the help of +an axiom, argue conversely that if all the predicates of \(x\) belong +to \(y\), all the second-order properties of \(x\) must also belong to +\(y\). Thus we cannot, without the help of an axiom, be sure that \(x\) +and \(y\) are identical if they have the same predicates. Leibniz's +identity of indiscernibles supplied this axiom. It should be observed +that by "indiscernibles" he cannot have meant two objects which agree +as to <i>all</i> their properties, for one of the properties of \(x\) +is to be identical with \(x\), and therefore this property would +necessarily belong to \(y\) if \(x\) and \(y\) agreed in <i>all</i> +their properties. Some limitation of the common properties necessary to +make things indiscernible is therefore implied by the necessity of an +axiom. For purposes of illustration (not of interpreting Leibniz) we +may suppose the common properties required for indiscernibility to be +limited to predicates. Then the identity of indiscernibles will state +that if \(x\) and \(y\) agree as to all their predicates, they are +identical. This can be proved if we assume the axiom of reducibility. +For, in that case, every property belongs to the same collection of +objects as is defined by some predicate. Hence there is some predicate +common and peculiar to the objects which are identical with \(x\). This +predicate belongs to \(x\), since \(x\) is identical with itself; hence +it belongs to \(y\), since y has all the predicates of \(x\); hence y +is identical with \(x\). It follows that we may <i>define</i> \(x\) and +\(y\) as identical when all the predicates of \(x\) belong to \(y\), +<i>i.e.</i> when (\(\phi)\colon \phi !x . \supset . \phi !y\). We +therefore adopt the following definition of identity<a id="FNanchor_21" href="#Footnote_21" class="fnanchor">[21]</a>: +\[ +x = y . = \colon (\phi) \colon \phi !x . \supset . \phi !y \qquad \text{Df}. +\]</p> + +<p>But apart from the axiom of reducibility, or some axiom equivalent +in this connection, we should be compelled to regard identity as +indefinable, and to admit (what seems impossible) that two objects may +agree in all their predicates without being identical.</p> + +<p>The axiom of reducibility is even more essential in the theory of +classes. It should be observed, in the first place, that if we assume +the existence of classes, the axiom of reducibility can be proved. +For in that case, given any function \(\phi \hat{z}\) of whatever +order, there is a class \(\alpha\) consisting of just those objects +which satisfy \(\phi \hat{z}\). Hence "\(\phi \hat{x}\)" is equivalent +to "\(x\) belongs to \(\alpha\)." But "\(x\) belongs to \(\alpha\)" +is a statement containing no apparent variable, and is therefore a +predicative function of \(x\). Hence if we assume the existence of<span class="pagenum" id="Page_61">[Pg 61]</span> +classes, the axiom of reducibility becomes unnecessary. The assumption +of the axiom of reducibility is therefore a smaller assumption than +the assumption that there are classes. This latter assumption has +hitherto been made unhesitatingly. However, both on the ground of the +contradictions, which require a more complicated treatment if classes +are assumed, and on the ground that it is always well to make the +smallest assumption required for proving our theorems, we prefer to +assume the axiom of reducibility rather than the existence of classes. +But in order to explain the use of the axiom in dealing with classes, +it is necessary first to explain the theory of classes, which is a +topic belonging to <a href="#CHAPTER_III">Chapter III</a>. We therefore postpone to that Chapter +the explanation of the use of our axiom in dealing with classes.</p> + +<p>It is worth while to note that all the purposes served by the axiom +of reducibility are equally well served if we assume that there is +always a function of the \(n\)th order (where \(n\) is fixed) which +is formally equivalent to \(\phi \hat{x}\), whatever may be the order +of \(\phi \hat{x}\). Here we shall mean by "a function of the \(n\)th +order" a function of the \(n\)th order relative to the arguments +to \(\phi \hat{x}\)\(\phi \hat{x}\); thus if these arguments are +absolutely of the \(m\)th order, we assume the existence of a function +formally equivalent to \(\phi \hat{x}\) whose absolute order is the +\(m + n\)th. The axiom of reducibility in the form assumed above +takes \(n = 1\), but this is not necessary to the use of the axiom. +It is also unnecessary that \(n\) should be the same for different +values of \(m\); what is necessary is that \(n\) should be constant so +long as \(m\) is constant. What is needed is that, where extensional +functions of functions are concerned, we should be able to deal with +any \(a\)-function by means of some formally equivalent function of a +given type, so as to be able to obtain results which would otherwise +require the illegitimate notion of "all \(a\)-functions"; but it does +not matter what the given type is. It does not appear, however, that +the axiom of reducibility is rendered appreciably more plausible by +being put in the above more general but more complicated form.</p> + +<p>The axiom of reducibility is equivalent to the assumption that "any +combination or disjunction of predicates<a id="FNanchor_22" href="#Footnote_22" class="fnanchor">[22]</a> is equivalent to a +single predicate," <i>i.e.</i> to the assumption that, if we assert +that \(x\) has all the predicates that satisfy a function \(f(\phi !\hat{z})\) +there is some one predicate which \(x\) will have whenever +our assertion is true, and will not have whenever it is false, and +similarly if we assert that \(x\) has some one of the predicates that +satisfy a function \(f(\phi ! \hat{z}\)). For by means of this +assumption, the order of a non-predicative function can be lowered by +one; hence, after some finite number of steps, we shall be able to get +from any non-predicative function to a formally equivalent predicative +function. It does not seem probable that<span class="pagenum" id="Page_62">[Pg 62]</span> the above assumption could +be substituted for the axiom of reducibility in symbolic deductions, +since its use would require the explicit introduction of the further +assumption that by a finite number of downward steps we can pass from +any function to a predicative function, and this assumption could not +well be made without developments that are scarcely possible at an +early stage. But on the above grounds it seems plain that in fact, if +the above alternative axiom is true, so is the axiom of reducibility. +The converse, which completes the proof of equivalence, is of course +evident.</p> + + +<p class="nindc space-above2"> +VII. <i>Reasons for Accepting the Axiom of Reducibility.</i></p> + + +<p><span class="pagenum" id="Page_63">[Pg 63]</span></p> + +<p>That the axiom of reducibility is self-evident is a proposition +which can hardly be maintained. But in fact self-evidence is never +more than a part of the reason for accepting an axiom, and is never +indispensable. The reason for accepting an axiom, as for accepting +any other proposition, is always largely inductive, namely that many +propositions which are nearly indubitable can be deduced from it, and +that no equally plausible way is known by which these propositions +could be true if the axiom were false, and nothing which is probably +false can be deduced from it. If the axiom is apparently self-evident, +that only means, practically, that it is nearly indubitable; for things +have been thought to be self-evident and have yet turned out to be +false. And if the axiom itself is nearly indubitable, that merely adds +to the inductive evidence derived from the fact that its consequences +are nearly indubitable: it does not provide new evidence of a radically +different kind. Infallibility is never attainable, and therefore some +element of doubt should always attach to every axiom and to all its +consequences. In formal logic, the element of doubt is less than in +most sciences, but it is not absent, as appears from the fact that +the paradoxes followed from premisses which were not previously known +to require limitations. In the case of the axiom of reducibility, the +inductive evidence in its favour is very strong, since the reasonings +which it permits and the results to which it leads are all such as +appear valid. But although it seems very improbable that the axiom +should turn out to be false, it is by no means improbable that it +should be found to be deducible from some other more fundamental and +more evident axiom. It is possible that the use of the vicious-circle +principle, as embodied in the above hierarchy of types, is more drastic +than it need be, and that by a less drastic use the necessity for +the axiom might be avoided. Such changes, however, would not render +anything false which had been asserted on the basis of the principles +explained above: they would merely provide easier proofs of the same +theorems. There would seem, therefore, to be but the slenderest ground +for fearing that the use of the axiom of reducibility may lead us into +error.</p> + + +<p class="nindc space-above2"> +VIII. <i>The Contradictions.</i></p> + + +<p>We are now in a position to show how the theory of types affects the +solution of the contradictions which have beset mathematical logic. +For this purpose, we shall begin by an enumeration of some of the more +important and illustrative of these contradictions, and shall then show +how they all embody vicious-circle fallacies, and are therefore all +avoided by the theory of types. It will be noticed that these paradoxes +do not relate exclusively to the ideas of number and quantity. +Accordingly no solution can be adequate which seeks to explain them +merely as the result of some illegitimate use of these ideas. The +solution must be sought in some such scrutiny of fundamental logical +ideas as has been attempted in the foregoing pages.</p> + +<p>(1) The oldest contradiction of the kind in question is the +<i>Epimenides</i>. Epimenides the Cretan said that all Cretans were +liars, and all other statements made by Cretans were certainly lies. +Was this a lie? The simplest form of this contradiction is afforded +by the man who says "I am lying"; if he is lying, he is speaking the +truth, and vice versa.</p> + +<p>(2) Let \(w\) be the class of all those classes which are not members +of themselves. Then, whatever class \(x\) may be, "\(x\) is a \(w\)" is +equivalent to "\(x\) is not an \(x\)." Hence, giving to \(x\) the value +\(w\), "\(w\) is a \(w\)" is equivalent to "\(w\) is not a \(w\)."</p> + +<p>(3) Let \(T\) be the relation which subsists between two relations +\(R\) and \(S\) whenever \(R\) does not have the relation \(R\) to +\(S\). Then, whatever relations \(R\) and \(S\) may be, "\(R\) has the +relation \(T\) to \(S\)" is equivalent to "\(R\) does not have the +relation \(R\) to \(S\)." Hence, giving the value \(T\) to both \(R\) +and \(S\), "\(T\) has the relation \(T\) to \(T\)" is equivalent to +"\(T\) does not have the relation \(T\) to \(T\)."</p> + +<p>(4) Burali-Forti's contradiction<a id="FNanchor_23" href="#Footnote_23" class="fnanchor">[23]</a> may be stated as follows: It can +be shown that every well-ordered series has an ordinal number, that the +series of ordinals up to and including any given ordinal exceeds the +given ordinal by one, and (on certain very natural assumptions) that +the series of all ordinals (in order of magnitude) is well-ordered. +It follows that the series of all ordinals has an ordinal number, +\(\Omega\) say. But in that case the series of all ordinals including +\(\Omega\) has the ordinal number \(\Omega + 1\), which must be greater +than \(\Omega\). Hence \(\Omega\) is not the ordinal number of all +ordinals.</p> + +<p>(5) The number of syllables in the English names of finite integers +tends to increase as the integers grow larger, and must gradually +increase indefinitely, since only a finite number of names can be +made with a given finite number of syllables. Hence the names of some +integers must consist of at least nineteen syllables, and among these +there must be a least. Hence "the least integer not nameable in fewer +than nineteen syllables"<span class="pagenum" id="Page_64">[Pg 64]</span> must denote a definite integer; in fact, it +denotes 111,777. But "the least integer not nameable in fewer than +nineteen syllables" is itself a name consisting of eighteen syllables; +hence the least integer not nameable in fewer than nineteen syllables +can be named in eighteen syllables, which is a contradiction<a id="FNanchor_24" href="#Footnote_24" class="fnanchor">[24]</a>.</p> + +<p>(6) Among transfinite ordinals some can be defined, while +others can not; for the total number of possible definitions is +\(\aleph_{0}\)<a id="FNanchor_25" href="#Footnote_25" class="fnanchor">[25]</a>, while the number of transfinite ordinals exceeds +\(\aleph_{0}\). Hence there must be indefinable ordinals, and among +these there must be a least. But this is defined as "the least +indefinable ordinal," which is a contradiction<a id="FNanchor_26" href="#Footnote_26" class="fnanchor">[26]</a>.</p> + +<p>(7) Richard's paradox<a id="FNanchor_27" href="#Footnote_27" class="fnanchor">[27]</a> is akin to that of the least indefinable +ordinal. It is as follows: Consider all decimals that can be defined +by means of a finite number of words; let \(E\) be the class of such +decimals. Then \(E\) has \(\aleph_{0}\) terms; hence its members can +be ordered as the 1st, 2nd, 3rd,.... Let \(N\) be a number defined as +follows. If the \(n\)th figure in the \(n\)th decimal is \(p\), let the +\(n\)th figure in \(N\) be \(p + 1\) (or 0, if \(p = 9\)). Then \(N\) +is different from all the members of \(E\), since, whatever finite +value \(n\) may have, the \(n\)th figure in \(N\) is different from +the \(n\)th figure in the nth of the decimals composing \(E\), and +therefore \(N\) is different from the \(n\)th decimal. Nevertheless we +have defined \(N\) in a finite number of words, and therefore \(N\) +ought to be a member of \(E\). Thus \(N\) both is and is not a member +of \(E\).</p> + +<p>In all the above contradictions (which are merely selections from an +indefinite number) there is a common characteristic, which we may +describe as self-reference or reflexiveness. The remark of Epimenides +must include itself in its own scope. If <i>all classes</i>, provided +they are not members of themselves, are members of \(w\), this must +also apply to \(w\); and similarly for the analogous relational +contradiction. In the cases of names and definitions, the paradoxes +result from considering non-nameability and indefinability as elements +in names and definitions. In the case of Burali-Forti's paradox, the +series whose ordinal number causes the difficulty is the series of +all ordinal numbers. In each contradiction something is said about +<i>all</i> cases of some kind, and from what is said a new case seems +to be generated,<span class="pagenum" id="Page_65">[Pg 65]</span> which both is and is not of the same kind as the +cases of which <i>all</i> were concerned in what was said. But this is +the characteristic of illegitimate totalities, as we defined them in +stating the vicious-circle principle. Hence all our contradictions are +illustrations of vicious-circle fallacies. It only remains to show, +therefore, that the illegitimate totalities involved are excluded by +the hierarchy of types which we have constructed.</p> + +<p>(1) When a man says "I am lying," we may interpret his statement as: +"There is a proposition which I am affirming and which is false." That +is to say, he is asserting the truth of some value of the function "I +assert \(p\), and \(p\) is false." But we saw that the word "false" +is ambiguous, and that, in order to make it unambiguous, we must +specify the order of falsehood, or, what comes to the same thing, the +order of the proposition to which falsehood is ascribed. We saw also +that, if \(p\) is a proposition of the \(n\)th order, a proposition +in which \(p\) occurs as an apparent variable is not of the \(n\)th +order, but of a higher order. Hence the kind of truth or falsehood +which can belong to the statement "there is a proposition \(p\) which +I am affirming and which has falsehood of the \(n\)th order" is truth +or falsehood of a higher order than the \(n\)th. Hence the statement +of Epimenides does not fall within its own scope, and therefore no +contradiction emerges.</p> + +<p>If we regard the statement "I am lying" as a compact way of +simultaneously making all the following statements: "I am asserting +a false proposition of the first order," "I am asserting a false +proposition of the second order," and so on, we find the following +curious state of things: As no proposition of the first order is being +asserted, the statement "I am asserting a false proposition of the +first order" is false. This statement is of the second order, hence the +statement "I am making a false statement of the second order" is true. +This is a statement of the third order, and is the only statement of +the third order which is being made. Hence the statement "I am making +a false statement of the third order" is false. Thus we see that the +statement "I am making a false statement of order \(2n + 1\)" is false, +while the statement "I am making a false statement of order \(2n\)" is +true. But in this state of things there is no contradiction.</p> + +<p>(2) In order to solve the contradiction about the class of classes +which are not members of themselves, we shall assume, what will be +explained in the next Chapter, that a proposition about a class is +always to be reduced to a statement about a function which defines the +class, <i>i.e.</i> about a function which is satisfied by the members +of the class and by no other arguments. Thus a class is an object +derived from a function and presupposing the function, just as, for +example, \((x).{\phi}x\) presupposes the function \(\phi\hat{x}\). +Hence a class cannot, by the vicious-circle principle, significantly be +the argument to its defining function, that is to say, if we denote<span class="pagenum" id="Page_66">[Pg 66]</span> +by "\(\hat{z}({\phi}z)\)" the class defined by \(\phi\hat{z}\), the +symbol "\(\phi\{\hat{z}({\phi}z)\}\)" must be meaningless. Hence a +class neither satisfies nor does not satisfy its defining function, +and therefore (as will appear more fully in <a href="#CHAPTER_III">Chapter III</a>) is neither +a member of itself nor not a member of itself. This is an immediate +consequence of the limitation to the possible arguments to a function +which was explained at the beginning of the present Chapter. Thus if +\(\alpha\) is a class, the statement "\(\alpha\) is not a member of +\(\alpha\)" is always meaningless, and there is therefore no sense +in the phrase "the class of those classes which are not members of +themselves." Hence the contradiction which results from supposing that +there is such a class disappears.</p> + +<p>(3) Exactly similar remarks apply to "the relation which holds between +\(R\) and \(S\) whenever \(R\) does not have the relation \(R\) to +\(S\)." Suppose the relation \(R\) is defined by a function \(\phi(x,y)\), +<i>i.e.</i> \(R\) holds between \(x\) and \(y\) whenever +\(\phi(x, y)\) is true, but not otherwise. Then in order to interpret +"\(R\) has the relation \(R\) to \(S\)" we shall have to suppose that +\(R\) and \(S\) can significantly be the arguments to \(\phi\). But +(assuming, as will appear in <a href="#CHAPTER_III">Chapter III</a>, that \(R\) presupposes its +defining function) this would require that \(\phi\) should be able to +take as argument an object which is defined in terms of \(\phi\), and +this no function can do, as we saw at the beginning of this Chapter. +Hence "\(R\) has the relation \(R\) to \(S\)" is meaningless, and the +contradiction ceases.</p> + +<p>(4) The solution of Burali-Forti's contradiction requires some further +developments for its solution. At this stage, it must suffice to +observe that a series is a relation, and an ordinal number is a class +of series. (These statements are justified in the body of the work.) +Hence a series of ordinal numbers is a relation between classes of +relations, and is of higher type than any of the series which are +members of the ordinal numbers in question. Burali-Forti's "ordinal +number of all ordinals" must be the ordinal number of all ordinals of +a given type, and must therefore be of higher type than any of these +ordinals. Hence it is not one of these ordinals, and there is no +contradiction in its being greater than any of them<a id="FNanchor_28" href="#Footnote_28" class="fnanchor">[28]</a>.</p> + +<p>(5) The paradox about "the least integer not nameable in fewer than +nineteen syllables" embodies, as is at once obvious, a vicious-circle +fallacy. For the word "nameable" refers to the totality of names, and +yet is allowed to occur in what professes to be one among names. Hence +there can be no such thing as a totality of names, in the sense in +which the paradox speaks of "names." It is easy to see that, in virtue +of the hierarchy of functions, the theory of types renders a totality +of "names" impossible. We may, in fact, distinguish names of different +orders as follows: (<i>a</i>) Elementary names will be such as are true +"proper names," <i>i.e.</i> conventional<span class="pagenum" id="Page_67">[Pg 67]</span> appellations not involving +any description. (<i>b</i>) First-order names will be such as involve +a description by means of a first-order function; that is to say, if +\(\phi !\hat{x}\) is a first-order function, "the term which satisfies +\(\phi !\hat{x}\)" will be a first-order name, though there will not +always be an object named by this name. (<i>c</i>) Second-order names +will be such as involve a description by means of a second-order +function; among such names will be those involving a reference to +the totality of first-order names. And so we can proceed through a +whole hierarchy. But at no stage can we give a meaning to the word +"nameable" unless we specify the order of names to be employed; and +any name in which the phrase "nameable by names of order \(n\)" occurs +is necessarily of a higher order than the \(n\)th. Thus the paradox +disappears.</p> + +<p>The solutions of the paradox about the least indefinable ordinal and +of Richard's paradox are closely analogous to the above. The notion of +"definable," which occurs in both, is nearly the same as "nameable," +which occurs in our fifth paradox: "definable" is what "nameable" +becomes when elementary names are excluded, <i>i.e</i>. "definable" +means "nameable by a name which is not elementary." But here there +is the same ambiguity as to type as there was before, and the same +need for the addition of words which specify the type to which the +definition is to belong. And however the type may be specified, +"the least ordinal not definable by definitions of this type" is a +definition of a higher type; and in Richard's paradox, when we confine +ourselves, as we must, to decimals that have a definition of a given +type, the number \(N\), which causes the paradox, is found to have a +definition which belongs to a higher type, and thus not to come within +the scope of our previous definitions.</p> + +<p>An indefinite number of other contradictions, of similar nature to the +above seven, can easily be manufactured. In all of them, the solution +is of the same kind. In all of them, the appearance of contradiction is +produced by the presence of some word which has systematic ambiguity of +type, such as <i>truth, falsehood, function, property, class, relation, +cardinal, ordinal, name, definition</i>. Any such word, if its typical +ambiguity is overlooked, will apparently generate a totality containing +members defined in terms of itself, and will thus give rise to +vicious-circle fallacies. In most cases, the conclusions of arguments +which involve vicious-circle fallacies will not be self-contradictory, +but wherever we have an illegitimate totality, a little ingenuity +will enable us to construct a vicious-circle fallacy leading to a +contradiction, which disappears as soon as the typically ambiguous +words are rendered typically definite, <i>i.e</i>. are determined as +belonging to this or that type.</p> + +<p><span class="pagenum" id="Page_68">[Pg 68]</span></p> + +<p>Thus the appearance of contradiction is always due to the presence of +words embodying a concealed typical ambiguity, and the solution of the +apparent contradiction lies in bringing the concealed ambiguity to +light.</p> + +<p>In spite of the contradictions which result from unnoticed typical +ambiguity, it is not desirable to avoid words and symbols which have +typical ambiguity. Such words and symbols embrace practically all the +ideas with which mathematics and mathematical logic are concerned: the +systematic ambiguity is the result of a systematic analogy. That is +to say, in almost all the reasonings which constitute mathematics and +mathematical logic, we are using ideas which may receive any one of an +infinite number of different typical determinations, any one of which +leaves the reasoning valid. Thus by employing typically ambiguous words +and symbols, we are able to make one chain of reasoning applicable to +any one of an infinite number of different cases, which would not be +possible if we were to forego the use of typically ambiguous words and +symbols.</p> + +<p>Among propositions wholly expressed in terms of typically ambiguous +notions practically the only ones which may differ, in respect of +truth or falsehood, according to the typical determination which they +receive, are existence-theorems. If we assume that the total number of +individuals is \(n\), then the total number of classes of individuals +is \(2^{n}\), the total number of classes of classes of individuals +is \(2^{2^{{}n}}\), and so on. Here \(n\) may be either finite or +infinite, and in either case \(2^{n} \gt n\). Thus cardinals greater +than \(n\) but not greater than \(2^{n}\) exist as applied to classes +of classes, but not as applied to classes of individuals, so that +whatever may be supposed to be the number of individuals, there will +be existence-theorems which hold for higher types but not for lower +types. Even here, however, so long as the number of individuals is not +asserted, but is merely assumed hypothetically, we may replace the type +of individuals by any other type, provided we make a corresponding +change in all the other types occurring in the same context. That +is, we may give the name "relative individuals" to the members of an +arbitrarily chosen type \(\tau\), and the name "relative classes of +individuals" to classes of "relative individuals," and so on. Thus so +long as only hypotheticals are concerned, in which existence-theorems +for one type are shown to be implied by existence-theorems for +another, only relative types are relevant even in existence-theorems. +This applies also to cases where the hypothesis (and therefore the +conclusion) is asserted, provided the assertion holds for any type, +however chosen. For example, any type has at least one member; hence +any type which consists of classes, of whatever order, has at least two +members. But the further pursuit of these topics must be left to the +body of the work.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_12" href="#FNanchor_12" class="label">[12]</a> +See the last section of the present Chapter. Cf. also H. +Poincaré, "Les mathématiques et la logique," <i>Revue de Métaphysique +et de Morale</i>, Mai 1906, p. 307.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_13" href="#FNanchor_13" class="label">[13]</a> +When the word "function" is used in the sequel, +"propositional function" is always meant. Other functions will not be +in question in the present Chapter.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_14" href="#FNanchor_14" class="label">[14]</a> +We shall speak in this Chapter of "values for \(\phi\hat{x}\)" +and of "values of \(\phi x\)," meaning in each case the same +thing, namely \(\phi a\), \(\phi b\), \(\phi c\), etc. The distinction +of phraseology serves to avoid ambiguity where several variables are +concerned, especially when one of them is a function.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_15" href="#FNanchor_15" class="label">[15]</a> +We use "always" as meaning "in all cases," not "at all +times." Similarly "sometimes" will mean "in some cases."</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_16" href="#FNanchor_16" class="label">[16]</a> +See <a href="#CHAPTER_III">Chapter III</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_17" href="#FNanchor_17" class="label">[17]</a> +Note that statements concerning the significance of +a phrase containing "\(\phi\hat{z}\)" concern the <i>symbol</i> +"\(\phi\hat{z}\)," and therefore do not fall under the rule that the +elimination of the functional ambiguity is necessary to significance. +Significance is a property of signs. Cf. <a href="#Page_43">p. 43</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_18" href="#FNanchor_18" class="label">[18]</a> +Cf. <a href="#CHAPTER_III">Chapter III</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_19" href="#FNanchor_19" class="label">[19]</a> +When we speak of "values of \(\phi\hat{z}\)" it is +\(\phi\), not \(z\), that is to be assigned. This follows from the +explanation in the note on <a href="#Page_42">p. 42</a>. When the function itself is the +variable, it is possible and simpler to write \(\phi\) rather than +\(\phi\hat{z}\), except in positions where it is necessary to emphasize +that an argument must be supplied to secure significance.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_20" href="#FNanchor_20" class="label">[20]</a> +When a (finite) set of predicates is given by actual +enumeration, their disjunction is a predicate, because no predicate +occurs as apparent variable in the disjunction.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_21" href="#FNanchor_21" class="label">[21]</a> +Note that in this definition the second sign of equality +is to be regarded as combining with "Df" to form one symbol; what is +defined is the sign of equality <i>not</i> followed by the letters +"Df."</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_22" href="#FNanchor_22" class="label">[22]</a> +Here the combination or disjunction is supposed to +be given intensionally. If given extensionally (<i>i.e.</i> by +enumeration), no assumption is required; but in this case the number of +predicates concerned must be finite.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_23" href="#FNanchor_23" class="label">[23]</a> +"Una questione sui numeri transfiniti," <i>Rendiconti del +circolo matematico di Palermo</i>, Vol. <span class="allsmcap">XI</span>. (1897). See *256.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_24" href="#FNanchor_24" class="label">[24]</a> +This contradiction was suggested to us by Mr G. G. Berry +of the Bodleian Library.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_25" href="#FNanchor_25" class="label">[25]</a> +\(\aleph_{0}\) is the number of finite integers. See +*123.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_26" href="#FNanchor_26" class="label">[26]</a> +Cf. König, "Ueber die Grundlagen der Mengenlehre und +das Kontinuumproblem," <i>Math. Annalen</i>, Vol. <span class="allsmcap">LXI</span>. (1905); A. C. +Dixon, "On 'well-ordered' aggregates," <i>Proc. London Math. Soc.</i> +Series 2, Vol. <span class="allsmcap">IV</span>. Part i. (1906); and E. W. Hobson, "On +the Arithmetic Continuum," <i>ibid</i>. The solution offered in the +last of these papers depends upon the variation of the "apparatus of +definition," and is thus in outline in agreement with the solution +adopted here. But it does not invalidate the statement in the text, if +"definition" is given a constant meaning.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_27" href="#FNanchor_27" class="label">[27]</a> +Cf. Poincaré, "Les mathématiques et la logique," +<i>Revue de Métaphysique et de Morale</i>, Mai 1906, especially +sections <span class="allsmcap">VII</span>. and <span class="allsmcap">IX</span>.; also Peano, <i>Revista de +Mathematica</i>, Vol. <span class="allsmcap">VIII</span>. No. 5 (1906), p. 149 ff.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_28" href="#FNanchor_28" class="label">[28]</a> +The solution of Burali-Forti's paradox by means of the +theory of types is given in detail in *256.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_69">[Pg 69]</span></p> +<h2 class="nobreak" id="CHAPTER_III">CHAPTER III.<br> +INCOMPLETE SYMBOLS.</h2> +</div> + + +<p>(1) <i>Descriptions.</i> By an "incomplete" symbol we mean a symbol +which is not supposed to have any meaning in isolation, but is only +defined in certain contexts. In ordinary mathematics, for example, +\(\frac{d}{dx}\) and \(\int_{a}^{b}\) are incomplete symbols: something +has to be supplied before we have anything significant. Such symbols +have what may be called a "definition in use." Thus if we put +\[ +\nabla^{2} = \frac{\partial^{2}}{{\partial}x^{2}} + \frac{\partial^{2}}{{\partial}y^{2}} + \frac{\partial^{2}}{{\partial}z^{2}} + \quad \text{Df}\text{,} +\] +we define the <i>use</i> of \(\nabla^{2}\), but \(\nabla^{2}\) by itself +remains without meaning. This distinguishes such symbols from what +(in a generalized sense) we may call <i>proper names</i>: "Socrates," +for example, stands for a certain man, and therefore has a meaning by +itself, without the need of any context. If we supply a context, as +in "Socrates is mortal," these words express a fact of which Socrates +himself is a constituent: there is a certain object, namely Socrates, +which does have the property of mortality, and this object is a +constituent of the complex fact which we assert when we say "Socrates +is mortal." But in other cases, this simple analysis fails us. Suppose +we say: "The round square does not exist." It seems plain that this is +a true proposition, yet we cannot regard it as denying the existence +of a certain object called "the round square." For if there were such +an object, it would exist: we cannot first assume that there is a +certain object, and then proceed to deny that there is such an object. +Whenever the grammatical subject of a proposition can be supposed not +to exist without rendering the proposition meaningless, it is plain +that the grammatical subject is not a proper name, <i>i.e.</i> not a +name directly representing some object. Thus in all such cases, the +proposition must be capable of being so analysed that what was the +grammatical subject shall have disappeared. Thus when we say "the round +square does not exist," we may, as a first attempt at such analysis, +substitute "it is false that there is an object \(x\) which is both +round and square." Generally, when "the so-and-so" is said not to +exist, we have a proposition of the form<a id="FNanchor_29" href="#Footnote_29" class="fnanchor">[29]</a> +\[ +\unicode{x201c}{\sim}\text{E}!({℩}x)({\phi}x),\unicode{x201d} +\]</p> + +<p>\[ +\textit{i.e.} \quad {\sim}\{(\exists c) \colon \phi x \ldotp \equiv_{x} \ldotp x = c\}\text{,} +\]<span class="pagenum" id="Page_70">[Pg 70]</span> +or some equivalent. Here the apparent grammatical subject +\(({℩}x)({\phi}x)\) has completely disappeared; thus in +"\({\sim}\text{E}!({℩}x)({\phi}x)\)," \(({℩}x)({\phi}x)\) is an +<i>incomplete</i> symbol.</p> + +<p>By an extension of the above argument, it can easily be shown that +\(({℩}x)({\phi}x)\) is <i>always</i> an incomplete symbol. Take, +for example, the following proposition: "Scott is the author of +Waverley." [Here "the author of Waverley" is \(({℩}x)(x ~\text{wrote Waverley})\).] +This proposition expresses an identity; thus if "the +author of Waverley" could be taken as a proper name, and supposed to +stand for some object \(c\), the proposition would be "Scott is \(c\)." +But if \(c\) is any one except Scott, this proposition is false; while +if \(c\) is Scott, the proposition is "Scott is Scott," which is +trivial, and plainly different from "Scott is the author of Waverley." +Generalizing, we see that the proposition +\[ +a = ({℩}x)({\phi}x) +\] +is one which may be true or may be false, but is never merely trivial, +like \(a = a\); whereas, if \(({℩}x)({\phi}x)\) were a proper name, +\(a = ({℩}x)({\phi}x)\) would necessarily be either false or the same +as the trivial proposition \(a = a\). We may express this by saying +that \(a = ({℩}x)({\phi}x)\) is not a value of the propositional +function \(a = y\), from which it follows that \({℩}x)(\phi x)\) is not +a value of \(y\). But since \(y\) may be anything, it follows that +(\({℩}x)({\phi}x)\) is nothing. Hence, since in use it has meaning, it +must be an incomplete symbol.</p> + +<p>It might be suggested that "Scott is the author of Waverley" asserts +that "Scott" and "the author of Waverley" are two names for the same +object. But a little reflection will show that this would be a mistake. +For if that were the meaning of "Scott is the author of Waverley," what +would be required for its truth would be that Scott should have been +<i>called</i> the author of Waverley: if he had been so called, the +proposition would be true, even if some one else had written Waverley; +while if no one called him so, the proposition would be false, even +if he had written Waverley. But in fact he was the author of Waverley +at a time when no one called him so, and he would not have been the +author if every one had called him so but some one else had written +Waverley. Thus the proposition "Scott is the author of Waverley" is +not a proposition about names, like "Napoleon is Bonaparte"; and this +illustrates the sense in which "the author of Waverley" differs from a +true proper name.</p> + +<p><span class="pagenum" id="Page_71">[Pg 71]</span></p> + +<p>Thus all phrases (other than propositions) containing the word +<i>the</i> (in the singular) are incomplete symbols: they have a +meaning in use, but not in isolation. For "the author of Waverley" +cannot mean the same as "Scott," or "Scott is the author of Waverley" +would mean the same as "Scott is Scott," which it plainly does not; +nor can "the author of Waverley" mean anything other than "Scott," or +"Scott is the author of Waverley" would be false. Hence "the author of +Waverley" means nothing.</p> + +<p>It follows from the above that we must not attempt to define +"(\({℩}x)({\phi}x)\)," but must define the <i>uses</i> of this symbol, +<i>i.e.</i> the propositions in whose symbolic expression it occurs. +Now in seeking to define the uses of this symbol, it is important to +observe the import of propositions in which it occurs. Take as an +illustration: "The author of Waverley was a poet." This implies (1) +that Waverley was written, (2) that it was written by one man, and not +in collaboration, (3) that the one man who wrote it was a poet. If +any one of these fails, the proposition is false. Thus "the author of +'Slawkenburgius on Noses' was a poet" is false, because no such book +was ever written; "the author of 'The Maid's Tragedy' was a poet" is +false, because this play was written by Beaumont and Fletcher jointly. +These two possibilities of falsehood do not arise if we say "Scott was +a poet." Thus our interpretation of the uses of \(({℩}x)({\phi}x)\) +must be such as to allow for them. Now taking \({\phi}x\) to replace +"\(x\) wrote Waverley," it is plain that any statement apparently about +(\({℩}x)({\phi}x)\) requires (1) (\(\exists x).({\phi}x)\) and (2) +\({\phi}x . {\phi}y . \supset_{x, y} . x = y\); here (1) states that at +least one object satisfies \({\phi}x\), while (2) states that at most +one object satisfies \({\phi}x\). The two together are equivalent to +\[ +\begin{align} +(\exists c) \colon &{\phi}x \ldotp \equiv_{x} \ldotp x = c\text{,}\\ +\text{which we defined as}\qquad &\text{E}!({℩}x)({\phi}x). +\end{align} +\] +Thus "\(\text{E}!({℩}x)({\phi}x)\)" must be part of what is affirmed +by any proposition about (\({℩}x)({\phi}x)\). If our proposition is +\(f\{({℩}x)({\phi}x)\}\), what is further affirmed is \(fc\), if +\({\phi}x \ldotp \equiv_{x} \ldotp x = c\). Thus we have +\[ +f\{({℩}x)({\phi}x)\} \ldotp = : (\exists c) : {\phi}x . \equiv_{x} . x = c:fc \quad \text{Df}\text{,} +\] +<i>i.e.</i> "the \(x\) satisfying \({\phi}x\) satisfies \(fx\)" is to +mean: "There is an object \(c\) such that \({\phi}x\) is true when, +and only when, \(x\) is \(c\), and \(fc\) is true," or, more exactly: +"There is a \(c\) such that '\({\phi}x\)' is always equivalent to +'\(x\) is \(c\),' and \(fc\)." In this, "\(({℩}x)({\phi}x)\)" has +completely disappeared; thus "\(({℩}x)({\phi}x)\)" is merely symbolic, +and does not directly represent an object, as single small Latin +letters are assumed to do<a id="FNanchor_30" href="#Footnote_30" class="fnanchor">[30]</a>.</p> + +<p>The proposition "\(a = ({℩}x)({\phi}x)\)" is easily shown to be +equivalent to "\({\phi}x . \equiv_{x} . x = a\)." For, by the +definition, it is +\[ +(\exists c) : {\phi}x . \equiv_{x} . x = c \colon a = c\text{,} +\] +<i>i.e.</i> "there is a \(c\) for which \({\phi}x . \equiv_{x} . x = c\), +and this \(c\) is \(a\)," which is equivalent to "\({\phi}x . \equiv_{x} . x = a\)." +Thus "Scott is the author of Waverley" is equivalent to: +\[ +\unicode{x201c}\unicode{x2018}x\,\, \text{wrote Waverley}\unicode{x2019}\,\, \text{is always equivalent to}\,\, \unicode{x2018}x\,\, \text{is Scott},\unicode{x2019}\unicode{x201d} +\] +<i>i.e.</i> "\(x\) wrote Waverley" is true when \(x\) is Scott and +false when \(x\) is not Scott.</p> + +<p>Thus although "\(({℩}x)({\phi}x)\)" has no meaning by itself, it may be +substituted for \(y\) in any propositional function \(fy\), and we get +a significant proposition, though not a value of \(fy\).</p> + +<p><span class="pagenum" id="Page_72">[Pg 72]</span></p> + +<p>When \(f\{({℩}x)(\phi x)\}\), as above defined, forms part of some +other proposition, we shall say that (\({℩}x)(\phi x)\) has a +<i>secondary</i> occurrence. When (\({℩}x)(\phi x)\) has a secondary +occurrence, a proposition in which it occurs may be true even when +(\({℩}x)(\phi x)\) does not exist. This applies, <i>e.g.</i> to the +proposition: "There is no such person as the King of France." We may +interpret this as +\[ +\begin{array}{l} +&\sim{\{\text{E}!({℩}x)(\phi x)\}},\\ +\text{or as}\quad &\sim{\{(\exists c) . c = ({℩}x)(\phi x)\}}, +\end{array} +\] +if "\(\phi x\)" stands for "\(x\) is King of France." In either case, +what is asserted is that a proposition \(p\) in which (\({℩}x)(\phi x)\) +occurs is false, and this proposition \(p\) is thus part of a +larger proposition. The same applies to such a proposition as the +following: "If France were a monarchy, the King of France would be of +the House of Orleans."</p> + +<p>It should be observed that such a proposition as +\[ +\sim{f\{({℩}x)(\phi x)\}} +\] +is ambiguous; it may deny \(f\{({℩}x)(\phi x)\}\), in which case it +will be true if (\({℩}x)(\phi x)\) does not exist, or it may mean +\[ +(\exists c) \colon \phi x . \equiv_{x} . x = c \colon \sim{fc}, +\] +in which case it can only be true if (\({℩}x)(\phi x)\) exists. In +ordinary language, the latter interpretation would usually be adopted. +For example, the proposition "the King of France is not bald" would +usually be rejected as false, being held to mean "the King of France +exists and is not bald," rather than "it is false that the King of +France exists and is bald." When (\({℩}x)(\phi x)\) exists, the two +interpretations of the ambiguity give equivalent results; but when +(\({℩}x)(\phi x)\) does not exist, one interpretation is true and +one is false. It is necessary to be able to distinguish these in our +notation; and generally, if we have such propositions as +\[ +\begin{array}{l} +\psi({℩}x)(\phi x) . \supset . p,\\ +p . \supset . \psi({℩}x)(\phi x),\\ +\psi({℩}x)(\phi x) . \supset . \chi({℩}x)(\phi x), +\end{array} +\] +<span class="pagenum" id="Page_73">[Pg 73]</span>and so on, we must be able by our notation to distinguish whether the +whole or only part of the proposition concerned is to be treated as the +"\(f\{({℩}x)(\phi x)\}\)" of our definition. For this purpose, we will +put "[(\(℩x)(\phi x)\)]" followed by dots at the beginning of the +part (or whole) which is to be taken as \(f({℩}x)(\phi x)\), the +dots being sufficiently numerous to bracket off the \(f({℩}x)(\phi x)\); +<i>i.e.</i> \(f({℩}x)(\phi x)\) is to be everything +following the dots until we reach an equal number of dots not +signifying a logical product, or a greater number signifying a logical +product, or the end of the sentence, or the end of a bracket enclosing +"[(\({℩}x)(\phi x)\)]." Thus +\[ +\begin{array}{l} +&[({℩}x)(\phi x)] . \psi({℩}x)(\phi x) . \supset . p\\ +\text{will mean} \qquad &(\exists c) \colon \phi x . \equiv_{x} . x = c \colon \psi c \colon \supset . p,\\ +\text{but}\qquad &[({℩}x)(\phi x)] \colon \psi({℩}x)(\phi x) . \supset . p\\ +\text{will mean}\qquad &(\exists c) \colon \phi x . \equiv_{x} . x = c \colon \psi c . \supset . p. +\end{array} +\] +It is important to distinguish these two, for if (\({℩}x)(\phi x)\) +does not exist, the first is true and the second false. Again +\[ +\begin{array}{l} +&[({℩}x)(\phi x)] . \sim{\psi({℩}x)(\phi x)}\\ +\text{will mean} \qquad &(\exists c) \colon \phi x . \equiv_{x} . x = c \colon \sim{\psi c},\\ +\text{while}\qquad &\sim{\{[({℩}x)(\phi x)] . \psi({℩}x)(\phi x)\}}\\ +\text{will mean}\qquad &\sim{\{(\exists c) \colon \phi x . \equiv_{x} . x = c \colon \sim{\psi c}\}}. +\end{array} +\] +Here again, when (\({℩}x)(\phi x)\) does not exist, the first is false +and the second true.</p> + +<p>In order to avoid this ambiguity in propositions containing +(\({℩}x)(\phi x)\), we amend our definition, or rather our notation, +putting +\[ +[({℩}x)(\phi x)] . f({℩}x)(\phi x) . = \colon (\exists c) \colon \phi x . \equiv_{x} . x = c \colon fc \quad \text{Df}. +\] +By means of this definition, we avoid any doubt as to the portion +of our whole asserted proposition which is to be treated as the +"\(f({℩}x)(\phi x)\)" of the definition. This portion will be called +the scope of (\({℩}x)(\phi x)\). Thus in +\[ +[({℩}x)(\phi x)] . f({℩}x)(\phi x) . \supset . p +\] +the scope of (\({℩}x)(\phi x)\) is \(f({℩}x)(\phi x)\); but in +\[ +\begin{array}{l} +&[({℩}x)(\phi x)] \colon f({℩}x)(\phi x) . \supset . p\\ +\text{the scope is} \qquad &f({℩}x)(\phi x) . \supset . p;\\ +\text{in} \qquad &\sim{\{[({℩}x)(\phi x)] . f({℩}x)(\phi x)\}} +\end{array} +\] +the scope is \(f({℩}x)(\phi x)\); but in +\[ +\begin{array}{l} +&[({℩}x)(\phi x)] . \sim{f({℩}x)(\phi x)}\\ +\text{the scope is} \qquad &\sim{f({℩}x)(\phi x)}. +\end{array} +\]</p> + +<p>It will be seen that when (\({℩}x)(\phi x)\) has the whole of the +proposition concerned for its scope, the proposition concerned cannot +be true unless \(\text{E} !({℩}x)(\phi x)\); but when (\({℩}x)(\phi x)\) +has only part of the proposition concerned for its scope, it may +often be true even when (\({℩}x)(\phi x)\) does not exist. It will be +seen further that when \(\text{E} !({℩}x)(\phi x)\), we may enlarge or +diminish the scope of (\({℩}x)(\phi x)\) as much as we please without +altering the truth-value of any proposition in which it occurs.</p> + +<p>If a proposition contains two descriptions, say (\({℩}x)(\phi x)\) and +(\({℩}x)(\psi x)\), we have to distinguish which of them has the larger +scope, <i>i.e.</i> we have to distinguish +\[ +\begin{array}{l} +\text{(1)}\qquad [({℩}x)(\phi x)] \colon [({℩}x)(\psi x)] . f{({℩}x)(\phi x), ({℩}x)(\psi x)},\\ +\text{(2)}\qquad [({℩}x)(\psi x)] \colon [({℩}x)(\phi x)] . f{({℩}x)(\phi x), ({℩}x)(\psi x)}. +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_74">[Pg 74]</span></p> + +<p>The first of these, eliminating (\({℩}x)(\phi x)\), becomes +\[ +\text{(3)}\qquad (\exists c) \colon \phi x.\equiv_{x}.x = c\colon [({℩}x)(\psi x)].f\{c, ({℩}x)(\psi x)\}, +\] +which, eliminating (\({℩}x)(\phi x)\) becomes +\[ +\text{(4)}\qquad (\exists c) \colon\ldotp \phi x.\equiv_{x}.x = c\colon\ldotp (\exists d) \colon \psi x.\equiv_{x}.x = c \colon f(c, d), +\] +and the same proposition results if, in (1), we eliminate first +(\({℩}x)(\psi x)\) and then (\({℩}x)(\phi x)\). Similarly (2) becomes, +when (\({℩}x)(\phi x)\) and (\({℩}x)(\psi x)\) are eliminated, +\[ +\text{(5)}\qquad (\exists d)\colon\ldotp \psi x.\equiv_{x}.x = d\colon\ldotp(\exists c) \colon \phi x.\equiv_{x}.x = c\colon f(c, d). +\]</p> + +<p>(4) and (5) are equivalent, so that the truth-value of a proposition +containing two descriptions is independent of the question which has +the larger scope.</p> + +<p>It will be found that, in most cases in which descriptions occur, their +scope is, in practice, the smallest proposition enclosed in dots or +other brackets in which they are contained. Thus for example +\[ +[({℩}x)(\phi x)].\psi({℩}x)(\phi x). \supset . [({℩}x)(\phi x)].\chi({℩}x)(\phi x) +\] +will occur much more frequently than +\[ +[({℩}x)(\phi x)] \colon \psi({℩}x)(\phi x) . \supset . \chi({℩}x)(\phi x). +\] +For this reason it is convenient to decide that, when the scope of an +occurrence of (\({℩}x)(\phi x)\) is the smallest proposition, enclosed +in dots or other brackets, in which the occurrence in question is +contained, the scope need not be indicated by "[(\({℩}x)(\phi x)\)]." +Thus <i>e.g.</i> +\[ +\begin{array}{l} +&p.\supset.a = ({℩}x)(\phi x)\\ +\text{will mean}\qquad &p.\supset.[({℩}x)(\phi x)].a = ({℩}x)(\phi x);\\ +\text{and}\qquad &p.\supset.(\exists a).a = ({℩}x)(\phi x)\\ +\text{will mean}\qquad &p.\supset.(\exists a).[({℩}x)(\phi x)].a = ({℩}x)(\phi x);\\ +\text{and}\qquad &p.\supset.a \neq ({℩}x)(\phi x)\\ +\text{will mean}\qquad &p.\supset.[({℩}x)(\phi x)].\sim{\{a = ({℩}x)(\phi x)\}};\\ +\text{but}\qquad &p.\supset.\sim{\{a = ({℩}x)(\phi x)}\}\\ +\text{will mean}\qquad &p.\supset.\sim{\{[({℩}x)(\phi x)].a = ({℩}x)(\phi x)}\}. +\end{array} +\]</p> + +<p>This convention enables us, in the vast majority of cases that actually +occur, to dispense with the explicit indication of the scope of a +descriptive symbol; and it will be found that the convention agrees +very closely with the tacit conventions of ordinary language on this +subject. Thus for example, if "(\({℩}x)(\phi x)\)" is "the so-and-so," +"\(a \neq ({℩}x)(\phi x)\)" is to be read "\(a\) is not the so-and-so," +which would ordinarily be regarded as implying that "the so-and-so" +exists; but "\({\sim}\{a=({℩}x)(\phi x)\}\)" is to be read "it is not +true that \(a\) is the so-and-so," which would generally be allowed +to hold if "the so-and-so" does not exist. Ordinary language is, of +course, rather loose and fluctuating in its implications on this +matter; but subject to the requirement of definiteness, our convention +seems to keep as near to ordinary language as possible.</p> + +<p><span class="pagenum" id="Page_75">[Pg 75]</span></p> + +<p>In the case when the smallest proposition enclosed in dots or +other brackets contains two or more descriptions, we shall assume, +in the absence of any indication to the contrary, that one which +typographically occurs earlier has a larger scope than one which +typographically occurs later. Thus +\[ +\begin{array}{l} +&({℩}x)(\phi x) = ({℩}x)(\psi x)\\ +\text{will mean}\qquad &(\exists c)\colon \phi x.\equiv_{x}.x = c\colon [({℩}x)(\psi x)].c = ({℩}x)(\psi x),\\ +\text{while}\qquad &({℩}x)(\psi x) = ({℩}x)(\phi x)\\ +\text{will mean} &(\exists d)\colon \psi x.\equiv_{x}.x = d\colon[(℩x)(\phi x)].({℩}x)(\phi x) = d. +\end{array} +\]</p> + +<p>These two propositions are easily shown to be equivalent.</p> + +<p>(2) <i>Classes</i>. The symbols for classes, like those for +descriptions, are, in our system, incomplete symbols: their uses are +defined, but they themselves are not assumed to mean anything at all. +That is to say, the uses of such symbols are so defined that, when +the <i>definiens</i> is substituted for the <i>definiendum</i>, there +no longer remains any symbol which could be supposed to represent a +class. Thus classes, so far as we introduce them, are merely symbolic +or linguistic conveniences, not genuine objects as their members are if +they are individuals.</p> + +<p>It is an old dispute whether formal logic should concern itself +mainly with intensions or with extensions. In general, logicians +whose training was mainly philosophical have decided for intensions, +while those whose training was mainly mathematical have decided for +extensions. The facts seem to be that, while mathematical logic +requires extensions, philosophical logic refuses to supply anything +except intensions. Our theory of classes recognizes and reconciles +these two apparently opposite facts, by showing that an extension +(which is the same as a class) is an incomplete symbol, whose use +always acquires its meaning through a reference to intension.</p> + +<p>In the case of descriptions, it was possible to <i>prove</i> that they +are incomplete symbols. In the case of classes, we do not know of any +equally definite proof, though arguments of more or less cogency can be +elicited from the ancient problem of the One and the Many<a id="FNanchor_31" href="#Footnote_31" class="fnanchor">[31]</a>. It is +not necessary for our purposes, however, to assert dogmatically that +there are no such things as classes. It is only necessary for us to +show that the incomplete symbols which we introduce as representatives +of classes yield all the propositions for the sake of which classes +might be thought essential. When this has been shown, the mere +principle of economy of primitive ideas leads to the non-introduction +of classes except as incomplete symbols.</p> + +<p><span class="pagenum" id="Page_76">[Pg 76]</span></p> + +<p>To explain the theory of classes, it is necessary first to explain +the distinction between <i>extensional</i> and <i>intensional</i> +functions. This is effected by the following definitions:</p> + +<p>The <i>truth-value</i> of a proposition is truth if it is true, and +falsehood if it is false. (This expression is due to Frege.)</p> + +<p>Two propositions are said to be <i>equivalent</i> when they have the +same truth-value, <i>i.e.</i> when they are both true or both false.</p> + +<p>Two propositional functions are said to be <i>formally</i> equivalent +when they are equivalent with every possible argument, <i>i.e.</i> +when any argument which satisfies the one satisfies the other, and +vice versa. Thus "\(\hat{x}\) is a man" is formally equivalent to +"\(\hat{x}\) is a featherless biped"; "\(\hat{x}\) is an even prime" is +formally equivalent to "\(\hat{x}\) is identical with 2."</p> + +<p>A function of a function is called <i>extensional</i> when its +truth-value with any argument is the same as with any formally +equivalent argument. That is to say, \(f(\phi ! \hat{z})\) is an +extensional function of \(\phi ! \hat{z}\) if, provided \(\psi ! \hat{z}\) +is formally equivalent to \(\phi ! \hat{z}\), \(f(\phi ! \hat{z})\) +is equivalent to \(f(\psi ! \hat{z})\). Here the apparent +variables \(\phi\) and \(\psi\) are necessarily of the type from which +arguments can significantly be supplied to \(f\). We find no need to +use as apparent variables any functions of non-predicative types; +accordingly in the sequel all extensional functions considered are in +fact functions of predicative functions<a id="FNanchor_32" href="#Footnote_32" class="fnanchor">[32]</a>.</p> + +<p>A function of a function is called <i>intensional</i> when it is not +extensional.</p> + +<p>The nature and importance of the distinction between intensional and +extensional functions will be made clearer by some illustrations. The +proposition "'\(x\) is a man' always implies '\(x\) is a mortal'" is an +extensional function of the function "\(\hat{x}\) is a man," because we +may substitute, for "\(x\) is a man," "\(x\) is a featherless biped," +or any other statement which applies to the same objects to which +"\(x\) is a man" applies, and to no others. But the proposition "\(A\) +believes that '\(x\) is a man' always implies '\(x\) is a mortal'" +is an intensional function of "\(\hat{x}\) is a man," because \(A\) +may never have considered the question whether featherless bipeds +are mortal, or may believe wrongly that there are featherless bipeds +which are not mortal. Thus even if "\(x\) is a featherless biped" is +formally equivalent to "\(x\) is a man," it by no means follows that +a person who believes that all men are mortal must believe that all +featherless bipeds are mortal, since he may have never thought about +featherless bipeds, or have supposed that featherless bipeds were not +always men. Again the proposition "the number of arguments that satisfy +the function \(\phi ! \hat{z}\) is \(n\)" is an extensional function +of \(\phi ! \hat{z}\), because its truth or falsehood is unchanged +if we substitute for \(\phi ! \hat{z}\) any other function which is +true whenever \(\phi ! \hat{z}\) is true, and false whenever \(\phi ! \hat{z}\) +is false. But the proposition "\(A\) asserts that the number +of arguments satisfying \(\phi ! \hat{z}\) is n" is an intensional +function of \(\phi ! \hat{z}\),<span class="pagenum" id="Page_77">[Pg 77]</span> since, if \(A\) asserts this +concerning \(\phi ! \hat{z}\), he certainly cannot assert it concerning +all predicative functions that are equivalent to \(\phi ! \hat{z}\), +because life is too short. Again, consider the proposition "two white +men claim to have reached the North Pole." This proposition states +"two arguments satisfy the function '\(\hat{x}\) is a white man who +claims to have reached the North Pole.'" The truth or falsehood of this +proposition is unaffected if we substitute for "\(\hat{x}\) is a white +man who claims to have reached the North Pole" any other statement +which holds of the same arguments, and of no others. Hence it is an +extensional function. But the proposition "it is a strange coincidence +that two white men should claim to have reached the North Pole," which +states "it is a strange coincidence that two arguments should satisfy +the function '\(\hat{x}\) is a white man who claims to have reached +the North Pole,'" is not equivalent to "it is a strange coincidence +that two arguments should satisfy the function '\(\hat{x}\) is Dr Cook +or Commander Peary.'" Thus "it is a strange coincidence that \(\phi +! \hat{x}\) should be satisfied by two arguments" is an intensional +function of \(\phi ! \hat{x}\).</p> + +<p>The above instances illustrate the fact that the functions of functions +with which mathematics is specially concerned are extensional, and that +intensional functions of functions only occur where non-mathematical +ideas are introduced, such as what somebody believes or affirms, or the +emotions aroused by some fact. Hence it is natural, in a mathematical +logic, to lay special stress on <i>extensional</i> functions of +functions.</p> + +<p>When two functions are formally equivalent, we may say that they +<i>have the same extension</i>. In this definition, we are in close +agreement with usage. We do not assume that there is such a thing +as an extension: we merely define the whole phrase "having the same +extension." We may now say that an extensional function of a function +is one whose truth or falsehood depends only upon the extension of its +argument. In such a case, it is convenient to regard the statement +concerned as being about the extension. Since extensional functions are +many and important, it is natural to regard the extension as an object, +called a <i>class</i>, which is supposed to be the subject of all the +equivalent statements about various formally equivalent functions. Thus +<i>e.g.</i> if we say "there were twelve Apostles," it is natural to +regard this statement as attributing the property of being twelve to +a certain collection of men, namely those who were Apostles, rather +than as attributing the property of being satisfied by twelve arguments +to the function "\(\hat{x}\) was an Apostle." This view is encouraged +by the feeling that there is something which is identical in the case +of two functions which "have the same extension." And if we take such +simple problems as "how many combinations can be made of \(n\) things?" +it seems at first sight necessary that each "combination" should be a +single object which can be counted as one. This, however, is certainly +not necessary technically, and we see no reason to suppose that it is +true<span class="pagenum" id="Page_78">[Pg 78]</span> philosophically. The technical procedure by which the apparent +difficulty is overcome is as follows.</p> + +<p>We have seen that an extensional function of a function may be regarded +as a function of the class determined by the argument-function, but +that an intensional function cannot be so regarded. In order to +obviate the necessity of giving different treatment to intensional +and extensional functions of functions, we construct an extensional +function derived from any function of a predicative function \(\psi !\hat{z}\), +and having the property of being equivalent to the function +from which it is derived, provided this function is extensional, +as well as the property of being significant (by the help of the +systematic ambiguity of equivalence) with any argument \(\phi \hat{z}\) +whose arguments are of the same type as those of \(\psi !\hat{z}\). The +derived function, written "\(f\{\hat{z}(\phi z)\}\)," is defined as +follows: Given a function \(f(\psi ! \hat{z}\)), our derived function +is to be "there is a predicative function which is formally equivalent +to \(\phi \hat{z}\) and satisfies \(f\)." If \(\phi \hat{z}\) is a +predicative function, our derived function will be true whenever +\(f(\phi \hat{z}\)) is true. If \(f(\phi \hat{z}\)) is an extensional +function, and \(\phi \hat{z}\) is a predicative function, our derived +function will not be true unless \(f(\phi \hat{z}\)) is true; thus in +this case, our derived function is equivalent to \(f(\phi \hat{z}\)). +If \(f(\phi \hat{z}\)) is not an extensional function, and if \(\phi \hat{z}\) +is a predicative function, our derived function may sometimes +be true when the original function is false. But in any case the +derived function is always extensional.</p> + +<p>In order that the derived function should be significant for any +function \(\phi \hat{z}\), of whatever order, provided it takes +arguments of the right type, it is necessary and sufficient that +\(f(\psi \hat{z}\)) should be significant, where \(\psi !\hat{z}\) +is any <i>predicative</i> function. The reason of this is that we +only require, concerning an argument \(\phi \hat{z}\), the hypothesis +that it is formally equivalent to some predicative function \(\psi +!\hat{z}\), and formal equivalence has the same kind of systematic +ambiguity as to type that belongs to truth and falsehood, and can +therefore hold between functions of any two different orders, +provided the functions take arguments of the same type. Thus by means +of our derived function we have not merely provided extensional +functions everywhere in place of intensional functions, but we have +<i>practically</i> removed the necessity for considering differences of +type among functions whose arguments are of the same type. This effects +the same kind of simplification in our hierarchy as would result from +never considering any but predicative functions.</p> + +<p>If \(f(\psi !\hat{z}\)) can be built up by means of the primitive ideas +of disjunction, negation, (\(x).\phi x\), and (\(\exists x).\phi x\), +as is the case with all the functions of functions that explicitly +occur in the present work, it will be found that, in virtue of the +systematic ambiguity of the above primitive ideas, any function \(\phi\hat{z}\) +whose arguments are of the same type as those of \(\psi!\hat{z}\) +can significantly be substituted for \(\psi !\hat{z}\) +in f without any other symbolic change. Thus in<span class="pagenum" id="Page_79">[Pg 79]</span> such a case what +is symbolically, though not really, the same function \(f\) can +receive as arguments functions of various different types. If, with +a given argument \(\phi \hat{z}\), the function \(f(\phi \hat{z}\)), +so interpreted, is equivalent to \(f(\psi !\hat{z}\)) whenever +\(\psi ! \hat{z}\) is formally equivalent to \(\phi \hat{z}\), then +\(f\{\hat{z}(\phi z)\}\) is equivalent to \(f(\phi \hat{z}\)) provided +there is any predicative function formally equivalent to \(\phi +\hat{z}\). At this point, we make use of the axiom of reducibility, +according to which there always is a predicative function formally +equivalent to \(\phi \hat{z}\).</p> + +<p>As was explained above, it is convenient to regard an extensional +function of a function as having for its argument not the function, but +the class determined by the function. Now we have seen that our derived +function is always extensional. Hence if our original function was +\(f(\phi !\hat{z}\)), we write the derived function \(f\{\hat{z}(\phi z)\}\), +where "\(\hat{z}(\phi z\))" may be read "the class of arguments +which satisfy \(\phi \hat{z}\)," or more simply "the class determined +by \(\phi \hat{z}\)." Thus "\(f\{\hat{z}(\phi z)\}\)" will mean: +"There is a predicative function \(\psi !\hat{z}\) which is formally +equivalent to \(\phi \hat{z}\) and is such that \(f(\psi !\hat{z}\)) +is true." This is in reality a function of \(\phi \hat{z}\), but we +treat it symbolically as if it had an argument \(\hat{z}(\phi z\)). +By the help of the axiom of reducibility, we find that the usual +properties of classes result. For example, two formally equivalent +functions determine the same class, and conversely, two functions which +determine the same class are formally equivalent. Also to say that +\(x\) is a member of \(\hat{z}(\phi z\)), <i>i.e.</i> of the class +determined by \(\phi \hat{z}\), is true when \(\phi x\) is true, and +false when \(\phi x\) is false. Thus all the mathematical purposes for +which classes might seem to be required are fulfilled by the purely +symbolic objects \(\hat{z}(\phi z\)) provided we assume the axiom of +reducibility.</p> + +<p><span class="pagenum" id="Page_80">[Pg 80]</span></p> + +<p>In virtue of the axiom of reducibility, if \(\phi \hat{z}\) is +any function, there is a formally equivalent predicative function +\(\psi ! \hat{z}\); then the class \(\hat{z}(\phi z\)) is identical +with the class \(\hat{z}(\psi !z\)), so that every class can be +defined by a <i>predicative</i> function. Hence the totality of +the <i>classes</i> to which a given term can be significantly said +to belong or not to belong is a legitimate totality, although the +totality of <i>functions</i> which a given term can be significantly +said to satisfy or not to satisfy is not a legitimate totality. The +classes to which a given term \(a\) belongs or does not belong are +the classes defined by \(a\)-functions; they are also the classes +defined by <i>predicative</i> \(a\)-functions. Let us call them +\(a\)-classes. Then "\(a\)-classes" form a legitimate totality, derived +from that of predicative \(a\)-functions. Hence many kinds of general +statements become possible which would otherwise involve vicious-circle +paradoxes. These general statements are none of them such as lead to +contradictions, and many of them such as it is very hard to suppose +illegitimate. The fact that they are rendered possible by the axiom +of reducibility, and that they would otherwise be excluded by the +vicious-circle principle, is to be regarded as an argument in favour of +the axiom of reducibility.</p> + +<p>The above definition of "the class defined by the function \(\phi\hat{z}\)," +or rather, of any proposition in which this phrase occurs, +is, in symbols, as follows: +\[ +f{\hat{z}(\phi z)}.=\colon(\exists \psi)\colon \phi x.\equiv_{x}.\psi !x\colon f\{\psi !\hat{z}\}\,\quad \text{Df}. +\] +In order to recommend this definition, we shall enumerate five +requisites which a definition of classes must satisfy, and we shall +then show that the above definition satisfies these five requisites.</p> + +<p>We require of classes, if they are to serve the purposes for which +they are commonly employed, that they shall have certain properties, +which may be enumerated as follows. (1) Every propositional function +must determine a class, which may be regarded as the collection of +all the arguments satisfying the function in question. This principle +must hold when the function is satisfied by an infinite number of +arguments as well as when it is satisfied by a finite number. It +must hold also when no arguments satisfy the function; <i>i.e.</i> +the "null-class" must be just as good a class as any other. (2) Two +propositional functions which are formally equivalent, <i>i.e.</i> such +that any argument which satisfies either satisfies the other, must +determine the same class; that is to say, a class must be something +wholly determined by its membership, so that <i>e.g.</i> the class +"featherless bipeds" is identical with the class "men," and the class +"even primes" is identical with the class "numbers identical with 2." +(3) Conversely, two propositional functions which determine the same +class must be formally equivalent; in other words, when the class is +given, the membership is determinate: two different sets of objects +cannot yield the same class. (4) In the same sense in which there are +classes (whatever this sense may be), or in some closely analogous +sense, there must also be classes of classes. Thus for example "the +combinations of \(n\) things \(m\) at a time," where the \(n\) things +form a given class, is a class of classes; each combination of \(m\) +things is a class, and each such class is a member of the specified +set of combinations, which set is therefore a class whose members are +classes. Again, the class of unit classes, or of couples, is absolutely +indispensable; the former is the number 1, the latter the number 2. +Thus without classes of classes, arithmetic becomes impossible. (5) +It must under all circumstances be meaningless to suppose a class +identical with one of its own members. For if such a supposition +had any meaning, "\(\alpha \in \alpha\)" would be a significant +propositional function<a id="FNanchor_33" href="#Footnote_33" class="fnanchor">[33]</a>, and so would "\(\alpha \sim{\in \alpha}\)." +Hence, by (1) and (4), there would be a class of all classes satisfying +the function "\(\alpha \sim{\in \alpha}\)." If we call this class +\(\kappa\), we shall have +\[ +\alpha \in \kappa.\equiv_{\alpha}. \alpha \sim{\in \alpha}. +\] +Since, by our hypothesis, "\(\kappa \in \kappa\)" is supposed +significant, the above equivalence, which holds with all possible +values of \(\alpha\), holds with the value \(\kappa\), <i>i.e.</i> +\[ +\kappa \in \kappa.\equiv. \kappa \sim{\in \kappa}. +\]<span class="pagenum" id="Page_81">[Pg 81]</span> +But this is a contradiction<a id="FNanchor_34" href="#Footnote_34" class="fnanchor">[34]</a>. Hence "\(\alpha \in \alpha\)" and +"\(\alpha \sim{\in \alpha}\)" must always be meaningless. In general, +there is nothing surprising about this conclusion, but it has two +consequences which deserve special notice. In the first place, a class +consisting of only one member must not be identical with that one +member, <i>i.e</i>. we must not have \(\iota ʻx=x\). For we have \(x \in \iota ʻx\), +and therefore, if \(x = \iota ʻx\), we have \(\iota ʻx \in \iota ʻx\), +which, we saw, must be meaningless. It follows that +"\(x = \iota ʻx\)" must be absolutely meaningless, not simply false. In +the second place, it might appear as if the class of all classes were +a class, <i>i.e.</i> as if (writing "Cls" for "class") "\(\text{Cls}\,\in\, \text{Cls}\)" +were a true proposition. But this combination of symbols must be +meaningless; unless, indeed, an ambiguity exists in the meaning of +"Cls," so that, in "\(\text{Cls}\, \in\, \text{Cls}\)," the first "Cls" +can be supposed to have a different meaning from the second.</p> + +<p>As regards the above requisites, it is plain, to begin with, that, +in accordance with our definition, every propositional function +\(\phi \hat{z}\) determines a class \(\hat{z}(\phi z\)). Assuming +the axiom of reducibility, there must always be true propositions +about \(\hat{z}(\phi z\)), <i>i.e.</i> true propositions of the form +\(f\{\hat{z}(\phi z)\}\). For suppose \(\phi \hat{z}\) is formally +equivalent to \(\psi !\hat{z}\), and suppose \(\psi !\hat{z}\) satisfies +some function \(f\). Then \(\hat{z}(\phi z\)) also satisfies \(f\). +Hence, given any function \(\phi \hat{z}\), there are true propositions +of the form \(f\{\hat{z}(\phi z)\}\), <i>i.e.</i> true propositions in +which "the class determined by \(\phi \hat{z}\)" is grammatically the +subject. This shows that our definition fulfils the first of our five +requisites.</p> + +<p>The second and third requisites together demand that the classes +\(\hat{z}(\phi z\)) and \(\hat{z}(\psi z\)) should be identical when, +and only when, their defining functions are formally equivalent, +<i>i.e.</i> that we should have +\[ +\hat{z}(\phi z)=\hat{z}(\psi z).\equiv\colon \phi x.\equiv_{x}.\psi x. +\] +Here the meaning of "\(\hat{z}(\phi z)\) = \(\hat{z}(\psi z)\)" is to +be derived, by means of a two-fold application of the definition of +\(f\{\hat{z}(\phi z)\}\), from the definition of +\[ + \unicode{x201c}\chi !\hat{z} = \theta!\hat{z}, \unicode{x201d} +\] +\[ +\text{which is}\quad \chi !\hat{z}= \theta!.\hat{z}=\colon (f)\colon f!\chi !\hat{z}.\supset.f!\theta !\hat{z}\, \quad\text{Df} +\] +by the general definition of identity.</p> + +<p>In interpreting "\(\hat{z}(\phi z = \hat{z}(\psi z)\)," we will adopt +the convention which we adopted in regard to (\({℩}x)(\phi x\)) and +(\({℩}x)(\psi x)\), namely that the incomplete symbol which occurs +first is to have the larger scope. Thus \(\hat{z}(\phi z = \hat{z}(\psi z)\) +becomes, by our definition, +\[ +(\exists \chi) \colon \phi x . \equiv_{x} . \chi !x \colon \chi ! \hat{z} = \hat{z}(\psi z), +\] +which, by eliminating \(\hat{z}(\psi z\)), becomes +\[ +(\exists \chi) \colon\ldotp \phi x.\equiv_{x}.\chi !x \colon\ldotp (\exists \theta)\colon \psi x.\equiv_{x}.\theta !x\colon \chi !\hat{z} = \theta !\hat{z}, +\] +which is equivalent to +\[ +(\exists \chi , \theta)\colon \phi x.\equiv_{x}.\chi !x\colon \psi x.\equiv_{x}. \theta !x\colon \chi !\hat{z} = \theta !\hat{z}, +\]<span class="pagenum" id="Page_82">[Pg 82]</span> +which, again, is equivalent to +\[ +(\exists \chi) \colon \phi x \ldotp \equiv_{x} \ldotp \chi!x \colon \psi x \ldotp \equiv_{x} \ldotp \chi!x, +\] +which, in virtue of the axiom of reducibility, is equivalent to +\[ +\phi x \ldotp \equiv_{x} \ldotp \psi x. +\] +Thus our definition of the use of \(\hat{z}(\phi z)\) is such as to +satisfy the conditions (2) and (3) which we laid down for classes, +<i>i.e.</i> we have +\[ +\vdash \colon\ldotp \hat{z}(\phi z) = \hat{z}(\psi z) \ldotp \equiv \colon \phi x \ldotp \equiv_{x} \ldotp \psi x. +\]</p> + +<p>Before considering classes of classes, it will be well to define +membership of a class, <i>i.e.</i> to define the symbol "\(x \in\hat{z} (\phi z)\)," +which may be read "\(x\) is a member of the +class determined by \(\phi\hat{z}\)." Since this is a function of the +form \(f\{\hat{z}(\phi z)\}\), it must be derived, by means of our +general definition of such functions, from the corresponding function +\(f\{\psi!\hat{z}\}\). We therefore put +\[ +x \in \psi!\hat{z} \ldotp = \ldotp \psi!x \quad\text{Df}. +\] +This definition is only needed in order to give a meaning to +"\(x\in \hat{z}(\phi z)\)"; the meaning it gives is, in virtue of the +definition of \(f\{\hat{z}(\phi z)\}\), +\[ +(\exists\psi) \colon \phi y \ldotp \equiv_{y} \ldotp \psi!y \colon \psi!x. +\] +It thus appears that "\(x \in \hat{z}(\phi z)\)" implies \(\phi x\), +since it implies \(\psi!x\), and \(\psi!x\) is equivalent to \(\phi x\); +also, in virtue of the axiom of reducibility, \(\phi x\) implies +"\(x \in \hat{z}(\phi z)\)," since there is a predicative function +\(\psi\) formally equivalent to \(\phi\), and \(x\) must satisfy +\(\psi\), since \(x\) (<i>ex hypothesi</i>) satisfies \(\phi\). Thus in +virtue of the axiom of reducibility we have +\[ +\vdash \colon x \in \hat{z}(\phi z) \ldotp \equiv \ldotp \phi x, +\] +<i>i.e.</i> \(x\) is a member of the class \(\hat{z}(\phi z)\) when, +and only when, \(x\) satisfies the function \(\phi\) which defines the +class.</p> + +<p>We have next to consider how to interpret a class of classes. As we +have defined \(f\{\hat{z}(\phi z)\}\), we shall naturally regard a +class of classes as consisting of those values of \(\hat{z}(\phi z)\) +which satisfy \(f\{\hat{z}(\phi z)\}\). Let us write \(\alpha\) for +\(\hat{z}(\phi z)\); then we may write \(\hat{\alpha}(f\alpha)\) for +the class of values of \(\alpha\) which satisfy \(f\alpha\)<a id="FNanchor_35" href="#Footnote_35" class="fnanchor">[35]</a>. We +shall apply the same definition, and put +\[ +F\{\hat{\alpha}(f\alpha)\} \ldotp = \colon (\exists g) \colon f\beta \ldotp \equiv_{\beta} \ldotp g!\beta \colon F\{g!\hat{\alpha}\} \quad \text{Df}, +\] +where "\(\beta\)" stands for any expression of the form +\(\hat{z}(\psi!z)\).</p> + +<p>Let us take "\(\gamma \in \hat{\alpha}(f\alpha)\)" as an instance of +\(F\{\hat{\alpha}(f\alpha)\}\). Then +\[ +\vdash \colon\ldotp \gamma \in \hat{\alpha} (f\alpha) \ldotp \equiv \colon (\exists g) : f\beta \ldotp \equiv_{\beta} \ldotp g!\beta \colon \gamma\in g!\hat{\alpha}. +\] +Just as we put +\[ +x \in\psi! \hat{z} \ldotp = \ldotp \psi!x \quad \text{Df}, +\] +so we put +\[ +\gamma \in g! \hat{\alpha} \ldotp = \ldotp g!\gamma \quad \text{Df}. +\]</p> + +<p>Thus we find +\[ +\vdash \colon\ldotp \gamma \in \hat{\alpha} (f\alpha) \ldotp \equiv : (\exists g) : f\beta \ldotp \equiv_{\beta} \ldotp g!\beta : g!\gamma. +\]</p> + +<p><span class="pagenum" id="Page_83">[Pg 83]</span></p> + +<p>If we now extend the axiom of reducibility so as to apply to functions +of functions, <i>i.e.</i> if we assume +\[ +(\exists g)\colon f(\psi !\hat{z}).\equiv_{\psi}.g!(\psi !\hat{z}), +\] +we easily deduce +\[ +\begin{array}{l} +&\vdash\colon(\exists g)\colon f{\{\hat{z}(\psi !z)\}}.\equiv_{\psi}.g!\{\hat{z}(\psi !z)\},\\ +\textit{i.e.}\quad &\vdash\colon(\exists g)\colon f \beta .\equiv_{\beta}.g!\beta.\\ +\text{Thus}\quad &\vdash \colon \gamma \in \hat{\alpha}(f \alpha).\equiv.f \gamma. +\end{array} +\]</p> + +<p>Thus every function which can take classes as arguments, <i>i.e.</i> +every function of functions, determines a class of classes, whose +members are those classes which satisfy the determining function. Thus +the theory of classes of classes offers no difficulty.</p> + +<p>We have next to consider our fifth requisite, namely that +"\(\hat{z}(\phi z)\in \hat{z}(\phi z\))" is to be meaningless. Applying +our definition of \(f\{\hat{z}(\phi z)\}\), we find that if this +collection of symbols had a meaning, it would mean +\[ +(\exists \psi)\colon \phi x.\equiv_{x}. \psi!x \colon \psi !\hat{z} \in \psi !\hat{z}, +\] +<i>i.e.</i> in virtue of the definition +\[ +\begin{array}{l} +&x \in \psi!\hat{z}.=. \psi !x \quad \text{Df},\\ +\text{it would mean}\quad &(\exists \psi)\colon \phi x.\equiv_{x}. \psi!x\colon \psi !(\psi !\hat{z}). +\end{array} +\] +But here the symbol "\(\psi!(\psi !\hat{z})\)" occurs, which assigns a +function as argument to itself. Such a symbol is always meaningless, +for the reasons explained at the beginning of Chapter II (<a href="#Page_41">pp. 41</a>-<a href="#Page_43">43</a>). +Hence "\(\hat{z}(\phi z) \in \hat{z}(\phi z)\)" is meaningless, and our +fifth and last requisite is fulfilled.</p> + +<p>As in the case of \(f({℩}x)(\phi x\)), so in that of \(f\{\hat{z}(\phi z)\}\), +there is an ambiguity as to the scope of \(\hat{z}(\phi z)\) +if it occurs in a proposition which itself is part of a larger +proposition. But in the case of classes, since we always have the axiom +of reducibility, namely +\[ +(\exists \psi) \colon \phi x.\equiv_{x}.\psi !x, +\] +which takes the place of \(\text{E}!(℩x)(\phi x)\), it follows that the +truth-value of any proposition in which \(\hat{z}(\phi z)\) occurs is +the same whatever scope we may give to \(\hat{z}(\phi z)\), provided +the proposition is an extensional function of whatever functions it +may contain. Hence we may adopt the convention that the scope is to be +always the smallest proposition enclosed in dots or brackets in which +\(\hat{z}(\phi z)\) occurs. If at any time a larger scope is required, +we may indicate it by "[\(\hat{z}(\phi z)\)]" followed by dots, in the +same way as we did for \([({℩}x)(\phi x)]\).</p> + +<p><span class="pagenum" id="Page_84">[Pg 84]</span></p> + +<p>Similarly when two class symbols occur, <i>e.g.</i> in a proposition +of the form \(f\{\hat{z}(\phi z),\hat{z}(\psi z)\}\), we need not +remember rules for the scopes of the two symbols, since all choices +give equivalent results, as it is easy to prove. For the preliminary +propositions a rule is desirable, so we can decide that the class +symbol which occurs first in the order of writing is to have the larger +scope.</p> + +<p>The representation of a class by a single letter \(\alpha \) can now +be understood. For the denotation of \(\alpha \) is ambiguous, in so +far as it is undecided as to which of the symbols \(\hat{z}(\phi z)\), +\(\hat{z}(\psi z)\), \(\hat{z}(\chi z)\), etc. it is to stand for, +where \(\phi \hat{z}\), \(\psi \hat{z}\), \(\chi \hat{z}\), etc. are +the various determining functions of the class. According to the choice +made, different propositions result. But all the resulting propositions +are equivalent by virtue of the easily proved proposition: +\[ + \unicode{x201c}\vdash :\phi x\equiv_{x}\psi x.\supset .f\{\hat{z}(\phi z)\}\equiv f\{\hat{z}(\psi z)\}. \unicode{x201d} +\] +Hence unless we wish to discuss the determining function itself, +so that the notion of a class is really not properly present, the +ambiguity in the denotation of \(\alpha\) is entirely immaterial, +though, as we shall see immediately, we are led to limit ourselves +to predicative determining functions. Thus "\(f(\alpha)\)," where +\(\alpha\) is a variable class, is really "\(f\{\hat{z}(\phi z)\}\)," +where \(\phi\) is a variable function, that is, it is +\[ +\unicode{x201c}(\exists \psi ).\phi x\equiv_{x}\psi !x.f\{\psi !\hat{z}\}, \unicode{x201d} +\] +where \(\phi\) is a variable function. But here a difficulty arises +which is removed by a limitation to our practice and by the axiom +of reducibility. For the determining functions \(\phi \hat{z}\), +\(\psi \hat{z}\), etc. will be of different types, though the axiom +of reducibility secures that some are predicative functions. Then, in +interpreting \(\alpha\) as a variable in terms of the variation of any +determining function, we shall be led into errors unless we confine +ourselves to predicative determining functions. These errors especially +arise in the transition to total variation (cf. <a href="#Page_15">pp. 15</a>, <a href="#Page_16">16</a>). Accordingly +\[ +f\alpha =.(\exists \psi ).\phi !x\equiv_{x}\psi !x.f\{\psi !\hat{z}\}\quad \text{Df.} +\] +It is the peculiarity of a definition of the use of a single letter +[viz. \(\alpha\)] for a variable incomplete symbol that it, though +in a sense a real variable, occurs only in the <i>definiendum</i>, +while "\(\phi \)," though a real variable, occurs only in the +<i>definiens</i>.</p> + +<p>Thus "\(f\hat{\alpha }\)" stands for +\[ +\unicode{x201c}(\exists \psi ).\hat{\phi }!x\equiv_{x}\psi !x.f\{\psi !\hat{z}\},\unicode{x201d} +\] +and "(\(\alpha).f\alpha\)" stands for +\[ + \unicode{x201c}(\phi ):(\exists \psi ).\phi !x\equiv_{x}\psi !x.f\{\psi !\hat{z}\}. \unicode{x201d} +\] +Accordingly, in mathematical reasoning, we can dismiss the whole +apparatus of functions and think only of classes as "quasi-things," +capable of immediate representation by a single name. The advantages +are two-fold: (1) classes are determined by their membership, so that +to one set of members there is one class, (2) the "type" of a class is +entirely defined by the type of its members.</p> + +<p>Also a predicative function of a class can be defined thus +\[ +f!\alpha =.(\exists \psi ).\phi !x\equiv_{x}\psi !x.f!\{\psi !\hat{z}\}\quad \text{Df.} +\] +Thus a predicative function of a class is always a predicative function of any +predicative determining function of the class, though the converse does not +hold.</p> + +<p><span class="pagenum" id="Page_85">[Pg 85]</span></p> + +<p>(3) <i>Relations</i>. With regard to relations, we have a theory +strictly analogous to that which we have just explained as regards +classes. Relations in extension, like classes, are incomplete symbols. +We require a division of functions of two variables into predicative +and non-predicative functions, again for reasons which have been +explained in <a href="#CHAPTER_II">Chapter II</a>. We use the notation "\(\phi !(x,y)\)" for a +<i>predicative</i> function of \(x\) and \(y\).</p> + +<p>We use "\(\phi !(\hat{x},\hat{z})\)" for the function as opposed to its +values; and we use "\(\hat{x}\hat{y} \phi (x,y)\)" for the relation (in +extension) determined by \(\phi(x,y)\). We put +\[ +f\{\hat{x}\hat{y} \phi(x,y)\}.=\colon(\exists \psi)\colon \phi(x,y).\equiv_{x,y}.\psi !(x,y)\colon f\{\psi!(\hat{x},\hat{y}\} \quad \text{Df}. +\] +Thus even when \(f\{\psi !(\hat{x},\hat{y})\}\) is not an extensional +function of \(\psi\), \(f\{\hat{x}\hat{y} \phi(x,y)\}\) is an +extensional function of \(\phi\). Hence, just as in the case of +classes, we deduce +\[ +\vdash \colon\ldotp\hat{x}\hat{y} \phi(x,y)=\hat{x}\hat{y} \psi(x,y).\equiv\colon \phi(x,y).\equiv_{x,y}.\psi(x,y), +\] +<i>i.e.</i> a relation is determined by its extension, and vice versa.</p> + +<p>On the analogy of the definition of "\(x \in \psi!\hat{z}\)," we put<a id="FNanchor_36" href="#Footnote_36" class="fnanchor">[36]</a> +\[ +x\{\psi!(\hat{x},\hat{y})\}y.=.\psi !(x,y) \quad \text{Df}. +\]</p> + +<p>This definition, like that of "\(x \in \psi!\hat{z}\)," is not +introduced for its own sake, but in order to give a meaning to +\[ +x\{\hat{x}\hat{y} \phi(x,y)\}y. +\] +This meaning, in virtue of our definitions, is +\[ +\begin{array}{l} +&(\exists \psi)\colon \phi(x,y).\equiv_{x,y}.\psi !(x,y)\colon x\{\psi !(\hat{x},\hat{y})\}y,\\ +\textit{i.e.}\quad &(\exists \psi)\colon \phi(x,y).\equiv_{x,y}. \psi !(x,y)\colon \psi !(x,y), +\end{array} +\] +and this, in virtue of the axiom of reducibility +\[ +\unicode{x201c}(\exists \psi)\colon \phi(x,y).\equiv_{x,y}. \psi !(x,y),\unicode{x201d} +\] +is equivalent to \(\phi(x,y)\).</p> + +<p>Thus we have always +\[ +\vdash \colon x\{\hat{x}\hat{y} \phi(x,y)\}y.\equiv.\phi(x,y). +\]</p> + +<p>Whenever the determining function of a relation is not relevant, we may +replace \(\hat{x}\hat{y} \phi (x,y)\) by a single capital letter. In +virtue of the propositions given above, +\[ +\begin{array}{l} +&\vdash \colon\ldotp R=S.\equiv \colon xRy.\equiv_{x,y}.xSy,\\ +&\vdash \colon\ldotp R=\hat{x}\hat{y} \phi(x,y).\equiv \colon xRy.\equiv_{x,y}.\phi(x,y),\\ +\text{and}\quad &\vdash .R=\hat{x}\hat{y}(xRy). +\end{array} +\]</p> + +<p>Classes of relations, and relations of relations, can be dealt with as +classes of classes were dealt with above.</p> + +<p><span class="pagenum" id="Page_86">[Pg 86]</span></p> + +<p>Just as a class must not be capable of being or not being a member +of itself, so a relation must neither be nor not be referent or +relatum with respect to itself. This turns out to be equivalent to the +assertion that \(\phi!\hat{x},\hat{y}\) cannot significantly be either +of the arguments \(x\) or \(y\) in \(\phi!(x,y)\). This principle, again, +results from the limitation to the possible arguments to a function +explained at the beginning of <a href="#CHAPTER_II">Chapter II</a>.</p> + +<p>We may sum up this whole discussion on incomplete symbols as follows.</p> + +<p>The use of the symbol "(\({℩}x)(\phi x)\)" as if in "\(f({℩}x)(\phi x)\)" +it <i>directly</i> represented an argument to the function +\(f\hat{z}\) is rendered possible by the theorems +\[ +\begin{array}{l} +\vdash \colon\ldotp \text{E}!(℩x)(\phi x).\supset \colon (x).fx.\supset.f(℩x)(\phi x),\\ +\vdash \colon(℩x)(\phi x)=(℩x)(\psi x).\supset.f(℩x)((\phi x)\equiv f(℩x)(\psi x),\\ +\vdash \colon \text{E}!(℩x)(\phi x).\supset.(℩x)(\phi x)=(℩x)(\phi x),\\ +\vdash \colon(℩x)(\phi x)=(℩x)(\psi x).\equiv.(℩x)(\psi x)=(℩x)(\phi x),\\ +\vdash \colon(℩x)(\phi x)=(℩x)(\psi x).(℩x)(\psi x)=(℩x)(\chi x).\supset.(℩x)(\phi x)=(℩x)(\chi x). +\end{array} +\]</p> + +<p>The use of the symbol "\(\hat{x}(\phi x)\)" (or of a single +letter, such as \(\alpha\), to represent such a symbol) as if, in +"\(f\{\hat{x}(\phi x)\}\)," it <i>directly</i> represented an argument +\(\alpha\) to a function \(f\hat{\alpha}\), is rendered possible by the +theorems +\[ +\begin{array}{l} +\vdash \colon(\alpha).f_{\alpha}.\supset.f\{\hat{x}(\phi x)\},\\ +\vdash \colon \hat{x}(\phi x)=\hat{x}(\psi x).\supset.f\{\hat{x}(\phi x)\}\equiv f\{\hat{x}(\psi x)\},\\ +\vdash .\hat{x}(\phi x)=\hat{x}(\phi x),\\ +\vdash \colon \hat{x}(\phi x)=\hat{x}(\psi x).\equiv.\hat{x}(\psi x)=\hat{x}(\phi x),\\ +\vdash \colon \hat{x}(\phi x)=\hat{x}(\psi x).\hat{x}(\psi x)=\hat{x}(\chi x).\supset.\hat{x}(\phi x)=\hat{x}(\chi x). +\end{array} +\]</p> + +<p>Throughout these propositions the types must be supposed to be properly +adjusted, where ambiguity is possible.</p> + +<p>The use of the symbol "\(\hat{x}\hat{y}\{\phi (x,y)\}\)"(or of a +single letter, such as \(R\), to represent such a symbol) as if, in +"\(f\{\hat{x}\hat{y} \phi (x,y)\}\)," it directly represented an +argument \(R\) to a function \(f\hat{R}\), is rendered possible by the +theorems +\[ +\begin{array}{l} +\vdash \colon(R).fR.\supset.f\{\hat{x}\hat{y} \phi(x,y)\},\\ +\vdash \colon \hat{x}\hat{y} \phi(x,y)=\hat{x}\hat{y} \psi(x,y).\supset.f\{\hat{x}\hat{y}\phi (x,y)\}\equiv f\{\hat{x}\hat{y} \psi(x,y)\},\\ +\vdash .\hat{x}\hat{y} \phi(x,y)=\hat{x}\hat{y} \phi(x,y),\\ +\vdash \colon \hat{x}\hat{y} \phi(x,y)=\hat{x}\hat{y} \psi(x,y).\equiv.\hat{x}\hat{y} \psi(x,y)=\hat{x}\hat{y} \phi(x,y),\\ +\vdash \colon \hat{x}\hat{y} \phi(x,y)=\hat{x}\hat{y} \psi(x,y).\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y} \chi(x,y).\\ +\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\supset.\hat{x}\hat{y} \phi(x,y)=\hat{x}\hat{y} \chi(x,y). +\end{array} +\] +Throughout these propositions the types must be supposed to be properly +adjusted where ambiguity is possible.</p> + +<p><span class="pagenum" id="Page_87">[Pg 87]</span></p> + +<p>It follows from these three groups of theorems that these incomplete +symbols are obedient to the same formal rules of identity as symbols +which directly represent objects, so long as we only consider the +<i>equivalence</i> of the resulting variable (or constant) values of +propositional functions and not their identity. This consideration +of the <i>identity</i> of propositions never enters into our formal +reasoning.</p> + +<p>Similarly the <i>limitations</i> to the use of these symbols can be +summed up as follows. In the case of (\({℩}x)(\phi x)\), the chief way +in which its incompleteness is relevant is that we do not always have +\[ +(x).fx.\supset.f({℩}x)(\phi x), +\] +<i>i.e.</i> a function which is always true may nevertheless not be +true of (\({℩}x)(\phi x)\). This is possible because \(f({℩}x)(\phi x)\) +is not a value of \(f\hat{x}\), so that even when all values +of \(f\hat{x}\) are true, \(f({℩}x)(\phi x)\) may not be true. This +happens when (\({℩}x)(\phi x)\) does not exist. Thus for example we +have (\(x).x=x\), but we do not have +\[ +\begin{array}{l} +&\text{the round square} = \text{the round square}.\\ +\text{The inference} &\qquad\qquad\quad(x).fx.\supset.f({℩}x)(\phi x) +\end{array} +\] +is only valid when \(\text{E}!({℩}x)(\phi x)\). As soon as we know +\(\text{E}!({℩}x)(\phi x)\), the fact that (\({℩}x)(\phi x\)) is an +incomplete symbol becomes irrelevant so long as we confine ourselves +to truth-functions<a id="FNanchor_37" href="#Footnote_37" class="fnanchor">[37]</a> of whatever proposition is its scope. But even +when \(\text{E}!({℩}x)(\phi x)\), the incompleteness of (\({℩}x)(\phi x)\) +may be relevant when we pass outside truth-functions. For example, +George IV wished to know whether Scott was the author of Waverley, +<i>i.e.</i> he wished to know whether a proposition of the form +"\(c=({℩}x)(\phi x)\)" was true. But there was no proposition of the +form "\(c=y\)" concerning which he wished to know if it was true.</p> + +<p>In regard to classes, the relevance of their incompleteness is somewhat +different. It may be illustrated by the fact that we may have +\[ +\begin{array}{l} +&\hat{z}(\phi z)=\psi !\hat{z}.\hat{z}(\phi z)=\chi !\hat{z}\\ +\text{without having} &\qquad\qquad\quad \psi!\hat{z}=\chi!\hat{z}. +\end{array} +\] +For, by a direct application of the definitions, we find that +\[ +\vdash \colon\hat{z}(\phi z)=\psi !\hat{z}.\equiv.\phi x\equiv_{x}\psi !x. +\] +Thus we shall have +\[ +\vdash \colon \phi x\equiv_{x}\psi !x.\phi x\equiv_{x}\chi !x.\supset.\hat{z}(\phi z)=\psi !\hat{z}.\hat{z}(\phi z)=\chi !\hat{z}, +\] +but we shall not necessarily have \(\psi!\hat{z} = \chi !\hat{z}\) +under these circumstances, for two functions may well be formally +equivalent without being identical; for example, +\[ +x=\text{Scott}.\equiv_{x}.x=\text{the author of Waverley}, +\]<span class="pagenum" id="Page_88">[Pg 88]</span> +but the function "\(\hat{z}\)=the author of Waverley" has the property +that George IV wished to know whether its value with the argument +"Scott" was true, whereas the function "\(\hat{z}\)=Scott" has no such +property, and therefore the two functions are not identical. Hence +there is a propositional function, namely +\[ +x=y.x=z.\supset.y=z, +\] +which holds without any exception, and yet does not hold when for \(x\) +we substitute a class, and for \(y\) and \(z\) we substitute functions. +This is only possible because a class is an incomplete symbol, and +therefore "\(\hat{z}(\phi z)=\psi !\hat{z}\)" is not a value of +"\(x=y\)."</p> + +<p>It will be observed that "\(\theta!\hat{z}=\psi !\hat{z}\)" is not +an extensional function of \(\psi!\hat{z}\). Thus the scope of +\(\hat{z}(\phi z\)) is relevant in interpreting the product +\[ +\hat{z}(\phi z)=\psi !\hat{z}.\hat{z}(\phi z)=\chi !\hat{z}. +\] +If we take the whole of the product as the scope of \(\hat{z}(\phi z)\), +the product is equivalent to +\[ +\begin{array}{l} +&(\exists \theta)\colon \phi x\equiv_{x}\theta !x.\theta !\hat{z}=\psi !\hat{z}.\theta!\hat{z}=\chi !\hat{z},\\ +\text{and this}\, \textit{does}\, \text{imply} &\psi !\hat{z}=\chi !\hat{z}. +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_89">[Pg 89]</span></p> + +<p>We may say generally that the fact that \(\hat{z}(\phi z\)) is an +incomplete symbol is not relevant so long as we confine ourselves to +extensional functions of functions, but is apt to become relevant for +other functions of functions.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_29" href="#FNanchor_29" class="label">[29]</a> +Cf. <a href="#Page_31">pp. 31</a>, <a href="#Page_32">32</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_30" href="#FNanchor_30" class="label">[30]</a> +We shall generally write "\(f({℩}x)({\phi}x)\)" rather +than "\(f\{({℩}x)({\phi}x)\}\)" in future.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_31" href="#FNanchor_31" class="label">[31]</a> +Briefly, these arguments reduce to the following: If +there is such an object as a class, it must be in some sense <i>one</i> +object. Yet it is only of classes that <i>many</i> can be predicated. +Hence, if we admit classes as objects, we must suppose that the same +object can be both one and many, which seems impossible.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_32" href="#FNanchor_32" class="label">[32]</a> +Cf. <a href="#Page_56">p. 56</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_33" href="#FNanchor_33" class="label">[33]</a> +As explained in Chapter I (<a href="#Page_25">pp. 25</a>, <a href="#Page_26">26</a>), "\(x \in \alpha\)" +means "\(x\) is a member of the class \(\alpha\)," or, more shortly, +"\(x\) is an \(\alpha\)." The definition of this expression in terms of +our theory of classes will be given shortly.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_34" href="#FNanchor_34" class="label">[34]</a> +This is the second of the contradictions discussed at the +end of <a href="#CHAPTER_II">Chapter II</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_35" href="#FNanchor_35" class="label">[35]</a> +The use of a single letter, such as \(\alpha\) or +\(\beta\), to represent a variable class, will be further explained +shortly.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_36" href="#FNanchor_36" class="label">[36]</a> +This definition raises certain questions as to the two +senses of a relation, which are dealt with in <a href="#*21">*21</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_37" href="#FNanchor_37" class="label">[37]</a> +Cf. <a href="#Page_8">p. 8</a>.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="PART_I">PART I.<br> +<br> +MATHEMATICAL LOGIC.</h2> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_91">[Pg 91]</span></p> +<h2 class="nobreak" id="SUMMARY_OF_PART_I"><span class="allsmcap">SUMMARY OF PART I.</span></h2> +</div> + + +<p>IN this Part, we shall deal with such topics as belong traditionally +to symbolic logic, or deserve to belong to it in virtue of their +generality. We shall, that is to say, establish such properties of +propositions, propositional functions, classes and relations as are +likely to be required in any mathematical reasoning, and not merely in +this or that branch of mathematics.</p> + +<p>The subjects treated in Part I may be viewed in two aspects: (1) as +a deductive chain depending on the primitive propositions, (2) as +a formal calculus. Taking the first view first: We begin, in <a href="#*1">*1</a>, +with certain axioms as to deduction of one proposition or asserted +propositional function from another. From these primitive propositions, +in <a href="#SECTION_A_a">Section A</a>, we deduce various propositions which are all concerned +with four ways of obtaining new propositions from given propositions, +namely negation, disjunction, joint assertion and implication, of which +the last two can be defined in terms of the first two. Throughout +this first section, although, as will be shown at the beginning of +<a href="#SECTION_B_a">Section B</a>, our propositions, symbolically unchanged, will apply to any +propositions as values of our variables, yet it will be supposed that +our variable propositions are all what we shall call <i>elementary</i> +propositions, <i>i.e.</i> such as contain no reference, explicit or +implicit, to any totality. This restriction is imposed on account +of the distinction between different <i>types</i> of propositions, +explained in <a href="#CHAPTER_II">Chapter II</a> of the Introduction. Its importance and +purpose, however, are purely philosophical, and so long as only +mathematical purposes are considered, it is unnecessary to remember +this preliminary restriction to elementary propositions, which is +symbolically removed at the beginning of the next section.</p> + +<p><a href="#SECTION_B_a">Section B</a> deals, to begin with, with the relations of propositions +containing apparent variables (<i>i.e.</i> involving the notions of +"all" or "some") to each other and to propositions not containing +apparent variables. We show that, where propositions containing +apparent variables are concerned, we can define negation, disjunction, +joint assertion and implication in such a way that their properties +shall be exactly analogous to the properties of the corresponding ideas +as applied to elementary propositions. We show also that <i>formal +implication</i>, <i>i.e.</i> "\((x) . {\phi}x \supset {\psi}x\)" +considered as a relation of \(\phi\hat{x}\) to \(\psi\hat{x}\), has +many properties analogous to those of <i>material implication</i>, +<i>i.e.</i> "\(p \supset q\)"<span class="pagenum" id="Page_92">[Pg 92]</span> considered as a relation of \(p\) +and \(q\). We then consider <i>predicative</i> functions and the +axiom of <i>reducibility</i>, which are vital in the employment of +<i>functions</i> as apparent variables. An example of such employment +is afforded by <i>identity</i>, which is the next topic considered +in <a href="#SECTION_B_a">Section B</a>. Finally, this section deals with <i>descriptions</i>, +<i>i.e.</i> phrases of the form "the so-and-so" (in the singular). It +is shown that the appearance of a grammatical subject "the so-and-so" +is deceptive, and that such propositions, fully stated, contain no such +subject, but contain instead an apparent variable.</p> + +<p><a href="#SECTION_C_a">Section C</a> deals with classes, and with relations in so far as they are +analogous to classes. Classes and relations, like descriptions, are +shown to be "incomplete symbols" (cf. Introduction, <a href="#CHAPTER_III">Chapter III</a>), and +it is shown that a proposition which is grammatically about a class is +to be regarded as really concerned with a propositional function and +an apparent variable whose values are <i>predicative</i> propositional +functions (with a similar result for relations). The remainder of +<a href="#SECTION_C_a">Section C</a> deals with the calculus of classes, and with the calculus of +relations in so far as it is analogous to that of classes.</p> + +<p><a href="#SECTION_D_a">Section D</a> deals with those properties of relations which have no +analogues for classes. In this section, a number of ideas and notations +are introduced which are constantly needed throughout the rest of the +work. Most of the properties of relations which have analogues in the +theory of classes are comparatively unimportant, while those that have +no such analogues are of the very greatest utility. It is partly for +this reason that emphasis on the calculus-aspect of symbolic logic has +proved a hindrance, hitherto, to the proper development of the theory +of relations.</p> + +<p><a href="#SECTION_E_a">Section E</a>, finally, extends the notions of the addition and +multiplication of classes or relations to cases where the summands or +factors are not individually given, but are given as the members of +some class. The advantage obtained by this extension is that it enables +us to deal with an infinite number of summands or factors.</p> + +<p>Considered as a formal calculus, mathematical logic has three analogous +branches, namely (1) the calculus of propositions, (2) the calculus of +classes, (3) the calculus of relations. Of these, (1) is dealt with +in <a href="#SECTION_A_a">Section A</a>, while (2) and (3), in so far as they are analogous, are +dealt with in <a href="#SECTION_C_a">Section C</a>. We have, for each of the three, the four +analogous ideas of negation, addition, multiplication, and implication +or inclusion. Of these, negation is analogous to the negative in +ordinary algebra, and implication or inclusion is analogous to the +relation "less than or equal to" in ordinary algebra. But the analogy +must not be pressed, as it has important limitations. The sum of two +propositions is their disjunction, the sum of two classes is the +class of terms belonging to one or other, the sum of two relations +is the relation consisting in the fact that one or other of the two +relations holds. The sum of a class of classes is the class of all +terms belonging to some one or other of the classes, and the sum of a +class of relations is the relation consisting in the fact that some +one relation of the class holds. The product of two propositions is +their joint assertion, the product of two classes is their common +part, the product of two relations is the relation consisting in the +fact that both the relations hold. The product of a class of classes +is the part common to all of them, and the product of a class of +relations is the relation consisting in the fact that all relations +of the class in question hold. The inclusion of one class in another +consists in the fact that all members of the one are members of the +other, while the inclusion of one relation in another consists in the +fact that every pair of terms which has the one relation also has the +other relation. It is then shown that the properties of negation, +addition, multiplication and inclusion are exactly analogous for +classes and relations, and are, with certain exceptions, analogous to +the properties of negation, addition, multiplication and implication +for propositions. (The exceptions arise chiefly from the fact that +"\(p\) implies \(q\)" is itself a proposition, and can therefore imply +and be implied, while "\(\alpha\) is contained in \(\beta\)," where +\(\alpha\) and \(\beta\) are classes, is not a class, and can therefore +neither contain nor be contained in another class \(\gamma\).) But +classes have certain properties not possessed by propositions: these +arise from the fact that classes have not a <i>two-fold</i> division +corresponding to the division of propositions into true and false, but +a <i>threefold</i> division, namely into (1) the universal class, which +contains the whole of a certain type, (2) the null-class, which has +no members, (3) all other classes, which neither contain nothing nor +contain everything of the appropriate type. The resulting properties +of classes, which are not analogous to properties of propositions, are +dealt with in <a href="#*24">*24</a>. And just as classes have properties not analogous +to any properties of propositions, so relations have properties not +analogous to any properties of classes, though all the properties of +classes have analogues among relations. The special properties of +relations are much more numerous and important than the properties +belonging to classes but not to propositions. These special properties +of relations therefore occupy a whole section, namely <a href="#SECTION_D_a">Section D</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_94">[Pg 94]</span></p> +<h2 class="nobreak" id="SECTION_A_a">SECTION A.<br> +<br> +THE THEORY OF DEDUCTION.</h2> +</div> + + +<p>THE purpose of the present section is to set forth the first stage +of the deduction of pure mathematics from its logical foundations. +This first stage is necessarily concerned with deduction itself, +<i>i.e.</i> with the principles by which conclusions are inferred from +premisses. If it is our purpose to make all our assumptions explicit, +and to effect the deduction of all our other propositions from these +assumptions, it is obvious that the first assumptions we need are those +that are required to make deduction possible. Symbolic logic is often +regarded as consisting of two coordinate parts, the theory of classes +and the theory of propositions. But from our point of view these two +parts are not coordinate; for in the theory of classes we deduce one +proposition from another by means of principles belonging to the theory +of propositions, whereas in the theory of propositions we nowhere +require the theory of classes. Hence, in a deductive system, the theory +of propositions necessarily precedes the theory of classes.</p> + +<p>But the subject to be treated in what follows is not quite properly +described as the theory of <i>propositions</i>. It is in fact the +theory of how one proposition can be inferred from another. Now +in order that one proposition may be inferred from another, it is +necessary that the two should have that relation which makes the one +a consequence of the other. When a proposition \(q\) is a consequence +of a proposition \(p\), we say that \(p\) <i>implies</i> \(q\). Thus +deduction depends upon the relation of implication, and every deductive +system must contain among its premisses as many of the properties of +implication as are necessary to legitimate the ordinary procedure of +deduction. In the present section, certain propositions will be stated +as premisses, and it will be shown that they are sufficient for all +common forms of inference. It will not be shown that they are all +<i>necessary</i>, and it is possible that the number of them might +be diminished. All that is affirmed concerning the premisses is (1) +that they are true, (2) that they are sufficient for the theory of +deduction, (3) that we do not know how to diminish their number. But +with regard to (2), there must always be some element of doubt, since +it is hard to be sure that one never uses some principle unconsciously. +The habit of being rigidly guided by formal symbolic rules is a +safeguard against unconscious assumptions; but even this safeguard is +not always adequate.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_95">[Pg 95]</span></p> +<h2 class="nobreak" id="*1">1. PRIMITIVE IDEAS AND PROPOSITIONS.</h2> +</div> + +<p>Since all definitions of terms are effected by means of other terms, +every system of definitions which is not circular must start from a +certain apparatus of undefined terms. It is to some extent optional +what ideas we take as undefined in mathematics; the motives guiding +our choice will be (1) to make the number of undefined ideas as small +as possible, (2) as between two systems in which the number is equal, +to choose the one which seems the simpler and easier. We know no way +of proving that such and such a system of undefined ideas contains as +few as will give such and such results<a id="FNanchor_38" href="#Footnote_38" class="fnanchor">[38]</a>. Hence we can only say that +such and such ideas are undefined in such and such a system, not that +they are indefinable. Following Peano, we shall call the undefined +ideas and the undemonstrated propositions <i>primitive</i> ideas +and <i>primitive</i> propositions respectively. The primitive ideas +are <i>explained</i> by means of descriptions intended to point out +to the reader what is meant; but the explanations do not constitute +definitions, because they really involve the ideas they explain.</p> + +<p>In the present number, we shall first enumerate the primitive ideas +required in this section; then we shall define <i>implication</i>; and +then we shall enunciate the primitive propositions required in this +section. Every definition or proposition in the work has a number, +for purposes of reference. Following Peano, we use numbers having a +decimal as well as an integral part, in order to be able to insert +new propositions between any two. A change in the integral part of +the number will be used to correspond to a new chapter. Definitions +will generally have numbers whose decimal part is less than ·1, and +will be usually put at the beginning of chapters. In references, the +integral parts of the numbers of propositions will be distinguished +by being preceded by a star; thus "*1·01" will mean the definition +or proposition so numbered, and "*1" will mean the chapter in which +propositions have numbers whose integral part is 1, <i>i.e.</i> the +present chapter. Chapters will generally be called "numbers."</p> + + +<p class="nindc space-above2"> +<span class="allsmcap">PRIMITIVE IDEAS.</span></p> + +<p>(1) <i>Elementary propositions</i>. By an "elementary" proposition +we mean one which does not involve any variables, or, in other +language, one which does not involve such words as "all," "some," +"the" or equivalents for such words. A proposition such as "this +is red," where "this" is something given<span class="pagenum" id="Page_96">[Pg 96]</span> in sensation, will be +elementary. Any combination of given elementary propositions by means +of negation, disjunction or conjunction (see below) will be elementary. +In the primitive propositions of the present number, and therefore +in the deductions from these primitive propositions in <a href="#*2">*2</a>—<a href="#*5">*5</a>, the +letters \(p\), \(q\), \(r\), \(s\) will be used to denote elementary +propositions.</p> + +<p>(2) <i>Elementary propositional functions</i>. By an "elementary +propositional function" we shall mean an expression containing an +undetermined constituent, <i>i.e.</i> a variable, or several such +constituents, and such that, when the undetermined constituent or +constituents are determined, <i>i.e.</i> when values are assigned to +the variable or variables, the resulting value of the expression in +question is an elementary proposition. Thus if \(p\) is an undetermined +elementary proposition, "not-\(p\)" is an elementary propositional +function.</p> + +<p>We shall show in <a href="#*9">*9</a> how to extend the results of this and the following +numbers (<a href="#*1">*1</a>—<a href="#*5">*5</a>) to propositions which are not elementary.</p> + +<p>(3) <i>Assertion</i>. Any proposition may be either asserted or +merely considered. If I say "Caesar died," I assert the proposition +"Caesar died," if I say "'Caesar died' is a proposition," I make a +different assertion, and "Caesar died" is no longer asserted, but +merely considered. Similarly in a hypothetical proposition, <i>e.g.</i> +"if \(a = b\), then \(b = a\)," we have two unasserted propositions, +namely "\(a = b\)" and "\(b = a\)," while what is asserted is that +the first of these implies the second. In language, we indicate +when a proposition is merely considered by "<i>if</i> so-and-so" or +"<i>that</i> so-and-so" or merely by inverted commas. In symbols, if +\(p\) is a proposition, \(p\) by itself will stand for the unasserted +proposition, while the asserted proposition will be designated by +\[ +\unicode{x201c}\vdash.p.\unicode{x201d} +\] +The sign "\(\vdash\)" is called the assertion-sign<a id="FNanchor_39" href="#Footnote_39" class="fnanchor">[39]</a>; it may be read +"it is true that" (although philosophically this is not exactly what +it means). The dots after the assertion-sign indicate its range; that +is to say, everything following is asserted until we reach either an +equal number of dots preceding a sign of implication or the end of the +sentence. Thus "\(\vdash \colon p.\supset.q\)" means "it is true that +\(p\) implies \(q\)," whereas "\(\vdash.p.\supset\vdash.q\)" means +"\(p\) is true; therefore \(q\) is true<a id="FNanchor_40" href="#Footnote_40" class="fnanchor">[40]</a>." The first of these does +not necessarily involve the truth either of \(p\) or of \(q\), while +the second involves the truth of both.</p> + +<p>(4) <i>Assertion of a propositional function</i>. Besides the assertion +of definite propositions, we need what we shall call "assertion of a +propositional function." The general notion of asserting <i>any</i> +propositional function is not used until <a href="#*9">*9</a>, but we use at once the +notion of asserting various special elementary propositional functions. +Let \(\phi x\) be a propositional function whose argument is \(x\); +then we may assert \(\phi x\) without assigning a value to \(x\).<span class="pagenum" id="Page_97">[Pg 97]</span> +This is done, for example, when the law of identity is asserted in +the form "\(A\) is \(A\)." Here \(A\) is left undetermined, because, +however \(A\) may be determined, the result will be true. Thus when +we assert \(\phi x\), leaving \(x\) undetermined, we are asserting an +ambiguous value of our function. This is only legitimate if, however +the ambiguity may be determined, the result will be true. Thus take, as +an illustration, the primitive proposition <a href="#*1·2">*1·2</a> below, namely +\[ +\unicode{x201c}\vdash \colon p \lor p.\supset.p,\unicode{x201d} +\] +<i>i.e.</i> "'\(p\) or \(p\)' implies \(p\)." Here p may be <i>any</i> +elementary proposition: by leaving \(p\) undetermined, we obtain +an assertion which can be applied to any particular elementary +proposition. Such assertions are like the particular enunciations in +Euclid: when it is said "let \(ABC\) be an isosceles triangle; then the +angles at the base will be equal," what is said applies to <i>any</i> +isosceles triangle; it is stated concerning <i>one</i> triangle, but +not concerning a definite one. All the assertions in the present +work, with a very few exceptions, assert propositional functions, not +definite propositions.</p> + +<p>As a matter of fact, no constant elementary proposition will occur in +the present work, or can occur in any work which employs only logical +ideas. The ideas and propositions of logic are all <i>general</i>: an +assertion (for example) which is true of Socrates but not of Plato, +will not belong to logic<a id="FNanchor_41" href="#Footnote_41" class="fnanchor">[41]</a>, and if an assertion which is true of +both is to occur in logic, it must not be made concerning either, but +concerning a variable \(x\). In order to obtain, in logic, a definite +proposition instead of a propositional function, it is necessary to +take some propositional function and assert that it is true always or +sometimes, <i>i.e.</i> with all possible values of the variable or with +some possible value. Thus, giving the name "individual" to whatever +there is that is neither a proposition nor a function, the proposition +"every individual is identical with itself" or the proposition "there +are individuals" will be a proposition belonging to logic. But these +propositions are not elementary.</p> + +<p>(5) <i>Negation</i>. If \(p\) is any proposition, the proposition +"not-\(p\)," or "\(p\) is false," will be represented by "\(\sim{p}\)." +For the present, \(p\) must be an <i>elementary</i> proposition.</p> + +<p>(6) <i>Disjunction</i>. If \(p\) and \(q\) are any propositions, +the proposition "\(p\) or \(q\)," <i>i.e.</i> "either \(p\) is true +or \(q\) is true," where the alternatives are to be not mutually +exclusive, will be represented by +\[ +\unicode{x201c}p \lor q.\unicode{x201d} +\] +This is called the <i>disjunction</i> or the <i>logical sum</i> of +\(p\) and \(q\). Thus "\(\sim{p \lor q}\)" will mean "\(p\) is false or +\(q\) is true"; \(\sim{(p \lor q)}\) will mean "it is false that either +\(p\) or \(q\) is true," which is equivalent to "\(p\) and \(q\) are +both false";<span class="pagenum" id="Page_98">[Pg 98]</span> \(\sim{(\sim{p} \lor \sim{q}}\)) will mean "it is false +that either \(p\) is false or \(q\) is false," which is equivalent to +"\(p\) and \(q\) are both true"; and so on. For the present, \(p\) and +\(q\) must be elementary propositions.</p> + +<p>The above are all the primitive ideas required in the theory of +deduction. Other primitive ideas will be introduced in <a href="#SECTION_B_a">Section B</a>.</p> + +<p><i>Definition of Implication</i>. When a proposition \(q\) follows +from a proposition \(p\), so that if \(p\) is true, \(q\) must also be +true, we say that \(p\) <i>implies</i> \(q\). The idea of implication, +in the form in which we require it, can be defined. The meaning to be +given to implication in what follows may at first sight appear somewhat +artificial; but although there are other legitimate meanings, the one +here adopted is very much more convenient for our purposes than any of +its rivals. The essential property that we require of implication is +this: "What is implied by a true proposition is true." It is in virtue +of this property that implication yields proofs. But this property by +no means determines whether anything, and if so what, is implied by a +false proposition. What it does determine is that, if \(p\) implies +\(q\), then it cannot be the case that \(p\) is true and \(q\) is +false, <i>i.e.</i> it must be the case that either \(p\) is false or +\(q\) is true. The most convenient interpretation of implication is to +say, conversely, that if either \(p\) is false or \(q\) is true, then +"\(p\) implies \(q\)" is to be true. Hence "\(p\) implies \(q\)" is to +be defined to mean: "Either \(p\) is false or \(q\) is true." Hence we +put:</p> + +<p class="nind"> +<b><a id="*1·01">*1·01</a></b>. \(p\supset q.=.\sim {p} \lor q \qquad \text{Df}.\)</p> + +<p>Here the letters "Df" stand for "definition." They and the sign of +equality together are to be regarded as forming one symbol, standing +for "is defined to mean<a id="FNanchor_42" href="#Footnote_42" class="fnanchor">[42]</a>." Whatever comes to the left of the sign +of equality is defined to mean the same as what comes to the right of +it. Definition is not among the primitive ideas, because definitions +are concerned solely with the symbolism, not with what is symbolised; +they are introduced for practical convenience, and are theoretically +unnecessary.</p> + +<p>In virtue of the above definition, when "\(p \supset q\)" holds, +then either \(p\) is false or \(q\) is true; hence if \(p\) is true, +\(q\) must be true. Thus the above definition preserves the essential +characteristic of implication; it gives, in fact, the most general +meaning compatible with the preservation of this characteristic.</p> + + +<p class="nindc space-above2"> +<span class="allsmcap">PRIMITIVE PROPOSITIONS.</span></p> + +<p class="nind"> +<b><a id="*1·1">*1·1</a>.</b> Anything implied by a true elementary proposition is true. +Pp<a id="FNanchor_43" href="#Footnote_43" class="fnanchor">[43]</a>.</p> + +<p>The above principle will be extended in <a href="#*9">*9</a> to propositions which +are not elementary. It is not the same as "<i>if</i> \(p\) is true, +then <i>if</i> \(p\) implies \(q\), \(q\) is<span class="pagenum" id="Page_99">[Pg 99]</span> true." This is a true +proposition, but it holds equally when \(p\) is not true and when \(p\) +does not imply \(q\). It does not, like the principle we are concerned +with, enable us to assert \(q\) simply, without any hypothesis. We +cannot express the principle symbolically, partly because any symbolism +in which \(p\) is variable only gives the <i>hypothesis</i> that \(p\) +is true, not the fact that it is true<a id="FNanchor_44" href="#Footnote_44" class="fnanchor">[44]</a>.</p> + +<p>The above principle is used whenever we have to deduce a +<i>proposition</i> from a <i>proposition</i>. But the immense majority +of the assertions in the present work are assertions of propositional +functions, <i>i.e.</i> they contain an undetermined variable. Since the +assertion of a propositional function is a different primitive idea +from the assertion of a proposition, we require a primitive proposition +different from <a href="#*1·1">*1·1</a>, though allied to it, to enable us to deduce the +assertion of a propositional function "\(\psi x\)" from the assertions +of the two propositional functions "\(\phi x\)" and "\(\phi x \supset\psi x\)." +This primitive proposition is as follows:</p> + +<p class="nind"> +<b><a id="*1·11">*1·11</a></b>. When \(\phi x\) can be asserted, where \(x\) is a real +variable, and \(\phi x \supset \psi x\) can be asserted, where \(x\) is +a real variable, then \(\psi x\) can be asserted, where \(x\) is a real +variable. Pp.</p> + +<p>This principle is also to be assumed for functions of several variables.</p> + +<p>Part of the importance of the above primitive proposition is due to the +fact that it expresses in the symbolism a result following from the +theory of types, which requires symbolic recognition. Suppose we have +the two assertions of <i>propositional functions</i> "\(\vdash.\phi x\)" +and "\(\vdash.\phi x \supset \psi x\)"; then the "\(x\)" in \(\phi x\) +is not absolutely anything, but anything for which as argument +the function "\(\phi x\)" is significant; similarly in "\(\phi x \supset \psi x\)" +the \(x\) is anything for which "\(\phi x \supset \psi x\)" +is significant. Apart from some axiom, we do not know that +the \(x\)'s for which "\(\phi x \supset \psi x\)" is significant are +the same as those for which "\(\phi x\)" is significant. The primitive +proposition <a href="#*1·11">*1·11</a>, by securing that, as the result of the assertions of +the <i>propositional functions</i> "\(\phi x\)" and "\(\phi x \supset \psi x\)" +the propositional function \(\psi x\) can also be asserted, +secures partial symbolic recognition, in the form most useful in +actual deductions, of an important principle which follows from the +theory of types, namely that, if there is any one argument a for which +both "\(\phi a\)" and "\(\psi a\)" are significant, then the range of +arguments for which "\(\phi x\)" is significant is the same as the +range of arguments for which "\(\psi x\)" is significant. It is obvious +that, if the propositional function "\(\phi x \supset \psi x\)" can be +asserted, there must be arguments a for which "\(\phi a \supset \psi a\)" +is significant, and for which, therefore, "\(\phi a\)" and "\(\psi a\)" +must be significant. Hence, by our principle, the values of \(x\) +for which "\(\phi x\)" is significant are the same as those for which +"\(\psi x\)" is significant, <i>i.e.</i> the type of possible arguments +for \(\phi \hat{x}\) (cf. <a href="#Page_15">p. 15</a>) is the same as that of possible +arguments for \(\psi \hat{x}\). The<span class="pagenum" id="Page_100">[Pg 100]</span> primitive proposition <a href="#*1·11">*1·11</a>, since +it states a practically important consequence of this fact, is called +the "axiom of identification of type."</p> + +<p>Another consequence of the principle that, if there is an argument +\(a\) for which both \(\phi a\) and \(\psi a\) are significant, then +\(\phi x\) is significant whenever \(\psi x\) is significant, and +vice versa, will be given in the "axiom of identification of real +variables," introduced in *<a href="#*1·72">*1·72</a>. These two propositions, <a href="#*1·11">*1·11</a> +and <a href="#*1·72">*1·72</a>, give what is symbolically essential to the conduct of +demonstrations in accordance with the theory of types.</p> + +<p>The above proposition <a href="#*1·11">*1·11</a> is used in every inference from one +asserted propositional function to another. We will illustrate the use +of this proposition by setting forth at length the way in which it is +first used, in the proof of <a href="#*2·06">*2·06</a>. That proposition is +\[ +\unicode{x201c}\vdash \colon\ldotp p\supset q.\supset \colon q\supset r.\supset .p\supset r.\unicode{x201d} +\] +We have already proved, in <a href="#*2·05">*2·05</a>, the proposition +\[ +\vdash \colon\ldotp q\supset r.\supset \colon p\supset q.\supset.p\supset r. +\] +It is obvious that <a href="#*2·06">*2·06</a> results from <a href="#*2·05">*2·05</a> by means of <a href="#*2·04">*2·04</a>, which is +\[ +\vdash \colon\ldotp p.\supset.q\supset r \colon \supset \colon q.\supset.p\supset r. +\] +For if, in this proposition, we replace \(p\) by \(q \supset r\), \(q\) +by \(p \supset q\), and \(r\) by \(p \supset r\), we obtain, as an +instance of <a href="#*2·04">*2·04</a>, the proposition +\[ +\vdash \colon\colon q \supset r.\supset \colon p\supset q.\supset.p\supset r\colon\ldotp\supset\colon\ldotp p\supset q.\supset \colon q\supset r.\supset.p\supset r \qquad \text{(1)}, +\] +and here the hypothesis is asserted by <a href="#*2·05">*2·05</a>. Thus our primitive +proposition *1·11 enables us to assert the conclusion.</p> + +<p class="nind"> +<b><a id="*1·2">*1·2</a></b>. \(\vdash \colon p \lor p.\supset.p \quad \text{Pp}.\)</p> + +<p>This proposition states: "If either \(p\) is true or \(p\) is true, +then \(p\) is true." It is called the "principle of tautology," and +will be quoted by the abbreviated title of "Taut." It is convenient, +for purposes of reference, to give names to a few of the more important +propositions; in general, propositions will be referred to by their +numbers.</p> + +<p class="nind"> +<b><a id="*1·3">*1·3</a></b>. \(\vdash \colon q.\supset .p \lor q \qquad \text{Pp}.\)</p> + +<p>This principle states: "If \(q\) is true, then '\(p\) or \(q\)' +is true." Thus <i>e.g.</i> if \(q\) is "to-day is Wednesday" and +\(p\) is "to-day is Tuesday," the principle states: "If to-day is +Wednesday, then to-day is either Tuesday or Wednesday." It is called +the "principle of addition," because it states that if a proposition +is true, any alternative may be added without making it false. The +principle will be referred to as "Add."</p> + +<p class="nind"> +<b>*1·4.</b> \(\vdash \colon p \lor q.\supset.p \lor p \qquad \text{Pp}.\)</p> + +<p><span class="pagenum" id="Page_101">[Pg 101]</span></p> + +<p>This principle states that "\(p\) or \(q\)" implies "\(q\) or \(p\)." +It states the permutative law for logical addition of propositions, and +will be called the "principle of permutation." It will be referred to +as "Perm."</p> + +<p class="nind"> +<b>1·5.</b> \(\vdash \colon p \lor (q \lor r).\supset.q \lor (p\lor r) \qquad \text{Pp}\).</p> + +<p>This principle states: "If either \(p\) is true, or '\(q\) or \(r\)' is +true, then either \(q\) is true, or '\(p\) or \(r\)' is true." It is a +form of the associative law for logical addition, and will be called +the "associative principle." It will be referred to as "Assoc." The +proposition +\[ +p \lor (q\lor r).\supset.(p\lor q) \lor r, +\] +which would be the natural form for the associative law, has less +deductive power, and is therefore not taken as a primitive proposition.</p> + +<p class="nind"> +<b><a id="*1·6">*1·6</a></b> \(\vdash \colon\ldotp q\supset r.\supset\colon p \lor q.\supset.p \lor r \qquad \text{Pp}\).</p> + +<p>This principle states: "If \(q\) implies \(r\), then '\(p\) or +\(q\)' implies '\(p\) or \(r\).'" In other words, in an implication, +an alternative may be added to both premiss and conclusion without +impairing the truth of the implication. The principle will be called +the "principle of summation," and will be referred to as "Sum."</p> + +<p class="nind"> +<b><a id="*1·7">*1·7</a></b>. If \(p\) is an elementary proposition, \(\sim{p}\) is an +elementary proposition. Pp.</p> + + +<p class="nind"> +<b><a id="*1·71">*1·71</a>.</b> If \(p\) and \(q\) are elementary propositions, \(p \lor +q\) is an elementary proposition. Pp.</p> + +<p class="nind"> +<b><a id="*1·72">*1·72</a>.</b> If \(\phi p\) and \(\psi p\) are elementary propositional +functions which take elementary propositions as arguments, \(\phi p +\lor \psi p\) is an elementary propositional function. Pp.</p> + +<p>This axiom is to apply also to functions of two or more variables. It +is called the "axiom of identification of real variables." It will +be observed that if \(\phi\) and \(\psi\) are functions which take +arguments of different types, there is no such function as "\(\phi x\lor \psi x\)," +because \(\phi\) and \(\psi\) cannot significantly have +the same argument. A more general form of the above axiom will be given +in <a href="#*9">*9</a>.</p> + +<p>The use of the above axioms will generally be tacit. It is only through +them and the axioms of *9 that the theory of types explained in the +Introduction becomes relevant, and any view of logic which justifies +these axioms justifies such subsequent reasoning as employs the theory +of types.</p> + +<p>This completes the list of primitive propositions required for the +theory of deduction as applied to elementary propositions.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_102">[Pg 102]</span></p> +<h2 class="nobreak" id="*2">*2. IMMEDIATE CONSEQUENCES OF THE PRIMITIVE PROPOSITIONS.</h2> +</div> + + +<p><i>Summary of</i> *2.</p> + +<p>The proofs of the earlier of the propositions of this number consist +simply in noticing that they are instances of the general rules given +in <a href="#*1">*1</a>. In such cases, these rules are not premisses, since they +assert any instance of themselves, not something other than their +instances. Hence when a general rule is adduced in early proofs, it +will be adduced in brackets<a id="FNanchor_45" href="#Footnote_45" class="fnanchor">[45]</a>, with indications, when required, as +to the changes of letters from those given in the rule to those in +the case considered. Thus "Taut \(\dfrac{\sim{p}}{p}\)" will mean +what "Taut" becomes when \(\sim{p}\) is written in place of \(p\). If +"Taut \(\dfrac{\sim{p}}{p}\)" is enclosed in square brackets before +an asserted proposition, that means that, in accordance with "Taut," +we are asserting what "Taut" becomes when \(\sim{p}\) is written in +place of \(p\). The recognition that a certain proposition is an +instance of some general proposition previously proved or assumed is +essential to the process of deduction from general rules, but cannot +itself be erected into a general rule, since the application required +is particular, and no general rule can <i>explicitly</i> include a +particular application.</p> + +<p>Again, when two different sets of symbols express the same proposition +in virtue of a definition, say <a href="#*1·01">*1·01</a>, and one of these, which we will +call (1), has been asserted, the assertion of the other is made by +writing "[(1).(*1·01)]" before it, meaning that, in virtue of *1·01, +the new set of symbols asserts the same proposition as was asserted in +(1). A reference to a definition is distinguished from a reference to a +previous proposition by being enclosed in round brackets.</p> + +<p>The propositions in this number are all, or nearly all, actually needed +in deducing mathematics from our primitive propositions. Although +certain abbreviating processes will be gradually introduced, proofs +will be given very fully, because the importance of the present subject +lies, not in the propositions themselves, but (1) in the fact that +they follow from the primitive propositions, (2) in the fact that +the subject is the easiest, simplest, and most elementary example of +the symbolic method of dealing with the principles of mathematics +generally. Later portions—the theories of classes, relations, cardinal +numbers, series, ordinal numbers, geometry, etc.—all employ the same +method, but with an increasing complexity in the entities and functions +considered.</p> + +<p><span class="pagenum" id="Page_103">[Pg 103]</span></p> + +<p>The most important propositions proved in the present number are the +following:</p> + +<p class="nind"> +<b><a id="*2·02">*2·02</a>.</b> \(\vdash \colon q.\supset.p\supset q\)</p> + +<p><i>I.e.</i> \(q\) implies that \(p\) implies \(q\), <i>i.e.</i> a true +proposition is implied by any proposition. This proposition is called +the "principle of simplification" (referred to as "Simp"), because, +as will appear later, it enables us to pass from the joint assertion +of \(q\) and \(p\) to the assertion of \(q\) simply. When the special +meaning which we have given to implication is remembered, it will be +seen that this proposition is obvious.</p> + +<p class="nind"> +<b>*2·03.</b> \(\vdash \colon p\supset \sim{q}.\supset.q\supset \sim{p}\)</p> + +<p class="nind"> +<b>*2·15.</b> \(\vdash \colon \sim{p}\supset q.\supset.\sim{q}\supset p\)</p> + +<p class="nind"> +<b>*2·16.</b> \(\vdash \colon p\supset q.\supset.\sim{q}\supset\sim{p}\)</p> + +<p class="nind"> +<b>*2·17.</b> \(\vdash \colon\sim{q}\supset\sim{p}.\supset.p\supset q\)</p> + +<p>These four analogous propositions constitute the "principle of +transposition," referred to as "Transp." They lead to the rule that in +an implication the two sides may be interchanged by turning negative +into positive and positive into negative. They are thus analogous +to the algebraical rule that the two sides of an equation may be +interchanged by changing the signs.</p> + +<p class="nind"> +<b>*2·04.</b> \(\vdash \colon\ldotp p.\supset.q\supset r:\supset:q.\supset.p\supset r\)</p> + +<p>This is called the "commutative principle" and referred to as "Comm." +It states that, if \(r\) follows from \(q\) provided \(p\) is true, +then \(r\) follows from \(p\) provided \(q\) is true.</p> + +<p class="nind"> +<b>*2·05.</b> \(\vdash \colon\ldotp q\supset r.\supset \colon p\supset q.\supset.p\supset r\)</p> + +<p class="nind"> +<b>*2·06.</b> \(\vdash \colon\ldotp p\supset q.\supset \colon q\supset r.\supset.p\supset r\)</p> + +<p>These two propositions are the source of the syllogism in Barbara (as +will be shown later) and are therefore called the "principle of the +syllogism" (referred to as "Syll"). The first states that, if \(r\) +follows from \(q\), then if \(q\) follows from \(p\), \(r\) follows +from \(p\). The second states the same thing with the premisses +interchanged.</p> + +<p class="nind"> +<b>*2·08.</b> \(\vdash.p\supset p\)</p> + +<p><i>I.e.</i> any proposition implies itself. This is called the +"principle of identity" and referred to as "Id." It is not the same as +the "law of identity" ("\(x\) is identical with \(x\)"), but the law of +identity is inferred from it (cf. <a href="#*13·15">*13·15</a>).</p> + +<p class="nind"> +<b>*2·21.</b> \(\vdash \colon\sim{p}.\supset.p\supset q\)</p> + +<p><i>I.e.</i> a false proposition implies any proposition.</p> + +<p><span class="pagenum" id="Page_104">[Pg 104]</span></p> + +<p>The later propositions of the present number are mostly subsumed +under propositions in <a href="#*3">*3</a> or <a href="#*4">*4</a>, which give the same results in more +compendious forms. We now proceed to formal deductions.</p> + +<hr class="tb"> + +<p class="nind"> +<b>*2·01.</b> \(\vdash \colon p \supset \sim{p} . \supset . \sim{p}\)</p> + +<p>This proposition states that, if \(p\) implies its own falsehood, then +\(p\) is false. It is called the "principle of the<i> reductio ad +absurdum</i>," and will be referred to as "Abs."<a id="FNanchor_46" href="#Footnote_46" class="fnanchor">[46]</a> The proof is as +follows (where "<i>Dem.</i>" is short for demonstration"):</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{Taut}\, \frac{\sim{p}}{p}\right] &\vdash \colon \sim{p} \lor \sim{p} . \supset . \sim{p} \qquad \text{(1)}\\ +[\text{(1).(*1·01)}] &\vdash \colon p \supset \sim{p} . \supset . \sim{p} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·02.</b> \(\vdash \colon q . \supset . p \supset q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\left[\text{Add}\, \frac{\sim{p}}{p}\right] &\vdash \colon q . \supset . \sim{p} \lor q \qquad \text{(1)}\\ +[\text{(1).(*1·01)}] &\vdash \colon q . \supset . p \supset q +\end{array} +\]</p> + +<p class="nind"> +<b>*2·03.</b> \(\vdash \colon p \supset \sim{q} . \supset . q \supset \sim{p}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\left[\text{Perm}\, \frac{\sim{p}, \sim{q}}{p, q}\right] &\vdash \colon \sim{p} \lor \sim{q} . \supset . \sim{q} \lor \sim{p} \qquad \text{(1)}\\ +[\text{(1).(*1·01)}] &\vdash \colon p \supset \sim{q} . \supset . q \supset \sim{p} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·04">*2·04</a>.</b> \(\vdash \colon\ldotp p . \supset . q \supset r \colon \supset \colon q . \supset . p \supset r\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\left[\text{Assoc}\, \frac{\sim{p}, \sim{q}}{p, q}\right] &\vdash \colon\ldotp \sim{p} \lor (\sim{q} \lor r) . \supset . \sim{q} \lor (\sim{p} \lor r) \qquad \text{(1)}\\ +[\text{(1).(*1·01)}] &\vdash \colon\ldotp p . \supset . q \supset r \colon \supset \colon q . \supset . p \supset r +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·05">*2·05</a>.</b> \(\vdash \colon\ldotp q \supset r . \supset \colon p \supset q . \supset . p \supset r\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\left[\text{Sum} \frac{\sim{p}}{p}\right] &\vdash \colon\ldotp q \supset r . \supset \colon \sim{p} \lor q . \supset . \sim{p} \lor r \qquad \text{(1)}\\ +[\text{(1).(*1·01)}] &\vdash \colon\ldotp q \supset r . \supset \colon p \supset q . \supset . p \supset r +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·06">*2·06</a>.</b> \(\vdash \colon\ldotp p \supset q . \supset \colon q \supset r . \supset . p \supset r\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\left[\text{Comm} \frac{q \supset r, p \supset q, p \supset r}{p,\,\,\, q,\,\,\, r}\right] &\vdash \colon\colon q \supset r . \supset \colon p \supset q . \supset . p \supset r \colon\ldotp\\ +&\supset \colon\ldotp p \supset q . \supset \colon q \supset r . \supset . p \supset r \qquad \text{(1)}\\ +[\text{*2·05}] &\vdash \colon\ldotp q \supset r . \supset \colon p \supset q . \supset . p \supset r \qquad \text{(2)}\\ +[\text{(1).(2).*1·11}] &\vdash \colon\ldotp p \supset q . \supset \colon q \supset r . \supset . p \supset r +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_105">[Pg 105]</span></p> + +<p>In the last line of this proof, "(1).(2).*1·11" means that we are +inferring in accordance with <a href="#*1·11">*1·11</a>, having before us a proposition, +namely \(p\supset q\ldotp \supset :q\supset r\ldotp\supset \ldotp p\supset r\), +which, by (1), is implied by \(q\supset r\ldotp\supset :p\supset q\ldotp\supset \ldotp p\supset r\), +which, by (2), is true. In general, in such cases, we shall omit +the reference to *1·11.</p> + +<p>The above two propositions will both be referred to as the "principle +of the syllogism" (shortened to "Syll"), because, as will appear later, +the syllogism in Barbara is derived from them.</p> + +<p class="nind"> +<b>*2·07</b> \(\vdash:p\ldotp\supset \ldotp p\lor p \quad\left[\text{*1·3}\, \frac{p}{q}\right]\)</p> + +<p>Here we put nothing beyond "*1·3 \(\dfrac{p}{q}\)," because the +proposition to be proved is what <a href="#*1·3">*1·3</a> becomes when \(p\) is written in +place of \(q\).</p> + +<p class="nind"> +<b>*2·08</b> \(\vdash\ldotp p\supset p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +{\Large [}\text{*2·05} \frac{p\lor p,p}{q,r}{\Large ]} &\vdash\colon\colon p\lor p\ldotp\supset \ldotp p:\supset \colon\ldotp p\ldotp\supset \ldotp p\lor p:\supset \ldotp p\supset p &\qquad \text{(1)}\\ +[\text{Taut}] &\vdash:p\lor p\ldotp\supset \ldotp p &\qquad \text{(2)}\\ +[\text{(1).(2).*1·11}] &\vdash\colon\ldotp p\ldotp\supset \ldotp p\lor p:\supset \ldotp p\supset p &\qquad \text{(3)}\\ +[\text{2·07}] &\vdash:p\ldotp\supset \ldotp p\lor p &\qquad \text{(4)}\\ +[\text{(3).(4).*1·11}] &\vdash\ldotp p\supset p +\end{array} +\]</p> + +<p class="nind"> +<b>*2·1</b> \(\vdash.\sim p\lor p \quad[\text{Id. (*1·01)}]\)</p> + +<p class="nind"> +<b><a id="*2·11">*2·11</a></b> \(\vdash.p\lor \sim p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +{\LARGE [}\text{Perm}\, \frac{\sim p,p}{p,q}{\LARGE ]} &\vdash:\sim p\lor p\ldotp\supset \ldotp p\lor \sim p &\qquad \text{(1)}\\ +[\text{(1).*2·1.*1·11}] &\vdash\ldotp p\lor \sim p +\end{array} +\]</p> + +<p>This is the law of excluded middle.</p> + +<p class="nind"> +<b><a id="*2·12">*2·12</a></b> \(\vdash\ldotp p\supset \sim (\sim p)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +{\Large [}\text{*2·11}\, \frac{\sim p}{p}{\Large ]} &\vdash.\sim p\lor \sim (\sim p) &(1)\\ +[\text{(1).(*1·01)}] &\vdash\ldotp p\supset \sim (\sim p) +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_106">[Pg 106]</span></p> + +<p class="nind"> +<b>*2·13.</b> \(\vdash.p \lor {\sim}\{{{\sim}({\sim}p)\}}\)</p> + +<p>This proposition is a lemma for <a href="#*2·14">*2·14</a>, which, with <a href="#*2·12">*2·12</a>, constitutes +the principle of double negation.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{Sum}\, \frac{{\sim}p{\sim}\{{{\sim}({\sim}p)\}}}{q,\,\,\,\,r}\right] &\vdash \colon\ldotp{\sim}p.\supset.\sim\{{{\sim}({\sim}p)\}}.\supset \colon +p \lor {\sim}p.\supset.p \lor {\sim}\{{{\sim}({\sim}p)\}} &\qquad \text{(1)}\\ +\left[\text{*2·12} \quad \frac{{\sim}p}{p}\right] &\vdash\colon:{\sim}p.\supset.{\sim}\{{{\sim}({\sim}p)}\} &\qquad \text{(2)}\\ +[\text{(1).(2).*1·11}] &\vdash\colon p\lor{\sim}p.\supset.p\lor{\sim}\{{{\sim}({\sim}p)\}} &\qquad \text{(3)}\\ +[\text{(3).*2·11.*1·11}] &\vdash\colon p\lor{\sim}\{{{\sim}({\sim}p)\}} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·14">*2·14</a>.</b> \(\vdash.{\sim}({\sim}p) \supset p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{Perm}\, \frac{{\sim}\{{{\sim}({\sim}p)\}}}{q}\right] &\vdash\colon p \lor {\sim}\{{{\sim}({\sim}p)\}}.\supset.{\sim}\{{{\sim}({\sim}p)\}}\lor p &\qquad \text{(1)}\\ +[\text{(1).*2·13.*1·11}] &\vdash\colon{\sim}\{{{\sim}({\sim}p)\}} \lor p &\qquad \text{(2)}\\ +[\text{(2).(*1·01)}] &\vdash\colon{\sim}({\sim}p)\supset p +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·15">*2·15</a>.</b> \(\vdash \colon {\sim}p \supset q.\supset.{\sim}q \supset p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{*2·05}\, \frac{{\sim}p,{\sim}({\sim}q)}{p,\,\,\,r}\right] &\vdash\colon\ldotp q\supset{\sim}({\sim}q).\supset\colon{\sim}p\supset q.\supset.{\sim}p\supset{\sim}({\sim}q) &\qquad \text{(1)}\\ +\left[\text{*2·12}\, \frac{q}{p}\right]& \vdash\colon q \supset{\sim}({\sim}q) \qquad\qquad\qquad\qquad &\qquad\text{(2)}\\ +[\text{(1).(2).*1·11}] &\vdash\colon\supset {\sim}p.\supset.q.\supset.{\sim}p\supset {\sim}({\sim}q) &\qquad \text{(3)}\\ +\left[\text{*2·03}\, \frac{{\sim}p,{\sim}q}{p,\,\,\,q}\right] &\vdash\colon{\sim}p\supset{\sim}({\sim}q).\supset.{\sim}q\supset{\sim}({\sim}p) +&\qquad \text{(4)}\\ +\left[\text{*2·05}\, \frac{{\sim}q,{\sim}({\sim}p)}{p,\,\,q,\,\,r}\right] &\vdash\colon\ldotp{\sim}({\sim}p)\supset p.\supset\colon{\sim}q\supset{\sim}({\sim}p).\supset.{\sim}q\supset p &\qquad \text{(5)}\\ +[\text{(5).*2·14.*1·11}] &\vdash\colon{\sim}q\supset{\sim}({\sim}p).\supset.{\sim}q\supset p &\qquad \text{(6)}\\ +\left[\text{*2·05}\ \frac{{\sim}p\supset q,{\sim}p\supset{\sim}({\sim}q),{\sim}q\supset{\sim}({\sim}p)}{p,\,\,\,q,\,\,\,r}\right] &\vdash\colon\colon\\ +\qquad\qquad\qquad&{\sim}p\supset{\sim}({\sim}q).\supset.{\sim}q\supset{\sim}({\sim}p)\colon\supset\colon\ldotp\\ +\qquad\qquad\qquad&{\sim}p\supset q.\supset.{\sim}p\supset{\sim}({\sim}q)\colon\supset\colon{\sim}p\supset q.\supset.{\sim}q\supset{\sim}({\sim}p) \qquad &\qquad\text{(7)}\\ +[(\text{4).(7).*1·11}] &\vdash\colon\ldotp{\sim}p\supset q.\supset.{\sim}p\supset{\sim}({\sim}q)\colon\supset\colon\\ +\qquad\qquad\qquad&{\sim}p\supset q.\supset.{\sim}q\supset{\sim}({\sim}p) \qquad &\qquad\text{(8)}\\ +[\text{(3).(8).*1·11}] &\vdash:{\sim}p\supset q.\supset.{\sim}q\supset{\sim}({\sim}p) &\qquad \text{(9)}\\ +\left[\text{*2·05}\, \frac{{\sim}p\supset q,{\sim}q\supset{\sim}({\sim}p),{\sim}q\supset p}{p,\,\,q,\,\,r}\right] &\vdash\colon\colon{\sim}q\supset{\sim}({\sim}p).\supset.{\sim}q\supset p:\\ +&\supset\colon\ldotp{\sim}p\supset q.\supset.{\sim}q\supset{\sim}({\sim}p):\supset:{\sim}p\supset q.\supset.{\sim}q\supset p &\qquad \text{(10)}\\ +[\text{(6).(10).*1·11}] &\vdash\colon\ldotp{\sim}p\supset q.\supset.{\sim}q\supset{\sim}({\sim}p):\supset:\\ +\qquad\qquad\qquad&{\sim}p\supset q.\supset.{\sim}q\supset p &\qquad \text{(11)}\\ +[\text{(9).(11).*1·11}] &\vdash:{\sim}p\supset q.\supset.{\sim}q\supset p +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_107">[Pg 107]</span></p> + +<p><i>Note on the proof of</i> <a href="#*2·15">*2·15</a>. In the above proof, it will be +seen that (3), (4), (6) are respectively of the forms \(p_{1}\supset p_{2}\), +\(p_{2}\supset p_{3}\), \(p_{3}\supset p_{4}\), where +\(p_{1}\supset p_{4}\) is the proposition to be proved. From +\(p_{1}\supset p_{2}\), \(p_{2}\supset p_{3}\), \(p_{3}\supset p_{4}\) +the proposition \(p_{1}\supset p_{4}\) results by repeated applications +of <a href="#*2·05">*2·05</a> or <a href="#*2·06">*2·06</a> (both of which are called "Syll"). It is tedious +and unnecessary to repeat this process every time it is used; it will +therefore be abbreviated into +\[ +\unicode{x201c}[\text{Syll}]\, \vdash.(a).(b).(c).\supset \vdash.(d),\unicode{x201d} +\] +where (<i>a</i>) is of the form \(p_{1}\supset p_{2}\), (<i>b</i>) of +the form \(p_{2}\supset p_{3}\), (<i>c</i>) of the form \(p_{3}\supset p_{4}\), +and (<i>d</i>) of the form \(p_{1}\supset p_{4}\). The same +abbreviation will be applied to a sorites of any length.</p> + +<p>Also where we have "\(\vdash.p_{1}\)" and "\(\vdash.p_{1}\supset p_{2}\)," +and \(p_{2}\) is the proposition to be proved, it is +convenient to write simply +\[ +\begin{array}{l} +\unicode{x201c}&\vdash.p_{1}.\supset\\ +[\text{etc}.]\qquad &\vdash.p_{2},\unicode{x201d} +\end{array} +\] +where "etc." will be a reference to the previous propositions in +virtue of which the implication "\(p_{1}\supset p_{2}\)" holds. This +form embodies the use of <a href="#*1·11">*1·11</a> or <a href="#*1·1">*1·1</a>, and makes many proofs at once +shorter and easier to follow. It is used in the first two lines of the +following proof.</p> + +<p class="nind"> +<b><a id="*2·16">*2·16</a>.</b> \(\vdash\colon p\supset q.\supset .{\sim}q\supset {\sim}p\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +[\text{*2·12}] &\vdash.q\supset{\sim}({\sim}q).\supset\\ +[\text{*2·05}] &\vdash\colon p\supset q.\supset.p\supset {\sim}({\sim}q) &\qquad \text{(1)}\\ +\left[\text{*2·03}\, \frac{{\sim}q}{q}\right] &\vdash\colon p{\sim}({\sim}q).\supset.{\sim}q\supset{\sim}p &\qquad \text{(2)}\\ +[\text{Syll}] &\vdash.(1).(2).\supset\vdash\colon p\supset q.\supset.{\sim}q\supset{\sim}p +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_108">[Pg 108]</span></p> + +<p><i>Note.</i> The proposition to be proved will be called +"Prop," and when a proof ends, like that of <a href="#*2·16">*2·16</a>, by an +implication between asserted propositions, of which the +consequent is the proposition to be proved, we shall write +"\(\vdash.\text{etc}.\supset\vdash.\text{Prop}\)". Thus +"\(\supset\vdash.\text{Prop}\)" ends a proof, and more or less +corresponds to "<span class="allsmcap">Q.E.D.</span>"</p> + +<p class="nind"> +<b><a id="*2·17">*2·17</a>.</b> \(\vdash\colon {\sim}q\supset{\sim}p.\supset.p\supset q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{*2·03} \frac{{\sim}q\, p}{p,\,\,q}\right] &\vdash\colon{\sim}q\supset{\sim}p.\supset.p\supset{\sim}({\sim}q) &\qquad \text{(1)}\\ +[\text{*2·14}] &\vdash\colon{\sim}({\sim}q)\supset q\colon \supset\\ +[\text{*2·05}] &\vdash\colon p\supset{\sim}({\sim}q).\supset.p\supset q &\qquad \text{(2)}\\ +[\text{Syll}] &\vdash.(1).(2).\supset\vdash.\, \text{Prop} +\end{array} +\]</p> + +<p><a href="#*2·15">*2·15</a>, <a href="#*2·16">*2·16</a> and <a href="#*2·17">*2·17</a> are forms of the principle of transposition, and +will be all referred to as "Transp."</p> + +<p class="nind"> +<b>*2·18.</b> \(\vdash\colon{\sim}p\supset p.\supset.p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{*2·12}] &\vdash.p\supset{\sim}({\sim}p).\supset\\ +[\text{*2·05}] &\vdash.{\sim}p\supset p.\supset.{\sim}p\supset{\sim}({\sim}p) &\qquad \text{(1)}\\ +\left[\text{*2·01} \frac{{\sim}p}{p}\right] &\vdash\colon{\sim}p\supset{\sim}({\sim}p).\supset.{\sim}({\sim}p) &\qquad \text{(2)}\\ +[\text{Syll}] &\vdash.(1).(2).\supset \vdash\colon{\sim}p\supset p.\supset.{\sim}({\sim}p) &\qquad \text{(3)}\\ +[\text{*2·14}] &\vdash.{\sim}({\sim}p)\supset p &\qquad \text{(4)}\\ +[\text{Syll}] &\vdash.(3).(4).\supset\vdash.\, \text{Prop} +\end{array} +\]</p> + +<p>This is the complement of the principle of the <i>reductio ad +absurdum</i>. It states that a proposition which follows from the +hypothesis of its own falsehood is true.</p> + +<p class="nind"> +<b><a id="*2·2">*2·2</a>.</b> \(\vdash\colon p.\supset.p \lor q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{Add}.\supset \vdash\colon p.\supset.q \lor p &\qquad \text{(1)}\\ +[\text{Perm}]\vdash\colon q \lor p.\supset.p \lor q &\qquad \text{(2)}\\ +[\text{Syll}]\vdash.(1).(2).\supset\vdash.\, \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·21.</b> \(\vdash\colon{\sim}p.\supset.p\supset q \quad\left[\text{*2·2}\, \frac{{\sim}p}{p}\right]\)</p> + +<p>The above two propositions are very frequently used.</p> + +<p class="nind"> +<b>*2·24.</b> \(\vdash\colon p.\supset.{\sim}p\supset q \quad[\text{*2·21.Comm}]\)</p> + +<p><span class="pagenum" id="Page_109">[Pg 109]</span></p> + +<p class="nind"> +<b>*2·25.</b> \(\vdash \colon\ldotp p\colon \lor \colon p \lor q.\supset.q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*2·1}.\supset \vdash \colon{\sim}(p \lor q).\lor.(p \lor q)\colon\\ +[\text{Assoc}]\, \supset \vdash \colon p.\lor.{{\sim}(p \lor q).\lor .q}\colon\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·26.</b> \(\vdash \colon\ldotp {\sim}p\colon \lor \colon p\supset q.\supset.q \quad\left[\text{*2·25}\, \frac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b>*2·27.</b> \(\vdash \colon\ldotp p.\supset\colon p\supset q.\supset.q\, \quad[\text{*2·26}]\)</p> + +<p class="nind"> +<b>*2·3.</b> \(\vdash \colon p \lor (q \lor r).\supset.p \lor(r \lor q)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{Perm}\,\frac{q,\,r}{p,\,q}\right] &\vdash \colon q \lor r.\supset.r\lor q\colon\\ +\left[\text{Sum}\, \frac{q \lor r,\,r \lor q}{q,\,\,r}\right] &\supset\vdash \colon p \lor(q \lor r).\supset.p \lor(r \lor q)\\ +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·31">*2·31</a>.</b> \(\vdash \colon p \lor (q \lor r).\supset.(p \lor q) \lor r\)</p> + +<p>This proposition and <a href="#*2·32">*2·32</a> together constitute the associative law +for logical addition of propositions. In the proof, the following +abbreviation (constantly used hereafter) will be employed<a id="FNanchor_47" href="#Footnote_47" class="fnanchor">[47]</a>: When we +have a series of propositions of the form \(a \supset b\), \(b \supset c\), +\(c \supset d\), all asserted, and "\(a \supset d\)" is the +proposition to be proved, the proof in full is as follows:</p> + +<p>\[ +\begin{array}{l} +&[\text{Syll}] &\vdash \colon\ldotp a\supset b.\supset \colon b\supset c.\supset.a\supset c &\qquad \text{(1)}\\ +&&\vdash \colon.\supset.b &\qquad \text{(2)}\\ +&[\text{(1).(2).*1·11}] &\vdash \colon b \supset c.\supset.a\supset c &\qquad \text{(3)}\\ +&&\vdash \colon b.\supset.c &\qquad \text{(4)}\\ +&[\text{(3).(4).*1·11}] &\vdash \colon a.\supset.c &\qquad \text{(5)}\\ +&[\text{Syll}] &\vdash \colon\ldotp a\supset c.\supset\colon c\supset d.\supset.a \supset d &\qquad \text{(6)}\\ +&[\text{(5).(6).*1·11}] &\vdash \colon c\supset d.\supset.a\supset d &\qquad \text{(7)}\\ +&&\vdash \colon c.\supset.d &\qquad \text{(8)}\\ +&[\text{(7).(8).*1·11}] &\vdash \colon a.\supset.d +\end{array} +\]</p> + +<p>It is tedious to write out this process in full; we therefore write +simply</p> + +<p>\[ +\begin{array}{l} +&\vdash \colon a.\supset.b.\\ +&[\text{etc.}]\,\supset.c.\\ +&[\text{etc.}]\,\supset.d:\supset\vdash.\text{Prop}, +\end{array} +\] +where "\(a \supset d\)" is the proposition to be proved. We indicate on +the left by references in square brackets the propositions in virtue +of which the successive implications hold. We put one dot (not two) +after "\(b\)," to show<span class="pagenum" id="Page_110">[Pg 110]</span> that it is \(b\), not "\(a\supset b\)," that +implies \(c\). But we put two dots after \(d\), to show that now the +whole proposition "\(a\supset d\)" is concerned. If "\(a\supset d\)" is +not the proposition to be proved, but is to be used subsequently in the +proof, we put</p> + +<p>\[ +\begin{array}{l} +\vdash\colon a.\supset.b.\\ +[\text{etc.}]\supset.c.\\ +[\text{etc.}]\supset.d &\qquad \text{(1)}, +\end{array} +\] +and then "(1)" means "\(a\supset d.\)" The proof of <a href="#*2·31">*2·31</a> is as follows:</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{*2·3}] \vdash\colon p \lor (q\lor r).&\supset.p \lor(r \lor q).\\ +\left[\text{Assoc}\, \frac{r,\,q}{q,\,r}\right] &\supset.r \lor (p \lor q).\\ +\left[\text{Perm}\, \frac{r,\,p \lor q}{p,\,q}\right] &\supset.(p \lor q) \lor r \colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·32">*2·32</a>.</b> \(\vdash\colon (p\lor q) \lor r.\supset.p \lor (q \lor r)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\left[\text{Perm}\, \frac{p \lor q,\,r}{p,\,q}\right] \vdash\colon(p \lor q)\lor r.&\supset.r \lor (p\lor q)\\ +\left[\text{Assoc}\, \frac{r,\,p,\,q}{p,\,q,\,r}\right] &\supset.p \lor(r \lor q)\\ +[\text{*2·3}] &\supset.p \lor (q \lor r)\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·33.</b> \(p\lor q \lor r.=.(p \lor q) \lor r\, \quad\text{Df}\)</p> + +<p>This definition serves only for the avoidance of brackets.</p> + +<p class="nind"> +<b><a id="*2·36">*2·36</a>.</b> \(\vdash\colon\ldotp q\supset r.\supset\colon p \lor q.\supset.r \lor p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{Perm}] &\vdash\colon p\ lor r.\supset.r \lor p\colon\\ +\left[\text{Syll} \frac{p \lor q,\,p \lor r,\,r \lor p}{p,\,q,\,r}\right] &\supset\vdash\colon\ldotp p\lor q.\supset.p\lor r\colon\supset\colon p \lor q.\supset.r \lor p &\qquad \text{(1)}\\ +[Sum] &\vdash\colon\ldotp p q\supset r.\supset\colon p \lor q.\supset.p\lor r &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Syll}.\supset\vdash.\, \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*2·37">*2·37</a>.</b> \[\begin{align}\vdash\colon\ldotp q\supset r.\supset\colon q \lor p.&\supset.p \lor r\\ +&[\text{Syll.Perm.Sum}]\end{align}\]</p> + +<p class="nind"> +<b><a id="*2·38">*2·38</a>.</b> \[\begin{align}\vdash\colon\ldotp q\supset r.\supset\colon q \lor p.&\supset.r\lor p\\ +&[\text{Syll.Perm.Sum}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_111">[Pg 111]</span></p> + +<p>The proofs of *2·37·38 are exactly analogous to that of <a href="#*2·36">*2·36</a>. (We use +"*2·37·38" as an abbreviation for "<a href="#*2·37">*2·37</a> and <a href="#*2·38">*2·38</a>." Such abbreviations +will be used throughout.)</p> + +<p>The use of a general principle of deduction, such as either form +of "Syll," in a proof, is different from the use of the particular +premisses to which the principle of deduction is applied. The principle +of deduction gives the general rule according to which the inference is +made, but is not itself a premiss in the inference. If we treated it +as a premiss, we should need either it or some other general rule to +enable us to infer the desired conclusion, and thus we should gradually +acquire an increasing accumulation of premisses without ever being able +to make any inference. Thus when a general rule is adduced in drawing +an inference, as when we write "\([\text{Syll}]\vdash.(1).(2).\supset +\vdash.\text{Prop},\)" the mention of "Syll" is only required in order +to remind the reader how the inference is drawn.</p> + +<p>The rule of inference may, however, also occur as one of the ordinary +premisses, that is to say, in the case of "Syll" for example, the +proposition "\(p\supset q.\supset \colon q\supset r.\supset.p\supset r\)" +may be one of those to which our rules of deduction are applied, +and it is then an ordinary premiss. The distinction between the two +uses of principles of deduction is of some philosophical importance, +and in the above proofs we have indicated it by putting the rule of +inference in square brackets. It is, however, practically inconvenient +to continue to distinguish in the manner of the reference. We shall +therefore henceforth both adduce ordinary premisses in square brackets +where convenient, and adduce rules of inference, along with other +propositions, in asserted premisses, <i>i.e.</i> we shall write +<i>e.g.</i></p> + +<p>\[ +\begin{array}{l} +&\unicode{x201c}\vdash.(1).(2).\text{Syll}.\supset\vdash.\text{Prop}\unicode{x201d}\\ +\text{rather than}\qquad &\unicode{x201c}\text{Syll}\vdash.(1).(2).\supset\vdash.\text{Prop}\unicode{x201d} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·4.</b> \(\vdash\colon\ldotp p.\lor.p\lor q:\supset.p\lor q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*2·31}.\supset\vdash\colon\ldotp p.\lor.p \lor q\colon &\supset\colon p\lor p.\lor q\colon\\ +[\text{Taut.*2·38}] &\supset\colon p\lor q\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·41.</b> \(\vdash\colon\ldotp q.\lor.p\lor q\colon\supset p\lor q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{Assoc}\, \frac{q,\,p,\,q}{p,\,q,\,r}\right] \vdash\colon\ldotp q.\lor.p\lor q\colon&\supset\colon p.\lor .q\lor q\colon\\ +[\text{Taut.Sum}] &\supset\colon p \lor q\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·42.</b> \(\vdash\colon\ldotp{\sim}p.\lor.p\supset q\colon \supset.p\supset q \quad\left[\text{*2·4}\, \tfrac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b>*2·43.</b> \(\vdash\colon\ldotp p.\supset.p\supset q:\supset.p\supset q +\quad [\text{*2·42}]\)</p> + +<p class="nind"> +<b>*2·45.</b> \(\vdash\colon{\sim}(p\lor q).\supset.{\sim}p \quad [\text{*2·2.Transp}]\)</p> + +<p class="nind"> +<b>*2·46.</b> \(\vdash\colon{\sim}(p \lor q).\supset.{\sim}q \quad [\text{*1·3.Transp}]\)</p> + +<p><span class="pagenum" id="Page_112">[Pg 112]</span></p> + +<p class="nind"> +<b>*2·47.</b> \(\vdash\colon{\sim}(p \lor q).\supset.{\sim}p \lor q \quad\left[\text{*2·45.*2·2}\, \frac{{\sim}p}{p}.\text{Syll}\right]\)</p> + +<p class="nind"> +<b>*2·48.</b> \(\vdash\colon{\sim}(p\lor q).\supset.p \lor {\sim}q \quad\left[\text{*2·46.*1·3}\, \frac{{\sim}q}{q}.\text{Syll}\right]\)</p> + +<p class="nind"> +<b>*2·49.</b> \(\vdash\colon{\sim}(p\lor q).\supset.{\sim}p \lor{\sim}q \quad\left[\text{*2·45.*2·2}\, \frac{{\sim}p,\,{\sim}q}{p,\,q}.\text{Syll}\right]\)</p> + +<p class="nind"> +<b>*2·5.</b> \(\vdash\colon{\sim}(p\supset q).\supset.{\sim}p\supset q \quad\left[\text{*2·47}\, \frac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b>*2·51.</b> \(\vdash\colon{\sim}(p\supset q).\supset.p\supset{\sim}q \quad\left[\text{*2·48}\, \frac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b>*2·52.</b> \(\vdash\colon{\sim}(p\supset q).\supset.{\sim}p\supset{\sim}q \quad\left[\text{*2·49}\, \frac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b>*2·521.</b> \(\vdash\colon{\sim}(p\supset q).\supset.q\supset p \quad[\text{*2·52·17}]\)</p> + +<p class="nind"> +<b>*2·53.</b> \(\vdash\colon p\lor q.\supset.{\sim}p\supset q\)</p> + + +<p><i>Dem.</i> +\[ +\vdash.\text{*2·12·38}.\supset\vdash\colon p \lor q.\supset.{\sim}({\sim}p)\lor q\colon\supset\vdash. \quad\text{Prop} +\]</p> + +<p class="nind"> +<b>*2·54.</b> \(\vdash\colon{\sim}p\supset q.\supset.p\lor q \quad[\text{*2·14·38}]\)</p> + +<p class="nind"> +<b>*2·55.</b> \(\vdash\colon\ldotp{\sim}p.\supset.\colon p\lor q.\supset.q \quad[\text{*2·53.Comm}]\)</p> + +<p class="nind"> +<b>*2·56.</b> \(\vdash\colon\ldotp{\sim}q.\supset\colon p\lor q.\supset.p \quad\left[\text{*2·55}\, \frac{q,\,p}{p,\,q}.\,\text{Perm}\right]\)</p> + +<p class="nind"> +<b>*2·6.</b> \(\vdash\colon\ldotp{\sim}p\supset q.\supset\colon p\supset q.\supset.q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{*2·38}] &\vdash\colon\ldotp{\sim}p\supset q.\supset\colon{\sim}p \lor q.\supset.q \lor q &\qquad \text{(1)}\\ +[\text{Taut.Syll}] &\vdash\colon\ldotp{\sim}p \lor q.\supset.q\lor q\colon\supset\colon{\sim}p \lor q.\supset.q &\qquad \text{(2)}\\ +\vdash.(1).(2).\text{Syll}.&\supset\vdash\colon\ldotp{\sim}p\supset q.\supset\colon{\sim}p \lor q.\supset.q\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·61.</b> \(\vdash\colon\ldotp p\supset q.\supset\colon{\sim}p\supset q.\supset.q \quad[\text{*2·6.Comm}]\)</p> + +<p class="nind"> +<b>*2·62.</b> \(\vdash\colon\ldotp p \lor q.\supset\colon p\supset q.\supset.q \quad[\text{*2·53·6.Syll}]\)</p> + +<p class="nind"> +<b>*2·621.</b> \(\vdash\colon\ldotp p\supset q.\supset\colon p \lor q.\supset.q \quad[\text{*2·62.Comm}]\)</p> + +<p class="nind"> +<b>*2·63.</b> \(\vdash\colon\ldotp p \lor q.\supset\colon{\sim}p \lor q.\supset.q \quad[\text{*2·62}]\)</p> + +<p class="nind"> +<b>*2·64.</b> \(\vdash\colon\ldotp p \lor q.\supset\colon p\lor{\sim}q.\supset.p \quad\left[\text{*2·63}\, \frac{q,\,p}{p,\,q}.\text{Perm}\right]\)</p> + +<p class="nind"> +<b>*2·65.</b> \(\vdash\colon\ldotp p\supset q.\supset\colon p\supset{\sim}q.\supset.{\sim}p \quad\left[\text{*2·64}\, \frac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b>*2·67.</b> \(\vdash\colon\ldotp p\lor q.\supset.q\colon\supset.p\supset q\)</p> + + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{*2·54.Syll}] &\vdash\colon\ldotp p \lor q.\supset.q\colon\supset:{\sim}p\supset q.\supset.q &\qquad \text{(1)}\\ +[\text{*2·24.Syll}] &\vdash\colon\ldotp{\sim}p\supset q.\supset.q\colon\supset.p\supset q &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Syll}.\supset\vdash.\, \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_113">[Pg 113]</span></p> + +<p class="nind"> +<b>*2·68.</b> \(\vdash\colon\ldotp p\supset q.\supset.q\colon\supset.p\lor q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\left[\text{*2·67} \frac{{\sim}p}{p}\right] &\vdash\colon\ldotp p\supset q.\supset.q\colon\supset.{\sim}p\supset q \qquad \text{(1)}\\ +\vdash.(1).\text{*2·54}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"> +<b>*2·69.</b> \(\vdash\colon\ldotp p\supset q.\supset.q\colon\supset\colon q\supset p.\supset.p \quad\left[\text{*2·68.Perm.*2·62}\, \frac{q,\,p}{p,\,q}\right]\)</p> + +<p class="nind"> +<b>*2·73.</b> \(\vdash\colon\ldotp p\supset q.\supset\colon p \lor q\lor r.\supset.q\lor r \quad[\text{*2·621·38}]\)</p> + +<p class="nind"> +<b>*2·74.</b> \(\vdash\colon\ldotp q\supset p.\supset\colon p\lor q\lor r.\supset.p\lor r \quad\left[\text{*2·73}\, \frac{q,\,p}{p,\,q}.\text{Assoc.Syll}\right]\)</p> + +<p class="nind"> +<b><a id="*2·75">*2·75</a>.</b> \(\vdash\colon\colon p\lor q.\supset\colon\ldotp p.\lor.q\supset r\colon\supset.p\lor r \quad\left[\text{*2·74}\, \frac{{\sim}q}{q}.\text{*2·53·31}\right]\)</p> + +<p class="nind"> +<b>*2·76.</b> \(\vdash\colon\ldotp p.\lor.q\supset r\colon\supset\colon p\lor q.\supset.p\lor r \quad[\text{*2·75.Comm}]\)</p> + +<p class="nind"> +<b>*2·77.</b> \(\vdash\colon\ldotp p.\supset.q\supset r\colon\supset\colon p\supset q.\supset.p\supset r \quad\left[\text{*2·76}\, \frac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b>*2·8.</b> \(\vdash\colon\ldotp q\lor r.\supset\colon{\sim}r\lor s.\supset.q\lor s\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*2·53.Perm}.\supset\vdash\colon\ldotp q\lor r.&\supset\colon{\sim}r\supset q\colon\\ +[\text{*2·38}] &\supset\colon{\sim}r\lor s.\supset.q\lor s\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·81.</b> \(\vdash\colon\colon q.\supset.r\supset s\colon\supset\colon\ldotp p\lor q.\supset\colon p\lor r.\supset.p\lor s\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{Sum}.\supset\vdash\colon\colon q.\supset.r\supset s\colon\supset\colon\ldotp &p\lor q.\supset\colon p.\lor.r\supset s &\qquad \text{(1)}\\ +\vdash.\text{*2·76.Syll}.\supset\vdash\colon\colon p\lor q.\supset\colon& p.\lor.r\supset s\colon\ldotp\supset\colon\ldotp\\ +&p\lor q.\supset\colon p\lor r.\supset.p\lor s &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·82.</b> \[\begin{align}\vdash\colon\ldotp p\lor q\lor r.&\supset\colon p\lor {\sim}r\lor s.\supset.p\lor q\lor s\\ +&\left[\text{*2·8.*2·81}\, \frac{q\lor r,{\sim}r\lor s,q \lor s}{q,\,r,\,s}\right]\end{align}\]</p> + +<p class="nind"> +<b>*2·83.</b> \[\begin{align}\vdash\colon\colon p.\supset.q\supset r\colon\supset\colon\ldotp p.\supset.r\supset s\colon&\supset\colon p.\supset.q\supset s\\ +&\left[\text{*2·82}\, \frac{{\sim}p,{\sim}q}{p,q}\right]\end{align}\]</p> + +<p class="nind"> +<b>*2·85.</b> \(\vdash\colon\ldotp p\lor q.\supset.p\lor r\colon\supset\colon p.\lor.q\supset r\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{Add.Syll}] \vdash\colon\ldotp p\lor q.&\supset.r\colon\supset.q\supset r &\qquad \text{(1)}\\ +\vdash.\text{*2·55}.\supset\vdash\colon\colon{\sim}p.&\supset\colon\ldotp p\lor r.\supset.r\colon\ldotp\\ +[\text{Syll}] &\supset\colon\ldotp p\lor q.\supset.p\lor r\colon\supset\colon p\lor q.\supset.r\colon\ldotp\\ +[\text{(1).*2·83}] \supset\colon\ldotp p\lor q.\supset.&p\lor r\colon\supset\colon q\supset r &\qquad \text{(2)}\\ +\vdash.(2).\text{Comm}.\supset\vdash\colon\ldotp &p\lor q.\supset.p \lor r\colon\supset\colon{\sim}p.\supset.q\supset r\colon\\ +[\text{*2·54}] &\supset\colon p.\lor.q\supset r\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*2·86.</b> \(\vdash\colon\ldotp p\supset q.\supset.p\supset r\colon\supset\colon p.\supset.q\supset r \quad\left[\text{*2·85}\, \frac{{\sim}p}{p}\right]\)</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_38" href="#FNanchor_38" class="label">[38]</a> +The recognized methods of proving independence are +not applicable, without reserve, to fundamentals. Cf. <i>Principles +of Mathematics</i>, § 17. What is there said concerning primitive +propositions applies with even greater force to primitive ideas.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_39" href="#FNanchor_39" class="label">[39]</a> +We have adopted both the idea and the symbol of assertion +from Frege.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_40" href="#FNanchor_40" class="label">[40]</a> +Cf. <i>Principles of Mathematics</i>, § 38.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_41" href="#FNanchor_41" class="label">[41]</a> +When we say that a proposition "belongs to logic," we +mean that it can be expressed in terms of the primitive ideas of logic. +We do not mean that logic <i>applies</i> to it, for that would of +course be true of any proposition.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_42" href="#FNanchor_42" class="label">[42]</a> +The sign of equality not followed by the letters "Df" +will have a different meaning, to be defined later.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_43" href="#FNanchor_43" class="label">[43]</a> +The letters "Pp" stand for "primitive proposition," as +with Peano.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_44" href="#FNanchor_44" class="label">[44]</a> +For further remarks on this principle, cf. <i>Principles +of Mathematics</i>, § 38.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_45" href="#FNanchor_45" class="label">[45]</a> +Later on we shall cease to mark the distinction between a +premiss and a rule according to which an inference is conducted. It is +only in early proofs that this distinction is important.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_46" href="#FNanchor_46" class="label">[46]</a> +There is an interesting historical article on this +principle by Vailati, "A proposito d'un passo del Teeteto e di +una dimostrazione di Euclide," <i>Rivista di Filosofia e scienze +affine</i>, 1904.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_47" href="#FNanchor_47" class="label">[47]</a> +This abbreviation applies to the same type of cases as +those concerned in the note to <a href="#*2·15">*2·15</a>, but is often more convenient than +the abbreviation explained in that note.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_114">[Pg 114]</span></p> +<h2 class="nobreak" id="*3">*3. THE LOGICAL PRODUCT OF TWO PROPOSITIONS.</h2> +</div> + + +<p><i>Summary of</i> *3.</p> + +<p>The logical product of two propositions \(p\) and \(q\) is practically +the proposition "\(p\) and \(q\) are both true." But this as it stands +would have to be a new primitive idea. We therefore take as the logical +product the proposition \({\sim}({\sim}p\lor {\sim}q)\), <i>i.e.</i> +"it is false that either \(p\) is false or \(q\) is false," which is +obviously true when and only when \(p\) and \(q\) are both true. Thus +we put</p> + +<p class="nind"> +<b><a id="*3·01">*3·01</a>.</b> \(p.q .=. {\sim}({\sim}p \lor {\sim}q) \quad \text{Df}\)</p> + +<p>where "\(p . q\)" is the logical product of \(p\) and \(q\).</p> + +<p class="nind"> +<b><a id="*3·02">*3·02</a>.</b> \(p \supset q \supset r.=. p \supset q . q \supset r \quad \text{Df}\)</p> + +<p>This definition serves merely to abbreviate proofs.</p> + +<p>When we are given two asserted propositional functions "\(\vdash \ldotp{\phi}x\)" +and "\(\vdash \ldotp {\psi}x\)," we shall have "\(\vdash\ldotp {\phi}x \ldotp {\psi}x\)" +whenever \(\phi\) and \(\psi\) take arguments of the same type. This +will be proved for any functions in *9; for the present, we are +confined to <i>elementary</i> propositional functions of elementary +propositions. In this case, the result is proved as follows:</p> + +<p>By <a href="#*1·7">*1·7</a>, \({\sim}{\phi}p\) and \({\sim}{\psi}p\) are elementary +propositional functions, and therefore, by <a href="#*1·72">*1·72</a>, \({\sim}{\phi}p \lor{\sim}{\psi}p\) +is an elementary propositional function. Hence by <a href="#*2·11">*2·11</a>, +\[ +\vdash : {\sim}{\phi} \lor {\sim}{\psi}p .\lor. {\sim}({\sim}{\phi}p \lor {\sim}{\psi}p)\text{.} +\]</p> + +<p>Hence by <a href="#*2·32">*2·32</a> and <a href="#*1·01">*1·01</a>, +\[ +\vdash \colon\ldotp {\phi}p .\supset : {\psi}p .\supset. {\sim}({\sim}{\phi}p \lor {\sim}{\psi}p)\text{,} +\] +<i>i.e.</i> by <a href="#*3·01">*3·01</a>, +\[ +\vdash \colon\ldotp {\phi}p . \supset: {\psi}p .\supset. {\phi}p . {\psi}p\text{.} +\]</p> + +<p>Hence by <a href="#*1·11">*1·11</a>, when we have "\(\vdash . {\phi}p\)" and "\(\vdash .{\psi}p\)" +we have "\(\vdash . {\phi}p . {\psi}p\)." This proposition +is <a href="#*3·03">*3·03</a>. It is to be understood, like <a href="#*1·72">*1·72</a>, as applying also to +functions of two or more variables.</p> + +<p>The above is the practically most useful form of the axiom of +identification of real variables (cf. <a href="#*1·72">*1·72</a>). In practice, when the +restriction to <i>elementary</i> propositions and propositional +functions has been removed, a convenient means by which two functions +can often be recognized as taking arguments of the same type is the +following:</p> + +<p>If \({\phi}x\) contains, in any way, a constituent \(\chi(x, y, z,\ldots)\) +and \({\psi}x\) contains, in any way, a constituent \(\chi(x, u, v, \ldots)\), +then both \({\phi}x\) and \({\psi}x\) take arguments<span class="pagenum" id="Page_115">[Pg 115]</span> +of the type of the argument \(x\) in \(\chi(x, y, z, ...)\), and +therefore both \(\phi x\) and \(\psi x\) take arguments of the same +type. Hence, in such a case, if both \(\phi x\) and \(\psi x\) can be +asserted, so can \(\phi x\ldotp \psi x\).</p> + +<p>As an example of the use of this proposition, take the proof of <a href="#*3·47">*3·47</a>. +We there prove +\[ +\begin{aligned} +&\vdash \colon\ldotp p\supset r\ldotp q\supset s\ldotp\supset :p\ldotp q\ldotp\supset \ldotp q\ldotp r &\qquad \text{(1)}\\ +\text{and}\quad &\vdash \colon\ldotp p\supset r\ldotp q\supset s\ldotp \supset :q\ldotp r\ldotp\supset \ldotp r\ldotp s &\qquad \text{(2)}\\ +\end{aligned} +\] +and what we wish to prove is +\[ +p\supset r\ldotp q\supset s\ldotp\supset :p\ldotp q\ldotp\supset \ldotp r\ldotp s, +\] +which is <a href="#*3·47">*3·47</a>. Now in (1) and (2), \(p\), \(q\), \(r\), \(s\) are +elementary propositions (as everywhere in Section A); hence by *1·7·71, +applied repeatedly, "\(p\supset r\ldotp q\supset r\ldotp\supset:p\ldotp q\ldotp\supset \ldotp q\ldotp r\)" +and "\(p\supset r\ldotp q\supset s\ldotp\supset :q\ldotp r\ldotp\supset \ldotp r\ldotp s\)" +are elementary propositional functions. Hence by <a href="#*3·03">*3·03</a>, we have +\[ +\vdash \colon\colon p\supset r\ldotp q\supset s\ldotp\supset :p\ldotp q\ldotp\supset \ldotp q\ldotp r \colon\ldotp p\supset r\ldotp q\supset s\ldotp\supset :q\ldotp r\ldotp\supset \ldotp r\ldotp s, +\] +whence the result follows by <a href="#*3·43">*3·43</a> and <a href="#*3·33">*3·33</a>.</p> + +<p>The principal propositions of the present number are the following:</p> + +<p class="nind"> +<b>*3·2</b> \(\vdash \colon\ldotp p\ldotp\supset :q\ldotp\supset \ldotp p\ldotp q\)</p> + +<p><i>I.e</i>. "\(p\) implies that \(q\) implies \(p\ldotp q\)," +<i>i.e</i>. if each of two propositions is true, so is their logical +product.</p> + +<p class="nind"> +<b><a id="*3·26">*3·26</a></b> \(\vdash :p\ldotp q\ldotp \supset \ldotp p\)</p> + +<p class="nind"> +<b>*3·27</b> \(\vdash :p\ldotp q\ldotp\supset \ldotp q\)</p> + +<p><i>I.e</i>. if the logical product of two propositions is true, then +each of the two propositions severally is true.</p> + +<p class="nind"> +<b>*3·3</b> \(\vdash \colon\ldotp p\ldotp q\ldotp\supset \ldotp r:\supset :p\ldotp\supset \ldotp q\supset r\)</p> + +<p><i>I.e</i>. if \(p\) and \(q\) jointly imply \(r\), then \(p\) implies +that \(q\) implies \(r\). This principle (following Peano) will be +called "exportation," because \(q\) is "exported" from the hypothesis. +It will be referred to as "Exp."</p> + +<p class="nind"> +<b>*3·31</b> \(\vdash \colon\ldotp p\ldotp\supset \ldotp q\supset r:\supset :p\ldotp q\ldotp\supset \ldotp r\)</p> + +<p>This is the correlative of the above, and will be called (following +Peano) "importation" (referred to as "Imp").</p> + +<p class="nind"> +<b>*3·35</b> \(\vdash :p\ldotp p\supset q\ldotp\supset \ldotp q\)</p> + +<p><i>I.e</i>. "if \(p\) is true, and \(q\) follows from it, then \(q\) is +true." This will be called the "principle of assertion" (referred to as +"Ass"). It differs from *1·1 by the fact that it does not apply only +when \(p\) really is true, but requires merely the <i>hypothesis</i> +that p is true.</p> + +<p class="nind"> +<b>*3·43</b> \(\vdash \colon\ldotp p\supset q\ldotp p\supset r\ldotp\supset :p\ldotp\supset \ldotp q\ldotp r\)</p> + +<p><span class="pagenum" id="Page_116">[Pg 116]</span></p> + +<p><i>I.e</i>. if a proposition implies each of two propositions, then it +implies their logical product. This is called by Peano the "principle +of composition." It will be referred to as "Comp."</p> + +<p class="nind"> +<b>*3·45.</b> \(\vdash\colon\ldotp p \,\supset:\, q \,\supset.\, p \,.\, r \,\supset\, q \,.\, r\)</p> + +<p><i>I.e.</i> both sides of an implication may be multiplied by a common +factor. This is called by Peano the "principle of the factor." It will +be referred to as "Fact."</p> + +<p class="nind"> +<b>*3·47.</b> \(\vdash\colon\ldotp p \supset r . q \supset s. \supset: p . q. \supset. r . s\)</p> + +<p><i>I.e.</i> if \(p\) implies \(q\) and \(r\) implies \(s\), then +\(p\) and \(q\) jointly imply \(r\) and \(s\) jointly. The law of +contradiction, "\(\vdash.{\sim}(p.{\sim}p\))," is proved in this number +(<a href="#*3·24">*3·24</a>); but in spite of its fame we have found few occasions for its +use.</p> + +<hr class="tb"> + +<p class="nind"> +<b>*3·01.</b> \(p . q . = {\sim} ({\sim} p \lor {\sim} q) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*3·02.</b> \(p\supset q\supset r.=.p\supset q.q \supset r \quad \text{Df}\)</p> + +<p class="nind"> +<b><a id="*3·03">*3·03</a>.</b> Given two asserted elementary propositional functions +"\(\vdash.\phi p\)" and "\(\vdash.\psi p\)" whose arguments are +elementary propositions, we have \(\vdash.\phi p.\psi p\).</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash .\text{*1·7·72.*2·11} . \supset \vdash : {\sim} \psi p \lor {\sim} \psi p . \lor . {\sim}({\sim} \psi p \lor {\sim} \psi p) &\qquad \text{(1)} \\ +\vdash .\text{(1).*2·32.(*1·01)} . \supset \vdash \vdash \psi p . \supset : \psi p . \supset . {\sim} ({\sim} \psi p \lor {\sim} \psi p) &\qquad \text{(2)} \\ +\vdash .\text{(2).(*3·03)} . \supset \vdash \vdash \psi p . \supset : \psi p . \supset . \psi p . \psi p &\qquad \text{(3)} \\ +\vdash .\text{(3).*1·11} . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*3·1.</b> \(\vdash : p . q . \supset . {\sim} ({\sim} p \lor {\sim} q) +\quad [\text{Id. (*3·01)}]\)</p> + +<p class="nind"> +<b>*3·11.</b> \(\vdash : {\sim} ({\sim} p \lor {\sim} q) . \supset . p . q +\quad [\text{Id.} (*3·01)]\)</p> + +<p class="nind"> +<b>*3·12.</b> \(\vdash : {\sim} p . \lor . {\sim} q . \lor . p . q +\quad \left[\text{*2·11} \frac{{\sim} p \lor {\sim} q}{p}\right]\)</p> + +<p class="nind"> +<b>*3·13.</b> \(\vdash : {\sim} (p . q) . \supset . {\sim} p \lor {\sim} q +\quad [\text{*3·11.Transp}]\)</p> + +<p class="nind"> +<b>*3·14.</b> \(\vdash : {\sim} p \lor {\sim} q . \supset . {\sim} (p . q) +\quad [\text{*3·1.Transp}]\)</p> + +<p class="nind"> +<b>*3·2.</b> \(\vdash\colon\ldotp p . \supset : q . \supset . p . q +\quad [\text{*3·12}] \)</p> + +<p class="nind"> +<b>*3·21.</b> \(\vdash\colon\ldotp q . \supset : p . \supset . p . q +\quad [\text{*3·2.Comm}]\)</p> + +<p class="nind"> +<b>*3·22.</b> \(\vdash : p . q . \supset . q . p\)</p> + +<p>This is one form of the commutative law for logical multiplication. A +more complete form is given in <a href="#*4·3">*4·3</a>.</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +[\text{*3·13} \frac{q,p}{p,q}] \vdash : \sim (q . p) . &\supset . \sim q \lor \sim p. \\ +[\text{Perm}] &\supset . \sim p \lor \sim q. \\ +[\text{*3·14}] &\supset . \sim (p . q) &\qquad \text{(1)} \\ +\vdash . \text{(1) . Transp}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_117">[Pg 117]</span></p> + +<p>Note that, in the above proof, "(1)" stands for the proposition +\[ +\unicode{x201c}\sim (q\ldotp p)\ldotp\supset\ldotp\sim (p\ldotp q),\unicode{x201d} +\] +as was explained in the proof of <a href="#*2·31">*2·31</a>.</p> + +<p class="nind"> +<b><a id="*3·24">*3·24</a>.</b> \(\vdash\ldotp\sim (p\ldotp\sim p)\)</p> + +<p><i>Dem.</i>\[ +\begin{align*} +\left [\text{*2·11}\, \frac{\sim p}{p}\right] &\vdash\ldotp \sim p\lor\sim (\sim p)\ldotp \supset\\ +\left [\text{*3·14}\, \frac{\sim p}{q}\right] &\vdash\ldotp\sim (p\ldotp\sim p)\\ +\end{align*} +\]</p> + +<p>The above is the law of contradiction.</p> + +<p class="nind"> +<b>*3·26.</b> \(\vdash\colon p\ldotp q\ldotp\supset\ldotp p\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\left[\text{*2·02}\, \frac{q,\,p}{p,\,q}\right] &\vdash\colon p\ldotp\supset\ldotp q\supset p &\qquad \text{(1)}\\ +[\text{(1).*1·01})] &\vdash\colon\sim p\ldotp \lor\ldotp \sim q\lor p\colon\\ +[\text{*2·31}] \supset&\vdash\colon \sim p\lor\sim q\ldotp\lor\ldotp p\colon\\ +\left [\text{*2·53} \frac{\sim p\lor\sim q,\,p}{p,\,q}\right] \supset &\vdash \colon\sim (\sim p\lor\sim q)\ldotp \supset\ldotp p &\qquad \text{(2)}\\ +[\text{(2).*3·01)}] &\vdash\colon p\ldotp q\ldotp\supset\ldotp p +\end{array} +\]</p> + +<p class="nind"> +<b>*3·27.</b> \(\vdash\colon p\ldotp q\ldotp\supset\ldotp q\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +[\text{*3·22}] \vdash\colon p\ldotp q\ldotp&\supset\ldotp q\ldotp p\ldotp \\ +\left[\text{*3·26} \frac{q,\,p}{p,\,q} \right] & \supset\ldotp q\colon\supset\vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p>*3·26·27 will both be called the "principle of simplification," like +<a href="#*2·02">*2·02</a>, from which they are deduced. They will be referred to as "Simp."</p> + +<p class="nind"> +<b>*3·3.</b> \(\vdash\colon\ldotp p\ldotp q\ldotp \supset \ldotp r\colon \supset \colon p\ldotp \supset \ldotp q \supset r\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +[\text{Id.}(\text{*3·01})] \vdash\colon\ldotp p\ldotp q\ldotp \supset \ldotp r\colon &\supset \colon\sim (\sim p \lor \sim q)\ldotp \supset \ldotp r\colon\\ +[\text{Transp}] &\supset\colon\sim r\ldotp \supset\ldotp\sim p\ lor \sim q\colon\\ +[\text{Id.}(\text{*1·01})] &\supset\colon\sim r\ldotp \supset \ldotp p \supset \sim q\colon\\ +[\text{Comm}] &\supset\colon p\ldotp \supset \ldotp \sim r\supset \sim q\colon\\ +[\text{Transp.Syll}] &\supset\colon p\ldotp \supset \ldotp q\supset r\colon\ldotp \supset\vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*3·31.</b> \(\vdash\colon\ldotp p\ldotp \supset \ldotp q \supset r\colon \supset \colon p\ldotp q\ldotp \supset\ldotp r\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +[\text{Id.}(\text{*1·01})] \vdash\colon\ldotp p\ldotp \supset \ldotp q \supset r\colon &\supset \colon \sim p\ldotp \lor \ldotp \sim q\lor r\colon\\ +[\text{*2·31}] &\supset\colon\sim p\lor\sim q\ldotp \lor \ldotp r\colon\\ +\left[\text{*2·53}\, \frac{\sim p\lor \sim q,\,r}{p,\,q}\right] &\supset \colon\sim (\sim p\lor \sim q)\ldotp \supset \ldotp r\colon\\ +[\text{Id.(*3·01)}] &\supset \colon p\ldotp q\ldotp \supset\ldotp r\colon\ldotp \supset\vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_118">[Pg 118]</span></p> + +<p class="nind"> +<b><a id="*3·33">*3·33</a>.</b> \(\vdash :p\supset q .q\supset r. \supset .p\supset r \quad[\text{Syll. Imp}]\)</p> + +<p class="nind"> +<b>*3·34.</b> \(\vdash:q\supset r.p\supset q.\supset.p\supset r \quad[\text{Syll. Imp}]\)</p> + +<p>These two propositions will hereafter be referred to as "Syll"; they +are usually more convenient than either <a href="#*2·05">*2·05</a> or <a href="#*2·06">*2·06</a>.</p> + +<p class="nind"> +<b>*3·35.</b> \(\vdash : p. p \supset q . \supset . q \quad[\text{*2·27. Imp}]\)</p> + +<p class="nind"> +<b>*3·37.</b> \(\vdash \colon\ldotp p . q . \supset . r :\supset: p. {\sim} r. \supset . {\sim} q\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash . \text{Transp} . &\supset \vdash : q\supset r . \supset . {\sim}r \supset {\sim}q:\\ +[\text{Syll}] &\supset\vdash\colon\ldotp p.\supset.q\supset r:\supset:p.\supset.{\sim}r\supset{\sim}q &\qquad \text{(1)}\\ +\vdash . \text{Exp}. &\supset\vdash\colon\ldotp p . q . \supset . r :\supset: p .\supset. q\supset r &\qquad \text{(2)}\\ +\vdash . \text{Imp}. &\supset\vdash\colon\ldotp p.\supset.{\sim}r\supset{\sim}q :\supset: p.{\sim}r .\supset. {\sim}q &\qquad \text{(3)}\\ +\vdash . \text{(2).(1).(3).Syll}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This is another form of transposition.</p> + +<p class="nind"> +<b>*3·4.</b> \(\vdash:p.q.\supset.p\supset q \quad[\text{*2·51.Transp.(*1·01. *3·01)}]\)</p> + +<p class="nind"> +<b>*3·41.</b> \(\vdash \colon\ldotp p \supset r . \supset : p . q . \supset . r \quad[\text{*3·26.Syll}]\)</p> + +<p class="nind"> +<b>*3·42.</b> \(\vdash \colon\ldotp q \supset r . \supset : p . q . \supset . r \quad[\text{*3·27.Syll}]\)</p> + +<p class="nind"> +<b><a id="*3·43">*3·43</a>.</b> \(\vdash\colon\ldotp p\supset q.p\supset r. \supset : p . \supset . q . r\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash . \text{*3·2}.\supset \vdash \colon\ldotp q . \supset : r . \supset . q . r &\qquad \text{(1)}\\ +\vdash . \text{(1). Syll}. \supset \vdash \colon\colon p \supset q . &\supset \colon\ldotp p . \supset : r. \supset . q . r\colon\ldotp \\ +[\text{*2·77}] &\supset \colon\ldotp p \supset r . \supset : p . \supset . q . r &\qquad \text{(2)}\\ +\vdash . \text{(2). Imp}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*3·44">*3·44</a>.</b> \(\vdash \colon\ldotp q \supset p . r \supset p . \supset : q \lor r . \supset . p\)</p> + +<p>This principle is analogous to <a href="#*3·43">*3·43</a>. The analogy between *3·43 and +<a href="#*3·44">*3·44</a> is of a sort which generally subsists between formulae concerning +products and formulae concerning sums.</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash . \text{Syll}. \supset \vdash\colon\ldotp {\sim}q\supset r . r\supset p . &\supset:{\sim}q\supset p:\\ +[\text{*2·6}] &\supset :q\supset p . \supset . p &\qquad \text{(1)}\\ +\vdash .\text{(1).Exp}. \supset \vdash\colon\colon {\sim}q \supset r . &\supset\colon\ldotp r\supset p .\supset: q\supset p . \supset . p\colon\ldotp\\ +[\text{Comm.Imp}] &\supset\colon\ldotp q \supset p . r\supset p . \supset . p &\qquad \text{(2)}\\ +\vdash.\text{(2).Comm}. &\supset \vdash\colon\ldotp q \supset p . r \supset p. \supset: {\sim}q \supset r .\supset. p\colon\ldotp \\ +[\text{*2·53.Syll}] &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_119">[Pg 119]</span></p> + +<p class="nind"> +<b>*3·45.</b> \(\vdash\colon\ldotp p\supset q.\supset :p.r.\supset .q.r\)</p> + +<p>This principle shows that we may multiply both sides of an implication +by a common factor; hence it is called by Peano the "principle of +the factor." We shall refer to it as "Fact." It is the analogue, for +multiplication, of the primitive proposition <a href="#*1·6">*1·6</a>.</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash.\text{Syll}\, \frac{{\sim}r}{r} .\supset \vdash\colon\ldotp p\supset q.&\supset :q\supset {\sim}r.\supset .p\supset {\sim}r:\\ +[\text{Transp}] &\supset :{\sim}(p\supset {\sim}r).\supset .{\sim}(q\supset {\sim}r)\colon\ldotp \\ +[\text{Id.(*1·01.*3·01)}] \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*3·47">*3·47</a>.</b> \(\vdash\colon\ldotp p\supset r.q\supset s.\supset :p.q.\supset .r.s\)</p> + +<p>This proposition, or rather its analogue for classes, was proved by +Leibniz, and evidently pleased him, since he calls it "præclarum +theorema<a id="FNanchor_48" href="#Footnote_48" class="fnanchor">[48]</a>."</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash.\text{*3·26}.\supset \vdash\colon\ldotp p\supset r.q\supset s.&\supset :p\supset r:\\ +[\text{Fact}] &\supset :p.q.\supset .r.q:\\ +[\text{*3·22}] &\supset :p.q.\supset .q.r &\qquad \text{(1)}\\ +\vdash. \text{*3·27}.\supset \vdash\colon\ldotp p\supset r.q\supset s.&\supset :q\supset s:\\ +[\text{Fact}] &\supset :q.r.\supset .s.r:\\ +[\text{*3·22}] &\supset :q.r.\supset .r.s &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*3·03.*2·83}.\supset \\ +\qquad\vdash\colon\ldotp p\supset r.q\supset s.\supset :p.q.\supset .r.s\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*3·48.</b> \(\vdash\colon\ldotp p\supset r.q\supset s.\supset :p\lor q.\supset .r\lor s\)</p> + +<p>This theorem is the analogue of <a href="#*3·47">*3·47</a>.</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash.\text{*3·26}.\supset \vdash\colon\ldotp p\supset r.q\supset s.&\supset :p\supset r:\\ +[\text{Sum}] &\supset :p\lor q.\supset .r\lor q:\\ +[\text{Perm}] &\supset :p\lor q.\supset .q\lor r &\qquad \text{(1)}\\ +\vdash.\text{*3·27}.\supset \vdash\colon\ldotp p\supset r.q\supset s.&\supset :q\supset s:\\ +[\text{Sum}] &\supset :q\lor r.\supset .s\lor r:\\ +[\text{Perm}] &\supset :q\lor r.\supset .r\lor s &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*2·83}.\supset \\ +\qquad\vdash\colon\ldotp p\supset r.q\supset s.\supset :p\lor q.\supset .r\lor s\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_48" href="#FNanchor_48" class="label">[48]</a> +<i>Philosophical works</i>, Gerhardt's edition, Vol. +<span class="allsmcap">VII</span>. p. 223.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_120">[Pg 120]</span></p> +<h2 class="nobreak" id="*4">*4. EQUIVALENCE AND FORMAL RULES.</h2> +</div> + + +<p><i>Summary of</i> *4.</p> + +<p>In this number, we shall be concerned with rules analogous, more or +less, to those of ordinary algebra. It is from these rules that the +usual "calculus of formal logic" starts. Treated as a "calculus," the +rules of deduction are capable of many other interpretations. But all +other interpretations depend upon the one here considered, since in all +of them we deduce consequences from our rules, and thus presuppose the +theory of deduction. One very simple interpretation of the "calculus" +is as follows: The entities considered are to be numbers which are all +either \(0\) or \(1\); "\(p \supset q\)" is to have the value \(0\) +if \(p\) is \(1\) and \(q\) is \(0\); otherwise it is to have the +value \(1\); \({\sim}p\) is to be \(1\) if \(p\) is \(0\), and \(0\) +if \(p\) is \(1\); \(p . q\) is to be \(1\) if \(p\) and \(q\) are +both \(1\), and is to be \(0\) in any other case; \(p \lor q\) is to +be \(0\) if \(p\) and \(q\) are both \(0\), and is to be \(1\) in any +other case; and the assertion-sign is to mean that what follows has the +value \(1\). Symbolic logic considered as a calculus has undoubtedly +much interest on its own account; but in our opinion this aspect has +hitherto been too much emphasized, at the expense of the aspect in +which symbolic logic is merely the most elementary part of mathematics, +and the logical prerequisite of all the rest. For this reason, we shall +only deal briefly with what is required for the algebra of symbolic +logic.</p> + +<p>When each of two propositions implies the other, we say that the two +are <i>equivalent</i>, which we write "\(p \equiv q\)." We put</p> + +<p class="nind"> +<b>*4·01.</b> \(p \equiv q .=. p \supset q . q \supset p \quad \text{Df}\)</p> + +<p>It is obvious that two propositions are equivalent when, and only when, +both are true or both are false. Following Frege, we shall call the +<i>truth-value of a proposition</i> truth if it is true, and falsehood +if it is false. Thus two propositions are equivalent when they have the +same truth-value.</p> + +<p>It should be observed that, if \(p \equiv q\), \(q\) may be substituted +for \(p\) without altering the truth-value of any function of \(p\) +which involves no primitive ideas except those enumerated in <a href="#*1">*1</a>. This +can be proved in each separate case, but not generally, because we have +no means of specifying (with our apparatus of primitive ideas) that a +function is one which can be built up out<span class="pagenum" id="Page_121">[Pg 121]</span> of these ideas alone. We +shall give the name of a <i>truth-function</i> to a function \(f(p)\) +whose argument is a proposition, and whose truth-value depends only +upon the truth-value of its argument. All the functions of propositions +with which we shall be specially concerned will be truth-functions, +<i>i.e.</i> we shall have +\[ +p\equiv q.\supset.f(p)\equiv f(q). +\] +The reason of this is, that the functions of propositions with which +we deal are all built up by means of the primitive ideas of <a href="#*1">*1</a>. But it +is not a universal characteristic of functions of propositions to be +truth-functions. For example, "\(A\) believes \(p\)" may be true for +one true value of \(p\) and false for another.</p> + +<p>The principal propositions of this number are the following:</p> + +<p class="nind"> +<b>*4·1.</b> \(\vdash:p\supset q.\equiv.{\sim} q\supset{\sim} p\)</p> + +<p class="nind"> +<b>*4·11.</b> \(\vdash:p\equiv q.\equiv.{\sim} p\equiv{\sim} q\)</p> + +<p>These are both forms of the "principle of transposition."</p> + +<p class="nind"> +<b><a id="*4·13">*4·13</a>.</b> \(\vdash.p\equiv{\sim}({\sim} p)\)</p> + +<p>This is the principle of double negation, <i>i.e.</i> a proposition is +equivalent to the falsehood of its negation.</p> + +<p class="nind"> +<b>*4·2.</b> \(\vdash.p\equiv p\)</p> + +<p class="nind"> +<b>*4·21.</b> \(\vdash:p\equiv q.\equiv.q\equiv p\)</p> + +<p class="nind"> +<b>*4·22.</b> \(\vdash:p\equiv q.q\equiv r.\supset.p\equiv r\)</p> + +<p>These propositions assert that equivalence is <i>reflexive</i>, +<i>symmetrical</i> and <i>transitive</i>.</p> + +<p class="nind"> +<b><a id="*4·24">*4·24</a>.</b> \(\vdash:p.\equiv.p.p\)</p> + +<p class="nind"> +<b>*4·25.</b> \(\vdash:p.\equiv.p\lor p\)</p> + +<p><i>I.e.</i> \(p\) is equivalent to "\(p\) and \(p\)" and to "\(p\) or +\(p\)," which are two forms of the <i>law of tautology</i>, and are the +source of the principal differences between the algebra of symbolic +logic and ordinary algebra.</p> + +<p class="nind"> +<b><a id="*4·3">*4·3</a>.</b> \(\vdash:p.q.\equiv.q.p\)</p> + +<p>This is the commutative law for the product of propositions.</p> + +<p class="nind"> +<b>*4·31.</b> \(\vdash:p\lor q.\equiv.q\lor p\)</p> + +<p>This is the commutative law for the sum of propositions.</p> + +<p>The associative laws for multiplication and addition of propositions, +namely</p> + +<p class="nind"> +<b>*4·32.</b> \(\vdash:(p.q).r.\equiv.p.(q.r)\)</p> + +<p class="nind"> +<b>*4·33.</b> \(\vdash:(p\lor q)\lor r.\equiv.p\lor (q\lor r)\)</p> + +<p>The distributive law in the two forms</p> + +<p><span class="pagenum" id="Page_122">[Pg 122]</span></p> + +<p class="nind"> +<b>*4·4.</b> \(\vdash\colon\ldotp p.q\lor r.\equiv:p.q.\lor.p.r\)</p> + +<p class="nind"> +<b>*4·41.</b> \(\vdash\colon\ldotp p.\lor.q.r:\equiv.p\lor q.p\lor r\)</p> + +<p>The second of these forms has no analogue in ordinary algebra.</p> + +<p class="nind"> +<b>*4·71.</b> \(\vdash\colon\ldotp p\supset q.\equiv:p.\equiv.p.q\)</p> + +<p><i>I.e.</i> \(p\) implies \(q\) when, and only when, \(p\) is +equivalent to \(p.q\). This proposition is used constantly; it enables +us to replace any implication by an equivalence.</p> + +<p class="nind"> +<b>*4·73.</b> \(\vdash\colon\ldotp q.\supset:p.\equiv.p.q\)</p> + +<p><i>I.e.</i> a true factor may be dropped from or added to a proposition +without altering the truth-value of the proposition.</p> + +<hr class="tb"> + +<p class="nind"> +<b>*4·01.</b> \(p\equiv q.=.p\supset q.q\supset p\quad \text{Df}\)</p> + +<p class="nind"> +<b>*4·02.</b> \(p\equiv q\equiv r.=.p\equiv q.q\equiv r \quad\text{Df}\)</p> + +<p>This definition serves merely to provide a convenient abbreviation.</p> + +<p class="nind"> +<b>*4·1.</b> \(\vdash:p\supset q.\equiv.{\sim} q\supset{\sim} p \quad[\text{*2·16·17}]\)</p> + +<p class="nind"> +<b>*4·11.</b> \(\vdash:p\equiv q.\equiv.{\sim} p\equiv{\sim} q \quad[\text{*2·16·17.*3·47·22}]\)</p> + +<p class="nind"> +<b>*4·12.</b> \(\vdash:p\equiv{\sim} q.\equiv.q\equiv{\sim} p \quad[\text{*2·03·15}]\)</p> + +<p class="nind"> +<b>*4·13.</b> \(\vdash.p\equiv{\sim}({\sim} p) \quad\text{[*2·12·14}]\)</p> + +<p class="nind"> +<b>*4·14.</b> \(\vdash\colon\ldotp p.q.\supset.r:\equiv:p.{\sim} r.\supset.{\sim} q \quad[\text{*3·37.*4·13}]\)</p> + +<p class="nind"> +<b>*4·15.</b> \(\vdash\colon\ldotp p.q.\supset.{\sim} r:\equiv:q.r.\supset.{\sim} p \quad[\text{*3·22.*4·13·14}]\)</p> + +<p class="nind"> +<b><a id="*4·2">*4·2</a>.</b> \(\vdash.p\equiv p \quad[\text{Id.*3·2}]\)</p> + +<p class="nind"> +<b><a id="*4·21">*4·21</a>.</b> \(\vdash:p\equiv q.\equiv.q\equiv p \quad[\text{*3·22}]\)</p> + +<p class="nind"> +<b><a id="*4·22">*4·22</a>.</b> \(\vdash:p\equiv q.q\equiv r.\supset.p\equiv r\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash .\text{*3·26.} &\supset \vdash : p \equiv q . q \equiv r . \supset . p \equiv q \\ +[\text{*3·26}] &\supset . p \supset q &\qquad \text{(1)} \\ +\vdash .\text{*3·27}. &\supset \vdash : p \equiv q . q \equiv r . \supset . q \equiv r \\ +[\text{*3·26}] &\supset . q \supset r &\qquad \text{(2)} \\ +\vdash . \text{(1) . (2) .*2·83}. &\supset \vdash : p \equiv q . q \equiv r . \supset . p \supset r &\qquad \text{(3)} \\ +\vdash .\text{*3·27}. &\supset \vdash : p \equiv q . q \equiv r . \supset . q \equiv r \\ +[\text{*3·27}] &\supset . r \supset q &\qquad \text{(4)} \\ +\vdash .\text{*3·26}. &\supset \vdash : p \equiv q . q \equiv r . \supset . p \equiv q \\ +[\text{*3·27}] &\supset . q \supset p &\qquad \text{(5)} \\ +\vdash . \text{(4) . (5) .*2·83}. &\supset \vdash : p \equiv q . q \equiv r . \supset . r \supset p &\qquad \text{(6)} \\ +\vdash . \text{(3) . (6) . Comp}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_123">[Pg 123]</span></p> + +<p><i>Note.</i> The above three propositions show that the relation of +equivalence is reflexive (<a href="#*4·2">*4·2</a>), symmetrical (<a href="#*4·21">*4·21</a>), and transitive +(<a href="#*4·22">*4·22</a>). Implication is reflexive and transitive, but not symmetrical. +The properties of being symmetrical, transitive, and (at least within a +certain field) reflexive are essential to any relation which is to have +the formal characters of equality.</p> + +<p class="nind"> +<b>*4·24.</b> \(\vdash:p.\equiv.p.p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*3·26}.&\supset\vdash:p.p.\supset.p &\qquad \text{(1)}\\ +&\vdash.\text{*3·2}.&\supset\vdash\colon\ldotp p.\supset:p.\supset.p.p.:\\ +&[\text{*2·43}] &\supset\vdash:p.\supset.p.p &\qquad \text{(2)}\\ +&\vdash.\text{(1).(2).*3·2}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*4·25.</b> \(\vdash:p.\equiv.p\lor p \quad\left[\text{Taut.Add}\,\frac{p}{q}\right]\)</p> + +<p><i>Note.</i> *4·24·25 are two forms of the <i>law of tautology</i>, +which is what chiefly distinguishes the algebra of symbolic logic from +ordinary algebra.</p> + +<p class="nind"> +<b>*4·3.</b> \(\vdash:p.q.\equiv.q.p \quad[\text{*3·22}]\)</p> + +<p><i>Note.</i> Whenever we have, whatever values \(p\) and \(q\) may have, +\[ +\phi(p,q).\supset.\phi(q,p), +\] +we have also +\[ +\phi(p,q).\equiv.\phi(q,p). +\] +For \(\{\phi(p,q).\supset.\phi(q,p)\} \frac{q,\,p}{p,\,q} .\supset:\phi(q,p).\supset.\phi(p,q).\)</p> + +<p class="nind"> +<b>*4·31.</b> \(\vdash:p\lor q.\equiv.q\lor p \quad[\text{Perm}]\)</p> + +<p class="nind"> +<b>*4·32.</b> \(\vdash:(p.q).r.\equiv.p.(q.r)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·15}. \supset\vdash\colon\ldotp p.q.\supset.{\sim}r:&\equiv:q.r.\supset.{\sim}p:\\ +[\text{*4·12}] &\equiv:p.\supset.{\sim}(q.r) &\qquad \text{(1)}\\ +\vdash.\text{(1).*4·11}.\supset\vdash:{\sim}(p.q.\supset.{\sim}r).&\equiv.{\sim}{p.\supset.{\sim}(q.r)}:\\ +[\text{(*1·01.*3·01)}]\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><i>Note.</i> Here "(1)" stands for "\(\vdash\colon\ldotp p.q.\supset.{\sim}r:\equiv:p.\supset.{\sim}(q.r)\)," +which is obtained from the above steps by <a href="#*4·22">*4·22</a>. The use of *4·22 will +often be tacit, as above. The principle is the same as that explained +in respect of implication in <a href="#*2·31">*2·31</a>.</p> + +<p class="nind"> +<b>*4·33.</b> \(\vdash:(p\lor q)\lor r.\equiv.p\lor (q\lor r) \quad[\text{*2·31·32}]\)</p> + +<p><span class="pagenum" id="Page_124">[Pg 124]</span></p> + +<p>The above are the associative laws for multiplication and addition. To +avoid brackets, we introduce the following definition:</p> + +<p class="nind"> +<b>*4·34.</b> \(p.q.r.=.(p.q).r \quad\text{Df}\)</p> + +<p class="nind"> +<b>*4·36.</b> \(\vdash\colon\ldotp p\equiv q.\supset:p.r.\equiv .q.r \quad[\text{Fact. *3·47}]\)</p> + +<p class="nind"> +<b>*4·37.</b> \(\vdash\colon\ldotp p\equiv q.\supset:p\lor r.\equiv .q\lor r \quad[\text{Sum. *3·47}]\)</p> + +<p class="nind"> +<b>*4·38.</b> \(\vdash\colon\ldotp p\equiv r.q\equiv s.\supset:p.q.\equiv .r.s \quad[\text{*3·47. *4·32. *3·22}]\)</p> + +<p class="nind"> +<b>*4·39.</b> \(\vdash\colon\ldotp p\equiv r.q\equiv s.\supset:p\lor q.\equiv .r\lor s \quad[\text{*3·48·47. *4·32. *3·22}]\)</p> + +<p class="nind"> +<b>*4·4.</b> \(\vdash\colon\ldotp p.q\lor r.\equiv :p.q.\lor .p.r\)</p> + +<p>This is the first form of the distributive law.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*3·2}. &\supset\vdash\colon\colon p.\supset:q.\supset.p.q\colon\ldotp p.\supset:r.\supset.p.r\colon\colon \\ +[\text{Comp}] &\supset\vdash\colon\colon p.\supset\colon\ldotp q.\supset.p.q:r.\supset.p.r\colon\ldotp \\ +[\text{*3·48}] &\qquad\quad\supset\colon\ldotp q\lor r.\supset:p.q.\lor .p.r &\qquad \text{(1)}\\ +\vdash.\text{(1).Imp}. & \supset\vdash\colon\ldotp p.q\lor r.\supset:p.q.\lor .p.r &\qquad \text{(2)}\\ +\vdash.\text{*3·26}. &\supset\vdash\colon\ldotp p.q.\supset.p:p.r.\supset.p\colon\ldotp \\ +[\text{*3·44}] &\supset\vdash\colon\ldotp p.q.\lor .p.r:\supset.p &\qquad \text{(3)}\\ +\vdash.\text{*3·27}. &\supset\vdash\colon\ldotp p.q.\supset.q:p.r.\supset.r\colon\ldotp \\ +[\text{*3·48}] &\supset\vdash\colon\ldotp p.q.\lor .p.r:\supset.q\lor r &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).Comp}. &\supset\vdash\colon\ldotp p.q.\lor .p.r:\supset.p.q\lor r &\qquad \text{(5)}\\ +\vdash.\text{(2).(5)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*4·41.</b> \(\vdash\colon\ldotp p.\lor .q.r:\equiv .p\lor q.p\lor r\)</p> + +<p>This is the second form of the distributive law—a form to which there +is nothing analogous in ordinary algebra. By the conventions as to +dots, "\(p.\lor .q.r\)" means "\(p\lor (q.r)\)."</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*3·26.Sum}. &\supset\vdash\colon\ldotp p.\lor .q.r:\supset.p\lor q &\qquad \text{(1)}\\ +\vdash.\text{*3·27.Sum}. &\supset\vdash\colon\ldotp p.\lor .q.r:\supset.p\lor r &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Comp}. &\supset\vdash\colon\ldotp p.\lor .q.r:\supset.p\lor q.p\lor r &\qquad \text{(3)}\\ +\vdash.\text{*2·53.*3·47}. \supset\vdash\colon\ldotp p\lor q.p\lor r.\supset:{\sim}p&\supset q.{\sim}p\supset r:\\ +[\text{Comp}] &\supset:{\sim}p.\supset.q.r:\\ +[\text{*2·54}] &\supset:p.\lor .q.r &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*4·42.</b> \(\vdash\colon\ldotp p.\equiv :p.q.\lor .p.{\sim}q\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*3·21}. & \supset\vdash\colon\ldotp q\lor {\sim}q.\supset:p.\supset.p.q\lor {\sim}q\colon\ldotp \\ +[\text{*2·11}] & \supset\vdash:p.\supset.p.q\lor {\sim}q &\qquad \text{(1)}\\& \supset\vdash:p.q\lor {\sim}q.\supset.p &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon\ldotp p.&\equiv :p.q\lor {\sim}q:\\ +[\text{*4·4}] &\equiv :p.q.\lor .p.{\sim}q\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_125">[Pg 125]</span></p> + +<p class="nind"> +<b><a id="*4·43">*4·43</a>.</b> \(\vdash\colon\ldotp p.\equiv :p\lor q.p\lor {\sim}q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*2·2}. &\supset\vdash:p.\supset.p\lor q:p.\supset.p\lor{\sim}q:\\ +[\text{Comp}] &\supset\vdash:p.\supset.p\lor q.p\lor {\sim}q &\qquad \text{(1)}\\ +\vdash.\text{*2·65} \frac{{\sim}p}{p}. &\supset\vdash\colon\ldotp {\sim}p\supset q.\supset:{\sim}p\supset{\sim}q.\supset.p\colon\ldotp\\ +[\text{Imp}]& \supset\vdash\colon\ldotp {\sim}p\supset q.{\sim}p\supset{\sim}q.\supset.p\colon\ldotp \\ +[\text{*2·53.*3·47}] &\supset\vdash\colon\ldotp p\lor q.p\lor {\sim}q.\supset.p &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*4·44">*4·44</a>.</b> \(\vdash\colon\ldotp p.\equiv :p.\lor.p.q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*2·2}. &\supset\vdash\colon\ldotp p.\supset:p.\lor.p.q &\qquad \text{(1)}\\ +\vdash.\text{Id.*3·26}. &\supset\vdash\colon\ldotp p\supset p:p.q.\supset.p\colon\ldotp \\ +[\text{*3·44}] &\supset\vdash\colon\ldotp p.\lor.p.q:\supset.p &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*4·45.</b> \(\vdash:p.\equiv .p.p\lor q \quad[\text{*3·26.*2·2}]\)</p> + +<p>The following formulae are due to De Morgan, or rather, are the +propositional analogues of formulae given by De Morgan for classes. The +first of them, it will be observed, merely embodies our definition of +the logical product.</p> + +<p class="nind"> +<b>*4·5.</b> \(\vdash: p.q.\equiv .{\sim}({\sim}p\lor{\sim}q) \quad[\text{*4·2. (*3·01)}]\)</p> + +<p class="nind"> +<b>*4·51.</b> \(\vdash: {\sim}(p.q).\equiv .{\sim}p\lor{\sim}q \quad[\text{*4·5·12}]\)</p> + +<p class="nind"> +<b>*4·52.</b> \(\vdash: p.{\sim}q.\equiv .{\sim}({\sim}p\lor q) \quad\left[\text{*4·5}\, \frac{{\sim}q}{q}.\text{*4·13}\right]\)</p> + +<p class="nind"> +<b>*4·53.</b> \(\vdash: {\sim}(p.{\sim}q).\equiv .{\sim}p\lor q \quad[\text{*4·52·12}]\)</p> + +<p class="nind"> +<b>*4·54.</b> \(\vdash: {\sim}p.q.\equiv .{\sim}(p\lor{\sim}q) \quad\left[\text{*4·5}\, \frac{{\sim}p}{p}.\text{*4·13}\right]\)</p> + +<p class="nind"> +<b>*4·55.</b> \(\vdash: {\sim}({\sim}p.q).\equiv .p\lor{\sim}q \quad[\text{*4·54·12}]\)</p> + +<p class="nind"> +<b>*4·56.</b> \(\vdash: {\sim}p.{\sim}q.\equiv .{\sim}(p\lor q) \quad\left[\text{*4·54} \frac{{\sim}q}{q}.\text{*4·13}\right]\)</p> + +<p class="nind"> +<b>*4·57.</b> \(\vdash: {\sim}({\sim}p.{\sim}q).\equiv .p\lor q \quad[\text{*4·56·12}]\)</p> + +<p><span class="pagenum" id="Page_126">[Pg 126]</span></p> + +<p>The following formulae are obtained immediately from the above. They +are important as showing how to transform implications into sums or +into denials of products, and vice versa. It will be observed that the +first of them merely embodies the definition <a href="#*1·01">*1·01</a>.</p> + +<p class="nind"> +<b>*4·6.</b> \(\vdash: p\supset q.\equiv.{\sim}p\lor q \quad[\text{*4·2.(*1·01)}]\)</p> + +<p class="nind"> +<b>*4·61.</b> \(\vdash: {\sim}(p\supset q).\equiv.p.{\sim}q \quad[\text{*4·6·11·52}]\)</p> + +<p class="nind"> +<b>*4·62.</b> \(\vdash: p\supset {\sim}q.\equiv.{\sim}p\lor {\sim}q \quad\left[\text{*4·6}\, \frac{{\sim}q}{q}\right]\)</p> + +<p class="nind"> +<b><a id="*4·63">*4·63</a>.</b> \(\vdash: {\sim}(p\supset {\sim}q).\equiv.p.q \quad[\text{*4·62·11·5}]\)</p> + +<p class="nind"> +<b>*4·64.</b> \(\vdash: {\sim}p\supset q.\equiv.p\lor q \quad[\text{*2·53·54}]\)</p> + +<p class="nind"> +<b>*4·65.</b> \(\vdash: {\sim}({\sim}p\supset q).\equiv.{\sim}p.{\sim}q \quad[\text{*4·64·11·56}]\)</p> + +<p class="nind"> +<b>*4·66.</b> \(\vdash: {\sim}p\supset {\sim}q.\equiv.p\lor {\sim}q \quad\left[\text{*4·64}\, \frac{{\sim}q}{q}\right]\)</p> + +<p class="nind"> +<b>*4·67.</b> \(\vdash:{\sim}({\sim}p\supset {\sim}q).\equiv.{\sim}p.q \quad[\text{*4·66·11·54}]\)</p> + +<p class="nind"> +<b>*4·7.</b> \(\vdash\colon\ldotp p\supset q.\equiv:p.\supset .p.q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin {array}{l} +\vdash.\text{*3·27.Syll}. &\supset \vdash\colon\ldotp p.\supset .p.q:\supset .p\supset q &\qquad \text{(1)}\\ +\vdash.\text{Comp} &\supset \vdash\colon\ldotp p\supset p.p\supset q.\supset :p.\supset .p.q\colon\ldotp \\ +[\text{Exp}] &\supset \vdash\colon\colon p\supset p.\supset \colon\ldotp p\supset q.\supset :p.\supset .p.q\colon\colon \\ +[\text{Id}] &\supset \vdash\colon\ldotp p\supset q.\supset :p.\supset .p.q &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*4·71.</b> \(\vdash\colon\ldotp p\supset q.\equiv:p.\equiv.p.q\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*3·21}. &\supset \vdash\colon\colon p.q.\supset .p:\supset \colon\ldotp p.\supset .p.q:\supset :p.\equiv.p.q\colon\colon\\ +[\text{*3·26}] &\supset \vdash\colon\ldotp p.\supset .p.q:\supset :p.\equiv.p.q &\qquad \text{(1)}\\ +\vdash.\text{*3·26}. &\supset \vdash\colon\ldotp p.\equiv.p.q:\supset :p.\supset .p.q &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset \vdash\colon\ldotp p.\supset .p.q:\equiv:p.\equiv.p.q &\qquad \text{(3)}\\ +\vdash.\text{(3).*4·7·22}.&\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is constantly used. It enables us to transform +every implication into an equivalence, which is an advantage if we +wish to assimilate symbolic logic as far as possible to ordinary +algebra. But when symbolic logic is regarded as an instrument of proof, +we need implications, and it is usually inconvenient to substitute +equivalences. Similar remarks apply to the following proposition.</p> + +<p class="nind"> +<b>*4·72.</b> \(\vdash\colon\ldotp p\supset q.\equiv:q.\equiv.p\lor q\)</p> + +<p><i>Dem.</i>\[ +\begin {array}{l} +\vdash.\text{*4·1}.\supset \vdash\colon\ldotp p\supset q.&\equiv:{\sim}q\supset {\sim}p:\\ +\left[\text{*4·71}\, \frac{{\sim}q,\,{\sim}p}{p,\,q}\right] &\equiv:{\sim}q.\equiv.{\sim}q.{\sim}p:\\ +[\text{*4·12}] &\equiv:q.\equiv.{\sim}({\sim}q.{\sim}p):\\ +[\text{*4·57}] &\equiv:q.\equiv.q\lor p:\\ +[\text{*4·31}] &\equiv:q.\equiv.p\lor q\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_127">[Pg 127]</span></p> + +<p class="nind"> +<b>*4·73.</b> \(\vdash\colon\ldotp q.\supset :p.\equiv.p.q \quad[\text{Simp.*4·71}]\)</p> + +<p>This proposition is very useful, since it shows that a true factor may +be omitted from a product without altering its truth or falsehood, just +as a true hypothesis may be omitted from an implication.</p> + +<p class="nind"> +<b>*4·74.</b> \(\vdash\colon\ldotp {\sim}p.\supset :q.\equiv.p\lor q \quad[\text{*2·21.*4·72}]\)</p> + +<p class="nind"> +<b>*4·76.</b> \(\vdash\colon\ldotp p\supset q.p\supset r.\equiv:p.\supset .q.r \quad\left[\text{*4·41}\, \frac{{\sim}p}{p}.\text{(*1·01)}\right]\)</p> + +<p class="nind"> +<b><a id="*4·77">*4·77</a>.</b> \(\vdash\colon\ldotp q\supset p.r\supset p.\equiv:q\lor r.\supset .p \quad[\text{*3·44. Add. *2·2}]\)</p> + +<p class="nind"> +<b><a id="*4·78">*4·78</a>.</b> \(\vdash\colon\ldotp p\supset q.\lor .p\supset r:\equiv:p.\supset .q\lor r\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*4·2.(*1·01)}.\supset \vdash\colon\ldotp p\supset q.\lor .p\supset r:&\equiv:{\sim}p\lor q.\lor .{\sim}p\lor r:\\ +[\text{*4·33}] &\equiv.{\sim}p.\lor .q\lor {\sim}p\lor r:\\ +[\text{*4·31·37}] &\equiv:{\sim}p.\lor .{\sim}p\lor q\lor r:\\ +[\text{*4·33}] &\equiv:{\sim}p\lor {\sim}p.\lor .q\lor r:\\ +[\text{*4·25·37}] &\equiv:{\sim}p.\lor .q\lor r:\\ +[\text{*4·2.(*1·01)}] &\equiv:p.\supset .q\lor r\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*4·79">*4·79</a>.</b> \(\vdash\colon\ldotp q\supset p.\lor .r\supset p:\equiv:q.r.\supset .p\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*4·1·39}.\supset \vdash\colon\ldotp q\supset p.\lor .r\supset p:&\equiv:{\sim}p\supset {\sim}q.\lor .{\sim}p\supset {\sim}r:\\ +[\text{*4·78}] &\equiv:{\sim}p.\supset .{\sim}q\lor {\sim}r:\\ +[\text{*2·15}] &\equiv:{\sim}({\sim}q\lor {\sim}r).\supset .p:\\ +[\text{*4·2.(*3·01)}] &\equiv:q.r.\supset .p\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><i>Note.</i> The analogues, for classes, of *4·78·79 are false. Take, +<i>e.g.</i> <a href="#*4·78">*4·78</a>, and put \(p\) = English people, \(q\) = men, \(r\) = +women. Then \(p\) is contained in \(q\) or \(r\), but is not contained +in \(q\) and is not contained in \(r\).</p> + +<p class="nind"> +<b>*4·8.</b> \(\vdash:p\supset {\sim}p.\equiv.{\sim}p \quad[\text{*2·01. Simp}]\)</p> + +<p class="nind"> +<b>*4·81.</b> \(\vdash:{\sim}p\supset p.\equiv.p \quad[\text{*2·18. Simp}]\)</p> + +<p class="nind"> +<b>*4·82.</b> \(\vdash:p\supset q.p\supset {\sim}q.\equiv.{\sim}p \quad[\text{*2·65. Imp.*2·21. Comp}]\)</p> + +<p class="nind"> +<b>*4·83.</b> \(\vdash:p\supset q.{\sim}p\supset q.\equiv.q \quad[\text{*2·61.Imp. Simp. Comp}]\)</p> + +<p><i>Note.</i> *4·82·83 may also be obtained from <a href="#*4·43">*4·43</a>, of which they +are virtually other forms.</p> + +<p class="nind"> +<b>*4·84.</b> \(\vdash\colon\ldotp p\equiv q.\supset :p\supset r.\equiv.q\supset r \quad[\text{*2·06.*3·47}]\)</p> + +<p class="nind"> +<b>*4·85.</b> \(\vdash\colon\ldotp p\equiv q.\supset :r\supset p.\equiv.r\supset q \quad[\text{*2·05.*3·47}]\)</p> + +<p class="nind"> +<b>*4·86.</b> \(\vdash\colon\ldotp p\equiv q.\supset :p\equiv r.\equiv.q\equiv r \quad[\text{*4·21·22}]\)</p> + +<p class="nind"> +<b><a id="*4·87">*4·87</a>.</b> \[\begin{align}\vdash\colon\ldotp p.q.\supset .r:\equiv:p.\supset .q\supset &r:\equiv:q.\supset .p\supset r:\equiv:q.p.\supset .r\\ +&\quad[\text{Exp. Comm. Imp}]\end{align}\]</p> + +<p><a href="#*4·87">*4·87</a> embodies in one proposition the principles of exportation and +importation and the commutative principle.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_128">[Pg 128]</span></p> +<h2 class="nobreak" id="*5">*5. MISCELLANEOUS PROPOSITIONS.</h2> +</div> + + +<p><i>Summary of</i> *5.</p> + +<p>The present number consists chiefly of propositions of two sorts: +(1) those which will be required as lemmas in one or more subsequent +proofs, (2) those which are on their own account illustrative, or would +be important in other developments than those that we wish to make. +A few of the propositions of this number, however, will be used very +frequently. These are:</p> + +<p class="nind"> +<b>*5·1.</b> \(\vdash : p . q .\supset. p \equiv q\)</p> + +<p><i>I.e.</i> two propositions are equivalent if they are both true. (The +statement that two propositions are equivalent if they are both false +is <a href="#*5·21">*5·21</a>.)</p> + +<p class="nind"> +<b>*5·32.</b> \(\vdash \colon\ldotp p .\supset. q \equiv r :\equiv: p . q .\equiv. p.r\)</p> + +<p><i>I.e.</i> to say that, on the hypothesis \(p\), \(q\) and \(r\) are +equivalent, is equivalent to saying that the joint assertion of \(p\) +and \(q\) is equivalent to the joint assertion of \(p\) and \(r\). This +is a very useful rule in inference.</p> + +<p class="nind"> +<b>*5·6.</b> \(\vdash \colon\ldotp p . {\sim}q .\supset. r :\equiv: p.\supset.q \lor r\)</p> + +<p><i>I.e.</i> "\(p\) and not-\(q\) imply \(r\)" is equivalent to "\(p\) +implies \(q\) or \(r\)."</p> + +<p>Among propositions never subsequently referred to, but inserted for +their intrinsic interest, are the following: *5·11·12·13·14, which +state that, given any two propositions \(p\), \(q\), either \(p\) or +\({\sim}p\) must imply \(q\), and \(p\) must imply either \(q\) or +not-\(q\), and either \(p\) implies \(q\) or \(q\) implies \(p\); and +given any third proposition \(r\), either \(p\) implies \(q\) or \(q\) +implies \(r\)<a id="FNanchor_49" href="#Footnote_49" class="fnanchor">[49]</a>.</p> + +<p>Other propositions not subsequently referred to are *5·22·23·24; in +these it is shown that two propositions are not equivalent when, and +only when, one is true and the other false, and that two propositions +are equivalent when, and only when, both are true or both false. +It follows (<a href="#*5·24">*5·24</a>) that the negation of "\(p . q .\lor. {\sim}p . +{\sim}q\)" is equivalent to "\(p . {\sim}q .\lor. q .{\sim}p\)." +*5·54·55 state that both the product and the sum of \(p\) and \(q\) are +equivalent, respectively, either to \(p\) or to \(q\).</p> + +<p>The proofs of the following propositions are all easy, and we shall +therefore often merely indicate the propositions used in the proofs.</p> + +<p><span class="pagenum" id="Page_129">[Pg 129]</span></p> + +<hr class="tb"> + +<p class="nind"> +<b>*5·1.</b> \(\vdash:p.q.\supset .p\equiv q \quad[\text{*3·4·22}]\)</p> + +<p class="nind"> +<b>*5·11.</b> \(\vdash:p\supset q.\lor .{\sim}p\supset q \quad[\text{*2·5·54}]\)</p> + +<p class="nind"> +<b>*5·12.</b> \(\vdash:p\supset q.\lor .p\supset {\sim}q \quad[\text{*2·51·54}]\)</p> + +<p class="nind"> +<b>*5·13.</b> \(\vdash:p\supset q.\lor .q\supset p \quad[\text{*2·521}]\)</p> + +<p class="nind"> +<b>*5·14.</b> \(\vdash:p\supset q.\lor .q\supset r \quad[\text{Simp. Transp.*2·21}]\)</p> + +<p class="nind"> +<b>*5·15.</b> \(\vdash:p\equiv q.\lor .p\equiv {\sim}q\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*4·61}.&\supset \vdash:{\sim}(p\supset q).\supset .p.{\sim}q.\\ +[\text{*5·1}] &\qquad\qquad\supset .p\equiv {\sim}q:\\ +[\text{*2·54}] &\supset \vdash:p\supset q.\lor .p\equiv {\sim}q &\qquad \text{(1)}\\ +\vdash.\text{*4·61}.&\supset \vdash:{\sim}(q\supset p).\supset .q.{\sim}p.\\ +[\text{*5·1}] &\qquad\qquad\supset .q\equiv {\sim}p.\\ +[\text{*4·12}] &\qquad\qquad\supset .p\equiv {\sim}q:\\ +[\text{*2·54}] &\supset \vdash:q\supset p.\lor .p\equiv {\sim}q &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·41}.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*5·16.</b> \(\vdash.{\sim}(p\equiv q.p\equiv {\sim}q)\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*3·26}.\supset \vdash:p\equiv q.p\supset {\sim}q.\supset .p\supset q.p\supset {\sim}q.\\ +[\text{*4·82}] \supset .{\sim}p &\qquad \text{(1)}\\ +\vdash.\text{*3·27}.\supset \vdash:p\equiv q.p\supset {\sim}q.\supset .q\supset p.p\supset {\sim}q.\\ +[\text{Syll}] \supset .q\supset {\sim}q.\\ +[\text{Abs}] \supset .{\sim}q &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Comp}.\supset \vdash:p\equiv q.p\supset {\sim}q.\supset .{\sim}p.{\sim}q.\\ +\left[\text{*4·65} \frac{q,\,p}{p,\,q}\right] \supset .{\sim}({\sim}q\supset p) &\qquad \text{(3)}\\ +\vdash.\text{(3).Exp}.\supset \vdash\colon\ldotp p\equiv q.\supset :p\supset {\sim}q.\supset .{\sim}({\sim}q\supset p):\\ +[\text{Id.(*1·01)}] \supset :{\sim}(p\supset {\sim}q).\lor .{\sim}({\sim}q\supset p):\\ +[\text{*4·51.(*4·01)}] \supset :{\sim}(p\equiv {\sim}q)\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*5·17.</b> \(\vdash:p\lor q.{\sim}(p.q).\equiv .p\equiv {\sim}q\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*4·64·21}. &\supset \vdash:p\lor q.\equiv .{\sim}q\supset p &\qquad \text{(1)}\\ +\vdash.\text{*4·63.Transp}. &\supset \vdash:{\sim}(p.q).\equiv .p\supset {\sim}q &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·38·21}.&\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_130">[Pg 130]</span></p> + +<p class="nind"> +<b>*5·18.</b> \(\vdash:p\equiv q.\equiv .{\sim}(p\equiv {\sim}q) \quad\left[\text{*5·15·16.*5·17}\, \frac{p\equiv q,\,p\equiv {\sim}q}{p,\,\,\,q}\right]\)</p> + +<p class="nind"> +<b>*5·19.</b> \(\vdash.{\sim}(p\equiv {\sim}p) \quad\left[\text{*5·18}\, \frac{p}{q}.\text{*4·2}\right]\)</p> + +<p class="nind"> +<b><a id="*5·21">*5·21</a>.</b> \(\vdash:{\sim}p.{\sim}q.\supset .p\equiv q \quad[\text{*5·1.*4·11}]\)</p> + +<p class="nind"> +<b>*5·22.</b> \(\vdash\colon\ldotp {\sim}(p\equiv q).\equiv :p.{\sim}q.\lor .q.{\sim}p \quad[\text{*4·61·51·39}]\)</p> + +<p class="nind"> +<b>*5·23.</b> \(\vdash\colon\ldotp p\equiv q.\equiv :p.q.\lor .{\sim}p.{\sim}q \quad\left[\text{*5·18.*5·22}\, \frac{{\sim}q}{q}.\text{*4·13·36}\right]\)</p> + +<p class="nind"> +<b><a id="*5·24">*5·24</a>.</b> \(\vdash\colon\ldotp {\sim}(p.q.\lor .{\sim}p.{\sim}q).\equiv :p.{\sim}q.\lor .q.{\sim}p \quad[\text{*5·22·23}]\)</p> + +<p class="nind"> +<b><a id="*5·25">*5·25</a>.</b> \(\vdash\colon\ldotp p\lor q.\equiv :p\supset q.\supset .q \quad[\text{*2·62·68}]\)</p> + +<p>From <a href="#*5·25">*5·25</a> it appears that we might have taken implication, instead of +disjunction, as a primitive idea, and have defined "\(p\lor q\)" as +meaning "\(p\supset q.\supset .q\)." This course, however, requires +more primitive propositions than are required by the method we have +adopted.</p> + +<p class="nind"> +<b>*5·3.</b> \(\vdash\colon\ldotp p.q.\supset .r:\equiv :p.q.\supset .p.r \quad[\text{Simp. Comp. Syll}]\)</p> + +<p class="nind"> +<b>*5·31.</b> \(\vdash\colon\ldotp r.p\supset q:\supset :p.\supset .q.r \quad[\text{Simp. Comp}]\)</p> + +<p class="nind"> +<b>*5·32.</b> \(\vdash\colon\ldotp p.\supset .q\equiv r:\equiv :p.q.\equiv .p.r \quad[\text{*4·76.*3·3·31.*5·3}]\)</p> + +<p>This proposition is constantly required in subsequent proofs.</p> + +<p class="nind"> +<b>*5·33.</b> \(\vdash\colon\ldotp p.q\supset r.\equiv :p:p.q.\supset .r \quad[\text{*4·73·84.*5·32}]\)</p> + +<p class="nind"> +<b>*5·35.</b> \(\vdash\colon\ldotp p\supset q.p\supset r.\supset :p.\supset .q\equiv r \quad[\text{Comp. *5·1}]\)</p> + +<p class="nind"> +<b>*5·36.</b> \(\vdash:p.p\equiv q.\equiv .q.p\equiv q \quad[\text{Ass. *4·38}]\)</p> + +<p class="nind"> +<b>*5·4.</b> \(\vdash\colon\ldotp p.\supset .p\supset q:\equiv .p\supset q \quad[\text{Simp. *2·43}]\)</p> + +<p class="nind"> +<b>*5·41.</b> \(\vdash\colon\ldotp p\supset q.\supset .p\supset r:\equiv :p.\supset .q\supset r \quad[\text{*2·77·86}]\)</p> + +<p class="nind"> +<b>*5·42.</b> \(\vdash\colon\colon p.\supset .q\supset r:\equiv \colon\ldotp p.\supset :q.\supset .p.r \quad[\text{*5·3.*4·87}]\)</p> + +<p class="nind"> +<b>*5·44.</b> \(\vdash\colon\colon p\supset q.\supset \colon\ldotp p\supset r.\equiv :p.\supset .q.r \quad[\text{*4·76.*5·3·32}]\)</p> + +<p class="nind"> +<b>*5·5.</b> \(\vdash\colon\ldotp p.\supset :p\supset q.\equiv .q \quad[\text{Ass. Exp. Simp}]\)</p> + +<p class="nind"> +<b>*5·501.</b> \(\vdash\colon\ldotp p.\supset :q.\equiv .p\equiv q \quad[\text{*5·1.Exp. Ass}]\)</p> + +<p class="nind"> +<b>*5·53.</b> \(\vdash\colon\ldotp p\lor q\lor r.\supset .s:\equiv :p\supset s.q\supset s.r\supset s \quad[\text{*4·77}]\)</p> + +<p class="nind"> +<b>*5·54.</b> \(\vdash\colon\ldotp p.q.\equiv .p:\lor :p.q.\equiv .q \quad[\text{*4·73.*4·44.Transp. *5·1}]\)</p> + +<p class="nind"> +<b>*5·55.</b> \(\vdash\colon\ldotp p\lor q.\equiv .p:\lor :p\lor q.\equiv .q \quad[\text{*1·3.*5·1.*4·74}]\)</p> + +<p class="nind"> +<b>*5·6.</b> \(\vdash\colon\ldotp p.{\sim}q.\supset .r:\equiv :p.\supset .q\lor r \quad\left[\text{*4·87}\, \frac{{\sim}q}{q}.\text{*4·64·85}\right]\)</p> + +<p class="nind"> +<b>*5·61.</b> \(\vdash:p\lor q.{\sim}q.\equiv .p.{\sim}q \quad[\text{*4·74.*5·32}]\)</p> + +<p class="nind"> +<b>*5·62.</b> \(\vdash\colon\ldotp p.q.\lor .{\sim}q:\equiv .p\lor {\sim}q \quad\left[\text{*4·7}\, \frac{q,\,p}{p,\,q}\right]\)</p> + +<p><span class="pagenum" id="Page_131">[Pg 131]</span></p> + +<p class="nind"> +<b>*5·63.</b> \(\vdash\colon\ldotp p\lor q.\equiv :p.\lor .{\sim}p.q \quad\left[\text{*5·62}\, \frac{{\sim}p,\,q}{q,\,p}\right]\)</p> + +<p class="nind"> +<b>*5·7.</b> \(\vdash\colon\ldotp p\lor r.\equiv .q\lor r:\equiv :r.\lor .p\equiv q \quad[\text{*4·74.*1·3.*5·1.*4·37}]\)</p> + +<p class="nind"> +<b>*5·71.</b> \(\vdash\colon\ldotp q\supset {\sim}r.\supset :p\lor q.r.\equiv .p.r\)</p> + +<p>In the following proof, as always henceforth, "\(\text{Hp}\)" means the +hypothesis of the proposition to be proved.</p> + +<p><i>Dem.</i>\[ +\begin {array}{l} +\vdash.\text{*4·4}. &\supset \vdash\colon\ldotp p\lor q.r.\equiv :p.r.\lor .q.r &\qquad \text{(1)}\\ +\vdash.\text{*4·62·51}.&\supset \vdash\colon\colon \text{Hp}.\supset \colon\ldotp {\sim}(q.r)\colon\ldotp \\ +[\text{*4·74}] &\supset \colon\ldotp p.r.\lor .q.r:\equiv :p.r &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·22}.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*5·74.</b> \(\vdash\colon\ldotp p.\supset .q\equiv r:\equiv :p\supset q.\equiv .p\supset r\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*5·41}.\supset \vdash\colon\colon p\supset q.\supset .&p\supset r:\equiv :p.\supset .q\supset r\colon\ldotp \\ +& p\supset r.\supset .p\supset q:\equiv :p.\supset .r\supset q &\qquad \text{(1)}\\ +\vdash.\text{(1).*4·38}.\supset \vdash\colon\colon p\supset q.&\equiv .p\supset r.\equiv \colon\ldotp p.\supset .q\supset r:p.\supset .r\supset q\colon\ldotp\\ +[\text{*4·76}] &\equiv \colon\ldotp p.\supset .q\equiv r\colon\colon \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*5·75.</b> \(\vdash\colon\ldotp r\supset {\sim}q:p.\equiv .q\lor r:\supset :p.{\sim}q.\equiv .r\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*5·6}. &\supset \vdash\colon\ldotp \text{Hp}.\supset :p.{\sim}q.\supset .r &\qquad \text{(1)}\\ +\vdash.\text{*3·27}.&\supset \vdash\colon\ldotp \text{Hp}.\supset :q\lor r.\supset .p:\\ +[\text{*4·77}] &\supset :r\supset p &\qquad \text{(2)}\\ +\vdash.\text{*3·26}.&\supset \vdash\colon\ldotp \text{Hp}.\supset :r\supset {\sim}q &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).Comp}.&\supset \vdash\colon\ldotp \text{Hp}.\supset :r\supset p.r\supset {\sim}q:\\ +[\text{Comp}] &\supset :r.\supset .p.{\sim}q &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).Comp}.&\supset \vdash\colon\ldotp \text{Hp}.\supset :p.{\sim}q.\equiv .r\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_49" href="#FNanchor_49" class="label">[49]</a> +Cf. Schröder, <i>Vorlesungen über Algebra der Logik</i>, +Zweiter Band (Leipzig, 1891), pp. 270-271, where the apparent oddity of +the above proposition is explained.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_132">[Pg 132]</span></p> +<h2 class="nobreak" id="SECTION_B_a">SECTION B.<br> +THEORY OF APPARENT VARIABLES.</h2> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="*9">*9. EXTENSION OF THE THEORY OF DEDUCTION FROM LOWER TO +HIGHER TYPES OF PROPOSITIONS.</h2> +</div> + + +<p><i>Summary of</i> *9.</p> + +<p>In the present number, we introduce two new primitive ideas, which may +be expressed as "\({\phi}x\) is always<a id="FNanchor_50" href="#Footnote_50" class="fnanchor">[50]</a> true" and "\({\phi}x\) is +sometimes<a id="FNanchor_51" href="#Footnote_51" class="fnanchor">[51]</a> true," or, more correctly, as "\({\phi}x\) always" and +"\({\phi}x\) sometimes." When we assert "\({\phi}x\) always," we are +asserting all values of \(\phi\hat{x}\), where "\(\phi\hat{x}\)" means +the function itself, as opposed to an ambiguous value of the function +(cf. <a href="#Page_15">pp. 15</a>, <a href="#Page_42">42</a>); we are not asserting that \({\phi}x\) is true for +all values of \(x\), because, in accordance with the theory of types, +there are values of \(x\) for which "\({\phi}x\)" is meaningless; for +example, the function \(\phi\hat{x}\) itself must be such a value. We +shall denote "\({\phi}x\) always" by the notation +\[ +(x) . {\phi}x, +\] +where the "\((x)\)" will be followed by a sufficiently large number of +dots to cover the function of which "all values" are concerned. The +form in which such propositions most frequently occur is the "formal +implication," <i>i.e.</i> such a proposition as +\[ +(x): {\phi}x .\supset. {\psi}x\text{,} +\] +<i>i.e.</i> "\({\phi}x\) always implies \({\psi}x\)." This is the form +in which we express the universal affirmative "all objects having the +property \(\phi\) have the property \(\psi\)."</p> + +<p>We shall denote "\({\phi}x\) sometimes" by the notation +\[ +(\exists x). {\phi}x\text{.} +\] +Here "\(\exists\)" stands for "there exists," and the whole symbol may +be read "there exists an \(x\) such that \({\phi}x\)."</p> + +<p>In a proposition of either of the two forms \((x).{\phi}x\), (\(\exists x).{\phi}x\), +the \(x\) is called an <i>apparent variable</i>. +A proposition which contains no apparent variables is called +"elementary," and a function, all whose values are<span class="pagenum" id="Page_133">[Pg 133]</span> elementary +propositions, is called an elementary function. For reasons explained +in <a href="#CHAPTER_II">Chapter II</a> of the Introduction, it would seem that negation and +disjunction and their derivatives must have a different meaning when +applied to elementary propositions from that which they have when +applied to such propositions as \((x).\phi x\) or \((\exists x).\phi x\). +If \(\phi \hat{x}\) is an elementary function, we will in this +number call \((x).\phi x\) and \((\exists x).\phi x\) "first-order +propositions." Then in virtue of the fact that disjunction and +negation do not have the same meanings as applied to elementary or to +first-order propositions, it follows that, in asserting the primitive +propositions of <a href="#*1">*1</a>, we must either confine them, in their application, +to propositions of a single type, or we must regard them as the +simultaneous assertion of a number of different primitive propositions, +corresponding to the different meanings of "disjunction" and +"negation." Likewise in regard to the primitive ideas of disjunction +and negation, we must either, in the primitive propositions of <a href="#*1">*1</a>, +confine them to disjunctions and negations of elementary propositions, +or we must regard them as really each multiple, so that in regard +to each type of propositions we shall need a new primitive idea of +negation and a new primitive idea of disjunction. In the present +number, we shall show how, when the primitive ideas of negation +and disjunction are restricted to elementary propositions, and the +\(p\), \(q\), \(r\) of <a href="#*1">*1</a>—<a href="#*5">*5</a> are therefore necessarily elementary +propositions, it is possible to obtain definitions of the negation and +disjunction of first-order propositions, and proofs of the analogues, +for first-order propositions, of the primitive propositions <a href="#*1·2">*1·2</a>—<a href="#*1·6">·6</a>. +(<a href="#*1·1">*1·1</a> and <a href="#*1·11">*1·11</a> have to be assumed afresh for first-order propositions, +and the analogues of *1·7·71·72 require a fresh treatment.) It follows +that the analogues of the propositions of <a href="#*2">*2</a>—<a href="#*5">*5</a> follow by merely +repeating previous proofs. It follows also that the theory of deduction +can be extended from first-order propositions to such as contain two +apparent variables, by merely repeating the process which extends +the theory of deduction from elementary to first-order propositions. +Thus by merely repeating the process set forth in the present +number, propositions of any order can be reached. Hence negation and +disjunction may be treated in practice as if there were no difference +in these ideas as applied to different types; that is to say, when +"\({\sim}p\)" or "\(p \lor q\)" occurs, it is unnecessary in practice +to know what is the type of \(p\) or \(q\), since the properties +of negation and disjunction assumed in <a href="#*1">*1</a> (which are alone used in +proving other properties) can be asserted, without formal change, of +propositions of any order or, in the case of \(p \lor q\), of any two +orders. The limitation, in practice, to the treatment of negation or +disjunction as single ideas, the same in all types, would only arise +if we ever wished to assume that there is some one function of \(p\) +whose value is always \({\sim}p\), whatever may be the order of \(p\), +or that there is some one function of \(p\) and \(q\) whose value is +always \(p \lor q\), whatever may be the orders of \(p\) and \(q\). +Such an assumption is not involved so long as \(p\) (and \(q\)) remain +<i>real</i> variables,<span class="pagenum" id="Page_134">[Pg 134]</span> since, in that case, there is no need to give +the same meaning to negation and disjunction for different values +of \(p\) (and \(q\)), when these different values are of different +types. But if \(p\) (or \(q\)) is going to be turned into an apparent +variable, then, since our two primitive ideas \((x).\phi x\) and +\((\exists x).\phi x\) both demand some definite function \(\phi\), +and restrict the apparent variable to possible arguments for \(\phi\), +it follows that negation and disjunction must, wherever they occur in +the expression in which \(p\) (or \(q\)) is an apparent variable, be +restricted to the kind of negation or disjunction appropriate to a +given type or pair of types. Thus, to take an instance, if we assert +the law of excluded middle in the form +\[ +\unicode{x201c}\vdash.p \lor {\sim}p\unicode{x201d} +\] +there is no need to place any restriction upon \(p\): we may give +to \(p\) a value of any order, and then give to the negation and +disjunction involved those meanings which are appropriate to that +order. But if we assert +\[ +\unicode{x201c}\vdash.(p).p \lor {\sim}p\unicode{x201d} +\] +it is necessary, if our symbol is to be significant, that "\(p \lor{\sim}p\)" +should be the value, for the argument \(p\), of a function +\(\phi p\); and this is only possible if the negation and disjunction +involved have meanings fixed in advance, and if, therefore, \(p\) is +limited to one type. Thus the assertion of the law of excluded middle +in the form involving a real variable is more general than in the form +involving an apparent variable. Similar remarks apply generally where +the variable is the argument to a typically ambiguous function.</p> + +<p>In what follows the single letters p and q will represent +<i>elementary</i> propositions, and so will "\(\phi x\)," "\(\psi x\)," +etc. We shall show how, assuming the primitive ideas and propositions +of <a href="#*1">*1</a> as applied to elementary propositions, we can define and prove +analogous ideas and propositions as applied to propositions of the +forms \((x).\phi x\) and \((\exists x).\phi x\). By mere repetition of +the analogous process, it will then follow that analogous ideas and +propositions can be defined and proved for propositions of any order; +whence, further, it follows that, in all that concerns disjunction +and negation, so long as propositions do not appear as apparent +variables, we may wholly ignore the distinction between different +types of propositions and between different meanings of negation and +disjunction. Since we never have occasion, in practice, to consider +propositions as apparent variables, it follows that the hierarchy of +propositions (as opposed to the hierarchy of functions) will never be +relevant in practice after the present number.</p> + +<p>The purpose and interest of the present number are purely +philosophical, namely to show how, by means of certain primitive +propositions, we can deduce the theory of deduction for propositions +containing apparent variables from the theory of deduction for +elementary propositions. From the purely technical point of view, +the distinction between elementary and other propositions may be +ignored, so long as propositions do not appear as apparent variables; +we may then regard the primitive propositions of <a href="#*1">*1</a> as applying<span class="pagenum" id="Page_135">[Pg 135]</span> to +propositions of any type, and proceed as in <a href="#*10">*10</a>, where the purely +technical development is resumed.</p> + +<p>It should be observed that although, in the present number, we prove +that the analogues of the primitive propositions of <a href="#*1">*1</a>, if they hold +for propositions containing \(n\) apparent variables, also hold for +such as contain \(n+1\), yet we must not suppose that mathematical +induction may be used to infer that the analogues of the primitive +propositions of <a href="#*1">*1</a> hold for propositions containing any number of +apparent variables. Mathematical induction is a method of proof which +is not yet applicable, and is (as will appear) incapable of being used +freely until the theory of propositions containing apparent variables +has been established. What we are enabled to do, by means of the +propositions in the present number, is to prove our desired result for +any assigned number of apparent variables—say ten—by ten applications +of the same proof. Thus we can prove, concerning any assigned +proposition, that it obeys the analogues of the primitive propositions +of <a href="#*1">*1</a>, but we can only do this by proceeding step by step, not by any +such compendious method as mathematical induction would afford. The +fact that higher types can only be reached step by step is essential, +since to proceed otherwise we should need an apparent variable which +would wander from type to type, which would contradict the principle +upon which types are built up.</p> + +<hr class="tb"> + +<p><i>Definition of Negation.</i> We have first to define the negations of +(\(x) \ldotp \phi x\) and (\(\exists x) \ldotp \phi x\). We define +the negation of \((x)\ldotp \phi x\) as \((\exists x)\ldotp \sim \phi x\), +<i>i.e.</i> "it is not the case that \(\phi x\) is always true" +is to mean "it is the case that \(\text{not-}\phi x\) is sometimes +true." Similarly the negation of (\(\exists x) \ldotp \phi x\) is to be +defined as \((x)\ldotp \sim \phi x\). Thus we put</p> + +<p class="nind"> +<b>*9·01.</b> \(\sim \{(x)\ldotp \phi x\}\ldotp =\ldotp (\exists x)\ldotp \sim \phi x \quad \text{Df}\)</p> + +<p class="nind"> +<b>*9·02.</b> \(\sim \{(\exists x)\ldotp \phi x\}\ldotp =\ldotp (x)\ldotp \sim \phi x \quad \text{Df}\)</p> + +<p>To avoid brackets, we shall write \(\sim (x)\ldotp \phi x\) in place of +\(\sim \{(x)\ldotp \phi x\}\), and \(\sim (\exists x)\ldotp \phi x\) in +place of \(\sim \{(\exists x)\ldotp \phi x\}\). Thus:</p> + +<p class="nind"> +<b>*9·011.</b> \(\sim (x)\ldotp \phi x\ldotp =\ldotp \sim \{(x)\ldotp \phi x\} \quad \text{Df}\)</p> + +<p class="nind"> +<b>*9·021.</b> \(\sim (\exists x)\ldotp \phi x\ldotp =\ldotp \sim \{(\exists x)\ldotp \phi x\} \quad \text{Df}\)</p> + +<p><i>Definition of Disjunction.</i> To define disjunction when one or +both of the propositions concerned is of the first order, we have to +distinguish six cases, as follows:</p> + +<p class="nind"> +<b><a id="*9·03">*9·03</a>.</b> (\(x)\ldotp \phi x\ldotp \lor \ldotp p\colon=\ldotp (x)\ldotp \phi x \lor p \quad \text{Df}\)</p> + +<p class="nind"> +<b>*9·04.</b> \(p\ldotp \lor \ldotp (x)\ldotp \phi x\colon=\ldotp (x)\ldotp p \lor \phi x \quad \text{Df}\)</p> + +<p class="nind"> +<b><a id="*9·05">*9·05</a>.</b> (\(\exists x)\ldotp \phi x\ldotp \lor \ldotp p\colon=\ldotp (\exists x)\ldotp \phi x \lor p \quad \text{Df}\)</p> + +<p class="nind"> +<b>*9·06.</b> \(p\ldotp \lor \ldotp (\exists x)\ldotp \phi x\colon=\ldotp (\exists x)\ldotp p \lor \phi x \quad \text{Df}\)</p> + +<p class="nind"> +<b>*9·07.</b> (\(x)\ldotp \phi x\ldotp \lor \ldotp (\exists y)\ldotp \psi y \colon=\colon(x)\colon(\exists y)\ldotp \phi x \lor \psi y \quad \text{Df}\)</p> + +<p class="nind"> +<b>*9·08.</b> (\(\exists y)\ldotp \psi y\ldotp \lor \ldotp (x)\ldotp \phi x\colon=\colon(x)\colon(\exists y)\ldotp \psi y \lor \phi x \quad \text{Df}\)</p> + +<p><span class="pagenum" id="Page_136">[Pg 136]</span></p> + +<p>(The definitions *9·07·08 are to apply also when \(\phi\) and \(\psi\) +are not both elementary functions.)</p> + +<p>In virtue of these definitions, the true scope of an apparent +variable is always the whole of the asserted proposition in which +it occurs, even when, typographically, its scope appears to be only +part of the asserted proposition. Thus when (\(\exists x).{\phi}x\) +or \((x).{\phi}x\) <i>appears</i> as <i>part</i> of an asserted +proposition, it does not really occur, since the scope of the apparent +variable really extends to the whole asserted proposition. It will be +shown, however, that, so far as the theory of deduction is concerned, +(\(\exists x).{\phi}x\) and (\(x).{\phi}x\) behave like propositions +not containing apparent variables.</p> + +<p>The definitions of implication, the logical product, and equivalence +are to be transferred unchanged to (\(x).{\phi}x\) and (\(\exists +x).{\phi}x\).</p> + +<p>The above definitions can be repeated for successive types, and thus +reach propositions of any type.</p> + +<p><i>Primitive Propositions.</i> The primitive propositions required +are six in number, and may be divided into three sets of two. We have +first two propositions which effect the passage from elementary to +first-order propositions, namely</p> + +<p class="nind"> +<b><a id="*9·1">*9·1</a>.</b> \(\vdash: \phi x .\supset. (\exists z).\phi z \quad \text{Pp}\)</p> + +<p class="nind"> +<b><a id="*9·11">*9·11</a>.</b> \(\vdash: \phi x \lor \phi y .\supset. (\exists z) . \phi z \quad \text{Pp}\)</p> + +<p>Of these, the first states that, if \(\phi x\) is true, then there is a +value of \(\phi\hat{z}\) which is true; <i>i.e.</i> if we can find an +instance of a function which is true, then the function is "sometimes +true." (When we speak of a function as "sometimes" true, we do not +mean to assert that there is <i>more</i> than one argument for which +it is true, but only that there is <i>at least</i> one.) Practically, +the above primitive proposition gives the only method of proving +"existence-theorems": in order to prove such theorems, it is necessary +(and sufficient) to find some instance in which an object possesses +the property in question. If we were to assume what may be called +"existence-axioms," <i>i.e.</i> axioms stating (\(\exists z).\phi z\) +for some particular \(\phi\), these axioms would give other methods +of proving existence. Instances of such axioms are the multiplicative +axiom (<a href="#*88">*88</a>) and the axiom of infinity (defined in *120·03). But we have +not assumed any such axioms in the present work.</p> + +<p>The second of the above primitive propositions is only used once, +in proving (\(\exists z).\phi z .\lor. (\exists z).\phi z :\supset.(\exists z). \phi z\), +which is the analogue of <a href="#*1·2">*1·2</a> (namely \(p \lor p .\supset. p)\) +when \(p\) is replaced by (\(\exists z).\phi z\). The +effect of this primitive proposition is to emphasize the ambiguity of +the \(z\) required in order to secure (\(\exists z).\phi z\). We have, +of course, in virtue of <a href="#*9·1">*9·1</a>, +\[ +\phi x .\supset. (\exists z).\phi z ~ \text{and} ~ \phi y .\supset. (\exists z).\phi z. +\] +But if we try to infer from these that \(\phi x \lor \phi y .\supset.(\exists z).\phi z\), +we must use the<span class="pagenum" id="Page_137">[Pg 137]</span> proposition \(q \supset p.r \supset p .\supset. q \lor r \supset p\), +where \(p\) is (\(\exists z).\phi z\). Now it will be found, on +referring to <a href="#*4·77">*4·77</a> and the propositions used in its proof, that this +proposition depends upon <a href="#*1·2">*1·2</a>, <i>i.e.</i> \(p \lor p .\supset. p\). +Hence it cannot be used by us to prove (\(\exists x).\phi x .\lor. +(\exists x).\phi x :\supset. (\exists x).\phi x\), and thus we are +compelled to assume the primitive proposition <a href="#*9·11">*9·11</a>.</p> + +<p>We have next two propositions concerned with <i>inference</i> to +or from propositions containing apparent variables, as opposed to +implication. First, we have, for the new meaning of implication +resulting from the above definitions of negation and disjunction, the +analogue of <a href="#*1·1">*1·1</a>, namely</p> + +<p class="nind"> +<b>*9·12.</b> \(\text{What is implied by a true premiss is true.} \quad \text{Pp}\text{.}\)</p> + +<p>That is to say, given "\(\vdash .p\)" and "\(\vdash. p \supset q\)," +we may proceed to "\(\vdash. q\)," even when the propositions \(p\) +and \(q\) are not elementary. Also, as in <a href="#*1·11">*1·11</a>, we may proceed +from "\(\vdash. \phi x\)" and "\(\vdash. \phi x \supset \psi x\)" +to "\(\vdash.\psi x\)" where \(x\) is a real variable, and \(\phi\) +and \(\psi\) are not necessarily elementary functions. It is in this +latter form that the axiom is usually needed. It is to be assumed for +functions of several variables as well as for functions of one variable.</p> + +<p>We have next the primitive proposition which permits the passage +from a real to an apparent variable, namely "when \(\phi y\) may be +asserted, where \(y\) may be any possible argument, then (\(x).\phi x\) +may be asserted." In other words, when \(\phi y\) is true however +\(y\) may be chosen among possible arguments, then (\(x).\phi x\) is +true, <i>i.e.</i> all values of \(\phi\) are true. That is to say, +if we can assert a wholly ambiguous value \(\phi y\), that must be +because all values are true. We may express this primitive proposition +by the words: "What is true in <i>any</i> case, however the case may +be selected, is true in <i>all</i> cases." We cannot symbolise this +proposition, because if we put +\[ +\unicode{x201c}\vdash:\phi y .\supset. (x).\phi x \unicode{x201d} +\] +that means: "However \(y\) may be chosen, \(\phi y\) implies (\(x).\phi x\)," +which is in general false. What we mean is: "If \(\phi y\) is +true however \(y\) may be chosen, then (\(x).\phi x\) is true." But we +have not supplied a symbol for the mere <i>hypothesis</i> of what is +<i>asserted</i> in "\(\vdash.\phi y\)," where \(y\) is a real variable, +and it is not worth while to supply such a symbol, because it would be +very rarely required. If, for the moment, we use the symbol \([\phi y]\) +to express this hypothesis, then our primitive proposition is +\[ +\vdash:[\phi y] .\supset. (x).\phi x \quad \text{Pp}\text{.} +\] +In practice, this primitive proposition is only used for +<i>inference</i>, not for implication; that is to say, when we actually +have an assertion containing a real variable, it enables us to turn +this real variable into an apparent variable by placing it in brackets +immediately after the assertion-sign, followed by enough dots to reach +to the end of the assertion. This process will be called "turning +a real variable into an apparent variable." Thus we may assert our +primitive proposition, for technical use, in the form:</p> + +<p><span class="pagenum" id="Page_138">[Pg 138]</span></p> + +<p class="nind"> +<b><a id="*9·13">*9·13</a>.</b> In any assertion containing a real variable, this real +variable may be turned into an apparent variable of which all possible +values are asserted to satisfy the function in question. Pp.</p> + +<p>We have next two primitive propositions concerned with types. These +require some preliminary explanations.</p> + +<p><i>Primitive Idea: Individual.</i> We say that \(x\) is "individual" if +\(x\) is neither a proposition nor a function (cf. <a href="#Page_53">pp. 53</a>, <a href="#Page_54">54</a>).</p> + +<p class="nind"> +<b><a id="*9·131">*9·131</a>.</b> <i>Definition of "being of the same type."</i> The +following is a step-by-step definition, the definition for higher types +presupposing that for lower types. We say that \(u\) and \(v\) "are of +the same type" if (1) both are individuals, (2) both are elementary +functions taking arguments of the same type, (3) \(u\) is a function +and \(v\) is its negation, (4) \(u\) is \(\phi \hat{x}\) or \(\psi\hat{x}\) +and \(v\) is \(\phi \hat{x} \lor \psi \hat{x}\), where \(\phi \hat{x}\) +and \(\psi \hat{x}\) are elementary functions, (5) \(u\) +is \((y). \phi (\hat{x},y)\) and \(v\) is (\(z). \psi (\hat{x},z)\), +where \(\phi (\hat{x}, \hat{y})\), \(\psi (\hat{x}, \hat{y})\) are +of the same type, (6) both are elementary propositions, (7) \(u\) is +a proposition and \(v\) is \({\sim}u\), or (8) \(u\) is (\(x).\phi x\) +and \(v\) is (\(y).\psi y\), where \(\phi \hat{x}\) and \(\psi \hat{x}\) +are of the same type.</p> + +<p>Our primitive propositions are:</p> + +<p class="nind"> +<b><a id="*9·14">*9·14</a>.</b> If "\(\phi x\)" is significant, then if \(x\) is of the +same type as \(a\), "\(\phi a\)" is significant, and vice versa. Pp. +(Cf. note on *10·121, <a href="#Page_146">p. 146</a>.)</p> + +<p class="nind"> +<b><a id="*9·15">*9·15</a>.</b> If, for some \(a\), there is a proposition \(\phi a\), +then there is a function \(\phi \hat{x}\), and vice versa. Pp.</p> + +<p>It will be seen that, in virtue of the definitions,</p> + +<p>\[ +\begin{align} +&(x).\phi x .\supset. p &~\text{means}~ {\sim}(x).\phi x .\lor. p, ~\textit{i.e.}~ (\exists x).{\sim}\phi x .\lor. p,\\ +& &\textit{i.e.}~ (\exists x).{\sim}\phi x \lor p, ~\textit{i.e.}~ (\exists x).\phi x \supset p\\ +&(\exists x).\phi x .\supset. p &~\text{means}~ {\sim}(\exists x).\phi x .\lor. p, ~\textit{i.e.}~ (x).{\sim}\phi x .\lor. p,\\ +& &\textit{i.e.}~ (x).{\sim}\phi x \lor p, ~\textit{i.e.}~ (x).\phi x\supset p\\ +\end{align} +\] +In order to prove that (\(x).\phi x\) and (\(\exists x).\phi x\) obey +the same rules of deduction as \(\phi x\), we have to prove that +propositions of the forms (\(x).\phi x\) and (\(\exists x).\phi x\) +may replace one or more of the propositions \(p\), \(q\), \(r\) in +<a href="#*1·2">*1·2</a>—<a href="#*1·6">·6</a>. When this has been proved, the previous proofs of subsequent +propositions in <a href="#*2">*2</a>—<a href="#*5">*5</a> become applicable. These proofs are given below. +Certain other propositions, required in the proofs, are also proved.</p> + +<p class="nind"> +<b><a id="*9·2">*9·2</a>.</b> \(\vdash:(x). \phi x .\supset. \phi y\)</p> + +<p>The above proposition states the principle of deduction from the +general to the particular, <i>i.e.</i> "what holds in all cases, holds +in any one case."</p> + +<p><i>Dem.</i> +\[ +\begin{align} +&\vdash.\text{*2·1.} \supset \vdash. {\sim}\phi y \lor \phi y& \quad &(1)\\ +&\vdash.\text{*9·1.} \supset \vdash: {\sim} \phi y \lor \phi y & .\supset.(\exists x). {\sim}\phi x \lor \phi y \quad &(2)\\ +&\vdash.\text{(1).(2).*1·11.} \supset &\vdash.(\exists x). {\sim}\phi x \lor \phi y \quad &(3)\\ +&\text{[(3).(*9·05)]} &\vdash:(\exists x).{\sim}\phi x .\lor. \phi y \quad &(4)\\ +&\text{[(4).(*9·01.*1·01)]} &\vdash:(x). \phi x .\supset.\phi y \quad &\\ +\end{align} +\]</p> + +<p><span class="pagenum" id="Page_139">[Pg 139]</span></p> + +<p>In the second line of the above proof, "\({\sim}\phi y \lor \phi y\)" +is taken as the value, for the argument \(y\), of the function +"\({\sim}\phi x \lor \phi y\)," where \(x\) is the argument. A similar +method of using <a href="#*9·1">*9·1</a> is employed in most of the following proofs.</p> + +<p><a href="#*1·11">*1·11</a> is used, as in the third line of the above proof, in almost all +steps except such as are mere applications of definitions. Hence it +will not be further referred to, unless in cases where its employment +is obscure or specially important.</p> + +<p class="nind"> +<b><a id="*9·21">*9·21</a>.</b> \(\vdash \colon\ldotp (x) . \phi x \supset \psi x . \supset : (x) . \phi x . \supset . (x) . \psi x\)</p> + +<p><i>I.e.</i> if \(\phi x\) always implies \(\psi x\), then "\(\phi x\) +always" implies "\(\psi x\) always." The use of this proposition is +constant throughout the remainder of this work.</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash . \text{*2·08}. &\supset \vdash : \phi z \supset \psi z . \supset . \phi z \supset \psi z &\qquad \text{(1)}\\ +\vdash . \text{(1) . *9·1}. &\supset \vdash : (\exists y) : \phi z \supset \psi z . \supset . \phi y \supset \psi z &\qquad \text{(2)}\\ +\vdash .\text{(2). *9·1}. &\supset \vdash \colon\ldotp (\exists x) \colon\ldotp (\exists y) : \phi x \supset \psi x . \supset . \phi y \supset \psi z &\qquad \text{(3)}\\ +\vdash . \text{(3). *9·13}. &\supset \vdash \colon\colon (z) \colon\colon (\exists x) \colon\ldotp (\exists y) : \phi x \supset \psi x . \supset . \phi y \supset \psi z &\qquad \text{(4)}\\ +[\text{(4).(*9·06)}] &\vdash \colon\colon (z) \colon\colon (\exists x) \colon\ldotp \phi x \supset \psi x . \supset : (\exists y) . \phi y \supset \psi z &\qquad \text{(5)}\\ +[\text{(5).(*1·01.*9·08)}] &\vdash \colon\ldotp (\exists x) . {\sim}(\phi x \supset \psi x) : \lor : (z) : (\exists y) . {\sim}\phi y \lor \psi z &\qquad \text{(6)}\\ +[\text{(6).(*9·08)}] &\vdash \colon\ldotp (\exists x) . {\sim}(\phi x \supset \psi x) : \lor : (\exists y) . {\sim}\phi y . \lor . (z) . \psi z &\qquad \text{(7)}\\ +[\text{(7).(*1·01)}] & \vdash \colon\ldotp (x) . \phi x \supset \psi x . \supset : (y) . \phi y . \supset . (z) . \psi z +\end{array} +\]</p> + +<p>This is the proposition to be proved, since "(\(y) . \phi y\)" is the +same proposition as "(\(x) . \phi x\)," and "(\(z) . \psi z\)" is the +same proposition as "(\(x) . \psi x\)."</p> + +<p class="nind"> +<b><a id="*9·22">*9·22</a>.</b> \(\vdash \colon\ldotp (x) . \phi x \supset \psi x . \supset : (\exists x) . \phi x . \supset . (\exists x) . \psi x\)</p> + +<p><i>I.e.</i> if \(\phi x\) always implies \(\psi x\), then if \(\phi x\) +is sometimes true, so is \(\psi x\). This proposition, like <a href="#*9·21">*9·21</a>, is +constantly used in the sequel.</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash . \text{*2·08}.&\supset \vdash : \phi y \supset \psi y . \supset . \phi y \supset \psi y &\qquad \text{(1)}\\ +\vdash .\text{(1) . *9·1}. & \supset \vdash : (\exists z) : \phi y \supset \psi y . \supset . \phi y \supset \psi z &\qquad \text{(2)}\\ +\vdash . \text{(2) . *9·1}. &\supset \vdash \colon\ldotp (\exists x) \colon\ldotp (\exists z) : \phi x \supset \psi x . \supset . \phi y \supset \psi z &\qquad \text{(3)}\\ +\vdash .\text{(3) . *9·13}. & \supset \vdash \colon\colon (y) \colon\colon (\exists x) \colon\ldotp (\exists z) : \phi x \supset \psi x . \supset . \phi y \supset \psi z &\qquad \text{(4)}\\ +[\text{(4).(*9·06)}] &\vdash \colon\colon (y) \colon\colon (\exists x) \colon\ldotp \phi x \supset \psi x . \supset : (\exists z) . \phi y \supset \psi z &\qquad \text{(5)}\\ +[\text{(5).(*1·01.*9·08)}] & \vdash \colon\colon (\exists x) . {\sim}(\phi x \supset \psi x) : \lor : (y) : (\exists z) . \phi y \supset \psi z &\qquad \text{(6)}\\ +[\text{(6).(*1·01.*9·07)}] &\vdash \colon\colon (\exists x) . {\sim}(\phi x \supset \psi x) : \lor : (y) . {\sim}\phi y . \lor . (\exists z) . \psi z &\qquad \text{(7)}\\ +[\text{(7).(*1·01.*9·01·02)}] & \vdash \colon\ldotp (x) . \phi x \supset \psi x . \supset : (\exists y) . \phi y . \supset . (\exists z) . \psi z +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_140">[Pg 140]</span></p> + +<p>This is the proposition to be proved, because (\(\exists y).\phi y\) is +the same proposition as (\(\exists x).\phi x\), and (\(\exists z).\psi z\) +is the same proposition as (\(\exists x).\psi x\).</p> + +<p class="nind"> +<b>*9·23.</b> \(\vdash:(x).\phi x.\supset .(x).\phi x \quad[\text{Id.*9·13·21}]\)</p> + +<p class="nind"> +<b>*9·24.</b> \(\vdash:(\exists x).\phi x.\supset .(\exists x).\phi x \quad[\text{Id.*9·13·22}]\)</p> + +<p class="nind"> +<b><a id="*9·25">*9·25</a>.</b> \(\vdash\colon\ldotp (x).p\lor \phi x.\supset :p.\lor .(x).\phi x \quad[\text{*9·23.(*9·04)}]\)</p> + +<p>We are now in a position to prove the analogues of <a href="#*1·2">*1·2</a>—<a href="#*1·6">·6</a>, replacing +one of the letters \(p\), \(q\), \(r\) in those propositions by +(\(x).\phi x\) or (\(\exists x).\phi x\). The proofs are given below.</p> + +<p class="nind"> +<b>*9·3.</b> \(\vdash\colon\ldotp (x).\phi x.\lor .(x).\phi x:\supset .(x).\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·2}. &\supset \vdash.\phi x\lor \phi x.\supset .\phi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*9·1}. &\supset \vdash:(\exists y):\phi x\lor \phi y.\supset .\phi x &\qquad \text{(2)}\\ +\vdash.\text{(2).*9·13}. &\supset \vdash\colon\ldotp (x)\colon\ldotp (\exists y):\phi x\lor \phi y.\supset .\phi x &\qquad \text{(3)}\\ +[\text{(3).(*9·05·01·04)}] &\vdash\colon\ldotp (x)\colon\ldotp \phi x.\lor .(y).\phi y:\supset .\phi x &\qquad \text{(4)}\\ +\vdash.\text{(4).*9·21}. &\supset \vdash\colon\ldotp (x):\phi x.\lor .(y).\phi y:\supset .(x).\phi x &\qquad \text{(5)}\\ +[\text{(5).(*9·03)}] &\vdash\colon\ldotp (x).\phi x.\lor .(y).\phi y:\supset .(x).\phi x\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*9·31.</b> \(\vdash\colon\ldotp (\exists x).\phi x.\lor .(\exists x).\phi x:\supset .(\exists x).\phi x\)</p> + +<p>This is the only proposition which employs <a href="#*9·11">*9·11</a>.</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*9·11·13}. &\supset \vdash:(y):\phi x\lor \phi y.\supset .(\exists z).\phi z &\qquad \text{(1)}\\ +[\text{(1).(*9·03·02)}] &\vdash:(\exists y).\phi x\lor \phi y.\supset .(\exists z).\phi z &\qquad \text{(2)}\\ +\vdash.\text{(2).*9·13}.&\supset \vdash:(x):(\exists y).\phi x\lor \phi y.\supset .(\exists z).\phi z &\qquad \text{(3)}\\ +[\text{(3).(*9·03·02)}] &\vdash\colon\ldotp (\exists x):(\exists y).\phi x\lor \phi y:\supset .(\exists z).\phi z &\qquad \text{(4)}\\ +[\text{(4).(*9·05·06)}] &\vdash\colon\ldotp (\exists x).\phi x.\lor .(\exists y).\phi y:\supset .(\exists z).\phi z +\end{array} +\]</p> + +<p class="nind"> +<b>*9·32.</b> \(\vdash\colon\ldotp q.\supset :(x).\phi x.\lor .q\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·3}. &\supset \vdash\colon\ldotp q.\supset :\phi x.\lor .q &\qquad \text{(1)}\\ +\vdash.\text{(1).*9·13}.&\supset \vdash\colon\ldotp (x)\colon\ldotp q.\supset :\phi x.\lor .q\\ +[\text{*9·25}] &\supset \vdash\colon\ldotp q.\supset :(x):\phi x.\lor .q &\qquad \text{(2)}\\ +[\text{(2).(*9·03)}] &\vdash\colon\ldotp q.\supset :(x).\phi x.\lor .q +\end{array} +\]</p> + +<p class="nind"> +<b>*9·33.</b> \(\vdash\colon\ldotp q.\supset :(\exists x).\phi x.\lor .q \quad[\text{Proof as above}]\)</p> + +<p><span class="pagenum" id="Page_141">[Pg 141]</span></p> + +<p class="nind"> +<b>*9·34.</b> \(\vdash\colon\ldotp (x).\phi x.\supset :p.\lor .(x).\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·3}. &\supset \vdash:\phi x.\supset .p\lor \phi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*9·13}. &\supset \vdash:(x):\phi x.\supset .p\lor \phi x &\qquad \text{(2)}\\ +\vdash.\text{(2).*9·21}. &\supset \vdash:(x).\phi x.\supset .(x).p\lor \phi x &\qquad \text{(3)}\\ +\vdash.\text{(3).(*9·04)}. &\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*9·35.</b> \(\vdash\colon\ldotp (\exists x).\phi x.\supset :p.\lor .(\exists x).\phi x \quad[\text{Proof as above}]\)</p> + +<p class="nind"> +<b>*9·36.</b> \(\vdash\colon\ldotp p.\lor .(x).\phi x:\supset :(x).\phi x.\lor .p\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·4}. &\supset \vdash:p\lor \phi x.\supset .\phi x\lor p &\qquad \text{(1)}\\ +\vdash.\text{(1).*9·13·21}. &\supset \vdash:(x).p\lor \phi x.\supset .(x).\phi x\lor p &\qquad \text{(2)}\\ +\vdash.\text{(2).(*9·03·04)}.&\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*9·361.</b> \(\vdash\colon\ldotp (x).\phi x.\lor .p:\supset :p.\lor .(x).\phi x \quad[\text{Similar proof}]\)</p> + +<p class="nind"> +<b>*9·37.</b> \(\vdash\colon\ldotp p.\lor .(\exists x).\phi x:\supset :(\exists x).\phi x.\lor .p \quad[\text{Similar proof}]\)</p> + +<p class="nind"> +<b>*9·371.</b> \(\vdash\colon\ldotp (\exists x).\phi x.\lor .p:\supset :p.\lor .(\exists x).\phi x \quad[\text{Similar proof}]\)</p> + +<p class="nind"> +<b>*9·4.</b> \(\vdash\colon\colon p:\lor :q.\lor .(x).\phi x\colon\ldotp \supset \colon\ldotp q:\lor :p.\lor .(x).\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·5.*9·21}. &\supset \vdash\colon\ldotp (x):p.\lor .q\lor \phi x:\supset :(x):q.\lor .p\lor \phi x &\qquad \text{(1)}\\ +\vdash.\text{(1).(*9·04)}.&\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*9·401.</b> \(\vdash\colon\colon p:\lor :q.\lor .(\exists x).\phi x\colon\ldotp \supset \colon\ldotp q:\lor :p.\lor .(\exists x).\phi x \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·41.</b> \(\vdash\colon\colon p:\lor :(x).\phi x.\lor .r\colon\ldotp \supset \colon\ldotp (x).\phi x:\lor :p\lor r \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·411.</b> \(\vdash\colon\colon p:\lor :(\exists x).\phi x.\lor .r\colon\ldotp \supset \colon\ldotp (\exists x).\phi x:\lor :p\lor r \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·42.</b> \(\vdash\colon\colon (x).\phi x:\lor :q\lor r\colon\ldotp \supset \colon\ldotp q:\lor :(x).\phi x.\lor .r \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·421.</b> \(\vdash\colon\colon (\exists x).\phi x:\lor :q\lor r\colon\ldotp \supset \colon\ldotp q:\lor :(\exists x).\phi x.\lor .r \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·5.</b> \(\vdash\colon\colon p\supset q.\supset \colon\ldotp p.\lor .(x).\phi x:\supset :q.\lor .(x).\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·6}. &\supset \vdash\colon\ldotp p\supset q.\supset :p\lor \phi y.\supset .q\lor \phi y &\qquad \text{(1)}\\ +\vdash.\text{(1).*9·1.(*9·06)}. &\supset \vdash\colon\ldotp p\supset q.\supset :(\exists x):p\lor \phi x.\supset .q\lor \phi y &\qquad \text{(1)}\\ +\vdash.\text{(2).*9·13.(*9·04)}.&\supset \vdash\colon\colon p\supset q.\supset \colon\ldotp (y)\colon\ldotp (\exists x):p\lor \phi x.\supset .q\lor \phi y &\qquad \text{(3)}\\ +[\text{(3).(*9·08)}] &\vdash\colon\colon p\supset q.\supset \colon\ldotp (\exists x).{\sim}(p\lor \phi x).\lor .(y).q\lor \phi y &\qquad \text{(4)}\\ +[\text{(4).(*9·01)}] &\vdash\colon\colon p\supset q.\supset \colon\ldotp (x).p\lor \phi x.\supset .(y).q\lor \phi y &\qquad \text{(5)}\\ +[\text{(5).(*9·04)}] &\vdash\colon\colon p\supset q.\supset \colon\ldotp p.\lor .(x).\phi x:\supset :q.\lor .(y).\phi y +\end{array} +\]</p> + +<p class="nind"> +<b>*9·501.</b> \(\vdash\colon\colon p\supset q.\supset \colon\ldotp p.\lor .(\exists x).\phi x:\supset :q.\lor .(\exists x).\phi x \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·51.</b> \(\vdash\colon\colon p.\supset .(x).\phi x:\supset \colon\ldotp p\lor r.\supset :(x).\phi x.\lor .r\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·6}. &\supset \vdash\colon\ldotp p\supset \phi x.\supset :p\lor r.\supset .\phi x\lor r &\qquad \text{(1)}\\ +\vdash.\text{(1).*9·13·21}. &\supset \vdash\colon\colon (x).p\supset \phi x.\supset \colon\ldotp (x):p\lor r.\supset .\phi x\lor r &\qquad \text{(2)}\\ +\vdash.\text{(2).(*9·03·04)}.&\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_142">[Pg 142]</span></p> + +<p class="nind"> +<b>*9·511.</b> \(\vdash\colon\colon p.\supset .(\exists x).\phi x:\supset \colon\ldotp p\lor r.\supset :(\exists x).\phi x.\lor .r \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·52.</b> \(\vdash\colon\colon (x).\phi x.\supset .q:\supset \colon\ldotp (x).\phi x.\lor .r:\supset .q\lor r\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*1·6}. &\supset \vdash\colon\ldotp \phi x\supset q.\supset :\phi x\lor r.\supset .q\lor r &\qquad \text{(1)}\\ +\vdash.\text{(1).*9·13·22}. &\supset \vdash\colon\colon (\exists x).\phi x\supset q.\supset \colon\ldotp (\exists x):\phi x\lor r.\supset .q\lor r &\qquad \text{(2)}\\ +\vdash.\text{(2).(*9·05·01)}. &\supset \vdash\colon\colon (x).\phi x.\supset .q:\supset \colon\ldotp (x).\phi x\lor r.\supset .q\lor r &\qquad \text{(3)}\\ +\vdash.\text{(3).(*9·03)}. &\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*9·521.</b> \(\vdash\colon\colon (\exists x).\phi x.\supset .q:\supset \colon\ldotp (\exists x).\phi x.\lor .r:\supset .q\lor r \quad[\text{As above}]\)</p> + +<p class="nind"> +<b>*9·6.</b> \[\begin{align}(x).\phi x,\, {\sim}(x).\phi x\,\, (\exists x).\phi x\,\, \text{and}\,\, {\sim}(\exists x).\phi x\,\, &\text{are of the same type}.\\ +&[\text{*9·131, (7) and (8)}]\end{align}\]</p> + +<p class="nind"> +<b>*9·61.</b> If \(\phi\hat{x}\) and \(\psi\hat{x}\) are elementary +functions of the same type, there is a function \(\phi\hat{x} \lor \psi\hat{x}\).</p> + +<p><i>Dem.</i></p> + +<p>By *9·14·15, there is an \(a\) for which "\(\psi a\)," and therefore +"\(\phi a\)," are significant, and therefore so is "\(\phi a\lor \psi a\)," +by the primitive idea of disjunction. Hence the result by <a href="#*9·15">*9·15</a>.</p> + +<p>The same proof holds for functions of any number of variables.</p> + +<p class="nind"> +<b>*9·62.</b> If \(\phi(\hat{x},\hat{y})\) and \(\psi\hat{z}\) are +elementary functions, and the \(x\)-argument to \(\phi\) is of the same +type as the argument to \(\psi\), there are functions +\[ +(y).\phi(\hat{x},y). \lor .\psi\hat{x}, (\exists y).\phi(\hat{x},y). \lor .\psi\hat{x}. +\]</p> + +<p><i>Dem.</i></p> + +<p>By <a href="#*9·15">*9·15</a>, there are propositions \(\phi(x,b)\) and \(\psi a\), where +by hypothesis \(x\) and \(a\) are of the same type. Hence by <a href="#*9·14">*9·14</a> +there is a proposition \(\phi(a,b)\), and therefore, by the primitive +idea of disjunction, there is a proposition \(\phi(a,b)\lor \psi a\), +and therefore, by *9·15 and <a href="#*9·03">*9·03</a>, there is a proposition +(\(y).\phi(a,y).\lor .\psi a\). Similarly there is a proposition +(\(\exists y).\phi(a,y).\lor .\psi a\). Hence the result, by *9·15.</p> + +<p class="nind"> +<b>*9·63.</b> If \(\phi(\hat{x},\hat{y})\), \(\psi\hat{x},\hat{y})\) +are elementary functions of the same type, there are functions +(\(y).\phi(\hat{x},y).\lor .(z).\psi(\hat{x},z)\), etc. [Proof as above]</p> + +<p>We have now completed the proof that, in the primitive propositions +of *1, any one of the propositions that occur may be replaced by +(\(x).\phi x\) or (\(\exists x).\phi x\). It follows that, by merely +repeating the proofs, we can show that any other of the propositions +that occur in these propositions can be simultaneously replaced by +(\(x).\psi x\) or (\(\exists x).\psi x\). Thus all the primitive +propositions of <a href="#*1">*1</a>, and therefore all the propositions of <a href="#*2">*2</a>—<a href="#*5">*5</a>, hold +equally when some or all of the propositions concerned are of one of +the forms (\(x).\phi x\), (\(\exists x).\phi x\), which was to be +proved.</p> + +<p>It follows, by mere repetition of the proofs, that the propositions +of <a href="#*1">*1</a>—<a href="#*5">*5</a> hold when \(p\), \(q\), \(r\) are replaced by propositions +containing any number of apparent variables.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_50" href="#FNanchor_50" class="label">[50]</a> +We use "always" as meaning "in all cases," not "at all +times." A similar remark applies to "sometimes."</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_51" href="#FNanchor_51" class="label">[51]</a> +As above.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_143">[Pg 143]</span></p> +<h2 class="nobreak" id="*10">*10. THEORY OF PROPOSITIONS CONTAINING ONE APPARENT +VARIABLE.</h2> +</div> + + +<p><i>Summary of</i> *10.</p> + +<p>The chief purpose of the propositions of this number is to extend +to formal implications (<i>i.e.</i> to propositions of the form +\((x).{\phi}x \supset {\psi}x\)) as many as possible of the +propositions proved previously for material implications, <i>i.e.</i> +for propositions of the form \(p \supset q\). Thus <i>e.g.</i> we have +proved in <a href="#*3·33">*3·33</a> that +\[ +p \supset q . q \supset r .\supset. p \supset r\text{.} +\]</p> + +<p>\[ +\begin{align} +\text{Put}\quad &p &=~ &\text{Socrates is a Greek}\text{,}\\ +&q &=~ &\text{Socrates is a man}\text{,}\\ +&r &=~ &\text{Socrates is a mortal}\text{.}\\ +\end{align} +\] +Then we have "if 'Socrates is a Greek' implies 'Socrates is a man,' +and 'Socrates is a man' implies 'Socrates is a mortal,' it follows +that 'Socrates is a Greek' implies 'Socrates is a mortal.'" But this +does not of itself prove that if all Greeks are men, and all men are +mortals, then all Greeks are mortals.</p> + +<p>\[ +\begin{align} +\text{Putting} \quad &{\phi}x &.=.~ &x~ \text{is a Greek}\text{,}\\ +&{\psi}x &.=.~ &x~ \text{is a man}\text{,}\\ +&{\chi}x &.=.~ &x~ \text{is a mortal}\text{,}\\ +\end{align} +\] +we have to prove +\[ +(x).{\phi}x \supset {\psi}x : (x).{\psi}x \supset {\chi}x :\supset: (x).{\phi}x \supset {\chi}x\text{.} +\] +It is such propositions that have to be proved in the present +number. It will be seen that formal implication \(((x).{\phi}x \supset {\psi}x\)) +is a relation of two functions \(\phi\hat{x}\) +and \(\psi\hat{x}\). Many of the formal properties of this relation +are analogous to properties of the relation "\(p \supset q\)" which +expresses material implication; it is such analogues that are to be +proved in this number.</p> + +<p>We shall assume in this number, what has been proved in <a href="#*9">*9</a>, that +the propositions of <a href="#*1">*1</a>—<a href="#*5">*5</a> can be applied to such propositions as +(\(x).{\phi}x\) and (\(\exists x) . {\phi}x\). Instead of the method +adopted in *9, it is possible to take negation and disjunction as +new primitive ideas, as applied to propositions containing apparent +variables, and to assume that, with the new meanings of negation and +disjunction, the primitive propositions of <a href="#*1">*1</a> still hold. If this<span class="pagenum" id="Page_144">[Pg 144]</span> +method is adopted, we need not take \((\exists x).\phi x\) as a +primitive idea, but may put</p> + +<p class="nind"> +<b><a id="*10·01">*10·01</a>.</b> (\(\exists x).\phi x.=.{\sim}(x).{\sim} \phi x\quad\text{Df}\)</p> + +<p>In order to make it clear how this alternative method can be developed, +we shall, in the present number, assume nothing of what has been proved +in <a href="#*9">*9</a> except certain propositions which, in the alternative method, +will be primitive propositions, and (what in part characterizes the +alternative method) the applicability to propositions containing +apparent variables of analogues of the primitive ideas and propositions +of <a href="#*1">*1</a>, and therefore of their consequences as set forth in <a href="#*2">*2</a>—<a href="#*5">*5</a>.</p> + +<p>The two following definitions merely serve to introduce a notation +which is often more convenient than the notation \((x).\phi x\supset\psi x\) +or \((x).\phi x\equiv\psi x\).</p> + +<p class="nind"> +<b>*10·02.</b> \(\phi x\supset_{x}\psi x.=.(x).\phi x\supset\psi x\,\quad\text{Df}\)</p> + +<p class="nind"> +<b>*10·03.</b> \(\phi x\equiv_{x}\psi x.=.(x).\phi x\equiv \psi x\,\quad\text{Df}\)</p> + +<p>The first of these notations is due to Peano, who, however, has no +notation for (\(x).\phi x\) except in the special case of a formal +implication.</p> + +<p>The following propositions (*10·1·11·12·121·122) have already been +given in <a href="#*9">*9</a>. <a href="#*10·1">*10·1</a> is *9·2*10·11 is <a href="#*9·13">*9·13</a>, <a href="#*10·12">*10·12</a> is <a href="#*9·25">*9·25</a>, <a href="#*10·121">*10·121</a> +is <a href="#*9·14">*9·14</a>, and <a href="#*10·122">*10·122</a> is <a href="#*9·15">*9·15</a>. These five propositions must all be +taken as primitive propositions in the alternative method; on the other +hand, <a href="#*9·1">*9·1</a> and <a href="#*9·11">*9·11</a> are not required as primitive propositions in the +alternative method.</p> + +<p>The propositions of the present number are very much used throughout +the rest of the work. The propositions most used are the following:</p> + +<p class="nind"> +<b><a id="*10·1">*10·1</a>.</b> \(\vdash:(x).\phi x.\supset.\phi y\)</p> + +<p><i>I.e.</i> what is true in all cases is true in any one case.</p> + +<p class="nind"> +<b>*10·11.</b> If \(\phi y\) is true whatever possible argument +\(y\) may be, then (\(x).\phi x\) is true. In other words, whenever +the propositional function \(\phi y\) can be asserted, so can the +proposition (\(x).\phi x\).</p> + +<p class="nind"> +<b>*10·21.</b> \(\vdash\colon\ldotp (x).p\supset\phi x.\equiv:p.\supset.(x).\phi x\)</p> + +<p class="nind"> +<b>*10·22.</b> \(\vdash\colon\ldotp (x).\phi x:(x).\psi x:\equiv.(x).\phi x.\psi x\)</p> + +<p>The conditions of significance in this proposition demand that \(\phi\) +and \(\psi\) should take arguments of the same type.</p> + +<p class="nind"> +<b>*10·23.</b> \(\vdash \colon\ldotp (x).\phi x\supset p.\equiv \colon(\exists x).\phi x.\supset.p\)</p> + +<p><i>I.e.</i> if \(\phi x\) always implies \(p\), then if \(\phi x\) is +ever true, \(p\) is true.</p> + +<p class="nind"> +<b>*10·24.</b> \(\vdash:\phi y.\supset.(\exists x).\phi x\)</p> + +<p><span class="pagenum" id="Page_145">[Pg 145]</span></p> + +<p><i>I.e.</i> if \(\phi y\) is true, then there is an \(x\) for +which \(\phi x\) is true. This is the sole method of proving +existence-theorems.</p> + +<p class="nind"> +<b>*10·27.</b> \(\vdash\colon\ldotp (z).\phi z\supset\psi z.\supset:(z).\phi z.\supset.(z).\psi z\)</p> + +<p><i>I.e.</i> if \(\phi z\) always implies \(\psi z\), then "\(\phi z\) +always" implies "\(\psi z\) always." The three following propositions, +which are equally useful, are analogous to <a href="#*10·27">*10·27</a>.</p> + +<p class="nind"> +<b>*10·271.</b> \(\vdash\colon\ldotp (z).\phi z\equiv\psi z.\supset:(z).\phi z.\equiv.(z).\psi z\)</p> + +<p class="nind"> +<b>*10·28.</b> \(\vdash\colon\ldotp (x).\phi x\supset\psi x.\supset:(\exists x).\phi x.\supset.(\exists x).\psi x\)</p> + +<p class="nind"> +<b>*10·281.</b> \(\vdash\colon\ldotp (x).\phi x\equiv\psi x.\supset:(\exists x).\phi x.\equiv.(\exists x).\psi x\)</p> + +<p class="nind"> +<b>*10·35.</b> \(\vdash\colon\ldotp (\exists x).p.\phi x.\equiv:p:(\exists x).\phi x\)</p> + +<p class="nind"> +<b>*10·42.</b> \(\vdash\colon\ldotp (\exists x).\phi x.\lor .(\exists x).\psi x:\equiv.(\exists x).\phi x \lor \psi x\)</p> + +<p class="nind"> +<b>*10·5.</b> \(\vdash\colon\ldotp (\exists x).\phi x.\psi x.\supset:(\exists x).\phi x:(\exists x).\psi x\)</p> + +<p>It should be noticed that whereas <a href="#*10·42">*10·42</a> expresses an equivalence, +<a href="#*10·5">*10·5</a> only expresses an implication. This is the source of many +subsequent differences between formulae concerning addition and +formulae concerning multiplication.</p> + +<p class="nind"> +<b>*10·51.</b> \(\vdash\colon\ldotp {\sim}\{(\exists x).\phi x.\psi x\}.\equiv:\phi x.\supset_{x}.{\sim}\psi x\)</p> + +<p>This proposition is analogous to +\[ +\vdash:{\sim}(p.q).\equiv.p\supset{\sim}q +\] +which results from <a href="#*4·63">*4·63</a> by transposition.</p> + +<p>Of the remaining propositions of this number, some are employed fairly +often, while others are lemmas which are used only once or twice, +sometimes at a much later stage.</p> + +<hr class="tb"> + +<p class="nind"><b>*10·01.</b> (\(\exists x).\phi x.=.{\sim}(x).{\sim}\phi x \quad\text{Df}\)</p> + +<p>This definition is only to be used when we discard the method of *9 in +favour of the alternative method already explained. In either case we +have +\[ +\vdash:(\exists x).\phi x.\equiv.{\sim}(x).{\sim}\phi x. +\]</p> + +<p class="nind"> +<b>*10·02.</b> \(\phi x\supset_{x}\psi x.=.(x).\phi x\supset\psi x \quad\text{Df}\)</p> + +<p class="nind"> +<b>*10·03.</b> \(\phi x\equiv_{x}\psi x.=.(x).\phi x\equiv\psi x \quad\text{Df}\)</p> + +<p class="nind"> +<b>*10·1.</b> \(\vdash:(x).\phi x.\supset \phi y \quad[\text{*9·2}]\)</p> + +<p class="nind"> +<b><a id="*10·11">*10·11</a>.</b> If \(\phi y\) is true whatever possible argument \(y\) +may be, then (\(x).\phi x\) is true. [<a href="#*9·13">*9·13</a>]</p> + +<p><span class="pagenum" id="Page_146">[Pg 146]</span></p> + +<p>This proposition is, in a sense, the converse of <a href="#*10·1">*10·1</a>. *10·1 may be +stated: "What is true of all is true of any," while <a href="#*10·11">*10·11</a> may be +stated: "What is true of any, however chosen, is true of all."</p> + +<p class="nind"> +<b><a id="*10·12">*10·12</a>.</b> \(\vdash\colon\ldotp (x).p\lor\phi x.\supset:p.\lor.(x).\phi x \quad[\text{*9·25}]\)</p> + +<p>According to the definitions in <a href="#*9">*9</a>, this proposition is a mere +example of "\(q\supset q\)," since by definition the two sides of the +implication are different symbols for the same proposition. According +to the alternative method, on the contrary, <a href="#*10·12">*10·12</a> is a substantial +proposition.</p> + +<p class="nind"> +<b><a id="*10·121">*10·121</a>.</b> If "\(\phi x\)" is significant, then if \(a\) is of the +same type as \(x\), "\(\phi a\)" is significant, and vice versa. [<a href="#*9·14">*9·14</a>]</p> + +<p>It follows from this proposition that two arguments to the same +function must be of the same type; for if \(x\) and a are arguments +to \(\phi\hat{x}\), "\(\phi x\)" and "\(\phi a\)" are significant, +and therefore \(x\) and \(a\) are of the same type. Thus the above +primitive proposition embodies the outcome of our discussion of the +vicious-circle paradoxes in <a href="#CHAPTER_II">Chapter II</a> of the Introduction.</p> + +<p class="nind"> +<b><a id="*10·122">*10·122</a>.</b> If, for some \(a\), there is a proposition \(\phi a\), +then there is a function \(\phi\hat{x}\), and vice versa. [<a href="#*9·15">*9·15</a>]</p> + +<p class="nind"> +<b><a id="*10·13">*10·13</a>.</b> If \(\phi\hat{x}\) and \(\psi\hat{x}\) take arguments of +the same type, and we have "\(\vdash.\phi x\)" and "\(\vdash.\psi x\)," +we shall have "\(\vdash.\phi x.\psi x\)."</p> + +<p><i>Dem.</i></p> + +<p>By repeated use of 9·61·62·63·131 (3), there is a function +\({\sim}\phi\hat{x}\lor{\sim}\psi\hat{x}\). Hence by <a href="#*2·11">*2·11</a> and <a href="#*3·01">*3·01</a>, +\[ +\begin{array}{l} +\vdash:{\sim}\phi x\lor{\sim}\psi x.\lor.\phi x.\psi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*2·32.(*1·01)}.\supset\vdash\colon\ldotp \phi x.\supset:\psi x.\supset.\phi x.\psi x &\qquad \text{(2)}\\ +\vdash.\text{(2).*9·12}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*10·14">*10·14</a>.</b> \(\vdash\colon\ldotp (x).\phi x:(x).\psi x:\supset.\phi y.\psi y\)</p> + +<p>This proposition is true whenever it is significant, but it is not +always significant when its hypothesis is significant. For the thesis +demands that \(\phi\) and \(\psi\) should take arguments of the same +type, while the hypothesis does not demand this. Hence, if it is to +be applied when \(\phi\) and \(\psi\) are given, or when \(\psi\) is +given as a function of \(\phi\) or vice versa, we must not argue from +the hypothesis to the thesis unless, in the supposed case, \(\phi\) and +\(\psi\) take arguments of the same type.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash:(x).\phi x.\supset.\phi y &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash:(x).\psi x.\supset.\psi y &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*10·13}.&\supset\vdash:(x).\phi x.\supset.\phi y:(x).\psi x.\supset.\psi y:\\ +[\text{*3·47}] &\supset\vdash\colon\ldotp (x).\phi x:(x).\psi x:\supset.\phi y.\psi y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_147">[Pg 147]</span></p> + +<p class="nind"> +<b><a id="*10·2">*10·2</a>.</b> \(\vdash\colon\ldotp (x).p\lor \phi x.\equiv:p.\lor .(x)\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1.*1·6}.&\supset\vdash\colon\ldotp p.\lor .(x).\phi x:\supset.p\lor \phi y\colon\ldotp \\ +[\text{*10·11}] &\supset\vdash\colon\ldotp (y)\colon\ldotp p.\lor .(x).\phi x:\supset.p\lor \phi y\colon\ldotp \\ +[\text{*10·12}] &\supset\vdash\colon\ldotp p.\lor .(x).\phi x:\supset.(y).p\lor \phi y &\qquad \text{(1)}\\ +\vdash.\text{*10·12}. &\supset\vdash\colon\ldotp (y).p\lor \phi y.\supset:p.\lor .(x).\phi x &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop}. +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*10·21">*10·21</a>.</b> \(\vdash\colon\ldotp (x).p\supset\phi x.\equiv:p.\supset.(x).\phi x \quad\left[\text{*10·2}\, \frac{{\sim}p}{p}\right]\)</p> + +<p>This proposition is much more used than <a href="#*10·2">*10·2</a>.</p> + +<p class="nind"> +<b>*10·22.</b> \(\vdash\colon\ldotp (x).\phi x.\psi x.\equiv:(x).\phi x:(x).\psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash:(x).\phi x.\psi x.\supset.\phi y.\psi y. &\qquad \text{(1)}\\ +[\text{*3·26}] &\qquad\qquad\qquad\quad\supset.\phi y:\\ +[\text{*10·11}] &\supset\vdash\colon\ldotp (y):(x).\phi x.\psi x.\supset.\phi y\colon\ldotp \\ +[\text{*10·21}] &\supset\vdash\colon\ldotp (x).\phi x.\psi x.\supset.(y).\phi y &\qquad \text{(2)}\\ +\vdash.\text{(1).*3·27}. &\supset\vdash\colon\ldotp (x).\phi x.\psi x.\supset.\psi z\colon\ldotp \\ +[\text{*10·11}] &\supset\vdash\colon\ldotp (z):(x).\phi x.\psi x.\supset.\psi z\colon\ldotp \\ +[\text{*10·21}] &\supset\vdash\colon\ldotp (x).\phi x.\psi x.\supset.(z).\psi z &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).Comp}. &\supset\vdash\colon\ldotp (x).\phi x.\psi x.\supset:(y).\phi y:(z).\psi z &\qquad \text{(4)}\\ +\vdash.\text{*10·14·11}. &\supset\vdash\colon\ldotp (y)\colon\ldotp (x).\phi x:(x).\psi x:\supset.\phi y.\psi y\colon\ldotp \\ +[\text{*10·21}] &\supset\vdash\colon\ldotp (x).\phi x:(x).\psi x:\supset.(y).\phi y.\psi y &\qquad \text{(5)}\\ +\vdash.\text{(4).(5)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is true whenever it is significant; but, as was +pointed out in connexion with <a href="#*10·14">*10·14</a>, it is not always significant when +"(\(x).\phi x:(x).\psi x\)" is significant.</p> + +<p class="nind"><b><a id="*10·221">*10·221</a>.</b> If \(\phi x\) contains a constituent \(\chi(x, y, z, ...)\) +and \(\psi x\) contains a constituent \(\chi(x, u, v, \ldots)\), +where \(\chi\) is an elementary function and \(y, z, \ldots u, v, \ldots\) +are either constants or apparent variables, +then \(\phi\hat{x}\) and \(\psi\hat{x}\) take arguments of the +same type. This can be proved in each particular case, though not +generally, provided that, in obtaining \(\phi\) and \(\psi\) from +\(\chi\), \(\chi\) is only submitted to negations, disjunctions and +generalizations. The process may be illustrated by an example. Suppose +\(\phi x\) is (\(y).\chi(x,y).\supset.\theta x\), and \(\psi x\) +is \(fx.\supset.(y).\chi(x,y)\). By the definitions of <a href="#*9">*9</a>, \(\phi x\) +is (\(\exists y).{\sim}\chi(x,y)\lor \theta x\), and \(\psi x\) +is (\(y).{\sim}fx\lor \chi(x,y)\). Hence since the primitive ideas +(\(x).Fx\) and (\(\exists x).Fx\) only apply to functions, there +are functions \({\sim}\chi(\hat{x},\hat{y})\lor \theta\hat{x}\), +\({\sim}f\hat{x}\lor \chi(\hat{x},\hat{y})\). Hence there is a +proposition \({\sim}\chi(a,b)\lor \theta a\). Hence, since "\(p\lor q\)" +and "\({\sim}p\)" are only significant<span class="pagenum" id="Page_148">[Pg 148]</span> when \(p\) and \(q\) +are propositions, there is a proposition \(\chi(a, b)\). Similarly, +for some \(u\) and \(v\), there are propositions \({\sim}fu \lor \chi(u,v)\) +and \(\chi(u, v)\). Hence by <a href="#*9·14">*9·14</a>, \(u\) and \(a\), \(v\) +and \(b\) are respectively of the same type, and (again by *9·14) there +is a proposition \({\sim}fa\lor \chi(a, b)\). Hence (<a href="#*9·15">*9·15</a>) there are +functions \({\sim}\chi(a, \hat{y}) \lor \theta a\), \({\sim}fa \lor \chi(a, \hat{y})\), +therefore there are propositions +\[ +(\exists y).{\sim}\chi(a, y) \lor \theta a, (y).{\sim}fa \lor \chi(a, y), +\] +<i>i.e.</i> there are propositions \(\phi a\), \(\psi a\), which was to +be proved. This process can be applied similarly in any other instance.</p> + +<p class="nind"> +<b><a id="*10·23">*10·23</a>.</b> \(\vdash\colon\ldotp (x).\phi x\supset p.\equiv:(\exists x).\phi x.\supset.p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·2.(*9·03)}.\supset\vdash\colon\ldotp (x).{\sim}\phi x \lor p.&\equiv:(x).{\sim}\phi x. \lor .p:\\ +[\text{(*9·02)}] &\equiv.(\exists x).\phi x.\supset.p &\qquad \text{(1)}\\ +\vdash.\text{(1).(*1·01)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the above proof, we employ the definitions of <a href="#*9">*9</a>. In the alternative +method, in which (\(\exists x).\phi x\) is defined in accordance with +<a href="#*10·01">*10·01</a>, the proof proceeds as follows.</p> + +<p class="nind"> +<b>*10·23.</b> \(\vdash\colon\ldotp (x).\phi x\supset p.\equiv:(\exists x).\phi x.\supset.p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{Transp.(*10·01)}.&\supset\vdash\colon\ldotp (\exists x).\phi x.\supset.p:\equiv:{\sim}p.\supset.(x).{\sim}\phi x:\\ +[\text{*10·21}] &\equiv:(x):{\sim}p.\supset.{\sim}\phi x: &\qquad \text{(1)}\\ +[\text{*10·1}] &\supset:{\sim}p.\supset.{\sim}\phi x\\ +[\text{Transp}] &\supset:\phi x.\supset.p\colon\ldotp \\ +[\text{*10·11}] &\supset\vdash\colon\ldotp (x)\colon\ldotp (\exists x).\phi x.\supset.p:\supset:\phi x.\supset.p\colon\ldotp \\ +[\text{*10·21}] &\supset\vdash\colon\ldotp (\exists x).\phi x.\supset.p:\supset:(x):\phi x.\supset.p &\qquad \text{(2)}\\ +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp (x):\phi x.\supset.p:\supset:\phi x\supset p:\\ +[\text{Transp}] &\supset:{\sim}p.\supset.{\sim}\phi x\colon\ldotp \\ +[\text{*10·11·21}] &\supset\vdash\colon\ldotp (x):\phi x.\supset.p:\supset:(x):{\sim}p.\supset.{\sim}\phi x:\\ +[\text{(1)}] &\supset :(\exists x).\phi x.\supset.p &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_149">[Pg 149]</span></p> + +<p>Whenever we have an asserted proposition of the form \(p\supset\phi x\), +we can pass by *10·11·21 to an asserted proposition +\(p.\supset.(x).\phi x\). This passage is constantly required, as in +the last line but one of the above proof. It will be indicated merely +by the reference "*10·11·21," and the two steps which it requires will +not be separately put down.</p> + +<p class="nind"> +<b><a id="*10·24">*10·24</a>.</b> \(\vdash:\phi y.\supset.(\exists x).\phi x\)</p> + +<p>This is <a href="#*9·1">*9·1</a>. In the alternative method, the proof is as follows.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.*\text{10·1}. &\supset\vdash:(x).{\sim}\phi x.\supset.{\sim}\phi y:\\ +[\text{Transp}] &\supset\vdash:\phi y.\supset.{\sim}(x).{\sim}\phi x:\\ +[\text{(*10·01)}] &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·25.</b> \(\vdash:(x).\phi x.\supset.(\exists x).\phi x \quad[\text{*10·1·24}]\)</p> + +<p class="nind"> +<b><a id="*10·251">*10·251</a>.</b> \(\vdash:(x).{\sim}\phi x.\supset.{\sim}\{(x).\phi x\} \quad[\text{*10·25.Transp}]\)</p> + +<p class="nind"> +<b>*10·252.</b> \(\vdash:{\sim}\{(\exists x).\phi x\}.\equiv.(x).{\sim}\phi x \quad[\text{*4·2.(*9·02)}]\)</p> + +<p class="nind"> +<b>*10·253.</b> \(\vdash:{\sim}\{(x).\phi x\}.\equiv.(\exists x).{\sim}\phi x \quad[\text{*4·2.(*9·01)}]\)</p> + +<p>In the alternative method, in which (\(\exists x).\phi x\) is defined +as in <a href="#*10·01">*10·01</a>, the proofs of *10·252·253 are as follows.</p> + +<p class="nind"> +<b>*10·252.</b> \(\vdash:{\sim}\{(\exists x).\phi x\}.\equiv.(x).{\sim}\phi x \quad[\text{*4·13.(*10·01)}]\)</p> + +<p class="nind"> +<b>*10·253.</b> \(\vdash:{\sim}\{(x).\phi x\}.\equiv.(\exists x).{\sim}\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash:(x).\phi x.\supset.\phi y.\\ +[\text{*2·12}] &\qquad\qquad\quad\supset.{\sim}({\sim}\phi y):\\ +[\text{*10·11·21}] &\supset\vdash:(x).\phi x.\supset.(y).{\sim}({\sim}\phi y):\\ +[\text{Transp}] &\supset\vdash:{\sim}\{(y).{\sim}({\sim}\phi y)\}.\supset.{\sim}\{(x).\phi x\}:\\ +[\text{(*10·01)}] &\supset\vdash:(\exists y).{\sim}\phi y. \supset.{\sim}\{(x).\phi x\} &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash:(y).{\sim}({\sim}\phi y). \supset.{\sim}({\sim}\phi x).\\ +[\text{*2·14}] &\qquad\qquad\quad\supset.\phi x:\\ +[\text{*10·11·21}] &\supset\vdash:(y).{\sim}({\sim}\phi y). \supset.(x).\phi x:\\ +[\text{Transp}] &\supset\vdash:{\sim}\{(x).\phi x\}. \supset.{\sim}\{(y).{\sim}({\sim}\phi y)\}.\\ +[\text{(*10·01)}] &\qquad\qquad\quad\supset.(\exists y).{\sim}\phi y &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)} &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·26.</b> \(\vdash\colon\ldotp (z).\phi z\supset\psi z:\phi x:\supset.\psi x \quad[\text{*10·1. Imp}]\)</p> + +<p>This is one form of the syllogism in Barbara. <i>E.g.</i> put \(\phi z.=.z\) +is a man, \(\psi z.=.z\) is mortal, \(x\) = Socrates. Then the +proposition becomes:</p> + +<p>"If all men are mortal, and Socrates is a man, then Socrates is mortal."</p> + +<p>Another form of the syllogism in Barbara is given in <a href="#*10·3">*10·3</a>. The two +forms, formerly wrongly identified, were first distinguished by Peano +and Frege.</p> + +<p class="nind"> +<b><a id="*10·27">*10·27</a>.</b> \(\vdash\colon\ldotp (z).\phi z\supset\psi z.\supset:(z).\phi z.\supset.(z).\psi z\)</p> + +<p>This is <a href="#*9·21">*9·21</a>. In the alternative method, the proof is as follows.</p> + +<p><span class="pagenum" id="Page_150">[Pg 150]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·14}. & \supset\vdash\colon\ldotp (z).\phi z\supset\psi z:(z).\phi z:\supset.\phi y\supset\psi y.\phi y.\\ +[\text{Ass}] &\supset.\psi y\colon\ldotp \\ +[\text{*10·1}] &\supset\vdash\colon\ldotp (y)\colon\ldotp (z).\phi z\supset\psi z:(z).\phi z:\supset.\psi y\colon\ldotp \\ +[\text{*10·21}] &\supset\vdash\colon\ldotp (z).\phi z\supset\psi z:(z).\phi z:\supset.(y).\psi y &\qquad \text{(1)}\\ +\vdash.\text{(1).Exp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*10·271">*10·271</a>.</b> \(\vdash\colon\ldotp (z).\phi z\equiv\psi z.\supset:(z).\phi z.\equiv.(z).\psi z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:(z).\phi z\supset\psi z:\\ +[\text{*10·27}] &\qquad\qquad \supset:(z).\phi z.\supset.(z).\psi z &\qquad \text{(1)}\\ +\vdash.\text{*10·22}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:(z).\psi z\supset\phi z:\\ +[\text{*10·27}] & \supset:(z).\psi z.\supset.(z).\phi z &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Comp}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*10·28">*10·28</a>.</b> \(\vdash\colon\ldotp (x).\phi x\supset\psi x.\supset:(\exists x).\phi x.\supset.(\exists x).\psi x\)</p> + +<p>This is <a href="#*9·22">*9·22</a>. In the alternative method, the proof is as follows.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp (x).\phi x\supset\psi x.\supset.\phi y\supset\psi y.\\ +[\text{Transp}] &\qquad\qquad\qquad\supset.{\sim}\psi y\supset{\sim}\phi y\colon\ldotp \\ +[\text{*10·11·21}]&\supset\vdash\colon\ldotp (x).\phi x\supset\psi x.\supset:(y).{\sim}\psi y\supset{\sim}\phi y:\\ +[\text{*10·27}] &\qquad\qquad\qquad\supset:(y).{\sim}\psi y.\supset.(y).{\sim}\phi y:\\ +[\text{Transp}] &\qquad\qquad\qquad\supset:(\exists y).\phi y.\supset.(\exists y).\psi y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*10·281">*10·281</a>.</b> \(\vdash\colon\ldotp (x).\phi x\equiv\psi x.\supset:(\exists x).\phi x.\equiv.(\exists x).\psi x \quad[\text{*10·22·28.Comp}]\)</p> + +<p class="nind"> +<b>*10·29.</b> \(\vdash\colon\ldotp (x).\phi x\supset\psi x:(x).\phi x\supset\chi x:\equiv:(x):\phi x.\supset.\psi x.\chi x.\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}. &\supset\vdash\colon\ldotp (x).\phi x\supset\psi x:(x).\phi x\supset\chi x:\\ +&\qquad\qquad\quad\equiv:(x):\phi x\supset\psi x.\phi x\supset\chi x &\qquad \text{(1)}\\ +\vdash.\text{*4·76}. &\supset\vdash\colon\ldotp \phi x\supset\psi x.\phi x\supset\chi x.\equiv:\phi x.\supset.\psi x.\chi x\colon\ldotp \\ +[\text{*10·11}] &\supset\vdash\colon\ldotp (x)\colon\ldotp \phi x\supset\psi x.\phi x\supset\chi x.\equiv:\phi x.\supset.\psi x.\chi x\colon\ldotp\\ +[\text{*10·271}] &\supset\vdash\colon\ldotp (x):\phi x\supset\psi x.\phi x\supset\chi x:\equiv:(x):\phi x.\supset.\psi x.\chi x &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This is an extension of the principle of composition.</p> + +<p class="nind"> +<b><a id="*10·3">*10·3</a>.</b> \(\vdash\colon\ldotp (x).\phi x\supset\psi x:(x).\psi x\supset\chi x:\supset.(x).\phi x\supset\chi x\)</p> + +<p>This is the second form of the syllogism in Barbara.</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash.\text{*10·22·221}.\supset\vdash:\text{Hp}.&\supset.(x).\phi x\supset\psi x.\psi x\supset\chi x.\\ +[\text{Syll.*10·27}] &\supset.(x).\phi x\supset\chi x:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_151">[Pg 151]</span></p> + +<p class="nind"> +<b><a id="*10·301">*10·301</a>.</b> \(\vdash\colon\ldotp (x).\phi x\equiv\psi x:(x).\psi x\equiv\chi x:\supset.(x).\phi x\equiv\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22·221}.\supset\vdash\colon\ldotp \text{Hp}. &\supset:(x).\phi x\equiv\psi x.\psi x\equiv\chi x:\\ +[\text{*4·22.*10·27}] &\supset:(x).\phi x\equiv\chi x\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the second line of the proofs of <a href="#*10·3">*10·3</a> and <a href="#*10·301">*10·301</a>, we abbreviate +the process of proof in a way which is often convenient. In *10·3, the +full process would be as follows: +\[ +\begin{array}{l} +\vdash.\text{Syll}. &\supset\vdash:\phi x\supset\psi x.\psi x\supset\chi x.\supset.\phi x\supset\chi x:\\ +[\text{*10·11}] &\supset\vdash:(x):\phi x\supset\psi x.\psi x\supset\chi x.\supset.\phi x\supset\chi x:\\ +[\text{*10·27}] &\supset\vdash:(x).\phi x\supset\psi x.\psi x\supset\chi x.\supset.(x).\phi x\supset\chi x +\end{array} +\]</p> + +<p>The above two propositions show that formal implication and formal +equivalence are transitive relations between functions.</p> + +<p class="nind"> +<b>*10·31.</b> \(\vdash\colon\ldotp (x).\phi x\supset\psi x.\supset:(x):\phi x.\chi x.\supset.\psi x.\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{Fact.*10·11}. &\supset\vdash\colon\ldotp (x)\colon\ldotp \phi x\supset\psi x.\supset:\phi x.\chi x.\supset.\psi x.\chi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·27}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·311.</b> \(\vdash\colon\ldotp (x).\phi x\equiv\psi x.\supset:(x)\phi x.\chi x.\equiv.\psi x.\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·36.*10·11}. &\supset\vdash\colon\ldotp (x)\colon\ldotp \phi x\equiv\psi x.\supset:\phi x.\chi x.\equiv.\psi x.\chi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·27}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above two propositions are extensions of the principle of the +factor.</p> + +<p class="nind"> +<b>*10·32.</b> \(\vdash:\phi x\equiv_{x}\psi x.\equiv.\psi x\equiv_{x}\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}. \supset\vdash:\phi x&\equiv_{x}\psi x.\equiv.\phi x\supset_{x}\psi x.\psi x\supset_{x}\phi x.\\ +[\text{*4·3}] &\equiv.\psi x\supset_{x}\phi x.\phi x\supset_{x}\psi x.\\ +[\text{*10·22}] &\equiv.\psi x\equiv_{x}\phi x:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that formal equivalence is symmetrical.</p> + +<p class="nind"> +<b>*10·321.</b> \(\vdash:\phi x\equiv_{x}\psi x.\phi x\equiv_{x}\chi x.\supset.\psi x\equiv_{x}\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·32.Fact}.&\supset\vdash:\text{Hp}. \supset.\psi x\equiv_{x}\phi x.\phi x\equiv_{x}\chi x.\\ +[\text{*10·301}] &\supset.\psi x\equiv_{x}\chi x:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·322.</b> \(\vdash:\psi x\equiv_{x}\phi x.\chi x\equiv_{x}\phi x.\supset.\psi x\equiv_{x}\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·32}.&\supset\vdash:\text{Hp}. \supset.\psi x\equiv_{x}\phi x.\phi x\equiv_{x}\chi x.\\ +[\text{*10·301}] &\supset.\psi x\equiv_{x}\chi x:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_152">[Pg 152]</span></p> + +<p class="nind"> +<b>*10·33.</b> \(\vdash\colon\ldotp (x):\phi x.p:\equiv:(x).\phi x:p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp (x):\phi x.p:\supset.\phi y.p. &\qquad \text{(1)}\\ +[\text{*3·27}] &\supset.p &\qquad \text{(2)}\\ +\vdash.\text{(1).*3·26}. &\supset\vdash\colon\ldotp (x):\phi x.p:\supset.\phi y:\\ +[\text{*10·11·21}] &\supset\vdash\colon\ldotp (x):\phi x.p:\supset.(y).\phi y &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash\colon\ldotp (x):\phi x.p:\supset:(y).\phi y:p &\qquad \text{(4)}\\ +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp (y).\phi y. \supset.\phi x\colon\ldotp \\ +[\text{Fact}] &\supset\vdash\colon\ldotp (y).\phi y:p:\supset.\phi x.p\colon\ldotp \\ +[\text{*10·11·21}] &\supset\vdash\colon\ldotp (y).\phi y:p:\supset:(x):\phi x.p &\qquad \text{(5)}\\ +\vdash.\text{(4).(5)}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·34.</b> \(\vdash\colon\ldotp (\exists x).\phi x\supset p.\equiv:(x).\phi x.\supset.p\)</p> + +<p>This follows immediately from *9·05·01 and <a href="#*1·01">*1·01</a>. In the alternative +method, the proof is as follows.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·2.(*10·01)}.\supset\\ +\vdash\colon\ldotp (\exists x).\phi x\supset p. &\equiv:{\sim}\{(x).{\sim}(\phi x\supset p)\}:\\ +[\text{*4·61.*10·271}] &\equiv:{\sim}\{(x):\phi x.{\sim}p\}:\\ +[\text{*10·33}] &\equiv:{\sim}\{(x).\phi x:{\sim}p\}:\\ +[\text{*4·53}] &\equiv:{\sim}\{(x).\phi x\}.\lor .p:\\ +[\text{*4·6}] &\equiv:(x).\phi x.\supset.p +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*10·35">*10·35</a>.</b> \(\vdash\colon\ldotp (\exists x).p.\phi x.\equiv:p:(\exists x).\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*3·26}. &\supset\vdash:p.\phi x.\supset.p:\\ +[\text{*10·11}] &\supset\vdash:(x):p.\phi x.\supset.p:\\ +[\text{*10·23}] &\supset\vdash:(\exists x).p.\phi x.\supset.p &\qquad \text{(1)}\\ +\vdash.\text{*3·27}. &\supset\vdash:p.\phi x.\supset.\phi x:\\ +[\text{*10·11}] &\supset\vdash:(x):p.\phi x.\supset.\phi x:\\ +[\text{*10·28}] &\supset\vdash:(\exists x).p.\phi x.\supset.(\exists x).\phi x &\qquad \text{(2)}\\ +\vdash.\text{*3·2}. &\supset\vdash\colon\ldotp p.\supset:\phi x.\supset.p.\phi x.\\ +[\text{*10·11·21}] \supset\vdash\colon\ldotp p.&\supset:(x):\phi x.\supset.p.\phi x:\\ +[\text{*10·28}] &\supset:(\exists x).\phi x.\supset.(\exists x).p.\phi x &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).Imp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_153">[Pg 153]</span></p> + +<p class="nind"> +<b>*10·36.</b> \(\vdash\colon\ldotp (\exists x).\phi x\lor p.\equiv:(\exists x).\phi x.\lor .p\)</p> + +<p>This follows immediately from <a href="#*9·05">*9·05</a>. In the alternative method, the +proof is as follows.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·64} &\supset\vdash:\phi x\lor p.\equiv.{\sim}\phi x\supset p:\\ +[\text{*10·11}] &\supset\vdash:(x):\phi x\lor p.\equiv.{\sim}\phi x\supset p:\\ +[\text{*10·281}] \supset\vdash\colon\ldotp (\exists x).\phi x\lor p.&\equiv:(\exists x).{\sim}\phi x\supset p:\\ +[\text{*10·34}] &\equiv:(x).{\sim}\phi x.\supset.p:\\ +[\text{*4·6.(*10·01)}] &\equiv:(\exists x).\phi x.\lor .p\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is only required in order to lead to the +following:</p> + +<p class="nind"> +<b>*10·37.</b> \(\vdash\colon\ldotp (\exists x).p\supset\phi x.\equiv:p.\supset.(\exists x).\phi x \quad\left[\text{*10·36}\, \frac{{\sim}p}{p}\right]\)</p> + +<p class="nind"> +<b><a id="*10·39">*10·39</a>.</b> \(\vdash\colon\ldotp \phi x\supset_{x}\chi x:\psi x\supset_{x}\theta x:\supset:\phi x.\psi x.\supset_{x}.\chi x.\theta x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(x):\phi x\supset\chi x.\psi x\supset\theta x:\\ +[\text{*3·47.*10·27}] &\supset:(x):\phi x.\psi x.\supset.\chi x.\theta x\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is only true when the conclusion is significant; the +significance of the hypothesis does not insure that of the conclusion. +On the conditions of significance, see the remarks on <a href="#*10·4">*10·4</a>, below.</p> + +<p class="nind"> +<b><a id="*10·4">*10·4</a>.</b> \(\vdash\colon\ldotp \phi x\equiv_{x}\chi x.\psi x\equiv_{x}\theta x.\supset:\phi x.\psi x.\equiv_{x}.\chi x.\theta x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}. & \supset\vdash\colon\ldotp \text{Hp}.\supset:\phi x\supset_{x}\chi x.\psi x\supset_{x}\theta x:\\ +[\text{*10·39}] &\supset:\phi x.\psi x.\supset_{x}.\chi x.\theta x &\qquad \text{(1)}\\ +\text{Similarly} &\vdash\colon\ldotp \text{Hp}.\supset:\chi x.\theta x.\supset_{x}.\phi x.\psi x &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Comp}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\phi x.\psi x.\supset_{x}.\chi x.\theta x:\chi x.\theta x.\supset_{x}.\phi x.\psi x:\\ +[\text{*10·22}] &\qquad\qquad\supset:\phi x.\psi x.\equiv_{x}.\chi x.\theta x\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_154">[Pg 154]</span></p> + +<p>In <a href="#*10·4">*10·4</a> and many later propositions, as in <a href="#*10·39">*10·39</a>, the conclusion may +be not significant when the hypothesis is true. Hence, in order that it +may be legitimate to use *10·4 in <i>inference</i>, <i>i.e.</i> to pass +from the <i>assertion</i> of the hypothesis to the <i>assertion</i> of +the conclusion, the functions \(\phi\), \(\psi\), \(\chi\), \(\theta\) +must be such as to have overlapping ranges of significance. In virtue +of <a href="#*10·221">*10·221</a>, this is secured if they are of the forms \(F\{x, \chi(x, \hat{y}, \hat{z}, ...)\}\), +\(f\{x, \chi (x, \hat{y}, \hat{z},\ldots)\}\), +\(G\{x, \chi (x, \hat{y}, \hat{z}, \ldots)\}\), \(g\{x, \chi (x, \hat{y}, \hat{z}, \ldots)\}\). +It is also secured if \(\phi\) and \(\psi\) or \(\phi\) and \(\theta\) +or \(\chi\) and \(\psi\) or \(\chi\) and \(\theta\) are of such forms, +for \(\phi\) and \(\chi\) must have overlapping ranges of significance +if the hypothesis is to be significant, and so must \(\psi\) and +\(\theta\).</p> + +<p class="nind"> +<b><a id="*10·41">*10·41</a>.</b> \(\vdash\colon\ldotp (x).\phi x.\lor .(x).\psi x:\supset.(x).\phi x\lor \psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash:(x).\phi x.\supset.\phi y.\\ +[\text{*2·2}] &\qquad\qquad\quad\supset.\phi y\lor \psi y &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash:(x).\psi x.\supset.\psi y.\\ +[\text{*1·3}] &\qquad\qquad\quad\supset.\phi y\lor \psi y &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*10·13}. &\supset\vdash\colon\ldotp (x).\phi x.\supset.\phi y\lor \psi y:(x).\psi x.\supset.\phi y\lor \psi y\colon\ldotp \\ +[\text{*3·44}] &\supset\vdash\colon\ldotp (x).\phi x.\lor .(x).\psi x:\supset.\phi y\lor \psi y\\ +[\text{*10·11·21}] &\supset\vdash\colon\ldotp (x).\phi x.\lor .(x).\psi x:\supset.(y).\phi y\lor \psi y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Observe that in the above proof the uses of <a href="#*2·2">*2·2</a> and <a href="#*1·3">*1·3</a> are only +legitimate if \(\phi y\) and \(\psi y\) have overlapping ranges +of significance, for otherwise, if \(y\) is such that there is a +proposition \(\phi y\), it is such that there is no proposition \(\psi y\), +and conversely.</p> + +<p class="nind"> +<b>*10·411.</b> \(\vdash\colon\ldotp \phi x\equiv_{x}\chi x.\psi x\equiv_{x}\theta x.\supset:\phi x\lor \psi x.\equiv_{x}.\chi x\lor \theta x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·14}.&\supset\vdash\colon\ldotp \text{Hp}. \supset:\phi x\equiv\chi x.\psi x\equiv\theta x:\\ +[\text{*4·39}] &\supset:\phi x\lor \psi x.\equiv.\chi x\lor \theta x &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·412.</b> \(\vdash:\phi x\equiv_{x}\psi x.\equiv.{\sim}\phi x\equiv_{x}{\sim}\psi x \quad[\text{*4·11.*10·11·271}]\)</p> + +<p class="nind"> +<b>*10·413.</b> \(\vdash\colon\ldotp \phi x\equiv_{x}\chi x.\psi x\equiv_{x}\theta x.\supset:\phi x\supset\psi x.\equiv_{x}.\chi x\supset\theta x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·411·412}.\supset\vdash\colon\ldotp \text{Hp}. &\supset:{\sim}\phi x\lor \psi x.\equiv_{x}.{\sim}\chi x\lor \theta x\\ +[\text{(*1·01)}] &\supset:\phi x\supset\psi x.\equiv_{x}.\chi x\supset\theta x\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·414.</b> \(\vdash\colon\ldotp \phi x\equiv_{x}\chi x.\psi x\equiv_{x}\theta x.\supset:\phi x\equiv\psi x.\equiv_{x}.\chi x\equiv\theta x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·413}\, \frac{\psi,\,\phi,\,\theta,\,\chi}{\phi,\,\psi,\,\chi,\,\theta}.\text{*10·32}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:\psi x\supset\phi x.\equiv_{x}.\theta + x\supset\chi x &\qquad \text{(1)}\\ +\vdash.\text{*10·413.(1).*10·4}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The propositions *10·413·414 are chiefly used in cases where either +\(\chi\) is replaced by \(\phi\) or \(\theta\) is replaced by \(\psi\), +in which case half the hypothesis becomes superfluous, being true by +<a href="#*4·2">*4·2</a>.</p> + +<p class="nind"> +<b><a id="*10·42">*10·42</a>.</b> \(\vdash\colon\ldotp (\exists x).\phi x.\lor .(\exists x).\psi x:\equiv.(\exists x).\phi x\lor \psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}. &\supset\vdash\colon\ldotp (x).{\sim}\phi x:(x).{\sim}\psi x:\equiv.(x).{\sim}\phi x.{\sim}\psi x\colon\ldotp \\ +[\text{*4·11}] &\supset\vdash\colon\ldotp {\sim}\{(x).{\sim}\phi x:(x).{\sim}\psi x\}.\equiv.{\sim}\{(x).{\sim}\phi x.{\sim}\psi x\}\colon\ldotp \\ +[\text{*4·51·56.*10·271}] &\supset\vdash\colon\ldotp {\sim}\{(x).{\sim}\phi x\}.\lor .{\sim}\{(x).{\sim}\psi x\}:\\ +&\qquad\qquad\qquad\qquad\equiv.{\sim}\{(x).{\sim}(\phi x\lor \psi x)\}\colon\ldotp \\ +[\text{*10·253}] &\supset\vdash\colon\ldotp (\exists x).\phi x.\lor .(\exists x).\psi x:\equiv.(\exists x).\phi x\lor \psi x\colon\ldotp +\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_155">[Pg 155]</span></p> + +<p>This proposition is very frequently used. It should be contrasted with +<a href="#*10·5">*10·5</a>, in which we have only an implication, not an equivalence.</p> + +<p class="nind"> +<b>*10·43.</b> \(\vdash : \phi z \equiv_{z} \psi z . \phi x . \equiv . \phi z \equiv_{z} \psi z . \psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*10·1}. &\supset \vdash : \phi z \equiv_{z} \psi z . \supset . \phi x \equiv \psi x &\qquad \text{(1)}\\ +\vdash .\text{(1) . *5·32}.& \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*10·5">*10·5</a>.</b> \(\vdash \colon\ldotp (\exists x) . \phi x . \psi x . \supset : (\exists x) . \phi x : (\exists x) . \psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*3·26 . *10·11}. &\supset \vdash : (x) : \phi x . \psi x . \supset . \phi x:\\ +[\text{*10·28}] & \supset \vdash : (\exists x) . \phi x . \psi x . \supset . (\exists x) . \phi x &\qquad \text{(1)}\\ +\vdash .\text{*3·27 . *10·11}. &\supset \colon\ldotp (x) : \phi x . \psi x . \supset . \psi x :\\ +[\text{*10·28}] & \supset \vdash : (\exists x) . \phi x . \psi x . \supset . (\exists x) . \psi x &\qquad \text{(2)}\\ +\vdash .\text{(1).(2).Comp}. &\supset \vdash \colon\ldotp \text{Prop} +\end{array} +\]</p> + +<p>The converse of the above proposition is false. The fact that +this proposition states an implication, while <a href="#*10·42">*10·42</a> states an +equivalence, is the source of many subsequent differences between +formulae concerning logical addition and formulae concerning logical +multiplication.</p> + +<p class="nind"> +<b>*10·51.</b> \(\vdash \colon\ldotp {\sim}\{(\exists x) . \phi x . \psi x\} . \equiv : \phi x . \supset_{x} . {\sim}\psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*10·252}. \supset \vdash \colon\ldotp {\sim}\{(\exists x) . \phi x . \psi x\} . &\equiv : (x) . {\sim}(\phi x . \psi x) :\\ +[\text{*4·51·62.*10·271}] & \equiv : (x) : \phi x . \supset . {\sim}\psi x \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·52.</b> \(\vdash \colon\ldotp (\exists x) . \phi x . \supset : (x) . \phi x \supset p . \equiv . p\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*5·5}. \supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp p . &\equiv : (\exists x) . \phi x . \supset . p :\\ +[\text{*10·23}] &\equiv : (x) . \phi x \supset p \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·53.</b> \(\vdash \colon\ldotp {\sim}(\exists x) . \phi x . \supset : \phi x . \supset_{x} . \psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*2·21 . *10·11}. \supset\\ +\vdash \colon\ldotp (x) \colon\ldotp {\sim}\phi x . \supset : \phi x . \supset . \psi x \colon\ldotp \\ +[\text{*10·27}] \supset \vdash \colon\ldotp (x) . {\sim}\phi x . \supset : (x) : \phi x . \supset . \psi x \colon\ldotp \\ +[\text{*10·252}] \supset \vdash \colon\ldotp {\sim}(\exists x) . \phi x . \supset : (x) : \phi x . \supset . \psi x \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*10·541.</b> \(\vdash \colon\colon \phi y . \supset_{y} . p \lor \psi y : \equiv : p . \lor . \phi y \supset_{y} \psi y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*4·2 . (*1·01)}. \supset \vdash \colon\ldotp \phi y . \supset_{y} . p \lor \psi y : &\equiv : (y) . {\sim}\phi y \lor p \lor \psi y :\\ +[\text{Assoc.*10·271}] &\equiv : (y) . p \lor {\sim}\phi y \lor \psi y :\\ +[\text{*10·2}] &\equiv : p . \lor . (y) . {\sim}\phi y \lor \psi y :\\ +[\text{(*1·01)}] & \equiv : p . \lor . \phi y \supset_{y} \psi y \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_156">[Pg 156]</span></p> + +<p>The above proposition is only needed in order to lead to the following:</p> + +<p class="nind"> +<b>*10·542.</b> \(\vdash\colon\ldotp \phi y.\supset_{y}.p\supset\psi y:\equiv:p.\supset.\phi y\supset_{y}\psi y \quad\left[\text{*10·541}\, \frac{{\sim}p}{p}\right]\)</p> + +<p>This proposition is a lemma for <a href="#*84·43">*84·43</a>.</p> + +<p class="nind"> +<b>*10·55.</b> \(\vdash\colon\ldotp (\exists x).\phi x.\psi x:\phi x\supset_{x}\psi x:\equiv:(\exists x).\phi x:\phi x\supset_{x}\psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·71}.&\supset\vdash\colon\ldotp \phi x\supset\psi x.\supset:\phi x.\psi x.\equiv.\phi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·27}.&\supset\\ +&\vdash\colon\ldotp \phi x\supset_{x}\psi x.\supset:(x):\phi x.\psi x.\equiv.\phi x:\\ +[\text{*10·281}] &\supset:(\exists x).\phi x.\psi x.\equiv.(\exists x).\phi x &\qquad \text{(2)}\\ +\vdash.\text{(2).*5·32}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is a lemma for *117·12·121.</p> + +<p class="nind"> +<b>*10·56.</b> \(\vdash\colon\ldotp \phi x\supset_{x}.\psi x:(\exists x).\phi x.\chi x:\supset.(\exists x).\psi x.\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·31}. \supset\vdash\colon\ldotp \phi x&\supset_{x}.\psi x:\supset:\phi x.\chi x.\supset_{x}.\psi x.\chi x:\\ +[\text{*10·28}] &\supset:(\exists x).\phi x.\chi x.\supset.(\exists x).\psi x.\chi x &\qquad \text{(1)}\\ +\vdash.\text{(1).Imp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition and <a href="#*10·57">*10·57</a> are used in the theory of series (Part V).</p> + +<p class="nind"> +<b><a id="*10·57">*10·57</a>.</b> \(\vdash\colon\ldotp \phi x.\supset_{x}.\psi x\lor \chi x:\supset:\phi x\supset_{x}\psi x.\lor .(\exists x).\phi x.\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·51.Fact}.\supset +\vdash\colon\ldotp \phi x.\supset_{x}.\psi x\lor \chi x:{\sim}(\exists x).\phi x.\chi x:&\supset:\phi x.\supset_{x}.\psi x\lor \chi x:\phi x.\supset_{x}.{\sim}\chi x:\\ +[\text{*10·29}] &\supset:\phi x.\supset_{x}.\psi x\lor \chi x.{\sim}\chi x:\\ +[\text{*5·61}] &\supset:\phi x.\supset_{x}.\psi x &&\qquad \text{(1)}\\ +\vdash.\text{(1).*5·6}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_157">[Pg 157]</span></p> + +<h2 class="nobreak" id="*11">*11. THEORY OF TWO APPARENT VARIABLES.</h2> +</div> + + +<p><i>Summary of</i> *11.</p> + +<p>In this number, the propositions proved for one variable in *10 are to +be extended to two variables, with the addition of a few propositions +having no analogues for one variable, such as *11·2·21·23·24 and +*11·53·55·6·7. "\(\phi(x,y)\)" stands for a proposition containing +\(x\) and containing \(y\); when \(x\) and \(y\) are unassigned, +\(\phi(x,y)\) is a propositional function of \(x\) and \(y\). The +definition <a href="#*11·01">*11·01</a> shows that "the truth of all values of \(\phi(x,y)\)" +does not need to be taken as a new primitive idea, but is definable +in terms of "the truth of all values of \(\psi x\)." The reason is +that, when \(x\) is assigned, \(\phi(x,y)\) becomes a function of one +variable, namely \(y\), whence it follows that, for every possible +value of \(x\), "(\(y).\phi(x,y)\)" embodies merely the primitive idea +introduced in <a href="#*9">*9</a>. But "(\(y).\phi(x,y)\)" is again only a function of +one variable, namely \(x\), since \(y\) has here become an apparent +variable. Hence the definition *11·01 below is legitimate. We put:</p> + +<p class="nind"> +<b><a id="*11·01">*11·01</a>.</b> (\(x,y).\phi(x,y).=:(x):(y).\phi(x,y) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*11·02.</b> (\(x,y,z).\phi(x,y,z).=:(x):(y,z).\phi(x,y,z) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*11·03.</b> (\(\exists x,y).\phi(x,y).=:(\exists x):(\exists y).\phi(x,y) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*11·04.</b> (\(\exists x,y,z).\phi(x,y,z).=:(\exists x):(\exists y,z).\phi(x,y,z) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*11·05.</b> \(\phi(x,y) .\supset_{x,y}. \psi(x,y) :=: (x,y):\phi(x,y) .\supset. \psi(x,y) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*11·06.</b> \(\phi(x,y) .\equiv_{x,y}. \psi(x,y) :=: (x,y):\phi(x,y) .\equiv. \psi(x,y) \quad \text{Df}\)</p> + +<p>All the above definitions are supposed extended to any number of +variables that may occur.</p> + +<p>The propositions of this section can all be extended to any finite +number of variables; as the analogy is exact, it is not necessary to +carry the process beyond two variables in our proofs.</p> + +<p>In addition to the definition <a href="#*11·01">*11·01</a>, we need the primitive proposition +that "whatever possible argument \(x\) may be, \(\phi(x,y)\) is true +whatever possible argument \(y\) may be" implies the corresponding +statement with \(x\) and \(y\) interchanged. Either may be taken as the +meaning of "\(\phi(x,y)\) is true whatever possible arguments \(x\) and +\(y\) may be."</p> + +<p><span class="pagenum" id="Page_158">[Pg 158]</span></p> + +<p>The propositions of the present number are somewhat less used than +those of <a href="#*10">*10</a>, but some of them are used frequently. Such are the +following:</p> + +<p class="nind"> +<b>*11·1.</b> \(\vdash:(x,y)\ldotp\phi (x,y)\ldotp \supset .\phi (z,w)\)</p> + +<p class="nind"> +<b><a id="*11·11">*11·11</a>.</b> If \( \phi (z,w) \) is true whatever possible arguments +\(z\) and \(w\) may be, then (\(x,y)\ldotp\phi (x,y)\) is true.</p> + +<p>These two propositions are the analogues of *10·1·11.</p> + +<p class="nind"> +<b>*11·2.</b> \(\vdash :(x,y)\ldotp\phi (x,y)\ldotp\equiv \ldotp(y,x)\ldotp\phi (x,y)\)</p> + +<p><i>I.e.</i> to say that "for all possible values of \(x\), \(\phi(x,y)\) +is true for all possible values of \(y\)" is equivalent to +saying "for all possible values of \(y\), \(\phi (x,y)\) is true for +all possible values of \(x\)."</p> + +<p class="nind"> +<b>*11·3.</b> \(\vdash \colon \ldotp p \ldotp\supset \ldotp (x,y)\ldotp\phi (x,y):\equiv :(x,y):p\ldotp\supset \ldotp\phi (x,y)\)</p> + +<p>This is the analogue of <a href="#*10·21">*10·21</a>.</p> + +<p class="nind"> +<b>*11·32.</b> \(\vdash \colon \ldotp (x,y):\phi (x,y).\supset \ldotp\psi (x,y):\supset:(x,y)\ldotp\phi (x,y)\ldotp\supset\ldotp (x,y)\ldotp\psi (x,y)\)</p> + +<p><i>I.e.</i> "if \(\phi (x,y)\) always implies \(\psi (x,y)\), then +'\(\phi (x,y)\) always' implies '\(\psi(x,y)\) always.'" This is the +analogue of <a href="#*10·21">*10·21</a>. *11·33·34·341 are respectively the analogues of +*10·271·28·281, and are also much used.</p> + +<p class="nind"> +<b>*11·35.</b> \(\vdash \colon \ldotp (x,y)\colon \phi (x,y)\ldotp\supset\ldotp p\colon\equiv\colon (\exists x,y)\ldotp \phi(x,y)\ldotp\supset\ldotp p\)</p> + +<p><i>I.e.</i> if \(\phi (x,y)\) always implies \(p\), then if \(\phi(x,y)\) +is ever true, \(p\) is true. This is the analogue of <a href="#*10·23">*10·23</a>.</p> + +<p class="nind"> +<b>*11·45.</b> \(\vdash \colon\ldotp (\exists x,y)\colon p\ldotp \phi (x,y)\colon \equiv\colon p\colon (\exists x,y)\ldotp \phi (x,y)\)</p> + +<p>This is the analogue of <a href="#*10·35">*10·35</a>.</p> + +<p class="nind"> +<b>*11·54.</b> \(\vdash\colon\ldotp (\exists x,y)\ldotp\phi x\ldotp\psi y\ldotp \equiv\colon (\exists x)\ldotp\phi x\colon (\exists y).\psi y\)</p> + +<p>This proposition is useful because it analyses a proposition containing +two apparent variables into two propositions which each contain only +one. "\(\phi x\ldotp\psi y\)" is a function of two variables, but is +compounded of two functions of one variable each. Such a function +is like a conic which is two straight lines: it may be called an +"analysable" function.</p> + +<p class="nind"> +<b>*11·55.</b> \(\vdash\colon\ldotp (\exists x,y)\ldotp \phi x\ldotp \psi (x,y)\ldotp +\equiv\colon (\exists x)\colon \phi x\colon (\exists y).\psi (x,y)\)</p> + +<p><i>I.e.</i> to say "there are values of \(x\) and \(y\) for which +\(\phi x\ldotp \psi (x,y)\) is true" is equivalent to saying "there is +a value of \(x\) for which \(\phi x\) is true and for which there is a +value of \(y\) such that \(\psi (x,y)\) is true."</p> + +<p class="nind"> +<b>*11·6.</b> \(\vdash\colon\colon (\exists x)\colon\ldotp (\exists y)\ldotp \phi (x,y)\ldotp \psi y\colon\chi x\colon\ldotp\equiv\colon\ldotp (\exists y)\colon\ldotp (\exists x)\ldotp \phi (x,y)\ldotp \chi x\colon \psi y\)</p> + +<p>This gives a transformation which is useful in many proofs.</p> + +<p class="nind"> +<b>*11·62.</b> \(\vdash\colon\colon\phi x\ldotp \psi (x,y)\ldotp \supset_{x,y}\ldotp \chi (x,y)\colon\equiv\colon\ldotp\phi x\ldotp\supset_{x}\colon\psi (x,y)\ldotp \supset_{y}.\chi(x,y)\)</p> + +<p>This transformation also is often useful.</p> + +<hr class="tb"> + +<p><span class="pagenum" id="Page_159">[Pg 159]</span></p> + +<p class="nind"> +<b>*11·01.</b> (\(x,y).\phi(x,y).=:(x):(y).\phi(x,y) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*11·02.</b> (\(x,y,z).\phi(x,y,z).=:(x):(y,z).\phi(x,y,z) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*11·03.</b> (\(\exists x,y).\phi(x,y).=:(\exists x):(\exists y).\phi(x,y) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*11·04.</b> (\(\exists x,y,z).\phi(x,y,z).=:(\exists x):(\exists y,z).\phi(x,y,z) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*11·05.</b> \(\phi(x,y).\supset_{x,y}.\psi(x,y):=:(x,y):\phi(x,y).\supset.\psi(x,y) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*11·06.</b> \(\phi(x,y).\equiv_{x,y}.\psi(x,y):=:(x,y):\phi(x,y).\equiv.\psi(x,y) \quad\text{Df}\)</p> + +<p>with similar definitions for any number of variables.</p> + +<p class="nind"> +<b><a id="*11·07">*11·07</a>.</b> "Whatever possible argument \(x\) may be, \(\phi(x,y)\) is true +whatever possible argument \(y\) may be" implies the corresponding +statement with \(x\) and \(y\) interchanged. Pp.</p> + +<p class="nind"> +<b>*11·1.</b> \(\vdash:(x,y).\phi(x,y).\supset.\phi(z,w)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}.\supset\vdash:\text{Hp}.&\supset.(y).\phi(z,y).\\ +[\text{*10·1}] \supset.\phi(z,w):&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·11.</b> If \(\phi(x,y)\) is true whatever possible arguments +\(z\) and \(w\) may be, then (\(x,y).\phi(x,y)\) is true.</p> + +<p><i>Dem.</i></p> + +<p>By <a href="#*10·11">*10·11</a>, the hypothesis implies that (\(y).\phi(z,y)\) is true +whatever possible argument \(z\) may be; and this, by *10·11, implies +(\(x,y).\phi(x,y)\).</p> + +<p class="nind"> +<b>*11·12.</b> \(\vdash\colon\ldotp(x,y).p\lor\phi(x,y).\supset:p.\lor.(x,y).\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{10·12}.\supset\vdash\colon\ldotp(y).p\lor\phi(x,y). &\supset:p.\lor.(y).\phi(x,y)\colon\ldotp\\ +[\text{*10·11·27}]\supset\vdash\colon\ldotp(x,y).p\lor\phi(x,y).&\supset:(x):p.\lor.(y).\phi(x,y):\\ +[\text{*10·12}] &\supset:p.\lor.(x,y).\phi(x,y)\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is only used for proving <a href="#*11·2">*11·2</a>.</p> + +<p class="nind"> +<b><a id="*11·13">*11·13</a>.</b> If \(\phi(\hat{x},\hat{y})\), \(\psi(\hat{x},\hat{y})\) +take their first and second arguments respectively of the same type, +and we have "\(\vdash.\phi(x,y)\)" and "\(\vdash.\psi(x,y)\)," we shall +have "\(\vdash.\phi(x,y).\psi(x,y)\)." [Proof as in <a href="#*10·13">*10·13</a>]</p> + +<p class="nind"> +<b><a id="*11·14">*11·14</a>.</b> \(\vdash\colon\ldotp(x,y).\phi(x,y):(x,y).\psi(x,y):\supset:\phi(z,w).\psi(z,w)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·14}.\supset\vdash\colon\ldotp\text{Hp}.&\supset:(y).\phi(z,y):(y).\psi(z,y)\\ +[\text{*10·14}] &\supset:\phi(z,w).\psi(z,w)\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_160">[Pg 160]</span></p> + +<p>This proposition, like <a href="#*10·14">*10·14</a>, is not always significant when its +hypothesis is true. <a href="#*11·13">*11·13</a>, on the contrary, is always significant +when its hypothesis is true. For this reason, *11·13 may always be +safely used in <i>inference</i>, whereas <a href="#*11·14">*11·14</a> can only be used +in <i>inference</i> (<i>i.e.</i> for the actual assertion of the +conclusion when the hypothesis is asserted) if it is known that the +conclusion is significant.</p> + +<p class="nind"> +<b><a id="*11·2">*11·2</a>.</b> \(\vdash:(x,y).\phi(x,y).\equiv.(y,x).\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{11·1}. \supset\vdash:(x,y).\phi(x,y).&\supset.\phi(z,w) &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·07·11}. \supset\vdash\colon\ldotp(w,z):(x,y).\phi(x,y).&\supset.\phi(z,w) &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·12}. \frac{{\sim}\{(x,y).\phi(x,y)\}}{p} .\supset\\ +&\vdash\colon\ldotp(x,y).\phi(x,y).\supset.(w,z).\phi(z,w) &\qquad \text{(3)}\\ +\text{Similarly} &\vdash\colon\ldotp(w,z).\phi(z,w).\supset.(x,y).\phi(x,y) &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Note that "(\(w,z).\phi(z,w)\)" is the same proposition as +"(\(y,x).\phi(x,y)\)"; a proposition is not a function of any apparent +variable which occurs in it.</p> + +<p class="nind"> +<b>*11·21.</b> \(\vdash:(x,y,z).\phi(x,y,z).\equiv.(y,z,x).\phi(x,y,z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{(*11·01·02)}] \vdash\colon\colon(x,y,z).\phi(x,y,z).&\equiv\colon\ldotp(x)\colon\ldotp(y):(z).\phi(x,y,z)\colon\ldotp\\ +[\text{*11·2}] &\equiv\colon\ldotp(y)\colon\ldotp(x):(z).\phi(x,y,z)\colon\ldotp\\ +[\text{*11·2.*10·271}] &\equiv\colon\ldotp(y)\colon\ldotp(z):(x).\phi(x,y,z)\colon\ldotp\\ +[\text{(*11·01·02)}] &\equiv\colon\ldotp(y,z,x).\phi(x,y,z)\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·22.</b> \(\vdash:(\exists x,y).\phi(x,y).\equiv.{\sim}\{(x,y).{\sim}\phi(x,y)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·252.Transp.(*11·03)}.\supset\\ +\vdash:(\exists x,y).\phi(x,y). &\equiv.{\sim}\{(x):{\sim}(\exists y).\phi(x,y)\}.\\ +[\text{*10·252·271}] &\equiv.{\sim}\{(x):(y).{\sim}\phi(x,y)\}.\\ +[\text{(*11·01)}] &\equiv.{\sim}\{(x,y).{\sim}\phi(x,y)\}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·23.</b> \(\vdash:(\exists x,y).\phi(x,y).\equiv.(\exists y,x).\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·22}. \supset\vdash:(\exists x,y).\phi(x,y).&\equiv.{\sim}\{(x,y).{\sim}\phi(x,y)\}.\\ +[\text{*11·2.Transp}] &\equiv.{\sim}\{(y,x).{\sim}\phi(x,y)\}.\\ +[\text{*11·22}] &\equiv.(\exists y,x).\phi(x,y):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·24.</b> \(\vdash:(\exists x,y,z).\phi(x,y,z).\equiv.(\exists y,z,x).\phi(x,y,z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +[\text{(*11·03·04)}] \vdash\colon\colon(\exists x,y,z).\phi(x,y,z).&\equiv\colon\ldotp(\exists x)\colon\ldotp(\exists y):(\exists z).\phi(x,y,z)\colon\ldotp\\ +[\text{*11·23}] &\equiv\colon\ldotp(\exists y)\colon\ldotp(\exists x):(\exists z).\phi(x,y,z)\colon\ldotp\\ +[\text{*11·23.*10·281}] &\equiv\colon\ldotp(\exists y)\colon\ldotp(\exists z):(\exists x).\phi(x,y,z)\colon\ldotp\\ +[\text{(*11·03·04)}] &\equiv\colon\ldotp(\exists y,z,x).\phi(x,y,z)\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·25.</b> \(\vdash:{\sim}\{(\exists x,y).\phi(x,y)\}.\equiv.(x,y).{\sim}\phi(x,y) \quad[\text{*11·22.Transp}]\)</p> + +<p><span class="pagenum" id="Page_161">[Pg 161]</span></p> + +<p class="nind"> +<b>*11·26.</b> \(\vdash\colon\ldotp(\exists x):(y).\phi(x,y):\supset:(y):(\exists x).\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1·28}.\supset\vdash\colon\ldotp(\exists x):(y).\phi(x,y):\supset:(\exists x).\phi(x,y) &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Note that the converse of this proposition is false. <i>E.g.</i> let +\(\phi(x,y)\) be the propositional function "if \(y\) is a proper fraction, +then \(x\) is a proper fraction greater than \(y\)." Then for all +values of \(y\) we have (\(\exists x).\phi(x,y)\), so that (\(y):(\exists x).\phi(x,y)\) +is satisfied. In fact "(\(y):(\exists x).\phi(x,y)\)" +expresses the proposition: "If \(y\) is a proper fraction, then there +is always a proper fraction greater than \(y\)." But "(\(\exists x):(y).\phi(x,y)\)" +expresses the proposition: "There is a proper fraction which is greater +than any proper fraction," which is false.</p> + +<p class="nind"> +<b>*11·27.</b> \[\begin{align}\vdash\colon\ldotp(\exists x,y):(\exists z).\phi(x,y,z):&\equiv:(\exists x):(\exists y,z).\phi(x,y,z):\\ +&\equiv:(\exists x,y,z).\phi(x,y,z)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·2.(*11·03)}.\supset +\vdash\colon\colon(\exists x,y):(\exists z).\phi(x,y,z):\equiv\colon\ldotp(\exists x)\colon\ldotp(\exists y):(\exists z).\phi(x,y,z) &\qquad \text{(1)}\\ +\vdash.\text{*4·2.(*11·03)}.\supset\\ +\vdash\colon\ldotp(\exists y):(\exists z).\phi(x,y,z):\equiv:(\exists y,z).\phi(x,y,z) &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·281}.\supset\\ +\vdash\colon\colon(\exists x)\colon\ldotp(\exists y):(\exists z).\phi(x,y,z)\colon\ldotp\equiv\colon\ldotp(\exists x):(\exists y,z).\phi(x,y,z) &\qquad \text{(3)}\\ +\vdash.\text{(1).(3).(*11·04)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>All the propositions of <a href="#*10">*10</a> have analogues which hold for two or more +variables. The more important of these are proved in what follows.</p> + +<p class="nind"> +<b><a id="*11·3">*11·3</a>.</b> \(\vdash\colon\ldotp p.\supset.(x,y).\phi(x,y):\equiv:(x,y):p.\supset.\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·21}.\supset\vdash\colon\ldotp p.\supset.(x,y).\phi(x,y):&\equiv:(x):p.\supset.(y).\phi(x,y):\\ +[\text{*10·21·271}] &\equiv:(x,y):p.\supset.\phi(x,y)\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·31.</b> \(\vdash\colon\ldotp(x,y).\phi(x,y):(x,y).\psi(x,y):\equiv:(x,y):\phi(x,y).\psi(x,y)\)</p> + +<p>Here the conditions of significance on the right-hand side require that +\(\phi\) and \(\psi\) should take arguments of the same types.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}.\supset\vdash\colon\colon(x,y).&\phi(x,y):(x,y).\psi(x,y):\\ +&\equiv\colon\ldotp(x)\colon\ldotp(y).\phi(x,y):(y).\psi(x,y)\colon\ldotp\\ +[\text{*10·22·271}] &\equiv\colon\ldotp(x,y):\phi(x,y).\psi(x,y)\colon\colon\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>The proofs of most of the following propositions are conducted exactly +as those of *11·3·31 are conducted: the analogous proposition in <a href="#*10">*10</a> is +used<span class="pagenum" id="Page_162">[Pg 162]</span> twice, together with <a href="#*10·27">*10·27</a> or <a href="#*10·271">*10·271</a> or <a href="#*10·28">*10·28</a> or <a href="#*10·281">*10·281</a> as +the case may be. When proofs conform to this pattern we shall merely +give references to the propositions used.</p> + +<p class="nind"> +<b>*11·311.</b> If \(\phi(\hat{x},\hat{y})\), \(\psi(\hat{x},\hat{y})\) +take arguments of the same type, and we have "\(\vdash.\phi(x,y)\)" and +"\(\vdash.\psi(x,y)\)," we shall have "\(\vdash.\phi(x,y).\psi(x,y)\)." +[Proof as in <a href="#*10·13">*10·13</a>.]</p> + +<p class="nind"> +<b>*11·32.</b> \(\vdash\colon\ldotp(x,y):\phi(x,y).\supset.\psi(x,y):\supset:(x,y).\phi(x,y).\supset.(x,y).\psi(x,y) +\quad[\text{*10·27}]\)</p> + +<p class="nind"> +<b>*11·33.</b> \(\vdash\colon\ldotp(x,y):\phi(x,y).\equiv.\psi(x,y):\supset:(x,y).\phi(x,y).\equiv.(x,y).\psi(x,y) +\quad[\text{*10·271}]\)</p> + +<p class="nind"> +<b>*11·34.</b> \[\begin{align}\vdash\colon\ldotp(x,y):\phi(x,y).&\supset.\psi(x,y):\supset:\\ +&(\exists x,y).\phi(x,y).\supset.(\exists x,y).\psi(x,y) \quad[\text{*10·27·28}]\end{align}\]</p> + +<p class="nind"> +<b>*11·341.</b> \[\begin{align}\vdash\colon\ldotp(x,y):\phi(x,y).&\equiv.\psi(x,y):\supset:\\ +&(\exists x,y).\phi(x,y).\equiv.(\exists x,y).\psi(x,y) \quad[\text{*10·271·281}]\end{align}\]</p> + +<p class="nind"> +<b>*11·35.</b> \(\vdash\colon\ldotp(x,y):\phi(x,y).\supset.p:\equiv:(\exists x,y).\phi(x,y).\supset.p \quad[\text{*10·23·271}]\)</p> + +<p class="nind"> +<b>*11·36.</b> \(\vdash:\phi(z,w).\supset.(\exists x,y).\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·1}.\supset\vdash:(x,y).{\sim}\phi(x,y).\supset.{\sim}\phi(z,w) &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·37.</b> \[\begin{align}\vdash\colon\colon(x,y):\phi(x,y).\supset.\psi(x,y)&\colon\ldotp(x,y):\psi(x,y).\supset.\chi(x,y)\colon\ldotp\\ +&\supset:(x,y):\phi(x,y).\supset.\chi(x,y)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>In the following demonstration, "\(\text{Hp}\)" means the hypothesis +of the proposition to be proved. We shall employ this abbreviation, +whenever convenient, in all cases where the proposition to be proved is +a hypothetical, <i>i.e.</i> is of the form "\(p\supset q\)." Similarly +"\(\text{Hp (1)}\)" will mean "the hypothesis of (1)," and so on. +\[ +\begin{array}{l} +\vdash.\text{*11·31}.&\supset\vdash\colon\colon\text{Hp}.\supset\colon\ldotp(x,y)\colon\ldotp\phi(x,y).\supset.\psi(x,y):\psi(x,y).\supset.\chi(x,y) &\qquad \text{(1)}\\ +\vdash.\text{Syll.*11·11}.&\supset\vdash\colon\ldotp(x,y)\colon\ldotp\phi(x,y).\supset.\psi(x,y):\psi(x,y).\supset.\chi(x,y):\\ +&\supset:\phi(x,y).\supset.\chi(x,y)\colon\ldotp\\ +[\text{*11·32}] &\supset\vdash\colon\ldotp(x,y):\phi(x,y).\supset.\psi(x,y):\psi(x,y).\supset.\chi(x,y):\\ +&\supset:(x,y):\phi(x,y).\supset.\chi(x,y) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Syll}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_163">[Pg 163]</span></p> + +<p>The above is a type of proof which recurs frequently in what follows. +Proofs conforming to this pattern will be indicated only by the numbers +of the propositions used.</p> + +<p class="nind"> +<b>*11·371.</b> \[\begin{align}\vdash\colon\colon(x,y):\phi(x,y).&\equiv.\psi(x,y)\colon\ldotp(x,y):\psi(x,y).\equiv.\chi(x,y)\colon\ldotp\\ +&\supset\colon\ldotp(x,y):\phi(x,y).\equiv.\chi(x,y) \quad[\text{*11·31·11·33}]\end{align}\]</p> + +<p class="nind"> +<b>*11·38.</b> \[\begin{align}\vdash\colon\colon&(x,y):\phi(x,y).\supset.\psi(x,y)\colon\ldotp\supset\colon\ldotp\\ +&(x,y):\phi(x,y).\chi(x,y).\supset.\psi(x,y).\chi(x,y) \quad[\text{Fact.*11·11·32}]\end{align}\]</p> + +<p class="nind"> +<b>*11·39.</b> \[\begin{align}\vdash\colon\colon&(x,y):\phi(x,y).\supset.\psi(x,y)\colon\ldotp(x,y):\chi(x,y).\supset.\theta(x,y)\colon\ldotp\supset\colon\ldotp\\ +&(x,y):\phi(x,y).\chi(x,y).\supset.\psi(x,y).\theta(x,y)) \quad[\text{*3·47.*11·11·32}]\end{align}\]</p> + +<p class="nind"> +<b>*11·391.</b> \[\begin{align}\vdash\colon\colon(x,y):\phi(x,y).\supset.&\psi(x,y)\colon\ldotp(x,y):\phi(x,y).\supset.\chi(x,y)\colon\ldotp\\ +&\equiv:(x,y):\phi(x,y).\supset.\psi(x,y).\chi(x,y)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·76}. \supset\vdash\colon\ldotp\phi(x,y)&.\supset.\psi(x,y):\phi(x,y).\supset.\chi(x,y):\\ +&\equiv:\phi(x,y).\supset.\psi(x,y).\chi(x,y)\colon\ldotp\\ +[\text{*11·11·33}] \supset\vdash\colon\ldotp(x,y):\phi(x,y)&.\supset.\psi(x,y):\phi(x,y).\supset.\chi(x,y):\\ +&\equiv:(x,y):\phi(x,y).\supset.\psi(x,y).\chi(x,y)\colon\colon\\ +[\text{*11·31}] \supset\vdash\colon\colon(x,y):\phi(x,y)&.\supset.\psi(x,y)\colon\ldotp(x,y):\phi(x,y).\supset.\chi(x,y)\colon\ldotp\\ +&\equiv:(x,y):\phi(x,y).\supset.\psi(x,y).\chi(x,y)\colon\colon\\ +\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·4.</b> \[\begin{align}\vdash\colon\colon&(x,y):\phi(x,y).\equiv.\psi(x,y)\colon\ldotp(x,y):\chi(x,y).\equiv.\theta(x,y)\colon\ldotp\supset\colon\ldotp\\ +&(x,y):\phi(x,y).\chi(x,y).\equiv.\psi(x,y).\theta(x,y)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·31}.\supset\vdash\colon\colon\text{Hp}.&\supset\colon\ldotp(x,y)\colon\ldotp\phi(x,y).\equiv.\psi(x,y):\chi(x,y).\equiv.\theta(x,y)\colon\ldotp\\ +[\text{*4·38.*11·11·32}] &\supset\colon\ldotp(x,y):\phi(x,y).\chi(x,y).\equiv.\psi(x,y).\theta(x,y)\colon\colon\\ +&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·401.</b> \[\begin{align}\vdash\colon\colon&(x,y):\phi(x,y).\equiv.\psi(x,y):\supset\colon\ldotp\\ +&(x,y):\phi(x,y).\chi(x,y).\equiv.\psi(x,y).\chi(x,y) \quad\left[\text{*11·4}\, \frac{\chi}{\theta}.\text{Id}\right]\end{align}\]</p> + +<p class="nind"> +<b>*11·41.</b> \[\begin{align}\vdash\colon\ldotp(\exists x,y).\phi(x,y)&:\lor:(\exists x,y).\psi(x,y):\\ +&\equiv:(\exists x,y):\phi(x,y).\lor.\psi(x,y) \quad[\text{*10·42·281}]\end{align}\]</p> + +<p class="nind"> +<b>*11·42.</b> \(\vdash\colon\ldotp(\exists x,y).\phi(x,y).\psi(x,y).\supset:(\exists x,y).\phi(x,y):(\exists x,y).\psi(x,y) +\quad[\text{*10·5}]\)</p> + +<p class="nind"> +<b>*11·421.</b> \[\begin{align}\vdash\colon\ldotp(x,y).\phi(x,y).\lor.(x,y).&\psi(x,y):\supset:(x,y):\phi(x,y).\lor.\psi(x,y)\\ +&\left[\text{*11·42}\, \frac{{\sim}\phi,\,{\sim}\psi}{\phi,\,\psi}.\, \text{Transp.*4·56}\right]\end{align}\]</p> + +<p class="nind"> +<b>*11·43.</b> \(\vdash\colon\ldotp(\exists x,y):\phi(x,y).\supset.p:\equiv:(x,y).\phi(x,y).\supset.p \quad[\text{*10·34·281}]\)</p> + +<p class="nind"> +<b>*11·44.</b> \(\vdash\colon\ldotp(x,y):\phi(x,y).\lor.p:\equiv:(x,y).\phi(x,y).\lor.p \quad[\text{*10·2·271}]\)</p> + +<p><span class="pagenum" id="Page_164">[Pg 164]</span></p> + +<p class="nind"> +<b>*11·45.</b> \(\vdash \colon\ldotp (\exists x, y) : p . \phi (x, y) : \equiv : p : (\exists x, y) . \phi(x, y) \quad[\text{*10·35·281}]\)</p> + +<p class="nind"> +<b>*11·46.</b> \(\vdash \colon\ldotp (\exists x, y) : p . \supset . \phi (x, y) : \equiv : p . \supset . (\exists x, y) . \phi(x, y) \quad[\text{*10·37·281}]\)</p> + +<p class="nind"> +<b>*11·47.</b> \(\vdash \colon\ldotp (x, y) : p . \phi(x, y) : \equiv : p : (x, y) . \phi(x, y) \quad[\text{*10·33·271}]\)</p> + +<p class="nind"> +<b>*11·5.</b> \(\vdash \colon\ldotp (\exists x) : {\sim}\{(y) . \phi(x, y)\} : \equiv : {\sim}\{(x, y) . \phi(x, y)\} : \equiv : (\exists x, y) . {\sim}\phi(x, y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . \text{*10·253}. \supset \vdash \colon\ldotp (\exists x) : {\sim}\{(y) . \phi (x, y)\} : &\equiv : {\sim}\{(x) : (y) . \phi(x, y)\} :\\ +[\text{(*11·01)}] &\equiv : {\sim}\{(x, y) . \phi(x, y)\} &\qquad \text{(1)}\\ +\vdash . \text{*10·253}. \supset \vdash : {\sim}\{(y) . \phi(x, y)\} . &\equiv . (\exists y) . {\sim}\phi(x, y) :\\ +[\text{*10·11·281}] \supset \vdash \colon\ldotp (\exists x) : {\sim}\{(y) . \phi (x, y)\} : &\equiv : (\exists x) : (\exists y) . {\sim}\phi(x, y) :\\ +[\text{(*11·03)}] & \equiv : (\exists x, y) . {\sim}\phi(x, y) &\qquad \text{(2)}\\ +\vdash . \text{(1).(2)}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·51.</b> \(\vdash \colon\ldotp (\exists x) : (y) . \phi(x, y) : \equiv : {\sim}{(x) : (\exists y) . {\sim}\phi(x, y)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*10·252.Transp}. \supset \vdash \colon\ldotp (\exists x) : (y) . \phi(x, y) : &\equiv : {\sim}[(x) : {\sim}(y) . \phi(x, y)] &\qquad \text{(1)}\\ +\vdash .\text{*10·253}. \supset \vdash \colon\ldotp {\sim}(y) . \phi(x, y) . & \equiv : (\exists y) . {\sim}\phi(x, y) \colon\ldotp\\ +[\text{*10·11·271}] \supset \vdash \colon\ldotp (x) : {\sim}(y) . \phi(x, y) : &\equiv : (x) : (\exists y) . {\sim}\phi(x, y) \colon\ldotp\\ +[\text{Transp}]. \supset \vdash \colon\ldotp {\sim}[(x) : {\sim}\{(y) . \phi(x, y)\}]. & \equiv : {\sim}\{(x) : (\exists y) . {\sim}\phi(x, y)\} &\qquad \text{(2)}\\ +\vdash .\text{(1).(2)}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·52.</b> \(\vdash \colon\ldotp (\exists x, y) . \phi(x, y) . \psi(x, y) . \equiv . {\sim}{(x, y) : \phi(x, y) . \supset . {\sim}\psi(x, y)}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*4·51·62}. \supset\\ +\vdash \colon\ldotp {\sim}\{\phi(x,y) . \psi(x,y)\} . & \equiv : \phi(x, y) . \supset . {\sim}\psi(x, y) &\qquad \text{(1)}\\ +\vdash .\text{(1).*11·11·33}. \supset\\ +\vdash \colon\ldotp (x,y) . {\sim}\{\phi(x,y) . \psi(x,y)\} : &\equiv : (x, y) : \phi(x, y) . \supset . {\sim}\psi(x, y) &\qquad \text{(2)}\\ +\vdash : \text{(2).Transp.*11·22}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11.521.</b> \[\begin{align}\vdash \colon\ldotp {\sim}(\exists x,y) . \phi(x, y) . {\sim}\psi(x, y) . &\equiv : (x, y) : \phi(x, y) . \supset . \psi(x, y)\\ +&\left[\text{*11·52.Transp}.\, \frac{{\sim}\psi(x,y)}{\psi(x, y)}\right]\end{align}\]</p> + +<p class="nind"> +<b>*11.53.</b> \(\vdash \colon\ldotp (x, y) . \phi x \supset \psi y . \equiv : (\exists x) . \phi x . \supset . (y) . \psi y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*10·21.271}. \supset \vdash \colon\ldotp (x, y) . \phi x \supset \psi y . &\equiv : (x) : \phi x . \supset . (y) . \psi y :\\ +[\text{*10·23}] &\equiv : (\exists x) . \phi x . \supset . (y) . \psi y \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·54.</b> \(\vdash \colon\ldotp (\exists x, y) . \phi x . \psi y . \equiv : (\exists x) . \phi x : (\exists y) . \psi y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*10·35}. \supset \vdash \colon\ldotp (\exists y) . \phi x . \psi y . &\equiv : \phi x : (\exists y) . \psi y \colon\ldotp\\ +[\text{*10·11·281}] \supset \vdash \colon\ldotp (\exists x, y) . \phi x . \psi y . &\equiv : (\exists x) : \phi x : (\exists y) . \psi y :\\ +[\text{*10·35}] & \equiv : (\exists x) . \phi x : (\exists y) . \psi y \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This proposition is very often used.</p> + +<p><span class="pagenum" id="Page_165">[Pg 165]</span></p> + +<p class="nind"> +<b>*11·55.</b> \(\vdash\colon\ldotp(\exists x,y).\phi x.\psi(x,y).\equiv:(\exists x):\phi x:(\exists y).\psi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·35}.\supset\vdash\colon\ldotp(\exists y).\phi x.\psi(x,y). &\equiv:\phi x:(\exists y).\psi(x,y)\colon\ldotp\\ +[\text{*10·11}] \supset\vdash\colon\ldotp(x)\colon\ldotp(\exists y).\phi x.\psi(x,y).&\equiv:\phi x:(\exists y).\psi(x,y)\colon\ldotp\\ +[\text{*10·281}] \supset\vdash\colon\ldotp(\exists x):(\exists y).\phi x.\psi(x,y).&\equiv:(\exists x):\phi x:(\exists y).\psi(x,y)\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is very often used.</p> + +<p class="nind"> +<b>*11·56.</b> \(\vdash\colon\ldotp(x).\phi x:(y).\psi y:\equiv:(x,y).\phi x.\psi y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·33}.\supset\vdash\colon\colon(x).\phi x:(y).\psi y:&\equiv\colon\ldotp(x)\colon\ldotp\phi x:(y).\psi y &\qquad \text{(1)}\\ +\vdash.\text{*10·33}.\supset\vdash\colon\ldotp \phi x:(y).\psi y:&\equiv:(y).\phi x.\psi y\colon\ldotp\\ +[\text{*10·11}] \supset\vdash\colon\ldotp(x)\colon\ldotp\phi x:(y).\psi y:&\equiv:(y).\phi x.\psi y\colon\ldotp\\ +[\text{*10·271}] \supset\vdash\colon\colon(x)\colon\ldotp\phi x:(y).\psi y\colon\ldotp&\equiv:(x):(y).\phi x.\psi y:\\ +[\text{(*11·01)}] &\equiv:(x,y).\phi x.\psi y &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·57.</b> \(\vdash:(x).\phi x.\equiv.(x,y).\phi x.\phi y \quad[\text{*11·56.*4·24}]\)</p> + +<p>The use of <a href="#*4·24">*4·24</a> here depends upon the fact that (\(x).\phi x\) and +(\(y).\phi y\) are the same proposition.</p> + +<p class="nind"> +<b>*11·58.</b> \(\vdash:(\exists x).\phi x.\equiv.(\exists x,y).\phi x.\phi y \quad[\text{*11·54.*4·24}]\)</p> + +<p class="nind"> +<b>*11·59.</b> \(\vdash\colon\ldotp\phi x.\supset_{x}.\psi x:\equiv:\phi x.\phi y.\supset_{x,y}.\psi x.\psi y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·57}.\supset\vdash\colon\ldotp\phi x.\supset_{x}.\psi x:\equiv:(x,y):\phi x.\supset.\psi x:\phi y.\supset.\psi y:\\ +[\text{*3·47.*11·32}] \supset:(x,y):\phi x.\phi y.\supset.\psi x.\psi y &\qquad \text{(1)}\\ +\vdash.\text{*11·1}.\supset\vdash\colon\ldotp(x,y):\phi x.\phi y.\supset.\psi x.\psi y:\supset:\phi x.\phi y.\supset.\psi x.\psi y &\qquad \text{(2)}\\ +\vdash.\text{(2)}\, \frac{x}{y}.\text{*4·24}.\supset\vdash\colon\ldotp\text{Hp (2)}.\supset:\phi x.\supset.\psi x &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·21}.\supset\\ +\vdash\colon\ldotp(x,y):\phi x.\phi y.\supset.\psi x.\psi y:\supset:\phi x.\supset_{x}.\psi x &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·6.</b> \(\vdash\colon\colon(\exists x)\colon\ldotp(\exists y).\phi(x,y).\psi y:\chi x\colon\ldotp\equiv\colon\ldotp(\exists y)\colon\ldotp(\exists x).\phi(x,y).\chi x:\psi y\)</p> + +<p>This proposition is very frequently employed in subsequent proofs.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·35}. \supset\vdash\colon\ldotp(\exists y).\phi(x,y).\psi y:\chi x:&\equiv:(\exists y):\phi(x,y).\psi y.\chi x\colon\ldotp\\ +[\text{*10·11·281}]\supset\vdash\colon\colon(\exists x)\colon\ldotp(\exists y).\phi(x,y).\psi y:\chi x:\\ +&\equiv\colon\ldotp(\exists x)\colon\ldotp(\exists y).\phi(x,y).\psi y.\chi x\colon\ldotp\\ +[\text{*11·23}] &\equiv\colon\ldotp(\exists y)\colon\ldotp(\exists x).\phi(x,y).\psi y.\chi x\colon\ldotp\\ +[\text{*11·341.Perm}] &\equiv\colon\ldotp(\exists y)\colon\ldotp(\exists x).\phi(x,y).\chi x.\psi y\colon\ldotp\\ +[\text{*10·35·281}] &\equiv\colon\ldotp(\exists y)\colon\ldotp(\exists x).\phi(x,y).\chi x:\psi y\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_166">[Pg 166]</span></p> + +<p class="nind"> +<b>*11·61.</b> \(\vdash\colon\ldotp(\exists y):\phi x.\supset_{x}.\psi(x,y):\supset:\phi x.\supset_{x}.(\exists y).\psi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·26}. &\supset\vdash\colon\colon\text{Hp}.\supset\colon\ldotp(x)\colon\ldotp(\exists y):\phi x.\supset.\psi(x,y) &\qquad \text{(1)}\\ +\vdash.\text{*10·37}. &\supset\vdash\colon\ldotp(\exists y):\phi x.\supset.\psi(x,y):\supset:\phi x.\supset.(\exists y).\psi(x,y)\colon\ldotp\\ +[\text{*10·11·27}] &\supset\vdash\colon\colon(x)\colon\ldotp(\exists y):\phi x.\supset.\psi(x,y)\colon\ldotp\supset\colon\ldotp(x):\phi x.\supset.(\exists y).\psi(x,y) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·62.</b> \(\vdash\colon\colon\phi x.\psi(x,y).\supset_{x,y}.\chi(x,y):\equiv\colon\ldotp\phi x.\supset_{x}:\psi(x,y).\supset_{y}.\chi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·87.*11·11·33}. \supset\\ +\vdash\colon\colon\phi x.\psi(x,y).\supset_{x,y}.\chi(x,y):&\equiv\colon\ldotp(x,y)\colon\ldotp\phi x.\supset:\psi(x,y).\supset.\chi(x,y)\\ +[\text{*10·21·11·271}] &\equiv\colon\ldotp(x)\colon\ldotp\phi x.\supset:(y):\psi(x,y).\supset.\chi(x,y)\colon\colon\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·63.</b> \(\vdash\colon\ldotp{\sim}(\exists x,y).\phi(x,y).\supset:\phi(x,y).\supset_{x,y}.\psi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*2·21.*11·11}. &\supset\vdash\colon\ldotp(x,y)\colon\ldotp{\sim}\phi(x,y).\supset:\phi(x,y).\supset.\psi(x,y)\colon\ldotp\\ +[\text{*11·32}] &\supset\vdash\colon\ldotp(x,y).{\sim}\phi(x,y).\supset:(x,y):\phi(x,y).\supset.\psi(x,y)\colon\ldotp\\ +[\text{*11·25}] &\supset\vdash\colon\ldotp{\sim}(\exists x,y).\phi(x,y).\supset:(x,y):\phi(x,y).\supset.\psi(x,y)\colon\ldotp\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*11·7.</b> \(\vdash\colon\ldotp(\exists x,y):\phi(x,y).\lor.\phi(y,x):\equiv.(\exists x,y).\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·41}. \supset\vdash\colon\ldotp(\exists x,y):\phi(x,y).&\lor.\phi(y,x):\\ +&\equiv:(\exists x,y).\phi(x,y).\lor.(\exists x,y).\phi(y,x):\\ +[\text{*11·23}] &\equiv:(\exists x,y).\phi(x,y).\lor.(\exists y,x).\phi(y,x):\\ +[\text{*4·25}] &\equiv:(\exists x,y).\phi(x,y)\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the last line of the above proof, use is made of the fact that +\[ +(\exists x,y).\phi(x,y)\, \text{and}\, (\exists y,x).\phi(y,x) +\] +are the same proposition.</p> + +<p>The first use of the following proposition occurs in the proof of +*234·12. Its utility lies in its enabling us to pass from a hypothesis +\[ +\phi z.\chi w.\supset_{z,w}.\psi z.\theta w, +\] +containing two apparent variables, to the product of two hypotheses +each containing only one.</p> + +<p><span class="pagenum" id="Page_167">[Pg 167]</span></p> + +<p class="nind"> +<b>*11·71.</b> \[\begin{align}\vdash\colon\colon(\exists z).\phi z:&(\exists w).\chi w:\supset\colon\ldotp\\ +&\phi z.\supset_{z}.\psi z:\chi w.\supset_{w}.\theta w:\equiv:\phi z.\chi w.\supset_{z,w}.\psi z.\theta w\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1.*3·47}.\supset\vdash\colon\ldotp\phi z.\supset_{z}.\psi z:\chi w.\supset_{w}.\theta w:\\ +&\supset:\phi z.\chi w.\supset.\psi z.\theta w &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·3}.\supset\vdash\colon\ldotp\phi z.\supset_{z}.\psi z:\chi w.\supset_{w}.\theta w:\\ +&\supset:\phi z.\chi w.\supset_{z,w}.\psi z.\theta w &\qquad \text{(2)}\\ +\vdash.\text{*10·1}.\supset\vdash\colon\colon\phi z.\chi w.\supset_{z,w}.\psi z.\theta w:\supset\colon\ldotp\phi z.\chi w.\supset_{w}.\psi z.\theta w\colon\ldotp\\ +[\text{*10·28}] &\supset\colon\ldotp(\exists w).\phi z.\chi w.\supset.(\exists w).\psi z.\theta w\colon\ldotp\\ +[\text{*10·35}] &\supset\colon\ldotp\phi z:(\exists w):\chi w:\supset:\psi z:(\exists w):\theta w &\qquad \text{(3)}\\ +\vdash.\text{(3).Comm.*3·26}.\supset\vdash\colon\colon(\exists w).\chi w:\supset\colon\ldotp\phi z.\chi w.\supset_{z,w}.\psi z.\theta w:\\ +&\supset:\phi z.\supset.\psi z &\qquad \text{(4)}\\ +\vdash.\text{(4).*10·11·21}.\supset\vdash\colon\colon(\exists w).\chi w.\supset\colon\ldotp\phi z.\chi w.\supset_{z,w}.\psi z.\theta w:\\ +&\supset:\phi z.\supset_{z}.\psi z &\qquad \text{(5)}\\ +\text{Similarly} \vdash\colon\colon(\exists z).\phi z.\supset\colon\ldotp\phi z.\chi w.\supset_{z,w}.\psi z.\theta w:\\ +&\supset:\chi w.\supset_{w}.\theta w &\qquad \text{(6)}\\ +\vdash.\text{(5).(6).*3·47.Comp}.\supset\\ +\vdash\colon\colon\text{Hp}.\supset\colon\ldotp\phi z.\chi w.\supset_{z,w}.\psi z.\theta w:\supset:\phi z.\supset_{z}.\psi z:\chi w.\supset_{w}.\theta w &&\qquad \text{(7)}\\ +\vdash.\text{(2).(7)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_168">[Pg 168]</span></p> + +<h2 class="nobreak" id="*12">*12. THE HIERARCHY OF TYPES AND THE AXIOM OF REDUCIBILITY.</h2> +</div> + + +<p>The primitive idea "\((x).{\phi}x\)" has been explained to mean +"\({\phi}x\) is always true," <i>i.e.</i> "all values of \({\phi}x\) +are true." But whatever function \(\phi\) may be, there will be +arguments \(x\) with which \({\phi}x\) is meaningless, <i>i.e.</i> with +which as arguments \(\phi\) does not have any value. The arguments +with which \({\phi}x\) has values form what we will call the "range of +significance" of \({\phi}x\). A "<i>type</i>" is defined as the range +of significance of some function. In virtue of <a href="#*9·14">*9·14</a>, if \({\phi}x\), +\({\phi}y\), and \({\psi}x\) are significant, <i>i.e.</i> either true +or false, so is \({\psi}y\). From this it follows that two types +which have a common member coincide, and that two different types +are mutually exclusive. Any proposition of the form \((x).{\phi}x\), +<i>i.e.</i> any proposition containing an apparent variable, determines +some type as the range of the apparent variable, the type being fixed +by the function \(\phi\).</p> + +<p>The division of objects into types is necessitated by the +vicious-circle fallacies which otherwise arise<a id="FNanchor_52" href="#Footnote_52" class="fnanchor">[52]</a>. These fallacies +show that there must be no totalities which, if legitimate, would +contain members defined in terms of themselves. Hence any expression +containing an apparent variable must not be in the range of that +variable, <i>i.e.</i> must belong to a different type. Thus the +apparent variables contained or presupposed in an expression are what +determines its type. This is the guiding principle in what follows.</p> + +<p>As explained in <a href="#*9">*9</a>, propositions containing variables are generated +from propositional functions which do not contain these apparent +variables, by the process of asserting all or some values of such +functions. Suppose \({\phi}a\) is a proposition containing \(a\); +we will give the name of <i>generalization</i> to the process which +turns \({\phi}a\) into \((x).{\phi}x\) or \((\exists x).{\phi}x\), +and we will give the name of <i>generalized propositions</i> to all +such as contain apparent variables. It is plain that propositions +containing apparent variables presuppose others not containing +apparent variables, from which they can be derived by generalization. +Propositions which contain no apparent variables we call <i>elementary +propositions</i><a id="FNanchor_53" href="#Footnote_53" class="fnanchor">[53]</a>, and the terms of such propositions, other than +functions, we call <i>individuals</i>. Then individuals form the first +type.</p> + +<p><span class="pagenum" id="Page_169">[Pg 169]</span></p> + +<p>It is unnecessary, in practice, to know what objects belong to the +lowest type, or even whether the lowest type of variable occurring +in a given context is that of individuals or some other. For in +practice only the <i>relative</i> types of variables are relevant; +thus the lowest type occurring in a given context may be called that +of individuals, so far as that context is concerned. It follows that +the above account of individuals is not essential to the truth of what +follows; all that is essential is the way in which other types are +generated from individuals, however the type of individuals may be +constituted.</p> + +<p>By applying the process of generalization to individuals occurring in +elementary propositions, we obtain new propositions. The legitimacy of +this process requires only that no individuals should be propositions. +That this is so, is to be secured by the meaning we give to the word +<i>individual</i>. We may explain an individual as something which +exists on its own account; it is then obviously not a proposition, +since propositions, as explained in Chapter II of the Introduction (<a href="#Page_46">p. 46</a>), +are incomplete symbols, having no meaning except in use. Hence in +applying the process of generalization to individuals we run no risk of +incurring reflexive fallacies. We will give the name of <i>first-order +propositions</i> to such as contain one or more apparent variables +whose possible values are individuals, but contain no other apparent +variables. First-order propositions are not all of the same type, +since, as was explained in <a href="#*9">*9</a>, two propositions which do not contain +the same number of apparent variables cannot be of the same type. But +owing to the systematic ambiguity of negation and disjunction, their +differences of type may usually be ignored in practice. No reflexive +fallacies will result, since no first-order proposition involves any +totality except that of individuals.</p> + +<p>Let us denote by "\(\phi!\hat{x}\)" or "\(\phi!(\hat{x},\hat{y})\)" +or etc. an elementary function whose argument or arguments are +individual. We will call such a function a <i>predicative function of +an individual</i>. Such functions, together with those derived from +them by generalization, will be called <i>first-order functions</i>. In +practice we may without risk of reflexive fallacies treat first-order +functions as a type, since the only totality they involve is that of +individuals, and, by means of the systematic ambiguity of negation +and disjunction, any function of a first-order function which will +concern us will be significant whatever first-order function is taken +as argument, provided the right meanings are given to the negations and +disjunctions involved.</p> + +<p>For the sake of clearness, we will repeat in somewhat different terms +our account of what is meant by a first-order function. Let us give +the name of <i>matrix</i> to any function, of however many variables, +which does not involve any apparent variables. Then any possible +function other than a matrix is derived from a matrix by means of +generalization, <i>i.e.</i> by considering the proposition which +asserts that the function in question is true with all<span class="pagenum" id="Page_170">[Pg 170]</span> possible values +or with some value of one of the arguments, the other argument or +arguments remaining undetermined. Thus <i>e.g.</i> from the function +\(\phi(x,y)\) we shall be able to derive the four functions +\[ +(x).\phi(x,y),\quad (\exists x).\phi(x,y),\quad (y).\phi(x,y),\quad (\exists y).\phi(x,y), +\] +of which the two first are functions of \(y\), while the two last are +functions of \(x\). (All <i>propositions</i>, with the exception of +such as are values of matrices, are also derived from matrices by the +above process of generalization. In order to obtain a proposition from +a matrix containing \(n\) variables, without assigning values to any of +the variables, it is necessary to turn all the variables into apparent +variables. Thus if \(\phi(x,y)\) is a matrix, (\(x,y).\phi(x,y)\) is +a proposition.) We will give the name <i>first-order matrices</i> to +such as have only individuals for their arguments, and we will give the +name of <i>first-order functions</i> (of any number of variables) to +such as either are first-order matrices or are derived from first-order +matrices by generalization applied to some (not all) of the arguments +to such matrices. First-order <i>propositions</i> will be such as +result from applying generalization to <i>all</i> the arguments to a +first-order matrix.</p> + +<p>As we have already stated, the notation "\(\phi!\hat{z}\)" is used +for any elementary function of one variable. Thus "\(\phi!x\)" +represents any value of any elementary function of one variable. It +will be seen that "\(\phi!x\)" is a function of two variables, namely +\(\phi!\hat{z}\) and \(x\). Since it contains no apparent variable, it +is a matrix, but since it contains a variable (namely \(\phi!\hat{z})\) +which is not an individual, it is not a first-order matrix. The same +applies to \(\phi!a\), where a is some definite constant. We can build +up a number of new matrices, such as +\[ +\begin{split} +{\sim} \phi!a,\quad {\sim} \phi!x,\quad \phi!x\lor \phi!y,\quad \phi!x\lor \psi!x,\quad \phi!x\lor\psi!y,\\ +\phi!x.\supset.\psi!x,\quad \phi!x.\psi!x,\quad \phi!x\lor\psi!y\lor\chi!z,\quad\text{ and so on.} +\end{split} +\] +All these are matrices which involve first-order functions among their +arguments. Such matrices we will call <i>second-order matrices</i>. +From these matrices, by applying generalization to their arguments, +whether to such as are functions or to such (if any) as are +individuals, we obtain new functions and propositions. Such functions +(together with second-order matrices) will be called <i>second-order +functions</i>, and such propositions will be called <i>second-order +propositions</i>. Thus we are led to the following definitions:</p> + +<p>A <i>second-order matrix</i> is one which has at least one first-order +matrix among its arguments, but has no arguments other than first-order +matrices and individuals.</p> + +<p>A <i>second-order function</i> is one which either is a second-order +matrix or results from one by applying generalization to some (not all) +of the arguments to a second-order matrix.</p> + +<p>A <i>second-order proposition</i> is one which results from a +second-order matrix by applying generalization to all its arguments.</p> + +<p><span class="pagenum" id="Page_171">[Pg 171]</span></p> + +<p>In addition to the above illustrations of second-order matrices, we may +give the following examples of second-order functions:</p> + +<p>(1) Functions in which the argument is \(\phi!\hat{z}: (x).\phi!x\), +(\(\exists x).\phi!x\), \(\phi!a.\supset.\phi!b\), where \(a\) and +\(b\) are constants, \(\phi!x.\supset_{x}.g!x\), where \(g!\hat{z}\) is +a constant function, and so on.</p> + +<p>(2) Functions in which the arguments are \(\phi!\hat{z}\) and +\(\psi!\hat{z}\): +\[ +\phi!x.\supset_{x}.\psi!x,\quad \phi!x.\equiv_{x}.\psi!x,\quad (\exists x).\phi x.\psi x,\quad \phi!a.\supset.\psi!b, +\] +where \(a\) and \(b\) are constants, and so on.</p> + +<p>(3) Functions in which the argument is an individual +\(x: (\phi).\phi!x\), \(\exists \phi).\phi!x\), +\(\phi!x.\supset_{\phi}.\phi!a\), where \(a\) is constant, and so on.</p> + +<p>(4) Functions in which the arguments are \(\phi!\hat{z}\) and \(x:\phi!x\), +\(\phi!x.\supset.\phi!a\), where \(a\) is constant, +(\(\exists \psi):\phi!x.\equiv.\psi!x\) and so on.</p> + +<p>Examples of second-order functions might, of course, be multiplied +indefinitely, but the above seem sufficient for purposes of +illustration.</p> + +<p>A second-order matrix of one variable will be called a <i>predicative +second-order function of one variable</i> or a <i>predicative function +of a first-order matrix</i>. Thus \(\phi!a\), \({\sim}\phi!a\) and +\(\phi!a\supset\phi!b\) are predicative functions of \(\phi!\hat{z}\). +Similarly a function of several variables of which at least one +is a first-order matrix, while the rest are either individuals or +first-order matrices, will be called <i>predicative</i> if it is a +matrix.</p> + +<p>It will be seen, however, that a second-order function may have only +individuals for its arguments; instances were given just now under +the heading (3). Such functions we shall not call predicative, since +predicative functions of individuals have already been defined as +being such as are of the first order. Thus the order of a function is +not determined by the order of its argument or arguments; indeed, the +function may be of any order superior to the order or orders of its +arguments.</p> + +<p>A variable matrix whose argument is \(\phi!\hat{z}\) will be denoted +by \(f!\phi!\hat{z}\), and generally, a matrix whose arguments are +\(\phi!\hat{z}\), \(\psi!\hat{z}\), ... \(x\), \(y\), ... (where there +is at least one function among the arguments) will be denoted by +\[ +f!(\phi!\hat{z}, \psi!\hat{z}, ... x, y, ...). +\] +Such a matrix is not of the first or second order, since it contains +the new variable \(f\) whose values are second-order matrices. +We proceed to construct new matrices as we did with the matrix +\(\phi!\hat{x}\); these constitute <i>third-order matrices</i>. These +together with the functions derived from them by generalization are +called <i>third-order functions</i>, and the propositions derived +from third-order matrices by generalization are called <i>third-order +propositions</i>.</p> + +<p><span class="pagenum" id="Page_172">[Pg 172]</span></p> + +<p>In this way we can proceed indefinitely to matrices, functions and +propositions of higher and higher orders. We introduce the following +definition:</p> + +<p>A function is said to be <i>predicative</i> when it is a matrix. It +will be observed that, in a hierarchy in which all the variables are +individuals or matrices, a matrix is the same thing as an elementary +function (cf. <a href="#Page_132">pp. 132</a>, <a href="#Page_133">133</a>).</p> + +<p>"Matrix" or "predicative function" is a primitive idea.</p> + +<p>The fact that a function is predicative is indicated, as above, by a +note of exclamation after the functional letter.</p> + +<p>The variables occurring in the present work, from this point onwards, +will all be either individuals or matrices of some order in the above +hierarchy. Propositions, which have occurred hitherto as variables, +will no longer do so except in a few isolated cases of which no +subsequent use is made. In practice, for the reasons explained on <a href="#Page_169">p. 169</a>, +a function of a matrix may be regarded as capable of any argument +which is a function of the same order and takes arguments of the same +type.</p> + +<p>In practice, we never need to know the absolute types of our variables, +but only their <i>relative</i> types. That is to say, if we prove +any proposition on the assumption that one of our variables is an +individual, and another is a function of order \(n\), the proof will +still hold if, in place of an individual, we take a function of order +\(m\), and in place of our function of order \(n\) we take a function +of order \(n + m\), with corresponding changes for any other variables +that may be involved. This results from the assumption that our +primitive propositions are to apply to variables of any order.</p> + +<p>We shall use small Latin letters (other than \(p\), \(q\), \(r\), +\(s)\) for variables of the lowest type concerned in any context. For +functions, we shall use the letters \(\phi\), \(\psi\), \(\chi\), +\(\theta\), \(f\), \(g\), \(F\) (except that, at a later stage, \(F\) +will be defined as a constant relation, and \(\theta\) will be defined +as the order-type of the continuum).</p> + +<p>We shall explain later a different hierarchy, that of classes and +relations, which is derived from the functional hierarchy explained +above, but is more convenient in practice.</p> + +<p>When any predicative function, say \(\phi!\hat{z}\), occurs as +apparent variable, it would be strictly more correct to indicate the +fact by placing "(\(\phi!\hat{z})\)" before what follows, as thus: +"(\(\phi!\hat{z}).f(\phi!\hat{z})\)." But for the sake of brevity we +write simply "(\(\phi)\)" instead of "(\(\phi!\hat{z})\\)." Since what +follows the \(\phi\) in brackets must always contain \(\phi\) with +arguments supplied, no confusion can result from this practice.</p> + +<p>It should be observed that, in virtue of the manner in which our +hierarchy of functions was generated, non-predicative functions always +result from such as are predicative by means of generalization. Hence +it is unnecessary to introduce a special notation for non-predicative +functions of a given order<span class="pagenum" id="Page_173">[Pg 173]</span> and taking arguments of a given order. +For example, second-order functions of an individual \(x\) are always +derived by generalization from a matrix +\[ +f!(\phi!\hat{z},\, \psi!\hat{z},\, \ldots x,\, y,\, z,\, ...), +\] +where the functions \(f\), \(\phi\), \(\psi\), ... are predicative. It +is possible, therefore, without loss of generality, to use no apparent +variables except such as are predicative.</p> + +<p>We require, however, a means of symbolising a function whose order is +not assigned. We shall use "\(\phi x\)" or "\(f(\chi!\hat{z})\)" or +etc. to express a function (\(\phi\) or \(f\)) whose order, relatively +to its argument, is not given. Such a function cannot be made into an +apparent variable, unless we suppose its order previously fixed. As the +only purpose of the notation is to avoid the necessity of fixing the +order, such a function will not be used as an apparent variable; the +only functions which will be so used will be predicative functions, +because, as we have just seen, this restriction involves no loss of +generality.</p> + +<p>We have now to state and explain the <i>axiom of reducibility</i>.</p> + +<p>It is important to observe that, since there are various types of +propositions and functions, and since generalization can only be +applied within some one type (or, by means of systematic ambiguity, +within some well-defined and completed set of types), all phrases +referring to "all propositions" or "all functions," or to "some +(undetermined) proposition" or "some (undetermined) function," are +<i>prima facie</i> meaningless, though in certain cases they are +capable of an unobjectionable interpretation. Contradictions arise from +the use of such phrases in cases where no innocent meaning can be found.</p> + +<p>If mathematics is to be possible, it is absolutely necessary (as +explained in the Introduction, <a href="#CHAPTER_II">Chapter II</a>) that we should have some +method of making statements which will usually be equivalent to what +we have in mind when we (inaccurately) speak of "all properties of +\(x\)." (A "property of \(x\)" may be defined as a propositional +function satisfied by \(x\).) Hence we must find, if possible, some +method of reducing the order of a propositional function without +affecting the truth or falsehood of its values. This seems to be what +common-sense effects by the admission of <i>classes</i>. Given any +propositional function \(\psi x\) of whatever order, this is assumed +to be equivalent, for all values of \(x\), to a statement of the form +"\(x\) belongs to the class \(\alpha\)." Now assuming that there is +such an entity as the class \(\alpha\), this statement is of the first +order, since it involves no allusion to a variable function. Indeed +its only practical advantage over the original statement \(\psi x\) +is that it is of the first order. There is no advantage in assuming +that there really are such things as classes, and the contradiction +about the classes which are not members of themselves shows that, if +there are classes, they must be something radically different from +individuals. It would seem that the sole purpose which classes serve, +and one main reason which makes them linguistically convenient, is<span class="pagenum" id="Page_174">[Pg 174]</span> +that they provide a method of reducing the order of a propositional +function. We shall, therefore, not assume anything of what may seem to +be involved in the common-sense admission of classes, except this, that +every propositional function is equivalent, for all its values, to some +predicative function of the same argument or arguments.</p> + +<p>This assumption with regard to functions is to be made whatever may be +the type of their arguments. Let \(fu\) be a function, of any order, +of an argument \(u\), which may itself be either an individual or a +function of any order. If \(f\) is a matrix, we write the function in +the form \(f!u\); in such a case we call \(f\) a <i>predicative</i> +function. Thus a predicative function of an individual is a first-order +function; and for higher types of arguments, predicative functions take +the place that first-order functions take in respect of individuals. We +assume, then, that every function of one variable is equivalent, for +all its values, to some predicative function of the same argument. This +assumption seems to be the essence of the usual assumption of classes; +at any rate, it retains as much of classes as we have any use for, +and little enough to avoid the contradictions which a less grudging +admission of classes is apt to entail. We will call this assumption the +<i>axiom of classes</i>, or the <i>axiom of reducibility</i>.</p> + +<p>We shall assume similarly that every function of two variables is +equivalent, for all its values, to a predicative function of those +variables, <i>i.e.</i> to a matrix. This assumption is what seems to +be meant by saying that any statement about two variables defines +a relation between them. We will call this assumption the <i>axiom +of relations</i> or (like the previous axiom) the <i>axiom of +reducibility</i>.</p> + +<p>In dealing with relations between more than two terms, similar +assumptions would be needed for three, four, ... variables. But these +assumptions are not indispensable for our purpose, and are therefore +not made in this work.</p> + +<p>Stated in symbols, the two forms of the axiom of reducibility are as +follows:</p> + +<p class="nind"> +<b><a id="*12·1">*12·1</a></b>. \(\vdash:(\exists f):\phi x.\equiv_{x}.f!x\quad\text{Pp}\)</p> + +<p class="nind"> +<b><a id="*12·11">*12·11</a></b>. \(\vdash:(\exists f):\phi(x,y).\equiv_{x,y}.f!(x,y)\quad\text{Pp}\)</p> + +<p>We call two functions \(\phi\hat{x}\), \(\psi\hat{x}\) formally +equivalent when \(\phi x.\equiv_{x}.\psi x\), and similarly we call +\(\phi(\hat{x},\hat{y})\) and \(\psi(\hat{x},\hat{y})\) formally +equivalent when +\[ +\phi(x,y).\equiv_{x,y}.\psi(x,y). +\] +Thus the above axioms state that any function of one or two variables +is formally equivalent to some <i>predicative</i> function of one or +two variables, as the case may be.</p> + +<p><span class="pagenum" id="Page_175">[Pg 175]</span></p> + +<p>Of the above two axioms, the first is chiefly needed in the theory of +classes (<a href="#*20">*20</a>), and the second in the theory of relations (<a href="#*21">*21</a>). But the +first is also essential to the theory of identity, if identity is to be +defined (as we have done, in <a href="#*13·01">*13·01</a>); its use in the theory of identity +is embodied in the proof of <a href="#*13·101">*13·101</a>, below.</p> + +<p>We may sum up what has been said in the present number as follows:</p> + +<p>(1) A function of the first order is one which involves no variables +except individuals, whether as apparent variables or as arguments.</p> + +<p>(2) A function of the (\(n + 1)\)th order is one which has at least one +argument or apparent variable of order \(n\), and contains no argument +or apparent variable which is not either an individual or a first-order +function or a second-order function or ... or a function of order \(n\).</p> + +<p>(3) A predicative function is one which contains no apparent variables, +<i>i.e.</i> is a matrix. It is possible, without loss of generality, to +use no variables except matrices and individuals, so long as variable +<i>propositions</i> are not required.</p> + +<p>(4) Any function of one argument or of two is formally equivalent to a +predicative function of the same argument or arguments.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_52" href="#FNanchor_52" class="label">[52]</a> +Cf. Introduction, <a href="#CHAPTER_II">Chapter II</a>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_53" href="#FNanchor_53" class="label">[53]</a> +Cf. <a href="#Page_95">pp. 95</a>, <a href="#Page_96">96</a>.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_176">[Pg 176]</span></p> +<h2 class="nobreak" id="*13">*13. IDENTITY.</h2> +</div> + + +<p><i>Summary of</i> *13.</p> + +<p>The propositional function "\(x\) is identical with \(y\)" will +be written "\(x = y\)." We shall find that this use of the sign +of equality covers all the common uses of equality that occur in +mathematics. The definition is as follows:</p> + +<p class="nind"> +<b><a id="*13·01">*13·01</a>.</b> \(x = y .=: (\phi):{\phi}!x.\supset. {\phi}!y \quad \text{Df}\)</p> + +<p>This definition states that \(x\) and \(y\) are to be called identical +when every predicative function satisfied by \(x\) is also satisfied by +\(y\). We cannot state that <i>every</i> function satisfied by \(x\) is +to be satisfied by \(y\), because \(x\) satisfies functions of various +orders, and these cannot all be covered by one apparent variable. +But in virtue of the axiom of reducibility it follows that, if \(x = y\) +and \(x\) satisfies \({\psi}x\), where \(\psi\) is any function, +predicative or non-predicative, then \(y\) also satisfies \({\psi}y\) +(cf. <a href="#*13·101">*13·101</a>, below). Hence in effect the definition is as powerful as +it would be if it could be extended to cover <i>all</i> functions of +\(x\).</p> + +<p>Note that the second sign of equality in the above definition is +combined with "\(\text{Df}\)," and thus is not really the same symbol +as the sign of equality which is defined. Thus the definition is not +circular, although at first sight it appears so.</p> + +<p>The propositions of the present number are constantly referred to. Most +of them are self-evident, and the proofs offer no difficulty. The most +important of the propositions of this number are the following:</p> + +<p class="nind"> +<b><a id="*13·101">*13·101</a>.</b> \(\vdash : x = y .\supset. {\psi}x \supset {\psi}y\)</p> + +<p><i>I.e.</i> if \(x\) and \(y\) are identical, any property of \(x\) is +a property of \(y\).</p> + +<p class="nind"> +<b><a id="*13·12">*13·12</a>.</b> \(\vdash : x = y .\supset. {\psi}x \equiv {\psi}y\)</p> + +<p>This includes <a href="#*13·101">*13·101</a> together with the fact that if \(x\) and \(y\) +are identical any property of \(y\) is a property of \(x\).</p> + +<p class="nind"> +<b>*13·15·16·17</b>, which state that identity is reflexive, symmetrical and transitive.</p> + +<p class="nind"> +<b>*13·191.</b> \(\vdash \colon\ldotp y = x .\supset_{y}. {\phi}y :\equiv. {\phi}x\)</p> + +<p><span class="pagenum" id="Page_177">[Pg 177]</span></p> + +<p><i>I.e.</i> to state that everything that is identical with \(x\) has a +certain property is equivalent to stating that \(x\) has that property.</p> + +<p class="nind"> +<b>*13·195.</b> \(\vdash:(\exists y).y=x.\phi y.\equiv.\phi x\)</p> + +<p><i>I.e.</i> to state that something identical with \(x\) has a certain +property is equivalent to saying that \(x\) has that property.</p> + +<p class="nind"> +<b>*13·22.</b> \(\vdash:(\exists z,w).z=x.w=y.\phi(z,w).\equiv.\phi(x,y)\)</p> + +<p>This is the analogue of <a href="#*13·195">*13·195</a> for two variables.</p> + +<hr class="tb"> + +<p class="nind"> +<b>*13·01.</b> \(x=y.=:(\phi):\phi!x.\supset.\phi!y \quad\text{Df}\)</p> + +<p>The following definitions embody abbreviations which are often +convenient.</p> + +<p class="nind"> +<b>*13·02.</b> \(x\neq y.=.{\sim}(x=y) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*13·03.</b> \(x=y=z.=.x=y.y=z \quad\text{Df}\)</p> + +<p class="nind"> +<b>*13·1.</b> \(\vdash\colon\ldotp x=y.\equiv:\phi!x.\supset_{\phi}.\phi!y \quad[\text{*4·2.(*13·01).(*10·02)}]\)</p> + +<p class="nind"> +<b>*13·101.</b> \(\vdash:x=y.\supset.\psi x\supset\psi y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*12·1}.\quad\supset\vdash\colon\ldotp(\exists \phi)\colon\ldotp\psi x.\equiv.\phi!x:\psi y.\equiv.\phi!y &\qquad \text{(1)}\\ +&\vdash.\text{*13·1}.\quad\supset\vdash\colon\colon\text{Hp}.\supset\colon\ldotp \phi!x.\supset_{\phi}.\phi!y\colon\ldotp\\ +&[\text{*4·84·85.*10·27}] \quad\supset\colon\ldotp\psi x.\equiv.\phi!x:\psi y.\equiv.\phi!y:\supset_{\phi}:\psi x.\supset.\psi y\colon\ldotp\\ +&[\text{*10·23}] \quad\supset\colon\ldotp(\exists \phi):\psi x.\equiv.\phi!x:\psi y.\equiv.\phi!y:\supset:\psi x.\supset.\psi y &\qquad \text{(2)}\\ +&\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In virtue of this proposition, if \(x=y\), \(y\) satisfies any +function, whether predicative or non-predicative, which is satisfied +by \(x\). It will be observed that the proof uses the axiom of +reducibility (<a href="#*12·1">*12·1</a>). But for this axiom, two terms \(x\) and \(y\) +might agree in respect of all predicative functions, but not in respect +of all non-predicative functions. We should thus be led to identities +of different degrees, according to the degree of the functions in +respect of which \(x\) and \(y\) agreed. Strict identity would, in this +case, have to be taken as a primitive idea, and <a href="#*13·101">*13·101</a> would have to +be a primitive proposition, as would also *13·15·16·17.</p> + +<p class="nind"> +<b>*13·11.</b> \(\vdash\colon\ldotp x=y.\equiv:\phi!x.\equiv_{\phi}.\phi!y\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash.\text{*10·22}. &\supset\vdash\colon\ldotp\phi!x.\equiv_{\phi}.\phi!y:\supset:\phi!x.\supset_{\phi}.\phi!y:\\ +[\text{*13·1}] &\supset:x=y &\qquad \text{(1)}\\ +\vdash.\text{*13·101}. &\supset\vdash\colon\ldotp x=y.\supset.\phi!x\supset\phi!y &\qquad \text{(2)}\\ +\vdash.\text{*13·101.*1·7}. &\supset\vdash\colon\ldotp x=y.\supset.{\sim}\phi!x\supset{\sim}\phi!y.\\ +[\text{Transp}] &\supset.\phi!y\supset\phi!x &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).Comp}.&\supset\vdash\colon x=y.\supset.\phi!x\equiv\phi!y:\\ +[\text{*10·11·21}] &\supset\vdash\colon\ldotp x=y.\supset:\phi!x.\equiv_{\phi}.\phi!y &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_178">[Pg 178]</span></p> + +<p class="nind"> +<b>*13·12.</b> \(\vdash : x = y . \supset . \psi x \equiv \psi y\)</p> + +<p><i>Dem.</i>\[ +\begin{array}{l} +\vdash .\text{*13·101 . Comp}. \supset \vdash : x = y . \supset . \psi x &\supset \psi y . {\sim}\psi x \supset {\sim}\psi y .\\ +[\text{Transp}] &\supset . \psi x \equiv \psi y : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*13·13.</b> \(\vdash : \psi x . x = y . \supset . \psi y \quad[\text{*13·101 . Comm . Imp}]\)</p> + +<p class="nind"> +<b>*13·14.</b> \(\vdash : \psi x . {\sim}\psi y . \supset . x \neq y \quad[\text{*13·13 . *4·14}]\)</p> + +<p class="nind"> +<b><a id="*13·15">*13·15</a>.</b> \(\vdash . x = x \quad[\text{Id . *10·11 . *13·1}]\)</p> + +<p class="nind"> +<b><a id="*13·16">*13·16</a>.</b> \(\vdash : x = y . \equiv . y = x \quad[\text{*13·11 . *10·32}]\)</p> + +<p class="nind"> +<b><a id="*13·17">*13·17</a>.</b> \(\vdash : x = y . y = z . \supset . x = z\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*13·1}. \supset\vdash \colon\colon \text{Hp} . &\supset \colon\ldotp \phi ! x . \supset_\phi . \phi ! y : \phi ! y . \supset_\phi . \phi ! z \colon\ldotp\\ +[\text{*10·3}] &\supset \colon\ldotp \phi ! x . \supset_\phi . \phi ! z \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>In the above use of <a href="#*10·3">*10·3</a>, \(\phi ! x\), \(\phi ! y\), \(\phi ! z\) +are regarded as three different functions of \(\phi\), and \(\phi\) +replaces the \(x\) of *10·3.</p> + +<p>The above three propositions show that identity is reflexive (<a href="#*13·15">*13·15</a>), +symmetrical (<a href="#*13·16">*13·16</a>), and transitive (<a href="#*13·17">*13·17</a>). These are the three +marks of relations having the formal properties which we associate +commonly with the sign of equality.</p> + +<p class="nind"> +<b>*13·171.</b> \(\vdash : x = y . x = z . \supset . y = z \quad[\text{*13·16·17}]\)</p> + +<p class="nind"> +<b>*13·172.</b> \(\vdash : y = x . z = x . \supset . y = z \quad[\text{*13·16·17}]\)</p> + +<p class="nind"> +<b>*13·18.</b> \(\vdash : x = y . x \neq z . \supset . y \neq z \quad[\text{*13·17 . *4·14}]\)</p> + +<p class="nind"> +<b>*13·181.</b> \(\vdash : x = y . y \neq z . \supset . x \neq z \quad[\text{*13·171 . *4·14}]\)</p> + +<p class="nind"> +<b>*13·182.</b> \(\vdash \colon\ldotp x = y . \supset : z = x . \equiv . z = y \quad[\text{*13·17·172 . Exp . Comp}]\)</p> + +<p class="nind"> +<b>*13·183.</b> \(\vdash \colon\ldotp x = y . \equiv : z = x . \equiv_z . z = y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*13·182 . *10·11·21}. &\supset \vdash \colon\ldotp x = y . \supset : z = x . \equiv_z . z = y &\qquad \text{(1)}\\ +\vdash .\text{*10·1}. &\supset \vdash \colon\ldotp z = x . \equiv_z . z = y : \supset : x = x . \supset . x = y :\\ +[\text{*13·15}] &\supset : x = y &\qquad \text{(2)}\\ +\vdash .\text{(1) . (2)}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*13·19.</b> \(\vdash . (\exists y) . y = x \quad[\text{*13·15 . *10·24}]\)</p> + +<p class="nind"> +<b><a id="*13·191">*13·191</a>.</b> \(\vdash \colon\ldotp y = x . \supset_y . \phi y : \equiv . \phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*10·1}. &\supset \vdash \colon\ldotp y = x . \supset_y . \phi y : \supset : x = x . \supset . \phi x :\\ +[\text{*13·15}] &\supset : \phi x &\qquad \text{(1)}\\ +\vdash .\text{*13·12}. &\supset \vdash \colon\ldotp y = x . \supset : \phi x . \supset . \phi y \colon\ldotp\\ +[\text{Comm}] &\supset \vdash \colon\ldotp \phi x . \supset : y = x . \supset . \phi y \colon\ldotp\\ +[\text{*10·11·21}] &\supset \vdash \colon\ldotp \phi x . \supset : y = x . \supset_y . \phi y &\qquad \text{(2)}\\ +\vdash .\text{(1) . (2)}.& \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This proposition is constantly used in subsequent proofs.</p> + +<p><span class="pagenum" id="Page_179">[Pg 179]</span></p> + +<p class="nind"> +<b>*13·192.</b> \(\vdash\colon\ldotp(\exists c):x=b.\equiv_{x}.x=c:\psi c:\equiv.\psi b\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·2.*3·2.} &\supset\vdash\colon\colon\psi b.\supset\colon\ldotp x=b.\equiv_{x}.x=b:\psi b\colon\ldotp\\ +[\text{*10·24}] &\supset\colon\ldotp(\exists c):x=b.\equiv_{x}.x=c:\psi c &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp x=b.\equiv_{x}.x=c:\psi c:\supset:b=b.\equiv.b=c:\psi c:\\ +[\text{*5·501.*13·15}] &\supset:b=c.\psi c:\\ +[\text{*13·13}] &\supset:\psi b &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23}. &\supset\vdash\colon\ldotp(\exists c):x=b.\equiv_{x}.x=c:\psi c:\supset.\psi b &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is useful in the theory of descriptions (<a href="#*14">*14</a>).</p> + +<p class="nind"> +<b>*13·193.</b> \(\vdash:\phi x.x=y.\equiv.\phi y.x=y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{Simp}. &\supset\vdash:\phi x.x=y.\supset.x=y &\qquad \text{(1)}\\ +\vdash.\text{*13·13}. &\supset\vdash:\phi x.x=y.\supset.\phi y &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Comp}. &\supset\vdash:\phi x.x=y.\supset.\phi y.x=y &\qquad \text{(3)}\\ +\vdash.\text{*13·16.Fact}. &\supset\vdash:\phi y.x=y.\supset.\phi y.y=x.\\ +[\text{(3)} \frac{y,\,x}{x,\,y}] & \supset.\phi x.y=x.\\ +[\text{*13·16.Fact}] &\supset.\phi x.x=y &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is very often used.</p> + +<p class="nind"> +<b>*13·194.</b> \(\vdash:\phi x.x=y.\equiv.\phi x.\phi y.x=y \quad[\text{*13·13.*4·71}]\)</p> + +<p>This proposition is used in <a href="#*37·65">*37·65</a> and *101·14.</p> + +<p class="nind"> +<b><a id="*13·195">*13·195</a>.</b> \(\vdash:(\exists y).y=x.\phi y.\equiv.\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*3·2.*13·15}. & \supset\vdash:\phi x.\supset.x=x.\phi x.\\ +[\text{*10·24}] &\supset.(\exists y).y=x.\phi y &\qquad \text{(1)}\\ +\vdash.\text{*13·13.*10·11}. &\supset\vdash\colon\ldotp(y):y=x.\phi y.\supset.\phi x:\\ +[\text{*10·23}] &\supset\vdash\colon\ldotp(\exists y).y=x.\phi y.\supset.\phi x &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. & \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The use of this proposition in subsequent proofs is very frequent.</p> + +<p class="nind"> +<b>*13·196.</b> \(\vdash\colon\ldotp{\sim}\phi x.\equiv:\phi y.\supset_{y}.y\neq x \quad[\text{*13·195.Transp.*10·51}]\)</p> + +<p class="nind"> +<b>*13·21.</b> \(\vdash\colon\ldotp z=x.w=y.\supset_{z,w}.\phi(z,w):\equiv.\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·62}.\supset\\ +\vdash\colon\colon z=x.w=y.\supset_{z,w}.\phi(z,w):&\equiv\colon\ldotp z=x.\supset_{z}:w=y.\supset_{w}.\phi(z,w)\colon\ldotp\\ +[\text{*13·191}] & \equiv\colon\ldotp w=y.\supset_{w}.\phi(x,w)\colon\ldotp\\ +[\text{*13·191}] &\equiv\colon\ldotp\phi(x,y)\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is the analogue, for two variables, of <a href="#*13·191">*13·191</a>.</p> + +<p><span class="pagenum" id="Page_180">[Pg 180]</span></p> + +<p class="nind"> +<b>*13·22.</b> \(\vdash:(\exists z,w).z=x.w=y.\phi(z,w).\equiv.\phi(x,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·55}.\supset\vdash\colon\ldotp(\exists z,w).z=x.&w=y.\phi(z,w).\\ +&\equiv:(\exists z):z=x:(\exists w).w=y.\phi(z,w):\\ +[\text{*13·195}] &\equiv:(\exists w).w=y.\phi(x,w):\\ +[\text{*13·195}] &\equiv:\phi(x,y)\colon\ldotp\supset\vdash.\text{Prop} +\end{array}\\ +\]</p> + +<p>This proposition is the analogue, for two variables, of <a href="#*13·195">*13·195</a>. It is +frequently used, especially in the theory of couples (<a href="#*54">*54</a>, <a href="#*55">*55</a>, <a href="#*56">*56</a>).</p> + +<p>The following proposition is useful in the theory of types. Its purpose +is to show that, if \(a\) is any argument for which "\(\phi a\)" is +significant, <i>i.e.</i> for which we have \(\phi a\lor{\sim}\phi a\), +then "\(\phi x\)" is significant when, and only when, \(x\) is either +identical with a or not identical with \(a\). It follows (as will be +proved in <a href="#*20·81">*20·81</a>) that, if "\(\phi a\)" and "\(\psi a\)" are both +significant, the class of values of \(x\) for which "\(\phi x\)" is +significant is the same as the class of those for which "\(\psi x\)" +is significant, <i>i.e.</i> two types which have a common member are +identical.</p> + +<p>In the following proof, the chief point to observe is the use of +<a href="#*10·221">*10·221</a>. There are two variables, \(a\) and \(x\), to be identified. +In the first use, we depend upon the fact that \(\phi a\) and \(x=a\) +both occur in both (4) and (5): the occurrence of \(\phi a\) in both +justifies the identification of the two \(a\)'s, and when these +have been identified, the occurrence of \(x=a\) in both justifies +the identification of the two \(x\)'s. (Unless the \(a\)'s had been +already identified, this would not be legitimate, because "\(x=a\)" is +typically ambiguous if neither \(x\) nor \(a\) is of given type.) The +second use of *10·221 is justified by the fact that both \(\phi a\) and +\(\phi x\) occur in both (2) and (6).</p> + +<p class="nind"> +<b><a id="*13·3">*13·3</a>.</b> \(\vdash\colon\colon\phi a\lor{\sim}\phi a.\supset\colon\ldotp\phi x\lor{\sim}\phi x.\equiv:x=a.\lor.x\neq a\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*2·11}. & \supset\vdash.\phi x\lor{\sim}\phi x &\qquad \text{(1)}\\ +\vdash.\text{(1).Simp}. &\supset\vdash:\phi a\lor{\sim}\phi a.\supset.\phi x\lor{\sim}\phi x &\qquad \text{(2)}\\ +\vdash.\text{*2·11}. &\supset\vdash:x=a.\lor.x\neq a &\qquad \text{(3)}\\ +\vdash.\text{(3).Simp}. & \supset\vdash\colon\ldotp\phi a\lor{\sim}\phi a.\supset:x=a.\lor.x\neq a &\qquad \text{(4)}\\ +\vdash.\text{*13·101.Comm}.&\supset\vdash\colon\ldotp\phi a\lor{\sim}\phi a.\supset:x=a.\supset.\phi x\lor{\sim}\phi x &\qquad \text{(5)}\\ +\vdash.\text{(4).(5).*10·13·221}.&\supset\\ +\vdash\colon\colon\phi a\lor{\sim}\phi a.&\supset:x=a.\lor.x\neq a\colon\ldotp\phi a\lor{\sim}\phi a.\supset:x=a.\supset.\phi x\lor{\sim}\phi x &\qquad \text{(6)}\\ +\vdash.\text{(2).(6).*10·13·221}.&\supset\\ +\vdash\colon\colon\phi a\lor{\sim}\phi a.&\supset.\phi x\lor{\sim}\phi x\colon\ldotp\phi a\lor{\sim}\phi a.\supset:x=a.\lor.x\neq a\colon\ldotp\\ +&\phi a\lor{\sim}\phi a.\supset:x=a.\supset.\phi x\lor{\sim}\phi x &\qquad \text{(7)}\\ +\vdash.\text{(7).Simp}.&\supset\\ +\vdash\colon\colon\phi a\lor{\sim}\phi a.&\supset.\phi x\lor{\sim}\phi x\colon\ldotp\phi a\lor{\sim}\phi a.\supset:x=a.\lor.x\neq a &\qquad \text{(8)}\\ +\vdash.\text{(8).*5·35}. &\supset\vdash\colon\colon\phi a\lor{\sim}\phi a.\supset\colon\ldotp\phi x\lor{\sim}\phi x.\equiv:x=a.\lor.x\neq a\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_181">[Pg 181]</span></p> + +<h2 class="nobreak" id="*14">*14. DESCRIPTIONS.</h2> +</div> + + +<p><i>Summary of</i> *14.</p> + +<p>A <i>description</i> is a phrase of the form "the term which etc.," +or, more explicitly, "the term \(x\) which satisfies \(\phi\hat{x}\)," +where \(\phi\hat{x}\), is some function satisfied by one and only +one argument. For reasons explained in the Introduction (<a href="#CHAPTER_III">Chapter III</a>), +we do not define "the \(x\) which satisfies \(\phi\hat{x}\)," +but we define any proposition in which this phrase occurs. Thus when +we say: "The term \(x\) which satisfies \(\phi\hat{x}\) satisfies +\(\psi\hat{x}\)," we shall mean: "There is a term \(b\) such that +\({\phi}x\) is true when, and only when, \(x\) is \(b\), and +\({\psi}b\) is true." That is, writing "\((℩x)({\phi}x)\)" for "the +term \(x\) which satisfies \({\phi}x\)," \(\psi(℩x)({\phi}x)\) is to +mean +\[ +(\exists b): {\phi}x .\equiv_{x}. x = b: {\psi}b\text{.} +\] +This, however, is not yet quite adequate as a definition, for when +(\({℩}x)({\phi}x)\) occurs in a proposition which is part of a +larger proposition, there is doubt whether the smaller or the larger +proposition is to be taken as the "\(\psi(℩x)({\phi}x)\)." Take, for +example, \(\psi({℩}x)({\phi}x) .\supset. p\). This may be either +\[ +\begin{align} +&(\exists b):\phi x .\equiv_{x}. x = b :\psi b :\supset. p \\ +\text{or} \quad &(\exists b) \colon\ldotp \phi x .\equiv_{x}. x = b: \psi b .\supset.p. \\ +\end{align} +\] +If "(\(\exists b): {\phi}x .\equiv_{x}. x = b\)" is false, the first of these must be true, while the +second must be false. Thus it is very necessary to distinguish them.</p> + +<p>The proposition which is to be treated as the "\(\psi({℩}x)({\phi}x)\)" +will be called the <i>scope</i> of (\({℩}x)({\phi}x)\). Thus +in the first of the above two propositions, the scope of +(\({℩}x)({\phi}x)\) is \(\psi({℩}x)({\phi}x)\), while in the second +it is \(\psi({℩}x)({\phi}x) .\supset. p\). In order to avoid +ambiguities as to scope, we shall indicate the scope by writing +"\([({℩}x)({\phi}x)]\)" at the beginning of the scope, followed by +enough dots to extend to the end of the scope. Thus of the above two +propositions the first is +\[ +[({℩} x)(\phi x)].\psi({℩} x)(\phi x) .\supset. p +\] +while the second is +\[ +[({℩} x)(\phi x)]:\psi({℩} x)(\phi x) .\supset. p. +\] +Thus we arrive at the following definition:</p> + +<p class="nind"> +<b><a id="*14·01">*14·01</a>.</b> \([({℩} x)(\phi x)].\psi({℩} x)(\phi x).=:(\exists b):\phi x .\equiv_{x}. x = b:\psi b \quad \text{Df}\)</p> + +<p>It will be found in practice that the scope usually required is +the smallest proposition enclosed in dots or brackets in which +"(\({℩}x)({\phi}x)\)" occurs. Hence<span class="pagenum" id="Page_182">[Pg 182]</span> when this scope is to be given +to (\({℩}x)(\phi x)\), we shall usually omit explicit mention of the +scope. Thus <i>e.g</i>. we shall have +\[ +\begin{aligned} +a\neq (℩x)(\phi x)\ldotp &=:~ (\exists b):\phi x\ldotp \equiv_{x}\ldotp x=b:a\neq b,\\ +\sim \{a=(℩x)(\phi x)\}\ldotp &=\ldotp \sim\{(\exists b):\phi x\ldotp \equiv_{x}\ldotp x=b:a=b\}. +\end{aligned} +\] +Of these the first necessarily implies (\(\exists b):\phi x\ldotp\equiv_{x}\ldotp x=b\), +while the second does not. We put</p> + +<p class="nind"> +<b><a id="*14·02">*14·02</a>.</b> \(\text{E}!({℩}x)(\phi x)\ldotp =:(\exists b):\phi x\ldotp \equiv_{x}\ldotp x=b\quad\text{Df}\)</p> + +<p>This defines: "The \(x\) satisfying \(\phi \hat{x}\) exists," which +holds when, and only when, \(\phi \hat{x}\) is satisfied by one value +of \(x\) and by no other value.</p> + +<p>When two or more descriptions occur in the same proposition, there is +need of avoiding ambiguity as to which has the larger scope. For this +purpose, we put</p> + +<p class="nind"> +<b><a id="*14·03">*14·03</a>.</b> \[\begin{align}[({℩}x)(\phi x),\,&({℩}x)(\psi x)]\ldotp f\{({℩}x)(\phi x),({℩}x)(\psi x)\}\ldotp =:\\ +&[({℩}x)(\phi x)]:[({℩}x)(\psi x)]\ldotp f\{({℩}x)(\phi x),({℩}x)(\psi x)\}\quad\text{Df}\end{align}\]</p> + +<p>It will be shown <a href="#*14·113">*14·113</a> that the truth-value of a proposition +containing two descriptions is unaffected by the question which has +the larger scope. Hence we shall in general adopt the convention that +the description occurring first typographically is to have the larger +scope, unless the contrary is expressly indicated. Thus <i>e.g</i>. +\[ +(℩x)(\phi x)=(℩x)(\psi x) +\] +will mean +\[ +(\exists b):\phi x\ldotp \equiv_{x}\ldotp x=b:b=(℩x)(\psi x), +\] +<i>i.e</i>. +\[ +(\exists b)\colon\ldotp \phi x\ldotp \equiv_{x}\ldotp x=b\colon\ldotp (\exists c):\psi x\ldotp \equiv_{x}\ldotp x=c:b=c. +\] +By this convention we are able almost always to avoid explicit +indication of the order of elimination of two or more descriptions. If, +however, we require a larger scope for the later description, we put</p> + +<p class="nind"> +<b>*14·04</b> \[\begin{align}[({℩}x)(\psi x)]\ldotp &f\{({℩}x)(\phi x),\,({℩}x)(\psi x)\}\ldotp =\ldotp\\ +&[({℩}x)(\psi x),\,({℩}x)(\phi x)]\ldotp f\{({℩}x)(\phi x),\,({℩}x)(\psi x)\}\quad\text{Df}\end{align}\]</p> + +<p>Whenever we have \(\text{E}!({℩}x)(\phi x),~({℩}x)(\phi x)\) behaves, +formally, like an ordinary argument to any function in which it may +occur. This fact is embodied in the following proposition:</p> + +<p class="nind"> +<b>*14·18</b> \(\vdash \colon\ldotp \text{E}!({℩}x)(\phi x)\ldotp \supset :(x)\ldotp \psi x\ldotp \supset \ldotp \psi ({℩}x)(\phi x)\)</p> + +<p>That is to say, when (\({℩}x)(\phi x)\) exists, it has any property +which belongs to everything. This does not hold when (\({℩}x)(\phi x)\) +does not exist; for example, the present King of France does not have +the property of being either bald or not bald.</p> + +<p>If (\({℩}x)(\phi x)\) has any property whatever, it must exist. This +fact is stated in the proposition:</p> + +<p class="nind"> +<b>*14·21</b> \(\vdash :\psi (℩x)(\phi x)\ldotp \supset \ldotp \text{E}!({℩}x)(\phi x)\)</p> + +<p>This proposition is obvious, since "E\(!({℩}x)(\phi x)\)" is, by +the definitions, part<span class="pagenum" id="Page_183">[Pg 183]</span> of "\(\psi(℩x)(\phi x)\)." When, in ordinary +language or in philosophy, something is said to "exist," it is always +something <i>described</i>, <i>i.e.</i> it is not something immediately +presented, like a taste or a patch of colour, but something like +"matter" or "mind" or "Homer" (meaning "the author of the Homeric +poems"), which is known by description as "the so-and-so," and is thus +of the form (\(℩x)(\phi x)\). Thus in all such cases, the existence of +the (grammatical) subject (\(℩x)(\phi x)\) can be analytically inferred +from any true proposition having this grammatical subject. It would +seem that the word "existence" cannot be significantly applied to +subjects immediately given; <i>i.e.</i> not only does our definition +give no meaning to "\(\text{E}!x\)," but there is no reason, in +philosophy, to suppose that a meaning of existence could be found which +would be applicable to immediately given subjects.</p> + +<p>Besides the above, the following are among the more useful propositions +of the present number.</p> + +<p class="nind"> +<b>*14·202.</b> \(\vdash\colon\ldotp\phi x.\equiv_{x}.x=b:\equiv:({℩}x)(\phi x)=b:\equiv:\phi x.\equiv_{x}.b=x:\equiv:b=({℩}x)(\phi x)\)</p> + +<p>From the first equivalence in the above, it follows that</p> + +<p class="nind"> +<b>*14·204.</b> \(\vdash:\text{E}!({℩}x)(\phi x).\equiv.(\exists b).({℩}x)(\phi x)=b\)</p> + +<p><i>I.e.</i> (\({℩}x)(\phi x)\) exists when there is something which +(\({℩}x)(\phi x)\) is.</p> + +<p>We have</p> + +<p class="nind"> +<b>*14·205.</b> \(\vdash:\psi({℩}x)(\phi x).\equiv.(\exists b).b=({℩}x)(\phi x).\psi b\)</p> + +<p><i>I.e.</i> (\({℩}x)(\phi x)\) has the property \(\psi\) when there +is something which is (\({℩}x)(\phi x)\) and which has the property +\(\psi\).</p> + +<p>We have to prove that such symbols as "(\({℩}x)(\phi x)\)" obey the +same rules with regard to identity as symbols which directly represent +objects. To this, however, there is one partial exception, for instead +of having +\[ +({℩}x)(\phi x)=(℩x)(\phi x), +\] +we only have</p> + +<p class="nind"> +<b>*14·28.</b> \(\vdash:\text{E}!({℩}x)(\phi x).\equiv.({℩}x)(\phi x)=({℩}x)(\phi x)\)</p> + +<p><i>I.e.</i> "(\({℩}x)(\phi x)\)" only satisfies the reflexive property +of identity if (\({℩}x)(\phi x)\) exists.</p> + +<p>The symmetrical property of identity holds for such symbols as +(\({℩}x)(\phi x)\), without the need of assuming existence, <i>i.e.</i> +we have</p> + +<p class="nind"> +<b>*14·13.</b> \(\vdash:a=({℩}x)(\phi x).\equiv.({℩}x)(\phi x)=a\)</p> + +<p class="nind"> +<b>*14·131.</b> \(\vdash:({℩}x)(\phi x)=({℩}x)(\psi x).\equiv.({℩}x)(\psi x)=({℩}x)(\phi x)\)</p> + +<p>Similarly the transitive property of identity holds without the need of +assuming existence. This is proved in *14·14·142·144.</p> + +<p><span class="pagenum" id="Page_184">[Pg 184]</span></p> + +<hr class="tb"> + +<p class="nind"> +<b>*14·01.</b> \([({℩}x)(\phi x)].\psi ({℩}x)(\phi x).=:(\exists b):\phi x.\equiv_{x}.x=b:\psi b \quad\text{Df}\)</p> + +<p class="nind"> +<b>*14·02.</b> \(\text{E}!({℩}x)(\phi x).=:(\exists b):\phi x.\equiv_{x}.x=b \quad\text{Df}\)</p> + +<p class="nind"> +<b>*14·03.</b> \[\begin{align}[({℩}x)(\phi x),({℩}x)&(\psi x)].f\{({℩}x)(\phi x),\,({℩}x)(\psi x)\}.=:\\ +&[({℩}x)(\phi x)]:[({℩}x)(\psi x)].f\{({℩}x)(\phi x),({℩}x)(\psi x)\} \quad\text{Df}\end{align}\]</p> + +<p class="nind"> +<b>*14·04.</b> \[\begin{align}[({℩}x)(\psi x)].&f\{({℩}x)(\phi x),\,({℩}x)(\psi x)\}.=.\\ +&[({℩}x)(\psi x),({℩}x)(\phi x)].f\{({℩}x)(\phi x),\,({℩}x)(\psi x)\} \quad\text{Df}\end{align}\]</p> + +<p class="nind"> +<b><a id="*14·1">*14·1</a>.</b> \[\begin{align}\vdash\colon\ldotp[({℩}x)(\phi x)].\psi(℩x)(\phi x).\equiv:(\exists b):\phi x.\equiv_{x}.&x=b:\psi b +&[\text{*4·2.(*14·01)}]\end{align}\]</p> + +<p>In virtue of our conventions as to the scope intended when no scope +is explicitly indicated, the above proposition is the same as the +following:</p> + +<p class="nind"> +<b>*14·101.</b> \(\vdash\colon\ldotp \psi({℩}x)(\phi x).\equiv:(\exists b):\phi x.\equiv_{x}.x=b:\psi b \quad[\text{*14·1}]\)</p> + +<p class="nind"> +<b><a id="*14·11">*14·11</a>.</b> \(\vdash\colon\ldotp\text{E}!({℩}x)(\phi x).\equiv:(\exists b):\phi x.\equiv_{x}.x=b \quad[\text{*4·2.(*14·02)}]\)</p> + +<p class="nind"> +<b>*14·111.</b> \[\begin{align}\vdash\colon\ldotp[({℩}x)(\psi x)].&f\{({℩}x)(\phi x),\,({℩}x)(\psi x)\}.\equiv:\\ +&(\exists b,c):\phi x.\equiv_{x}.x=b:\psi x.\equiv_{x}.x=c:f(b,c)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·2.(*14·04·03)}.\supset\\ +\vdash\colon\colon[(℩x)(\psi x)].f\{({℩}x)(\phi x),({℩}x)(\psi x)\}.\equiv\colon\ldotp\\ +\qquad\qquad[({℩}x)(\psi x)]:[({℩}x)(\phi x)].f\{({℩}x)(\phi x),({℩}x)(\psi x)\}\colon\ldotp\\ +[\text{*14·1}] \equiv\colon\ldotp[(℩x)(\psi x)]\colon\ldotp(\exists b):\phi x.\equiv_{x}.x=b:f\{b,({℩}x)(\psi x)\}\colon\ldotp\\ +[\text{*14·1}] \equiv\colon\ldotp(\exists c)\colon\ldotp\psi x.\equiv_{x}.x=c\colon\ldotp(\exists b):\phi x.\equiv_{x}.x=b:f(b,c)\colon\ldotp\\ +[\text{*11·55}] \equiv\colon\ldotp(\exists b,c):\phi x.\equiv_{x}.x=c:\psi x.\equiv_{x}.x=c:f(b,c)\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·112.</b> \[\begin{align}\vdash\colon\ldotp f\{(℩x)&(\phi x),\,({℩}x)(\psi x)\}.\equiv:\\ +&(\exists b,c):\phi x.\equiv_{x}.x=b:\psi x.\equiv_{x}.x=c:f(b,c)\\ +[\text{Proof as in *14·111}]\end{align}\]</p> + +<p>In the above proposition, we assume the convention explained on <a href="#Page_182">p. 182</a>, +after the statement of <a href="#*14·03">*14·03</a>.</p> + +<p class="nind"> +<b><a id="*14·113">*14·113</a>.</b> \[\begin{align}&\vdash:[({℩}x)(\psi x)].f\{({℩}x)(\phi x),({℩}x)(\psi x)\}.\equiv.f\{({℩}x)(\phi x),\,({℩}x)(\psi x)\}\\ +&[\text{*14·111·112}]\end{align}\]</p> + +<p>This proposition shows that when two descriptions occur in the same +proposition, the truth-value of the proposition is unaffected by the +question which has the larger scope.</p> + +<p class="nind"> +<b>*14·12.</b> \(\vdash\colon\ldotp\text{E}!({℩}x)(\phi x).\supset:\phi x.\phi y.\supset_{x,y}.x=y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·11} &\supset\vdash\colon\ldotp\text{Hp}.\supset:(\exists b):\phi x.\equiv_{x}.x=b &\qquad \text{(1)}\\ +\vdash.\text{*4·38.*10·1.*11·11·3}.&\supset\\ +\vdash\colon\ldotp\phi x.\equiv_{x}.x=b:&\supset:\phi x.\phi y.\equiv_{x,y}.x=b.y=b.\\ +[\text{*13·172}] &\supset_{x,y}.x=y &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23}. &\supset\vdash\colon\ldotp(\exists b):\phi x.\equiv_{x}.x=b:\supset:\phi x.\phi y.\supset_{x,y}.x=y &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}. & \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_185">[Pg 185]</span></p> + +<p class="nind"> +<b>*14·121.</b> \(\vdash\colon\ldotp\phi x.\equiv_{x}.x=b:\phi x.\equiv_{x}.x=c:\supset.b=c\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}.\supset\vdash\colon\ldotp\text{Hp}.&\supset:\phi b.\equiv.b=b:\phi b.\equiv.b=c:\\ +[\text{*13·15}] &\supset:\phi b:\phi b.\equiv.b=c:\\ +[\text{Ass}] &\supset:b=c\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*14·122">*14·122</a>.</b> \[\begin{align}\vdash\colon\ldotp\phi x.&\equiv_{x}.x=b:\equiv:\phi x.\supset_{x}.x=b:\phi b:\\ +&\equiv:\phi x.\supset_{x}.x=b:(\exists x).\phi x\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·22}. &\supset\vdash\colon\ldotp\phi x.\equiv_{x}.x=b:\equiv:\phi x.\supset_{x}.x=b:x=b.\supset_{x}.\phi x:\\ +[\text{*13·191}] &\equiv:\phi x.\supset_{x}.x=b:\phi b &\qquad \text{(1)}\\ +\vdash.\text{*4·71}. &\supset\vdash\colon\ldotp\phi x.\supset.x=b:\supset:\phi x.\equiv.\phi x.x=b\colon\ldotp\\ +[\text{*10·11·27}] &\supset\vdash\colon\ldotp\phi x.\supset_{x}.x=b:\supset:\phi x.\equiv_{x}.\phi x.x=b:\\ +[\text{*10·281}] & \supset:(\exists x).\phi x.\equiv.(\exists x).\phi x.x=b.\\ +[\text{*13·195}] &\equiv.\phi b &\qquad \text{(2)}\\ +\vdash.\text{(2).*5·32}.&\supset\vdash\colon\ldotp\phi x.\supset_{x}.x=b:(\exists x).\phi x:\equiv:\phi x.\supset_{x}.x=b:\phi b &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The two following propositions (*14·123·124) are placed here because of +the analogy with <a href="#*14·122">*14·122</a>, but they are not used until we come to the +theory of couples (<a href="#*55">*55</a> and <a href="#*56">*56</a>).</p> + +<p class="nind"> +<b>*14·123.</b> \[\begin{align}\vdash\colon\ldotp \phi(z,w).\equiv_{z,w}.z&=x.w=y:\\ +&\equiv:\phi(z,w).\supset_{z,w}.z=x.w=y:\phi(x,y):\\ +&\equiv:\phi(z,w).\supset_{z,w}.z=x.w=y:(\exists z,w).\phi(z,w)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*11·31}. &\supset\vdash\colon\ldotp\phi(z,w).\equiv_{z,w}.z=x.w=y:\\ +&\equiv:\phi(z,w).\supset_{z,w}.z=x.w=y:z=x.w=y.\supset_{z,w}.\phi(z,w):\\ +[\text{*13·21}] &\equiv:\phi(z,w).\supset_{z,w}.z=x.w=y:\phi(x,y) &\qquad \text{(1)}\\ +\vdash.\text{*4·71}. &\supset\vdash\colon\ldotp\phi(z,w).\supset.z=x.w=y:\\ +&\supset:\phi(z,w).\equiv.\phi(z,w).z=x.w=y\colon\ldotp\\ +[\text{*11·11·32}] &\supset\vdash\colon\ldotp\phi(z,w).\supset_{z,w}.z=x.w=y:\\ +&\supset:\phi(z,w).\equiv_{z,w}.\phi(z,w).z=x.w=y:\\ +[\text{*11·341}] &\supset:(\exists z,w).\phi(z,w).\equiv.(\exists z,w).\phi(z,w).z=x.w=y.\\ +[\text{*13·22}] &\equiv.\phi(x,y) &\qquad \text{(2)}\\ +\vdash.\text{(2).*5·32}.&\supset\vdash\colon\ldotp\phi(z,w).\supset_{z,w}.z=x.w=y:(\exists z,w).\phi(z,w):\\ +&\equiv:\phi(z,w).\supset_{z,w}.z=x.w=y:\phi(x,y) &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}. & \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_186">[Pg 186]</span></p> + +<p class="nind"> +<b>*14·124.</b> \[\begin{align}\vdash\colon\ldotp(\exists x,y):&\phi(z,w).\equiv_{z,w}.z=x.w=y:\\ +&\equiv:(\exists x,y).\phi(x,y):\phi(z,w).\phi(u,v).\supset_{z,w,u,v}.z=u.w=v\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·123.*3·27}.&\supset +\vdash\colon\ldotp(\exists x,y):\phi(z,w).\equiv_{z,w}.z=x.w=y:\supset.(\exists x,y).\phi(x,y) &\qquad \text{(1)}\\ +\vdash.\text{*11·1.*3·47}.&\supset\vdash\colon\ldotp\phi(z,w).\equiv_{z,w}.z=x.w=y:\\ +&\supset:\phi(z,w).\phi(u,v).\supset.z=x.w=y.u=x.v=y.\\ +[\text{*13·172}] &\supset.z=u.w=v &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·35}.\supset\\ +&\vdash\colon\ldotp(\exists x,y):\phi(z,w).\equiv_{z,w}.z=x.w=y:\\ +&\supset:\phi(z,w).\phi(u,v).\supset.z=u.w=v &\qquad \text{(3)}\\ +\vdash.\text{(3).*11·11·3}.\supset\\ +&\vdash\colon\ldotp(\exists x,y):\phi(z,w).\equiv_{z,w}.z=x.w=y:\\ +&\supset:\phi(z,w).\phi(u,v).\supset_{z,w,u,v}.z=u.w=v &\qquad \text{(4)}\\ +\vdash.\text{*11·1}.&\supset\vdash\colon\ldotp\phi(x,y):\phi(z,w).\phi(u,v).\supset_{z,w,u,v}.z=u.w=v:\\ +&\supset:\phi(x,y):\phi(z,w).\phi(x,y).\supset_{z,w}.z=x.w=y:\\ +[\text{*5·33}] &\supset:\phi(x,y):\phi(z,w).\supset_{z,w}.z=x.w=y:\\ +[\text{*14·123}] &\supset:\phi(z,w).\equiv_{z,w}.z=x.w=y &\qquad \text{(5)}\\ +\vdash.\text{(5).*11·11·34·45}.\supset\\ +&\vdash\colon\ldotp(\exists x,y).\phi(x,y):\phi(z,w).\phi(u,v).\supset_{z,w,u,v}.z=u.w=v:\\ +&\supset:(\exists x,y):\phi(z,w).\equiv_{z,w}.z=x.w=y &\qquad \text{(6)}\\ +\vdash.\text{(1).(4).(6)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*14·13">*14·13</a>.</b> \(\vdash:a=({℩}x)(\phi x).\equiv.({℩}x)(\phi x)=a\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}. &\supset\vdash\colon\ldotp a=({℩}x)(\phi x).\equiv:(\exists b):\phi x.\equiv_{x}.x=b:a=b &\qquad \text{(1)}\\ +\vdash.\text{*13·16.*4·36}.&\supset\vdash\colon\ldotp\phi x.\equiv_{x}.x=b:a=b:\equiv:\phi x.\equiv_{x}.x=b:b=a:\\ +[\text{*10·11·281}] &\supset\vdash\colon\ldotp(\exists b):\phi x.\equiv_{x}.x=b:a=b:\\ +&\equiv:(\exists b):\phi x.\equiv_{x}.x=b:b=a:\\ +[\text{*14·1}] &\equiv:({℩}x)(\phi x)=a &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is not an <i>immediate</i> consequence of <a href="#*13·16">*13·16</a>, +because "\(a=({℩}x)(\phi x)\)" is not a value of the function +"\(x=y\)." Similar remarks apply to the following propositions.</p> + +<p class="nind"> +<b>*14·131.</b> \(\vdash:({℩}x)(\phi x)=(℩x)(\psi x).\equiv.({℩}x)(\psi x)=({℩}x)(\phi x)\)</p> + +<p><span class="pagenum" id="Page_187">[Pg 187]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}. &\supset\vdash\colon\colon({℩}x)(\phi x)=({℩}x)(\psi x).\equiv\colon\ldotp(\exists b):\phi x.\equiv_{x}.x=b:b=({℩}x)(\psi x)\colon\ldotp\\ +[\text{*14·1}] &\equiv\colon\ldotp(\exists b)\colon\ldotp\phi x.\equiv_{x}.x=b\colon\ldotp(\exists c):\psi x.\equiv_{x}.x=c:b=c\colon\ldotp\\ +[\text{*11·6}] &\equiv\colon\ldotp (\exists c)\colon\ldotp \psi x.\equiv_{x}.x=c\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:b=c\colon\ldotp\\ +[\text{*14·1}] &\equiv\colon\ldotp (\exists c)\colon\ldotp \psi x.\equiv_{x}.x=c:({℩}x)(\phi x)=c\colon\ldotp\\ +[\text{*14·13}] &\equiv\colon\ldotp (\exists c)\colon\ldotp \psi x.\equiv_{x}.x=c:c=({℩}x)(\phi x)\colon\ldotp \\ +[\text{*14·1}] &\equiv\colon\ldotp (℩x)(\psi x)=({℩}x)(\phi x)\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition, in accordance with our convention, the +descriptive expression (\({℩}x)(\phi x)\) is eliminated before +(\({℩}x)(\psi x)\), because it occurs first in "(\({℩}x)(\phi x)=({℩}x)(\psi x)\)"; +but in "(\({℩}x)(\psi x)=({℩}x)(\phi x)\)," +(\({℩}x)(\psi x)\) is to be first eliminated. The order of elimination +makes no difference to the truth-value, as was proved in <a href="#*14·113">*14·113</a>.</p> + +<p>The above proposition may also be proved as follows: +\[ +\begin{array}{l} +\vdash.\text{*14·111}.\supset\vdash\colon\ldotp (℩x)(\phi x)&=({℩}x)(\psi x).\\ +&\equiv:(\exists b,c):\phi x.\equiv_{x}.x=b:\psi x.\equiv_{x}.x=c:b=c:\\ +[\text{*4·3.*13·6.*11·11·341}] &\equiv:(\exists b,c):\psi x.\equiv_{x}.x=c:\phi x.\equiv_{x}.x=b:c=b:\\ +[\text{*11·2.*14·111}] &\equiv:(℩x)(\psi x)=({℩}x)(\phi x)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·14.</b> \(\vdash:a=b.b=({℩}x)(\phi x).\supset.a=({℩}x)(\phi x) \quad[\text{*13·13}]\)</p> + +<p class="nind"> +<b>*14·142.</b> \(\vdash:a=({℩}x)(\phi x).({℩}x)(\phi x)=({℩}x)(\psi x).\supset.a=({℩}x)(\psi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}.&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:a=b\colon\ldotp\\ +&(\exists c):\phi x.\equiv_{x}.x=c:c=({℩}x)(\psi x)\colon\ldotp\\ +[\text{*13·195}] &\supset\colon\ldotp \phi x.\equiv_{x}.x=a\colon\ldotp (\exists c):\phi x.\equiv_{x}.x=c:c=({℩}x)(\psi x)\colon\ldotp\\ +[\text{*10·35}] &\supset\colon\ldotp (\exists c)\colon\ldotp \phi x.\equiv_{x}.x=a:\phi x.\equiv_{x}.x=c:c=({℩}x)(\psi x)\colon\ldotp\\ +[\text{*14·121}] &\supset\colon\ldotp (\exists c)\colon\ldotp \phi x.\equiv_{x}.x=a:a=c:c=({℩}x)(\psi x)\colon\ldotp\\ +[\text{*3·27.*13·195}] &\supset\colon\ldotp a=({℩}x)(\phi x)\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·144.</b> \(\vdash:({℩}x)(\phi x)=({℩}x)(\psi x).({℩}x)(\psi x)=({℩}x)(\chi x).\supset.({℩}x)(\phi x)=({℩}x)(\chi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·111}.&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (\exists a,b):\phi x.\equiv_{x}.x=a:\psi x.\equiv_{x}.x=b:a=b\colon\ldotp\\ +&\qquad(\exists c,d):\psi x.\equiv_{x}.x=c:\chi x.\equiv_{x}.x=d:c=d\colon\ldotp\\ +[\text{*13·195}] &\supset\colon\ldotp (\exists a):\phi x.\equiv_{x}.x=a:\psi x.\equiv_{x}.x=a\colon\ldotp\\ +&\qquad(\exists c):\psi x.\equiv_{x}.x=c:\chi x.\equiv_{x}.x=c\colon\ldotp\\ +[\text{*11·54}] &\supset\colon\ldotp (\exists a,c):\phi x.\equiv_{x}.x=a:\psi x.\equiv_{x}.x=a:\\ +&\qquad\psi x.\equiv_{x}.x=c:\chi x.\equiv_{x}.x=c\colon\ldotp\\ +[\text{*14·121.*11·42}] &\supset\colon\ldotp (\exists a,c):\phi x.\equiv_{x}.x=a:\chi x.\equiv_{x}.x=c:a=c\colon\ldotp\\ +[\text{*14·111}] &\supset\colon\ldotp ({℩}x)(\phi x)=(℩x)(\chi x)\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·145.</b> \(\vdash:a=({℩}x)(\phi x).a=({℩}x)(\psi x).\supset.({℩}x)(\phi x)=({℩}x)(\psi x)\)</p> + +<p><span class="pagenum" id="Page_188">[Pg 188]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}. \supset\vdash\colon\ldotp a=({℩}x)(\phi x).&\equiv:(\exists b):\phi x.\equiv_{x}.x=b:a=b:\\ +[\text{*13·195}] &\equiv:\phi x.\equiv_{x}.x=a &\qquad \text{(1)}\\ +\vdash . \text{(1) . *14·1}. \supset \vdash \colon\colon \text{Hp} . &\equiv \colon\ldotp \phi x . \equiv_{x} . x = a \colon\ldotp (\exists b) : \psi x . \equiv_{x}. x = b : a = b \colon\ldotp\\ +[\text{*10·35}] &\equiv \colon\ldotp (\exists b) \colon\ldotp \phi x . \equiv_{x} . x = a : \psi x . \equiv_{x} . x = b : a = b \colon\ldotp \\ +[\text{*14·111}] & \supset \colon\ldotp ({℩}x)(\phi x) = (℩x)(\psi x) \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·15.</b> \(\vdash \colon\ldotp ({℩}x)(\phi x) = b . \supset : \psi \{({℩}x)(\phi x)\} . \equiv . \psi b\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash . \text{*14·1}. &\supset\\ +\vdash \colon\colon \text{Hp} . &\supset \colon\ldotp (\exists c) : \phi x . \equiv_{x} . x = c : c = b \colon\ldotp \\ +[\text{*13·195}] &\supset \colon\ldotp \phi x . \equiv_{x} . x \equiv b &&\qquad \text{(1)}\\ +\vdash . \text{(1) . *14·1}. &\supset\\ +\vdash \colon\colon \text{Hp} . \supset \colon\ldotp \psi \{({℩}x)(\phi x)\} .& \equiv : (\exists c) : x = b . \equiv_{x} . x = c : \psi c :\\ +[\text{*13·192}] &\equiv : \psi b \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·16.</b> \(\vdash \colon\ldotp ({℩}x)(\phi x) = ({℩}x)(\psi x) . \supset : \chi \{({℩}x)(\phi x)\} . \equiv . \chi \{({℩}x)(\psi x)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*14·1}. &\supset \vdash \colon\ldotp \text{Hp}. \supset : (\exists b) : \phi x . \equiv_{x} . x = b : b = ({℩}x)(\psi x) &\qquad \text{(1)}\\ +\vdash .\text{*14·1}. &\supset \vdash \colon\colon \phi x . \equiv_{x} . x = b : \supset \colon\ldotp\\ +&\chi \{(℩x)(\phi x)\} . \equiv : (\exists c) : x = b . \equiv_{x} . x = c : \chi c :\\ +[\text{*13·192}] &\equiv : \chi b &\qquad \text{(2)}\\ +\vdash . \text{*14·13·15}. &\supset \vdash \colon\ldotp b = ({℩}x)(\psi x) . \supset : \chi b . \equiv . \chi \{({℩}x)(\psi x)\} &\qquad \text{(3)}\\ +\vdash .\text{(2) . (3)}. &\supset \vdash \colon\ldotp \phi x . \equiv_{x} . x = b : b = ({℩}x)(\psi x) :\\ +&\supset : \chi \{({℩}x)(\phi x)\} . \equiv . \chi \{({℩}x)(\psi x)\} &\qquad \text{(4)}\\ +\vdash . \text{(1) . (4) . *10·1·23}. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·17.</b> \(\vdash \colon\ldotp ({℩}x)(\phi x) = b . \equiv : \psi ! ({℩}x)(\phi x) . \equiv_{\psi} . \psi ! b\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*14·15 . *10·11·21}. \supset\\ +\vdash \colon\ldotp ({℩}x)(\phi x) = b . \supset : \psi ! ({℩}x)(\phi x) . \equiv_{\psi} . \psi ! b &\qquad \text{(1)}\\ +\vdash .\text{*10·1 . *4·22}. \supset \vdash \colon\colon \chi ! x . \equiv_{x} . x = b : \psi ! ({℩}x)(\phi x) . \equiv_{\psi} . \psi ! b :\\ +\supset : ({℩}x)(\phi x) = b . \equiv . b = b :\\ +[\text{*13·15}] \supset : ({℩}x)(\phi x) = b &\qquad \text{(2)}\\ +\vdash .\text{(2) . Exp}. *10·11·23 . \supset\\ +\vdash \colon\colon (\exists \chi) : \chi ! x . \equiv_{x} . x = b : \supset \colon\ldotp \psi ! ({℩}x)(\phi x) . \equiv_{\psi} + . \psi ! b : \supset . ({℩}x)(\phi x) = b &\qquad \text{(3)}\\ +\vdash . \text{*12·1}. \supset \vdash : (\exists \chi) : \chi ! x . \equiv_{x} . x = b &\qquad \text{(4)}\\ +\vdash .\text{(3) . (4)}. \supset \vdash \colon\ldotp \psi ! ({℩}x)(\phi x) . \equiv_{\psi} .\psi ! b : \supset . ({℩}x)(\phi x) = b &\qquad \text{(5)}\\ +\vdash .\text{(1) . (5)}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>It should be observed that we do <i>not</i> have +\[ +({℩}x)(\phi x) = b . \equiv : \psi ! ({℩}x)(\phi x) . \supset_{\psi} . \psi ! b +\] +for, if \({\sim}\text{E} ! ({℩}x)(\phi x), \psi ! ({℩}x)(\phi x)\) is +always false, and therefore +\[ +\psi ! ({℩}x)(\phi x) . \supset_{\psi} . \psi ! b +\] +holds for all values of \(b\). But we do have</p> + +<p><span class="pagenum" id="Page_189">[Pg 189]</span></p> + +<p class="nind"> +<b>*14·171.</b> \(\vdash\colon\ldotp ({℩}x)(\phi x)=b.\equiv:\psi!b.\supset_{\psi}.\psi!({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·17}. &\supset\vdash\colon\ldotp ({℩}x)(\phi x)=b.\supset:\psi!b.\supset_{\psi}.\psi!({℩}x)(\phi x) &\qquad \text{(1)}\\ +\vdash.\text{*10·1.*12·1}. &\supset\vdash\colon\ldotp \psi!b.\supset_{\psi}.\psi!({℩}x)(\phi x):\supset:b=b.\supset.({℩}x)(\phi x)=b:\\ +[\text{*13·15}] &\supset:({℩}x)(\phi x)=b &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*14·18">*14·18</a>.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\supset:(x).\psi x.\supset.\psi({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash:(x).\psi x.\supset.\psi b:\\ +[\text{Fact}] &\supset\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:(x).\psi x:\supset:\phi x.\equiv_{x}.x=b:\psi b:\\ +[\text{*10·11·28}] &\supset\vdash\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:(x).\psi x:\supset:(\exists b):\phi x.\equiv_{x}.x=b:\psi b\colon\ldotp \\ +[\text{*10·35}] &\supset\vdash\colon\colon (\exists b):\phi x.\equiv_{x}.x=b\colon\ldotp (x).\psi x\colon\ldotp \supset:(\exists b):\phi x.\equiv_{x}.x=b:\psi b\colon\ldotp\\ +[\text{*14·1·11}] &\supset\vdash\colon\ldotp \text{E}!({℩}x)(\phi x):(x).\psi x:\supset:\psi({℩}x)(\phi x)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition shows that, provided (\({℩}x)(\phi x)\) exists, +it has (speaking formally) all the logical properties of symbols which +directly represent objects. Hence when (\({℩}x)(\phi x)\) exists, +the fact that it is an incomplete symbol becomes irrelevant to the +truth-values of logical propositions in which it occurs.</p> + +<p class="nind"> +<b>*14·2.</b> \(\vdash.({℩}x)(x=a)=a\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·101}. \supset\vdash\colon\ldotp ({℩}x)(x=a)=a.\equiv:(\exists b):x=a.\equiv_{x}.x=b:b=a:\\ +[\text{*13·195}] \equiv:x=a.\equiv_{x}.x=a &\qquad \text{(1)}\\ +\vdash.\text{(1).Id}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·201.</b> \(\vdash:\text{E}!({℩}x)(\phi x).\supset.(\exists x).\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{ll} +\vdash.\text{*14·11}. \supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists b):\phi x.\equiv_{x}.x=b:\\ +[\text{*10·1}] \supset:(\exists b):\phi b.\equiv.b=b:\\ +[\text{*13·15}] \supset:(\exists b).\phi b\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*14·202">*14·202</a>.</b> \(\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\equiv:({℩}x)(\phi x)=b:\equiv:\phi x.\equiv_{x}.b=x:\equiv:b=({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}. \supset\vdash\colon\ldotp ({℩}x)(\phi x)=b.&\equiv:(\exists c):\phi x.\equiv_{x}.x=c:c=b:\\ +[\text{*13·195}] &\equiv:\phi x.\equiv_{x}.x=b\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>[The second half is proved in the same way as the first half.]</p> + +<p class="nind"> +<b>*14·203.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\equiv:(\exists x).\phi x:\phi x.\phi y.\supset_{x,y}.x=y\)</p> + +<p><span class="pagenum" id="Page_190">[Pg 190]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·12·201}. &\supset\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\supset:(\exists x).\phi x:\phi x.\phi y.\supset_{x,y}.x=y &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp \phi b:\phi x.\phi y.\supset_{x,y}.x=y:\supset:\phi b:\phi x.\phi b.\supset_{x}.x=b:\\ +[\text{*5·33}] & \supset:\phi b:\phi x.\supset_{x}.x=b:\\ +[\text{*13·191}] &\supset:x=b.\supset_{x}.\phi x:\\ +&\qquad\quad\phi x.\supset_{x}.x=b:\\ +[\text{*10·22}] &\supset:\phi x.\equiv_{x}.x=b &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·1·28}. &\supset\vdash\colon\ldotp (\exists b):\phi b:\phi x.\phi y.\supset_{x,y}.x=y:\supset:(\exists b):\phi x.\equiv_{x}.x=b\colon\ldotp\\ +[\text{*10·35}] & \supset\vdash\colon\ldotp (\exists b).\phi b:\phi x.\phi y.\supset_{x,y}.x=y:\supset:(\exists b):\phi x.\equiv_{x}.x=b:\\ +[\text{*14·11}] &\supset:\text{E}!({℩}x)(\phi x) &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·204.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\equiv:(\exists b).({℩}x)(\phi x)=b\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·202.*10·11}.\supset\\ +\vdash\colon\ldotp (b)\colon\ldotp \phi x.\equiv_{x}.x=b:\equiv:({℩}x)(\phi x)=b\colon\ldotp \supset\\ +[\text{*10·281}] \vdash\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:\equiv:(\exists b).({℩}x)(\phi x)=b &\qquad \text{(1)}\\ +\vdash.\text{(1).*14·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·205.</b> \(\vdash:\psi({℩}x)(\phi x).\equiv.(\exists b).b=({℩}x)(\phi x).\psi b \quad[\text{*14·202·1}]\)</p> + +<p class="nind"> +<b><a id="*14·21">*14·21</a>.</b> \(\vdash:\psi({℩}x)(\phi x).\supset.\text{E}!({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}.\supset\\ +\vdash\colon\ldotp \psi\{({℩}x)(\phi x)\}.&\supset:(\exists b):\phi x.\equiv_{x}.x=b:\psi b:\\ +[\text{*10·5}] &\supset:(\exists b):\phi x.\equiv_{x}.x=b:\\ +[\text{*14·11}] &\supset:\text{E}!({℩}x)(\phi x)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that if any true statement can be made about +(\({℩}x)(\phi x)\), then (\({℩}x)(\phi x)\) must exist. Its use +throughout the remainder of the work will be very frequent.</p> + +<p>When (\({℩}x)(\phi x)\) does not exist, there are still true +propositions in which "(\({℩}x)(\phi x)\)" occurs, but it has, in such +propositions, a <i>secondary</i> occurrence, in the sense explained in +<a href="#CHAPTER_III">Chapter III</a> of the Introduction, <i>i.e.</i> the asserted proposition +concerned is not of the form \(\psi({℩}x)(\phi x)\), but of the form +\(f\{\psi({℩}x)(\phi x)\}\), in other words, the proposition which is +the scope of (\({℩}x)(\phi x)\) is only part of the whole asserted +proposition.</p> + +<p class="nind"> +<b>*14·22.</b> \(\vdash:\text{E}!({℩}x)(\phi x).\equiv.\phi({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·122}. &\supset\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\supset.\phi b &\qquad \text{(1)}\\ +\vdash.\text{(1).*4·71}. &\supset\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\equiv:\phi x.\equiv_{x}.x=b:\phi b\colon\ldotp \\ +[\text{*10·11·281}] &\supset\vdash\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:\equiv:(\exists b):\phi x.\equiv_{x}.x=b:\phi b\colon\ldotp \\ +[\text{*14·11·101}] &\supset\vdash:\text{E}!({℩}x)(\phi x).\equiv.\phi({℩}x)(\phi x):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_191">[Pg 191]</span></p> + +<p>As an instance of the above proposition, we may take the following: +"The proposition 'the author of Waverley existed' is equivalent to +'the man who wrote Waverley wrote Waverley.'" Thus such a proposition +as "the man who wrote Waverley wrote Waverley" does not embody a +logically necessary truth, since it would be false if Waverley had not +been written, or had been written by two men in collaboration. For +example, "the man who squared the circle squared the circle" is a false +proposition.</p> + +<p class="nind"> +<b>*14·23.</b> \(\vdash: \text{E}!({℩}x)(\phi x.\psi x).\equiv.\phi \{({℩}x)(\phi x.\psi x)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·22}.&\supset\vdash\colon\ldotp \text{E}!({℩}x)(\phi x.\psi x).\\ +&\equiv:[({℩}x)(\phi x.\psi x)]:\phi \{({℩}x)(\phi x.\psi x)\}\psi \{({℩}x)(\phi x.\psi x)\}\\ +[\text{*10·5.*3·26}] &\supset:\phi\{({℩}x)(\phi x.\psi x)\} &\qquad \text{(1)}\\ +\vdash.\text{*14·21}.&\supset\vdash:\phi \{({℩}x)(\phi x.\psi x)\}.\supset.\text{E}!({℩}x)(\phi x.\psi x) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Note that in the second line of the above proof <a href="#*10·5">*10·5</a>, not only <a href="#*3·26">*3·26</a>, +is required. For the scope of the descriptive symbol (\({℩}x)(\phi x.\psi x)\) +is the whole product \(\phi \{({℩}x)(\phi x.\psi x)\}\psi\{({℩}x)(\phi x.\psi x)\}\), +so that, applying <a href="#*14·1">*14·1</a>, the proposition on the right in the first line +becomes +\[ +(\exists b):\phi x.\psi x.\equiv_{x}.x=b:\phi b.\psi b +\] +which, by <a href="#*10·5">*10·5</a> and <a href="#*3·26">*3·26</a>, implies +\[ +\begin{array}{l} +(\exists b):&\phi x.\psi x.\equiv_{x}.x=b:\phi b,\\ +\textit{i.e.} &\phi\{({℩}x)(\phi x.\psi x)\}. +\end{array} +\]</p> + +<p class="nind"> +<b>*14·24.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\equiv:[({℩}x)(\phi x)]:\phi y.\equiv_{y}.y=({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}.\supset\vdash\colon\ldotp [({℩}x)(\phi x)]:\phi y.&\equiv_{y}.y=(℩x)(\phi x):\\ +&\equiv:(\exists b):\phi y.\equiv_{y}.y=b:\phi y.\equiv_{y}.y=b:\\ +[\text{*4·24.*10·281}] &\equiv:(\exists b):\phi y.\equiv_{y}.y=b:\\ +[\text{*14·11}] & \equiv:\text{E}!({℩}x)(\phi x)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition should be compared with <a href="#*14·241">*14·241</a>, where, in virtue of +the smaller scope of (\({℩}x)(\phi x)\), we get an implication instead +of an equivalence.</p> + +<p class="nind"> +<b><a id="*14·241">*14·241</a>.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\supset:\phi y.\equiv_{y}.y=({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·203}.&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \phi y.\phi x.\supset.y=x\colon\ldotp \\ +[\text{Exp}] &\supset\colon\ldotp \phi y.\supset:\phi x.\supset.y=x\colon\colon \\ +[\text{*10·11·21}] &\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \phi y.\supset:\phi x.\supset_{x}.y=x\colon\ldotp\\ +[\text{*4·71}] \supset\colon\ldotp \phi y.&\equiv:\phi y:\phi x.\supset_{x}.y=x:\\ +[\text{*13·191}] &\equiv:y=x.\supset_{x}.\phi x:\phi x.\supset_{x}.y=x:\\ +[\text{*10·22}] &\equiv:\phi x.\equiv_{x}.y=x\\ +[\text{*14·202}] &\equiv:y=({℩}x)(\phi x)\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_192">[Pg 192]</span></p> + +<p class="nind"> +<b><a id="*14·242">*14·242</a>.</b> \(\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\supset:\psi b.\equiv.\psi({℩}x)(\phi x) \quad[\text{*14·202·15}]\)</p> + +<p class="nind"> +<b>*14·25.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\supset:\phi x\supset_{x}\psi x.\equiv.\psi({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·84.*10·27·271}.\supset\vdash\colon\colon \phi x.&\equiv_{x}.x=b:\supset\colon\ldotp \phi x\supset_{x}\psi x.\equiv:x=b.\supset_{x}.\psi x:\\ +[\text{*13·191}] &\equiv:\psi b:\\ +[\text{*14·242}] &\equiv.\psi({℩}x)(\phi x) &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·23}.&\supset\vdash\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:\\ +&\supset:\phi x\supset_{x}\psi x.\equiv.\psi({℩}x)(\phi x) &\qquad \text{(2)}\\ +\vdash.\text{(2).*14·11}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·26.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\supset:(\exists x).\phi x.\psi x.\equiv.\psi \{({℩}x)(\phi x)\}.\equiv.\phi x\supset_{x}\psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·11}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:(\exists b):\phi x.\equiv_{x}.x=b &\qquad \text{(1)}\\ +\vdash.\text{*10·311}.&\supset\vdash\colon\colon \phi x.\equiv_{x}.x=b:\supset\colon\ldotp \phi x.\psi x.\equiv_{x}.x=b.\psi x\colon\ldotp \\ +[\text{*10·281}] & \supset\colon\ldotp (\exists x).\phi x.\psi x.\equiv.(\exists x).x=b.\psi x.\\ +[\text{*13·195}] &\equiv.\psi b\\ +[\text{*14·242}] &\equiv.\psi \{({℩}x)(\phi x)\} &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23}.\supset\\ +\vdash\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:&\supset:(\exists x).\phi x.\psi x.\equiv.\psi \{({℩}x)(\phi x)\} &\qquad \text{(3)}\\ +\vdash.\text{(1).(3).*14·25}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·27.</b> \(\vdash\colon\ldotp \text{E}!({℩}x)(\phi x).\supset:\phi x\equiv_{x}\psi x.\equiv.({℩}x)(\phi x)=({℩}x)(\psi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·86·21}. &\supset\vdash\colon\colon \phi x.\equiv_{x}.x=b:\supset\colon\ldotp \phi x.\equiv.\psi x:\equiv:\psi x.\equiv.x=b &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·27}.&\supset\vdash\colon\colon \phi x.\equiv_{x}.x=b:\supset\colon\ldotp (x)\colon\ldotp \phi x.\equiv.\psi x:\equiv:\psi x.\equiv.x=b\colon\ldotp\\ +[\text{*10·271}] &\supset\colon\ldotp \phi x.\equiv_{x}.\psi x:\equiv:\psi x.\equiv_{x}.x=b:\\ +[\text{*14·202}] &\equiv:b=({℩}x)(\psi x)\\ +[\text{*14·242}] &\equiv:({℩}x)(\phi x)=({℩}x)(\psi x) &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23.*14·11}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*14·271.</b> \(\vdash\colon\ldotp \phi x.\equiv_{x}.\psi x:\supset:\text{E}!({℩}x)(\phi x).\equiv.\text{E}!({℩}x)(\psi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·86}. &\supset\vdash\colon\colon \phi x\equiv\psi x.\supset\colon\ldotp \phi x.\equiv.x=b:\equiv:\psi x.\equiv.x=b\colon\colon \\ +[\text{*10·11·27}]&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (x)\colon\ldotp \phi x.\equiv.x=b:\equiv:\psi x.\equiv.x=b\colon\ldotp \\ +[\text{*10·271}] &\supset\colon\ldotp (x):\phi x.\equiv.x=b:\equiv:(x):\psi x.\equiv.x=b\colon\colon \\ +[\text{*10·11·21}]&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (b)\colon\ldotp \phi x.\equiv_{x}.x=b:\equiv:\psi x.\equiv_{x}.x=b\colon\ldotp \\ +[\text{*10·281}] &\supset\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:\equiv:(\exists b):\psi x.\equiv_{x}.x=b\colon\colon \\ +&\qquad\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_193">[Pg 193]</span></p> + +<p class="nind"> +<b>*14·272.</b> \(\vdash\colon\ldotp \phi x.\equiv_{x}.\psi x:\supset:\chi(℩x)(\phi x).\equiv.\chi({℩}x)(\psi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·86}. &\supset\vdash\colon\colon \phi x\equiv\psi x.\supset\colon\ldotp \phi x.\equiv.x=b:\equiv:\psi x.\equiv.x=b\colon\ldotp \\ +[\text{*10·11·414}]&\supset\vdash\colon\colon \text{Hp}. \supset\colon\ldotp \phi x.\equiv_{x}.x=b:\equiv:\psi x.\equiv_{x}.x=b\colon\ldotp \\ +[\text{Fact}] &\supset\colon\ldotp \phi x.\equiv_{x}.x=b:\chi b:\equiv:\psi x.\equiv_{x}.x=b:\chi b\colon\ldotp \\ +[\text{*10·11·21}] &\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (b)\colon\ldotp \phi x.\equiv_{x}.x=b:\chi b:\equiv:\psi x.\equiv_{x}.x=b:\chi b\colon\ldotp \\ +[\text{*10·281}] &\supset\colon\ldotp (\exists b)\colon\ldotp \phi x.\equiv_{x}.x=b:\chi b:\equiv\\ +&:(\exists b):\psi x.\equiv_{x}.x=b:\chi b\colon\ldotp \\ +[\text{*14·101}] &\supset\colon\ldotp \chi({℩}x)(\phi x).\equiv.\chi({℩}x)(\psi x)\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above two propositions show that \(\text{E}!({℩}x)(\phi x)\) and +\(\chi({℩}x)(\phi x)\) are "extensional" properties of \(\phi\hat{x}\), +<i>i.e.</i> their truth-value is unchanged by the substitution, for +\(\phi\hat{x}\), of any formally equivalent function \(\psi\hat{x}\).</p> + +<p class="nind"> +<b><a id="*14·28">*14·28</a>.</b> \(\vdash:\text{E}!({℩}x)(\phi x).\equiv.(℩x)(\phi x)=({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*13·15.*4·73}.&\supset\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\equiv:\phi x.\equiv_{x}.x=b:b=b &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·281}.&\supset\\ +&\vdash\colon\ldotp (\exists b):\phi x.\equiv_{x}.x=b:\equiv:(\exists b):\phi x.\equiv_{x}.x=b:b=b &\qquad \text{(2)}\\ +\vdash.\text{(2).*14·1·11}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition states that (\({℩}x)(\phi x)\) is identical with +itself whenever it exists, but not otherwise. Thus for example the +proposition "the present King of France is the present King of France" +is false.</p> + +<p>The purpose of the following propositions is to show that, when +\(\text{E}!({℩}x)(\phi x)\), the scope of (\({℩}x)(\phi x)\) does not +matter to the truth-value of any proposition in which (\({℩}x)(\phi x)\) +occurs. This proposition cannot be proved generally, but it can +be proved in each particular case. The following propositions show the +method, which proceeds always by means of <a href="#*14·242">*14·242</a>, <a href="#*10·23">*10·23</a> and <a href="#*14·11">*14·11</a>. +The proposition can be proved generally when (\({℩}x)(\phi x)\) +occurs in the form \(\chi({℩}x)(\phi x)\), and \(\chi({℩}x)(\phi x)\) +occurs in what we may call a "truth-function," <i>i.e.</i> a function +whose truth or falsehood depends only upon the truth or falsehood of +its argument or arguments. This covers all the cases with which we are +ever concerned. That is to say, if \(\chi({℩}x)(\phi x)\) occurs in +any of the ways which can be generated by the processes of <a href="#*1">*1</a>—<a href="#*11">*11</a>, +then, provided \(\text{E}!({℩}x)(\phi x)\), the truth-value of +\(f\{[({℩}x)(\phi x)].\chi({℩}x)(\phi x)\}\) is the same as that of +\[ +[({℩}x)(\phi x)].f\{\chi({℩}x)(\phi x)\}. +\] +<span class="pagenum" id="Page_194">[Pg 194]</span>This is proved in the following proposition. In this proposition, +however, the use of propositions as apparent variables involves an +apparatus not required elsewhere, and we have therefore not used this +proposition in subsequent proofs.</p> + +<p class="nind"> +<b><a id="*14·3">*14·3</a>.</b> \[\begin{align}&\vdash\colon\ldotp p\equiv q.\supset_{p,q}.f(p)\equiv f(q):\text{E}!({℩}x)(\phi x):\supset:\\ +&f\{[({℩}x)(\phi x)].\chi({℩}x)(\phi x)\}.\equiv.[({℩}x)(\phi x)].f\{\chi(℩x)(\phi x)\}\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·242}.\supset\\ +\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\supset:[(℩x)(\phi x)].\chi(℩x)(\phi x).\equiv.\chi b &\qquad \text{(1)}\\ +\vdash.\text{(1)}.\supset\vdash\colon\ldotp p\equiv q.\supset_{p,q}.f(p)\equiv f(q):\phi x.\equiv_{x}.x=b:\supset:\\ +\qquad\qquad f\{[(℩x)(\phi x)].\chi(℩x)(\phi x)\}.\equiv.f(\chi b) &\qquad \text{(2)}\\ +\vdash.\text{*14·242}.\supset\\ +\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\supset:[(℩x)(\phi x)].f\{\chi(℩x)(\phi x)\}.\equiv.f(\chi b) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\\ +\vdash\colon\ldotp p\equiv q.\supset_{p,q}.f(p)\equiv f(q):\phi x.\equiv_{x}.x=b:\supset:\\ +\qquad\qquad f\{[(℩x)(\phi x)].\chi(℩x)(\phi x)\}.\equiv.[(℩x)(\phi x)].f\{\chi(℩x)(\phi x)\} &\qquad \text{(4)}\\ +\vdash.\text{(4).*10·23.*14·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are immediate applications of the above. +They are, however, independently proved, because <a href="#*14·3">*14·3</a> introduces +propositions (\(p\), \(q\) namely) as apparent variables, which we have +not done elsewhere, and cannot do legitimately without the explicit +introduction of the hierarchy of propositions with a reducibility-axiom +such as <a href="#*12·1">*12·1</a>.</p> + +<p class="nind"> +<b><a id="*14·31">*14·31</a>.</b> \[\begin{align}\vdash\colon\colon \text{E}!(℩x)(\phi x).\supset\colon\ldotp [(℩x)(\phi x)].&p\lor \chi(℩x)(\phi x).\\ +&\equiv:p.\lor.[(℩x)(\phi x)].\chi(℩x)(\phi x)\end{align}\]</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·242}.&\supset\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\supset:[(℩x)(\phi x)].p\lor\chi(℩x)(\phi x).\equiv.p\lor\chi b &\qquad \text{(1)}\\ +\vdash.\text{*14·242}.&\supset\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\supset:[(℩x)(\phi x)].\chi(℩x)(\phi x).\equiv.\chi b:\\ +[\text{*4·37}] & \supset:p\lor[(℩x)(\phi x)]\chi(℩x)(\phi x).\equiv.p\lor\chi b &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp \phi x.\equiv_{x}.x=b:\supset:[(℩x)(\phi x)].p\lor\chi(℩x)(\phi x).\\ +&\equiv.p\lor[(℩x)(\phi x)]\chi(℩x)(\phi x) &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·23.*14·11}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are proved in precisely the same way as +<a href="#*14·31">*14·31</a>; hence we shall merely give references to the propositions used +in the proofs.</p> + +<p class="nind"> +<b>*14·32.</b> \[\begin{align}\vdash\colon\ldotp \text{E}!(℩x)(\phi x).\equiv:[(℩x)(\phi x)].{\sim}&\chi(℩x)(\phi x).\\ +&\equiv.{\sim}\{[(℩x)(\phi x)].\chi(℩x)(\phi x)\}\\ +\quad[\text{*14·242.*4·11.*10·23.*14·11}]\end{align}\]</p> + +<p>The equivalence asserted here fails when \({\sim}\text{E}!(℩x)(\phi x)\). +Thus, for example, let \(\phi y\) be "\(y\) is King of France." +Then (\(℩x)(\phi x)\) = the King of France. Let \(\chi y\) be "\(y\) +is bald." Then \([(℩x)(\phi x)].{\sim}\chi(℩x)(\phi x).=.\) the<span class="pagenum" id="Page_195">[Pg 195]</span> +King of France exists and is not bald; but \({\sim}{[(℩x)(\phi x)].\chi(℩x)(\phi x)}.=.\) +it is false that the King of France exists and is bald. Of these the +first is false, the second true. Either might be meant by "the King of +France is not bald," which is ambiguous; but it would be more natural +to take the first (false) interpretation as the meaning of the words. +If the King of France existed, the two would be equivalent; thus as +applied to the King of England, both are true or both false.</p> + +<p class="nind"> +<b>*14·33.</b> \[\begin{align}\vdash\colon\colon \text{E}!(℩x)(\phi x).\supset\colon\ldotp [(℩x)(\phi x)].&p\supset\chi(℩x)(\phi x).\\ +&\equiv:p.\supset.[(℩x)(\phi x)].\chi(℩x)(\phi x)\\ +\qquad[\text{*14·242.*4·85.*10·23.*14·11}]\end{align}\]</p> + +<p class="nind"> +<b>*14·331.</b> \[\begin{align}\vdash\colon\colon \text{E}!(℩x)(\phi x).\supset\colon\ldotp [(℩x)(\phi x)].&\chi(℩x)(\phi x)\supset p.\\ +&\equiv:[(℩x)(\phi x)].\chi(℩x)(\phi x).\supset.p\\ +\qquad[\text{*4·84.*14·242.*10·23.*14·11}]\end{align}\]</p> + +<p class="nind"> +<b>*14·332.</b> \[\begin{align}\vdash\colon\colon \text{E}!(℩x)(\phi x).\supset\colon\ldotp [(℩x)(\phi x)].p&\equiv\chi(℩x)(\phi x).\equiv\\ +&:p.\equiv.[(℩x)(\phi x)].\chi (℩x)(\phi x)\\ +\qquad[\text{*4·86.*14·242.*10·23.*14·11}]\end{align}\]</p> + +<p class="nind"> +<b><a id="*14·34">*14·34</a>.</b> \(\vdash\colon\ldotp p:[(℩x)(\phi x)].\chi(℩x)(\phi x):\equiv:[(℩x)(\phi x)]:p.\chi(℩x)(\phi x)\)</p> + +<p>This proposition does not require the hypothesis \(\text{E}!(℩x)(\phi x)\).</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}.\supset\\ +\vdash\colon\ldotp p:[(℩x)(\phi x)].\chi (℩x)(\phi x):\equiv:p:(\exists b):\phi x.&\equiv_{x}.x=b:\chi b:\\ +[\text{*10·35}] &\equiv:(\exists b):p:\phi x.\equiv_{x}.x=b:\chi b:\\ +[\text{*14·1}] &\equiv:[(℩x)(\phi x)]:p.\chi(℩x)(\phi x)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Propositions of the above type might be continued indefinitely, but +as they are proved on a uniform plan, it is unnecessary to go beyond +the fundamental cases of \(p\lor q\), \({\sim}p\), \(p\supset q\) and +\(p.q.\)</p> + +<p>It should be observed that the proposition in which (\(℩x)(\phi x)\) +has the larger scope always implies the corresponding one in which +it has the smaller scope, but the converse implication only holds if +either (<i>a</i>) we have \(\text{E}!(℩x)(\phi x)\) or (<i>b</i>) +the proposition in which (\(℩x)(\phi x)\) has the smaller scope +implies \(\text{E}!(℩x)(\phi x)\). The second case occurs in <a href="#*14·34">*14·34</a>, +and is the reason why we get an equivalence without the hypothesis +\(\text{E}!(℩x)(\phi x)\). The proposition in which (\(℩x)(\phi x)\) +has the larger scope always implies \(\text{E}!(℩x)(\phi x)\), in +virtue of <a href="#*14·21">*14·21</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_196">[Pg 196]</span></p> +<h2 class="nobreak" id="SECTION_C_a">SECTION C.<br> +CLASSES AND RELATIONS.</h2> +</div> + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<h2 class="nobreak" id="*20">*20. GENERAL THEORY OF CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *20.</p> + +<p>The following theory of classes, although it provides a notation to +represent them, avoids the assumption that there are such things +as classes. This it does by merely defining propositions in whose +expression the symbols representing classes occur, just as, in <a href="#*14">*14</a>, we +defined propositions containing descriptions.</p> + +<p>The characteristics of a class are that it consists of all the terms +satisfying some propositional function, so that every propositional +function determines a class, and two functions which are formally +equivalent (<i>i.e.</i> such that whenever either is true, the other +is true also) determine the same class, while conversely two functions +which determine the same class are formally equivalent. When two +functions are formally equivalent, we shall say that they have the +same <i>extension</i>. The incomplete symbols which take the place of +classes serve the purpose of technically providing something identical +in the case of two functions having the same extension; without +something to represent classes, we cannot, for example, count the +combinations that can be formed out of a given set of objects.</p> + +<p>Propositions in which a function \(\phi\) occurs may depend, for their +truth-value, upon the particular function \(\phi\), or they may depend +only upon the <i>extension</i> of \(\phi\). In the former case, we +will call the proposition concerned an <i>intensional</i> function +of \(\phi\); in the latter case, an <i>extensional</i> function of +\(\phi\). Thus, for example, (\(x) . {\phi}x\) or (\(\exists x). {\phi}x\) +is an extensional function of \(\phi\), because, if \(\phi\) is +formally equivalent to \(\psi\), <i>i.e.</i> if \({\phi}x .\equiv_{x}. {\psi}x\), +we have (\(x).{\phi}x .\equiv. (x) . {\psi}x\) +and (\(\exists x) . {\phi}x .\equiv. (\exists x) . {\psi}x\). But on +the other hand "I believe (\(x) . {\phi}x\)" is an <i>intensional</i> +function, because, even if \({\phi}x .\equiv_{x}. {\psi}x\), it by no +means follows that I believe (\(x) . {\psi}x\) provided I believe +(\(x) . {\phi}x\). The mark of an extensional function \(f\) of a +function \({\phi}!\hat{z}\) is +\[ +{\phi}!x .\equiv_{x}. {\psi}!x :\supset_{\phi,\psi}:f({\phi}!\hat{z}) .\equiv. f({\psi}!\hat{z})\text{.} +\]<span class="pagenum" id="Page_197">[Pg 197]</span> +(We write "\(\phi!\hat{z}\)" when we wish to speak of the function +itself as opposed to its argument.) The functions of functions with +which mathematics is specially concerned are all extensional.</p> + +<p>When a function of \(\phi!\hat{z}\) is extensional, it may be regarded +as being about the class determined by \(\phi!\hat{z}\), since its +truth-value remains unchanged so long as the class is unchanged. +Hence we require, for the theory of classes, a method of obtaining an +extensional function from any given function of a function. This is +effected by the following definition:</p> + +<p class="nind"> +<b><a id="*20·01">*20·01</a>.</b> \(f\{\hat{z}(\psi z)\}.=:(\exists \phi):\phi!x.\equiv_{x}.\psi x:f\{\phi!\hat{z}\}\quad\text{Df}\)</p> + +<p>Here \(f\{\hat{z}(\psi z)\}\) is in reality a function of +\(\phi!\hat{z}\), which is defined whenever \(f\{\phi!\hat{z}\}\) +is significant for predicative functions \(\phi!\hat{z}\). But it +is convenient to regard \(f\{\hat{z}(\psi z)\}\) as though it had +an argument \(\hat{z}(\psi z\)), which we will call "the class +determined by the function \(\psi!\hat{z}\)." It will be proved +shortly that \(f\{\hat{z}(\psi z)\}\) is always an <i>extensional</i> +function of \(\psi!\hat{z}\), and that, applying the definition of +identity (<a href="#*13·01">*13·01</a>) to the fictitious objects \(\hat{z}(\phi z\)) and +\(\hat{z}(\psi z\)), we have +\[ +\hat{z}(\phi z)=\hat{z}(\psi z).\equiv:(x):\phi x.\equiv.\psi x. +\] +This last is the distinguishing characteristic of classes, and +justifies us in treating \(\hat{z}(\psi z\)) as the class determined by +\(\psi!\hat{z}\).</p> + +<p>With regard to the scope of \(\hat{z}(\psi z\)), and to the order +of elimination of two such expressions, we shall adopt the same +conventions as were explained in <a href="#*14">*14</a> for (\(℩x)(\phi x)\). The +condition corresponding to +\[ +\exists!(℩x)(\psi x)\,\, \text{is}\,\, (\exists \phi):\phi!x.\equiv_{x}.\psi x, +\] +which is always satisfied because of <a href="#*12·1">*12·1</a>.</p> + +<p>Following Peano, we shall use the notation +\[ +x\in\hat{z}(\psi z) +\] +to express "\(x\) is a member of the class determined by +\(\psi!\hat{z}\)." We therefore introduce the following definition:</p> + + +<p class="nind"> +<b><a id="*20·02">*20·02</a>.</b> \(x\in(\phi!\hat{z}).=.\phi!x \quad\text{Df}\)</p> + +<p>In this form, the definition is never used; it is introduced for the +sake of the proposition +\[ +\vdash\colon\ldotp x\in\hat{z}(\psi z).\equiv:(\exists \phi):\psi y.\equiv_{y}.\phi!y:\phi!x +\] +which results from <a href="#*20·02">*20·02</a> and <a href="#*20·01">*20·01</a>, and leads to +\[ +\vdash\colon x\in\hat{z}(\psi z).\equiv.\psi x +\] +by the help of <a href="#*12·1">*12·1</a>.</p> + +<p>We shall use small Greek letters (other than \(\epsilon\), \(\iota\), +\(\pi\), \(\phi\), \(\psi\), \(\chi\), \(\theta\)) to represent +classes, <i>i.e.</i> to stand for symbols of the form \(\hat{z}(\phi z)\) +or \(\hat{z}!(\phi z)\). When a small Greek letter occurs as +apparent variable, it is to be understood to stand for a<span class="pagenum" id="Page_198">[Pg 198]</span> symbol of +the form \(\hat{z}(\phi!z)\), where \(\phi\) is properly the apparent +variable concerned. The use of single letters in place of such symbols +as \(\hat{z}(\phi!z)\) or \(\hat{z}!(\phi!z)\) is practically almost +indispensable, since otherwise the notation rapidly becomes intolerably +cumbrous. Thus "\(x\in\alpha\)" will mean "\(x\) is a member of the +class \(\alpha\)," and may be used wherever no special defining +function of the class \(\alpha\) is in question.</p> + +<p>The following definition defines what is meant by a class.</p> + +<p class="nind"> +<b><a id="*20·03">*20·03</a>.</b> \(\text{Cls}=\hat{\alpha}\{(\exists\phi).\alpha=\hat{z}!(\phi!z)\} \quad\text{Df}\)</p> + +<p>Note that the expression +"\(\hat{\alpha}\{(\exists\phi).\alpha=\hat{z}!(\phi!z)\}\)" has +no meaning in isolation: we have merely defined (in <a href="#*20·01">*20·01</a>) +certain uses of such expressions. What the above definition +decides is that the symbol "Cls" may replace the symbol +"\(\hat{\alpha}\{(\exists\phi).\alpha=\hat{z}!(\phi!z)\}\)," wherever +the latter occurs, and that the meaning of the combination of symbols +concerned is to be unchanged thereby. Thus "Cls," also, has no meaning +in isolation, but merely in certain uses.</p> + +<p>The above definition, like many future definitions, is ambiguous as +to type. The Latin letter \(z\), according to our conventions, is to +represent the lowest type concerned; thus \(\phi\) is of the type +next above this. It is convenient to speak of a class as being of the +same type as its defining function; thus \(\alpha\) is of the type +next above that of \(z\), and "Cls" is of the type next above that of +\(\alpha\). Thus the type of "Cls" is fixed relatively to the lowest +type concerned; but if, in two different contexts, different types are +the lowest concerned, the meaning of "Cls" will be different in these +two contexts. The meaning of "Cls" only becomes definite when the +lowest type concerned is specified.</p> + +<p>Equality between classes is defined by applying <a href="#*13·01">*13·01</a>, symbolically +unchanged, to their defining functions, and then using <a href="#*20·01">*20·01</a>.</p> + +<p>The propositions of the present number may be divided into three sets. +First, we have those that deal with the fundamental properties of +classes; these end with <a href="#*20·43">*20·43</a>. Then we have a set of propositions +dealing with both classes and descriptions; these extend from <a href="#*20·5">*20·5</a> +to <a href="#*20·59">*20·59</a> (with the exception of *20·53·54). Lastly, we have a set of +propositions designed to prove that classes of classes have all the +same formal properties as classes of individuals.</p> + +<p>In the first set, the principal propositions are the following.</p> + +<p class="nind"> +<b>*20·15.</b> \(\vdash\colon\ldotp \psi x.\equiv_{x}.\chi x:\equiv.\hat{z}(\psi z)=\hat{z}(\chi z)\)</p> + +<p><i>I.e.</i> two classes are identical when, and only when, their +defining functions are formally equivalent. This is the principal +property of classes.</p> + +<p class="nind"> +<b>*20·31.</b> \(\vdash\colon\ldotp \hat{z}(\psi z)=\hat{z}(\chi z).\equiv:x\in\hat{z}(\psi z).\equiv_{x}.x\in\hat{z}(\chi z)\)</p> + +<p><span class="pagenum" id="Page_199">[Pg 199]</span></p> + +<p><i>I.e.</i> two classes are identical when, and only when, they have +the same members.</p> + +<p class="nind"> +<b>*20·43.</b> \(\vdash\colon\ldotp \alpha=\beta.\equiv:x\in \alpha.\equiv_{x}.x\in \beta\)</p> + +<p>This is the same proposition as <a href="#*20·31">*20·31</a>, merely employing Greek letters +in place of \(\hat{z}(\psi z)\) and \(\hat{z}(\chi z)\).</p> + +<p class="nind"> +<b>*20·18.</b> \(\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\psi z).\supset:f\{\hat{z}(\phi z)\}.\equiv.f\{\hat{z}(\psi z)\}\)</p> + +<p><i>I.e.</i> if two classes are identical, any property of either +belongs also to the other. This is the analogue of <a href="#*13·12">*13·12</a>.</p> + +<p class="nind"> +<b>*20·2·21·22</b>, which prove that identity between classes is +reflexive, symmetrical and transitive.</p> + +<p class="nind"> +<b>*20·3.</b> \(\vdash:x\in \hat{z}(\psi z).\equiv.\psi x\)</p> + +<p><i>I.e.</i> a term belongs to a class when, and only when, it satisfies +the defining function of the class.</p> + +<p>In the second set of propositions (<a href="#*20·3">*20·3</a>—<a href="#*20·59">·59</a>), we show that, under +suitable circumstances, expressions such as (\(℩x)(\phi x)\) may be +substituted for \(x\) in <a href="#*20·3">*20·3</a> and various other propositions of +the first set, and we prove a few properties of such expressions as +"(\(℩ \alpha)(f \alpha)\)," <i>i.e.</i> "the class which satisfies +the function \(f\)." Here it is to be remembered that "\(\alpha\)" +stands for "\(\hat{z}(\phi z)\)," and that "\(f \alpha\)" therefore +stands for "\(f\{\hat{z}(\phi z)\}\)." This is, in reality, a function +of \(\phi\hat{z}\), namely the extensional function associated with +\(f(\psi!\hat{z})\) by means of <a href="#*20·01">*20·01</a>. Thus an expression containing a +variable class is always an abbreviation for an expression containing a +variable function.</p> + +<p>In the third set of propositions, we prove that variable classes +satisfy all the primitive propositions assumed for variable individuals +or functions, whence it follows, by merely repeating the proofs of the +first set of propositions (<a href="#*20·1">*20·1</a>—<a href="#*20·43">·43</a>), that classes of classes have +all the formal properties of classes of individuals or functions. We +shall never have occasion explicitly to consider classes of functions, +but classes of classes will occur constantly—for example, every +cardinal number will be defined as a class of classes. Classes of +relations, which will also frequently occur, will be considered in <a href="#*21">*21</a>.</p> + +<hr class="tb"> + +<p class="nind"> +<b>*20·01.</b> \(f\{\hat{z}(\psi z)\}.=:(\exists \phi):\phi!x.\equiv_{x}.\psi x:f(\phi!\hat{z}) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·02.</b> \(x\in (\phi!\hat{z}).=.\phi!x \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·03.</b> \(\text{Cls}=\hat{\alpha}\{(\exists \phi).\alpha=\hat{z}(\phi!z)\} \quad\text{Df}\)</p> + +<p>The three following definitions serve merely for purposes of +abbreviation.</p> + +<p class="nind"> +<b>*20·04.</b> \(x,\,y\in \alpha.=.x\in \alpha.y\in \alpha \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·05.</b> \(x,\,y,\,z\in \alpha.=.x,\,y\in \alpha.z\in \alpha \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·06.</b> \(x{\sim}\in \alpha.=.{\sim}(x\in \alpha) \quad\text{Df}\)</p> + +<p><span class="pagenum" id="Page_200">[Pg 200]</span></p> + +<p>The following definitions merely extend to symbols representing classes +the definitions which have already been given for other symbols, with +the smallest possible modifications.</p> + +<p class="nind"> +<b><a id="*20·07">*20·07</a>.</b> (\(\alpha).f\alpha.=.(\phi).f\{\hat{z}(\phi!z)\} \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·071.</b> (\(\exists \alpha).f\alpha.=.(\exists \phi).f\{\hat{z}(\phi!z)\} \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·072.</b> \([(℩\alpha)(\phi\alpha)].f(℩\alpha)(\phi\alpha).=:(\exists \gamma):\phi\alpha.\equiv_{\alpha}.\alpha=\gamma:f\gamma \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·08.</b> \(f\{\hat{\alpha}(\psi\alpha)\}.=:(\exists \phi):\psi\alpha.\equiv_{\alpha}.\phi!\alpha:f(\phi!\hat{\alpha}) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*20·081.</b> \(\alpha\in \psi!\hat{\alpha}.=.\psi!\alpha \quad\text{Df}\)</p> + +<p>The propositions which follow give the most general properties of +classes.</p> + +<p class="nind"> +<b><a id="*20·1">*20·1</a>.</b> \(\vdash\colon\ldotp f\{\hat{z}(\psi z)\}.\equiv:(\exists \phi):\phi!x.\equiv_{x}.\psi x:f\{\phi!\hat{z}\} \quad[\text{*4·2.(*20·01)}]\)</p> + +<p class="nind"> +<b>*20·11.</b> \(\vdash\colon\ldotp \psi x.\equiv_{x}.\chi x:\supset:f\{\hat{z}(\psi z)\}.\equiv.f\{\hat{z}(\chi z)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·86}.&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \phi!x.\equiv_{x}.\psi x:\equiv_{\phi}:\phi!x.\equiv_{x}.\chi x\colon\ldotp\\ +[\text{*4·36}] & \supset\colon\ldotp \phi!x.\equiv_{x}.\psi x:f\{\phi!\hat{z}\}:\equiv_{\phi}:\phi!x.\equiv_{x}.\chi x:f{\phi!\hat{z}}\colon\ldotp\\ +[\text{*10·281}] & \supset\colon\ldotp (\exists \phi):\phi!x.\equiv_{x}.\psi x:f\{\phi!\hat{z}\}:\\ +&\equiv:(\exists \phi):\phi!x.\equiv_{x}.\chi x:f\{\phi!\hat{z}\}\colon\ldotp\\ +[\text{*20·1}] &\supset\colon\ldotp f\{\hat{z}(\psi z)\}.\equiv.f\{\hat{z}(\chi z)\}\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proves that every proposition about a class expresses an +extensional property of the determining function of the class, +and therefore does not depend for its truth or falsehood upon the +particular function selected for determining the class, but only upon +the extension of the determining function.</p> + +<p class="nind"> +<b>*20·111.</b> \(\vdash\colon\ldotp f(\phi!\hat{z}).\equiv_{\phi}.g(\phi!\hat{z}):\supset:f\{\hat{z}(\phi!z)\}.\equiv_{\phi}.g\{\hat{z}(\phi!z)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{Fact}. &\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \phi!x.\equiv_{x}.\psi!x:f(\psi!\hat{z}):\equiv:\phi!x.\equiv_{x}.\psi!x:g(\psi!\hat{z})\colon\colon\\ +[\text{*10·11·21}] &\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \phi!x.\equiv_{x}.\psi!x:f(\psi!\hat{z}):\equiv_{\psi}:\phi!x.\equiv_{x}.\psi!x:g(\psi!\hat{z})\colon\ldotp\\ +[\text{*10·281}] &\supset\colon\ldotp (\exists \psi):\phi!x.\equiv_{x}.\psi!x:f(\psi!\hat{z}):\equiv:(\exists \psi):\phi!x.\equiv_{x}.\psi!x:g(\psi!\hat{z})\colon\ldotp\\ +[\text{*20·1}] & \supset\colon\ldotp f\{\hat{z}(\phi!x)\}.\equiv.g\{\hat{z}(\phi!x)\} &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*20·112">*20·112</a>.</b> \(\vdash\colon\ldotp (\exists g)\colon\ldotp f\{\hat{z}(\phi!z)\}.\equiv_{\phi}.g!\{\hat{z}(\phi!z)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*12·1}.\supset\vdash\colon\ldotp (\exists g):f(\phi!\hat{z}).\equiv_{\phi}.g!(\phi!\hat{z}) &\qquad \text{(1)}\\ +&\vdash.\text{(1).*20·111}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Thus the axiom of reducibility still holds for classes as arguments.</p> + +<p><span class="pagenum" id="Page_201">[Pg 201]</span></p> + +<p class="nind"> +<b>*20·12.</b> \(\vdash:(\exists \phi):\phi!x.\equiv_{x}.\psi x:f\{\hat{z}(\psi z)\}.\equiv.f\{\hat{z}(\phi!z)\} \quad[\text{*20·11.*12·1}]\)</p> + +<p class="nind"> +<b><a id="*20·13">*20·13</a>.</b> \(\vdash\colon\ldotp \psi x.\equiv_{x}.\chi x:\supset.\hat{z}(\psi z)=\hat{z}(\chi z)\)</p> + +<p>The meaning of "\(\hat{z}(\psi z)=\hat{z}(\chi z)\)" is obtained by a +double application of <a href="#*20·01">*20·01</a> to <a href="#*13·01">*13·01</a>, remembering the convention that +\(\hat{z}(\psi z)\) is to have a larger scope than \(\hat{z}(\chi z)\) +because it occurs first.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·1}.&\supset\vdash\colon\colon \hat{z}(\psi z)=\hat{z}(\chi z).\equiv\colon\ldotp (\exists \phi):\psi x.\equiv_{x}.\phi!x:\phi!\hat{z}=\hat{z}(\chi z)\colon\ldotp \\ +[\text{*20·1}] &\equiv\colon\ldotp (\exists \phi,\theta)\colon\ldotp \psi x.\equiv_{x}.\phi!x:\chi x.\equiv_{x}.\theta!x:\phi!\hat{z}=\theta!\hat{z} &\qquad \text{(1)}\\ +\vdash.\text{*12·1.*10·321}.&\supset\\ +&\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (\exists \phi):\psi x.\equiv_{x}.\phi!x:\chi x.\equiv_{x}.\phi!x\colon\ldotp \\ +[\text{*13·195}]&\supset\colon\ldotp (\exists \phi,\theta)\colon\ldotp \psi x.\equiv_{x}.\phi!x:\chi x.\equiv_{x}.\theta!x:\phi!\hat{z}=\theta!\hat{z} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·14.</b> \(\vdash\colon\ldotp \hat{z}(\psi z)=\hat{z}(\chi z).\supset:\psi x.\equiv_{x}.\chi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·1}.&\supset\vdash\colon\colon \hat{z}(\psi z)=\hat{z}(\chi z).\equiv\colon\ldotp (\exists \phi):\psi x.\equiv_{x}.\phi!x:\phi!\hat{z}=\hat{z}(\chi z)\colon\ldotp\\ +[\text{*20·1}] & \equiv\colon\ldotp (\exists \phi,\theta)\colon\ldotp \psi x.\equiv_{x}.\phi!x:\chi x.\equiv_{x}.\theta!x:\phi!\hat{z}=\theta!\hat{z}\colon\ldotp \\ +[\text{*13·195}] & \equiv\colon\ldotp (\exists \phi)\colon\ldotp \psi x.\equiv_{x}.\phi!x:\chi x.\equiv_{x}.\phi!x\colon\ldotp \\ +[\text{*10·322}] &\supset\colon\ldotp \psi x.\equiv_{x}.\chi x\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is the converse of <a href="#*20·13">*20·13</a>.</p> + +<p class="nind"> +<b>*20·15.</b> \(\vdash\colon\ldotp \psi x.\equiv_{x}.\chi x:\equiv.\hat{z}(\psi z)=\hat{z}(\chi z) \quad[\text{*20·13·14}]\)</p> + +<p>This proposition states that two functions determine the same class +when, and only when, they are formally equivalent, <i>i.e.</i> are +satisfied by the same set of values. This is the essential property of +classes, and gives the justification of the definition <a href="#*20·01">*20·01</a>.</p> + +<p class="nind"> +<b><a id="*20·151">*20·151</a>.</b> \(\vdash.(\exists \phi).\hat{z}(\psi z)=\hat{z}(\phi!z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·15}. &\supset\vdash\colon\ldotp \psi x.\equiv_{x}.\phi!x:\supset.\hat{z}(\psi z)=\hat{z}(\phi!z)\colon\ldotp \\ +[\text{*10·11·28}] &\supset\vdash\colon\ldotp (\exists \phi):\psi x.\equiv_{x}.\phi!x:\supset.(\exists \phi).\hat{z}(\psi z)=\hat{z}(\phi!z) &\qquad \text{(1)}\\ +\vdash.\text{(1).*12·1}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In virtue of this proposition, all classes can be obtained from +predicative functions. This fact is especially important when classes +are used as apparent variables. For in that case, according to the +definitions *20·07·071, the apparent variable really involved is a +predicative function. In virtue of <a href="#*20·151">*20·151</a>, this places no limitation +upon the classes concerned, except the limitation which inevitably +results from the nature of their membership.</p> + +<p><span class="pagenum" id="Page_202">[Pg 202]</span></p> + +<p>A class, therefore, unlike a function, has its order completely +determined by the order of its possible members, <i>i.e.</i> of the +arguments which render its defining function significant.</p> + +<p class="nind"> +<b>*20·16.</b> \(\vdash:(\exists \phi):f\{\hat{z}(\psi z)\}.\equiv.f{\hat{z}(\phi!z)} \quad[\text{*20·12}]\)</p> + +<p class="nind"> +<b><a id="*20·17">*20·17</a>.</b> \(\vdash:(\phi).f\{\hat{z}(\phi!z)\}.\supset.f\{\hat{z}(\psi z)\} \quad[\text{*20·16.*10·1}]\)</p> + +<p class="nind"> +<b>*20·18.</b> \(\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\psi z).\supset:f\{\hat{z}(\phi z)\}.\equiv.f\{\hat{z}(\psi z)\} \quad[\text{*20·11·15}]\)</p> + +<p class="nind"> +<b>*20·19.</b> \(\vdash\colon\ldotp \hat{z}(\psi z)=\hat{z}(\chi z).\equiv:(f):f!\hat{z}(\psi z).\supset.f!\hat{z}(\chi z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·18.*10·11·21}.&\supset\vdash\colon\ldotp \hat{z}(\psi z)=\hat{z}(\chi z).\supset:\\ +&\qquad(f):f!\hat{z}(\psi z).\supset.f!\hat{z}(\chi z) &\qquad \text{(1)}\\ +\vdash.\text{*20·18·15}.&\supset\vdash\colon\colon \phi!x.\equiv_{x}.\psi x:\theta!x.\equiv_{x}.\chi x:f!\hat{z}(\psi z).\supset.f!\hat{z}(\chi z):\supset:\\ +&\qquad f!\hat{z}(\phi!z).\supset.f!\hat{z}(\theta!z) &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·27·33}.&\supset\\ +&\vdash\colon\colon \phi!x.\equiv_{x}.\psi x:\theta!x.\equiv_{x}.\chi x\colon\ldotp (f):f!\hat{z}(\psi z).\supset.f!\hat{z}(\chi z)\colon\ldotp \supset\colon\ldotp +(f):f!\hat{z}(\phi!z).\supset.f!\hat{z}(\theta!z)\colon\ldotp \\ +[\text{*20·112.*10·1}] &\supset\colon\ldotp \phi!x.\equiv_{x}.\phi!x:\supset:\phi!x.\equiv_{x}.\theta!x\colon\ldotp \\ +[\text{*4·2}] & \supset\colon\ldotp \phi!x.\equiv_{x}.\theta!x\colon\ldotp \\ +[\text{*10·301·32.Hp}] &\supset\colon\ldotp \psi x.\equiv_{x}.\chi x\colon\ldotp \\ +[\text{*20·15}] & \supset\colon\ldotp \hat{z}(\psi z)=\hat{z}(\chi z) &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·23·35}.&\supset\\ +&\vdash\colon\colon (\exists \phi,\theta):\phi!x.\equiv_{x}.\psi x:\theta!x.\equiv_{x}.\chi x\colon\ldotp (f):f!\hat{z}(\psi z).\supset.f!\hat{z}(\chi z)\colon\ldotp \\ +&\supset.\hat{z}(\psi z)=\hat{z}(\chi z) &\qquad \text{(4)}\\ +\vdash.\text{(4).*12·1}.&\supset\vdash\colon\ldotp (f):f!\hat{z}(\psi z).\supset.f!\hat{z}(\chi z):\supset.\hat{z}(\psi z)=\hat{z}(\chi z) &\qquad \text{(5)}\\ +\vdash.\text{(1).(5)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·191.</b> \[\begin{align}\vdash\colon\ldotp \hat{z}(\psi z)=\hat{z}(\chi z).\equiv:(f):f!\hat{z}(\psi z).\equiv.&f!\hat{z}(\chi z)\\ +&\quad[\text{*20·18·19.*10·22}]\end{align}\]</p> + +<p class="nind"> +<b>*20·2.</b> \(\vdash.\hat{z}(\phi z)=\hat{z}(\phi z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·15}.&\supset\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\phi z).\equiv:\phi x.\equiv_{x}.\phi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*4·2.*10·11}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·21.</b> \(\vdash:\hat{z}(\phi z)=\hat{z}(\psi z).\equiv.\hat{z}(\psi z)=\hat{z}(\phi z) \quad[\text{*20·15.*10·32}]\)</p> + +<p class="nind"> +<b>*20·22.</b> \[\begin{align}\vdash:\hat{z}(\phi z)=\hat{z}(\psi z).\hat{z}(\psi z)=\hat{z}(\chi z).\supset.\hat{z}(\phi z)&=\hat{z}(\chi z)\\ +&\quad[\text{*20·15.*10·301}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_203">[Pg 203]</span></p> + +<p>The above propositions are not <i>immediate</i> consequences of +*13·15·16·17, for a reason analogous to that explained in the note +to <a href="#*14·13">*14·13</a>, namely because \(f\{\hat{z}(\phi z)\}\) is not a value of +\(fx\), and therefore in particular "\(\hat{z}(\phi z)=\hat{z}(\psi z)\)" +is not a value of "\(x=y\)."</p> + +<p class="nind"> +<b>*20·23.</b> \(\vdash:\hat{z}(\phi z)=\hat{z}(\psi z).\hat{z}(\phi z)=\hat{z}(\chi z).\supset.\hat{z}(\psi z)=\hat{z}(\chi z) \quad[\text{*20·21·22}]\)</p> + +<p class="nind"> +<b>*20·24.</b> \(\vdash:\hat{z}(\psi z)=\hat{z}(\phi z).\hat{z}(\chi z)=\hat{z}(\phi z).\supset.\hat{z}(\psi z)=\hat{z}(\chi z) \quad[\text{*20·21·22}]\)</p> + +<p class="nind"> +<b><a id="*20·25">*20·25</a>.</b> \(\vdash\colon\ldotp \alpha=\hat{z}(\phi z).\equiv_{\alpha}.\alpha=\hat{z}(\psi z):\equiv.\hat{z}(\phi z)=\hat{z}(\psi z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp \alpha=\hat{z}(\phi z).\equiv_{\alpha}.\alpha=\hat{z}(\psi z):\supset:\\ +&\qquad\qquad\qquad\hat{z}(\phi z)=\hat{z}(\phi z).\equiv.\hat{z}(\phi z)=\hat{z}(\psi z):\\ +[\text{*20·2}] &\qquad\qquad\qquad\supset:\hat{z}(\phi z)=\hat{z}(\psi z) &\qquad \text{(1)}\\ +\vdash.\text{*20·22}. & \supset\vdash:\alpha=\hat{z}(\phi z).\hat{z}(\phi z)=\hat{z}(\psi z).\supset.\alpha=\hat{z}(\psi z):\\ +[\text{Exp.Comm}] &\supset\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\psi z).\supset:\alpha=\hat{z}(\phi z).\supset.\alpha=\hat{z}(\psi z) &\qquad \text{(2)}\\ +\vdash.\text{*20·24}. &\supset\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\psi z).\alpha=\hat{z}(\psi z).\supset.\alpha=\hat{z}(\phi z)\colon\ldotp \\ +[\text{Exp}] &\supset\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\psi z).\supset:\alpha=\hat{z}(\psi z).\supset.\alpha=\hat{z}(\phi z) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\psi z).\supset:\alpha=\hat{z}(\phi z).\equiv.\alpha=\hat{z}(\psi z)\colon\ldotp \\ +[\text{*10·11·21}]&\supset\vdash\colon\ldotp \hat{z}(\phi z)=\hat{z}(\psi z).\supset:\alpha=\hat{z}(\phi z).\equiv_{\alpha}.\alpha=\hat{z}(\psi z) &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*20·3">*20·3</a>.</b> \(\vdash:x\in \hat{z}(\psi z).\equiv.\psi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·1}.\supset\\ +\vdash\colon\colon x\in \hat{z}(\psi z).&\equiv\colon\ldotp (\exists \phi)\colon\ldotp \psi y.\equiv_{y}.\phi!y:x\in (\phi!\hat{z})\colon\ldotp \\ +[\text{(*20·02)}] &\equiv\colon\ldotp (\exists \phi)\colon\ldotp \psi y.\equiv_{y}.\phi!y:\phi!x\colon\ldotp \\ +[\text{*10·43}] & \equiv\colon\ldotp (\exists \phi)\colon\ldotp \psi y.\equiv_{y}.\phi!y:\psi x\colon\ldotp \\ +[\text{*10·35}] &\equiv\colon\ldotp (\exists \phi):\psi y.\equiv_{y}.\phi!y\colon\ldotp \psi x\colon\ldotp \\ +[\text{*12·1}] & \equiv\colon\ldotp \psi x\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that \(x\) is a member of the class determined +by \(\psi\) when, and only when, \(x\) satisfies \(\psi\).</p> + +<p class="nind"> +<b><a id="*20·31">*20·31</a>.</b> \(\vdash\colon\ldotp \hat{z}(\psi z)=\hat{z}(\chi z).\equiv:x\in \hat{z}(\psi z).\equiv_{x}.x\in \hat{z}(\chi z) \quad[\text{*20·15·3}]\)</p> + +<p class="nind"> +<b><a id="*20·32">*20·32</a>.</b> \(\vdash.\hat{x}\{x\in \hat{z}(\phi z)\}=\hat{z}(\phi z) \quad[\text{*20·3·15}]\)</p> + +<p class="nind"> +<b>*20·33.</b> \(\vdash\colon\ldotp \alpha=\hat{z}(\phi z).\equiv:x\in \alpha.\equiv_{x}.\phi x\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·31}. &\supset\vdash\colon\ldotp \alpha=\hat{z}(\phi z).\equiv:x\in \alpha.\equiv_{x}.x\in \hat{z}(\phi z) &\qquad \text{(1)}\\ +\vdash.\text{(1).*20·3}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Here \(\alpha\) is written in place of some expression of the form +\(\hat{z}(\psi z)\). The use of the single Greek letter is more +convenient whenever the determining function is irrelevant.</p> + +<p class="nind"> +<b>*20·34.</b> \(\vdash\colon\ldotp x=y.\equiv:x\in \alpha.\supset_{\alpha}.y\in \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·2.(*20·07)}.\supset\vdash\colon\ldotp x\in \alpha.\supset_{\alpha}.y\in \alpha:&\equiv:x\in \hat{z}(\phi!z).\supset_{\phi}.y\in \hat{z}(\phi!z):\\ +[\text{*20·3}] &\equiv:\phi!x.\supset_{\phi}.\phi!y:\\ +[\text{*13·1}] &\equiv:x=y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_204">[Pg 204]</span></p> + +<p>The above proposition and <a href="#*20·25">*20·25</a> illustrate the use of Greek letters as +apparent variables.</p> + +<p class="nind"> +<b>*20·35.</b> \(\vdash\colon\ldotp x=y.\equiv:x\in \alpha.\equiv_{\alpha}.y\in \alpha \quad[\text{*20·3.*13·11}]\)</p> + +<p class="nind"> +<b>*20·4.</b> \(\vdash:\alpha\in \text{Cls}.\equiv.(\exists \phi).\alpha=\hat{z}(\phi!z) \quad[\text{*20·3.(*20·03)}]\)</p> + +<p class="nind"> +<b><a id="*20·41">*20·41</a>.</b> \(\vdash.\hat{z}(\psi z)\in \text{Cls} \quad[\text{*20·4·151}]\)</p> + +<p class="nind"> +<b>*20·42.</b> \(\vdash.\hat{z}(z\in \alpha)=\alpha\)</p> + +<p>A Greek letter, such as \(\alpha\), is merely an abbreviation for an +expression of the form \(\hat{z}(\phi z)\), thus this proposition is +<a href="#*20·32">*20·32</a> repeated.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·3.*10·11}. &\supset\vdash:x\in \hat{z}(\psi z).\equiv_{x}.\psi x:\\ +[\text{*20·15}] &\supset\vdash.\hat{x}\{x\in \hat{z}(\psi z)\}=\hat{x}(\psi x).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*20·43">*20·43</a>.</b> \(\vdash\colon\ldotp \alpha=\beta.\equiv:x\in \alpha.\equiv_{x}.x\in \beta \quad[\text{*20·31}]\)</p> + +<p>The following propositions deal with cases in which both classes and +descriptions occur. In such cases, we shall, in the absence of any +indication to the contrary, adopt the convention that the descriptions +are to have a larger scope than the classes, in applying the +definitions <a href="#*14·01">*14·01</a> and <a href="#*20·01">*20·01</a>.</p> + +<p class="nind"> +<b><a id="*20·5">*20·5</a>.</b> \(\vdash:(℩x)(\phi x)\in \hat{z}(\psi z).\equiv.\psi\{(℩x)(\phi x)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}. \supset\vdash\colon\colon (℩x)(\phi x)\in \hat{z}(\psi z).&\equiv\colon\ldotp (\exists c):\phi x.\equiv_{x}.x=c:c\in \hat{z}(\psi z)\colon\ldotp\\ +[\text{*20·3}] &\equiv\colon\ldotp (\exists c):\phi x.\equiv_{x}.x=c:\psi c\colon\ldotp \\ +[\text{*14·1}] &\equiv\colon\ldotp \psi\{(℩x)(\phi x)\}\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·51.</b> \(\vdash\colon\ldotp (℩x)(\phi x)=b.\equiv:(℩x)(\phi x)\in \alpha.\equiv_{\alpha}.b\in \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·5·3}.\supset\\ +&\vdash\colon\ldotp (℩x)(\phi x)\in \hat{z}(\psi!z).\equiv.b\in \hat{z}(\psi!z):\equiv:\psi!(℩x)(\phi x).\equiv.\psi!b\colon\ldotp \supset\\ +[\text{*10·11}] &\vdash\colon\ldotp (℩x)(\phi x)\in \alpha.\equiv_{\alpha}.b\in \alpha:\equiv:\psi!(℩x)(\phi x).\equiv_{\psi}.\psi!b:\\ +[\text{*14·17}] &\equiv:(℩x)(\phi x)=b\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·52.</b> \(\vdash\colon\ldotp \text{E}!(℩x)(\phi x).\equiv:(\exists b):(℩x)(\phi x)\in \alpha.\equiv_{\alpha}.b\in \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*20·51.*10·11·281}.\supset\\ +&\vdash\colon\ldotp (\exists b).(℩x)(\phi x)=b.\equiv:(\exists b):(℩x)(\phi x)\in \alpha.\equiv_{\alpha}.b\in \alpha &\qquad \text{(1)}\\ +&\vdash.\text{(1).*14·204}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·53.</b> \(\vdash\colon\ldotp \beta=\alpha.\supset_{\beta}.\phi\beta:\equiv.\phi\alpha\)</p> + +<p>This is the analogue of <a href="#*13·191">*13·191</a>.</p> + +<p><span class="pagenum" id="Page_205">[Pg 205]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp \beta=\alpha.\supset_{\beta}.\phi\beta:\supset:\alpha=\alpha.\supset.\phi\alpha:\\ +[\text{*20·2}] &\qquad\qquad\qquad\supset:\phi\alpha &\qquad \text{(1)}\\ +\vdash.\text{*20·18·21}.&\supset\vdash\colon\ldotp \beta=\alpha.\supset:\phi\alpha.\supset.\phi\beta\colon\ldotp \\ +[\text{Comm}] &\supset\vdash\colon\ldotp \phi\alpha.\supset:\beta=\alpha.\supset.\phi\beta\colon\ldotp \\ +[\text{*10·11·21}] &\supset\vdash\colon\ldotp \phi\alpha.\supset:\beta=\alpha.\supset_{\beta}.\phi\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·54.</b> \(\vdash:(\exists \beta).\beta=\alpha.\phi\beta.\equiv.\phi\alpha\)</p> + +<p>This proposition is the analogue of <a href="#*13·195">*13·195</a>.</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·18.*10·11}.&\supset\vdash:\beta=\alpha.\phi\beta.\supset_{\beta}.\phi\alpha:\\ +[\text{*10·23}] &\supset\vdash:(\exists \beta).\beta=\alpha.\phi\beta.\supset.\phi\alpha &\qquad \text{(1)}\\ +\vdash.\text{*20·2.*3·2}. &\supset\vdash:\phi\alpha.\supset.\alpha=\alpha.\phi\alpha.\\ +[\text{*10·24}] &\qquad\qquad\supset.(\exists \beta).\beta=\alpha.\phi\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·55.</b> \(\vdash.\hat{z}(\phi z)=(℩\alpha)(x\in \alpha.\equiv_{x}.\phi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·33}.&\supset\vdash\colon\ldotp x\in \alpha.\equiv_{x}.\phi x:\equiv_{\alpha}.\alpha=\hat{z}(\phi z)\colon\ldotp \\ +[\text{*20·54}] &\supset\vdash\colon\ldotp (\exists \beta)\colon\ldotp x\in \alpha.\equiv_{x}.\phi x:\equiv_{\alpha}.\alpha=\beta\colon\ldotp \hat{z}(\phi z)=\beta\colon\ldotp \\ +[\text{*14·1}] &\supset\vdash.\hat{z}(\phi z)=(℩\alpha)(x\in \alpha.\equiv_{x}.\phi x).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·56.</b> \(\vdash.\text{E}!(℩\alpha)(x\in \alpha.\equiv_{x}.\phi x) \quad[\text{*20·55.*14·21}]\)</p> + +<p class="nind"> +<b><a id="*20·57">*20·57</a>.</b> \(\vdash\colon\ldotp \hat{z}(\phi z)=(℩\alpha)(f\alpha).\supset:g\{\hat{z}(\phi z)\}.\equiv.g\{(℩\alpha)(f\alpha)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·1}. \supset\vdash\colon\colon \text{Hp}.&\equiv\colon\ldotp (\exists \beta):f\alpha.\equiv_{\alpha}.\alpha=\beta:\hat{z}(\phi z)=\beta\colon\ldotp \\ +[\text{*20·54}] &\equiv\colon\ldotp f\alpha.\equiv_{\alpha}.\alpha=\hat{z}(\phi z) &\qquad \text{(1)}\\ +\vdash.\text{*14·1}. \supset\vdash\colon\ldotp g\{(℩\alpha)(f\alpha)\}.&\equiv:(\exists \beta):f\alpha.\equiv_{\alpha}.\alpha=\beta:g\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp g\{(℩\alpha)(f\alpha)\}.&\equiv:(\exists \beta):\alpha=\hat{z}(\phi z).\equiv_{\alpha}.\alpha=\beta:g\beta:\\ +[\text{*13·183}] &\equiv:(\exists \beta).\hat{z}(\phi z)=\beta.g\beta:\\ +[\text{*20·54}] &\equiv:g\{\hat{z}(\phi z)\}\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*20·58.</b> \(\vdash.\hat{z}(\phi z)=(℩\alpha)\{\alpha=\hat{z}(\phi z)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·2.*10·11}.&\supset\vdash:\alpha=\hat{z}(\phi z).\equiv_{\alpha}.\alpha=\hat{z}(\phi z):\\ +[\text{*20·54}] &\supset\vdash\colon\ldotp (\exists \beta)\colon\ldotp \alpha=\hat{z}(\phi z).\equiv_{\alpha}.\alpha=\beta:\hat{z}(\phi z)=\beta\colon\ldotp \\ +[\text{*14·1}] &\supset\vdash.\hat{z}(\phi z)=(℩\alpha){\alpha=\hat{z}(\phi z)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_206">[Pg 206]</span></p> + +<p class="nind"> +<b><a id="*20·59">*20·59</a>.</b> \(\vdash:\hat{z}(\phi z)=(℩\alpha)(f\alpha).\equiv.(℩\alpha)(f\alpha)=\hat{z}(\phi z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·1}.\supset\vdash\colon\ldotp \hat{z}(\phi z)=(℩\alpha)(f\alpha)&.\equiv:(\exists \psi):\phi x.\equiv_{x}.\psi!x:\psi!\hat{z}=(℩\alpha)(f\alpha):\\ +[\text{*14·13}] & \equiv:(\exists \psi):\phi x.\equiv_{x}.\psi!x:(℩\alpha)(f\alpha)=\psi!\hat{z}:\\ +[\text{*20·1}] &\equiv:(℩\alpha)(f\alpha)=\hat{z}(\phi z)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the following propositions, we shall prove that classes have all +the formal properties of individuals, and have the same relations to +classes of classes as individuals have to classes of individuals. +It is only necessary to prove the analogues of our primitive +propositions, and of our definitions in cases where their analogues +are not themselves definitions. We shall take the propositions +*10·1·11·12·121·122, rather than those of <a href="#*9">*9</a>, and we shall prove the +analogue of <a href="#*10·01">*10·01</a>. As was pointed out in <a href="#*10">*10</a>, we shall thus have +proved everything upon which subsequent proofs depend. The analogues +of *20·01·02 and of <a href="#*14·01">*14·01</a> remain definitions, but those of <a href="#*10·01">*10·01</a> +and <a href="#*13·01">*13·01</a> become propositions to be proved. <a href="#*9·131">*9·131</a> must be extended +by the definition: Two classes are "of the same type" when they +have predicative defining functions of the same type. In addition +to these, we have to prove the analogues of *10·1·11·12·121·122, +<a href="#*11·07">*11·07</a> and *12·1·11. When these have been proved, the analogues of +other propositions follow by merely repeating previous proofs. These +analogues will, therefore, be quoted by the numbers of the original +propositions whose analogues they are.</p> + +<p class="nind"> +<b><a id="*20·6">*20·6</a>.</b> \(\vdash:(\exists \alpha).f\alpha.\equiv.{\sim}\{(\alpha).{\sim}f\alpha\}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*4·2.(*20·071)}.\supset\\ +&\vdash:(\exists \alpha).f\alpha.&\equiv.(\exists \phi).f\{\hat{z}(\phi!z)\}.\\ +&[\text{(*10·01)}] &\equiv.{\sim}[(\phi).{\sim}f\{\hat{z}(\phi!z)\}].\\ +&[\text{(*20·07)}] &\equiv.{\sim}\{(\alpha).{\sim}f\alpha\}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This is the analogue of <a href="#*10·01">*10·01</a>.</p> + +<p class="nind"> +<b>*20·61.</b> \(\vdash:(\alpha).f\alpha.\supset.f\beta\)</p> + +<p><i>Dem.</i> +\[ +\vdash.\text{*10·1.(*20·07)}.\supset\vdash:(\alpha).f\alpha.\supset.f\{\hat{z}(\phi!z)\}:\supset\vdash.\text{Prop} +\]</p> + +<p>This is the analogue of <a href="#*10·1">*10·1</a>.</p> + +<p>In practice we also need +\[ +\vdash:(\alpha).f\alpha.\supset.f\{\hat{z}(\psi z)\}. +\] +This is <a href="#*20·17">*20·17</a>.</p> + +<p>We need further \(\vdash.(\exists \alpha).\hat{z}(\psi z)=\alpha.\)</p> + +<p>This is <a href="#*20·41">*20·41</a>.</p> + +<p><span class="pagenum" id="Page_207">[Pg 207]</span></p> + +<p class="nind"><b><a id="*20·62">*20·62</a></b>. When \(f\beta\) is true, whatever possible +argument of the form \(\hat{z}(\phi!z)\) \(\beta\) may be, then +(\(\alpha).f\alpha\) is true.</p> + +<p>This is the analogue of <a href="#*10·11">*10·11</a>.</p> + +<p><i>Dem.</i></p> + +<p>\(\vdash.\text{*10·11}.\supset.\) when \(f\{\hat{z}(\phi!z)\}\) +is true, whatever possible argument \(\phi\) may be, then +(\(\phi).f\{\hat{z}(\phi!z)\}\) is true, <i>i.e.</i> (by <a href="#*20·07">*20·07</a>), +(\(\alpha).f\alpha\) is true.</p> + +<p class="nind"> +<b>*20·63.</b> \(\vdash\colon\ldotp (\alpha).p\lor f\alpha.\supset :p.\lor .(\alpha).f\alpha\)</p> + +<p>This is the analogue of <a href="#*10·12">*10·12</a>.</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*4·2.(*20·07)}.\supset \\ +\vdash\colon\ldotp (\alpha).p.\lor f\alpha.&\equiv :(\phi).p\lor f\{\hat{z}(\phi!z)\}:\\ +[\text{*10·12}] &\equiv :p.\lor .(\phi).f\{\hat{z}(\phi!z)\}:\\ +[\text{(*20·07)}] &\equiv :p.\lor .(\alpha).f\alpha\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*20·631">*20·631</a>.</b> If "\(f\alpha\)" is significant, then if \(\beta\) is +of the same type as \(\alpha\), "\(f\beta\)" is significant, and vice +versa.</p> + +<p>This is the analogue of <a href="#*10·121">*10·121</a>.</p> + +<p><i>Dem.</i></p> + +<p>By <a href="#*20·151">*20·151</a>, \(\alpha\) is of the form \(\hat{z}(\phi!z)\), and +therefore, by <a href="#*20·01">*20·01</a>, \(f\alpha\) is a function of \(\phi!\hat{z}\). +Similarly \(\beta\) is of the form \(\hat{z}(\psi!z)\), and \(f\beta\) +is a function of \(\psi!\hat{z}\). Hence by applying <a href="#*10·121">*10·121</a> to +\(\phi!\hat{z}\) and \(\psi!\hat{z}\) the result follows.</p> + +<p class="nind"> +<b><a id="*20·632">*20·632</a>.</b> If, for some \(\alpha\), there is a proposition +\(f\alpha\), then there is a function \(f\hat{\alpha}\), and vice versa.</p> + +<p><i>Dem.</i></p> + +<p>By the definition in <a href="#*20·01">*20·01</a>, \(f\{\hat{z}(\psi!z)\}\) is a function of +\(\psi!\hat{z}\). Hence the proposition follows from <a href="#*10·122">*10·122</a>.</p> + +<p class="nind"> +<b><a id="*20·633">*20·633</a>.</b> "Whatever possible class \(\alpha\) may be, +\(f(\alpha,\beta)\) is true whatever possible class \(\beta\) may be" +implies the corresponding statement with \(\alpha\) and \(\beta\) +interchanged except in "\(f(\alpha,\beta)\)." (The corresponding +exception is to be understood in <a href="#*11·07">*11·07</a>.)</p> + +<p>This is the analogue of *11·07, and follows at once from *11·07 +because \(f(\alpha,\beta)\) is a function of the defining functions of +\(\alpha\) and \(\beta\).</p> + +<p class="nind"> +<b>*20·64.</b> \(\vdash\colon\ldotp (\alpha).f\alpha:(\alpha).g\alpha:\supset .f\beta.g\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*4·2.(*20·07)}.&\supset \\ +\vdash\colon\ldotp (\alpha).f\alpha:(\alpha).g\alpha:&\equiv :(\phi).f\{\hat{z}(\phi!z)\}:(\phi).g\{\hat{z}(\phi!z)\}:\\ +[\text{*10·14}] &\supset :f\{\hat{z}(\psi!z)\}.g\{\hat{z}(\psi!z)\}\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_208">[Pg 208]</span></p> + +<p>Observe that "\(\beta\)" is merely an abbreviation for any symbol of +the form \(\hat{z}(\psi!z)\). This is why nothing further is required +in the above proof.</p> + +<p>The above proposition is the analogue of <a href="#*10·14">*10·14</a>. Like that proposition, +it requires, for the significance of the conclusion, that \(f\) and +\(g\) should be functions which take arguments of the same type. This +is not required for the significance of the hypothesis. Hence, though +the above proposition is true whenever it is significant, it is not +true whenever its hypothesis is significant.</p> + +<p class="nind"> +<b>*20·7.</b> \(\vdash:(\exists g):f\alpha.\equiv_{\alpha}.g!\alpha \quad[\text{*20·112}]\)</p> + +<p>This is the analogue of <a href="#*12·1">*12·1</a>.</p> + +<p class="nind"> +<b><a id="*20·701">*20·701</a>.</b> \(\vdash:(\exists g):f\{\hat{z}(\phi!z),x\}.\equiv_{\phi,x}.g!\{\hat{z}(\phi!z),x\}\)</p> + +<p>[The proof proceeds as in <a href="#*20·112">*20·112</a>, using <a href="#*12·11">*12·11</a> instead of *12·1.]</p> + +<p class="nind"> +<b>*20·702.</b> \(\vdash:(\exists g):f\{x,\hat{z}(\phi!z)\}.\equiv_{\phi,x}.g!\{x,\hat{z}(\phi!z)\}\)</p> + +<p>[Proof as in <a href="#*20·701">*20·701</a>.]</p> + +<p class="nind"> +<b>*20·703.</b> \(\vdash:(\exists g):f\{\hat{z}(\phi!z),\hat{z}(\psi!z)\}.\equiv_{\phi,\psi}.g!\{\hat{z}(\phi!z),\hat{z}(\psi!z)\}\)</p> + +<p><i>Dem.</i> +\[ +\begin {array}{l} +\vdash.\text{*10·311}.&\supset\vdash\colon\ldotp f\{\chi!\hat{z},\theta!\hat{z}\}.\equiv_{\chi,\theta}.g!\{\chi!\hat{z},\theta!\hat{z}\}:\supset:\\ +&\phi!x\equiv_{x}\chi!x.\psi!x\equiv_{x}\theta!x.f\{\chi!\hat{z},\theta!\hat{z}\}.\equiv_{\chi,\theta}.\\ +&\phi!x\equiv_{x}\chi!x.\psi!x\equiv_{x}\theta!x.g!\{\chi!\hat{z},\theta!\hat{z}\} &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·3·341}.&\supset\\ +\vdash\colon\ldotp \text{Hp(1)}. &\supset:(\exists \chi,\theta).\phi!x\equiv_{x}\chi!x.\psi!x\equiv_{x}\theta!x.f\{\chi!\hat{z},\theta!\hat{z}\}.\equiv_{\phi,\psi}.\\ +&(\exists \chi,\theta).\phi!x\equiv _{x}\chi!x.\psi!x\equiv _{x}\theta!x.g!\{\chi!\hat{z},\theta!\hat{z}\}:\\ +[\text{*20·1.*10·35}]&\supset:f\{\hat{z}(\phi!z),\hat{z}(\psi!z)\}.\equiv_{\phi,\psi}.g!\{\phi!\hat{z},\psi!\hat{z}\} &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·281}.&\supset\\ +\vdash\colon\ldotp &(\exists g):f\{\chi!\hat{z},\theta!\hat{z}\}.\equiv_{\chi,\theta}.g!\{\chi!\hat{z},\theta!\hat{z}\}:\supset:\\ +&(\exists g):f\{\hat{z}(\phi!z),\hat{z}(\psi!z)\}.\equiv_{\phi,\psi}.g!\{\hat{z}(\phi!z),\hat{z}(\psi!z)\} &\qquad \text{(3)}\\ +\vdash.\text{(3).*12·11}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>*20·701·702·703 give the analogues, for classes, of <a href="#*12·11">*12·11</a>.</p> + +<p class="nind"> +<b>*20·71.</b> \(\vdash\colon\ldotp \alpha=\beta.\equiv :g!\alpha.\supset_{g}.g!\beta \quad[\text{*20·19}]\)</p> + +<p>This is the analogue of <a href="#*13·01">*13·01</a>.</p> + +<p><span class="pagenum" id="Page_209">[Pg 209]</span></p> + +<p>This completes the proof that all propositions hitherto given apply to +classes as well as to individuals. Precisely similar reasoning extends +this result to classes of classes, classes of classes of classes, etc.</p> + +<p>From the above propositions it appears that, although expressions such +as \(\hat{z}(\phi z)\) have no meaning in isolation, yet those of their +formal properties with which we have been hitherto concerned are the +same as the corresponding properties of symbols which have a meaning +in isolation. Hence nothing in the apparatus hitherto introduced +requires us to determine whether a given symbol stands for a class or +not, unless the symbol occurs in a way in which only a class can occur +significantly. This is an important result, which enables us to give +much greater generality to our propositions than would otherwise be +possible.</p> + +<p>The two following propositions (*20·8·81) are consequences of <a href="#*13·3">*13·3</a>. +The "type" of any object \(x\) will be defined in <a href="#*63">*63</a> as the class of +terms either identical with \(x\) or not identical with \(x\). We may +define the "type of the arguments to \(\phi\hat{z}\)" as the class of +arguments \(x\) for which "\(\phi x\)" is significant, <i>i.e.</i> the +class \(\hat{x}(\phi x \lor {\sim} \phi x)\). Then the first of the +following propositions shows that if "\(\phi a\)" is significant, the +type of the arguments to \(\phi\hat{z}\) is the type of \(a\); the +second proposition shows that, if "\(\phi a\)" and "\(\psi a\)" are +both significant, the type of the arguments to \(\phi\hat{z}\) is the +same as the type of the arguments to \(\psi\hat{z}\), because each is +the type of \(a\). <a href="#*20·8">*20·8</a> will be used in <a href="#*63·11">*63·11</a>, which is a fundamental +proposition in the theory of relative types.</p> + +<p class="nind"> +<b><a id="*20·8">*20·8</a>.</b> \(\vdash:\phi a \lor {\sim}\phi a.\supset.\hat{x}(\phi x \lor {\sim} \phi x)=\hat{x}(x=a.\lor.x \neq a)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*13·3.*10·11·21}.\supset\\ +\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \phi x \lor {\sim} \phi x.\equiv_{x}:x=a.\lor.x \neq a\colon\ldotp \\ +[\text{*20·15}]\supset\colon\ldotp \hat{x}(\phi x \lor {\sim} \phi x)=\hat{x}(x=a.\lor.x \neq a)\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*20·81">*20·81</a>.</b> \(\vdash:\phi a \lor {\sim} \phi a.\psi a \lor {\sim} \psi a.\supset.\hat{x}(\phi x \lor {\sim}\phi x)=\hat{x}(\psi x \lor {\sim} \psi x)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*20·8}.\supset\vdash:\text{Hp}.\supset.\hat{x}(\phi x \lor {\sim} \phi x)=\hat{x}(x=a.\lor.x \neq a) &\qquad \text{(1)}\\ +\vdash.\text{*20·8}.\supset\vdash:\text{Hp}.\supset.\hat{x}(\psi x \lor {\sim} \psi x)=\hat{x}(x=a.\lor.x \neq a) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*10·121·13.Comp}.\supset\\ +\vdash:\text{Hp}.\supset.\hat{x}(\phi x \lor {\sim} \phi x)=\hat{x}(x=a.\lor.x \neq a).\hat{x}(\psi x \lor {\sim} \psi x)=\hat{x}(x=a.\lor.x \neq a).\\ +[\text{*20·24}]\supset.\hat{x}(\phi x \lor {\sim} \phi x)=\hat{x}(\psi x \lor {\sim} \psi x):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the third line of the above proof, the use of <a href="#*10·121">*10·121</a> depends upon +the fact that the "\(a\)" in both (1) and (2) must be such as to render +the hypothesis significant, <i>i.e.</i> such as to render +\[ +\unicode{x201c}\phi a \lor {\sim}\phi a.\psi a \lor {\sim} \psi a\unicode{x201d} +\]<span class="pagenum" id="Page_210">[Pg 210]</span> +significant. Hence the "\(a\)" in (1) and the "a" in (2) must be of the +same type, by <a href="#*10·121">*10·121</a>, and hence by <a href="#*10·13">*10·13</a> we can assert the product of +(1) and (2), identifying the two "\(a\)'s."</p> + +<p>Since a type is the range of significance of a function, if \(\phi x\) +is a function which is always true, \(\hat{z}(\phi z)\) must be +a type. For if a function is always true, the arguments for which it +is true are the same as the arguments for which it is significant; +hence \(\hat{z}(\phi z)\) is the range of significance of \(\phi x\), +if (\(x) . \phi x\) holds. Thus any class \(\alpha\) is a type +if (\(x).x\in \alpha\). It follows that, whatever function \(\phi\) +may be, \(\hat{x}(\phi x \lor {\sim}\phi x)\) is a type; and in +particular, \(\hat{x}(x=a.\lor.x \neq a)\) is a type. Since \(a\) is +a member of this class, this class is the type to which a belongs. In +virtue of <a href="#*20·8">*20·8</a>, if \(\phi a\) is significant, the type to which a +belongs is the class of arguments for which \(\phi x\) is significant, +<i>i.e.</i> \(\hat{x}(\phi x \lor {\sim}\phi x)\). And if there is any +argument a for which \(\phi a\) and \(\psi a\) are both significant, +then \(\phi\hat{x}\) and \(\psi\hat{x}\) have the same range of +significance, in virtue of <a href="#*20·81">*20·81</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_211">[Pg 211]</span></p> +<h2 class="nobreak" id="*21">*21. GENERAL THEORY OF RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *21.</p> + +<p>The definitions and propositions of this number are exactly analogous +to those of <a href="#*20">*20</a>, from which they differ by being concerned with +functions of two variables instead of one. A <i>relation</i>, as we +shall use the word, will be understood in extension: it may be regarded +as the class of couples (\(x,y)\) for which some given function +\(\psi(x,y)\) is true. Its relation to the function \(\psi(\hat{x},\hat{y})\) +is just like that of the class to its determining function. +We put</p> + +<p class="nind"> +<b><a id="*21·01">*21·01</a>.</b> \(f\{\hat{x}\hat{y}\psi(x,y)\} +.=: (\exists \phi):{\phi}!(x,y).\equiv_{x,y}.\psi(x,y): f\{{\phi}!(\hat{u}, \hat{v})\} \quad \text{Df}\)</p> + +<p>Here "\(\hat{x}\hat{y}\psi(x, y)\)" has no meaning in isolation, but +only in certain of its uses. In <a href="#*21·01">*21·01</a> the <i>alphabetical</i> order +of \(u\) and \(v\) corresponds to the <i>typographical</i> order of +\(\hat{x}\) and \(\hat{y}\) in \(f\{\hat{x}\hat{y}\psi(x,y)\}\), so that +\[ +f\{\hat{y}\hat{x}\psi(x,y)\} .=: (\exists \phi):{\phi}!(x,y).\equiv_{x,y}.\psi(x,y):f\{{\phi}!(\hat{v}, \hat{u})\} \quad \text{Df} +\] +This is important in relation to the substitution-convention below.</p> + +<p>It will be shown that +\[ +\hat{x}\hat{y}\psi(x,y) = \hat{x}\hat{y}\chi(x,y) .\equiv:\psi(x,y).\equiv_{x,y}.\chi(x,y)\text{,} +\] +<i>i.e.</i> that two relations, as above defined, are identical when, +and only when, they are satisfied by the same pairs of arguments.</p> + +<p>For substitution in \({\phi}!(\hat{x}, \hat{y})\) and +\({\phi}!(\hat{y}, \hat{x})\), we adopt the convention that when a +function (as opposed to its values) is represented in a form involving +\(\hat{x}\) and \(\hat{y}\), or any other two letters of the alphabet, +the value of this function for the arguments \(a\) and \(b\) is to be +found by substituting \(a\) for \(\hat{x}\) and \(b\) for \(\hat{y}\), +while the value for the arguments \(b\) and \(a\) is to be found by +substituting \(b\) for \(\hat{x}\) and \(a\) for \(\hat{y}\). That +is, the argument mentioned first is to be substituted for the letter +which comes first in the alphabet, and the argument mentioned second +for the later letter; thus the mode of substitution depends upon the +<i>alphabetical</i> order of the letters which have circumflexes and +the <i>typographical</i> order of the other letters.</p> + +<p><span class="pagenum" id="Page_212">[Pg 212]</span></p> + +<p>The above convention as to order is presupposed in the following +definition, where \(a\) is the first argument mentioned and \(b\) the +second:</p> + +<p class="nind"> +<b>*21·02.</b> \(a\{\phi!(\hat{x},\hat{y})\}b.=.\phi!(a,b) \quad\text{Df}\)</p> + +<p>Hence, following the convention,</p> + +<p>\[ +b\{\phi!(\hat{x},\hat{y})\}a.=.\phi!(b,a) \quad\text{Df}\\ +a\{\phi!(\hat{y},\hat{x})\}b.=.\phi!(b,a) \quad\text{Df}\\ +b\{\phi!(\hat{y},\hat{x})\}a.=.\phi!(a,b) \quad\text{Df} +\] +This definition is not used as it stands, but is introduced for the +sake of +\[ +a\{\hat{x}\hat{y}\psi(x,y)\}b.\equiv:(\exists \phi):\phi!(x,y).\equiv_{x,y}.\psi(x,y):\phi!(a,b) +\] +which results from *21·01·02. We shall use capital Latin +letters to represent variable expressions of the form +\(\hat{x}\hat{y}\phi!(x,y)\), just as we used Greek letters for +variable expressions of the form \(\hat{z}(\phi!z)\). If a capital +Latin letter, say \(R\), is used as an apparent variable, it is +supposed that the \(R\) which occurs in the form "(\(R)\)" or +"(\(\exists R)\)" is to be replaced by "(\(\phi)\)" or "(\(\exists\phi)\)," +while the \(R\) which occurs later is to be replaced by +"\(\hat{x}\hat{y}\phi!(x,y)\)." In fact we put +\[ +(R).fR.=.(\phi).f\{\hat{x}\hat{y}\phi!(x,y)\} \quad\text{Df}. +\] +The use of single letters for such expressions as +\(\hat{x}\hat{y}\phi(x,y)\) is a practically indispensable convenience.</p> + +<p>The following is the definition of the class of relations:</p> + +<p class="nind"> +<b>*21·03.</b> \(\text{Rel}=\hat{R}{(\exists \phi).R=\hat{x}\hat{y}\phi!(x,y)} \quad\text{Df}\)</p> + +<p>Similar remarks apply to it as to the definition of "Cls" (<a href="#*20·03">*20·03</a>).</p> + +<p>In virtue of the definitions *21·01·02 and the convention as to +capital Latin letters, the notation "\(xRy\)" will mean "\(x\) has the +relation \(R\) to \(y\)." This notation is practically convenient, and +will, after the preliminaries, wholly replace the cumbrous notation +\(x\{\hat{x}\hat{y}\phi(x,y)\}y\).</p> + +<p>The proofs of the propositions of this number are usually omitted, +since they are exactly analogous to those of <a href="#*20">*20</a>, merely substituting +<a href="#*12·11">*12·11</a> for <a href="#*12·1">*12·1</a>, and propositions in <a href="#*11">*11</a> for propositions in <a href="#*10">*10</a>.</p> + +<p>The propositions of this number, like those of <a href="#*20">*20</a>, fall into three +sections. Those of the second section are seldom referred to. Those +of the third section, extending to relations the formal properties +hitherto assumed or proved for individuals and functions, are not +explicitly referred to in the sequel, but are constantly relevant, +namely whenever a proposition which has been assumed or proved for +individuals and functions is applied to relations. The principal +propositions of the first section are the following.</p> + +<p class="nind"> +<b>*21·15.</b> \(\vdash\colon\ldotp\psi(x,y).\equiv_{x,y}.\chi(x,y):\equiv.\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y)\)</p> + +<p><i>I.e.</i> two relations are identical when, and only when, their +defining functions are formally equivalent.</p> + +<p class="nind"> +<b>*21·31.</b> \(\vdash\colon\ldotp\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y).\equiv:x\{\hat{x}\hat{y}\psi(x,y)\}y.\equiv_{x,y}.x\{\hat{x}\hat{y}\chi(x,y)\}y\)</p> + +<p><span class="pagenum" id="Page_213">[Pg 213]</span></p> + +<p><i>I.e.</i> two relations are identical when, and only when, they hold +between the same pairs of terms. The same fact is expressed by the +following proposition:</p> + +<p class="nind"> +<b><a id="*21·43">*21·43</a>.</b> \(\vdash\colon\ldotp R=S.\equiv:xRy.\equiv_{x,y}.xSy\)</p> + +<p class="nind"> +<b>*21·2·21·22</b> show that identity of relations is reflexive, +symmetrical and transitive.</p> + +<p class="nind"> +<b>*21·3.</b> \(\vdash:x{\hat{x}\hat{y}\psi(x,y)}y.\equiv.\psi(x,y)\)</p> + +<p><i>I.e.</i> two terms have a given relation when, and only when, they +satisfy its defining function.</p> + +<p class="nind"> +<b>*21·151.</b> \(\vdash.(\exists \phi).\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\phi!(x,y)\)</p> + +<p class="nind"> +<i>I.e.</i> every relation can be defined by a predicative function. +Hence when, using <a href="#*21·07">*21·07</a> or <a href="#*21·071">*21·071</a>, we have a relation as apparent +variable, and are therefore confined to predicative defining functions, +there is no loss of generality.</p> + +<hr class="tb"> + +<p class="nind"> +<b>*21·01.</b> \(f\{\hat{x}\hat{y}\psi(x,y)\}.=:(\exists \phi):\phi!(x,y).\equiv_{x,y}.\psi(x,y):f\{\phi!(\hat{u},\hat{v})\} \quad\text{Df}\)</p> + +<p>On the convention as to order in *21·01·02, cf. <a href="#Page_211">p. 211</a>, and thus relate +\(\hat{u}\), \(\hat{v}\) to \(\hat{x}\), \(\hat{y}\) so that +\[ +f\{\hat{y}\hat{x}\psi(x,y)\}.=:(\exists \phi):\phi!(x,y).\equiv_{x,y}.\psi(x,y):f\{\phi!(\hat{v},\hat{u})\} \quad\text{Df} +\]</p> + +<p class="nind"> +<b>*21·02.</b> \(a\{\phi!(\hat{x},\hat{y})\}b.=.\phi!(a,b) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*21·03.</b> \(\text{Rel}=\hat{R}\{(\exists \phi).R=\hat{x}\hat{y}\phi!(x,y)\} \quad\text{Df}\)</p> + +<p>The following definitions merely extend to relations, with as little +modification as possible, the definitions already given for other +symbols.</p> + +<p class="nind"> +<b><a id="*21·07">*21·07</a>.</b> (\(R).fR.=.(\phi).f\{\hat{x}\hat{y}\phi!(x,y)\} \quad\text{Df}\)</p> + +<p class="nind"> +<b><a id="*21·071">*21·071</a>.</b> (\(\exists R).fR.=.(\exists \phi).f\{\hat{x}\hat{y}\phi!(x,y)\} \quad\text{Df}\)</p> + +<p class="nind"> +<b>*21·072.</b> \([({℩}R)(\phi R)].f({℩}R)(\phi R).=:(\exists S):\phi R.\equiv_{R}.R=S:fS \quad\text{Df}\)</p> + +<p class="nind"> +<b>*21·08.</b> \(f\{\hat{R}\hat{S}\psi(R,S)\}.=:(\exists \phi):\psi(R,S).\equiv_{R,S}.\phi!(R,S):f\{\phi!(\hat{R},\hat{S})\} \quad\text{Df}\)</p> + +<p class="nind"> +<b>*21·081.</b> \(P\{\phi!(\hat{R},\hat{S})\}Q.=.\phi!(P,Q) \quad\text{Df}\)</p> + +<p>The convention as to typographic and alphabetic order is here retained.</p> + +<p class="nind"> +<b>*21·082.</b> \(f\{\hat{R}(\psi R)\}.=:(\exists \phi):\psi R.\equiv_{R}.\phi!R:f(\phi!\hat{R}) \quad\text{Df}\)</p> + +<p class="nind"> +<b>*21·083.</b> \(R\in \phi!\hat{R}.=.\phi!R \quad\text{Df}\)</p> + +<p class="nind"> +<b>*21·1.</b> \[\begin{align}&\vdash\colon\ldotp f\{\hat{x}\hat{y}\psi(x,y)\}.\equiv:(\exists \phi):\phi!(x,y).\equiv_{x,y}.\psi(x,y):f\{\phi!(\hat{u},\hat{v})\}\\ +&\quad[\text{*4·2.(*21·01)}]\end{align}\]</p> + +<p class="nind"> +<b>*21·11.</b> \[\begin{align}&\vdash\colon\ldotp\psi(x,y).\equiv_{x,y}.\chi(x,y):\supset:f\{\hat{x}\hat{y}\psi(x,y)\}.\equiv.f\{\hat{x}\hat{y}\chi(x,y)\}\\ +&\quad[\text{*4·86·36.*10·281.*21·1}]\end{align}\]</p> + +<p>This proposition proves that every proposition about a relation +expresses an extensional property of the determining function.</p> + +<p class="nind"> +<b>*21·111.</b> \[\begin{align}&\vdash\colon\ldotp f\{\phi!(\hat{x},\hat{y})\}.\equiv_{\phi}.g\{\phi!(x,y)\}:\supset:f\{\hat{x}\hat{y}\phi!(x,y)\}.\equiv_{\phi}.g\{\hat{x}\hat{y}\phi!(x,y)\}\\ +&\quad[\text{Fact.*11·11·3.*10·281.*21·1}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_214">[Pg 214]</span></p> + +<p class="nind"> +<b>*21·112</b>. \(\vdash\colon\ldotp(\exists g)\colon\ldotp f\{\hat{x}\hat{y}\phi!(x,y)\}.\equiv_{\phi}.g!\{\hat{x}\hat{y}\phi!(x,y)\} \quad[\text{*12·1.*21·111}]\)</p> + +<p>It is <a href="#*12·1">*12·1</a>, not <a href="#*12·11">*12·11</a>, which is required in this proposition, because +we are concerned with a function (\(f)\) of <i>one</i> variable, namely +\(\phi\), although that one variable is itself a function of two +variables.</p> + +<p class="nind"> +<b>*21·12.</b> \[\begin{align}&\vdash\colon\ldotp (\exists \phi)\colon\ldotp \phi!(x,y).\equiv_{x,y}.\psi(x,y):f\{\hat{x}\hat{y}\psi(x,y)\}.\equiv.f\{\hat{x}\hat{y}\phi!(x,y)\}\\ +&\quad[\text{*21·11.*12·11}]\end{align}\]</p> + +<p>This is the first use of the primitive proposition <a href="#*12·11">*12·11</a>, except in +*20·701·702·703.</p> + +<p class="nind"> +<b>*21·13.</b> \[\begin{align}&\vdash\colon\ldotp \psi(x,y).\equiv_{x,y}.\chi(x,y):\supset.\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y)\\ +&\quad[\text{*21·1.*12·11.*13·195}]\end{align}\]</p> + +<p class="nind"> +<b>*21·14.</b> \[\begin{align}&\vdash\colon\ldotp \hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y).\supset:\psi(x,y).\equiv_{x,y}.\chi(x,y)\\ +&\quad[\text{Proof as in *20·14}]\end{align}\]</p> + +<p class="nind"> +<b>*21·15.</b> \(\vdash\colon\ldotp \psi(x,y).\equiv_{x,y}.\chi(x,y):\equiv.\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y) \quad[\text{*21·13·14}]\)</p> + +<p>This proposition states that two double functions determine the same +relation when, and only when, they are formally equivalent, <i>i.e.</i> +are satisfied by the same pairs of arguments. This is a fundamental +property of relations as defined above (<a href="#*21·01">*21·01</a>).</p> + +<p class="nind"> +<b>*21·151.</b> \(\vdash.(\exists \phi).\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\phi!(x,y) \quad[\text{*21·15.*12·11}]\)</p> + +<p class="nind"> +<b>*21·16.</b> \(\vdash:(\exists \phi):f\{\hat{x}\hat{y}\psi(x,y)\}.\equiv.f\{\hat{x}\hat{y}\phi!(x,y)\} \quad[\text{*21·12}]\)</p> + +<p class="nind"> +<b>*21·17.</b> \(\vdash:(\phi).f\{\hat{x}\hat{y}\phi!(x,y)\}.\supset.f\{\hat{x}\hat{y}\psi(x,y)\} \quad[\text{*21·16.*10·1}]\)</p> + +<p class="nind"><b>*21·18.</b> \[\begin{align}\vdash\colon\ldotp \hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\psi(x,y).\supset:f\{\hat{x}\hat{y}\phi(x,y)\}.&\equiv.f\{\hat{x}\hat{y}\psi(x,y)\}\\ +&\quad[\text{*21·11·15}]\end{align}\]</p> + +<p class="nind"> +<b>*21·19.</b> \[\begin{align}&\vdash\colon\ldotp \hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y).\equiv:(f):f!\hat{x}\hat{y}\psi(x,y).\supset.f!\hat{x}\hat{y}\chi(x,y)\\ +&\quad[\text{*21·18.*10·11·21.*21·1.*10·35.(*13·01).*21·112.*10·301}]\end{align}\]</p> + +<p class="nind"> +<b>*21·191.</b> \[\begin{align}\vdash\colon\ldotp \hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y).&\equiv:(f):f!\hat{x}\hat{y}\psi(x,y).\equiv.f!\hat{x}\hat{y}\chi(x,y)\\ +&\quad[\text{*21·18·19}]\end{align}\]</p> + +<p class="nind"> +<b>*21·2.</b> \(\vdash.\hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\phi(x,y) \quad[\text{*21·15.*4·2}]\)</p> + +<p class="nind"> +<b>*21·21.</b> \(\vdash:\hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\psi(x,y).\equiv.\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\phi(x,y) \quad[\text{*21·15.*10·32}]\)</p> + +<p class="nind"> +<b>*21·22.</b> \[\begin{align}\vdash:\hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\psi(x,y).&\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y).\supset.\\ +&\hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\chi(x,y) \quad[\text{*21·15.*10·301}]\end{align}\]</p> + +<p class="nind"> +<b>*21·23.</b> \[\begin{align}\vdash:\hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\psi(x,y).&\hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\chi(x,y).\supset.\\ +&\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y) \quad[\text{*21·21·22}]\end{align}\]</p> + +<p class="nind"> +<b>*21·24.</b> \[\begin{align}\vdash:\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\phi(x,y).&\hat{x}\hat{y}\chi(x,y)=\hat{x}\hat{y}\phi(x,y).\supset.\\ +&\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x,y) \quad[\text{*21·21·22}]\end{align}\]</p> + +<p class="nind"> +<b>*21·3.</b> \(\vdash:x{\hat{x}\hat{y}\psi(x,y)}y.\equiv.\psi(x,y) \quad[\text{*21·1·02.*10·43·35.*12·11}]\)</p> + +<p><span class="pagenum" id="Page_215">[Pg 215]</span></p> + +<p>This shows that \(x\) has to y the relation determined by \(\psi\) +when, and only when, \(x\) and \(y\) satisfy \(\psi(x, y)\).</p> + +<p>Note that the primitive proposition <a href="#*12·11">*12·11</a> is again required here.</p> + +<p class="nind"> +<b>*21·31.</b> \[\begin{align}\vdash\colon\ldotp \hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\chi(x, y).\equiv:x\{\hat{x}\hat{y}\psi(x,y)\}y.&\equiv_{x,y}.x\{\hat{x}\hat{y}\chi(x, y)\}y\\ +&\quad[\text{*21·15·3}]\end{align}\]</p> + +<p class="nind"> +<b>*21·32.</b> \(\vdash.\hat{x}\hat{y}[x\{\hat{x}\hat{y}\phi(x,y)\}y]=\hat{x}\hat{y}\phi(x,y) \quad[\text{*21·3·15}]\)</p> + +<p class="nind"> +<b>*21·33.</b> \(\vdash\colon\ldotp R=\hat{x}\hat{y}\phi(x, y).\equiv:xRy.\equiv_{x, y}.\phi(x, y) \quad[\text{*21·31·3}]\)</p> + +<p>Here \(R\) is written for some expression of the form +\(\hat{x}\hat{y}\psi(x,y)\). The use of a single capital letter for a +relation is convenient whenever the determining function is irrelevant.</p> + +<p class="nind"> +<b>*21·4.</b> \(\vdash:R\in \text{Rel}.\equiv.(\exists \phi).R=\hat{x}\hat{y}\phi!(x,y) \quad[\text{*20·3.(*21·03)}]\)</p> + +<p class="nind"> +<b>*21·41.</b> \(\vdash.\hat{x}\hat{y}\phi(x, y)\in \text{Rel} \quad[\text{*21·4·151}]\)</p> + +<p class="nind"> +<b>*21·42.</b> \(\vdash.\hat{x}\hat{y}(xRy)=R \quad[\text{*21·3·15}]\)</p> + +<p class="nind"> +<b>*21·43.</b> \(\vdash\colon\ldotp R=S.\equiv:xRy.\equiv_{x,y}.xSy \quad[\text{*21·15·3}]\)</p> + +<p class="nind"> +<b>*20·5·51·52</b> have no analogues in the theory of relations.</p> + +<p class="nind"> +<b>*21·53.</b> \(\vdash\colon\ldotp S=R.\supset_{S}.\phi S:\equiv.\phi R \quad[\text{*10·1.*21·2·18·21.Comm.*10·11·21}]\)</p> + +<p class="nind"> +<b>*21·54.</b> \(\vdash\colon\ldotp (\exists S).S=R.\phi S.\equiv.\phi R \quad[\text{*21·18.*10·11·23.*21·2.*10·24}]\)</p> + +<p class="nind"> +<b>*21·55.</b> \(\vdash.\hat{x}\hat{y}\phi(x,y)=(℩R)\{xRy.\equiv_{x, y}.\phi(x,y)\} \quad[\text{*21·33·54.*14·1}]\)</p> + +<p class="nind"> +<b>*21·56.</b> \(\vdash.\text{E}!(℩R)\{xRy.\equiv_{x,y}.\phi(x,y)\} \quad[\text{*21·55.*14·21}]\)</p> + +<p class="nind"> +<b>*21·57.</b> \[\begin{align}\vdash\colon\ldotp \hat{x}\hat{y}\phi(x,y)=(℩R)(fR).\supset:g\{\hat{x}\hat{y}\phi(x,y)\}.&\equiv.g\{(℩R)(fR)\}\\ +&\quad[\text{*14·1.*21·54.*13·183}]\end{align}\]</p> + +<p class="nind"> +<b>*21·58.</b> \(\vdash.\hat{x}\hat{y}\phi(x,y)=({℩}R)\{R=\hat{x}\hat{y}\phi(x,y)\} \quad[\text{*4·2.*10·11.*21·54.*14·1}]\)</p> + +<p>The following propositions are the analogues of <a href="#*20·6">*20·6</a> ff., and have a +similar purpose.</p> + +<p class="nind"> +<b>*21·6.</b> \(\vdash:(\exists R).fR.\equiv.{\sim}\{(R).{\sim}fR\} \quad[\text{Proof as in *20·6}]\)</p> + +<p class="nind"> +<b>*21·61.</b> \(\vdash:(R).fR.\supset.fS \quad[\text{Proof as in *20·61}]\)</p> + +<p class="nind"> +<b>*21·62.</b> When \(fR\) is true, whatever possible argument of the form \(\hat{x}\hat{y}\phi!(x,y)\) +\(R\) may be, (\(R).fR\) is true. [Proof as in <a href="#*20·62">*20·62</a>]</p> + +<p class="nind"> +<b>*21·63.</b> \(\vdash\colon\ldotp (R).p\lor fR.\supset:p.\lor.(R).fR \quad[\text{Proof as in *20·63}]\)</p> + +<p class="nind"> +<b>*21·631.</b> If "\(fR\)" is significant, then if \(S\) is of the +same type as \(R\), "\(fS\)" is significant, and vice versa.</p> + +<p>[Proof as in <a href="#*20·631">*20·631</a>]</p> + +<p><span class="pagenum" id="Page_216">[Pg 216]</span></p> + +<p class="nind"> +<b>*21·632.</b> If, for some \(R\), there is a proposition \(fR\), then +there is a function \(f\hat{R}\), and vice versa.</p> + +<p>[Proof as in <a href="#*20·632">*20·632</a>]</p> + +<p class="nind"> +<b>*21·633.</b> "Whatever possible relation \(R\) may be, \(f(R,S)\) +is true whatever possible relation \(S\) may be" implies "whatever +possible relation \(S\) may be, \(f(R,S)\) is true whatever possible +relation \(R\) may be."</p> + +<p>[Proof as in <a href="#*20·633">*20·633</a>]</p> + +<p class="nind"> +<b>*21·64.</b> \(\vdash\colon\ldotp (R).fR:(R).gR:\supset.fS.gS \quad[\text{Proof as in *20·64}]\)</p> + +<p class="nind"> +<b>*21·7.</b> \(\vdash:(\exists g):fR.\equiv_{R}.g!R \quad[\text{Proof as in *20·7}]\)</p> + +<p class="nind"> +<b>*21·701.</b> \(\vdash:(\exists g):f(R, x).\equiv_{R,x}.g!(R, x) \quad[\text{Proof as in *20·701}]\)</p> + +<p class="nind"> +<b>*21·702.</b> \(\vdash:(\exists g):f(x, R).\equiv_{R,x}.g!(R, x) \quad[\text{Proof as in *20·702}]\)</p> + +<p class="nind"> +<b>*21·703.</b> \(\vdash:(\exists g):f(R, S).\equiv_{R,S}.g!(R, S) \quad[\text{Proof as in *20·703}]\)</p> + +<p class="nind"> +<b>*21·704.</b> \(\vdash:(\exists g):f(R, \alpha).\equiv_{R,\alpha}.g!(R, \alpha) \quad[\text{Proof as in *20·703}]\)</p> + +<p class="nind"> +<b>*21·705.</b> \(\vdash:(\exists g):f(\alpha, R).\equiv_{\alpha,R}.g!(\alpha, R) \quad[\text{Proof as in *20·703}]\)</p> + +<p class="nind"> +<b>*21·71.</b> \(\vdash\colon\ldotp R=S.\equiv:g!R.\supset_{g}.g!S \quad[\text{Proof as in *20·71}]\)</p> + +<p>From the above propositions it appears that relations, like classes, +have all the formal properties which they would have if they were +symbols having a meaning in isolation. Hence unless a symbol occurs +in a way in which only a relation can occur significantly, we do not +need to decide whether it stands for a relation or not. This result, +like the corresponding result for classes mentioned at the end of *20, +is important as giving greater generality to our propositions than +they would otherwise possess. The results obtained in <a href="#*20">*20</a> and <a href="#*21">*21</a> for +classes and relations whose members or terms are neither classes nor +relations can be extended, by mere repetition of the proofs, to classes +of classes, classes of relations, relations of classes, relations of +relations, and so on.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_217">[Pg 217]</span></p> +<h2 class="nobreak" id="*22">*22. CALCULUS OF CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *22.</p> + +<p>In this number we reach what was historically the starting-point of +symbolic logic. The Greek letters used (except \(\phi\), \(\psi\), +\(\chi\), \(\theta\)) are always to stand for expressions of the form +\(\hat{x}({\phi}!x)\), or, where the Greek letters are not apparent +variables, \(\hat{x}({\phi}x)\). The small Latin letters may either +be such as have a meaning in isolation, or may represent classes or +relations; this is possible in virtue of the notes at the ends of <a href="#*20">*20</a> +and <a href="#*21">*21</a>. We put:</p> + +<p class="nind"> +<b>*22·01.</b> \(\alpha \subset \beta .=: x \in \alpha .\supset_{x}. x \in \beta \quad \text{Df}\)</p> + +<p>This defines "the class \(\alpha\) is contained in the class +\(\beta\)," or "all \(\alpha\)'s are \(\beta\)'s."</p> + +<p class="nind"> +<b>*22·02.</b> \(\alpha \cap \beta = \hat{x}(x \in \alpha . x \in \beta) \quad \text{Df}\)</p> + +<p>This defines the logical product or common part of two classes +\(\alpha\) and \(\beta\).</p> + +<p class="nind"> +<b>*22·03.</b> \(\alpha \cup \beta = \hat{x}(x \in \alpha .\lor. x \in \beta) \quad \text{Df}\)</p> + +<p>This defines the logical sum of two classes; it is the class consisting +of all the members of one together with all the members of the other.</p> + +<p class="nind"> +<b>*22·04.</b> \(-\alpha = \hat{x}(x {\sim}{\in} \alpha) \quad \text{Df}\)</p> + +<p>This defines the negation of a class. It is read "not-\(\alpha\)." It +does not contain every object \(x\) concerning which "\(x \in \alpha\)" +is <i>not true</i>, but only those objects concerning which "\(x \in \alpha\)" +is <i>false</i>; <i>i.e.</i> it excludes those objects for +which "\(x \in \alpha\)" is meaningless. Thus it consists of all +objects, of the type next below \(\alpha\), which are not members of +\(\alpha\); but it does not contain objects of any other type but this.</p> + +<p class="nind"> +<b>*22·05.</b> \(\alpha - \beta = \alpha \cap -\beta \quad \text{Df}\)</p> + +<p>This definition gives an abbreviation which is often convenient.</p> + +<p>The postulates required for the algebra of logic have been enumerated +by Huntington<a id="FNanchor_54" href="#Footnote_54" class="fnanchor">[54]</a>. In our notation, they are as follows.</p> + +<p><span class="pagenum" id="Page_218">[Pg 218]</span></p> + +<p>We assume a class \(K\), with two rules of combination, namely +\(\cup\) and \(\cap\); and we then require the following ten postulates:</p> + +<p>I <i>a.</i> \(a \cup b\) is in the class whenever \(a\) and \(b\) are +in the class.</p> + +<p>I <i>b.</i> \(a \cap b\) is in the class whenever \(a\) and \(b\) are +in the class.</p> + +<p>II <i>a.</i> There is an element \(\Lambda\) such that \(a \cup \Lambda = a\) +for every element \(a\).</p> + +<p>II <i>b.</i> There is an element \(\text{V}\) such that \(a \cap\text{V} = a\) +for every element \(a\).</p> + +<p>III <i>a.</i> \(a \cup b = b \cup a\) whenever \(a\), \(b\), \(a \cup b\) +and \(b \cup a\) are in the class.</p> + +<p>III <i>b.</i> \(a \cap b = b \cap a\) whenever \(a\), \(b\), \(a \cap b\) +and \(b \cap a\) are in the class.</p> + +<p>IV <i>a.</i> \(a \cup (b \cap c) = (a \cup b) \cap (a \cup c)\) +whenever \(a\), \(b\), \(c\), \(a \cup b\), \(a \cup c\), \(b \cap c\), +\(a \cup (b \cap c)\), and \((a \cup b) \cap (a \cup c)\) are in the +class.</p> + +<p>IV <i>b.</i> \(a \cap (b \cup c) = (a \cap b) \cup (a \cap c)\) +whenever \(a\), \(b\), \(c\), \(a \cap b\), \(a \cap c\), \(b \cup c\), +\(a \cap (b \cup c)\), and (\(a \cap b) \cup (a \cap c)\) are in the +class.</p> + +<p>V. If the elements \(\Lambda\) and \(\text{V}\) in postulates II +<i>a</i> and II <i>b</i> exist and are unique, then for every element +\(a\) there is an element \(-a\) such that \(a \cup -a = \text{V}\) and +\(a \cap -a = \Lambda\).</p> + +<p>VI. There are at least two elements, \(x\) and \(y\), in the class, +such that \(x \neq y\).</p> + +<p>The form of the above postulates is such that they are mutually +independent, <i>i.e.</i> any nine of them are satisfied by +interpretations of the symbols which do not satisfy the remaining one.</p> + +<p>For our purposes, "\(K\)" must be replaced by "\(\text{Cls}\)." +\(\Lambda\) and \(\text{V}\) will be the null-class and the universal +class, which are defined in <a href="#*24">*24</a>. Then the above ten postulates are +proved below, as follows:</p> + +<p>I <i>a</i>. in *22·37, namely "\(\vdash . \alpha \cup \beta \in \text{Cls}\)"</p> + +<p>I <i>b</i>. in *22·36, namely "\(\vdash . \alpha \cap \beta \in \text{Cls}\)"</p> + +<p>II <i>a</i>. in *24·24, namely "\(\vdash . \alpha \cup \Lambda = \alpha\)"</p> + +<p>II <i>b</i>. in *24·26, namely "\(\vdash . a \cap \text{V} = \alpha\)"</p> + +<p>III <i>a</i>. in *22·57, namely "\(\vdash . a \cup \beta = \beta \cup \alpha\)"</p> + +<p>III <i>b</i>. in *22·51, namely "\(\vdash . \alpha \cap \beta = \beta \cap \alpha\)"</p> + +<p>IV <i>a</i>. in *22·69, namely "\(\vdash . (\alpha \cup \beta) \cap (\alpha \cup \gamma) = \alpha \cup (\beta \cap \gamma)\)"</p> + +<p>IV <i>b</i>. in *22·68, namely "\(\vdash . (\alpha \cap \beta) \cup (\alpha \cap \gamma) = \alpha \cap (\beta \cup \gamma)\)"</p> + +<p>V. in *24·21·22, namely "\(\vdash . \alpha \cap -\alpha = \Lambda\)" and "\(\vdash . \alpha \cup -\alpha = \text{V}\)"</p> + +<p>VI. in *24·1, namely "\(\vdash . \Lambda \neq \text{V}\)"</p> + +<p>Hence, assuming Huntington's analysis of the postulates for the formal +algebra of logic, the propositions proved in what follows suffice +to establish that this algebra holds for classes. The corresponding +propositions of <a href="#*23">*23</a> and <a href="#*25">*25</a> prove that it holds for relations, +substituting \(\text{Rel}\), \(\unicode{x228d}\), \(\dot{\cap}\), +\(\dot{\Lambda}\), \(\dot{\text{V}}\) for \(\text{Cls}\), \(\cup\), +\(\cap\), \(\Lambda\), \(\text{V}\).</p> + +<p><span class="pagenum" id="Page_219">[Pg 219]</span></p> + +<p>The principal propositions of the present number are the following:</p> + +<p>(1) Those embodying the formal rules:</p> + +<p class="nind"> +<b>*22·51.</b> \(\vdash. \alpha \cap \beta = \beta \cap \alpha\)</p> + +<p class="nind"> +<b>*22·57.</b> \(\vdash. \alpha \cup \beta = \beta \cup \alpha\)</p> + +<p>These embody the commutative law.</p> + +<p class="nind"> +<b>*22·52.</b> \(\vdash. (\alpha \cap \beta) \cap \gamma = \alpha \cap (\beta \cap \gamma)\)</p> + +<p class="nind"> +<b>*22·7.</b> \(\vdash.(\alpha \cup \beta) \cup \gamma = \alpha \cup (\beta \cup \gamma)\)</p> + +<p>These embody the associative law.</p> + +<p class="nind"> +<b>*22·5.</b> \(\vdash. \alpha \cap \alpha = \alpha\)</p> + +<p class="nind"> +<b>*22·56.</b> \(\vdash. \alpha \cup \alpha = \alpha\)</p> + +<p>These embody the law of tautology.</p> + +<p class="nind"> +<b>*22·68.</b> \(\vdash. (\alpha \cap \beta) \cup (\alpha \cap \gamma) = \alpha \cap (\beta \cup \gamma)\)</p> + +<p class="nind"> +<b>*22·69.</b> \(\vdash. (\alpha \cup \beta) \cap (\alpha \cup \gamma) = \alpha \cup (\beta \cap \gamma)\)</p> + +<p>These embody the distributive law. It will be seen that the second +results from the first by everywhere interchanging the signs of +addition and multiplication.</p> + +<p class="nind"> +<b>*22·8.</b> \(\vdash. -(-\alpha) = \alpha\)</p> + +<p>This is the principle of double negation.</p> + +<p class="nind"> +<b>*22·81.</b> \(\vdash: \alpha \subset \beta .\equiv. -\beta \subset -\alpha\)</p> + +<p>This is the principle of transposition.</p> + +<p>(2) Other useful propositions:</p> + +<p class="nind"> +<b>*22·44.</b> \(\vdash: \alpha \subset \beta . \beta \subset \gamma .\supset. \alpha \subset \gamma\)</p> + +<p class="nind"> +<b>*22·441.</b> \(\vdash: \alpha \subset \beta . x \in \alpha .\supset. x \in \beta\)</p> + +<p>These embody the two forms of the syllogism in Barbara.</p> + +<p class="nind"> +<b>*22·62.</b> \(\vdash: \alpha \subset \beta .\equiv. \alpha \cup \beta = \beta\)</p> + +<p class="nind"> +<b>*22·621.</b> \(\vdash: \alpha \subset \beta .\equiv. \alpha \cap \beta = \alpha\)</p> + +<p>These two propositions enable us to transform any inclusion (\(\alpha \subset \beta)\) +into an equation.</p> + +<p class="nind"> +<b>*22·91.</b> \(\vdash. \alpha \cup \beta = \alpha \cup (\beta - \alpha)\)</p> + +<p><i>I.e.</i> "\(\alpha\) or \(\beta\)" is identical with "\(\alpha\) or +the part of \(\beta\) which is excluded from \(\alpha\)."</p> + +<hr class="tb"> + +<p class="nind"> +<b>*22·01.</b> \(\alpha \subset \beta .=: x \in \alpha .\supset_{x}. x \in \beta \quad \text{Df}\)</p> + +<p class="nind"> +<b>*22·02.</b> \(\alpha \cap \beta = \hat{x}(x \in \alpha . x \in \beta) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*22·03.</b> \(\alpha \cup \beta = \hat{x}(x \in \alpha .\lor. x \in \beta) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*22·04.</b> \(-\alpha = \hat{x}(x \mathop{{\sim}{\in}} \alpha) \quad \text{Df}\)</p> + +<p><span class="pagenum" id="Page_220">[Pg 220]</span></p> + +<p class="nind"> +<b>*22·05.</b> \(\alpha-\beta=\alpha\cap -\beta \quad\text{Df}\)</p> + +<p class="nind"> +<b>*22·1.</b> \(\vdash\colon\ldotp \alpha\subset\beta.\equiv:x\in \alpha.\supset_{x}.x\in \beta \quad[\text{*4·2.(*22·01)}]\)</p> + +<p class="nind"> +<b>*22·2.</b> \(\vdash.\alpha\cap \beta=\hat{x}(x\in \alpha.x\in \beta) \quad[\text{*20·2.(*22·02)}]\)</p> + +<p class="nind"> +<b>*22·3.</b> \(\vdash.\alpha\cup \beta=\hat{x}(x\in \alpha.\lor .x\in \beta) \quad[\text{*20·2.(*22·03)}]\)</p> + +<p class="nind"> +<b>*22·31.</b> \(\vdash.-\alpha=\hat{x}(x{\sim}\in \alpha) \quad[\text{*20·2.(*22·04)}]\)</p> + +<p class="nind"> +<b>*22·32.</b> \(\vdash.\alpha-\beta=\hat{x}(x\in \alpha.x{\sim}\in \beta) \quad[\text{*20·2.(*22·05).*22·2.*20·32}]\)</p> + +<p class="nind"> +<b>*22·33.</b> \(\vdash:x\in \alpha\cap \beta.\equiv.x\in \alpha.x\in \beta \quad[\text{*20·3.*22·2}]\)</p> + +<p class="nind"> +<b>*22·34.</b> \(\vdash\colon\ldotp x\in \alpha\cup \beta.\equiv:x\in \alpha.\lor .x\in \beta \quad[\text{*20·3.*22·3}]\)</p> + +<p class="nind"> +<b>*22·35.</b> \(\vdash:x\in -\alpha.\equiv.x{\sim}\in \alpha \quad[\text{*20·3.*22·31}]\)</p> + +<p class="nind"> +<b>*22·351.</b> \(\vdash.-\alpha \neq \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·35.*5·19}.&\supset\vdash:{\sim}\{x\in -\alpha.\equiv.x\in \alpha\}:\\ +[\text{*10·11}] &\supset\vdash:(x):{\sim}\{x\in -\alpha.\equiv.x\in \alpha\}:\\ +[\text{*10·251}] &\supset\vdash:{\sim}\{(x):x\in -\alpha.\equiv.x\in \alpha\}:\\ +[\text{*20·43.Transp}]&\supset\vdash:{\sim}(-\alpha=\alpha):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in proving that the null-class is not +identical with the class containing everything (<a href="#*24·1">*24·1</a>), which is used +to show that at least two classes exist. Our axioms do not suffice +to prove that more than one <i>individual</i> exists, but they +prove the existence of at least two <i>classes</i> and at least two +<i>relations</i>.</p> + +<p class="nind"> +<b>*22·36.</b> \(\vdash.\alpha\cap \beta\in \text{Cls} \quad[\text{*20·41}]\)</p> + +<p class="nind"> +<b>*22·37.</b> \(\vdash.\alpha\cup \beta\in \text{Cls} \quad[\text{*20·41}]\)</p> + +<p class="nind"> +<b>*22·38.</b> \(\vdash.-\alpha\in \text{Cls} \quad[\text{*20·41}]\)</p> + +<p class="nind"> +<b>*22·39.</b> \(\vdash.\hat{z}(\phi z)\cap \hat{z}(\psi z)=\hat{z}(\phi z.\psi z)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·33}. \supset\vdash:x\in \hat{z}(\phi z)\cap \hat{z}(\psi z).&\equiv.x\in \hat{z}(\phi z).x\in \hat{z}(\psi z).\\ +[\text{*20·3}] &\equiv.\phi x.\psi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*20·33}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·391.</b> \(\vdash.\hat{z}(\phi z)\cup \hat{z}(\psi z)=\hat{z}(\phi z\lor \psi z) \quad[\text{Similar proof}]\)</p> + +<p class="nind"> +<b>*22·392.</b> \(\vdash.-\hat{z}(\phi z)=\hat{z}({\sim}\phi z) \quad[\text{Similar proof}]\)</p> + +<p class="nind"> +<b>*22·4.</b> \(\vdash\colon\ldotp \alpha\subset\beta.\beta\subset\alpha.\equiv:x\in \alpha.\equiv_{x}.x\in \beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·1}.&\supset\vdash\colon\colon \alpha\subset\beta.\equiv:x\in \alpha.\supset_{x}.x\in \beta\colon\ldotp \beta\subset\alpha.\equiv:x\in \beta.\supset_{x}.x\in + \alpha\colon\ldotp\\ +[\text{*4·38}] &\supset\vdash\colon\colon \alpha\subset\beta.\beta\subset\alpha.\equiv\colon\ldotp x\in \alpha.\supset_{x}.x\in \beta:x\in \beta.\supset_{x}.x\in + \alpha\colon\ldotp \\ +[\text{*10·22}] &\qquad\qquad\qquad\qquad\equiv\colon\ldotp x\in \alpha.\equiv_{x}.x\in \beta\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·41.</b> \(\vdash:\alpha\subset\beta.\beta\subset\alpha.\equiv.\alpha=\beta \quad[\text{*22·4.*20·43}]\)</p> + +<p><span class="pagenum" id="Page_221">[Pg 221]</span></p> + +<p class="nind"> +<b>*22·42.</b> \(\vdash . \alpha \subset \alpha \quad[\text{Id.*10·11}]\)</p> + +<p class="nind"> +<b>*22·43.</b> \(\vdash : \alpha \cap \beta \subset \alpha \quad[\text{*3·26.*10·11}]\)</p> + +<p class="nind"> +<b>*22·44.</b> \(\vdash : \alpha \subset \beta . \beta \subset \gamma . \supset . \alpha \subset \gamma \quad[\text{*10·3}]\)</p> + +<p>This is one form of the syllogism in Barbara. Another form is the +following:</p> + +<p class="nind"> +<b>*22·441.</b> \(\vdash : \alpha \subset \beta . x \in \alpha . \supset . x \in \beta \quad[\text{*10·1.Imp}]\)</p> + +<p class="nind"> +<b>*22·45.</b> \(\vdash : \alpha \subset \beta . \alpha \subset \gamma . \equiv . \alpha \subset \beta \cap \gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash .\text{*22·1}. \supset \vdash \colon\ldotp \alpha \subset \beta . \alpha \subset \gamma . &\equiv : x \in \alpha . \supset_{x} . x \in \beta : x \in \alpha . \supset_{x} + . x \in \gamma :\\ +[\text{*10·29}] &\equiv : x \in \alpha . \supset_{x} . x \in \beta . x \in \gamma :\\ +[\text{*22·33.*10·413}] & \equiv : x \in \alpha . \supset_{x} . x \in \beta \cap \gamma \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·46.</b> \(\vdash : x \in \alpha . \alpha \subset \beta . \supset . x \in \beta \quad[\text{*22·441 . Perm}]\)</p> + +<p class="nind"> +<b>*22·47.</b> \(\vdash : \alpha \subset \gamma . \supset . \alpha \cap \beta \subset \gamma \quad[\text{*22·43·44}]\)</p> + +<p class="nind"> +<b>*22·48.</b> \(\vdash : \alpha \subset \beta . \supset . \alpha \cap \gamma \subset \beta \cap \gamma \quad[\text{*10·31}]\)</p> + +<p class="nind"> +<b>*22·481.</b> \(\vdash : \alpha = \beta . \supset . \alpha \cap \gamma = \beta \cap \gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·41}. \supset \colon\ldotp \text{Hp} . &\supset : \alpha \subset \beta . \beta \subset \alpha :\\ +[\text{*22·48}] & \supset : \alpha \cap \gamma \subset \beta \cap \gamma . \beta \cap \gamma \subset \alpha \cap \gamma :\\ +[\text{*22·41}] &\supset : \alpha \cap \gamma = \beta \cap \gamma \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·49.</b> \(\vdash : \alpha \subset \beta . \gamma \subset \delta . \supset . \alpha \cap \gamma \subset \beta \cap \delta \quad[\text{*10·39}]\)</p> + +<p class="nind"> +<b>*22·5.</b> \(\vdash . \alpha \cap \alpha = \alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·33}. \supset \vdash \colon\ldotp x \in \alpha \cap \alpha . &\equiv : x \in \alpha . x \in \alpha :\\ +[\text{*4·24}] & \equiv : x \in \alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*20·43}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The above is the law of tautology for the logical multiplication of +classes.</p> + +<p class="nind"> +<b>*22·51.</b> \(\vdash . \alpha \cap \beta = \beta \cap \alpha \quad[\text{*22·33.*4·3.*10·11.*20·43}]\)</p> + +<p class="nind"> +<b>*22·52.</b> \(\vdash . (\alpha \cap \beta) \cap \gamma = \alpha \cap (\beta \cap \gamma) \quad[\text{*22·33.*4·32.*10·11.*20·43}]\)</p> + +<p>Thus logical multiplication of classes obeys the commutative and +associative laws. References to *22·33·34·35 and to <a href="#*20·43">*20·43</a> will in +future often be omitted.</p> + +<p class="nind"> +<b>*22·53.</b> \(\alpha \cap \beta \cap \gamma = (\alpha \cap \beta) \cap \gamma \quad\text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets.</p> + +<p class="nind"> +<b>*22·54.</b> \(\vdash \colon\ldotp \alpha = \beta . \supset : \alpha \subset \gamma . \equiv . \beta \subset \gamma \quad[\text{*20·18}]\)</p> + +<p class="nind"> +<b>*22·55.</b> \(\vdash \colon\ldotp \alpha = \beta . \supset : \gamma \subset \alpha . \equiv . \gamma \subset \beta \quad[\text{*20·18}]\)</p> + +<p class="nind"> +<b>*22·551.</b> \(\vdash : \alpha = \beta . \supset . \alpha \cup \gamma = \beta \cup \gamma \quad[\text{*10·411}]\)</p> + +<p class="nind"> +<b>*22·56.</b> \(\vdash . \alpha \cup \alpha = \alpha \quad[\text{*4·25.*10·11}]\)</p> + +<p><span class="pagenum" id="Page_222">[Pg 222]</span></p> + +<p>The above is the law of tautology for the logical addition of classes.</p> + +<p class="nind"> +<b>*22·57.</b> \(\vdash.\alpha\cup\beta=\beta\cup\alpha \quad[\text{*4·31.*10·11}]\)</p> + +<p class="nind"> +<b>*22·58.</b> \(\vdash.\alpha\subset\alpha\cup\beta.\beta\subset\alpha\cup\beta \quad[\text{*1·3.*2·2}]\)</p> + +<p class="nind"> +<b>*22·59.</b> \(\vdash:\alpha\subset\gamma.\beta\subset\gamma.\equiv.\alpha\cup\beta\subset\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·1}.\supset\vdash\colon\colon\text{Hp}.&\equiv\colon\ldotp x\in\alpha.\supset_{x}.x\in\gamma:x\in\beta.\supset_{x}.x\in\gamma\colon\ldotp\\ +[\text{*10*22}] &\equiv\colon\ldotp(x)\colon\ldotp x\in\alpha.\supset.x\in\gamma:x\in\beta.\supset.x\in\gamma\colon\ldotp\\ +[\text{*4·77.*10·271}] &\equiv\colon\ldotp(x)\colon\ldotp x\in\alpha.\lor.x\in\beta:\supset.x\in\gamma\colon\ldotp\\ +[\text{*22·34.*10·413}] &\equiv\colon\ldotp(x):x\in\alpha\cup\beta.\supset.x\in\gamma\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The analogue of <a href="#*4·78">*4·78</a>, <i>i.e.</i> +\[ +\alpha\subset\beta.\lor.\alpha\subset\gamma:\equiv.\alpha\subset\beta\cup\gamma +\] +is false. We have only +\[ +\alpha\subset\beta.\lor.\alpha\subset\gamma:\supset.\alpha\subset\beta\cup\gamma +\]</p> + +<p>A similar remark applies to the analogue of <a href="#*4·79">*4·79</a>. Cf. *22·64·65.</p> + +<p class="nind"> +<b><a id="*22·6">*22·6</a>.</b> \(\vdash\colon\ldotp x\in\alpha\cup\beta.\equiv:\alpha\subset\gamma.\beta\subset\gamma.\supset_{\gamma}.x\in\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·59}.&\supset\vdash\colon\ldotp\alpha\subset\gamma.\beta\subset\gamma.\supset:x\in\alpha\cup\beta.\supset.x\in\gamma\colon\ldotp\\ +[\text{Comm}] &\supset\vdash\colon\ldotp x\in\alpha\cup\beta.\supset:\alpha\subset\gamma.\beta\subset\gamma.\supset.x\in\gamma\colon\ldotp\\ +[\text{*10·11·21}]&\supset\vdash\colon\ldotp x\in\alpha\cup\beta.\supset:\alpha\subset\gamma.\beta\subset\gamma.\supset_{\gamma}.x\in\gamma &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp\alpha\subset\gamma.\beta\subset\gamma.\supset_{\gamma}.x\in\gamma:\supset:\alpha\subset\alpha\cup\beta.\beta\subset\alpha\cup\beta.\supset.x\in\alpha\cup\beta:\\ +[\text{*22·58}] &\supset:x\in\alpha\cup\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·61.</b> \(\vdash:\alpha\subset\beta.\supset.\alpha\subset\beta\cup\gamma \quad[\text{*22*44·58}]\)</p> + +<p class="nind"> +<b><a id="*22·62">*22·62</a>.</b> \(\vdash:\alpha\subset\beta.\equiv.\alpha\cup\beta=\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·72}. \supset\vdash\colon\colon x\in\alpha.\supset.x\in\beta:&\equiv\colon\ldotp x\in\alpha.\lor.x\in\beta:\equiv.x\in\beta\colon\ldotp\\ +[\text{*22·34}] &\equiv\colon\ldotp x\in\alpha\cup\beta.\equiv.x\in\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·271}.\supset\vdash\colon\colon\alpha\subset\beta.&\equiv\colon\ldotp x\in\alpha\cup\beta.\equiv_{x}.x\in\beta\colon\ldotp\\ +[\text{*20·43}] &\equiv\colon\ldotp\alpha\cup\beta=\beta\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b><a id="*22·621">*22·621</a>.</b> \(\vdash:\alpha\subset\beta.\equiv.\alpha\cap\beta=\alpha \quad[\text{*4·71}]\)</p> + +<p>The proof proceeds as in <a href="#*22·62">*22·62</a>. The proposition <a href="#*22·621">*22·621</a> is one of the +most useful propositions in the present number.</p> + +<p class="nind"><b><a id="*22·63">*22·63</a>.</b> \(\vdash:\alpha\cup(\alpha\cap\beta)=\alpha \quad[\text{*4·44}]\)</p> + +<p>The process of obtaining <a href="#*22·63">*22·63</a> from <a href="#*4·44">*4·44</a> is of the same kind as the +process employed in the proofs that have been written out in this +number.<span class="pagenum" id="Page_223">[Pg 223]</span> Hence only *4·44 is referred to. We shall similarly restrict +references for later propositions in this number. The process is always +roughly as follows: \(p\), \(q\), \(r\) are replaced by \(x\in\alpha\), +\(x\in\beta\), \(x\in\gamma\); then <a href="#*10·11">*10·11</a> is applied, and such further +propositions of <a href="#*10">*10</a> as may be required, together with *22·33·34·35.</p> + +<p class="nind"> +<b>*22·631.</b> \(\vdash.\alpha\cap(\alpha\cup\beta)=\alpha \quad[\text{*22·58·621}]\)</p> + +<p class="nind"> +<b>*22·632.</b> \(\vdash:\alpha=\beta.\supset.\alpha=\alpha\cap\beta \quad[\text{*22·42·621}]\)</p> + +<p class="nind"> +<b>*22·633.</b> \(\vdash:\alpha\subset\beta.\supset.\alpha\cup\gamma=(\alpha\cap\beta)\cup\gamma \quad[\text{*22·551·621}]\)</p> + +<p class="nind"> +<b>*22·64.</b> \(\vdash\colon\ldotp\alpha\subset\gamma.\lor.\beta\subset\gamma:\supset.\alpha\cap\beta\subset\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·47·51}.&\supset\vdash:\alpha\subset\gamma.\supset.\alpha\cap\beta\subset\gamma:\beta\subset\gamma.\supset.\alpha\cap\beta\subset\gamma &\qquad \text{(1)}\\ +\vdash.\text{(1).*4·77}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The converse of this proposition does not hold, because the converse of +<a href="#*10·41">*10·41</a> does not hold.</p> + +<p class="nind"> +<b>*22·65.</b> \(\vdash\colon\ldotp\alpha\subset\beta.\lor.\alpha\subset\gamma:\supset.\alpha\subset\beta\cup\gamma \quad[\text{*22·61·57.*4·77}]\)</p> + +<p>Here again the converse is untrue.</p> + +<p class="nind"> +<b>*22·66.</b> \(\vdash:\alpha\subset\beta.\supset.\alpha\cup\gamma\subset\beta\cup\gamma \quad[\text{*2·38}]\)</p> + +<p class="nind"> +<b>*22·68.</b> \(\vdash.(\alpha\cap\beta)\cup(\alpha\cap\gamma)=\alpha\cap(\beta\cup\gamma)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·34}.\supset\vdash\colon\colon x\in\{(\alpha\cap\beta)\cup(\alpha\cap\gamma)\}.&\equiv\colon\ldotp x\in\alpha\cap\beta.\lor.x\in\alpha\cap\gamma\colon\ldotp\\ +[\text{*22·33}] &\equiv\colon\ldotp x\in\alpha.x\in\beta.\lor.x\in\alpha.x\in\gamma\colon\ldotp\\ +[\text{*4·4}] &\equiv\colon\ldotp x\in\alpha:x\in\beta.\lor.x\in\gamma\colon\ldotp\\ +[\text{*22·34}] &\equiv\colon\ldotp x\in\alpha.x\in\beta\cup\gamma\colon\ldotp\\ +[\text{*22·33}] &\equiv\colon\ldotp x\in\alpha\cap(\beta\cup\gamma) &\qquad \text{(1)}\\ +\vdash.(1).*10·11.*20·43.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·69.</b> \(\vdash.(\alpha\cup\beta)\cap(\alpha\cup\gamma)=\alpha\cup(\beta\cap\gamma) \quad[\text{Similar proof, by *4·41}]\)</p> + +<p>The above propositions *22·68·69 are the two forms of the distributive +law. Note that either results from the other by interchanging the signs +of addition and multiplication.</p> + +<p class="nind"> +<b>*22·7.</b> \(\vdash.(\alpha\cup\beta)\cup\gamma=\alpha\cup(\beta\cup\gamma) \quad[\text{*4·33}]\)</p> + +<p class="nind"> +<b>*22·71.</b> \(\alpha\cup\beta\cup\gamma=(\alpha\cup\beta)\cup\gamma \quad\text{Df}\)</p> + +<p class="nind"> +<b>*22·72.</b> \(\vdash:\alpha\subset\gamma.\beta\subset\delta.\supset.\alpha\cup\beta\subset\gamma\cup\delta \quad[\text{*3·48}]\)</p> + +<p class="nind"> +<b>*22·73.</b> \(\vdash:\alpha=\gamma.\beta=\delta.\supset.\alpha\cup\beta=\gamma\cup\delta \quad[\text{*10·411}]\)</p> + +<p class="nind"> +<b>*22·74.</b> \(\vdash:\alpha\cap\beta\subset\gamma.\alpha\cap\gamma\subset\beta.\equiv.\alpha\cap\beta=\alpha\cap\gamma\)</p> + +<p><span class="pagenum" id="Page_224">[Pg 224]</span></p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*22·43.*4·73}. \supset\vdash:\alpha\cap\beta\subset\gamma.&\equiv.\alpha\cap\beta\subset\alpha.\alpha\cap\beta\subset\gamma.\\ +[\text{*22·45}] &\equiv.\alpha\cap\beta\subset\alpha\cap\gamma &\qquad \text{(1)}\\ +\vdash.\text{(1)}\, \frac{\gamma,\,\beta}{\beta,\,\gamma}. \supset\vdash:\alpha\cap\gamma\subset\beta.&\equiv.\alpha\cap\gamma\subset\alpha\cap\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·38}. \supset\vdash:\alpha\cap\beta\subset\gamma.\alpha\cap\gamma\subset\beta.&\equiv.\alpha\cap\beta\subset\alpha\cap\gamma.\alpha\cap\gamma\subset\alpha\cap\beta.\\ +[\text{*22·41}] &\equiv.\alpha\cap\beta=\alpha\cap\gamma:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·8.</b> \(\vdash.-(-\alpha)=\alpha \quad[\text{*4·13}]\)</p> + +<p class="nind"> +<b>*22·81.</b> \(\vdash:\alpha\subset\beta.\equiv.-\beta\subset-\alpha \quad[\text{*4·1}]\)</p> + +<p class="nind"> +<b>*22·811.</b> \(\vdash:\alpha\subset-\beta.\equiv.\beta\subset-\alpha \quad[\text{*4·1.*22·8}]\)</p> + +<p class="nind"> +<b>*22·82.</b> \(\vdash:\alpha\cap\beta\subset\gamma.\equiv.\alpha-\gamma\subset-\beta \quad[\text{*4·14}]\)</p> + +<p class="nind"> +<b>*22·83.</b> \(\vdash:\alpha=\beta.\equiv.-\alpha=-\beta \quad[\text{*4·11}]\)</p> + +<p class="nind"> +<b>*22·831.</b> \(\vdash:\alpha=-\beta.\equiv.\beta=-\alpha \quad[\text{*4·12}]\)</p> + +<p class="nind"> +<b>*22·84.</b> \(\vdash.-(\alpha\cap\beta)=-\alpha\lor-\beta \quad[\text{*4·51}]\)</p> + +<p class="nind"> +<b>*22·85.</b> \(\vdash.\alpha\cap\beta=-(-\alpha\lor-\beta) \quad[\text{*22·84·831}]\)</p> + +<p class="nind"> +<b>*22·86.</b> \(\vdash.-(-\alpha\cap-\beta)=\alpha\cup\beta \quad[\text{*4·57}]\)</p> + +<p class="nind"> +<b>*22·87.</b> \(\vdash.-\alpha\cap-\beta=-(\alpha\cup\beta) \quad[\text{*22·86·831}]\)</p> + +<p>*22·84·85·86·87 are De Morgan's formulae.</p> + +<p class="nind"> +<b>*22·88.</b> \(\vdash.(x).x\in(\alpha\cup-\alpha) \quad[\text{*2·11}]\)</p> + +<p>This is a form of the law of excluded middle.</p> + +<p class="nind"> +<b>*22·89.</b> \(\vdash.(x).x{\sim}\in(\alpha-\alpha) \quad[\text{*3·24}]\)</p> + +<p>This is a form of the law of contradiction.</p> + +<p class="nind"> +<b>*22·9.</b> \(\vdash.(\alpha\cup\beta)-\beta=\alpha-\beta \quad[\text{*5·61}]\)</p> + +<p class="nind"> +<b>*22·91.</b> \(\vdash.\alpha\cup\beta=\alpha\cup(\beta-\alpha)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*5·63}. \supset\vdash\colon\ldotp x\in\alpha.\lor.x\in\beta:&\equiv:x\in\alpha.\lor.x\in\beta.x{\sim}\in\alpha\colon\ldotp\\ +[\text{*22·33·34·35}] \supset\vdash\colon\ldotp x\in\alpha\cup\beta.&\equiv:x\in\alpha.\lor.x\in(\beta-\alpha):\\ +[\text{*22·34}] &\equiv:x\in\alpha\cup(\beta-\alpha) &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*20·43}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"> +<b>*22·92.</b> \(\vdash:\alpha\subset\beta.\supset.\beta=\alpha\cup(\beta-\alpha) \quad[\text{*22·91·62}]\)</p> + +<p class="nind"> +<b>*22·93.</b> \(\vdash.\alpha-\beta=\alpha-(\alpha\cap\beta)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·73.Transp}. \supset\vdash\colon\ldotp x\in\alpha.\supset:x{\sim}\in\beta.&\equiv.{\sim}(x\in\alpha.x\in\beta).\\ +[\text{*22.33}] &\equiv.x{\sim}\in(\alpha\cap\beta)\colon\ldotp\\ +[\text{*5·32}] &\supset\vdash\colon\ldotp x\in\alpha.x{\sim}\in\beta.\equiv.x\in\alpha.x{\sim}\in(\alpha\cap\beta)\colon\ldotp\\ +[\text{*22·35·33}] &\supset\vdash:x\in\alpha-\beta.\equiv.x\in\alpha-(\alpha\cap\beta):\\ +[\text{*10·11.*20·43}]&\supset\vdash.\alpha-\beta=\alpha-(\alpha\cap\beta).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_225">[Pg 225]</span></p> + +<p class="nind"> +<b>*22·94.</b> \(\vdash:(\alpha).f\alpha.\equiv.(\alpha).f(-\alpha)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash:(\alpha).f\alpha.\supset.f(-\alpha)\\ +[\text{*10·11·21}] &\supset\vdash:(\alpha).f\alpha.\supset.(\alpha).f(-\alpha) &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash:(\alpha).f(-\alpha).\supset.f\{-(-\alpha)\}.\\ +[\text{*22·8.*20·18}] &\supset.f\alpha\\ +[\text{*10·11·21}] &\supset\vdash:(\alpha).f(-\alpha).\supset.(\alpha).f\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in connection with mathematical induction, in +<a href="#*90·102">*90·102</a>, which is required for the proof of <a href="#*90·132">*90·132</a>, which is one of +the fundamental propositions in the theory of mathematical induction.</p> + +<p class="nind"> +<b>*22·95.</b> \(\vdash:(\exists \alpha).f\alpha.\equiv.(\exists \alpha).f(-\alpha)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash:\text{*22·94}.\supset\vdash:(\alpha).{\sim}f\alpha.\equiv.(\alpha).{\sim}f(-\alpha) &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp.*20·6}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_54" href="#FNanchor_54" class="label">[54]</a> +<i>Trans. Amer. Math. Soc.</i> Vol. 5, July 1904, p. 292.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_226">[Pg 226]</span></p> +<h2 class="nobreak" id="*23">*23. CALCULUS OF RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *23.</p> + +<p>The definitions and propositions of this number are to be exact +analogues of those of <a href="#*22">*22</a>. Properties of relations which have no +analogues for classes will not be dealt with till Section D. Proofs +will be omitted in the present number, as they are precisely analogous +to those of analogous propositions in *22. In this number, as always +in future, capital Latin letters stand for expressions of the form +\(\hat{x}\hat{y}{\phi}!(x, y)\), or, where they are not being used as +apparent variables, for \(\hat{x}\hat{y}{\phi}(x, y)\). The principal +propositions of this number are the analogues of those of *22.</p> + +<hr class="tb"> + +<p class="nind"> +<b>*23·01.</b> \(R \unicode{x2abd} S.=:xRy.\supset_{x,y}.xSy \quad \text{Df}\)</p> + +<p class="nind"> +<b>*23·02.</b> \(\dot{\cap} S = \hat{x}\hat{y}(xRy.xSy) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*23·03.</b> \(R \unicode{x228d} S = \hat{x}\hat{y}(xRy .\lor. xSy) \quad \text{Df}\)</p> + +<p class="nind"> +<b>*23·04.</b> \(\unicode{x2238} R = \hat{x}\hat{y}\{{\sim}(xRy)\} \quad \text{Df}\)</p> + +<p class="nind"> +<b>*23·05.</b> \(R \unicode{x2238}S = R \dot{\cap} \unicode{x2238}S \quad \text{Df}\)</p> + +<p>Similar remarks apply to these definitions as to those of <a href="#*22">*22</a>.</p> + +<p class="nind"> +<b>*23·1.</b> \(\vdash \colon\ldotp R \unicode{x2abd} S .\equiv: xRy .\supset_{x,y}. xSy\)</p> + +<p class="nind"> +<b>*23·2.</b> \(\vdash . R \dot{\cap} S = \hat{x}\hat{y}(xRy . xSy)\)</p> + +<p class="nind"> +<b>*23·3.</b> \(\vdash . R \unicode{x228d} S = \hat{x}\hat{y}(xRy .\lor. xSy)\)</p> + +<p class="nind"> +<b>*23·31.</b> \(\vdash . \unicode{x2238}R = \hat{x}\hat{y}\{{\sim}(xRy)\}\)</p> + +<p class="nind"> +<b>*23·32.</b> \(\vdash . R \unicode{x2238} S = \hat{x}\hat{y}\{xRy . {\sim}(xSy)\}\)</p> + +<p class="nind"> +<b>*23·33.</b> \(\vdash :x(R \dot{\cap} S)y .\equiv. xRy . xSy\)</p> + +<p class="nind"> +<b>*23·34.</b> \(\vdash \colon\ldotp x(R \unicode{x228d} S)y .\equiv: x R y .\lor. x S y\)</p> + +<p class="nind"> +<b>*23·35.</b> \(\vdash : x \unicode{x2238}R y .\equiv. {\sim}(x R y)\)</p> + +<p class="nind"> +<b><a id="*23·351">*23·351</a>.</b> \(\vdash . \unicode{x2238}R \neq R\)</p> + +<p class="nind"> +<b>*23·36.</b> \(\vdash . R \dot{\cap} S \in \text{Rel}\)</p> + +<p class="nind"> +<b>*23·37.</b> \(\vdash . R \unicode{x228d} S \in \text{Rel}\)</p> + +<p class="nind"> +<b>*23·38.</b> \(\vdash . \unicode{x2238}R \in \text{Rel}\)</p> + +<p><span class="pagenum" id="Page_227">[Pg 227]</span></p> + +<p class="nind"> +<b>*23·39.</b> \(\vdash.\hat{x}\hat{y}\phi(x,y)\dot{\cap}\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\{\phi(x,y).\psi(x,y)\}\)</p> + +<p class="nind"> +<b>*23·391.</b> \(\vdash.\hat{x}\hat{y}\phi(x,y)\unicode{x228d}\hat{x}\hat{y}\psi(x,y)=\hat{x}\hat{y}\{\phi(x,y).\lor.\psi(x,y)\}\)</p> + +<p class="nind"> +<b>*23·392.</b> \(\vdash.\dot{-}\hat{x}\hat{y}\phi(x,y)=\hat{x}\hat{y}\{{\sim}\phi(x,y)\}\)</p> + +<p class="nind"> +<b>*23·4.</b> \(\vdash\colon\ldotp R\unicode{x2abd}S.S\unicode{x2abd}R.\equiv:xRy.\equiv_{x,y}.xSy\)</p> + +<p class="nind"> +<b>*23·41.</b> \(\vdash:R\unicode{x2abd}S.S\unicode{x2abd}R.\equiv.R=S\)</p> + +<p class="nind"> +<b>*23·42.</b> \(\vdash.R\unicode{x2abd}R\)</p> + +<p class="nind"> +<b>*23·43.</b> \(\vdash.R\dot{\cap}S\unicode{x2abd}R\)</p> + +<p class="nind"> +<b>*23·44.</b> \(\vdash:R\unicode{x2abd}S.S\unicode{x2abd}T.\supset.R\unicode{x2abd}T\)</p> + +<p class="nind"> +<b>*23·441.</b> \(\vdash:R\unicode{x2abd}S.xRy.\supset.xSy\)</p> + +<p class="nind"> +<b>*23·45.</b> \(\vdash:R\unicode{x2abd}S.R\unicode{x2abd}T.\supset.R\unicode{x2abd}S\dot{\cap}T\)</p> + +<p class="nind"> +<b>*23·46.</b> \(\vdash:xRy.R\unicode{x2abd}S.\supset.xSy\)</p> + +<p class="nind"> +<b>*23·47.</b> \(\vdash:R\unicode{x2abd}T.\supset.R\dot{\cap}S\unicode{x2abd}T\)</p> + +<p class="nind"> +<b>*23·48.</b> \(\vdash:R\unicode{x2abd}S.\supset.R\dot{\cap}T\unicode{x2abd}S\dot{\cap}T\)</p> + +<p class="nind"> +<b>*23·481.</b> \(\vdash:R=S.\supset.R\dot{\cap}T=S\dot{\cap}T\)</p> + +<p class="nind"> +<b>*23·49.</b> \(\vdash:P\unicode{x2abd}Q.R\unicode{x2abd}S.\supset.P\dot{\cap}R\unicode{x2abd}Q\dot{\cap}S\)</p> + +<p class="nind"> +<b>*23·5.</b> \(\vdash.R\dot{\cap}R=R\)</p> + +<p class="nind"> +<b>*23·51.</b> \(\vdash.R\dot{\cap}S=S\dot{\cap}R\)</p> + +<p class="nind"> +<b>*23·52.</b> \(\vdash.(R\dot{\cap}S)\dot{\cap}T=R\dot{\cap}(S\dot{\cap}T)\)</p> + +<p class="nind"> +<b>*23·53.</b> \(R\dot{\cap}S\dot{\cap}T=(R\dot{\cap}S)\dot{\cap}T \quad\text{Df}\)</p> + +<p class="nind"> +<b>*23·54.</b> \(\vdash\colon\ldotp R=S.\supset:R\unicode{x2abd}T.\equiv.S\unicode{x2abd}T\)</p> + +<p class="nind"> +<b>*23·55.</b> \(\vdash\colon\ldotp R=S.\supset:T\unicode{x2abd}R.\equiv.T\unicode{x2abd}S\)</p> + +<p class="nind"> +<b>*23·551.</b> \(\vdash:R=S.\supset.R\unicode{x228d}T=S\unicode{x228d}T\)</p> + +<p class="nind"> +<b>*23·56.</b> \(\vdash.R\unicode{x228d}R=R\)</p> + +<p class="nind"> +<b>*23·57.</b> \(\vdash.R\unicode{x228d}S=S\unicode{x228d}R\)</p> + +<p class="nind"> +<b>*23·58.</b> \(\vdash.R\unicode{x2abd}R\unicode{x228d}S.S\unicode{x2abd}R\unicode{x228d}S\)</p> + +<p class="nind"> +<b>*23·59.</b> \(\vdash:R\unicode{x2abd}T.S\unicode{x2abd}T.\equiv.R\unicode{x228d}S\unicode{x2abd}T\)</p> + +<p class="nind"> +<b>*23·6.</b> \(\vdash\colon\ldotp x(R\unicode{x228d}S)y.\equiv:R\unicode{x2abd}T.S\unicode{x2abd}T.\supset_{T}.xTy\)</p> + +<p class="nind"> +<b>*23·61.</b> \(\vdash:R\unicode{x2abd}S.\supset.R\unicode{x2abd}S\unicode{x228d}T\)</p> + +<p class="nind"> +<b>*23·62.</b> \(\vdash:R\unicode{x2abd}S.\equiv.R\unicode{x228d}S=S\)</p> + +<p class="nind"> +<b>*23·621.</b> \(\vdash:R\unicode{x2abd}S.\equiv.R\dot{\cap}S=R\)</p> + +<p class="nind"> +<b>*23·63.</b> \(\vdash.R\unicode{x228d}(R\dot{\cap}S)=R\)</p> + +<p class="nind"> +<b>*23·631.</b> \(\vdash.R\dot{\cap}(R\unicode{x228d}S)=R\)</p> + +<p class="nind"> +<b>*23·632.</b> \(\vdash:R=S.\supset.R=R\dot{\cap}S\)</p> + +<p class="nind"> +<b>*23·633.</b> \(\vdash:R\unicode{x2abd}S.\supset.R\unicode{x228d}T=(R\dot{\cap}S)\unicode{x228d}T\)</p> + +<p><span class="pagenum" id="Page_228">[Pg 228]</span></p> + +<p class="nind"> +<b>*23·64.</b> \(\vdash\colon\ldotp R\unicode{x2abd}T.\lor.S\unicode{x2abd}T:\supset.R\dot{\cap}S\unicode{x2abd}T\)</p> + +<p class="nind"> +<b>*23·65.</b> \(\vdash\colon\ldotp R\unicode{x2abd}S.\lor.R\unicode{x2abd}T:\supset.R\unicode{x2abd}S\unicode{x228d}T\)</p> + +<p class="nind"> +<b>*23·66.</b> \(\vdash:R\unicode{x2abd}S.\supset.R\unicode{x228d}T\unicode{x2abd}S\unicode{x228d}T\)</p> + +<p class="nind"> +<b>*23·68.</b> \(\vdash.(R\dot{\cap}S)\unicode{x228d}(R\dot{\cap}T)=R\dot{\cap}(S\unicode{x228d}T)\)</p> + +<p class="nind"> +<b>*23·69.</b> \(\vdash.(R\unicode{x228d}S)\dot{\cap}(R\unicode{x228d}T)=R\unicode{x228d}(S\dot{\cap}T)\)</p> + +<p class="nind"> +<b>*23·7.</b> \(\vdash.(R\unicode{x228d}S)\unicode{x228d}T=R\unicode{x228d}(S\unicode{x228d}T)\)</p> + +<p class="nind"> +<b>*23·71.</b> \(R\unicode{x228d}S\unicode{x228d}T=(R\unicode{x228d}S)\unicode{x228d}T \quad{\text{Df}}\)</p> + +<p class="nind"> +<b>*23·72.</b> \(\vdash:P\unicode{x2abd}R.Q\unicode{x2abd}S.\supset.P\unicode{x228d}Q\unicode{x2abd}R\unicode{x228d}S\)</p> + +<p class="nind"> +<b>*23·73.</b> \(\vdash:P=R.Q=S.\supset.P\unicode{x228d}Q=R\unicode{x228d}S\)</p> + +<p class="nind"> +<b>*23·74.</b> \(\vdash:P\dot{\cap}Q\unicode{x2abd}R.P\dot{\cap}R\unicode{x2abd}Q.\equiv.P\dot{\cap}Q=P\dot{\cap}R\)</p> + +<p class="nind"> +<b>*23·8.</b> \(\vdash.\dot{-}(\dot{-}R)=R\)</p> + +<p class="nind"> +<b>*23·81.</b> \(\vdash:R\unicode{x2abd}S.\equiv.\dot{-}S\unicode{x2abd}\dot{-}R\)</p> + +<p class="nind"> +<b>*23·811.</b> \(\vdash:R\unicode{x2abd}\dot{-}S.\equiv.S\unicode{x2abd}\dot{-}R\)</p> + +<p class="nind"> +<b>*23·82.</b> \(\vdash:R\dot{\cap}S\unicode{x2abd}T.\equiv.R\dot{-}T\unicode{x2abd}\dot{-}S\)</p> + +<p class="nind"> +<b>*23·83.</b> \(\vdash:R=S.\equiv.\dot{-}R=\dot{-}S\)</p> + +<p class="nind"> +<b>*23·831.</b> \(\vdash:R=\dot{-}S.\equiv.S=\dot{-}R\)</p> + +<p class="nind"> +<b>*23·84.</b> \(\vdash.\dot{-}(R\dot{\cap}S)=\dot{-}R\unicode{x228d}\dot{-}S\)</p> + +<p class="nind"> +<b>*23·85.</b> \(\vdash.R\dot{\cap}S=\dot{-}(\dot{-}R\unicode{x228d}\dot{-}S)\)</p> + +<p class="nind"> +<b>*23·86.</b> \(\vdash.\dot{-}(\dot{-}R\dot{\cap}\dot{-}S)=R\unicode{x228d}S\)</p> + +<p class="nind"> +<b>*23·87.</b> \(\vdash.\dot{-}R\dot{\cap}\dot{-}S=\dot{-}(R\unicode{x228d}S)\)</p> + +<p class="nind"> +<b>*23·88.</b> \(\vdash.(x,y).x(R\unicode{x228d}\dot{-}R)y\)</p> + +<p class="nind"> +<b>*23·89.</b> \(\vdash.(x,y).{\sim}\{x(R\dot{-}R)y\}\)</p> + +<p class="nind"> +<b>*23·9.</b> \(\vdash.(R\unicode{x228d}S)\dot{-}S=R\dot{-}S\)</p> + +<p class="nind"> +<b>*23·91.</b> \(\vdash.R\unicode{x228d}S=R\unicode{x228d}(S\dot{-}R)\)</p> + +<p class="nind"> +<b>*23·92.</b> \(\vdash:R\unicode{x2abd}S.\supset.S=R\unicode{x228d}(S\dot{-}R)\)</p> + +<p class="nind"> +<b>*23·93.</b> \(\vdash.R\dot{-}S=R\dot{-}(R\dot{\cap}S)\)</p> + +<p class="nind"> +<b>*23·94.</b> \(\vdash:(R).fR.\equiv.(R).f(\dot{-}R)\)</p> + +<p class="nind"> +<b>*23·95.</b> \(\vdash:(\exists R).fR.\equiv.(\exists R).f(\dot{-}R)\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_229">[Pg 229]</span></p> + +<h2 class="nobreak" id="*24">*24. THE UNIVERSAL CLASS, THE NULL-CLASS, AND THE EXISTENCE +OF CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *24.</p> + +<p>The universal class, denoted by \(\text{V}\), is the class of all +objects of the type which, in the given context, is being denoted by +small Latin letters, <i>i.e.</i> of the lowest type concerned. Thus +\(\text{V}\), like "\(\text{Cls}\)," is ambiguous as to type. Its +definition is as follows:</p> + +<p class="nind"><b>*24·01.</b> \(\text{V} = \hat{x}(x = x) \quad \text{Df}\)</p> + +<p>Any other property possessed by everything would do as well as "\(x = x\)," +but this is the only such property which we have hitherto studied.</p> + +<p>The null-class, denoted by \(\Lambda\), is the class which has no +members. Like \(\text{V}\), it is ambiguous as to type. We use the +same symbol, \(\Lambda\), for null-classes of various types; but these +null-classes differ. The type of \(\Lambda\) is determined by that of +the terms \(x\) concerning which "\(x \in \Lambda\)" is false: whatever +\(x\) may be, "\(x \in \Lambda\)" will not represent a <i>true</i> +proposition, but unless \(x\) is of the appropriate type, "\(x \in \Lambda\)" +will be meaningless, not false. Thus \(\Lambda\) is of the +type next above that of an \(x\) concerning which "\(x \in \Lambda\)" +is significant and false. The definition of \(\Lambda\) is</p> + +<p class="nind"><b>*24·02.</b> \(\Lambda = -\text{V} \quad \text{Df}\)</p> + +<p>When a class \(\alpha\) is not null, so that it has one or more +members, it is said to <i>exist</i>. (This sense of "existence" must +not be confused with that defined in <a href="#*14·02">*14·02</a>.) We write "\(\exists!\alpha\)" +for "\(\alpha\) exists." The definition is</p> + +<p class="nind"><b>*24·03.</b> \(\exists! \alpha .=. (\exists x) . x \in \alpha \quad \text{Df}\)</p> + +<p>In the present number, we shall deal first with the properties of +\(\Lambda\) and \(\text{V}\), then with those of existence. In +comparing the algebra of symbolic logic with ordinary algebra, +\(\Lambda\) takes the place of \(0\), while \(\text{V}\) combines the +properties of \(1\) and of \(\infty\).</p> + +<p>Among the more important properties of \(\Lambda\) and \(\text{V}\) which are +proved in this number are the following:</p> + +<p class="nind"><b><a id="*24·1">*24·1</a>.</b> \(\vdash . \Lambda \neq \text{V}\)</p> + +<p><i>I.e.</i> "nothing is not everything." This is useful as giving us +the existence of at least two classes. If the monistic philosophers +were right in maintaining that only one individual exists, there would +be only two classes,<span class="pagenum" id="Page_230">[Pg 230]</span> \(\Lambda\) and \(\text{V}\), \(\text{V}\) being +(in that case) the class whose only member is the one individual. Our +primitive propositions do not require the existence of more than one +individual.</p> + +<p class="nind"><b>*24·102·103</b> show that any function which is always true +determines the universal class, and any function which is always false +determines the null-class.</p> + +<p class="nind"><b>*24·21·22</b> give forms of the laws of contradiction and excluded +middle, namely "nothing is both \(\alpha\) and not-\(\alpha\)" +(\(\alpha\cap - \alpha = \Lambda)\) and "everything is either +\(\alpha\) or not-\(\alpha\)" (\(\alpha\cap - \alpha = \text{V})\).</p> + +<p class="nind"><b>*24·23·24·26·27</b> give the properties of \(\text{V}\) and +\(\Lambda\) with respect to addition and multiplication, namely: +multiplication by \(\text{V}\) and addition of \(\Lambda\) make +no change in a class (*24·26·24); addition of \(\text{V}\) gives +\(\text{V}\), and multiplication by \(\Lambda\) gives \(\Lambda\) +(*24·27·23). It will be observed that the properties of \(\Lambda\) +and \(\text{V}\) result from each other by interchanging addition and +multiplication.</p> + +<p class="nind"><b><a id="*24·3">*24·3</a>.</b> \(\vdash\colon \alpha\subset\beta.\equiv.\alpha - \beta = \Lambda\)</p> + +<p><i>I.e.</i> "\(\alpha\) is contained in \(\beta\)" is equivalent to +"nothing is \(\alpha\) but not \(\beta\)."</p> + +<p class="nind"><b>*24·311.</b> \(\vdash\colon \alpha\subset - \beta.\equiv.\alpha\cap\beta = \Lambda\)</p> + +<p><i>I.e.</i> "no \(\alpha\) is a \(\beta\)" is equivalent to "nothing is +both \(\alpha\) and \(\beta\)."</p> + +<p class="nind"><b>*24·411.</b> \(\vdash\colon\beta\subset\alpha.\supset\alpha = \beta\cup(\alpha - \beta)\)</p> + +<p class="nind"><b>*24·43.</b> \(\vdash\colon\alpha - \beta\subset\gamma.\equiv.\alpha\subset\beta\cup\gamma\)</p> + +<p>As a rule, propositions concerning \(\text{V}\) are much less used than +the correlative propositions concerning \(\Lambda\).</p> + +<p>The properties of the existence of classes result from those of +\(\Lambda\), owing to the fact that \(\exists!\alpha\) is the +contradictory of \(\alpha = \Lambda\), as is proved in <a href="#*24·54">*24·54</a>. Thus we +have, in virtue of <a href="#*24·3">*24·3</a>,</p> + +<p class="nind"><b>*24·55.</b> \(\vdash\colon{\sim}(\alpha\subset\beta).\equiv.\exists!\alpha - \beta\)</p> + +<p><i>I.e.</i> "not all \(\alpha\)'s are \(\beta\)'s" is equivalent +to "there are \(\alpha\)'s which are not \(\beta\)'s." This is the +familiar proposition of formal logic, that the contradictory of the +universal affirmative is the particular negative.</p> + +<p>We have</p> + +<p class="nind"><b>*24·56.</b> \(\vdash\colon\ldotp \exists!(\alpha\cup\beta).\equiv:\exists!\alpha.\lor.\exists!\beta\)</p> + +<p class="nind"><b>*24·561.</b> \(\vdash\colon\exists!(\alpha\cap\beta).\supset.\exists!\alpha.\exists!\beta\)</p> + +<p><i>I.e.</i> if a sum exists, then one of the summands exists, and vice +versa; and if a product exists, both the factors exist (but not vice +versa).</p> + +<p>The proofs of propositions in the present number offer no difficulty.</p> + +<p><span class="pagenum" id="Page_231">[Pg 231]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*24·01.</b> \(\text{V} =\hat{x}(x=x) \quad\text{Df}\)</p> + +<p class="nind"><b>*24·02.</b> \(\Lambda =- \text{V} \quad\text{Df}\)</p> + +<p class="nind"><b>*24·03.</b> \(\exists!\alpha.=.(\exists x).x \in \alpha \quad\text{Df} \quad\text{Df}\)</p> + +<p class="nind"><b>*24·1.</b> \(\vdash.\Lambda\neq\text{V} \quad[\text{*22·351.(*24·02)}]\)</p> + +<p class="nind"><b>*24·101.</b> \(\vdash.\text{V} = -\Lambda \quad[\text{*22·831.(*24·02)}]\)</p> + +<p class="nind"><b>*24·102.</b> \(\vdash\colon(x).\phi x.\equiv.\hat{z}(\phi z) = \text{V}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*13·15.*5·501}.&\supset\vdash\colon\ldotp \phi x.\equiv\colon\phi x.\equiv.x=x\colon\ldotp\\ +&[\text{*10·11·271}] &\supset\vdash\colon\ldotp (x).\phi x.\equiv\colon(x)\colon\phi x.\equiv.x = x\colon\\ +&[\text{*20·15}] &\equiv\colon\hat{z}(\phi z)=\hat{z}(x=x)\colon\\ +&[\text{*24·01)}] &\equiv\colon\hat{z}(\phi z)=\text{V}\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Thus any function which is always true determines the universal class, +and vice versa.</p> + +<p class="nind"><b>*24·103.</b> \(\vdash\colon(x).{\sim}\phi x.\equiv.\hat{z}(\phi z)=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*24·102}.\supset\vdash\colon\ldotp(x).{\sim}\phi x.&\equiv\colon\hat{z}({\sim}\phi z)=\text{V}\colon\\ +&[\text{*22·392}] &\equiv\colon-\hat{z}(\phi z)=\text{V}\colon\\ +&[\text{*22·831}] &\equiv\colon\hat{z}(\phi z)=-\text{V}\colon\\ +&[\text{(*24·02)}] &\equiv\colon\hat{z}(\phi z)=\Lambda\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·104.</b> \(\vdash.(x).x\in\text{V}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*20·3}.\supset\vdash\colon x\in\text{V}.\equiv.x=x \qquad \text{(1)}\\ +&\vdash.(1).\text{*13·15.*10·11·271}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·105.</b> \(\vdash.(x).x{\sim}\in\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*22·35}.&\supset\vdash\colon x\in\Lambda.\equiv.x{\sim}\in\text{V}\colon\\ +&[\text{*4·12}] &\supset\vdash\colon x{\sim}\in\text{V}.\equiv.x\in\text{V} \qquad \text{(1)}\\ +&\vdash.[\text{(1).*10·11·271.*24·104}].&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·11.</b> \(\vdash.(\alpha).\alpha\subset\text{V}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +&\vdash.\text{*24·104.*10·1}.&\supset\vdash.x\in\text{V}.\\ +&[\text{Simp}] &\supset\vdash\colon x\in\alpha.\supset.x\in\text{V}\colon\\ +&[\text{*10·11.*22·1}] &\supset\vdash\colon \alpha\subset\text{V}\\ +&[\text{*10·11}] &\supset\vdash\colon(\alpha).\alpha\subset\text{V}\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·12.</b> \(\vdash.(\alpha).\Lambda\subset\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*24·105.*10·1}. \supset\vdash.x{\sim}\in\Lambda.\\ +&[\text{*2·21}] \supset\vdash\colon x\in\Lambda.\supset.x\in\alpha \qquad \text{(1)}\\ +&\vdash.(1).\text{*10·11.*22·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_232">[Pg 232]</span></p> + +<p class="nind"><b>*24·13.</b> \(\vdash\colon \alpha=\Lambda.\equiv.\alpha\subset\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·12.*4·73}.\supset\vdash\colon \alpha\subset\Lambda.&\equiv.\alpha\subset\Lambda.\Lambda\subset\alpha.\\ +[\text{*22·41}] &\equiv.\alpha = \Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·14.</b> \(\vdash\colon(x).x \in \alpha.\equiv.\alpha=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·102}.\supset\vdash\colon(x).x \in \alpha.&\equiv.\hat{x}(x \in \alpha)=\text{V}.\\ +[\text{*20·32}] &\equiv.\alpha = \text{V}\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·141.</b> \(\vdash\colon\text{V}\subset\alpha.\equiv.\text{V}=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·11.*4·73}.\supset\vdash\colon\text{V}\subset \alpha.&\equiv.\alpha\subset\text{V}.\text{V}\subset\alpha.\\ +[\text{*22·41}] &\equiv.\alpha=\text{V}\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·15.</b> \(\vdash\colon(x).x{\sim}\in\alpha.\equiv.\alpha=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·103}.\supset\vdash\colon(x).x{\sim}\in\alpha.&\equiv.\hat{x}(x \in \alpha)=\Lambda.\\ +[*20·32] &\equiv.\alpha =\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·17.</b> \(\vdash\colon \alpha=\text{V}.\equiv.- \alpha=\Lambda \quad[\text{*22·83.(*24·02)}]\)</p> + +<p class="nind"><b>*24·21.</b> \(\vdash.\alpha\cap - \alpha=\text{V} \quad[\text{*24·103.*22·89}]\)</p> + +<p class="nind"><b>*24·22.</b> \(\vdash.\alpha\cup - \alpha=\text{V} \quad[\text{*22·88.*24·102}]\)</p> + +<p class="nind"><b>*24·23.</b> \(\vdash.\alpha\cap\Lambda=\Lambda \quad[\text{*24·12.*22·621}]\)</p> + +<p class="nind"><b>*24·24.</b> \(\vdash.\alpha\cup\Lambda=\alpha \quad[\text{*24·12.*22·62}]\)</p> + +<p>The above two propositions (*24·23·24) exhibit the algebraical analogy +of \(\Lambda\) to zero.</p> + +<p class="nind"><b>*24·26.</b> \(\vdash.\alpha\cap\text{V}=\alpha \quad[\text{*22·621.*24·11}]\)</p> + +<p>This exhibits the analogy of \(\text{V}\) to 1.</p> + +<p class="nind"><b>*24·27.</b> \(\vdash.\alpha\cup\text{V}=\text{V} \quad[\text{*22·62.*24·11}]\)</p> + +<p>This exhibits the analogy of \(\text{V}\) to \(\infty\).</p> + +<p class="nind"><b>*24·3.</b> \(\vdash\colon\alpha\subset\beta.\equiv.\alpha - \beta=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·53·6}.\supset\\ +\vdash\colon\ldotp x \in \alpha.\supset.x \in \beta:&\equiv:{\sim}(x \in \alpha.x{\sim}\beta)\colon\\ +[\text{*22·35}] &\equiv\colon{\sim}(x \in \alpha.x \in - \beta)\colon\\ +[\text{*22·33}] &\equiv\colon{\sim}(x \in \alpha - \beta) \qquad\quad \text{(1)}\\ +\vdash.\text{(1).*10·11·271}.\supset\\ +\vdash\colon \alpha\subset\beta.\equiv.(x).{\sim}(x \in \alpha - \beta).\\ +[\text{*24·15}] \equiv.\alpha - \beta =\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is very frequently used.</p> + +<p><span class="pagenum" id="Page_233">[Pg 233]</span></p> + +<p class="nind"><b>*24·31.</b> \(\vdash\colon \alpha\subset\beta.\equiv.-\alpha\cup\beta=\text{V}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*4·6}. &\supset\vdash\colon\ldotp x \in \alpha.\supset.x\in \beta\colon\equiv:x{\sim}\in \alpha.\lor.x\in \beta\colon\ldotp\\ +[\text{*10·11·271}] &\supset\vdash\colon\ldotp \alpha\subset\beta.\equiv\colon(x)\colon x{\sim}\in \alpha.\lor.x\in \beta\colon\\ +[\text{*22·35}] &\equiv\colon(x):x\in - \alpha.\lor.x\in \beta\colon\\ +[\text{*22·34}] &\equiv\colon(x).x\in(-\alpha\cup\beta)\colon\\ +[\text{*24·14}] &\equiv\colon -\alpha\cup\beta=\text{V}\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is the correlative of <a href="#*24·3">*24·3</a>, but, unlike that +proposition, it is not useful in the sequel. Every proposition +concerning \(\Lambda\) has a correlative concerning \(\text{V}\), but +we shall often not give these correlatives, since they are seldom +required for subsequent proofs.</p> + +<p class="nind"><b>*24·311.</b> \(\vdash\colon \alpha\subset - \beta.\equiv.\alpha\cap\beta=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·35}.\supset\vdash\colon\ldotp x\in \alpha.\supset.x\in - \beta\colon&\equiv\colon x\in \alpha.\supset.x{\sim}\in \beta\colon\\ +[\text{*4·51·62}] &\equiv\colon{\sim}(x\in \alpha.x\in \beta)\colon\\ +[\text{*22·33}] &\equiv\colon{\sim}(x\in \alpha\cap\beta) \quad\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·271}.\supset\vdash\colon \alpha\subset - \beta.&\equiv.(x).x{\sim}\in \alpha\cap\beta.\\ +[\text{*24·15}]&\equiv.\alpha\cap\beta=\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·312.</b> \(\vdash\colon - \alpha\subset\beta.\equiv.\alpha\cup\beta=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·35}.\supset\vdash\colon\ldotp - \alpha\subset\beta.&\equiv:x{\sim}\in \alpha.\supset_{x}.x\in \beta\colon\\ +[\text{*4·64}] &\equiv\colon(x):x\in \alpha.\lor.x\in \beta\colon\\ +[\text{*22·34}] &\equiv\colon(x).x\in \alpha\cup\beta\colon\\ +[\text{*24·14}] &\equiv\colon\alpha\cup\beta=\text{V}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·313.</b> \(\vdash\colon\alpha\cap\beta=\Lambda.\equiv.\alpha=\alpha -\beta \quad[\text{*24·311.*22·621}]\)</p> + +<p class="nind"><b>*24·32.</b> \(\vdash\colon\ldotp\alpha\cup\beta=\Lambda.\equiv.\alpha=\Lambda.\beta=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·13}.\supset\vdash\colon\ldotp \alpha\cup\beta=\Lambda.&\equiv\colon\alpha\cup\beta\subset\Lambda\colon\\ +[\text{*22·59}] &\equiv\colon \alpha\subset\Lambda.\beta\subset\Lambda\colon\\ +[\text{*24·13}] &\equiv\colon \alpha=\Lambda.\beta=\Lambda\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·33.</b> \(\vdash\colon\alpha=\text{V}.\supset.\alpha\cup\beta=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·551}.\supset\vdash\colon\text{Hp}.\supset.\alpha\cup\beta&=\text{V}\cup\beta\\ +[\text{*24·27.*22·57}] &=\text{V}\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·34.</b> \(\vdash\colon \alpha=\Lambda.\supset.\alpha\cap\beta=\Lambda \quad[\text{*22·481.*24·23}]\)</p> + +<p class="nind"><b>*24·35.</b> \(\vdash\colon \alpha=\text{V}.\supset.\alpha\cap\beta=\beta \quad[\text{*22·481.*24·26}]\)</p> + +<p class="nind"><b>*24·36.</b> \(\vdash\colon \alpha=\Lambda.\supset.\alpha\cup\beta=\beta \quad[\text{*22·551.*24·24}]\)</p> + +<p><span class="pagenum" id="Page_234">[Pg 234]</span></p> + +<p class="nind"><b>*24·37.</b> \(\vdash\colon\ldotp \alpha\cap\beta=\Lambda.\equiv\colon x\in \alpha.y\in \beta.\supset_{x,y}.x\neq y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·15}.\supset\vdash\colon\ldotp \alpha\cap\beta=\Lambda.&\equiv\colon(x).x{\sim}\in(\alpha\cap\beta)\colon\\ +[\text{*22·33}] &\equiv\colon(x).{\sim}(x\in \alpha.x\in \beta)\colon\\ +[\text{*13·191}] &\equiv\colon(x,y):x=y.\supset.{\sim}(x\in \alpha.y\in \beta)\colon\\ +[\text{Transp}] &\equiv\colon(x,y)\colon x\in \alpha.y\in \beta.\supset.x\neq y\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·38.</b> \(\vdash\colon\ldotp \alpha\cap\beta=\Lambda.\supset\colon \alpha\neq\beta.\lor.\alpha=\Lambda.\beta=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·481}.\supset\vdash\colon \alpha\cap\beta=\Lambda.\alpha=\beta.&\supset.\alpha\cap\alpha=\Lambda.\\ +[\text{*22·5}] &\supset.\alpha=\Lambda.\\ +[\text{*20·23}] &\supset.\alpha=\Lambda.\beta=\Lambda \qquad \text{(1)}\\ +\vdash.\text{(1).Exp}.\supset\vdash\colon\ldotp \alpha\cap\beta=\Lambda.&\supset\colon \alpha=\beta.\supset.\alpha=\Lambda.\beta=\Lambda\colon\\ +[\text{*4·6}]&\supset\colon\alpha\neq\beta.\lor.\alpha=\Lambda.\beta=\Lambda\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·39.</b> \(\vdash\colon\ldotp \alpha\cap\beta=\Lambda.\equiv\colon x\in \alpha.\supset_{x}.x{\sim}\in\beta \quad[\text{*24·311.*22·35}]\)</p> + +<p class="nind"><b>*24·4.</b> \(\vdash\colon \alpha\cap\beta=\Lambda.\equiv.(\alpha\cup\beta)- \alpha=\beta.\equiv.(\alpha\cup\beta)- \beta=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·311}.\supset\vdash\colon \alpha\cap\beta=\Lambda.&\equiv.\beta\subset - \alpha.\\ +[\text{*22·621}] &\equiv.\beta - \alpha=\beta\\ +[\text{*22·9}] &\equiv.(\alpha\cup\beta)- \alpha=\beta \qquad \text{(1)}\\ +\vdash.(1)\frac{\beta,\,\alpha}{\alpha,\,\beta}.\supset\vdash\colon\beta\cap\alpha=\Lambda.&\equiv.(\beta\cup\alpha)- \beta= \alpha\colon\\ +[\text{*22·51·57}] \supset\vdash\colon \alpha\cap\beta=\Lambda.&\equiv.(\alpha\cup\beta)- \beta= \alpha \qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·401.</b> \(\vdash\colon \beta\subset\alpha.\supset.(\beta\cup\gamma)- \alpha=\gamma- \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·68}. &\supset\vdash.(\beta\cup\gamma)- \alpha&=(\beta - \alpha)\cup(\gamma - \alpha) &\qquad\qquad\qquad \text{(1)}\\ +\vdash.\text{*24·3}. &\supset\vdash\colon\text{Hp}.\supset.\beta - \alpha&=\Lambda &\qquad\qquad\qquad \text{(2)}\\ +\vdash.(1).(2).&\supset\vdash\colon\text{Hp}.\supset.(\beta\cup\gamma)- \alpha&=\Lambda\cup(\gamma - \alpha)\\ +[\text{*24·24}] &&=\gamma - \alpha\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·402.</b> \(\vdash\colon\alpha\cap\beta=\Lambda.\xi\subset\alpha.\eta\subset\beta.\supset.\xi\cap\eta=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*22·49}.\supset\colon\text{Hp}.&\supset.\xi\cap\eta\subset\alpha\cap\beta\\ +&[\text{*22·55}] &\supset.\xi\cap\eta\subset\Lambda.\\ +&[\text{*24·13}] &\supset.\xi\cap\eta=\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_235">[Pg 235]</span></p> + +<p class="nind"><b>*24·41.</b> \(\vdash.\alpha =(\alpha\cap\beta)\cup(\alpha - \beta\))</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·68}.\supset\vdash.(\alpha\cap\beta)\cup(\alpha - \beta)&=\alpha\cap(\beta\cup - \beta)\\ +[\text{*24·22}] &=\alpha\cap\text{V}\\ +[\text{*24·26}] &=\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·411.</b> \(\vdash\colon\beta\subset\alpha.\supset.\alpha =\beta\cup(\alpha - \beta\))</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·633}\frac{\beta,\,\alpha,\,\alpha - \beta}{\alpha,\,\beta,\,\gamma}.\supset\vdash\colon\beta\subset\alpha.\supset.\beta\cup(\alpha - \beta)&=(\alpha\cap\beta)\cup(\alpha - \beta)\\ +[\text{*24·41}] &=\alpha\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·412.</b> \(\vdash\colon\beta\subset\alpha.\gamma\subset\beta.\supset.(\alpha - \beta)\cup(\beta - \gamma)= \alpha - \gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·41}.\supset\vdash\colon\text{Hp}.\supset.(\alpha - \beta)\cup(\beta - \gamma)&=(\alpha - \beta\cap\gamma)\cup(\alpha - \beta - \gamma)\cup(\beta - \gamma)\\ +[\text{*24·3·23}] &=(\alpha - \beta - \gamma)\cup(\beta - \gamma)\\ +[\text{*22·68}] &=\{(\alpha - \beta)\cup\beta\} - \gamma\\ +[\text{*24·411}] &=\alpha - \gamma\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in *234·181, in the theory of continuous +functions.</p> + +<p class="nind"><b>*24·42.</b> \(\vdash\colon\alpha\cap\beta\subset\gamma.\alpha - \beta\subset\gamma.\equiv.\alpha\subset\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·59}.\supset\vdash\colon \alpha\cap\beta\subset\gamma.\alpha - \beta \subset\gamma.&\equiv.(\alpha\cap\beta)\cup(\alpha - \beta)\subset\gamma.\\ +[\text{*24·411}] \equiv.\alpha\subset\gamma\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·43.</b> \(\vdash\colon \alpha - \beta\subset\gamma.\equiv.\alpha\subset\beta\cup\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*5·6}.&\supset\vdash\colon\colon x\in \alpha.x{\sim}\in \beta.\supset.x \in\gamma\colon&\equiv\colon\ldotp x\in \alpha.\supset\colon x\in \beta.\lor.x\in\gamma\colon\ldotp\\ +[\text{*22·35·33}] &\supset\vdash\colon\colon x\in\alpha - \beta.\supset.x \in\gamma\colon&\equiv\colon\ldotp x\in\alpha.\supset\colon x\in\beta.\lor.x\in\gamma\colon\ldotp\\ +[\text{*22·34}] &&\equiv\colon\ldotp x\in\alpha.\supset.x\in(\beta\cup\gamma) &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·271}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·431.</b> \(\vdash.(\alpha\cup\gamma)\cap(\beta\cup - \gamma)=(\alpha\cap\gamma)\cup(\alpha - \gamma)\cup(\beta\cap\gamma\))</p> + +<p>This and the following proposition are lemmas for <a href="#*24·44">*24·44</a>.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·68}.\supset\vdash.(\alpha\cup\gamma)\cap(\beta\cup - \gamma)&=\{(\alpha\cup\gamma)\cap\beta\}\cup\{(\alpha\cup\gamma)\cap-\gamma\}\\ +[\text{*22·68}] &=(\alpha\cap\gamma)\cup(\gamma\cap\beta)\cup(\alpha - \gamma)\cup(\gamma - \gamma)\\ +[\text{*24·21}] &=(\alpha\cap\gamma)\cup(\gamma\cap\beta)\cup(\alpha - \gamma)\cup\Lambda\\ +[\text{*24·24}] &=(\alpha\cap\gamma)\cup(\gamma\cap\beta)\cup(\alpha - \gamma)\\ +[\text{*22·51·57}] &=(\alpha\cap\gamma)\cup(\alpha - \gamma)\cup(\beta\cap - \gamma).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_236">[Pg 236]</span></p> + +<p class="nind"><b>*24·432.</b> \(\vdash.(\alpha - \gamma)\cup(\beta\cap\gamma)=(\alpha\cap\beta)\cup(\alpha - \gamma)\cup(\beta\cap\gamma)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·22·35}.&\supset\vdash.\alpha\cap\beta=(\alpha\cap\beta)\cap(\gamma\cup - \gamma)\\ +[\text{*22·68}] &=(\alpha\cap\beta\cap\gamma)\cup(\alpha\cap\beta - \gamma)\\ +[\text{*22·51}] &=(\alpha\cap\beta\cap\gamma)\cup(\alpha\cap - \gamma\cap\beta).\\ +[\text{*22·551}] &\supset\vdash.(\alpha\cap\beta)\cup(\alpha - \gamma)=(\alpha\cap\beta\cap\gamma)\cup(\alpha\cap - \gamma\cap\beta)\cup(\alpha - \gamma)\\ +[\text{*22·63}] &=(\alpha\cap\beta\cap\gamma)\cup(\alpha - \gamma)\\ +[\text{*22·57}] &=(\alpha - \gamma)\cup(\alpha\cap\beta\cap\gamma).\\ +[\text{*22·551}] &\supset\vdash.(\alpha\cap\beta)\cup(\alpha - \gamma)\cup(\beta\cap\gamma)=(\alpha - \gamma)\cup(\alpha\cap\beta\cap\gamma)\cup(\beta\cap\gamma)\\ +[\text{*22·63}] &=(\alpha - \gamma)\cup(\beta\cap\gamma).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*24·44">*24·44</a>.</b> \(\vdash.(\alpha\cup\gamma)\cap(\beta\cup - \gamma)=(\alpha\cap - \gamma)\cup(\beta\cap\gamma) \quad[\text{*24·431·432}]\)</p> + +<p class="nind"><b>*24·45.</b> \(\vdash\colon(\alpha\cap\gamma)\cup(\beta - \gamma)=\Lambda.\equiv.\beta\subset\gamma\subset.\gamma\subset - \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·32}.\supset\vdash\colon(\alpha\cap\gamma)\cup(\beta - \gamma)=\Lambda.&\equiv.\alpha\cap\gamma=\Lambda.\beta - \gamma=\Lambda.\\ +[\text{*24·3·311}] &\equiv.\gamma\subset - \alpha.\beta\subset\gamma\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·46.</b> \(\vdash\colon(\alpha\cap\gamma)\cup(\beta - \gamma)=\Lambda.\supset.\alpha\cap\beta=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·45.*22·44}.\supset\vdash\colon\text{Hp}.&\supset.\beta\subset - \alpha.\\ +[\text{*22·811}] &\supset.\alpha\subset - \beta.\\ +[\text{*24·311}] &\supset.\alpha\cap\beta=\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions, down to <a href="#*24·495">*24·495</a> inclusive, are lemmas +inserted for use in much later propositions, most of them being only +used a few times.</p> + +<p class="nind"><b>*24·47.</b> \(\vdash\colon\alpha\cap\beta=\Lambda.\alpha\cup\beta=\gamma.\equiv.\alpha\subset\gamma.\beta=\gamma - \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·311}.\supset\vdash\colon \alpha\cap\beta=\Lambda.&\equiv.\beta\subset - \alpha &\qquad \text{(1)}\\ +\vdash.\text{*22·41}. \supset\vdash\colon \alpha\cup\beta=\gamma.&\equiv.\alpha\cup\beta\subset\gamma.\gamma\subset\alpha\cup\beta.\\ +[\text{*22·59.*24·43}] &\equiv.\alpha\subset\gamma.\beta\subset\gamma.\gamma - \alpha\subset\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon \alpha\cap\beta=\Lambda.\alpha\cup\beta=\gamma.&\equiv.\beta\subset - \alpha.\alpha\subset\gamma.\beta\subset\gamma.\gamma-\alpha\subset\beta.\\ +[\text{*4·3}] &\equiv.\alpha\subset\gamma.\beta\subset\gamma.\beta\subset - \alpha.\gamma-\alpha\subset\beta.\\ +[\text{*22·45}] &\equiv.\alpha\subset\gamma.\beta\subset\gamma - \alpha.\gamma-\alpha\subset\beta.\\ +[\text{*22·41}] &\equiv.\alpha\subset\gamma.\beta=\gamma-\alpha\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·48.</b> \[\begin{align}\vdash\colon\ldotp \xi\subset\alpha.\xi'\subset\alpha.\eta\subset\beta.\eta'\subset\beta.\alpha\cap\beta&=\Lambda.\supset\colon\\ +&\xi\cup\eta=\xi'\cup\eta'.\equiv.\xi=\xi'.\eta=\eta'\end{align}\]</p> + +<p><span class="pagenum" id="Page_237">[Pg 237]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·73}. &\supset\vdash\colon \xi=\xi'.\eta=\eta'.\supset.\xi\cup\eta=\xi'\cup\eta' &\qquad \text{(1)}\\ +\vdash.\text{*22·481}. &\supset\vdash\colon\ldotp \xi\cup\eta=\xi'\cup\eta'.\supset\colon(\xi\cup\eta)\cap\alpha=(\xi'\cup\eta')\cap\alpha\colon +[\text{*22·68}] &\supset\colon(\xi\cap\alpha)\cup(\eta\cap\alpha)=(\xi'\cap\alpha)\cup(\eta'\cap\alpha) &\qquad \text{(2)}\\ +\vdash.\text{*22·621}. &\supset\vdash\colon\xi\subset\alpha.\supset.\xi\cap\alpha=\xi\colon\xi'\subset\alpha.\supset.\xi'\cap\alpha=\xi'\colon\\ +[\text{*3·47}] &\supset\vdash\colon\xi\subset\alpha.\xi'\subset\alpha.\supset.\xi\cap\alpha=\xi.xi'\cap\alpha=\xi' &\qquad \text{(3)}\\ +\vdash.\text{*22·48}. &\supset\vdash\colon\eta\subset\beta.\supset.\eta\cap\alpha\supset\alpha\cap\beta\colon\\ +[\text{*22·55}] &\supset\vdash\colon\eta\subset\beta.\alpha\cap\beta=\lor.\supset.\eta\cap\alpha\subset\Lambda\\ +[\text{*24·13}] &\supset.\eta\cap\alpha=\Lambda &\qquad \text{(4)}\\ +\text{Similarly} &\vdash\colon\eta'\subset\beta.\alpha\cap\beta=\Lambda.\supset.\eta'\cap\alpha=\Lambda &\qquad \text{(5)}\\ +\vdash.\text{(3).(4)}. &\supset\vdash\colon\ldotp\text{Hp}.\supset\colon(\xi\cap\alpha)\cup(\eta\cap\alpha)=\xi\cup\Lambda\\ +[\text{*24·24}] &=\xi &\qquad \text{(6)}\\ +\vdash.\text{(3).(5)}. &\supset\vdash\colon\ldotp\text{Hp}.\supset\colon(\xi'\cap\alpha)\cup(\eta'\cap\alpha)=\xi'\cup\Lambda\\ +[\text{*24·24}] &=\xi' &\qquad \text{(7)}\\ +\vdash.\text{(2).(6).(7)}.&\supset\vdash\colon\ldotp\text{Hp}\colon\xi\cup\eta=\xi'\cup\eta'.\supset.\xi=\xi' &\qquad \text{(8)}\\ +\text{Similarly} &\vdash\colon\ldotp\text{Hp}.\supset\colon\xi\cup\eta=\xi'\cup\eta'.\supset.\eta=\eta' &\qquad \text{(9)}\\ +\vdash.\text{(1).(8).(9)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition, besides being used in the next two, is used +in the theory of couples (<a href="#*54·6">*54·6</a>), in the theory of greater and less +(*117·632), and in the chapter on the ordering of classes by the +principle of first differences (*170·68).</p> + +<p class="nind"><b>*24·481</b>. \(\vdash\colon\ldotp \alpha\cap\beta=\Lambda.\alpha\cap\gamma=\Lambda.\supset\colon \alpha\cup\beta=\alpha\cup\gamma.\equiv.\beta=\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·48} \frac{\alpha,\,-\alpha,\,\alpha,\,\alpha,\,\beta,\,\gamma}{\alpha,\,\beta,\,\xi,\,\xi',\,\eta,\,\eta'}.\supset\\ +\vdash\colon\ldotp \alpha\subset\alpha.\alpha\subset\alpha.\beta\subset - \alpha.\gamma\subset - \alpha.\alpha - \alpha=\Lambda.\supset\colon\\ +&\alpha\cup\beta=\alpha\cup\gamma.\equiv.\alpha=\alpha.\beta=\gamma &\qquad \text{(1)}\\ +\vdash.\text{*22·42.*24·21}.\supset\\ +\vdash\colon\ldotp \alpha\subset\alpha.\alpha\subset\alpha.\beta\subset - \alpha.\gamma\subset - \alpha.\alpha - \alpha=\Lambda.&\equiv.\beta\subset - \alpha.\gamma\subset - \alpha.\\ +[\text{*24·311}] &\equiv.\alpha\cap\beta=\Lambda.\alpha\cap\gamma=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*20·2.*4·73}.\supset\vdash\colon \alpha=\alpha.\beta=\gamma.\equiv.\beta=\gamma &&\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the theory of selections (<a href="#*83·74">*83·74</a>), +in the theory of greater and less (*117·582), and in the theory of +transfinite induction (*257).</p> + +<p class="nind"><b>*24·482</b>. \[\begin{align}&\vdash\colon\ldotp \xi\subset\alpha.\eta\subset\beta.\alpha\cap\beta=\Lambda.\supset\colon \xi\cup\eta=\alpha\cup\beta.\equiv.\xi=\alpha.\eta=\beta\\ +&\quad\left[\text{*24·48} \frac{\alpha,\,\beta}{\xi',\,\eta'}.\text{*22·42}\right]\end{align}\]</p> + +<p><span class="pagenum" id="Page_238">[Pg 238]</span></p> + +<p>The above proposition is used in the theory of convergence (*232·34).</p> + +<p class="nind"><b>*24·49.</b> \(\vdash\colon\ldotp\alpha\cap\beta=\Lambda.\supset\colon\alpha\subset\beta\cup\gamma.\equiv.\alpha\subset\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·621}.&\supset\vdash\colon \alpha\subset\beta\cup\gamma.\equiv.\alpha=\alpha\cap(\beta\cup\gamma)\\ +[\text{*22·68}] &=(\alpha\cap\beta)\cup(\alpha\cap\gamma) &\qquad \text{(1)}\\ +\vdash.*24·24. &\supset\vdash\colon \alpha\cap\beta=\Lambda.\supset.(\alpha\cap\beta)\cup(\alpha\cap\gamma)=\alpha\cap\gamma &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp\text{Hp}.\supset\colon\alpha\subset\beta\cup\gamma.\equiv.\alpha=\alpha\cap\gamma.\\ +[\text{*22·621}] &\equiv.\alpha\subset\gamma\colon:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·491.</b> \[\begin{align}\vdash\colon\beta\cap\gamma=\Lambda.&\alpha\subset\beta\cup\gamma.\\ +&\supset.\alpha - \beta=\alpha\cap\gamma.\alpha - \gamma=\alpha\cap\beta.\alpha=(\alpha - \beta)\cup(\alpha - \gamma)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·621}. &\supset\vdash\colon\text{Hp}.\supset.\alpha=\alpha\cap(\beta\cup\gamma).\\ +[\text{*22·481}] &\supset.\alpha - \gamma=\alpha\cap(\beta\cup\gamma)-\gamma\\ +[\text{*24·4}] & =\alpha\cap\beta &\qquad \text{(1)}\\ +\text{Similarly} &\vdash\colon\text{Hp}.\supset.\alpha - \beta=\alpha\cap\gamma &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash\colon\text{Hp}.\supset.(\alpha - \beta)\cup(\alpha - \gamma)=(\alpha\cap\gamma)\cup(\alpha\cap\beta)\\ +[\text{*22·68}] &=\alpha\cap(\gamma\cup\beta)\\ +[\text{*22·621}] &=\alpha &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the theory of selections (*83·63·65) +and in the theory of segments of a series (*211·84).</p> + +<p class="nind"><b>*24·492.</b> \(\vdash\colon\beta\subset\alpha.\alpha - \beta=\gamma.\supset.\alpha - \gamma=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·481}.\supset\vdash\colon\text{Hp}.\supset.\alpha - \gamma&=\alpha-(\alpha - \beta)\\ +[\text{*22·8·86}] &=\alpha\cap(-\alpha\cup\beta)\\ +[\text{*22·8·9}] &=\alpha\cap\beta\\ +[\text{*22·621}] &=\beta\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used fairly frequently, especially in the +theory of series. It is first used in <a href="#*93·273">*93·273</a>, in the theory of +"generations."</p> + +<p class="nind"><b>*24·493.</b> \(\vdash\colon\beta\cap\gamma=\Lambda.\supset.\alpha=(\alpha - \beta)\cup(\alpha - \gamma)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·84.*24·17}. \supset\vdash\colon\text{Hp}.&\supset.- \beta\cup - \gamma=\lor.\\ +[\text{*24·26}] &\supset.\alpha=\alpha\cap(- \beta\cup - \gamma)\\ +[\text{*22·68}] &=(\alpha - \beta)\cup(\alpha - \gamma)\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·494.</b> \(\vdash\colon\xi\subset\alpha.\eta\subset\beta.\alpha\cap\beta=\Lambda.\supset.(\xi\cup\eta)- \alpha=\eta.(\xi\cup\eta)- \beta=\xi\)</p> + +<p><span class="pagenum" id="Page_239">[Pg 239]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·3}. &\supset\vdash\colon\text{Hp}.\supset.\xi - \alpha=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*24·311}. &\supset\vdash\colon\text{Hp}.\supset.\beta\subset - \alpha.\\ +[\text{*22·44}] &\supset.\eta\subset - \alpha.\\ +[\text{*22·621}] &\supset.\eta - \alpha = \eta &\qquad \text{(2)}\\ +\vdash.\text{*22·68}. &\supset\vdash.(\xi\cup\eta) - \alpha = (\xi - \alpha)\cup(\eta - \alpha) &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*24·24}.&\supset\vdash\colon\text{Hp}.\supset.(\xi \cup \eta) - \alpha = \eta &\qquad \text{(4)}\\ +\text{Similarly} &\vdash\colon\text{Hp}.\supset.(\xi\cup\eta) - \beta = \xi &\qquad \text{(5)}\\ +\vdash.\text{(4).(5)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in the theory of selections (<a href="#*83·63">*83·63</a> and +<a href="#*88·45">*88·45</a>).</p> + +<p class="nind"><b><a id="*24·495">*24·495</a>.</b> \(\vdash\colon\alpha\cap\gamma=\Lambda.\supset.(\alpha\cup\gamma)-(\beta\cup\gamma)=\alpha - \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·87·68}.\supset\\ +\vdash.(\alpha\cup\gamma)-(\beta\cup\gamma)&=(\alpha - \beta - \gamma)\cup(\gamma - \beta - \gamma)\\ +[\text{*24·21}] &= \alpha - \beta - \gamma &\qquad \text{(1)}\\ +\vdash.\text{*24·311.*22·621}.&\supset\vdash\colon\text{Hp}.\supset.\alpha - \gamma = \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the theory of minimum points +(*205·83·832·84).</p> + +<p>In the remainder of this number we shall be concerned with the +existence of classes. Many of the properties of the existence of +classes follow from the fact that to say a class exists is equivalent +to saying that the class is not equal to the null-class. This is proved +in <a href="#*24·54">*24·54</a>.</p> + +<p class="nind"><b>*24·5.</b> \(\vdash\colon\exists!\alpha.\equiv.(\exists x).x \in \alpha \quad[\text{*4·2.(*24·03)}]\)</p> + +<p class="nind"><b>*24*51</b>. \(\vdash\colon{\sim}\exists!\alpha.\equiv.\alpha=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·5}.\supset\vdash\colon{\sim}\exists!\alpha.&\equiv.{\sim}\{(\exists x).x \in \alpha\}.\\ +[\text{*10·252}] &\equiv.(x).x{\sim}\in \alpha.\\ +[\text{*24·15}] &\equiv.\alpha=\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*24·52">*24·52</a>.</b> \(\vdash.\exists!\text{V} \quad [\text{*24·51·1.Transp}]\)</p> + +<p>This proposition states that the class of all objects of the type in +question is not null, but has at least one member. The assumption +that there is something, which is equivalent to this proposition, +is implicit in the proposition <a href="#*10·1">*10·1</a>, that what is true always is +true in any instance. This would not hold if there were no instances +of anything; hence it implies the existence of something. It will +be observed that the above proposition (<a href="#*24·52">*24·52</a>) depends on <a href="#*24·1">*24·1</a>, +which depends on <a href="#*23·351">*23·351</a>, which depends on <a href="#*10·251">*10·251</a>, which depends on +<a href="#*10·24">*10·24</a>, which depends on *10·1 or on <a href="#*9·1">*9·1</a>. The assumption that there +is something is involved in the use of the real variable, which would +otherwise be meaningless. This is made explicit in *9·1, and in the +proof of <a href="#*9·2">*9·2</a>, which is the same proposition as *10·1.</p> + +<p class="nind"><b>*24·53.</b> \(\vdash.{\sim}\exists!\Lambda \quad[\text{*24·51.*20·2}]\)</p> + +<p class="nind"><b><a id="*24·54">*24·54</a>.</b> \(\vdash\colon\exists!\alpha.\equiv.\alpha\neq\Lambda \quad[\text{*24·51.Transp}]\)</p> + +<p><span class="pagenum" id="Page_240">[Pg 240]</span></p> + +<p class="nind"><b>*24·55.</b> \(\vdash\colon{\sim}(\alpha\supset\beta).\equiv.\exists!\alpha - \beta \quad[\text{*24·3.Transp.*24·54]}\)</p> + +<p class="nind"><b>*24·56.</b> \(\vdash\colon\ldotp \exists!(\alpha\cup\beta).\equiv\colon\exists!\alpha.\lor.\exists!\beta \quad[\text{*10·42.*22·34}]\)</p> + +<p class="nind"><b>*24·561.</b>\(\vdash\colon \exists!(\alpha\cap\beta).\supset.\exists!\alpha.\exists!\beta \quad[\text{*10·5.*22·33}]\)</p> + +<p class="nind"><b>*24·57.</b> \(\vdash\colon\ldotp \alpha\cap\beta=\Lambda.\supset\colon\exists!\alpha.\supset.\alpha\neq \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·481}. \supset\vdash\colon \alpha\cap\beta=\Lambda.\alpha=\beta.&\supset.\alpha\cap\alpha=\Lambda.\\ +[\text{*22·5}] &\supset.\alpha=\Lambda \\ +[\text{*24·51}] &\supset.{\sim}\exists!\alpha \qquad\quad \text{(1)}\\ +\vdash.\text{(1).Exp.Transp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·571.</b> \(\vdash\colon\exists!\alpha.\alpha=\beta.\supset.\exists!(\alpha\cap\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash\colon.\text{*24·57.Comm}. \supset\vdash\colon\ldotp\exists!\alpha.&\supset\colon\alpha\cap\beta=\Lambda.\supset.\alpha\neq\beta\colon \\ +[\text{Transp}] &\supset\colon\alpha=\beta.\supset.\alpha\cap\beta\neq\Lambda\\ +[\text{*24·54}] &\supset.\exists!(\alpha\cap\beta) \qquad\qquad\qquad \text{(1)}\\ +\vdash.\text{(1).Imp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*24·58.</b> \(\vdash\colon\ldotp\alpha\supset\beta.\supset\colon\exists!\alpha.\supset.\exists!\beta \qquad\quad[\text{*10·28}]\)</p> + +<p class="nind"><b>*24·6.</b> \(\vdash\colon\ldotp\alpha\supset\beta.\supset\colon\alpha\neq\beta.\equiv.\exists!\beta - \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·41.Transp}. &\supset\vdash\colon\ldotp\text{Hp}.\supset:\alpha\neq\beta.\supset.{\sim}(\beta\supset\alpha).\\ +[\text{*24·55}] &\supset.\exists!\beta - \alpha &\qquad \text{(1)}\\ +\vdash.\text{*24·21} &\supset\vdash\colon \alpha = \beta.\supset.\beta - \alpha = \Lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).Transp.*24·54}. &\supset\vdash\colon\exists!\beta - \alpha.\supset.\alpha \neq \beta &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<p class="nind"><b>*24·61.</b> \(\vdash\colon{\sim}\exists!\beta.\supset.\alpha\cup\beta=\alpha \quad[\text{*24·51·24}]\)</p> + +<p class="nind"><b>*24·62.</b> \(\vdash\colon{\sim}\exists!\beta.\supset.\alpha\cap\beta=\Lambda \quad[\text{*24·51·23}]\)</p> + +<p class="nind"><b>*24·63.</b> \(\vdash\colon\ldotp\Lambda{\sim}\in\kappa.\equiv:\alpha\in\kappa.\supset_{\alpha}.\exists!\alpha\)</p> + + +<p>In this proposition, the conditions of significance require that +\(\kappa\) should be a class of classes. The condition "\(\alpha \in\kappa.\supset_{\alpha}.\exists!\alpha\)" +is one required as hypothesis in many propositions. In virtue of the +above proposition, this hypothesis may be replaced by "\(\Lambda{\sim}\in\kappa\)."</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*13·191}.\supset\vdash\colon\ldotp\Lambda{\sim}\in\kappa.&\equiv\colon \alpha = \Lambda.\supset_{\alpha}.\alpha{\sim}\in\kappa\colon\\ +[\text{Transp}]&\equiv\colon\alpha\in\kappa.\supset_{\alpha}.\alpha\neq\Lambda\colon\\ +[\text{*24·54}]&\equiv\colon\alpha\in\kappa.\supset_{\alpha}.\exists!\alpha \colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is frequently used in later parts of the work. We +often have to deal with classes of existent classes, and the most +convenient form in which to state that all the members of a class of +classes exist is "\(\Lambda{\sim}\in\kappa\)."</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_241">[Pg 241]</span></p> +<h2 class="nobreak" id="*25">*25. THE UNIVERSAL RELATION, THE NULL RELATION, AND THE +EXISTENCE OF RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *25.</p> + +<p>This number contains the analogues, for relations, of the definitions +and propositions of <a href="#*24">*24</a>. Proofs will not be given, as they proceed +precisely as in *24.</p> + +<p>The universal relation, denoted by \(\dot{\text{V}}\), is the relation +which holds between any two terms whatever of the appropriate types, +whatever these may be in the given context. The null relation, +\(\dot{\Lambda}\), is the relation which does not hold between any +pair of terms whatever, its type being fixed by the types of the terms +concerning which the denial that it holds is significant. A relation +\(R\) is said to <i>exist</i> when there is at least one pair of terms +between which it holds; "\(R\) exists" is written "\(\dot{\exists}!R\)."</p> + +<p>The propositions of this number are much less often referred to +than those of *24, but for the sake of uniformity we have given the +analogues of all propositions in *24, with the same numeration (except +for the integral part).</p> + +<p>All the remarks made in *24 apply, <i>mutatis mutandis</i>, in the +present number.</p> + +<hr class="tb"> + +<p class="nind"><b>*25·01.</b> \(\dot{\text{V}} = \hat{x}\hat{y}(x = x . y = y) \quad \text{Df}\)</p> + +<p class="nind"><b>*25·02.</b> \(\dot{\Lambda} = \dot{-}\dot{\text{V}} \quad \text{Df}\)</p> + +<p class="nind"><b>*25·03.</b> \(\dot{\exists}!R .=. (\exists x,y) . xRy \quad \text{Df}\)</p> + +<p class="nind"><b>*25·1.</b> \(\vdash. \dot{\Lambda} \neq \dot{\text{V}}\)</p> + +<p class="nind"><b>*25·101.</b> \(\vdash. \dot{\text{V}} = \dot{-}\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·102.</b> \(\vdash: (x,y).\phi(x,y) .\equiv. \hat{x}\hat{y}\phi(x,y) = \dot{\text{V}}\)</p> + +<p class="nind"><b>*25·103.</b> \(\vdash: (x,y).{\sim}\phi(x,y) .\equiv. \hat{x}\hat{y}\phi(x,y) = \dot{\Lambda}\)</p> + +<p class="nind"><b>*25·104.</b> \(\vdash. (x,y).x\dot{\text{V}}y\)</p> + +<p class="nind"><b>*25·105.</b> \(\vdash.(x,y).{\sim}(x \dot{\Lambda} y)\)</p> + +<p class="nind"><b>*25·11.</b> \(\vdash. (R).R \unicode{x2abd} \dot{\text{V}}\)</p> + +<p class="nind"><b>*25·12.</b> \(\vdash. (R).\dot{\Lambda} \unicode{x2abd} R\)</p> + +<p><span class="pagenum" id="Page_242">[Pg 242]</span></p> + +<p class="nind"><b>*25·13.</b> \(\vdash:R=\dot{\Lambda}.\equiv.R\unicode{x2abd}\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·14.</b> \(\vdash:(x,y).xRy.\equiv.R=\dot{\text{V}}\)</p> + +<p class="nind"><b>*25·141.</b> \(\vdash:\dot{\text{V}}\unicode{x2abd}R.\equiv.\dot{\text{V}}=R\)</p> + +<p class="nind"><b>*25·15.</b> \(\vdash:(x,y).{\sim}(xRy).\equiv.R=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·17.</b> \(\vdash:R=\dot{\text{V}}.\equiv.\dot{-}R=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·21.</b> \(\vdash.R\dot{\cap}\dot{-}R=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·22.</b> \(\vdash.R\unicode{x228d}\dot{-}R=\dot{\text{V}}\)</p> + +<p class="nind"><b>*25·23.</b> \(\vdash.R\dot{\cap}\dot{\Lambda}=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·24.</b> \(\vdash.R\unicode{x228d}\dot{\Lambda}=R\)</p> + +<p class="nind"><b>*25·26.</b> \(\vdash.R\dot{\cap}\dot{\text{V}}=R\)</p> + +<p class="nind"><b>*25·27.</b> \(\vdash.R\unicode{x228d}\dot{\text{V}}=\dot{\text{V}}\)</p> + +<p class="nind"><b>*25·3.</b> \(\vdash:R\unicode{x2abd}S.\equiv.R\dot{-}S=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·31.</b> \(\vdash:R\unicode{x2abd}S.\equiv.\dot{-}R\unicode{x228d}S=\dot{\text{V}}\)</p> + +<p class="nind"><b>*25·311.</b> \(\vdash:R\unicode{x2abd}\dot{-}S.\equiv.R\dot{\cap}S=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·312.</b> \(\vdash:\dot{-}R\unicode{x2abd}S.\equiv.R\unicode{x228d}S=\dot{\text{V}}\)</p> + +<p class="nind"><b>*25·313.</b> \(\vdash:R\dot{\cap}S=\dot{\Lambda}.\equiv.R\dot{-}S=R\)</p> + +<p class="nind"><b>*25·32.</b> \(\vdash:R\unicode{x228d}S=\dot{\Lambda}.\equiv.R=\dot{\Lambda}.S=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·33.</b> \(\vdash:R=\dot{\text{V}}.\supset.R\unicode{x228d}S=\dot{\text{V}}\)</p> + +<p class="nind"><b>*25·34.</b> \(\vdash:R=\dot{\Lambda}.\supset.R\dot{\cap}S=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·35.</b> \(\vdash:R=\dot{\text{V}}.\supset.R\dot{\cap}S=S\)</p> + +<p class="nind"><b>*25·36.</b> \(\vdash:R=\dot{\Lambda}.\supset.R\unicode{x228d}S=S\)</p> + +<p class="nind"><b>*25·37.</b> \(\vdash\colon\ldotp R\dot{\cap}S=\dot{\Lambda}.\equiv\colon\ldotp xRy.zSw.\supset_{x,y,z,w}:x\neq z.\lor.y\neq w\)</p> + +<p class="nind"><b>*25·38.</b> \(\vdash\colon\ldotp R\dot{\cap}S=\dot{\Lambda}.\supset:R\neq S.\lor.R=\dot{\Lambda}.S=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·39.</b> \(\vdash\colon\ldotp R\dot{\cap}S=\dot{\Lambda}.\equiv:xRy.\supset_{x,y}.{\sim}(xSy)\)</p> + +<p class="nind"><b>*25·4.</b> \(\vdash:P\dot{\cap}Q=\dot{\Lambda}.\equiv.(P\unicode{x228d}Q)\dot{-}P=Q.\equiv.(P\unicode{x228d}Q)\dot{-}Q=P\)</p> + +<p class="nind"><b>*25·401.</b> \(\vdash:Q\unicode{x2abd}P.\supset.(Q\unicode{x228d}R)\dot{-}P=R\dot{-}P\)</p> + +<p class="nind"><b>*25·402.</b> \(\vdash:P\dot{\cap}Q=\dot{\Lambda}.R\unicode{x2abd}P.S\unicode{x2abd}Q.\supset.R\dot{\cap}S=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·41.</b> \(\vdash.R=(R\dot{\cap}S)\unicode{x228d}(R\dot{-}S)\)</p> + +<p class="nind"><b>*25·411.</b> \(\vdash:S\unicode{x2abd}R.\supset.R=S\unicode{x228d}(R\dot{-}S)\)</p> + +<p class="nind"><b>*25·412.</b> \(\vdash:Q\unicode{x2abd}P.S\unicode{x2abd}Q.\supset.(P\dot{-}Q)\unicode{x228d}(Q\dot{-}S)=P\dot{-}S\)</p> + +<p class="nind"><b>*25·42.</b> \(\vdash:P\dot{\cap}Q\unicode{x2abd}R.P\dot{-}Q\unicode{x2abd}R.\equiv.P\unicode{x2abd}R\)</p> + +<p class="nind"><b>*25·43.</b> \(\vdash:P\dot{-}Q\unicode{x2abd}R.\equiv.P\unicode{x2abd}Q\unicode{x228d}R\)</p> + +<p class="nind"><b>*25·431.</b> \(\vdash.(P\unicode{x228d}R)\dot{\cap}(Q\unicode{x228d}\dot{-}R)=(P\dot{\cap}Q)\unicode{x228d}(P\dot{-}R)\unicode{x228d}(Q\dot{\cap}R)\)</p> + +<p class="nind"><b>*25·432.</b> \(\vdash.(P\dot{-}R)\unicode{x228d}(Q\dot{\cap}R)=(P\dot{\cap}Q)\unicode{x228d}(P\dot{-}R)\unicode{x228d}(Q\dot{\cap}R)\)</p> + +<p class="nind"><b>*25·44.</b> \(\vdash.(P\unicode{x228d}R)\dot{\cap}(Q\unicode{x228d}\dot{-}R)=(P\dot{\cap}\dot{-}R)\unicode{x228d}(Q\dot{\cap}R)\)</p> + +<p><span class="pagenum" id="Page_243">[Pg 243]</span></p> + +<p class="nind"><b>*25·45.</b> \(\vdash:(P\dot{\cap}R)\unicode{x228d}(Q\dot{-}R)=\dot{\Lambda}.\equiv.Q\unicode{x2abd}R.R\unicode{x2abd}\dot{-}P\)</p> + +<p class="nind"><b>*25·46.</b> \(\vdash:(P\dot{\cap}R)\unicode{x228d}(Q\dot{-}R)=\dot{\Lambda}.\supset.P\dot{\cap}Q=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·47.</b> \(\vdash:(P\dot{\cap}Q)=\dot{\Lambda}.P\unicode{x228d}Q=R.\equiv.P\unicode{x2abd}R.Q=R\dot{-}P\)</p> + +<p class="nind"><b>*25·48.</b> \[\begin{align}\vdash\colon\ldotp R\unicode{x2abd}P.R'\unicode{x2abd}P.S\unicode{x2abd}Q.S'\unicode{x2abd}Q.&P\dot{\cap}Q=\dot{\Lambda}.\supset:\\ +&R\unicode{x228d}S=R'\unicode{x228d}S'.\equiv.R=R'.S=S'\end{align}\]</p> + +<p class="nind"><b>*25·481.</b> \(\vdash\colon\ldotp P\dot{\cap}Q=\dot{\Lambda}.P\dot{\cap}R=\dot{\Lambda}.\supset:P\unicode{x228d}Q=P\unicode{x228d}R.\equiv.Q=R\)</p> + +<p class="nind"><b>*25·482.</b> \(\vdash\colon\ldotp R\unicode{x2abd}P.S\unicode{x2abd}Q.P\dot{\cap}Q=\dot{\Lambda}.\supset:R\unicode{x228d}S=P\unicode{x228d}Q.\equiv.R=P.S=Q\)</p> + +<p class="nind"><b>*25·49.</b> \(\vdash\colon\ldotp P\dot{\cap}Q=\dot{\Lambda}.\supset:P\unicode{x2abd}Q\unicode{x228d}R.\equiv.P\unicode{x2abd}R\)</p> + +<p class="nind"><b>*25·491.</b> \[\begin{align}\vdash:Q\dot{\cap}R=\dot{\Lambda}.&P\unicode{x2abd}Q\unicode{x228d}R.\supset.\\ +&P\dot{-}Q=P\dot{\cap}R.P\dot{-}R=P\dot{\cap}Q.P=(P\dot{-}Q)\unicode{x228d}(P\dot{-}R)\end{align}\]</p> + +<p class="nind"><b>*25·492.</b> \(\vdash:Q\unicode{x2abd}P.P\dot{-}Q=R.\supset.P\dot{-}R=Q\)</p> + +<p class="nind"><b>*25·493.</b> \(\vdash:Q\dot{\cap}R=\dot{\Lambda}.\supset.P=(P\dot{-}Q)\unicode{x228d}(P\dot{-}R)\)</p> + +<p class="nind"><b>*25·494.</b> \(\vdash:R\unicode{x2abd}P.S\unicode{x2abd}Q.P\dot{\cap}Q=\dot{\Lambda}.\supset.(R\unicode{x228d}S)\dot{-}P=S.(R\unicode{x228d}S)\dot{-}Q=R\)</p> + +<p class="nind"><b>*25·495.</b> \(\vdash:P\dot{\cap}R=\dot{\Lambda}.\supset.(P\unicode{x228d}R)\dot{-}(Q\unicode{x228d}R)=P\dot{-}Q\)</p> + +<p class="nind"><b>*25·5.</b> \(\vdash:\dot{\exists}!R.\equiv.(\exists x,y).xRy\)</p> + +<p class="nind"><b>*25·51.</b> \(\vdash:{\sim}\dot{\exists}!R.\equiv.R=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·52.</b> \(\vdash.\dot{\exists}!\dot{\text{V}}\)</p> + +<p class="nind"><b>*25·53.</b> \(\vdash.{\sim}\dot{\exists}!\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·54.</b> \(\vdash:\dot{\exists}!R.\equiv.R\neq\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·55.</b> \(\vdash:{\sim}(R\unicode{x2abd}S).\equiv.\dot{\exists}!R\dot{-}S\)</p> + +<p class="nind"><b>*25·56.</b> \(\vdash\colon\ldotp\dot{\exists}!R\unicode{x228d}S.\equiv:\dot{\exists}!R.\lor.\dot{\exists}!S\)</p> + +<p class="nind"><b>*25·561.</b> \(\vdash:\dot{\exists}!(R\dot{\cap}S).\supset.\dot{\exists}!R.\dot{\exists}!S\)</p> + +<p class="nind"><b>*25·57.</b> \(\vdash\colon\ldotp R\dot{\cap}S=\dot{\Lambda}.\supset:\dot{\exists}!R.\supset.R\neq S\)</p> + +<p class="nind"><b>*25·571.</b> \(\vdash:\dot{\exists}!R.R=S.\supset.\dot{\exists}!(R\dot{\cap}S)\)</p> + +<p class="nind"><b>*25·58.</b> \(\vdash\colon\ldotp R\unicode{x2abd}S.\supset:\dot{\exists}!R.\supset.\dot{\exists}!S\)</p> + +<p class="nind"><b>*25·6.</b> \(\vdash\colon\ldotp R\unicode{x2abd}S.\supset:R\neq S.\equiv.\dot{\exists}!S\dot{-}R\)</p> + +<p class="nind"><b>*25·61.</b> \(\vdash:{\sim}\dot{\exists}!S.\supset.R\unicode{x228d}S=R\)</p> + +<p class="nind"><b>*25·62.</b> \(\vdash:{\sim}\dot{\exists}!S.\supset.R\dot{\cap}S=\dot{\Lambda}\)</p> + +<p class="nind"><b>*25·63.</b> \(\vdash\colon\ldotp\dot{\Lambda}{\sim}\in\kappa.\equiv:R\in\kappa.\supset_{R}.\dot{\exists}!R\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_244">[Pg 244]</span></p> +<h2 class="nobreak" id="SECTION_D_a">SECTION D.<br> +LOGIC OF RELATIONS.</h2> +</div> + + +<p>In the present section we shall be concerned with such of the general +properties of relations as have no analogues in the theory of classes. +The notations introduced in this section will be used constantly +throughout the rest of the work, and the ideas expressed in the +definitions will be found to be of fundamental importance.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_245">[Pg 245]</span></p> +<h2 class="nobreak" id="*30">*30. DESCRIPTIVE FUNCTIONS.</h2> +</div> + + +<p><i>Summary of</i> *30.</p> + +<p>The functions hitherto considered, with the exception of a few +particular functions such as \(\alpha \cap \beta\), have been +propositional, <i>i.e.</i> have had propositions for their values. +But the ordinary functions of mathematics, such as \(x^{2}\), \(\sin x\), +\(\log x\), are not propositional. Functions of this kind always +mean "the term having such and such a relation to \(x\)." For this +reason they may be called <i>descriptive</i> functions, because +they <i>describe</i> a certain term by means of its relation to +their argument. Thus "\(\sin \dfrac{\pi}{2}\)" describes the number +\(1\); yet propositions in which \(\sin \dfrac{\pi}{2}\) occurs are +not the same as they would be if \(1\) were substituted for +\(\sin\dfrac{\pi}{2}\). This appears <i>e.g.</i> from the proposition +"\(\sin\dfrac{\pi}{2} = 1\)," which conveys valuable information, +whereas "\(1 = 1\)" is trivial. Descriptive functions, like +descriptions in general, have no meaning by themselves, but only as +constituents of propositions<a id="FNanchor_55" href="#Footnote_55" class="fnanchor">[55]</a>.</p> + +<p>The general definition of a descriptive function is:</p> + +<p class="nind"><b><a id="*30·01">*30·01</a>.</b> \(Rʻy=({℩}x)(xRy) \quad \text{Df}\)</p> + +<p>That is, "\(Rʻy\)" is to mean "the term \(x\) which has the relation +\(R\) to \(y\)." If there are several terms or none having the relation +\(R\) to \(y\), all propositions about \(Rʻy\), <i>i.e.</i> all +propositions of the form "\(\phi(Rʻy)\)," will be false. The apostrophe +in "\(Rʻy\)" may be read "of." Thus if \(R\) is the relation of father +to son, "\(Rʻy\)" means "the father of \(y\)." If \(R\) is the relation +of son to father, "\(Rʻy\)" means "the son of \(y\)"; in this case, all +propositions of the form "\(\phi(Rʻy)\)" will be false unless \(y\) has +one son and no more.</p> + +<p>All the functions that occur in ordinary mathematics are instances +of the above definition; all are obtained in the above manner from +some relation. Thus in our notation "\(Rʻy\)" takes the place of what +would commonly be "\(fy\)," this latter notation being reserved for +<i>propositional</i> functions. We should write "\(\sin ʻy\)" in place +of "\(\sin y\)," using "\(\sin\)" to express the relation of \(x\) to +\(y\) when \(x = \sin y\).</p> + +<p>A definition such as \(Rʻy = (℩x)(xRy)\), where the meaning given to +the term defined is a description, must be understood to mean that the +term defined (in this case \(Rʻy\)) and the description assigned as +its meaning (in<span class="pagenum" id="Page_246">[Pg 246]</span> this case (\({℩} x)(xRy))\) are to be interchangeable +in use: the definition is, in a sense, more purely symbolic than other +definitions, since the description assigned as the meaning has itself +no meaning except in use. It would perhaps be more formally correct to +write +\[ +f(Rʻy) .=. f\{({℩} x)(xRy)\} \quad \text{Df}\text{.} +\]</p> + +<p>But even this definition would not be quite complete, because it +omits mention of the <i>scope</i> of the two descriptions \(Rʻy\) and +\((℩x)(xRy)\). Thus the complete form would be +\[ +[Rʻy].f(Rʻy) .=. [({℩}x)(xRy)] . f\{({℩}x)(xRy)\} \quad \text{Df}\text{.} +\] +But it is unnecessary to adopt this form of definition, provided it +is understood that the definition <a href="#*30·01">*30·01</a> means that "\(Rʻy\)" may +be written for "(\({℩}x)(xRy)\)" <i>everywhere</i>, <i>i.e.</i> in +indications of scope as well as elsewhere. The use of the definition +occurs always in accordance with the proposition: +\[ +\vdash: [Rʻy].f(Rʻy) .\equiv. [({℩}x)(xRy)] . f({℩}x)(xRy)\text{,} +\] +which is <a href="#*30·1">*30·1</a>, below.</p> + +<p>It is to be observed that *30·01 does not necessarily involve +\[ +Rʻy = ({℩}x)(xRy)\text{.} +\] +For this, by the definition, is equivalent to +\[ +({℩}x)(xRy) = ({℩}x)(xRy)\text{,} +\] +which, by <a href="#*14·28">*14·28</a>, only holds when \(\text{E}!({℩}x)(xRy)\), <i>i.e.</i> +when there is one term, and no more, which has the relation \(R\) to +\(y\).</p> + +<p>All the conventions as to scope explained in <a href="#*14">*14</a> are to be transferred +to \(Rʻx\), <i>i.e.</i>, in the absence of any contrary indication, the +scope of \(Rʻx\) is to be the smallest proposition, enclosed in dots or +other brackets, in which the \(Rʻx\) in question occurs.</p> + +<p>We put</p> + +<p class="nind"><b><a id="*30·02">*30·02</a>.</b> \(RʻSʻy = Rʻ(Sʻy) \quad \text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets. It is to +be interpreted as meaning +\[ +[RʻSʻy].f(RʻSʻy) .=. [Rʻ(Sʻy)].f\{Rʻ(Sʻy)\} \quad \text{Df}\text{.} +\] +In future, we shall often define a new expression as having a +descriptive phrase for its meaning; in such a case, the definition is +always to be interpreted as above. That is, any proposition in which +the new expression occurs is to be the proposition which is obtained +by substituting the old expression for the new one wherever the latter +occurs.</p> + +<p><span class="pagenum" id="Page_247">[Pg 247]</span></p> + +<p>\(Rʻ(Sʻy)\), in the above, is to be interpreted by first treating +\(Sʻy\) as if it were not a descriptive symbol, and applying <a href="#*30·01">*30·01</a> and +<a href="#*14·01">*14·01</a> or <a href="#*14·02">*14·02</a> to \(Rʻ(Sʻy)\), +and by then applying *30·01 and *14·01 or *14·02 to \(Sʻy\).</p> + +<p>The majority of the propositions of the present number are immediate +consequences of the corresponding propositions in *14. Thus <a href="#*14·31">*14·31</a>—<a href="#*14·34">·34</a> +and <a href="#*14·113">*14·113</a> lead immediately to <a href="#*30·12">*30·12</a>—<a href="#*30·16">·16</a>, which show that, either +always or when \(Rʻy\) exists, the "scope" of \(Rʻy\) or of \(Rʻy\) and +\(Sʻy\) makes no difference to the truth-values of such propositions as +we are concerned with. We have</p> + +<p class="nind"><b>*30·18.</b> \(\vdash\colon\ldotp\exists!Rʻy:(z).\phi z:\supset.\phi(Rʻy)\)</p> + +<p>so that what holds of everything holds of \(Rʻy\), provided \(Rʻy\) +exists. This results immediately from <a href="#*14·18">*14·18</a>, and shows that, provided +\(Rʻy\) exists, the fact that "\(Rʻy\)" is an incomplete symbol does +not prevent its being substituted as a value of \(z\) whenever we have +(\(z).\phi z\), or an assertion of the propositional function \(\phi +z\).</p> + +<p>One of the most used propositions of this number is:</p> + +<p class="nind"><b>*30·3.</b> \(\vdash\colon\ldotp x=Rʻy.\equiv:zRy.\equiv_{z}.z=x\)</p> + +<p>which results immediately from <a href="#*14·202">*14·202</a>. The following analogous +proposition results from the above by means of <a href="#*14·122">*14·122</a>:</p> + +<p class="nind"><b>*30·31.</b> \(\vdash\colon\ldotp x=Rʻy.\equiv:xRy:zRy.\supset_{z}.z=x\)</p> + +<p><i>I.e.</i> "\(x=Rʻy\)" involves, in addition to "\(xRy\)," the +statement that whatever has the relation \(R\) to \(y\) is identical +with \(x\).</p> + +<p>A proposition constantly referred to is:</p> + +<p class="nind"><b>*30·37.</b> \(\vdash:\exists!Rʻy.y=z.\supset.Rʻy=Rʻz\)</p> + +<p>In the hypothesis, \(\exists!Rʻy\) might be replaced by +\(\exists!Rʻz\), but one or other of them is essential. For, by <a href="#*14·21">*14·21</a>, +"\(Rʻy=Rʻz\)" implies \(\exists!Rʻy\) and \(\exists!Rʻz\) (these are +equivalent when \(y=z\)), and therefore cannot be true when \(Rʻy\) and +\(Rʻz\) do not exist.</p> + +<p>The use of *30·37 is chiefly in cases where \(y\) or \(z\) or both are +replaced by descriptive functions. Suppose, for example, that \(z\) is +replaced by \(Sʻw\). By <a href="#*30·18">*30·18</a>, we may substitute \(Sʻw\) for \(z\) if +\(Sʻw\) exists. By <a href="#*14·21">*14·21</a>, both sides of the implication in <a href="#*30·37">*30·37</a> will +become false if \(Sʻw\) does not exist, and therefore the implication +will still hold. Hence whether \(Sʻw\) exists or not, we may substitute +it for \(z\) and obtain +\[ +\vdash:\exists!Rʻy.y=Sʻw.\supset.Rʻy=RʻSʻw. +\] +In like manner, if we replace \(y\) by \(Tʻv\), we obtain +\[ +\vdash:\exists x!RʻTʻv.Tʻv=Sʻw.\supset.RʻTʻv=RʻSʻw. +\]</p> + +<p>A very important proposition is:</p> + +<p class="nind"><b>*30·4.</b> \(\vdash\colon\ldotp\exists!Rʻy.\supset:a=Rʻy.\equiv.aRy\)</p> + +<p>This proposition states that, provided \(Rʻy\) exists, to say that +\(a\) is <i>the</i> term which has the relation \(R\) to \(y\) is +equivalent to saying that a has the relation \(R\) to \(y\). Thus for +example "\(a\) is the occupier of the house \(y\)" is equivalent to +"\(a\) occupies the house \(y\)," "\(a\) is the writer of Waverley" is +equivalent to<span class="pagenum" id="Page_248">[Pg 248]</span> "\(a\) wrote Waverley," "\(a\) is the father of \(y\)" +is equivalent to "\(a\) begot \(y\)." But we cannot argue from "John +Smith inhabits London" to "John Smith is <i>the</i> inhabitant of +London."</p> + +<p>We shall introduce in this and subsequent sections many constant +relations for which \(\exists!Rʻy\) is always true. When \(R\) is such +that \(\exists!Rʻy\) is always true, we have, in virtue of <a href="#*30·4">*30·4</a>, +\[ +a=Rʻy.\equiv.aRy +\] +for every possible value of \(y\). The following proposition is useful +in cases where both \(R\) and \(S\) are such that \(Rʻy\) and \(Sʻy\) +always exist:</p> + +<p class="nind"><b>*30·41.</b> \(\vdash\colon\ldotp(y).Rʻy=Sʻy.\equiv:(y).\exists!Rʻy:R=S\)</p> + +<p>Thus if we know that \(Rʻy\) and \(Sʻy\) are always identical, we know +not only that \(R\) and \(S\) are identical, but also that \(Rʻy\) (and +therefore \(Sʻy\)) always exists.</p> + +<hr class="tb"> + +<p class="nind"><b>*30·01.</b> \(Rʻy=({℩}x)(xRy) \quad\text{Df}\)</p> + +<p class="nind"><b>*30·02.</b> \(RʻSʻy=Rʻ(Sʻy) \quad\text{Df}\)</p> + +<p>In interpreting \(Rʻ(Sʻy)\), \(Sʻy\) is to be treated as an ordinary +symbol until \(Rʻ(Sʻy)\) has been eliminated by <a href="#*30·01">*30·01</a> and <a href="#*14·01">*14·01</a> or +<a href="#*14·02">*14·02</a>, and then the above definitions are to be applied to \(Sʻy\).</p> + +<p class="nind"><b><a id="*30·1">*30·1</a>.</b> \(\vdash:[Rʻy].f(Rʻy).\equiv.[({℩}x)(xRy)].f({℩}x)(xRy) \quad[\text{*4·2.(*30·01)}]\)</p> + +<p class="nind"><b>*30·11.</b> \(\vdash\colon\ldotp[Rʻy].f(Rʻy).\equiv:(\exists b):xRy.\equiv_{x}.x=b:fb \quad[\text{*30·1.*14·1}]\)</p> + +<p>The following propositions are immediate applications of <a href="#*14·31">*14·31</a> ff., +made in accordance with *30·1.</p> + +<p class="nind"><b><a id="*30·12">*30·12</a>.</b> \(\vdash\colon\colon\exists!Rʻy.\supset\colon\ldotp[Rʻy].p\lor\chi(Rʻy).\equiv:p.\lor.[Rʻy].\chi(Rʻy) +\quad[\text{*14·31}]\)</p> + +<p class="nind"><b>*30·13.</b> \(\vdash\colon\colon\exists!Rʻy.\supset\colon\ldotp[Rʻy].{\sim}\chi(Rʻy).\equiv.{\sim}\{[Rʻy].\chi(Rʻy)\} \quad[\text{*14·32}]\)</p> + +<p class="nind"><b>*30·14.</b> \(\vdash\colon\colon\exists!Rʻy.\supset\colon\ldotp[Rʻy].p\supset\chi(Rʻy).\equiv:p.\supset.[Rʻy].\chi(Rʻy) +\quad[\text{*14·33}]\)</p> + +<p class="nind"><b>*30·141.</b> \(\vdash\colon\colon\exists!Rʻy.\supset\colon\ldotp[Rʻy].\chi(Rʻy)\supset p.\equiv:[Rʻy].\chi(Rʻy).\supset.p +\quad[\text{*14·331}]\)</p> + +<p class="nind"><b>*30·142.</b> \(\vdash\colon\colon\exists!Rʻy.\supset\colon\ldotp[Rʻy].p\equiv \chi(Rʻy).\equiv:p.\equiv.[Rʻy].\chi(Rʻy) +\quad[\text{*14·332}]\)</p> + +<p class="nind"><b>*30·15.</b> \(\vdash\colon\ldotp p:[Rʻy].\chi(Rʻy):\equiv:[Rʻy].p.\chi(Rʻy) \quad[\text{*14·34}]\)</p> + +<p>The following two propositions are immediate consequences of +*14·113·112.</p> + +<p class="nind"><b><a id="*30·16">*30·16</a>.</b> \(\vdash\colon [Rʻy].f(Rʻy,Sʻz).\equiv.[Sʻz].f(Rʻy,Sʻz) \quad[\text{*14·113}]\)</p> + +<p class="nind"><b>*30·17.</b> \[\begin{align}\vdash\colon\ldotp[Rʻy].&f(Rʻy,Sʻz).\equiv:\\ +&(\exists b,c):xRy.\equiv_{x}.x=b:xSz.\equiv_{x}.x=c:f(b,c) \quad[\text{*14·112}]\end{align}\]</p> + +<p class="nind"><b><a id="*30·18">*30·18</a>.</b> \(\vdash\colon\ldotp\exists!Rʻy:(z).\phi z:\supset.\phi(Rʻy) \quad[\text{*14·18}]\)</p> + +<p><span class="pagenum" id="Page_249">[Pg 249]</span></p> + +<p class="nind"><b>*30·19.</b> \(\vdash\colon\ldotp Rʻy=b.\supset:\psi(Rʻy).\equiv.\psi b \quad[\text{*14·15}]\)</p> + +<p class="nind"><b>*30·2.</b> \(\vdash\colon\ldotp\text{E}!Rʻy.\equiv:(\exists b):xRy.\equiv_{x}.x=b \quad[\text{*4·2.*14·11.(*30·01)}]\)</p> + +<p>In proving *30·2, we have to use the definition <a href="#*30·01">*30·01</a>, not <a href="#*30·1">*30·1</a>, +because \(\text{E}!({℩}x)(\phi x)\) is not of the form \(f({℩}x)(\phi x)\). This appears if we attempt to apply the definition <a href="#*14·01">*14·01</a> to +\(\text{E}!({℩}x)(\phi x)\), which leads to an expression containing +the meaningless constituent \(\text{E}!b\). But by the definition +*30·01, every typographical occurrence of the symbol "\(Rʻy\)" means +what results when this symbol is replaced by "(\({℩}x)(xRy)\)" hence +"\(\text{E}!Rʻy\)" means "\(\text{E}!({℩}x)(xRy)\)."</p> + +<p class="nind"><b>*30·21.</b> \[\begin{align}&\vdash\colon\colon\text{E}!Rʻy.\equiv\colon\ldotp(\exists x).xRy:xRy.zRy.\supset_{x,z}.x=z\\ +&\quad[\text{*14·203.(*30·01)}]\end{align}\]</p> + +<p class="nind"><b>*30·22.</b> \(\vdash:\text{E}!Rʻy.\equiv.Rʻy=({℩}x)(xRy) \quad[\text{*14·28.(*30·01)}]\)</p> + +<p>Note that we do not necessarily have +\[ +Rʻy=({℩}x)(xRy), +\] +which is only true when \(\text{E}!Rʻy\).</p> + +<p class="nind"><b>*30·3.</b> \(\vdash\colon\ldotp x=Rʻy.\equiv:zRy.\equiv_{z}.z=x \quad[\text{*14·202}]\)</p> + +<p class="nind"><b>*30·31.</b> \(\vdash\colon\ldotp x=Rʻy.\equiv:xRy:zRy.\supset_{z}.z=x \quad[\text{*14·122.*30·3}]\)</p> + +<p class="nind"><b>*30·32.</b> \(\vdash:\text{E}!Rʻy.\equiv.(Rʻy)Ry \quad[\text{*14·22}]\)</p> + +<p class="nind"><b>*30·33.</b> \[\begin{align}&\vdash\colon\colon\text{E}!Rʻy.\supset\colon\ldotp \psi(Rʻy):\equiv:(\exists x).xRy.\psi x:\equiv:xRy.\supset_{x}.\psi x\\ +&\quad[\text{*14·26}]\end{align}\]</p> + +<p class="nind"><b>*30·34.</b> \(\vdash\colon\ldotp xRy.\equiv_{x}.xSy:\supset:\text{E}!Rʻy.\equiv.\text{E}!Sʻy \quad[\text{*14·271}]\)</p> + +<p class="nind"><b>*30·341.</b> \(\vdash\colon\ldotp xRy.\equiv_{x}.xSy:\supset:\text{E}!Rʻy.\equiv.Rʻy=Sʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}. &\supset\vdash:Rʻy=Sʻy.\supset.\text{E}!Rʻy &\qquad \text{(1)}\\ +\vdash.\text{.*14·27.Comm}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:\text{E}!Rʻy.\supset.Rʻy=Sʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*30·35.</b> \(\vdash\colon\ldotp R=S.\supset:\text{E}!Rʻy.\equiv.\text{E}!Sʻy \quad[\text{*30·34.*21·43}]\)</p> + +<p class="nind"><b>*30·36.</b> \(\vdash:\text{E}!Rʻy.R=S.\supset.Rʻy=Sʻy \quad[\text{*14·27.Imp.*21·43}]\)</p> + +<p class="nind"><b><a id="*30·37">*30·37</a>.</b> \(\vdash:\text{E}!Rʻy.y=z.\supset.Rʻy=Rʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·28}.&\supset\vdash:\text{E}!Rʻy.\supset.Rʻy=Rʻy &\qquad \text{(1)}\\ +\vdash.\text{*13·12}.&\supset\vdash\colon\ldotp y=z.\supset:Rʻy=Rʻy.\equiv.Rʻy=Rʻz &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Ass}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is very frequently used.</p> + +<p class="nind"><b><a id="*30·4">*30·4</a>.</b> \(\vdash\colon\ldotp\text{E}!Rʻy.\supset:a=Rʻy.\equiv.aRy \quad[\text{*14·241}]\)</p> + +<p>This is a very important proposition, of which the use is constant.</p> + +<p><span class="pagenum" id="Page_250">[Pg 250]</span></p> + +<p class="nind"><b>*30·41.</b> \(\vdash\colon\ldotp (y).Rʻy=Sʻy.\equiv:(y).\text{E}!Rʻy:R=S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21.*10·11·27}.&\supset\vdash:(y).Rʻy=Sʻy.\supset.(y).\text{E}!Rʻy &\qquad \text{(1)}\\ +\vdash.\text{*14·13·142}. &\supset\vdash\colon\ldotp (y).Rʻy=Sʻy.\supset:(x,y):x=Rʻy.\equiv.x=Sʻy:\\ +[\text{(1).*30·4}] &\qquad\qquad\qquad\qquad\supset:(x,y):xRy.\equiv.xSy:\\ +[\text{*21·43}] &\qquad\qquad\qquad\qquad\supset:R=S &\qquad \text{(2)}\\ +\vdash.\text{*30·36}. &\supset\vdash:\text{E}!Rʻy.R=S.\supset.Rʻy=Sʻy:\\ +[\text{*10·11·27·35}] &\supset\vdash\colon\ldotp (y).\text{E}!Rʻy:R=S:\supset.(y).Rʻy=Sʻy &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*30·42.</b> \(\vdash\colon\ldotp (y).\text{E}!Rʻy.\supset:(y).Rʻy=Sʻy.\equiv.R=S \quad[\text{*30·41}]\)</p> + +<p>The hypothesis (\(y).\text{E}!Rʻy\) is fulfilled by a number of +important special relations, of which examples will occur in the +subsequent numbers of the present section.</p> + +<p class="nind"><b>*30·5.</b> \(\vdash:\text{E}!PʻQʻz.\supset.\text{E}!Qʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·2}.\supset\vdash\colon\ldotp \text{E}!PʻQʻz.&\equiv:(\exists b):xP(Qʻz).\equiv_{x}.x=b:\\ +[\text{*10·1}] &\supset:(\exists b):bP(Qʻz).\equiv.b=b:\\ +[\text{*13·15}] &\supset:(\exists b).bP(Qʻz):\\ +[\text{*14·21}] &\supset:\text{E}!Qʻz\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*30·501.</b> \(\vdash:\phi(PʻQʻz).\equiv.(\exists b,c).c=Qʻz.b=Pʻc.\phi b\)</p> + +<p>On the meaning of "\(\phi(PʻQʻz)\)," see note to the definition <a href="#*30·02">*30·02</a>.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·1·122}.\supset\vdash\colon\colon \phi(PʻQʻz).&\equiv\colon\ldotp (\exists b):bP(Qʻz):xP(Qʻz).\supset_{x}.x=b:\phi b\colon\ldotp \\ +[\text{*14·205}] &\equiv\colon\ldotp (\exists b)\colon\ldotp (\exists c):c=Qʻz:bPc:xPc.\supset_{x}.x=b:\phi b\colon\ldotp \\ +[\text{*14·122·202}] &\equiv\colon\ldotp (\exists b,c).c=Qʻz.b=Pʻc.\phi b\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*30·51.</b> \(\vdash:b=PʻQʻz.\equiv.(\exists c).b=Pʻc.c=Qʻz \quad[\text{*30·501.*13·195}]\)</p> + +<p class="nind"><b>*30·52.</b> \(\vdash:\text{E}!PʻQʻz.\equiv.(\exists b,c).b=Pʻc.c=Qʻz \quad[\text{*30·51.*14·204}]\)</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_55" href="#FNanchor_55" class="label">[55]</a> +Cf. <a href="#*14">*14</a>, above.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_251">[Pg 251]</span></p> +<h2 class="nobreak" id="*31">*31. CONVERSES OF RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *31.</p> + +<p>If \(R\) is a relation, the relation which \(y\) has to \(x\) when +\(x R y\) is called the <i>converse</i> of \(R\). Thus <i>greater</i> +is the converse of <i>less</i>, <i>before</i> of <i>after</i>, +<i>husband</i> of <i>wife</i>. The converse of identity is identity, +and the converse of diversity is diversity. The converse of \(R\) is +written \(\breve{R}\) (read "\(R\)-converse"). When \(R = \breve{R}\), +\(R\) is called a <i>symmetrical</i> relation, otherwise it is called +<i>not-symmetrical</i>. When \(R\) is incompatible with \(\breve{R}\), +\(R\) is called <i>asymmetrical</i>. Thus "cousin" is symmetrical, +"brother" is not-symmetrical (because when \(x\) is the brother of +\(y\), \(y\) may be either the brother or the sister of \(x)\), and +"husband" is asymmetrical.</p> + +<p>The relation of \(\breve{R}\) to \(R\) is called "\(\text{Cnv}\)." It +will be shown that every relation has one, and only one, converse; +hence, applying the notation of <a href="#*30">*30</a>, that one is \(\text{Cnv}ʻR\). +Thus \(\breve{R} = \text{Cnv}ʻR\). We have thus two notations for the +converse of \(R\); the second is more convenient for the converse of a +relation not denoted by a single letter.</p> + +<p>The more important propositions of the present number are the following:</p> + +<p class="nind"><b>*31·13.</b> \(\vdash . \text{E}! \text{Cnv}ʻP\)</p> + +<p><i>I.e.</i> any relation \(P\) has a converse. Hence the relation +"\(\text{Cnv}\)" verifies the hypothesis \((y).\text{E}!Rʻy\), +<i>i.e.</i> we have \((P).\text{E}!\text{Cnv}ʻP\).</p> + +<p class="nind"><b>*31·32.</b> \(\vdash : P = Q .\equiv. \breve{P} = \breve{Q}\)</p> + +<p><i>I.e.</i> two relations are identical when, and only when, their +converses are identical.</p> + +<p class="nind"><b>*31·33.</b> \(\vdash . \text{Cnv}ʻ\text{Cnv}ʻP = P\)</p> + +<p><i>I.e.</i> any relation is the converse of its converse.</p> + +<p><span class="pagenum" id="Page_252">[Pg 252]</span></p> + +<p>Very many of the subsequent uses of the notion of the converse of a +relation require only the propositions which embody the definitions of +\(\breve{P}\) and \(\text{Cnv}\), namely</p> + +<p class="nind"><b>*31·11.</b> \(\vdash:x\breve{P}y.\equiv.yPx\)</p> + +<p>and</p> + +<p class="nind"><b>*31·131.</b> \(\vdash:x(\text{Cnv}ʻP)y.\equiv.yPx\)</p> + +<hr class="tb"> + +<p class="nind"><b>*31·01.</b> \(\text{Cnv}=\hat{Q}\hat{P}\{xQy.\equiv_{x,y}.yPx\} \quad\text{Df}\)</p> + +<p class="nind"><b>*31·02.</b> \(\breve{P}=\hat{x}\hat{y}(yPx) \quad\text{Df}\)</p> + +<p class="nind"><b>*31·1.</b> \(\vdash\colon\ldotp Q\, \text{Cnv}\, P.\equiv:xQy.\equiv_{x,y}.yPx \quad[\text{*21·3.(*31·01)}]\)</p> + +<p class="nind"><b>*31·101.</b> \(\vdash:Q\, \text{Cnv}\, P.R\, \text{Cnv}\, P.\supset.Q=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·1}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:xQy.\equiv_{x,y}.yPx:xRy.\equiv_{x,y}.yPx:\\ +[\text{*11·371}] & \supset:xQy.\equiv_{x,y}.xRy:\\ +[\text{*21·43}] &\supset:Q=R\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·11.</b> \(\vdash:x\breve{P}y.\equiv.yPx \quad[\text{*21·3.(*31·02)}]\)</p> + +<p class="nind"><b>*31·111.</b> \(\vdash.\breve{P}\,\text{Cnv} P \quad[\text{*31·1·11}]\)</p> + +<p class="nind"><b>*31·12.</b> \(\vdash.\breve{P}=\text{Cnv}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·101}.&\supset\vdash:Q \text{Cnv} P.\breve{P}\,\text{Cnv}\, P.\supset.Q=\breve{P}:\\ +[\text{*31·111}]&\supset\vdash:Q\, \text{Cnv}\, P.\supset.Q=\breve{P} &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*31·111}.&\supset\\ +&\vdash:\breve{P} \text{Cnv} P:Q\, \text{Cnv}\, P.\supset_{Q}.Q=\breve{P}:\\ +[\text{*30·31}] &\supset\vdash.\breve{P}=\text{Cnv}ʻP +\end{array} +\]</p> + +<p class="nind"><b>*31·13.</b> \(\vdash.\text{E}!\text{Cnv}ʻP \quad[\text{*14·21.*31·12}]\)</p> + +<p class="nind"><b>*31·131.</b> \(\vdash:x(\text{Cnv}ʻP)y.\equiv.yPx \quad[\text{*31·11·12.*21·43}]\)</p> + +<p class="nind"><b>*31·132.</b> \(\vdash:Q\, \text{Cnv}\, P.\equiv.Q=\text{Cnv}ʻP.\equiv.Q=\breve{P} \quad[\text{*30·4.*31·13·12}]\)</p> + +<p class="nind"><b>*31·14.</b> \(\vdash.\text{Cnv}ʻ(P\dot{\cap}Q)=\text{Cnv}ʻP\dot{\cap}\text{Cnv}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\vdash:x\{\text{Cnv}ʻ(P\dot{\cap}Q)\}y.&\equiv.y(P\dot{\cap}Q)x.\\ +[\text{*21·33}] &\equiv.yPx.yQx.\\ +[\text{*31·131}] &\equiv.x(\text{Cnv}ʻP)y.x(\text{Cnv}ʻQ)y.\\ +[\text{*21·33}] &\equiv.x\{\text{Cnv}ʻP\dot{\cap}\text{Cnv}ʻQ\}y &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·15.</b> \(\vdash.\text{Cnv}ʻ(P\unicode{x228d}Q)=\text{Cnv}ʻP\unicode{x228d}\text{Cnv}ʻQ \quad[\text{Similar proof}]\)</p> + +<p><span class="pagenum" id="Page_253">[Pg 253]</span></p> + +<p class="nind"><b>*31·16.</b> \(\vdash.\text{Cnv}ʻ\dot{-}P=\dot{-}(\text{Cnv}ʻP)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\vdash:x(\text{Cnv}ʻ\dot{-}P)y.&\equiv.y\dot{-}Px.\\ +[\text{*23·35}] &\equiv.{\sim}(yPx).\\ +[\text{*31·131}] & \equiv.{\sim}{x(\text{Cnv}ʻP)y}.\\ +[\text{*21·35}] & \equiv.x{\dot{-}(\text{Cnv}ʻP)}y &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·17.</b> \(\vdash\colon\ldotp y=\breve{P}ʻx.\equiv:xPz.\equiv_{z}.z=y \quad[\text{*30·3.*31·11}]\)</p> + +<p class="nind"><b>*31·18.</b> \(\vdash\colon\ldotp \text{E}!\breve{P}ʻx.\equiv:(\exists y):xPz.\equiv_{z}.z=y \quad[\text{*30·2.*31·11}]\)</p> + +<p class="nind"><b>*31·21.</b> \(\vdash.\text{Cnv}ʻ\dot{\Lambda}=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*31·131}. \supset\vdash:x(\text{Cnv}ʻ\dot{\Lambda})y.\equiv.y\dot{\Lambda}x:\\ +&[\text{*25·105}] \supset\vdash.{\sim}x(\text{Cnv}ʻ\dot{\Lambda})y &\qquad \text{(1)}\\ +&\vdash.\text{(1).*11·11.*25·15}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·22.</b> \(\vdash.\text{Cnv}ʻ\dot{\text{V}}=\dot{\text{V}} \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*31·23.</b> \(\vdash:\breve{P}=\dot{\text{V}}.\equiv.P=\dot{\text{V}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*25·14}.\supset\vdash:\breve{P}=\dot{\text{V}}.&\equiv.(x,y).x\breve{P}y.\\ +[\text{*31·11.*11·33}] &\equiv.(x,y).yPx.\\ +[\text{*11·2}] &\equiv.(y,x).yPx.\\ +[\text{*25·14}] &\equiv.P=\dot{\text{V}}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·24.</b> \(\vdash:\breve{P}=\dot{\Lambda}.\equiv.P=\dot{\Lambda} \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*31·32.</b> \(\vdash:P=Q.\equiv.\breve{P}=\breve{Q}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·43}.\supset\vdash\colon\ldotp P=Q.&\equiv:xPy.\equiv_{x,y}.xQy:\\ +[\text{*4·86·21.*31·11}] &\equiv:y\breve{P}x.\equiv_{x,y}.y\breve{Q}x:\\ +[\text{*11·2}] &\equiv:y\breve{P}x.\equiv_{y,x}.y\breve{Q}x:\\ +[\text{*21·43}] &\equiv:\breve{P}=\breve{Q}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·33.</b> \(\vdash.\text{Cnv}ʻ\text{Cnv}ʻP=P\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\vdash:x(\text{Cnv}ʻ\text{Cnv}ʻP)y.&\equiv.y(\text{Cnv}ʻP)x.\\ +[\text{*31·131}] &\equiv.xPy &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_254">[Pg 254]</span></p> + +<p class="nind"><b>*31·34.</b> \(\vdash:P=\breve{Q}.\equiv.Q=\breve{P}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·32}.\supset\vdash:P=\breve{Q}.\equiv.\breve{P}&=\text{Cnv}ʻ\breve{Q}\\ +[\text{*31·12·32}] &=\text{Cnv}ʻ\text{Cnv}ʻQ\\ +[\text{*31·33}] &=Q:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·4.</b> \(\vdash:P\unicode{x2abd}Q.\equiv.\breve{P}\unicode{x2abd}\breve{Q} \quad[\text{*31·11.*11·33}]\)</p> + +<p class="nind"><b>*31·41.</b> \(\vdash:P\unicode{x2abd}\breve{Q}.\equiv.\breve{P}\unicode{x2abd}Q \quad[\text{*31·4·33·12}]\)</p> + +<p class="nind"><b>*31·5.</b> \(\vdash:\dot{\exists}!P.\equiv.\dot{\exists}!\breve{P} \quad[\text{*31·24.Transp.*25·54}]\)</p> + +<p class="nind"><b>*31·51.</b> \(\vdash:(P).f\breve{P}.\equiv.(P).fP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·1}. &\supset\vdash:(P).fP.\supset.f\breve{P}:\\ +[\text{*10·11·21}]&\supset\vdash:(P).fP.\supset.(P).f\breve{P} &\qquad \text{(1)}\\ +\vdash.\text{*10·1.*31·12}.&\supset\\ +&\vdash:(P).f\breve{P}.\supset.f(\text{Cnv}ʻ\breve{P}).\\ +[\text{*31·33·12}] &\supset.fP:\\ +[\text{*10·11·21}]&\supset\vdash:(P).f\breve{P}.\supset.(P).fP &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*31·52.</b> \(\vdash:(\exists P).f\breve{P}.\equiv.(\exists P).fP \quad[\text{*31·51.Transp}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_255">[Pg 255]</span></p> + +<h2 class="nobreak" id="*32">*32. REFERENTS AND RELATA OF A GIVEN TERM WITH RESPECT +TO A GIVEN RELATION.</h2> +</div> + + +<p><i>Summary of</i> *32.</p> + +<p>Given any relation \(R\), the class of terms which have the relation +\(R\) to a given term \(y\) are called the <i>referents</i> of \(y\), +and the class of terms to which a given term \(x\) has the relation +\(R\) are called the <i>relata</i> of \(x\). We shall denote by +\(\overrightarrow{R}\) the relation of the class of referents of \(y\) +to \(y\), and by \(\overleftarrow{R}\) the relation of the class of +relata of \(x\) to \(x\). It is convenient also to have a notation +for the relations of \(\overrightarrow{R}\) and \(\overleftarrow{R}\) +to \(R\). We shall denote the relation of \(\overrightarrow{R}\) +to \(R\) by "\(\text{sg}\)," where "\(\text{sg}\)" stands for +"sagitta." Similarly we shall denote by "\(\text{gs}\)" the relation +of \(\overleftarrow{R}\) to \(R\), to suggest an arrow running from +right to left instead of from left to right. \(\overrightarrow{R}\) +and \(\overleftarrow{R}\) are chiefly useful for the sake +of the descriptive functions to which they give rise; thus +\(\overrightarrow{R}ʻy = \hat{x}(xRy)\) and \(\overleftarrow{R}ʻx= \hat{y}(xRy)\). +Thus <i>e.g.</i> if \(R\) is the relation of +parent to son, \(\overrightarrow{R}ʻy =\) the parents of \(y\), +\(\overleftarrow{R}ʻx =\) the sons of \(x\). If \(R\) is the relation +of less to greater among numbers of any kind, \(\overrightarrow{R}ʻy =\) +numbers less than \(y\), and \(\overleftarrow{R}ʻx =\) numbers +greater than \(x\). When \(Rʻy\) exists, \(\overrightarrow{R}ʻy\) is +the class whose only member is \(Rʻy\). But when there are many terms +having the relation \(R\) to \(y\), \(\overrightarrow{R}ʻy\), which is +the class of those terms, supplies a notation which cannot be supplied +by \(Rʻy\). And similarly if there are many terms to which \(x\) has +the relation \(R\), \(\overleftarrow{R}ʻx\) supplies the notation for +these terms. Thus for example let \(R\) be the relation "\(\sin\)," +<i>i.e.</i> the relation which \(x\) has to \(y\) when \(x = \sin y\). +Then "\(\overleftarrow{\sin}ʻx\)" represents all values of \(y\) +such that \(x = \sin y\), <i>i.e.</i> all values of \(\sin^{-1}x\) or +\(\arcsin x\). Unlike the usual symbol, it is not ambiguous, since +instead of representing some one of these values, it represents the +class of them.</p> + +<p>The definitions of \(\overrightarrow{R}\), \(\overleftarrow{R}\), +\(\text{sg}\), \(\text{gs}\) are as follows:</p> + +<p class="nind"><b>*32·01.</b> \(\overrightarrow{R} = \hat{\alpha}\hat{y}\{\alpha = \hat{x}(xRy)\} \quad \text{Df}\)</p> + +<p class="nind"><b>*32·02.</b> \(\overleftarrow{R} = \hat{\beta}\hat{x}\{\beta = \hat{y}(xRy)\} \quad \text{Df}\)</p> + +<p><span class="pagenum" id="Page_256">[Pg 256]</span></p> + +<p class="nind"><b>*32·03.</b> \(\text{sg} = \hat{A}\hat{R}(A = \overrightarrow{R}) \quad \text{Df}\)</p> + +<p class="nind"><b>*32·04.</b> \(\text{gs} = \hat{A}\hat{R}(A = \overleftarrow{R}) \quad \text{Df}\)</p> + +<p>In virtue of the above definitions, we shall have \(\text{sg}ʻR =\overrightarrow{R}\), +\(\text{gs}ʻR=\overleftarrow{R}\). This gives an +alternative notation which is convenient in dealing with a relation not +represented by a single letter.</p> + +<p>It should be observed that if \(R\) is a homogeneous relation +(<i>i.e.</i> one in which referents and relata are of the same +type), then \(\overrightarrow{R}\) and \(\overleftarrow{R}\) are not +homogeneous, but relate a class to objects of the type of its members.</p> + +<p>In virtue of the definitions of \(\overrightarrow{R}\) and +\(\overleftarrow{R}\), we shall have</p> + +<p class="nind"><b>*32·13.</b> \(\vdash. \overrightarrow{R}ʻy = \hat{x}(xRy)\)</p> + +<p class="nind"><b>*32·131.</b> \(\vdash. \overleftarrow{R}ʻx = \hat{y}(xRy)\)</p> + +<p>Thus by <a href="#*14·21">*14·21</a>, we always have \(\text{E}!\overrightarrow{R}ʻy\) +and \(\text{E}!\overleftarrow{R}ʻx\). Thus whatever relation +\(R\) may be, we have \((y).\text{E}!\overrightarrow{R}ʻy\) +and \((x).\text{E}!\overleftarrow{R}ʻx\). We do not in general +have \((y). \exists! \overrightarrow{R}ʻy\) or (\(x). \exists!\overleftarrow{R}ʻx\). +Thus taking \(R\) to be the relation of parent and child, +\(\overrightarrow{R}ʻy = \text{the parents of} ~ y\) and +\(\overleftarrow{R}ʻx = \text{the children of} ~ x\). Thus +\(\overleftarrow{R}ʻx = \Lambda\), <i>i.e.</i> +\({\sim}\exists! \overleftarrow{R}ʻx\) when \(x\) is childless, and +\(\overrightarrow{R}ʻy = \Lambda\), <i>i.e.</i> \({\sim}\exists!\overrightarrow{R}ʻy\), +when \(y\) is Adam or Eve. The two sorts +of existence, \(\text{E}!\overrightarrow{R}ʻy\) and \(\exists!\overrightarrow{R}ʻy\), +can both be <i>significantly</i> predicated +of \(\overrightarrow{R}ʻy\), because "\(\overrightarrow{R}ʻy\)" is a +descriptive function whose value is a class; and the same applies to +\(\overleftarrow{R}ʻx\). It will be seen that (by *14·21) +\(\exists!\overrightarrow{R}ʻy .\supset. \text{E}!\overrightarrow{R}ʻy\), +but the converse implication does not hold in general.</p> + +<p>We have</p> + +<p class="nind"><b>*32·16.</b> \(\vdash: \overrightarrow{R} = \overrightarrow{S} .\equiv. \overleftarrow{R} = \overleftarrow{S} .\equiv. R = S\)</p> + +<p>Also by *32·18·181, +\[ +\vdash: x \in \overrightarrow{R}ʻy .\equiv. xRy .\equiv. y \in \overleftarrow{R}ʻx\text{.} +\]</p> + +<p>Thus by the use of \(\overrightarrow{R}ʻy\) or \(\overleftarrow{R}ʻx\), +every statement of the form "\(xRy\)" can be reduced to a statement +asserting membership of a class. Since, however, the class in question +is given by a descriptive function, and descriptive functions are +defined by means of relations, we do not thus obtain a method of +reducing the theory of relations to the theory of classes.</p> + +<hr class="tb"> + +<p class="nind"><b>*32·01.</b> \(\overrightarrow{R} = \hat{\alpha}\hat{y}\{\alpha = \hat{x}(x R y)\} \quad \text{Df}\)</p> + +<p class="nind"><b>*32·02.</b> \(\overleftarrow{R} = \hat{\beta}\hat{x}\{\beta = \hat{y}(xRy)\} \quad \text{Df}\)</p> + +<p><span class="pagenum" id="Page_257">[Pg 257]</span></p> + +<p class="nind"><b>*32·03.</b> \(\text{sg}=\hat{A}\hat{R}(A=\overrightarrow{R}) \quad\text{Df}\)</p> + +<p class="nind"><b>*32·04.</b> \(\text{gs}=\hat{A}\hat{R}(A=\overleftarrow{R}) \quad\text{Df}\)</p> + +<p class="nind"><b>*32·1.</b> \(\vdash:\alpha\overrightarrow{R}y.\equiv.\alpha=\hat{x}(xRy) \quad[\text{*21·3.(*32·01)}]\)</p> + +<p class="nind"><b>*32·101.</b> \(\vdash:\beta\overleftarrow{R}x.\equiv.\beta=\hat{y}(xRy) \quad[\text{*21·3.(*32·02)}]\)</p> + +<p class="nind"><b>*32·11.</b> \(\vdash.\hat{x}(xRy)=\overrightarrow{R}ʻy \quad[\text{*32·1.*30·3}]\)</p> + +<p class="nind"><b>*32·111.</b> \(\vdash.\hat{y}(xRy)=\overleftarrow{R}ʻx \quad[\text{*32·101.*30·3}]\)</p> + +<p class="nind"><b>*32·12.</b> \(\vdash.\text{E}!\overrightarrow{R}ʻy \quad[\text{*32·11.*14·21}]\)</p> + +<p class="nind"><b>*32·121.</b> \(\vdash.\text{E}!\overleftarrow{R}ʻx \quad[\text{*32·111.*14·21}]\)</p> + +<p>"\(\text{E}!\overrightarrow{R}ʻy\)" must not be confounded with +"\(\exists !\overrightarrow{R}ʻy\)." The former means that there is +such a class as \(\overrightarrow{R}ʻy\), which, as we have just seen, +is always true; the latter means that \(\overrightarrow{R}ʻy\) is +not null, which is only true if \(y\) is a term to which some other +term has the relation \(R\). Note that, by <a href="#*14·21">*14·21</a>, both \(\exists!\overrightarrow{R}ʻy\) +and \({\sim}\exists !\overrightarrow{R}ʻy\) +imply \(\text{E}!\overrightarrow{R}ʻy\). The contradictory of \(\exists!\overrightarrow{R}ʻy\) +is not \({\sim}\exists !\overrightarrow{R}ʻy\), +but \({\sim}\{[\overrightarrow{R}ʻy].\exists !\overrightarrow{R}ʻy\}\). +This last would not imply \(\text{E}!\overrightarrow{R}ʻy\), but for +the fact that \(\text{E}!\overrightarrow{R}ʻy\) is always true.</p> + +<p class="nind"><b>*32·13.</b> \(\vdash.\overrightarrow{R}ʻy=\hat{x}(xRy) \quad[\text{*32·11.*20·59}]\)</p> + +<p class="nind"><b>*32·131.</b> \(\vdash.\overleftarrow{R}ʻx=\hat{y}(xRy) \quad[\text{*32·111.*20·59}]\)</p> + +<p class="nind"><b>*32·132.</b> \(\vdash:\alpha\overrightarrow{R}y.\equiv.\alpha=\overrightarrow{R}ʻy.\equiv.\alpha=\hat{x}(xRy) \quad[\text{*32·1·13.*20·57}]\)</p> + +<p class="nind"><b>*32·133.</b> \(\vdash:\beta\overleftarrow{R}x.\equiv.\beta=\overleftarrow{R}ʻx.\equiv.\beta=\hat{y}(xRy) \quad[\text{*32·101·131.*20·57}]\)</p> + +<p>The use of <a href="#*20·57">*20·57</a> will in general be tacit. It happens constantly that +we have propositions such as *32·13, in which a descriptive expression +is shown to be identical with a class. In such cases, whenever the +properties of the class are asserted of the descriptive expression, +*20·57 is relevant.</p> + +<p class="nind"><b>*32·14.</b> \(\vdash:\overrightarrow{R}=\overrightarrow{S}.\equiv.R=S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·43}.\supset\vdash\colon\colon \overrightarrow{R}=\overrightarrow{S}.&\equiv\colon\ldotp \alpha\overrightarrow{R}y.\equiv_{\alpha,y}.\alpha\overrightarrow{S}y\colon\ldotp + \\ +[\text{*32·1}] & \equiv\colon\ldotp \alpha=\hat{x}(xRy).\equiv_{\alpha,y}.\alpha=\hat{x}(xSy)\colon\ldotp \\ +[\text{*11·2}] &\equiv\colon\ldotp (y)\colon\ldotp \alpha=\hat{x}(xRy).\equiv_{\alpha}.\alpha=\hat{x}(xSy)\colon\ldotp \\ +[\text{*20·25}] &\equiv\colon\ldotp (y):\hat{x}(xRy)=\hat{x}(xSy)\colon\ldotp \\ +[\text{*20·15}] &\equiv\colon\ldotp (y)\colon\ldotp (x):xRy.\equiv.xSy\colon\ldotp \\ +[\text{*11·2}] & \equiv\colon\ldotp (x,y):xRy.\equiv.xSy\colon\ldotp \\ +[\text{*21·43}] &\equiv\colon\ldotp R=S\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_258">[Pg 258]</span></p> + +<p class="nind"><b>*32·15.</b> \(\vdash:\overleftarrow{R}=\overleftarrow{S}.\equiv.R=S \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*32·16.</b> \(\vdash:\overrightarrow{R}=\overrightarrow{S}.\equiv.\overleftarrow{R}=\overleftarrow{S}.\equiv.R=S \quad[\text{*32·14·15}]\)</p> + +<p class="nind"><b>*32·18.</b> \(\vdash:x\in \overrightarrow{R}ʻy.\equiv.xRy \quad[\text{*32·13.*20·33}]\)</p> + +<p class="nind"><b>*32·181.</b> \(\vdash:y\in \overleftarrow{R}ʻx.\equiv.xRy \quad[\text{*32·131.*20·33}]\)</p> + +<p class="nind"><b>*32·182.</b> \(\vdash:x\in \overrightarrow{R}ʻy.\equiv.y\in \overleftarrow{R}ʻx \quad[\text{*32·18·181}]\)</p> + +<p>The transformation from "\(xRy\)" to "\(x\in \overrightarrow{R}ʻy\)" +is one commonly effected in language. <i>E.g.</i> suppose "\(xRy\)" is +"\(x\) loves \(y\)," then "\(x\in \overrightarrow{R}ʻy\)" is "\(x\) is +a lover of \(y\)."</p> + +<p class="nind"><b>*32·19.</b> \(\vdash:R\unicode{x2abd}S.\supset.\overrightarrow{R}ʻy\subset\overrightarrow{S}ʻy.\overleftarrow{R}ʻx\subset\overleftarrow{S}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:x\in \overrightarrow{R}ʻy.\supset_{x}.x\in \overrightarrow{S}ʻy:\\ +[\text{*22·1}] &\supset:\overrightarrow{R}ʻy\subset\overrightarrow{S}ʻy &\qquad \text{(1)}\\ +\vdash.\text{*32·181}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \overleftarrow{R}ʻx.\supset_{y}.y\in \overleftarrow{S}ʻx:\\ +[\text{*22·1}] &\supset:\overleftarrow{R}ʻx\subset\overleftarrow{S}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*32·2.</b> \(\vdash:A \text{sg} R.\equiv.A=\overrightarrow{R} \quad[\text{*21·3.(*32·03)}]\)</p> + +<p class="nind"><b>*32·201.</b> \(\vdash:A \text{gs} R.\equiv.A=\overleftarrow{R} \quad[\text{*21·3.(*32·04)}]\)</p> + +<p class="nind"><b>*32·21.</b> \(\vdash.\overrightarrow{R}=\text{sg}ʻR \quad[\text{*32·2.*30·3}]\)</p> + +<p class="nind"><b>*32·211.</b> \(\vdash.\overleftarrow{R}=\text{gs}ʻR \quad[\text{*32·201.*30·3}]\)</p> + +<p class="nind"><b>*32·22.</b> \(\vdash.\text{E}!\text{sg}ʻR \quad[\text{*32·21.*14·21}]\)</p> + +<p class="nind"><b>*32·221.</b> \(\vdash.\text{E}!gsʻR \quad[\text{*32·211.*14·21}]\)</p> + +<p class="nind"><b>*32·23.</b> \(\vdash.\text{sg}ʻR=\overrightarrow{R} \quad[\text{*32·21.*21·2·57}]\)</p> + +<p class="nind"><b>*32·231.</b> \(\vdash.\text{gs}ʻR=\overleftarrow{R} \quad[\text{*32·211.*21·2·57}]\)</p> + +<p class="nind"><b>*32·24.</b> \(\vdash.\text{sg}ʻ\breve{R}=\text{gs}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·23.(*32·01)}. &\supset\vdash.\text{sg}ʻ\breve{R}=\hat{\alpha}\hat{y}\{\alpha=\hat{x}(x\breve{R}y)\}.\\ +[\text{*21·33}] &\supset\vdash:\alpha(\text{sg}ʻ\breve{R})y.\equiv.\alpha=\hat{x}(x\breve{R}y).\\ +[\text{*31·11.*20·15}] & \qquad\qquad\qquad\equiv.\alpha=\hat{x}(yRx).\\ +[\text{*32·101}] &\qquad\qquad\qquad\equiv.\alpha\overleftarrow{R}x.\\ +[\text{*32·211}] &\qquad\qquad\qquad\equiv.\alpha({gs}ʻR)x &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_259">[Pg 259]</span></p> + +<p class="nind"><b>*32·241.</b> \(\vdash.\text{gs}ʻ\breve{R}=\text{sg}ʻR \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*32·25.</b> \(\vdash:A \text{sg} R.\equiv.A=\text{sg}ʻR \quad[\text{*30·4.*32·22}]\)</p> + +<p class="nind"><b>*32·251.</b> \(\vdash:A \text{gs} R.\equiv.A=\text{gs}ʻR \quad[\text{*30·4.*32·221}]\)</p> + +<p class="nind"><b><a id="*32·3">*32·3</a>.</b> \(\vdash.{\text{sg}ʻ(R\dot{\cap}S)}ʻy=\overrightarrow{R}ʻy\cap \overrightarrow{S}ʻy\)</p> + +<p>Notice that we do not have +\[ +\text{sg}ʻ(R\dot{\cap}S)=\text{sg}ʻR\dot{\cap}\text{sg}ʻS. +\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·23·13}.\supset\vdash.\{\text{sg}ʻ(R\dot{\cap}S)\}ʻy&=\hat{x}\{x(R\dot{\cap}S)y\}\\ +[\text{*23·33}] & =\hat{x}(xRy.xSy)\\ +[\text{*22·39}] &=\hat{x}(xRy)\cap \hat{x}(xSy)\\ +[\text{*32·13}] & =\overrightarrow{R}ʻy\cap \overrightarrow{S}ʻy.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*32·31.</b> \(\vdash.\{\text{gs}ʻ(R\dot{\cap}S)\}ʻx=\overleftarrow{R}ʻx\cap \overleftarrow{R}ʻx\)</p> + +<p class="nind"><b>*32·32.</b> \(\vdash.\{\text{sg}ʻ(R\unicode{x228d}S)\}ʻy=\overrightarrow{R}ʻy\cup \overrightarrow{S}ʻy\)</p> + +<p class="nind"><b>*32·33.</b> \(\vdash.\{\text{gs}ʻ(R\unicode{x228d}S)\}ʻx=\overleftarrow{R}ʻx\cup \overleftarrow{R}ʻx\)</p> + +<p class="nind"><b>*32·34.</b> \(\vdash.\{\text{sg}ʻ(\dot{-}R)\}ʻy=-\overrightarrow{R}ʻy\)</p> + +<p class="nind"><b>*32·35.</b> \(\vdash.\{\text{gs}ʻ(\dot{-}R)\}ʻx=-\overleftarrow{R}ʻx\)</p> + +<p>The proofs of the above propositions are similar to that of <a href="#*32·3">*32·3</a>.</p> + +<p class="nind"><b>*32·4.</b> \(\vdash\colon\ldotp \text{E}!Rʻz.\equiv:\exists !\overrightarrow{R}ʻz:x,y\in \overrightarrow{R}ʻz.\supset_{x,y}.x=y \quad[\text{*30·21.*32·18}]\)</p> + +<p class="nind"><b>*32·41.</b> \(\vdash\colon\ldotp \text{E}!Sʻy.\supset:\overrightarrow{R}ʻy=\overrightarrow{S}ʻy.\equiv.Rʻy=Sʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·86}. \supset\vdash\colon\colon xSy.\equiv_{x}.x=b:\supset\colon\ldotp \\ +\qquad\qquad\qquad xRy.\equiv_{x}.xSy:\equiv:xRy.\equiv_{x}.x=b &\qquad \text{(1)}\\ +\vdash.\text{(1).*5·32}.\supset\vdash\colon\ldotp xSy.\equiv_{x}.x=b:xRy.\equiv_{x}.xSy:\equiv:\\ +\qquad\qquad\qquad xSy.\equiv_{x}.x=b:xRy.\equiv_{x}.x=b &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·281.*32·18·181}.\supset\\ +\vdash\colon\ldotp (\exists b):xSy.\equiv_{x}.x=b:\overrightarrow{R}ʻy=\overrightarrow{S}ʻy:\equiv:(\exists b):xSy.\equiv_{x}.x=b:xRy.\equiv_{x}.x=b:\\ +[\text{*30·3.*14·13}] \qquad\equiv:(\exists b):xSy.\equiv_{x}.x=b:Rʻy=b:\\ +[\text{*14·101}] \qquad\qquad\equiv:Rʻy=Sʻy &\qquad \text{(3)}\\ +\vdash.\text{(3).*30·2}.\supset\vdash\colon\ldotp \text{E}!Sʻy.\overrightarrow{R}ʻy=\overrightarrow{S}ʻy.\equiv.Rʻy=Sʻy\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*32·42.</b> \(\vdash\colon\ldotp \overrightarrow{R}ʻy=\overrightarrow{S}ʻy.\supset:\text{E}!Rʻy.\equiv.\text{E}!Sʻy \quad[\text{*30·34.*32·18}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_260">[Pg 260]</span></p> +<h2 class="nobreak" id="*33">*33. DOMAINS, CONVERSE DOMAINS, AND FIELDS OF RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *33.</p> + +<p>If \(R\) is any relation, the <i>domain</i> of \(R\), which we denote +by \(\text{D}ʻR\), is the class of terms which have the relation \(R\) +to something or other; the <i>converse domain</i>, \(\text{ᗡ}ʻR\), +is the class of terms to which something or other has the relation +\(R\); and the <i>field</i>, \(CʻR\), is the sum of the domain and the +converse domain. (Note that the field is only significant when \(R\) is +a <i>homogeneous</i> relation.)</p> + +<p>The above notations \(\text{D}ʻR\), \(\text{ᗡ}ʻR\), \(CʻR\) are +derivative from the notations \(\text{D}\), \(\text{ᗡ}\), \(C\) for the +relations, to a relation, of its domain, converse domain, and field +respectively. We are to have +\[ +\begin{align} +\text{D}ʻR &= \hat{x}\{(\exists y).xRy\}\\ +\text{ᗡ}ʻR &= \hat{y}\{(\exists x).xRy\}\\ +CʻR &= \hat{x}\{(\exists y):xRy .\lor. yRx\}\text{;}\\ +\end{align} +\] +hence we define \(\text{D}\), \(\text{ᗡ}\), \(C\) as follows:</p> + +<p class="nind"><b>*33·01.</b> \(\text{D} = \hat{\alpha}\hat{R}[\alpha = \hat{x}\{(\exists y).xRy\}] \quad \text{Df}\)</p> + +<p class="nind"><b>*33·02.</b> \(\text{ᗡ} = \hat{\beta}\hat{R}[\beta = \hat{y}\{(\exists x).xRy\}] \quad \text{Df}\)</p> + +<p class="nind"><b>*33·03.</b> \(C = \hat{\gamma}\hat{R}[\gamma = \hat{x}\{(\exists y):xRy .\lor. yRx\}] \quad \text{Df}\)</p> + +<p>The letter \(C\) is chosen as the initial of the word "campus." We +require one other definition, namely of the relation of \(x\) to \(R\) +when \(x\) is a member of the field of \(R\). This relation, which we +will call \(F\), is defined as follows:</p> + +<p class="nind"><b>*33·04.</b> \(F = \hat{x}\hat{R}\{(\exists y):xRy .\lor. yRx\} \quad \text{Df}\)</p> + +<p>We shall find that \(C = \overrightarrow{F}\). \(\breve{\text{D}}\) +will be the relation of a relation to its domain, +\(\overleftarrow{\text{D}}ʻ\alpha\) will be the class of relations +having \(\alpha\) for their domain. Similar remarks apply to +\(\text{ᗡ}\) and \(C\). The <i>field</i> of a relation is specially +important in connection with series.</p> + +<p>The propositions of this number are constantly used throughout the +remainder of the work. The ideas of the domain, converse domain, and +field are very general, and have somewhat different uses for relations +of different<span class="pagenum" id="Page_261">[Pg 261]</span> kinds. Consider first the sort of relation that gives +rise to a descriptive function \(Rʻy\). For this we require that +\(Rʻy\) should exist whenever there is anything having the relation +\(R\) to \(y\), <i>i.e.</i> that there should never be more than one +term having the relation \(R\) to a given term \(y\). In this case, +the values of \(y\) for which \(Rʻy\) exists will constitute the +"converse domain" of \(R\), <i>i.e.</i> \(\text{ᗡ}ʻR\), and the values +which \(Rʻy\) assumes for various values of \(y\) will constitute +the "domain" of \(R\), <i>i.e.</i> \(\text{D}ʻR\). Thus the converse +domain is the class of possible arguments for the descriptive function +\(Rʻy\), and the domain is the class of all values of the function. +Thus, for example, if \(R\) is the relation of the square of an integer +\(y\) to \(y\), then \(Rʻy\)=the square of \(y\), provided \(y\) is +an integer. In this case, \(\text{ᗡ}ʻR\) is the class of integers, +and \(\text{D}ʻR\) is the class of perfect squares. Or again, suppose +\(R\) is the relation of wife to husband; then \(Rʻy\)=the wife of +\(y\), \(\text{ᗡ}ʻR\)=married men, \(\text{D}ʻR\)=married women. In +such cases, the <i>field</i> usually has little importance; and if +the values of the function \(Rʻy\) are not of the same type as its +arguments, <i>i.e.</i> if the relation \(R\) is not <i>homogeneous</i>, +the field is meaningless. Thus, for example, if \(R\) is a homogeneous +relation, \(\overrightarrow{R}\) and \(\overleftarrow{R}\) are +not homogeneous, and therefore "\(Cʻ\overrightarrow{R}\)" and +"\(Cʻ\overleftarrow{R}\)" are meaningless.</p> + +<p>Let us next suppose that \(R\) is the sort of relation that generates +a series, say the relation of less to greater among integers. Then +\(\text{D}ʻR\)= all integers that are less than some other integer = +all integers, \(\text{ᗡ}ʻR\)= all integers that are greater than some +other integer = all integers except 0. In this case, \(CʻR\)= all +integers that are either greater or less than some other integer = all +integers. Generally, if \(R\) generates a series, \(\text{D}ʻR\) = all +members of the series except the last (if any), \(\text{ᗡ}ʻR\) = all +members of the series except the first (if any), and \(CʻR\) = all +members of the series. In this case, "\(xFR\)" expresses the fact that +\(x\) is a member of the series. Thus when \(R\) generates a series, +\(CʻR\) becomes important, and the relation \(F\) is likely to be +useful.</p> + +<p>We shall have occasion to deal with many relations having some of +the properties of series, and with many propositions which, though +only important in connection with serial relations, hold much more +generally. In such cases, the field of a relation is likely to be +important. Thus in the section on Induction (Part II, <a href="#SECTION_E_b">Section E</a>), where +we are preparing the way for the construction of serial relations +by means of a certain kind of non-serial relation, and throughout +relation-arithmetic (Part IV), the fields of relations will occur +constantly. But in the earlier parts of the work, it is chiefly domains +and converse domains that occur.</p> + +<p>Among the more important properties of domains, converse domains and +fields, which are proved in the present number, are the following.</p> + +<p><span class="pagenum" id="Page_262">[Pg 262]</span></p> + +<p>We have always \(\text{E}!\text{D}ʻR\), \(\text{E}!\text{ᗡ}ʻR\), +\(\text{E}!CʻR\) (*33·12·121·122). (The last of these, however, is only +significant when \(R\) is homogeneous.)</p> + +<p class="nind"><b>*33·13.</b> \(\vdash:x\in \text{D}ʻR.\equiv.(\exists y).xRy\)</p> + +<p class="nind"><b>*33·131.</b> \(\vdash:y\in \text{ᗡ}ʻR.\equiv.(\exists x).xRy\)</p> + +<p class="nind"><b>*33·132.</b> \(\vdash\colon\ldotp x\in CʻR.\equiv:(\exists y):xRy.\lor .yRx\)</p> + +<p class="nind"><b>*33·14.</b> \(\vdash:xRy.\supset.x\in \text{D}ʻR.y\in \text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*33·16.</b> \(\vdash.CʻR=\text{D}ʻR\cup \text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*33·2·21·22.</b> The converse domain of a relation is the domain of +its converse, the domain of a relation is the converse domain of its +converse, and the field of a relation is the field of its converse.</p> + +<p class="nind"><b>*33·24.</b> \(\vdash:\exists !\text{D}ʻR.\equiv.\exists !\text{ᗡ}ʻR.\equiv.\exists !CʻR.\equiv.\dot{\exists}!R\)</p> + +<p class="nind"><b>*33·4.</b> \(\vdash.\text{D}ʻR=\hat{x}\{\exists !\overleftarrow{R}ʻx\}\)</p> + +<p>with corresponding propositions (*33·41·42) for \(\text{ᗡ}ʻR\) and +\(CʻR\).</p> + +<p class="nind"><b>*33·43.</b> \(\vdash:\text{E}!Rʻy.\supset.y\in \text{ᗡ}ʻR.Rʻy\in \text{D}ʻR\)</p> + +<p class="nind"><b>*33·431.</b> \(\vdash:(y).\text{E}!Rʻy.\supset.(\beta).\beta\subset\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*33·5.</b> \(\vdash.C=\overrightarrow{F}\)</p> + +<p class="nind"><b>*33·51.</b> \(\vdash:x\in CʻR.\equiv.xFR\)</p> + +<p>The proofs of propositions concerning \(\text{ᗡ}\) and \(C\) are +usually similar to those for \(\text{D}\), and are therefore often +omitted.</p> + +<hr class="tb"> + +<p class="nind"><b>*33·01.</b> \(\text{D}=\hat{\alpha}\hat{R}[\alpha=\hat{x}\{(\exists y).xRy\}] \quad\text{Df}\)</p> + +<p class="nind"><b>*33·02.</b> \(\text{ᗡ}=\hat{\beta}\hat{R}[\hat{\beta}=\hat{y}\{(\exists x).xRy\}] \quad\text{Df}\)</p> + +<p class="nind"><b>*33·03.</b> \(C=\hat{\gamma}\hat{R}[\gamma=\hat{x}\{(\exists y):xRy.\lor .yRx\}] \quad\text{Df}\)</p> + +<p class="nind"><b>*33·04.</b> \(F=\hat{x}\hat{R}\{(\exists y):xRy.\lor .yRx\} \quad\text{Df}\)</p> + +<p class="nind"><b>*33·1.</b> \(\vdash:\alpha\text{D}R.\equiv.\alpha=\hat{x}\{(\exists y).xRy\} \quad[\text{*21·3.(*33·01)}]\)</p> + +<p class="nind"><b>*33·101.</b> \(\vdash:\beta\text{ᗡ}R.\equiv.\beta=\hat{y}\{(\exists x).xRy\}\)</p> + +<p class="nind"><b>*33·102.</b> \(\vdash:\gamma CR.\equiv.\gamma=\hat{x}\{(\exists y):xRy.\lor .yRx\}\)</p> + +<p class="nind"><b>*33·103.</b> \(\vdash\colon\ldotp xFR.\equiv:(\exists y):xRy.\lor .yRx\)</p> + +<p class="nind"><b>*33·11.</b> \(\vdash.\text{D}ʻR=\hat{x}\{(\exists y).xRy\} \quad[\text{*33·1.*30·3.*20·59}]\)</p> + +<p class="nind"><b>*33·111.</b> \(\vdash.\text{ᗡ}ʻR=\hat{y}\{(\exists x).xRy\}\)</p> + +<p class="nind"><b>*33·112.</b> \(\vdash.CʻR=\hat{x}\{(\exists y):xRy.\lor .yRx\}\)</p> + +<p class="nind"><b>*33·12.</b> \(\vdash.\text{E}!\text{D}ʻR \quad[\text{*33·11.*14·21}]\)</p> + +<p class="nind"><b>*33·121.</b> \(\vdash.\text{E}!\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*33·122.</b> \(\vdash.\text{E}!CʻR\)</p> + +<p class="nind"><b>*33·123.</b> \(\vdash:\alpha\text{D}R.\equiv.\alpha=\text{D}ʻR \quad[\text{*30·4.*33·12}]\)</p> + +<p class="nind"><b>*33·124.</b> \(\vdash:\beta\text{ᗡ}R.\equiv.\beta=\text{ᗡ}ʻR \quad[\text{*30·4.*33·121}]\)</p> + +<p><span class="pagenum" id="Page_263">[Pg 263]</span></p> + +<p class="nind"><b>*33·125.</b> \(\vdash:\gamma CR.\equiv.\gamma=CʻR \quad[\text{*30·4.*32·123}]\)</p> + +<p class="nind"><b>*33·13.</b> \(\vdash:x\in \text{D}ʻR.\equiv.(\exists y).xRy \quad[\text{*33·11.*20·3·57}]\)</p> + +<p class="nind"><b>*33·131.</b> \(\vdash:y\in \text{ᗡ}ʻR.\equiv.(\exists x).xRy\)</p> + +<p class="nind"><b>*33·132.</b> \(\vdash\colon\ldotp x\in CʻR.\equiv:(\exists y):xRy.\lor .yRx\)</p> + +<p class="nind"><b>*33·14.</b> \(\vdash:xRy.\supset.x\in \text{D}ʻR.y\in \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·24}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists y).xRy:(\exists x).xRy:\\ +[\text{*33·13·131}] & \supset:x\in \text{D}ʻR.y\in \text{ᗡ}ʻR\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·15.</b> \(\vdash.\overrightarrow{R}ʻy\subset\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18}.\supset\vdash:x\in \overrightarrow{R}ʻy.&\supset_{x}.xRy.\\ +[\text{*10·24}] & \supset_{x}.(\exists y).xRy.\\ +[\text{*33·13}] &\supset_{x}.x\in \text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33.151.</b> \(\vdash.\overleftarrow{R}ʻx\subset\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*33·152.</b> \(\vdash.\overrightarrow{R}ʻx\cup \overleftarrow{R}ʻx\subset CʻR\)</p> + +<p class="nind"><b>*33·16.</b> \(\vdash.CʻR=\text{D}ʻR\cup \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·132.*10·42}.\supset\\ +\vdash\colon\ldotp x\in CʻR.\equiv:(\exists y).xRy.\lor .(\exists y).yRx:\\ +[\text{*33·13·131}]\qquad\equiv:x\in \text{D}ʻR.\lor .x\in \text{ᗡ}ʻR:\\ +[\text{*22·34}] \qquad\qquad\equiv:x\in \text{D}ʻR\cup \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*20·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·161.</b> \(\vdash.\text{D}ʻR\subset CʻR.\text{ᗡ}ʻR\subset CʻR \quad[\text{*33·16.*22·58}]\)</p> + +<p class="nind"><b>*33·17.</b> \(\vdash:xRy.\supset.x,y\in CʻR \quad[\text{*33·14·161}]\)</p> + +<p class="nind"><b>*33·18.</b> \(\vdash:\text{D}ʻR=\text{ᗡ}ʻR.\supset.\text{D}ʻR=CʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·56}.\supset\vdash:\text{D}ʻR=\text{ᗡ}ʻR.\supset.\text{D}ʻR&=\text{D}ʻR\cup \text{ᗡ}ʻR\\ +[\text{*33·16}] &=CʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·181.</b> \(\vdash:\text{ᗡ}ʻR\subset \text{D}ʻR.\equiv.\text{D}ʻR=CʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·62}.\supset\vdash:\text{ᗡ}ʻR\subset\text{D}ʻR.\equiv.\text{D}ʻR&=\text{D}ʻR\cup \text{ᗡ}ʻR\\ +[\text{*33·16}] &=CʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·182.</b> \(\vdash:\text{D}ʻR\subset\text{ᗡ}ʻR.\equiv.\text{ᗡ}ʻR=CʻR \quad[\text{Similar proof}]\)</p> + +<p>If \(R\) is the sort of relation which generates a series, +so that "\(xRy\)" may be read "\(x\) precedes \(y\)," then +\(\text{ᗡ}ʻR\subset\text{D}ʻR\) is the condition that the series may<span class="pagenum" id="Page_264">[Pg 264]</span> +have no last term, since it states that every term which follows some +term precedes some other term, and is therefore not the last of the +series.</p> + +<p class="nind"><b>*33·2.</b> \(\vdash.\text{ᗡ}ʻR=\text{D}ʻ\breve{R}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11.*10·11}.&\supset\vdash:xRy.\equiv_{x}.y\breve{R}x:\\ +[\text{*10·281}] &\supset\vdash:(\exists x).xRy.\equiv.(\exists x).y\breve{R}x:\\ +[\text{*33·13·131}] &\supset\vdash:y\in \text{ᗡ}ʻR.\equiv.y\in \text{D}ʻ\breve{R} &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*20·43}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·21.</b> \(\vdash.\text{D}ʻR=\text{ᗡ}ʻ\breve{R} \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*33·22.</b> \(\vdash.CʻR=Cʻ\breve{R}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·16·2·21}.\supset\vdash.CʻR&=\text{ᗡ}ʻ\breve{R}\cup\text{D}ʻ\breve{R}\\ +[\text{*33·16}] &=Cʻ\breve{R}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·24.</b> \(\vdash:\exists !\text{D}ʻR.\equiv.\exists !\text{ᗡ}ʻR.\equiv.\exists !CʻR.\equiv.\dot{\exists}!R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}. &\supset\vdash\colon\ldotp \exists !\text{D}ʻR.\equiv:(\exists x):(\exists y).xRy:\\ +[\text{*25·5.(*11·03)}] & \equiv:\dot{\exists}!R &\qquad \text{(1)}\\ +\vdash.\text{*33·131}.&\supset\vdash\colon\ldotp \exists !\text{ᗡ}ʻR.\equiv:(\exists y):(\exists x).xRy:\\ +[\text{*11·2}] &\equiv:(\exists x,y).xRy:\\ +[\text{*25·5}] &\equiv:\dot{\exists}!R &\qquad \text{(2)}\\ +\vdash.\text{*33·132}.&\supset\vdash\colon\colon \exists !CʻR.\equiv\colon\ldotp (\exists x)\colon\ldotp (\exists y):xRy.\lor.yRx\colon\ldotp \\ +[\text{*11·7}] &\equiv\colon\ldotp (\exists x,y).xRy\colon\ldotp \\ +[\text{*25·5}] & \equiv\colon\ldotp \dot{\exists}!R &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·241.</b> \[\begin{align}&\vdash:\text{D}ʻR=\Lambda.\equiv.\text{ᗡ}ʻR=\Lambda.\equiv.CʻR=\Lambda.\equiv.R=\dot{\Lambda}\\ +&[\text{*33·24.Transp.*24·51.*25·51}]\end{align}\]</p> + +<p class="nind"><b>*33·25.</b> \(\vdash.\text{D}ʻ(R\dot{\cap}S)\subset\text{D}ʻR\cap \text{D}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}.&\supset\vdash\colon\ldotp x\in \text{D}ʻ(R\dot{\cap}S).\equiv:(\exists y).x(R\dot{\cap}S)y:\\ +[\text{*21·33.*10·281}] &\equiv:(\exists y).xRy.xSy:\\ +[\text{*10·5}] &\supset:(\exists y).xRy:(\exists y).xSy:\\ +[\text{*33·13}] & \supset:x\in \text{D}ʻR.x\in \text{D}ʻS:\\ +[\text{*21·33}] &\supset:x\in \text{D}ʻR\cap \text{D}ʻS &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_265">[Pg 265]</span></p> + +<p class="nind"><b>*33·251.</b> \(\vdash.\text{ᗡ}ʻ(R\dot{\cap}S)\subset\text{ᗡ}ʻR\cap \text{ᗡ}ʻS \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*33·252.</b> \(\vdash.Cʻ(R\dot{\cap}S)\subset CʻR\cap CʻS \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*33·26.</b> \(\vdash.\text{D}ʻ(R\unicode{x228d}S)=\text{D}ʻR\cup \text{D}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*33·13}.\supset \vdash\colon\ldotp x\in \text{D}ʻ(R\unicode{x228d}S).&\equiv:(\exists y).x(R\unicode{x228d}S)y:\\ +[\text{*23·34.*10·281}] &\equiv:(\exists y):xRy.\lor.xSy:\\ +[\text{*10·42}] &\equiv:(\exists y).xRy:\lor:(\exists y).xSy:\\ +[\text{*33·13}] &\equiv:x\in \text{D}ʻR.\lor.x\in \text{D}ʻS:\\ +[\text{*22·34}] &\equiv:x\in \text{D}ʻR\cup \text{D}ʻS &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*20·43}.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·261.</b> \(\vdash.\text{ᗡ}ʻ(R\unicode{x228d}S)=\text{ᗡ}ʻR\cup \text{ᗡ}ʻS \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*33·262.</b> \(\vdash.Cʻ(R\unicode{x228d}S)=CʻR\cup CʻS \quad[\text{*33·26·261·16}]\)</p> + +<p class="nind"><b>*33·263.</b> \(\vdash:R\unicode{x2abd}S.\supset .\text{D}ʻR\subset\text{D}ʻS\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*23·1}.\supset \vdash\colon\ldotp \text{Hp}.&\supset :xRy.\supset _{x,y}.xSy:\\ +[\text{*10*28*27}] &\supset :(x):(\exists y).xRy.\supset .(\exists y).xSy:\\ +[\text{*33·13}] & \supset :(x):x\in \text{D}ʻR.\supset .x\in \text{D}ʻS:\\ +[\text{*23·1}] &\supset :\text{D}ʻR\subset\text{D}ʻS\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·264.</b> \(\vdash:R\unicode{x2abd}S.\supset .\text{ᗡ}ʻR\subset\text{ᗡ}ʻS \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*33·265.</b> \(\vdash:R\unicode{x2abd}S.\supset .CʻR\subset CʻS \quad[\text{*33·263·264·16.*22·72}]\)</p> + +<p class="nind"><b>*33·27.</b> \(\vdash.CʻR=\text{D}ʻ(R\unicode{x228d}\breve{R})\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*33·16·2}.\supset \vdash.CʻR&=\text{D}ʻR\cup \text{D}ʻ\breve{R}\\ +[\text{*33·26}] &=\text{D}ʻ(R\unicode{x228d}\breve{R}).\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·271.</b> \(\vdash.CʻR=\text{ᗡ}ʻ(R\unicode{x228d}\breve{R}) \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*33·272.</b> \(\vdash.\text{D}ʻ(R\unicode{x228d}\breve{R})=\text{ᗡ}ʻ(R\unicode{x228d}\breve{R})=Cʻ(R\unicode{x228d}\breve{R})=CʻR \quad[\text{*33·27·271·16}]\)</p> + +<p class="nind"><b>*33·28.</b> \(\vdash.\text{D}ʻ\text{V}=\text{ᗡ}ʻ\dot{\text{V}}=Cʻ\text{V}=\text{V}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*10·25.*25·104}.&\supset \vdash\colon\ldotp (x):(\exists y).x\dot{\text{V}}y\colon\ldotp (x)\colon\ldotp (\exists y).y\dot{\text{V}}x\colon\ldotp \\ +[\text{*33·13·131}] &\supset \vdash\colon\ldotp (x).x\in \text{D}ʻ\dot{\text{V}}:(x).x\in \text{ᗡ}ʻ\dot{\text{V}}\colon\ldotp \\ +[\text{*24·14}] &\supset \vdash:\text{D}ʻ\dot{\text{V}}=\text{V}.\text{ᗡ}ʻ\dot{\text{V}}=\text{V} &\qquad \text{(1)}\\ +[\text{*33·16}] &\supset \vdash.Cʻ\dot{\text{V}}=\text{V}\cup \text{V}\\ +[\text{*22·56}] &\qquad\qquad =\text{V} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_266">[Pg 266]</span></p> + +<p class="nind"><b>*33·29.</b> \(\vdash.\text{D}ʻ\Lambda=\text{ᗡ}ʻ\Lambda=Cʻ\dot{\Lambda}=\Lambda \quad[\text{*33·241.*21·2}]\)</p> + +<p class="nind"><b>*33·3.</b> \(\vdash\colon\ldotp \alpha\subset\text{D}ʻR.\equiv:x\in \alpha.\supset_{x}.\exists !\overleftarrow{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·181}.\supset\vdash\colon\ldotp x\in \alpha.\supset_{x}.\exists !\overleftarrow{R}ʻx:&\equiv:x\in \alpha.\supset_{x}.(\exists y).xRy:\\ +[\text{*33·13}] &\equiv:x\in \alpha.\supset_{x}.x\in \text{D}ʻR\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·31.</b> \(\vdash\colon\ldotp \beta\subset\text{ᗡ}ʻR.\equiv:y\in \beta.\supset_{y}.\exists !\overrightarrow{R}ʻy \quad[\text{Proof as in *33·3}]\)</p> + +<p>The three following propositions are used in the theory of selections +(<a href="#*80">*80</a>, <a href="#*83">*83</a> and <a href="#*85">*85</a>). The second of them is also used in the theory of +greater and less (*117) and in the theory of transitive relations +(*201).</p> + +<p class="nind"><b>*33·32.</b> \(\vdash:\text{D}ʻR\cap \text{D}ʻS=\Lambda.\supset.R\dot{\cap}S=\Lambda\)</p> + +<p>The converse of this proposition is not true.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·33}. \supset\vdash:x(R\dot{\cap}S)y.&\supset.xRy.xSy.\\ +[\text{*33·14.*22·33}] &\supset.x\in \text{D}ʻR\cap \text{D}ʻS\\ +[\text{*10·24}] & \supset.\exists !\text{D}ʻR\cap \text{D}ʻS &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp}. &\supset\vdash:\text{D}ʻR\cap \text{D}ʻS=\Lambda.\supset.{\sim}\{x(R\dot{\cap}S)y\} &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·3}.&\supset\vdash:\text{D}ʻR\cap \text{D}ʻS=\Lambda.\supset.(x,y).{\sim}\{x(R\dot{\cap}S)y\}.\\ +[\text{*25·15}] & \supset.R\dot{\cap}S=\Lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·33.</b> \(\vdash:\text{ᗡ}ʻR\cap \text{ᗡ}ʻS=\Lambda.\supset.R\dot{\cap}S=\dot{\Lambda} \quad[\text{Proof as in *33·32}]\)</p> + +<p class="nind"><b>*33·34.</b> \(\vdash:CʻR\cap CʻS=\Lambda.\supset.R\dot{\cap}S=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·161.*22·49}.&\supset\vdash.\text{D}ʻR\cap \text{D}ʻS\subset CʻR\cap CʻS.\\ +[\text{*24·13}] &\supset\vdash:CʻR\cap CʻS=\Lambda.\supset.\text{D}ʻR\cap \text{D}ʻS=\Lambda.\\ +[\text{*33·32}] &\supset.R\dot{\cap}S=\dot{\Lambda}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·35.</b> \(\vdash\colon\ldotp \text{D}ʻR\subset\alpha.\equiv:xRy.\supset_{x,y}.x\in \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}.\supset\vdash\colon\ldotp \text{D}ʻR\subset\alpha.&\equiv:(\exists y).xRy.\supset_{x}.x\in \alpha:\\ +[\text{*10·23}] &\equiv:xRy.\supset_{x,y}.x\in \alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·351.</b> \(\vdash\colon\ldotp \text{ᗡ}ʻR\subset\alpha.\equiv:xRy.\supset_{x,y}.y\in \alpha \quad[\text{Proof as in *33·35}]\)</p> + +<p class="nind"><b>*33·352.</b> \(\vdash\colon\ldotp CʻR\subset\alpha.\equiv:xRy.\supset_{x,y}.x,y\in \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·16.*22·59}.\supset\\ +\vdash\colon\ldotp CʻR\subset\alpha.&\equiv:\text{D}ʻR\subset\alpha.\text{ᗡ}ʻR\subset\alpha:\\ +[\text{*33·35·351}]&\equiv:xRy.\supset_{x,y}.x\in \alpha:xRy.\supset_{x,y}.y\in \alpha:\\ +[\text{*11·391}] & \equiv:xRy.\supset_{x,y}.x,y\in \alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_267">[Pg 267]</span></p> + +<p>The two following propositions (*33·4·41) are very frequently used.</p> + +<p class="nind"><b>*33·4.</b> \(\vdash.\text{D}ʻR=\hat{x}\{\exists !\overleftarrow{R}ʻx\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}.\supset\vdash:x\in \text{D}ʻR.&\equiv.(\exists y).xRy.\\ +[\text{*32·181}] &\equiv.(\exists y).y\in \overleftarrow{R}ʻx.\\ +[\text{*24·5}] &\equiv.\exists !\overleftarrow{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*20·33}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·41.</b> \(\vdash.\text{ᗡ}ʻR=\hat{y}\{\exists !\overrightarrow{R}ʻy\} \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*33·42.</b> \(\vdash.CʻR=\hat{x}\{\exists !(\overrightarrow{R}ʻx\cup \overleftarrow{R}ʻx)\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·4·41·16}.\supset\vdash.CʻR&=\hat{x}\{\exists !\overrightarrow{R}ʻx\}\cup \hat{x}\{\exists !\overleftarrow{R}ʻx\}\\ +[\text{*22·391}] &=\hat{x}\{\exists !\overrightarrow{R}ʻx.\lor .\exists !\overleftarrow{R}ʻx\}\\ +[\text{*24·56.*20·15}] &=\hat{x}\{\exists !(\overrightarrow{R}ʻx\cup \overleftarrow{R}ʻx)\}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·43.</b> \(\vdash:\text{E}!Rʻy.\supset.y\in \text{ᗡ}ʻR.Rʻy\in \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·32}.\supset\vdash:\text{E}!Rʻy.&\supset.(Rʻy)Ry.\\ +[\text{*33·14}] & \supset.y\in \text{ᗡ}ʻR.Rʻy\in \text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·431.</b> \(\vdash:(y).\text{E}!Rʻy.\supset.(\beta).\beta\subset\text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·43}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \text{ᗡ}ʻR.\\ +[\text{Simp}] &\qquad\qquad\supset:y\in \beta.\supset.y\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21}.&\supset\vdash:\text{Hp}.\supset.\beta\subset\text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·21}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·432.</b> \(\vdash:(y).\text{E}!Rʻy.\supset.\text{ᗡ}ʻR=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·43.*10·11·27}.\supset\vdash:\text{Hp}.&\supset.(y).y\in \text{ᗡ}ʻR.\\ +[\text{*24·14}] &\supset.\text{ᗡ}ʻR=\text{V}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·44.</b> \(\vdash:\text{E}!\breve{R}ʻx.\supset.x\in \text{D}ʻR.\breve{R}ʻx\in \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·43}\, \frac{\breve{R}}{R} .\supset\vdash:\text{Hp}.&\supset.x\in \text{ᗡ}ʻ\breve{R}.\breve{R}ʻx\in \text{D}ʻ\breve{R}\\ +[\text{*33·2·21}] &\supset.x\in \text{D}ʻR.\breve{R}ʻx\in \text{ᗡ}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·45.</b> \(\vdash\colon\ldotp y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.\supset_{y}.Rʻy=Sʻy:\supset.R=S\)</p> + +<p><span class="pagenum" id="Page_268">[Pg 268]</span></p> + +<p>Note that by our conventions as to denoting expressions, the scope of +both \(Rʻy\) and \(Sʻy\) in the above is "\(Rʻy=Sʻy\)," and \(Rʻy\) is +to have the larger scope.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·11}.\supset\vdash\colon\colon Rʻy&=Sʻy.\equiv\colon\ldotp (\exists b):xRy.\equiv_{x}.x=b:b=Sʻy\colon\ldotp \\ +[\text{*30·11}] &\equiv\colon\ldotp (\exists b)\colon\ldotp xRy.\equiv_{x}.x=b\colon\ldotp (\exists c):xSy.\equiv_{x}.x=c:b=c\colon\ldotp \\ +[\text{*13·195}] &\equiv\colon\ldotp (\exists b):xRy.\equiv_{x}.x=b:xSy.\equiv_{x}.x=b\colon\ldotp \\ +[\text{*10·322}] &\supset\colon\ldotp xRy.\equiv_{x}.xSy &\qquad \text{(1)}\\ +\vdash.\text{(1)}.\supset\vdash\colon\colon \text{Hp}. &\supset\colon\ldotp y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.\supset:xRy.\equiv.xSy\\ +[\text{*5·32}] &\supset\colon\ldotp y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.xRy.\equiv.y\in \text{ᗡ}ʻR\subset \text{ᗡ}ʻS.xSy\colon\ldotp \\ +[\text{*33·14.*4·71}] &\supset\colon\ldotp xRy.\equiv.xSy &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·3}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(x,y):xRy.\equiv.xSy:\\ +[\text{*21·43}] &\supset:R=S\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·46.</b> \(\vdash\colon\ldotp x\in \text{D}ʻR\cup \text{D}ʻS.\supset_{x}.\breve{R}ʻx=\breve{S}ʻx:\supset.R=S \quad[\text{Proof as in *33·45}]\)</p> + +<p class="nind"><b>*33·47.</b> \(\vdash\colon\ldotp y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.\supset_{y}.\overrightarrow{R}ʻy=\overrightarrow{S}ʻy:\supset.R=S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·41.Transp}. &\supset\vdash:y{\sim}\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS. \supset\overrightarrow{R}ʻy=\Lambda.\overrightarrow{S}ʻy=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*13·172.*4·83}.&\supset\vdash:\text{Hp}.\supset.(y).\overrightarrow{R}ʻy=\overrightarrow{S}ʻy.\\ +[\text{*30·41}] &\qquad\qquad\supset.\overrightarrow{R}=\overrightarrow{S}\\ +[\text{*32·14}] &\qquad\qquad\supset.R=S:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·48.</b> \(\vdash\colon\ldotp x\in \text{D}ʻR\cup \text{D}ʻS.\supset_{x}.\overleftarrow{R}ʻx=\overleftarrow{S}ʻx:\supset.R=S \quad[\text{Proof as in *33·47}]\)</p> + +<p class="nind"><b>*33·5.</b> \(\vdash.C=\overrightarrow{F}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·1}.\supset\vdash\colon\ldotp \alpha\overrightarrow{F}R.&\equiv.\alpha=\hat{x}(xFR)\\ +[\text{*33·103}] &=\hat{x}\{(\exists y):xRy.\lor .yRx\}\\ +[\text{*33·102}] &\equiv.\alpha CR &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·51.</b> \(\vdash:x\in CʻR.\equiv.xFR \quad[\text{*33·132·103}]\)</p> + +<p>\(F\) is useful in ordinal arithmetic, where we are concerned with +a series generated by a relation \(P\), and "\(xFP\)" expresses the +fact that \(x\) is a member of this series. The above two propositions +(*33·5·51) will be much used in Part IV, where we deal with the +foundations of ordinal arithmetic, but will not often be referred to +elsewhere.</p> + +<p class="nind"><b>*33·6.</b> \(\vdash:R\in \overleftarrow{\text{D}}ʻ\alpha.\equiv.\alpha=\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·181}.\supset\vdash:R\in \overleftarrow{\text{D}}ʻ\alpha.&\equiv.\alpha \text{D}R.\\ +[\text{*33·123}] &\equiv.\alpha=\text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*33·61.</b> \(\vdash:R\in \overleftarrow{ᗡ}ʻ\alpha.\equiv.\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*33·62.</b> \(\vdash:R\in \overleftarrow{C}ʻ\alpha.\equiv.CʻR\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_269">[Pg 269]</span></p> +<h2 class="nobreak" id="*34">*34. THE RELATIVE PRODUCT OF TWO RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *34.</p> + +<p>The relative product of two relations \(R\) and \(S\) is the relation +which holds between \(x\) and \(z\) when there is an intermediate term +\(y\) such that \(x\) has the relation \(R\) to \(y\) and \(y\) has +the relation \(S\) to \(z\). Thus <i>e.g.</i> the relative product +of <i>brother</i> and <i>father</i> is <i>paternal uncle</i>; the +relative product of <i>father</i> and <i>father</i> is <i>paternal +grandfather</i>; and so on. The relative product of \(R\) and \(S\) is +denoted by "\(R \mid S\)"; the definition is:</p> + +<p class="nind"><b>*34·01.</b> \(R \mid S = \hat{x}\hat{z}\{(\exists y).xRy.ySz\} \quad \text{Df}\)</p> + +<p>This definition is only significant when \(\text{ᗡ}ʻR\) and +\(\text{D}ʻS\) belong to the same type.</p> + +<p>The relative product of \(R\) and \(R\) is called the square of \(R\); +we put</p> + +<p class="nind"><b>*34·02.</b> \(R^{2} = R \mid R \quad \text{Df}\)</p> + +<p class="nind"><b>*34·03.</b> \(R^{3} = R^{2} \mid R \quad \text{Df}\)</p> + +<p>The most useful propositions in the present number are the following:</p> + +<p class="nind"><b>*34·2.</b> \(\vdash . \text{Cnv}ʻ(R \mid S) = \breve{S} \mid \breve{R}\)</p> + +<p><i>I.e.</i> the converse of a relative product is obtained by turning +each factor into its converse and reversing the order of the factors.</p> + +<p class="nind"><b><a id="*34·21">*34·21</a>.</b> \(\vdash . (P \mid Q) \mid R = P \mid (Q \mid R)\)</p> + +<p><i>I.e.</i> the relative product obeys the associative law.</p> + +<p class="nind"><b>*34·25.</b> \(\vdash . P \mid (Q\unicode{x228d}R) = (P \mid Q) \unicode{x228d} (P \mid R)\)</p> + +<p class="nind"><b>*34·26.</b> \(\vdash . (P \unicode{x228d} Q) \mid R = (P \mid R) \unicode{x228d} (Q \mid R)\)</p> + +<p><i>I.e.</i> the relative product obeys the distributive law +with respect to the logical addition of relations. (For logical +multiplication instead of logical addition, we only get inclusion +instead of identity; cf. *34·23·24.)</p> + +<p class="nind"><b>*34·34.</b> \(\vdash : R \unicode{x2abd} P . S \unicode{x2abd} Q .\supset. R \mid S \unicode{x2abd} P \mid Q\)</p> + +<p class="nind"><b>*34·36.</b> \(\vdash . \text{D}ʻ(P \mid Q) \subset \text{D}ʻP.\text{ᗡ}ʻ(P \mid Q)\subset\text{ᗡ}ʻQ\)</p> + +<p class="nind"><b>*34·41.</b> \(\vdash : \text{E}!PʻQʻz .\supset. PʻQʻz = (P \mid Q)ʻz\)</p> + +<p><span class="pagenum" id="Page_270">[Pg 270]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*34·01.</b>\( R\mid S=\hat{x}\hat{z}\{(\exists y).xRy.ySz\} \quad\text{Df}\)</p> + +<p class="nind"><b>*34·02.</b> \(R^{2}=R\mid R \quad\text{Df}\)</p> + +<p class="nind"><b>*34·03.</b> \(R^{3}=R^{2}\mid R \quad\text{Df}\)</p> + +<p class="nind"><b>*34·1.</b> \(\vdash:x(R\mid S)z.\equiv.(\exists y).xRy.ySz \quad[\text{*21·3.(*34·01)}]\)</p> + +<p class="nind"><b>*34·11.</b> \(\vdash:x(R\mid S)z.\equiv.\exists !(\overleftarrow{R}ʻx\cap \overrightarrow{S}ʻz)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1.*32·18·181}.\supset\\ +\vdash:x(R\mid S)z.&\equiv.(\exists y).y\in \overleftarrow{R}ʻx.y\in \overrightarrow{S}ʻz.\\ +[\text{*22·33}] & \equiv.(\exists y).y\in \overleftarrow{R}ʻx\cap \overrightarrow{S}ʻz.\\ +[\text{*24·5}] & \equiv.\exists !(\overleftarrow{R}ʻx\cap \overrightarrow{S}ʻz):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·12.</b> \(\vdash.R\mid S=\hat{x}\hat{z}\{\exists !(\overleftarrow{R}ʻx\cap \overrightarrow{S}ʻx)\} \quad[\text{*21·33.*34·11}]\)</p> + +<p class="nind"><b>*34·2.</b> \(\vdash.\text{Cnv}ʻ(R\mid S)=\breve{S}\mid \breve{R}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\vdash:x\{\text{Cnv}ʻ(R\mid S)\}z.&\equiv.z(R\mid S)x.\\ +[\text{*34·1}] &\equiv.(\exists y).zRy.ySx.\\ +[\text{*34·11}] &\equiv.(\exists y).y\breve{R}z.x\breve{S}y.\\ +[\text{*34·1}] &\equiv.x(\breve{S}\mid \breve{R})z &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·202.</b> \(\vdash.R\mid S=(\text{Cnv}ʻ\breve{R})\mid S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\vdash:x(\text{Cnv}ʻ\breve{R})y.ySz.&\equiv.y\breve{R}x.ySz.\\ +[\text{*31·11}] &\equiv.xRy.ySz &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·281.*34·1}.\supset\vdash:x\{(\text{Cnv}ʻ\breve{R})\mid S\}z.&\equiv.x(R\mid S)z &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·203.</b> \(\vdash.R\mid S=R\mid (\text{Cnv}ʻ\breve{S}) \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*34·21.</b> \(\vdash.(P\mid Q)\mid R=P\mid (Q\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1.*10·281}.\supset\vdash\colon\colon(\exists z).x(P\mid Q)z.zRw.&\equiv\colon\ldotp (\exists z):(\exists y).xPy.yQz:zRw\colon\ldotp \\ +[\text{*11·6}] &\equiv\colon\ldotp (\exists y)\colon\ldotp xPy:(\exists z).yQz.zRw\colon\ldotp \\ +[\text{*34·1.*10·281}] & \equiv\colon\ldotp (\exists y).xPy.y(Q\mid R)w &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*34·1.*21·43}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·22.</b> \(P\mid Q\mid R=(P\mid Q)\mid R \quad\text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets.</p> + +<p><span class="pagenum" id="Page_271">[Pg 271]</span></p> + +<p class="nind"><b>*34·23.</b> \(\vdash.P\mid (Q\dot{\cap}R)\unicode{x2abd}(P\mid Q)\dot{\cap}(P\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{34·1}.\supset +\vdash\colon\ldotp x\{P\mid (Q\dot{\cap}R)\}y.&\equiv:(\exists z).xPz.z(Q\dot{\cap}R)y:\\ +[\text{*23·33}] &\equiv:(\exists z).xPz.zQy.zRy:\\ +[\text{*10·5}] &\supset:(\exists z).xPz.zQy:(\exists z).xPz.zRy:\\ +[\text{*34·1}] &\supset:x(P\mid Q)y.x(P\mid R)y:\\ +[\text{*23·33}] &\supset:x\{(P\mid Q)\dot{\cap}(P\mid R)\}y &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The converse of the above is not true.</p> + +<p class="nind"><b>*34·24.</b> \(\vdash.(P\dot{\cap}Q)\mid R\unicode{x2abd}(P\mid R)\dot{\cap}(Q\mid R) \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*34·25.</b> \(\vdash.P\mid (Q\unicode{x228d}R)=(P\mid Q)\unicode{x228d}(P\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·34.*10·281}.\supset\\ +\vdash\colon\ldotp (\exists z).xPz.z(Q\unicode{x228d}R)y.&\equiv:(\exists z):xPz:zQy.\lor .zRy:\\ +[\text{*4·4.*10·281}] &\equiv:(\exists z):xPz.zQy.\lor .xPz.zRy:\\ +[\text{*10·42}] & \equiv:(\exists z).xPz.zQy:\lor :(\exists z).xPz.zRy:\\ +[\text{*34·1}] &\equiv:x(P\mid Q)y.\lor .x(P\mid R)y:\\ +[\text{*23·34}] &\equiv:x(P\mid Q\unicode{x228d}P\mid R)y &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11.*34·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·26.</b> \(\vdash.(P\unicode{x228d}Q)\mid R=(P\mid R)\unicode{x228d}(Q\mid R) \quad[\text{Similar proof}]\)</p> + +<p>The above two forms of the distributive law, and the associative law +(<a href="#*34·21">*34·21</a>), are the only ones of the usual formal laws that hold for the +relative product. The commutative law, in particular, does not hold in +general.</p> + +<p class="nind"><b>*34·27.</b> \(\vdash:R=R'.\supset.R\mid P=R'\mid P\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·43}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:(x,y):xRy.\equiv.xR'y:\\ +[\text{*11·401}] &\supset:(x,y):xRy.yPz.\equiv_{z}.xR'y.yPz:\\ +[\text{*10·281}] & \supset:(x):(\exists y).xRy.yPz.\equiv_{z}.(\exists y).xR'y.yPz:\\ +[\text{*21·15}] &\supset:R\mid P=R'\mid P\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·28.</b> \(\vdash:R=R'.\supset.P\mid R=P\mid R' \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*34·29.</b> \(\vdash:R=R'.\supset.P\mid R\mid Q=P\mid R'\mid Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·27}.\supset\vdash:\text{Hp}.&\supset.R\mid Q=R'\mid Q.\\ +[\text{*34·28}] &\supset.P\mid R\mid Q=P\mid R'\mid Q:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_272">[Pg 272]</span></p> + +<p>In proving the equality of two relations, say \(R\) and \(S\), we +usually establish first an asserted proposition of the form +\[ +\begin{aligned} +xRy.&\equiv.xSy\\ +\text{or}\qquad\qquad \text{Hp}.\supset:xRy.&\equiv.xSy. +\end{aligned} +\]</p> + +<p>We then proceed by <a href="#*11·11">*11·11</a> (together with <a href="#*11·3">*11·3</a> in the second case) to +\[ +(x,y):xRy.\equiv.xSy\, \text{or Hp}.\supset:(x,y):xRy.\equiv.xSy, +\] +whence the result follows by <a href="#*21·43">*21·43</a>. We shall in future omit these +steps, and write "\(\supset\vdash.\text{Prop}\)" after we have +established +\[ +xRy.\equiv.xSy\, \text{or Hp}.\supset:xRy.\equiv.xSy. +\] +A similar ellipsis will be made in proving the equality of classes.</p> + +<p class="nind"><b>*34·3.</b> \(\vdash:\dot{\exists}!(P\mid Q).\equiv.\exists !(\text{ᗡ}ʻP\cap \text{D}ʻQ)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*25·5}.\supset\\ +\vdash\colon\colon \dot{\exists}!(P\mid Q).&\equiv\colon\ldotp (\exists x,y).x(P\mid Q)y\colon\ldotp \\ +[\text{*34·1}] &\equiv\colon\ldotp (\exists x,y):(\exists z).xPz.zQy\colon\ldotp \\ +[\text{*11·27}] &\equiv\colon\ldotp (\exists x,y,z).xPz.zQy\colon\ldotp \\ +[\text{*11·24}] &\equiv\colon\ldotp (\exists z,x,y).xPz.zQy\colon\ldotp \\ +[\text{*11·27}] & \equiv\colon\ldotp (\exists z)\colon\ldotp (\exists x,y).xPz.zQy\colon\ldotp \\ +[\text{*11·54}] &\equiv\colon\ldotp (\exists z)\colon\ldotp (\exists x).xPz:(\exists y).zQy\colon\ldotp \\ +[\text{*33·13·131}] &\equiv\colon\ldotp (\exists z)\colon\ldotp z\in \text{ᗡ}ʻP.z\in \text{D}ʻQ\colon\ldotp \\ +[\text{*22·33}] & \equiv\colon\ldotp (\exists z)\colon\ldotp z\in \text{ᗡ}ʻP\cap \text{D}ʻQ\colon\ldotp \\ +[\text{*24·5}] &\equiv\colon\ldotp \exists !(\text{ᗡ}ʻP\cap \text{D}ʻQ)\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·301.</b> \(\vdash:\text{ᗡ}ʻP\cap \text{D}ʻQ=\Lambda.\equiv.P\mid Q=\Lambda \quad[\text{*34·3.Transp}]\)</p> + +<p class="nind"><b>*34·302.</b> \(\vdash:CʻP\cap CʻQ=\Lambda.\supset.P\mid Q=\dot{\Lambda}.Q\mid P=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·16}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻP\cap \text{D}ʻQ=\Lambda.\text{ᗡ}ʻQ\cap \text{D}ʻP=\Lambda.\\ +[\text{*34·301}] &\supset.P\mid Q=\Lambda.Q\mid P=\dot{\Lambda}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·31.</b> \(\vdash:\dot{\exists}!(P\mid Q).\supset.\dot{\exists}!P.\dot{\exists}!Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·3}.\supset\vdash:\text{Hp}.&\supset.\exists !(\text{ᗡ}ʻP\cap \text{D}ʻQ).\\ +[\text{*24·561}] &\supset.\exists !\text{ᗡ}ʻP.\exists !\text{D}ʻQ.\\ +[\text{*33·24}] &\supset.\dot{\exists}!P.\dot{\exists}!Q:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·32.</b> \(\vdash\colon\ldotp P=\Lambda.\lor .Q=\dot{\Lambda}:\supset.P\mid Q=\Lambda \quad[\text{*34·31.Transp.*25·51}]\)</p> + +<p><span class="pagenum" id="Page_273">[Pg 273]</span></p> + +<p class="nind"><b>*34·33.</b> \(\vdash : x \in \text{D}ʻR . \equiv .x(R\mid \breve{R})x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*33·13} . \supset \vdash : x \in \text{D}ʻR . &\equiv . (\exists y) . xRy .\\ +[\text{*4·24}] &\equiv . (\exists y) . xRy . xRy .\\ +[\text{*31·11}] &\equiv . (\exists y) . xRy . y\breve{R}x.\\ +[\text{*34·1}] & \equiv . x(R\mid \breve{R})x : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·34.</b> \(\vdash : R \unicode{x2abd} P . S \unicode{x2abd} Q . \supset . R\mid S \unicode{x2abd} P Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*23·1} . \supset \vdash \colon\ldotp \text{Hp} . &\supset : xRy . \supset_{x,y} . xPy : ySz . \supset_{y,z} . yQz :\\ +[\text{*11·2.*10·1·41}] &\supset : xRy . \supset . xPy : ySz . \supset . yQz :\\ +[\text{*3·47}] &\supset : xRy . ySz . \supset . xPy . yQz &\qquad \text{(1)}\\ +\vdash .\text{ (1) . *10·11·21·28} . &\supset\\ +\vdash \colon\ldotp \text{Hp} . &\supset : (\exists y) . xRy . ySz . \supset . (\exists y) . xPy . yQz :\\ +[\text{*34·1}] &\supset : x(R S)z . \supset . x(P Q)z &\qquad \text{(2)}\\ +\vdash . \text{(2) . *11·11·3} . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·35.</b> \(\vdash : \dot{\exists} ! R . \text{ᗡ}ʻR \subset \text{D}ʻP . \supset . \dot{\exists} ! R\mid P\)</p> + + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*33·24} . \supset \vdash : \text{Hp} . &\supset . \exists ! \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash . \text{*22·621} . \supset \vdash : \text{Hp} . &\supset . \text{ᗡ}ʻR = \text{ᗡ}ʻR \cap \text{D}ʻP &\qquad \text{(2)}\\ +\vdash . \text{(1). (2)} . \supset \vdash : \text{Hp} . &\supset . \exists ! \text{ᗡ}ʻR \cap \text{D}ʻP.\\ +[\text{*34·3}] &\supset . \dot{\exists} ! R\mid P : \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·351.</b> \(\vdash : \dot{\exists} ! R . \text{D}ʻR \subset \text{ᗡ}ʻP . \supset . \dot{\exists} ! P\mid R \quad[\text{Proof as in *34·35}]\)</p> + +<p class="nind"><b>*34·36.</b> \(\vdash . \text{D}ʻ(P\mid Q) \subset \text{D}ʻP . \text{ᗡ}ʻ(P\mid Q) \subset \text{ᗡ}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*33·13} . \supset \vdash \colon\ldotp x \in \text{D}ʻ(P\mid Q) . &\supset : (\exists z) . x(P\mid Q)z :\\ +[\text{*34·1}] &\supset : (\exists z, y) . xPy . yQz :\\ +[\text{*11·23}] & \supset : (\exists y, z) . xPy . yQz :\\ +[\text{*11·55.*10·5}] &\supset : (\exists y) . xPy :\\ +[\text{*33·13}] &\supset : x \in \text{D}ʻP &\qquad \text{(1)}\\ +\text{Similarly}\qquad \vdash \colon\ldotp z \in \text{ᗡ}ʻ(P\mid Q) . &\supset : z \in \text{ᗡ}ʻP &\qquad \text{(2)}\\ +\vdash . \text{(1) . (2) . *10·11} . &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following proposition is a lemma for <a href="#*95·31">*95·31</a>.</p> + +<p class="nind"><b>*34·361.</b> \(\vdash : \dot{\exists} ! R . \text{D}ʻR \subset \text{ᗡ}ʻP . \text{ᗡ}ʻR \subset \text{D}ʻQ . \supset . \dot{\exists} ! P\mid R\mid Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*34·35} . \supset \vdash : \text{Hp} . &\supset . \dot{\exists} ! R\mid Q &\qquad \text{(1)}\\ +\vdash . \text{*34·36} . &\supset \vdash : \text{Hp} . \supset . \text{D}ʻ(R\mid Q) \subset \text{ᗡ}ʻP &\qquad \text{(2)}\\ +\vdash . \text{(1). (2). *34·351}. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_274">[Pg 274]</span></p> + +<p class="nind"><b>*34·37.</b> \(\vdash.Cʻ(P\mid Q)\subset \text{D}ʻP\cup \text{ᗡ}ʻQ \quad[\text{*34·36.*33·161.*22·72}]\)</p> + +<p class="nind"><b>*34·38.</b> \(\vdash.Cʻ(P\mid Q)\subset CʻP\cup CʻQ \quad[\text{*34·37.*33·161.*22·72}]\)</p> + +<p class="nind"><b>*34·4.</b> \(\vdash:b=Pʻc.c=Qʻz.\supset.b=(P\mid Q)ʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·31}.\supset\vdash:\text{Hp}.&\supset.bPc.cQz.\\ +[\text{*34·1}] &\supset.b(P\mid Q)z &\qquad \text{(1)}\\ +\vdash.\text{*30·31}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:yQz.\supset_{y}.y=c:\\ +[\text{Fact}] & \supset:xPy.yQz.\supset_{x,y}.xPy.y=c.\\ +[\text{*13·13}] &\qquad\qquad\qquad\supset_{x,y}.xPc &\qquad \text{(2)}\\ +\vdash.\text{*30·31}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:xPc.\supset_{x}.x=b &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}\supset\vdash\colon\ldotp \text{Hp}.&\supset:xPy.yQz.\supset_{x,y}.x=b:\\ +[\text{*10·23}] &\supset:(\exists y).xPy.yQz.\supset_{x}.x=b:\\ +[\text{*34·1}] &\supset:x(P\mid Q)z.\supset_{x}.x=b &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*30·31}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·41.</b> \(\vdash:\text{E}!PʻQʻz.\supset.PʻQʻz=(P\mid Q)ʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·52}.\supset\vdash:\text{Hp}.&\supset.(\exists b,c).b=Pʻc.c=Qʻz.\\ +[\text{*30·51.*34·4}] &\supset.(\exists b).b=PʻQʻz.b=(P\mid Q)ʻz.\\ +[\text{*14·145}] &\supset.PʻQʻz=(P\mid Q)ʻz:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is no longer true if we change the hypothesis +into \(\text{E}!(P\mid Q)ʻz\), since (\(P\mid Q)ʻz\) may exist when +\(PʻQʻz\) does not. Suppose, <i>e.g.</i>, that \(Q\)is the relation +of child to father, and \(P\) the relation of daughter to father. +Then (\(P\mid Q)ʻz\)= the granddaughter of \(z\), but \(PʻQʻz\)= the +daughter of the child of \(z\). The first exists whenever \(z\) has +only one granddaughter, while the second requires further that \(z\) +should have only one child.</p> + +<p>For the same reason we do not have +\[ +b=(P\mid Q)ʻz.\supset.(\exists c).b=Pʻc.c=Qʻz. +\] +This will hold if \(P\), \(Q\) are one-many relations (cf. <a href="#*71">*71</a>), but +not in general otherwise.</p> + +<p class="nind"><b>*34·42.</b> \(\vdash:(z).Rʻz=PʻQʻz.\supset.R=P\mid Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:(z).\text{E}!Rʻz:(z).\text{E}!PʻQʻz &\qquad \text{(1)}\\ +\vdash.\text{(1).*34·41}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(z).Rʻz=(P\mid Q)ʻz:\\ +[\text{*30·42.(1)}] & \supset:R=P\mid Q\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_275">[Pg 275]</span></p> + +<p class="nind"><b>*34·5.</b> \(\vdash:xR{^{2}}y.\equiv.(\exists z).xRz.zRy \quad[\text{*34·1.(*34·02)}]\)</p> + +<p class="nind"><b>*34·51.</b> \(\vdash:xR{^{3}}y.\equiv.(\exists z,w).xRz.zRw.wRy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1.(*34·03)}.\supset\\ +\vdash\colon\ldotp xR{^{3}}y.&\equiv:(\exists w).xR{^{2}}w.wRy:\\ +[\text{*34·5}] &\equiv:(\exists w):(\exists z).xRz.zRw:wRy:\\ +[\text{*11·55}]&\equiv:(\exists w,z).xRz.zRw.wRy:\\ +[\text{*11·2}] &\equiv:(\exists z,w).xRz.zRw.wRy\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·52.</b> \(\vdash.R^{3}=R\mid R^{2} \quad[\text{*34·21}]\)</p> + +<p class="nind"><b>*34·53.</b> \(\vdash:\dot{\exists}!R^{2}.\equiv.\exists !\text{D}ʻR\cap \text{ᗡ}ʻR \quad[\text{*34·3}]\)</p> + +<p class="nind"><b>*34·531.</b> \(\vdash:\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda.\equiv.R^{2}=\dot{\Lambda} \quad[\text{*34·53.Transp}]\)</p> + +<p class="nind"><b>*34·54.</b> \(\vdash:xRx.\supset.xR{^{2}}x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·24}.\supset\vdash:xRx.&\supset.xRx.xRx.\\ +[\text{*10·24}] &\supset.(\exists y).xRy.yRx.\\ +[\text{*34·5}] & \supset.xR^{2}x:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·55.</b> \(\vdash\colon\ldotp R{^{2}}\unicode{x2abd}S.\equiv:xRy.yRz.\supset_{x,y,z}.xSz \quad[\text{*34·5.*10·23}]\)</p> + +<p class="nind"><b>*34·56.</b> \(\vdash.\text{D}ʻR^{2}\subset\text{D}ʻR.\text{ᗡ}ʻR{^{2}}\subset\text{ᗡ}ʻR.CʻR{^{2}}\subset CʻR \quad[\text{*34·36·38}]\)</p> + +<p class="nind"><b>*34·6.</b> \(\vdash.(R\dot{\cap}S){^{2}}\unicode{x2abd}R^{2}\dot{\cap}S^{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·5}.\supset\vdash\colon\ldotp x(R\dot{\cap}S){^{2}}y.&\equiv:(\exists z).x(R\dot{\cap}S)z.z(R\dot{\cap}S)y:\\ +[\text{*23·33.*10·281}] &\equiv:(\exists z).xRz.xSz.zRy.zSy:\\ +[\text{*4·3.*10·281}] &\equiv:(\exists z).xRz.zRy.xSz.zSy:\\ +[\text{*10·5}] &\supset:(\exists z).xRz.zRy:(\exists z).xSz.zSy:\\ +[\text{*34·5}] & \supset:xR{^{2}}y.xS{^{2}}y:\\ +[\text{*23·33}] & \supset:x(R{^{2}}\dot{\cap}S{^{2}})y &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·62.</b> \(\vdash.(R\unicode{x228d}S)^{2}=R{^{2}}\unicode{x228d}R\mid S\unicode{x228d}S\mid R\unicode{x228d}S^{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·26}.\supset\vdash.(R\unicode{x228d}S)^{2}&=R\mid (R\unicode{x228d}S)\unicode{x228d}S\mid (R\unicode{x228d}S)\\ +[\text{*34·25}] & =R{^{2}}\unicode{x228d}R\mid S\unicode{x228d}S\mid R\unicode{x228d}S^{2}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is a lemma for *160·51, as is also <a href="#*34·73">*34·73</a>, which +employs the above proposition.</p> + +<p class="nind"><b>*34·63.</b> \(\vdash.\text{Cnv}ʻ(R^{2})=(\text{Cnv}ʻR)^{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\\ +\vdash\colon\ldotp x\{\text{Cnv}ʻ(R^{2})\}y.&\equiv:yR{^{2}}x:\\ +[\text{*34·5}] &\equiv:(\exists z).yRz.zRx:\\ +[\text{*31·131.*10·281}]&\equiv:(\exists z).x\breve{R}z.z\breve{R}y:\\ +[\text{*31·131.*34·5}] &\equiv:x(\text{Cnv}ʻR){^{2}}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_276">[Pg 276]</span></p> + +<p class="nind"><b>*34·7.</b> \(\vdash.\text{Cnv}(S\mid \breve{S})=S\mid \breve{S}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·2}.\supset\vdash.\text{Cnv}(S\mid \breve{S})&=(\text{Cnv}\breve{S})\mid \breve{S}\\ +[\text{*34·202}] & =S\mid \breve{S}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Thus \(S\mid \breve{S}\) is always a symmetrical relation, <i>i.e.</i> +one which is equal to its converse.</p> + +<p class="nind"><b>*34·701.</b> \(\vdash.\text{Cnv}(\breve{S}\mid S)=\breve{S}\mid S \quad[\text{*34·2·203}]\)</p> + +<p class="nind"><b>*34·702.</b> \(\vdash.Cʻ(S\mid \breve{S})=\text{D}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·37}.\supset\vdash.Cʻ(S\mid \breve{S})&\subset\text{D}ʻS\cup \text{ᗡ}ʻ\breve{S}\\ +[\text{*33·21}] &\subset\text{D}ʻS &\qquad \text{(1)}\\ +\vdash.\text{*33·13}.\supset\vdash:x\in \text{D}ʻS.&\supset.(\exists y).xSy.\\ +[\text{*31·11}] &\supset.(\exists y).xSy.y\breve{S}x.\\ +[\text{*34·1}] &\supset.x(S\mid \breve{S})x.\\ +[\text{*33·17}] &\supset.x\in Cʻ(S\mid \breve{S})&\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*10·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·703.</b> \(\vdash.Cʻ(\breve{S}\mid S)=\text{ᗡ}ʻS \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b><a id="*34·73">*34·73</a>.</b> \(\vdash:CʻP\cap CʻQ=\Lambda.\supset.(P\unicode{x228d}Q)^{2}=P^{2}\unicode{x228d}Q^{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·302}.\supset\vdash:\text{Hp}.&\supset.P\mid Q=\dot{\Lambda}.Q\mid P=\dot{\Lambda}.\\ +[\text{*25·24}] &\supset.P{^{2}}\unicode{x228d}Q^{2}=P^{2}\unicode{x228d}P\mid Q\unicode{x228d}Q\mid P\unicode{x228d}Q^{2}\\ +[\text{*34·62}] &\qquad\qquad=(P\unicode{x228d}Q)^{2}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·8.</b> \(\vdash:R=\breve{R}.R^{2}\unicode{x2abd}R.\supset.R=R^{2}=R\mid \breve{R}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·28}. &\supset\vdash:R=\breve{R}.\supset.R^{2}=R\mid \breve{R} &\qquad \text{(1)}\\ +\vdash.\text{*34·33.*33·14}.&\supset\vdash:xRy.\supset.x(R\mid \breve{R})x &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash\colon\ldotp R=\breve{R}.\supset:xRy.\supset.xR^{2}x &\qquad \text{(3)}\\ +\vdash.\text{(3).*23·1}. &\supset\vdash\colon\ldotp R=\breve{R}.R{^{2}}\unicode{x2abd}R.\supset:xRy.\supset.xRx:\\ +[\text{*4·7}] &\qquad\qquad \supset:xRy.\supset.xRx.xRy.\\ +[\text{*10·24.*34·5}] &\qquad\qquad \supset.xR^{2}y &\qquad \text{(4)}\\ +\vdash.\text{(4).*11·11·3}. &\supset\vdash:\text{Hp}.\supset.R\unicode{x2abd}R^{2} &\qquad \text{(5)}\\ +\vdash.\text{*3·27}. \supset\vdash:\text{Hp}.&\supset.R{^{2}}\unicode{x2abd}R &\qquad \text{(6)}\\ +\vdash.\text{(5).(6).*23·41}.&\supset\vdash:\text{Hp}.\supset.R=R^{2} &\qquad \text{(7)}\\ +\vdash.\text{(1).(7)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_277">[Pg 277]</span></p> + +<p>The hypothesis of the above proposition is the hypothesis that \(R\) is +symmetrical (\(R=\breve{R}\)) and transitive (\(R^{2}\unicode{x2abd}R)\). +These are the formal properties of those relations which can suitably +be regarded as expressing equality in some respect.</p> + +<p class="nind"><b>*34·81.</b> \(\vdash:R=\breve{R}.R^{2}\unicode{x2abd}R.\equiv.R=\breve{R}.R^{2}=R \quad[\text{*34·8.*4·71}]\)</p> + +<p>The following propositions are lemmas for <a href="#*34·85">*34·85</a>, which is used in +<a href="#*72·64">*72·64</a>:</p> + +<p class="nind"><b>*34·82.</b> \(\vdash\colon\ldotp R=\breve{R}.R^{2}\unicode{x2abd}R.\supset:x\in \text{D}ʻR.\equiv.xRx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·33}. &\supset\vdash:x\in \text{D}ʻR.\equiv.x(R\mid \breve{R})x &\qquad \text{(1)}\\ +\vdash.\text{*34·8}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:x(R\mid \breve{R})x.\equiv.xRx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·83.</b> \(\vdash:R=\breve{R}.R^{2}\unicode{x2abd}R.xRy.\supset.\overleftarrow{R}ʻx=\overleftarrow{R}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:yRx:\\ +[\text{*3·2}] &\supset:xRz.\supset.yRx.xRz.\\ +[\text{*34·55.Hp}] &\qquad\qquad\qquad \supset.yRz &\qquad \text{(1)}\\ +\vdash.\text{*3·2}. & \supset\vdash\colon\ldotp \text{Hp}.\supset:yRz.\supset.xRy.yRz.\\ +[\text{*34·55.Hp}] &\qquad\qquad\qquad\supset.xRz &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:xRz.\equiv.yRz:\\ +[\text{*10·11·21.*20·15.*32·111}]&\supset:\overleftarrow{R}ʻx=\overleftarrow{R}ʻy\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·84.</b> \(\vdash:R=\breve{R}.R^{2}\unicode{x2abd}R.y\in \text{D}ʻR.\overleftarrow{R}ʻx=\overleftarrow{R}ʻy.\supset.xRy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·82}. &\supset\vdash:\text{Hp}.\supset.yRy &\qquad \text{(1)}\\ +\vdash.\text{*32·181.*20·31}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:xRz.\equiv_{z}.yRz:\\ +[\text{*10·1}] &\qquad\qquad\supset:xRy.\equiv.yRy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*34·841.</b> \(\vdash:R=\breve{R}.R^{2}\unicode{x2abd}R.x\in \text{D}ʻR.\overleftarrow{R}ʻx=\overleftarrow{R}ʻy.\supset.xRy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·84}\, \frac{y,\,x}{x,\,y} .\supset\vdash:\text{Hp}.&\supset.yRx.\\ +[\text{*31·11.Hp}]&\supset.xRy:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*34·85">*34·85</a>.</b> \[\begin{align}&\vdash\colon\ldotp R=\breve{R}.R^{2}\unicode{x2abd}R.\supset:xRy.\equiv.x\in \text{D}ʻR.\overleftarrow{R}ʻx=\overleftarrow{R}ʻy\\ +&\quad[\text{*34·83·841.*33·14}]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_278">[Pg 278]</span></p> +<h2 class="nobreak" id="*35">*35. RELATIONS WITH LIMITED DOMAINS AND CONVERSE DOMAINS.</h2> +</div> + + +<p><i>Summary of</i> *35.</p> + +<p>In this section, we have to consider the relation derived from a given +relation \(R\) by limiting either its domain or its converse domain +to members of some assigned class. A relation \(R\) with its domain +limited to members of \(\alpha\) is written "\(\alpha \upharpoonleft R\)"; +with its converse domain limited to members of \(\beta\), it +is written "\(R \upharpoonright \beta\)"; with both limitations, it +is written "\(\alpha \upharpoonleft R \upharpoonright \beta\)." Thus +<i>e.g.</i> "brother" and "sister" express the same relation (that of a +common parentage), with the domain limited in the first case to males, +in the second to females. "The relation of white employers to coloured +employees" is a relation limited both as to its domain and as to its +converse domain. We put</p> + +<p class="nind"><b>*35·01.</b> \(\alpha \upharpoonright R =\hat{x}\hat{y}(x \in \alpha . xRy) \quad \text{Df}\)</p> + +<p>with similar definitions for \(R \upharpoonright \alpha\) and \(\alpha +\upharpoonleft R \upharpoonright \beta\).</p> + +<p>A particularly important case is the case in which the same limitation +is imposed on the domain and on the converse domain, <i>i.e.</i> +where we have a relation of the form "\(\alpha \upharpoonleft R\upharpoonright \alpha\)." +In this case, the limitation to members +of \(\alpha\) may be more briefly stated as being imposed on +the <i>field</i>. For this case, it is convenient to adopt "\(R\unicode{x294f} \alpha\)" +as an alternative notation. This case will be considered in <a href="#*36">*36</a>.</p> + +<p>It is convenient to consider in the present connection the relation +between \(x\) and \(y\) which is constituted by x being a member of +\(\alpha\) and \(y\) being a member of \(\beta\). This relation will be +denoted by "\(\alpha \uparrow \beta\)." Thus we put</p> + +<p class="nind"><b><a id="*35·04">*35·04</a>.</b> \(\alpha \uparrow \beta = \hat{x}\hat{y}(x \in \alpha . y \in \beta) \quad \text{Df}\)</p> + +<p>The chief importance of relations with limited <i>fields</i> arises +in the theory of series. Given a series generated by a relation +\(R\), let \(\alpha\) be a class consisting of part of this series. +Then \(\alpha\) is the field of the relation \(\alpha \upharpoonleft R \upharpoonright \alpha\) +<span class="pagenum" id="Page_279">[Pg 279]</span>or \(R \unicode{x294f} \alpha\), and it is this relation which is the +generating relation of the series of members of \(\alpha\) in the same +order which they have as parts of the original series. Thus parts of a +series, considered not merely as classes but as series, are dealt with +by means of serial relations with limited fields.</p> + +<p>Relations with limited <i>domains</i> are not nearly so much used +as relations with limited <i>converse domains</i>. Relations with +limited converse domains play a great part in arithmetic, especially +in establishing the formal laws. What is wanted in such cases is a +one-one relation correlating two classes or two series. That is, +we want a relation such that not only does \(Rʻy\) exist whenever +\(y\in \text{ᗡ}ʻR\), but also \(\breve{R}ʻx\) exists whenever \(x\in +\text{D}ʻR\). The kind of relation which is most frequently found +to effect such a correlation is some such relation as \(\text{D}\) +or \(\text{ᗡ}\) or \(C\), or some other constant relation for +which we always have \(\text{E}!Rʻy\), with its converse domain so +limited that, subject to the limitation, only one value of \(y\) +gives any given value of \(Rʻy\). Thus for example let \(\lambda\) +be a class of relations no two of which have the same domain; then +\(\text{D}\upharpoonright \lambda\) will give a one-one correlation +of these relations with their domains: if \(R\), \(S\in \lambda\), we +shall have +\[ +\text{D}ʻR=\text{D}ʻS.\supset.R=S. +\] +We shall also have \(\text{D}ʻR=(\text{D}\upharpoonright \lambda)ʻR\) +and \(\text{D}ʻS=(\text{D}\upharpoonright \lambda)ʻS\). Moreover the +converse domain of \(\text{D}\upharpoonright \lambda\) is \(\lambda\), +and the domain of \(\text{D}\upharpoonright \lambda\) is the class +of domains of members of \(\lambda\). Thus \(\text{D}\upharpoonright\lambda\) +gives a one-one correlation of \(\lambda\) with the domains +of members of \(\lambda\). It is chiefly in such ways that relations +with limited converse domains are useful.</p> + +<p>For purposes of reference, a great many propositions are given in the +present number, but the propositions that will be used frequently are +comparatively few. Among these are the following:</p> + +<p class="nind"><b>*35·21.</b> \(\vdash.\alpha\upharpoonleft R\upharpoonright \beta=(\alpha\upharpoonleft R)\upharpoonright \beta=\alpha\upharpoonleft (R\upharpoonright \beta)\)</p> + +<p class="nind"><b>*35·31.</b> \(\vdash.(R\upharpoonright \alpha)\upharpoonright \beta=R\upharpoonright (\alpha\cap \beta)\)</p> + +<p class="nind"><b>*35·354.</b> \(\vdash.(R\upharpoonright \alpha)\mid S=R\mid \alpha\upharpoonleft S\)</p> + +<p><i>I.e.</i> in a relative product it makes no difference whether we +limit the converse domain of the first factor, or the domain of the +second.</p> + +<p class="nind"><b>*35·412.</b> \(\vdash.R\upharpoonright (\beta\cup \beta') = R\upharpoonright \beta\unicode{x228d}R\upharpoonright \beta'\)</p> + +<p class="nind"><b>*35·452.</b> \(\vdash:\text{ᗡ}ʻR\subset\beta.\supset.R\upharpoonright \beta=R\)</p> + +<p class="nind"><b>*35·48.</b> \(\vdash:\text{ᗡ}ʻP\subset\alpha.\supset.P\mid (\alpha\upharpoonleft R)=P\mid R\)</p> + +<p class="nind"><b>*35·52.</b> \(\vdash.\text{Cnv}ʻ(R\upharpoonright \beta)=\beta\upharpoonleft \breve{R}\)</p> + +<p class="nind"><b>*35·61.</b> \(\vdash.\text{D}ʻ(\alpha\upharpoonleft R)=\alpha\cap \text{D}ʻR\)</p> + +<p class="nind"><b>*35·64.</b> \(\vdash.\text{ᗡ}ʻ(R\upharpoonright \beta)=\beta\cap \text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*35·65.</b> \(\vdash:\beta\subset\text{ᗡ}ʻR.\supset.\text{ᗡ}ʻ(R\upharpoonright \beta)=\beta\)</p> + +<p>The hypothesis \(\beta\subset\text{ᗡ}ʻR\) is fulfilled in the great +majority of cases in which we have occasion to use \(R\upharpoonright\beta\).</p> + +<p class="nind"><b>*35·66.</b> \(\vdash:\text{ᗡ}ʻR\subset\beta.\equiv.R\upharpoonright \beta=R\)</p> + +<p class="nind"><b>*35·7.</b> \(\vdash:\phi\{(R\upharpoonright \beta)ʻy\}.\equiv.y\in \beta.\phi(Rʻy)\)</p> + +<p><span class="pagenum" id="Page_280">[Pg 280]</span></p> + +<p>This proposition is used very frequently, owing to the fact that limitation +of the converse domain is chiefly applied to such relations as give rise to +descriptive functions (<i>e.g.</i> \(\text{D}\), \(\text{ᗡ}\), \(C\)).</p> + +<p class="nind"><b>*35·71.</b> \(\vdash\colon\ldotp y\in \beta.\supset_{y}.Rʻy=Sʻy:\supset.R\upharpoonright \beta=S\upharpoonright \beta\)</p> + +<p>This proposition is useful for a reason similar to that which makes +*35·7 useful.</p> + +<p class="nind"><b>*35·82.</b> \(\vdash.\alpha\uparrow \beta=\alpha\upharpoonleft \dot{\text{V}}\upharpoonright \beta\)</p> + +<p>Owing to this proposition, the properties of \(\alpha\uparrow\beta\) +can be deduced from the already proved properties of +\(\alpha\upharpoonleft R\upharpoonright \beta\), by putting \(R =\dot{\text{V}}\).</p> + +<p>The relation "\(\alpha\uparrow \beta\)" is what may be called an +"analysable" relation, <i>i.e.</i> it holds between \(x\) and y when +\(x\in \alpha\) and \(y\in \beta\), <i>i.e.</i> when \(x\) has a +property independent of \(y\), and \(y\) has a property independent of +\(x\).</p> + +<p class="nind"><b>*35·85.</b> \(\vdash:\exists !\beta.\supset.\text{D}ʻ(\alpha\uparrow \beta)=\alpha\)</p> + +<p class="nind"><b>*35·86.</b> \(\vdash:\exists !\alpha.\supset.\text{ᗡ}ʻ(\alpha\uparrow \beta)=\beta\)</p> + +<p>If either \(\alpha\) or \(\beta\) is null, so is \(\alpha\uparrow\beta\) +(<a href="#*35·88">*35·88</a>).</p> + +<hr class="tb"> + +<p class="nind"><b>*35·01.</b> \(\alpha\upharpoonleft R=\hat{x}\hat{y}(x\in \alpha.xRy) \quad\text{Df}\)</p> + +<p class="nind"><b>*35·02.</b> \(R\upharpoonright \beta=\hat{x}\hat{y}(xRy.y\in \beta) \quad\text{Df}\)</p> + +<p class="nind"><b>*35·03.</b> \(\alpha\upharpoonleft R\upharpoonright \beta=\hat{x}\hat{y}(x\in \alpha.xRy.y\in \beta)\quad\text{Df}\)</p> + +<p class="nind"><b>*35·04.</b> \(\alpha\uparrow \beta=\hat{x}\hat{y}(x\in \alpha.y\in \beta) \quad\text{Df}\)</p> + +<p class="nind"><b>*35·05.</b> \(Rʻx\uparrow \beta=(Rʻx)\uparrow \beta \quad\text{Df}\)</p> + +<p>The last definition serves merely for the avoidance of brackets.</p> + +<p class="nind"><b>*35·1.</b> \(\vdash:x(\alpha\upharpoonleft R)y.\equiv.x\in \alpha.xRy \quad[\text{*21·3.(*35·01)}]\)</p> + +<p class="nind"><b>*35·101.</b> \(\vdash:x(R\upharpoonright \beta)y.\equiv.xRy.y\in \beta\)</p> + +<p class="nind"><b>*35·102.</b> \(\vdash:x(\alpha\upharpoonleft R\upharpoonright \beta)y.\equiv.x\in \alpha.xRy.y\in \beta\)</p> + +<p class="nind"><b>*35·103.</b> \(\vdash:x(\alpha\uparrow \beta)y.\equiv.x\in \alpha.y\in \beta\)</p> + +<p class="nind"><b>*35·11.</b> \(\vdash.\alpha\upharpoonleft R\upharpoonright \beta=(\alpha\upharpoonleft R)\dot{\cap}(R\upharpoonright \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·102}.\supset\vdash:x(\alpha\upharpoonleft R\upharpoonright \beta)y.&\equiv.x\in \alpha.xRy.y\in \beta.\\ +[\text{*4·24}] &\equiv.x\in \alpha.xRy.xRy.y\in \beta.\\ +[\text{*35·1·101}] & \equiv.x(\alpha\upharpoonleft R)y.x(R\upharpoonright \beta)y.\\ +[\text{*23·33}] & \equiv.x\{(\alpha\upharpoonleft R)\dot{\cap}(R\upharpoonright \beta)\}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·12.</b> \(\vdash.(\alpha\upharpoonleft R)\dot{\cap}(S\upharpoonright \beta)=\alpha\upharpoonleft (R\dot{\cap}S)\upharpoonright \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·33}.\supset\vdash:x\{(\alpha\upharpoonleft R)\dot{\cap}(S\upharpoonright \beta)\}y.&\equiv.x(\alpha\upharpoonleft R)y.x(S\upharpoonright \beta)y.\\ +[\text{*35·1·101}] &\equiv.x\in \alpha.xRy.xSy.y\in \beta.\\ +[\text{*23·33}] &\equiv.x\in \alpha.x(R\dot{\cap}S)y.y\in \beta.\\ +[\text{*35·102}]& \equiv.x\{\alpha\upharpoonleft (R\dot{\cap}S)\upharpoonright \beta\}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_281">[Pg 281]</span></p> + +<p class="nind"><b>*35·13.</b> \(\vdash.(\alpha\upharpoonleft R)\dot{\cap}(\beta\upharpoonleft S)=(\alpha \cap \beta)\upharpoonleft (R\dot{\cap}S)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*23·33.\supset\vdash:x\{(\alpha\upharpoonleft R)\dot{\cap}(\beta\upharpoonleft S)\}y.&\equiv.x(\alpha\upharpoonleft R)y.x(\beta\upharpoonleft S)y.\\ +[*35·1] &\equiv.x\in \alpha.xRy.x\in \beta.xSy.\\ +[*22·33.*23·33] &\equiv.x\in (\alpha \cap \beta).x(R\dot{\cap}S)y.\\ +[*35·1] & \equiv.x\{(\alpha \cap \beta)\upharpoonleft (R\dot{\cap}S)\}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·14.</b> \(\vdash.(R\upharpoonright \alpha)\dot{\cap}(S\upharpoonright \beta)=(R\dot{\cap}S)\upharpoonright (\alpha \cap \beta) \quad[\text{Similar proof to *35·13}]\)</p> + +<p class="nind"><b>*35·15.</b> + \(\vdash.(\alpha\upharpoonleft R\upharpoonright \beta)\dot{\cap}(\alpha'\upharpoonleft S\upharpoonright \beta')=(\alpha \cap \alpha')\upharpoonleft (R\dot{\cap}S)\upharpoonright (\beta \cap \beta')\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*35·11.\supset\\ +\vdash.(\alpha\upharpoonleft R\upharpoonright \beta)\dot{\cap}(\alpha'\upharpoonleft S\upharpoonright \beta')&=(\alpha\upharpoonleft R)\dot{\cap}(R\upharpoonright \beta)\dot{\cap}(\alpha'\upharpoonleft S)\dot{\cap}(S\upharpoonright \beta')\\ +[*35·13·14] &=\{(\alpha \cap \alpha')\upharpoonleft (R\dot{\cap}S)\}\dot{\cap}\{(R\dot{\cap}S)\upharpoonright (\beta \cap \beta')\}\\ +[*35·11] & =\{(\alpha \cap \alpha')\upharpoonleft (R\dot{\cap}S)\upharpoonright (\beta \cap \beta')\}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·16.</b> \(\vdash.(\alpha\upharpoonleft R)\dot{\cap}S=\alpha\upharpoonleft (R\dot{\cap}S)=R\dot{\cap}\alpha\upharpoonleft S \quad[\text{Similar proof to *35·13}]\)</p> + +<p class="nind"><b>*35·17.</b> \(\vdash.(R\upharpoonright \beta)\dot{\cap}S=(R\dot{\cap}S)\upharpoonright \beta=R\dot{\cap}S\upharpoonright \beta \quad[\text{Similar proof to *35·13}]\)</p> + +<p class="nind"><b>*35·18.</b> + \[\begin{align}\vdash.(\alpha\upharpoonleft R\upharpoonright \beta)\dot{\cap}S=\alpha\upharpoonleft (R\dot{\cap}S)\upharpoonright \beta=R\dot{\cap}\alpha&\upharpoonleft S\upharpoonright \beta\\ +&\qquad\qquad\qquad[\text{Similar proof to *35·15}]\end{align}\]</p> + +<p class="nind"><b>*35·21.</b> \(\vdash.\alpha\upharpoonleft R\upharpoonright \beta=(\alpha\upharpoonleft R)\upharpoonright \beta=\alpha\upharpoonleft (R\upharpoonright \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*35·102.\supset\vdash:x(\alpha\upharpoonleft R\upharpoonright \beta)y.&\equiv.x\in \alpha.xRy.y\in \beta.\\ +[*35·1] &\equiv.x(\alpha\upharpoonleft R)y.y\in \beta.\\ +[*35·101] &\equiv.x\{(\alpha\upharpoonleft R)\upharpoonright \beta\}y &\qquad \text{(1)}\\ +\vdash.*35·102.\supset\vdash:x(\alpha\upharpoonleft R\upharpoonright \beta)y.&\equiv.x\in \alpha.xRy.y\in \beta.\\ +[*35·101] & \equiv.x\in \alpha.x(R\upharpoonright \beta)y.\\ +[*35·1] &\equiv.x\{\alpha\upharpoonleft (R\upharpoonright \beta)\}y &\qquad \text{(2)}\\ +\vdash.(1).(2).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·22.</b> \(\vdash.(\alpha\upharpoonleft R)\mid S=\alpha\upharpoonleft (R\mid S)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*34·1.\supset\vdash\colon\ldotp x\{(\alpha\upharpoonleft R)\mid S\}y.&\equiv:(\exists z).x(\alpha\upharpoonleft R)z.zSy:\\ +[*35·1] &\equiv:(\exists z).x\in \alpha.xRz.zSy:\\ +[*10·35] &\equiv:x\in \alpha:(\exists z).xRz.zSy.\\ +[*34·1] &\equiv:x\in \alpha.x(R\mid S)y:\\ +[*35·1] &\equiv:x\{\alpha\upharpoonleft (R\mid S)\}y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·23.</b> \(\vdash.S\mid (R\upharpoonright \beta)=(S\mid R)\upharpoonright \beta \quad[\text{Similar proof to *35·22}]\)</p> + +<p class="nind"><b>*35·24.</b> \(\alpha\upharpoonleft R\mid S=(\alpha\upharpoonleft R)\mid S \quad\text{Df}\)</p> + +<p class="nind"><b>*35·25.</b> \(S\mid R\upharpoonright \beta=(S\mid R)\upharpoonright \beta \quad\text{Df}\)</p> + +<p><span class="pagenum" id="Page_282">[Pg 282]</span></p> + +<p class="nind"><b>*35·26.</b> + \[\begin{align}\vdash.(\alpha\upharpoonleft R)\mid (S\upharpoonright \beta)&=\alpha\upharpoonleft (R\mid S)\upharpoonright \beta=\{\alpha\upharpoonleft (R\mid S)\}\upharpoonright \beta=\alpha\upharpoonleft \{(R\mid S)\upharpoonright \beta\}\\ +&=\{(\alpha\upharpoonleft R)\mid S)\}\upharpoonright \beta=\alpha\upharpoonleft \{R\mid (S\upharpoonright \beta)\}\\ +&=(\alpha\upharpoonleft R\mid S)\upharpoonright \beta=\alpha\upharpoonleft (R\mid S\upharpoonright \beta)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1}.\supset\vdash\colon\ldotp x\{(\alpha\upharpoonleft R)\mid (S\upharpoonright \beta)\}y.&\equiv:(\exists z).x(\alpha\upharpoonleft R)z.z(S\upharpoonright \beta)y:\\ +[\text{*35·1·101}]& \equiv:(\exists z).x\in \alpha.xRz.zSy.y\in \beta:\\ +[\text{*10·35}] & \equiv:x\in \alpha.y\in \beta:(\exists z).xRz.zSy:\\ +[\text{*34·1}] & \equiv:x\in \alpha.x(R\mid S)y.y\in \beta:\\ +[\text{*35·102}]& \equiv:x\{\alpha\upharpoonleft (R\mid S)\upharpoonright \beta\}y &\qquad \text{(1)}\\ +\vdash.\text{(1).*35·21·22·23.(*35·24·25)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·27.</b> \(\alpha\upharpoonleft R\mid S\upharpoonright \beta=(\alpha\upharpoonleft R\mid S)\upharpoonright \beta \quad\text{Df}\)</p> + +<p class="nind"><b>*35·31.</b> \(\vdash.(R\upharpoonright \alpha)\upharpoonright \beta=R\upharpoonright (\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·101}.\supset\vdash:x\{(R\upharpoonright \alpha)\upharpoonright \beta\}y.&\equiv.x(R\upharpoonright \alpha)y.y\in \beta.\\ +[\text{*35·101}] &\equiv.xRy.y\in \alpha.y\in \beta.\\ +[\text{*22·33}] &\equiv.xRy.y\in \alpha\cap \beta.\\ +[\text{*35·101}]& \equiv.x\{R\upharpoonright (\alpha\cap \beta)\}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·32.</b> \(\vdash.\alpha\upharpoonleft (\beta\upharpoonleft R)=(\alpha\cap \beta)\upharpoonleft R \quad[\text{Proof similar to that of *35·31}]\)</p> + +<p class="nind"><b>*35·33.</b> + \(\vdash.(\alpha\upharpoonleft R\upharpoonright \beta)\upharpoonright \gamma=\{\alpha\upharpoonleft R\upharpoonright (\beta\cap \gamma)\} \quad[\text{Proof similar to that of *35·31}]\)</p> + +<p class="nind"><b>*35·34.</b> + \(\vdash.\alpha\upharpoonleft (\beta\upharpoonleft R\upharpoonright \gamma)=\{(\alpha\cap \beta)\upharpoonleft R\upharpoonright \gamma\} \quad[\text{Proof similar to that of *35·31}]\)</p> + +<p class="nind"><b>*35·35.</b> \(\vdash.\alpha\upharpoonleft R=(\alpha\cap \text{D}ʻR)\upharpoonleft R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1}.\supset\vdash:x(\alpha\upharpoonleft R)y.&\equiv.x\in \alpha.xRy.\\ +[\text{*33·14}] &\equiv.x\in \alpha.x\in \text{D}ʻR.xRy.\\ +[\text{*22·33.*35·1}] & \equiv.x\{(\alpha\cap \text{D}ʻR)\upharpoonleft R\}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·351.</b> \(\vdash.R\upharpoonright \beta=R\upharpoonright (\beta\cap \text{ᗡ}ʻR) \quad[\text{Proof as in *35·35}]\)</p> + +<p class="nind"><b>*35·352.</b> \(\vdash.\alpha\upharpoonleft R\upharpoonright \beta=(\alpha\cap \text{D}ʻR)\upharpoonleft R\upharpoonright (\beta\cap \text{ᗡ}ʻR) \quad[\text{Proof as in *35·35}]\)</p> + +<p class="nind"><b>*35·354.</b> \(\vdash.(R\upharpoonright \alpha)\mid S=R\mid \alpha\upharpoonleft S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1.*35·101}.&\supset\\ +\vdash:x\{(R\upharpoonright \alpha)\mid S\}z.&\equiv.(\exists y).xRy.y\in \alpha.ySz.\\ +[\text{*35·1}] &\equiv.(\exists y).xRy.y(\alpha\upharpoonleft S)z.\\ +[\text{*34·1}] & \equiv.x\{R\mid (\alpha\upharpoonleft S)\}z:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·41.</b> \(\vdash.(\alpha\cup \alpha')\upharpoonleft R=\alpha\upharpoonleft R\unicode{x228d}\alpha'\upharpoonleft R \quad[\text{*35·1.*22·34}]\)</p> + +<p class="nind"><b>*35·412.</b> \(\vdash.R\upharpoonright (\beta\cup \beta')=R\upharpoonright \beta\unicode{x228d}R\upharpoonright \beta' \quad[\text{*35·101.*22·34}]\)</p> + +<p><span class="pagenum" id="Page_283">[Pg 283]</span></p> + +<p class="nind"><b>*35·413.</b> + \[\begin{align}\vdash.(\alpha\cup\alpha')\upharpoonleft R\upharpoonright (\beta\cup\beta')&=(\alpha\upharpoonleft R\upharpoonright \beta)\unicode{x228d}(\alpha\upharpoonleft R\upharpoonright \beta')\\ +&\unicode{x228d}(\alpha'\upharpoonleft R\upharpoonright \beta)\unicode{x228d}(\alpha'\upharpoonleft R\upharpoonright \beta') \quad[\text{*35·102.*22·34}]\end{align}\]</p> + +<p class="nind"><b>*35·42.</b> \(\vdash.\alpha\upharpoonleft (R\unicode{x228d}S)=(\alpha\upharpoonleft R)\unicode{x228d}(\alpha\upharpoonleft S) \quad[\text{*35·1.*23·34}]\)</p> + +<p class="nind"><b>*35·421.</b> \(\vdash.(R\unicode{x228d}S)\upharpoonright \beta=(R\upharpoonright \beta)\unicode{x228d}(S\upharpoonright \beta) \quad[\text{*35·101.*23·34}]\)</p> + +<p class="nind"><b>*35·422.</b> + \(\vdash.\alpha\upharpoonleft (R\unicode{x228d}S)\upharpoonright \beta=(\alpha\upharpoonleft R\upharpoonright \beta)\unicode{x228d}(\alpha\upharpoonleft S\upharpoonright \beta) \quad[\text{*35·102.*23·34}]\)</p> + +<p class="nind"><b>*35·43.</b> \(\vdash:\alpha\subset\beta.\supset.\alpha\upharpoonleft R\unicode{x2abd}\beta\upharpoonleft R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1}.\supset\vdash\colon\ldotp \alpha\subset\beta.\supset:x(\alpha\upharpoonleft R)y.&\equiv.x\in \alpha.xRy.\\ +[\text{*22·1}]&\supset.x\in \beta.xRy.\\ +[\text{*35·1}] & \supset.x(\beta\upharpoonleft R)y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·431.</b> \(\vdash:\beta\subset\gamma.\supset.R\upharpoonright \beta\unicode{x2abd}R\upharpoonright \gamma \quad[\text{Proof similar to that of *35·43}]\)</p> + +<p class="nind"><b>*35·432.</b> \[\begin{align}\vdash:\alpha\subset\gamma.\beta\subset\delta.\supset.\alpha\upharpoonleft R\upharpoonright \beta&\unicode{x2abd}\gamma\upharpoonleft R\upharpoonright \delta\\ +&[\text{Proof similar to that of *35·43}]\end{align}\]</p> + +<p class="nind"><b>*35·44.</b> \(\vdash.\alpha\upharpoonleft R\unicode{x2abd}R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1}.\supset\vdash:x(\alpha\upharpoonleft R)y.&\supset.x\in \alpha.xRy.\\ +[\text{*3·27}] & \supset.xRy:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·441.</b> \(\vdash.R\upharpoonright \beta\unicode{x2abd}R \quad[\text{Proof similar to that of *35·44}]\)</p> + +<p class="nind"><b>*35·442.</b> \(\vdash.\alpha\upharpoonleft R\upharpoonright \beta\unicode{x2abd}R \quad[\text{Proof similar to that of *35·44}]\)</p> + +<p class="nind"><b>*35·451.</b> \(\vdash:\text{D}ʻR\subset\alpha.\supset.\alpha\upharpoonleft R=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·71}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:x\in \text{D}ʻR.\equiv.x\in \text{D}ʻR.x\in \alpha:\\ +[\text{*4·36}] &\supset:x\in \text{D}ʻR.xRy.\equiv.x\in \text{D}ʻR.xRy.x\in \alpha &\qquad \text{(1)}\\ +\vdash.\text{*33·14.*4·71}. &\supset\vdash:xRy.\equiv.x\in \text{D}ʻR.xRy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:xRy.\equiv.xRy.x\in \alpha.\\ +[\text{*35·1}] &\qquad\qquad \equiv.x(\alpha\upharpoonleft R)y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·452.</b> \(\vdash:\text{ᗡ}ʻR\subset\beta.\supset.R\upharpoonright \beta=R \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*35·453.</b> \(\vdash:\text{D}ʻR\subset\alpha.\supset.\alpha\upharpoonleft R\upharpoonright \beta=R\upharpoonright \beta \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*35·454.</b> \(\vdash:\text{ᗡ}ʻR\subset\beta.\supset.\alpha\upharpoonleft R\upharpoonright \beta=\alpha\upharpoonleft R \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*35·46.</b> \(\vdash:R\unicode{x2abd}S.\supset.\alpha\upharpoonleft R\unicode{x2abd}\alpha\upharpoonleft S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·1}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:xRy.\supset.xSy:\\ +[\text{Fact}] &\supset:x\in \alpha.xRy.\supset.x\in \alpha.xSy:\\ +[\text{*35·1}] &\supset:x(\alpha\upharpoonleft R)y.\supset.x(\alpha\upharpoonleft S)y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·461.</b> \(\vdash:R\unicode{x2abd}S.\supset.R\upharpoonright \beta\unicode{x2abd}S\upharpoonright \beta \quad[\text{Similar proof}]\)</p> + +<p><span class="pagenum" id="Page_284">[Pg 284]</span></p> + +<p class="nind"><b>*35·462.</b> \(\vdash:R\unicode{x2abd}S.\supset.\alpha\upharpoonleft R\upharpoonright \beta\unicode{x2abd}\alpha\upharpoonleft S\upharpoonright \beta \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*35·471.</b> \(\vdash:\text{ᗡ}ʻP\cap \alpha=\Lambda.\supset.P\mid (\alpha\upharpoonleft R)=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1}.\supset\vdash:x\{P\mid (\alpha\upharpoonleft R)\}z.&\supset.(\exists y).xPy.y(\alpha\upharpoonleft R)z.\\ +[\text{*35·1}] &\supset.(\exists y).xPy.y\in \alpha.yRz.\\ +[\text{*33·14.*10·5}] & \supset.(\exists y).y\in \text{ᗡ}ʻP.y\in \alpha.\\ +[\text{*22·33.*24·5}] & \supset.\exists !\text{ᗡ}ʻP\cap \alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp.*24·51}.&\supset\\ +\vdash:\text{ᗡ}ʻP\cap \alpha=\Lambda.&\supset.{\sim}x\{P\mid (\alpha\upharpoonleft R)\}z:\\ +[\text{*11·11·3}] &\supset\vdash:\text{ᗡ}ʻP\cap \alpha=\Lambda.\supset.(x,z).{\sim}x\{P\mid (\alpha\upharpoonleft R)\}z.\\ +[\text{*25·15}] & \supset.P\mid (\alpha\upharpoonleft R)=\dot{\Lambda}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·472.</b> \(\vdash:\text{D}ʻP\cap \alpha=\Lambda.\supset.(R\upharpoonright \alpha)\mid P=\dot{\Lambda}\)</p> + +<p class="nind"><b>*35·473.</b> \(\vdash:\text{ᗡ}ʻP\cap \alpha=\Lambda.\supset.P\mid (\alpha\upharpoonleft R\upharpoonright \beta)=\dot{\Lambda}\)</p> + +<p class="nind"><b>*35·474.</b> \(\vdash:\text{D}ʻP\cap \beta=\Lambda.\supset.(\alpha\upharpoonleft R\upharpoonright \beta)\mid P=\dot{\Lambda}\)</p> + +<p class="nind"><b>*35·48.</b> \(\vdash:\text{ᗡ}ʻP\subset\alpha.\supset.P\mid (\alpha\upharpoonleft R)=P\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·1}. \supset\vdash\colon\ldotp \text{Hp}.&\supset:y\in \text{ᗡ}ʻP.\supset_{y}.y\in \alpha:\\ +[\text{*4·71}] &\supset:y\in \text{ᗡ}ʻP.y\in \alpha.\equiv_{y}.y\in \text{ᗡ}ʻP:\\ +[\text{*10·311}] &\supset:xPy.y\in \text{ᗡ}ʻP.y\in \alpha.\equiv_{y}.xPy.y\in \text{ᗡ}ʻP &\qquad \text{(1)}\\ +\vdash.\text{*33·14.*4·71}.&\supset\vdash:xPy.y\in \text{ᗡ}ʻP.\equiv.xPy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:xPy.y\in \alpha.\equiv_{y}.xPy:\\ +[\text{*10·311}] &\supset:xPy.y\in \alpha.yRz.\equiv_{y}.xPy.yRz:\\ +[\text{*35·1}] & \supset:xPy.y(\alpha\upharpoonleft R)z.\equiv_{y}.xPy.yRz:\\ +[\text{*10·281}] & \supset:(\exists y).xPy.y(\alpha\upharpoonleft R)z.\equiv.(\exists y).xPy.yRz:\\ +[\text{*34·1}] &\supset:x(P\mid \alpha\upharpoonleft R)z.\equiv.x(P\mid R)z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·481.</b> \(\vdash:\text{D}ʻR\subset\beta.\supset.(P\upharpoonright \beta)\mid R=P\mid R \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*35·51.</b> \(\vdash.\text{Cnv}ʻ(\alpha\upharpoonleft R)=\breve{R}\upharpoonright \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\vdash:x\{\text{Cnv}ʻ(\alpha\upharpoonleft R)\}y.&\equiv.y(\alpha\upharpoonleft R)x.\\ +[\text{*35·1}] &\equiv.y\in \alpha.yRx.\\ +[\text{*31·11}] & \equiv.x\breve{R}y.y\in \alpha.\\ +[\text{*35·101}] & \equiv.x(\breve{R}\upharpoonright \alpha)y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·52.</b> \(\vdash.\text{Cnv}ʻ(R\upharpoonright \beta)=\beta\upharpoonleft \breve{R} \quad[\text{Proof similar to that of *35·51}]\)</p> + +<p class="nind"><b>*35·53.</b> \(\vdash.\text{Cnv}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)=\beta\upharpoonleft \breve{R}\upharpoonright \alpha \quad[\text{Proof similar to that of *35·51}]\)</p> + +<p><span class="pagenum" id="Page_285">[Pg 285]</span></p> + +<p class="nind"><b>*35·61.</b> \(\vdash.\text{D}ʻ(\alpha\upharpoonleft R)=\alpha\cap \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}.\supset\vdash\colon\ldotp x\in \text{D}ʻ(\alpha\upharpoonleft R).&\equiv:(\exists y).x(\alpha\upharpoonleft R)y:\\ +[\text{*35·1}] &\equiv:(\exists y).x\in \alpha.xRy:\\ +[\text{*10·35}] &\equiv:x\in \alpha:(\exists y).xRy:\\ +[\text{*33·13}] &\equiv:x\in \alpha.x\in \text{D}ʻR:\\ +[\text{*22·33}] & \equiv:x\in (\alpha\cap \text{D}ʻR)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·62.</b> \(\vdash:\alpha\subset\text{D}ʻR.\supset.\text{D}ʻ(\alpha\upharpoonleft R)=\alpha \quad[\text{*35·61.*22·621}]\)</p> + +<p class="nind"><b>*35·63.</b> \(\vdash:\text{D}ʻR\subset\alpha.\equiv.\alpha\upharpoonleft R=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·61}.\supset\vdash:\alpha\upharpoonleft R=R.&\supset.\alpha\cap \text{D}ʻR=\text{D}ʻR\\ +[\text{*22·621}] & \supset.\text{D}ʻR\subset\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*35·451}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·64.</b> \(\vdash.\text{ᗡ}ʻ(R\upharpoonright \beta)=\beta\cap \text{ᗡ}ʻR \quad[\text{Proof as in *35·61}]\)</p> + +<p class="nind"><b>*35·641.</b> \(\vdash:\alpha\cap \text{D}ʻR=\Lambda.\supset.\alpha\upharpoonleft R=\dot{\Lambda} \quad[\text{*35·61.*33·241}]\)</p> + +<p class="nind"><b>*35·642.</b> \(\vdash:\alpha\cap \text{ᗡ}ʻR=\Lambda.\supset.R\upharpoonright \alpha=\dot{\Lambda} \quad[\text{*35·64.*33·241}]\)</p> + +<p class="nind"><b>*35·643.</b> \(\vdash:\alpha\cap \text{D}ʻR=\Lambda.\supset.\alpha\upharpoonleft (R\unicode{x228d}S)=\alpha\upharpoonleft S \quad[\text{*35·641·42}]\)</p> + +<p class="nind"><b>*35·644.</b> \(\vdash:\alpha\cap \text{ᗡ}ʻR=\Lambda.\supset.(R\unicode{x228d}S)\upharpoonright \alpha=S\upharpoonright \alpha \quad[\text{*35·642·421}]\)</p> + +<p class="nind"><b>*35·65.</b> \(\vdash:\beta\subset\text{ᗡ}ʻR.\supset.\text{ᗡ}ʻ(R\upharpoonright \beta)=\beta \quad[\text{*35·64.*22·621}]\)</p> + +<p class="nind"><b>*35·66.</b> \(\vdash:\text{ᗡ}ʻR\subset\beta.\equiv.R\upharpoonright \beta=R \quad[\text{Proof as in *35·63}]\)</p> + +<p class="nind"><b>*35·671.</b> \(\vdash.\text{D}ʻ(R\mid S)=\text{D}ʻ(R\upharpoonright \text{D}ʻS)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}.\supset\vdash\colon\ldotp x\in \text{D}ʻ(R\mid S).&\equiv:(\exists y).x(R\mid S)y:\\ +[\text{*34·1}] & \equiv:(\exists y,z).xRz.zSy:\\ +[\text{*11·23}] &\equiv:(\exists z,y).xRz.zSy:\\ +[\text{*10·35}] &\equiv:(\exists z):xRz:(\exists y).zSy:\\ +[\text{*33·13}] & \equiv:(\exists z).xRz.z\in \text{D}ʻS:\\ +[\text{*35·101}] & \equiv:(\exists z).x(R\upharpoonright \text{D}ʻS)z:\\ +[\text{*33·13}] & \equiv:x\in \text{D}ʻ(R\upharpoonright \text{D}ʻS)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·672.</b> \(\vdash.\text{ᗡ}ʻ(R\mid S)=\text{ᗡ}ʻ(\text{ᗡ}ʻR\upharpoonleft S) \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*35·68.</b> \(\vdash:\alpha\cap \beta=\Lambda.\supset.(\alpha\upharpoonleft R\upharpoonright \beta)^{2}=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·61·64·21}.\supset\vdash.\text{D}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)&\subset\alpha.\text{ᗡ}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)\subset\beta.\\ +[\text{*22·49.*24·13}]\supset\vdash:\alpha\cap \beta=\Lambda.&\supset.\text{D}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)\cap \text{ᗡ}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)=\Lambda.\\ +[\text{*34·531}] & \supset.(\alpha\upharpoonleft R\upharpoonright \beta)^{2}=\dot{\Lambda}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_286">[Pg 286]</span></p> + +<p class="nind"><b>*35·7.</b> \(\vdash:\phi\{(R\upharpoonright \beta)ʻy\}.\equiv.y\in \beta.\phi(Rʻy)\)</p> + +<p>This proposition is very often used in the later parts of the work.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}. \supset\vdash:\phi\{(R\upharpoonright \beta)ʻy\}.&\supset.\text{E}!(R\upharpoonright \beta)ʻy.\\ +[\text{*33·43}] &\supset.y\in \text{ᗡ}ʻ(R\upharpoonright \beta).\\ +[\text{*35·64}] &\supset.y\in \beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*4·71}.\supset\vdash:\phi\{(R\upharpoonright \beta)ʻy\}.&\equiv.y\in \beta.\phi\{(R\upharpoonright \beta)ʻy\} &\qquad \text{(2)}\\ +\vdash.\text{*4·73.*35·101}.\supset\vdash\colon\ldotp y\in \beta. &\supset:x(R\upharpoonright \beta)y.\equiv_{x}.xRy:\\ +[\text{*14·272}] &\supset:\phi\{(R\upharpoonright \beta)ʻy\}.\equiv.\phi(Rʻy) &\qquad \text{(3)}\\ +\vdash.\text{(3).*5·32}.\supset\vdash:y\in \beta.\phi\{(R\upharpoonright \beta)ʻy\}.&\equiv.y\in \beta.\phi(Rʻy) &\qquad \text{(4)}\\ +\vdash.\text{(2).(4)}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·71.</b> \(\vdash\colon\ldotp y\in \beta.\supset_{y}.Rʻy=Sʻy:\supset.R\upharpoonright \beta=S\upharpoonright \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·7}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:y\in \beta.\supset_{y}.y\in \beta.Rʻy=Sʻy:\\ +[\text{*35·7}] &\supset:y\in \beta.\supset_{y}.(R\upharpoonright \beta)ʻy=(S\upharpoonright \beta)ʻy:\\ +[\text{*35·64}] &\supset:y\in \text{ᗡ}ʻ(R\upharpoonright \beta)\cup \text{ᗡ}ʻ(S\upharpoonright \beta).\supset_{y}.(R\upharpoonright \beta)ʻy=(S\upharpoonright \beta)ʻy:\\ +[\text{*33·45}] &\supset:R\upharpoonright \beta=S\upharpoonright \beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·75.</b> \(\vdash.\Lambda\upharpoonleft R=R\upharpoonright \Lambda=\Lambda\upharpoonleft R\upharpoonright \beta=\alpha\upharpoonleft R\upharpoonright \Lambda=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·61}. &\supset\vdash.\text{D}ʻ(\Lambda\upharpoonleft R)=\Lambda.\\ +[\text{*33·241}] &\supset\vdash.\Lambda\upharpoonleft R=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash.\text{*35·64}.& \supset\vdash.\text{ᗡ}ʻ(R\upharpoonright \Lambda)=\Lambda.\\ +[\text{*33·241}] & \supset\vdash.R\upharpoonright \Lambda=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash.\text{*35·441·21}.&\supset\vdash.\Lambda\upharpoonleft R\upharpoonright \beta\unicode{x2abd}\Lambda\upharpoonleft R.\\ +[\text{(1).*25·13}] &\supset\vdash.\Lambda\upharpoonleft R\upharpoonright \beta=\dot{\Lambda} &\qquad \text{(3)}\\ +\vdash.\text{*35·44·21}.& \supset\vdash.\alpha\upharpoonleft R\upharpoonright \Lambda\unicode{x2abd}R\upharpoonright \Lambda.\\ +[\text{(2).*25·13}] &\supset\vdash.\alpha\upharpoonleft R\upharpoonright \Lambda=\dot{\Lambda} &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(3).(4)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·76.</b> \(\vdash.\text{V}\upharpoonleft R=R\upharpoonright \text{V}=\text{V}\upharpoonleft R\upharpoonright \text{V}=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1}. \supset\vdash:x(\text{V}\upharpoonleft R)y. &\equiv.x\in \text{V}.xRy.\\ +[\text{*24·104.*4·73}] &\equiv.xRy &\qquad \text{(1)}\\ +\vdash.\text{*35·101}.\supset\vdash:x(R\upharpoonright \text{V})y. &\equiv.xRy.y\in \text{V}.\\ +[\text{*24·104.*4·73}]&\equiv.xRy &\qquad \text{(2)}\\ +\vdash.\text{*35·102}.\supset\vdash:x(\text{V}\upharpoonleft R\upharpoonright \text{V})y.&\equiv.x\in \text{V}.xRy.y\in \text{V}.\\ +[\text{*24·104.*4·73}] &\equiv.xRy &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_287">[Pg 287]</span></p> + +<p>The rest of this number, down to <a href="#*35·93">*35·93</a> exclusive, is concerned with +\(\alpha\uparrow \beta\), except *35·81·812.</p> + +<p class="nind"><b>*35·81.</b> \(\vdash:x(\alpha\upharpoonleft \dot{\text{V}})y.\equiv.x\in \alpha \quad[\text{*35·1.*25·104}]\)</p> + +<p class="nind"><b>*35·812.</b> \(\vdash:x(\dot{\text{V}}\upharpoonright \beta)y.\equiv.y\in \beta \quad[\text{*35·101.*25·104}]\)</p> + +<p class="nind"><b>*35·82.</b> \(\vdash.\alpha\uparrow \beta=\alpha\upharpoonleft \dot{\text{V}}\upharpoonright \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·103}.\supset\vdash:x(\alpha\uparrow \beta)y.&\equiv.x\in \alpha.y\in \beta.\\ +[\text{*25·104}] & \equiv.x\in \alpha.x\dot{\text{V}}y.y\in \beta.\\ +[\text{*35·102}] &\equiv.x(\alpha\upharpoonleft \dot{\text{V}}\upharpoonright \beta)y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·822.</b> \(\vdash.\alpha\upharpoonleft R\upharpoonright \beta=R\dot{\cap}(\alpha\uparrow \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·102}.\supset\vdash:x(\alpha\upharpoonleft R\upharpoonright \beta)y.&\equiv.x\in \alpha.xRy.y\in \beta.\\ +[\text{*4·3}]&\equiv.xRy.x\in \alpha.y\in \beta.\\ +[\text{*35·103}] &\equiv.xRy.x(\alpha\uparrow \beta)y.\\ +[\text{*23·33}]& \equiv.x\{R\dot{\cap}(\alpha\uparrow \beta)\}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*35·83">*35·83</a>.</b> \(\vdash:\text{D}ʻR\subset \alpha.\text{ᗡ}ʻR\subset \beta.\equiv.R\unicode{x2abd}\alpha\uparrow \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·14}. &\supset\vdash\colon\ldotp xRy.\supset:x\in \text{D}ʻR.y\in \text{ᗡ}ʻR:\\ +[\text{*22·46}]&\quad\qquad\quad\supset:\text{D}ʻR\subset \alpha.\text{ᗡ}ʻR\subset \beta.\supset.x\in \alpha.y\in \beta &\qquad \text{(1)}\\ +\vdash.\text{(1).Comm}.&\supset\vdash\colon\ldotp \text{D}ʻR\subset \alpha.\text{ᗡ}ʻR\subset \beta.\supset:xRy.\supset.x\in \alpha.y\in \beta.\\ +[\text{*35·103}] & \qquad\qquad\qquad\qquad\qquad\qquad\quad\supset.x(\alpha\uparrow \beta)y &\qquad \text{(2)}\\ +\vdash.\text{*35·103}. &\supset\vdash\colon\ldotp R\unicode{x2abd}\alpha\uparrow \beta.\supset:xRy.\supset_{x,y}.x\in \alpha.y\in \beta:\\ +[\text{*33·35·351}] &\qquad\qquad\quad\quad\supset:\text{D}ʻR\subset \alpha.\text{ᗡ}ʻR\subset \beta &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·831.</b> \(\vdash.\dot{-}(\alpha\uparrow \beta)=(-\alpha\uparrow \beta)\unicode{x228d}(\alpha\uparrow -\beta)\unicode{x228d}(-\alpha\uparrow -\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·35}.\supset\vdash\colon\colon x{\dot{-}(\alpha\uparrow \beta)}y.&\equiv\colon\ldotp {\sim}\{x(\alpha\uparrow \beta)y\}\colon\ldotp \\ +[\text{*35·103}] &\equiv\colon\ldotp {\sim}(x\in \alpha.y\in \beta)\colon\ldotp \\ +[\text{*4·51}] &\equiv\colon\ldotp x{\sim}\in \alpha.\lor .y{\sim}\in \beta\colon\ldotp \\ +[\text{*4·42}] & \equiv\colon\ldotp x{\sim}\in \alpha:y\in \beta.\lor .y{\sim}\in \beta\colon\ldotp \lor \colon\ldotp x\in \alpha.\lor .x{\sim}\in \alpha:y{\sim}\in \beta\colon\ldotp \\ +[\text{*4·4}] & \equiv\colon\ldotp x{\sim}\in \alpha.y\in \beta.\lor .x{\sim}\in \alpha.y{\sim}\in \beta.\lor .x\in \alpha.y{\sim}\in \beta.\lor .x{\sim}\in \alpha.y{\sim}\in \beta\colon\ldotp \\ +[\text{*4·25·31·37}]&\equiv\colon\ldotp x{\sim}\in \alpha.y\in \beta.\lor .x\in \alpha.y{\sim}\in \beta.\lor .x{\sim}\in \alpha.y{\sim}\in \beta\colon\ldotp \\ +[\text{*22·35}] &\equiv\colon\ldotp x\in -\alpha.y\in \beta.\lor .x\in \alpha.y\in -\beta.\lor .x\in -\alpha.y\in -\beta\colon\ldotp \\ +[\text{*35·103}] &\equiv\colon\ldotp x(-\alpha\uparrow \beta)y.\lor .x(\alpha\uparrow -\beta)y.\lor .x(-\alpha\uparrow -\beta)y\colon\ldotp \\ +[\text{*23·34}] &\equiv\colon\ldotp x\{(-\alpha\uparrow \beta)\unicode{x228d}(\alpha\uparrow -\beta)\unicode{x228d}(-\alpha\uparrow -\beta)\}y\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_288">[Pg 288]</span></p> + +<p class="nind"><b>*35·832.</b> \[\begin{align}&\dot{-}(\alpha\upharpoonleft R\upharpoonright \beta)=(-\alpha\uparrow \beta)\unicode{x228d}(\alpha\uparrow -\beta)\unicode{x228d}(-\alpha\uparrow -\beta)\unicode{x228d}\dot{-}R\\ +&\quad[\text{*35·822·831.Transp.*23·84}]\end{align}\]</p> + +<p class="nind"><b>*35·834.</b> \(\vdash.(\alpha\uparrow \beta)\dot{\cap}(\gamma\uparrow \delta)=(\alpha\cap \gamma)\uparrow (\beta\cap \delta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·103}.\supset\\ +\vdash:x\{(\alpha\uparrow \beta)\dot{\cap}(\gamma\uparrow \delta)\}y.&\equiv.x\in \alpha.y\in \beta.x\in \gamma.y\in \delta.\\ +[\text{*22·33.*35·103}] & \equiv.x\{(\alpha\cap \gamma)\uparrow (\beta\cap \delta)\}y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·84.</b> \(\vdash.\text{Cnv}ʻ(\alpha\uparrow \beta)=\beta\uparrow \alpha \quad[\text{*35·103.*31·131}]\)</p> + +<p class="nind"><b>*35·85.</b> \(\vdash:\exists !\beta.\supset.\text{D}ʻ(\alpha\uparrow \beta)=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·103.*10·281}.\supset\\ +\vdash\colon\ldotp (\exists y).x(\alpha\uparrow \beta)y.&\equiv:(\exists y).x\in \alpha.y\in \beta:\\ +[\text{*10·35}] &\equiv:x\in \alpha:(\exists y).y\in \beta:\\ +[\text{*24·5}] & \equiv:x\in \alpha.\exists !\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*33·13.*10·35}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·86.</b> \(\vdash:\exists !\alpha.\supset.\text{ᗡ}ʻ(\alpha\uparrow \beta)=\beta \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*35·87.</b> \(\vdash:\dot{\exists}!(\alpha\uparrow \beta).\equiv.\exists !\alpha.\exists !\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·103}.\supset\vdash\colon\ldotp \dot{\exists}!(\alpha\uparrow \beta).&\equiv:(\exists x,y).x\in \alpha.y\in \beta:\\ +[\text{*11·54}] &\equiv:(\exists x).x\in \alpha:(\exists y).y\in \beta:\\ +[\text{*24·5}] &\equiv:\exists !\alpha.\exists !\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*35·88">*35·88</a>.</b> \[\begin{align}&\vdash\colon\ldotp \alpha\uparrow \beta=\dot{\Lambda}.\equiv:\alpha=\Lambda.\lor.\beta=\Lambda\\ +&\quad[\text{*35·87.Transp.*24·51.*25·51}]\end{align}\]</p> + +<p class="nind"><b>*35·881.</b> \(\vdash:\text{ᗡ}ʻR\subset\alpha.\supset.R\mid (\alpha\uparrow \beta)=\text{D}ʻR\uparrow \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1.*35·103}.\supset\\ +\vdash:x\{R\mid (\alpha\uparrow \beta)\}y.\equiv.(\exists z).xRz.z\in \alpha.y\in \beta &&\qquad \text{(1)}\\ +\vdash.\text{*33·14}.\supset\vdash\colon\ldotp \text{ᗡ}ʻR\subset\alpha.&\supset:xRz.\supset.z\in \alpha:\\ +[\text{*4·73}] &\supset:xRz.\equiv.xRz.z\in \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp x\{R\mid (\alpha\uparrow \beta)\}y.&\equiv:(\exists z).xRz.y\in \beta:\\ +[\text{*10·35}] &\equiv:(\exists z).xRz:y\in \beta:\\ +[\text{*33·13}] &\equiv:x\in \text{D}ʻR.y\in \beta:\\ +[\text{*35·103}] & \equiv:x(\text{D}ʻR\uparrow \beta)y\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·882.</b> \(\vdash:\text{D}ʻR\subset\beta.\supset.(\alpha\uparrow \beta)\mid R=\alpha\uparrow \text{ᗡ}ʻR \quad[\text{Similar proof}]\)</p> + +<p><span class="pagenum" id="Page_289">[Pg 289]</span></p> + +<p class="nind"><b>*35·89.</b> + \(\vdash:\exists !\beta.\supset.(\alpha\uparrow \beta)\mid (\beta\uparrow \gamma)=(\alpha\uparrow \gamma):{\sim}\exists !\beta.\supset.(\alpha\uparrow \beta)\mid (\beta\uparrow \gamma)=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1}.\supset\vdash\colon\ldotp x\{(\alpha\uparrow \beta)\mid (\beta\uparrow \gamma)\}z.\\ +&\equiv:(\exists y).x(\alpha\uparrow \beta)y.y(\beta\uparrow \gamma)z:\\ +[\text{*35·103}] & \equiv:(\exists y).x\in \alpha.y\in \beta.y\in \beta.z\in \gamma:\\ +[\text{*4·24}] &\equiv:(\exists y).x\in \alpha.y\in \beta.z\in \gamma:\\ +[\text{*10·35}] & \equiv:\exists !\beta:x\in \alpha.z\in \gamma:\\ +[\text{*35·103}] & \equiv:\exists !\beta:x(\alpha\uparrow \gamma)z &\qquad \text{(1)}\\ +\vdash.\text{(1)}.\supset\vdash\colon\colon \exists !\beta.\supset:x\{(\alpha\uparrow \beta)\mid (\beta\uparrow \gamma)\}z.\equiv.x(\alpha\uparrow \gamma)z\colon\ldotp \\ +\qquad\qquad{\sim}\exists !\beta.\supset:{\sim}[x\{(\alpha\uparrow \beta)\mid (\beta\uparrow \gamma)\}z]\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·891.</b> \(\vdash\colon\ldotp \exists !\beta.\lor .{\sim}\exists !\alpha:\supset.(\alpha\uparrow \beta)\mid (\beta\uparrow \alpha)=(\alpha\uparrow \alpha)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·88}.\supset\vdash:{\sim}\exists !\alpha.&\supset.\alpha\uparrow \alpha=\dot{\Lambda}.\alpha\uparrow \beta=\dot{\Lambda}.\\ +[\text{*34·32}] &\supset.\alpha\uparrow \alpha=\dot{\Lambda}.(\alpha\uparrow \beta)\mid (\beta\uparrow \alpha)=\dot{\Lambda}.\\ +[\text{*21·24}] &\supset.(\alpha\uparrow \alpha)=(\alpha\uparrow \beta)\mid (\beta\uparrow \alpha) &\qquad \text{(1)}\\ +\vdash.\text{(1).*35·89}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·892.</b> \(\vdash:(\alpha\uparrow \alpha)^{2}=(\alpha\uparrow \alpha) \quad\left[\text{*35·891}\, \frac{\alpha}{\beta}\right]\)</p> + +<p class="nind"><b>*35·895.</b> \(\vdash:\alpha \cap \beta=\Lambda.\supset.(\alpha\uparrow \beta)^{2}=\dot{\Lambda} \quad[\text{*35·68·82}]\)</p> + +<p class="nind"><b>*35·9.</b> \(\vdash.\text{D}ʻ(\alpha\uparrow \alpha)=\text{ᗡ}ʻ(\alpha\uparrow \alpha)=Cʻ(\alpha\uparrow \alpha)=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·85·86}. & \supset\vdash:\exists !\alpha.\supset.\text{D}ʻ(\alpha\uparrow \alpha)=\alpha.\text{D}ʻ(\alpha\uparrow \alpha)=\alpha &\qquad \text{(1)}\\ +\vdash.\text{*35·88}. \supset\vdash:{\sim}\exists !\alpha.&\supset.{\sim}\dot{\exists}!(\alpha\uparrow \alpha).\\ +[\text{*33·29}] &\supset.\text{D}ʻ(\alpha\uparrow \alpha)=\Lambda.\text{ᗡ}ʻ(\alpha\uparrow \alpha)=\Lambda.\\ +[\text{*24·51}]&\supset.\text{D}ʻ(\alpha\uparrow \alpha)=\alpha.\text{ᗡ}ʻ(\alpha\uparrow \alpha)=\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·83}.&\supset\vdash.\text{D}ʻ(\alpha\uparrow \alpha)=\text{ᗡ}ʻ(\alpha\uparrow \alpha)=\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·91.</b> \(\vdash:R\unicode{x2abd}\alpha\uparrow \alpha.\equiv.CʻR\subset\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·103}. \supset\vdash\colon\ldotp R\unicode{x2abd}\alpha\uparrow \alpha.&\equiv:xRy.\supset_{x,y}.x,y\in \alpha:\\ +[\text{*33·352}] & \equiv:CʻR\subset\alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·92.</b> \(\vdash\colon\ldotp (\exists \alpha).P=\alpha\uparrow \alpha.\supset:R\unicode{x2abd}P.\equiv.CʻR\subset CʻP \quad[\text{*35·9·91}]\)</p> + +<p><span class="pagenum" id="Page_290">[Pg 290]</span></p> + +<p class="nind"><b><a id="*35·93">*35·93</a>.</b> \(\vdash:(R).\phi(\text{D}ʻR).\equiv.(\alpha).\phi\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·12.*14·18}.&\supset\vdash:(\alpha).\phi\alpha.\supset.\phi(\text{D}ʻR):\\ +[\text{*10·11·21}] &\supset\vdash:(\alpha).\phi\alpha.\supset.(R).\phi(\text{D}ʻR) &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash:(R).\phi(\text{D}ʻR).\supset.\phi\{\text{D}ʻ(\alpha\uparrow \alpha)\}.\\ +[\text{*35·9}] &\qquad\qquad\qquad\qquad\supset.\phi\alpha:\\ +[\text{*10·11·21}] &\supset\vdash:(R).\phi(\text{D}ʻR).\supset.(\alpha).\phi\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*35·931.</b> \(\vdash:(R).\phi(\text{ᗡ}ʻR).\equiv.(\alpha).\phi\alpha \quad[\text{Proof as in *35·93}]\)</p> + +<p class="nind"><b>*35·932.</b> \(\vdash:(R).\phi(CʻR).\equiv.(\alpha).\phi\alpha \quad[\text{Proof as in *35·93}]\)</p> + +<p class="nind"><b>*35·94.</b> \(\vdash:(\exists R).\phi(\text{D}ʻR).\equiv.(\exists \alpha).\phi\alpha \quad[\text{*35·93.Transp}]\)</p> + +<p class="nind"><b>*35·941.</b> \(\vdash:(\exists R).\phi(\text{ᗡ}ʻR).\equiv.(\exists \alpha).\phi\alpha \quad[\text{*35·931. Transp}]\)</p> + +<p class="nind"><b>*35·942.</b> \(\vdash:(\exists R).\phi(CʻR).\equiv.(\exists \alpha).\phi\alpha \quad[\text{*35·932.Transp}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_291">[Pg 291]</span></p> +<h2 class="nobreak" id="*36">*36. RELATIONS WITH LIMITED FIELDS.</h2> +</div> + + +<p><i>Summary of</i> *36.</p> + +<p>In this number we are concerned with the special case in which the +same limitation is imposed upon the domain and the converse domain +of a relation. In this case, the same result is achieved by imposing +the limitation on the field. It is convenient to be able to regard +\(\alpha \upharpoonleft P \upharpoonright \alpha\) as a descriptive +function of \(\alpha\) or of \(P\), which we secure by the notation +\(P \unicode{x294f} \alpha\), whence, as will be explained in <a href="#*38">*38</a>, \(P +\unicode{x294f} ʻ\alpha\) and \(\unicode{x294f} \alphaʻP\) will both +mean \(P \unicode{x294f} \alpha\). If \(P\) is a serial relation, and +\(\alpha \subset CʻP\), "\(P \unicode{x294f} \alpha\)" will stand for +"the terms of \(\alpha\) arranged in the order determined by \(P\)," +or, as we may call it briefly, "\(\alpha\) in the \(P\)-order." +\(P\unicode{x294f} \alpha\) is defined as follows:</p> + +<p class="nind"><b>*36·01.</b> \(P \unicode{x294f} \alpha = \alpha \upharpoonleft P \upharpoonright \alpha \quad \text{Df}\)</p> + +<p>We thus have</p> + +<p class="nind"><b>*36·13.</b> \(\vdash: x (P \unicode{x294f} \alpha) y .\equiv. x,y \in \alpha . x P y\)</p> + +<p>Most of the propositions concerning \(P \unicode{x294f} \alpha\) +demand that \(P\) should have some at least of the characteristics +of a <i>serial</i> relation. Hence the propositions concerning +\(P\unicode{x294f} \alpha\) which can be given in the present number are, +for the most part, not the most useful propositions concerning +\(P\unicode{x294f} \alpha\). The most useful propositions in the present +number are the following:</p> + +<p class="nind"><b>*36·25.</b> \(\vdash: CʻP \subset \alpha .\equiv. P \unicode{x294f} \alpha = P\)</p> + +<p class="nind"><b>*36·29.</b> \(\vdash. P \unicode{x294f} \alpha = P \dot{\cap} \alpha \uparrow \alpha\)</p> + +<p class="nind"><b>*36·3.</b> \(\vdash. P \unicode{x294f} \alpha = P \unicode{x294f} (\alpha \cap CʻP)\)</p> + +<p class="nind"><b>*36·33.</b> \(\vdash. P \unicode{x294f} CʻP = P\)</p> + +<hr class="tb"> + +<p class="nind"><b>*36·01.</b> \(P \unicode{x294f} \alpha = \alpha \upharpoonleft P \upharpoonright \alpha \quad \text{Df}\)</p> + +<p class="nind"><b>*36·11.</b> \(\vdash. P \unicode{x294f} \alpha = \alpha \upharpoonleft P \upharpoonright \alpha \quad\text{[(*36·01)]} \)</p> + +<p class="nind"><b>*36·13.</b> \(\vdash: x (P \unicode{x294f} \alpha) y .\equiv. x,y \in \alpha . xPy \quad\text{[*36·11.*35·102]}\)</p> + +<p>The following propositions are obtained from those of <a href="#*35">*35</a> by means of +*36·11, which, as it is used in each case, is not referred to again.</p> + +<p class="nind"><b>*36·2.</b> \(\vdash. P \unicode{x294f} \alpha \dot{\cap} Q \unicode{x294f} \beta = (P \dot{\cap} Q) \unicode{x294f} (\alpha \cap \beta) \quad \text{[*35·15]} \)</p> + +<p class="nind"><b>*36·201.</b> \(\vdash. P \unicode{x294f} \alpha \dot{\cap} P \unicode{x294f} \beta = P \unicode{x294f} (\alpha \cap \beta) \quad \text{[*36·2]} \)</p> + +<p class="nind"><b>*36·202.</b> \(\vdash. P \unicode{x294f} \alpha \dot{\cap} Q \unicode{x294f} \alpha = (P \dot{\cap} Q) \unicode{x294f} \alpha \quad \text{[*36·2]} \)</p> + +<p class="nind"><b>*36·203.</b> \(\vdash. P \unicode{x294f} \alpha \dot{\cap} Q = (P \dot{\cap} Q) \unicode{x294f} \alpha \quad \text{[*35·18]} \)</p> + +<p class="nind"><b>*36·21.</b> \(\vdash.(P \unicode{x294f} \alpha) \unicode{x294f} \beta = P \unicode{x294f}(\alpha \cap \beta) \quad \text{[*35·33·34]} \)</p> + +<p><span class="pagenum" id="Page_292">[Pg 292]</span></p> + +<p class="nind"><b>*36·22.</b> \(\vdash.(P\unicode{x0294f}\alpha)\mid (Q\unicode{x0294f}\alpha)\unicode{x0294f}(P\mid Q)\unicode{x0294f}\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*36·13.*34·1.}\supset\vdash:x\{(P\unicode{x0294f}\alpha)\mid (Q\unicode{x0294f}\alpha)\}z.&\equiv.(\exists y).x,y,z\in \alpha.xPy.yQz.\\ +[\text{*10·5}] &\supset.(\exists y).x,z\in \alpha.xPy.yQz &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·35.*34·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*36·23.</b> \(\vdash.(P\unicode{x228d}Q)\unicode{x0294f}\alpha=P\unicode{x0294f}\alpha\unicode{x228d}Q\unicode{x0294f}\alpha \quad[\text{*35·422}]\)</p> + +<p class="nind"><b>*36·24.</b> \(\vdash:\alpha\subset \beta.\supset.P\unicode{x0294f}\alpha\unicode{x0294f}P\unicode{x0294f}\beta \quad[\text{*35·432}]\)</p> + +<p class="nind"><b>*36·241.</b> \(\vdash:P\unicode{x0294f}Q.\supset.P\unicode{x0294f}\alpha\unicode{x0294f}Q\unicode{x0294f}\alpha \quad[\text{*35·462}]\)</p> + +<p class="nind"><b>*36·25.</b> \(\vdash:CʻP\subset \alpha.\equiv.P\unicode{x0294f}\alpha=P\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*36·13.*4·7}.\supset\vdash\colon\ldotp P\unicode{x0294f}\alpha=P.&\equiv:xPy.\supset_{x,y}.x,y\in \alpha:\\ +[\text{*33·352}] &\equiv:CʻP\subset \alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*36·26.</b> \(\vdash:CʻP\cap \alpha=\Lambda.\supset.P\mid (Q\unicode{x0294f}\alpha)=\dot{\Lambda}.(Q\unicode{x0294f}\alpha)\mid P=\dot{\Lambda} \quad[\text{*35·473·474}]\)</p> + +<p class="nind"><b>*36·27.</b> \(\vdash:P\unicode{x0294f}\Lambda=\dot{\Lambda} \quad[\text{*35·75}]\)</p> + +<p class="nind"><b>*36·28.</b> \(\vdash.P\unicode{x0294f}\text{V}=P \quad[\text{*35·76}]\)</p> + +<p class="nind"><b>*36·29.</b> \(\vdash.P\unicode{x0294f}\alpha=P\dot{\cap}\alpha\uparrow \alpha \quad[\text{*35·822}]\)</p> + +<p class="nind"><b>*36·3.</b> \(\vdash.P\unicode{x0294f}\alpha=P\unicode{x0294f}(\alpha\cap CʻP)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·17.*4·71}.&\supset\vdash:xPy.\equiv.x,y\in CʻP.xPy:\\ +[\text{Fact}] &\supset\vdash:x,y\in \alpha.xPy.\equiv.x,y\in \alpha.x,y\in CʻP.xPy.\\ +[\text{*22·33}] &\qquad\qquad\qquad\qquad\equiv.x,y\in \alpha\cap CʻP.xPy.\\ +[\text{*36·13}] &\qquad\qquad\qquad\qquad\equiv.x\{P\unicode{x0294f}(\alpha\cap CʻP)\}y &\qquad \text{(1)}\\ +\vdash.\text{(1).*36·13}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*36·31.</b> \(\vdash:\alpha\cap CʻP=\Lambda.\supset.P\unicode{x0294f}a=\dot{\Lambda} \quad[\text{*36·3·27}]\)</p> + +<p class="nind"><b>*36·32.</b> \(\vdash:\alpha\cap CʻP=\beta\cap CʻP.\supset.P\unicode{x0294f}\alpha=P\unicode{x0294f}\beta \quad[\text{*36·3}]\)</p> + +<p class="nind"><b>*36·33.</b> \(\vdash.P\unicode{x0294f}CʻP=P \quad[\text{*36·25}]\)</p> + +<p class="nind"><b>*36·34.</b> \(\vdash.\text{Cnv}ʻP\unicode{x0294f}\alpha=(\breve{P})\unicode{x0294f}\alpha \quad[\text{*35·53}]\)</p> + +<p class="nind"><b>*36·35.</b> \(\vdash.(P\unicode{x0294f}\alpha)^{2}\unicode{x0294f}(P^{2})\unicode{x0294f}\alpha \quad[\text{*36·22}]\)</p> + +<p class="nind"><b>*36·4.</b> \(\vdash\colon\ldotp \alpha\cap \text{D}ʻR=\Lambda.\lor .\alpha\cap \text{ᗡ}ʻR=\Lambda:\supset.(R\unicode{x228d}S)\unicode{x0294f}\alpha=S\unicode{x0294f}\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·643}.\supset\vdash:\alpha\cap \text{D}ʻR=\Lambda.&\supset.\alpha\upharpoonleft (R\unicode{x228d}S)=\alpha\upharpoonleft S.\\ +[\text{*35·21}] &\supset.(R\unicode{x228d}S)\unicode{x0294f}\alpha=S\unicode{x0294f}\alpha &\qquad \text{(1)}\\ +\text{Similarly} &\vdash:\alpha\cap \text{ᗡ}ʻR=\Lambda.\supset.(R\unicode{x228d}S)\unicode{x0294f}\alpha=S\unicode{x0294f}\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_293">[Pg 293]</span></p> +<h2 class="nobreak" id="*37">*37. PLURAL DESCRIPTIVE FUNCTIONS.</h2> +</div> + + +<p><i>Summary of</i> *37.</p> + +<p>In this number, we introduce what may be regarded as the plural of +\(Rʻy\). "\(Rʻy\)" was defined to mean "the term which has the relation +\(R\) to \(y\)." We now introduce the notation "\(Rʻʻ\beta\)" to mean +"the terms which have the relation \(R\) to members of \(\beta\)." +Thus if \(\beta\) is the class of great men, and \(R\) is the relation +of wife to husband, \(Rʻʻ\beta\) will mean "wives of great men." If +\(\beta\) is the class of fractions of the form \(1-\tfrac{1}{2}^{n}\) +for integral values of \(n\), and \(R\) is the relation "less than," +\(Rʻʻ\beta\) will be the class of fractions each of which is less than +some member of this class of fractions, <i>i.e.</i> \(Rʻʻ\beta\) will +be the class of proper fractions. Generally, \(Rʻʻ\beta\) is the class +of those referents which have relata that are members of \(\beta\).</p> + +<p>We require also a notation for the relation of \(Rʻʻ\beta\) to +\(\beta\). This relation we will call \(R_{\in}\). Thus \(R_{\in}\) is +the relation which holds between two classes \(\alpha\) and \(\beta\) +when \(\alpha\) consists of all terms which have the relation \(R\) to +some member of \(\beta\).</p> + +<p>A specially important case arises when \(Rʻy\) always exists if +\(y\in \beta\). In this case, \(Rʻʻ\beta\) is the class of all terms of +the form \(Rʻy\) when \(y \in \beta\). We will denote the hypothesis +that \(Rʻy\) always exists if \(y \in \beta\) by the notation +\(\text{E}‼Rʻʻ\beta\), meaning "the \(R\)'s of \(\beta\)'s exist."</p> + +<p>The definitions are as follows:</p> + +<p class="nind"><b>*37·01.</b> \(Rʻʻ\beta = \hat{x}\{(\exists y) . y \in \beta . x R y\} \quad \text{Df}\)</p> + +<p class="nind"><b>*37·02.</b> \(R_{\in} = \hat{\alpha}\hat{\beta}(\alpha = Rʻʻ\beta) \quad \text{Df}\)</p> + +<p class="nind"><b><a id="*37·03">*37·03</a>.</b> \(\breve{R}_{\in} = \text{Cnv}ʻ(R_{\in}) \quad \text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets. +Without it, "\(\breve{R}_{\in}\)" would be ambiguous as between +(\(\breve{R})_{\in}\) and \(\text{Cnv}ʻ(R_{\in})\), which are not +equal. In all cases in which a suffix occurs, we shall adopt the same +convention, <i>i.e.</i> we shall always put +\[ +\breve{R}_{\text{suffix}} = \text{Cnv}ʻ(R_{\text{suffix}}). +\]</p> + +<p><span class="pagenum" id="Page_294">[Pg 294]</span></p> + +<p class="nind"><b>*37·04.</b> \(Rʻʻʻ\kappa = R_{\in}ʻʻ\kappa \quad \text{Df}\)</p> + +<p>Thus \(Rʻʻʻ\kappa\) consists of all classes which have the relation +\(R_{\in}\) to some member of \(\kappa\). \(Rʻʻʻ\kappa\) is only +significant when \(\kappa\) is a class of classes relatively to members +of the converse domain of \(R\); in this case, \(Rʻʻʻ\kappa\) is a +class of classes relatively to members of the domain of \(R\).</p> + +<p class="nind"><b><a id="*37·05">*37·05</a>.</b> \(\text{E}!!Rʻʻ\beta .=: y \in \beta .\supset_{y}. \text{E}!Rʻy \quad \text{Df}\)</p> + +<p>Here the symbol "\(\text{E}!!Rʻʻ\beta\)" must be treated as a whole, +<i>i.e.</i> we must not regard it as making an assertion about +\(Rʻʻ\beta\). If \(Rʻʻ\beta = \alpha\), we must not suppose that we +shall be able to put "\(\text{E}!!\alpha\)," which would be nonsense, +just as "\(\text{E}!x\)" is nonsense even when \(x = Rʻy\) and +\(\text{E}!Rʻy\).</p> + +<p>The notation \(Rʻʻ\alpha\), introduced in the present number, is +extremely useful, and embodies a very important idea. Its use is +somewhat different according to the kind of relation concerned. +Consider first the kind of relation which leads to a descriptive +function, say \(\text{D}\). If \(\lambda\) is a class of relations, +\(\text{D}ʻʻ\lambda\) is the class of the domains of these relations. +In this case, \(\text{D}ʻʻ\lambda\) is a class each of whose members +is of the form \(\text{D}ʻR\), where \(R \in \lambda\). Again, let +us denote by "\(\times n\)" the relation of \(m\) to \(m \times n\); +then if we denote by "\(NC\)" the class of cardinal numbers, \(\times nʻʻNC\) +will denote all numbers that result from multiplying a cardinal +number by \(n\), <i>i.e.</i> all multiples of \(n\). Thus <i>e.g.</i> +\(\times 2ʻʻNC\) will be the class of even numbers. If \(R\) is a +correlation between two classes \(\alpha\) and \(\beta\), <i>i.e.</i> a +relation such that, if \(y \in \beta\), \(Rʻy\) exists and is a member +of \(\alpha\), while conversely, if \(x \in \alpha\), \(\breve{R}ʻx\) +exists and is a member of \(\beta\), then \(\alpha = Rʻʻ\beta\), and +we may regard \(R\) as a transformation applied to each member of +\(\beta\) and giving rise to a member of \(\alpha\). It is by means of +such transformations that two classes are shown to be <i>similar</i>, +<i>i.e.</i> to have the same (cardinal) number of terms.</p> + +<p><span class="pagenum" id="Page_295">[Pg 295]</span></p> + +<p>In the case of serial relations, the utility of the notation +\(Rʻʻ\beta\) is somewhat different. Suppose, for example, that \(R\) is +the relation of less to greater among real numbers. Then if \(\beta\) +is any class of real numbers, \(Rʻʻ\beta\) will be the segment of +real numbers determined by \(\beta\), <i>i.e.</i> the class of real +numbers which are less than the limit or maximum of \(\beta\). In any +series, if \(\beta\) is a class contained in the series and \(R\) is +the generating relation of the series, \(Rʻʻ\beta\) is the segment +determined by \(\beta\). If \(\beta\) has either a limit or a maximum, +say \(x\), \(Rʻʻ\beta\) will be \(\overrightarrow{R}ʻx\). But if +\(\beta\) has neither a limit nor a maximum, \(Rʻʻ\beta\) will be what +we may call an "irrational" segment of the series. We shall see at a +later stage that the real numbers may be identified with the segments +of the series of rationals, <i>i.e.</i> if \(R\) is the relation of +less to greater among rationals, the real numbers will be all classes +such as \(Rʻʻ\beta\), for different values of \(\beta\). The real +numbers which correspond to rationals will be those resulting from a +\(\beta\) which has a limit or maximum; the irrationals will be those +resulting from a \(\beta\) which has no limit or maximum.</p> + +<p>The present number may be divided into various sections, as follows: +(1) First, we have various elementary properties of the terms defined +at the beginning of the number; this section ends with <a href="#*37·29">*37·29</a>. (2) We +have next a set of propositions dealing with relative products, and +with such symbols as \(PʻʻQʻʻ\gamma\), \(PʻʻQʻʻʻ\kappa\), and so on. +The central proposition here is</p> + +<p class="nind"><b>*37·33.</b> \(\vdash.(P\mid Q)ʻʻ\gamma=PʻʻQʻʻ\gamma\)</p> + +<p>By the definition, \(Qʻʻʻ\kappa=Q_{\in}ʻʻ\kappa\). Thus +\(PʻʻQʻʻʻ\kappa=(P\mid Q_{\in})ʻʻ\kappa.\) This connects propositions +concerning such symbols as \(PʻʻQʻʻʻ\kappa\) with propositions +concerning relative products. This second section consists of +the propositions from <a href="#*37·3">*37·3</a> to <a href="#*37·39">*37·39</a>. (3) We have next a set of +propositions on relations with limited domains and converse domains. +The chief of these are</p> + +<p class="nind"><b>*37·401.</b> \(\vdash.\text{D}ʻ(R\upharpoonright \beta)=Rʻʻ\beta\)</p> + +<p class="nind"><b>*37·412.</b> \(\vdash.(R\upharpoonright \alpha)ʻʻ\beta=Rʻʻ(\alpha\cap \beta)\)</p> + +<p class="nind"><b>*37·41.</b> \(\vdash.\text{D}ʻ(R\unicode{x0294f}\alpha)=\alpha\cap Rʻʻ\alpha.\text{ᗡ}ʻ(R\unicode{x0294f}\alpha)=\alpha\cap \breve{R}ʻʻ\alpha\)</p> + +<p>These propositions on relations with limited domains and converse +domains, together with certain others naturally connected with +them, extend from <a href="#*37·4">*37·4</a> to <a href="#*37·52">*37·52</a>. (4) We next have a number of +very important propositions on the consequences of the hypothesis +\(\text{E}‼Rʻʻ\beta\), <i>i.e.</i> the hypothesis that, for any +argument which is a member of \(\beta\), \(R\) gives rise to a +descriptive function \(Rʻy\). The chief proposition in this section is</p> + +<p class="nind"><b>*37·6.</b> \(\vdash:\text{E}‼Rʻʻ\beta.\supset.Rʻʻ\beta=\hat{x}\{(\exists y).y\in \beta.x=Rʻy\}\)</p> + +<p>Propositions with the hypothesis \(\text{E}‼Rʻʻ\beta\) are applied to +the cases of \(\overrightarrow{R}\) and \(\overleftarrow{R}\), in which +the hypothesis is verified. This section extends from <a href="#*37·6">*37·6</a> to <a href="#*37·791">*37·791</a>. +(5) Finally, we have three propositions on the relative product of +\(\alpha\uparrow \beta\) with other relations. These propositions are +useful in relation-arithmetic (Part IV).</p> + +<p>The propositions of the present number which are most used in the +sequel, apart from those already mentioned, are the following (omitting +such as merely embody definitions):</p> + +<p class="nind"><b>*37·15.</b> \(\vdash.Rʻʻ\alpha\subset\text{D}ʻR\)</p> + +<p class="nind"><b>*37·16.</b> \(\vdash.\breve{R}ʻʻ\alpha\subset\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*37·2.</b> \(\vdash:\alpha\subset\beta.\supset.Pʻʻ\alpha\subset Pʻʻ\beta\)</p> + +<p class="nind"><b>*37·22.</b> \(\vdash.Pʻʻ(\alpha\cup\beta)=Pʻʻ\alpha\cup Pʻʻ\beta\)</p> + +<p class="nind"><b>*37·25.</b> \(\vdash.\text{D}ʻR=Rʻʻ\text{ᗡ}ʻR.\text{ᗡ}ʻR=\breve{R}ʻʻ\text{D}ʻR\)</p> + +<p class="nind"><b>*37·26.</b> \(\vdash.Rʻʻ\beta=Rʻʻ(\beta\cap \text{ᗡ}ʻR)\)</p> + +<p class="nind"><b>*37·265.</b> \(\vdash.Rʻʻ\alpha=Rʻʻ(\alpha\cap CʻR).\breve{R}ʻʻ\alpha=\breve{R}ʻʻ(\alpha\cap CʻR)\)</p> + +<p><span class="pagenum" id="Page_296">[Pg 296]</span></p> + +<p class="nind"><b>*37·29.</b> \(\vdash.Rʻʻ\Lambda=\Lambda.\breve{R}ʻʻ\Lambda=\Lambda\)</p> + +<p class="nind"><b>*37·32.</b> \(\vdash.\text{D}ʻ(P\mid Q)=Pʻʻ\text{D}ʻQ.\text{ᗡ}ʻ(P\mid Q)=\breve{Q}ʻʻ\text{ᗡ}ʻP\)</p> + +<p class="nind"><b>*37·45.</b> \(\vdash\colon\ldotp (y).\text{E}!Rʻy.\supset:\exists !Rʻʻ\beta.\equiv.\exists !\beta\)</p> + +<p class="nind"><b>*37·46.</b> \(\vdash:x\in Rʻʻ\alpha.\equiv.\exists !\alpha\cap \overleftarrow{R}ʻx\)</p> + +<p class="nind"><b>*37·61.</b> \(\vdash\colon\colon \text{E}‼Rʻʻ\beta.\supset\colon\ldotp Rʻʻ\beta\subset \alpha.\equiv:y\in \beta.\supset_{y}.Rʻy\in \alpha\)</p> + +<p>For example, let \(R\) be the relation of father to son, \(\beta\) +the class of Etonians, \(\alpha\) the class of rich men; then +"\(Rʻʻ\beta\subset \alpha\)" states "all fathers of Etonians are rich," +while "\(y\in \beta.\supset_{y}.Rʻy\in \alpha\)" states "if a boy is an +Etonian, his father must be rich." In virtue of the above proposition, +these two statements are equivalent.</p> + +<p class="nind"><b>*37·62.</b> \(\vdash:\text{E}!Rʻy.y\in \alpha.\supset.Rʻy\in Rʻʻ\alpha\)</p> + +<p class="nind"><b>*37·63.</b> \(\vdash\colon\colon \text{E}‼Rʻʻ\alpha.\supset\colon\ldotp x\in Rʻʻ\alpha.\supset_{x}.\psi x:\equiv:y\in \alpha.\supset_{y}.\psi(Rʻy)\)</p> + +<hr class="tb"> + +<p class="nind"><b>*37·01.</b> \(Rʻʻ\beta=\hat{x}\{(\exists y).y\in \beta.xRy\} \quad\text{Df}\)</p> + +<p class="nind"><b>*37·02.</b> \(R_{\in}=\hat{\alpha}\hat{\beta}(\alpha=Rʻʻ\beta) \quad\text{Df}\)</p> + +<p class="nind"><b>*37·03.</b> \(\breve{R}_{\in}=\text{Cnv}ʻR_{\in} \quad\text{Df}\)</p> + +<p class="nind"><b>*37·04.</b> \(Rʻʻʻ\kappa=R_{\in}ʻʻ\kappa \quad\text{Df}\)</p> + +<p class="nind"><b>*37·05.</b> \(\text{E}‼Rʻʻ\beta.=:y\in \beta.\supset_{y}.\text{E}!Rʻy \quad\text{Df}\)</p> + +<p class="nind"><b>*37·1.</b> \(\vdash:x\in Rʻʻ\beta.\equiv.(\exists y).y\in \beta.xRy \quad[\text{*20·3.(*37·01)}]\)</p> + +<p class="nind"><b>*37·101.</b> \(\vdash:\alpha R_{\in}\beta.\equiv.\alpha=Rʻʻ\beta \quad[\text{*21·3.(*37·02)}]\)</p> + +<p class="nind"><b>*37·102.</b> \(\vdash:\alpha(\breve{R})_{\in}\beta.\equiv.\alpha=\breve{R}ʻʻ\beta \quad[\text{*37·101}]\)</p> + +<p class="nind"><b>*37·103.</b> \[\begin{align}&\vdash:\alpha\in Rʻʻʻ\kappa.\equiv.(\exists \beta).\beta\in \kappa.\alpha=Rʻʻ\beta.\equiv.\alpha\in R_{\in}ʻʻ\kappa\\ +&[\text{*37·1·101.(*37·04)}]\end{align}\]</p> + +<p class="nind"><b>*37·104.</b> \(\vdash\colon\ldotp \text{E}‼Rʻʻ\beta.\equiv:y\in \beta.\supset_{y}.\text{E}!Rʻy \quad[\text{*4·2.(*37·05)}]\)</p> + +<p class="nind"><b>*37·105.</b> \(\vdash:x\in \breve{R}ʻʻ\beta.\equiv.(\exists y).y\in \beta.yRx \quad[\text{*37·1.*31·11}]\)</p> + +<p class="nind"><b>*37·106.</b> \(\vdash\colon\ldotp \text{E}!Rʻx.\supset:x\in \breve{R}ʻʻ\beta.\equiv.Rʻx\in \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·105.*30·4}.\supset\vdash\colon\ldotp Hp.\supset:x\in \breve{R}ʻʻ\beta.&\equiv.(\exists y).y\in \beta.y=Rʻx.\\ +[\text{*14·205}] &\equiv.Rʻx\in \beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·11.</b> \(\vdash.R_{\in}ʻ\beta=Rʻʻ\beta \quad[\text{*37·101.*30·3}]\)</p> + +<p class="nind"><b>*37·111.</b> \(\vdash.\text{E}!R_{\in}ʻ\beta \quad[\text{*37·11.*14·21}]\)</p> + +<p><span class="pagenum" id="Page_297">[Pg 297]</span></p> + +<p class="nind"><b>*37·12.</b> \(\vdash:(\beta).Rʻʻ\beta=Qʻ\beta.\equiv.R_{\in}=Q \quad[\text{*30·42.*37·111·11}]\)</p> + +<p class="nind"><b>*37·13.</b> \(\vdash:P=Q.\supset.Pʻʻ\beta=Qʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·43}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:xPy.\equiv_{x,y}.xQy:\\ +[\text{Fact}] &\supset:y\in \beta.xPy.\equiv_{x,y}.y\in \beta.xQy:\\ +[\text{*10·281}]&\supset:(\exists y).y\in \beta.xPy.\equiv_{x}.(\exists y).y\in \beta.xQy:\\ +[\text{*37·1}] &\supset:x\in Pʻʻ\beta.\equiv_{x}.x\in Qʻʻ\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·131.</b> \(\vdash:P=Q.\supset.P_{\in}=Q_{\in}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·13}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\alpha=Pʻʻ\beta.\equiv_{\alpha,\beta}.\alpha=Qʻʻ\beta:\\ +[\text{*37·101}] &\supset:\alpha P_{\in}\beta.\equiv_{\alpha,\beta}.\alpha Q_{\in}\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·14.</b> \(\vdash:P=Q.\equiv.P_{\in}=Q_{\in}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·101.*21·15}.\supset\\ +\vdash\colon\ldotp P_{\in}=Q_{\in}. &\equiv:\alpha=Pʻʻ\beta.\equiv_{\alpha,\beta}.\alpha=Qʻʻ\beta:\\ +[\text{*13·183}] &\equiv:(\beta).Pʻʻ\beta=Qʻʻ\beta:\\ +[\text{*37·1.*20·15}] &\equiv:(\beta,x):(\exists y).y\in \beta.xPy.\equiv.(\exists y).y\in \beta.xQy:\\ +[\text{*10·1}]&\supset:(x):(\exists y).y\in \hat{z}(z=w).xPy.\equiv.(\exists y).y\in \hat{z}(z=w).xQy:\\ +[\text{*20·3}] &\supset:(x):(\exists y).y=w.xPy.\equiv.(\exists y).y=w.xQy:\\ +[\text{*13·195}]&\supset:(x):xPw.\equiv.xQw &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21.*11·2}.\supset\\ +\vdash\colon\ldotp P_{\in}=Q_{\in}.&\supset:(x,w):xPw.\equiv.xQw:\\ +[\text{*21·43}] &\supset:P=Q &\qquad \text{(2)}\\ +\vdash.\text{(2).*37·131}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·15.</b> \(\vdash.Rʻʻ\alpha\subset \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash:x\in Rʻʻ\alpha.&\supset.(\exists y).y\in \alpha.xRy.\\ +[\text{*10·5}] &\supset.(\exists y).xRy.\\ +[\text{*33·13}] &\supset.x\in \text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·16.</b> \(\vdash.\breve{R}ʻʻ\alpha\subset \text{ᗡ}ʻR \quad\left[\text{*37·15}\, \frac{\breve{R}}{R}.\text{*33·2}\right]\)</p> + +<p class="nind"><b>*37·17.</b> \(\vdash\colon\ldotp Rʻʻ\beta\subset \alpha.\equiv:y\in \beta.xRy.\supset_{x,y}.x\in \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash\colon\ldotp Rʻʻ\beta\subset \alpha.&\equiv:(\exists y).y\in \beta.xRy.\supset_{x}.x\in \alpha:\\ +[\text{*10·23}] &\equiv:y\in \beta.xRy.\supset_{x,y}.x\in \alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_298">[Pg 298]</span></p> + +<p class="nind"><b>*37·171.</b> \(\vdash\colon\ldotp \breve{R}ʻʻ\alpha\subset \beta.\equiv:x\in \alpha.xRy.\supset_{x,y}.y\in \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·105}.\supset\vdash\colon\ldotp \breve{R}ʻʻ\alpha\subset \beta.&\equiv:(\exists x).x\in \alpha.xRy.\supset_{y}.y\in \beta:\\ +[\text{*10·23}] &\equiv:x\in \alpha.xRy.\supset_{x,y}.y\in \beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·18.</b> \(\vdash:y\in \beta.\supset.\overrightarrow{R}ʻy\subset Rʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x\in \overrightarrow{R}ʻy.&\supset.xRy.y\in \beta.\\ +[\text{*37·1}] &\supset.x\in Rʻʻ\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·181.</b> \(\vdash:x\in \alpha.\supset.\overleftarrow{R}ʻx\subset \breve{R}ʻʻ\alpha \quad[\text{Proof as in *37·18}]\)</p> + +<p class="nind"><b>*37·2.</b> \(\vdash:\alpha\subset \beta.\supset.Pʻʻ\alpha\subset Pʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·1}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:y\in \alpha.\supset_{y}.y\in \beta:\\ +[\text{*10·31}] &\supset:y\in \alpha.xPy.\supset_{y}.y\in \beta.xPy:\\ +[\text{*10·28}] &\supset:(\exists y).y\in \alpha.xPy.\supset.(\exists y).y\in \beta.xPy:\\ +[\text{*37·1}] &\supset:x\in \alpha Pʻʻ\alpha.\supset.x\in Pʻʻ\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition (*37·2) is one of the forms of asyllogistic +inference due to Leibniz's teacher Jungius. The instance given by +Jungius is: "Circulus est figura; ergo qui circulum describit, is +figuram describit<a id="FNanchor_56" href="#Footnote_56" class="fnanchor">[56]</a>." Here the class of circles is our \(\alpha\), +the class of figures is our \(\beta\), and the relation of describing +is our \(P\).</p> + +<p class="nind"><b>*37·201.</b> \(\vdash:P\unicode{x2abd}Q.\supset.Pʻʻ\alpha\subset Qʻʻ\alpha \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*37·202.</b> \(\vdash:a\subset \beta.P\unicode{x2abd}Q.\supset.Pʻʻ\alpha\subset Qʻʻ\beta \quad[\text{*37·2·201}]\)</p> + +<p class="nind"><b><a id="*37·21">*37·21</a>.</b> \(\vdash.Pʻʻ(\alpha\cap \beta)\subset Pʻʻ\alpha\cap Pʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash\colon\ldotp x\in Pʻʻ(\alpha\cap \beta).&\equiv:(\exists y).y\in \alpha\cap \beta.xPy:\\ +[\text{*22·33}] &\equiv:(\exists y).y\in \alpha.y\in \beta.xPy:\\ +[\text{*10·5}] &\supset:(\exists y).y\in \alpha.xPy:(\exists y).y\in \beta.xPy:\\ +[\text{*37·1}] &\supset:x\in Pʻʻ\alpha.x\in Pʻʻ\beta:\\ +[\text{*22·33}] &\supset:x\in Pʻʻ\alpha\cap Pʻʻ\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·211.</b> \(\vdash.(P\dot{\cap}Q)ʻʻ\alpha\subset Pʻʻ\alpha\cap Qʻʻ\alpha \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*37·212.</b> \(\vdash.(P\dot{\cap}Q)ʻʻ(\alpha\cap \beta)\subset Pʻʻ\alpha\cap Pʻʻ\beta\cap Qʻʻ\alpha\cap Qʻʻ\beta \quad[\text{*37·21·211}]\)</p> + +<p class="nind"><b>*37·22.</b> \(\vdash.Pʻʻ(\alpha\cup \beta)=Pʻʻ\alpha\cup Pʻʻ\beta\)</p> + +<p><span class="pagenum" id="Page_299">[Pg 299]</span></p> + +<p>This proposition is very frequently used. The fact that here we have +identity, while in *37·21 we only have inclusion, is due to the +fact that <a href="#*10·42">*10·42</a> states an equivalence, while <a href="#*10·5">*10·5</a> only states an +implication.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash\colon\ldotp x\in Pʻʻ(\alpha\cup \beta).&\equiv:(\exists y).y\in \alpha\cup \beta.xPy:\\ +[\text{*22·34}] &\equiv:(\exists y):y\in \alpha.\lor .y\in \beta:xPy:\\ +[\text{*4·4}] &\equiv:(\exists y):y\in \alpha.xPy.\lor .y\in \beta.xPy:\\ +[\text{*10·42}] &\equiv:(\exists y).y\in \alpha.xPy:\lor :(\exists y).y\in \beta.xPy:\\ +[\text{*37·1}] & \equiv:x\in Pʻʻ\alpha.\lor .x\in Pʻʻ\beta:\\ +[\text{*22·34}]&\equiv:x\in Pʻʻ\alpha\cup Pʻʻ\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·221.</b> \(\vdash.(P\unicode{x228d}Q)ʻʻ\alpha=Pʻʻ\alpha\cup Qʻʻ\alpha \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*37·222.</b> \(\vdash.(P\unicode{x228d}Q)ʻʻ(\alpha\cup \beta)=Pʻʻ\alpha\cup Pʻʻ\beta\cup Qʻʻ\alpha\cup Qʻʻ\beta \quad[\text{*37·22·221}]\)</p> + +<p class="nind"><b>*37·23.</b> \(\vdash.DʻR_{\in}=\hat{\alpha}\{(\exists \beta).\alpha=Rʻʻ\beta\} \quad[\text{*37·101.*33·11}]\)</p> + +<p class="nind"><b>*37·231.</b> \(\vdash.\text{ᗡ}ʻR_{\in}=\text{Cls}\)</p> + +<p>The type of "\(\text{Cls}\)" here is that type whose members are of +the same type as \(\text{ᗡ}ʻR\). In the proof, use is made of the +convention that a Greek letter always stands for an expression of the +form \(\hat{z}(\phi!z)\).</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·101}.&\supset\vdash:\alpha R_{\in}\hat{z}(\phi!z).\equiv.\alpha=Rʻʻ\hat{z}(\phi!z):\\ +[\text{*10·11·281}] &\supset\vdash:(\exists \alpha).\alpha R_{\in}\hat{z}(\phi!z).\equiv.(\exists \alpha).\alpha=Rʻʻ\hat{z}(\phi!z):\\ +[\text{*33·131}] &\supset\vdash:\hat{z}(\phi!z)\in \text{ᗡ}ʻR_{\in}.\equiv.(\exists \alpha).\alpha=Rʻʻ\hat{z}(\phi!z) &\qquad \text{(1)}\\ +\vdash.\text{*20·2.(*37·01)}. &\supset\vdash:\hat{x}{(\exists y).y\in \hat{z}(\phi!z).xRy}=Rʻʻ\hat{z}(\phi!z):\\ +[\text{*10·11·24}] &\supset\vdash:(\phi):(\exists \alpha).\alpha=Rʻʻ\hat{z}(\phi!z) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*2·02}. &\supset\vdash:\hat{z}(\phi!z)\in \text{Cls}.\supset.\hat{z}(\phi!z)\in \text{ᗡ}ʻR_{\in} &\qquad \text{(3)}\\ +\vdash.\text{*20·41.*2·02}.&\supset\vdash:\hat{z}(\phi!z)\in \text{ᗡ}ʻR_{\in}.\supset.\hat{z}(\phi!z)\in \text{Cls} &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>As appears in the above proof, it is necessary, when a proposition +containing "\(\text{Cls}\)" is to be proved, to abandon the notation +with Greek letters, and revert to the explicit functional notation.</p> + +<p class="nind"><b>*37·24.</b> \(\vdash:\alpha\in \text{D}ʻR_{\in}.\supset.\alpha\subset \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13.*37·101}.\supset\vdash\colon\colon \alpha\in \text{D}ʻR_{\in}.&\equiv\colon\ldotp (\exists \beta).\alpha=Rʻʻ\beta\colon\ldotp\\ +[\text{*20·33.*37·1}] & \equiv\colon\ldotp (\exists \beta):x\in \alpha.\equiv_{x}.(\exists y).y\in \beta.xRy\colon\ldotp \\ +[\text{*11·61}] &\supset\colon\ldotp x\in \alpha.\supset_{x}:(\exists \beta,y).y\in \beta.xRy:\\ +[\text{*11·23}] &\qquad\quad\quad\supset_{x}:(\exists x,\beta).y\in \beta.xRy:\\ +[\text{*11·55}] &\qquad\qquad\supset_{x}:(\exists y):xRy:(\exists \beta).y\in \beta:\\ +[\text{*10·5}] &\qquad\qquad\supset_{x}:(\exists y).xRy:\\ +[\text{*33·13}] &\qquad\qquad\supset_{x}:x\in \text{D}ʻR\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_300">[Pg 300]</span></p> + +<p class="nind"><b>*37·25.</b> \(\vdash.\text{D}ʻR=Rʻʻ\text{ᗡ}ʻR.\text{ᗡ}ʻR=\breve{R}ʻʻ\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}. \supset\vdash:x\in \text{D}ʻR.&\equiv.(\exists y).xRy.\\ +[\text{*33·14.*4·71}] &\equiv.(\exists y).y\in \text{ᗡ}ʻR.xRy.\\ +[\text{*37·1}] &\equiv.x\in Rʻʻ\text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*33·131}.\supset\vdash:y\in \text{ᗡ}ʻR.&\equiv.(\exists x).xRy.\\ +[\text{*33·14.*4·71}] &\equiv.(\exists x).x\in \text{D}ʻR.xRy.\\ +[\text{*37·105}] &\equiv.y\in \breve{R}ʻʻ\text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·26.</b> \(\vdash.Rʻʻ\beta=Rʻʻ(\beta\cap \text{ᗡ}ʻR)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash\colon\ldotp x\in Rʻʻ\beta.&\equiv:(\exists y).y\in \beta.xRy:\\ +[\text{*33·14.*4·71}] &\equiv:(\exists y).y\in \beta.y\in \text{ᗡ}ʻR.xRy:\\ +[\text{*22·33}]&\equiv:(\exists y).y\in \beta\cap \text{ᗡ}ʻR.xRy:\\ +[\text{*37·1}]&\equiv:x\in Rʻʻ(\beta\cap \text{ᗡ}ʻR)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·261.</b> \(\vdash.\breve{R}ʻʻ\beta=\breve{R}ʻʻ(\beta\cap \text{D}ʻR) \quad[\text{*37·26.*33·21}]\)</p> + +<p class="nind"><b>*37·262.</b> \(\vdash:\alpha\cap \text{ᗡ}ʻR=\beta\cap \text{ᗡ}ʻR.\supset.Rʻʻ\alpha=Rʻʻ\beta \quad[\text{*37·26}]\)</p> + +<p class="nind"><b>*37·263.</b> \(\vdash:\alpha\cap \text{D}ʻR=\beta\cap \text{D}ʻR.\supset.\breve{R}ʻʻ\alpha=\breve{R}ʻʻ\beta \quad[\text{*37·261}]\)</p> + +<p class="nind"><b><a id="*37·264">*37·264</a>.</b> \(\vdash:\exists !\alpha\cap Rʻʻ\beta.\equiv.(\exists x,y).x\in \alpha.y\in \beta.xRy.\equiv.\exists !\beta\cap \breve{R}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·33.*37·1}.\supset\vdash\colon\ldotp \exists !\alpha\cap Rʻʻ\beta.&\equiv:(\exists x):x\in \alpha:(\exists y).y\in \beta.xRy: &\qquad \text{(1)}\\ +[\text{*11·55}] &\equiv:(\exists x,y).x\in \alpha.y\in \beta.xRy &\qquad \text{(2)}\\ +\vdash.\text{(1).*11·6}. \supset\vdash\colon\ldotp \exists !\alpha\cap Rʻʻ\beta.&\equiv:(\exists y):y\in \beta:(\exists x).x\in \alpha.xRy:\\ +[\text{*37·105}] &\equiv:(\exists y).y\in \beta.y\in \breve{R}ʻʻ\alpha:\\ +[\text{*22·33}] &\equiv:\exists !\beta\cap \breve{R}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·265.</b> \(\vdash.Rʻʻ\alpha=Rʻʻ(\alpha\cap CʻR).\breve{R}ʻʻ\alpha=\breve{R}ʻʻ(\alpha\cap CʻR)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·161.*22·621}.&\supset\vdash.\text{ᗡ}ʻR=CʻR\cap \text{ᗡ}ʻR.\\ +[\text{*22·481}] &\supset\vdash.\alpha\cap \text{ᗡ}ʻR=\alpha\cap CʻR\cap \text{ᗡ}ʻR.\\ +[\text{*37·262}] &\supset\vdash.Rʻʻ\alpha=Rʻʻ(\alpha\cap CʻR) &\qquad \text{(1)}\\ +\vdash.\text{(1).*33·22}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_301">[Pg 301]</span></p> + +<p class="nind"><b>*37·27.</b> \(\vdash:\text{ᗡ}ʻR\subset \beta.\supset.\text{D}ʻR=Rʻʻ\beta \quad[\text{*22·621.*37·25·26}]\)</p> + +<p class="nind"><b>*37·271.</b> \(\vdash:\text{D}ʻR\subset \alpha.\supset.\text{ᗡ}ʻR=\breve{R}ʻʻ\alpha \quad[\text{*22·621.*37·25·261}]\)</p> + +<p class="nind"><b>*37·28.</b> \(\vdash.Rʻʻ\text{V}=\text{D}ʻR.\breve{R}ʻʻ\text{V}=\text{ᗡ}ʻR \quad[\text{*37·27·271.*24·11}]\)</p> + +<p class="nind"><b><a id="*37·29">*37·29</a>.</b> \(\vdash.Rʻʻ\Lambda=\Lambda.\breve{R}ʻ\Lambda=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·5}.&\supset\vdash:(\exists y).y\in \Lambda.xRy.\supset.(\exists y).y\in \Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp.*24·53}.&\supset\vdash.~(\exists y).y\in \Lambda.xRy.\\ +[\text{*37·1}] &\supset\vdash.{\sim}\exists ! Rʻʻ\Lambda.\\ +[\text{*24·51}] &\supset\vdash.Rʻʻ\Lambda=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(2)}\, \frac{\breve{R}}{R}. &\supset\vdash.\breve{R}ʻʻ\Lambda=\Lambda &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*37·3">*37·3</a>.</b> \(\vdash.\{\text{sg}ʻ(P\mid Q)\}ʻz=Pʻʻ\overrightarrow{Q}ʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·23·13}.\supset\\ +\vdash.\{\text{sg}ʻ(P\mid Q)\}ʻz&=\hat{x}\{x(P\mid Q)z\}\\ +[\text{*34·1}] &=\hat{x}\{(\exists y).xPy.yQz\}\\ +[\text{*32·18}] &=\hat{x}\{(\exists y).xPy.y\in \overrightarrow{Q}ʻz\}\\ +[\text{(*37·01)}] &= Pʻʻ\overrightarrow{Q}ʻz.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·301.</b> \(\vdash.\{\text{gs}ʻ(P\mid Q)\}ʻx=\breve{Q}ʻʻ\overleftarrow{P}ʻx \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*37·302.</b> \[\begin{align}&\vdash:R=P\mid Q.\supset.\overrightarrow{R}ʻz=Pʻʻ\overrightarrow{Q}ʻz.\overleftarrow{R}ʻx=\breve{Q}ʻʻ\overleftarrow{P}ʻx\\ +&[\text{*37·3·301.*32·23·231·16}]\end{align}\]</p> + +<p class="nind"><b>*37·31.</b> \(\vdash.\text{sg}ʻ(P\mid Q)=P_{\in}\mid \overrightarrow{Q}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·11·3}. &\supset\vdash.(z).\{\text{sg}ʻ(P\mid Q)\}ʻz=P_{\in}ʻ\overrightarrow{Q}ʻz &\qquad \text{(1)}\\ +\vdash.\text{(1).*34·42}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·311.</b> \(\vdash.\text{gs}ʻ(P\mid Q)=(\breve{Q}_{\in}\mid \overleftarrow{P} \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*37·32.</b> \(\vdash.\text{D}ʻ(P\mid Q)=Pʻʻ\text{D}ʻQ. \text{ᗡ}ʻ(P\mid Q)=\breve{Q}ʻʻ\text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13.*34·1}.\supset\\ +\vdash\colon\ldotp x\in \text{D}ʻ(P\mid Q).&\equiv:(\exists z):(\exists y).xPy.yQz:\\ +[\text{*11·23}] &\equiv:(\exists y):(\exists z)·xPy.yQz:\\ +[\text{*11·55}] &\equiv:(\exists y):xPy:(\exists z).yQz:\\ +[\text{*33·13}] &\equiv:(\exists y).xPy.y\in \text{D}ʻQ:\\ +[\text{*37·1}] &\equiv:x\in Pʻʻ\text{D}ʻQ &\qquad \text{(1)}\\ +\end{array} +\]<span class="pagenum" id="Page_302">[Pg 302]</span> +\[ +\begin{array}{l} +\vdash.\text{(1).*10·11.*20·43}.\supset\\ +\vdash.\text{D}ʻ(P\mid Q)&=Pʻʻ\text{D}ʻQ &\qquad \text{(2)}\\ +\vdash.\text{*33·2}. \supset\vdash.\text{ᗡ}ʻ(P\mid Q)&=\text{D}ʻ\text{Cnv}ʻ(P\mid Q)\\ +[\text{*34·2}] &=\text{D}ʻ(\breve{Q}\mid \breve{P})\\ +[\text{(2)}] &=\breve{Q}ʻʻ\text{D}ʻ\breve{P}\\ +[\text{*33·2}] &=\breve{Q}ʻʻ\text{ᗡ}ʻP &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·321.</b> \(\vdash:\text{ᗡ}ʻP\subset\text{D}ʻQ.\supset.\text{D}ʻ(P\mid Q)=\text{D}ʻP \quad[\text{*37·32·27}]\)</p> + +<p class="nind"><b>*37·322.</b> \(\vdash:\text{D}ʻQ\subset\text{ᗡ}ʻP.\supset.\text{ᗡ}ʻ(P\mid Q)=\text{ᗡ}ʻQ \quad[\text{*37·32·271}]\)</p> + +<p class="nind"><b>*37·323.</b> \(\vdash:\text{ᗡ}ʻP=\text{D}ʻQ.\supset.\text{D}ʻ(P\mid Q)=\text{D}ʻP.\text{ᗡ}ʻ(P\mid Q)=\text{ᗡ}ʻQ \quad[\text{*37·321·322}]\)</p> + +<p class="nind"><b>*37·33.</b> \(\vdash.(P\mid Q)ʻʻ\gamma=PʻʻQʻʻ\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash\colon\ldotp x\in (P\mid Q)ʻʻ\gamma.&\equiv:(\exists z).z\in \gamma.x(P\mid Q)z:\\ +[\text{*34·1.*11·55}] &\equiv:(\exists z,y).z\in \gamma.xPy.yQz:\\ +[\text{*11·23}] &\equiv:(\exists y,z).xPy.yQz.z\in \gamma:\\ +[\text{*11·55}] &\equiv:(\exists y):xPy:(\exists z).yQz.z\in \gamma:\\ +[\text{*37·1}] &\equiv:(\exists y).xPy.y\in Qʻʻ\gamma:\\ +[\text{*37·1}] &\equiv:x\in PʻʻQʻʻ\gamma\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·34.</b> \(\vdash.(P\mid Q)_{\in}=P_{\in}\mid Q_{\in}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·11}.\supset\vdash.(P\mid Q)_{\in}ʻ\gamma&=(P\mid Q)ʻʻ\gamma\\ +[\text{*37·33}] &=PʻʻQʻʻ\gamma\\ +[\text{*37·11}] &=P_{\in}ʻQ_{\in}ʻ\gamma &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*34·42}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·341.</b> \(\vdash.{\text{Cnv}ʻ(P\mid Q)}_{\in}=(\breve{Q})_{\in}\mid (\breve{P})_{\in} \quad[\text{*34·2.*37·34}]\)</p> + +<p class="nind"><b>*37·35.</b> \(\vdash:(z).Rʻz=PʻQʻz.\supset.(\gamma).Rʻʻ\gamma=PʻʻQʻʻ\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·42}.\supset\vdash:\text{Hp}.\supset.R&=P\mid Q.\\ +[\text{*37·13}] \supset.Rʻʻ\gamma&=(P\mid Q)ʻʻ\gamma\\ +[\text{*37·33}] &=PʻʻQʻʻ\gamma:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·351.</b> \[\begin{align}&\vdash:(\alpha).Rʻ\alpha=PʻQʻʻ\alpha.\supset.(\kappa).Rʻʻ\kappa=PʻʻQʻʻʻ\kappa\\ +&\left[\text{*37·35} \frac{Q_{\in}}{Q}.\text{*37·11.(*37·04)}\right]\end{align}\]</p> + +<p><span class="pagenum" id="Page_303">[Pg 303]</span></p> + +<p class="nind"><b>*37·352.</b> \[\begin{align}&\vdash:(\alpha).Rʻʻ\alpha=PʻQʻʻ\alpha.\supset.(\kappa).Rʻʻʻ\kappa=PʻʻQʻʻʻ\kappa\\ +&\left[\text{*37·351}\, \frac{R_{\in}}{R}.\text{*37·11.(*37·04)}\right]\end{align}\]</p> + +<p class="nind"><b>*37·353.</b> \(\vdash:(z).RʻSʻz=PʻQʻz.\supset.(\gamma).RʻʻSʻʻ\gamma=PʻʻQʻʻ\gamma\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*14·21}.\supset\vdash:\text{Hp}.&\supset.(z).\text{E}!RʻSʻz.\\ +[\text{*34·41}] &\supset.(z).RʻSʻz=(R\mid S)ʻz.\\ +[\text{*14·131·144}] &\supset.(z).(R\mid S)ʻz=PʻQʻz.\\ +[\text{*37·35}] &\supset.(\gamma).(R\mid S)ʻʻ\gamma=PʻʻQʻʻ\gamma.\\ +[\text{*37·33}] &\supset.(\gamma).RʻʻSʻʻ\gamma=PʻʻQʻʻ\gamma:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·354.</b> \(\vdash.(\alpha).RʻSʻ\alpha=PʻQʻʻ\alpha.\supset.(\kappa).RʻʻSʻʻ\kappa=PʻʻQʻʻʻ\kappa \quad\left[\text{*37·353}\, \frac{Q_{\in}}{Q}\right]\)</p> + +<p class="nind"><b>*37·355.</b> \(\vdash:(z).RʻSʻz=PʻʻQʻz.\supset.(\gamma).RʻʻSʻʻ\gamma=PʻʻʻQʻʻ\gamma \quad\left[\text{*37·353}\, \frac{P_{\in}}{P}\right]\)</p> + +<p class="nind"><b>*37·36.</b> \(\vdash.\text{D}ʻR^{2}=Rʻʻ\text{D}ʻR.\text{ᗡ}ʻR^{2}=\breve{R}ʻʻ\text{ᗡ}ʻR \quad[\text{*37·32}]\)</p> + +<p class="nind"><b>*37·37.</b> \(\vdash.(R^{2})_{\in}=(R_{\in})^{2} \quad[\text{*37·34}]\)</p> + +<p class="nind"><b>*37·371.</b> \(R_{\in}^{2}=(R_{\in})^{2} \quad\text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets. Like +<a href="#*37·03">*37·03</a>, this definition will be extended to all suffixes.</p> + +<p class="nind"><b>*37·38.</b> \(\vdash.\overrightarrow{R}^{2}ʻx=Rʻʻ\overrightarrow{R}ʻx \quad[\text{*37·3}]\)</p> + +<p class="nind"><b><a id="*37·39">*37·39</a>.</b> \(\vdash.R^{2}ʻʻ\alpha=RʻʻRʻʻ\alpha \quad[\text{*37·33}]\)</p> + +<p class="nind"><b><a id="*37·4">*37·4</a>.</b> \(\vdash.\text{ᗡ}ʻ(\alpha\upharpoonleft R)=\breve{R}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*33·131.*35·1}.\supset\vdash:y\in \text{ᗡ}ʻ(\alpha\upharpoonleft R).&\equiv.(\exists x).x\in \alpha.xRy.\\ +[\text{*37·105}] &\equiv.y\in \breve{R}ʻʻ\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·401.</b> \(\vdash.\text{D}ʻ(R\upharpoonright \beta)=Rʻʻ\beta \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*37·402.</b> \(\vdash.\text{D}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)=\alpha\cap Rʻʻ\beta.\text{ᗡ}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)=\beta\cap \breve{R}ʻʻ\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*33·13.*35·102}.\supset\\ +\vdash\colon\ldotp x\in \text{D}ʻ(\alpha\upharpoonleft R\upharpoonright \beta).&\equiv:(\exists y).x\in \alpha.xRy.y\in \beta:\\ +[\text{*10·35}] &\equiv:x\in \alpha:(\exists y).xRy.y\in \beta:\\ +[\text{*37·1}] &\equiv:x\in \alpha.x\in Rʻʻ\beta:\\ +[\text{*22·33}] &\equiv:x\in \alpha\cap Rʻʻ\beta &\qquad \text{(1)}\\ +\text{Similarly}\\ +\vdash:y\in \text{ᗡ}ʻ(\alpha\upharpoonleft R\upharpoonright \beta).&\equiv.y\in \beta\cap \breve{R}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·41.</b> \(\vdash.\text{D}ʻ(R\unicode{x0294f}\alpha)=\alpha\cap Rʻʻ\alpha.\text{ᗡ}ʻ(R\unicode{x0294f}\alpha)=\alpha\cap \breve{R}ʻʻ\alpha \quad[\text{*37·402.*36·11}]\)</p> + +<p><span class="pagenum" id="Page_304">[Pg 304]</span></p> + +<p class="nind"><b>*37·411.</b> \(\vdash.(\alpha\upharpoonleft R)ʻʻ\beta=\text{D}ʻ(\alpha\upharpoonleft R\upharpoonright \beta)=\alpha\cap Rʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·401}.\supset\vdash.(\alpha\upharpoonleft R)ʻʻ\beta&=\text{D}ʻ(\alpha\upharpoonleft R)\upharpoonright \beta\\ +[\text{*35·21}] &=\text{D}ʻ(\alpha\upharpoonleft R\upharpoonright \beta) &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·402}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·412.</b> \(\vdash.(R\upharpoonright \alpha)ʻʻ\beta=Rʻʻ(\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·401}.\supset\vdash.(R\upharpoonright \alpha)ʻʻ\beta&=\text{D}ʻ(R\upharpoonright \alpha)\upharpoonright \beta\\ +[\text{*35·31}] &=\text{D}ʻR\upharpoonright (\alpha\cap \beta)\\ +[\text{*37·401}] &=Rʻʻ(\alpha\cap \beta).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·413.</b> \(\vdash.(R\unicode{x0294f}\alpha)ʻʻ\beta=\alpha\cap Rʻʻ(\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·411.*35·21}.\supset\vdash.(R\unicode{x0294f}\alpha)ʻʻ\beta&=\alpha\cap (R\upharpoonright \alpha)ʻʻ\beta\\ +[\text{*37·412}] &=\alpha\cap Rʻʻ(\alpha\cap \beta).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·42.</b> \(\vdash:Rʻʻ\beta\subset \alpha.\supset.(\alpha\upharpoonleft R)ʻʻ\beta=Rʻʻ\beta \quad[\text{*37·411.*22·621}]\)</p> + +<p class="nind"><b>*37·421.</b> \(\vdash:\beta\subset \alpha.\supset.(R\upharpoonright \alpha)ʻʻ\beta=Rʻʻ\beta \quad[\text{*37·412.*22·621}]\)</p> + +<p class="nind"><b>*37·43.</b> \(\vdash\colon\ldotp \beta\subset \text{ᗡ}ʻR.\supset:\exists !Rʻʻ\beta.\equiv.\exists !\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·401.*35·65}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:Rʻʻ\beta=\text{D}ʻ(R\upharpoonright \beta).\beta=\text{ᗡ}ʻ(R\upharpoonright \beta) &\qquad \text{(1)}\\ +\vdash.\text{(1).*33·24}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·431.</b> \(\vdash\colon\ldotp \alpha\subset \text{D}ʻR.\supset:\exists !\breve{R}ʻʻ\alpha.\equiv.\exists !\alpha \quad[\text{Proof as in *37·43}]\)</p> + +<p class="nind"><b>*37·44.</b> \(\vdash\colon\ldotp \text{ᗡ}ʻR=V.\supset:\exists !Rʻʻ\beta.\equiv.\exists !\beta \quad[\text{*37·43.*24·11}]\)</p> + +<p class="nind"><b>*37·441.</b> \(\vdash\colon\ldotp \text{D}ʻR=V.\supset:\exists !\breve{R}ʻʻ\alpha.\equiv.\exists !\alpha \quad[\text{Proof as in *37·44}]\)</p> + +<p class="nind"><b>*37·45.</b> \(\vdash\colon\ldotp (y).\text{E}!Rʻy.\supset:\exists !Rʻʻ\beta.\equiv.\exists !\beta \quad[\text{*33·431.*37·43}]\)</p> + +<p class="nind"><b>*37·451.</b> \(\vdash\colon\ldotp (x).\text{E}!\breve{R}ʻx.\supset:\exists !\breve{R}ʻʻ\alpha.\equiv.\exists !\alpha \quad[\text{Proof as in *37·45}]\)</p> + +<p class="nind"><b>*37·46.</b> \(\vdash:x\in Rʻʻ\alpha.\equiv.\exists !\alpha\cap \overleftarrow{R}ʻx \quad[\text{*37·1.*32·181}]\)</p> + +<p class="nind"><b>*37·461.</b> \(\vdash:x{\sim}\in Rʻʻ\alpha.\equiv.\alpha\cap \overleftarrow{R}ʻx=\Lambda.\equiv.\overleftarrow{R}ʻx\subset -\alpha \quad[\text{*37·46.*24·311}]\)</p> + +<p class="nind"><b>*37·462.</b> \(\vdash:x{\sim}\in \breve{R}ʻʻ\alpha.\equiv.\alpha\cap \overrightarrow{R}ʻx=\Lambda.\equiv.\overrightarrow{R}ʻx\subset -\alpha \quad[\text{*37·461.*32·241}]\)</p> + +<p class="nind"><b>*37·47.</b> \(\vdash:\exists !\alpha.\equiv.\exists !Rʻʻʻ\alpha.\equiv.\exists !\breve{R}ʻʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·45·111}.&\supset\vdash:\exists !\alpha.\equiv.\exists !R_{\in}ʻʻ\alpha.\\ +[\text{(*37·04)}] &\qquad\qquad\equiv.\exists !Rʻʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1)}\,\frac{\breve{R}}{R}. &\supset\vdash:\exists !\alpha.\equiv.\exists !\breve{R}ʻʻʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_305">[Pg 305]</span></p> + +<p class="nind"><b>*37·5.</b> \(\vdash:(\beta).Pʻʻ\beta=Qʻ\beta.\supset.(\kappa).Pʻʻʻ\kappa=Qʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·12}.\supset\vdash:\text{Hp}.&\supset.P_{\in}=Q.\\ +[\text{*37·13}] &\supset.P_{\in}ʻʻ\kappa=Qʻʻ\kappa.\\ +[\text{(*37·04)}] &\supset.Pʻʻʻ\kappa=Qʻʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·501.</b> \(\vdash.\beta\cap \text{ᗡ}ʻR\subset \breve{R}ʻʻRʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1.*10·24}.&\supset\vdash:y\in \beta.xRy.\supset.x\in Rʻʻ\beta:\\ +[\text{Exp.*10·11·21}]&\supset\vdash\colon\ldotp y\in \beta.\supset:xRy.\supset_{x}.x\in Rʻʻ\beta:\\ +[\text{*4·7}] &\supset:xRy.\supset_{x}.xRy.x\in Rʻʻ\beta:\\ +[\text{*10·28}] &\supset:(\exists x).xRy.\supset.(\exists x).xRy.x\in Rʻʻ\beta:\\ +[\text{*33·131.*37·105}]&\supset:y\in \text{ᗡ}ʻR.\supset.y\in RʻʻRʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).Imp.*22·33}.\supset\\ +&\vdash:y\in \beta\cap \text{ᗡ}ʻR.\supset.y\in \breve{R}ʻʻRʻʻ\beta:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·502.</b> \(\vdash.\alpha\cap \text{D}ʻR\subset Rʻʻ\breve{R}ʻʻ\alpha \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*37·51.</b> \(\vdash:\beta\subset \text{ᗡ}ʻR.\equiv.\beta\subset \breve{R}ʻʻRʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·501.*22·621}.&\supset\vdash:\beta\subset \text{ᗡ}ʻR.\supset.\beta\subset \breve{R}ʻʻRʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*37·16}. &\supset\vdash:\beta\subset \breve{R}ʻʻRʻʻ\beta.\supset.\beta\subset \text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*37·52">*37·52</a>.</b> \(\vdash:\alpha\subset \text{D}ʻR.\equiv.\alpha\subset Rʻʻ\breve{R}ʻʻ\alpha \quad[\text{Similar proof}]\)</p> + +<p>The following propositions, down to <a href="#*37·7">*37·7</a> exclusive, are concerned +with the special properties of \(Rʻʻ\beta\) which result from the +hypothesis \(\text{E}‼Rʻʻ\beta\), defined in <a href="#*37·05">*37·05</a>. The hypothesis +\(\text{E}‼Rʻʻ\beta\) is important, because it has many consequences +and is satisfied in many cases with which we wish to deal.</p> + +<p class="nind"><b><a id="*37·6">*37·6</a>.</b> \(\vdash:\text{E}‼Rʻʻ\beta.\supset.Rʻʻ\beta=\hat{x}\{(\exists y).y\in \beta.x=Rʻy\}\)</p> + +<p>This proposition is very important, and is used constantly.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·104}.\supset\vdash\colon\colon \text{Hp}.&\supset\colon\ldotp y\in \beta.\supset_{y}:\text{E}!Rʻy:\\ +[\text{*30·4}] &\supset_{y}:x=Rʻy.\equiv.xRy\colon\ldotp \\ +[\text{*5·32}] &\supset\colon\ldotp y\in \beta.x=Rʻy.\equiv_{y}.y\in \beta.xRy\colon\ldotp \\ +[\text{*10·281}] \supset\colon\ldotp (\exists y).y\in \beta.x=Rʻy.&\equiv.(\exists y).y\in \beta.xRy.\\ +[\text{*37·1}] &\equiv.x\in Rʻʻ\beta &&\qquad \text{(1)}\\ +\vdash.\text{(1). *10·11·21.*20·33}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_306">[Pg 306]</span></p> + +<p class="nind"><b>*37·601.</b> \(\vdash:(x).\text{E}!Rʻx.\supset.Rʻʻ\text{V}=\hat{x}\{(\exists y).x=Rʻy\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*2·02.*10·11·27}.\supset\\ +\vdash\colon\ldotp \text{Hp}. \supset:x\in \text{V}.\supset_{x}.E!Rʻx:\\ +[\text{*37·104}]&\supset:E‼Rʻʻ\text{V}:\\ +[\text{*37·6}] &\supset:Rʻʻ\text{V}=\hat{x}{(\exists y).y\in \text{V}.x=Rʻy} &\qquad \text{(1)}\\ +\vdash.\text{*24·104.*4·73}.&\supset\vdash:y\in \text{V}.x=Rʻy.\equiv.x=Rʻy:\\ +[\text{*10·11·281}] &\supset\vdash:(\exists y).y\in \text{V}.x=Rʻy.\equiv.(\exists y).x=Rʻy:\\ +[\text{*20·15}] &\supset\vdash:\hat{x}\{(\exists y).y\in \text{V}.x=Rʻy\}=\hat{x}\{(\exists y).x=Rʻy\} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·61.</b> \(\vdash\colon\colon \text{E}‼Rʻʻ\beta.\supset\colon\ldotp Rʻʻ\beta\subset\alpha.\equiv:y\in \beta.\supset_{y}.Rʻy\in \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·17}. &\supset\vdash\colon\colon Rʻʻ\beta\subset\alpha.\equiv\colon\ldotp y\in \beta.xRy.\supset_{x,y}.x\in \alpha\colon\ldotp \\ +[\text{*11·2·62}] & \equiv\colon\ldotp y\in \beta.\supset_{y}:xRy.\supset_{x}.x\in \alpha &\qquad \text{(1)}\\ +\vdash.\text{*37·104}.&\supset\vdash\colon\colon.\text{Hp}.\supset\colon\colon y\in \beta.\supset_{y}\colon\ldotp E!Rʻy\colon\ldotp \\ +[\text{*30·33}] &\supset_{y}\colon\ldotp Rʻy\in \alpha.\equiv:xRy.\supset_{x}.x\in \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp Rʻʻ\beta\subset\alpha.\equiv:y\in \beta.\supset_{y}.Rʻy\in \alpha\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·62.</b> \(\vdash:\text{E}!Rʻy.y\in \alpha.\supset.Rʻy\in Rʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·33}.\supset\\ +\vdash\colon\colon \text{E}!Rʻy.&\supset\colon\ldotp Rʻy\in Rʻʻ\alpha.\equiv:xRy.\supset_{x}.x\in Rʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*3·2}. &\supset\vdash\colon\ldotp y\in \alpha.\supset:xRy.\supset.y\in \alpha.xRy.\\ +[\text{*10·24.*37·1}] &\supset.x\in Rʻʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·21}.&\supset\vdash\colon\ldotp y\in \alpha.\supset:xRy.\supset_{x}.x\in Rʻʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_307">[Pg 307]</span></p> + +<p>The above is the type of inference concerning which Jevons says<a id="FNanchor_57" href="#Footnote_57" class="fnanchor">[57]</a>: +"I remember the late Prof. De Morgan remarking that all Aristotle's +logic could not prove that 'Because a horse is an animal, the head of a +horse is the head of an animal.'" It must be confessed that this was a +merit in Aristotle's logic, since the proposed inference is fallacious +without the added premiss "\(\text{E}!\) the head of the horse in +question." <i>E.g.</i> it does not hold for an oyster or a hydra. But +with the addition \(\text{E}!Rʻy\), the above proposition gives an +important and common type of asyllogistic inference.</p> + +<p class="nind"><b>*37·63.</b> \(\vdash\colon\colon \text{E}‼Rʻʻ\alpha.\supset\colon\ldotp x\in Rʻʻ\alpha.\supset_{x}.\psi x:\equiv:y\in \alpha.\supset_{y}.\psi(Rʻy)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}. \supset\vdash\colon\colon x\in Rʻʻ\alpha.\supset_{x}.\psi x:&\equiv\colon\ldotp (\exists y).y\in \alpha.xRy.\supset_{x}.\psi x\colon\ldotp\\ +[\text{*10·23}] &\equiv\colon\ldotp y\in \alpha.xRy.\supset_{x,y}.\psi x\colon\ldotp \\ +[\text{*11·2·62}] &\equiv\colon\ldotp y\in \alpha.\supset_{y}:xRy.\supset_{x}.\psi x &\qquad \text{(1)}\\ +\vdash.\text{*37·104}.\supset\vdash\colon\colon\ldotp\text{Hp}.\supset\colon\colon y\in \alpha.\supset_{y}\colon\ldotp \text{E}!Rʻy\colon\ldotp \\ +[\text{*30·33}] \qquad\qquad\qquad\qquad\qquad\supset_{y}\colon\ldotp \psi(Rʻy).\equiv:xRy.\supset_{x}.\psi x &&\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is very frequently used.</p> + +<p class="nind"><b>*37·64.</b> \(\vdash\colon\ldotp \text{E}‼Rʻʻ\alpha.\supset:(\exists y).y\in \alpha.\psi(Rʻy).\equiv.(\exists x).x\in Rʻʻ\alpha.\psi x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·33}.\supset\vdash\colon\colon \text{Hp}.&\supset\colon\ldotp y\in \alpha.\supset:\psi(Rʻy).\equiv.(\exists x).xRy.\psi x\colon\ldotp \\ +[\text{*5·32}] &\supset\colon\ldotp y\in \alpha.\psi(Rʻy).\equiv:y\in \alpha:(\exists x).xRy.\psi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21·281}.\supset\\ +\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (\exists y).y\in \alpha.\psi(Rʻy).&\equiv:(\exists y):y\in \alpha:(\exists x).xRy.\psi x:\\ +[\text{*11·6}] &\equiv:(\exists x):(\exists y).y\in \alpha.xRy:\psi x:\\ +[\text{*37·1}] &\equiv:(\exists x).x\in Rʻʻ\alpha.\psi x\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*37·65">*37·65</a>.</b> \(\vdash:\text{E}‼Rʻʻ\beta.\alpha\subset Rʻʻ\beta.\supset.\alpha=Rʻʻ(\breve{R}ʻʻ\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·21.*3·27}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp y\in \beta.\supset_{y}:zRy.xRy.\supset.z=x &&\qquad \text{(1)}\\ +\vdash.\text{*37·1}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +x\in Rʻʻ(\breve{R}ʻʻ\alpha\cap \beta).&\equiv.(\exists y).y\in \breve{R}ʻʻ\alpha\cap \beta.xRy.\\ +[\text{*37·105.*11·55}] &\equiv.(\exists y,z).z\in \alpha.zRy.y\in \beta.xRy.\\ +[\text{(1).*4·71}] &\equiv.(\exists y,z).z\in \alpha.zRy.y\in \beta.xRy.z=x.\\ +[\text{*13·194}] &\equiv.(\exists y,z).z\in \alpha.y\in \beta.xRy.z=x.\\ +[\text{*13·195}] &\equiv.(\exists y).x\in \alpha.y\in \beta.xRy.\\ +[\text{*10·35.*37·1}] &\equiv.x\in \alpha.x\in Rʻʻ\beta.\\ +[\text{*4·71.Hp}] &\equiv.x\in \alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·66.</b> \(\vdash\colon\ldotp \text{E}‼Rʻʻ\beta.\supset:\alpha\subset Rʻʻ\beta.\equiv.(\exists \gamma).\gamma\subset \beta.\alpha=Rʻʻ\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·65.Exp}.*13·195.*22·43.\supset\\ +&\vdash\colon\ldotp \text{Hp}.\supset:\alpha\subset Rʻʻ\beta.\supset.(\exists \gamma).\gamma\subset \beta.\alpha=Rʻʻ\gamma &\qquad \text{(1)}\\ +\vdash.\text{*37·2.*13·13}.&\supset\vdash:\gamma\subset \beta.\alpha=Rʻʻ\gamma.\supset.\alpha\subset Rʻʻ\beta:\\ +[\text{*10·11·23}] &\supset\vdash:(\exists y).\gamma\subset \beta.\alpha=Rʻʻ\gamma.\supset.\alpha\subset Rʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_308">[Pg 308]</span></p> + +<p class="nind"><b>*37·67.</b> \(\vdash\colon\ldotp z\in \gamma.\supset_{z}.\text{E}!RʻSʻz:\supset. RʻʻSʻʻ\gamma=\hat{x}\{(\exists z). z\in \gamma. x=RʻSʻz\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·41}. &\supset\vdash:\text{Hp}. z\in \gamma. \supset_{z}. RʻSʻz=(R\mid S)ʻz &\qquad \text{(1)}\\ +\vdash.\text{(1). *14·21}. &\supset\vdash:\text{Hp}. z\in \gamma. \supset_{z}. E!(R\mid S)ʻz &\qquad \text{(2)}\\ +\vdash.\text{(2).*37·6}. &\supset\vdash:\text{Hp}. \supset. (R\mid S)ʻʻ\gamma=\hat{x}\{(\exists z). z\in \gamma. x=(R\mid S)ʻ\gamma\}\\ +[\text{(1)}] &\qquad\qquad\qquad\qquad\quad=\hat{x}\{(\exists z). z\in \gamma. x=RʻSʻ\gamma\} &\qquad \text{(3)}\\ +\vdash. \text{*37·33}. &\supset\vdash. RʻʻSʻʻ\gamma = (R\mid S)ʻʻ\gamma &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}. &\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·68.</b> \(\vdash\colon\ldotp z\in \gamma. \supset_{z}. PʻQʻz = Rʻz:\supset. PʻʻQʻʻ\gamma = Rʻʻ\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}. \supset\vdash: \text{Hp}. z\in \gamma. &\supset. \text{E}! PʻQʻz. \text{E}! Rʻz.\\ +[\text{*34·41}] &\supset. PʻQʻz = (P\mid Q)ʻz.\text{E}!Rʻz. &\qquad \text{(1)}\\ +[\text{*14·21·131·144.Hp}] &\supset. \text{E}! (P\mid Q)ʻz. (P\mid Q)ʻz = Rʻz &\qquad \text{(2)}\\ +\vdash.\text{*37·33}. &\supset\vdash. PʻʻQʻʻ\gamma = (P\mid Q)ʻʻ\gamma &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*37·6}.\supset\\ +\vdash: \text{Hp}. \supset.PʻʻQʻʻ\gamma &= \hat{x}\{(\exists z). z\in \gamma. x=(P\mid Q)ʻz\}\\ +[\text{(2)}] &= \hat{x}\{(\exists z). z\in \gamma. x=Rʻz\}\\ +[\text{*37·6.(1)}] &= Rʻʻz:\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·69.</b> \(\vdash\colon\ldotp y\in \beta. \supset_{y}. Rʻy = Sʻy: \supset. Rʻʻ\beta = Sʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}. \supset\vdash\colon\colon \text{Hp}. &\supset\colon\ldotp y\in \beta. \supset. \text{E}!Rʻy. \text{E}!Sʻy\colon\ldotp &\qquad \text{(1)}\\ +[\text{*30·4}] &\supset\colon\ldotp y\in \beta. \supset:xRy. \equiv. x=Rʻy.\\ +[\text{*14·142}] &\equiv. x=Sʻy.\\ +[\text{*30·4.(1)}] &\equiv. xSy\colon\ldotp \\ +[\text{*5·32}] &\supset\colon\ldotp y\in \beta. xRy. \equiv. y\in \beta. xSy &\qquad \text{(2)}\\ +\vdash. \text{(2). *10·11·21·281}.&\supset\\ +\vdash\colon\ldotp \text{Hp}.& \supset: (\exists y). y\in \beta. xRy. \equiv. (\exists y). y\in \beta. xSy:\\ +[\text{*37·1}] &\supset: x\in Rʻʻ\beta. \equiv. x\in Sʻʻ\beta\colon\ldotp \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p>A specially important case of \(Rʻʻ\beta\) is +\(\overrightarrow{R}ʻʻ\beta\) or \(\overleftarrow{R}ʻʻ\beta\). This +case will be further studied later (in <a href="#*70">*70</a>); for the present, we +shall only give a few preliminary propositions about it. It will be +observed that the hypothesis \(\text{E}‼\overrightarrow{R}ʻʻ\beta\) or +\(\text{E}‼\overleftarrow{R}ʻʻ\beta\) is always verified, in virtue of +*32·12·121. Hence the following applications of <a href="#*37·6">*37·6</a> ff.:</p> + +<p class="nind"><b><a id="*37·7">*37·7</a>.</b> \(\vdash. \overrightarrow{R}ʻʻ\beta=\hat{\alpha}\{(\exists y) . y\in \beta. \alpha= \overrightarrow{R}ʻy\} \quad[\text{*37·6 . *32·12}]\)</p> + +<p class="nind"><b>*37·701.</b> \(\vdash. \overleftarrow{R}ʻʻ\alpha = \hat{\beta}\{(\exists x) . x\in \alpha. \beta= \overleftarrow{R}ʻx\} \quad[\text{*37·6 . *32·121}]\)</p> + +<p><span class="pagenum" id="Page_309">[Pg 309]</span></p> + +<p class="nind"><b>*37·702.</b> \(\vdash\colon\ldotp \overrightarrow{R}ʻʻ\beta\subset \kappa.\equiv:y\in \beta.\supset_{y}.\overrightarrow{R}ʻy\in \kappa \quad[\text{*37·61}]\)</p> + +<p class="nind"><b>*37·703.</b> \(\vdash\colon\ldotp \overleftarrow{R}ʻʻ\beta\subset \kappa.\equiv:x\in \beta.\supset_{x}.\overleftarrow{R}ʻx\in \kappa \quad[\text{*37·61}]\)</p> + +<p class="nind"><b>*37·704.</b> \(\vdash:y\in \alpha.\supset.\overrightarrow{R}ʻy\in \overrightarrow{R}ʻʻ\alpha \quad[\text{*37·62.*32·12}]\)</p> + +<p class="nind"><b>*37·705.</b> \(\vdash:x\in \alpha.\supset.\overleftarrow{R}ʻx\in \overleftarrow{R}ʻʻ\alpha \quad[\text{*37·62.*32·121}]\)</p> + +<p class="nind"><b>*37·706.</b> \(\vdash\colon\ldotp \alpha\in \overrightarrow{R}ʻʻ\beta.\supset_{\alpha}.\psi\alpha:\equiv:y\in \beta.\supset_{y}.\psi(\overrightarrow{R}ʻy) + \quad[\text{*37·63}]\)</p> + +<p class="nind"><b>*37·707.</b> \(\vdash\colon\ldotp \beta\in \overleftarrow{R}ʻʻ\alpha.\supset_{\beta}.\psi\beta:\equiv:x\in \alpha.\supset_{x}.\psi(\overleftarrow{R}ʻx) + \quad[\text{*37·63}]\)</p> + +<p class="nind"><b>*37·708.</b> \(\vdash\colon\ldotp (\exists \alpha).\alpha\in \overrightarrow{R}ʻʻ\beta.\psi\alpha.\equiv.(\exists y).y\in \beta.\psi(\overrightarrow{R}ʻy) \quad[\text{*37·64}]\)</p> + +<p class="nind"><b>*37·709.</b> \(\vdash\colon\ldotp (\exists \alpha).\alpha\in \overleftarrow{R}ʻʻ\beta.\psi\alpha.\equiv.(\exists x).x\in \beta.\psi(\overleftarrow{R}ʻx) \quad[\text{*37·64}]\)</p> + +<p class="nind"><b>*37·71.</b> \(\vdash:\kappa\subset \overrightarrow{R}ʻʻ\beta.\supset.\kappa=\overrightarrow{R}ʻʻ\{(\text{Cnv}ʻ\overrightarrow{R})ʻʻ\kappa\cap \beta\} \quad[\text{*37·65}]\)</p> + +<p class="nind"><b>*37·711.</b> \(\vdash:\kappa\subset \overleftarrow{R}ʻʻ\beta.\supset.\kappa=\overleftarrow{R}ʻʻ\{(\text{Cnv}ʻ\overleftarrow{R})ʻʻ\kappa\cap \beta\} \quad[\text{*37·65}]\)</p> + +<p class="nind"><b>*37·712.</b> \(\vdash:\kappa\subset \overrightarrow{R}ʻʻ\beta.\equiv.(\exists \gamma).\gamma\subset \beta.\kappa=\overrightarrow{R}ʻʻ\gamma \quad[\text{*37·66}]\)</p> + +<p class="nind"><b>*37·713.</b> \(\vdash:\kappa\subset \overleftarrow{R}ʻʻ\beta.\equiv.(\exists \gamma).\gamma\subset \beta.\kappa=\overleftarrow{R}ʻʻ\gamma \quad[\text{*37·66}]\)</p> + +<p class="nind"><b>*37·72.</b> \(\vdash:R=P\mid Q.\supset.\overrightarrow{R}ʻʻ\gamma=Pʻʻʻ\overrightarrow{Q}ʻʻ\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·11·302}.\supset\vdash:\text{Hp}.&\supset.(z).P_{\in}ʻ\overrightarrow{Q}ʻz=\overrightarrow{R}ʻz.\\ +[\text{*37·68}] &\supset.P_{\in}ʻʻ\overrightarrow{Q}ʻʻ\gamma=\overrightarrow{R}ʻʻ\gamma.\\ +[\text{(*37·04)}] &\supset.Pʻʻʻ\overrightarrow{Q}ʻʻ\gamma=\overrightarrow{R}ʻʻ\gamma:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·721.</b> \(\vdash:R=P\mid Q.\supset.\overleftarrow{R}ʻʻ\gamma=\breve{Q}ʻʻʻ\overleftarrow{P}ʻʻ\gamma \quad[\text{Proof as in *37·72}]\)</p> + +<p class="nind"><b>*37·73.</b> \(\vdash:\exists !\beta.\equiv.\exists !\overrightarrow{R}ʻʻ\beta.\equiv.\exists !\overleftarrow{R}ʻʻ\beta \quad[\text{*37·45.*32·12·121}]\)</p> + +<p class="nind"><b>*37·731.</b> \(\vdash:\beta=\Lambda.\equiv.\overrightarrow{R}ʻʻ\beta=\Lambda.\equiv.\overleftarrow{R}ʻʻ\beta=\Lambda \quad[\text{*37·73.Transp}]\)</p> + +<p><span class="pagenum" id="Page_310">[Pg 310]</span></p> + +<p>Observe that the \(\Lambda\)'s which occur in this proposition will not +be all of the same type. <i>E.g.</i> if \(R\) relates individuals to +individuals, the first \(\Lambda\) will be the class of no individuals, +while the second and third will be the class of no classes. Thus +the ambiguity which attaches to the type of \(\Lambda\) must be +differently determined for different occurrences of \(\Lambda\) in +this proposition. In general, when this is the case with our ambiguous +symbols, we shall adopt a notation which indicates the fact. But when +the ambiguous symbol is \(\Lambda\), it seems hardly worth while.</p> + +<p class="nind"><b>*37·74.</b> \(\vdash\colon\ldotp \beta\subset\text{ᗡ}ʻR.\equiv:\alpha\in \overrightarrow{R}ʻʻ\beta.\supset_{\alpha}.\exists !\alpha\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*37·706}.\supset\vdash\colon\ldotp \alpha\in \overrightarrow{R}ʻʻ\beta.\supset_{\alpha}.\exists !\alpha:&\equiv:y\in \beta.\supset_{y}.\exists + !\overrightarrow{R}ʻy:\\ +[\text{*33·31}]&\equiv:\beta\subset\text{ᗡ}ʻR\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·75.</b> \(\vdash\colon\ldotp \alpha\subset\text{D}ʻR.\equiv:\beta\in \overleftarrow{R}ʻʻ\alpha.\supset_{\beta}.\exists !\beta \quad[\text{Proof as in *37·74}]\)</p> + +<p class="nind"><b>*37·76.</b> \(\vdash.\overrightarrow{R}ʻʻ\beta\subset\text{Cls}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*37·7}.\supset\vdash\colon\ldotp \alpha\in \overrightarrow{R}ʻʻ\beta.&\supset:(\exists y).y\in \beta.a=\overrightarrow{R}ʻy:\\ +[\text{*10·5}] &\supset:(\exists y).\alpha=\overrightarrow{R}ʻy:\\ +[\text{*32·13}] &\supset:(\exists y).\alpha=\hat{x}(xRy):\\ +[\text{*20·16}] &\supset:(\exists \phi).\alpha=\hat{x}(\phi!x):\\ +[\text{*20·4}] &\supset:\alpha\in \text{Cls}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·761.</b> \(\vdash.\overleftarrow{R}ʻʻ\alpha \subset \text{Cls} \quad[\text{Proof as in *37·76}]\)</p> + +<p class="nind"><b>*37·77.</b> \(\vdash: \alpha\in \overrightarrow{R}ʻʻ\text{ᗡ}ʻR.\supset_{\alpha}.\exists !\alpha \quad[\text{*37·74.*22·42}]\)</p> + +<p class="nind"><b>*37·771.</b> \(\vdash: \beta\in \overleftarrow{R}ʻʻ\text{D}ʻR.\supset_{\beta}.\exists !\beta \quad[\text{Proof as in *37·77}]\)</p> + +<p class="nind"><b>*37·772.</b> \(\vdash. \Lambda{\sim}\in \overrightarrow{R}ʻʻ\text{ᗡ}ʻR \quad[\text{*37·77.*24·63}]\)</p> + +<p class="nind"><b>*37·773.</b> \(\vdash. \Lambda{\sim}\in \overleftarrow{R}ʻʻ\text{D}ʻR \quad[\text{*37·771.*24·63}]\)</p> + +<p class="nind"><b>*37·78.</b> \(\vdash. \text{D}ʻ\overrightarrow{R}=\overrightarrow{R}ʻʻ\text{V} \quad[\text{*37·28}]\)</p> + +<p class="nind"><b>*37·781.</b> \(\vdash. \text{D}ʻ\overleftarrow{R}=\overleftarrow{R}ʻʻ\text{V} \quad[\text{*37·28}]\)</p> + +<p class="nind"><b>*37·79.</b> \(\vdash. \overrightarrow{R}ʻʻ\text{V}= \hat{\alpha}\{\exists y).\alpha=\overrightarrow{R}ʻy\} \quad[\text{*37·601.*32·12}]\)</p> + +<p class="nind"><b><a id="*37·791">*37·791</a>.</b> \(\vdash. \overleftarrow{R}ʻʻ\text{V}= \hat{\beta}\{(\exists x).\beta=\overleftarrow{R}ʻx\} \quad[\text{*37·601.*32·121}]\)</p> + +<p class="nind"><b>*37·8.</b> \(\vdash. (\alpha\uparrow \beta)\mid S=\alpha\uparrow \breve{S}ʻʻ\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash. \text{*35·103.*34·1}.\supset\vdash: x \{(\alpha\uparrow \beta)\mid S\}z.&\equiv.(\exists y).x\in \alpha.y\in \beta.ySz.\\ +[\text{*10·35.*37·105}] &\equiv.x\in \alpha.z\in \breve{S}ʻʻ\beta.\\ +[\text{*35·103}] &\equiv.x(\alpha\uparrow \breve{S}ʻʻ\beta)z:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*37·81.</b> \(\vdash.R\mid (\alpha\uparrow \beta)=(Rʻʻ\alpha)\uparrow \beta \quad[\text{Proof as in *37·8}]\)</p> + +<p class="nind"><b>*37·82.</b> \(\vdash.R\mid (\alpha\uparrow \beta)\mid S=(Rʻʻ\alpha)\uparrow (\breve{S}ʻʻ\beta) \quad[\text{*37·8·81}]\)</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_56" href="#FNanchor_56" class="label">[56]</a> +We quote from Couturat, <i>La Logique de Leibniz</i>, +Chapter <span class="allsmcap">III</span>, § 15 (p. 75 n.).</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_57" href="#FNanchor_57" class="label">[57]</a> +<i>Principles of Science</i>, chap. <span class="allsmcap">I</span>. (p. 18 of edition +of 1887).</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_311">[Pg 311]</span></p> +<h2 class="nobreak" id="*38">*38. RELATIONS AND CLASSES DERIVED FROM A DOUBLE +DESCRIPTIVE FUNCTION.</h2> +</div> + + +<p><i>Summary of</i> *38.</p> + +<p>A double descriptive function is a non-propositional function of two +arguments, such as \(\alpha \cap \beta\), \(\alpha \cup \beta\), +\(R \dot{\cap} S\), \(R \unicode{x228d} S\), \(R \mid S\), \(\alpha\upharpoonleft R\), +\(R \upharpoonright \alpha\), \(R \unicode{x294f}\alpha\). The +propositions of the present number apply to all such functions, +assuming the notation to be (as in the above instances) a functional +sign placed between the two arguments. In order to deal with all +analogous cases at once, we shall in this number adopt the notation +\[ +x \unicode{x2640} y\text{,} +\] +where "\(\unicode{x2640}\)" stands for any such sign as \(\cap\), +\(\cup\), \(\dot{\cap}\), \(\unicode{x228d}\), \(\mid\), +\(\upharpoonleft\), \(\upharpoonright\), \(\unicode{x294f}\), or any +functional sign to be hereafter defined and satisfying the condition +\[ +(x, y) . \text{E}!(x \unicode{x2640} y)\text{.} +\] +The derived relations and classes with which we shall be concerned +may be illustrated by taking the case of \(\alpha \cap \beta\). The +relation of \(\alpha \cap \beta\) to \(\beta\) will be written \(\alpha\cap\), +and the relation of \(\alpha \cap \beta\) to \(\alpha\) will be +written \(\cap \beta\). Thus we shall have +\[ +\vdash . \alpha \cap \beta = \alpha \cap ʻ\beta = \cap \betaʻ\alpha\text{.} +\] +The utility of this notation is chiefly due to the possibility of such +notations as \(\alpha \cap ʻʻ\kappa\) and \(\cap \beta ʻʻ\kappa\). For +example, take such a phrase as "the foreign members of English Clubs." +Then if we put \(\alpha\) = foreigners, \(\kappa\) = English Clubs, we +have +\[ +\alpha \cap ʻʻ\kappa = \text{the classes of foreign members of the various English Clubs.} +\] +Or again, let \(\alpha\) be a conic, and \(\kappa\) a pencil of lines; +then +\[ +\alpha \cap ʻʻ\kappa = \text{the various pairs of points in which members of \(\kappa\) meet \(\alpha\).} +\] +In this case, since \(\alpha \cap \beta = \beta \cap \alpha\), we have +\(\alpha \cap = \cap \alpha\). But when the function concerned is not +commutative, this does not hold. Thus for example we do not have \(R +\mid = \mid R\).</p> + +<p>The notations of this number will be frequently applied hereafter to +\(R \mid S\). In accordance with what was said above, we write \(R\mid\) +for the relation of \(R \mid S\) to \(S\), and \(\mid S\) for the +relation of \(R \mid S\) to \(R\). Hence we have +\[ +R \mid ʻS = \mid SʻR = R \mid S\text{.} +\] +Hence \(\mid Sʻʻ\lambda\) will be the class of relations obtained by +taking members of \(\lambda\)<span class="pagenum" id="Page_312">[Pg 312]</span> and relatively multiplying them by +\(S\). Thus if \(\lambda \) were the class of relations first cousin, +second cousin, etc., and \(S\) were the relation of parent to child, +\(\mid Sʻʻ\lambda \) would be the class of relations first cousin once +removed, second cousin once removed, etc., taken in the sense which +goes from the older to the younger generation.</p> + +<p>It is often convenient to be able to exhibit \(\mid Sʻʻ\lambda \) and +kindred expressions as descriptive functions of the first argument +instead of the second. For this purpose we put +\[ +\lambda \mid_{,,}S=\mid Sʻʻ\lambda +\] +with similar notations for other descriptive double functions. We then +have, just as in the case of \(R \mid S\), +\[ +\lambda \mid_{,,}ʻS=\mid_{,,}Sʻ\lambda =\lambda \mid_{,,}S. +\] +This enables us to form the class \(\lambda \mid_{,,}ʻʻ\mu +\). This class is chiefly useful because the members of its members +(<i>i.e</i>. \(sʻ\lambda \mid_{,,}ʻʻ\mu \), as we shall define it +in <a href="#*40">*40</a>) constitute the class of all products \(R \mid S\) that can be +formed of a member of \(\lambda\) and a member of \(\mu\).</p> + +<p>Thus we are led to three general definitions for descriptive double +functions, namely (if \(x\unicode{x2640} y\) be any such function) +\[ +\begin{aligned} +&x\unicode{x2640} \text{ is the relation of }x\unicode{x2640} y\text{ to }&y\text{ for any }&y,\\ +&\unicode{x2640} y &x \quad \quad \,\,\,\,\,\,\,\,&x,\\ +&\alpha \unicode{x2640}_{,,} y\text{ is the class of values of }&x\unicode{x2640} y\text{ when }&x\text{ is an }\alpha . +\end{aligned} +\] +Since \(\alpha \unicode{x2640}_{,,}y\) is again a descriptive double +function, the first two of the above definitions can be applied to it. +The third definition, for typographical reasons, cannot be applied +conveniently, though theoretically it is of course applicable. The +relations \(x\unicode{x2640}\) and \(\unicode{x2640} y\) represent the +general idea contained in some of the uses in mathematics of the term +'operation,' <i>e.g</i>. +1 is the operation of adding 1.</p> + +<p>The uses of the notations introduced in the present number occur +chiefly in arithmetic (Parts III and IV). Few propositions can be given +at this stage, since most of the important uses of the notation here +introduced depend upon the substitution of some special function for +the general function "\(\unicode{x2640}\)" here used. In the present +number, the propositions given are all immediate consequences of the +definitions.</p> + +<hr class="tb"> + +<p class="nind"><b>*38·01.</b> \(x\unicode{x2640} =\hat{u}\hat{y}(u=x\unicode{x2640} y)\quad \text{Df}\)</p> + +<p class="nind"><b>*38·02.</b> \(\unicode{x2640} y=\hat{u}\hat{x}(u=x\unicode{x2640} y)\quad \text{Df}\)</p> + +<p class="nind"><b>*38·03.</b> \(\alpha \unicode{x2640}_{,,}y=\unicode{x2640} yʻʻ\alpha\quad \text{Df}\)</p> + +<p class="nind"><b>*38·1.</b> \(\vdash :u(x\unicode{x2640} )y\ldotp \equiv \ldotp u=x\unicode{x2640} y\quad [\text{(*38·01)}]\)</p> + +<p><span class="pagenum" id="Page_313">[Pg 313]</span></p> + +<p class="nind"><b>*38·101.</b> \(\vdash:u(\unicode{x2640}y)x.\equiv.u=x\unicode{x2640}y \quad[\text{(*38·02)}]\)</p> + +<p class="nind"><b>*38·11.</b> \(\vdash.x\unicode{x2640}ʻy=\unicode{x2640}yʻx=x\unicode{x2640}y \quad[\text{*38·1·101.*30·3}]\)</p> + +<p class="nind"><b>*38·12.</b> \(\vdash.\text{E}!x\unicode{x2640}ʻy.\text{E}!\unicode{x2640}yʻx \quad[\text{*38·11.*14·21}]\)</p> + +<p class="nind"><b>*38·13.</b> \(\vdash:u\in x\unicode{x2640}ʻʻ\alpha.\equiv.(\exists y).y\in \alpha.u=x\unicode{x2640}y \quad[\text{*38·1.*37·1}]\)</p> + +<p class="nind"><b>*38·131.</b> \(\vdash:u\in \unicode{x2640}yʻʻ\alpha.=.(\exists x).x\in \alpha.u=x\unicode{x2640}y \quad[\text{*38·101.*37·1}]\)</p> + +<p class="nind"><b>*38·2.</b> \(\vdash.\alpha\unicode{x2640}_{,,}y=\unicode{x2640}yʻʻ\alpha \quad[\text{(*38·03)}]\)</p> + +<p class="nind"><b>*38·21.</b> \(\vdash.\alpha\unicode{x2640}_{,,}y=\hat{u}\{(\exists x).x\in \alpha.u=x\unicode{x2640}y\} \quad[\text{*38·2·131}]\)</p> + +<p class="nind"><b>*38·22.</b> \(\vdash.\alpha\unicode{x2640}_{,,}ʻy=\unicode{x2640}_{,,}yʻ\alpha=\alpha\unicode{x2640}_{,,}y \quad[\text{*38·11}]\)</p> + +<p class="nind"><b>*38·23.</b> \(\vdash.\text{E}!\alpha\unicode{x2640}_{,,}ʻy.\text{E}!\unicode{x2640}_{,,}yʻ\alpha \quad[\text{*38·22.*14·21}]\)</p> + +<p class="nind"><b>*38·24.</b> \(\vdash:\exists !\alpha\unicode{x2640}_{,,}y.\equiv.\exists !\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*38·2.*37·29.Transp}.&\supset\vdash:\exists !\alpha\unicode{x2640}_{,,}y.\supset.\exists !\alpha &\qquad \text{(1)}\\ +\vdash.\text{*38·21}. &\supset\vdash:x\in \alpha.\supset.(x\unicode{x2640}y)\in \alpha\unicode{x2640}_{,,}y.\\ +[\text{*10·24}] &\supset.\exists !\alpha\unicode{x2640}_{,,}y &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*38·3.</b> \[\begin{align}&\vdash.\alpha\unicode{x2640}_{,,}ʻʻ\beta=\hat{\gamma}\{(\exists y).y\in \beta.\gamma=\alpha\unicode{x2640}_{,,}\gamma\}=\hat{\gamma}\{(\exists + y).\gamma\in \beta.\gamma=\unicode{x2640}yʻʻ\alpha\}\\ +&[\text{*38·13·2}]\end{align}\]</p> + +<p class="nind"><b>*38·31.</b> \[\begin{align}&\vdash.\unicode{x2640}_{,,}yʻʻ\kappa=\hat{\gamma}\{(\exists \alpha).\alpha\in \kappa.\gamma=\alpha\unicode{x2640}_{,,}y\}=\hat{\gamma}\{(\exists + \alpha).\alpha\in \kappa.\gamma=\unicode{x2640}yʻʻ\alpha\}=\unicode{x2640}yʻʻʻ\kappa\\ +&[\text{*38·131·2.*37·103}]\end{align}\]</p> +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_314">[Pg 314]</span></p> + +<h2 class="nobreak" id="NOTE_TO_SECTION_D">NOTE TO SECTION D.</h2> +</div> + + +<p><i>General Observations on Relations.</i> The notion of "relation" +is so general that it is important to realize the different sorts of +relations to which the notations defined in the preceding section may +be applied. It often happens that a proposition which holds for any +relation is only important for relations of certain kinds; hence it is +desirable that the reader should have in mind some of the principal +kinds of relations. Of the various uses to which different sorts of +relations may be put, there are three which are specially important, +namely (1) to give rise to descriptive functions, (2) to establish +correlations between different classes, (3) to generate series. Let us +consider these in succession.</p> + +<p>(1) In order that a relation \(R\) may give rise to a descriptive +function, it must be such that the referent is unique when the +relatum is given. Thus, for example, the relations \(\text{Cnv}\), +\(\overrightarrow{R}\), \(\overleftarrow{R}\), \(\text{D}\), +\(\text{ᗡ}\), \(C\), \(R_{\in}\), defined above, all give rise to +descriptive functions. In general, if \(R\) gives rise to a descriptive +function, there will be a certain class, namely \(\text{ᗡ}ʻR\), to +which the argument of the function must belong in order that the +function may have a value for that argument. For example, taking +the sine as an illustration, and writing "\({\sin}ʻy\)" instead of +"\(\sin y\)," \(y\) must be a number in order that \({\sin}ʻy\) may +exist. Then \(\breve{\sin}\) is the relation of \(y\) to \(x\) when +\(x = {\sin}ʻy\). If we put \(\alpha =\) numbers between \(-\pi/2\) +and \(\pi/2\), both included, \(\sin \upharpoonright \alpha\) will +be the relation of \(x\) to \(y\) when \(x = {\sin}ʻy\) and \(-\pi/2\le y \le \pi/2\). +The converse of this relation, which is \(\alpha \upharpoonleft \breve{\sin}\), +will also give rise to a descriptive function; thus +\((\alpha \upharpoonleft \breve{\sin})ʻx =\) that +value of \(\sin^{-1}x\) which lies between \(-\pi/2\) and \(\pi/2\). +This illustrates a case which arises very frequently, namely, that +a relation \(R\) does not, as it stands, give rise to a descriptive +function, but does do so when its domain or converse domain is suitably +limited. Thus for example the relation "parent" does not give rise to +a descriptive function, but does do so when its domain is limited to +males or limited to females. The relation "square root," similarly, +gives rise to a descriptive function when its domain is limited +to positive numbers, or limited to negative numbers. The relation +"wife" gives rise to a descriptive function when its converse domain +is limited to Christian men, but not when Mohammedans are included. +The domain<span class="pagenum" id="Page_315">[Pg 315]</span> of a relation which gives rise to a descriptive function +without limiting its domain or converse domain consists of all +possible values of the function; the converse domain consists of all +possible arguments to the function. Again, if \(R\) gives rise to a +descriptive function, \(\overleftarrow{R}ʻx\) will be the class of +those arguments for which the value of the function is \(x\). Thus +\(\overleftarrow{\sin}ʻx\) consists of all numbers whose sine is \(x\), +<i>i.e.</i> all values of \(\sin^{-1}x\). Again, \(\sinʻʻ\alpha\) will +be the sines of the various members of \(\alpha\). If \(\alpha\) is a +class of numbers, then, by the notation of *38, \(2\timesʻʻ\alpha\) +will be the doubles of those numbers, \(3\timesʻʻ\alpha\) the trebles +of them, and so on. To take another illustration, let \(\alpha\) be a +pencil of lines, and let \(Rʻx\) be the intersection of a line \(x\) +with a given transversal. Then \(Rʻʻ\alpha\) will be the intersections +of lines belonging to the pencil with the transversal.</p> + +<p>(2) Relations which establish a correlation between two classes are +really a particular case of relations giving rise to descriptive +functions, namely the case in which the converse relation also +gives rise to a descriptive function. In this case, the relation is +"one-one," <i>i.e.</i> given the referent, the relatum is determinate, +and vice versa. A relation which is to be conceived as a correlation +will generally be denoted by \(S\) or \(T\). In such cases, we are +as a rule less interested in the particular terms \(x\) and \(y\) +for which \(xRy\), than in classes of such terms. We generally, in +such cases, have some class \(\beta\) contained in the converse +domain of our relation \(S\), and we have a class \(\alpha\) such +that \(\alpha=Sʻʻ\beta\). In this case, the relation \(S\) correlates +the members of \(\alpha\) and the members of \(\beta\). We shall +have also \(\beta=\breve{S}ʻʻ\alpha\), so that, for such a relation, +the correlation is reciprocal. Such relations are fundamental in +arithmetic, since they are used in defining what is meant by saying +that two classes (or series) have the same cardinal (or ordinal) number +of terms.</p> + +<p><span class="pagenum" id="Page_316">[Pg 316]</span></p> + +<p>(3) Relations which give rise to series will in general be denoted +by \(P\) or \(Q\), and in propositions whose chief importance +lies in their application to series we shall also, as a rule, +denote a variable relation by \(P\) or \(Q\). When \(P\) is used, +it may be read as "precedes." Then \(\breve{P}\) may be read +"follows," \(\overrightarrow{P}ʻx\) may be read "predecessors of +\(x\)," \(\overleftarrow{P}ʻx\) may be read "followers of \(x\)." +\(\text{D}ʻP\) will be all members of the series generated by \(P\) +except the last (if any), \(\text{ᗡ}ʻP\) will be all members of the +series except the first (if any), \(CʻP\) will be all the members of +the series. \(Pʻʻ\alpha\) will consist of all terms preceding some +member of \(\alpha\). Suppose, for example, that our series is the +series of real numbers, and that \(\alpha\) is the class of members of +an ascending series \(x_{1}\), \(x_{2}\), \(x_{3}\), ... \(x_{\nu}\), ... +Then \(Pʻʻ\alpha\) will be the segment of the real numbers defined +by this series, <i>i.e.</i> it will be all the predecessors of the +limit of the series. (In the event of the series \(x_{1}\), \(x_{2}\), +\(x_{3}\), ... \(x_{\nu}\), ... growing without limit, \(Pʻʻ\alpha\) +will be the whole series of real numbers.)</p> + +<p>It very often happens that a relation has more or less of a serial +character, without having all the characteristics necessary for +generating series. Take, for example, the relation of son to father. +It is obvious that by means of this relation series can be generated +which start from any man and end with Adam. But these series are not +the field of the relation in question; moreover this relation is not +<i>transitive</i>, <i>i.e.</i> a son of a son of \(x\) is not a son of +\(x\). If, however, we substitute for "son" the relation "descendant +in the direct male line" (which can be defined in terms of "son" by +the method explained in <a href="#*90">*90</a> and <a href="#*91">*91</a>), and if we limit the converse +domain of this relation to ancestors of \(x\) in the direct male line, +we obtain a new relation which <i>is</i> serial, and has for its +field \(x\) and all his ancestors in the direct male line. Again, one +relation may generate a number of series, as for example the relation +"\(x\) is east of \(y\)." If \(x\) and \(y\) are points on the earth's +surface, and in the eastern hemisphere, this relation generates one +series for every parallel of latitude. By confining the field of the +relation further to one parallel of latitude, we obtain a relation +which generates a series. (The reason for confining \(x\) and \(y\) +to one hemisphere is to insure that the relation shall be transitive, +since otherwise we might have \(x\) east of \(y\) and \(y\) east of +\(z\), but \(x\) west of \(z\).)</p> + +<p>A relation may have the characteristics of all the three kinds of +relations, provided we include in the third kind all those which +lead to series by some such limitations as those just described. For +example, the relation \(+1\), <i>i.e.</i> (in virtue of the notation of +<a href="#*38">*38</a>) the relation of \(x+1\) to \(x\), where \(x\) is supposed to be +a finite cardinal integer, has the characteristics of all three kinds +of relations. In the first place, it leads to the descriptive function +(\(+1)ʻx\), <i>i.e.</i> \(x+1\). In the second place, it correlates +with any class \(\alpha\) of numbers the class obtained by adding +1 to each member of \(\alpha\), <i>i.e.</i> (\(+1)ʻʻ\alpha\). This +correlation may be used to prove that the number of finite integers is +infinite (in one of the two senses of the word "infinite"); for if we +take as our class \(\alpha\) all the natural numbers including 0, the +class (\(+1)ʻʻ\alpha\) consists of all the natural numbers except 0, so +that the natural numbers can be correlated with a proper part<a id="FNanchor_58" href="#Footnote_58" class="fnanchor">[58]</a> of +themselves. Again the relation \(+1\) may be used, like that of father +to son, to generate a series, namely the usual series of the natural +numbers in order of magnitude, in which each has to its immediate +predecessor the relation \(+1\). Thus this relation partakes of the +characteristics of all three kinds of relations.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_58" href="#FNanchor_58" class="label">[58]</a> +<i>I.e.</i> a part not the whole. On this definition of +infinity, see *124.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_317">[Pg 317]</span></p> +<h2 class="nobreak" id="SECTION_E_a">SECTION E.<br> +PRODUCTS AND SUMS OF CLASSES.</h2> +</div> + +<p><i>Summary of Section E.</i></p> + + +<p>In the present section, we make an extension of \(\alpha\cap\beta\), +\(\alpha\cup\beta\), \(R\dot{\cap} S\), \(R\unicode{x228d}S\). Given +a class of classes, say \(\kappa\), the product of \(\kappa\) (which +is denoted by \(pʻ\kappa\) is the common part of all the members of +\(\kappa\), <i>i.e.</i> the class consisting of those terms which +belong to every member of \(\kappa\). The definition is +\[ +pʻ\kappa =\hat{x}(\alpha \in \kappa .\supset_\alpha .x\in \alpha ) \quad\text{Df}. +\] +If \(\kappa\) has only two members, \(\alpha\) and \(\beta\) +say, \(pʻ\kappa = \alpha\cap\beta\). If \(\kappa\) has three +members, \(\alpha\), \(\beta\), \(\gamma\), then \(pʻ\kappa =\alpha\cap\beta\cap\gamma\); +and so on. But this process can only +be continued to a finite number of terms, whereas the definition of +\(pʻ\kappa\) does not require that \(\kappa\) should be finite. This +notion is chiefly important in connection with the lower limits of +series. For example, let \(\lambda\) be the class of rational numbers +whose square is greater than 2, and let "\(xMy\)" mean "\(x \lt y\), +where \(x\) and \(y\) are rationals." Then if \(x\in\lambda\), +\(\overrightarrow{M}ʻx\) will be the class of rationals less than +x. Thus \(\overrightarrow{M}ʻʻ\lambda\) will be the class of such +classes as \(\overrightarrow{M}ʻʻx\), where \(x\in\lambda\). Thus +the product of \(\overrightarrow{M}ʻʻ\lambda\), which we call +\(pʻ\overrightarrow{M}ʻʻ\lambda\), will be the class of rationals +which are less than every member of \(\lambda\), <i>i.e.</i> the +class of rationals whose squares are less than 2. Each member of +\(\overrightarrow{M}ʻʻ\lambda\) is a segment of the series of +rationals, and \(pʻ\overrightarrow{M}ʻʻ\lambda\) is the lower limit of +these segments. It is thus that we prove the existence of lower limits +of series of segments.</p> + +<p>Similarly the <i>sum</i> of a class of classes \(\kappa\) is defined as +the class consisting of all terms belonging to <i>some</i> member of +\(\kappa\); <i>i.e.</i> +\[ +sʻ\kappa =\hat{x}\{(\exists \alpha ).\alpha \in \kappa .x\in \alpha\} \quad\text{Df}, +\] +<i>i.e.</i> \(x\) belongs to the sum of \(\kappa\) if \(x\) belongs +to some \(\kappa\). This notion plays the same part for upper limits +of series of segments as \(pʻ\kappa\) plays for lower limits. It has, +however, many more other uses than \(pʻ\kappa\), and is altogether +a more important conception. Thus in cardinal arithmetic, if no two +members of \(\kappa\) have any term in common, the arithmetical sum of +the numbers of members possessed by the various members of \(\kappa\) +is the number of members possessed by \(sʻ\kappa\).</p> + +<p><span class="pagenum" id="Page_318">[Pg 318]</span></p> + +<p>The product of a class of relations (\(\lambda\) say) is the relation +which holds between \(x\) and \(y\) when \(x\) and \(y\) have every +relation of the class \(\lambda\). The definition is +\[ +\dot{p}ʻ\lambda = \hat{x}\hat{y} (R\in\lambda . \supset_{R} . xRy) \quad\text{Df}. +\] +The properties of \(\dot{p}ʻ\lambda\) are analogous to those of +\(pʻ\kappa\), but its uses are fewer.</p> + +<p>The sum of a class of relations (\(\lambda\) say) is the relation which +holds between \(x\) and \(y\) whenever there is a relation of the class +\(\lambda\) which holds between \(x\) and \(y\). The definition is +\[ +\dot{s}ʻ\lambda =\hat{x}\hat{y} \{(\exists R) . R\in\lambda . xRy\} \quad\text{Df}. +\] +This conception, though less important than \(sʻ\kappa\), is more +important than \(\dot{p}ʻ\lambda\). The summation of series and ordinal +numbers depends upon it, though the connection is less immediate than +that of the summation of cardinal numbers with \(sʻ\kappa\).</p> + +<p>Instead of defining \(pʻ\kappa\), \(sʻ\kappa\), \(\dot{p}ʻ\lambda\), +\(\dot{s}ʻ\lambda\), it would be formally more correct to define \(p\), +\(s\), \(\dot{p}\) and \(\dot{s}\), which are the relations giving rise +to the above descriptive functions. Thus we should have +\[ +p = \hat{\beta}\hat{\kappa}\{\beta=\hat{x}(\alpha \in \kappa . \supset_{\alpha} . x \in \alpha)\} \quad\text{Df}, +\] +whence we should proceed to +\[ +\begin{array}{l} +&\vdash : \beta p\kappa . \equiv .\beta = \hat{x}(\alpha\in\kappa . \supset_{\alpha} . x\in\alpha),\\ +&\vdash . pʻ\kappa = \hat{x}(\alpha\in\kappa . \supset_{\alpha} . x\in\alpha)\\ +\text{and}\qquad &\vdash . \text{E}!pʻ\kappa. +\end{array} +\]</p> + +<p>But in cases where the relation, as opposed to the descriptive +function, is very seldom required, it is simpler and easier to give +the definition of the descriptive function in the first instance. In +such cases, the relation is always tacitly assumed to be also defined; +<i>i.e.</i> when we give a definition of the form +\[ +Rʻx = Sʻx \quad\text{Df}, +\] +where \(S\) is some previously defined relation, we always assume that +this definition is to be regarded as derived from +\[ +R = \hat{u}\hat{x} (u = Sʻx) \quad\text{Df}. +\]</p> + +<p>In addition to products and sums, we deal, in the present section, +with certain properties of the relations \(R \mid\) and \(\mid S\), +the meanings of which result from the notation introduced in <a href="#*38">*38</a>. +Such relations are very useful in arithmetic. The reason for dealing +with them in the present section is that a large proportion of the +propositions to be proved involve sums of classes of classes or +relations.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_319">[Pg 319]</span></p> +<h2 class="nobreak" id="*40">*40. PRODUCTS AND SUMS OF CLASSES OF CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *40.</p> + + +<p>In this number, we introduce the two notations (explained above) +\[ +\begin{aligned} +&pʻ\kappa = \hat{x} (\alpha\in\kappa . \supset_{\alpha} . x\in\alpha) \quad\text{Df}\\ +&sʻ\kappa = \hat{x} \{(\exists\alpha) . \alpha\in\kappa . x\in \alpha\} \quad\text{Df} +\end{aligned} +\] +Both these notions will be found increasingly useful as we proceed, +but \(sʻ\kappa\) remains more useful than \(pʻ\kappa\) throughout. It +is required for the significance of \(pʻ\kappa\) and \(sʻ\kappa\) that +\(\kappa\) should be a class of classes.</p> + +<p>In the present number, the most useful propositions are the following:</p> + +<p class="nind"><b>*40·12.</b> \(\vdash : \alpha \in \kappa . \supset . pʻ\kappa \subset \alpha\)</p> + +<p><i>I.e.</i> the product of \(\kappa\) is contained in every member of \(\kappa\).</p> + +<p class="nind"><b>*40·13.</b> \(\vdash : \alpha \in \kappa . \supset . \alpha \subset sʻ\kappa\)</p> + +<p><i>I.e.</i> every member of \(\kappa\) is contained in the sum of +\(\kappa\).</p> + +<p class="nind"><b>*40·15.</b> \(\vdash \colon\ldotp \beta \subset pʻ\kappa . \equiv : \gamma \in \kappa . \supset_{\gamma} . \beta \subset \gamma\)</p> + +<p><i>I.e.</i> \(\beta\) is contained in the product of \(\kappa\) if +\(\beta\) is contained in every member of \(\kappa\), and vice versa.</p> + +<p class="nind"><b>*40·151.</b> \(\vdash \colon\ldotp sʻ\kappa \subset \beta . \equiv : \gamma \in \kappa . \supset_{\gamma} . \gamma \subset \beta\)</p> + +<p><i>I.e.</i> the sum of \(\kappa\) is contained in \(\beta\) if every +member of \(\kappa\) is contained in \(\beta\), and vice versa.</p> + +<p class="nind"><b>*40·2.</b> \(\vdash : \kappa = \Lambda . \supset . pʻ\kappa = \text{V}\)</p> + +<p><i>I.e.</i> the product of the null-class of classes is the universal +class. This may seem paradoxical at first sight, but it is really not +so. The fewer members \(\kappa\) has, the larger, speaking generally, +\(pʻ\kappa\) becomes. If \(\kappa\) has no members, then \(\kappa\) has +no members to which a given term \(x\) does not belong, and therefore +\(x\) belongs to \(pʻ\kappa\).</p> + +<p class="nind"><b>*40·23.</b> \(\vdash : \exists ! \kappa . \supset . pʻ\kappa \subset sʻ\kappa\)</p> + +<p><i>I.e.</i> unless \(\kappa\) is null, its product is contained in its +sum.</p> + +<p class="nind"><b>*40·38.</b> \(\vdash . Rʻʻsʻ\kappa = sʻRʻʻʻ\kappa\)</p> + +<p>This proposition is very often used in arithmetic. What it states +is as follows: Given a class of classes \(\kappa\), take its sum, +\(sʻ\kappa\), and then consider all the<span class="pagenum" id="Page_320">[Pg 320]</span> terms that have the +relation \(R\) to some member of \(sʻ\kappa\); this gives the class +\(Rʻʻsʻ\kappa\); next, take each separate member of \(\kappa\), say +\(\alpha\), and form the class \(Rʻʻ\alpha\), consisting of all terms +having the relation \(R\) to some member of \(\alpha\). The class of +all such classes as \(Rʻʻ\alpha\), for various \(\alpha\)'s which are +members of \(\kappa\), is \(Rʻʻʻ\kappa\); the sum of this class, by the +above proposition, is the same as \(Rʻʻsʻ\kappa\).</p> + +<p class="nind"><b>*40·4.</b> \(\vdash \colon\ldotp \text{E} ‼ Rʻʻ\beta . \supset . sʻRʻʻ\beta = \hat{x} \{(\exists y) . y\in\beta . x\in Rʻy\}\)</p> + +<p>This proposition requires, for significance, that \(Rʻy\) should always +be a class. The proposition states that, if \(Rʻy\) always exists when +\(y\in\beta\), then the sum of all classes which have the relation +\(R\) to some member of \(\beta\) consists of all members of such +classes as \(Rʻy\), where \(y\in\beta\).</p> + +<p class="nind"><b>*40·5.</b> \(\vdash . sʻ\overrightarrow{R}ʻʻ\beta = Rʻʻ\beta\)</p> + +<p>This proposition results from *40·4 by substituting +\(\overrightarrow{R}\) for \(R\) in that proposition.</p> + +<p class="nind"><b>*40·51.</b> \(\vdash . pʻ\overrightarrow{R}ʻʻ\beta = \hat{x} \{y\in\beta . \supset_{y} . xRy\}\)</p> + +<p>In virtue of <a href="#*40·5">*40·5</a>, \(pʻ\overrightarrow{R}ʻʻ\beta\) is +correlative to \(Rʻʻ\beta\). Thus if \(R\) is a serial relation, +\(pʻ\overrightarrow{R}ʻʻ\beta\) consists of terms preceding the whole +of \(\beta\), and \(Rʻʻ\beta\) consists of terms preceding part of +\(\beta\). If \(\beta\) has a lower limit, it will be the upper limit +or maximum of \(pʻ\overrightarrow{R}ʻʻ\beta\); if \(\beta\) has an upper +limit, it will be the upper limit of \(Rʻʻ\beta\).</p> + +<p class="nind"><b>*40·61.</b> \(\vdash : \exists ! \beta . \supset . pʻ\overrightarrow{R}ʻʻ\beta \subset Rʻʻ\beta . pʻ\overleftarrow{R}ʻʻ\beta \subset \breve{R}ʻʻ\beta\)</p> + +<p>In this proposition the hypothesis is essential, since, if \(\beta =\Lambda\), +\(pʻ\overrightarrow{R}ʻʻ\beta = \text{V}\) and \(Rʻʻ\beta = \Lambda\).</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*40·01">*40·01</a>.</b> \(pʻ\kappa = \hat{x} (\alpha\in\kappa . \supset_{\alpha} . x\in\alpha) \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*40·02">*40·02</a>.</b> \(sʻ\kappa = \hat{x} {(\exists\alpha) . \alpha\in\kappa . x\in\alpha} \quad\text{Df}\)</p> + +<p class="nind"><b>*40·1.</b> \(\vdash \colon\ldotp x\in pʻ\kappa . \equiv : \alpha\in\kappa . \supset_{\alpha} . x\in\alpha \quad[\text{*20·3.(*40·01)}]\)</p> + +<p class="nind"><b>*40·11.</b> \(\vdash : x\in sʻ\kappa . \equiv . (\exists\alpha). \alpha\in\kappa . x\in\alpha \quad[\text{*20·3.(*40·02)}]\)</p> + +<p class="nind"><b>*40·12.</b> \(\vdash : \alpha\in\kappa . \supset . pʻ\kappa \subset \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*40·1.*10·1}. &\supset \vdash \colon\ldotp x\in pʻ\kappa . \supset : \alpha\in\kappa . \subset . x\in\alpha\ldotp\colon\\ +[\text{Comm}] &\supset \vdash \colon\ldotp \alpha\in\kappa . \supset : x\in pʻ\kappa . \supset . x\in\alpha \qquad \text{(1)}\\ +\vdash .\text{(1).*10·11·21.*22·1}. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·13.</b> \(\vdash : \alpha\in\kappa . \supset . \alpha \subset sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*40·11.*10·24}. &\supset \vdash : \alpha\in\kappa . x\in\alpha . \supset . x\in sʻ\kappa :\\ +[\text{Exp}] &\supset \vdash \colon\ldotp \alpha\in\kappa . \supset : x\in\alpha . \supset . x\in sʻ\kappa \qquad \text{(1)}\\ +\vdash. \text{(1).*10·11·21.*22·1}. &\supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_321">[Pg 321]</span></p> + +<p class="nind"><b>*40·14.</b> \(\vdash : \alpha \in \kappa . x \in pʻ\kappa . \supset . x \in \alpha \quad[\text{*40·12 . Imp}]\)</p> + +<p class="nind"><b>*40·141.</b> \(\vdash : \alpha \in \kappa . x \in \alpha . \supset . x \in sʻ\kappa \quad[\text{*40·11 . *10·24}]\)</p> + +<p class="nind"><b>*40·15.</b> \(\vdash \colon\ldotp \beta \subset pʻ\kappa . \equiv : \gamma \in \kappa . \supset_{\gamma} . \beta \subset \gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*40·1}. \supset \vdash \colon\colon \beta \subset pʻ\kappa : &\equiv \colon\ldotp x \in \beta . \supset_{x} : \gamma \in \kappa . \supset_{\gamma} + . x \in \gamma \colon\ldotp\\ +[\text{*11·62}] &\equiv \colon\ldotp (x, \gamma) : x \in \beta . \gamma \in \kappa . \supset . x \in \gamma \colon\ldotp\\ +[\text{*4·3·84.*11·33}] &\equiv \colon\ldotp (x, \gamma) : \gamma \in \kappa . x \in \beta . \supset . x \in \gamma :\\ +[\text{*11·2·62}] &\equiv \colon\ldotp \gamma \in \kappa . \supset_{\gamma} : x \in \beta . \supset_{x} . x \in \gamma \colon\ldotp\\ +[\text{*22·1}] &\equiv \colon\ldotp \gamma \in \kappa . \supset_{\gamma} . \beta \subset \gamma \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·151.</b> \(\vdash \colon\ldotp sʻ\kappa \subset \beta . \equiv : \gamma \in \kappa . \supset_{\gamma} . \gamma \subset \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*40·11}. \supset \vdash \colon\colon sʻ\kappa \subset \beta . &\equiv \colon\ldotp (\exists\gamma) . \gamma \in \kappa . x \in \gamma . \supset_{x} + . x \in \beta \colon\ldotp\\ +[\text{*10·23}] &\equiv \colon\ldotp (\gamma, x) \colon\ldotp \gamma \in \kappa . x \in \gamma . \supset . x \in \beta \colon\ldotp\\ +[\text{*11·62}] &\equiv \colon\ldotp (\gamma) \colon\ldotp \gamma \in \kappa . \supset : (x) : x \in \gamma . \supset . x \in \beta \colon\ldotp\\ +[\text{*22·1}] &\equiv \colon\ldotp \gamma \in \kappa . \supset_{\gamma} : \gamma \subset \beta \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This proposition is frequently used.</p> + +<p class="nind"><b>*40·16.</b> \(\vdash : \kappa \subset \lambda . \supset . pʻ\lambda \subset pʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*10·1}. &\supset \vdash \colon\colon \text{Hp} . \supset \colon\ldotp \gamma \in \kappa . \supset . \gamma \in \lambda \colon\ldotp\\ +[\text{Syll}] &\supset \colon\ldotp \gamma \in \lambda . \supset . x \in \gamma : \supset : \gamma \in \kappa . \supset . x \in \gamma &\qquad \text{(1)}\\ +\vdash .\text{(1) . *10·11·21}. &\supset\\ +\vdash \colon\colon \text{Hp} . &\supset \colon\ldotp (\gamma) \colon\ldotp \gamma \in \lambda . \supset . x \in \gamma : \supset : \gamma \in \kappa . \supset . x \in \gamma \colon\ldotp\\ +[\text{*10·27}] &\supset \colon\ldotp (\gamma) : \gamma \in \lambda . \supset . x \in \gamma : \supset : (\gamma) : \gamma \in \kappa . \supset . x \in \gamma \colon\ldotp\\ +[\text{*40·1}] &\supset \colon\ldotp x \in pʻ\lambda . \supset . x \in pʻ\kappa &\qquad \text{(2)}\\ +\vdash .\text{(2) . *10·11·21}. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·161.</b> \(\vdash : \kappa \subset \lambda . \supset . sʻ\kappa \subset sʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*10·1}.& \supset \vdash \colon\ldotp \text{Hp} . \supset : \gamma \in \kappa . \supset . \gamma \in \lambda :\\ +[\text{Fact}] &\supset : \gamma \in \kappa . x \in \gamma . \supset . \gamma \in \lambda . x \in \gamma:\\ +[\text{*10·11·28}] &\supset : (\exists\gamma) . \gamma \in \kappa . x \in \gamma . \supset . (\exists\gamma) . \gamma \in \lambda . x \in \gamma:\\ +[\text{*40·11}] &\supset : x \in sʻ\kappa . \supset . x \in sʻ\lambda &\qquad \text{(1)}\\ +\vdash .\text{(1).*10·11·21}. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·17.</b> \(\vdash . pʻ\kappa \cup pʻ\lambda \subset pʻ(\kappa \cap \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*22·34}. &\supset \vdash \colon\colon x \in pʻ\kappa \cup pʻ\lambda . \equiv \colon\ldotp x \in pʻ\kappa . \lor . x \in pʻ\lambda \colon\ldotp\\ +[\text{*40·1}] &\equiv \colon\ldotp \gamma \in \kappa . \supset_{\gamma} . x \in \gamma : \lor : \gamma \in \lambda . \supset_{\gamma} . x \in \gamma \colon\ldotp\\ +[\text{*10·41}] &\supset \colon\ldotp (\gamma) \colon\ldotp \gamma \in \kappa . \supset . x \in \gamma : \lor : \gamma \in \lambda . \supset . x \in \gamma \colon\ldotp\\ +[\text{*4·79}] &\supset \colon\ldotp (\gamma) : \gamma \in \kappa . \gamma \in \lambda . \supset . x \in \gamma \colon\ldotp\\ +[\text{*22·33}] &\supset \colon\ldotp (\gamma) : \gamma \in \kappa \cap \lambda . \supset . x \in \gamma \colon\ldotp\\ +[\text{*40·1}] &\supset \colon\ldotp x \in pʻ(\kappa \cap \lambda) &\qquad \text{(1)}\\ +\vdash . \text{(1) . *10·11}. &\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_322">[Pg 322]</span></p> + +<p class="nind"><b>*40·171.</b> \(\vdash.sʻ\kappa\cup sʻ\lambda=sʻ(\kappa\cup\lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·34}.&\supset\vdash\colon\colon x\in sʻ\kappa\cup sʻ\lambda.\equiv\colon\ldotp x\in sʻ\kappa.\lor.x\in sʻ\lambda\colon\ldotp\\ +[\text{*40·11}] &\equiv\colon\ldotp(\exists\gamma).\gamma\in\kappa.x\in\gamma:\lor:(\exists\gamma).\gamma\in\lambda.x\in\gamma\colon\ldotp\\ +[\text{*10·42}] &\equiv\colon\ldotp(\exists\gamma):\gamma\in\kappa.x\in \gamma.\lor.\gamma\in\lambda.x\in\gamma\colon\ldotp\\ +[\text{*4·4}] &\equiv\colon\ldotp(\exists\gamma)\colon\ldotp\gamma\in\kappa.\lor.\gamma\in\lambda:x\in\gamma\colon\ldotp\\ +[\text{*22·34}] &\equiv\colon\ldotp(\exists\gamma).\gamma\in\kappa\cup\lambda.x\in\gamma\colon\ldotp\\ +[\text{*40·11}] &\equiv\colon\ldotp x\in sʻ(\kappa\cup\lambda)\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·18.</b> \(\vdash.pʻ(\kappa\cup\lambda)=pʻ\kappa\cap pʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·1}.\supset\vdash\colon\colon &x\in pʻ(\kappa\cup\lambda).\equiv\colon\ldotp\gamma\in\kappa\cup\lambda.\supset_{\gamma}.x\in\gamma\colon\ldotp\\ +[\text{*22·34}] &\equiv\colon\ldotp(\gamma)\colon\ldotp\gamma\in\kappa.\lor.y\in\lambda:\supset.x\in\gamma\colon\ldotp\\ +[\text{*4·77}] &\equiv\colon\ldotp(\gamma)\colon\ldotp\gamma\in\kappa.\supset.x\in\gamma:\gamma\in\lambda.\supset.x\in\gamma\colon\ldotp\\ +[\text{*10·22·221}]&\equiv\colon\ldotp(\gamma):\gamma\in\kappa.\supset.x\in\gamma\colon\ldotp(\gamma):\gamma\in\lambda.\supset.x\in\gamma\colon\ldotp\\ +[\text{*40·1}] &\equiv\colon\ldotp x\in pʻ\kappa.x\in pʻ\lambda\colon\ldotp\\ +[\text{*22·33}] &\equiv\colon\ldotp x\in pʻ\kappa\cap pʻ\lambda\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·181.</b> \(\vdash.sʻ(\kappa\cap\lambda)\subset sʻ\kappa\cap sʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·11}.&\supset\vdash\colon\colon x\in sʻ(\kappa\cap\lambda).\equiv\colon\ldotp(\exists\gamma).\gamma\in\kappa\cap\lambda.x\in\gamma\colon\ldotp\\ +[\text{*22·33}] &\equiv\colon\ldotp(\exists\gamma).\gamma\in\kappa.\gamma\in\lambda.x\in\gamma\colon\ldotp\\ +[\text{*10·5}] &\supset\colon\ldotp(\exists\gamma).\gamma\in\kappa.x\in\gamma:(\exists\gamma).\gamma\in\lambda.x\in\gamma\colon\ldotp\\ +[\text{*40·11.*22·33}] &\supset\colon\ldotp x\in sʻ\kappa\cap sʻ\lambda\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·19.</b> \(\vdash\colon\colon x\in sʻ\kappa.\equiv\colon\ldotp\gamma\in\kappa.\supset_{\gamma}.\gamma\subset\beta:\supset_{\beta}.x\in\beta\)</p> + +<p>This proposition is the extension of <a href="#*22·6">*22·6</a>.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·151}.&\supset\\ +&\vdash\colon\colon\gamma\in\kappa.\supset_{\gamma}.\gamma\subset\beta:\supset_{\beta}.x\in\beta\colon\ldotp\equiv\colon\ldotp sʻ\kappa\subset\beta.\supset_{\beta}.x\in\beta + &\qquad \text{(1)}\\ +\vdash.\text{*10·1}. &\supset\vdash\colon\ldotp sʻ\kappa\subset\beta.\supset_{\beta}.x\in\beta:\supset:sʻ\kappa\subset sʻ\kappa.\supset.x\in sʻ\kappa:\\ +[\text{*22·42}] & \supset:x\in sʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{*22·46}. &\supset\vdash\colon\ldotp x\in sʻ\kappa.sʻ\kappa\subset\beta.\supset.x\in\beta\colon\ldotp\\ +[\text{Exp}] & \supset\vdash\colon\ldotp x\in sʻ\kappa.\supset:sʻ\kappa\subset\beta.\supset.x\in\beta\colon\ldotp\\ +[\text{*10·11·21}]&\supset\vdash\colon\ldotp x\in sʻ\kappa.\supset:sʻ\kappa\subset\beta.\supset_{\beta}.x\in\beta &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.& \supset\vdash\colon\ldotp sʻ\kappa\subset\beta.\supset_{\beta}.x\in\beta:\equiv.x\in sʻ\kappa &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_323">[Pg 323]</span></p> + +<p class="nind"><b>*40·2</b>. \(\vdash\colon\kappa=\Lambda.\supset.pʻ\kappa=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*24·5·51}. &\supset\vdash\colon\ldotp\text{Hp}.\supset\colon{\sim}(\exists\alpha).\alpha\in\kappa\colon\\ +&[\text{*10·53}] &\supset\colon(\alpha)\colon\alpha\in\kappa.\supset.x\in\alpha\colon\\ +&[\text{*40·1}] &\supset\colon x\in pʻ\kappa \qquad\qquad\qquad\quad \text{(1)}\\ +&\vdash.\text{(1).*10·11·21}.&\supset\vdash\colon\text{Hp}.\supset.(x).x\in pʻ\kappa.\\ +&[\text{*24·14}] &\supset.pʻ\kappa=V:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·21.</b> \(\vdash\colon\kappa=\Lambda.\supset.sʻ\kappa=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*24·51}. &\supset\vdash\colon\text{Hp}.\supset.{\sim}(\exists \alpha).\alpha\in\kappa.\\ +&[\text{*10·5.Transp}] &\supset.{\sim}(\exists \alpha).\alpha\in\kappa.x\in\alpha.\\ +&[\text{*40·11.Transp}] &\supset.x{\sim}\in sʻ\kappa \qquad\qquad\qquad \text{(1)}\\ +&\vdash.\text{(1).*10·11·21}.&\supset\vdash\colon\text{Hp}.\supset.(x).x{\sim}\in sʻ\kappa.\\ +&[\text{*24·15}] &\supset.sʻ\kappa=\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition, the two \(\Lambda\)'s are of different types, +since \(\kappa\) is of the type next above that of \(sʻ\kappa\). Thus +it would be more correct to write +\[ +\vdash\colon\kappa=\Lambda\cap\text{Cls}.\supset.sʻ\kappa=\Lambda\cap\text{V}. +\]</p> + +<p>But in the case of \(\Lambda\) it is not very important to keep the +types distinct.</p> + +<p class="nind"><b>*40·22.</b> \(\vdash\colon\colon\Lambda\in\kappa.\supset.pʻ\kappa=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*40·12}.\supset\vdash\colon\text{Hp}.&\supset.pʻ\kappa\subset\Lambda.\\ +&[\text{*24·13}] &\supset.pʻ\kappa=\Lambda\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In this proposition, the two \(\Lambda\)'s are of the same type.</p> + +<p class="nind"><b>*40·221.</b> \(\vdash\colon=\text{V}\in\kappa.\supset.sʻ\kappa=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*40·13}.\supset\vdash\colon\text{Hp}.&\supset.\text{V}\subset sʻ\kappa.\\ +&[\text{*24·141}] &\supset.sʻ\kappa=\text{V}\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Observe that the hypothesis \(\exists!\kappa\) is essential to this +proposition, since when \(\kappa=\Lambda\), \(pʻ\kappa=\text{V}\) and +\(sʻ\kappa=\Lambda\). Thus +\[ +\vdash\colon\exists\!\kappa.\equiv.pʻ\kappa\subset sʻ\kappa. +\]</p> + +<p class="nind"><b>*40·23.</b> \(\vdash\colon\exists!\kappa.\supset.pʻ\kappa\subset sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*40·12·13}.&\supset\vdash\colon\alpha\in\kappa.\supset.pʻ\kappa\subset\alpha.\alpha\subset sʻ\kappa.\\ +&[\text{*22·44}] &\supset.pʻ\kappa\subset sʻ\kappa\colon\\ +&[\text{*10·11·23}] &\supset\vdash\colon(\exists \alpha).\alpha\in\kappa.\supset.pʻ\kappa\subset sʻ\kappa\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Observe that the hypothesis \(\exists!\kappa\) is essential to this +proposition, since when \(\kappa=\Lambda\), \(pʻ\kappa=\text{V}\) and +\(sʻ\kappa=\Lambda\). Thus +\[ +\vdash\colon\exists\!\kappa.\equiv.pʻ\kappa\subset sʻ\kappa. +\]</p> + +<p><span class="pagenum" id="Page_324">[Pg 324]</span></p> + +<p class="nind"><b>*40·24.</b> \(\vdash\colon\ldotp\exists!\kappa\colon\gamma\in\kappa.\supset_{\gamma}.\beta\subset\gamma\colon\supset.\beta\subsetʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*40·15}. \supset\vdash\colon\ldotp\gamma\in\kappa.&\supset_{\gamma}.\beta\subset\gamma\colon\supset.\beta\subset pʻ\kappa &\qquad \text{(1)}\\ +&\vdash.\text{*40·23}. \supset\vdash\colon\exists!\kappa.&\supset.pʻ\kappa\subset sʻ\kappa &\qquad \text{(2)}\\ +&\vdash.\text{(1).(2)}.\supset\vdash\colon\text{Hp}.&\supset.\beta\subset pʻ\kappa.pʻ\kappa\subset sʻ\kappa.\\ +&[\text{*22·44}] &\supset.\beta\subset sʻ\kappa\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the proof of *215·25.</p> + +<p class="nind"><b>*40·25.</b> \(\vdash\colon x\in sʻ\kappa.\equiv.\exists!\kappa\cap\hat{a}(x \in\alpha)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*22·33} \supset\vdash\colon\exists!\kappa\cap\hat{a}(x\in\alpha).&\equiv.(\exists \gamma).\gamma\in\kappa.\gamma\in\hat{a}(x \in\alpha).\\ +&[\text{*20·3}] &\equiv.(\exists \gamma).\gamma\in\kappa.\gamma\in\gamma.\\ +&[\text{*40·11}] &\equiv.x\in sʻ\kappa\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·26.</b> \(\vdash\colon\exists!sʻ\kappa.\equiv.(\exists \alpha).\alpha\in\kappa.\exists!\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*40·11}.\supset\vdash\colon\ldotp\exists!sʻ\kappa.&\equiv\colon(\exists x)\colon(\exists \alpha).\alpha\in\kappa.x\in\alpha\colon\\ +&[\text{*11·23·55}] &\equiv\colon(\exists \alpha)\colon\alpha\in\kappa\colon(\exists x).x\in\alpha\colon\\ +&[\text{*24·5}] &\equiv\colon(\exists \alpha)\colon\alpha\in\kappa\colon\exists!\alpha\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in the proof of *216·51.</p> + +<p class="nind"><b>*40·27.</b> \(\vdash\colon\ldotp \alpha\cap sʻ\kappa=\Lambda.\equiv\colon \gamma\in\kappa.\supset_{\gamma}.\alpha\cap\gamma=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*24·311}.\supset\\ +&\vdash\colon\colon\alpha\cap sʻ\kappa=\Lambda.&\equiv\colon\ldotp sʻ\kappa\subset - \alpha\colon\ldotp\\ +&[\text{*22·1·35}]&\equiv\colon\ldotp x\in sʻ\kappa.\supset_{x}.x{\sim}\in\alpha\colon\ldotp\\ +&[\text{*40·1}] &\equiv\colon\ldotp(\exists\gamma).\gamma\in\kappa.x\in\gamma.\supset_{x}.x{\sim}\in\alpha\colon\ldotp\\ +&[\text{*10·23}] &\equiv\colon\ldotp \gamma\in\kappa.x\in\gamma.\supset_{x,\gamma}.x{\sim}\in\alpha\colon\ldotp\\ +&[\text{*11·2·62}]&\equiv\colon\ldotp \gamma\in\kappa.\supset_{\gamma}\colon x\in\gamma.\supset_{x}.x{\sim}\in\alpha\colon\ldotp\\ +&[\text{*24·39}] &\equiv\colon\ldotp \gamma\in\kappa.\supset_{\gamma}.\alpha\cap\gamma=\lor\colon\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are only significant when \(R\) is +a relation whose domain consists of classes, for they concern +\(pʻRʻʻ\alpha\) or \(sʻRʻʻ\alpha\), and therefore require that +\(Rʻʻ\alpha\) should be a class of classes.</p> + +<p class="nind"><b><a id="*40·3">*40·3</a>.</b> \(\vdash.pʻRʻʻ(\alpha\cup\beta)=pʻRʻʻ\alpha\cap pʻRʻʻ\beta \quad[\text{*37·22.*40·18}]\)</p> + +<p class="nind"><b>*40·31.</b> \(\vdash.sʻRʻʻ(\alpha\cup\beta)=sʻRʻʻ\alpha\cup sʻRʻʻ\beta \quad[\text{*37·22.*40·171}]\)</p> + +<p class="nind"><b>*40·32.</b> \(\vdash.pʻRʻʻ\alpha\cup pʻRʻʻ\beta\subset pʻRʻʻ(\alpha\cap\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*37·21}.&\supset\vdash.Rʻʻ(\alpha\cap\beta)\subset Rʻʻ\alpha\cap Rʻʻ\beta.\\ +&[\text{*40·16}] &\supset\vdash.pʻ(Rʻʻ\alpha\cap Rʻʻ\beta)\subset pʻRʻʻ(\alpha\cap\beta) &\qquad \text{(1)}\\ +&\vdash.\text{*40·17}.&\supset\vdash.pʻRʻʻ\alpha\cup pʻRʻʻ\beta\subset pʻ(Rʻʻ\alpha\cup Rʻʻ\beta) &\qquad \text{(2)}\\ +&\vdash.\text{(1).(2).*22·44}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_325">[Pg 325]</span></p> + +<p class="nind"><b>*40·33.</b> \(\vdash . sʻRʻʻ(\alpha \cap \beta) \subset sʻRʻʻ\alpha \cap sʻRʻʻ\beta \quad[\text{*37·21.*40·161.*40·181}]\)</p> + +<p>The following propositions no longer require that the domain of \(R\) +should be composed of classes.</p> + +<p class="nind"><b>*40·35.</b> \(\vdash . pʻRʻʻʻ\kappa = \hat{x}\{\beta \in \kappa . \supset_{\beta} . x \in Rʻʻ\beta\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash .*40·1.\supset \vdash \colon\ldotp x \in pʻRʻʻʻ\kappa &\equiv : \gamma \in Rʻʻʻ\kappa . \supset_{\gamma} . x \in \gamma : \\ +&[\text{*37·103}] &\equiv : (\exists \beta) . \beta \in \kappa . \gamma = Rʻʻ\beta . \supset_{\gamma} . x \in \gamma : \\ +&[\text{*10·23}] &\equiv : \beta \in \kappa . \gamma = Rʻʻ\beta . \supset_{\beta,\gamma} . x \in \gamma : \\ +&[\text{*13·191}] &\equiv : \beta \in \kappa . \supset_{\beta} . x \in Rʻʻ\beta &\qquad \text{(1)} \\ +&\vdash .(1).*10·11.*20·3.\supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·36.</b> \(\vdash . sʻRʻʻʻ\kappa = \hat{x}\{(\exists \beta) . \beta \in \kappa . x \in Rʻʻ\beta \} \quad [\text{Similar proof}]\)</p> + +<p class="nind"><b><a id="*40·37">*40·37</a>.</b> \(\vdash . Rʻʻpʻ\kappa \subset pʻRʻʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash .\text{*37·1}.&\supset \vdash \colon\colon x \in Rʻʻpʻ\kappa \equiv : (\exists y) . y \in pʻ\kappa . xRy : \\ +&[\text{*40·1}] &\equiv\colon\ldotp (\exists y) : \beta \in \kappa . \supset_{\beta} . y \in \beta : xRy : \\ +&[\text{*10·33}] &\equiv\colon\ldotp (\exists y) (\beta) : \beta \in \kappa . \supset . y \in \beta : xRy : \\ +&[\text{*11·26}] &\supset\colon\ldotp (\beta) (\exists y) : \beta \in \kappa . \supset . y \in \beta : xRy : \\ +&[\text{*5·31}] &\supset\colon\ldotp (\beta) (\exists y) : \beta \in \kappa . \supset . y \in \beta . xRy : \\ +&[\text{*10·37}] &\supset\colon\ldotp (\beta) \beta \in \kappa . \supset (\exists y) . y \in \beta . xRy : \\ +&[\text{*37·1}] &\supset\colon\ldotp (\beta) : \beta \in \kappa . \supset . x \in Rʻʻ\beta : \\ +&[\text{*40·35}] &\supset \colon\ldotp x \in pʻRʻʻʻ\kappa \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·38.</b> \(\vdash . Rʻʻsʻ\kappa = sʻRʻʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash .*37·1.\supset \vdash \colon\colon x \in Rʻʻsʻ\kappa &\equiv \colon\ldotp (\exists y) . y \in sʻ\kappa . xRy \colon\ldotp \\ +&[\text{*40·11}] &\equiv \colon\ldotp (\exists y)\colon\ldotp (\exists \alpha) . \alpha \in \kappa . y \in \alpha : xRy \colon\ldotp \\ +&[\text{*11·6}] &\equiv \colon\ldotp (\exists \alpha) \alpha \in \kappa : (\exists y) . y \in \alpha . xRy \colon\ldotp \\ +&[\text{*37·1}] &\equiv \colon\ldotp (\exists \alpha) . \alpha \in \kappa . x \in Rʻʻ\alpha \colon\ldotp \\ +&[\text{*40·36}] &\equiv \colon\ldotp x \in sʻRʻʻʻ\kappa \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>This proposition is frequently used in the proofs of arithmetical +propositions.</p> + +<p class="nind"><b>*40·4.</b> \(\vdash : \text{E} !! Rʻʻ\beta . \supset . sʻRʻʻ\beta = \hat{x}\{ (\exists y) . y \in \beta . x \in Rʻy \}\)</p> + +<p>This proposition is only significant when \(\text{D}ʻR \subset\text{Cls}\).</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash .\text{*37·6}.\supset \vdash : \text{Hp} . \supset . Rʻʻ\beta = \hat{\alpha}\{ (\exists y) . y \in \beta . \alpha = Rʻy \} &\qquad \text{(1)} \\ +&\vdash .\text{(1).*40·11}.\supset \\ +&\vdash \colon\colon \text{Hp} . \supset \colon\ldotp x \in sʻRʻʻ\beta. \equiv : (\exists \alpha) : (\exists y) . y \in \beta . \alpha = Rʻy : x \in \alpha : \\ +&[\text{*11·6}] \equiv : (\exists y) : y \in \beta : (\exists \alpha) . \alpha = Rʻy . x \in \alpha : \\ +&[\text{*14·205}] \equiv : (\exists y) . y \in \beta . x \in Rʻy \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_326">[Pg 326]</span></p> + +<p class="nind"><b>*40·41.</b> \(\vdash : E‼Rʻʻ\beta .\supset. pʻRʻʻ\beta = \hat{x}\{ y \in \beta \supset_{y} x \in Rʻy\} \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*40·42.</b> \(\vdash : (x).Rʻx = Pʻx \cup Qʻx \supset sʻRʻʻ\alpha = sʻ(Pʻʻ\alpha \cup Qʻʻ\alpha) = sʻPʻʻ\alpha \cup sʻQʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash .*14·21. &\supset \vdash : \text{Hp}. \supset . (x). \text{E}!Rʻx. E!Pʻx. E!Qʻx &\qquad \text{(1)}\\ +&\vdash .(1).*40·4. &\supset \vdash : \text{Hp}. \supset . sʻRʻʻ\alpha = \hat{x}\{ (\exists y). y \in \alpha. x \in Rʻy \}\\ +&[\text{Hp}] &= \hat{x}\{ (\exists y) · y \in \alpha. x \in Pʻy \cup Qʻy \}\\ +&[\text{*22·34}] &= \hat{x}\{ (\exists y). y \in \alpha. x \in Pʻy \lor x \in Qʻy \}\\ +&[\text{*4·4.*10·42}] &= \hat{x}\{ (\exists y): y \in \alpha: x \in Pʻy. \lor. (\exists y). y \in \alpha. x \in Qʻy \}\\ +&[\text{(1).*40·4}] &= \hat{x}\{ x \in sʻPʻʻ\alpha \lor x \in sʻQʻʻ\alpha \}\\ +&[\text{*20·42.*22·34}] &= sʻPʻʻ\alpha \cup sʻQʻʻ\alpha\\ +&[\text{*40·171}] &= sʻ(Pʻʻ\alpha \cup Qʻʻ\alpha) \supset \vdash. \text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in <a href="#*40·57">*40·57</a>, where we take \(R = C\), +\(P = \text{D}\), \(Q = \text{ᗡ}\).</p> + +<p class="nind"><b>*40·43.</b> \(\vdash\colon\colon \text{E} ‼ Rʻʻ\beta \supset\colon\ldotp sʻRʻʻ\beta \subset \alpha. \equiv: y \in \beta \supset_{y}. Rʻy \subset \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*37·63}. \supset \vdash\colon\colon \text{Hp}. \supset y \in \beta \supset_{y} Rʻy \subset \alpha: &\equiv: \gamma \in Rʻʻ\beta \supset_{\gamma} + \gamma \subset \alpha\\ +[\text{*40·151}] &\equiv: sʻRʻʻ\beta \subset \alpha\colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·44.</b> \(\vdash \text{E}‼Rʻʻ\beta \supset \; \alpha \subset pʻRʻʻ\beta \equiv y \in \beta \supset_{y} \alpha \subset Rʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*37·63}. \supset \vdash\colon\colon \text{Hp}. \supset\colon\ldotp y \in \beta \supset_{y} \alpha \subset Rʻy: &\equiv: \gamma \in Rʻʻ\beta \supset_{\gamma} + \alpha \subset: \gamma\\ +[\text{*40·15}] &\equiv: \alpha \subset pʻRʻʻ\beta\colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in the proof of *84·44.</p> + +<p class="nind"><b>*40·45.</b> \(\vdash \colon\ldotp y \in \beta \supset_{y} Rʻy \subset Sʻy \supset sʻRʻʻ\beta \subset sʻSʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*14·21}. \supset \vdash\colon\ldotp \text{Hp}. &\supset: \text{E}‼Sʻʻ\beta. \text{E}‼Rʻʻ\beta: &\qquad \text{(1)}\\ +[\text{*37·62.*40·13}] &\supset: y \in \beta \supset_{y}. Sʻy \subset sʻSʻʻ\beta:\\ +[\text{Hp}] &\supset: y \in \beta \supset_{y}. Rʻy \subset sʻSʻʻ\beta\\ +[\text{*40·43.(1)}] &\supset: sʻRʻʻ\beta \subset sʻSʻʻ\beta\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in the proof of <a href="#*94·402">*94·402</a>.</p> + +<p class="nind"><b>*40·451.</b> \(\vdash \colon\ldotp y \in \beta \supset_{y} Rʻy \subset Sʻy \supset pʻRʻʻ\beta \subset pʻSʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*14·21.*37·62.*40·12}. \supset \vdash\colon\ldotp \text{Hp}. \supset: y \in \beta. \supset. pCRʻʻ\beta \subset Rʻy.\\ +[\text{Hp}] \supset. pʻRʻʻ\beta \subset Sʻy.\\ +[\text{*40·44}] \supset: pʻRʻʻ\beta \subset pʻSʻʻ\beta\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_327">[Pg 327]</span></p> + +<p class="nind"><b><a id="*40·5">*40·5</a>.</b> \(\vdash.\, sʻ\overrightarrow{R}ʻʻ\beta = Rʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·12.*40·4}.\supset\vdash. sʻ\overrightarrow{R}ʻʻ\beta +&= \hat{x}\{(\exists y).\, y \in \beta. x \in \overrightarrow{R}ʻy\}\\ +[\text{*32·18}] &= \hat{x}\{(\exists y). y \in \beta. x R y\}\\ +[\text{(*37·01)}] &= Rʻʻ\beta.\supset\vdash.\, \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·51.</b> \(\vdash.\, pʻ\overrightarrow{R}ʻʻ\beta= \hat{x}\{y \in \beta.\, \supset_{y}. x R y\} +\quad [\text{*32·12.*40·41.*32·18}]\)</p> + +<p>\(pʻ\overrightarrow{R}ʻʻ\beta\) is the class of terms each of which +has the relation \(R\) to <i>every</i> member of \(\beta\), just as +\(Rʻʻ\beta\) is the class of terms each of which has the relation +\(R\) to <i>some</i> member of \(\beta\). In the theory of series, +\(pʻ\overrightarrow{R}ʻʻ\beta\) plays an important part, correlative to +that played by \(Rʻʻ\beta\) (which is \(sʻ\overrightarrow{R}ʻʻ\beta\), +by <a href="#*40·5">*40·5</a>). If \(\beta\) is a class contained in a series whose +generating relation is \(R\), then \(pʻ\overrightarrow{R}ʻʻ\beta\) will +be the predecessors of all members of \(\beta\), while \(Rʻʻ\beta\) +will be the predecessors of some \(\beta\).</p> + +<p class="nind"><b>*40·52.</b> \(\vdash. sʻ\overleftarrow{R}ʻʻ\beta = \overline{R}ʻʻ\beta +\quad [\text{Proof as in *40·5}]\)</p> + +<p class="nind"><b>*40·53.</b> \(\vdash.\, pʻ\overleftarrow{R}ʻʻ\beta= \hat{y}\{x \in \beta.\, \supset_{x}. x R y\} +\quad [\text{Proof as in *40·51}]\)</p> + +<p class="nind"><b>*40·54.</b> \(\vdash. pʻ\overrightarrow{R}ʻʻ\beta= \hat{x}(\beta \subset \overleftarrow{R}ʻx) +\quad [\text{*40·51.*32·181}]\)</p> + +<p class="nind"><b>*40·55.</b> \(\vdash. pʻ\overleftarrow{R}ʻʻ\alpha= \hat{y}(\alpha \subset \overrightarrow{R}ʻy) +\quad [\text{*40·53.*32·18}]\)</p> + +<p>From this point onwards to <a href="#*40·69">*40·69</a>, the propositions are inserted on +account of their use in the theory of series.</p> + +<p class="nind"><b>*40·56.</b> \(\vdash.\, sʻCʻʻ\lambda = Fʻʻ\lambda \quad [*33·5.*40·5]\)</p> + +<p>In the above proposition, the conditions of significance require that +\(\lambda\) should be a class of relations.</p> + +<p class="nind"><b><a id="*40·57">*40·57</a>.</b> \(\vdash.\, sʻCʻʻ\lambda = sʻ(\text{D}ʻʻ\lambda \cup \text{ᗡ}ʻʻ\lambda)= sʻDʻʻ\lambda \cup sʻ\text{ᗡ}ʻʻ\lambda +\quad [\text{*40·42.*33·16}]\)</p> + +<p class="nind"><b>*40·6.</b> \(\vdash.\, pʻ\overrightarrow{R}ʻʻ\Lambda = \text{V}. pʻ\overleftarrow{R}ʻʻ\Lambda = \text{V} +\quad [\text{*37·29.*40·2}]\)</p> + +<p class="nind"><b>*40·61.</b> \(\vdash: \exists!\beta. \supset. pʻ\overrightarrow{R}ʻʻ\beta \subset Rʻʻ\beta. +pʻ\overleftarrow{R}ʻʻ\beta \subset \overline{R}ʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·73}. \supset\vdash: \text{Hp}.\supset.\exists!\overrightarrow{R}ʻʻ\beta.\\ +[\text{*40·23}] \supset. pʻ\overrightarrow{R}ʻʻ\beta \subset sʻ\overrightarrow{R}ʻʻ\beta.\\ +[\text{*40·5}] \supset. pʻ\overrightarrow{R}ʻʻ\beta \subset Rʻʻ\beta &\qquad \text{(1)}\\ +\text{Similarly},\qquad \vdash: \text{Hp}.\,\supset. pʻ\overleftarrow{R}ʻʻ\beta \subset \overline{R}ʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_328">[Pg 328]</span></p> + +<p class="nind"><b>*40·62.</b>\[\begin{align}&\vdash:\exists!\beta.\supset.pʻ\overrightarrow{R}ʻʻ\beta\subset CʻR.pʻ\overleftarrow{R}ʻʻ\beta\subset CʻR\\ +&[\text{*40·61.*37·15·16.*33·161}]\end{align}\]</p> + +<p>The two following propositions (*40·63·64) are used in proving <a href="#*40·65">*40·65</a>, +which is used in *204·63.</p> + +<p class="nind"><b>*40·63.</b> \(\vdash:\exists!\beta-\text{ᗡ}ʻR.\supset.pʻ\overrightarrow{R}ʻʻ\beta=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·41.Transp.}\supset\vdash:x{\sim}\in \text{ᗡ}ʻR.\supset.\overrightarrow{R}ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*37·704}.\supset\vdash:x\in\beta.\supset.\overrightarrow{R}ʻx\in\overrightarrow{R}ʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*22·32}.\supset\vdash:x\in\beta-\text{ᗡ}ʻR.\supset.\overrightarrow{R}ʻx\in\overrightarrow{R}ʻʻ\beta.\overrightarrow{R}ʻx=\Lambda.\\ +[\text{*20·57}]\quad\supset.\Lambda\in\overrightarrow{R}ʻʻ\beta.\\ +[\text{*40·22}]\quad\supset.pʻ\overrightarrow{R}ʻʻ\beta=\Lambda &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·23}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*40·64.</b> \(\vdash:\exists!\beta-\text{D}ʻR.\supset.pʻ\overleftarrow{R}ʻʻ\beta=\Lambda +\quad [\text{Proof as in *40·63}]\)</p> + +<p class="nind"><b><a id="*40·65">*40·65</a>.</b> \(\vdash:\exists!\beta-CʻR.\supset.pʻ\overrightarrow{R}ʻʻ\beta=\Lambda.pʻ\overleftarrow{R}ʻʻ\beta=\Lambda +\quad [\text{*40·63·64.*33·16}]\)</p> + +<p class="nind"><b>*40·66.</b> \(\vdash\;\;\alpha\subset pʻ\overrightarrow{R}ʻʻ\beta.\equiv:x\in\alpha.y\in\beta.\supset_{x,y}.xRy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·51}.\supset\vdash\colon\colon \alpha\subset pʻ\overrightarrow{R}ʻʻ\beta. +\equiv\colon\ldotp \alpha\subset \hat{x}(y\in\beta.\supset_{y}.xRy)\colon\ldotp \\ +[\text{*20·3}]\equiv\colon\ldotp x\in\alpha.\supset_{x}:y\in\beta.\supset_{y}.xRy\colon\ldotp \\ +[\text{*11·62}]\equiv\colon\ldotp (x,y) x\in\alpha.y\in\beta.\supset.xRy +\quad\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*40·67">*40·67</a>.</b> \(\vdash\;\;\beta\subset pʻ\overleftarrow{R}ʻʻ\alpha.\equiv:x\in\alpha.y\in\beta.\supset_{x,y}.xRy\equiv.\alpha\subset pʻ\overrightarrow{R}ʻʻ\beta +\quad [\text{Proof as in *40·66}]\)</p> + +<p class="nind"><b>*40·68.</b> \( +\vdash.\alpha\cap pʻ\overleftarrow{P}ʻʻ\alpha\subset\breve{P}ʻʻpʻ\overleftarrow{P}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·53}.\supset\vdash x\in\alpha\cap pʻ\overleftarrow{P}ʻʻ\alpha +\supset:x\in\alpha:y\in\alpha.\supset_{y}.yPx:\\ +[\text{*10·26}]\quad\supset:xPx:y\in\alpha.\supset_{y}.yPx:\\ +[\text{*10·24}]\quad\supset:(\exists z):zPx:y\in\alpha.\supset_{y}.yPz:\\ +[*40·53.*37·105]\quad\supset:x\in\breve{P}ʻʻpʻ\overleftarrow{P}ʻʻ\alpha +\quad\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in the theory of series (*206·2).</p> + +<p class="nind"><b>*40·681.</b> \(\vdash.\alpha\cap pʻ\overrightarrow{P}ʻʻ\alpha\subset Pʻʻpʻ\overrightarrow{P}ʻʻ\alpha +\quad [\text{Proof as in *40·68}]\)</p> + +<p>The following proposition is used in *211·56.</p> + +<p><span class="pagenum" id="Page_329">[Pg 329]</span></p> + +<p class="nind"><b>*40·682.</b> \(\vdash : \exists! \alpha \cap pʻ \overleftarrow{P} ʻʻ \beta. \supset. \beta \subset Pʻʻ \alpha \)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*40·53} . \supset: \vdash\colon\ldotp \text{Hp}. \supset: (\exists x): x \in \alpha y \in \beta \supset_{y} y P x: \\ +[\text{*5·31}] \quad \supset: (\exists x): y \in \beta .\supset_{y}. x \in \alpha . y P x: \\ +[\text{*11·61}] \quad \supset: y \in \beta .\supset_{y}. (\exists x) x \in \alpha . y P x. \\ +[\text{*37·1}] \quad \supset_{y}. y \in Pʻʻ \alpha\colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*40·69">*40·69</a>.</b> \(\vdash : \exists! Cʻ \overrightarrow{P} \cap pʻ \overrightarrow{P} ʻʻ \alpha .\equiv. \dot{\exists}! P . \exists! pʻ \overrightarrow{P} ʻʻ \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*33·24 .*24·561}. \supset \vdash : \exists! Cʻ P \cap pʻ \overleftarrow{P} ʻʻ \alpha .\supset. \exists! P . \exists! pʻ \overrightarrow{P} ʻʻ \alpha &\qquad \text{(1)} \\ +\vdash .\text{*40·62}. \qquad\supset \vdash : \exists! \alpha . \exists! pʻ \overrightarrow{P} ʻʻ \alpha .\supset. \exists! Cʻ P \cap pʻ \overleftarrow{P} ʻʻ \alpha &\qquad \text{(2)} \\ +\vdash .\text{*40·6}. \qquad \supset \vdash: \alpha = \Lambda \supset: Cʻ P \cap pʻ \overleftarrow{P} ʻʻ \alpha = Cʻ P: \\ +[\text{*33·24}] \qquad \supset: \exists! P. \supset. \exists! Cʻ P \cap pʻ \overleftarrow{P} ʻʻ \alpha &\qquad \text{(3)} \\ +\vdash .\text{(2).(3).*4·83}. \supset \vdash: \exists! P . \exists! pʻ \overleftarrow{P}ʻʻ \alpha .\supset. \exists! Cʻ P \cap pʻ \overleftarrow{P} ʻʻ \alpha &\qquad \text{(4)} \\ +\vdash .\text{(1).(4)}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>The above propositions concerning \(pʻ\overrightarrow{P}ʻʻ\beta\) +and \(pʻ\overleftarrow{P}ʻʻ\beta\) of course have analogues for +\(sʻ\overrightarrow{P}ʻʻ\beta\) and \(sʻ\overleftarrow{P}ʻʻ\beta\). +But owing to <a href="#*40·5">*40·5</a>, these analogues are more simply stated as +properties of \(Rʻʻ\beta\) and \(\breve{R}ʻʻ\beta\). Thus, for example, +<a href="#*37·264">*37·264</a> is the analogue of <a href="#*40·67">*40·67</a>. The above propositions concerning +\(pʻ\overrightarrow{P}ʻʻ\beta\) and \(pʻ\overleftarrow{P}ʻʻ\beta\) will +be used in the theory of series, but until we reach that stage they +will seldom be referred to.</p> + +<p class="nind"><b><a id="*40·7">*40·7</a>.</b> \(\vdash . sʻ \alpha \unicode{x2640}_{,,} ʻʻ \beta = \hat{z} \{( \exists x, y ). x \in \alpha . y \in \beta . z = x \, \unicode{x2640} y\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*40·11 .*38·3} \supset\\ +\vdash . sʻ \alpha \unicode{x2640} ʻʻ \beta = \hat{z} \{( \exists \gamma, y ). y \in \beta . \gamma = \unicode{x2640}yʻʻ \alpha . z \in \gamma\} \\ +[\text{*38·131}] \quad = \hat{z} \{( \exists \gamma, x, y). y \in \beta . \gamma = \unicode{x2640} yʻʻ \alpha . x \in \alpha . z = x \unicode{x2640} y \}\\ +[\text{*13·19}] \qquad = \hat{z} \{( \exists x, y). x \in \alpha . y \in \beta . z = x \unicode{x2640} y \} \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_330">[Pg 330]</span></p> + +<p>This proposition is of considerable importance, since it gives +a compact form for the class of all values of the function +\(x\unicode{x2640}y\) obtained by taking \(x\) in the class \(\alpha\) +and \(y\) in the class\(\beta\). Thus, for example, suppose \(\alpha\) +is the class of numbers which are multiples of 3, and \(\beta\) is +the class of numbers which are multiples of 5, and \(x \times y\) +represents the arithmetical product of \(x\) and \(y\), then \(sʻ\alpha\times ʻʻ \beta\) +will be the class of products of multiples of 3 and +multiples of 5, <i>i.e.</i> the class of multiples of 15. Again suppose +\(\alpha\) and \(\beta\) are both classes of relations; then +\(sʻ\alpha\mid_{,,} ʻʻ \beta\) will be all relative products +\(R \mid S\) obtained by choosing \(R\) in the class \(\alpha\) +and \(S\) in the class \(\beta\).</p> + +<p class="nind"><b>*40·71.</b> \(\vdash . sʻ\unicode{x2640}_{,,}yʻʻ\kappa = (sʻ\kappa)\unicode{x2640}_{,,}y = \unicode{x2640}ʻʻ sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{aligned} +&\vdash .\text{*40·38 .*38·31}. &\supset \vdash . sʻ\unicode{x2640}_{,,}yʻʻ\kappa &= \unicode{x2640} yʻʻ sʻ\kappa \\ +&[\text{*38·2}] &&= (sʻ\kappa)\unicode{x2640}_{,,}y . \supset \vdash . \text{Prop} +\end{aligned} +\]</p> + +<p>The hypothesis \(\breve{R}ʻʻ\alpha \subset \alpha\), which appears +in *40·8·81, is one which plays an important part at a later stage. +In the theory of induction (Part II, Section E) it characterizes a +<i>hereditary</i> class, and in the theory of series it characterizes +an <i>upper section</i> (when combined with \(\alpha \subset CʻR\)).</p> + +<p class="nind"><b>*40·8.</b> \(\vdash\colon\ldotp \alpha \epsilon \kappa . \supset_{\alpha}. \breve{R}ʻʻ\alpha \subset \alpha : +\supset . \breve{R}ʻʻ sʻ\kappa \subset sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash .\text{*37·171}. &\supset \vdash\colon\colon \text{Hp} . \supset\colon\ldotp \alpha \in \kappa . \supset_{\alpha} : x \in \alpha . x R y . \supset_{x,y} + . y \in \alpha\colon\ldotp \\ +&[\text{*11·62}] &\supset\colon\ldotp \alpha \in \kappa . x \in \alpha . x R y . \supset_{\alpha,x,y} . y \in \alpha\colon\ldotp \\ +&[\text{*40·13}] &\supset_{\alpha,x,y} . y \in s' \kappa\colon\ldotp \\ +&[\text{*40·11.*10·23}] &\supset\colon\ldotp x \in sʻ \kappa . x R y . \supset_{x,y} . y \in sʻ \kappa \colon\ldotp \\ +&[\text{*37·171}] &\supset\colon\ldotp \breve{R}ʻʻ sʻ \kappa \subset sʻ \kappa \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p>*40·81. \(\vdash\colon\ldotp \alpha \epsilon \kappa . \supset_{\alpha}. \breve{R}ʻʻ \alpha \subset \alpha : \supset \breve{R}ʻʻ pʻ \kappa \subset pʻ \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·171}.\supset \vdash\colon\colon\ldotp\text{Hp}.&\supset\colon\colon\alpha\in\kappa.\supset:x\in\alpha.xRy.\supset.y\in\alpha\colon\colon\\ +[\text{Exp.Comm}] &\supset\colon\colon xRy.\supset\colon\ldotp\alpha\in\kappa.\supset:x\in\alpha.\supset.y\in\alpha\colon\ldotp\\ +[\text{*2·77}] &\supset\colon\ldotp\alpha\in\kappa.\supset.x\in\alpha:\supset:\alpha\in\kappa.\supset.y\in\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21·27}.\supset\\ +&\qquad\vdash\colon\colon\ldotp\text{Hp}.\supset\colon\colon xRy.\supset\colon\ldotp\alpha\in\kappa.\supset_{\alpha}.x\in\alpha:\supset:\alpha\in\kappa.\supset_{\alpha}.y\in\alpha\colon\ldotp\\ +&\qquad\qquad\supset\colon\ldotp x\in pʻ\kappa.\supset.y\in pʻ\kappa\colon\colon\\ +[\text{Imp}] &\qquad\qquad\supset\colon\colon x\in pʻ\kappa.xRy.\supset.y\in pʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(2).*37·171}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_331">[Pg 331]</span></p> +<h2 class="nobreak" id="*41">*41. THE PRODUCT AND SUM OF A CLASS OF RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *41.</p> + +<p>The propositions to be given in this number, down to <a href="#*41·3">*41·3</a> exclusive, are +the analogues of those of <a href="#*40">*40</a>, excluding those from <a href="#*40·3">*40·3</a> onwards, which +have no analogues. Proofs will not be given, in this number, when they are +exactly analogous to those of propositions with the same decimal part in *40. +The smaller importance of \(\dot{p}ʻ\lambda\) and \(\dot{s}ʻ\lambda\), as compared with \(pʻ\lambda\) and \(sʻ\lambda\), is +illustrated by the smaller number of propositions in <a href="#*41">*41</a> as compared +with *40.</p> + +<p>Our definitions are</p> + +<p class="nind"><b>*41·01.</b> \(\dot{p}ʻ\lambda = \hat{x}\hat{y}\{R \in \lambda .\supset_{R}. xRy\} \quad \text{Df}\)</p> + +<p class="nind"><b>*41·02.</b> \(\dot{s}ʻ\lambda = \hat{x}\hat{y}\{(\exists R) . R \in \lambda . xRy\} \quad \text{Df}\)</p> + +<p>Of the propositions preceding *41·3, which are analogues of propositions +in *40, the only two that are frequently used are</p> + +<p class="nind"><b>*41·13.</b> \(\vdash : R \in \lambda .\supset. R \unicode{x2abd} \dot{s}ʻ\lambda\)</p> + +<p class="nind"><b>*41·151.</b> \(\vdash \colon\ldotp \dot{s}ʻ\lambda \unicode{x2abd} S .\equiv: R \in \lambda.\supset_{R} . R \unicode{x2abd} S\)</p> + +<p>Of the remaining propositions of this number, which have no analogues +in *40, the most important are *41·43·44·45, namely +\[ +\text{D}ʻ\dot{s}ʻ\lambda = sʻ\text{D}ʻʻ\lambda, \text{ᗡ}ʻ\dot{s}ʻ\lambda = sʻ\text{ᗡ}ʻʻ\lambda, Cʻ\dot{s}ʻ\lambda = sʻCʻʻ\lambda. +\] +These propositions are constantly required in the theory of selections +(Part II, Section D) and in relation-arithmetic. Most of the other +propositions of this number are used only once or not at all.</p> + +<hr class="tb"> + +<p class="nind"><b>*41·01.</b> \(\dot{p}ʻ\lambda = \hat{x}\hat{y}(R \in \lambda .\supset_{R}. xRy) \quad \text{Df}\)</p> + +<p class="nind"><b>*41.02.</b> \(\dot{s}ʻ\lambda = \hat{x}\hat{y}\{(\exists R).R \in \lambda . xRy\} \quad \text{Df}\)</p> + +<p class="nind"><b>*41·1.</b> \(\vdash \colon\ldotp x(\dot{p}ʻ\lambda)y .\equiv: R \in \lambda .\supset_{R}. xRy\)</p> + +<p class="nind"><b>*41·11.</b> \(\vdash :x(\dot{s}ʻ\lambda)y .\equiv. (\exists R) . R \in \lambda . xRy\)</p> + +<p class="nind"><b>*41·12.</b> \(\vdash : R \in \lambda .\supset. \dot{p}ʻ\lambda \unicode{x2abd} R\)</p> + +<p class="nind"><b>*41·13.</b> \(\vdash : R \in \lambda .\supset. R \unicode{x2abd} \dot{s}ʻ\lambda\)</p> + +<p><span class="pagenum" id="Page_332">[Pg 332]</span></p> + +<p class="nind"><b>*41·14.</b> \(\vdash:R\in \lambda.x(\dot{p}ʻ\lambda)y.\supset.xRy\)</p> + +<p class="nind"><b>*41·141.</b> \(\vdash:R\in \lambda.xRy.\supset.x(\dot{s}ʻ\lambda)y\)</p> + +<p class="nind"><b>*41·15.</b> \(\vdash\colon\ldotp S\unicode{x2abd}\dot{p}ʻ\lambda.\equiv:R\in \lambda.\supset_{R}.S\unicode{x2abd}R\)</p> + +<p class="nind"><b>*41·151.</b> \(\vdash\colon\ldotp \dot{s}ʻ\lambda\unicode{x2abd}S.\equiv:R\in \lambda.\supset_{R}.R\unicode{x2abd}S\)</p> + +<p class="nind"><b>*41·16.</b> \(\vdash:\lambda\subset \mu.\supset.\dot{p}ʻ\mu\unicode{x2abd}\dot{p}ʻ\lambda\)</p> + +<p class="nind"><b>*41·161.</b> \(\vdash:\lambda\subset \mu.\supset.\dot{s}ʻ\lambda\unicode{x2abd}\dot{s}ʻ\mu\)</p> + +<p class="nind"><b>*41·17.</b> \(\vdash.\dot{p}ʻ\lambda\unicode{x228d}\dot{p}ʻ\mu\unicode{x2abd}\dot{p}ʻ(\lambda\cap \mu)\)</p> + +<p class="nind"><b>*41·171.</b> \(\vdash.\dot{s}ʻ\lambda\unicode{x228d}\dot{s}ʻ\mu=\dot{s}ʻ(\lambda\cup \mu)\)</p> + +<p class="nind"><b>*41·18.</b> \(\vdash.\dot{p}ʻ(\lambda\cup \mu)=\dot{p}ʻ\lambda\dot{\cap}\dot{p}ʻ\mu\)</p> + +<p class="nind"><b>*41·181.</b> \(\vdash.\dot{s}ʻ(\lambda\cap \mu)\unicode{x2abd}\dot{s}ʻ\lambda\dot{\cap}\dot{s}ʻ\mu\)</p> + +<p class="nind"><b>*41·19.</b> \(\vdash\colon\colon x(\dot{s}ʻ\lambda)y.\equiv\colon\ldotp R\in \lambda.\supset_{R}.R\unicode{x2abd}S:\supset_{S}.xSy\)</p> + +<p class="nind"><b>*41·2.</b> \(\vdash:\lambda=\Lambda.\supset.\dot{p}ʻ\lambda=\dot{\text{V}}\)</p> + +<p class="nind"><b>*41·21.</b> \(\vdash:\lambda=\Lambda.\supset.\dot{s}ʻ\lambda=\dot{\Lambda}\)</p> + +<p class="nind"><b>*41·22.</b> \(\vdash:\dot{\Lambda}\in \lambda.\supset.\dot{p}ʻ\lambda=\dot{\Lambda}\)</p> + +<p class="nind"><b>*41·221.</b> \(\vdash:\dot{\text{V}}\in \lambda.\supset.\dot{s}ʻ\lambda=\dot{\text{V}}\)</p> + +<p class="nind"><b>*41·23.</b> \(\vdash:\exists !\lambda.\supset.\dot{p}ʻ\lambda\unicode{x2abd}\dot{s}ʻ\lambda\)</p> + +<p class="nind"><b>*41·24.</b> \(\vdash\colon\ldotp \exists !\lambda:R\in \lambda.\supset_{R}.S\unicode{x2abd}R:\supset.S\unicode{x2abd}\dot{s}ʻ\lambda\)</p> + +<p class="nind"><b>*41·25.</b> \(\vdash:x(\dot{s}ʻ\lambda)y.\equiv.\exists !\lambda\cap \hat{R}(xRy)\)</p> + +<p class="nind"><b>*41·26.</b> \(\vdash:\dot{\exists}!\dot{s}ʻ\lambda.\equiv.(\exists R).R\in \lambda.\dot{\exists}!R\)</p> + +<p class="nind"><b>*41·27.</b> \(\vdash\colon\ldotp P\dot{\cap}\dot{s}ʻ\lambda=\dot{\Lambda}.\equiv:R\in \lambda.\supset_{R}.P\dot{\cap}R=\dot{\Lambda}\)</p> + +<p class="nind"><b><a id="*41·3">*41·3</a>.</b> \(\vdash.\text{Cnv}ʻ\dot{p}ʻ\lambda =\dot{p}ʻ\text{Cnv}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\\ +\vdash\colon\ldotp y(\text{Cnv}ʻ\dot{p}ʻ\lambda)x.&\equiv:x(\dot{p}ʻ\lambda)y:\\ +[\text{*41·1}] &\equiv:R\in \lambda.\supset_{R}.xRy:\\ +[\text{*31·131}]&\equiv:R\in \lambda.\supset_{R}.y(\text{Cnv}ʻR)x:\\ +[\text{*37·63.*31·13}] &\equiv:P\in \text{Cnv}ʻʻ\lambda.\supset_{P}.yPx:\\ +[\text{*41·1}] &\equiv:y(\dot{p}ʻ\text{Cnv}ʻʻ\lambda)x\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*41·31.</b> \(\vdash.\text{Cnv}ʻ\dot{s}ʻ\lambda=\dot{s}ʻ\text{Cnv}ʻʻ\lambda \quad[\text{Proof as in *41·3}]\)</p> + +<p class="nind"><b>*41·32.</b> \(\vdash.\text{Cnv}ʻʻ\dot{p}ʻʻ\kappa =\dot{p}ʻʻ\text{Cnv}ʻʻʻ\kappa \quad[\text{*41·3.*37·354}]\)</p> + +<p class="nind"><b>*41·33.</b> \(\vdash.\text{Cnv}ʻʻ\dot{s}ʻʻ\kappa =\dot{s}ʻʻ\text{Cnv}ʻʻʻ\kappa \quad[\text{*41·31.*37·354}]\)</p> + +<p class="nind"><b>*41·34.</b> \(\vdash.\dot{s}\alpha\upharpoonleft ʻʻ\lambda=\alpha\upharpoonleft \dot{s}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11.*38·13.*13·195}.\supset\vdash\colon\ldotp x(\dot{s}ʻ\alpha\upharpoonleft ʻʻ\lambda)y.&\equiv:(\exists P).P\in \lambda.x(\alpha\upharpoonleft P)y:\\ +[\text{*35·1}] &\equiv:(\exists P).P\in \lambda.x\in \alpha.xPy:\\ +[\text{*10·35}] &\equiv:x\in \alpha:(\exists P).P\in \lambda.xPy:\\ +[\text{*41·11.*35·1}] &\equiv:x(\alpha\upharpoonleft \dot{s}ʻ\lambda)y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_333">[Pg 333]</span></p> + +<p class="nind"><b>*41·341.</b> \(\vdash. \dot{s}ʻ\upharpoonright \alphaʻʻ\lambda = (\dot{s}ʻ\lambda)\upharpoonright \alpha \quad[\text{Proof as in *41·34}]\)</p> + +<p class="nind"><b>*41·342.</b> \(\vdash. \dot{s}ʻ\unicode{x0294f} \alphaʻʻ\lambda = (\dot{s}ʻ\lambda)\unicode{x0294f}\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*36·11.*35·21}.\supset\vdash.\dot{s}ʻ\unicode{x0294f} \alphaʻʻ\lambda &= \dot{s}ʻ\alpha\upharpoonleft ʻʻ\upharpoonright \alphaʻʻ\lambda\\ +[\text{*41·34}] & = \alpha\upharpoonleft (\dot{s}ʻ\upharpoonright \alphaʻʻ\lambda)\\ +[\text{*41·341}] &= \alpha\upharpoonleft (\dot{s}ʻ\lambda)\upharpoonright \alpha\\ +[\text{*36·11}] & = (\dot{s}ʻ\lambda)\unicode{x0294f}\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in <a href="#*85·22">*85·22</a>.</p> + +<p class="nind"><b>*41·35.</b> \(\vdash. \dot{s}ʻM\upharpoonright ʻʻ\kappa = M\upharpoonright sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11.*38·13}.\supset\vdash: x(\dot{s}ʻM\upharpoonright ʻʻ\kappa)y. &\equiv .(\exists \alpha).\alpha\in \kappa.x(M\upharpoonright \alpha)y.\\ +[\text{*35·101}] &\equiv .(\exists \alpha).\alpha\in \kappa.y\in \alpha.xMy.\\ +[\text{*40·11.*35·101}] &\equiv .x(M\upharpoonright sʻ\kappa)y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*41·351.</b> \(\vdash. \dot{s}ʻ\upharpoonleft Mʻʻ\kappa = (sʻ\kappa)\upharpoonleft M \quad[\text{Proof as in *41·35}]\)</p> + +<p class="nind"><b>*41·4.</b> \(\vdash. \text{D}ʻ\dot{p}ʻ\lambda\subset pʻ\text{D}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*33·13} . \supset\\ +\vdash \colon\colon x\in \text{D}ʻ\dot{p}ʻ\lambda.&\equiv \colon\ldotp (\exists y).x(\dot{p}ʻ\lambda)y\colon\ldotp \\ +[\text{*41·1}] &\equiv \colon\ldotp (\exists y):R\in \lambda.\supset_{R}.xRy\colon\ldotp \\ +[\text{*11·61}] &\supset\colon\ldotp R\in \lambda.\supset_{R}.(\exists y).xRy\colon\ldotp \\ +[\text{*33·13}] &\supset\colon\ldotp R\in \lambda.\supset_{R}.x\in \text{D}ʻR\colon\ldotp \\ +[\text{*40·41.*33·12}] &\supset\colon\ldotp x\in pʻ\text{D}ʻʻ\lambda\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*41·41.</b> \(\vdash.\text{ᗡ}ʻ\dot{p}ʻ\lambda\subset pʻ\text{ᗡ}ʻʻ\lambda \quad[\text{Proof as in *41·4}]\)</p> + +<p class="nind"><b>*41·42.</b> \(\vdash.Cʻ\dot{p}ʻ\lambda\subset pʻCʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*33·132}.\supset\vdash\colon\colon\ldotp x\in Cʻ\dot{p}ʻ\lambda.&\equiv \colon\colon (\exists y):x(\dot{p}ʻ\lambda)y.\lor .y(\dot{p}ʻ\lambda)x\colon\colon\\ +[\text{*41·1}] &\equiv\colon\colon (\exists y)\colon\colon R\in \lambda.\supset_{R}.xRy:\lor :R\in \lambda.\supset_{R}.yRx\colon\colon \\ +[\text{*10·41·221}] &\supset\colon\colon (\exists y)\colon\colon (R)\colon\ldotp R\in \lambda.\supset.xRy:\lor :R\in \lambda.\supset.yRx\colon\colon\\ +[\text{*4·78}] &\supset\colon\colon (\exists y)\colon\colon (R)\colon\ldotp R\in \lambda.\supset:xRy.\lor .yRx\colon\colon \\ +[\text{*11·61}] &\supset\colon\colon (R)\colon\colon R\in \lambda.\supset: (\exists y): xRy .\lor . yRx:\\ +[\text{*33·132}] &\supset:x\in CʻR\colon\colon \\ +[\text{*40·41.*33·122}] & \supset\colon\colon x\in pʻCʻʻ\lambda\colon\colon .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*41·43.</b> \(\vdash.\text{D}ʻ\dot{s}ʻ\lambda = sʻ\text{D}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13} .\supset\vdash\colon\ldotp x\in \text{D}ʻ\dot{s}ʻ\lambda. &\equiv: (\exists y).x(\dot{s}ʻ\lambda)y:\\ +[\text{*41·11}] &\equiv: (\exists y):(\exists R).R\in \lambda.xRy:\\ +[\text{*11·23·55}] &\equiv: (\exists R):R\in \lambda:(\exists y).xRy:\\ +[\text{*33·13}] &\equiv:(\exists R). R\in \lambda.x\in \text{D}ʻR:\\ +[\text{*40·4.*33·12}] &\equiv: x\in sʻ\text{D}ʻʻ\lambda\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_334">[Pg 334]</span></p> + +<p class="nind"><b>*41·44.</b> \(\vdash.\text{ᗡ}ʻ\dot{s}ʻ\lambda=sʻ\text{ᗡ}ʻʻ\lambda \quad[\text{Proof as in *41·43}]\)</p> + +<p class="nind"><b>*41·45.</b> \(\vdash.Cʻ\dot{s}ʻ\lambda=sʻCʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·16}.\supset\vdash.Cʻ\dot{s}ʻ\lambda&=\text{D}ʻ\dot{s}ʻ\lambda\cup \text{ᗡ}ʻ\dot{s}ʻ\lambda\\ +[\text{*41·43·44}] &=sʻ\text{D}ʻʻ\lambda\cup sʻ\text{ᗡ}ʻʻ\lambda\\ +[\text{*40·57}] &=sʻCʻʻ\lambda.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*41·5.</b> \(\vdash.\dot{p}ʻ\lambda\mid \dot{p}ʻ\mu\unicode{x2abd}\dot{p}ʻ(sʻ\lambda\mid_{,,}ʻʻ\mu)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1}.\supset\\ +\vdash\colon\colon x(\dot{p}ʻ\lambda\mid \dot{p}ʻ\mu)z.&\equiv\colon\ldotp (\exists y).x(\dot{p}ʻ\lambda)y.y(\dot{p}ʻ\mu)z\colon\ldotp \\ +[\text{*41·1}] &\equiv\colon\ldotp (\exists y)\colon\ldotp P\in \lambda.\supset{_P}.xPy:Q\in \mu.\supset{_Q}.yQz\colon\ldotp \\ +[\text{*11·56}] &\equiv\colon\ldotp (\exists y)\colon\ldotp (P,Q):P\in \lambda.\supset.xPy:Q\in \mu.\supset.yQz\colon\ldotp \\ +[\text{*11·37·39}] &\supset\colon\ldotp (\exists y)\colon\ldotp (P,Q):P\in \lambda.Q\in \mu.\supset.xPy.yQz\colon\ldotp \\ +[\text{*11·61}] &\supset\colon\ldotp (P,Q)\colon\ldotp P\in \lambda.Q\in \mu.\supset.(\exists y).xPy.yQz.\\ +[\text{*34·1}] &\supset.x(P\mid Q)z\colon\ldotp \\ +[\text{*13·191}] &\supset\colon\ldotp (P,Q,R)\colon\ldotp P\in \lambda.Q\in \mu.R=P\mid Q.\supset.xRz\colon\ldotp \\ +[\text{*11·21·35}] &\supset\colon\ldotp (R):(\exists P,Q).P\in \lambda.Q\in \mu.R=P\mid Q.\supset.xRz:\\ +[\text{*40·7}] &\supset\colon\ldotp (R):R\in sʻ\lambda\mid_{,,}ʻʻ\mu.\supset.xRz\colon\ldotp \\ +[\text{*41·1}] &\supset\colon\ldotp x(\dot{p}ʻsʻ\lambda\mid_{,,}ʻʻ\mu)z\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*41·51.</b> \(\vdash.\dot{s}ʻ\lambda\mid \dot{s}ʻ\mu=\dot{s}ʻsʻ\lambda\mid_{,,}ʻʻ\mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*34·1.\supset\\ +\vdash\colon\colon x(\dot{s}ʻ\lambda\mid \dot{s}ʻ\mu)z.&\equiv\colon\ldotp (\exists y).x(\dot{s}ʻ\lambda)y.y(\dot{s}ʻ\mu)z\colon\ldotp\\ +[*41·11] & \equiv\colon\ldotp (\exists y)\colon\ldotp (\exists P).P\in \lambda.xPy:(\exists Q).Q\in \mu.yQz\colon\ldotp \\ +[*11·54] &\equiv\colon\ldotp (\exists y)\colon\ldotp (\exists P,Q):P\in \lambda.xPy.Q\in \mu.yQz\colon\ldotp \\ +[*11·24·27] &\equiv\colon\ldotp (\exists P,Q)\colon\ldotp (\exists y).P\in \lambda.xPy.Q\in \mu.yQz\colon\ldotp \\ +[*10·35] &\equiv\colon\ldotp (\exists P,Q)\colon\ldotp P\in \lambda.Q\in \mu:(\exists y).xPy.yQz\colon\ldotp \\ +[*34·1] &\equiv\colon\ldotp (\exists P,Q):P\in \lambda.Q\in \mu.x(P\mid Q)z\colon\ldotp \\ +[*13·195] &\equiv\colon\ldotp (\exists P,Q,R).P\in \lambda.Q\in \mu.R=P\mid Q.xRz\colon\ldotp \\ +[*11·24.*40·7] &\equiv\colon\ldotp (\exists R).R\in sʻ\lambda\mid_{,,}ʻʻ\mu.xRz\colon\ldotp \\ +[*41·11] &\equiv\colon\ldotp x(\dot{s}ʻsʻ\lambda\mid_{,,}ʻʻ\mu)z\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition, which is used in <a href="#*92·31">*92·31</a>, states that, if +\(\lambda\) and \(\mu\) are classes of relations, the relative product +of the relational sum of \(\lambda\) and the relational sum of \(\mu\) +is the relational sum of all the relative products formed of a member +of \(\lambda\) and a member of \(\mu\).</p> + +<p>The following proposition is used in <a href="#*96·111">*96·111</a>.</p> + +<p><span class="pagenum" id="Page_335">[Pg 335]</span></p> + +<p class="nind"><b>*41·52.</b> \(\vdash\colon\ldotp \alpha\upharpoonleft \dot{s}ʻ\lambda\unicode{x2abd}Q.\equiv:P\in \lambda.\supset_{P}.\alpha\upharpoonleft P\unicode{x2abd}Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1.*41·11}.\supset\\ +\vdash\colon\colon \alpha\upharpoonleft \dot{s}ʻ\lambda\unicode{x2abd}Q.&\equiv\colon\ldotp x\in \alpha:(\exists P).P\in \lambda.xPy:\supset_{x,y}.xQy\colon\ldotp\\ +[\text{*10·35·23}] &\equiv\colon\ldotp x\in \alpha.P\in \lambda.xPy.\supset_{P,x,y}.xQy\colon\ldotp \\ +[\text{*35·1}] &\equiv\colon\ldotp P\in \lambda.x(\alpha\upharpoonleft P)y.\supset_{P,x,y}.xQy\colon\ldotp \\ +[\text{*11·62}] &\equiv\colon\ldotp P\in \lambda.\supset_{P}.\alpha\upharpoonleft P\unicode{x2abd}Q\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in *162·32 and in *166·461.</p> + +<p class="nind"><b>*41·6.</b> \(\vdash\colon\ldotp y\in \beta.\supset_{y}.Pʻy=Qʻy\unicode{x228d}Rʻy:\supset.\dot{s}ʻPʻʻ\beta=\dot{s}ʻQʻʻ\beta\unicode{x228d}\dot{s}ʻRʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·6.*14·21.*41.·11.*13·195}.\supset\\ +\vdash\colon\colon \text{Hp}.\supset\colon\ldotp u(\dot{s}ʻPʻʻ\beta)v.&\equiv:(\exists y).y\in \beta.u(Pʻy)v:\\ +[\text{Hp}] &\equiv:(\exists y).y\in \beta.u(Qʻy\unicode{x228d}Rʻy)v:\\ +[\text{*23·34.*10·42}] &\equiv:(\exists y).y\in \beta.u(Qʻy)v.\lor.(\exists y).y\in \beta.u(Rʻy)v:\\ +[\text{*37·6.*41·11}] &\equiv:u(\dot{s}ʻQʻʻ\beta)v.\lor.u(\dot{s}ʻRʻʻ\beta)v\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_336">[Pg 336]</span></p> +<h2 class="nobreak" id="*42">*42. MISCELLANEOUS PROPOSITIONS.</h2> +</div> + + +<p><i>Summary of</i> *42.</p> + +<p>The present number contains various propositions concerning products +and sums of classes. They are concerned chiefly with classes of classes +of classes, or with relations of relations of relations. These are +required respectively in cardinal and in ordinal arithmetic. Thus +<a href="#*42·1">*42·1</a> is used in *112 and *113, which are concerned with cardinal +addition and multiplication, while *42·12·2 are used in *160 and +*162, which are concerned with ordinal addition. <a href="#*42·22">*42·22</a>, though not +explicitly referred to, is useful in facilitating the comprehension +of propositions on series of series of series, or rather on relations +between relations between relations, which are required in connection +with the associative law of multiplication in relation-arithmetic.</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*42·1">*42·1</a>.</b> \(\vdash . sʻsʻʻ\kappa = sʻsʻ\kappa\)</p> + +<p>Here \(\kappa\) must, for significance, be a class of classes +of classes. The proposition states that if we take each member, +\(\alpha\), of \(\kappa\), and form \(sʻ\alpha\), and then form the +sum of all the classes so obtained, the result is the same as if we +form the sum of the sum of \(\kappa\). This is the associative law for +\(s\), and is (as will appear later) the source of the associative law +of addition in cardinal arithmetic. The way in which this proposition +comes to be the associative law for \(s\) may be seen as follows: +Suppose \(\kappa\) consists of two classes, \(\alpha\) and \(\beta\); +suppose \(\alpha\) in turn consists of the two classes \(\xi\) and +\(\eta\), and \(\beta\) of the two classes \(\xi'\) and \(\eta'\). Then +\(sʻ\alpha = \xi \cup \eta . sʻ\beta = \xi' \cup \eta'\). (This will +be proved later.) Thus \(sʻʻ\kappa\) has two members, one of which is +\(\xi \cup \eta\), while the other is \(\xi' \cup \eta'\). Thus +\[ +sʻsʻʻ\kappa = (\xi \cup \eta) \cup (\xi' \cup \eta'). +\] +But \(sʻ\kappa\) has four members, namely \(\xi\), \(\eta\), \(\xi'\), +\(\eta'\). Thus \(sʻsʻ\kappa = \xi \cup \eta \cup \xi' \cup \eta'\).</p> + +<p class="nind">Thus our proposition leads to +\[ +(\xi \cup \eta) \cup (\xi' \cup \eta') = \xi \cup \eta \cup \xi' \cup \eta'\text{,} +\] +which is obviously a case of the associative law.</p> + +<p><span class="pagenum" id="Page_337">[Pg 337]</span></p> + +<p>Our proposition states the associative law generally, including the +case where the number of brackets, or of summands in any bracket, is +infinite. The proof is as follows.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·4}.\supset\vdash\colon\colon x\in sʻsʻʻ\kappa.&\equiv\colon\ldotp (\exists \alpha).\alpha\in \kappa.x\in sʻ\alpha\colon\ldotp \\ +[\text{*40·11}] &\equiv\colon\ldotp (\exists \alpha):\alpha\in \kappa:(\exists \xi).\xi\in \alpha.x\in \xi\colon\ldotp \\ +[\text{*11·6}] &\equiv\colon\ldotp (\exists \xi)\colon\ldotp (\exists \alpha).\alpha\in \kappa.\xi\in \alpha:x\in \xi\colon\ldotp \\ +[\text{*40·11}] &\equiv\colon\ldotp (\exists \xi).\xi\in sʻ\kappa.x\in \xi\colon\ldotp \\ +[\text{*40·11}] &\equiv\colon\ldotp x\in sʻsʻ\kappa\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*42·11.</b> \(\vdash.pʻpʻʻ\kappa=pʻsʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·41}.\supset\vdash\colon\ldotp x\in pʻpʻʻ\kappa.&\equiv:\beta\in \kappa.\supset_{\beta}.x\in pʻ\beta:\\ +[\text{*40·1.*11·62}] &\equiv:\beta\in \kappa.\gamma\in \beta.\supset_{\beta,\gamma}.x\in \gamma:\\ +[\text{*11·2.*10·23}] &\equiv:(\exists \beta).\beta\in \kappa.\gamma\in \beta.\supset_{\gamma}.x\in \gamma:\\ +[\text{*40·11}] &\equiv:\gamma\in sʻ\kappa.\supset_{\gamma}.x\in \gamma:\\ +[\text{*40·1}] &\equiv:x\in pʻsʻ\kappa\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This is the associative law for products. Supposing again, for +illustration, that \(\kappa\) consists of the two classes \(\alpha\), +\(\beta\), while \(\alpha\) consists of the two classes \(\xi\), +\(\eta\) and \beta of the two classes \(\xi'\), \(\eta'\), then +\(pʻʻ\kappa\) consists of the two classes \(\xi\cap \eta\) and +\(\xi'\cap \eta'\), so that \(pʻpʻʻ\kappa=(\xi\cap \eta)\cap (\xi'\cap +\eta')\), while \(pʻsʻ\kappa=\xi\cap \eta\cap \xi'\cap \eta'\). Thus +our proposition becomes +\[ +(\xi\cap \eta)\cap (\xi'\cap \eta') = \xi\cap \eta\cap \xi'\cap \eta'. +\]</p> + +<p>A descriptive function \(Rʻ\kappa\) whose arguments are classes or +classes of classes may be said to obey the associative law provided +\[ +RʻRʻʻ\kappa=Rʻsʻ\kappa. +\]</p> + +<p>This equation may be interpreted as follows: Given a class \(\alpha\), +divide it into any number of subordinate classes, so that no member is +left out, though one member may belong to two or more classes. Let the +classes into which \(\alpha\) is divided make up the class \(\kappa\), +so that \(\kappa\) is a class of classes, and \(sʻ\kappa=\alpha\). Then +the above equation asserts that if we first form the \(R's\) of the +various sub-classes of \(\alpha\), and then the \(R\) of the resulting +class, the result is the same as if we formed the \(R\) of \(\alpha\) +directly.</p> + +<p>In some cases—for example, that of arithmetical addition of +cardinals—the above equation holds only when no two members of +\(\kappa\) have a common term, <i>i.e.</i> when the parts into which +\(\alpha\) is divided are mutually exclusive.</p> + +<p><span class="pagenum" id="Page_338">[Pg 338]</span></p> + +<p>For a descriptive function whose arguments are relations of relations, +we shall find another form for the associative law; this form plays in +ordinal arithmetic a part analogous to that played by the above form in +cardinal arithmetic.</p> + +<p class="nind"><b>*42·12.</b> \(\vdash.\dot{s}ʻ\dot{s}ʻʻ\lambda=\dot{s}ʻsʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11}. \supset\vdash:x(\dot{s}ʻ\dot{s}ʻʻ\lambda)y.&\equiv.(\exists \mu).\mu\in \lambda.x(\dot{s}ʻ\mu)y.\\ +[\text{*41·11}] &\equiv.(\exists \mu,P).\mu\in \lambda.P\in \mu.xPy.\\ +[\text{*40·11}] &\equiv.(\exists P).P\in sʻ\lambda.xPy.\\ +[\text{*41·11}] &\equiv.x(\dot{s}ʻsʻ\lambda)y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*42·13.</b> \(\vdash.\dot{p}ʻ\dot{p}ʻʻ\lambda=\dot{p}ʻsʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·1}. \supset\vdash\colon\ldotp x(\dot{p}ʻ\dot{p}ʻʻ\lambda)y.&\equiv:\mu\in \lambda.\supset_{\mu}.x(\dot{p}ʻ\mu)y:\\ +[\text{*41·1}] &\equiv:\mu\in \lambda.R\in \mu.\supset_{\mu,R}.xRy:\\ +[\text{*11·2.*10·23}] &\equiv:(\exists \mu).\mu\in \lambda.R\in \mu.\supset_{R}.xRy:\\ +[\text{*40·11}] &\equiv:R\in sʻ\lambda.\supset_{R}.xRy:\\ +[\text{*41·1}] &\equiv:x(\dot{p}ʻsʻ\lambda)y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*42·2.</b> \(\vdash.Cʻ\dot{s}ʻCʻP=sʻCʻʻCʻP=FʻʻCʻP=\overrightarrow{F}^{2}ʻP\)</p> + +<p>This proposition assumes that \(P\) is a relation between relations. +For example, suppose we have a series of series, whose generating +relations are ordered by the relation \(P\). Then \(CʻP\) is the +class of these generating relations; \(\dot{s}ʻCʻP\) is the relation +"one or other of the generating relations which compose \(CʻP\)," +and \(Cʻ\dot{s}ʻCʻP\) is the class of all the terms occurring in any +of the series. \(CʻʻCʻP\) is the fields of the various series, and +\(sʻCʻʻCʻP\) is again all the terms occurring in any of the series. +\(FʻʻCʻP\) is all the terms belonging to fields of series which are +members of \(CʻP\), and \(\overrightarrow{F}^{2}ʻP\) is all members of +fields of members of the field of \(P\); each of these again is all the +terms occurring in any of the series. The proof is as follows:</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·45}.\supset\vdash.Cʻ\dot{s}ʻCʻP&=sʻCʻʻCʻP &\qquad \text{(1)}\\ +\vdash.\text{*40·56}.\supset\vdash.sʻCʻʻCʻP&=FʻʻCʻP &\qquad \text{(2)}\\ +\vdash.\text{*33·5}.\supset\vdash.FʻʻCʻP&=Fʻʻ\overrightarrow{F}ʻP\\ +[\text{*37·38}] & =\overrightarrow{F}^{2}ʻP &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_339">[Pg 339]</span></p> + +<p>The following propositions apply to a relation of relations of +relations. These propositions are useful for proving associative laws +in ordinal arithmetic, since these laws deal with series of series of +series, and series of series of series are most simply constituted by +supposing the generating relations of the constituent series to be +ordered by relations which are themselves ordered by a relation \(P\).</p> + +<p class="nind"><b>*42·21.</b> \(\vdash.sʻCʻʻʻCʻʻCʻP=CʻʻsʻCʻʻCʻP=CʻʻCʻ\dot{s}ʻCʻP=CʻʻFʻʻCʻP=Cʻʻ\overrightarrow{F}^{2}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·38}.&\supset\vdash.sʻCʻʻʻCʻʻCʻP=CʻʻsʻCʻʻCʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*42·2}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*42·22">*42·22</a>.</b> \[\begin{align}\vdash.sʻsʻCʻʻʻCʻʻCʻP&=sʻCʻʻsʻCʻʻCʻP=sʻCʻʻCʻ\dot{s}ʻCʻP\\ +&=Cʻ\dot{s}ʻCʻ\dot{s}ʻCʻP=sʻCʻʻFʻʻCʻP\\ +&=FʻʻFʻʻCʻP=Fʻʻ\overrightarrow{F}^{2}ʻP=\overrightarrow{F}^{3}ʻP\\ +&\quad[\text{*42·21.*41·45.*40·56.*42·2.*37·3}]\end{align}\]</p> + +<p>If \(P\), in the above proposition, is a relation which generates a +series of series of series, the above gives various forms for the +class of ultimate terms of these series. Thus suppose \(Q\in CʻP\); +then \(Q\) is a relation between generating relations of series. If +now \(R\in CʻQ\), \(R\) is the generating relation of a series which +we may regard as composed of individuals. The class of individuals so +obtainable may be expressed in any of the above forms, as well as in +others which are not given above.</p> + +<p class="nind"><b>*42·3.</b> \(\vdash.sʻsʻʻ\overrightarrow{R}ʻʻ\alpha=sʻRʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*42·1}.\supset\vdash.sʻsʻʻ\overrightarrow{R}ʻʻ\alpha&=sʻsʻ\overrightarrow{R}ʻʻ\alpha\\ +[\text{*40·5}] &=sʻRʻʻ\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*42·31.</b> \(\vdash.sʻsʻʻ\overleftarrow{R}ʻʻ\alpha=sʻ\breve{R}ʻʻ\alpha \quad[\text{Proof as in *42·3}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_340">[Pg 340]</span></p> +<h2 class="nobreak" id="*43">*43. THE RELATIONS OF A RELATIVE PRODUCT TO ITS FACTORS.</h2> +</div> + + +<p><i>Summary of</i> *43.</p> + +<p>The purpose of the present number is to give certain propositions on +the relation which holds between \(P\) and \(Q\) whenever \(P = Q \mid R\), +or whenever \(P = R \mid Q\), or whenever \(P = R \mid Q \mid S\), +where \(R\) and \(S\) are fixed. In virtue of the general definitions +of <a href="#*38">*38</a>, these relations are respectively \(\mid R\), \(R \mid\), and +(\(R \mid) \mid (\mid S)\). Such relations are of great utility both +in cardinal and in ordinal arithmetic; they are also much used in the +theory of induction (Part II, Section E). In place of the notation +(\(R \mid) \mid (\mid S)\), which is cumbrous, we adopt the more compact +notation \(R \parallel S\). If \(\lambda\) is a class of relations, +\(R \mid ʻʻ\lambda\) will be the class of relations \(R \mid P\) where +\(P \in \lambda\), \(\mid Rʻʻ\lambda\) will be the class of relations +\(P \mid R\) where \(P \in \lambda\), and (\(R \parallel S)ʻʻ\lambda\) will +be the class of relations \(R \mid P \mid S\) where \(P \in \lambda\). +These classes of relations are often required in subsequent work.</p> + +<p>In virtue of our definitions, we have</p> + +<p class="nind"><b>*43·112.</b> \(\vdash . (R \parallel S)ʻQ = R \mid Q \mid S\)</p> + +<p>The propositions most used in the present number (except such as merely +embody definitions) are the following:</p> + +<p class="nind"><b>*43·302.</b> \(\vdash . (P).P \in \text{ᗡ}ʻ(R \parallel S)\)</p> + +<p class="nind"><b>*43·411.</b> \(\vdash . \breve{R}ʻʻʻ\text{ᗡ}ʻʻ\lambda = \text{ᗡ}ʻʻ \mid Rʻʻ\lambda\)</p> + +<p class="nind"><b>*43·421.</b> \(\vdash . \dot{s}ʻ \mid Rʻʻ\lambda = (\dot{s}ʻ\lambda) \mid R\)</p> + +<p>The remaining propositions are used seldom, but their uses, when they +are used, are important.</p> + +<hr class="tb"> + +<p class="nind"><b>*43·01.</b> \(R \parallel S = (R \mid) \mid (\mid S) \quad \text{Df}\)</p> + + +<p><span class="pagenum" id="Page_341">[Pg 341]</span></p> + +<p>At a later stage (in *150) we shall introduce a simpler notation +for the special case of \(R \parallel \breve{R}\). The following +propositions are for the most part immediate consequences of the +definitions, and proofs are therefore usually omitted.</p> + +<p class="nind"><b>*43·1.</b> \(\vdash:P(R\mid )Q.\equiv.P=R\mid Q\)</p> + +<p class="nind"><b>*43·101.</b> \(\vdash:P(\mid R)Q.\equiv.P=Q\mid R\)</p> + +<p class="nind"><b>*43·102.</b> \(\vdash:P(R\Arrowvert S)Q.\equiv.P=R\mid Q\mid S\)</p> + +<p class="nind"><b>*43·11.</b> \(\vdash.R\mid ʻQ=R\mid Q\)</p> + +<p class="nind"><b>*43·111.</b> \(\vdash.\mid RʻQ=Q\mid R\)</p> + +<p class="nind"><b>*43·112.</b> \(\vdash.(R\Arrowvert S)ʻQ=R\mid Q\mid S\)</p> + +<p class="nind"><b>*43·12.</b> \(\vdash.\text{E}!R\mid ʻQ\)</p> + +<p class="nind"><b>*43·121.</b> \(\vdash.\text{E}!\mid RʻQ\)</p> + +<p class="nind"><b>*43·122.</b> \(\vdash.\text{E}!(R\Arrowvert S)ʻQ\)</p> + +<p class="nind"><b>*43·2.</b> \(\vdash.(R\mid )\mid (S\mid )=(R\mid S)\mid \)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·1}.\supset\vdash:L\{(R\mid )\mid (S\mid )\}N.&\equiv.(\exists M).L=R\mid M.M=S\mid N.\\ +[\text{*13·195.*34·21}] &\equiv.L=R\mid S\mid N.\\ +[\text{*43·1}] &\equiv.L\{(R\mid S)\mid \}N:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*43·201.</b> \(\vdash.(\mid R)\mid (\mid S)=\mid (S\mid R) \quad[\text{Proof as in *43·2}]\)</p> + +<p class="nind"><b>*43·202.</b> \(\vdash.(\mid R)\mid (S\mid )=(S\mid )\mid (\mid R)=S\Arrowvert R \quad[\text{Proof as in *43·2}]\)</p> + +<p class="nind"><b>*43·21.</b> \(\vdash.(P\mid Q)\mid (R\mid )=(P\mid R)\Arrowvert Q\)</p> + +<p class="nind"><b>*43·211.</b> \(\vdash.(R\mid )\mid (P\Arrowvert Q)=(R\mid P)\Arrowvert Q\)</p> + +<p class="nind"><b>*43·212.</b> \(\vdash.(P\Arrowvert Q)\mid (\mid R)=P\Arrowvert (R\mid Q)\)</p> + +<p class="nind"><b>*43·213.</b> \(\vdash.(\mid R)\mid (P\Arrowvert Q)=P\Arrowvert (Q\mid R)\)</p> + +<p class="nind"><b>*43·22.</b> \(\vdash.(P\Arrowvert Q)\mid (R\Arrowvert S)=(P\mid R)\Arrowvert (S\mid Q)\)</p> + +<p class="nind"><b>*43·3.</b> \(\vdash.(P).P\in \text{ᗡ}ʻR\mid \quad[\text{*43·12.*33·43}]\)</p> + +<p class="nind"><b>*43·301.</b> \(\vdash.(P).P\in \text{ᗡ}ʻ\mid R\)</p> + +<p class="nind"><b>*43·302.</b> \(\vdash.(P).P\in \text{ᗡ}ʻ(R\Arrowvert S)\)</p> + +<p class="nind"><b>*43·31.</b> \(\vdash.P\upharpoonright \text{ᗡ}ʻR\mid =P\upharpoonright CʻR\mid =P\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·12.*33·431}.&\supset\vdash.\text{ᗡ}ʻP\subset \text{ᗡ}ʻR\mid &\qquad \text{(1)}\\ +[\text{*33·161}] &\supset\vdash.\text{ᗡ}ʻP\subset CʻR\mid &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*35·45}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*43·311.</b> \(\vdash.P\upharpoonright \text{ᗡ}ʻ\mid R=P\upharpoonright Cʻ\mid R=P\)</p> + +<p class="nind"><b>*43·312.</b> \(\vdash.P\upharpoonright \text{ᗡ}ʻ(R\Arrowvert S)=P\upharpoonright Cʻ(R\Arrowvert S)=P\)</p> + +<p class="nind"><b>*43·34.</b> \(\vdash.R\mid ʻR=\mid RʻR=R^{2} \quad[\text{*43·11·111}]\)</p> + +<p class="nind"><b>*43·4.</b> \(\vdash.Rʻʻ\text{D}ʻP=\text{D}ʻR\mid ʻP \quad[\text{*37·32.*43·1}]\)</p> + +<p class="nind"><b>*43·401.</b> \(\vdash.\breve{R}ʻʻ\text{ᗡ}ʻP=\text{ᗡ}ʻ\mid RʻP \quad[\text{*37·32.*43·101}]\)</p> + +<p class="nind"><b>*43·41.</b> \(\vdash.Rʻʻʻ\text{D}ʻʻ\lambda=\text{D}ʻʻR\mid ʻʻ\lambda \quad[\text{*43·4.*37·355}]\)</p> + +<p><span class="pagenum" id="Page_342">[Pg 342]</span></p> + +<p class="nind"><b>*43·411.</b> \(\vdash.\breve{R}ʻʻʻ\text{ᗡ}ʻʻ\lambda=\text{ᗡ}ʻʻ\mid Rʻʻ\lambda \quad[\text{*43·401.*37·355}]\)</p> + +<p class="nind"><b>*43·42.</b> \(\vdash.\dot{s}ʻR\mid ʻʻ\lambda=R\mid \dot{s}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11.*37·1.*43·1}.\supset\\ +\vdash\colon\ldotp x(\dot{s}ʻR\mid ʻʻ\lambda)z.&\equiv:(\exists T).T\in \lambda.x(R\mid T)z:\\ +[\text{*34·1}] &\equiv:(\exists T):T\in \lambda:(\exists y).xRy.yTz:\\ +[\text{*11·6}] &\equiv:(\exists y):xRy:(\exists T).T\in \lambda.yTz:\\ +[\text{*41·11.*34·1}] &\equiv:x(R\mid \dot{s}ʻ\lambda)z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*43·421.</b> \(\vdash.\dot{s}ʻ\mid Rʻʻ\lambda=(\dot{s}ʻ\lambda)\mid R \quad[\text{Proof as in *43·42}]\)</p> + +<p class="nind"><b>*43·43.</b> \(\vdash.\dot{s}ʻ(R\Arrowvert S)ʻʻ\lambda=(R\Arrowvert S)ʻ\dot{s}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·33}.\supset\vdash.\dot{s}ʻ(R\Arrowvert S)ʻʻ\lambda&=\dot{s}ʻR\mid ʻʻ\mid Sʻʻ\lambda\\ +[\text{*43·42}] &=R\mid (\dot{s}ʻ\mid Sʻʻ\lambda)\\ +[\text{*43·421}] &=R\mid \dot{s}ʻ\lambda\mid S\\ +[\text{*43·112}] &=(R\Arrowvert S)ʻ\dot{s}ʻ\lambda.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*43·48.</b> \(\vdash:\text{D}ʻP\subset \alpha.\supset.Q\mid ʻP=(Q\upharpoonright \alpha)\mid ʻP \quad[\text{*35·481}]\)</p> + +<p class="nind"><b>*43·481.</b> \(\vdash:\text{ᗡ}ʻP\subset \beta.\supset.\mid RʻP=\mid (\beta\upharpoonleft R)ʻP \quad[\text{*35·48}]\)</p> + +<p class="nind"><b>*43·49.</b> \(\vdash:sʻ\text{D}ʻʻ\lambda\subset \alpha.\supset.(Q\mid )\upharpoonright \lambda=\{(Q\upharpoonright \alpha)\mid\}\upharpoonright \lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·43}.\supset\vdash\colon\ldotp \text{Hp}.\supset:P\in \lambda.&\supset.\text{D}ʻP\subset \alpha.\\ +[\text{*43·48}] &\supset.Q\mid ʻP=\{(Q\upharpoonright \alpha)\mid\}ʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*35·71}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*43·491.</b> \(\vdash:sʻ\text{ᗡ}ʻʻ\lambda\subset \beta.\supset.(\mid R)\upharpoonright \lambda=\{\mid (\alpha\upharpoonleft R)\}\upharpoonright \lambda \quad[\text{Proof as in *43·49}]\)</p> + +<p class="nind"><b>*43·5.</b> \[\begin{align}&\vdash:\text{D}ʻP\subset \alpha.\text{ᗡ}ʻP\subset \beta.\supset.(Q\Arrowvert R)ʻP=\{(Q\upharpoonright \alpha)\Arrowvert (\beta\upharpoonleft R)\}ʻP\\ +&\quad[\text{*35·48·481 .*43·112}]\end{align}\]</p> + +<p class="nind"><b>*43·51.</b> + \(\vdash:sʻDʻʻ\lambda\subset \alpha.sʻ\text{ᗡ}ʻʻ\lambda\subset \beta.\supset.(Q\Arrowvert R)\upharpoonright \lambda=\{(Q\upharpoonright \alpha)\Arrowvert (\beta\upharpoonleft R)\}\upharpoonright \lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·43}.\supset\vdash\colon\ldotp \text{Hp}.\supset:P\in \lambda.&\supset.\text{D}ʻP\subset \alpha.\text{ᗡ}ʻP\subset \beta.\\ +[\text{*43·5}] &\supset.(Q\Arrowvert R)ʻP=\{(Q\upharpoonright \alpha)\Arrowvert (\beta\upharpoonleft R)\}ʻP (1) +\vdash.\text{(1).*35·71}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the proof of <a href="#*74·773">*74·773</a>.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_343">[Pg 343]</span></p> +<h2 class="nobreak" id="PART_II">PART II.<br> +<br> +PROLEGOMENA TO CARDINAL ARITHMETIC.</h2> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_345">[Pg 345]</span></p> +<h2 class="nobreak" id="SUMMARY_OF_PART_II">SUMMARY OF PART II.</h2> +</div> + + +<p>THE objects to be studied in this Part are not sharply distinguished +from those studied in Part I. The difference is one of degree, the +objects in this Part being of somewhat less general importance than +those of Part I, and being studied more on account of their bearing +on cardinal arithmetic than on their own account. Although cardinal +arithmetic is the goal which determines our course in Part II, all the +objects studied will be found to be also required in ordinal arithmetic +and the theory of series. As this Part advances, the approach to +cardinal arithmetic becomes gradually more marked, until at last +nothing is lacking except the definition of cardinal numbers, with +which Part III opens.</p> + +<p><a href="#SECTION_A_b">Section A</a> of this Part deals with unit classes and couples. A +<i>unit</i> class is the class of terms identical with a given term, +<i>i.e.</i> the class whose only member is the given term. (As +explained in the Introduction, Chapter III, <a href="#Page_80">pp. 80</a> to <a href="#Page_83">83</a>, the class +whose only member is \(x\) is not identical with \(x\).) We define +\(1\) as the class of all unit classes, leaving it to Part III to show +that \(1\), so defined, is a cardinal number. In like manner, we define +a (cardinal or ordinal) couple, and then define \(2\) as the class of +all couples. The propositions on couples will not be much referred to +in the remainder of the present Part, since their use belongs chiefly +to arithmetic (Parts III and IV). On the other hand, the properties of +unit classes are constantly required in Sections C, D, E of this Part.</p> + +<p><a href="#SECTION_B_b">Section B</a> deals, first, with the class of sub-classes of a given class, +<i>i.e.</i> of classes contained in a given class. The sub-classes +of a given class are often important in arithmetic. Next we consider +the class of sub-relations of a given relation, <i>i.e.</i> relations +contained in a given relation. The propositions on this subject are +analogous to those on sub-classes, but less important. Next we consider +the question of "relative types," <i>i.e.</i> taking any object \(x\), +and calling its type \(tʻx\), we give a notation for expressing in +terms of \(tʻx\) the type of classes of which \(x\) is a member, or +of relations in which \(x\) may be either referent or relatum, and +so on. The notations introduced in this connection are very useful +in arithmetic, especially in connection with existence-theorems. But +the propositions of Section B are very seldom required in the later +sections of the present Part.</p> + +<p><span class="pagenum" id="Page_346">[Pg 346]</span></p> + +<p><a href="#SECTION_C_b">Section C</a>, which deals with one-many, many-one and one-one relations, +is very important, and is constantly relevant in the sequel. A relation +is one-many when no term has more than one referent, many-one if no +term has more than one relatum, and one-one if it is both one-many and +many-one. In this section, we define the notion of <i>similarity</i>, +upon which all cardinal arithmetic is based: two classes are said to +be <i>similar</i> when there is a one-one relation whose domain is the +one and whose converse domain is the other. We prove the elementary +properties of similarity, including the Schröder-Bernstein theorem, +namely: If \(\alpha\) is similar to part of \(\beta\), and \(\beta\) is +similar to part of \(\alpha\), then \(\alpha\) is similar to \(\beta\).</p> + +<p><a href="#SECTION_D_b">Section D</a> deals with the notion of <i>selections</i>, upon which both +cardinal and ordinal multiplication are based. A selection from a set +of classes is a class consisting of one member from each class of the +set. Thus a selective relation \(R\) may be defined as one which, for +a given class of classes \(\kappa\), makes \(Rʻ\alpha\) a member of +\(\alpha\) whenever \(\alpha\) is a member of \(\kappa\). More exactly, +a selective relation for a class of classes \(\kappa\) is one which +is one-many, which has \(\kappa\) for its converse domain, and is +such that, if \(x R \alpha\), then \(x \in \alpha\). Such a relation +may be called an \(\in\)-selector from \(\kappa\). More generally, +we may define a \(P\)-selector from \(\kappa\) as a relation which +is one-many, which has \(\kappa\) for its converse domain, and which +is contained in \(P\). The theory of selectors is very important in +arithmetic. But until we come to cardinal multiplication in Part III, +Section B, the propositions of this fourth section will seldom be +relevant.</p> + +<p><a href="#SECTION_E_b">Section E</a> deals with mathematical induction, not in the special +form in which it applies to finite integers (this is considered in +Part III, Section C), but in a general form in which it applies to +all relations. The propositions of this section are of very great +importance, primarily in the theory of finite and infinite (Part III, +Section C, and Part V, Section E), but also in many other subjects, +and especially in the derivation of series from one-many, many-one or +one-one relations—for example, in ordering the "rational" points of +a projective space by means of successive constructions of harmonic +points. The ideas involved in this section are somewhat complicated, +and we must refer the reader to the section itself for an account of +them.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_347">[Pg 347]</span></p> +<h2 class="nobreak" id="SECTION_A_b">SECTION A.<br> +UNIT CLASSES AND COUPLES.</h2> +</div> + + +<p><i>Summary of Section A.</i></p> + +<p>In this section we begin (<a href="#*50">*50</a>) by introducing a notation for the +<i>relation</i> of identity, as opposed to the <i>function</i> +"\(x=y\)"; that is, calling the relation of identity I, we put +\[ +I=\hat{x}\hat{y}(x=y) \quad\text{Df.} +\] +The purpose of this definition is chiefly convenience of notation. +The definition enables us to speak of \(\overrightarrow{I}\), +\(\text{D}ʻI\), \(I\mid R\), \(\alpha\upharpoonleft I\), \(Iʻʻ\alpha\), +etc., which we could not otherwise do.</p> + +<p>At the same time we introduce <i>diversity</i>, which is defined as the +negation of identity, and denoted by the letter \(J\). The properties +of \(I\) and \(J\) result immediately from <a href="#*13">*13</a>, since +\[ +xIy.\equiv.x=y. +\]</p> + +<p>We next introduce a very important notation, due to Peano, for the +class whose only member is \(x\). If we took a strictly and purely +extensional view of classes, we should naturally suppose this class to +be identical with \(x\). But in view of the theory of classes explained +in <a href="#*20">*20</a>, it is plain that \(x\) can never be identical with a class of +which it is a member, even when it is the only member of that class. +Peano uses the notation "\(\iota x\)" for the class whose only member +is \(x\); we shall alter this to "\(\iota ʻx\)," following our general +notation for descriptive functions. Thus we are to have +\[ +\iota ʻx=\hat{y}(y=x)=\hat{y}(yIx)=\overrightarrow{I}ʻx. +\] +Hence we take as our definition +\[ +\iota =\overrightarrow{I} \quad\text{Df}, +\] +since this definition gives the desired value of \(\iota ʻx\). The +properties of \(\iota\) are many and important.</p> + +<p>It is important to observe that "\(\breve{\iota}ʻ\alpha\)" means +"the only member of \(\alpha\)." Thus it exists when, and only when, +\(\alpha\) has one member and no more, in which case \(\alpha\) +is of the form \(\iota ʻx\), if x is its only member. Thus +"\(\breve{\iota}ʻ\alpha\)" means the same as "(\({℩}x)(x\in\alpha)\)," +and "\(\breve{\iota}ʻ\hat{z}(\phi z)\)" means the same as +"(\({℩}x)(\phi x)\)." What we call "\(\breve{\iota}ʻ\alpha\)" is +denoted, in Peano's notation, by "\({℩}\alpha\)."</p> + +<p><span class="pagenum" id="Page_348">[Pg 348]</span></p> + +<p>Classes of the form \(\iota ʻx\) are called <i>unit classes</i>, and +the class of all such classes is called 1. This is the cardinal number +1, according to the definition of cardinal numbers which will be given +in *100. The properties of 1, so far as they do not depend upon other +cardinals, or upon the fact that 1 is a cardinal, will be studied in +<a href="#*52">*52</a>.</p> + +<p>After a number (<a href="#*53">*53</a>) containing various propositions involving 1 or +\(\iota\), we pass to the consideration of cardinal couples (<a href="#*54">*54</a>) and +ordinal couples (<a href="#*55">*55</a>). A cardinal couple is a class \(\iota ʻx\cup +\iota ʻy\), where \(x\neq y\). The class of such couples is defined as +2, and will be shown at a later stage (*101) to be a cardinal number. +An ordinal couple, which, unlike a cardinal couple, involves an order +as between its members, is defined as a relation \(\iota ʻx\uparrow\iota ʻy\) +(cf. <a href="#*35·04">*35·04</a>), where we may either add \(x\neq y\) or not. +The properties of ordinal couples are in part analogous to those of +unit classes, in part to those of cardinal couples. In *56, we define +the ordinal number 2 (which we denote by \(2_{r}\), to distinguish +it from the cardinal 2) as the class of all ordinal couples +\(\iota ʻx\uparrow \iota ʻy\), where \(x\neq y\). It will be shown +at a later stage that this is an ordinal number according to our +definition of ordinal numbers (*153 and *251).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_349">[Pg 349]</span></p> +<h2 class="nobreak" id="*50">*50. IDENTITY AND DIVERSITY AS RELATIONS.</h2> +</div> + + +<p><i>Summary of *50.</i></p> + +<p>The purpose of the present number is primarily notational. For +notational reasons, we must be able to express identity and +diversity as relations, and not merely as propositional functions, +<i>i.e.</i> we require a notation for \(\hat{x}\hat{y}(x=y)\) and +\(\hat{x}\hat{y}(x\neq y)\). We therefore put +\[ +\begin{array}{l} +I=\hat{x}\hat{y}(x=y) & \text{Df,} \\ +J=\dot{-}I & \text{Df.} \\ +\end{array} +\]</p> + +<p>In spite of the fact that diversity is merely the negation of +identity, the kinds of propositions which employ diversity are quite +different from the kinds that employ identity. Identity as a relation +is required, to begin with, in the theory of unit classes, which is +our reason for treating of it at this stage. It is next required, +constantly, in the theory of mathematical induction (Part II, Section +E). It is required also in showing that cardinal and ordinal similarity +are reflexive. These are its principal uses.</p> + +<p>Diversity, on the other hand, is required almost exclusively in the +theory of series (Part V), and the first number in that theory will +be devoted to diversity. Until that stage, diversity will seldom be +referred to, with one important exception, namely in proving the +associative law of multiplication in relation-arithmetic (*174).</p> + +<p>The most important propositions on identity in the present number are +the following:</p> + +<p><b>50·16.</b> \(\vdash .Iʻʻ\alpha=\alpha\)</p> + +<p><b>50·4.</b> \(\vdash.R\mid I=I\mid R=R\)</p> + +<p><b>50·5.</b> \(\vdash.\alpha\upharpoonleft I=I\upharpoonright\alpha=\alpha\upharpoonleft I\upharpoonright \alpha\)</p> + +<p><b>50·51.</b> \(\vdash.\text{Cnv}ʻ(\alpha\upharpoonleft I)=\alpha\upharpoonleft I\)</p> + +<p><b>50·52.</b> \(\vdash.\text{D}ʻ(\alpha\upharpoonleft I)=\text{ᗡ}ʻ (\alpha\upharpoonleft I)=Cʻ(\alpha\upharpoonleft I)=\alpha\)</p> + +<p><b>50·62.</b> \(\vdash:\text{ᗡ}ʻR\subset \alpha.\supset.R \mid (I\upharpoonright\alpha)=R\)</p> + +<p><b>50·63.</b> \(\vdash:\text{D}ʻR\subset \alpha.\subset.I\upharpoonright\alpha \mid R=R\)</p> + +<p><span class="pagenum" id="Page_350">[Pg 350]</span></p> + +<p>The most important propositions on diversity in the present number are +the following:</p> + +<p class="nind"><b>*50·23</b>. \(\vdash: R \unicode{x2abd} J .\equiv. \breve{R} \unicode{x2abd} J\)</p> + +<p class="nind"><b>*50·24.</b> \(\vdash: R \unicode{x2abd} J .\equiv. (x){\sim}(x R x)\)</p> + +<p class="nind"><b>*50·43.</b> \(\vdash: R^{2} \unicode{x2abd}J .\equiv.R \dot{\cap} \breve{R} = \dot{\Lambda}\)</p> + +<p class="nind"><b>*50·45.</b> \(\vdash: R^{2} \unicode{x2abd} J .\supset. R \unicode{x2abd} J\)</p> + +<p class="nind"><b>*50·47.</b> \(\vdash\colon\ldotp R^{2} \unicode{x2abd} R.\supset: R \unicode{x2abd} J.\equiv. R^{2} \unicode{x2abd} J.\equiv. R \dot{\cap} \breve{R} = \dot{\Lambda}\)</p> + +<p>It will be observed that all these propositions are concerned with +\(R \unicode{x2abd} J\) or \(R^{2} \unicode{x2abd} J\), both of which are +satisfied if \(R\) is a <i>serial</i> relation. The hypothesis +\(R^{2}\unicode{x2abd} J\) or \(R \dot{\cap} \breve{R} = \dot{\Lambda}\) +characterizes an <i>asymmetrical</i> relation, <i>i.e.</i> one which, +if it holds between \(x\) and \(y\), cannot hold between \(y\) and +\(x\).</p> + +<hr class="tb"> + +<p class="nind"><b>*50·01.</b> \(I=\hat{x}\hat{y}(x=y) \quad\text{Df}\)</p> + +<p class="nind"><b>*50·02.</b> \(J=\dot{-}I \quad\text{Df}\)</p> + +<p>Most of the propositions of this number are obvious, and call for no +comment.</p> + +<p class="nind"><b>*50·1.</b> \(\vdash:xIy.\equiv.x=y \quad[\text{*21·3.(*50·01)}]\)</p> + +<p class="nind"><b>*50·11</b> \(\vdash:xJy.\equiv.x\neq y \quad[\text{*23·35.*50·1.(*50·02)}]\)</p> + +<p class="nind"><b>*50·12.</b> \(\vdash.J=\hat{x}\hat{y}(x\neq y) \quad[\text{*50·11.*21·33}]\)</p> + +<p class="nind"><b>*50·13.</b> \(\vdash.\dot{\exists}!I \quad[\text{*13·19.*10·24·281.*50·1}]\)</p> + +<p class="nind"><b>*50·14.</b> \(\vdash.Iʻy=y \quad[\text{*30·3.*50·1.*10·11}]\)</p> + +<p class="nind"><b>*50·15.</b> \(\vdash.(y).\exists!Iʻy \quad[\text{*50·14.*14·21.*10·11}]\)</p> + +<p class="nind"><b>*50·16.</b> \(\vdash.Iʻʻ\alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash\colon x\in Iʻʻ\alpha.&\equiv.(\exists y).y\in\alpha.xIy.\\ +[\text{*50.1}] &\equiv.(\exists y).y\in\alpha.x=y.\\ +[\text{*13·195}] &\equiv.x\in\alpha\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·17</b>. \(\vdash\colon\ldotp x\in\alpha.\supset_{x}.Rʻx=x\colon\supset.Rʻʻ\alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}.\supset\vdash\colon\text{Hp}.&\supset.\text{E}‼Rʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*50·14}.\supset\vdash\colon\ldotp\text{Hp}.&\supset\colon x\in\alpha.\supset_{x}.Rʻx=Iʻx:\\ +[\text{*37·69.(1)}] &\supset\colon Rʻʻ\alpha=Iʻʻ\alpha\colon\\ +[\text{*50·16}] &\supset\colon Rʻʻ\alpha=\alpha\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·2.</b> \(\vdash.I=\breve{I}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·1}.\supset\vdash\colon xIy.&\equiv.x=y.\\ +[\text{*13·16}] &\equiv.y=x.\\ +[\text{*50·1}] &\equiv.yIx.\\ +[\text{*31·11}] &\equiv.x\breve{I}y\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_351">[Pg 351]</span></p> + +<p class="nind"><b>*50·21.</b> \(\vdash.J=\breve{J}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·2.(*50·02)}.\supset\vdash.J&=\dot{-}I &\qquad \text{(1)} \\ +[\text{*50·2.*23·83}] & =\dot{-}\breve{I} \\ +[\text{*31·16}] & =\text{Cnv}ʻ\dot{-}I +[\text{(1).*31·32}] =\breve{J}.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·22.</b> \(\vdash\colon R \unicode{x2abd} I .\equiv. \breve{R} \unicode{x2abd} I \quad[\text{*31·4, *50·2}]\)</p> + +<p class="nind"><b>*50·23.</b> \(\vdash\colon R \unicode{x2abd} J .\equiv. \breve{R} \unicode{x2abd} J \quad[\text{*31·4, *50·21}]\)</p> + +<p class="nind"><b>*50·24.</b> \(\vdash\colon R \unicode{x2abd} J .\equiv. (x)\,{\sim}(xRx)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·11}.\supset\vdash\colon\ldotp R\unicode{x2abd}J.&\equiv:xRy.\supset_{x,y}.x\neq y:\\ +[\text{Transp}] & \equiv :x=y.\supset_{x,y}.{\sim}(xRy): \\ +[\text{*13·191}] & \equiv :(x)=.{\sim}(xRx)\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*50·3.</b> \(\vdash.(x).xIx \quad[\text{*50·1.*13·15}]\)</p> + +<p class="nind"><b>*50·31.</b> \(\vdash.\text{D}ʻI=\text{V}.\text{ᗡ}ʻI=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·3.*10·24}.&\supset\vdash\colon\ldotp (x):(\exists y).xIy\colon\ldotp (x):(\exists y).yIx\colon\ldotp \\ +[\text{*33·13·131}] & \supset\vdash :(x).x\in\text{D}ʻI:(x).x\in \text{ᗡ}ʻI:\\ +[\text{*24·14}] & \supset\vdash.\text{D}ʻI=\text{V}.\text{ᗡ}ʻI=\text{V}.\supset\vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·32</b>. \(\vdash.CʻI=\text{V} \quad[\text{*50·31.*33·16.*24·27}]\)</p> + +<p class="nind"><b><a id="*50·33">*50·33</a>.</b> \(\vdash:\dot{\exists}!J.\supset.\text{D}ʻ J=\text{V}.\text{ᗡ}ʻJ=\text{V}.CʻJ=\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*13·171.Transp}.&\supset\vdash\colon\ldotp y\neq z.&\supset :x\neq y.\lor.x\neq z\colon\ldotp \\ +[\text{*50·11}] & \supset\vdash\colon\ldotp yJz.&\supset :xJy.\lor.xJz: \\ +[\text{*33·14}] && \supset:x\in\text{D}ʻJ &\qquad \text{(1)} \\ +\vdash.\text{(1).*11·11·35}. \supset\vdash:\dot{\exists}!J. &\supset .x\in \text{D}ʻJ: \\ +[\text{*10·11·21}] & \supset\vdash:\dot{\exists}!J.&\supset .(x).x\in \text{D}ʻJ. \\ +[\text{*24·14}] &&\supset .\text{D}ʻJ=\text{V} &\qquad \text{(2)} \\ +\vdash.\text{(2).*50·21}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition (<a href="#*50·33">*50·33</a>), the hypothesis \(\dot{\exists}!J\) +is equivalent to the hypothesis that more than one object exists of +the type in question. This can be proved for all except the lowest +type. For the lowest type, we can only prove the existence of at +least one object: this is proved in <a href="#*24·52">*24·52</a>. For the next type, we can +prove the existence of at least two objects, namely \(\Lambda\) and +\(\text{V}\); these are distinct, by<a href="#*24·1">*24·1</a>. For a class of order \(n\), +we can prove the existence of \(2^{n}\) objects. But for the class of +individuals we cannot prove,<span class="pagenum" id="Page_352">[Pg 352]</span> from our primitive propositions, that +there is more than one object in the universe, and therefore we cannot +prove \(\dot{\exists}!J\). We might, of course, have included among our +primitive propositions the assumption that more than one individual +exists, or some assumption from which this would follow, such as +\[ +(\exists\phi,x,y).\phi!x.{\sim}\phi!y. +\] +But very few of the propositions which we might wish to prove depend +upon this assumption, and we have therefore excluded it. It should be +observed that most philosophers, being monists, deny this assumption.</p> + +<p class="nind"><b>*50·34.</b> \(\vdash.\dot{\exists}!J\unicode{x0294f}\text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*20·41.*22·38.(*24·01·02)}.&\supset\vdash.\Lambda,\text{V}\in\text{Cls}. \\ +[\text{*24·1}] &\supset\vdash.\Lambda\neq \text{V}.\Lambda,\text{V}\in\text{Cls}. \\ +[\text{*36·13.*50·11}] &\supset\vdash.\Lambda\{J\,\unicode{x0294f}\,\text{Cls}\}\text{V}. \\ +[\text{*10·24}] &\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*50·35.</b> \(\vdash.\dot{\exists}!J\unicode{x0294f}\text{Rel} \quad[\text{Proof as in *50·34}]\)</p> + +<p class="nind"><b>*50·4.</b> \(\vdash.R \mid I=I \mid R=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1}.\supset\vdash:x(R \mid I)z.&\equiv.(\exists y).xRy.yIz. \\ +[\text{*50·1}] & \equiv.(\exists y).xRy.y=z. \\ +[\text{*13·195}] & \equiv.xRz &\qquad \text{(1)} \\ +\vdash.\text{*34·1}.\supset\vdash:x(I \mid R)z.&\equiv.(\exists y).xIy.yRz. \\ +[\text{*50·1}] &\equiv.(\exists y).x=y.yRz. \\ +[\text{*13·195}] &\equiv.xRz &\qquad \text{(2)} \\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·41.</b> \(\vdash:R \mid \breve{P}\unicode{x2abd}J.\equiv.\breve{R} \mid P\unicode{x2abd}J.\equiv.R\dot{\cap}P=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1.*50·11}.\supset\vdash\colon\ldotp R \mid \breve{P}\unicode{x2abd} J.&\equiv:(\exists y).xRy.y\breve{P}z.\supset_{x,z}.x\neq z: \\ +[\text{*13·196}] & \equiv:(x):{\sim}(\exists y).xRy.y\breve{P}x: \\ +[\text{*10·252}] & \equiv:{\sim}(\exists x,y).xRy.y\breve{P}x: \\ +[\text{*31·11}] & \equiv:{\sim}(\exists x,y).xRy.xPy: \\ +[\text{*23·33.*25·51}] & \equiv:R\dot{\cap}P=\dot{\Lambda}: &\qquad \text{(1)} \\ +[\text{*31·14·24}] & \equiv:\breve{R}\dot{\cap}\breve{P}=\dot{\Lambda}: \\ +\left[\text{(1)}\, \frac{\breve{R},\,\breve{P}}{R,\,P}\right] & \equiv:\breve{R}\mid \text{Cnv}ʻ\breve{P}\unicode{x2abd} J: \\ +[\text{*34·203}] & \equiv:\breve{R}\mid P\unicode{x2abd} J &\qquad \text{(2)} \\ +\vdash.(1).(2).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_353">[Pg 353]</span></p> + +<p class="nind"><b>*50·42.</b> \(\vdash. I^{2} = I\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{ *34·5} .\supset\vdash : xI^{2} z.& \equiv . (\exists y) . xIy. yIz .\\ +[\text{*50·1}] &\equiv .(\exists y).xIy. y = z.\\ +[\text{*13·195}] &\equiv . xIz : \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·43.</b> \(\vdash: R^{2} \unicode{x2abd} J. \equiv . R\dot{\cap})\breve{R} = \dot{\Lambda} \quad\left[\text{*50·41}\, \frac{\breve{R}}{P}\right]\)</p> + +<p>This proposition is useful in the theory of series. +"\(R \dot{\cap}\breve{R} = \dot{\Lambda}\)" is the characteristic +of an <i>asymmetrical</i> relation.</p> + +<p class="nind"><b>*50·44.</b> \(\vdash: \dot{\exists}!(R \dot{\cap} I). \supset . \dot{\exists}! (R^{2} \dot{\cap} I)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*23·33.*50·1}. \supset\vdash : \dot{\exists}! (R \dot{\cap} I).& \equiv . (\exists x, y) . xRy . x = y.\\ +[\text{*13·195}] &\equiv . (\exists x) . xRx.\\ +[\text{*34·54}] &\supset . (\exists x) . xR^{2} x .\\ +[\text{*13·195}] &\supset . (\exists x, y) .xR^{2} y . x = y.\\ +[\text{*23·33.*50·1}]&\supset . \dot{\exists}! (R^{2} \dot{\cap} I) : \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·45.</b> \(\vdash: R^{2} \unicode{x2abd} J. \supset . R \unicode{x2abd} J \quad[\text{*50·44. Transp . *25·311}]\)</p> + +<p class="nind"><b>*50·46.</b> \(\vdash: R \dot{\cap}\breve{R} = \dot{\Lambda} .\supset. R \unicode{x2abd} J \quad[\text{*50·43·45}]\)</p> + +<p class="nind"><b>*50·47.</b> \(\vdash\colon\ldotp R^{2} \unicode{x2abd} R. \supset : R \unicode{x2abd} J. \equiv . R^{2} \unicode{x2abd} J. \equiv . R\dot{\cap} \breve{R} = \dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{ *23·44} . \supset\vdash \colon\ldotp \text{Hp} . \supset : R \unicode{x2abd} J.\supset . R^{2}\unicode{x2abd} J &\qquad \text{(1)}\\ +\vdash. \text{(1) . *50·45·43} . \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in the theory of series. If \(R\) is a +serial relation, we shall have \(R^{2} \unicode{x2abd} R\) and \(R \unicode{x2abd} J\).</p> + +<p class="nind"><b>*50·5.</b> \(\vdash. \alpha\upharpoonleft I = I\upharpoonright \alpha = \alpha\upharpoonleft I\upharpoonright \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1}. \supset\vdash: x (\alpha\upharpoonleft I)y. &\equiv . x\in \alpha. xIy.\\ +[\text{*50·1}] &\equiv . x\in \alpha. x = y.\\ +[\text{*13·193}] & \equiv . y\in \alpha. x = y.\\ +[\text{*50·1}] &\equiv . xIy . y\in \alpha.\\ +[\text{*35·101}] & \equiv.x(I\upharpoonright \alpha)y &\qquad \text{(1)}\\ +\vdash. \text{(1). *23·5} .\supset\vdash .\alpha\upharpoonleft I &= \alpha\upharpoonleft I\dot{\cap}I\upharpoonright \alpha\\ +[\text{*35·11}] & \equiv \alpha\upharpoonleft I\upharpoonright \alpha &\qquad \text{(2)}\\ +\vdash. \text{(1). (2)} .\supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_354">[Pg 354]</span></p> + +<p class="nind"><b>*50·51.</b> \(\vdash.\text{Cnv}ʻ(\alpha\upharpoonleft I)=\alpha\upharpoonleft I \quad[\text{*35·51.*50·2·5}]\)</p> + +<p class="nind"><b>*50·52.</b> \(\vdash.\text{D}ʻ(\alpha\upharpoonleft I)=\text{ᗡ}ʻ(\alpha\upharpoonleft I)=Cʻ(\alpha\upharpoonleft I)=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·61}.\supset\vdash.\text{D}ʻ(\alpha\upharpoonleft I)&=\alpha\cap \text{D}ʻI\\ +[\text{*50·31}] &=\alpha\cap \text{V}\\ +[\text{*24·26}] &=\alpha &\qquad \text{(1)}\\ +\text{Similarly} \vdash.\text{ᗡ}ʻ(\alpha\upharpoonleft I)&=\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*33·18}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·53.</b> \(\vdash.\alpha\upharpoonleft I\upharpoonright \beta=(\alpha\cap \beta)\upharpoonleft I=I\upharpoonright (\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·21.*50·5}.\supset\vdash.\alpha\upharpoonleft I\upharpoonright \beta&=\alpha\upharpoonleft (\beta\upharpoonleft I)\\ +[\text{*35·32}] & =(\alpha\cap \beta)\upharpoonleft I &\qquad \text{(1)}\\ +\vdash.\text{(1).*50·5}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·54.</b> \(\vdash.(\alpha\upharpoonleft I)^{2}=\alpha\upharpoonleft I\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·5}.\supset\vdash.(\alpha\upharpoonleft I)^{2}&=(\alpha\upharpoonleft I)\mid (I\upharpoonright \alpha)\\ +[\text{*35·12}] & =\alpha\upharpoonleft I^{2}\upharpoonright \alpha\\ +[\text{*50·42}] &=\alpha\upharpoonleft I\upharpoonright \alpha\\ +[\text{*50·5}] &=\alpha\upharpoonleft I.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·55.</b> \(\vdash:\alpha\cap \beta=\Lambda.\equiv.\alpha\uparrow \beta\unicode{x2abd}J\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·37.*50·11}.\supset\\ +\vdash\colon\ldotp \alpha\cap \beta=\Lambda.&\equiv:x\in \alpha.y\in \beta.\supset_{x,y}.xJy:\\ +[\text{*35·103}] &\equiv:\alpha\uparrow \beta\unicode{x2abd}J\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·56.</b> \(\vdash:\exists !(\alpha\cap \beta).\equiv.\dot{\exists}!\{(\alpha\uparrow \beta)\dot{\cap}I\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·55.Transp.*24·54}.\supset\\ +\vdash:\exists !(\alpha\cap \beta).&\equiv.{\sim}\{\alpha\uparrow \beta\unicode{x2abd}J\}.\\ +[\text{*25·55}] &\equiv.\dot{\exists}(\alpha\uparrow \beta)\dot{-}J.\\ +[\text{*23·831.(*50·02)}]&\equiv.\dot{\exists}!\{(\alpha\uparrow \beta)\dot{\cap}I\}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·57.</b> \(\vdash.I\dot{\cap}\alpha\upharpoonleft R=I\dot{\cap}R\upharpoonright \alpha=I\dot{\cap}\alpha\upharpoonleft R\upharpoonright \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·16}.\supset\vdash.I\dot{\cap}\alpha\upharpoonleft R&=\alpha\upharpoonleft I\dot{\cap}R\\ +[\text{*50·5}] & =I\upharpoonright \alpha\dot{\cap}R\\ +[\text{*35·17}] &=I\dot{\cap}R\upharpoonright \alpha &\qquad \text{(1)}\\ +[\text{*50·5}] & =\alpha\upharpoonleft I\upharpoonright \alpha\dot{\cap}R\\ +[\text{*35·16·17·21}] &=I\dot{\cap}\alpha\upharpoonleft R\upharpoonright \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_355">[Pg 355]</span></p> + +<p class="nind"><b>*50·58.</b> \(\vdash:\alpha\upharpoonleft R\unicode{x2abd}J.\equiv.R\upharpoonright \alpha\unicode{x2abd}J.\equiv.\alpha\upharpoonleft R\upharpoonright \alpha\unicode{x2abd}J\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·57}.\supset\vdash:I\dot{\cap}\alpha\upharpoonleft R=\dot{\Lambda}.\equiv.I\dot{\cap}R\upharpoonright \alpha=\dot{\Lambda}.\equiv.I\dot{\cap}\alpha\upharpoonleft R\upharpoonright \alpha=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash.\text{(1).*50·41}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·59.</b> \(\vdash.(I\upharpoonright \alpha)ʻʻ\beta=\alpha\cap \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·412}.\supset\vdash.(I\upharpoonright \alpha)ʻʻ\beta&=Iʻʻ(\alpha\cap \beta)\\ +[\text{*50·16}] & =\alpha\cap \beta.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·6.</b> \(\vdash.R\mid (I\upharpoonright \alpha)=R\upharpoonright \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·23}.\supset\vdash.R\mid (I\upharpoonright \alpha)&=(R\mid I)\upharpoonright \alpha\\ +[\text{*50·4}] &=R\upharpoonright \alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·61.</b> \(\vdash.I\upharpoonright \alpha\mid R=\alpha\upharpoonleft R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·354}.\supset\vdash.I\upharpoonright \alpha\mid R&=I\mid (\alpha\upharpoonleft R)\\ +[\text{*50·4}] & =\alpha\upharpoonleft R.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·62.</b> \(\vdash:\text{ᗡ}ʻR\subset\alpha.\supset.R\mid (I\upharpoonright \alpha)=R \quad[\text{*50·6.*35·452}]\)</p> + +<p class="nind"><b>*50·63.</b> \(\vdash:\text{D}ʻR\subset\alpha.\supset.I\upharpoonright \alpha\mid R=R \quad[\text{*50·61.*35·451}]\)</p> + +<p class="nind"><b>*50·64.</b> \(\vdash.R\mid (I\upharpoonright \text{ᗡ}ʻR)=R\mid (I\upharpoonright CʻR)=R \quad[\text{*50·62.*22·42.*33·161}]\)</p> + +<p class="nind"><b>*50·65.</b> \(\vdash.I\upharpoonright (\text{D}ʻR)\mid R=I\upharpoonright (CʻR)\mid R=R \quad[\text{*50·63.*22·42.*33·161}]\)</p> + +<p class="nind"><b>*50·7.</b> \(\vdash:\text{ᗡ}ʻR\subset\alpha.\supset.R\mid ʻI\upharpoonright \alpha=R \quad[\text{*50·62.*43·11}]\)</p> + +<p class="nind"><b>*50·71.</b> \(\vdash:\text{D}ʻR\subset\alpha.\supset.\mid RʻI\upharpoonright \alpha=R \quad[\text{*50·63.*43·111}]\)</p> + +<p class="nind"><b>*50·72.</b> \(\vdash.R\Vert ʻ(I\upharpoonright CʻR)=\mid Rʻ(I\upharpoonright CʻR)=R \quad[\text{*50·7·71}]\)</p> + +<p class="nind"><b>*50·73.</b> \(\vdash.R\Vert ʻI=\mid RʻI=R \quad[\text{*50·4.*43·11·111}]\)</p> + +<p class="nind"><b>*50·74.</b> \(\vdash.R\Vert I=R\mid\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·112}.\supset\vdash.(R\mid I)ʻQ&=R\mid Q\mid I\\ +[\text{*50·4}] & =R\mid Q\\ +[\text{*43·11}] & =R\mid ʻQ &\qquad \text{(1)}\\ +\vdash.\text{(1).*30·41}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·75.</b> \(\vdash.I\Vert R=\mid R \quad[\text{Proof as in *50·74}]\)</p> + +<p class="nind"><b>*50·76.</b> \(\vdash:P\mid =R\mid .\equiv.P=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·27.*30·41}.&\supset\vdash:P=R.\supset.P\mid =R\mid &\qquad \text{(1)}\\ +\vdash.\text{*50·73.*30·36}.&\supset\vdash:P\mid =R\mid .\supset.P=R &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*50·761.</b> \(\vdash:\mid P=\mid R.\equiv.P=R \quad[\text{Proof as in *50·76}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_356">[Pg 356]</span></p> +<h2 class="nobreak" id="*51">*51. UNIT CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *51.</p> + +<p>In this number we introduce a new descriptive function \(\iotaʻx\), +meaning "the class of terms which are identical with \(x\)," which is +the same thing as "the class whose only member is \(x\)." We are thus +to have +\[ +\iotaʻx = \hat{y} (y = x)\text{.} +\] +But \(\hat{y}(y = x) = \overrightarrow{I}ʻx\). Hence we secure what we +require by the following definition:</p> + +<p class="nind"><b>*51·01.</b> \(\iota = \overrightarrow{I} \quad \text{Df}\)</p> + +<p>As a matter of notation, it might be thought that +\(\overrightarrow{I}\) would do as well as \(\iota\), and that this +definition is superfluous. But we need also the converse of this +relation, and "\(\text{Cnv}ʻ\overrightarrow{I}\)" is not a sufficiently +convenient symbol.</p> + +<p>The propositions of this number are constantly used in what follows. It +should be observed that the class whose members are \(x\) and \(y\) is +\(\iota ʻx \cup \iotaʻy\), the class whose members are \(x\), \(y\), +\(z\) is \(\iota ʻx \cup \iotaʻy \cup \iotaʻz\), the class formed by +adding \(x\) to \(\alpha\) is \(\alpha \cup \iotaʻx\), and the class +formed by taking \(x\) away from \(\alpha\) is \(\alpha - \iotaʻx\). +(If \(x\) is not a member of \(\alpha\), this is equal to \(\alpha\).)</p> + +<p>The distinction between \(x\) and \(\iotaʻx\) is one of the merits of +Peano's symbolic logic, as well as of Frege's. On the basis of our +theory of classes, the necessity for the distinction is of course +obvious. But apart from this, the following consideration makes the +necessity apparent. Let \(\alpha\) be a class; then the class whose +only member is \(\alpha\) has only one member, namely \(\alpha\), while +\(\alpha\) may have many members. Hence the class whose only member is +\(\alpha\) cannot be identical with \(\alpha\)<a id="FNanchor_59" href="#Footnote_59" class="fnanchor">[59]</a>.</p> + +<p>The propositions of the present number which are most used are the +following:</p> + +<p class="nind"><b>*51·15.</b> \(\vdash : y \in \iotaʻx .\equiv. y = x\)</p> + +<p class="nind"><b>*51·16.</b> \(\vdash . x \in \iotaʻx\)</p> + +<p class="nind"><b><a id="*51·2">*51·2</a>.</b> \(\vdash : x \in \alpha .\equiv. \iotaʻx \subset \alpha\)</p> + +<p><span class="pagenum" id="Page_357">[Pg 357]</span></p> + +<p>This proposition is useful because it enables us to replace membership +of a class (\(x \in \alpha\)) by inclusion in the class (\(\iotaʻx\subset \alpha\)).</p> + +<p class="nind"><b>*51·211.</b> \(\vdash\colon x{\sim}\in\alpha.\equiv.\iotaʻx\cap \alpha=\Lambda\)</p> + +<p class="nind"><b>*51·221.</b> \(\vdash\colon x\in\alpha.\equiv.(\alpha-\iotaʻx)\cup \iotaʻx=\alpha\)</p> + +<p class="nind"><b>*51·222.</b> \(\vdash\colon x{\sim}\in\alpha.\equiv.\alpha-\iotaʻx=\alpha\)</p> + +<p class="nind"><b><a id="*51·23">*51·23</a>.</b> \(\vdash\colon\iotaʻx=\iotaʻy.\equiv.y\in\iotaʻx.\equiv.x\in\iotaʻy.\equiv.x=y\)</p> + +<p class="nind"><b>*51·4.</b> \(\vdash\colon\exists!\alpha.\alpha\subset\iotaʻx.\equiv.\alpha=\iotaʻx\)</p> + +<p><i>I.e.</i> an existent class contained in a unit class must be +identical with the unit class. From this proposition it will follow +that 0 is the only cardinal which is less than 1.</p> + +<p class="nind"><b>*51·51.</b> \(\vdash\colon\alpha=\iotaʻx.\equiv.x=\breve{\iota}ʻ\alpha.\equiv.x\breve{\iota}\alpha\)</p> + +<p>For classes, \(\breve{\iota}ʻ\alpha\) has the same uses that +(\({℩}x)(\phi x)\) has for functions; "\(\breve{\iota}ʻ\alpha\)" means +"the only member of \(\alpha\)." We have</p> + +<p class="nind"><b>*51·59.</b> \(\vdash:\psi{\breve{\iota}ʻ\hat{z}(\phi z)}.\equiv.\psi(℩x)(\phi x)\)</p> + +<hr class="tb"> + +<p class="nind"><b>*51·01.</b> \(\iota=\overrightarrow{I} \quad\text{Df}\)</p> + +<p class="nind"><b>*51·1.</b> \(\vdash\colon\alpha \iota x.\equiv.\alpha=\hat{y}(y=x)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·2.(*51·01)}.\supset\vdash\colon\alpha\iota x.&\equiv.\alpha\overrightarrow{I}x. \\ +[\text{*32·1}] & \equiv.\alpha=\hat{y}(yIx). \\ +[\text{*50·1}] &\equiv.\alpha=\hat{y}(y=x):\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·11.</b> \(\vdash.\iotaʻ x=\hat{y}(y=x) \quad[\text{*30·3.*51·1}]\)</p> + +<p class="nind"><b>*51·12.</b> \(\vdash._\text{E}!\iotaʻ x \quad[\text{*51·11.*14·21}]\)</p> + +<p class="nind"><b>*51·13.</b> \(\vdash\colon\alpha=\iotaʻ x.\equiv.\alpha=\hat{y}(y=x) \quad [\text{*20·57·2.*51·11}]\)</p> + +<p class="nind"><b>*51·131.</b> \(\vdash\colon\alpha\iota x.\equiv.\alpha=\iotaʻ x \quad[\text{*51·1·13}]\)</p> + +<p class="nind"><b>*51·14.</b> \(\vdash\colon\ldotp \alpha=\iotaʻ x.\equiv\colon y\in\alpha.\equiv_{y}.y=x \quad[\text{*51·13.*20·33}]\)</p> + +<p class="nind"><b>*51·141.</b> \[\begin{align}\vdash\colon\ldotp \alpha=\iotaʻ x.\equiv:\exists!\alpha\colon y\in\alpha.\supset_{y}.y&=x\colon\equiv\colon + x\in\alpha\colon y\in\alpha.\supset_{y}.y=x\\ +&\quad[\text{*51·14.*14·122}]\end{align}\]</p> + +<p class="nind"><b>*51·15.</b> \(\vdash:y\in\iotaʻ x.\equiv.y=x \quad[\text{*51·11.*20·33}]\)</p> + +<p class="nind"><b>*51·16.</b> \(\vdash.x\in\iotaʻ x \quad[\text{*51·15.*13·15}]\)</p> + +<p class="nind"><b>*51.161.</b> \(\vdash.\exists!\iotaʻ x \quad[\text{*51·16.*10·24}]\)</p> + +<p class="nind"><b>*51·17.</b> \(\vdash.\text{ᗡ}ʻ\iota=\text{V}\)</p> + + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·1.*20·2}.& \supset\vdash.\{\hat{y}(y=x)\}\iota x. \\ +[*10·24] & \supset\vdash.(\exists\alpha).\alpha\iota x. \\ +[\text{*33·131}] & \supset\vdash.x\in\text{ᗡ}ʻ\iota. \\ +[\text{*10·11}] & \supset\vdash.(x).x\in\text{ᗡ}ʻ\iota. \\ +[\text{*24·14}] & \supset\vdash.\text{ᗡ}ʻ\iota=\text{V} \\ +\end{array} +\]</p> + +<p>The above proposition is used in the theory of selections (<a href="#*83·71">*83·71</a>).</p> + +<p><span class="pagenum" id="Page_358">[Pg 358]</span></p> + +<p class="nind"><b>*51·2.</b> \(\vdash:x\in\alpha.\equiv.\iotaʻ x\supset \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*13·191}.\supset\vdash\colon\ldotp x\in\alpha.&\equiv:y=x.\supset_{y}.y\in\alpha: \\ +[\text{*51·15}] & \equiv:y\in\iotaʻ x.\supset_{y}.y\in\alpha: \\ +[\text{*22·1}] & \equiv:\iotaʻ x\supset \alpha\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p>The above proposition shows how to replace membership of a class by +inclusion in a class; thus for example it gives: +\[ +\begin{align} +\text{Socrates is a man}\,.\equiv.\,&\text{the class of terms identical with Socrates is included} +\\ +&\text{in the class of men}. +\end{align} +\]</p> + +<p>Before Peano and Frege, the relation of membership (\(\in\)) was +regarded as merely a particular case of the relation of inclusion +(\(\subset\)). For this reason, the traditional formal logic treated +such propositions as "Socrates is a man" as instances of the universal +affirmative \(A\), "All \(S\) is \(P\)," which is what we express by +"\(\alpha\subset\beta\)." This involved a confusion of fundamentally +different kinds of propositions, which greatly hindered the development +and usefulness of symbolic logic. But by means of the above proposition +(<a href="#*51·2">*51·2</a>), we can always obtain a proposition stating an inclusion +(namely "\(\iotaʻx\subset\alpha\)") which is equivalent to a given +proposition stating membership of a class (namely "\(x\in\alpha\)").</p> + +<p class="nind"><b>*51·21.</b> \(\vdash.x{\sim} \in\alpha -\iotaʻ x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·33·5}.\supset\vdash:x\in\alpha -\iotaʻ x.&\equiv.x\in\alpha.x{\sim}\in\iotaʻ x. \\ +[\text{*3·27}] & \supset.x{\sim} \in\iotaʻ x \qquad\qquad \text{(1)} \\ +\vdash.\text{(1).Transp.*51·16}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*51·211.</b> \(\vdash:x{\sim} \in\alpha.\equiv.\iotaʻ x\cap\alpha=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·39}.\supset\vdash\colon\ldotp \iotaʻ x\cap \alpha=\Lambda.&\equiv:y\in\iotaʻ x.\supset_{y}.y{\sim} \in\alpha: \\ +[\text{*51·15}] & \equiv:y=x.\supset_{y}.y{\sim} \in\alpha: \\ +[\text{*13·191}] & \equiv:x{\sim}\in\alpha\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·22.</b> \(\vdash:\alpha\cap \iotaʻ x=\Lambda.\alpha\cup \iotaʻ x=\beta.\equiv.x\in\beta.\alpha=\beta -\iotaʻ x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{ll} +\vdash.\text{*24·47}.\supset \\ +\vdash:\alpha\cap \iotaʻ x=\Lambda.\alpha\cup \iotaʻ x=\beta.&\equiv.\iotaʻ x\supset \beta.\alpha=\beta - \iotaʻ x. \\ +[\text{*51·2}] & \equiv.x\in\beta.\alpha=\beta - \iotaʻ x:\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·221.</b> \(\vdash:x\in\alpha.\equiv.(\alpha -\iotaʻ x)\cup \iotaʻ x=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·2}.\supset\vdash:x\in\alpha.&\equiv.\iotaʻ x\subset \alpha. \\ +[\text{*22·62}] & \equiv.\iotaʻ x\cup \alpha=\alpha. \\ +[\text{*22·91}] & \equiv.(\alpha - \iotaʻ x)\cup \iotaʻ x=\alpha:\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_359">[Pg 359]</span></p> + +<p class="nind"><b>*51·222.</b> \(\vdash:x{\sim} \in\alpha.\equiv.\alpha - \iotaʻ x=\alpha \quad[\text{*51·211.*24·313}]\)</p> + +<p class="nind"><b>*51·23.</b> \(\vdash:\iotaʻ x= \iotaʻ y.\equiv.y\in\iotaʻ x.\equiv.x\in\iotaʻ y.\equiv.x=y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*20·31.*51·15}.\supset \\ +\vdash\colon\ldotp \iotaʻ x=\iotaʻ y.&\equiv:z=x.\equiv_{z}.z=y: \\ +[\text{*13·183}] &\equiv:x=y: &\qquad \text{(1)} \\ +[\text{*51·15}] &\equiv:x\in\iotaʻ y: &\qquad \text{(2)} \\ +[\text{(1).*13·16}] &\equiv:y\in\iotaʻ x &\qquad \text{(3)} \\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·231.</b> \(\vdash:\iotaʻ x\cap \iotaʻ y=\Lambda.\equiv.x\neq y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·311}. \supset\vdash\colon\ldotp \iotaʻ x\cap \iotaʻ y=\Lambda.&\equiv:\iotaʻ x\subset-\iotaʻ y: \\ +[\text{*51·15}] & \equiv:z=x.\supset_{z}.z\neq y: \\ +[\text{*3·191}] & \equiv:x\neq y\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b><a id="*51·232">*51·232</a>.</b> \(\vdash\colon\ldotp z\in(\iotaʻ x\cup \iotaʻ y).\equiv:z=x.\lor.z=y \quad[\text{*22·34.*51·15}]\)</p> + +<p>This proposition states that a member of \(\iotaʻx\cup\iotaʻy\) +must be either \(x\) or \(y\), and vice versa, <i>i.e.</i> that +\(\iotaʻx\cup\iotaʻy\) is the class whose only members are \(x\) and +\(y\).</p> + +<p class="nind"><b>*51·233.</b> \[\begin{align}&\vdash\colon\colon \alpha=\iotaʻ x\cup \iotaʻ y.\supset\colon\ldotp(z)\colon\ldotp z\in\alpha.\equiv:z=x.\lor.z=y\\ +&[\text{*51·232.*10·11.*20·18}]\end{align}\]</p> + +<p class="nind"><b><a id="*51·234">*51·234</a>.</b> \(\vdash\colon\colon \alpha=\iotaʻ x\cup \iotaʻ y.\supset\colon\ldotp z\in\alpha.\supset_{z}.\phi z:\equiv.\phi x.\phi y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·233}. \supset\vdash\colon\colon\ldotp\text{Hp}.\supset\colon\colon z\in\alpha.\supset_{z}.\phi z:&\equiv\colon\ldotp z=x.\lor.z=y:\supset_{z}.\phi + z\colon\ldotp \\ +[\text{*4·77}] &\equiv\colon\ldotp(z)\colon\ldotp z=x.\supset.\phi z:z=y.\supset.\phi z\colon\ldotp \\ +[\text{*10·22}] &\equiv\colon\ldotp z=x.\supset_{z}.\phi z:z=y.\supset_{z}.\phi z\colon\ldotp \\ +[\text{*13·191}] &\equiv\colon\ldotp \phi x.\phi y\colon\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b><a id="*51·235">*51·235</a>.</b> \(\vdash\colon\colon \alpha=\iotaʻ x\cup \iotaʻ y.\supset\colon\ldotp(\exists z).z\in\alpha.\phi z.\equiv:\phi x.\lor.\phi y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·233}.\supset \\ +\vdash\colon\colon\text{Hp}.\supset\colon\ldotp(\exists z).z\in\alpha.\phi z.&\equiv:(\exists z):z=x.\lor.z=y:\phi z: \\ +[\text{*4·4}] & \equiv:(\exists z):z=x.\phi z.\lor.z=y.\phi z: \\ +[\text{*10·42}] & \equiv:(\exists z).z=x.\phi z.\lor.(\exists z).z=y.\phi z: \\ +[\text{*13·195}] & \equiv:\phi x.\lor.\phi y\colon\colon\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·236.</b> \(\vdash\colon\ldotp z\in \iotaʻ x\cup \beta.\equiv:z=x.\lor.z\in\beta \quad[(\text{*22·34.*51·15}]\)</p> + +<p class="nind"><b>*51·237.</b> \[\begin{align}&\vdash\colon\colon \alpha=\iotaʻ x\cup \beta.\supset\colon\ldotp(z)\colon\ldotp z\in\alpha.\equiv:z=x.\lor.z\in\beta\\ +&[(\text{*51·236.*10·11.*20·18}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_360">[Pg 360]</span></p> + +<p class="nind"><b>*51·238.</b> \(\vdash\colon\colon \alpha=\iotaʻ x\cup \beta.\supset\colon\ldotp z\in\alpha.\supset_{z}.\phi z:\equiv:\phi x:z\in\beta.\supset_{z}.\phi z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·237}.\supset\vdash\colon\colon\ldotp \text{Hp}.\supset\colon\colon z\in\alpha.\supset_{z}.\phi z:&\equiv\colon\ldotp z=x.\lor.z\in\beta:\supset_{z}.\phi + z\colon\ldotp \\ +[\text{*4·77}] & \equiv\colon\ldotp(z)\colon\ldotp z=x.\supset.\phi z:z\in\beta.\supset.\phi z\colon\ldotp \\ +[\text{*10·22}] & \equiv\colon\ldotp z=x.\supset_{z}.\phi z:z\in\beta.\supset_{z}.\phi z\colon\ldotp \\ +[\text{*13·191}] & \equiv\colon\ldotp \phi x:z\in\beta.\supset_{z}.\phi z\colon\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·239.</b> \(\vdash\colon\colon \alpha=\iotaʻ x\cup \beta.\supset\colon\ldotp(\exists z).z\in\alpha.\phi z.\equiv:\phi x.\lor.(\exists z).z\in\beta.\phi z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·237}.\supset \\ +\vdash\colon\colon\text{Hp}.\supset\colon\ldotp(\exists z).z\in\alpha.\phi z.&\equiv:(\exists z):z=x.\lor.z\in\beta:\phi z: \\ +[\text{*4·4}] & \equiv:(\exists z):z=x.\phi z.\lor.z\in\beta.\phi z: \\ +[\text{*10·42}] & \equiv:(\exists z).z=x.\phi z.\lor.(\exists z).z\in\beta.\phi z: \\ +[\text{*13·195}] & \equiv:\phi x.\lor.(\exists z).z\in\beta.\phi z\colon\colon\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·24.</b> \(\vdash\colon\ldotp \iotaʻ y\subset \iotaʻ x\cup \beta.\equiv:y=x.\lor.y\in\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·236}.\supset \\ +\vdash\colon\colon \iotaʻ y\supset \iotaʻ x\cup \beta.&\equiv\colon\ldotp z\in \iotaʻ y.\supset_{z}:z=x.\lor.z\in\beta\colon\ldotp \\ +[\text{*51·15}] & \equiv\colon\ldotp z=y.\supset_{z}:z=x.\lor.z\in\beta\colon\ldotp \\ +[\text{*13·191}] & \equiv\colon\ldotp y=x.\lor.y\in\beta\colon\colon\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·25.</b> \(\vdash:\alpha\supset \iotaʻ x\cup \beta.x{\sim}\in\alpha.\supset.\alpha\subset \beta \quad[\text{*51·211.*24·49}]\)</p> + +<p class="nind"><b>*51·3.</b> \(\vdash:y\in\alpha.y\neq x.\equiv.y\in\alpha -\iotaʻ x \quad[\text{*51·15.*22·33·35}]\)</p> + +<p class="nind"><b><a id="*51·31">*51·31</a>.</b> \(\vdash:\exists!\alpha\cap \iotaʻ x.\equiv.\iotaʻ x\subset \alpha.\equiv.\alpha\cap \iotaʻ x=\iotaʻ x.\equiv.x\in\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·33.*51·15}.\supset\vdash:\exists!\alpha\cap \iotaʻ x.&\equiv.(\exists y).y\in\alpha.y=x. \\ +[\text{*13·195}] & \equiv.x\in\alpha. &\qquad \text{(1)} \\ +[\text{*51·2}] & \equiv.\iotaʻ x\subset \alpha. &\qquad \text{(2)} \\ +[\text{*22·621}] & \equiv.\iotaʻ x=\iotaʻ x\cap \alpha &\qquad \text{(3)} \\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·34.</b> \(\vdash:x\in\alpha.\equiv.-\alpha\subset-\iotaʻ x \quad[\text{*51·2.*22·81}]\)</p> + +<p class="nind"><b>*51·35.</b> \(\vdash:x{\sim}\in\alpha.\equiv.\iotaʻ x\subset-\alpha \quad[\text{*51·2.* 22·35}]\)</p> + +<p class="nind"><b>*51·36.</b> \(\vdash:x\sim \in\alpha.\equiv.\alpha\subset-\iotaʻ x \quad[\text{*51·35.*22·811}]\)</p> + +<p>*51·36 is frequently used.</p> + +<p class="nind"><b>*51·37.</b> \(\vdash.\alpha=\hat{x}(\iotaʻ x\subset \alpha) \quad[\text{*51·2.*20·33}]\)</p> + +<p><span class="pagenum" id="Page_361">[Pg 361]</span></p> + +<p class="nind"><b><a id="*51·4">*51·4</a>.</b> \(\vdash:\exists!\alpha.\alpha\subset \iotaʻ x.\equiv.\alpha=\iotaʻ x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·5.*51·15}.\supset\vdash\colon\ldotp\exists!\alpha.\alpha\subset \iotaʻ x.&\equiv:(\exists y).y\in\alpha:y\in\alpha.\supset_{y}.y=x: \\ +[\text{*14·122}] & \equiv:y\in\alpha.\equiv_{y}.y=x: \\ +[\text{*51·11.*20·33}] & \equiv:\alpha=\iotaʻ x\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·401.</b> \(\vdash\colon\ldotp \alpha\subset \iotaʻ x.\equiv:\alpha=\Lambda.\lor.\alpha=\iotaʻ x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·4.*5·6}. &\supset\vdash\colon\ldotp \alpha\subset \iotaʻ x.\supset:\alpha=\Lambda.\lor.\alpha=\iotaʻ x &\qquad \text{(1)} \\ +\vdash.\text{*24·12.*22·42}.&\supset\vdash\colon\ldotp \alpha=\Lambda.\lor.\alpha=\iotaʻ x:\supset.\alpha\subset \iotaʻ x &\qquad \text{(2)} \\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p>This proposition shows that unit classes are the smallest existent +classes.</p> + +<p class="nind"><b><a id="*51·41">*51·41</a>.</b> \(\vdash:\iotaʻ x\cup \iotaʻ y=\iotaʻ x\cup \iotaʻ z.\equiv.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*20·2.*13·13}.&\supset\vdash:y=z.\supset.\iotaʻ x\cup \iotaʻ y=\iotaʻ x\cup \iotaʻ z &\qquad \text{(1)} \\ +\vdash.\text{*22·58}.&\supset\vdash\colon\ldotp \iotaʻ x\cup \iotaʻ y=\iotaʻx\cup \iotaʻ z.\supset:\iotaʻ y\subset \iotaʻ x\cup \iotaʻ z.\iotaʻ z\subset \iotaʻ x\cup \iotaʻ y: \\ +[\text{*51·16·232}] &\supset:y=x.\lor.y=z:z=x.\lor.z=y: \\ +[\text{*13·16.*4·41}] &\supset:y=x.z=x.\lor.y=z: \\ +[\text{*13·172.*2·621}] &\supset:y=z &\qquad \text{(2)} \\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p>The two following propositions are lemmas for *51·43.</p> + +<p class="nind"><b>*51·42.</b> \(\vdash\colon\ldotp \iotaʻ x\cup \iotaʻ y=\iotaʻ z\cup \iotaʻ w.\supset:x=z.y=w.\lor.x=w.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·232}.\supset \\ +\vdash\colon\colon \iotaʻ x\cup \iotaʻ y=\iotaʻz\cup \iotaʻ w.\equiv\colon\ldotp a=x.\lor.a=y:\equiv_{a}:a=z.\lor.a=w\colon\ldotp \\ +[\text{*10·1}] \quad\supset\colon\ldotp x=x.\lor.x=y:\equiv:x=z.\lor.x=w\colon\ldotp \\ +[\text{*13·15}] \quad\supset\colon\ldotp x=z.\lor.x=w &\qquad \text{(1)} \\ +\vdash.\text{*20·2.*13·13}.\supset\vdash:\iotaʻ x\cup \iotaʻ y=\iotaʻ z\cup \iotaʻ w.x=z.\supset.\iotaʻ x\cup \iotaʻ y=\iotaʻ x\cup \iotaʻ w. \\ +[\text{*51·41}] \quad\supset.y=w &\qquad \text{(2)} \\ +\text{Similarly}\quad\vdash:\iotaʻ x\cup \iotaʻ y=\iotaʻ z\cup \iotaʻ w.x=w.\supset.y=z &\qquad \text{(3)} \\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·421.</b> \(\vdash\colon\ldotp x=z.y=w.\lor.x=w.y=z:\supset.\iotaʻ x\cup \iotaʻ y=\iotaʻ z\cup \iotaʻ w \quad[\text{*51·41}]\)</p> + +<p class="nind"><b><a id="*51·43">*51·43</a>.</b> \[\begin{align}\vdash\colon\ldotp \iotaʻ x\cup \iotaʻ y=\iotaʻ z\cup \iotaʻw.\equiv:x=z.y=w.\lor.x=w.y=z\\ +&[\text{*51·42·421}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_362">[Pg 362]</span></p> + +<p>The following propositions are concerned with \(\breve{\iota}\), +<i>i.e.</i> with the relation of the only member of a unit class to +that class. If \(\alpha\) is a unit class, \(\breve{\iota}ʻ\alpha\) is +its only member. (\({℩}x)(\phi x)\) and \(\breve{\iota}ʻ\hat{z}(\phi z)\) +are equal whenever either exists, and any proposition about the +one is equivalent to the same proposition about the other.</p> + +<p class="nind"><b>*51·51.</b> \(\vdash:\alpha=\iotaʻ x.\equiv.x=\breve{\iota}ʻ\alpha.\equiv.x\breve{\iota}\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·131.*31·11}.&\qquad\supset\vdash:\alpha=\iotaʻ x.\equiv.x\breve{\iota}\alpha &\qquad \text{(1)}\\ +&\qquad\vdash.(1).\supset\vdash:x\breve{\iota}\alpha.y\breve{\iota}\alpha.\supset.\alpha=\iotaʻ x.\alpha=\iotaʻ y. \\ +[\text{*51·23.*20·57·2}] &\qquad\supset.x=y &\qquad \text{(2)} \\ +\vdash.\text{(2).Exp.*10·11.*4·71}.&\qquad\supset\vdash\colon\ldotp x\breve{\iota}\alpha.\equiv:x\breve{\iota}\alpha:y\breve{\iota}\alpha.\supset_{y}.x=y: \\ +[\text{*30·31}] &\qquad\equiv:x=\breve{\iota}ʻ\alpha &\qquad \text{(3)} \\ +\vdash.\text{(1).(3)}.&\qquad\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·511.</b> \(\vdash.\breve{\iota}ʻ \iotaʻ x=x \quad\left[\text{*51·51} \frac{\iotaʻ x}{\alpha}.\,\text{*20·2}\right]\)</p> + +<p class="nind"><b>*51·52.</b> \(\vdash:\text{E}!\breve{\iota}ʻ \alpha.\equiv.\alpha=\iotaʻ\breve{\iota}ʻ\alpha \quad\left[\text{*51·51}\, \frac{\breve{\iota}ʻ\alpha}{x}.\,\text{*14·21·18}\right]\)</p> + +<p class="nind"><b>*51·53.</b> \(\vdash:\text{E}!\breve{\iota}ʻ\alpha.\equiv.\breve{\iota}ʻ\alpha\in\alpha \quad[\text{*51·52·16.*14·21·18}]\)</p> + +<p class="nind"><b>*51·54.</b> \(\vdash:\text{E}!\breve{\iota}ʻ\alpha.\equiv.(\exists x).\alpha=\iotaʻ x \quad[\text{*51·51.*14·204}]\)</p> + +<p class="nind"><b>*51·55.</b> \(\vdash:\text{E}!\breve{\iota}ʻ\alpha.\equiv.\text{E}!(℩x)(x\in\alpha)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·54·14}.\supset\vdash\colon\ldotp \text{E}\breve{\iota}ʻ\alpha.&\equiv:(\exists x):y\in\alpha.\equiv_{y}.y=x: \\ +[\text{*14·11}] &\equiv:\text{E}!({℩}x)(x\in\alpha)\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·56.</b> \(\vdash:b=\breve{\iota}ʻ\hat{y}(\phi y).\equiv.\hat{y}(\phi y)=\iotaʻ b.\equiv.b=({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·51}.\supset\vdash\colon\ldotp b=\breve{\iota}ʻ\hat{y}(\phi y).&\equiv:\hat{y}(\phi y)=\iotaʻ b: &\qquad \text{(1)} \\ +[\text{*20·15.*51·11}] &\equiv:\phi y.\equiv_{y}.y=b: \\ +[\text{*14·202}] & \equiv:b=({℩}x)(\phi x)&\qquad \text{(2)} \\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*51·57.</b> \(\vdash:\text{E}!\breve{\iota}ʻ\hat{y}(\phi y).\equiv.\breve{\iota}ʻ\hat{y}(\phi y)=({℩}x)(\phi x).\equiv.\text{E}!({℩}x)(\phi x)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·204.*51·56}.\supset\vdash:\text{E}!\breve{\iota}ʻ\hat{y}(\phi y).&\equiv.\text{E}!({℩}x)(\phi x)&\qquad \text{(1)} \\ +\vdash.*14· 205.\supset\vdash:(℩x)(\phi x)=\breve{\iota}ʻ\hat{y}(\phi y).&\equiv.(\exists b).b=({℩}x)(\phi x).b=\breve{\iota}ʻ\hat{y}(\phi y).\\ +[\text{*51·56.*4·71}] & \equiv.(\exists b).b=({℩}x)(\phi x). \\ +[\text{*14·204·13}] &\equiv.\text{E}!({℩}x)(\phi x)&\qquad \text{(2)} \\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*51·58.</b> \(\vdash:\text{E}!\breve{\iota}ʻ\alpha.\equiv.\breve{\iota}ʻ\alpha=({℩}x)(x\in\alpha) \quad[\text{*51·57.*20·3.*14·272}]\)</p> + +<p class="nind"><b>*51·59.</b> \(\vdash:\psi{\breve{\iota}ʻ\hat{z}(\phi z)}.\equiv.\psi({℩}x)(\phi x) \quad[\text{*51· 56.*14·205}]\)</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"> +<a id="Footnote_59" href="#FNanchor_59" class="label">[59]</a> +This argument is due to Frege. See his article "Kritische +Beleuchtung einiger Punkte in E. Schröder's Vorlesungen über die +Algebra der Logik," <i>Archiv für Syst. Phil.</i>, vol. <span class="allsmcap">I</span>. p. +444 (1895).</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_363">[Pg 363]</span></p> +<h2 class="nobreak" id="*52">*52. THE CARDINAL NUMBER 1.</h2> +</div> + + +<p><i>Summary of *52.</i></p> + +<p>In this number, we introduce the cardinal number 1, defined as the +class of all unit classes. The fact that 1 so defined is a cardinal +number is not relevant at present, and cannot of course be proved until +"cardinal number" has been defined. For the present, therefore, 1 is to +be regarded simply as the class of all unit classes, unit classes being +such classes as are of the form \(\iotaʻx\) for some \(x\).</p> + +<p>Like \(\Lambda\) and \(\text{V}\), 1 is ambiguous as to type: it means +"all unit classes of the type in question." The symbol "\(1(\alpha)\)," +where \(\alpha\) is a type, will mean "all unit classes whose sole +members belong to the type \(\alpha\)" (cf. <a href="#*65">*65</a>). Thus <i>e.g.</i> +"\(\xi\in 1(\text{Indiv}\))" will mean "\(\xi\) is a class consisting +of one individual," if "\(\text{Indiv}\)" stands for the class of +individuals.</p> + +<p>The properties of 1 to be proved in the present number are what we +may call <i>logical</i> as opposed to <i>arithmetical</i> properties, +<i>i.e.</i> they are not concerned with the arithmetical operations +(addition, etc.) which can be performed with 1, but with the relations +of 1 to unit classes. The arithmetical properties of 1 will be +considered later, in Part III.</p> + +<p>The propositions of the present number which are most used are the +following:</p> + +<p class="nind"><b>*52·16.</b> \(\vdash\colon\ldotp \alpha\in\text{1}.\equiv:\exists!\alpha\colon x,y\in\alpha.\supset_{x,y}.x=y\)</p> + +<p><i>I.e.</i> \(\alpha\) is a unit class if, and only if, it is not null, +and all its members are identical.</p> + +<p class="nind"><b>*52·22.</b> \(\vdash .\iotaʻ x\in\text{1}\)</p> + +<p class="nind"><b><a id="*52·4">*52·4</a>.</b> \(\vdash\colon\ldotp \alpha\in\text{1}\cup \iotaʻ\Lambda.\equiv\colon x,y\in\alpha.\supset_{x,y}.x=y\)</p> + +<p>We shall define 0 as \(\iotaʻ\Lambda\). Thus the above proposition +states that a class has one member or none when, and only when, all its +members are identical.</p> + +<p class="nind"><b>*52·41.</b> \(\vdash\colon\exists!\alpha.\alpha{\sim}\in\text{1}.\equiv.({\exists}x,y).x,y\in\alpha.x\neq y\)</p> + +<p><span class="pagenum" id="Page_364">[Pg 364]</span></p> + +<p>This proposition is obtainable from *52·4 by transposition, <i>i.e.</i> +by negating each side of the equivalence.</p> + +<p class="nind"><b>*52·46</b> \(\vdash\colon\ldotp \alpha ,\beta \in 1\ldotp \supset : \alpha \subset \beta \ldotp \equiv \ldotp \alpha = \beta \ldotp \equiv \ldotp \exists !(\alpha \cap \beta )\)</p> + +<p><i>I.e</i>. two unit classes are identical when, and only when, one is +contained in the other, and when and only when they have a common part.</p> + +<hr class="tb"> + +<p class="nind"><b>*52·01</b> \(\hat{\alpha }\{(\exists x) \ldotp \alpha = \iota ʻx\}\quad \text{Df}\)</p> + +<p class="nind"><b>*52·1</b> \(\vdash: \alpha \in 1 \ldotp \equiv \ldotp (\exists x) \ldotp \alpha = \iota ʻx\qquad\qquad\quad [*20·3.(*52·01)]\)</p> + +<p class="nind"><b>*52·11</b> \(\vdash\colon\ldotp \alpha \in 1 \ldotp \equiv : (\exists x): y \in \alpha \ldotp \equiv _{y} \ldotp y = x\quad [*52·1.*51·14]\)</p> + +<p class="nind"><b>*52·12</b> \(\vdash: \hat{z} (\phi z) \in 1 \ldotp \equiv \ldotp \text{E}! ({℩} x) (\phi x)\)</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·11}.\supset \vdash \colon\ldotp \hat{z}(\phi z) \in 1\ldotp &\equiv :(\exists x):y \in \hat{z}(\phi z)\ldotp \equiv _{y}\ldotp y=x:\\ +[\text{*20·3}] &\equiv : (\exists x):\qquad \phi y\ldotp \equiv _{y}\ldotp y=x:\\ +[\text{*14·11}] &\equiv : \text{E}! ({℩}x)(\phi x)\colon\ldotp \supset \vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*52·13</b> \(\vdash \ldotp 1 = \text{D}ʻ\iota \)</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·131}. &\supset \vdash: \alpha = \iota ʻx\ldotp \equiv \ldotp \alpha \iota x :\\ +[\text{*10·11·281}] &\supset \vdash:(\exists x)\ldotp \alpha = \iota ʻx\ldotp \equiv \ldotp (\exists x)\ldotp \alpha \iota x:\\ +[\text{*52·1}]\supset \vdash :\alpha \in 1 \ldotp &\equiv \ldotp (\exists x) \ldotp \alpha \iota x\\ +[\text{*33·13}]&\equiv \ldotp \alpha \in \text{D}ʻ\iota : \supset \vdash \ldotp \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*52·14</b> \(\vdash.1 = \iota ʻʻ\text{V} \quad [*52·13.*37·28]\)</p> + +<p class="nind"><b>*52·15</b> \(\vdash: \alpha \in 1 \ldotp \equiv \ldotp \text{E}! \breve{\iota }ʻ\alpha \quad[\text{*51·54.*52·1}]\)</p> + +<p class="nind"><b>*52·16</b> \(\vdash\colon\ldotp \alpha \in 1 \ldotp \equiv :\exists !\alpha : x, y \in \alpha \ldotp \supset _{x,y} \ldotp x = y \quad[\text{*52·15.*51·55.*14·203}]\)</p> + +<p class="nind"><b>*52·17</b> \(\vdash: \alpha \in 1 \ldotp \equiv \ldotp \breve{\iota }ʻ\alpha = ({℩}x) (x \in \alpha ) \quad\text{[*51·58.*52·15}]\)</p> + +<p class="nind"><b>*52·171</b> \(\vdash: \alpha \in 1 \ldotp \equiv \ldotp \text{E}! ({℩}x) (x \in \alpha ) \quad[\text{*51·55.*52·15}]\)</p> + +<p class="nind"><b>*52·172</b> \(\vdash: \alpha \in 1 \ldotp \equiv \ldotp \iota ʻ\breve{\iota }ʻ\alpha = \alpha \quad[\text{*51·52,*52·15}]\)</p> + +<p class="nind"><b>*52·173</b> \(\vdash: \alpha \in 1 \ldotp \equiv \ldotp \breve{\iota }ʻ\alpha \in \alpha \quad[\text{*51·53.*52·5}]\)</p> + +<p class="nind"><b>*52·18</b> \(\vdash\colon\ldotp \alpha \in 1 \ldotp \equiv : (\exists x) : x \in \alpha : y \in \alpha \ldotp \supset _{y} \ldotp y = x\)</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·141}. \supset \vdash \colon\ldotp (\exists x) \ldotp \alpha = \iota ʻx\ldotp \equiv : (\exists x): x\in \alpha :y\in \alpha \ldotp \supset _{y} +\ldotp y = x &\qquad \text{(1)}\\ +\vdash.\text{(1).*52·1}. \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*52·181</b> \(\vdash\colon\ldotp \alpha \sim \in 1 \ldotp \equiv : x\in \alpha \ldotp \supset_{x}\ldotp (\exists y)y\in \alpha \ldotp y\neq x \quad[\text{*52·18.*10·51}]\)</p> + +<p class="nind"><b>*52·2</b> \(\vdash \ldotp 1 \subset \text{Cls}\)</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·1}. \supset \vdash :\alpha \in 1 \ldotp &\supset \ldotp (\exists x) \ldotp \alpha = \iota ʻx\ldotp \\ +[\text{*51·11}] &\supset \ldotp (\exists x) \ldotp \alpha = \hat{z}(z = x)\ldotp \\ +[\text{*20·54}] &\supset \ldotp (\exists x, \phi ) \ldotp \hat{z} (\phi !z) = \hat{z}(z=x)\ldotp \alpha = \hat{z} (\phi !z)\ldotp \\ +[\text{*10·5}] &\supset \ldotp (\exists \phi )\ldotp \alpha = \hat{z} (\phi !z)\ldotp \\ +[\text{*20·4}] &\supset \ldotp \alpha \epsilon \text{Cls} : \supset \vdash \ldotp \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_365">[Pg 365]</span></p> + +<p class="nind"><b>*52·21.</b> \(\vdash.\Lambda{\sim}\in\text{1}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·16}.&\supset\vdash:\alpha\in\text{1}.\supset_{\alpha}.\exists !\alpha\colon \\ +[\text{*24·63}] &\supset\vdash:\lambda{\sim}\in\text{1} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·22.</b> \(\vdash.\iotaʻ x\in\text{1} \quad[\text{*51·12.*14·28.*10·24.*52·1}]\)</p> + +<p class="nind"><b>*52·23.</b> \(\vdash.\exists!\text{1}.\exists!-\text{1}\)</p> + + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·22.*10·24}.&\supset\vdash.(\exists x).\iotaʻ x\in\text{1}. \\ +[\text{*20·54}] &\supset\vdash.(\exists x,\alpha).\alpha=\iotaʻ x.\alpha\in\text{1}. \\ +[\text{*10·5}] &\supset\vdash.(\exists \alpha).\alpha\in\text{1} &\qquad \text{(1)} \\ +\vdash.\text{*52·21.*2·35}.&\supset\vdash.\Lambda\in - 1. \\ +[\text{*10·24}] &\supset\vdash.(\exists \alpha).\alpha\in - 1 &\qquad \text{(2)} \\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·24.</b> \(\vdash.\text{1}\neq\Lambda\cap\text{Cls}.\text{1}\neq\Lambda\cap\text{Cls} \quad[\text{*52·23.*24·54.*24·17.Transp}]\)</p> + +<p class="nind"><b>*52·3.</b> \(\vdash.\iotaʻʻ\alpha\supset\text{1}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·22.*2·02}. &\supset\vdash:y\in\alpha.\supset.\iotaʻ y\in\text{1}: \\ +[\text{*51·12.*10·11.*37·61}]&\supset\vdash.\iotaʻʻ\alpha\subset\text{1} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·31.</b> \(\vdash:\kappa\supset\text{1}.\equiv.(\exists \alpha).\kappa=\iotaʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·14}.\supset\vdash:\kappa\supset\text{1}.&\equiv.\kappa\subset\iotaʻʻ\text{V}. \\ +[\text{*37·66.*51·12}] &\equiv.(\exists \alpha).\alpha\subset\text{V}.\kappa=\iotaʻʻ \alpha. \\ +[\text{*24·11}] & \equiv.(\exists \alpha).\kappa=\iotaʻʻ \alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*52·4.</b> \(\vdash\colon\ldotp \alpha\in\text{1}\cup \iotaʻ\Lambda.\equiv:x,y\in\alpha.\supset_{x,y}.x=y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·16.*24·54}.\supset\\ +\vdash\colon\ldotp \alpha\in\text{1}. &\equiv:\alpha\neq\Lambda:x,y\in\alpha.\supset_{x,y}.x=y\colon\ldotp \\ +[\text{*4·37}] \supset\vdash\colon\colon \alpha\in 1.\lor.\alpha=\Lambda:&\equiv\colon\ldotp \alpha=\Lambda\colon\ldotp\lor\colon\ldotp \alpha\neq \Lambda:x,y\in\alpha.\supset_{x,y}.x=y\colon\ldotp\\ +[\text{*5·63}] &\equiv\colon\ldotp \alpha=\Lambda:\lor:x,y\in\alpha.\supset_{x,y}.x=y &\qquad \text{(1)} \\ +\vdash.\text{2*4·51.*10·53.*11·62}.&\supset\vdash\colon\ldotp \alpha=\Lambda.\supset:x,y\in\alpha.\supset_{x,y}.x=y &\qquad \text{(2)}\\ +\vdash.(1).(2).\text{*4·72}. &\supset\colon\colon \alpha\in\text{1}.\lor.\alpha=\Lambda:\equiv\colon\ldotp x,y\in\alpha.\supset_{x,y}.x=y &\qquad \text{(3)} \\ +\vdash.\text{(3).*51·236}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_366">[Pg 366]</span></p> + +<p>This proposition is frequently useful. We shall define the number 0 as +\(\iotaʻ\Lambda\); thus the above proposition states that a class has +one member or none when, and only when, all its members are identical. +It will be seen that \(x,y\in\alpha.\supset_{x,y}.x=y\) does not imply +\(\exists!\alpha\), and therefore allows the possibility of \(\alpha\) +having no members.</p> + +<p class="nind"><b>*52·41.</b> \(\vdash:\exists !\alpha.\alpha{\sim}\in\text{1}.\equiv.(\exists x,y).x,y\in\alpha.x\neq y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·54}.\supset\vdash\colon\ldotp \exists !\alpha.\alpha{\sim} \in\text{1}.&\equiv:\alpha\neq\Lambda.\alpha{\sim}\in\text{1}: \\ +[\text{*4·56}] & \equiv:{\sim}\{\alpha\in\text{1}.\lor.\alpha=\Lambda\}: \\ +[\text{*51·236}] & \equiv:{\sim}(\alpha\in\text{1}\cup \iotaʻ\Lambda): \\ +[\text{*52·4.Transp}] & \equiv:{\sim}\{x,y\in\alpha.\supset_{x,y}.x=y\} \\ +[\text{*11·52}] & \equiv:( \exists x,y).x,y\in\alpha.x\neq y\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·42.</b> \(\vdash\colon\ldotp \alpha\in\text{1}.\supset: \exists !\alpha\cap\beta.\equiv.\alpha\cap\beta\in\text{1} \)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·31}. & \supset\vdash\colon\ldotp \exists !\iotaʻ x\cap \beta.\equiv.\iotaʻ x\cap \beta=\iotaʻ x\colon\ldotp \\ +[\text{*20·53}] & \supset\vdash\colon\ldotp \alpha=\iotaʻ x.\supset: \exists !\alpha\cap\beta.\equiv.\alpha\cap\beta=\iotaʻ x\colon\ldotp \\ +[\text{*10·11·28}] &\supset\vdash\colon\ldotp(\exists x).\alpha=\iotaʻ x.\supset:( \exists x): \exists !\alpha\cap\beta.\equiv.\alpha\cap\beta=\iotaʻ x: \\ +[\text{*10·37}] &\supset: \exists !\alpha\cap \beta.\supset.(\exists x).\alpha\cap\beta=\iotaʻ x &\qquad \text{(1)} \\ +\vdash.\text{(1).*52·1}.&\supset\vdash\colon\ldotp \alpha\in\text{1}.\supset: \exists !\alpha\cap\beta.\supset.\alpha\cap \beta\in\text{1} &\qquad \text{(2)} \\ +\vdash.\text{*52·16}. &\supset\vdash:\alpha\cap \beta\in\text{1}.\supset.\exists !\alpha\cap\beta &\qquad \text{(3)} \\ +\vdash.\text{(2).(3)}. & \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*52·43.</b> \(\vdash:\alpha\in\text{1}. \exists!\alpha\cap\beta.\equiv.\alpha\in\text{1}.\alpha\cap \beta\in\text{1} \quad[\text{*52· 42.*5·32}]\)</p> + +<p class="nind"><b>*52·44.</b> \(\vdash\colon\ldotp \alpha\in\text{1}.\supset: \exists !\alpha\cap\beta.\equiv.\alpha\subset\beta.\equiv.\alpha\cap\beta=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·31}. & \supset\vdash: \exists !\iotaʻ x\cap\beta.\equiv.\iotaʻ x\subset\beta: \\ +[\text{*13·13.Exp}] & \supset\vdash\colon\ldotp \alpha=\iotaʻ x.\supset: \exists !\alpha\cap\beta.\equiv.\alpha\subset\beta\colon\ldotp \\ +[\text{*10·11·23}] & \supset\vdash\colon\ldotp(\exists x).\alpha=\iotaʻ x.\supset: \exists!\alpha\cap\beta.\equiv.\alpha\subset\beta\colon\ldotp \\ +[\text{*52·1}] & \supset\vdash\colon\ldotp \alpha\in\text{1}.\supset: \exists !\alpha\cap\beta.\equiv.\alpha\subset\beta &\qquad \text{(1)} \\ +\vdash.\text{(1).*22·621}.&\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·45.</b> \(\vdash\colon\colon\alpha,\beta\in\text{1}.\supset\colon\ldotp\alpha\subset\beta\cup\gamma.\equiv:\alpha=\beta.\lor.\alpha\subset\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·236}\, \frac{x,\,y,\,\gamma}{z,\,x,\,\beta} .\supset\\ +&\vdash\colon\ldotp x\in\iotaʻy\cup\gamma.\equiv:x=y.\lor.x\in\gamma\colon\ldotp\\ +[\text{*51·2·23}] &\supset\vdash\colon\ldotp \iotaʻ x\subset \iotaʻ y\cup \gamma.\equiv:\iotaʻ x=\iotaʻ y.\lor.\iotaʻ x\subset \gamma\colon\ldotp \\ +[\text{*13·21}] &\supset\vdash\colon\colon \alpha=\iotaʻ x.\beta=\iotaʻ y.\supset\colon\ldotp \alpha\subset\beta\cup\gamma.\equiv:\alpha=\beta.\lor.\alpha\subset\gamma\colon\colon \\ +[\text{*11·11·35}] &\supset\vdash\colon\colon(\exists x,y).\alpha=\iotaʻ x.\beta=\iotaʻ y.\supset\colon\ldotp \alpha\subset\beta\cup \gamma.\equiv:\alpha=\beta.\lor.\alpha\subset\gamma &\qquad \text{(1)} \\ +\vdash.\text{(1).*52·1}.&\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p><b>52·46.</b> \(\vdash\colon\ldotp\alpha,\beta\in\text{1}.\supset:\alpha\subset\beta.\equiv.\alpha=\beta.\equiv. \exists !(\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·2·23}. &\supset\vdash:\iotaʻ x\subset \iotaʻ y.\equiv.\iotaʻ x=\iotaʻ y &\qquad \text{(1)} \\ +\vdash.\text{(1).*13·21}. &\supset\vdash\colon\ldotp \alpha=\iotaʻ x.\beta=\iotaʻ y.\supset:\alpha\subset\beta.\equiv.\alpha=\beta &\qquad \text{(2)} \\ +\vdash.\text{*(2).*11·11·35.*52·1}.&\supset\vdash\colon\ldotp \alpha,\beta\in\text{1}.\supset:\alpha\subset\beta.\equiv.\alpha=\beta &\qquad \text{(3)} \\ +\vdash.\text{*(3).*52·44}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_367">[Pg 367]</span></p> + +<p class="nind"><b>*52·6.</b> \(\vdash\colon\ldotp \alpha\in\text{1}.\supset:x\in\alpha.\equiv.\iotaʻ x=\alpha.\equiv.x=\breve{\iota}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·23}. &\supset\vdash:x\in\iotaʻ y.\equiv.\iotaʻ x=\iotaʻ y:\\ +[\text{*13·13.Exp}] & \supset\vdash\colon\ldotp \alpha=\iotaʻ y.\supset:x\in\alpha.\equiv.\iotaʻ x=\alpha\colon\ldotp\\ +[\text{*10·11·23.*52·1}]&\supset\vdash\colon\ldotp \alpha\in\text{1}. \supset:x\in\alpha.\equiv.\iotaʻ x=\alpha. &\qquad \text{(1)} \\ +[\text{*51·51}] &\equiv.x=\breve{\iota}ʻ\alpha &\qquad \text{(2)} \\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·601.</b> \(\vdash\colon\colon \alpha\in\text{1}.\supset\colon\ldotp \phi(\breve{\iota}ʻ\alpha).\equiv:x\in\alpha.\supset_{x}.\phi x:\equiv:(\exists x).x\in\alpha.\phi x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·15}.\supset\vdash\colon\ldotp\text{Hp}.\supset:\text{E}!\breve{\iota}ʻ \alpha: &\qquad \text{(1)} \\ +[\text{*30·4}] \supset:x\breve{\iota}\alpha.\equiv.x=\breve{\iota}ʻ \alpha.\\ +[\text{*52·6}] \equiv.x\in\alpha &\qquad \text{(2)} \\ +\vdash.\text{(1).*30·33}.\supset \\ +\vdash\colon\colon\text{Hp}.\supset\colon\colon \phi(\breve{\iota}ʻ \alpha).\equiv :x\breve{\iota}\alpha.\supset_{x}.\phi + x:\equiv:(\exists x).x\breve{\iota}\alpha.\phi x &\qquad \text{(3)} \\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*52·602.</b> \[\begin{align}\vdash\colon\ldotp\hat{z}(\phi z)\in\text{1}.\supset:\psi(℩x)(\phi x).\equiv.\phi x\supset_{x}\psi x.&\equiv.(\exists x).\phi x.\psi x\\ +&\quad[\text{*52·12.*14·26}]\end{align}\]</p> + +<p class="nind"><b>*52·61.</b> + \(\vdash\colon\ldotp \alpha\in\text{1}.\supset:\breve{\iota}ʻ \alpha\in\beta.\equiv.\alpha\subset \beta.\equiv.\exists!(\alpha\cap \beta) \quad\left[\text{*52·601} \frac{x\, \in \, \beta}{\phi x}\right]\)</p> + +<p class="nind"><b>*52·62.</b> \(\vdash\colon\ldotp \alpha,\beta\in\text{1}.\supset:\alpha=\beta.\equiv.\breve{\iota}ʻ \alpha=\breve{\iota}ʻ \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·601}.\supset\vdash\colon\colon\text{Hp}.\supset\colon\ldotp\breve{\iota}ʻ \alpha=\breve{\iota}ʻ \beta.&\equiv:x\in\alpha.\supset_{x}.x=\breve{\iota}ʻ \beta: \\ +[\text{*52·6}] & \equiv:x\in\alpha.\supset_{x}.x\in\beta:\\ +[\text{*52·46}] & \equiv:\alpha=\beta\colon\colon\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·63.</b> \(\vdash:\alpha,\beta\in\text{1}.\alpha\neq \beta.\supset.\alpha\cap \beta=\Lambda \quad[\text{*52·46.Transp}]\)</p> + +<p class="nind"><b>*52·64.</b> \(\vdash:\alpha\in\text{1}.\supset.\alpha\cap \beta\in\text{1}\cup \iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·43}. & \supset\vdash:\text{Hp}.\exists!\alpha\cap \beta.\supset.\alpha\cap \beta\in\text{1}: \\ +[\text{*5·6.*24·54}]&\supset\vdash\colon\ldotp\text{Hp}.\supset:\alpha\cap \beta=\Lambda.\lor.\alpha\cap \beta\in\text{1}: \\ +[\text{*51·236}] &\supset:\alpha\cap \beta\in\text{1}\cup \iotaʻ\Lambda\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*52·7.</b> \(\vdash\colon\ldotp \beta-\alpha\in\text{1}.\alpha\subset\xi.\xi\subset\beta.\supset:\xi=\alpha.\lor.\xi=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·41}. &\supset\vdash:\text{Hp}.\xi\subset\alpha.\supset.\xi=\alpha &\qquad \text{(1)}\\ +\vdash.\text{*24·55}. & \supset\vdash:{\sim}(\xi\subset \alpha).\supset.\exists!\xi-\alpha &\qquad \text{(2)}\\ +\vdash.\text{*22·48}.&\supset\vdash:\text{Hp}. \supset.\xi-\alpha\subset \beta-\alpha&\qquad \text{(3)} \\ +\vdash.\text{(2).(3)}. &\supset\vdash:\text{Hp}.{\sim}(\xi\subset \alpha).\supset.\exists!\xi-\alpha.\xi-\alpha\subset \beta-\alpha&\qquad \text{(4)} \\ +\vdash.\text{*52·1}. &\supset\vdash:\text{Hp}.\supset.(\exists x).\beta-\alpha=\iotaʻ x &\qquad \text{(5)} \\ +\vdash.\text{(4).(5).*51·4}.&\supset\vdash:\text{Hp}.\sim(\xi\subset \alpha).\supset.\xi-\alpha=\beta-\alpha.\\ +[\text{*24·411}] &\supset.\xi=\beta &\qquad \text{(6)} \\ +\vdash.\text{(1).(6)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_368">[Pg 368]</span></p> + +<h2 class="nobreak" id="*53">*53. MISCELLANEOUS PROPOSITIONS INVOLVING UNIT CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *53.</p> + + +<p>The propositions to be given in this number are mostly such as would +have come more naturally at an earlier stage, but could not be given +sooner because they involved unit classes. It is to be observed that +\(\iotaʻx\cup\iotaʻy\) is the class consisting of the members \(x\) and +\(y\), while \(\iotaʻx\uparrow\iotaʻy\) is the relation which holds +only between \(x\) and \(y\). If \(\alpha\) and \(\beta\) are classes, +\(\iotaʻ\alpha\cup\iotaʻ\beta\) is a class of classes, its members +being \(\alpha\) and \(\beta\). If \(R\) and \(S\) are relations, +\(\iotaʻR\uparrow\iotaʻS\) is a relation of relations; and so on.</p> + +<p>The present number begins by connecting products and sums \(pʻ\kappa\), +\(sʻ\kappa\), \(\dot{p}ʻ\lambda\), \(\dot{s}ʻ\lambda\), in cases +where the members of \(\kappa\) or \(\lambda\) are specified, with +the products or sums \(\alpha\cap\beta\), \(\alpha\cup\beta\), +\(R\dot{\cap}S\), \(R\unicode{x228d} S\). We have</p> + +<p class="nind"><b>*53·01.</b> \(\vdash.pʻ\iotaʻ\alpha=\alpha\)</p> + +<p class="nind"><b><a id="*53·1">*53·1</a>.</b> \(\vdash.pʻ(\iotaʻ\alpha\cup\iotaʻ\beta)=\alpha\cap\beta\)</p> + +<p class="nind"><b>*53·14.</b> \(\vdash.pʻ(\kappa\cup\iotaʻ\alpha) =pʻ\kappa\cap\alpha\)</p> + +<p>with similar propositions for \(s\), \(\dot{p}\) and \(\dot{s}\).</p> + +<p>We have next a set of propositions on sums and products of classes of +unit classes. The most important of these is</p> + +<p class="nind"><b>*53·22.</b> \(\vdash.sʻ\iotaʻʻ\alpha=\alpha\)</p> + +<p>We have next a proposition showing that the sum of \(\kappa\) is null +when, and only when, \(\kappa\) is either null or has the null-class +for its only member, <i>i.e.</i></p> + +<p class="nind"><b>*53·24.</b> \(\vdash\colon\ldotp sʻ\kappa=\Lambda.\equiv:\kappa=\Lambda\cap\text{Cls}.\lor.\kappa=\iotaʻ\Lambda\)</p> + +<p>(Here we write "\(\Lambda\cap\text{Cls}\)," to show that the +"\(\Lambda\)" in question is of the next type above that of the other +two \(\Lambda\)'s.)</p> + +<p>We have next various propositions on the relations of +\(\overrightarrow{R}ʻx\) and \(Rʻx\) and \(Rʻʻ\alpha\) in various +cases, first for a general relation \(R\), and then for the particular +relation \(s\) defined in <a href="#*40">*40</a>. Three of these propositions are very +frequently used, namely:</p> + +<p class="nind"><b>*53·3.</b> \(\vdash:\text{E}!Rʻx.\equiv.\overrightarrow{R}ʻ x\in\text{1}\)</p> + +<p class="nind"><b>*53·301.</b> \(\vdash.Rʻʻ\iotaʻx=\overrightarrow{R}ʻx\)</p> + +<p><span class="pagenum" id="Page_369">[Pg 369]</span></p> + +<p class="nind"><b>*53·31.</b> \(\vdash:\text{E}!Rʻ x.\supset.Rʻʻ\iotaʻ x=\iotaʻRʻ x=\overrightarrow{R}ʻ x\)</p> + +<p>The remaining propositions of this number are of less importance, and +are seldom referred to.</p> + +<hr class="tb"> + +<p class="nind"><b>*53·01.</b> \(\vdash.pʻ\iotaʻ\alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·1}.\supset\vdash\colon\ldotp x\in pʻ \iotaʻ \alpha.&\equiv:\beta\in\iotaʻ \alpha.\supset_{\beta}.x\in\beta:\\ +[\text{*51·15}] & \equiv:\beta=\alpha.\supset_{\beta}.x\in\beta:\\ +[\text{*13·191}] & \equiv:x\in\alpha\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*53·02.</b> \(\vdash.sʻ\iotaʻ\alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{ll} +\vdash.\text{*40·11} .\supset\vdash:x\in sʻ \iotaʻ \alpha.&\equiv.(\exists \beta).\beta\in\iotaʻ \alpha.x\in\beta.\\ +[\text{*51·15}] & \equiv.(\exists \beta=\alpha.x\in\beta.\\ +[\text{*13·195}] & \equiv.x\in\alpha:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*53·03.</b> \(\vdash.\dot{p}ʻ\iotaʻR=R \quad[\text{Proof as in *53·01}]\)</p> + +<p class="nind"><b>*53·04.</b> \(\vdash.\dot{s}ʻ\iotaʻR=R \iotaʻ \quad[\text{Proof as in *53·02}]\)</p> + +<p class="nind"><b>*53·1.</b> \(\vdash .pʻ (\iotaʻ\alpha\cup\iotaʻ\beta)=\alpha\cap \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·18}.\supset\vdash.pʻ(\iotaʻ \alpha\cup \iotaʻ \beta)&=pʻ \iotaʻ \alpha\cap pʻ \iotaʻ \beta \\ +[\text{*53·01}] & =\alpha\cap \beta.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p>This proposition can be extended to +\(\iotaʻ\alpha\cup\iotaʻ\beta\cup\iotaʻ\gamma\), etc. It +shows the connection (for finite classes of classes) between +the product \(pʻ\kappa\) and the product of the members +\(\alpha\cap\beta\cap\gamma\cap \ldots\ldotp\)</p> + +<p class="nind"><b>*53·11.</b> \(\vdash.sʻ(\iotaʻ\alpha\cup\iotaʻ\beta)=\alpha\cup\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·171}.\supset\vdash.sʻ(\iotaʻ \alpha\cup \iotaʻ \beta)&=sʻ \iotaʻ \alpha\cup sʻ \iotaʻ \beta \\ +[\text{*53·02}] & =\alpha\cup\beta.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Similar remarks apply to this proposition as to <a href="#*53·1">*53·1</a>.</p> + +<p class="nind"><b>*53·12.</b> \(\vdash.\dot{p}ʻ (\iotaʻ R\cup \iotaʻ S)=R\dot{\cap}S \quad[\text{*41·18.*53·03}]\)</p> + +<p>This proposition shows the connection between the product \(pʻ\kappa\) +for a class \(\kappa\) consisting of two relations \(R\) and \(S\), and +the product \(R\dot{\cap}S\). The proposition can be extended to the +product of any given finite class of relations.</p> + +<p class="nind"><b>*53·13.</b> \(\vdash.\dot{s}ʻ(\iotaʻ R\cup \iotaʻ S)=R\unicode{x228d} S \quad[\text{*41·171.*53·04}]\)</p> + +<p>Similar remarks apply to this proposition as to *53·12.</p> + +<p class="nind"><b>*53·14.</b> \(\vdash.pʻ (\kappa\cup\iotaʻ\alpha)=pʻ\kappa\cap\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·18}.\supset\vdash.pʻ (\kappa\cup \iotaʻ\alpha)&=pʻ\kappa\cap pʻ \iotaʻ\alpha\\ +[\text{*53·01}] &=pʻ\kappa\cap\alpha\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_370">[Pg 370]</span></p> + +<p class="nind"><b>*53·15.</b> \(\vdash.sʻ (\kappa\cup \iotaʻ \alpha)=sʻ \kappa\cup \alpha \quad[\text{Proof as in *53·14}]\)</p> + +<p class="nind"><b>*53·16.</b> \(\vdash.\dot{p}ʻ (\lambda\cup \iotaʻ R)=\dot{p}ʻ \lambda\dot{cap}R \quad[\text{Proof as in *53·14}]\)</p> + +<p class="nind"><b>*53·17.</b> \(\vdash.\dot{s}ʻ (\lambda\cup \iotaʻ R)=\dot{s}ʻ \lambda\unicode{x228d} R \quad[\text{Proof as in *53·14}]\)</p> + +<p>The above proposition and the next are both used in connection with +mathematical induction (<a href="#*91·55">*91·55</a> and <a href="#*97·46">*97·46</a> respectively).</p> + +<p class="nind"><b>*53·18.</b> \(\vdash.sʻ (\alpha-\iotaʻ\Lambda)=sʻ \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·221}.\supset\vdash:\Lambda\in\alpha. & \supset.(\alpha-\iotaʻ \Lambda)\cup \iotaʻ\Lambda=\alpha.\\ +[\text{*53·15}] & \supset.sʻ (\alpha-\iotaʻ \Lambda)\cup\Lambda=sʻ \alpha.\\ +[\text{*24·24}] & \supset.sʻ (\alpha-\iotaʻ \Lambda)=sʻ \alpha &\qquad \text{(1)} \\ +\vdash.\text{*51·222}.\supset\vdash:\Lambda{\sim}\in\alpha. & \supset.\alpha-\iotaʻ\Lambda=\alpha.\\ +[\text{*30·37}] & \supset.sʻ (\alpha-\iotaʻ \Lambda)=sʻ \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·181.</b> \(\vdash.\dot{s}ʻ (\lambda-\iotaʻ\dot{\Lambda})=\dot{s}ʻ \lambda \quad[\text{Proof as in *53·18}]\)</p> + +<p class="nind"><b>*53·2.</b> \(\vdash:\kappa\in\text{1}.\supset.\breve{\iota}ʻ\kappa=pʻ\kappa=sʻ\kappa\)</p> + +<p>This proposition requires, for significance, that \(\kappa\) should +be a class of classes. It is used in <a href="#*88·47">*88·47</a>, in the number on the +existence of selections and the multiplication axiom.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·601}.\supset\vdash\colon\colon\text{Hp}.&\supset\colon\ldotp x\in\breve{\iota}ʻ \kappa:\equiv:\alpha\in\kappa.\supset_{\alpha}.x\in\alpha:\equiv:(\exists + \alpha).\alpha\in\kappa.x\in\alpha &\qquad \text{(1)} \\ +\vdash.\text{(1).*40·1·11}. &\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·21.</b> \(\vdash:\lambda\in\text{1}.\supset.\breve{\iota}ʻ \lambda=\dot{p}ʻ \lambda=\dot{s}ʻ \lambda \quad[\text{Similar proof}]\)</p> + +<p>This proposition requires, for significance, that \(\lambda\) should be +a class of relations.</p> + +<p class="nind"><b>*53·22.</b> \(\vdash.sʻ \iotaʻʻ \alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*40·11}.\supset \vdash:x\in sʻ\iotaʻʻ\alpha.&\equiv.(\exists \gamma).\gamma\in\iotaʻʻ \alpha.x\in\gamma.\\ +[\text{*37·64.*51·12}] & \equiv.(\exists y).y\in\alpha.x\in\iotaʻy.\\ +[\text{*51·15}] & \equiv.(\exists y).y\in\alpha.x=y.\\ +[\text{*13·195}] & \equiv.x\in\alpha:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*53·221.</b> \(\vdash.\iotaʻʻ (\iotaʻ x\cup \iotaʻ y)=\iotaʻ \iotaʻ x\cup \iotaʻ \iotaʻ y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash\colon\ldotp \alpha\in\iotaʻʻ (\iotaʻ x\cup \iotaʻ y).&\equiv:(\exists z).z\in(\iotaʻ x\cup \iotaʻ y).\alpha\iota z: \\ +[\text{*51·131}] & \equiv:(\exists z).z\in(\iotaʻ x\cup \iotaʻ y).\alpha=\iotaʻ z: \\ +[\text{*51·235}] & \equiv:\alpha=\iotaʻ x.\lor.\alpha=\iotaʻ y:\\ +[\text{*51·232}] & \equiv:\alpha\in(\iotaʻ \iotaʻ x\cup \iotaʻ \iotaʻ y)\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_371">[Pg 371]</span></p> + +<p class="nind"><b>*53·222.</b> \(\vdash\colon\kappa=\iotaʻʻ\alpha.\supset.\alpha=\breve{\iota}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*13·12.*20·2}.\supset\vdash\colon\text{Hp}.\supset.\breve{\iota}ʻʻ\kappa&=\breve{\iota}ʻʻ\iota ʻ\alpha\\ +[\text{*51·511.*14·21.*37·67}] &=\hat{x}\{(\exists y).y\in\alpha.x=\breve{\iota}ʻ\iotaʻy\}\\ +[\text{*51·511}] &=\hat{x}\{(\exists y).y\in\alpha.x=y\}\\ +[*13·195] &=\alpha\colon\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*53·23">*53·23</a>.</b> \(\vdash\colon\kappa\subset 1.\supset.sʻ\kappa=\breve{\iota}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·31}.\supset\vdash\colon\text{Hp}.\equiv.(\exists \alpha).\kappa&=\iotaʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*53·22}.\supset\vdash\colon\kappa=\iotaʻʻ\alpha.\supset.sʻ\kappa&=\alpha\\ +[\text{*53·222}] &=\breve{\iota}ʻʻ\kappa&\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*10·11·23}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*53·231.</b> \(\vdash\colon\ldotp x\in\alpha.\supset_{x}.x=y:\equiv:\alpha=\Lambda.\lor.\alpha=\iotaʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·141}.&\supset\vdash\colon\ldotp\exists!\alpha\colon x\in\alpha.\supset_{x}.x=y:&\equiv\colon\alpha=\iotaʻy &\qquad \text{(1)}\\ +\vdash.\text{*10·53}.&\supset\vdash\colon\ldotp{\sim}\exists!\alpha.\supset\colon x\in\alpha\supset{x}.x&=y\colon\ldotp\\ +[\text{*4·71}] &\supset\vdash\colon\ldotp{\sim} \exists!\alpha:x\in\alpha.\supset_{x}.x=y:&\equiv.{\sim}\exists!\alpha.\\ +[\text{*24·51}] &&\equiv.\alpha=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·42·39}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*53·24.</b> \(\vdash\colon\ldotp sʻ\kappa=\Lambda.\equiv:\kappa=\Lambda\cap\text{Cls}.\lor.\kappa=\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·15.*40·11}.\supset\\ +\vdash\colon\ldotp sʻ\kappa=\Lambda. &\equiv:(x):{\sim}\{(\exists\alpha).\alpha\in\kappa.x\in\alpha\}\colon\\ +[\text{*10·51}] &\equiv:(x,\alpha):x\in\alpha.\supset.\alpha{\sim} \in\kappa\colon\\ +[\text{*11·2.*10·23}]&\equiv:(\exists x).x\in\alpha.\supset_{\alpha}.\alpha{\sim} \in\kappa\colon\\ +[\text{*24·54}] &\equiv:\alpha\neq\Lambda.\supset_{\alpha}.\alpha{\sim}\in\kappa\colon\\ +[\text{Transp}] &\equiv:\alpha\in\kappa.\supset_{\alpha}.\alpha=\Lambda\colon\\ +[\text{*53·231}] &\equiv:\kappa=\Lambda\cap\text{Cls}.\lor.\kappa=\iotaʻ\Lambda\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the enunciation and the last line of the proof of the above +proposition, we write "\(\kappa=\Lambda\cap\text{Cls}\)" rather than +"\(\kappa=\Lambda\)," because this \(\Lambda\) must be of the type next +above that of the \(\Lambda\) in "\(\kappa=\iotaʻ\Lambda\)."</p> + +<p>The following proposition is used in the theory of selections (<a href="#*83·731">*83·731</a>).</p> + +<p class="nind"><b>*53·25.</b> \(\vdash\colon\ldotp sʻ\kappa\cap sʻ\lambda=\Lambda.\subset\colon\kappa\cap\lambda=\Lambda\cap\text{Cls}.\lor.\kappa\cap\lambda=\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·181}.\supset\vdash\colon\ldotp\text{Hp}.&\supset:sʻ(\kappa\cap\lambda)=\Lambda\colon\\ +[\text{*53·24}] &\supset\colon\kappa\cap\lambda=\Lambda\cap\text{Cls}.\lor.\kappa\cap\lambda=\iotaʻ\Lambda\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_372">[Pg 372]</span></p> + +<p class="nind"><b>*53·3.</b> \(\vdash:\text{E}!Rʻx.\equiv.\overrightarrow{R}ʻ x\in\text{1}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{ll} +\vdash. \text{*30·2}.\supset\vdash\colon\ldotp\text{E}!Rʻ x. &\equiv:(\exists b):yRx.\equiv_{y}.y=b: \\ +[\text{*32·18.*51·15}] &\equiv:(\exists b):y\in\overrightarrow{R}ʻ x.\equiv_{y}.y\in\iotaʻ b: \\ +[\text{*20·31}] &\equiv:(\exists b).\overrightarrow{R}ʻ x=\iotaʻ b: \\ +[\text{*52·1}] &\equiv: \overrightarrow{R}ʻ x\in\text{1}\colon\ldotp\supset\vdash. \text{Prop} \\ +\end{array} +\]</p> + +<p>The above proposition is very frequently used.</p> + +<p class="nind"><b>*53·301.</b> \(\vdash.Rʻʻ \iotaʻ x =\overrightarrow{R}ʻ x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1.*51·15}.\supset\vdash:y\in Rʻʻ \iotaʻ x.&\equiv.(\exists z).z=x.yRz.\\ +[\text{*13·195}] &\equiv.yRx.\\ +[\text{*32·18}] &\equiv.y\in\overrightarrow{R}ʻ x:\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·302.</b> \(\vdash.Rʻʻ (\iotaʻ x \cup \iotaʻ y) =\overrightarrow{R}ʻ x \cup \overrightarrow{R}ʻ y \quad[\text{*37·22.*53·301}]\)</p> + +<p>The above proposition is used in the cardinal theory of exponentiation +(*116·71).</p> + +<p class="nind"><b>*53·31.</b> \(\vdash:\text{E}!Rʻ x.\supset.Rʻʻ \iotaʻ x = \iotaʻ Rʻ x = \overrightarrow{R}ʻ x\)</p> + +<p>The above proposition is one of which the subsequent use is frequent.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·11.*14·18}.\supset\vdash:\text{Hp}.\supset.\iotaʻ Rʻ x &= \hat{y}(y=Rʻ x)\\ +[\text{*30·4}] &= \hat{y} (yRx)\\ +[\text{*32·13}] &= \overrightarrow{R}ʻ x &\qquad \text{(1)}\\ +\vdash.\text{(1).*53·301}. \supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*53·32.</b> \(\vdash:\text{E}!Rʻ x.\text{E}!Rʻ y.\supset.Rʻʻ (\iotaʻ x \cup \iotaʻ y) = \iotaʻ Rʻ x \cup \iotaʻ Rʻ y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·22} .\supset\vdash.Rʻʻ (\iotaʻ x \cup \iotaʻ y) = Rʻʻ \iotaʻ x \cup Rʻʻ \iotaʻ y &\qquad \text{(1)} \\ +\vdash.\text{(1).*53·31}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·33.</b> \(\vdash. sʻʻ \iotaʻ \kappa = \iotaʻ sʻ \kappa \quad\left[\text{*53*31}\, \frac{s}{R}\right]\)</p> + +<p class="nind"><b>*53·34.</b> \(\vdash. sʻʻ (\iotaʻ \kappa \cup \iotaʻ \lambda) = \iotaʻ sʻ \kappa \cup \iotaʻ sʻ \lambda \quad\left[\text{*53*32}\, \frac{s}{R}\right]\)</p> + +<p class="nind"><b>*53·35.</b> \(\vdash. sʻ sʻʻ (\iotaʻ \kappa \cup \iotaʻ \lambda) = sʻ \kappa \cup sʻ \lambda = sʻ(\kappa \cup \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·34}. \supset\vdash.sʻ sʻʻ (\iotaʻ \kappa \cup \iotaʻ \lambda) &= sʻ (\iotaʻ sʻ \kappa \cup \iotaʻ sʻ \lambda) \\ +[\text{*53·11}] &= sʻ \kappa \cup \iotaʻ \lambda\\ +[\text{*40·171}] &= sʻ (\kappa \cup \lambda) .\supset\vdash. \text{Prop} \\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_373">[Pg 373]</span></p> + +<p>The above proposition may also be proved as follows:</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*42·1}.\supset\vdash.sʻ sʻʻ (\iotaʻ \kappa\cup \iotaʻ \lambda)&=sʻ sʻ(\iotaʻ \kappa\cup \iotaʻ \lambda)\\ +[\text{*53·11}] & =sʻ (\kappa\cup \lambda)\\ +[\text{*40·171}] & =sʻ \kappa\cup sʻ \lambda.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·4.</b> \(\vdash:x=Rʻ y.\equiv.\overrightarrow{R}ʻ y\in\text{1}.x\in\overrightarrow{R}ʻ y.\equiv.\iotaʻ x=\overrightarrow{R}ʻ y.\equiv.x=\breve{\iota}ʻ\overrightarrow{R}ʻ y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21.*4·71}.\supset\vdash:x=Rʻ y.&\equiv.\text{E}!Rʻ y.x=Rʻ y.\\ +[\text{*30·4.*5·32}] & \equiv.\text{E}!Rʻ y.xRy.\\ +[\text{*53·3.*32·18}] & \equiv.\overrightarrow{R}ʻ y\in\text{1}.x\in\overrightarrow{R}ʻ y. &\qquad \text{(1)} \\ +[\text{*52·6.*5·32}] & \equiv.\overrightarrow{R}ʻ y\in\text{1}.\iotaʻ x=\overrightarrow{R}ʻ y. \\ +[\text{*52·22}] & \equiv.\iotaʻ x=\overrightarrow{R}ʻ y.&\qquad \text{(2)} \\ +[\text{*51·51}] & \equiv.x=\breve{\iota}ʻ \overrightarrow{R}ʻ y &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·5.</b> \(\vdash:\exists!\alpha.\equiv.\alpha\in\text{Cls}-\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*20·41}.\supset\vdash:\exists !\hat{z}(\phi z).&\equiv.\hat{z}(\phi z)\in\text{Cls}.\exists!\hat{z}(\phi z). \\ +[\text{*24·54}] & \equiv.\hat{z}(\phi z)\in\text{Cls}.\hat{z}(\phi z)\neq \Lambda.\\ +[\text{*51·3}] & \equiv.\hat{z}(\phi z)\in\text{Cls}-\iotaʻ \Lambda:\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p>In the above proof, as usually where "Cls" or other type-symbols occur, +it is necessary to abandon the notation by Greek letters and revert to +the explicit notation.</p> + +<p class="nind"><b>*53·51.</b> \(\vdash:\dot{\exists}!R.\equiv.R\in\text{Rel}-\iotaʻ\dot{\Lambda} \quad[\text{Proof as in *53·5}]\)</p> + +<p class="nind"><b>*53·52.</b> \(\vdash:\alpha\in\kappa.\exists!\alpha.\equiv.\alpha\in\kappa-\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·54}.\supset\vdash:\alpha\in\kappa.\exists!\alpha.&\equiv.\alpha\in\kappa.\alpha\neq\Lambda.\\ +[\text{*51·3}] &\equiv.\alpha\in\kappa-\iotaʻ\Lambda:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*53·53.</b> \(\vdash:R\in\lambda.\dot{\exists}!R.\equiv.R\in\lambda-\iotaʻ\dot{\Lambda} \quad[\text{Proof as in *53·52}]\)</p> + +<p>The following propositions are inserted because of their +connection with the definition of \(\alpha\rightarrow\beta\) +in <a href="#*70">*70</a>. \(\overrightarrow{R}ʻʻ \text{ᗡ}ʻ R\) and +(\(\overrightarrow{R}ʻʻ\text{V}\) are both important classes.</p> + +<p class="nind"><b>*53·6.</b> \(\vdash:R=\dot{\Lambda}.\exists!\alpha.\supset.\overrightarrow{R}ʻʻ \alpha=\iotaʻ\Lambda.\overrightarrow{R}ʻʻ \alpha=\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·15·241.24·13}.\supset\vdash:\text{Hp}.\supset.\overrightarrow{R}ʻ x&=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·7}.\supset\vdash:\text{Hp}.\supset.\overrightarrow{R}ʻʻ\alpha&=\hat{\beta}\{\exists x).x\in\alpha.\beta=\Lambda\}\\ +[\text{*10·35}] &=\hat{\beta}\{\exists!\alpha.\beta=\Lambda\}\\ +[\text{*4·73}] &=\hat{\beta}(\beta=\Lambda)\\ +[\text{*51·11}] &=\iotaʻ \Lambda &\qquad \text{(2)}\\ +\text{Similarly}\quad \vdash:\text{Hp}.\supset.\overleftarrow{R}ʻʻ \alpha=\iotaʻ\Lambda &&\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*53·601.</b> \(\vdash:\exists!\alpha.\alpha\cap\text{ᗡ}ʻ R=\Lambda.\supset.\overrightarrow{R}ʻʻ \alpha=\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·41}. &\supset\vdash:\text{Hp}.x\in\alpha.\supset.\overrightarrow{R}ʻ x&=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·7}.&\supset\vdash:\text{Hp}.\supset. \overrightarrow{R}ʻʻ \alpha&=\hat{\beta}\{(\exists x).x\in\alpha.\beta=\Lambda\}\\ +[\text{*10·35}] &&=\hat{\beta}\{\exists!\alpha.\beta=\Lambda\}\\ +[\text{*4·73.*51·11}] &&=\iotaʻ\Lambda:\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·602.</b> \(\vdash:\exists!\alpha.\alpha\cap\text{D}ʻ R=\Lambda.\supset.\overleftarrow{R}ʻʻ \alpha=\iotaʻ \Lambda \quad[\text{Proof as in *53·601}]\)</p> + +<p class="nind"><b>*53·603.</b> \(\vdash:\exists!-\text{ᗡ}ʻ R.\supset.\overrightarrow{R}ʻʻ (-\text{ᗡ}ʻ R)=\iotaʻ \Lambda \quad[\text{*24·21.*53·601}]\)</p> + +<p class="nind"><b>*53·604.</b> \(\vdash:\exists!-\text{D}ʻ R.\supset.\overleftarrow{R}ʻʻ (-\text{D}ʻ R)=\iotaʻ\Lambda \quad[\text{*24·21.*53·602}]\)</p> + +<p class="nind"><b><a id="*53·61">*53·61</a>.</b> \(\vdash:\text{ᗡ}ʻ R\subset \alpha.\text{ᗡ}ʻR\neq \alpha.\supset.\overrightarrow{R}ʻʻ \alpha=\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R\cup \iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·92}. &\supset\vdash:\text{Hp}.\supset.\alpha=\text{ᗡ}ʻ R\cup(\alpha-\text{ᗡ}ʻ R) &\qquad \text{(1)}\\ +\vdash.\text{*24·6}. &\supset\vdash:\text{Hp}.\supset.\exists!\alpha-\text{ᗡ}ʻ R.\\ +[\text{*24·21.*53·601]} &\supset.\overrightarrow{R}ʻʻ(\alpha-\text{ᗡ}ʻR)=\iotaʻ\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).*37·22}.&\supset\vdash:\text{Hp}.\supset.\overrightarrow{R}ʻʻ \alpha=\overrightarrow{R}ʻʻ \text{ᗡ}ʻ R\cup\overrightarrow{R}ʻʻ (\alpha-\text{ᗡ}ʻ R)\\ +[(2)] &=\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R\cup \iotaʻ \Lambda:\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b>*53·611.</b> + \(\vdash:\text{D}ʻ R\subset \alpha.\text{D}ʻ R\neq \alpha.\supset.\overleftarrow{R}ʻʻ \alpha=\overleftarrow{R}ʻʻ \text{D}ʻ R\cup \iotaʻ\Lambda \quad[\text{Proof as in *53·61}]\)</p> + +<p class="nind"><b>*53·612.</b> \(\vdash:\text{ᗡ}ʻ R\neq\text{V}.\supset.\overrightarrow{R}ʻʻ\text{V}=\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R\cup \iotaʻ\Lambda \quad[\text{*53·61.*24·11}]\)</p> + +<p class="nind"><b>*53·613.</b> \(\vdash:\text{D}ʻ R\neq\text{V}.\supset.\overleftarrow{R}ʻʻ \text{V}=\overleftarrow{R}ʻʻ \text{ᗡ}ʻ R\cup \iotaʻ\Lambda \quad[\text{*53·611.*24·11}]\)</p> + +<p class="nind"><b>*53·614.</b> \(\vdash.\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R=\overrightarrow{R}ʻʻ\text{V}-\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·612.*22·68.*24·21}.\supset\\ +\vdash:\text{ᗡ}ʻ R\neq\text{V}.\supset.\overrightarrow{R}ʻʻ\text{V}-\iotaʻ\Lambda=\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R-\iotaʻ\Lambda &(1)\\ +\vdash.\text{*22·481}.\supset\vdash:\text{ᗡ}ʻ R=\text{V}.\supset.\overrightarrow{R}ʻʻ\text{V}-\iotaʻ\Lambda=\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R-\iotaʻ\Lambda &(2)\\ +\vdash.\text{*37·772.*51·36.*22·621}.\supset\vdash.\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R-\iotaʻ\Lambda=\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R &(3)\\ +\vdash.(1).(2).(3).\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*53·615.</b> \(\vdash.\overleftarrow{R}ʻʻ\text{D}ʻ R=\overleftarrow{R}ʻʻ\text{V}-\iotaʻ\Lambda \quad[\text{Proof as in *53·614}]\)</p> + +<p><span class="pagenum" id="Page_375">[Pg 375]</span></p> + +<p>The two following propositions are used in <a href="#*70·12">*70·12</a>.</p> + +<p class="nind"><b>*53·62.</b> \(\vdash:\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R\subset \gamma.\equiv.\overrightarrow{R}ʻʻ\text{V}\subset \gamma\cup \iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·614}.\supset\vdash:\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R\subset \gamma.&\equiv.\overrightarrow{R}ʻʻ\text{V}-\iotaʻ\Lambda\subset \gamma.\\ +[\text{*24·43}] &\equiv .\overrightarrow{R}ʻʻ\text{V}\subset \gamma\cup \iotaʻ\Lambda:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*53·621.</b> \(\vdash:\overleftarrow{R}ʻʻ\text{D}ʻ R\subset \gamma.\equiv.\overleftarrow{R}ʻʻ\text{V}\subset \gamma\cup \iotaʻ\Lambda \quad[\text{Proof as in *53·62}]\)</p> + +<p class="nind"><b>*53·63.</b> \(\vdash:\text{ᗡ}ʻ R\neq\text{V}.\supset.\text{D}ʻ\overrightarrow{R}=\overrightarrow{R}ʻʻ\text{ᗡ}ʻ R\cup \iotaʻ\Lambda \quad[\text{*37·78.*53·612}]\)</p> + +<p class="nind"><b>*53·631.</b> \(\vdash:\text{D}ʻ R\neq\text{V}.\supset.\text{D}ʻ\overleftarrow{R}=\overleftarrow{R}ʻʻ\text{D}ʻ R\cup \iotaʻ\Lambda \quad[\text{*37·781.*53·613}]\)</p> + +<p class="nind"><b>*53·64.</b> \(\vdash:\text{ᗡ}ʻR=\text{V}.\supset.\text{D}ʻ\overrightarrow{R}=\overrightarrow{R}ʻʻ \text{ᗡ}ʻR \quad[\text{*37·78}]\)</p> + +<p class="nind"><b><a id="*53·641">*53·641</a>.</b> \(\vdash:\text{D}ʻ R=\text{V}.\supset.\text{D}ʻ\overleftarrow{R}=\overleftarrow{R}ʻʻ\text{D}ʻ R \quad[\text{*37·781}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_376">[Pg 376]</span></p> +<h2 class="nobreak" id="*54">*54. CARDINAL COUPLES.</h2> +</div> + + +<p><i>Summary</i> of *54.</p> + +<p>Couples are of two kinds, namely (1) \(\iotaʻx\cup\iotaʻy\), +in which there is no order as between \(x\) and \(y\), and (2) +\(\iotaʻx\uparrow\iotaʻy\), in which there is an order. We may +distinguish these two kinds of couples as cardinal and ordinal +respectively, since (as will be shown hereafter) the class of all +couples of the form \(\iotaʻx\cup\iotaʻy\) (where \(x\neq y\)) is +the cardinal number 2, while the class of all couples of the form +\(\iotaʻx\uparrow\iotaʻy\) (where \(x\neq y\)) is the ordinal number +2, to which, for the sake of distinction, we assign the symbol +"\(2_{r}\)," where the suffix "\(r\)" stands for "relational," because +the ordinal 2 is a class of relations. In the present and the following +numbers, we shall define 2 and \(2_{r}\) as the classes of cardinal and +ordinal couples respectively, leaving it to a later stage to show that +2 and \(2_{r}\), so defined, are respectively a cardinal and an ordinal +number. An ordinal couple will also be called an <i>ordered</i> couple +or a <i>couple with sense</i>. Thus a couple with sense is a couple of +which one comes first and the other second.</p> + +<p>We introduce here the cardinal number 0, defined as \(\iotaʻ\Lambda\). +That 0 so defined is a cardinal number, will be proved at a later +stage; for the present, we postpone the proof that 0 so defined has the +arithmetical properties of zero.</p> + +<p>Cardinal couples are much less important, even in cardinal arithmetic, +than ordinal couples, which will be considered in the two following +numbers (<a href="#*55">*55</a> and <a href="#*56">*56</a>). It is necessary, however, to prove some of the +properties of cardinal couples, and this will be done in the present +number. Some properties of cardinal couples which have been already +proved are here repeated for convenience of reference. The definitions +of 0 and 2 are:</p> + +<p class="nind"><b>*54·01.</b> \(0=\iotaʻ\Lambda \quad\text{Df}\)</p> + +<p class="nind"><b>*54·02.</b> \(2=\hat{\alpha}\{(\exists x,y).x\neq y.\alpha=\iotaʻx\cup\iotaʻy\} \quad\text{Df}\)</p> + +<p>Most of the propositions of the present number, except those that +merely embody the definitions (*54·1·101·102) are used very seldom. The +following are among the most important.</p> + +<p class="nind"><b>*54·26.</b> \(\vdash\colon\iotaʻx\cup\iotaʻy\in 2.\equiv.x\neq y\)</p> + +<p class="nind"><b>*54·3.</b> \(\vdash.2=\hat{\alpha}\{(\exists x).x\in\alpha.\alpha-\iotaʻx\in 1\}\)</p> + +<p><span class="pagenum" id="Page_377">[Pg 377]</span></p> + +<p class="nind"><b>*54·4.</b> \(\vdash\colon\ldotp \beta\subset \iotaʻx\cup \iotaʻy.\equiv:\beta=\Lambda.\lor.\beta=\iotaʻx.\lor.\beta=\iotaʻy.\lor.\beta=\iotaʻx\cup \iotaʻy\)</p> + +<p class="nind"><b>*54·53.</b> \(\vdash:\alpha\in 2.x,y\in \alpha.x \neq y.\supset.\alpha=\iotaʻx\cup \iotaʻy\)</p> + +<p class="nind"><b>*54·56.</b> \(\vdash:\alpha{\sim}\in 0\cup 1\cup 2.\equiv.(\exists x,y,z).x,y,z\in \alpha.x\neq y.x\neq z.y\neq z\)</p> + +<hr class="tb"> + +<p class="nind"><b>*54·01.</b> \(0=\iotaʻ\Lambda \quad\text{Df}\)</p> + +<p class="nind"><b>*54·02.</b> \(2=\hat{\alpha}\{(\exists x,y).x\neq y.\alpha=\iotaʻx\cup \iotaʻy\} \quad\text{Df}\)</p> + +<p class="nind"><b>*54·1.</b> \(\vdash.0=\iotaʻ\Lambda \quad[\text{(*54·01)}]\)</p> + +<p class="nind"><b>*54·101.</b> \(\vdash:\alpha\in 2.\equiv.(\exists x,y).x \neq y.\alpha=\iotaʻx\cup \iotaʻy \quad[\text{(*54·02)}]\)</p> + +<p class="nind"><b>*54·102.</b> \(\vdash:\alpha\in 0.\equiv.\alpha=\Lambda \quad[\text{*54·1}]\)</p> + +<p>The two following propositions have already occurred in <a href="#*51">*51</a>, but are +here repeated, because they belong to the subject of the present number.</p> + +<p class="nind"><b>*54·21.</b> \(\vdash:\iotaʻx\cup \iotaʻy=\iotaʻx\cup \iotaʻz.\equiv.y=z \quad[\text{*51·41}]\)</p> + +<p class="nind"><b><a id="*54·22">*54·22</a>.</b> \(\vdash\colon\ldotp \iotaʻx\cup \iotaʻy=\iotaʻz\cup \iotaʻw.\equiv:x=z.y=w.\lor.x=w.y=z \quad[\text{*51·43}]\)</p> + +<p class="nind"><b>*54·25.</b> \(\vdash:\iotaʻx\cup \iotaʻy\in 1.\equiv.x=y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·46·1.*22·58}.\supset\vdash:\iotaʻx\cup \iotaʻy\in 1.&\supset.\iotaʻx\cup \iotaʻy=\iotaʻx.\iotaʻx\cup \iotaʻy=\iotaʻy.\\ +[\text{*20·23}] &\supset.\iotaʻx=\iotaʻy &\qquad \text{(1)}\\ +\vdash.\text{*22·56}.&\supset\vdash:\iotaʻx=\iotaʻy.\supset.\iotaʻx\cup \iotaʻy=\iotaʻx.\\ +[\text{*52·22}] &\supset.\iotaʻx\cup \iotaʻy\in 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash:\iotaʻx\cup \iotaʻy\in 1.&\equiv.\iotaʻx=\iotaʻy.\\ +[\text{*51·23}] &\equiv.x=y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*54·26.</b> \(\vdash:\iotaʻx\cup \iotaʻy\in 2.\equiv.x\neq y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*54·101}.&\supset\vdash\colon\colon \iotaʻx\cup \iotaʻy\in 2.\\ +&\equiv\colon\ldotp (\exists z,w).z\neq w.\iotaʻx\cup \iotaʻy=\iotaʻz\cup \iotaʻw\colon\ldotp \\ +[\text{*54·22}] &\equiv\colon\ldotp (\exists z,w):z\neq w:x=z.y=w.\lor.x=w.y=z\colon\ldotp \\ +[\text{*4·4.*11·41}]&\equiv\colon\ldotp (\exists z,w).z\neq w.x=z.y=w.\lor.(\exists z,w).z\neq w.x=w.y=z\colon\ldotp\\ +[\text{*13·22}] &\equiv\colon\ldotp x\neq y.\lor.y\neq x\colon\ldotp \\ +[\text{*13·16}] &\equiv\colon\ldotp x\neq y\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*54·27.</b> \(\vdash.\iotaʻx\cup \iotaʻy\in 1\cup 2 \quad[\text{*54·25·26}]\)</p> + +<p class="nind"><b>*54·271.</b> \(\vdash.1\cup 2=\hat{\alpha}\{(\exists x,y).\alpha=\iotaʻx\cup \iotaʻy\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·42}.\supset +\vdash\colon\ldotp \alpha=\iotaʻx\cup \iotaʻy.&\equiv:x=y.\alpha=\iotaʻx\cup \iotaʻy.\lor.x \neq y.\alpha=\iotaʻx\cup \iotaʻy &&\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·341·41}.&\supset\vdash\colon\ldotp (\exists x,y).\alpha=\iotaʻx\cup \iotaʻy.\\ +&\equiv:(\exists x,y).x=y.\alpha=\iotaʻx\cup \iotaʻy.\lor.(\exists x,y).x\neq y.\alpha=\iotaʻx\cup \iotaʻy:\\ +[\text{*13·195}] &\equiv:(\exists x).\alpha=\iotaʻx\cup \iotaʻx.\lor.(\exists x,y).x\neq y.\alpha=\iotaʻx\cup \iotaʻy:\\ +[\text{*22·56}] &\equiv:(\exists x).\alpha=\iotaʻx.\lor.(\exists x,y).x\neq y.\alpha=\iotaʻx\cup \iotaʻy:\\ +[\text{*52·1.*54·101}]&\equiv:\alpha\in 1.\lor.\alpha\in 2:\\ +[\text{*22·34}] &\equiv:\alpha\in 1\cup 2\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_378">[Pg 378]</span></p> + +<p class="nind"><b>*54·3.</b> \(\vdash.2=\hat{\alpha}\{(\exists x).x\in\alpha.\alpha-\iotaʻ x\in 1\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·1.*10·35}.\supset\\ +&\vdash:(\exists x).x\in\alpha.\alpha-\iotaʻ x\in 1.&\equiv.(\exists x,y).x\in\alpha.\alpha-\iotaʻ x=\iotaʻ y.\\ +\left[\text{*51·22}\, \frac{\iotaʻy,\,\alpha}{\alpha,\,\beta}\right]&&\equiv.(\exists x,y).\iotaʻ x\cap \iotaʻ y=\Lambda.\iotaʻ x\cup \iotaʻ y=\alpha.\\ +[\text{*51·231.*54·101}] && \equiv.\alpha\in 2:\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p class="nind"><b><a id="*54·4">*54·4</a>.</b> \(\vdash\colon\ldotp \beta\subset \iotaʻ x\cup \iotaʻ y.\equiv:\beta=\Lambda.\lor.\beta=\iotaʻ x.\lor.\beta=\iotaʻ y.\lor.\beta=\iotaʻ x\cup \iotaʻ y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·2}. \supset\vdash:x,y\in\beta.\supset.\iotaʻ x\cup \iotaʻ y\subset \beta:\\ +[\text{Fact}] \supset\vdash:\beta\subset \iotaʻ x\cup \iotaʻy.x,y\in\beta.\supset.\beta\subset \iotaʻ x\cup \iotaʻ y.\iotaʻ x\cup \iotaʻ y\subset \beta.\\ +[\text{*22·41}]\supset.\beta=\iotaʻ x\cup \iotaʻ y &\qquad \text{(1)}\\ +\vdash.\text{*51·25}. \supset\vdash\colon\ldotp \beta\subset \iotaʻ x\cup \iotaʻ y.y{\sim}\in\beta.\supset:\beta\subset \iotaʻ x:\\ +[\text{*51·401}] \supset:\beta=\Lambda.\lor.\beta=\iotaʻ x &\qquad \text{(2)}\\ +\text{Similarly}\vdash\colon\ldotp \beta\subset \iotaʻ x\cup \iotaʻ y.x{\sim} \in\beta.\supset:\beta=\Lambda.\lor.\beta=\iotaʻ y &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*3·48}.\supset \\ +\vdash\colon\ldotp \beta\subset \iotaʻ x\cup \iotaʻ y.{\sim}(x,y\in\beta).\supset:\beta=\Lambda.\lor.\beta=\iotaʻ x.\lor.\beta=\iotaʻ y &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*34·8}.\supset\\ +\vdash\colon\ldotp \beta\subset \iotaʻ x\cup \iotaʻ y.\supset:\beta=\Lambda.\lor.\beta=\iotaʻ x.\lor.\beta=\iotaʻ y.\lor.\beta=\iotaʻ x\cup \iotaʻ y &\qquad \text{(5)}\\ +\vdash.\text{*24·12.*22·58·42}.\supset \\ +\vdash\colon\ldotp \beta=\Lambda.\lor.\beta=\iotaʻ x.\lor.\beta=\iotaʻ y.\lor.\beta=\iotaʻ x\cup \iotaʻ y:\supset .\beta\subset \iotaʻ x\cup \iotaʻ y &\qquad \text{(6)} \\ +\vdash.\text{(5).(6)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>This proposition shows that a class contained in a couple is either the +null-class or a unit class or the couple itself, whence it will follow +that 0 and 1 are the only numbers which are less than 2.</p> + +<p class="nind"><b>*54.41.</b> \(\vdash\colon\colon \alpha\in 2.\supset\colon\ldotp \beta\subset \alpha.\supset:\beta=\Lambda.\lor.\beta\in 1.\lor.\beta\in 2\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·1}. \supset\vdash\colon\ldotp \beta=\iotaʻ x.\lor.\beta=\iotaʻ y:\supset.\beta\in 1 &\qquad \text{(1)}\\ +\vdash.\text{*54·26}.\supset\vdash\colon\ldotp x\neq y.\supset:\beta=\iotaʻ x\cup \iotaʻ y.\supset.\beta\in 2 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*54·4}.\supset\\ +\vdash\colon\colon x\neq y.\supset\colon\ldotp \beta\subset \iotaʻ x\cup \iotaʻ y.\supset:\beta=\Lambda.\lor.\beta\in 1.\lor.\beta\in 2\colon\colon\\ +[\text{*13·12}]\supset\vdash\colon\colon \alpha=\iotaʻ x\cup \iotaʻ y.x\neq y.\supset\colon\ldotp \beta\subset \alpha.\supset:\beta=\Lambda.\lor.\beta\in 1.\lor.\beta\in 2\colon\colon\\ +[\text{*11·11·35}]\supset\\ +\vdash\colon\ldotp(\exists x,y).\alpha=\iotaʻ x\cup \iotaʻ y.x\neq y.\supset\colon\ldotp \beta\subset \alpha:\beta=\Lambda.\lor.\beta\in 1.\lor.\beta\in 2 &\qquad \text{(3)}\\ +\vdash.\text{(3).*54·101}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*54·411.</b> \(\vdash\colon\ldotp \alpha\in 2.\supset:\beta\subset \alpha.\supset.\beta\in 0\cup 1\cup 2 \quad[\text{*54·41·02}]\)</p> + +<p><span class="pagenum" id="Page_379">[Pg 379]</span></p> + +<p class="nind"><b>*54·42.</b> \(\vdash\colon\colon\alpha\in 2 .\supset\colon\ldotp\beta\subset\alpha.\exists!\beta.\beta\neq\alpha .\equiv. \beta\in\iotaʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*54·4}. &\supset\vdash\colon\colon \alpha = \iotaʻx \cup \iotaʻy .\supset\colon\ldotp\\ +&\beta \subset \alpha.\exists!\beta.\equiv: \beta=\Lambda.\lor.\beta=\iotaʻx.\lor.\beta = \iotaʻy.\iota.\beta = \alpha:\exists!\beta:\\ +[\text{*24·53·56.*51·161}] &\equiv: \beta = \iotaʻx.\lor.\beta =\iotaʻy.\lor.\beta = \alpha &\qquad \text{(1)}\\ +\vdash.\text{*54·25. Transp. *52·22}. &\supset\vdash\colon x\neq y .\supset. \iotaʻx \cup \iotaʻy\neq \iotaʻx.\iotaʻx \cup \iotaʻy\neq \iotaʻy:\\ +[\text{*13·12}] &\supset\vdash\colon \alpha = \iotaʻx \cup \iotaʻy .x\neq y .\supset. \alpha\neq\iotaʻx.\alpha\neq\iotaʻy &\qquad \text{(2)}\\ +\vdash.\text{(1). (2)}. &\supset\vdash\colon\colon \alpha = \iotaʻx \cup \iotaʻy .x\neq y. \supset\colon\ldotp\\ +&\beta\subset\alpha.\exists!\beta.\beta\neq\alpha .\equiv: \beta=\iotaʻx.\lor.\beta=\iotaʻy:\\ +[\text{*51·235}] &\equiv :(\exists z).z\in\alpha.\beta=\iotaʻz:\\ +[\text{*37·6}] &\equiv :\beta \in \iotaʻʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(3).*11·11·35.*54·101}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*54·43.</b> \(\vdash\colon\ldotp \alpha,\beta\in 1 .\supset: \alpha\cap\beta =\Lambda .\equiv. \alpha\cup\beta \in 2\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*54·26}.\supset\vdash\colon\ldotp \alpha = \iotaʻx. \beta =\iotaʻy .\supset: \alpha\cup\beta \in 2.\equiv. x\neq y.\\ +[\text{*51·231}] \equiv. \iotaʻx \cap \iotaʻy = \Lambda.\\ +[\text{*13.12}] \equiv.\alpha\cap\beta = \Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·35}.\supset\\ +\vdash\colon\ldotp (\exists x,y).\alpha =\iotaʻx.\beta = \iotaʻy .\supset: \alpha\cup\beta \in 2.\equiv.\alpha\cap\beta = \Lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·54.*52·1}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>From this proposition it will follow, when arithmetical addition has +been defined, that 1 + 1 = 2.</p> + +<p class="nind"><b>*54·44.</b> \(\vdash\colon\ldotp z, w \in \iotaʻx \cup \iotaʻy . \supset_{z,w}.\phi(z,w) :\equiv. \phi(x,x).\phi(x,y).\phi(y,x).\phi(y,y)\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*51·234.*11·62}.&\supset\vdash\colon\ldotp z,w \in \iotaʻx \cup \iotaʻy .\supset_{z,w}. \phi(z,w): \equiv :\\ +&z \in \iotaʻx \cup \iotaʻy .\supset_{z}. \phi(z,x) . \phi (z,y):\\ +[\text{*51·234.*10·29}] &\equiv : \phi(x,x).\phi(x,y).\phi(y,x).\phi(y,y) \colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*54·441.</b> \(\vdash\colon\colon z,w \in \iotaʻx \cup \iotaʻy. z\neq w. \supset_{z,w}. \phi(z,w) :\equiv\colon\ldotp x = y: \lor :\phi(x,y).\phi(y,x)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*5·6}.&\supset\vdash\colon\colon z,w \in \iotaʻx \cup \iotaʻy .z\neq w. \supset_{z,w} .\phi(z,w): \equiv\colon\ldotp\\ +&z,w \in \iotaʻx \cup \iotaʻy . \supset_{z,w}: z = w .\lor .\phi (z, w)\colon\ldotp\\ +[\text{*54·44}] &\equiv: x = x.\lor.\phi(x, x):x = y.\lor.\phi(x, y) :\\ +&y = x. \lor. \phi(y, x) : y = y .\lor . \phi(y,y):\\ +[\text{*13·15}] &\equiv : x = y .\lor. \phi(x,y) : y = x . \lor . \phi(y,x):\\ +[\text{*13·16.*4·41}] &\equiv:x = y .\lor. \phi(x,y).\phi(y,x)\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_380">[Pg 380]</span></p> + +<p>This proposition is used in *163·42, in the theory of relations of +mutually exclusive relations.</p> + +<p class="nind"><b>*54·442.</b> \(\vdash\colon\colon x\neq y.\supset\colon\ldotp z,w\in \iotaʻ x\cup \iotaʻ y.z\neq w.\supset_{z,w}.\phi(z,w):\equiv.\phi(x,y).\phi(y,x) \quad[\text{*54·441}]\)</p> + +<p class="nind"><b>*54·443.</b> \[\begin{align}\vdash\colon\colon x\neq y:&\phi(x,y).\equiv.\phi(y,x):\supset\colon\ldotp\\ +&z,w\in \iotaʻ x\cup \iotaʻ y.z\neq w.\supset_{z,w}.\phi(z,w):\equiv.\phi(x,y) \quad[\text{*54·442}]\end{align}\]</p> + +<p class="nind"><b>*54·45.</b> \[\begin{align}\vdash\colon\ldotp(\exists z,w).&z,w\in \iotaʻ x\cup \iotaʻ y.\phi(z,w).\\ +&\equiv:\phi(x,x).\lor.\phi(x,y).\lor.\phi(y,x).\lor.\phi(y,y) \quad[\text{*51·235}]\end{align}\]</p> + +<p class="nind"><b>*54·451.</b> \[\begin{align}\vdash\colon\colon{\sim}\phi(x,x).{\sim} \phi(y,y).\supset\colon\ldotp(\exists z,w).&z,w\in \iotaʻ x\cup \iotaʻ y.\phi(z,w).\\ +&\equiv:\phi(x,y).\lor.\phi(y,x) \quad[\text{*54·45}]\end{align}\]</p> + +<p class="nind"><b>*54·452.</b> \[\begin{align}\vdash\colon\colon{\sim} \phi(x,x).&{\sim} \phi(y,y):\phi(x,y).\equiv.\phi(y,x):\supset:\\ +&(\exists z,w).z,w \in \iotaʻ x\cup \iotaʻ y.\phi(z,w).\equiv.\phi(x,y) \quad[\text{*54·451}]\end{align}\]</p> + +<p class="nind"><b>*54·46.</b> \(\vdash:( \exists z,w).z,w\in \iotaʻ x\cup \iotaʻ y.z\neq w.\equiv.x\neq y \quad[\text{*54·452.*13·15·16}]\)</p> + +<p class="nind"><b>*54·5.</b> \(\vdash\colon\ldotp \alpha\in 2.\supset:\alpha\subset\iotaʻ z\cup\iotaʻ w.\equiv.\alpha=\iotaʻ z\cup \iotaʻ w\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*54·4}.\supset\\ +\vdash\colon\ldotp \alpha\subset \iotaʻ z\cup \iotaʻ w.&\supset:\alpha=\lambda.\lor.\alpha=\iotaʻ z.\lor.\alpha=\iotaʻ w.\lor.\alpha=\iotaʻ z\cup \iotaʻ w &\qquad \text{(1)}\\ +\vdash.\text{*54·3.*24·54}. &\supset\vdash:\text{Hp}.\supset.\alpha\neq\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*54·26}\, \frac{z,z}{x,y}.\,\text{*13·15}. &\supset\vdash:\text{Hp}.\supset.\alpha\neq \iotaʻ z &\qquad \text{(3)}\\ +\vdash.\text{(3)}\,\frac{w}{z}. &\supset\vdash:\text{Hp}.\supset.\alpha\neq \iotaʻ w &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(3).(4).*2·53}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:\alpha\subset \iotaʻ z\cup \iotaʻ w.\supset.\alpha=\iotaʻ z\cup \iotaʻ w &\qquad \text{(5)}\\ +\vdash.\text{*22·42}. &\supset \vdash:\alpha=\iotaʻ z\cup \iotaʻ w.\supset.\alpha\subset \iotaʻ z\cup \iotaʻ w &\qquad \text{(6)}\\ +\vdash.\text{(5).(6)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*54·51.</b> \( \vdash\colon\ldotp \alpha\in 2.\beta\in 1\cup 2.\supset:\alpha\subset\beta.\equiv.\alpha=\beta\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*54·5}.\supset\vdash\colon\ldotp \alpha\in 2.\beta=\iotaʻ z\cup \iotaʻ w.\supset:\alpha\subset \beta.\equiv.\alpha=\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·35·45}.\supset\\ +\vdash\colon\ldotp \alpha\in 2:(\exists z,w).\beta=\iotaʻ z\cup \iotaʻ w:\supset:\alpha\subset \beta.\equiv.\alpha=\beta &\qquad \text{(2)}\\ +\vdash.\text{(2).*54·271}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*54·52.</b> \(\vdash\colon\ldotp \alpha,\beta\in 2.\supset:\alpha\subset \beta.\equiv.\alpha=\beta.\equiv.\beta\subset \alpha \quad[\text{*54·51}]\)</p> + +<p class="nind"><b>*54·53.</b> \(\vdash:\alpha\in 2.x,y\in\alpha.x\neq y.\supset.\alpha=\iotaʻ x\cup \iotaʻ y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·2}. &\supset\vdash:\text{Hp}.&\supset.\iotaʻ x\subset \alpha.\iotaʻ y\subset \alpha.\\ +[\text{*22· 59}] &&\supset.\iotaʻ x\cup \iotaʻ y\subset \alpha &\qquad \text{(1)}\\ +\vdash.\text{*54·26}. &\supset\vdash:\text{Hp}.&\supset.\iotaʻ x\cup \iotaʻ y\in 2 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*54·52}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_381">[Pg 381]</span></p> + +<p class="nind"><b>*54·531.</b> \(\vdash\colon\ldotp \alpha\in 2.\supset:x,y\in\alpha.x\neq y.\equiv.\alpha=\iotaʻ x\cup \iotaʻ y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*54·53.Exp}.&\supset\vdash\colon\ldotp \alpha\in 2.\supset:x,y\in\alpha.x\neq y.\supset.\alpha=\iotaʻ x\cup \iotaʻ y &\qquad \text{(1)}\\ +\vdash.\text{*54·26}. &\supset\vdash\colon\ldotp \alpha\in 2.\supset:\alpha=\iotaʻ x\cup \iotaʻ y.\supset.x\neq y &\qquad \text{(2)}\\ +\vdash.\text{*51·16}. &\supset\vdash:\alpha=\iotaʻ x\cup \iotaʻ y.\supset.x,y\in\alpha &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash\colon\ldotp \alpha\in 2.\supset:\alpha=\iotaʻ x\cup \iotaʻ y.\supset.x,y\in\alpha.x\neq y &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}. &\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*54·54.</b> \(\vdash\colon\ldotp \alpha\in 2.\equiv:x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:(\exists x,y).x,y\in\alpha.x\neq y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*54·531.*11·11·3}.&\supset\vdash\colon\ldotp \alpha\in 2.\supset:x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y &\qquad \text{(1)}\\ +\vdash.\text{*51·16.*54·101}. &\supset\vdash:\alpha\in 2.\supset.(\exists x,y).x,y\in\alpha.x\neq y &\qquad \text{(2)}\\ +\vdash.\text{*5·3.*3·27}. &\supset\vdash\colon\ldotp x,y\in\alpha.x\neq y.\supset .\alpha=\iotaʻ x\cup \iotaʻ y:\supset:\\ + & x,y\in\alpha.x\neq y.\supset.x\neq y.\alpha=\iotaʻ x\cup \iotaʻ y\colon\ldotp\\ +[\text{*11·11·32·34}] &\supset\vdash\colon\ldotp x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:\supset:\\ + & (\exists x,y).x,y\in\alpha.x\neq y.\supset.(\exists x,y).x\neq y.\alpha=\iotaʻ x\cup \iotaʻ y &\qquad \text{(3)}\\ +\vdash.\text{(3).Imp.*54· 101}.&\supset\vdash\colon\ldotp x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:\\ + & (\exists x,y).x,y\in\alpha.x\neq y:\supset.\alpha\in 2 &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(4)}. &\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>In the above proposition, "\(x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻx\cup \iotaʻy\)" +secures that \(\alpha\) has not <i>more</i> than two members, while +"(\(\exists x,y).x,y\in\alpha.x\neq y\)" secures that \(\alpha\) +has not <i>fewer</i> than two members.</p> + +<p class="nind"><b>*54·55.</b> \(\vdash.0\cup 1\cup 2=\hat{\alpha}{x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·42}. &\supset\vdash\colon\colon x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:\equiv\colon\ldotp\\ +& x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:{\sim}(\exists x,y).x,y\in\alpha.x\neq y\colon\ldotp \\ +& \lor\colon\ldotp x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:(\exists x,y).x,y\in\alpha.x\neq y &\qquad \text{(1)}\\ +\vdash.\text{*11·63}.&\supset\vdash\colon\ldotp{\sim}(\exists x,y).x,y\in\alpha.x\neq y.\supset:x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y\colon\ldotp\\ +[\text{*4·71}] & \supset\vdash\colon\ldotp x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:{\sim}( \exists x,y).x,y\in\alpha.x\neq y:\equiv:\\ + &{\sim}(\exists x,y).x,y\in\alpha.x\neq y:\\ +[\text{*11·521}] & \equiv:x,y\in\alpha.\supset_{x,y}.x=y:\\ +[\text{*52·4}] &\equiv:\alpha\in 0\cup 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*54·54}.\supset\\ +\vdash\colon\ldotp x,y\in\alpha.x\neq y.\supset_{x,y}.\alpha=\iotaʻ x\cup \iotaʻ y:&\equiv:\alpha\in 0\cup 1.\lor.\alpha\in 2:\\ +[\text{*22·34}] &\equiv:\alpha\in 0\cup 1\cup 2\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_382">[Pg 382]</span></p> + +<p class="nind"><b>*54·56.</b> \(\vdash:\alpha{\sim}\in 0\cup 1\cup 2.\equiv.(\exists x,y,z).x,y,z\in\alpha.x\neq y.x\neq z.y\neq z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*54·55.*11·52}.&\supset\\ +\vdash\colon\ldotp \alpha{\sim}\in 0\cup 1\cup 2.&\equiv:(\exists x,y).x,y\in\alpha.x\neq y.\alpha\neq \iotaʻ x\cup \iotaʻ y:\\ +[\text{*51·2.*22·59}] &\equiv:(\exists x,y).\iotaʻ x\cup \iotaʻ y\subset \alpha.x\neq y.\alpha\neq \iotaʻ x\cup \iotaʻ y:\\ +[\text{*24·6}] &\equiv:(\exists x,y).\iotaʻ x\cup \iotaʻ y\subset \alpha.x\neq y.\exists!\alpha-(\iotaʻ x\cup \iotaʻ y):\\ +[\text{*51·232.Transp}]&\equiv:(\exists x,y):\iotaʻ x\cup \iotaʻ y\subset \alpha.x\neq y:(\exists z).z\in\alpha.z\neq x.z\neq y:\\ +[\text{*51·2.*22·59}] &\equiv:(\exists x,y,z).x,y,z\in\alpha.x\neq y.x\neq z.y\neq z\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>In virtue of this proposition, a class which is neither null nor a unit +class nor a couple contains at least three distinct members. Hence it +will follow that any cardinal number other than 0 or 1 or 2 is equal to +or greater than 3. The above proposition is used in *104·43, which is +an existence-theorem of considerable importance in cardinal arithmetic.</p> + +<p class="nind"><b><a id="*54·6">*54·6</a>.</b> \(\vdash\colon\ldotp \alpha\cap \beta=\Lambda.x,xʻ\in\alpha.y,yʻ\in\beta.\supset: +\iotaʻ x\cup \iotaʻ y=\iotaʻ xʻ\cup \iotaʻ yʻ.\equiv.x=xʻ.y=yʻ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·2}.\supset\vdash\colon\ldotp\text{Hp}.&\supset:\iotaʻ x\subset \alpha.\iotaʻ x'\subset \alpha.\iotaʻ y\subset \beta.\iotaʻ yʻ\subset \beta.\alpha\cap \beta=\Lambda:\\ +[\text{*24·48}] & \supset:\iotaʻ x\cup \iotaʻ y=\iotaʻ xʻ\cup \iotaʻ yʻ.&\equiv.\iotaʻ x=\iotaʻ xʻ.\iotaʻ y=\iotaʻ yʻ.\\ +[\text{*51·23}] && \equiv.x=x'.y=yʻ\colon\ldotp\supset\vdash.\text{Prop} \\ +\end{array} +\]</p> + +<p>The above proposition is useful in dealing with sets of couples +formed of one member of a class \(\alpha\) and one member of a class +\(\beta\), where \(\alpha\) and \(\beta\) have no members in common. It +is used in the theory of cardinal multiplication (*113·148).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_383">[Pg 383]</span></p> +<h2 class="nobreak" id="*55">*55. ORDINAL COUPLES.</h2> +</div> + + +<p><i>Summary of</i> *55.</p> + +<p>Ordinal couples, which are now to be considered, are much more +important, even in cardinal arithmetic, than cardinal couples. Their +properties are in part analogous to those of cardinal couples, but +in part also to those of unit classes; for they are the smallest +existent relations, just as unit classes are the smallest existent +classes. The properties which are analogous to those of unit classes +do not demand that the two terms of the couple should be distinct, +<i>i.e.</i> they hold for \(\iotaʻ x\downarrow \iotaʻ x\) as well as +for \(\iotaʻ x\uparrow \iotaʻ x\) (where \(x\neq y\)); on the other +hand, the properties which are analogous to those of cardinal couples +do in general demand that the two terms of the ordinal couple should be +distinct.</p> + +<p>The notation \(\iotaʻ x\uparrow \iotaʻ x\) is cumbrous, and does not +readily enable us to exhibit the couple as a descriptive function of +\(x\) for the argument \(y\), or vice versa. We therefore introduce +a new symbol, "\(x\downarrow y\)," for the couple. In a couple +\(x\downarrow y\), we shall call \(x\) the referent of the couple, and +\(y\) the relatum. In virtue of the definitions in <a href="#*38">*38</a>, this gives +rise to two relations, \(x\downarrow\) and \(\downarrow y\); hence we +obtain the notations \(x\downarrowʻʻ\beta\), \(\downarrow yʻʻ\alpha\), +\(\alpha\downarrow_{,,}y\), \(\alpha\downarrow_{,,}ʻʻ\beta\) and so +on, which will be much used in the sequel. It should be observed +that \(x\downarrowʻʻ\beta\) means the class of ordinal couples in +which \(x\) is referent and a member of \(\beta\) is relatum, while +\(\downarrow yʻʻ\alpha\) or \(\alpha\downarrow_{,,}y\) denotes the +class of couples having \(y\) as relatum and a member of \(\alpha\) as +referent; \(\alpha\downarrow_{,,}ʻʻ\beta\) denotes all such classes +of couples as \(\downarrow yʻʻ\alpha\), where y is any member of +\(\beta\); and in virtue of <a href="#*40·7">*40·7</a>, \(sʻ\alpha\downarrow_{,,}ʻʻ\beta\) +denotes all ordinal couples of which the referent is a member of +\(\alpha\), while the relatum is a member of \(\beta\). This is a +very important class, which will be used to define the product of two +cardinal numbers; for it is evident that the number of members of +\(sʻ\alpha\downarrow_{,,}ʻʻ\beta\) is the product of the number of +members of \(\alpha\) and the number of members of \(\beta\).</p> + +<p><span class="pagenum" id="Page_384">[Pg 384]</span></p> + +<p>The first few propositions of the present number are immediate +consequences of the definition of \(x\downarrow y\) and the notations +introduced in <a href="#*38">*38</a>. We then proceed to various elementary properties +of the relation \(x\downarrow y\), of which the most used are the +following:</p> + +<p class="nind"><b>*55·13.</b> \(\vdash:z(x\downarrow y)w.\equiv.z=x.w=y\)</p> + +<p class="nind"><b>*55·15.</b> \(\vdash.\text{D}ʻ(x\downarrow y)=\iotaʻ x.\text{ᗡ}ʻ(x\downarrow y)=\iotaʻ y.Cʻ (x\downarrow y)=\iotaʻ x\cup \iotaʻ y\)</p> + +<p class="nind"><b>*55·16.</b> \(\vdash:\text{D}ʻ R=\iotaʻ x.\text{ᗡ}ʻ R=\iotaʻ y.\equiv.R=x\downarrow y\)</p> + +<p class="nind"><b>*55·202.</b> \(\vdash:x\downarrow y=z\downarrow w.\equiv.x=z.y=w.\equiv.y\downarrow x=w\downarrow z\)</p> + +<p>This proposition should be contrasted with <a href="#*54·22">*54·22</a>, as giving one +reason why ordinal couples are more useful in arithmetic than cardinal +couples. In virtue of the above proposition, when two ordinal couples +are identical, their referents are identical, and their relata are +identical.</p> + +<p>We proceed next to various properties of the relations \(x\downarrow\) +and \(\downarrow x\). These relations play a great part in +arithmetic. It will be observed that if two terms have the relation +\(x\downarrow\), the referent is a couple whose relatum is the +relatum in the relation \(x\downarrow\), <i>i.e.</i> when we have +\(R(x\downarrow)y\), we have \(R=x\downarrow y\) (cf. <a href="#*55·122">*55·122</a>). +Similar remarks apply to the relation \(x\downarrow\). The class +\(x\downarrowʻʻ\alpha\), consisting of all couples whose referent is a +member of \(\alpha\), while the relatum is \(x\), is important. We have</p> + +<p class="nind"><b>*55·232.</b> \(\vdash:\exists!\downarrow xʻʻ\alpha\cap\downarrow yʻʻ\beta.\equiv.x=y.\exists!\alpha\cap\beta\)</p> + +<p>This proposition is frequently useful.</p> + +<p>We proceed next (*55·3—·51) to give various properties of +\(x\downarrow y\) which are analogous to the properties of unit +classes. Among the more important of these properties are the following:</p> + +<p class="nind"><b>*55·3.</b> \(\vdash:xRy.\equiv.x\downarrow y\unicode{x2abd}R.\equiv.\dot{\exists}!(x\downarrow y)\dot{\cap}R\)</p> + +<p>This is the analogue of <a href="#*51·31">*51·31</a>.</p> + +<p class="nind"><b>*55·34.</b> \(\vdash:\dot{\exists}!R.R\unicode{x2abd}x\downarrow y.\equiv.R=x\downarrow y\)</p> + +<p>This is the analogue of <a href="#*51·4">*51·4</a>.</p> + +<p class="nind"><b>*55·5.</b> \[\begin{align}\vdash\colon\ldotp R &\unicode{x2abd}x\downarrow y\unicode{x228d}z w.\equiv:\\ +&R=\dot{\Lambda}.\lor.R=x\downarrow y.\lor.R=z\downarrow w.\lor.R=x\downarrow y\unicode{x228d}z\downarrow w\end{align}\]</p> + +<p>This is the analogue of <a href="#*54·4">*54·4</a>.</p> + +<p>We then proceed to such properties of ordinal couples as are not +analogous to those of unit classes. For connecting the cardinal number +2 with the ordinal number \(2_{r}\), we have the proposition</p> + +<p class="nind"><b>*55·54.</b> \(\vdash\colon\colon x\neq y.\supset\colon\ldotp Cʻ R=\iotaʻ x\cup \iotaʻ y.R\dot{\cap}\breve{R}=\dot{\Lambda}.\equiv:R=x\downarrow y.\lor.R=y\downarrow x\)</p> + +<p><span class="pagenum" id="Page_385">[Pg 385]</span></p> + +<p>This proposition shows that the only asymmetrical relations which +have a given cardinal couple \(\iotaʻx\cup\iotaʻy\) for their +field are the two corresponding ordinal couples \(x\downarrow)y\) +and \(y\downarrow x\). We have next a set of propositions on the +relative products of couples and other relations, <i>i.e.</i> on +\(R\mid(x\downarrow y)\), \(x\downarrow y)\mid S\), and +\(R\mid(x\downarrow y)\mid S\). These propositions are very +useful in arithmetic. The chief of them is</p> + +<p class="nind"><b>*55·61.</b> \(\vdash:\text{E}!Rʻ z.\text{E}!Sʻ w.\supset.(R\Arrowvert\breve{S})ʻ (z\downarrow w)=(Rʻ z)\downarrow(Sʻ w)\)</p> + +<p>Finally we have four propositions which belong, by their subject, to +<a href="#*43">*43</a>, but could not be given there, because the proofs make use of +ordinal couples.</p> + +<hr class="tb"> + +<p class="nind"><b>*55·01.</b> \(x\downarrow y=\iotaʻx\uparrow \iotaʻy \quad\text{Df}\)</p> + +<p class="nind"><b>*55·02.</b> \(Rʻx\downarrow y=Rʻ(x\downarrow y) \quad\text{Df}\)</p> + +<p>This definition serves merely for the avoidance of brackets.</p> + +<p class="nind"><b>*55·1.</b> \(\vdash.x\downarrow y=\iotaʻx\uparrow \iotaʻy \quad[\text{(*55·01)}]\)</p> + +<p class="nind"><b>*55·11.</b> \(\vdash.x\downarrowʻy=\downarrow \iotaʻx=x\downarrow y=\iotaʻx\uparrow\iotaʻy \quad[\text{*38·11.*55·1}]\)</p> + +<p class="nind"><b>*55·12.</b> \(\vdash.\text{E}!x\downarrowʻy \quad[\text{*55·11.*14·21}]\)</p> + +<p class="nind"><b>*55·121.</b> \(\vdash.\text{E}!x\downarrow yʻx\)</p> + +<p class="nind"><b><a id="*55·122">*55·122</a>.</b> \(\vdash:R(x\downarrow)y.\equiv.R=x\downarrow y \quad[\text{*55·11}]\)</p> + +<p class="nind"><b>*55·123.</b> \(\vdash:R(\downarrow y)x.\equiv.R=x\downarrow y \quad[\text{*55·11}]\)</p> + +<p class="nind"><b><a id="*55·13">*55·13</a>.</b> \(\vdash:z(x\downarrow y)w.\equiv.z=x.w=y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·103.*55·1}.\supset\vdash:z(x\downarrow y)w.&\equiv.z\in\iotaʻ x.w\in\iotaʻy.\\ +[\text{*51·15}] &\equiv.z=x.w=y:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·132.</b> \(\vdash.x(x\downarrow y)y \quad[\text{*55·13}]\)</p> + +<p class="nind"><b>*55·134.</b> \(\vdash.\dot{\exists}!(x\downarrow y) \quad[\text{*55·132}]\)</p> + +<p class="nind"><b>*55·14.</b> \(\vdash.x\downarrow y=\text{Cnv}ʻy\downarrow x \quad[\text{*55·13.*31·131}]\)</p> + +<p class="nind"><b>*55·15.</b> \(\vdash.\text{D}ʻx\downarrow y=\iotaʻx.\text{ᗡ}ʻx\downarrow y=\iotaʻy.Cʻx\downarrow y=\iotaʻx\cup\iotaʻy +\quad[\text{*35·85·86.*51·161}]\)</p> + +<p class="nind"><b>*55·16.</b> \(\vdash:\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy.\equiv.R=x\downarrow y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13·131.*51·15}.\supset\\ +\vdash\colon\colon\text{D}ʻ R=\iotaʻ x.\text{ᗡ}ʻ R=\iotaʻ y.&\equiv\colon\ldotp(\exists w).zRw.\equiv_{z}.z=x:(\exists z).zRw.\equiv_{w}.w=y\colon\ldotp\\ +[\text{*14·122}] &\equiv\colon\ldotp(\exists z,w).zRw:(\exists w).zRw.\supset_{z}.z=x:\\ + &(\exists w,z).zRw:(\exists z).zRw.\supset_{w}.w=y\colon\ldotp\\ +[\text{*11·23.*4·71}]&\equiv\colon\ldotp(\exists z,w).zRw:(\exists w).zRw.\supset_{z}.z=x:(\exists z).zRw.\supset_{w}.w=y\colon\ldotp\\ +[\text{*10·23}] &\equiv\colon\ldotp(\exists z,w).zRw:zRw.\supset_{z,w}.z=x:zRw.\supset_{z,w}.w=y\colon\ldotp\\ +[\text{*11·391}] &\equiv\colon\ldotp(\exists z,w).zRw:zRw.\supset_{z,w}.z=x.w=y\colon\ldotp\\ +[\text{*14·123}] &\equiv\colon\ldotp zRw.\equiv_{z,w}.z=x.w=y\colon\ldotp\\ +[\text{*55·13}] &\equiv\colon\ldotp zRw.\equiv_{z,w}.z(x\downarrow y)w\colon\ldotp\\ +[\text{*21·43}] &\equiv\colon\ldotp R=x\downarrow y\colon\colon\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>The above proposition is important, and will be frequently used.</p> + +<p><span class="pagenum" id="Page_386">[Pg 386]</span></p> + +<p class="nind"><b>*55·161.</b> \(\vdash.x\downarrow y=\breve{\iota}ʻ \hat{R}(\text{D}ʻ R=\iotaʻ x.\text{ᗡ}ʻ R=\iotaʻ y)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·16.*20·15}.\supset\\ +\vdash.\hat{R}(\text{D}ʻ R=\iotaʻ x.\text{ᗡ}ʻR=\iotaʻ y)&=\hat{R}(R=x\downarrow y)\\ +[\text{*51·11}] & =\iotaʻ (x\downarrow y) &\qquad \text{(1)}\\ +\vdash.\text{(1).*51·51}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·17.</b> \(\vdash.x\downarrow y=\breve{\iota}ʻ (\overleftarrow{\text{D}}ʻ\iotaʻ x\cap\overleftarrow{\text{ᗡ}}ʻ \iotaʻ y) \quad[\text{*55·161.*33·6·61}]\)</p> + +<p class="nind"><b>*55·2.</b> \(\vdash:x\downarrow y=x\downarrow z.\equiv.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*30·37.*55·11·12}.\supset\vdash:y=z.\supset.x\downarrow y=x\downarrow z &\qquad \text{(1)}\\ +&\vdash.\text{*30·37.*33·121}.\supset \\ +&\vdash:x\downarrow y=x\downarrow z.\supset.\text{ᗡ}ʻ x\downarrow y=\text{ᗡ}ʻ x\downarrow z.\\ +&[\text{*55·15}] \supset.\iotaʻ y=\iotaʻ z.\\ +&[\text{*51·23}] \supset.y=z &\qquad \text{(2)}\\ +&\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·201.</b> \(\vdash:x\downarrow z=y\downarrow z.\equiv.x=y\)</p> + +<p class="nind"><b>*55·202.</b> \(\vdash:x\downarrow y=z\downarrow w.\equiv.x=z.y=w.\equiv.y\downarrow x=w\downarrow z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·2·201}.\supset\\ +\vdash:x=z.y=w.&\supset.x\downarrow y=z\downarrow y.z\downarrow y=z\downarrow w.\\ +[\text{*13·17}] &\supset.x\downarrow y=z\downarrow w &\qquad \text{(1)}\\ +\vdash.\text{*30·37.*33·12·121}.\supset\\ +\vdash:x\downarrow y=z\downarrow w.&\supset.\text{D}ʻx\downarrow y=\text{D}ʻz\downarrow w.\text{ᗡ}ʻx\downarrow y=\text{ᗡ}ʻz\downarrow w. +\\ +[\text{*55·15}] &\supset.\iotaʻ x=\iotaʻ z.\iotaʻ y=\iotaʻ w.\\ +[\text{*51·23}] &\supset.x=z.y=w &\qquad \text{(2)}\\ +\vdash.(1).(2).\supset\\ +\vdash:x\downarrow y=z\downarrow w.\equiv.x=z.y=w &&\qquad \text{(3)}\\ +\text{Similarly}\\ +\vdash:y\downarrow x=w\downarrow z.\equiv.x=z.y=w &&\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is important.</p> + +<p class="nind"><b>*55·21.</b> \(\vdash.\text{ᗡ}ʻx\downarrow=V.\text{ᗡ}ʻ\downarrow x=\text{V} \quad[\text{*33·432.*55·12·121}]\)</p> + +<p class="nind"><b>*55·22.</b> \(\vdash.\text{D}ʻx\downarrow =\breve{R}\{(\exists y).R=x\downarrow y\} \quad[\text{*55·122}]\)</p> + +<p class="nind"><b>*55·221.</b> \(\vdash.\text{D}ʻ\downarrow x=\breve{R}\{(\exists y).R=y\downarrow x\} \quad[\text{*55·123}]\)</p> + +<p><span class="pagenum" id="Page_387">[Pg 387]</span></p> + +<p class="nind"><b>*55·222.</b> \(\vdash:R\in\text{D}ʻ x\downarrow.\equiv.\text{D}ʻ R=\iotaʻ x.\text{ᗡ}ʻ R\in 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·22·16}.\supset\vdash\colon\ldotp R\in\text{D}ʻ x\downarrow.&\equiv:(\exists y).\text{D}ʻR=\iotaʻ x.\text{ᗡ}ʻ R=\iotaʻ y:\\ +[\text{*10·35}] &\equiv:\text{D}ʻ R=\iotaʻ x:(\exists y).\text{ᗡ}ʻ R=\iotaʻ y:\\ +[\text{*52·1}] &\equiv:\text{D}ʻ R=\iotaʻ x.\text{ᗡ}ʻ R\in 1\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·223.</b> \(\vdash:R\in\text{D}ʻ \downarrow x.\equiv.\text{ᗡ}ʻ R=\iotaʻ x.\text{D}ʻ R\in 1 \quad[\text{Proof as in *55·222}]\)</p> + +<p class="nind"><b>*55·224.</b> \(\vdash.\text{D}ʻx\downarrow\cap \text{D}ʻ\downarrow y=\iotaʻ (x\downarrow y)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·222·223}.\supset\\ +\vdash:R\in\text{D}ʻ x\downarrow\cap \text{D}ʻ \downarrow y.&\equiv.\text{D}ʻR=\iotaʻ x.\text{ᗡ}ʻ R\in 1.\text{ᗡ}ʻ R=\iotaʻ y.\text{D}ʻR\in 1.\\ +[\text{*52·22.*4·71}]&\equiv.\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻ y.\\ +[\text{*55·16}] &\equiv.R=x\downarrow y.\\ +[\text{*51·15}] &\equiv.R\in\iotaʻ(x\downarrow y):\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·23.</b> \(\vdash.x\downarrow ʻʻ\alpha=\hat{R}{(\exists y).y\in\alpha.R=x\downarrow y} \quad[\text{*38·13}]\)</p> + +<p class="nind"><b>*55·231.</b> \(\vdash.\downarrow xʻʻ\alpha=\hat{R}{(\exists y).y\in\alpha.R=y\downarrow x} \quad[\text{*38·131}]\)</p> + +<p class="nind"><b>*55·232.</b> \(\vdash:\exists !\downarrow xʻʻ\alpha\cap \downarrow yʻʻ\beta.\equiv.x=y.\exists !\alpha\cap \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·231.*11·55}.\supset\\ +\vdash\colon\ldotp\exists !\downarrow xʻʻ\alpha\cap \downarrow yʻʻ \beta.&\equiv:(\exists R):(\exists z,w).z\in\alpha.R=z\downarrow x.w\in\beta.R=w\downarrow y:\\ +[\text{*13·195}] &\equiv:(\exists z,w).z\in\alpha.w\in\beta.z\downarrow x=w\downarrow y:\\ +[\text{*55·202}] &\equiv:(\exists z,w).z\in\alpha.w\in\beta.x=y.z=w:\\ +[\text{*13·195}] &\equiv:(\exists z).z\in\alpha\cap\beta.x=y:\\ +[\text{*10·35}] &\equiv:\exists !\alpha\cap \beta.x=y\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*55·233.</b> \(\vdash:x\neq y.\supset.\downarrow xʻʻ\alpha\cap \downarrow yʻʻ\beta=\Lambda \quad[\text{*55·232.Transp}]\)</p> + +<p>The above two propositions are frequently useful in arithmetic.</p> + +<p class="nind"><b>*55·24.</b> \(\vdash.\dot{s}ʻ x\downarrowʻʻ\alpha=\iotaʻ x\uparrow\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11}.\supset\\ +\vdash\colon\ldotp z(\dot{s}ʻ x\downarrow ʻʻ\alpha)w.&\equiv.(\exists R).R\in x\downarrow ʻʻ\alpha.zRw.\\ +[\text{*55·23}] &\equiv.(\exists R,y).y\in\alpha.R=x\downarrow y.zRw.\\ +[\text{*13·195}] &\equiv.(\exists y).y\in\alpha.z(x\downarrow y)w.\\ +[\text{*55·13}] &\equiv.(\exists y).y\in\alpha.z=x.w=y.\\ +[\text{*13·195}] &\equiv.z=x.w\in\alpha.\\ +[\text{*51·15.*35·103}]&\equiv.z(\iotaʻ x\uparrow\alpha)w\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*55·241.</b> \(\vdash.\dot{s}ʻ \downarrow xʻʻ \alpha=\alpha\uparrow\iotaʻ x \quad[\text{Proof as in *55·24}]\)</p> + +<p><span class="pagenum" id="Page_388">[Pg 388]</span></p> + +<p class="nind"><b>*55·25.</b> \(\vdash:\exists !\alpha.\supset.\text{D}ʻʻx\downarrow ʻʻ\alpha=\iotaʻ\iotaʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·67.*33·12.*55·12}.\supset\\ +\vdash:\beta\in\text{D}ʻʻx\downarrow ʻʻ \alpha.&\equiv.(\exists y).y\in\alpha.\beta=\text{D}ʻx\downarrow y.\\ +[\text{*55·15}] &\equiv.(\exists y).y\in\alpha.\beta=\iotaʻx.\\ +[\text{*10·35}] &\equiv.\exists !\alpha.\beta=\iotaʻx &\qquad \text{(1)}\\ +\vdash.\text{(1)}.\supset\vdash\colon\ldotp\text{Hp}.\supset:\beta\in\text{D}ʻʻ x\downarrow ʻʻ\alpha.&\equiv.\beta=\iotaʻx.\\ +[\text{*51·15}] &\equiv.\beta\in\iotaʻ\iotaʻx\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·251.</b> \(\vdash:\exists !\alpha.\supset.\text{ᗡ}ʻʻ\downarrow xʻʻ \alpha=\iotaʻ\iotaʻx \quad[\text{Proof as in *55·25}]\)</p> + +<p>This proposition is used in the theory of cardinal multiplication +(*113·142).</p> + +<p class="nind"><b>*55·26.</b> \(\vdash.\text{ᗡ}ʻʻ x\downarrow ʻʻ\alpha=\iotaʻʻ\alpha \quad[\text{*55·15.*37·35}]\)</p> + +<p class="nind"><b>*55·261.</b> \(\vdash.\text{D}ʻʻ \downarrow xʻʻ\alpha=\iotaʻʻ\alpha \quad[\text{*55·15.*37·35}]\)</p> + +<p class="nind"><b>*55·262.</b> \(\vdash.\downarrow xʻʻ\alpha=\downarrow yʻʻ\beta.\supset.\alpha=\beta \quad[\text{*55·261.*53·22}]\)</p> + +<p class="nind"><b>*55·27.</b> \(\vdash.Cʻʻ \downarrow xʻʻ\alpha=Cʻʻx\downarrow ʻʻ\alpha=\hat{\beta}\{(\exists y).y\in\alpha.\beta=\iotaʻx\cup\iotaʻy\} \quad[\text{*55·15}]\)</p> + +<p class="nind"><b>*55·28.</b> \(\vdash.\text{ᗡ}ʻx\downarrow y=\text{ᗡ}ʻx\downarrow z.\equiv.y=z.\equiv.x\downarrow y=x\downarrow z +\quad[\text{*55·15.*51·23.*55·2}]\)</p> + +<p class="nind"><b>*55·281.</b> \(\vdash.\text{D}ʻy\downarrow x=\text{D}ʻz\downarrow x.\equiv.y=z.\equiv.y\downarrow x=z\downarrow x\)</p> + +<p class="nind"><b>*55·282.</b> \(\vdash:Cʻx\downarrow y=Cʻx\downarrow z.\equiv.y=z.\equiv.x\downarrow y=x\downarrow z +\quad[\text{*55·15·2.*54·21}]\)</p> + +<p class="nind"><b>*55·283.</b> \(\vdash:Cʻy\downarrow x=Cʻz\downarrow x.\equiv.y=z.\equiv.y\downarrow x=z\downarrow x\)</p> + +<p class="nind"><b>*55·29.</b> \(\vdash.\text{ᗡ}\mid(x\downarrow)=\iota \quad[\text{*55·15.*34·42}]\)</p> + +<p class="nind"><b>*55·291.</b> \(\vdash.\text{D}\mid(\downarrow x)=\iota \quad[\text{*55·15.*34·42}]\)</p> + +<p class="nind"><b>*55·292.</b> \(\vdash.C\mid(x\downarrow)=C\mid(\downarrow x)=\hat{\alpha}\hat{y}(\alpha=\iotaʻx\cup\iotaʻy) \quad[\text{*55·15.*34·41}]\)</p> + +<p>The following propositions, down to <a href="#*55·51">*55·51</a> inclusive, give properties +of ordinal couples which are analogous to the properties of unit +classes.</p> + +<p class="nind"><b>*55·3.</b> \(\vdash:xRy.\equiv.x\downarrow y\unicode{x2abd}R.\equiv.\dot{\exists}!(x\downarrow y)\unicode{x228d} R \quad[\text{*13·21·22.*55·13}]\)</p> + +<p>The first half of this proposition is the analogue of <a href="#*51·2">*51·2</a>; like that +proposition, it gives a means of reducing propositions to the form of +inclusions. For the second half, compare <a href="#*51·31">*51·31</a>.</p> + +<p class="nind"><b>*55·31.</b> \(\vdash:x\downarrow y=z\downarrow w.\equiv.z(x\downarrow y)w.\equiv.x(z\downarrow w)y.\equiv.x=z.y=w\)</p> + +<p>This proposition is the analogue of <a href="#*51·23">*51·23</a>.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·16}.\supset\vdash:x\downarrow y=z\downarrow w.&\equiv.\text{D}ʻx\downarrow y=\iotaʻz.\text{ᗡ}ʻx\downarrow y=\iotaʻw.\\ +[\text{*55·15}] &\equiv.\iotaʻx=\iotaʻz.\iotaʻy=\iotaʻw.\\ +[\text{*51·23}] &\equiv.x=z.y=w. &\qquad \text{(1)}\\ +[\text{*55·13}] &\equiv.x(z\downarrow w)y. &\qquad \text{(2)}\\ +[\text{(1).*13·16}] &\equiv.z=x.w=y.\\ +[\text{*55·13}] &\equiv.z(x\downarrow y)w &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_389">[Pg 389]</span></p> + +<p class="nind"><b>*55·32.</b> \(\vdash\colon\ldotp x\downarrow y\dot{\cap}z\downarrow w=\dot{\Lambda}.\equiv:x\neq z.\lor.y\neq w\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·3}.\supset\vdash:\dot{\exists}!x\downarrow y\dot{\cap}z\downarrow w.&\equiv.x(z\downarrow w)y.\\ +[\text{*55·13}] &\equiv.x=z.y=w &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·33.</b> \(\vdash:xRy.\equiv.x\downarrow y\dot{\cap}R=x\downarrow y \quad[\text{*55·3.*23·621}]\)</p> + +<p class="nind"><b>*55·34.</b> \(\vdash:\dot{\exists}!R.R\unicode{x2abd}x\downarrow y.\equiv.R=x\downarrow y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·13}.\supset\vdash\colon\ldotp \dot{\exists}!R.R\unicode{x2abd}x\downarrow y.&\equiv:(\exists z,w).zRw:zRw.\supset_{z,w}.z=x.w=y:\\ +[\text{*14·123}] &\equiv:zRw.\equiv_{z,w}.z=x.w=y:\\ +[\text{*55·13}] &\equiv:zRw.\equiv_{z,w}.z(x\downarrow y)w\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·341.</b> \(\vdash\colon\ldotp R\unicode{x2abd}x\downarrow y.\equiv:R=\dot{\Lambda}.\lor.R=x\downarrow y\)</p> + + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·42}.\supset\colon\ldotp R\unicode{x2abd}x\downarrow y.&\equiv:R\unicode{x2abd}x\downarrow y.R=\dot{\Lambda}.\lor.R\unicode{x2abd}x\downarrow y.R\neq\dot{\Lambda}:\\ +[\text{*25·54}] &\equiv:R\unicode{x2abd}x\downarrow y.R=\Lambda.\lor.R\unicode{x2abd}x\downarrow y.\dot{\exists}!R:\\ +[\text{*55·34}] &\equiv:R\unicode{x2abd}x\downarrow y.R=\dot{\Lambda}.\lor.R=x\downarrow y:\\ +[\text{*25·12}] &\equiv:R=\dot{\Lambda}.\lor.R=x\downarrow y\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·35.</b> \(\vdash:R\dot{\cap}x\downarrow y=\Lambda.R\unicode{x228d}x\downarrow y=S.\equiv.xSy.R=S\dot{-}x\downarrow y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*25·47}.\supset\\ +\vdash:R\dot{\cap}x\downarrow y=\dot{\Lambda}.R\unicode{x228d}x\downarrow y=S.&\equiv.x\downarrow y\unicode{x2abd}S.R=S\dot{-}x\downarrow y.\\ +[\text{*55·3}] &\equiv.xSy.R=S\dot{-}x\downarrow y:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·36.</b> \(\vdash:xRy.\equiv.(R\dot{-}x\downarrow y)\unicode{x228d}x\downarrow y=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·3}.\supset\vdash:xRy.&\equiv.x\downarrow y\unicode{x2abd}R.\\ +[\text{*23·62}] &\equiv.x\downarrow y\unicode{x228d}R=R.\\ +[\text{*23·91}] &\equiv.{R\dot{-}x\downarrow y}\unicode{x228d}x\downarrow y=R:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·37.</b> \(\vdash:x\in\alpha.y\in\beta.\equiv.x\downarrow y\unicode{x2abd}\alpha\uparrow \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·103}.\supset\vdash:x\in\alpha.y\in\beta.&\equiv.x(\alpha\uparrow\beta)y.\\ +[\text{*55·3}] &\equiv.x\downarrow y\unicode{x2abd}\alpha\uparrow\beta:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>The following proposition is the analogue of <a href="#*51·232">*51·232</a>.</p> + +<p class="nind"><b>*55·4.</b> \[\begin{align}&\vdash\colon\ldotp a{x\downarrow y\unicode{x228d}z\downarrow w}b.\equiv:a=x.b=y.\lor.a=z.b=w\\ +&[\text{*55·13.*23·34}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_390">[Pg 390]</span></p> + +<p class="nind"><b>*55·41.</b> \(\vdash\colon\colon R=x\downarrow y\unicode{x228d}z\downarrow w.\supset\colon\ldotp aRb.\supset_{a,b}.\phi(a,b):\equiv.\phi(x,y).\phi(z,w)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·4}.\supset\vdash\colon\colon\ldotp\text{Hp}.\supset\colon\colon aRb.\supset_{a,b}.\phi(a,b):\equiv\colon\ldotp\\ +\qquad\qquad a=x.b=y.\lor.a=z.b=w:\supset_{a,b}.\phi(a,b)\colon\ldotp\\ +[\text{*4·77}] \equiv\colon\ldotp(a,b)\colon\ldotp a=x.b=y.\supset.\phi(a,b):a=z.b=w.\supset.\phi(a,b)\colon\ldotp\\ +[\text{*11·31}] \equiv\colon\ldotp(a,b):a=x.b=y.\supset.\phi(a,b)\colon\ldotp(a,b):a=z.b=w.\supset.\phi(a,b)\colon\ldotp\\ +[\text{*13·21}] \equiv\colon\ldotp \phi(x,y).\phi(z,w)\colon\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>The above proposition is the analogue of <a href="#*51·234">*51·234</a>. The following +proposition (*55·42) is the analogue of <a href="#*51·235">*51·235</a>.</p> + +<p class="nind"><b>*55·42.</b> \(\vdash\colon\colon R=x\downarrow y\unicode{x228d}z\downarrow w.\supset\colon\ldotp(\exists a,b).aRb.\phi(a,b).\equiv:\phi(x,y).\lor.\phi(z,w)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·4}.\supset\vdash\colon\colon\ldotp\text{Hp}.\supset\colon\colon(\exists a,b).aRb.\phi(a,b).\equiv\colon\ldotp\\ +\qquad\qquad (\exists a,b)\colon\ldotp a=x.b=y.\lor.a=z.b=w:\phi(a,b)\colon\ldotp\\ +[\text{*4·4}] \equiv\colon\ldotp(\exists a,b):a=x.b=y.\phi(a,b):\lor:a=z.b=w.\phi(a,b)\colon\ldotp\\ +[\text{*11·41}] \equiv\colon\ldotp(\exists a,b).a=x.b=y.\phi(a,b).\lor.(\exists a,b).a=z.b=w.\phi(a,b)\colon\ldotp\\ +[\text{*13·22}] \equiv\colon\ldotp \phi(x,y).\lor.\phi(z,w)\colon\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·43.</b> \(\vdash:x\downarrow y\unicode{x228d}z\downarrow w=x\downarrow y\unicode{x228d}c\downarrow d.\equiv.z=c.w=d.\equiv.z\downarrow w=c\downarrow d\)</p> + +<p>This proposition is the analogue of <a href="#*51·41">*51·41</a>.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·202.}&\supset\vdash:z=c.w=d.\supset.z\downarrow w=c\downarrow d.\\ +[\text{*23·551}] &\supset.x\downarrow y\unicode{x228d}z\downarrow w=x\downarrow y\unicode{x228d}c\downarrow d &\qquad \text{(1)}\\ +\vdash.\text{*23·58}. &\supset\vdash\colon\ldotp x\downarrow y\unicode{x228d}z\downarrow w=x\downarrow y\unicode{x228d}c\downarrow d.\supset:\\ +&z\downarrow w\unicode{x2abd}x\downarrow y\unicode{x228d}c\downarrow d.c\downarrow d\unicode{x2abd}x\downarrow y\unicode{x228d}z\downarrow w:\\ +[\text{*55·3·13.*23·34}]&\supset:z=x.w=y.\lor.z=c.w=d:c=x.d=y.\lor.c=z.d=w:\\ +[\text{*13·16}] &\supset:z=x.w=y.\lor.z=c.w=d:c=x.d=y.\lor.z=c.w=d:\\ +[\text{*4·41}] &\supset:z=x.w=y.c=x.d=y.\lor.z=c.w=d:\\ +[\text{*13·172}] &\supset:z=c.w=d &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash:x\downarrow y\unicode{x228d}z\downarrow w=x\downarrow y\unicode{x228d}c\downarrow d.\equiv.z=c.w=d &&\qquad \text{(3)}\\ +\vdash.\text{(3).*55·202}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·431.</b> \[\begin{align}\vdash\colon\ldotp x\downarrow y\unicode{x228d}z&\downarrow w=a\downarrow b\unicode{x228d}c\downarrow d.\supset:\\ +&x=a.y=b.z=c.w=d.\lor.x=c.y=d.z=a.w=b\end{align}\]</p> + +<p><span class="pagenum" id="Page_391">[Pg 391]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·4}.\supset\vdash\colon\colon\text{Hp}.&\equiv\colon\ldotp u=x.v=y.\lor.u=z.v=w:\\ +&\equiv_{u,v}:u=a.v=b.\lor.u=c.v=d\colon\ldotp\\ +[\text{*11·1}] & \supset\colon\ldotp x=x.y=y.\lor.x=z.y=w:\\ +& \equiv :x=a.y=b.\lor.x=c.y=d\colon\ldotp\\ +[\text{*13·15}] &\supset\colon\ldotp x=a.y=b.\lor.x=c.y=d &\qquad \text{(1)}\\ +\vdash.\text{*55·43}.&\supset\vdash\colon\ldotp x=a.y=b.\supset:x\downarrow y\unicode{x228d}z\downarrow w=a\downarrow b\unicode{x228d}z\downarrow w:\\ +[\text{*13·171}] & \supset:\text{Hp}.\supset.a\downarrow b\unicode{x228d}z\downarrow w=a\downarrow b\unicode{x228d}c\downarrow d.\\ +[\text{*55·43}] & \supset.z=c.w=d &\qquad \text{(2)}\\ +\vdash.\text{(2).Comm.*4·7}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:x=a.y=b.\supset.x=a.y=b.z=c.w=d &\qquad \text{(3)}\\ +\text{Similarly} & \vdash\colon\ldotp\text{Hp}.\supset:x=c.y=d.\supset.x=c.y=d.z=a.w=b &\qquad \text{(4)}\\ +\vdash.\text{(1).(3).(4)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·44.</b> \[\begin{align}\vdash\colon\ldotp x\downarrow y & \unicode{x228d}z\downarrow w=a\downarrow b\unicode{x228d}c\downarrow d.\\ +&\equiv:x=a.y=b.z=c.w=d.\lor.x=c.y=d.z=a.w=b:\\ +&\equiv:x\downarrow y=a\downarrow b.z\downarrow w=c\downarrow d.\lor.x\downarrow y=c\downarrow d.z\downarrow w=a\downarrow b\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·43}. &\supset\vdash:x=a.y=b.\supset.x\downarrow y\unicode{x228d}z\downarrow w=a\downarrow b\unicode{x228d}z\downarrow w:\\ +& z=c.w=d.\supset.a\downarrow b\unicode{x228d}z\downarrow w=a\downarrow b\unicode{x228d}c\downarrow d:\\ +[\text{*3·47.*13·17}] &\supset\vdash:x=a.y=b.z=c.w=d.\\ +& \supset.x\downarrow y\unicode{x228d}z\downarrow w=a\downarrow b\unicode{x228d}c\downarrow d &\qquad \text{(1)}\\ +\text{Similarly} &\vdash:x=c.y=d.z=a.w=b.\\ +& \supset.x\downarrow y\unicode{x228d}z\downarrow w=a\downarrow b\unicode{x228d}c\downarrow d &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*55·431·202}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>The above proposition is the analogue of <a href="#*51·43">*51·43</a>.</p> + +<p class="nind"><b>*55·5.</b> \[\begin{align}\vdash\colon\ldotp R\unicode{x2abd}x&\downarrow y\unicode{x228d}z\downarrow w.\\ +&\equiv:R=\dot{\Lambda}.\lor.R=x\downarrow y.\lor.R=z\downarrow w.\lor.R=x\downarrow y\unicode{x228d}z\downarrow w\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*25·12.*23·58·42}.\supset\\ +\vdash\colon\ldotp R=\dot{\Lambda}.\lor.R=x\downarrow y.\lor.R=z\downarrow w.\lor.R=x\downarrow y\unicode{x228d}z\downarrow w:\\ +\supset.R\unicode{x2abd}x\downarrow y\unicode{x228d}z\downarrow w &\qquad \text{(1)}\\ +\vdash.\text{*25·49}. \supset\vdash\colon\ldotp R\unicode{x2abd}x\downarrow y\unicode{x228d}z\downarrow w.R\dot{\cap}x\downarrow y=\Lambda.\supset:R\unicode{x2abd}z\downarrow w:\\ +[\text{*55·341}] \supset:R=\Lambda.\lor.R=z\downarrow w &\qquad \text{(2)}\\ +\vdash.\text{*25·43}. \supset\vdash\colon\ldotp R\unicode{x2abd}x\downarrow y\unicode{x228d}z\downarrow w.\supset:R\dot{-}x\downarrow y\unicode{x2abd}z\downarrow w:\\ +[\text{*55·341}] \supset:R\dot{-}x\downarrow y=\dot{\Lambda}.\lor.R\dot{-}x\downarrow y=z\downarrow w:\\ +[\text{*25·24.*23·551}] \supset:(R\dot{-}x\downarrow y)\unicode{x228d}x\downarrow y=x\downarrow y.\lor.\\ +(R\dot{-}x\downarrow y)\unicode{x228d}x\downarrow y=x\downarrow y\unicode{x228d}z\downarrow w &\qquad \text{(3)}\\ +\vdash.\text{*55·3·36}.\supset\vdash:\dot{\exists}!(R\dot{\cap}x\downarrow y).\supset.(R\dot{-}x\downarrow y)\unicode{x228d}x\downarrow y=R &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash\colon\ldotp R\unicode{x2abd}x\downarrow y\unicode{x228d}z\downarrow w.\dot{\exists}!(R\dot{\cap}x\downarrow y).\supset:\\ +R=x\downarrow y.\lor.R=x\downarrow y\unicode{x228d}z\downarrow w &\qquad \text{(5)}\\ +\vdash.\text{(2).(5)}.\supset\vdash\colon\ldotp R\unicode{x2abd}x\downarrow y\unicode{x228d}z\downarrow w.\supset:\\ + R=\dot{\Lambda}.\lor.R=x\downarrow y.\lor.R=z\downarrow w.\lor.R=x\downarrow y\unicode{x228d}z\downarrow w &\qquad \text{(6)}\\ +\vdash.\text{(1).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is the analogue of <a href="#*54·4">*54·4</a>.</p> + +<p><span class="pagenum" id="Page_392">[Pg 392]</span></p> + +<p class="nind"><b><a id="*55·51">*55·51</a>.</b> \(\vdash\colon\ldotp R \unicode{x2abd} x \downarrow y \unicode{x228d} S .\supset: x R y .\lor. R \unicode{x2abd} S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*55·3}. &\supset\vdash :\dot{\exists}!(R \dot{\cap} x\downarrow y). \supset. xRy &\qquad \text{(1)}\\ +\vdash. \text{*25·49}. &\supset \vdash : \text{Hp}. {\sim}\dot{\exists}!(R \dot{\cap} x \downarrow y).\supset. R\unicode{x2abd}S &\qquad \text{(2)}\\ +\vdash. \text{(1).(2)}. &\supset\vdash . \text{Prop}\\ +\end{array} +\]</p> + +<p>In the remainder of the present number, we are concerned with +properties of ordinal couples which have no analogues for unit classes.</p> + +<p class="nind"><b>*55·52.</b> + \(\vdash. (\iotaʻ x\cup \iotaʻ y)\uparrow(\iotaʻ z\cup \iotaʻ w)= x\downarrow z\unicode{x228d}x\downarrow w\unicode{x228d}y\downarrow z\unicode{x228d}y\downarrow w \quad[\text{*35·82·413}]\)</p> + +<p class="nind"><b>*55·521.</b> \(\vdash: x\neq y. \equiv .x\downarrow y\unicode{x2abd}J \quad[\text{*55·3.*50·11}]\)</p> + +<p class="nind"><b>*55·53.</b> \(\vdash\colon\ldotp x \neq y.\supset: CʻR = \iotaʻ x\cup \iotaʻ y .R \unicode{x2abd} J .\equiv.\dot{\exists}!R.R\unicode{x2abd}x\downarrow + y\unicode{x228d}y\downarrow x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*55·5}. &\supset\vdash \colon\ldotp \dot{\exists} !R.R\unicode{x2abd}x\downarrow y\unicode{x228d}y\downarrow x.\equiv:\\ +&R = x\downarrow y .\lor. R=y\downarrow x.\lor. R = x\downarrow y\unicode{x228d}y\downarrow x &\qquad \text{(1)}\\ +\vdash . \text{*55·15}. &\supset\vdash. Cʻx\downarrow y = \iotaʻ x\cup \iotaʻ y . Cʻ y\downarrow x = \iotaʻ x\cup \iotaʻ y &\qquad \text{(2)}\\ +\vdash. \text{(2).*33·262}. &\supset\vdash. Cʻ (x\downarrow y\unicode{x228d}y\downarrow x) = \iotaʻ x\cup \iotaʻ y &\qquad \text{(3)}\\ +\vdash. \text{*55·521}. &\supset\vdash: x\neq y .\supset. x \downarrow y\unicode{x2abd}J.y\downarrow x\unicode{x2abd}J. &\qquad \text{(4)}\\ +[\text{*23·59}] &\supset. x\downarrow y\unicode{x228d}y \downarrow x\unicode{x2abd}J &\qquad \text{(5)}\\ +\vdash.\text{(1).(2).&(3).(4).(5)}.&\supset\vdash\colon\ldotp\\ +&x\neq y .\supset:\dot{\exists}!R . R\unicode{x2abd}x\downarrow y\unicode{x228d}y\downarrow x.\supset. Cʻ R= \iotaʻ x\cup \iotaʻ y. R\unicode{x2abd}J &\qquad \text{(6)}\\ +\vdash.\text{*35·91}. &\supset\vdash : Cʻ R = \iotaʻ x\cup \iotaʻ y .\cup . R\unicode{x2abd}(\iotaʻ x\cup \iotaʻ y)\uparrow (\iotaʻ x\cup \iotaʻ y).\\ +[\text{*55·52}]& \supset. R\unicode{x2abd}x\downarrow x\unicode{x228d}x \downarrow y\unicode{x228d}y \downarrow x\unicode{x228d}y \downarrow y &\qquad \text{(7)}\\ +\vdash· \text{*50·24}. &\supset\vdash : R\unicode{x2abd}J. \supset . {\sim}(xRx).{\sim}(yRy).\\ +[\text{*55·3. Transp}] &\supset.R\dot{\cap}x\downarrow x = \dot{\Lambda}. R\dot{\cap} y \downarrow y = \dot{\Lambda} &\qquad \text{(8)}\\ +\vdash. \text{(7).(8).*25·49}.&\supset\vdash:Cʻ R = \iotaʻ x\cup \iotaʻ y. R\unicode{x2abd}J .\supset . R\unicode{x2abd}x \downarrow y\unicode{x228d}y \downarrow x &\qquad \text{(9)}\\ +\vdash. \text{*33·24.*51·161}. &\supset\vdash: Cʻ R = \iotaʻ x\cup \iotaʻ y . \supset .\dot{\exists} !R &\qquad \text{(10)}\\ +\vdash. \text{(9).(10)}.&\supset\vdash: Cʻ R = \iotaʻ x\cup \iotaʻ y . R\unicode{x2abd}J .\supset .\dot{\exists}!R.R\unicode{x2abd}x\downarrow + y\unicode{x228d}y\downarrow x &\qquad \text{(11)}\\ +\vdash. \text{(6).(11)}. \supset\vdash .\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·54.</b> \(\vdash\colon\colon x\neq y.\supset\colon\ldotp CʻR = \iotaʻ x\cup \iotaʻ y.R\dot{\cap}\breve{R} = \dot{\Lambda} . \equiv:R=x\downarrow y.\lor.R=y\downarrow x\)</p> + +<p><span class="pagenum" id="Page_393">[Pg 393]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*50·46.*4·71} .&\supset\vdash:R\dot{\cap}\breve{R} = \dot{\Lambda} .\equiv .R \unicode{x2abd} J. R\dot{\cap}\breve{R}= \dot{\Lambda} &\qquad \text{(1)}\\ +\vdash . \text{(1).*55·53} .&\supset\vdash\colon\colon x \neq y. \supset\colon\ldotp CʻR= \iotaʻ x\cup \iotaʻ y. R\dot{\cap}\breve{R}= \dot{\Lambda}.\\ +&\equiv :\dot{\exists} !R .R\unicode{x2abd}x\downarrow y\unicode{x228d}y\downarrow x.R\dot{\cap}\breve{R}= \dot{\Lambda} :\\ +[\text{*55·5·134}] &\equiv:R = x\downarrow y.\lor.R = y\downarrow x.\lor.R = x\downarrow y\unicode{x228d}y\downarrow x:R\dot{\cap}\breve{R}=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash. \text{*55·32}.\supset\vdash\colon\ldotp x\neq y. &\supset: x\downarrow y\dot{\cap}y\downarrow x= \dot{\Lambda}:\\ +[\text{*55·14}] &\supset: R = x \downarrow y .\supset. R\dot{\cap}\breve{R}= \dot{\Lambda} :\\ +&R = y\downarrow x .\supset. R\dot{\cap}\breve{R}= \dot{\Lambda} &\qquad \text{(3)}\\ +\vdash.\text{*55·14.*31·15·33}.&\supset\vdash:R=x\downarrow y\unicode{x228d}y\downarrow x.\supset.R=\breve{R}.\\ +[\text{*23·5}] &\supset.R\dot{\cap}\breve{R}=R.\\ +[\text{*55·134}] &\supset.\dot{\exists}!R\dot{\cap}\breve{R} &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).*4·71.*5·71}.\supset\\ +\vdash\colon\colon x\neq y.\supset\colon\ldotp R=x\downarrow y.\lor.R=y\downarrow x.\lor.R&=x\downarrow y\unicode{x228d}y\downarrow x:R\dot{\cap}\breve{R}=\dot{\Lambda}:\\ +&\equiv:R=x\downarrow y.\lor.R=y\downarrow x &\qquad \text{(5)}\\ +\vdash.\text{(2).(5)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·57.</b> \(\vdash.R\mid(x\downarrow y)=\overrightarrow{R}ʻ x\uparrow\iotaʻ y \quad[\text{*37·81.*55·1.*53·301}]\)</p> + +<p class="nind"><b>*55·571.</b> \(\vdash.(x\downarrow y)\mid S=\iotaʻ x\uparrow\overleftarrow{S}ʻ y\)</p> + +<p class="nind"><b>*55·572.</b> \(\vdash.R\mid(x\downarrow y)\mid S=\overrightarrow{R}ʻ x\uparrow\overleftarrow{S}ʻy \quad[\text{*55·571.*37·81}]\)</p> + +<p class="nind"><b>*55·573.</b> \(\vdash.R\mid(x\downarrow y)\mid\breve{S}=\overrightarrow{R}ʻ x\uparrow\overrightarrow{S}ʻy \quad\left[\text{*55·572}\, \frac{\breve{S}}{S}\right]\)</p> + +<p class="nind"><b>*55·58.</b> \(\vdash:\text{E}!Rʻ x.\supset.R\mid(x\downarrow y)=(Rʻx)\downarrow y \quad[\text{*55·57.*53·31.*55·1}]\)</p> + +<p class="nind"><b>*55·581.</b> \(\vdash:\text{E}!\breve{S}ʻ y.\supset.(x\downarrow y)\mid S=x\downarrow(\breve{S}ʻy)\)</p> + +<p class="nind"><b>*55·582.</b> \(\vdash:\text{E}!Rʻ x.\text{E}!\breve{S}ʻ y.\supset.R\mid(x\downarrow y)\mid S=(Rʻx)\downarrow(\breve{S}ʻ y) \quad[\text{*55·58·581}]\)</p> + +<p class="nind"><b>*55·583.</b> \(\vdash:\text{E}!Rʻ x.\text{E}!Sʻ y.\supset.R\mid(x\downarrow y)\mid\breve{S}=(Rʻx)\downarrow(Sʻy) \quad\left[\text{*55·582}\, \frac{\breve{S}}{S}\right]\)</p> + +<p>The above propositions are frequently useful in arithmetic. Their +use arises as follows. Let \(\alpha\), \(\beta\), \(\gamma\), +\(\delta\) be classes of which \(\alpha\) is correlated with +\(\gamma\) by the relation \(R\), and \(\beta\) with \(\delta\) +by the relation \(S\). Then if \(x\in\gamma.y\in\delta\), the +couple consisting of the correlate of \(x\) and the correlate of +\(y\) is (\(Rʻx)\downarrow(Sʻy)\), <i>i.e.</i>, by the above, +\(R\mid(x\downarrow y)\mid\breve{S}\), <i>i.e.</i> +(\(R\Arrowvert\breve{S})ʻ(x\downarrow y)\). Thus the relation +\(R\Arrowvert\breve{S}\) correlates the couples, in \(\alpha\) and +\(\beta\), composed of the correlates of terms in \(\gamma\) and +\(\delta\). The most useful form, in practice, of *55·583, is that +given below in *55·61.</p> + +<p class="nind"><b>*55·6.</b> \(\vdash.(R\Arrowvert\breve{S})ʻ (z\downarrow w)=\overrightarrow{R}ʻ z\uparrow\overrightarrow{S}ʻ w \quad[\text{*55·573.*43·112}]\)</p> + +<p class="nind"><b>*55·61.</b> \[\begin{align}&\vdash:\text{E}!Rʻ z.\text{E}!Sʻ w.\supset.(R\Arrowvert\breve{S})ʻ (z\downarrow w)=(Rʻz)\downarrow(Sʻw)\\ +&[\text{*55·583.*43·112}]\end{align}\]</p> + +<p class="nind"><b>*55·62.</b> \(\vdash:z\neq w.S=x\downarrow z\unicode{x228d}y\downarrow w.\supset.Sʻz=x.Sʻw=y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·13}. &\supset\vdash\colon\colon\text{Hp}.\supset\colon\ldotp uSz.\equiv:u=x.z=z.\lor.u=y.z=w &\qquad \text{(1)}\\ +\vdash.\text{(1).*13·15}.&\supset \vdash\colon\ldotp\text{Hp}.\supset:uSz.\equiv.u=x &\qquad \text{(2)}\\ +\text{Similarly} &\vdash\colon\ldotp\text{Hp}.\supset:uSw.\equiv.u=y &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*30·3}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_394">[Pg 394]</span></p> + +<p class="nind"><b>*55·621.</b> \[\begin{align}&\vdash:x\neq y.S=x\downarrow z\unicode{x228d}y\downarrow w.\supset.\breve{S}ʻ x=z.\breve{S}ʻ y=w\\ +&[\text{Proof as in *55·6}]\end{align}\]</p> + +<p>The four following propositions belong to <a href="#*43">*43</a>, but are inserted here +because the proof uses <a href="#*55·13">*55·13</a>.</p> + +<p class="nind"><b>*55·63.</b> \(\vdash:\dot{\exists}!Q\dot{\cap}S.P\Arrowvert Q=R\Arrowvert S.\supset.P=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·112}.\supset\vdash\colon\colon\text{Hp}.&\supset\colon\ldotp P\mid(y\downarrow z)\mid Q=R\mid(y\downarrow z)\mid S\colon\ldotp\\ +[\text{*34·1}] &\supset\colon\ldotp(\exists u,v).xPu.u(y\downarrow z)v.vQw.\equiv_{x,w}.\\ +& (\exists u,v).xRu.u(y\downarrow z)v.vSw\colon\ldotp\\ +[\text{*55·13.*13·22}] &\supset\colon\ldotp xPy.zQw.\equiv_{x,w}.xRy.zSw\colon\ldotp\\ +[\text{*4·73}] &\supset\colon\ldotp zQw.zSw.\supset_{w}:xPy.\equiv_{x}.xRy &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*11·35}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:xPy.\equiv_{x}.xRy &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·21}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·631.</b> \(\vdash:\dot{\exists}!P\dot{\cap}R.P\Arrowvert Q=R\Arrowvert S.\supset.Q=S \quad[\text{Proof as in *55·63}]\)</p> + +<p class="nind"><b>*55·632.</b> \(\vdash:P\Arrowvert Q=R\Arrowvert S.\dot{\exists}!P.\dot{\exists}!Q.\supset.\dot{\exists}!P\dot{\cap}R.\dot{\exists}!Q\dot{\cap}S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·13}. &\supset\vdash:xPy.zQw.&\supset.x\{P\mid(y\downarrow z)\mid Q\}w.\\ +[\text{*43·112}] & &\supset.x{(P\Arrowvert Q)ʻ (y\downarrow z)}w &\qquad \text{(1)}\\ +\vdash.(1).\supset\vdash\colon\ldotp \text{Hp}.&\supset:xPy.zQw.&\supset.x\{(R\Arrowvert S)ʻ (y\downarrow z)\}w.\\ +[\text{*43·112}] & &\supset.x\{R\mid(y\downarrow z)\mid S\}w.\\ +[\text{*34·1}] & &\supset.(\exists u,v).xRu.u(y\downarrow z)v.vSw.\\ +[\text{*55·13.*13·22}]& &\supset.xRy.zSw.\\ +[\text{*4·7}] & &\supset.x(P\dot{\cap}R)y.z(Q\dot{\cap}S)w\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*55·64.</b> \[\begin{align}&\vdash\colon\ldotp \dot{\exists}!P.\dot{\exists}!Q.\lor.\dot{\exists}!R.\dot{\exists}!S:\supset:P\Arrowvert Q=R\Arrowvert S.\equiv.P=R.Q=S\\ +&[\text{*55·63·631·632}]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_395">[Pg 395]</span></p> + +<h2 class="nobreak" id="*56">*56. THE ORDINAL NUMBER \(2_{r}\).</h2> +</div> + + +<p><i>Summary of</i> *56.</p> + +<p>In this number, we have to consider the class of those relations which +are each constituted by a single couple. In case the two members of +this couple are not identical, the class of such relations is (as will +be shown later) the ordinal number 2, which, to distinguish it from +the cardinal number 2, we denote by "\(2_{r}\)." (Here the suffix +is intended to suggest "relational.") The class of all relations +consisting of a single couple, without the restriction that the +two members of the couple are to be distinct, will be denoted by +"\(\dot{2}\)." This is not an ordinal number. It will be observed that +there is no ordinal number 1, because ordinal numbers apply to series, +and series must have more than one member if they have any members. +This will appear more fully when we come to deal with series.</p> + +<p>The properties of \(\dot{2}\) are largely analogous to those of 1, +while the properties of \(2_{r}\) are more analogous to those of 2.</p> + +<p>Most of the propositions of the present number are seldom referred to +in the sequel, but such references as occur are important. The most +useful propositions in the present number are the following.</p> + +<p class="nind"><b>*56·111.</b> \(\vdash:R\in 2_{r}.\equiv.\text{D}ʻ R,\text{ᗡ}ʻ R\in 1.\text{D}ʻ R\cap \text{ᗡ}ʻ R=\Lambda\)</p> + +<p class="nind"><b>*56·112.</b> \(\vdash:R\in 2_{r}.\equiv.\text{D}ʻ R,\text{ᗡ}ʻ R\in 1.Cʻ R\in 2\)</p> + +<p class="nind"><b>*56·113.</b> \(\vdash.\dot{2}-2_{r}=\hat{R}\{(\exists a).R=a\downarrow a\}\)</p> + +<p>Observe that "\(\breve{C}ʻʻ2\)" means "relations whose fields have two +terms."</p> + +<p class="nind"><b>*56·13.</b> \(\vdash.\dot{2}-2_{r}=\hat{R}\{(\exists a).R=a\downarrow a\}\)</p> + +<p class="nind"><b>*56·37.</b> \(\vdash:R\in 2_{r}.\equiv.Cʻ R\in 2.R\dot{\cap}\breve{R}=\dot{\Lambda}\)</p> + +<p><i>I.e.</i> \(2_{r}\) is the class of asymmetrical relations whose fields have two terms.</p> + +<p class="nind"><b>*56·381.</b> \(\vdash:Cʻ R=\iotaʻ x.\equiv.R=x\downarrow x\)</p> + +<p class="nind"><b>*56·39.</b> \(\vdash.\dot{2}-2_{r}=\breve{C}ʻʻ 1\)</p> + +<p><i>I.e.</i> the relations which are couples whose referent and relatum +are identical are the relations whose fields consist of a single term.</p> + +<p><span class="pagenum" id="Page_396">[Pg 396]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*56·01.</b> \(\dot{2}=\hat{R}\{(\exists x,y).R=x\downarrow y\} \text{Df}\)</p> + +<p class="nind"><b>*56·02.</b> \(2_{r}=\hat{R}\{(\exists x,y).x \neq y.R=x\downarrow y\} \text{Df}\)</p> + +<p class="nind"><b>*56·03.</b> \(0_{r}=\iotaʻ\dot{\Lambda} \text{Df}\)</p> + +<p class="nind"><b>*56·1.</b> \(\vdash:R\in \dot{2}.\equiv.(\exists x,y).R=x\downarrow y \quad[\text{*20·3.(*56·01)}]\)</p> + +<p class="nind"><b>*56·101.</b> \(\vdash:R\in \dot{2}.\equiv.\text{D}ʻR,\text{ᗡ}ʻR\in 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·16.*11·11·341}.\supset\\ +\vdash\colon\ldotp (\exists x,y).R=x\downarrow y.&\equiv:(\exists x,y).\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy:\\ +[\text{*11·54}] &\equiv:(\exists x).\text{D}ʻR=\iotaʻx:(\exists y).\text{ᗡ}ʻR=\iotaʻy:\\ +[\text{*52·1}] &\equiv:\text{D}ʻR,\text{ᗡ}ʻR\in 1 &\qquad \text{(1)}\\ +\vdash.\text{(1).*56·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·102.</b> \(\vdash.\dot{2}=\breve{\text{D}}ʻʻ1\cap \breve{\text{ᗡ}}ʻʻ1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·101.*37·106}.\supset\\ +\vdash:R\in \dot{2}.&\equiv.R\in \breve{\text{D}}ʻʻ1.R\in \breve{\text{ᗡ}}ʻʻ1.\\ +[\text{*22·33}] &\equiv.R\in \breve{\text{D}}ʻʻ1\cap \breve{\text{ᗡ}}ʻʻ1:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·103.</b> \(\vdash:R\in \dot{2}.\supset.\dot{\exists}!R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·101}.\supset\vdash:R\in \dot{2}.&\supset.\text{D}ʻR\in 1.\\ +[\text{*52·16}] &\supset.\exists !\text{D}ʻR.\\ +[\text{*33·24}] &\supset.\dot{\exists}!R:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·104.</b> \(\vdash:R\in 0_{r}.\equiv.R=\dot{\Lambda} \quad[\text{(*56·03)}]\)</p> + +<p class="nind"><b>*56·11.</b> \(\vdash:R\in 2_{r}.\equiv.(\exists x,y).x \neq y.R=x\downarrow y \quad[\text{*20·3.(*56·02)}]\)</p> + +<p class="nind"><b>*56·111.</b> \(\vdash:R\in 2_{r}.\equiv.\text{D}ʻR,\text{ᗡ}ʻR\in 1.\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·231.*55·16}.\supset\\ +\vdash:x \neq y.R=x\downarrow y.&\equiv.\iotaʻx\cap \iotaʻy=\Lambda.\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy.\\ +[\text{*13·193}] &\equiv.\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda.\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy &\qquad \text{(1)}\\ +\vdash.\text{(1).*56·11.*11·11·341}.\supset\\ +\vdash\colon\ldotp R\in 2_{r}.\equiv:(\exists x,y).\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda.\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy:\\ +[\text{*11·45}] &\equiv:\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda:(\exists x,y).\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy:\\ +[\text{*11·54}] &\equiv:\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda:(\exists x).\text{D}ʻR=\iotaʻx:(\exists y).\text{ᗡ}ʻR=\iotaʻy:\\ +[\text{*52·1}] &\equiv:\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda.\text{D}ʻR,\text{ᗡ}ʻR\in 1\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_397">[Pg 397]</span></p> + +<p class="nind"><b>*56·112.</b> \(\vdash:R\in 2_{r}.\equiv.\text{D}ʻR,\text{ᗡ}ʻR\in 1.CʻR\in 2\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·111.*54·43}.\supset\\ +\vdash:R\in 2_{r}.&\equiv.\text{D}ʻR,\text{ᗡ}ʻR\in 1.\text{D}ʻR\cup \text{ᗡ}ʻR\in 2.\\ +[\text{*33·16}] &\equiv.\text{D}ʻR,\text{ᗡ}ʻR\in 1.CʻR\in 2:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·113.</b> \(\vdash.2_{r}=\dot{2}\cap \breve{C}ʻʻ2\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·112·101}.\supset\vdash:R\in 2_{r}.&\equiv.R\in \dot{2}.CʻR\in 2.\\ +[\text{*37·106.*33·122}] &\equiv.R\in \dot{2}.R\in \breve{C}ʻʻ2.\\ +[\text{*22·33}] &\equiv.R\in \dot{2}\cap \breve{C}ʻʻ2:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·114.</b> \(\vdash.2_{r}=\breve{\text{D}}ʻʻ1\cap \breve{\text{ᗡ}}ʻʻ1\cap \breve{C}ʻʻ2 \quad[\text{*56·113·102}]\)</p> + +<p class="nind"><b>*56·12.</b> \(\vdash:R\in 2_{r}.\equiv.R\in \dot{2}.R\unicode{x2abd}J\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·3.*50·11}. \supset\vdash:x \neq y.&\equiv.x\downarrow y\unicode{x2abd}J:\\ +[\text{Fact}] \supset\vdash:R=x\downarrow y.x \neq y.&\equiv.R=x\downarrow y.x\downarrow y\unicode{x2abd}J.\\ +[\text{*13·193}] &\equiv.R=x\downarrow y.R\unicode{x2abd}J &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·341}.\supset\\ +\vdash\colon\ldotp (\exists x,y).R=x\downarrow y.x \neq y.&\equiv:(\exists x,y).R=x\downarrow y.R\unicode{x2abd}J:\\ +[\text{*11·45}] &\equiv:(\exists x,y).R=x\downarrow y:R\unicode{x2abd}J:\\ +[\text{*56·1}] &\equiv:R\in \dot{2}.R\unicode{x2abd}J &\qquad \text{(2)}\\ +\vdash.\text{(2).*56·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·121.</b> \(\vdash.2_{r}\unicode{x2abd}\dot{2} \quad[\text{*56·113}]\)</p> + +<p class="nind"><b>*56·122.</b> \(\vdash:R\in 2_{r}.\supset.\dot{\exists}!R \quad[\text{*56·121·103}]\)</p> + +<p class="nind"><b>*56·13.</b> \(\vdash.\dot{2}-2_{r}=\hat{R}\{(\exists a).R=a\downarrow a\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·11.*11·52.Transp}.\supset +\vdash:R{\sim}\in 2_{r}. &\equiv:R=x\downarrow y.\supset_{x,y}.x=y &\qquad \text{(1)}\\ +\vdash.\text{(1).*56·1}.\supset\\ +\vdash\colon\ldotp R\in \dot{2}-2_{r}.&\equiv:(\exists a,b).R=a\downarrow b:R=x\downarrow y.\supset_{x,y}.x=y:\\ +[\text{*11·45}] &\equiv:(\exists a,b):R=a\downarrow b:R=x\downarrow y.\supset_{x,y}.x=y:\\ +[\text{*13·193}] &\equiv:(\exists a,b):R=a\downarrow b:a\downarrow b=x\downarrow y.\supset_{x,y}.x=y:\\ +[\text{*55·202}] &\equiv:(\exists a,b):R=a\downarrow b:a=x.b=y.\supset_{x,y}.x=y:\\ +[\text{*13·21}] &\equiv:(\exists a,b).R=a\downarrow b.a=b:\\ +[\text{*13·195}] &\equiv:(\exists a).R=a\downarrow a\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>\(\dot{2}-2_{r}\) might be defined as the ordinal number 1, since it +is what we shall call a relation number (cf. *153). But we wish our +ordinal numbers to be<span class="pagenum" id="Page_398">[Pg 398]</span> classes of <i>serial</i> relations, and such +relations have the property of being contained in diversity. Hence if +we were to define \(\dot{2}-2_{r}\) as the ordinal number 1, we should +introduce a tiresome exception, from which trivial complications would +be introduced into ordinal arithmetic. We have, therefore, not adopted +this course.</p> + +<p class="nind"><b>*56·14.</b> \(\vdash.\text{D}ʻ(x\downarrow )=\dot{2}\cap \overleftarrow{\text{D}}ʻ\iotaʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·6}.\supset\vdash:\text{D}ʻR&=\iotaʻx.\equiv.R\in \overleftarrow{\text{D}}ʻ\iotaʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*56·1}.\supset\\ +\vdash\colon\ldotp R\in \dot{2}\cap \overleftarrow{\text{D}}ʻ\iotaʻx.&\equiv:(\exists z,y).R=z\downarrow y:\text{D}ʻR=\iotaʻx:\\ +[\text{*55·16}] &\equiv:(\exists z,y).\text{D}ʻR=\iotaʻz.\text{ᗡ}ʻR=\iotaʻy:\text{D}ʻR=\iotaʻx:\\ +[\text{*11·45}] &\equiv:(\exists z,y).\text{D}ʻR=\iotaʻz.\text{ᗡ}ʻR=\iotaʻy.\text{D}ʻR=\iotaʻx:\\ +[\text{*13·193}] &\equiv:(\exists z,y).\text{D}ʻR=\iotaʻz.\text{ᗡ}ʻR=\iotaʻy.\iotaʻz=\iotaʻx:\\ +[\text{*51·23}] &\equiv:(\exists z,y).\text{D}ʻR=\iotaʻz.\text{ᗡ}ʻR=\iotaʻy.z=x:\\ +[\text{*13·195}] &\equiv:(\exists y).\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy:\\ +[\text{*55·16}] &\equiv:(\exists y).R=x\downarrow y:\\ +[\text{*55·22}] &\equiv:R\in \text{D}ʻ(x\downarrow )\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·141.</b> \(\vdash.\text{D}ʻ\downarrow x=\dot{2}\cap \overleftarrow{\text{ᗡ}}ʻ\iotaʻx \quad[\text{Proof as in *56·14}]\)</p> + +<p class="nind"><b>*56·15.</b> \(\vdash.\text{D}ʻ(x\downarrow )-\iotaʻ(x\downarrow x)=2_{r}\cap \overleftarrow{\text{D}}ʻ\iotaʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·22·16}.\supset\vdash\colon\ldotp R\in \{\text{D}ʻ(x\downarrow )\}-\iotaʻ(x\downarrow x).\\ +&\equiv:(\exists y).\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy:{\sim}(\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻx):\\ +[\text{*10·35.*4·51.*5·61}]&\equiv:(\exists y).\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy.{\sim}(\text{ᗡ}ʻR=\iotaʻx):\\ +[\text{*13·193}] &\equiv:(\exists y).\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy.{\sim}(\iotaʻy=\iotaʻx):\\ +[\text{*51·23}] &\equiv:(\exists y).\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy.x \neq y:\\ +[\text{*13·195.*51·23}]&\equiv:(\exists z,y).z \neq y.\text{D}ʻR=\iotaʻz.\text{ᗡ}ʻR=\iotaʻy.\iotaʻz=\iotaʻx:\\ +[\text{*13·193}] &\equiv:(\exists z,y).z \neq y.\text{D}ʻR=\iotaʻz.\text{ᗡ}ʻR=\iotaʻy.\text{D}ʻR=\iotaʻx:\\ +[\text{*11·45}] &\equiv:(\exists z,y).z\neq y.\text{D}ʻR=\iotaʻz.\text{ᗡ}ʻR=\iotaʻy:\text{D}ʻR=\iotaʻx:\\ +[\text{*55·16.*33·6}] &\equiv:(\exists z,y).z\neq y.R=z\downarrow y:R\in \overleftarrow{\text{D}}ʻ\iotaʻx:\\ +[\text{*56·11.*22·33}]&\equiv:R\in 2_{r}\cap \overleftarrow{\text{D}}ʻ\iotaʻx\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·151.</b> \(\vdash.\text{D}ʻ(\downarrow x)-\iotaʻ(x\downarrow x)=2_{r}\cap \overleftarrow{\text{ᗡ}}ʻ\iotaʻx \quad[\text{Proof as in *56·15}]\)</p> + +<p class="nind"><b>*56·16.</b> \(\vdash.x\downarrow y\in \dot{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·2}.&\supset\vdash.x\downarrow y=x\downarrow y.\\ +[\text{*11·36}]&\supset\vdash.(\exists z,w).x\downarrow y=z\downarrow w.\\ +[\text{*56·1}] &\supset\vdash.x\downarrow y\in \dot{2}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_399">[Pg 399]</span></p> + +<p class="nind"><b>*56·17.</b> \(\vdash:x\downarrow y\in 2_{r}.\equiv.y\downarrow x\in 2_{r}.\equiv.x \neq y\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash*56·11.\supset +\vdash\colon\ldotp x\downarrow y\in 2_{r}.&\equiv:(\exists z,w).z \neq w.x\downarrow y=z\downarrow w:\\ +[*55·202] &\equiv:(\exists z,w).z \neq w.x=z.y=w:\\ +[*13·22] &\equiv:x \neq y &\qquad \text{(1)}\\ +\text{Similarly}\\ +\vdash:y\downarrow x\in 2_{r}.\equiv.x \neq y &&\qquad \text{(2)}\\ +\vdash.(1).(2).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·18.</b> \(\vdash:x{\sim}\in \alpha.\equiv.x\downarrow ʻʻ\alpha\subset 2_{r}.\equiv.\downarrow xʻʻ\alpha\subset 2_{r}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*13·196}.\supset\vdash\colon\ldotp x{\sim}\in \alpha.&\equiv:y\in \alpha.\supset_{y}.y \neq x:\\ +[*56·17] &\equiv:y\in \alpha.\supset_{y}.x\downarrow y\in 2_{r}:\\ +[*37·61.*38·12·11] &\equiv:x\downarrow ʻʻ\alpha\subset 2_{r} &\qquad \text{(1)}\\ +\text{Similarly}\qquad\qquad\vdash:x{\sim}\in \alpha.&\equiv.\downarrow xʻʻ\alpha\subset 2_{r}&\qquad \text{(2)}\\ +\vdash.(1).(2).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·19.</b> \(\vdash:R\in 2_{r}.x\in \text{D}ʻR.\equiv.(\exists y).x \neq y.R=x\downarrow y.\equiv.R\in x\downarrow ʻʻ-\iotaʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·11.*11·45}.\supset\vdash\colon\ldotp R\in 2_{r}.x\in DʻR.&\equiv:(\exists y,z).y \neq z.R=y\downarrow z.x\in DʻR:\\ +[\text{*55·15}] &\equiv:(\exists y,z).y \neq z.R=y\downarrow z.x\in \iotaʻy:\\ +[\text{*51·23}] &\equiv:(\exists y,z).y \neq z.R=y\downarrow z.x=y:\\ +[\text{*13·195}] &\equiv:(\exists z).x \neq z.R=x\downarrow z: &\qquad \text{(1)}\\ +[\text{*51·15}] &\equiv:(\exists z).z\in -\iotaʻx.R=x\downarrow z:\\ +[\text{*38·13}] &\equiv:R\in x\downarrow ʻʻ-\iotaʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·191.</b> \[\begin{align}&\vdash:R\in 2_{r}.x\in \text{ᗡ}ʻR.\equiv.(\exists y).x \neq y.R=y\downarrow x.\equiv.R\in \downarrow xʻʻ-\iotaʻx\\ +&[\text{Proof as in *56*19}]\end{align}\]</p> + +<p class="nind"><b>*56·2.</b> \(\vdash\colon\ldotp R\in \dot{2}.\equiv:(\exists x,y):zRw.\equiv_{z,w}.z=x.w=y \quad[\text{*55·13.*56·1}]\)</p> + +<p class="nind"><b>*56·21.</b> \(\vdash\colon\ldotp R\in \dot{2}.\equiv:\dot{\exists}!R:xRy.zRw.\supset_{x,y,z,w}.x=z.y=w \quad[\text{*56·2.*14·124}]\)</p> + +<p class="nind"><b>*56·22.</b> \(\vdash.\dot{\Lambda}{\sim}\in \dot{2} \quad[\text{*56·103.*25·53}]\)</p> + +<p class="nind"><b>*56·24.</b> \(\vdash.\exists !\dot{2}.\exists !-\dot{2} \quad[\text{*56·22·16.*10·24}]\)</p> + +<p class="nind"><b>*56·25.</b> \(\vdash.\dot{2} \neq \Lambda\cap \text{Rel}.\dot{2} \neq \text{V}\cap \text{Rel} \quad[\text{*56·24.*24·54·17}]\)</p> + +<p class="nind"><b>*56·26.</b> \(\vdash\colon\ldotp R\in \dot{2}\cup \iotaʻ\dot{\Lambda}.\equiv:xRy.zRw.\supset_{x,y,z,w}.x=z.y=w\)</p> + +<p>This proposition is the analogue of <a href="#*52·4">*52·4</a>.</p> + +<p><span class="pagenum" id="Page_400">[Pg 400]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·236}.\supset\vdash\colon\colon R\in \dot{2}\cup \iotaʻ\dot{\Lambda}.\\ +&\equiv\colon\ldotp R\in \dot{2}.\lor.R=\dot{\Lambda}\colon\ldotp \\ +[\text{*25·51}] &\equiv\colon\ldotp R\in \dot{2}.\lor.{\sim}\dot{\exists}!R\colon\ldotp \\ +[\text{*56·21}] &\equiv\colon\ldotp \dot{\exists}!R:xRy.zRw.\supset_{x,y,z,w}.x=z.y=w\colon\ldotp \lor\colon\ldotp {\sim}\dot{\exists}!R\colon\ldotp\\ +[\text{*5·62}] &\equiv\colon\ldotp xRy.zRw.\supset_{x,y,z,w}.x=z.y=w.\lor.{\sim}\dot{\exists}!R &\qquad \text{(1)}\\ +\vdash.\text{*11·36.Transp}.&\supset\vdash\colon\ldotp {\sim}\dot{\exists}!R.\supset:{\sim}(xRy).{\sim}(zRw):\\ +[\text{*2·21}] &\supset:xRy.\supset.x=z:zRw.\supset.y=w:\\ +[\text{*3·47}] &\supset:xRy.zRw.\supset.x=z.y=w &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·3}. &\supset\vdash\colon\ldotp {\sim}\dot{\exists}!R.\supset:xRy.zRw.\supset_{x,y,z,w}.x=z.y=w &\qquad \text{(3)}\\ +\vdash.\text{(1).(3).*4·72}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·261.</b> \(\vdash\colon\colon R\in \dot{2}.\supset\colon\ldotp S\unicode{x2abd}R.\equiv:S=\Lambda.\lor.S=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·341}.\supset\vdash\colon\colon R=x\downarrow y\supset\colon\ldotp S\unicode{x2abd}R.\equiv:S=\dot{\Lambda}.\lor.S=R &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·35.*56·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·262.</b> \(\vdash\colon\ldotp R\in \dot{2}.\supset:S\unicode{x2abd}R.\dot{\exists}!S.\equiv.S=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·22}.\supset\vdash\colon\ldotp R\in \dot{2}.\supset:S=R.\supset.S \neq \dot{\Lambda} &\qquad \text{(1)}\\ +\vdash.\text{(1).*5·75.*56·261}.\supset\\ +\vdash\colon\ldotp R\in \dot{2}.\supset:S\unicode{x2abd}R.S \neq \dot{\Lambda}.\equiv.S=R &\qquad \text{(2)}\\ +\vdash.\text{(2).*25·54}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·27.</b> \(\vdash\colon\ldotp R\in \dot{2}.\supset:\dot{\exists}!R\dot{\cap}S.\equiv.R\dot{\cap}S\in \dot{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·34.*23·43}.\supset\\ +\vdash\colon\ldotp R=x\downarrow y.\supset:\dot{\exists}!R\dot{\cap}S.&\equiv.R\dot{\cap}S=R.\\ +[\text{*56·16}] &\supset.R\dot{\cap}S\in \dot{2} &\qquad \text{(1)}\\ +\vdash.\text{*56·103}.\supset\vdash:R\dot{\cap}S\in \dot{2}.&\supset.\dot{\exists}!R\dot{\cap}S &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon\ldotp R=x\downarrow y.&\supset:\dot{\exists}!R\dot{\cap}S.\equiv.R\dot{\cap}S\in \dot{2} &\qquad \text{(3)}\\ +\vdash.\text{(3).*11·11·35.*56.1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·28.</b> \(\vdash\colon\ldotp R\in \dot{2}.\supset:\dot{\exists}!R\dot{\cap}S.\equiv.R\unicode{x2abd}S.\equiv.R\dot{\cap}S=R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·3}.\supset\vdash\colon\ldotp R=x\downarrow y.\supset:\dot{\exists}!R\dot{\cap}S.&\equiv.R\unicode{x2abd}S. &\qquad \text{(1)}\\ +[\text{*23·621}] & \equiv.R\dot{\cap}S=R &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*11·11·35.*56·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·281.</b> \(\vdash\colon\ldotp R\in 2_{r}.\supset:\dot{\exists}!R\dot{\cap}S.\equiv.R\unicode{x2abd}S.\equiv.R\dot{\cap}S=R.\equiv.R\dot{\cap}S\in 2_{r}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·121}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:R \in \dot{2}:\\ +[\text{*56·28}]&\supset:\dot{\exists}!R\dot{\cap}S.\equiv.R\unicode{x2abd}S.\equiv.R\dot{\cap}S=R &\qquad \text{(1)}\\ +\vdash.\text{*13·13}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:R\dot{\cap}S=R.\supset.R\dot{\cap}S\in 2_{r}:\\ +[\text{(1)}] &\supset:\dot{\exists}!R\dot{\cap}S.\supset.R\dot{\cap}S\in 2_{r} &\qquad \text{(2)}\\ +\vdash.\text{*56·122}.&\supset\vdash:R\dot{\cap}S \in 2_{r}.\supset.\dot{\exists}!R\dot{\cap}S &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\dot{\exists}!R\dot{\cap}S.\equiv.R\dot{\cap}S\in 2_{r} &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_401">[Pg 401]</span></p> + +<p class="nind"><b><a id="*56·29">*56·29</a>.</b> \(\vdash\colon\colon P,\, Q\in \dot{2}.\supset\colon\ldotp P\unicode{x2abd}Q\unicode{x228d}R.\equiv:P=Q.\lor.P\unicode{x2abd}R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·51}.\supset\\ +\vdash\colon\ldotp x\downarrow y\unicode{x2abd}z\downarrow w\unicode{x228d}R.\supset:x(z\downarrow w)y.\lor.x\downarrow y\unicode{x2abd}R:\\ +[\text{*55·31}] \supset:x\downarrow y=z\downarrow w.\lor.x\downarrow y\unicode{x2abd}R &\qquad \text{(1)}\\ +\vdash.\text{(1).*13·12}.\supset\\ +\vdash\colon\colon\ldotp P=x\downarrow y.\supset\colon\colon Q=z\downarrow w.\supset\colon\ldotp P\unicode{x2abd}Q\unicode{x228d}R.\supset:P=Q.\lor.P\unicode{x2abd}R &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·35.*56·1}.\supset\\ +\vdash\colon\colon\ldotp P\in \dot{2}.\supset\colon\colon Q=z\downarrow w.\supset\colon\ldotp P\unicode{x2abd}Q\unicode{x228d}R.\supset:P=Q.\lor.P\unicode{x2abd}R &\qquad \text{(3)}\\ +\vdash.\text{(3).*11·11·3·35.*56·1}.\supset\\ +\vdash\colon\colon\ldotp P\in \dot{2}.\supset\colon\colon Q\in \dot{2}.\supset\colon\ldotp P\unicode{x2abd}Q\unicode{x228d}R.\supset:P=Q.\lor.P\unicode{x2abd}R &\qquad \text{(4)}\\ +\vdash.\text{*23·58·61}.\supset\vdash\colon\ldotp P=Q.\lor.P\unicode{x2abd}R:\supset.P\unicode{x2abd}Q\unicode{x228d}R &\qquad \text{(5)}\\ +\vdash.\text{(4).Imp.(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·3.</b> \(\vdash\colon\ldotp P,\,Q\in \dot{2}.\supset:P\unicode{x2abd}Q.\equiv.P=Q.\equiv.\dot{\exists}!P\dot{\cap}Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·3·31}.\supset\\ +\vdash:x\downarrow y\unicode{x2abd}z\downarrow w.\equiv.x\downarrow y=z\downarrow w.\equiv.\dot{\exists}!(x\downarrow y)\dot{\cap}(z\downarrow w) &\qquad \text{(1)}\\ +\vdash.\text{(1).*13·12}.\supset\\ +\vdash\colon\ldotp P=x\downarrow y.Q=z\downarrow w.\supset:P\unicode{x2abd}Q.\equiv.P=Q.\equiv.\dot{\exists}!P\dot{\cap}Q &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·35.*56·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The steps from (2) to the conclusion are analogous to those from (2) +of <a href="#*56·29">*56·29</a> to the conclusion of *56·29. Analogous steps in succeeding +proofs will be merely indicated as above.</p> + +<p class="nind"><b>*56·31.</b> \(\vdash\colon\ldotp P,\,Q\in \dot{2}.\supset:P \neq Q.\equiv.P\dot{\cap}Q=\dot{\Lambda} \quad[\text{*56·3.Transp}]\)</p> + +<p class="nind"><b>*56·32.</b> \(\vdash:P\in \dot{2}.\supset.P\dot{\cap}Q\in \dot{2}\cup \iotaʻ\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·27}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\dot{\exists}!P\dot{\cap}Q.\supset.P\dot{\cap}Q\in \dot{2}:\\ +[\text{*2·54.*25·54}] &\supset:P\dot{\cap}Q=\dot{\Lambda}.\lor.P\dot{\cap}Q\in \dot{2}:\\ +[\text{*51·236}] &\supset:P\dot{\cap}Q\in \dot{2}\cup \iotaʻ\dot{\Lambda}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·33.</b> \(\vdash\colon\colon P,\, Q\in \dot{2}.\supset\colon\ldotp R\unicode{x2abd}P\unicode{x228d}Q.\equiv:R=\dot{\Lambda}.\lor.R=P.\lor.R=Q.\lor.R=P\unicode{x228d}Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·5.*13·12}.&\supset\vdash\colon\colon P=x\downarrow y.Q=z\downarrow w.\supset\colon\ldotp \\ +&R\unicode{x2abd}P\unicode{x228d}Q.\equiv:R=\dot{\Lambda}.\lor.R=P.\lor.R=Q.\lor.R=P\unicode{x228d}Q &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·35.*56·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_402">[Pg 402]</span></p> + +<p class="nind"><b>*56·34.</b> \(\vdash\colon\colon P,\,Q\in \dot{2}.P \neq Q.\supset\colon\ldotp R\unicode{x2abd}P\unicode{x228d}Q.\dot{\exists}!R.R\neq P\unicode{x228d}Q.\equiv:R=P.\lor.R=Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·33·103.*5·75.*25·54}.\supset\\ +\vdash\colon\colon P,\,Q\in \dot{2}.&\supset\colon\ldotp R\unicode{x2abd}P\unicode{x228d}Q.\dot{\exists}!R.\equiv:R=P.\lor.R=Q.\lor.R=P\unicode{x228d}Q &\qquad \text{(1)}\\ +\vdash.\text{*23·62}. &\supset\vdash:P=P\unicode{x228d}Q.\equiv.Q\unicode{x2abd}P:\\ +[\text{*56·3}] &\supset\vdash\colon\ldotp P,Q\in \dot{2}.\supset:P=P\unicode{x228d}Q.\equiv.P=Q:\\ +[\text{Transp}] &\supset:P\neq Q.\supset.P\neq P\unicode{x228d}Q\colon\ldotp \\ +[\text{*13·181}] &\supset\vdash\colon\ldotp P,Q\in \dot{2}.P\neq Q.\supset:R=P.\supset.R\neq P\unicode{x228d}Q &\qquad \text{(2)}\\ +\vdash.\text{(2)}\, \frac{Q,\,P}{P,\,Q}.&\supset\vdash\colon\ldotp P,Q\in \dot{2}.P \neq Q.\supset:R=Q.\supset.R\neq P\unicode{x228d}Q &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash\colon\colon P,Q\in \dot{2}.P \neq Q.&\supset\colon\ldotp R=P.\lor.R=Q:\supset.R\neq P\unicode{x228d}Q &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*5·75}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·35.</b> \(\vdash:CʻR\in 2.R\dot{\cap}\breve{R}=\dot{\Lambda}.\supset.R\in 2_{r}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·54}.\supset\\ +\vdash\colon\ldotp x \neq y.CʻR=\iotaʻx\cup \iotaʻy.R\dot{\cap}\breve{R}=\dot{\Lambda}.\supset:R=x\downarrow y.\lor.R=y\downarrow x:\\ +[\text{*56·17}] \supset:R\in 2_{r} &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·35.*54·101}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*56·36.</b> \(\vdash:R\in 2_{r}.\supset.CʻR\in 2.R\dot{\cap}\breve{R}=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·54}.\supset\\ +\vdash:x \neq y.R=x\downarrow y.\supset.x\neq y.CʻR=\iotaʻx\cup \iotaʻy.R\dot{\cap}\breve{R}=\dot{\Lambda} &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·34.*56·11}.\supset\\ +\vdash\colon\ldotp R\in 2_{r}.\supset:(\exists x,y).x \neq y.CʻR=\iotaʻx\cup \iotaʻy.R\dot{\cap}\breve{R}=\dot{\Lambda}:\\ +[\text{*54·101.*11·45}]\supset:CʻR\in 2.R\dot{\cap}\breve{R}=\dot{\Lambda}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition, in addition to being used in <a href="#*56·38">*56·38</a>, is used +in the elementary theory of series (*204·463).</p> + +<p class="nind"><b>*56·37.</b> \(\vdash:R\in 2_{r}.\equiv.CʻR\in 2.R\dot{\cap}\breve{R}=\dot{\Lambda} \quad[\text{*56·35·36}]\)</p> + +<p class="nind"><b><a id="*56·38">*56·38</a>.</b> \(\vdash.2_{r}=\breve{C}ʻʻ2\cap \hat{R}(R\dot{\cap}\breve{R}=\dot{\Lambda})\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·106.*33·122}.\supset\vdash:CʻR\in 2.&\equiv.R\in \breve{C}ʻʻ2 &\qquad \text{(1)}\\ +\vdash.\text{*20·3}. \supset\vdash:R\dot{\cap}\breve{R}=\dot{\Lambda}.&\equiv.R\in \hat{R}(R\dot{\cap}\breve{R}=\dot{\Lambda}) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*56·37}. \supset\vdash:R\in 2_{r}.&\equiv.R\in \breve{C}ʻʻ2.R\in \hat{R}(R\dot{\cap}\breve{R}=\dot{\Lambda}).\\ +[\text{*22·33}] &\equiv.R\in \breve{C}ʻʻ2\cap \hat{R}(R\dot{\cap}\breve{R}=\dot{\Lambda}):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_403">[Pg 403]</span></p> + +<p>This proposition is important as establishing the connection between +the cardinal and ordinal 2. It shows that the ordinal 2 consists of +those asymmetrical relations whose fields have (cardinal) 2 terms. It +is used in the theory of well-ordered series (*250·44).</p> + +<p>The following proposition, in addition to being used in <a href="#*56·39">*56·39</a>, is used +in relation-arithmetic (*165·38) and in the theory of series (*205·4).</p> + +<p class="nind"><b>*56·381.</b> \(\vdash:CʻR=\iotaʻx.\equiv.R=x\downarrow x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·24·161.*51·161}.&\supset\vdash:CʻR=\iotaʻx.\supset.\exists !\text{D}ʻR.\text{D}ʻR\subset \iotaʻx.\\ +[\text{*51·4}] &\supset.\text{D}ʻR=\iotaʻx &\qquad \text{(1)}\\ +\text{Similarly} &\vdash:CʻR=\iotaʻx.\supset.\text{ᗡ}ʻR=\iotaʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*55·16}.&\supset\vdash:CʻR=\iotaʻx.\supset.R=x\downarrow x &\qquad \text{(3)}\\ +\vdash.\text{*55·15}. &\supset\vdash:R=x\downarrow x.\supset.CʻR=\iotaʻx &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*56·39">*56·39</a>.</b> \(\vdash.\dot{2}-2_{r}=\breve{C}ʻʻ1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*56·381}.\supset\vdash:CʻR\in 1.&\equiv.(\exists x).R=x\downarrow x.\\ +[\text{*56·13}] &\equiv.R\in \dot{2}-2_{r} &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·106}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition establishes the connection between \(\dot{2}-2_{r}\) +and 1, showing that \(\dot{2}-2_{r}\) is the class of those relations +whose fields consist of a single term. It is used in the discussion +of \(0_{r}\) and \(2_{r}\) and \(\dot{2}-2_{r}\) as relation-numbers +(*153·301).</p> + +<p class="nind"><b>*56·4.</b> \(\vdash\colon\ldotp \mu\subset \dot{2}.\supset:x\downarrow y\in \mu.\equiv.x(\dot{s}ʻ\mu)y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x(\dot{s}ʻ\mu)y.&\equiv.(\exists R).R\in \dot{2}.R\in \mu.xRy.\\ +[\text{*56·1}] &\equiv.(\exists z,w).z\downarrow w\in \mu.x(z\downarrow w)y.\\ +[\text{*55·13}] &\equiv.(\exists z,w).z\downarrow w\in \mu.z=x.w=y.\\ +[\text{*13·22}] &\equiv.x\downarrow y\in \mu\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is the analogue of <a href="#*53·23">*53·23</a>. It is used in the number on +exponentiation in relation-arithmetic (*176·19).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_404">[Pg 404]</span></p> +<h2 class="nobreak" id="SECTION_B_b">SECTION B.<br> +<br> +SUB-CLASSES, SUB-RELATIONS, AND RELATIVE TYPES.</h2> +</div> + + +<p><i>Summary of Section B.</i></p> + +<p>In this section, we consider first the classes contained in a given +class and the relations contained in a given relation. If \(\alpha\) +is any class, the classes contained in \(\alpha\) are the members +of \(\hat{\beta}(\beta \subset \alpha)\); these are also called the +sub-classes of \(\alpha\), or (sometimes) the "parts" of \(\alpha\). +In this last usage, they are called "proper parts" when they are not +coextensive with \(\alpha\), this phrase being formed on the analogy of +"proper fractions." The sub-classes of \(\alpha\) are all the classes +that can be formed from members of \(\alpha\); they are the same thing +as the "combinations" of members of \(\alpha\) taken any number at +a time. If \(n\) is the number of members of \(\alpha\), \(2^{n}\) is +the number of sub-classes of \(\alpha\), whether \(n\) be finite or +infinite. The number of sub-classes of \(\alpha\) is always greater +than the number of members of \(\alpha\). On account of these and other +propositions, the class of sub-classes of a given class is an important +function of the class. If the class is \(\alpha\), we denote the class +of its sub-classes by "\(\text{Cl}ʻ\alpha\)." This is a descriptive +function, derived from the relation "\(\text{Cl}\)," defined as follows: +\[ +\text{Cl} = \hat{\kappa}\hat{\alpha}\{\kappa = \hat{\beta}(\beta \subset \alpha)\} \quad \text{Df}\text{.} +\]</p> + +<p>The sub-relations of a given relation are all the relations contained +in the given relation, <i>i.e.</i> all relations which imply the given +relation for all possible arguments. That is, if \(P\) is the given +relation, \(R\) is a sub-relation of \(P\) if \(R \unicode{x2abd} P\). +Thus denoting the class of sub-relations of \(P\) by "\(\text{Rl}ʻP\)," +we are to have +\[ +\text{Rl}ʻP = \hat{R}(R \unicode{x2abd} P)\text{;} +\] +hence we take as the definition of "\(\text{Rl}\)" the following: +\[ +\text{Rl} = \hat{\lambda}\hat{P}\{\lambda = \hat{R}(R \unicode{x2abd} P)\} \quad \text{Df}. +\] +Sub-relations have properties analogous to those of sub-classes, but +they are of somewhat less importance. It should, however, be observed +that when one series is contained in another, <i>i.e.</i> is obtained +by selecting some of the terms of the other series without changing +their order, then the generating relation of the one series is a +sub-relation of the generating relation of the other series. (It is not +the case that a sub-relation of the<span class="pagenum" id="Page_405">[Pg 405]</span> generating relation of a series +must generate a contained series, for its field may fall apart into +detached portions, or otherwise fail of being serial.)</p> + +<p>We shall also consider in this section (<a href="#*62">*62</a>) the relation of membership +of a class, <i>i.e.</i> the relation which \(x\) has to \(\alpha\) when +\(x \in \alpha\). This relation bears the same relation to "\(x \in \alpha\)" +as "\(I\)" bears to "\(x = y\)." Strictly speaking, we ought +to introduce a new notation for it, putting (say) +\[ +A = \hat{x}\hat{\alpha}(x \in \alpha) \quad \text{Df}\text{.} +\] +But as \(\in\), unlike "\(=\)," is a letter, and capable of being +conveniently used alone, it seems more desirable, from the point of +view of avoiding unnecessary duplication of symbols, to put +\[ +\in = \hat{x}\hat{\alpha}(x \in \alpha) \quad \text{Df}\text{.} +\] +Strictly speaking, this definition is faulty, since it gives two +different meanings to "\(\in\)." But practically this does not matter, +since the above definition gives +\[ +\vdash: x\in \alpha .\equiv. x \in \alpha\text{,} +\] +where the first \(\in\) has the meaning just defined, while the second +has the old meaning. Thus all that is really required of the above +definition, namely to give a meaning to formulae in which \(\in\) +occurs without referent or relatum, is effected without the danger of +any confusion that could lead to errors.</p> + +<p>The chief importance of \(\in\) as a relation arises from the fact +that relations contained in \(\in\) play a very important part in +arithmetic. Take, for example, the problem of selecting one term out of +each member of a class of classes: in this case we require a selecting +relation \(R\) which is such that whenever \(x R \alpha\), \(x\) is +a member of \(\alpha\), <i>i.e.</i> such that \(R \unicode{x2abd} +{\in}\). (This condition is only part of the definition of a selecting +relation; the complete definition is given in <a href="#*80">*80</a>.)</p> + +<p>Three numbers in this section (<a href="#*63">*63</a>, <a href="#*64">*64</a>, <a href="#*65">*65</a>) are devoted to the +discussion of <i>relative types</i>. Given a variable \(x\), we +often want to define the relative types of other variables, or of +ambiguous symbols, occurring in the same context; that is, we wish to +express the types of these other symbols in terms of that of \(x\). +We use "\(tʻx\)" for the type of \(x\), "\(t_{0}ʻ\alpha\)" for the +type in which \(\alpha\) is contained. Then \(t_{0}ʻ\alpha = \alpha \cup -\alpha\), +\(tʻx = \iotaʻx \cup -\iotaʻx = t_{0}ʻ\iotaʻx\), and +\(tʻ\alpha = t_{0}ʻ\text{Cl}ʻ\alpha = \text{Cl}ʻt_{0}ʻ\alpha\). Also we +introduce a notation (*65) for giving typical definiteness, relatively +to \(x\), to typically ambiguous symbols. This notation is very useful +in cardinal and ordinal arithmetic, since numbers are typically +ambiguous, and the failure to take account of this fact has led to +the contradictions concerning the greatest cardinal and the greatest +ordinal.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_406">[Pg 406]</span></p> +<h2 class="nobreak" id="*60">*60. THE SUB-CLASSES OF A GIVEN CLASS.</h2> +</div> + + +<p><i>Summary of</i> *60.</p> + +<p>Our definitions in this number are as follows:</p> + +<p class="nind"><b>*60·01.</b> \(\text{Cl}=\hat{\kappa}\hat{\alpha}{\kappa=\hat{\beta}(\beta\subset\alpha)} \quad\text{Df}\)</p> + +<p>This defines the relation to a class \(\alpha\) of the class of all its sub-classes.</p> + +<p class="nind"><b>*60·02.</b> \(\text{Cl ex}=\hat{\kappa}\hat{\alpha}\{\kappa=\hat{\beta}(\beta\subset \alpha.\exists !\beta)\} \quad\text{Df}\)</p> + +<p>This defines the relation to a class \(\alpha\) of the class of all its +<i>existent</i> sub-classes, <i>i.e.</i> of all its sub-classes except +\(\Lambda\). This is often required, as, for example, in the statement +of Zermelo's axiom: "Given any class \(\alpha\), there is a relation +\(R\) such that, if \(\beta\) is any existent sub-class of \(\alpha\), +\(Rʻ\beta\) is a member of \(\beta\)," <i>i.e.</i> +\[ +\unicode{x201c}(\exists R):\beta\in\text{Cl ex}ʻ\alpha.\supset_{\beta}.Rʻ \beta\in\beta.\unicode{x201d} +\] +This axiom, or its equivalent the multiplicative axiom, plays (as +will appear hereafter) an important part as the hypothesis to many +propositions in cardinal arithmetic.</p> + +<p class="nind"><b>*60·03.</b> \(\text{Cls}^{2}=\text{Cl}ʻ\text{Cls} \quad\text{Df}\)</p> + +<p>A \(\text{Cls}^{2}\) is a class whose members are classes.</p> + +<p class="nind"><b>*60·04.</b> \(\text{Cls}^{3} = \text{Cl}ʻ\text{Cls}^{2} \quad\text{Df}\)</p> + +<p>A \(\text{Cls}^{3}\) is a class whose members are classes whose members +are classes, <i>i.e.</i> a \(\text{Cls}^{3}\) is a class of classes of +classes.</p> + +<p>Apart from propositions which merely embody the definitions, the most +useful propositions in this number are the following:</p> + +<p class="nind"><b>*60·3.</b> \(\vdash.\Lambda\in\text{Cl}ʻ \alpha\)</p> + +<p class="nind"><b>*60·32.</b> \(\vdash.\text{Cl}ʻ\Lambda=\iotaʻ\Lambda\)</p> + +<p class="nind"><b>*60·34.</b> \(\vdash.\alpha\in\text{Cl}ʻ \alpha\)</p> + +<p class="nind"><b>*60·362.</b> \(\vdash.\text{Cl}ʻ \iotaʻ x=\iotaʻ\Lambda\cup \iotaʻ \iotaʻ x\)</p> + +<p><i>I.e.</i> \(\Lambda\) and \(\iotaʻ x\) are the only sub-classes of a +unit class \(\iotaʻ x\).</p> + +<p class="nind"><b>*60·5.</b> \(\vdash.sʻ\text{Cl}ʻ \alpha=\alpha\)</p> + +<p class="nind"><b>*60·57.</b> \(\vdash.\kappa\subset\text{Cl}ʻsʻ \kappa\)</p> + +<p class="nind"><b>*60·6.</b> \(\vdash:x\in\alpha.\supset.\iotaʻ x\in\text{Cl ex}ʻ \alpha\)</p> + +<p><span class="pagenum" id="Page_407">[Pg 407]</span></p> + +<p>The propositions of this number are chiefly useful in cardinal and +ordinal arithmetic, but uses also occur in the theory of series; hardly +any uses occur before cardinal arithmetic.</p> + +<hr class="tb"> + +<p class="nind"><b>*60·01.</b> \(\text{Cl}=\hat{\kappa}\hat{\alpha}\{\kappa=\hat{\beta}(\beta\subset \alpha)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*60·02.</b> \(\text{Cl ex}=\hat{\kappa}\hat{\alpha}\{\kappa=\hat{\beta}(\beta\subset \alpha.\exists !\beta)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*60·03.</b> \(\text{Cls}^{2}=\text{Cl}ʻ\text{Cls} \quad\text{Df}\)</p> + +<p class="nind"><b>*60·04.</b> \(\text{Cls}^{3} = \text{Cl}ʻ\text{Cls}^{2} \quad\text{Df}\)</p> + +<p class="nind"><b>*60·1.</b> \(\vdash:\kappa\,\text{Cl}\, \alpha.\equiv.\kappa=\hat{\beta}(\beta\subset \alpha) \quad\text{[*21·3.(*60·01)]}\)</p> + +<p class="nind"><b>*60·11.</b> \(\vdash:\kappa\,\text{Cl ex}\, \alpha.\equiv.\kappa=\hat{\beta}(\beta\subset \alpha.\exists !\beta) \quad\text{[*21·3.(*60·02)]}\)</p> + +<p class="nind"><b>*60·12.</b> \(\vdash.\text{Cl}ʻ \alpha=\hat{\beta}(\beta\subset \alpha) \quad\text{[*30·3.*60·1]}\)</p> + +<p class="nind"><b>*60·13.</b> \(\vdash.\text{Cl ex}ʻ \alpha=\hat{\beta}(\beta\subset \alpha.\exists !\beta) \quad\text{[*30·3.*60·11]}\)</p> + +<p class="nind"><b>*60·14.</b> \(\vdash.\exists !\text{Cl}ʻ \alpha \quad\text{[*60·12.*14·21]}\)</p> + +<p class="nind"><b>*60·15.</b> \(\vdash.\exists !\text{Cl ex}ʻ \alpha \quad\text{[*60·13.*14·21]}\)</p> + +<p class="nind"><b>*60·2.</b> \(\vdash:\beta\in\text{Cl}ʻ \alpha.\equiv.\beta\subset \alpha \quad\text{[*60·12.*20·33]}\)</p> + +<p class="nind"><b>*60·21.</b> \(\vdash:\beta\in\text{Cl ex}ʻ \alpha.\equiv.\beta\subset \alpha.\exists !\beta \quad\text{[*60·13.*20·33]}\)</p> + +<p class="nind"><b>*60·22.</b> \(\vdash:\beta\in\text{Cl ex}ʻ \alpha.\equiv.\beta\in\text{Cl}ʻ \alpha.\exists !\beta \quad\text{[*60·2·21]}\)</p> + +<p class="nind"><b>*60·23.</b> \(\vdash:\beta\in\text{Cl ex}ʻ \alpha.\equiv.\beta\in\text{Cl}ʻ \alpha-\iotaʻ \Lambda \quad\text{[*60·22.*53·52]}\)</p> + +<p class="nind"><b>*60·24.</b> \(\vdash.\text{Cl ex}ʻ \alpha=\text{Cl}ʻ \alpha-\iotaʻ\Lambda \quad\text{[*60·23.*20·43]}\)</p> + +<p class="nind"><b>*60·3.</b> \(\vdash.\Lambda\in\text{Cl}ʻ \alpha \quad\text{[*24·12.*60·2]}\)</p> + +<p class="nind"><b>*60·31.</b> \(\vdash.\exists !\text{Cl}ʻ \alpha \quad\text{[*60·3.*10·24]}\)</p> + +<p class="nind"><b>*60·32.</b> \(\vdash.\text{Cl}ʻ\Lambda=\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·2.*24·13}.\supset\vdash:\alpha\in\text{Cl}ʻ\Lambda.&\equiv.\alpha=\Lambda.\\ +[\text{*51·15}] &\equiv.\alpha\in\iotaʻ\Lambda:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·321.</b> \(\vdash:\alpha=\Lambda.\equiv.\text{Cl}ʻ \alpha=\iotaʻ \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·32}. \supset\vdash:\alpha=\Lambda.\supset.\text{Cl}ʻ \alpha=\iotaʻ \alpha &\qquad \text{(1)}\\ +\vdash.\text{*60·2.*51·15}.\supset\\ +\qquad\qquad\vdash\colon\ldotp\text{Cl}ʻ \alpha=\iotaʻ \alpha.\equiv:\beta\subset \alpha.\equiv_{\beta}.\beta=\alpha:\\ +[\text{*10·1}] \supset:\Lambda\subset \alpha.\equiv.\Lambda=\alpha:\\ +[\text{*24·12}] \supset:\Lambda=\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_408">[Pg 408]</span></p> + +<p class="nind"><b>*60·33.</b> \(\vdash.\text{Cl ex}ʻ\Lambda=\Lambda\cap\text{Cls}\)</p> + +<p>We write "\(\Lambda\cap\text{Cls}\)" on the right, to indicate that the +\(\Lambda\) concerned is of higher type than the \(\Lambda\) on the +left.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·22·32}.\supset\vdash:\beta\in\text{Cl ex}ʻ\Lambda.&\equiv.\beta\in\iotaʻ\Lambda.\exists !\beta.\\ +[\text{*51·15.*24·54}] &\equiv.\beta=\Lambda.\beta\neq\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*3·24}.\supset\vdash.\beta{\sim}\in\text{Cl ex}ʻ\Lambda &&\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11.*24·15}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·34.</b> \(\vdash.\alpha\in\text{Cl}ʻ\alpha \quad\text{[*22·42.*60·2]}\)</p> + +<p class="nind"><b>*60·35.</b> \(\vdash:\exists !\alpha.\supset.\alpha\in\text{Cl ex}ʻ\alpha \quad\text{[*60·22·34]}\)</p> + +<p class="nind"><b>*60·36.</b> \(\vdash:\exists !\alpha.\supset.\exists !\text{Cl ex}ʻ\alpha \quad\text{[*60·35.*10·24]}\)</p> + +<p class="nind"><b>*60·361.</b> \(\vdash:\exists !\alpha.\equiv.\exists !\text{Cl ex}ʻ\alpha \quad\text{[*60·36·33]}\)</p> + +<p class="nind"><b>*60·362.</b> \(\vdash.\text{Cl}ʻ \iotaʻ x=\iotaʻ \Lambda\cup \iotaʻ \iotaʻ x \quad\text{[*51·401.*60·2]}\)</p> + +<p class="nind"><b>*60·37.</b> \(\vdash.\text{Cl ex}ʻ \iotaʻ x=\iotaʻ \iotaʻ x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·21}.\supset\vdash:\beta\in\text{Cl ex}ʻ\iotaʻ x.&\equiv.\beta\subset \iotaʻ x.\exists !\beta.\\ +[\text{*51·4}] &\equiv.\beta=\iotaʻ x.\\ +[\text{*51·15}] &\equiv.\beta\in \iotaʻ \iotaʻ x:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b><a id="*60·371">*60·371</a>.</b> \(\vdash:\alpha\in 1.\supset.\text{Cl}ʻ\alpha\subset 0\cup 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·401}.\supset\vdash\colon\colon \alpha=\iotaʻ x.&\supset\colon\ldotp \beta\subset \alpha.\equiv:\beta=\Lambda.\lor.\beta=\iotaʻ x:\\ +[\text{*54·102.*52·22}] &\supset:\beta\in 0.\lor.\beta\in 1\colon\ldotp\\ +[\text{*60·2.*22·34}] & \supset\colon\ldotp \beta\in\text{Cl}ʻ \alpha.\supset.\beta\in 0\cup 1 &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·23.*52·1}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·38.</b> \(\vdash:\alpha\in 1.\supset.\text{Cl}ʻ \alpha\subset 0\cup 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·37}.&\supset\vdash:\alpha=\iotaʻ x.\supset.\text{Cl ex}ʻ\alpha=\iotaʻ \alpha:\\ +[\text{*10·11·23}] &\supset\vdash:(\exists x).\alpha=\iotaʻ x.\supset.\text{Cl ex}ʻ\alpha=\iotaʻ \alpha:\\ +[\text{*52·1}] &\supset\vdash:\alpha\in 1.\supset.\text{Cl ex}ʻ\alpha=\iotaʻ \alpha &\qquad \text{(1)}\\ +\vdash.\text{*60·361.*51·161}. &\supset\vdash:\text{Cl ex}ʻ\alpha=\iotaʻ \alpha.\supset.\exists !\alpha &\qquad \text{(2)}\\ +\vdash.\text{*60·21.*10·1}. \supset\vdash\colon\ldotp\text{Cl ex}ʻ\alpha=\iotaʻ \alpha.&\supset:\iotaʻ x\subset \alpha.\exists !\iotaʻ x.\exists.\iotaʻ x=\alpha:\\ +[\text{*51·161}] &\supset:\iotaʻ x\subset \alpha.\equiv.\iotaʻ x=\alpha:\\ +[\text{*51·2}] & \supset:x\in\alpha.\equiv.\iotaʻ x=\alpha &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·21·281}.\supset\vdash\colon\ldotp\text{Cl ex}ʻ \alpha=\iotaʻ\alpha.\supset:\exists !\alpha.&\equiv.(\exists x).\iotaʻ x=\alpha.\\ +[\text{*52·1}] &\equiv.\alpha\in 1:\\ +[\text{(2)}] &\supset:\alpha\in 1 &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_409">[Pg 409]</span></p> + +<p class="nind"><b>*60·39.</b> \(\vdash.\text{Cl}ʻ (\iotaʻ x\cup \iotaʻ y)=\iotaʻ\Lambda\cup \iotaʻ\iotaʻ x\cup \iotaʻ\iotaʻ y\cup \iotaʻ (\iotaʻ x\cup \iotaʻ y) \quad[\text{*54·4.*60·2}]\)</p> + +<p class="nind"><b>*60·391.</b> \(\vdash:\alpha\in 2.\supset.\text{Cl}ʻ\alpha\subset 0\cup 1\cup 2 \quad[\text{*54·411.*60·2}]\)</p> + +<p>This proposition is used in the theory of the continuity of functions +(*234·202).</p> + +<p class="nind"><b>*60·4.</b> \(\vdash:\beta\in\text{Cl}ʻ\alpha.\gamma\subset\beta.\supset.\gamma\in\text{Cl}ʻ\alpha \quad[\text{*60·2.*22·44}]\)</p> + +<p class="nind"><b>*60·41.</b> \(\vdash:\beta\in\text{Cl}ʻ\alpha.\supset.\beta\cap \gamma\in\text{Cl}ʻ\alpha \quad[\text{*60·4.*22·43}]\)</p> + +<p>The following proposition is used in the theory of well-ordered series +(*250·14).</p> + +<p class="nind"><b>*60·42.</b> \(\vdash:\beta\in\text{Cl}ʻ\alpha.\gamma\subset\beta.\exists !\gamma.\supset.\gamma\in\text{Cl ex}ʻ\alpha \quad[\text{*60·4·22}]\)</p> + +<p class="nind"><b>*60·43.</b> \(\vdash:\beta,\gamma\in\text{Cl}ʻ\alpha.\equiv.\beta\cup \gamma\in\text{Cl}ʻ\alpha \quad[\text{*22·59.*60·2}]\)</p> + +<p class="nind"><b>*60·44.</b> \(\vdash:\beta\in\text{Cl}ʻ\alpha.\gamma\in\text{Cl ex}ʻ\alpha.\supset.\beta\cup \gamma\in\text{Cl ex}ʻ\alpha \quad[\text{*60·43.*24·56.*60·22}]\)</p> + +<p>The following proposition is required in the theory of "first +differences" (*170·65).</p> + +<p class="nind"><b>*60·45.</b> \(\vdash:\rho\in\text{Cl}ʻ (\alpha\cup\beta).\equiv.(\exists \gamma,\delta).\gamma\in\text{Cl}ʻ \alpha.\delta\in\text{Cl}ʻ \beta.\rho=\gamma\cup\delta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·2.*2·621·68}.\supset\\ +\vdash:\rho\in\text{Cl}ʻ (\alpha\cup\beta).&\supset.\rho=(\rho\cap\alpha)\cup (\rho\cap\beta)&\qquad \text{(1)}\\ +\vdash.\text{*60·2.*22·43}.&\supset\vdash.\rho\cap \alpha\in\text{Cl}ʻ \alpha.\rho\cap \beta\in\text{Cl}ʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*10·24}.\supset\\ +\vdash:\rho\in\text{Cl}ʻ (\alpha\cup\beta).&\supset.(\exists \gamma,\delta).\gamma\in\text{Cl}ʻ \alpha.\delta\in\text{Cl}ʻ \beta.\rho=\gamma\cup\delta &\qquad \text{(3)}\\ +\vdash.\text{*60·2}.\supset\\ +\vdash:(\exists \gamma,\delta).\gamma\in\text{Cl}ʻ \alpha.\delta\in\text{Cl}ʻ\beta.\rho=\gamma\cup \delta.&\supset.(\exists \gamma,\delta).\gamma\subset \alpha.\delta\subset \beta.\rho=\gamma\cup\delta.\\ +[\text{*22·72}] & \supset.\rho\subset \alpha\cup\beta.\\ +[\text{*60·2}] & \supset.\rho\in\text{Cl}ʻ (\alpha\cup\beta) &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·5.</b> \(\vdash.sʻ\text{Cl}ʻ \alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·1.*60·2}.\supset\vdash:x\in sʻ \text{Cl}ʻ \alpha.&\equiv.(\exists \beta).\beta\subset \alpha.x\in\beta.&\qquad \text{(1)}\\ +[\text{*22·441}] &\supset.x\in\alpha &\qquad \text{(2)}\\ +\vdash.\text{*22·42}.&\supset\vdash:x\in\alpha.\supset.\alpha\subset \alpha.x\in\alpha.\\ +[\text{*10·24}]&\supset.(\exists \beta).\beta\subset \alpha.x\in\beta.\\ +[\text{(1)}] &\supset.x\in sʻ \text{Cl}ʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_410">[Pg 410]</span></p> + +<p class="nind"><b>*60·501.</b> \(\vdash.sʻ\text{Cl ex}ʻ \alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·11.*60·21}.&\supset\vdash:x\in sʻ\text{Cl ex}ʻ \alpha.\equiv.(\exists \beta).\beta\subset \alpha.\exists !\beta.x\in\beta. &\qquad \text{(1)}\\ +[\text{*22·441}] &\supset.x\in\alpha &\qquad \text{(2)}\\ +\vdash.\text{*22·42}. &\supset\vdash:x\in\alpha.\supset.\alpha\subset \alpha.x\in\alpha.\\ +[\text{*10·24.*24·5.*4·7}]&\supset.\alpha\subset \alpha.\exists !\alpha.x\in\alpha.\\ +[\text{*10·24}] &\supset.(\exists \beta).\beta\subset \alpha.\exists !\beta.x\in\beta.\\ +[\text{(1)}] &\supset.x\in sʻ\text{Cl ex}ʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>The above proposition is used in the theory of cardinal multiplication +(*115·17).</p> + +<p class="nind"><b>*60·51.</b> \(\vdash.pʻ\text{Cl}ʻ\alpha=\Lambda \quad[\text{*40·22.*60·3}]\)</p> + +<p>The following proposition is used in the cardinal theory of finite and +infinite (*124·541).</p> + +<p class="nind"><b>*60·52.</b> \(\vdash:sʻ \kappa\subset\beta.\equiv.\kappa\subset \text{Cl}ʻ \beta \quad[\text{*40·151.*60·2}]\)</p> + +<p class="nind"><b>*60·53.</b> \(\vdash:\beta\subset pʻ \kappa.\equiv.\beta\in pʻ\text{Cl}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·15.*60·2}.\supset\vdash\colon\ldotp \beta\subset pʻ \kappa.&\equiv:\gamma\in\kappa.\supset_{\gamma}.\beta\in\text{Cl}ʻ\gamma:\\ +[\text{*40·41.*60·14}] &\equiv:\beta\in pʻ\text{Cl}ʻʻ \kappa\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·54.</b> \(\vdash.\text{Cl}ʻ pʻ \kappa=pʻ\text{Cl}ʻʻ\kappa \quad[\text{*60·53·2}]\)</p> + +<p class="nind"><b>*60·55.</b> \(\vdash:\text{Cl}ʻ \alpha=\text{Cl}ʻ\beta.\equiv.\alpha=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37.*60·14}.&\supset\vdash:\alpha=\beta.\supset.\text{Cl}ʻ \alpha=\text{Cl}ʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*30·37}. &\supset\vdash:\text{Cl}ʻ \alpha=\text{Cl}ʻ\beta.\supset.sʻ\text{Cl}ʻ \alpha=sʻ\text{Cl}ʻ\beta.\\ +[\text{*60·5}] &\supset.\alpha=\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·56.</b> \(\vdash:\text{Cl ex}ʻ\alpha=\text{Cl ex}ʻ\beta.\equiv.\alpha=\beta \quad[\text{Proof as in *60·55}]\)</p> + +<p>The following proposition is used frequently.</p> + +<p class="nind"><b>*60·57.</b> \(\vdash.\kappa\subset\text{Cl}ʻ sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·13.*60·2}. &\supset\vdash:\alpha\in\kappa.\supset.\alpha\in\text{Cl}ʻ sʻ \kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11.*22·1}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·6.</b> \(\vdash:x\in\alpha.\supset.\iotaʻ x\in\text{Cl ex}ʻ\alpha \quad[\text{*51·2·161.*60·21}]\)</p> + +<p><span class="pagenum" id="Page_411">[Pg 411]</span></p> + +<p>The following proposition is used in connection with cardinal +multiplication and with greater and less (*115·17 and *117·66).</p> + +<p class="nind"><b><a id="*60·61">*60·61</a>.</b> \(\vdash.\iotaʻʻ \alpha\subset\text{Cl ex}ʻ \alpha \quad[\text{*37·61.*51·12.*60·6}]\)</p> + +<p class="nind"><b>*60·62.</b> \(\vdash:x,y\in\alpha.\supset.\iotaʻ x\cup \iotaʻ y\in\text{Cl ex}ʻ \alpha \quad[\text{*60·6·44}]\)</p> + +<p class="nind"><b>*60·7.</b> \(\vdash.\text{Cl}ʻ \alpha\in\text{Cls}^{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·2}.\supset\vdash:\beta\in\text{Cl}ʻ \alpha.&\equiv.\beta\subset\alpha.\\ +[\text{*22·1.*20·1·3}] &\equiv.(\exists \phi,\psi).\alpha=\hat{z}(\phi!z).\beta=\hat{z}(\psi!z).\psi!x\supset_{x}\phi!x.\\ +[\text{*10·5}] &\supset.(\exists \psi).\beta=\hat{z}(\psi!z).\\ +[\text{*20·4}] &\supset.\beta\in\text{Cls}\qquad\qquad\qquad\qquad\qquad\qquad \text{(1)}\\ +\vdash.\text{(1).*60·2.(*60·03)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*60·71.</b> \(\vdash.\text{Cls}^{2}=\text{Cl}ʻ\text{Cls} \quad[\text{(*60·03)}]\)</p> + +<p class="nind"><b>*60·72.</b> \(\vdash.\text{Cls}^{3}=\text{Cl}ʻ\text{Cls}^{2} \quad[\text{(*60·04)}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_412">[Pg 412]</span></p> +<h2 class="nobreak" id="*61">*61. THE SUB-RELATIONS OF A GIVEN RELATION.</h2> +</div> + + +<p><i>Summary of</i> *61.</p> + +<p>The propositions of this number (except that *61·371·372·373 +imperfectly correspond to <a href="#*60·371">*60·371</a>) are the analogues of those with +the same decimal part in <a href="#*60">*60</a>. Proofs are omitted, as they are exactly +analogous to those in *60. There are very few subsequent references to +the propositions of this number.</p> + +<hr class="tb"> + +<p class="nind"><b>*61·01.</b> \(\text{Rl}=\hat{\lambda}\hat{P}\{\lambda=\hat{R}(R\unicode{x2abd}P)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*61·02.</b> \(\text{Rl ex}=\hat{\lambda}\hat{P}\{\lambda=\hat{R}(R\unicode{x2abd}P.\dot{\exists}!R)\} \quad\text{Df}\)</p> + +<p class="nind"><b>*61·03.</b> \(\text{Rel}^{2}=\text{Rl}ʻ (\text{Rel}\uparrow\text{Rel}) \quad\text{Df}\)</p> + +<p class="nind"><b>*61·04.</b> \(\text{Rel}^{3}=\text{Rl}ʻ (\text{Rel}^{2}\uparrow\text{Rel}^{2}) \quad\text{Df}\)</p> + +<p class="nind"><b>*61·1.</b> \(\vdash:\lambda\text{Rl}\, P.\equiv.\lambda=\hat{R}(R\unicode{x2abd}P)\)</p> + +<p class="nind"><b>*61·11.</b> \(\vdash:\lambda\text{Rl ex}\, P.\equiv.\lambda=\hat{R}(R\unicode{x2abd}P.\dot{\exists}!R)\)</p> + +<p class="nind"><b>*61·12.</b> \(\vdash.\text{Rl}ʻ P=\hat{R}(R\unicode{x2abd}P)\)</p> + +<p class="nind"><b>*61·13.</b> \(\vdash.\text{Rl ex}ʻ P=\hat{R}(R\unicode{x2abd}P.\dot{\exists}!R)\)</p> + +<p class="nind"><b>*61·14.</b> \(\vdash.\exists !\text{Rl}ʻ P\)</p> + +<p class="nind"><b>*61·15.</b> \(\vdash.\exists !\text{Rl ex}ʻ P\)</p> + +<p class="nind"><b>*61·2.</b> \(\vdash:R\in\text{Rl}ʻP.\equiv.R\unicode{x2abd}P\)</p> + +<p class="nind"><b>*61·21.</b> \(\vdash:R\in\text{Rl ex}ʻ P.\equiv.R\unicode{x2abd}P.\dot{\exists}!R\)</p> + +<p class="nind"><b>*61·22.</b> \(\vdash:R\in\text{Rl ex}ʻ P.\equiv.R\in\text{Rl}ʻ P.\dot{\exists}!R\)</p> + +<p class="nind"><b>*61·23.</b> \(\vdash:R\in\text{Rl ex}ʻ P.\equiv.R\in\text{Rl}ʻ P-\iotaʻ\dot{\Lambda}\)</p> + +<p class="nind"><b>*61·24.</b> \(\vdash.\text{Rl ex}ʻ P=\text{Rl}ʻ P-\iotaʻ\dot{\Lambda}\)</p> + +<p class="nind"><b>*61·3.</b> \(\vdash.\dot{\Lambda}\in\text{Rl}ʻ P\)</p> + +<p class="nind"><b>*61·31.</b> \(\vdash.\dot{\Lambda}\in\text{Rl}ʻ P\)</p> + +<p class="nind"><b>*61·32.</b> \(\vdash.\text{Rl}ʻ\dot{\Lambda}=\iotaʻ\dot{\Lambda}\)</p> + +<p class="nind"><b>*61·321.</b> \(\vdash:P=\dot{\Lambda}.\equiv.\text{Rl}ʻ P=\iotaʻ P\)</p> + +<p class="nind"><b>*61·33.</b> \(\vdash.\text{Rl ex}ʻ\dot{\Lambda}=\Lambda\cap\text{Rel}\)</p> + +<p><span class="pagenum" id="Page_413">[Pg 413]</span></p> + +<p class="nind"><b>*61·34.</b> \(\vdash.P\in\text{Rl}ʻ P\)</p> + +<p class="nind"><b>*61·35.</b> \(\vdash:\dot{\exists}!P.\supset.P\in\text{Rl ex}ʻ P\)</p> + +<p class="nind"><b>*61·36.</b> \(\vdash:\dot{\exists}!P.\supset.\exists !\text{Rl ex}ʻ P\)</p> + +<p class="nind"><b>*61·361.</b> \(\vdash:\dot{\exists}!P.\equiv.\exists !\text{Rl ex}ʻ P\)</p> + +<p class="nind"><b>*61·362.</b> \(\vdash.\text{Rl}ʻ (x\downarrow y)=\iotaʻ\dot{\Lambda}\cup \iotaʻ (x\downarrow y)\)</p> + +<p class="nind"><b>*61·37.</b> \(\vdash.\text{Rl ex}ʻ (x\downarrow y)=\iotaʻ (x\downarrow y)\)</p> + +<p class="nind"><b>*61·371.</b> \(\vdash:R\in\dot{2}.\supset.\text{Rl}ʻ R=\iotaʻ\dot{\Lambda}\cup \iotaʻ R\)</p> + +<p class="nind"><b>*61·372.</b> \(\vdash:R\in\dot{2}.\supset.\text{Rl}ʻ R\subset 0_{r}\cup\dot{2}\)</p> + +<p class="nind"><b>*61·373.</b> \(\vdash:R\in 2_{r}.\supset.\text{Rl}ʻ R\subset 0_{r}\cup 2_{r}\)</p> + +<p class="nind"><b>*61·38.</b> \(\vdash:R\in\dot{2}.\equiv.\text{Rl ex}ʻ R=\iotaʻ R\)</p> + +<p class="nind"><b>*61·39.</b> + \(\vdash.\text{Rl}ʻ (x\downarrow y\unicode{x228d}z\downarrow w)=\iotaʻ\dot{\Lambda}\cup \iotaʻ(x\downarrow y)\cup \iotaʻ (z\downarrow w)\cup \iotaʻ(x\downarrow y\unicode{x228d}z\downarrow w)\)</p> + +<p class="nind"><b>*61·391.</b> \(\vdash:P,Q\in 2.\supset.\text{Rl}ʻ (P\unicode{x228d}Q)=\iotaʻ\dot{\Lambda}\cup \iotaʻ P\cup \iotaʻ Q\cup \iotaʻ (P\unicode{x228d}Q)\)</p> + +<p class="nind"><b>*61·4.</b> \(\vdash:Q\in\text{Rl}ʻ P.R\unicode{x2abd}Q.\supset.R\in\text{Rl}ʻ P\)</p> + +<p class="nind"><b>*61·41.</b> \(\vdash:Q\in\text{Rl}ʻ P.\supset.Q\dot{\cap}R\in\text{Rl}ʻ P\)</p> + +<p class="nind"><b>*61·42.</b> \(\vdash:Q\in\text{Rl}ʻ P.R\unicode{x2abd}Q.\dot{\exists}!R.\supset.R\in\text{Rl ex}ʻ P\)</p> + +<p class="nind"><b>*61·43.</b> \(\vdash:Q,R\in\text{Rl}ʻP.\equiv.Q\unicode{x228d}R\in\text{Rl}ʻ P\)</p> + +<p class="nind"><b>*61·44.</b> \(\vdash:Q\in\text{Rl}ʻ P.R\in\text{Rl ex}ʻ P.\supset.Q\unicode{x228d}R\in\text{Rl ex}ʻ P\)</p> + +<p class="nind"><b>*61·5.</b> \(\vdash.\dot{s}ʻ \text{Rl}ʻ P=P\)</p> + +<p class="nind"><b>*61·501.</b> \(\vdash.\dot{s}ʻ\text{Rl ex}ʻ P=P\)</p> + +<p class="nind"><b>*61·51.</b> \(\vdash.\dot{p}ʻ\text{Rl}ʻ P=\dot{\Lambda}\)</p> + +<p class="nind"><b>*61·52.</b> \(\vdash:\dot{s}ʻ \lambda\unicode{x2abd}Q.\equiv.\lambda\subset \text{Rl}ʻ Q\)</p> + +<p class="nind"><b>*61·53.</b> \(\vdash:Q\unicode{x2abd}\dot{p}ʻ \lambda.\equiv.Q\in pʻ\text{Rl}ʻʻ \lambda\)</p> + +<p class="nind"><b>*61·54.</b> \(\vdash.\text{Rl}ʻ\dot{p}ʻ \lambda=pʻ\text{Rl}ʻʻ \lambda\)</p> + +<p class="nind"><b>*61·55.</b> \(\vdash.\text{Rl}ʻ P=\text{Rl}ʻ Q.\equiv.P=Q\)</p> + +<p class="nind"><b>*61·56.</b> \(\vdash.\text{Rl ex}ʻ P=\text{Rl ex}ʻ Q.\equiv.P=Q\)</p> + +<p class="nind"><b>*61·6.</b> \(\vdash:xPy.\supset.x\downarrow y\in\text{Rl ex}ʻ P\)</p> + +<p>The analogue of <a href="#*60·61">*60·61</a> is not given, because we have no suitable +notation for expressing it.</p> + +<p class="nind"><b>*61·62.</b> \(\vdash:xPy.zPw.\supset.x\downarrow y\unicode{x228d}z\downarrow w\in\text{Rl ex}ʻ P\)</p> + +<p class="nind"><b>*61·7.</b> \(\vdash.\text{Rl}ʻ P\in\text{Cl}ʻ\text{Rel}\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_414">[Pg 414]</span></p> +<h2 class="nobreak" id="*62">*62. THE RELATION OF MEMBERSHIP OF A CLASS.</h2> +</div> + + +<p><i>Summary of</i> *62.</p> + +<p>When "\(x\in\alpha\)" was defined, in <a href="#*20">*20</a>, it was defined as a +propositional function; and this mode of definition was necessary, +because we had to treat of this function before treating of relations. +But for many purposes it is desirable to regard \(\in\) as a relation, +so that "\(x\in\alpha\)" becomes an instance of the notation "\(uRv\)." +This requires, strictly speaking, a change in the meaning of +"\(x\in\alpha\)," but it is a change which does not falsify any of the +previous propositions in which "\(x\in\alpha\)" occurs; for if we call +the new meaning "\(x\unicode{x03AD}\alpha\)," <i>i.e.</i> if we put +\[ +\begin{array}{l} +&\unicode{x03AD} = \hat{x}\hat{\alpha}(x\in\alpha) \quad\text{Df},\\ +\text{we have}\qquad &\vdash: x\unicode{x03AD}\alpha . \equiv . x\in\alpha. +\end{array} +\]</p> + +<p>Hence it is unnecessary in practice to have a new notation for the new +meaning, and we put simply +\[ +\epsilon = \hat{x}\hat{\alpha}(x\in\alpha) \quad\text{Df}. +\] +This definition, though strictly incorrect, is recommended by its +convenience, and by the fact that it cannot lead to any harmful +confusions. The new meaning of \(\epsilon\) may be taken as replacing +the old throughout the remainder of this work.</p> + +<p>The uses of the propositions of the present number occur almost +exclusively in the theory of selections from a class of classes +(<a href="#*83">*83</a>, <a href="#*84">*84</a>, <a href="#*85">*85</a> and <a href="#*88">*88</a>). Such selections are effected by means of +selective relations, part of whose definition is that they are +contained in \(\epsilon\). Hence the uses of the present number. +If \(\kappa\) is the class of classes from which a selection +is to be made, a selective relation will in fact be contained +in \(\epsilon\upharpoonright\kappa\); hence the properties of +\(\epsilon\upharpoonright\kappa\) become important. Some of these +properties are given in <a href="#*62·4">*62·4</a> ff.</p> + +<p>The most important propositions of the present number are the following:</p> + +<p class="nind"><b>*62.2.</b> \(\vdash.\overrightarrow{\epsilon}ʻ \alpha = \alpha\)</p> + +<p class="nind"><b>*62.231.</b> \(\vdash : \kappa\subset\text{ᗡ}ʻ \epsilon . \equiv . \Lambda{\sim}\epsilon\kappa\)</p> + +<p class="nind"><b>*62.26.</b> \(\vdash . R = \epsilon \mid \overrightarrow{R}\)</p> + +<p class="nind"><b>*62.3.</b> \(\vdash . \epsilonʻʻ \kappa = sʻ\kappa\)</p> + +<p class="nind"><b>*62.42.</b> \(\vdash : \Lambda{\sim}\epsilon\kappa . \supset . \text{ᗡ}ʻ \epsilon\upharpoonright\kappa = \kappa\)</p> + +<p><span class="pagenum" id="Page_415">[Pg 415]</span></p> + +<p class="nind"><b>*62·43.</b> \(\vdash.\text{D}ʻ\in \upharpoonright \kappa=sʻ\kappa\)</p> + +<p class="nind"><b>*62·55.</b> \(\vdash:\kappa\subset 1.\supset.\in \upharpoonright \kappa=\breve{\iota}\upharpoonright \kappa\)</p> + +<hr class="tb"> + +<p class="nind"><b>*62·01.</b> \(\in =\hat{x}\hat{\alpha}(x\in \alpha) \text{Df}\)</p> + +<p class="nind"><b>*62·1.</b> \(\vdash:x\in \alpha.\equiv.x\in \alpha \quad[\text{*21·3.(*62·01)}]\)</p> + +<p>In the above proposition, the first \(\in\) has the newly-defined +meaning, while the second has the old meaning. In virtue of the above +proposition, the new meaning may be substituted for the old in all +propositions hitherto proved concerning \(\in\), and may take the place +of the old meaning in all that follows.</p> + +<p class="nind"><b><a id="*62·2">*62·2</a>.</b> \(\vdash.\overrightarrow{\in}ʻ\alpha=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·13}.\supset\vdash.\overrightarrow{\in}ʻ\alpha&=\hat{x}(x\in \alpha)\\ +[\text{*20·42}] & =\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·21.</b> \(\vdash.\overleftarrow{\in}ʻx=\hat{\alpha}(x\in \alpha) \quad[\text{*32·131}]\)</p> + +<p>Thus \(\overleftarrow{\in}ʻx\) consists of the classes of which \(x\) +is a member.</p> + +<p class="nind"><b>*62·22.</b> \(\vdash.\text{D}ʻ\in =\text{V}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·104}.&\supset\vdash.(x).x\in \text{V}.\\ +[\text{*10·24}] &\supset\vdash:(x):(\exists \alpha).x\in \alpha:\\ +[\text{*33·13}] &\supset\vdash.(x).x\in \text{D}ʻ\in :\\ +[\text{*24·14}] &\supset\vdash.\text{D}ʻ\in =\text{V} +\end{array} +\]</p> + +<p class="nind"><b>*62·23.</b> \(\vdash.\text{ᗡ}ʻ\in =\text{Cls}-\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·5}.\supset\vdash:\alpha\in \text{Cls}-\iotaʻ\Lambda.&\equiv.\exists !\alpha.\\ +[\text{*33·131}] &\equiv.\alpha\in \text{ᗡ}ʻ\in :\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·231.</b> \(\vdash:\kappa\subset \text{ᗡ}ʻ\in .\equiv.\Lambda{\sim}\in \kappa \quad[\text{*24·63.*33·131}]\)</p> + +<p class="nind"><b>*62·24.</b> \(\vdash.\in \mid \breve{\in}=\dot{\text{V}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·104.*11·57}.&\supset\vdash.(x,y).x\in \text{V}.y\in \text{V}.\\ +[\text{*31·11}]&\supset\vdash.(x,y).x\in \text{V}.\text{V}\breve{\in}y.\\ +[\text{*10·24}] &\supset\vdash:(x,y):(\exists \alpha).x\in \alpha.\alpha\breve{\in}y:\\ +[\text{*34·1}] &\supset\vdash:(x,y):x\in \mid \breve{\in}y:\\ +[\text{*25·14}] &\supset\vdash.\in \mid \breve{\in}=\dot{\text{V}} +\end{array} +\]</p> + +<p class="nind"><b>*62·25.</b> \(\vdash.\breve{\in}\mid \in =\hat{\alpha}\hat{\beta}\{\exists !(\alpha\cap \beta)\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1.*31·11}.\supset\vdash:\alpha(\breve{\in}\mid \in )\beta.&\equiv.(\exists x).x\in \alpha.x\in \beta.\\ +[\text{*22·33}] &\equiv.\exists !(\alpha\cap \beta):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_416">[Pg 416]</span></p> + +<p class="nind"><b><a id="*62·26">*62·26</a>.</b> \(\vdash.R={\in}\mid \overrightarrow{R}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18}.\supset\vdash:xRy.&\equiv.x \in \overrightarrow{R}ʻy.\\ +[\text{*30·33.*32·12}] &\equiv.(\exists \alpha).x \in \alpha.\alpha\overrightarrow{R}y.\\ +[\text{*34·1}] &\equiv.x({\in}\mid \overrightarrow{R})y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*62·3">*62·3</a>.</b> \(\vdash.{\in}ʻʻ\kappa=sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}.\supset\vdash.{\in}ʻʻ\kappa&=\hat{x}\{(\exists \alpha).\alpha \in \kappa.x \in \alpha\}\\ +[\text{(*40·02)}] &=sʻ\kappa.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·31.</b> \(\vdash.\overrightarrow{{\in}^{2}}ʻ\kappa=sʻ\kappa\)</p> + +<p>Note that, since \({\in}\) is not a homogeneous relation, <i>i.e.</i> +not one in which referent and relatum belong to the same type, +\({\in}^{2}\) is strictly meaningless. For if we have +\(x \in \alpha.\alpha \in\kappa\), the two \({\in}\)'s have different +meanings, and do not therefore properly give \(x{\in}{^{2}}\kappa\). But +it is convenient to allow \({\in}^{2}\), on the understanding that the +ambiguity of \({\in}\) is to be differently determined for the two +factors in the product \({\in}\mid {\in}\), namely the second \({\in}\) +must make both referent and relatum belong to the next type above that +to which they respectively belong for the first \({\in}\).</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·13}.\supset\vdash.\overrightarrow{{\in}^{2}}ʻ\kappa&=\hat{x}(x{\in}^{2}\kappa)\\ +[\text{*34·5}] &=\hat{x}\{(\exists \alpha).x \in \alpha.\alpha \in \kappa\}\\ +[\text{(*40·02)}] &=sʻ\kappa +\end{array} +\]</p> + +<p class="nind"><b>*62·32.</b> \(\vdash.s={\in}_{\in}=\overrightarrow{{\in}^{2}} \quad[\text{*30·41.*62·3·31.*37·11}]\)</p> + +<p class="nind"><b>*62·33.</b> \(\vdash.\overrightarrow{\in}=I\upharpoonright \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*62·2.*30·3}.\supset\vdash:\beta\overrightarrow{\in}\alpha.&\equiv_{\beta}.\beta=\alpha.\\ +[\text{*20·41}] & \equiv_{\beta}.\beta=\alpha.\alpha \in \text{Cls}.\\ +[\text{*50·1.*35·101}] &\equiv_{\beta}.\beta(I\upharpoonright \text{Cls})\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The use of <a href="#*20·41">*20·41</a> in the above proof depends upon the fact that +\(\alpha\) is merely an abbreviation for an expression of the form +\(\hat{z}(\psi z)\).</p> + +<p class="nind"><b>*62·34.</b> \(\vdash.P_{\in}=\text{sg}ʻ(P\mid {\in})\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·101.(*37·01)}.\supset\vdash\colon\ldotp \alpha P_{\in}\beta.&\equiv:\alpha=\hat{x}\{(\exists y).y \in \beta.xPy\}\\ +[\text{*34·1}] &\qquad\qquad =\hat{x}{x(P\mid {\in})\beta}:\\ +[\text{*32·1·23}] &\equiv:\alpha\{\text{sg}ʻ(P\mid {\in})\}\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_417">[Pg 417]</span></p> + +<p class="nind"><b><a id="*62·4">*62·4</a>.</b> \(\vdash.{\in} \upharpoonright \kappa=\hat{x}\hat{\alpha}(x \in \alpha.\alpha\in \kappa) \quad[\text{*21·2.(*35·02)}]\)</p> + +<p>The relation \({\in}\upharpoonright \kappa\) is very important in +cardinal arithmetic, in connection with the problem of selection from +the members of \(\kappa\), <i>i.e.</i> of extracting one term out of +each of the members of \(\kappa\). A relation which is to effect this +selection must be contained in \({\in}\upharpoonright \kappa\).</p> + +<p class="nind"><b>*62·41.</b> \(\vdash.\text{ᗡ}ʻ{\in}\upharpoonright \kappa=\kappa-\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·101}.\supset \vdash:x({\in}\upharpoonright \kappa)\alpha.\equiv.x \in \alpha.\alpha \in \kappa:\\ +[\text{*10·11·281}]\qquad\qquad\supset \vdash\colon\ldotp (\exists x).x({\in}\upharpoonright \kappa)\alpha.&\equiv:(\exists x).x \in \alpha.\alpha \in \kappa:\\ +[\text{*10·35}] &\equiv:(\exists x).x \in\alpha:\alpha \in \kappa:\\ +[\text{*24·5}] &\equiv:\exists !\alpha.\alpha \in \kappa:\\ +[\text{*53·52}] &\equiv:\alpha \in\kappa-\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*33·131}.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·42.</b> \(\vdash:\Lambda{\sim} {\in}\kappa.\supset .\text{ᗡ}ʻ{\in}\upharpoonright \kappa=\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·36}.\supset \vdash:\text{Hp}.&\supset .\kappa\subset -\iotaʻ\Lambda.\\ +[\text{*22·621}] &\supset .\kappa=\kappa-\iotaʻ\Lambda.\\ +[\text{*62·41}] &\supset .\text{ᗡ}ʻ{\in}\upharpoonright \kappa=\kappa:\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·43.</b> \(\vdash.\text{D}ʻ{\in}\upharpoonright \kappa=sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·11}.\supset \vdash.\text{D}ʻ{\in}\upharpoonright \kappa&=\hat{x}\{(\exists \alpha).x({\in}\upharpoonright \kappa)\alpha\}\\ +[\text{*35·101}] &=\hat{x}\{(\exists \alpha).x \in \alpha.\alpha \in \kappa\}\\ +[\text{(*40·02)}] &=sʻ\kappa.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·44.</b> \(\vdash:R\unicode{x2abd}{\in}.\equiv.(\alpha).\overrightarrow{R}ʻ\alpha\subset \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·1}.\supset \vdash\colon\ldotp R\unicode{x2abd}{\in}.&\equiv:xR\alpha.\supset _{x,\alpha}.x \in \alpha:\\ +[\text{*32·18}] &\equiv:x{\in}\overrightarrow{R}ʻ\alpha.\supset_{x,\alpha}.x \in \alpha:\\ +[\text{*11·2.*22·1}] &\equiv:(\alpha).\overrightarrow{R}ʻ\alpha\subset \alpha\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·45.</b> \(\vdash\colon\ldotp R\unicode{x2abd}{\in}.\text{E}‼Rʻʻ\text{ᗡ}ʻR.\equiv:\alpha \in \text{ᗡ}ʻR.\supset_{\alpha}.Rʻ\alpha \in \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21.*4·71}.\supset \vdash\colon\ldotp Rʻ\alpha \in \alpha.&\equiv:\text{E}!Rʻ\alpha.Rʻ\alpha \in \alpha:\\ +[\text{*30·33.*5·32}] &\equiv:\text{E}!Rʻ\alpha:xR\alpha.\supset_{x}.x \in \alpha &\qquad \text{(1)}\\ +\qquad\qquad\qquad\vdash.\text{(1).*10·413}.\supset \vdash\colon\colon &\alpha \in \text{ᗡ}ʻR.\supset_{\alpha}.Rʻ\alpha \in \alpha:\equiv\colon\ldotp\\ +&\alpha \in \text{ᗡ}ʻR.\supset_{\alpha}:\text{E}!Rʻ\alpha:xR\alpha.\supset_{x}.x \in \alpha\colon\ldotp\\ +[\text{*10·29.*11·62}]&\equiv\colon\ldotp \alpha \in \text{ᗡ}ʻR.\supset _{\alpha}.\text{E}!Rʻ\alpha:\alpha \in \text{ᗡ}ʻR.xR\alpha.\supset _{\alpha,x}.x \in \alpha\colon\ldotp\\ +[\text{*33·14.*4·71}] &\equiv\colon\ldotp \alpha \in \text{ᗡ}ʻR.\supset _{\alpha}.\text{E}!Rʻ\alpha:xR\alpha.\supset_{\alpha,x}.x \in \alpha\colon\ldotp \\ +[\text{*37·104.*11·2}]&\equiv\colon\ldotp \text{E}‼Rʻʻ\text{ᗡ}ʻR.R\unicode{x2abd}{\in}\colon\colon \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_418">[Pg 418]</span></p> + +<p>This proposition is useful in the theory of selections. It is used in +the proof of <a href="#*83·27">*83·27</a>, and thence of <a href="#*83·28">*83·28</a>.</p> + +<p class="nind"><b>*62·5.</b> \(\vdash.\breve{\iota}\unicode{x2abd}{\in}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·21.*52·13}.&\supset\vdash.\text{ᗡ}ʻ\breve{\iota}=1.\\ +[\text{*52·173}] &\supset\vdash:\alpha\in \text{ᗡ}ʻ\breve{\iota}.\supset_{\alpha}.\breve{\iota}ʻ\alpha\in \alpha:\\ +[\text{*62·45}] &\supset\vdash.\breve{\iota}\unicode{x2abd}{\in} +\end{array} +\]</p> + +<p class="nind"><b>*62·51.</b> \(\vdash:\text{E}!\breve{\iota}ʻ\alpha.\supset.\breve{\iota}ʻ\alpha=\in ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·15·172}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\iotaʻ\breve{\iota}ʻ\alpha=\alpha:\\ +[\text{*51·15}] &\supset:x=\breve{\iota}ʻ\alpha.\equiv_{x}.x\in \alpha:\\ +[\text{*30·3}] &\supset:\breve{\iota}ʻ\alpha={\in}ʻ\alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·52.</b> \(\vdash:\text{E}!{\in}ʻ\alpha.\equiv.\alpha\in 1.\equiv.\text{E}!\breve{\iota}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·2}.\supset\vdash\colon\ldotp \text{E}!\in ʻ\alpha.&\equiv:(\exists b):x\in \alpha.\equiv_{x}.x=b:\\ +[\text{*52·11}] &\equiv:\alpha\in 1:\\ +[\text{*52·15}] &\equiv:\text{E}!\breve{\iota}ʻ\alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·53.</b> \(\vdash:\text{E}!{\in} ʻ\alpha.\supset.{\in} ʻ\alpha=\breve{\iota}ʻ\alpha \quad[\text{*62·51·52}]\)</p> + +<p class="nind"><b>*62·54.</b> \(\vdash:\alpha\in 1.\supset.{\in} ʻ\alpha=\breve{\iota}ʻ\alpha \quad[\text{*62·51·52}]\)</p> + +<p class="nind"><b>*62·55.</b> \(\vdash:\kappa\subset 1.\supset.\in \upharpoonright \kappa=\breve{\iota}\upharpoonright \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*62·54}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\alpha\in \kappa.\supset_{\alpha}.\in ʻ\alpha=\breve{\iota}ʻ\alpha:\\ +[\text{*35·71}] &\supset:\in \upharpoonright \kappa=\breve{\iota}\upharpoonright \kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·56.</b> \(\vdash.{\in} \upharpoonright \iotaʻʻ\alpha=\breve{\iota}\upharpoonright \iotaʻʻ\alpha=\alpha\upharpoonleft \breve{\iota}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·3.*62·55}.\supset\vdash.\in \upharpoonright \iotaʻʻ\alpha=\breve{\iota}\upharpoonright \iotaʻʻ\alpha &&\qquad \text{(1)}\\ +\vdash.\text{*35·101.*37·6}.\supset\vdash\colon\ldotp x(\breve{\iota}\upharpoonright \iotaʻʻ\alpha)\beta.&\equiv:x\breve{\iota}\beta:(\exists y).y\in \alpha.\beta=\iotaʻy:\\ +[\text{*51·51}] & \equiv:\beta=\iotaʻx:(\exists y).y\in \alpha.\beta=\iotaʻy:\\ +[\text{*10·35}] &\equiv:(\exists y).\beta=\iotaʻx.y\in \alpha.\beta=\iotaʻy:\\ +[\text{*13·193}] &\equiv:(\exists y).\beta=\iotaʻx.y\in \alpha.\iotaʻx=\iotaʻy:\\ +[\text{*51·23}] &\equiv:(\exists y).\beta=\iotaʻx.y\in \alpha.x=y:\\ +[\text{*13·195}] &\equiv:\beta=\iotaʻx.x\in \alpha:\\ +[\text{*51·51}] &\equiv:x\breve{\iota}\beta.x\in \alpha:\\ +[\text{*35·1}] &\equiv:x(\alpha\upharpoonleft \breve{\iota})\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*62·57.</b> \(\vdash.\breve{\iota}=\in \upharpoonright 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*62·55}.\supset\vdash.\in \upharpoonright 1&=\breve{\iota}\upharpoonright 1\\ +[\text{*52·13}] &=\breve{\iota}\upharpoonright \text{ᗡ}ʻ\breve{\iota}\\ +[\text{*35·452}] &=\breve{\iota}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_419">[Pg 419]</span></p> + +<h2 class="nobreak" id="*63">*63. RELATIVE TYPES OF CLASSES.</h2> +</div> + +<p><i>Summary of</i> *63.</p> + +<p>The notations introduced in this and the two following numbers serve to +express the type of one variable in terms of the type of another. They +are very useful in arithmetic, where it is necessary to take account +of types in order to avoid contradictions. The two chief notations are +"\(t_{0}ʻ\alpha\)," for the type in which \(\alpha\) is contained, and +"\(tʻx\)," for the type of which \(x\) is a member. We put</p> + +<p class="nind"><b>*63·02.</b> \(t_{0}ʻ\alpha = \alpha \cup -\alpha \quad \text{Df}\)</p> + +<p>This defines "the type of members of \(\alpha\)," or "the type which is +of the same type as \(\alpha\)." The characteristic of a type is that +if \(\tau\) is a type, we have +\[ +(x) . x \in \tau\text{,} +\] +and conversely, if \((x) . x \in \tau\), then \(\tau\) is a type. For +in that case, "\(x \in \tau\)" is true whenever it is significant, +<i>i.e.</i> whenever \(x\) belongs to the type which is the range of +significance of \(x\) in "\(x \in \tau\)." Consequently \(\tau\) is +this range of significance, <i>i.e.</i> is a type.</p> + +<p>Since we have (\(x) . x \in (\alpha \cup -\alpha)\), it follows that +\(\alpha \cup -\alpha\) is a type. It is not "the type of \(\alpha\)," +but "the type of the members of \(\alpha\)." (In case \(\alpha\) is +null, "the type of the members of \(\alpha\)" may be interpreted as +meaning "the type to which \(x\) belongs when '\(x \in \alpha\)' is +significant.") "The type of \(x\)," <i>i.e.</i> the type of which \(x\) +is a member, is defined as follows:</p> + +<p class="nind"><b>*63·01.</b> \(tʻx = \iotaʻx \cup -\iotaʻx \quad \text{Df}\)</p> + +<p>By what was said above, "\(t_{0}ʻ\iotaʻx\)" is the type of the members +of \(\iotaʻx\), <i>i.e.</i> the type of \(x\). By combining the +definitions of \(tʻx\) and \(t_{0}ʻ\alpha\), we obtain +\[ +\vdash . tʻx = t_{0}ʻ\iotaʻx\text{.} +\] +Thus +\[ +\vdash . x \in tʻx \quad \text{and} \quad \vdash : y \neq x .\supset. y \in tʻx. +\]</p> + +<p>In short, \(tʻx\) consists of everything either identical or not identical with \(x\), +that is, every \(y\) for which there is such a proposition, whether true or false, +as "\(y = x\)." We put "\(tʻx\)" here instead of "\(tʻ\alpha\)," because \(x\) need not be a +class, and is in fact subject to no limitation whatever, whereas "\(t_{0}ʻx\)" is +not significant unless \(x\) is a class, and therefore we write "\(t_{0}ʻ\alpha\)" rather +<span class="pagenum" id="Page_420">[Pg 420]</span>than "\(t_{0}ʻx\)."</p> + +<p>We put also</p> + +<p class="nind"><b>*63·011.</b> \(t^{1}ʻx = tʻx \quad \text{Df}\)</p> + +<p>This definition serves merely to bring \(tʻx\) notationally into line +with \(t_{0}ʻx\) and the types \(t^{2}ʻx, t^{3}ʻx, \ldots t_{2}ʻx, t_{3}ʻx, \ldots\) +defined below.</p> + +<p>In virtue of <a href="#*20·8">*20·8</a>, we have +\[ +\vdash: {\phi}a \lor {\sim}{\phi}a .\supset. \hat{x}({\phi}x \lor {\sim}{\phi}x) = tʻa\text{,} +\] +<i>i.e.</i> if "\({\phi}a\)" is significant, then the range of +significance of the function \({\phi}\hat{z}\) is the type of \(a\). It +follows that two ranges of significance which overlap are identical, +and two different ranges of significance have no member in common.</p> + +<p>It will be seen that \(\iotaʻx\) is always of the next type above that +of \(x\), and \(sʻ\kappa\) (if \(\kappa\) is a class of classes) is of +the next type below that of \(\kappa\). We put</p> + +<p class="nind"><b>*63·03.</b> \(t_{1}ʻ\kappa = t_{0}ʻsʻ\kappa \quad \text{Df}\)</p> + +<p class="nind"> +so that \(t_{1}ʻ\kappa\) is the type next below that in which +\(\kappa\) is contained. Thus if \(\kappa\) is a class of classes of +individuals, \(t_{1}ʻ\kappa\) is the class of individuals. We put also</p> + +<p class="nind"><b>*63·04.</b> \(t^{2}ʻx = tʻtʻx \quad \text{Df}\)</p> + +<p class="nind"><b>*63·041.</b> \(t^{3}ʻx = tʻt^{2}ʻx \quad \text{Df} ~ \text{and so on}\)</p> + +<p class="nind"><b>*63·05.</b> \(t_{2}ʻ\kappa = t_{1}ʻt_{1}ʻ\kappa \quad \text{Df}\)</p> + +<p class="nind"><b>*63·051.</b> \(t_{3}ʻ\kappa = t_{1}ʻt_{2}ʻ\kappa \quad \text{Df} ~ \text{and so on}\)</p> + +<p>Thus given any two objects which are members of any one of the +following: the type of \(x\), the type of the classes to which \(x\) +belongs, the type of the classes to which these classes belong, and so +on, we can express the type of either of our two objects by means of +its relation to the other object.</p> + +<p>The propositions of this and the two following numbers will hardly ever +be used until we come to cardinal arithmetic. They are used constantly +in the first section on cardinal arithmetic, and they are constantly +relevant in the first section on relation-arithmetic. Moreover they are +usually required for cardinal and ordinal existence-theorems.</p> + +<p>Among the most useful propositions of the present number are the +following:</p> + +<p class="nind"><b>*63·103.</b> \(\vdash. x \in tʻx\)</p> + +<p class="nind"><b>*63·105.</b> \(\vdash. \alpha \subset t_{0}ʻ\alpha\)</p> + +<p class="nind"><b><a id="*63·11">*63·11</a>.</b> \(\vdash: x \in t_{0}ʻ\alpha .\supset. tʻx = \alpha \cup -\alpha = t_{0}ʻ\alpha\)</p> + +<p><i>I.e.</i> if \(x\) either is or is not a member of \(\alpha\), +then the type of \(x\) is the type which contains \(\alpha\). This +proposition uses <a href="#*20·8">*20·8</a>.</p> + +<p class="nind"><b>*63·13.</b> \(\vdash: {\phi}x . {\phi}y .\supset. y \in tʻx\)</p> + +<p><i>I.e.</i> if there is any function satisfied by both \(x\) and \(y\), +then \(y\) is of the type of \(x\). It is necessary to the use of +this proposition that, if \(\phi\hat{z}\) is a typically<span class="pagenum" id="Page_421">[Pg 421]</span> ambiguous +function, it should receive the same typical determination for \(x\) +and for \(y\). For example, we have always \(x = x\) and \(y = y\); +but we must not regard these as values of one function \(\hat{z} = \hat{z}\), +because such a function is typically ambiguous. On the other +hand, \(x = a\) and \(y = a\) are values of one function \(\hat{z} = a\), +because here the presence of a renders the function typically +determinate.</p> + +<p class="nind"><b>*63·15.</b> \(\vdash. t_{0}ʻtʻx = tʻx\)</p> + +<p class="nind"><b>*63·19.</b> \(\vdash. tʻt_{0}ʻ\alpha = tʻ\alpha\)</p> + +<p class="nind"><b>*63·16.</b> \(\vdash: x \in tʻy .\equiv. y \in tʻx .\equiv. \exists! tʻx \cap tʻy .\equiv. tʻx = tʻy\)</p> + +<p>This proposition, which depends upon <a href="#*63·11">*63·11</a>, and thence upon <a href="#*20·8">*20·8</a> and +<a href="#*13·3">*13·3</a>, and thence upon *9·14·15, is vital to the whole theory of types.</p> + +<p class="nind"><b>*63·32.</b> \(\vdash. t_{1}ʻ\kappa = sʻt_{0}ʻ\kappa\)</p> + +<p class="nind"><b>*63·371.</b> \(\vdash: \beta \subset t_{0}ʻ\alpha .\equiv. \beta \in tʻ\alpha\)</p> + +<p class="nind"><b>*63·383.</b> \(\vdash. tʻt_{1}ʻ\kappa = t_{0}ʻ\kappa\)</p> + +<p>We shall have generally \(t^{m}ʻt^{n}ʻ\kappa = t^{m+n}ʻ\kappa\), where +we may count suffixes as negative indices, so that \(t^{m}ʻt_{n}ʻ\kappa = t^{m-n}ʻ\kappa\) +or \(t_{n-m}ʻ\kappa\) according as \(m\) or \(n\) is the greater.</p> + +<p class="nind"><b>*63·5.</b> \(\vdash: x \in t_{0}ʻ\alpha .\equiv. \alpha \in t^{2}ʻx .\equiv. \alpha \subset tʻx .\equiv. tʻx = t_{0}ʻ\alpha\)</p> + +<p>This proposition is used constantly.</p> + +<p class="nind"><b>*63·51.</b> \(\vdash: \alpha \in t_{0}ʻ\kappa .\equiv. \alpha \subset t_{1}ʻ\kappa .\equiv. \kappa \subset tʻ\alpha .\equiv. tʻ\alpha = t_{0}ʻ\kappa\)</p> + +<p class="nind"><b>*63·52.</b> \(\vdash: \alpha \in t_{1}ʻ\lambda .\equiv. \alpha \subset t_{2}ʻ\lambda .\equiv. \lambda \subset t^{2}ʻ\alpha .\equiv. tʻ\alpha = t_{1}ʻ\lambda + .\equiv. t^{2}ʻ\alpha = t_{0}ʻ\lambda\)</p> + +<p class="nind"><b>*63·53.</b> \(\vdash: x \in t_{0}ʻ\alpha .\equiv. t^{2}ʻx = tʻ\alpha .\equiv. tʻx = t_{0}ʻ\alpha\)</p> + +<p>The above four propositions, together with four similar ones +(*63·54·55·56·57), give transformations which enable us to express any +relation of type, as between class and members or members of members or +etc., that is likely to occur in practice.</p> + +<p class="nind"><b>*63·64.</b> \(\vdash. tʻ\beta = t_{0}ʻ\iotaʻʻ\beta\)</p> + +<p>This proposition is often used in the first section on cardinal +arithmetic.</p> + +<p class="nind"><b>*63·66.</b> \(\vdash. \text{Cl}ʻtʻx = t^{2}ʻx\)</p> + +<hr class="tb"> + +<p class="nind"><b>*63·01.</b> \(tʻx = \iotaʻx \cup - \iotaʻx \quad \text{Df}\)</p> + +<p class="nind"><b>*63·011.</b> \(t^{1}ʻx = tʻx \quad \text{Df}\)</p> + +<p class="nind"><b>*63·02.</b> \(t_{0}ʻ\alpha = \alpha \cup -\alpha \quad \text{Df}\)</p> + +<p class="nind"><b>*63·03.</b> \(t_{1}ʻ\kappa = t_{0}ʻsʻ\kappa \quad \text{Df}\)</p> + +<p class="nind"><b>*63·04.</b> \(t^{2}ʻx = tʻtʻx \quad \text{Df}\)</p> + +<p class="nind"><b>*63·041.</b> \(t^{3}ʻx = tʻt^{2}ʻx \quad \text{Df}\)</p> + +<p><span class="pagenum" id="Page_422">[Pg 422]</span></p> + +<p class="nind"><b>*63·05.</b> \(t_{2}ʻ\kappa = t_{1}ʻt_{1}ʻ\kappa \quad \text{Df}\)</p> + +<p class="nind"><b>*63·051.</b> \(t_{3}ʻ\kappa = t_{1}ʻt_{2}ʻ\kappa \quad\text{Df}\)</p> + +<p class="nind"><b>*63·1.</b> \(\vdash . (x) . x \in t_{0}ʻ\alpha \quad[\text{*22·88}]\)</p> + +<p class="nind"><b>*63·101.</b> \(\vdash . tʻx = t_{0}ʻ\iotaʻx = \iotaʻx \cup - \iotaʻx \quad[\text{*20·2 . (*63·01·02)}]\)</p> + +<p class="nind"><b>*63·102.</b> \(\vdash . (y) . y \in tʻx \quad[\text{*63·1·101}]\)</p> + +<p class="nind"><b>*63·103.</b> \(\vdash . x \in tʻx \quad[\text{*63·101 . *51·16}]\)</p> + +<p class="nind"><b>*63·104.</b> \(\vdash : \phi x . {\sim}\phi y . \supset . y \in tʻx \quad[\text{*63·101 . *13·14}]\)</p> + +<p class="nind"><b>*63·105.</b> \(\vdash . \alpha \subset t_{0}ʻ\alpha \quad[\text{*22·58}]\)</p> + +<p class="nind"><b>*63·106.</b> \(\vdash . t_{0}ʻ\alpha = t_{0}ʻ - \alpha \quad[\text{*22·8}]\)</p> + +<p class="nind"><b>*63·107.</b> \(\vdash \colon\ldotp (x) . \phi x : f(\phi y) : \supset . \phi y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*2·11 . *10·11} . &\supset \vdash . (y) . f(\phi y) \lor {\sim}f(\phi y) &\qquad \text{(1)}\\ +\vdash . \text{(1) . *10·13·221} . \qquad\qquad\qquad\qquad\supset \vdash \colon\ldotp (x) . \phi x . &\supset : \phi y . f(\phi y) \lor {\sim}f(\phi y) :\\ +[\text{*5·1}] &\supset : \phi y . \equiv . f(\phi y) \lor {\sim}f(\phi y) :\\ +[\text{*2·2}] &\supset : f(\phi y) . \supset . \phi y \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·108.</b> \(\vdash : f(y \in tʻx) . \supset . y \in tʻx \quad[\text{*63·107·102}]\)</p> + +<p class="nind"><b>*63·109.</b> \(\vdash : f(y \in t_{0}ʻ\alpha) . \supset . y \in t_{0}ʻ\alpha \quad[\text{*63·107·1}]\)</p> + +<p class="nind"><b>*63·11.</b> \(\vdash : x \in t_{0}ʻ\alpha . \supset . tʻx = \alpha \cup - \alpha = t_{0}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*22·34 . (*63·02)} . \supset \vdash \colon\ldotp \text{Hp} . &\supset : x \in \alpha . \lor . x{\sim}\in \alpha :\\ +[\text{*20·8}] &\supset : \hat{y}(y \in \alpha . \lor . y{\sim}\in \alpha) = \hat{y}(y = x . \lor . y \neq x) :\\ +[\text{*22·3·31.*51·15}] &\supset : \alpha \cup - \alpha = \iotaʻx \cup - \iotaʻx &\qquad \text{(1)}\\ +\vdash . \text{(1) . (*63·01·02)} . \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·12.</b> \(\vdash \colon\ldotp \phi x \lor {\sim}\phi x . \supset : \phi y \lor {\sim}\phi y . \equiv_{y} . y \in tʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*63·11 . *20·8} . \supset \vdash \colon\ldotp \text{Hp} . &\supset : tʻx = \hat{z}(\phi z) \cup - \hat{z}(\phi z) :\\ +[\text{*20·31 . *22·391·392}] &\supset : y \in tʻx . \equiv_{y} . \phi y \lor {\sim}\phi y \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·13.</b> \(\vdash : \phi x . \phi y . \supset . y \in tʻx \quad[\text{*63·12 . Imp . Add}]\)</p> + +<p class="nind"><b>*63·14.</b> \(\vdash : (x) . x \in \alpha . \supset . t_{0}ʻ\alpha = \alpha \quad[\text{*24·14·17·24 . (*63·02)}]\)</p> + +<p class="nind"><b>*63·15.</b> \(\vdash . t_{0}ʻtʻx = tʻx \quad[\text{*63·14·102}]\)</p> + +<p class="nind"><b>*63·151.</b> \(\vdash . t_{0}ʻt_{0}ʻ\alpha = t_{0}ʻ\alpha \quad[\text{*63·14·1}]\)</p> + +<p><span class="pagenum" id="Page_423">[Pg 423]</span></p> + +<p class="nind"><b>*63·152.</b> \(\vdash . x \in t_{0}ʻtʻx \quad[\text{*63·103·15}]\)</p> + +<p class="nind"><b>*63·16.</b> \(\vdash:x\in tʻy.\equiv.y\in tʻx.\equiv.\exists !tʻx\cap tʻy.\equiv.tʻx=tʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·101.*51·23}.&\supset\vdash:x\in tʻy.\equiv.y\in tʻx &\qquad \text{(1)}\\ +\vdash.\text{*63·13}. &\supset\vdash:(\exists z).z\in tʻx.z\in tʻy.\supset.y\in tʻx &\qquad \text{(2)}\\ +\vdash.\text{*63·103}.&\supset\vdash:y\in tʻx.\supset.y\in tʻx.y\in tʻy.\\ +[\text{*10·24}] &\supset.\exists !tʻx\cap tʻy &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash:y\in tʻx.\equiv.\exists !tʻx\cap tʻy &\qquad \text{(4)}\\ +\vdash.\text{*63·103}.&\supset\vdash:tʻx=tʻy.\supset.y\in tʻx &\qquad \text{(5)}\\ +\vdash.\text{*63·13}.&\supset\vdash:y\in tʻx.z\in tʻx.\supset.z\in tʻy &\qquad \text{(6)}\\ +\vdash.\text{*63·13}.&\supset\vdash:x\in tʻy.z\in tʻy.\supset.z\in tʻx:\\ +[\text{(1)}] & \supset\vdash:y\in tʻx.z\in tʻy.\supset.z\in tʻx &\qquad \text{(7)}\\ +\vdash.\text{(6).(7)}.&\supset\vdash\colon\ldotp y\in tʻx.\supset:z\in tʻx.\equiv.z\in tʻy &\qquad \text{(8)}\\ +\vdash.\text{(5).(8)}.&\supset\vdash\colon\ldotp y\in tʻx.\equiv.tʻx=tʻy &\qquad \text{(9)}\\ +\vdash.\text{(1).(4).(9)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·17.</b> \(\vdash:y\in tʻx.z\in tʻy.\supset.z\in tʻx \quad[\text{*63·16}]\)</p> + +<p class="nind"><b>*63·18.</b> \(\vdash.\exists !t_{0}ʻ\alpha \quad[\text{*10·25.*63·1}]\)</p> + +<p class="nind"><b>*63·181.</b> \(\vdash:\alpha\subset t_{0}ʻ\beta.\equiv.\beta\subset t_{0}ʻ\alpha.\equiv.\exists !t_{0}ʻ\alpha\cap + t_{0}ʻ\beta.\equiv.t_{0}ʻ\alpha=t_{0}ʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·105}. & \supset\vdash:t_{0}ʻ\alpha=t_{0}ʻ\beta.\supset.\alpha\subset t_{0}ʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{24·6}. &\supset\vdash\colon\ldotp \alpha\subset t_{0}ʻ\beta.\supset:\alpha=t_{0}ʻ\beta.\lor.\exists !t_{0}ʻ\beta-\alpha &\qquad \text{(2)}\\ +\vdash.\text{*63·151}. &\supset\vdash:\alpha=t_{0}ʻ\beta.\supset.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(3)}\\ +\vdash.\text{*63·11}. &\supset\vdash:x\in t_{0}ʻ\beta.x\in -\alpha.\supset.tʻx=t_{0}ʻ\beta.tʻx=t_{0}ʻ-\alpha.\\ +[\text{*63·106}] &\supset.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4)}.&\supset\vdash:\alpha\subset t_{0}ʻ\beta.\supset.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(5)}\\ +\vdash.\text{(1).(5)}. &\supset\vdash:\alpha\subset t_{0}ʻ\beta.\equiv.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(6)}\\ +\vdash.\text{(6)} \frac{\beta,\,\alpha}{\alpha,\,\beta}.&\supset\vdash:\beta\subset t_{0}ʻ\alpha.\equiv.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(7)}\\ +\vdash.\text{*63·11}. &\supset\vdash:x\in t_{0}ʻ\alpha\cap t_{0}ʻ\beta.\supset.tʻx=t_{0}ʻ\alpha.tʻx=t_{0}ʻ\beta.\\ +[\text{*13·171}] &\supset.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(8)}\\ +\vdash.\text{*63·18}.&\supset\vdash:t_{0}ʻ\alpha=t_{0}ʻ\beta.\supset.\exists !t_{0}ʻ\alpha\cap t_{0}ʻ\beta &\qquad \text{(9)}\\ +\vdash.\text{(8).(9)}.&\supset\vdash:\exists !t_{0}ʻ\alpha\cap t_{0}ʻ\beta.\equiv.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(10)}\\ +\vdash.\text{(6).(7).(10)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·182.</b> \(\vdash:\alpha\subset t_{0}ʻ\beta.\beta\subset t_{0}ʻ\gamma.\supset.\alpha\subset t_{0}ʻ\gamma \quad[\text{*63·181}]\)</p> + +<p class="nind"><b>*63·19.</b> \(\vdash.tʻt_{0}ʻ\alpha=tʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·105.*22·42}.&\supset\vdash.\alpha\subset t_{0}ʻ\alpha.t_{0}ʻ\alpha\subset t_{0}ʻ\alpha.\\ +[\text{*63·13}] &\supset\vdash.\alpha\in tʻt_{0}ʻ\alpha.\\ +[\text{*63·16}] &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_424">[Pg 424]</span></p> + +<p class="nind"><b>*63·191.</b> \(\vdash.t_{0}ʻ\alpha\in tʻ\alpha \quad[\text{*63·103·19}]\)</p> + +<p class="nind"><b>*63·2.</b> \(\vdash:x\in t_{0}ʻ\alpha.\alpha\in t_{0}ʻ\kappa.\supset.t^{2}ʻx=tʻ\alpha=t_{0}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·11}.&\supset\vdash:\text{Hp}.\supset.tʻx=t_{0}ʻ\alpha.tʻ\alpha=t_{0}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·19.(*63·04)}.&\supset\vdash:\text{Hp}.\supset.t^{2}ʻx=tʻ\alpha=t_{0}ʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·21.</b> \(\vdash:\alpha\subset tʻx.\equiv.t_{0}ʻ\alpha=tʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·181·15}.\supset\vdash:\alpha\subset tʻx.\equiv.t_{0}ʻ\alpha&=t_{0}ʻtʻx\\ +[\text{*63·15}] &=tʻx:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·22.</b> \(\vdash:\alpha\subset tʻx.\equiv.x\in t_{0}ʻ\alpha.\equiv.tʻx=t_{0}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·103}.&\supset\vdash:tʻx=t_{0}ʻ\alpha.\supset.x\in t_{0}ʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·11}.&\supset\vdash:x\in t_{0}ʻ\alpha.\equiv.tʻx=t_{0}ʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(2).*63·21}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·23.</b> \(\vdash:\alpha\subset tʻx.\kappa\subset tʻ\alpha.\supset.t^{2}ʻx=tʻ\alpha=t_{0}ʻ\kappa \quad[\text{*63·2·22}]\)</p> + +<p>Propositions of the same kind as the above can obviously be extended to +\(t^{3}ʻx\), etc.</p> + +<p class="nind"><b>*63·3.</b> \(\vdash:(\alpha).\alpha\in \kappa.\supset.(x).x\in sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·1}.\supset\vdash:\text{Hp}.&\supset.\text{V}\in \kappa.\\ +[\text{*40·221}] &\supset.sʻ\kappa=\text{V}.\\ +[\text{*24·14}] &\supset.(x).x\in sʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·31.</b> \(\vdash.sʻ(\kappa\cup -\kappa)=sʻ\kappa\cup -sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·171}. &\supset\vdash\colon\ldotp x\in sʻ(\kappa\cup -\kappa).\equiv:x\in sʻ\kappa.\lor.x\in sʻ-\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*22·88.*63·3}.&\supset\vdash:x\in sʻ\kappa.\lor.x\in sʻ-\kappa &\qquad \text{(2)}\\ +\vdash.\text{*22·88}. &\supset\vdash:x\in sʻ\kappa.\lor.x\in -sʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*10·221·13}.&\supset\\ +&\vdash\colon\ldotp x\in sʻ\kappa.\lor.x\in sʻ-\kappa:x\in sʻ\kappa.\lor.x\in -sʻ\kappa\colon\ldotp\\ +[\text{(1).*5·1}] &\supset\vdash\colon\ldotp x\in sʻ(\kappa\cup -\kappa).\equiv:x\in sʻ\kappa.\lor.x\in -sʻ\kappa\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Note that the use of <a href="#*10·221">*10·221</a> in the above proof depends upon the fact +that \(x\in sʻ\kappa\) occurs both in (2) and in (3), so that these are +both of the form \(f(x\in sʻ\kappa)\).</p> + +<p><span class="pagenum" id="Page_425">[Pg 425]</span></p> + +<p class="nind"><b>*63·32.</b> \(\vdash.t_{1}ʻ\kappa=sʻt_{0}ʻ\kappa \quad[\text{*63·31.(*63·02·03)}]\)</p> + +<p class="nind"><b>*63·321.</b> \(\vdash.t_{1}ʻ\kappa=t_{1}ʻt_{0}ʻ\kappa=t_{0}ʻt_{1}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*20·2.(*63·03)}.\supset\vdash.t_{1}ʻt_{0}ʻ\kappa&=t_{0}ʻsʻt_{0}ʻ\kappa\\ +[\text{*63·32}] &=t_{0}ʻt_{1}ʻ\kappa &\qquad \text{(1)}\\ +[\text{*20·2.(*63·03)}] &=t_{0}ʻt_{0}ʻsʻ\kappa\\ +[\text{*63·151}] &=t_{0}ʻsʻ\kappa\\ +[\text{*20·2.(*63·03)}] &=t_{1}ʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·33.</b> \(\vdash:t_{0}ʻ\kappa=t_{0}ʻ\lambda.\supset.t_{1}ʻ\kappa=t_{1}ʻ\lambda \quad[\text{*30·37.*63·32}]\)</p> + +<p class="nind"><b>*63·34.</b> \(\vdash.t_{1}ʻtʻ\alpha=t_{0}ʻ\alpha=sʻtʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·32}.\supset\vdash.t_{1}ʻtʻ\alpha&=sʻt_{0}ʻtʻ\alpha\\ +[\text{*63·15}] &=sʻtʻ\alpha &\qquad \text{(1)}\\ +[\text{*63·101}] &=sʻ(\iotaʻ\alpha\cup -\iotaʻ\alpha)\\ +[\text{*63·31}] &=sʻ\iotaʻ\alpha\cup -sʻ\iotaʻ\alpha\\ +[\text{*53·02}] &=\alpha\cup -\alpha\\ +[\text{(*63·02)}] &=t_{0}ʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·35.</b> \(\vdash:tʻ\alpha=tʻ\beta.\supset.t_{0}ʻ\alpha=t_{0}ʻ\beta \quad[\text{*30·37.*63·34}]\)</p> + +<p class="nind"><b>*63·36.</b> \(\vdash:tʻ\kappa=tʻ\lambda.\supset.t_{1}ʻ\kappa=t_{1}ʻ\lambda \quad[\text{*63·35·33}]\)</p> + +<p class="nind"><b>*63·361.</b> \(\vdash:t_{0}ʻ\alpha=t_{0}ʻ\beta.\supset.tʻ\alpha=tʻ\beta \quad[\text{*30·37.*63·19}]\)</p> + +<p class="nind"><b>*63·37.</b> \(\vdash:t_{0}ʻ\alpha=t_{0}ʻ\beta.\equiv.tʻ\alpha=tʻ\beta \quad[\text{*63·35·361}]\)</p> + +<p class="nind"><b>*63·371.</b> \(\vdash:\beta\subset t_{0}ʻ\alpha.\equiv.\beta\in tʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·181}.\supset\vdash:\beta\subset t_{0}ʻ\alpha.&\equiv.t_{0}ʻ\alpha=t_{0}ʻ\beta.\\ +[\text{*63·37}] &\equiv.tʻ\alpha=tʻ\beta.\\ +[\text{*63·16}] &\equiv.\beta\in tʻ\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·38.</b> \(\vdash:\alpha\in t_{0}ʻ\kappa.x\in t_{0}ʻ\alpha.\supset.tʻx=t_{0}ʻ\alpha=t_{1}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·11}. \supset\vdash:\text{Hp}.\supset.tʻx&=t_{0}ʻ\alpha.tʻ\alpha=t_{0}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·34}.\supset\vdash:\text{Hp}.\supset.t_{0}ʻ\alpha&=t_{1}ʻt_{0}ʻ\kappa\\ +[\text{*63·151·33}] &=t_{1}ʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·381.</b> \(\vdash:x\in t_{1}ʻ\kappa.\supset.tʻx=t_{1}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·38·105}.&\supset\vdash:\alpha\in t_{0}ʻ\kappa.x\in \alpha.\supset.tʻx=t_{1}ʻ\kappa:\\ +[\text{*10·11·23.*40·11}]&\supset\vdash:x\in sʻt_{0}ʻ\kappa.\supset.tʻx=t_{1}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·32}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_426">[Pg 426]</span></p> + +<p class="nind"><b>*63·382.</b> \(\vdash.\exists !t_{1}ʻ\kappa \quad[\text{*63·18.(*63·03)}]\)</p> + +<p class="nind"><b>*63·383.</b> \(\vdash.tʻt_{1}ʻ\kappa=t_{0}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·38·18.*10·11·23·35}.\supset\vdash:\alpha\in t_{0}ʻ\kappa.\supset.tʻt_{1}ʻ\kappa&=tʻt_{0}ʻ\alpha\\ +[\text{*63·19}] &=tʻ\alpha\\ +[\text{*63·11}] &=t_{0}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·23.*63·18}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·384.</b> \(\vdash:t_{1}ʻ\kappa=t_{1}ʻ\lambda.\supset.t_{0}ʻ\kappa=t_{0}ʻ\lambda.tʻ\kappa=tʻ\lambda \quad[\text{*63·383·37}]\)</p> + +<p class="nind"><b>*63·39.</b> \(\vdash:t_{1}ʻ\kappa=t_{1}ʻ\lambda.\equiv.t_{0}ʻ\kappa=t_{0}ʻ\lambda.\equiv.tʻ\kappa=tʻ\lambda \quad[\text{*63·33·384·37}]\)</p> + +<p class="nind"><b>*63·391.</b> \(\vdash:tʻx=tʻy.\equiv.t^{2}ʻx=t^{2}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·39}.\supset\vdash:t^{2}ʻx=t^{2}ʻy.&\equiv.t_{0}ʻtʻx=t_{0}ʻtʻy.\\ +[\text{*63·15}] &\equiv.tʻx=tʻy:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·392.</b> \(\vdash:t_{2}ʻ\kappa=t_{2}ʻ\lambda.\equiv.t_{1}ʻ\kappa=t_{1}ʻ\lambda.\equiv.t_{0}ʻ\kappa=t_{0}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·39}.\supset\vdash:t_{2}ʻ\kappa=t_{2}ʻ\lambda.&\equiv.t_{0}ʻt_{1}ʻ\kappa=t_{0}ʻt_{1}ʻ\lambda.\\ +[\text{*63·321}] &\equiv.t_{1}ʻ\kappa=t_{1}ʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·39}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·4.</b> \(\vdash:\alpha\in t_{0}ʻ\kappa.\kappa\in t_{0}ʻ\lambda.\supset.t_{0}ʻ\alpha=t_{1}ʻ\kappa=t_{2}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·38·18}.\supset\vdash:\text{Hp}.&\supset.t_{0}ʻ\alpha=t_{1}ʻ\kappa.t_{0}ʻ\kappa=t_{1}ʻ\lambda.\\ +[\text{*30·37.(*63·05)}] &\supset.t_{0}ʻ\alpha=t_{1}ʻ\kappa.t_{1}ʻt_{0}ʻ\kappa=t_{2}ʻ\lambda.\\ +[\text{*63·321}] & \supset.t_{0}ʻ\alpha=t_{1}ʻ\kappa.t_{1}ʻ\kappa=t_{2}ʻ\lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·41.</b> \(\vdash.tʻt_{2}ʻ\lambda=t_{1}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·4·18.*10·11·23·35}.\supset\vdash:\kappa\in t_{0}ʻ\lambda.\supset.tʻt_{2}ʻ\lambda&=tʻt_{1}ʻ\kappa\\ +[\text{*63·383}] &=t_{0}ʻ\kappa\\ +[\text{*63·38·18.*10·11·23·35}]&=t_{1}ʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·18}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·42.</b> \(\vdash.t^{2}ʻt_{2}ʻ\lambda=t_{0}ʻ\lambda \quad[\text{*30·37.*63·41·383}]\)</p> + +<p class="nind"><b>*63·43.</b> \(\vdash.t_{1}ʻt^{2}ʻx=tʻx \quad[\text{*63·34·15}]\)</p> + +<p class="nind"><b>*63·44.</b> \(\vdash.t_{2}ʻt^{2}ʻ\alpha=t_{0}ʻ\alpha \quad[\text{*63·43·34}]\)</p> + +<p>It is obvious that the analogues of the above propositions will hold +for \(t^{3}\) and \(t_{3}\), \(t^{4}\) and \(t_{4}\), etc. We shall not +prove these analogues, but if occasion arises we shall assume them, +referring to the corresponding propositions for \(t^{2}\) and \(t_{2}\).</p> + +<p class="nind"><b>*63·5.</b> \(\vdash:x\in t_{0}ʻ\alpha.\equiv.\alpha\in t^{2}ʻx.\equiv.\alpha\subset tʻx.\equiv.tʻx=t_{0}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·15}.\supset\vdash:\alpha\subset tʻx.&\equiv.\alpha\subset t_{0}ʻtʻx.\\ +[\text{*63·371}] &\equiv.\alpha\in t^{2}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·22}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_427">[Pg 427]</span></p> + +<p class="nind"><b>*63·51.</b> \(\vdash:\alpha\in t_{0}ʻ\kappa.\equiv.\alpha\subset t_{1}ʻ\kappa.\equiv.\kappa\subset tʻ\alpha.\equiv.tʻ\alpha=t_{0}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·2.(*63·03)}.\supset\vdash:\alpha\subset t_{1}ʻ\kappa.&\equiv.\alpha\subset t_{0}ʻsʻ\kappa.\\ +[\text{*63·371·19}] & \equiv.\alpha\in tʻt_{0}ʻsʻ\kappa.\\ +[\text{*4·2.(*63·03)}] &\equiv.\alpha\in tʻt_{1}ʻ\kappa.\\ +[\text{*63·383}] &\equiv.\alpha\in t_{0}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·5·22}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·52.</b> \(\vdash:\alpha\in t_{1}ʻ\lambda.\equiv.\alpha\subset t_{2}ʻ\lambda.\equiv.\lambda\subset + t^{2}ʻ\alpha.\equiv.tʻ\alpha=t_{1}ʻ\lambda.\equiv.t^{2}ʻ\alpha=t_{0}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·51} \frac{sʻ\lambda}{\kappa}.(*63·03).\supset\\ +\vdash:\alpha\in t_{1}ʻ\lambda.&\equiv.\alpha\subset t_{1}ʻsʻ\lambda.\\ +[\text{*63·321}] &\equiv.\alpha\subset t_{1}ʻt_{0}ʻsʻ\lambda.\\ +[\text{(*63·03·05)}] &\equiv.\alpha\subset t_{2}ʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{*63·321}.\supset\\ +\qquad\qquad\qquad\vdash:\alpha\in t_{1}ʻ\lambda.&\equiv.\alpha\in t_{0}ʻt_{1}ʻ\lambda.\\ +[\text{*63·22}] &\equiv.tʻ\alpha=t_{0}ʻt_{1}ʻ\lambda\\ +[\text{*63·321}] &=t_{1}ʻ\lambda. &\qquad \text{(2)}\\ +[\text{*63·391·41·42}]&\equiv.t^{2}ʻ\alpha=t_{0}ʻ\lambda. &\qquad \text{(3)}\\ +[\text{*63·15·181}] &\equiv.\lambda\subset t_{0}ʻt^{2}ʻ\alpha.\\ +[\text{*63·15}] &\equiv.\lambda\subset t^{2}ʻ\alpha &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·53.</b> \(\vdash:x\in t_{0}ʻ\alpha.\equiv.t^{2}ʻx=tʻ\alpha.\equiv.tʻx=t_{0}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37}.\supset\vdash:t^{2}ʻx=tʻ\alpha.&\supset.t_{1}ʻt^{2}ʻx=t_{1}ʻtʻ\alpha.\\ +[\text{*63·43·34}] &\supset.tʻx=t_{0}ʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*63·19}.&\supset\vdash:tʻx=t_{0}ʻ\alpha.\supset.t^{2}ʻx=tʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*63·5}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·54.</b> \(\vdash:\alpha\in t_{0}ʻ\kappa.\equiv.t_{0}ʻ\alpha=t_{1}ʻ\kappa.\equiv.tʻ\alpha=t_{0}ʻ\kappa.\equiv.t^{2}ʻ\alpha=tʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37}.\supset\vdash:tʻ\alpha=t_{0}ʻ\kappa.&\supset.t_{1}ʻtʻ\alpha=t_{1}ʻt_{0}ʻ\kappa.\\ +[\text{*63·34·321}] &\supset.t_{0}ʻ\alpha=t_{1}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*30·37}.&\supset\vdash:t_{0}ʻ\alpha=t_{1}ʻ\kappa.\supset.tʻt_{0}ʻ\alpha=tʻt_{1}ʻ\kappa.\\ +[\text{*63·19·383}] &\supset.tʻ\alpha=t_{0}ʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*63·51·53}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·55.</b> + \[\begin{align}&\vdash:\kappa\in t_{0}ʻ\lambda.\equiv.t_{1}ʻ\kappa=t_{2}ʻ\lambda.\equiv.t_{0}ʻ\kappa=t_{1}ʻ\lambda.\equiv.tʻ\kappa=t_{0}ʻ\lambda.\equiv.t^{2}ʻ\kappa=tʻ\lambda\\ +&[\text{Proof as in *63·54}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_428">[Pg 428]</span></p> + +<p class="nind"><b>*63·56.</b> \(\vdash:x\in t_{1}ʻ\kappa.\equiv.tʻx=t_{1}ʻ\kappa.\equiv.t^{2}ʻx=t_{0}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·321}. \supset\vdash:x\in t_{1}ʻ\kappa.&\equiv.x\in t_{0}ʻt_{1}ʻ\kappa.\\ +[\text{*63·53}] &\equiv.t^{2}ʻx=tʻt_{1}ʻ\kappa &\qquad \text{(1)}\\ +[\text{*63·383}] &=t_{0}ʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).*63·53}.\supset\vdash:x\in t_{1}ʻ\kappa.&\equiv.tʻx=t_{0}ʻt_{1}ʻ\kappa\\ +[\text{*63·321}] &=t_{1}ʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·57.</b> + \[\begin{align}&\vdash:\alpha\in t_{1}ʻ\lambda.\equiv.t_{0}ʻ\alpha=t_{2}ʻ\lambda.\equiv.tʻ\alpha=t_{1}ʻ\lambda.\equiv.t^{2}ʻ\alpha=t_{0}ʻ\lambda\\ +&[\text{Proof as in *63·56}]\end{align}\]</p> + +<p class="nind"><b>*63·61.</b> \(\vdash.t^{2}ʻx=tʻ\iotaʻx \quad[\text{*63·19·101}]\)</p> + +<p class="nind"><b>*63·62.</b> \(\vdash:x\in t_{0}ʻ\alpha.\supset.\iotaʻx\in tʻ\alpha.tʻ\iotaʻx=tʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·53}.\supset\vdash:\text{Hp}.&\supset.t^{2}ʻx=tʻ\alpha.\\ +[\text{*63·61}] &\supset.tʻ\iotaʻx=tʻ\alpha.\\ +[\text{*63·16}] &\supset.\iotaʻx\in tʻ\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·621.</b> \(\vdash:x\in \alpha.\supset.\iotaʻx\in tʻ\alpha.tʻ\iotaʻx=tʻ\alpha \quad[\text{*63·62.*63·105}]\)</p> + +<p class="nind"><b>*63·63.</b> \(\vdash:x\in t_{0}ʻ\alpha.\supset.\iotaʻ\iotaʻx\in t^{2}ʻ\alpha.tʻ\iotaʻ\iotaʻx=t^{2}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·101}.&\supset\vdash.tʻ\iotaʻx=t_{0}ʻ\iotaʻ\iotaʻx.\\ +[\text{*63·62}]\qquad\qquad\qquad\qquad\supset\vdash:\text{Hp}.&\supset.tʻ\alpha=t_{0}ʻ\iotaʻ\iotaʻx.\\ +[\text{*63·19}] &\supset.t^{2}ʻ\alpha=tʻ\iotaʻ\iotaʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·103}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·64.</b> \(\vdash.tʻ\beta=t_{0}ʻ\iotaʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·16.*37·62}.\supset\\ +\vdash:x\in \beta.&\supset.x\in \iotaʻx.\iotaʻx\in \iotaʻʻ\beta.\\ +[\text{*63·105·38}]&\supset.x\in t_{0}ʻ\iotaʻx.t_{0}ʻ\iotaʻx=t_{1}ʻ\iotaʻʻ\beta.\\ +[\text{*13·13}] &\supset.x\in t_{1}ʻ\iotaʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·51}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*63·65.</b> \(\vdash.\text{Cl}ʻt_{0}ʻ\alpha=tʻ\alpha \quad[\text{*63·371.*60·2}]\)</p> + +<p class="nind"><b>*63·66.</b> \(\vdash.\text{Cl}ʻtʻx=t^{2}ʻx \quad[\text{*63·5.*60·2}]\)</p> + +<p class="nind"><b>*63·661.</b> \(\vdash.tʻ\text{Cl}ʻ\alpha=t^{2}ʻ\alpha \quad[\text{*60·34.*63·105·53}]\)</p> + +<p class="nind"><b>*63·67.</b> \(\vdash.\text{Cl}ʻt_{1}ʻ\kappa=t_{0}ʻ\kappa \quad[\text{*63·51.*60·2}]\)</p> + +<p class="nind"><b>*63·68.</b> \(\vdash.\text{Cl}ʻt_{2}ʻ\kappa=t_{1}ʻ\kappa \quad[\text{*63·52.*60·2}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_429">[Pg 429]</span></p> +<h2 class="nobreak" id="*64">*64. RELATIVE TYPES OF RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *64.</p> + +<p>In the present number, we introduce notations defining the type of a +relation relatively to the types of its domain and converse domain, +when these types are given relatively to some fixed class \(\alpha\). +If \(R\) is any relation, it is of the same type as \(t_{0}ʻ\text{D}ʻR +\uparrow t_{0}ʻ\text{ᗡ}ʻR\). If \(\text{D}ʻR\) and \(\text{ᗡ}ʻR\) are both of the +same type as \(\alpha\), \(R\) is of the same type as \(t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha\), +which is of the same type as \(\alpha\uparrow \alpha\). +The type of \(t_{0}ʻ\alpha \uparrow t_{0}ʻ\alpha\) +we call \(t_{00}ʻ\alpha\), and the type of \(t^{m}ʻ\alpha \uparrow t^{n}ʻ\alpha\) +we call \(t^{mn}ʻ\alpha\), and the type of \(t_{m}ʻ\alpha \uparrow t_{n}ʻ\alpha\) +we call \(t_{mn}ʻ\alpha\), and the type of +\(t_{m}ʻ\alpha \uparrow t^{n}ʻ\alpha\) we call \(t_{m}^{n}ʻ\alpha\), +and the type of \(t^{m}ʻ\alpha \uparrow t_{n}ʻ\alpha\) we call +\({}^{m}t_{n}ʻ\alpha\). We thus have a means of expressing the type of +any relation \(R\) in terms of the type of \(\alpha\), provided the +types of the domain and converse domain of \(R\) are given relatively +to \(\alpha\).</p> + +<p>The most useful propositions of the present number are the following:</p> + +<p class="nind"><b>*64·16.</b> \(\vdash : R \unicode{x2abd} t_{0}ʻ\alpha \uparrow t_{0}ʻ\beta .\equiv. R \in tʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\beta)\)</p> + +<p class="nind"><b>*64·201.</b> \(\vdash : R \unicode{x2abd} S .\supset. R \in tʻS . tʻR = tʻS\)</p> + +<p class="nind"><b>*64·231.</b> \(\vdash:R \in tʻQ .\supset. \text{D}ʻR \in tʻ\text{D}ʻQ . \text{ᗡ}ʻR \in tʻ\text{ᗡ}ʻQ .CʻR \in tʻCʻQ\)</p> + +<p>Here "\(CʻR \in tʻCʻQ\)" will only be significant if \(R\) and \(Q\) +are homogeneous relations, which is not required by the rest of the +proposition. When \(R\) and \(Q\) are homogeneous relations we have</p> + +<p class="nind"><b>*64·24.</b> \(\vdash : R \in tʻQ .\equiv. CʻR \in tʻCʻQ .\equiv. t_{0}ʻCʻR=t_{0}ʻCʻQ\)</p> + +<p>This proposition is useful in connecting ordinal and cardinal existence-theorems.</p> + +<p class="nind"><b>*64·312.</b> \(\vdash. t^{22}ʻx = t^{11}ʻtʻx = t_{00}ʻt^{2}ʻx\)</p> + +<p class="nind"><b>*64·5.</b> \(\vdash. \text{Rl}ʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\beta) = tʻ(t_{0}ʻ\alpha \uparrow t_{0}ʻ\beta) = tʻ(\alpha \uparrow \beta)\)</p> + +<p>This proposition is frequently used. It states that the class of +relations whose referents are of the type of members of \(\alpha\) +while its relata are of the type of members of \(\beta\) (<i>i.e.</i> +the class of all relations contained in \(t_{0}ʻ\alpha \uparrow t_{0}ʻ\beta\)) +is the type of \(t_{0}ʻ\alpha \uparrow t_{0}ʻ\beta\) and +is also the type of \(\alpha \uparrow \beta\).</p> + +<p class="nind"><b>*64·55.</b> \(\vdash: CʻP \subset t_{0}ʻ\alpha .\equiv. P \in t_{00}ʻ\alpha\)</p> + +<p><span class="pagenum" id="Page_430">[Pg 430]</span></p> + +<p class="nind"><b>*64·57.</b> \(\vdash : CʻP \subset tʻx .\equiv. P \in t^{11}ʻx\)</p> + +<p>The propositions of the present number are mostly obvious, though +formal proofs are sometimes not very easily found. The use of the +propositions of this number occurs chiefly in the first section on +relation-arithmetic and in the proofs of existence-theorems in ordinal +arithmetic and the theory of ratio.</p> + +<hr class="tb"> + +<p class="nind"><b>*64.01.</b> \(t_{00}ʻ\alpha=tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.011.</b> \(t^{11}ʻx=tʻ(tʻx\uparrow tʻx) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.012.</b> \(t^{12}ʻx=tʻ(tʻx\uparrow t^{2}ʻx) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.013.</b> \(t^{21}ʻx=tʻ(t^{2}ʻx\uparrow tʻx) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.014.</b> \(t^{22}ʻx=tʻ(t^{2}ʻx\uparrow t^{2}ʻx) \quad\text{Df}\)</p> + +<p>\[\text{etc.}\]</p> + +<p class="nind"><b>*64.02.</b> \(t_{01}ʻ\alpha=tʻ(t_{0}ʻ\alpha\uparrow t_{1}ʻ\alpha) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.021.</b> \(t_{10}ʻ\alpha=tʻ(t_{1}ʻ\alpha\uparrow t_{0}ʻ\alpha) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.022.</b> \(t_{11}ʻ\alpha=tʻ(t_{1}ʻ\alpha\uparrow t_{1}ʻ\alpha) \quad\text{Df}\)</p> + +<p>\[\text{etc.}\]</p> + +<p class="nind"><b>*64.03.</b> \(t_{0}^{1}ʻ\alpha=tʻ(t_{0}ʻ\alpha\uparrow tʻ\alpha) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.031.</b> \(t_{1}^{1}ʻ\alpha=tʻ(t_{1}ʻ\alpha\uparrow tʻ\alpha) \quad\text{Df}\)</p> + +<p>\[\text{etc.}\]</p> + +<p class="nind"><b>*64.04.</b> \(^{1}t_{0}ʻ\alpha=tʻ(tʻ\alpha\uparrow t_{0}ʻ\alpha) \quad\text{Df}\)</p> + +<p class="nind"><b>*64.041.</b> \(^{1}t_{1}ʻ\alpha=tʻ(tʻ\alpha\uparrow t_{1}ʻ\alpha) \quad\text{Df}\)</p> + +<p>\[\text{etc.}\]</p> + +<p class="nind"><b>*64.1.</b> \(\vdash.\alpha\uparrow \alpha\in t_{00}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·2}. &\supset\vdash:\alpha=t_{0}ʻ\alpha.\supset.\alpha\uparrow\alpha=t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*35·9}. &\supset\vdash:\alpha\uparrow\alpha=t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha.\supset.\alpha=t_{0}ʻ\alpha:\\ +[\text{Transp}] &\supset\vdash:\alpha\neq t_{0}ʻ\alpha.\supset.\alpha\uparrow\alpha\neq t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp \alpha=t_{0}ʻ\alpha.\lor.\alpha\neq t_{0}ʻ\alpha:\supset:\alpha\uparrow\alpha=t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha.\lor.\alpha\uparrow\alpha\neq + t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(3).*51·15.*63·101·191}.&\supset\vdash:\alpha\uparrow\alpha=t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha.\lor.\alpha\uparrow\alpha \neq t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha + &\qquad \text{(4)}\\ +\vdash.\text{(4).*51·15.*63·101.(*64·01)}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64.11.</b> \(\vdash.t_{00}ʻ\alpha=tʻ(\alpha\uparrow\alpha) \quad[\text{*64.1.*63.16}]\)</p> + +<p class="nind"><b>*64.12.</b> \(\vdash.\alpha\uparrow\beta\in tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta\))</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·85·86.*63·18}.&\supset\vdash:&\alpha\uparrow \beta=t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\equiv.\alpha=t_{0}ʻ\alpha.\beta=t_{0}ʻ\beta + &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp}. &\supset\vdash:&\alpha=t_{0}ʻ\alpha.\beta=t_{0}ʻ\beta.\supset.\alpha\uparrow \beta=t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta:\\ +& &\alpha=t_{0}ʻ\alpha.\beta\neq t_{0}ʻ\beta.\supset.\alpha\uparrow \beta\neq t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta:\\ +[\text{*63·101.*51·15}] &\supset\vdash:&\alpha=t_{0}ʻ\alpha.\supset.\alpha\uparrow \beta\in tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta) &\qquad \text{(2)}\\ +\vdash.\text{(1).Transp}. &\supset\vdash:&\alpha\neq t_{0}ʻ\alpha.\supset.\alpha\uparrow \beta\neq(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta).\\ +[\text{*63·101.*51·15.Transp}] & &\supset.\alpha\uparrow \beta\in tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_431">[Pg 431]</span></p> + +<p class="nind"><b>*64·13.</b> \(\vdash.tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta)=tʻ(\alpha\uparrow \beta) \quad[\text{*64·12.*63·16}]\)</p> + +<p class="nind"><b>*64·14.</b> \(\vdash.(x,y).x(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta)y \quad[\text{*63·1.*35·103}]\)</p> + +<p class="nind"><b>*64·15.</b> \(\vdash.(R).R\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta \quad[\text{*64·14.*25·14·11}]\)</p> + +<p class="nind"><b>*64·16.</b> \(\vdash:R\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\equiv.R\in tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta\))</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*2·11}. \supset\vdash:R=t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\lor.R\neq t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta:\\ +[\text{*23·42}] \supset\vdash:R=t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.R\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\lor.R\neq t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta + &\qquad \text{(1)}\\ +\vdash.\text{(1).*64·15.*10·221·13}.\supset\\ +\vdash:R\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta:R=t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.R\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\lor.R\neq + t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(2).*5·1}.\supset\\ +\vdash\colon\ldotp R\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\equiv:R=t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.R\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\lor.R\neq + t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta:\\ +[\text{*23·42}] \equiv:R=t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta.\lor.R\neq t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>By putting \(t_{s}^{i}ʻ\alpha\) (where \(i\) and \(s\) are some +index and suffix which have been defined) for \(\alpha\) and +\(t_{s}^{i}ʻ\alpha\) for \(\beta\), the above propositions give results +applicable to any of the types defined at the beginning of this number, +because of \(t_{0}ʻt_{s}^{i}ʻ\alpha=t_{s}^{i}ʻ\alpha\).</p> + +<p class="nind"><b>*64·2.</b> \(\vdash.\dot{\exists}!R\dot{\cap}S.\supset.S\in tʻR.tʻR=tʻS \quad[\text{*63·13·16}]\)</p> + +<p class="nind"><b>*64·201.</b> \(\vdash:R\unicode{x2abd}S.\supset.R\in tʻS.tʻR=tʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*25·6}.\supset\vdash\colon\ldotp\text{Hp}.&\supset:R=S.\lor.\dot{\exists}!S\dot{-}R:\\ +[\text{*13·14}] &\supset:R=S.\lor.R\neq S\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·21.</b> \(\vdash:xRy.\supset.R\in tʻ(tʻx\uparrow tʻy\))</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·103.*35·103}.&\supset\vdash.x(tʻx\uparrow tʻy)y &\qquad \text{(1)}\\ +\vdash.(1). &\supset\vdash:\text{Hp}.\supset.\dot{\exists}!R\dot{\cap}(tʻx\uparrow tʻy) &\qquad \text{(2)}\\ +\vdash.\text{(2).*64·2}. &\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·22.</b> \(\vdash.R\in tʻ(t_{0}ʻ\text{D}ʻR\uparrow t_{0}ʻ\text{ᗡ}ʻR) \quad[\text{*64·16.*63·105.*35·83}]\)</p> + +<p class="nind"><b>*64·23.</b> \(\vdash.tʻR=tʻ\breve{s}ʻtʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{lr} +\vdash.\text{*63·103.*41·13}.&\supset\vdash.R\unicode{x2abd}\dot{s}ʻtʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*64·201}. &\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·231.</b> \(\vdash:R\in tʻQ.\supset.\text{D}ʻR\in tʻ\text{D}ʻQ.\text{ᗡ}ʻR\in tʻ\text{ᗡ}ʻQ.CʻR\in tʻCʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·12}.&\supset\vdash\colon\colon\ldotp\text{Hp}.\supset\colon\colon xRy.\supset_{x,y}\colon\ldotp xQy.\lor.{\sim}(xQy)\colon\colon\\ +[\text{*10·28}] &\supset\colon\colon(\exists y).xRy.\supset_{x}\colon\ldotp(\exists y).xQy.\lor.(\exists y).{\sim}(xQy)\colon\ldotp\\ +[\text{*5·63}] &\supset_{x}\colon\ldotp(\exists y).xQy\colon\ldotp\lor\colon\ldotp{\sim}(\exists y).xQy:(\exists y).{\sim}(xQy)\colon\ldotp\\ +[\text{*3·26}] &\supset_{x}\colon\ldotp(\exists y).xQy.\lor.{\sim}(\exists y).(xQy) &\qquad \text{(1)}\\ +\vdash.\text{(1).*33·13}. &\supset\vdash\colon\ldotp\text{Hp}.\supset:x\in\text{D}ʻ R.\supset_{x}.x\in\text{D}ʻQ\cup-\text{D}ʻQ:\\ +[\text{(*63·02)}] &\supset:\text{D}ʻR\subset t_{0}ʻ\text{D}ʻQ:\\ +[\text{*63·371}] &\supset:\text{D}ʻR\in tʻ\text{D}ʻQ &\qquad \text{(2)}\\ +\text{Similarly} \vdash:\text{Hp}.&\supset.\text{ᗡ}ʻ R\in tʻ\text{ᗡ}ʻQ.CʻR\in tʻCʻQ &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_432">[Pg 432]</span></p> + +<p class="nind"><b>*64·24.</b> \(\vdash:R\in tʻQ.\equiv.CʻR\in tʻCʻQ.\equiv.t_{0}ʻCʻR=t_{0}ʻCʻQ\)</p> + +<p>This proposition is only significant when \(R\) and \(Q\) are +homogeneous relations.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*64·22.*63·181}.&\supset\vdash.R\in tʻ(t_{0}ʻCʻR \uparrow t_{0}ʻCʻR).\\ +[\text{*13·12}] &\supset\vdash:t_{0}ʻCʻR=t_{0}ʻCʻQ.\supset.R\in tʻ(t_{0}ʻCʻ Q\uparrow t_{0}ʻCʻQ) &\qquad \text{(1)}\\ +\vdash.\text{*64·22.*63·181}.&\supset\vdash.Q\in tʻ(t_{0}ʻCʻQ\uparrow t_{0}ʻCʻQ) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*63·16}. &\supset\vdash:t_{0}ʻCʻR=t_{0}ʻCʻQ.\supset.R\in tʻQ &\qquad \text{(3)}\\ +\vdash.\text{(3).*64·231.*63·16·37}.&\supset +\vdash:R\in tʻQ.\equiv.t_{0}ʻCʻR=t_{0}ʻCʻQ.\equiv.CʻR\in tʻCʻ Q:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·3.</b> \(\vdash:t_{00}ʻ\alpha=t_{00}ʻ\beta.\equiv.\alpha\in tʻ\beta.\equiv.tʻ\alpha=tʻ\beta.\equiv.t_{0}ʻ\alpha=t_{0}ʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37.(*64·01)}. &\supset\vdash:t_{0}ʻ\alpha=t_{0}ʻ\beta.\supset.t_{00}ʻ\alpha=t_{00}ʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*64·1}. &\supset\vdash:t_{00}ʻ\alpha=t_{00}ʻ\beta.\supset.\alpha\uparrow\alpha\in t_{00}ʻ\beta.\\ +[\text{*64·16}] &\supset.\alpha\uparrow\alpha\unicode{x2abd}t_{0}ʻ\beta\uparrow t_{0}ʻ\beta.\\ +[\text{*35·9·91}] &\supset.\alpha\subset t_{0}ʻ\beta.\\ +[\text{*63·181}] &\supset.t_{0}ʻ\alpha=t_{0}ʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*63·16·37}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·31.</b> \(\vdash.t^{11}ʻx=t_{00}ʻtʻx \quad[\text{*63·15.(*64·01·011)}]\)</p> + +<p class="nind"><b>*64·311.</b> \(\vdash.t_{11}ʻ\alpha=t_{00}ʻt_{1}ʻ\alpha \quad[\text{*63·321.(*64·022·01)}]\)</p> + +<p class="nind"><b>*64·312.</b> \(\vdash.t^{22}ʻx=t^{11}ʻtʻx=t_{00}ʻt^{2}ʻx \quad[\text{*63·15.(*63·04).(*64·014·011·01)}]\)</p> + +<p class="nind"><b>*64·313.</b> \(\vdash.t_{22}ʻ\alpha=t_{11}ʻt_{1}ʻ\alpha=t_{00}ʻt_{2}ʻ\alpha \quad[\text{*63·321.(*63·05)}]\)</p> + +<p class="nind"><b>*64·32.</b> \[\begin{align}\vdash:t_{22}ʻ\alpha=t_{22}ʻ\beta.\equiv.t_{11}ʻ\alpha=t_{11}ʻ\beta.\equiv.t_{00}ʻ\alpha=t_{00}ʻ\beta.\equiv.t^{11}ʻ\alpha=t^{11}ʻ\beta.\\ +&\equiv.t^{22}ʻ\alpha=t^{22}ʻ\beta.\equiv.\alpha\in tʻ\beta.\equiv.tʻ\alpha=ʻ\beta\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*64·313·3}.\supset\vdash:t_{22}ʻ\alpha=t_{22}ʻ \beta.&\equiv.tʻt_{2}ʻ\alpha=tʻt_{2}ʻ \beta.\\ +[\text{*63·41·39}]·&\equiv.tʻ\alpha=tʻ \beta\\ +\end{array} +\]</p> + +<p>Similarly the other equivalences are proved.</p> + +<p class="nind"><b>*64·33.</b> + \[\begin{align}\vdash:\alpha\in t_{0}ʻ\mu.\equiv.t_{11}ʻ\alpha=t_{22}ʻ\mu.\equiv.t_{00}ʻ\alpha=t_{11}ʻ\mu.&\equiv.t^{11}ʻ\alpha=t_{00}ʻ\mu.\\ +&\equiv.t^{22}ʻ\alpha=t^{11}ʻ\mu.\equiv.tʻ\alpha=t_{0}ʻ\mu +\end{align}\]</p> + + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*64·11·313}.\supset\vdash:t_{11}ʻ\alpha=t_{22}ʻ\mu.&\equiv.t_{00}ʻ t_{1}ʻ\alpha=t_{00}ʻt_{2}ʻ\mu.\\ +[\text{*64·3}] &\equiv.tʻt_{1}ʻ\alpha=tʻt_{2}ʻ\mu.\\ +[\text{*63·383·41·55}] &\equiv.tʻ\alpha=t_{0}ʻ\mu &\qquad·\text{(1)}\\ +\end{array} +\]</p> + +<p>Similarly the other equivalences are proved.</p> + +<p class="nind"><b>*64·34.</b> + \[\begin{align}&\vdash:\alpha\in t_{1}ʻ\mu.\equiv.t_{00}ʻ\alpha=t_{22}ʻ\mu.\equiv.t^{11}ʻ\alpha=t_{11}ʻ\mu.\equiv.t^{22}ʻ\alpha=t_{00}ʻ\mu.\equiv.t^{2}ʻ\alpha=t_{0}ʻ\mu\\ +&[\text{Proof as in *64·33}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_433">[Pg 433]</span></p> + +<p class="nind"><b>*64·5.</b> \(\vdash.\text{Rl}ʻ (t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta)=tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta)=tʻ(ʻ\alpha\uparrow\beta) + \quad[\text{*64·13·16.*61·2}]\)</p> + +<p class="nind"><b>*64·51.</b> \(\vdash.x\downarrow y\in tʻ(tʻ x\uparrow tʻy) \quad[\text{*64·21.*55·132}]\)</p> + +<p class="nind"><b>*64·52.</b> \(\vdash:x\in t_{0}ʻ\alpha.y\in t_{0}ʻ\beta.\supset.x\downarrow y\in tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta) \quad[\text{*63·11.*64·51}]\)</p> + +<p class="nind"><b>*64·53.</b> \(\vdash:x\in t_{0}ʻ\alpha.\delta\subset t_{0}ʻ\beta.\supset.(\iotaʻx)\downarrow\delta\in tʻ(tʻ\alpha\uparrow tʻ\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*64·51}. &\supset\vdash.(\iotaʻx)\downarrow\delta\in tʻ(tʻ \iotaʻx\uparrow tʻ\delta) &\qquad \text{(1)}\\ +\vdash.\text{*63·62}. &\supset\vdash:\text{Hp}.\supset.tʻ\iotaʻx=tʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{*63·181·37}. &\supset\vdash:\text{Hp}.\supset.tʻ\delta=tʻ\beta &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p>This proposition is used in connection with cardinal addition (*110·18).</p> + +<p class="nind"><b>*64·54.</b> \[\begin{align}\vdash.\text{Rl}ʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha)=t_{00}ʻ\alpha=tʻ(\alpha\uparrow\alpha)=t_{0}ʻ \text{Rl}ʻ(\alpha\uparrow\alpha)\\ +&[\text{*64·5.*61·34.*63·105·11.(*64·01)}]\end{align}\]</p> + +<p class="nind"><b>*64·55.</b> \(\vdash:CʻP\subset t_{0}ʻ\alpha.\equiv.P\in t_{00}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·91}.\supset\vdash:CʻP\subset t_{0}ʻ\alpha.&\equiv.P\unicode{x2abd}t_{0}ʻ\alpha\uparrow t_{0}ʻ\alpha.\\ +[\text{*64·54}] & \equiv.P\in t_{00}ʻ\alpha:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·56.</b> \(\vdash.\text{Rl}ʻ (tʻx\uparrow tʻx)=t^{11}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*64·5.*63·15}.\supset\vdash.\text{Rl}ʻ(tʻx\uparrow tʻx)&=tʻ(tʻx\uparrow tʻx)\\ +[\text{(*64·011)}] &=t^{11}ʻx.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·57.</b> \(\vdash:CʻP\subset tʻx.\equiv.P\in t^{11}ʻx \quad[\text{*64·56.*35·91.*61·2}]\)</p> + +<p class="nind"><b>*64·6.</b> \(\vdash.tʻP=\text{Rl}ʻ (t_{0}ʻ\text{D}ʻP\uparrow t_{0}ʻ\text{ᗡ}ʻP)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·83.*63·105}.&\supset\vdash.P\unicode{x2abd}t_{0}ʻ\text{D}ʻ P\uparrow t_{0}ʻ\text{ᗡ}ʻP.\\ +[\text{*64·201}] &\supset\vdash.tʻP=tʻ(t_{0}ʻ\text{D}ʻ P\uparrow t_{0}ʻ\text{ᗡ}ʻP)\\ +[\text{*64·5}] &=\text{Rl}ʻ (t_{0}ʻ\text{D}ʻ P\uparrow t_{0}ʻ\text{ᗡ}ʻ P).\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·61.</b> \(\vdash:\text{D}ʻP\in tʻ\alpha.\text{ᗡ}ʻP\in tʻ\beta.\supset.tʻP=tʻ(\alpha\uparrow\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*63·16·35}.\supset\vdash:\text{Hp}.&\supset.t_{0}ʻ\text{D}ʻ P&=t_{0}ʻ\alpha.t_{0}ʻ\text{ᗡ}ʻP=t_{0}ʻ\beta.\\ +[\text{*64·6}] &\supset.tʻP&=tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta)\\ +[\text{*64·5}] && =tʻ(\alpha\uparrow\beta):\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·62.</b> \(\vdash:\text{D}ʻP\in tʻ\text{D}ʻQ.\text{ᗡ}ʻP\in tʻ\text{ᗡ}ʻQ.\equiv.P\in tʻQ.\equiv.tʻP=tʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*64·61}.\supset\vdash:\text{Hp}.\supset.tʻP&=tʻ(\text{D}ʻ Q\uparrow\text{ᗡ}ʻQ)\\ +[\text{*64·5·22.*63·16}] &=tʻQ &\qquad \text{(1)}\\ +\vdash.\text{(1).*64·231}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*64·63.</b> \(\vdash:\text{D}ʻP\in tʻ\alpha.\text{ᗡ}ʻP\in tʻ\beta.\equiv.tʻP=tʻ(\alpha\uparrow\beta).\equiv.P\in tʻ(\alpha\uparrow\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*64·5}.\supset\vdash:tʻP=tʻ(\alpha\uparrow\beta).&\supset.tʻP=tʻ(t_{0}ʻ\alpha\uparrow t_{0}ʻ\beta).\\ +[\text{*64·231.*35·85·86}] & \supset.\text{D}ʻP\in tʻt_{0}ʻ\alpha.\text{ᗡ}ʻ P\in tʻt_{0}ʻ\beta.\\ +[\text{*63·19}] & \supset.\text{D}ʻP\in tʻ\alpha.\text{ᗡ}ʻP\in tʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*64·61.*63·16}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_434">[Pg 434]</span></p> +<h2 class="nobreak" id="*65">*65. ON THE TYPICAL DEFINITION OF AMBIGUOUS SYMBOLS.</h2> +</div> + + +<p><i>Summary of</i> *65.</p> + +<p>In this number we are concerned with definitions and propositions in +which an ambiguous symbol is determined as belonging to some assigned +type. If "\(\alpha\)" is an ambiguous symbol representing a class +(such as \(\Lambda\) or \(\text{V}\) for example), "\(\alpha_{x}\)" +is to denote what \(\alpha\) becomes when its members are determined +as belonging to the type of \(x\), while "\(\alpha(x)\)" denotes what +\(\alpha\) becomes when its members are determined as belonging to the +type of \(tʻx\). Thus <i>e.g.</i> "\(\Lambda_{x}\)" will be everything +of the same type as \(x\), <i>i.e.</i> \(tʻx\); \(\text{V}(x)\) will +be \(tʻtʻx\). Similarly if "\(R\)" stands for a relation of ambiguous +type, such as \(\Lambda\) or \(\dot{V}\), \(R_{x}\) will denote +what \(R\) becomes when its domain is confined within the type of +\(x\); \(R_{x,y}\) will denote what \(R\) becomes when its domain and +converse domain are confined respectively within the types of \(x\) and +\(y\); \(R(x,y)\) will have the domain and converse domain confined +respectively to the types of \(tʻx\) and \(tʻy\); with analogous +meanings for \(R(x)\) and \(R_{x,y}\). Throughout this number, \(R\) +and \(\alpha\) do not stand for proper variables, but for typically +ambiguous symbols.</p> + +<p>The notations of the present number are used in the elementary parts of +the theory of cardinals and ordinals, <i>i.e.</i> in Part III, Section +A, and in Part IV, Section A. The only <i>proposition</i>, however, +which is much used, is</p> + +<p class="nind"><b>*65·13.</b> \(\vdash : \alpha = \beta_{x} . \equiv . \alpha = tʻ x \cap \beta . \equiv . \alpha \subset tʻ x . \alpha = \beta\)</p> + +<p>Here \(\beta\) is supposed to be a typically ambiguous symbol. The +first equivalence, "\(\alpha = \beta_{x} . \equiv . \alpha = tʻ x \cap\beta\)," +merely embodies the definition of \(\beta_{x}\) (<a href="#*65·01">*65·01</a>). It +is the second equivalence that is important. Let us, for the sake of +illustration, put 1 in place of \(\beta\). Then we are to have +\[ +\alpha = tʻ x \cap 1 . \equiv . \alpha\subset tʻ x . \alpha = 1. +\] +(Since 1 is a class of classes, we shall have to suppose that \(x\) +is a class.) Consider \(y\in\alpha\). If \(\alpha = tʻx\cap 1\), +\(y\in\alpha.\equiv. y\in tʻx . y\in 1\). But we have \((y) . y\in +tʻx\). Hence \(y\in\alpha.\equiv. y\in 1\), whence \(\alpha = 1\). +Also if \(\alpha = tʻx\cap 1\), of course \(\alpha\subset tʻx\). +Thus \(\alpha = tʻx\cap 1.\supset.\alpha\subset tʻx.\alpha = 1\). +The converse implication follows from <a href="#*22·621">*22·621</a>. The reason for the +proposition is that a symbol such as "1," if it occurs in such a +proposition as \(\alpha = tʻx\cap 1\), must, for significance, be +determined as meaning that 1 which is of the same type as \(\alpha\), +<i>i.e.</i> the class of all<span class="pagenum" id="Page_435">[Pg 435]</span> unit classes which are of the same type +as members of \(\alpha\). And similarly, when we put \(\alpha=1\), +that does not mean that \(\alpha\) is the class of all unit classes, +but only that it is the class of all unit classes of the appropriate +type, which, if \(\alpha\subset tʻx\), will be \(tʻ x\cap 1\). The +proposition "\(tʻ x\cap 1=1\)" is true whenever it is significant, but +\(tʻc x\cap 1\) is typically definite when \(x\) is given, whereas 1 +is typically ambiguous. The use of the above proposition lies in its +enabling us to substitute typically definite symbols for such as are +typically ambiguous.</p> + +<p>Another useful proposition is</p> + +<p class="nind"><b>*65·2.</b> \(\vdash.\text{sg}ʻ \{R_{(x,y)}\}=\overrightarrow{R}(x_{y})\)</p> + +<p>Here \(R\) is supposed to be a typically ambiguous symbol; the proposition +states that if \(R\) is typically defined as going from objects of type \(x\) to objects +of type \(y\), then \(\overrightarrow{R}\) must go from objects of type \(tʻx\) to objects of type \(y\). This +proposition is only used twice (*102·3 and *154·2), but both uses are of great +importance, the one in cardinal and the other in ordinal arithmetic.</p> + +<p>The only other proposition of this number which is subsequently used is</p> + +<p class="nind"><b>*65·3.</b> \(\vdash.R_{\beta}ʻʻ \mu=(Rʻʻ\mu)_{\beta}=Rʻʻ\mu\cap tʻ \beta\)</p> + +<p>This proposition is used in *102·84.</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*65·01">*65·01</a>.</b> \(\alpha_{x}=\alpha\cap tʻx \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*65·02">*65·02</a>.</b> \(\alpha(x)=\alpha\cap tʻtʻx \quad\text{Df}\)</p> + +<p class="nind"><b>*65·03.</b> \(R_{x}=(tʻx)\upharpoonleft R \quad\text{Df}\)</p> + +<p class="nind"><b>*65·04.</b> \(R(x)=(t^{2}ʻx)\upharpoonleft R \quad\text{Df}\)</p> + +<p class="nind"><b>*65·1.</b> \(R_{(x,y)}=(tʻx)\upharpoonleft R\upharpoonright(tʻy) \quad\text{Df}\)</p> + +<p class="nind"><b>*65·11.</b> \(R(x_{y})=(t^{2}ʻx)\upharpoonleft R\upharpoonright(tʻy) \quad\text{Df}\)</p> + +<p class="nind"><b>*65·12.</b> \(R(x,y)=(t^{2}ʻx)\upharpoonleft R\upharpoonright(t^{2}ʻy) \quad\text{Df}\)</p> + +<p class="nind"><b>*65·13.</b> \(\vdash:\alpha=\beta_{x}.\equiv.\alpha=tʻx\cap\beta.\equiv.\alpha\subset tʻx.\alpha=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·2.(*65·01)}. &\supset\vdash:\alpha=\beta_{x}.\equiv.\alpha=tʻ x\cap\beta &\qquad \text{(1)}\\ +\vdash.\text{*22·621.*13·13}.&\supset\vdash:\alpha\subset tʻ x.\alpha=\beta.\supset.\alpha=tʻx\cap\beta &\qquad \text{(2)}\\ +\vdash.\text{*22·43}. &\supset\vdash:\alpha=tʻx\cap\beta.\supset.\alpha\subset tʻx.\alpha\subset\beta. &\qquad \text{(3)}\\ +[\text{*63·13}] &\supset.\beta\in tʻtʻx.\\ +[\text{*63·371·15}] &\supset.\beta\subset tʻx.\\ +[\text{*22·621}] &\supset.\beta=tʻ x\cap\beta &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}. &\supset\vdash:\alpha=tʻx\cap\beta.\supset.\alpha\subset tʻx.\alpha=\beta &\qquad \text{(5)}\\ +\vdash.\text{(1).(2).(5)}.&\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*65·14.</b> \(\vdash:x\in t_{0}ʻ\alpha.\supset.\gamma(x)=\gamma_{\alpha} \quad[\text{*63·53.(*65·01·02)}]\)</p> + +<p class="nind"><b>*65·15.</b> \(\vdash:x\in t_{0}ʻ\alpha.\supset.R(x)=R_{\alpha}.R(x_{y})=R_{(\alpha,\gamma)} \quad[\text{*63·53.(*65·03·04·1·11)}]\)</p> + +<p><span class="pagenum" id="Page_436">[Pg 436]</span></p> + +<p class="nind"><b>*65·16.</b> \(\vdash:x\in t_{0}ʻ\alpha.y\in t_{0}ʻ\beta.\supset.R(x,y)=R(x_{\beta})=R_{(\alpha,\beta)} \quad[\text{*63·53.(*65·1·11·12)}]\)</p> + +<p class="nind"><b>*65·2.</b> \(\vdash.\text{sg}ʻ \{R_{(x,y)}\}=\overrightarrow{R}(x_{y})\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·1·23.(*65·1)}.\supset\\ +\vdash:\alpha[\text{sg}ʻ\{R_{(x,y)}\}]w.&\equiv.\alpha=\hat{z}\{z\in tʻ x.w\in tʻ y.zRw\}.\\ +[\text{*22·39.*20·42}] &\equiv.\alpha=tʻ x\cap \hat{z}(w\in tʻ y.zRw).\\ +[\text{*65·13}] &\equiv.\alpha\subset tʻ x.\alpha=\hat{z}(w\in tʻ y.zRw)&\qquad \text{(1)}\\ +\vdash.\text{*20·33}.\supset\vdash\colon\colon \alpha=\hat{z}(w\in tʻ y.zRw).\equiv\colon\ldotp z\in\alpha.\equiv_{z}.w\in tʻ y.zRw\colon\ldotp\\ +[\text{*63·108}] &\equiv\colon\ldotp w\in tʻ y:z\in\alpha.\equiv_{z}.w\in tʻ y.zRw\colon\ldotp\\ +[\text{*4·73}] &\equiv\colon\ldotp w\in tʻy:z\in\alpha.\equiv_{z}.zRw\colon\ldotp\\ +[\text{*20·33.*32·1}]&\equiv\colon\ldotp w\in tʻ y.\alpha\overrightarrow{R}w &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*63·5}.\supset\vdash:\alpha[\text{sg}ʻ\{R_{(x,y)}\}]w.\equiv.\alpha\in t^{2}ʻx.w\in tʻ y.\alpha\overrightarrow{R}w.\\ +[\text{*35·102.(*65·11)}] \equiv.\alpha{\overrightarrow R\{_{(x_{y})}}\}w:\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b><a id="*65·21">*65·21</a>.</b> \(\vdash.R_{(x,y)}=\{R_{(x,y)}\}_{(x,y)}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21· 2.(*65·1)}.\supset\vdash.\{R_{(x,y)}\}_{(x,y)}&=tʻ x\upharpoonleft\{tʻ x\upharpoonleft R\upharpoonright tʻ y\}\upharpoonright tʻ y\\ +[\text{*35·33·34}] &=tʻ x\upharpoonleft R\upharpoonright tʻ y\\ +[\text{(*65·1)}] &=R_{(x,y)}.\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*65·22.</b> \(\vdash.R(x,y)=\{R(x,y)\}(x,y)\)</p> + +<p>This and the following three propositions are proved as <a href="#*65·21">*65·21</a> is proved.</p> + +<p class="nind"><b>*65·23.</b> \(\vdash.R(x_{y})=\{R(x_{y})\}(x_{y})\)</p> + +<p class="nind"><b>*65·24.</b> \(\vdash.R_{x}=(R_{x})_{x}\)</p> + +<p class="nind"><b>*65·25.</b> \(\vdash.R(x)=\{R(x)\}(x)\)</p> + +<p class="nind"><b>*65·3.</b> \(\vdash.R_{\beta}ʻʻ \mu=(Rʻʻ \mu)_{\beta}=Rʻʻ \mu\cap tʻ\beta\)</p> + +<p>Dem.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1.(*65·03)}.\supset\vdash.R_{\beta}ʻʻ \mu&=\hat{x}\{(\exists y).y\in\mu.xRy.x\in tʻ\beta\}\\ +[\text{*22·39.(*37·01)}] &=Rʻʻ\mu\cap tʻ\beta &\qquad \text{(1)}\\ +[\text{*65·01)}] &=(Rʻʻ\mu)_{\beta} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_437">[Pg 437]</span></p> +<h2 class="nobreak" id="SECTION_C_b">SECTION C.<br> +<br> +ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS.</h2> +</div> + + +<p><i>Summary of Section C.</i></p> + +<p>In the present section we have to consider three very important +classes of relations, of which the use in arithmetic is constant. A +<i>one-many</i> relation is a relation \(R\) such that, if \(y\) is any +member of \(\text{ᗡ}ʻ R\), there is one, and only one, term \(x\) which +has the relation \(R\) to \(y\), <i>i.e.</i> \(\overrightarrow{R}ʻ y\in 1\). +Thus the relation of father to son is one-many, because every +son has one father and no more. The relation of husband to wife is +one-many except in countries which practise polyandry. (It is one-many +in monogamous as well as in polygamous countries, because, according +to the definition, nothing is fixed as to the number of relata for a +given referent, and there <i>may</i> be only one relatum for each given +referent without the relation ceasing to be one-many according to the +definition.) The relation in algebra of \(x^{2}\) to \(x\) is one-many, +but that of \(x\) to \(x^{2}\) is not, because there are two different +values of \(x\) that give the same value of \(x^{2}\).</p> + +<p>When a relation \(R\) is one-many, \(Rʻy\) exists whenever +\(y\in\text{ᗡ}ʻ R\), and vice versa; <i>i.e.</i> we have +\[ +R \in \text{one-many}.\equiv:y\in\text{ᗡ}ʻ R.\supset_{y}.\text{E}!Rʻ y. +\]</p> + +<p>Thus relations which give descriptive functions that are existent +whenever their arguments belong to the converse domains of the +relations in question are one-many relations. Hence \(\text{Cnv}\), +\(\text{D}\), \(\text{ᗡ}\), \(C\), \(\overrightarrow{R}\), +\(\overleftarrow{R}\), \(\text{sg}\), \(\text{gs}\), \(R_{\epsilon}\), +\(p\), \(s\), \(\dot{p}\), \(\dot{s}\), \(I\), \(\iota\), +\(\breve{\iota}\), \(\text{Cl}\), \(\text{Rl}\) are all of them +one-many relations.</p> + +<p>When \(R\) is a one-many relation, \(Rʻy\) is a one-valued function; +conversely, every one-valued function is derivable from a one-many +relation. A <i>many</i>-valued function of \(y\) is a member of +\(\overrightarrow{R}ʻy\), where \(\overrightarrow{R}ʻy\) is not a +unit class, and any one of its members is regarded as a value of the +function for the argument y; but a <i>one</i>-valued function of \(y\) +is the single term \(Rʻy\) which is obtained when \(R\) is one-many. +Thus for example the sine would, in our notation, appear as a relation, +<i>i.e.</i> we should put +\[ +\begin{aligned} +\text{sin} &= \hat{x}\hat{y}\left\{x=y - \frac{y^{3}}{3!} + \frac{y^{5}}{5!} - \ldots\right\} \qquad\text{Df},\\ +\text{whence}\qquad\text{sin}ʻy &= y - \frac{y^{3}}{3!} + \frac{y^{5}}{5!} - \ldots, +\end{aligned} +\]<span class="pagenum" id="Page_438">[Pg 438]</span> +so that "\(\sin ʻy\)" has the usual meaning of \(\sin y\). Then +instead of \(\sin^{-1}x\), we should have \(\overleftarrow{\sin}ʻx\), +which would be the class of values of \(\sin^{-1}x\); and instead of +"\(y = \sin^{-1}x\)," which is a misleading notation because +\(y = \sin^{-1}x\) and \(z = \sin^{-1}x\) do not imply \(y = z\), we should +have \(y \in \overleftarrow{\sin}ʻx\). Similar remarks would apply to +any of the other functions that occur in analysis.</p> + +<p>A relation \(R\) is called <i>many-one</i> when, if \(x\) is any member +of \(\text{D}ʻR\), there is one, and only one, term \(y\) to which +\(x\) has the relation \(R\), <i>i.e.</i> \(\overleftarrow{R}ʻx \in +1\). Thus many-one relations are the converses of one-many relations. +When a relation \(R\) is many-one, \(\breve{R}ʻx\) exists whenever \(x +\in \text{D}ʻR\).</p> + +<p>A relation is called <i>one-one</i> when it is both one-many and +many-one, or, what comes to the same, when both it and its converse are +one-many. Of the one-many relations above enumerated, \(\text{Cnv}\), +\(\text{sg}\), \(\text{gs}\), \(I\), \(\iota\), \(\breve{\iota}\), +\(\text{Cl}\), \(\text{Rl}\) are one-one.</p> + +<p>Two classes \(\alpha\), \(\beta\) are said to be <i>similar</i> when +there is a one-one relation \(R\) such that \(\text{D}ʻR = \alpha . \text{ᗡ}ʻR += \beta\), <i>i.e.</i> when their terms can be connected one to one, +so that no term of either is omitted or repeated. We write "\(\alpha +\mathop{\text{ sm }} \beta\)" for "\(\alpha\) is similar to \(\beta\)." +When two classes are similar, the cardinal numbers of their terms are +the same; it is this fact chiefly that makes one-one relations of +fundamental importance in cardinal arithmetic.</p> + +<p>According to the above, a relation is one-many when +\[ +y \in \text{ᗡ}ʻR .\supset_{y}. \overrightarrow{R}ʻy \in 1\text{,} +\] +\[ +\textit{i.e.} ~ \text{when} \quad \overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset 1. +\]</p> + +<p>Similarly a relation is many-one when +\[ +\overleftarrow{R}ʻʻ\text{D}ʻR \subset 1\text{,} +\] +and a relation is one-one when both conditions are +fulfilled. The classes \(\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\), +\(\overleftarrow{R}ʻʻ\text{D}ʻR\), which appear here, are often +important; some of their properties have already been given in +*37·77·771·772·773 and in <a href="#*53·61">*53·61</a> to <a href="#*53·641">*53·641</a>.</p> + +<p>It is convenient to regard one-many, many-one and one-one relations +as particular cases of relations which, for some given \(\alpha\) and +\(\beta\), have +\[ +\overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset \alpha . \overleftarrow{R}ʻʻ\text{D}ʻR \subset \beta\text{.} +\] +\[ +\text{We put} \quad \alpha \rightarrow \beta = \hat{R}\{\overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset \alpha.\overleftarrow{R}ʻʻ\text{D}ʻR \subset \beta\} \quad \text{Df}\text{.} +\]</p> + +<p>Hence, without a new definition, "\(1 \rightarrow 1\)" becomes the +class of one-one relations; also, as will be shown, "\(1 \rightarrow +\text{Cls}\)" becomes the class of one-many relations, and +"\(\text{Cls} \rightarrow 1\)" becomes the class of many-one relations. +Although it is chiefly these three special values of \(\alpha\rightarrow \beta\) +that are important, we shall begin by a general study of classes of +relations of the form \(\alpha \rightarrow \beta\).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_439">[Pg 439]</span></p> +<h2 class="nobreak" id="*70">*70. RELATIONS WHOSE CLASSES OF REFERENTS AND OF RELATA +BELONG TO GIVEN CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *70.</p> + +<p>If \(\alpha\) and \(\beta\) are two given classes of classes, a +relation \(R\) is said to belong to the class \(\alpha \rightarrow\beta\) +if \(\overrightarrow{R}ʻy \in \alpha\) whenever \(y \in\text{ᗡ}ʻR\), +and \(\overleftarrow{R}ʻx \in \beta\) whenever \(x\in \text{D}ʻR\). +If only one of these conditions is to be imposed, this result is +secured by replacing the class involved in the other condition by +"\(\text{Cls}\)," since "\(\overrightarrow{R}ʻy \in \text{Cls}\)" +always holds, and so does "\(\overleftarrow{R}ʻx \in \text{Cls}\)," +and therefore neither imposes any limitation on \(R\). In the most +important cases, \(\alpha\) and \(\beta\) are either both cardinal +numbers, or one is a cardinal number while the other is \(\text{Cls}\).</p> + +<p>In virtue of *37·702·703, the conditions above mentioned as imposed +upon \(R\) by membership of \(\alpha \rightarrow \beta\) are equivalent +to +\[ +\overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset \alpha.\overleftarrow{R}ʻʻ\text{D}ʻR \subset \beta\text{.} +\]</p> + +<p>This form is used in the definition (<a href="#*70·01">*70·01</a>).</p> + +<p>The propositions of the present number are hardly ever used except +in <a href="#*71">*71</a>, where \(\alpha\) and \(\beta\) are both replaced by \(1\) or +\(\text{Cls}\). The most useful propositions are</p> + +<p class="nind"><b>*70·1.</b> \(\vdash: R \in \alpha \rightarrow \beta .\equiv. \overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset \alpha . \overleftarrow{R}ʻʻ\text{D}ʻR \subset\beta\)</p> + +<p>(This merely embodies the definition.)</p> + +<p class="nind"><b>*70·13.</b> \(\vdash \colon\ldotp R \in \alpha \rightarrow \beta .\equiv: (y).\overrightarrow{R}ʻy \in \alpha \cup \iotaʻ\Lambda:(x).\overleftarrow{R}ʻx \in \beta \cup \iotaʻ\Lambda\)</p> + +<p class="nind"><b>*70·22.</b> \(\vdash. \beta \rightarrow \alpha = \text{Cnv}ʻʻ(\alpha \rightarrow \beta)\)</p> + +<p class="nind"><b>*70·4.</b> \(\vdash . \alpha \rightarrow \text{Cls} = \hat{R}(\overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset \alpha)\)</p> + +<p class="nind"><b>*70·41.</b> \(\vdash . \text{Cls} \rightarrow \beta = \hat{R}(\overleftarrow{R}ʻʻ\text{D}ʻR \subset \beta)\)</p> + +<p class="nind"><b>*70·42.</b> \(\vdash . \alpha \rightarrow \beta = (\alpha \rightarrow \text{Cls}) \cap (\text{Cls} \rightarrow \beta)\)</p> + +<p class="nind"><b>*70·54.</b> \(\vdash:\text{ᗡ}ʻR \cap \text{ᗡ}ʻS = \Lambda.R,\,S \in \alpha \rightarrow \text{Cls} .\supset. R \unicode{x228d} S \in \alpha \rightarrow \text{Cls}\)</p> + +<p>with similar propositions for \(\text{Cls} \rightarrow \beta\) and \(\alpha \rightarrow \beta\).</p> + +<p class="nind"><b>*70·62.</b> \(\vdash: R \in \alpha \rightarrow \text{Cls} .\supset. R \upharpoonright \gamma \in \alpha \rightarrow \text{Cls}\)</p> + +<p>with a similar proposition for \(\text{Cls} \rightarrow \beta\).</p> + +<p><span class="pagenum" id="Page_440">[Pg 440]</span></p> + +<hr class="tb"> + +<p class="nind"><b><a id="*70·01">*70·01</a>.</b> \(\alpha\rightarrow\beta = \breve{R} (\overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset\alpha. \overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta) \quad\text{Df}\)</p> + +<p class="nind"><b>*70·1.</b> \(\vdash\colon R \in \alpha\rightarrow\beta.\equiv. \overrightarrow{R}ʻʻ\text{ᗡ}ʻR \subset\alpha. \overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta \quad[\text{*20·3.(*70·01)}]\)</p> + +<p class="nind"><b>*70·11.</b> + \[\begin{align}\vdash\colon\ldotp R \in\alpha\rightarrow\beta.\equiv: y\in\text{ᗡ}ʻR. \supset_{y}.\overrightarrow{R}ʻy\in\alpha:x\in\text{D}ʻR.&\supset_{x}.\overleftarrow{R}ʻx\in\beta\\ +&[\text{*37·702·703.*70·1}]\end{align}\]</p> + +<p class="nind"><b><a id="*70·12">*70·12</a>.</b> + \(\vdash\colon R \in\alpha\rightarrow\beta.\equiv. \overrightarrow{R}ʻʻ\text{V} \subset \alpha\cup\iotaʻ\Lambda.\overleftarrow{R}ʻʻ\text{V} \subset \beta\cup\iotaʻ\Lambda \quad[\text{*70·1.*53·62·621}]\)</p> + +<p class="nind"><b>*70·13.</b> \(\vdash\colon\ldotp R \in\alpha\rightarrow\beta.\equiv: (y). \overrightarrow{R}ʻy\in \alpha\cup\iotaʻ\Lambda : (x). \overleftarrow{R}ʻx\in\beta\cup\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*37·702}.&\supset\vdash\colon\ldotp \overrightarrow{R}ʻʻ\text{V} \subset \alpha\cup\iotaʻ\Lambda . \equiv: y \in\text{V} . \supset_{y}. + \overrightarrow{R}ʻy\in\alpha\cup\iotaʻ\Lambda:\\ +[\text{*24·104.*5·5}] &\equiv: (y). \overrightarrow{R}ʻy \in \alpha\cup\iotaʻ\Lambda &\qquad \text{(1)}\\ +\text{Similarly} &\vdash\colon\ldotp \overleftarrow{R}ʻʻ\text{V} \subset \beta\cup\iotaʻ\Lambda . \equiv : (x).\overleftarrow{R}ʻx \in\beta\cup\iotaʻ\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*70·12}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·14.</b> + \[\begin{align}&\vdash\colon\colon R\in\alpha\rightarrow\beta. \equiv \colon\ldotp (y): \overrightarrow{R}ʻy\in\alpha .\lor. \overrightarrow{R}ʻy=\Lambda\colon\ldotp(x):\overleftarrow{R}ʻx\in\beta.\lor.\overleftarrow{R}ʻx=\Lambda\\ +&[\text{*70·13.*51·236}]\end{align}\]</p> + +<p class="nind"><b>*70·15.</b> \[\begin{align}&\vdash\colon\ldotp R\in\alpha\rightarrow\beta.\equiv: \exists!\overrightarrow{R}ʻy. \supset_{y}.\overrightarrow{R}ʻy\in\alpha + : \exists! \overleftarrow{R}ʻx .\supset_{x} .\overleftarrow{R}ʻx\in\beta\\ +&[\text{*24·51.*4·6.*70·14}]\end{align}\]</p> + +<p class="nind"><b>*70·16.</b> + \(\vdash\colon R\in\alpha\rightarrow\beta.\equiv. \text{D}ʻ\overrightarrow{R} \subset \alpha\cup\iotaʻ\Lambda.\text{D}ʻ\overleftarrow{R} \subset \beta\cup\iotaʻ\Lambda \quad[\text{*37·78·781.*70·12}]\)</p> + +<p class="nind"><b>*70·17.</b> \(\vdash\colon\colon \Lambda\in\alpha .\supset\colon\ldotp R\in\alpha\rightarrow\beta.\equiv: (y). \overrightarrow{R}ʻy\in\alpha: \exists!\overleftarrow{R}ʻx.\supset_{x}. + \overleftarrow{R}ʻx\in\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·2.*22·62}.&\supset\vdash\colon\text{Hp}. \supset . \alpha = \alpha\cup\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*70·13}. &\supset\\ +&\vdash\colon\colon\text{Hp}. \supset\colon\ldotp R\in\alpha\rightarrow\beta. \equiv : (y).\overrightarrow{R}ʻy\in\alpha: (x).\overleftarrow{R}ʻx\in\beta\cup\iotaʻ\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*51·236}. &\supset\vdash\colon\ldotp \overleftarrow{R}ʻx\in\beta\cup\iotaʻ\Lambda . \equiv : \overleftarrow{R}ʻx\in\beta . \lor . \overleftarrow{R}ʻx=\Lambda:\\ +[\text{*24·51.*4·6}] &\equiv: \exists!\overleftarrow{R}ʻx.\supset . \overleftarrow{R}ʻx\in\beta &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·171.</b> \[\begin{align}&\vdash\colon\colon \Lambda\in\beta.\supset\colon\ldotp R\in\alpha\rightarrow\beta .\equiv: \exists! \overrightarrow{R}ʻy.\supset_{y}.\overrightarrow{R}ʻy\in\alpha: + (x). \overleftarrow{R}ʻx\in\beta\\ +&[\text{Proof as in *70·17}]\end{align}\]</p> + +<p class="nind"><b>*70·18.</b> + \[\begin{align}&\vdash\colon\colon \Lambda\in\alpha.\Lambda\in\beta.\supset\colon\ldotp R\in\alpha\rightarrow\beta .\equiv:(y). \overrightarrow{R}ʻy\in\alpha:(x).\overleftarrow{R}ʻx\in\beta\\ +&[\text{Proof as in *70·17}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_441">[Pg 441]</span></p> + +<p class="nind"><b>*70·2.</b> \(\vdash.\alpha\rightarrow\beta=(\alpha\cup\iotaʻ\Lambda)\rightarrow\beta=\alpha\rightarrow(\beta\cup\iotaʻ\Lambda)=(\alpha\cup\iotaʻ\Lambda)\rightarrow(\beta\cup\iotaʻ\Lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·58·62}.\supset\vdash.(\alpha\cup\iotaʻ\Lambda)\cup\iotaʻ\Lambda=\alpha\cup\iotaʻ\Lambda.(\beta\cup\iotaʻ\Lambda)\cup\iotaʻ\Lambda=\beta\cup\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*70·12.(1)}.\supset\vdash:R\in\alpha\rightarrow\beta.\equiv.\overrightarrow{R}ʻʻ\text{V}\subset(\alpha\cup\iotaʻ\Lambda)\cup\iotaʻ\Lambda.\overleftarrow{R}ʻʻ\text{V}\subset\beta\cup\iotaʻ\Lambda.\\ +[\text{*70·12}] \equiv.R\in(\alpha\cup\iotaʻ\Lambda)\rightarrow\beta. &\qquad \text{(2)}\\ +[\text{*70·12.(1)}] \equiv.\overrightarrow{R}ʻʻ\text{V}\subset(\alpha\cup\iotaʻ\Lambda)\cup\iotaʻ\Lambda.\overleftarrow{R}ʻʻ\text{V}\subset(\beta\cup\iotaʻ\Lambda)\cup\iotaʻ\Lambda.\\ +[\text{*70·12}] \equiv.R\in(\alpha\cup\iotaʻ\Lambda)\rightarrow(\beta\cup\iotaʻ\Lambda). &\qquad \text{(3)}\\ +[\text{*70·12.(1)}] \equiv.\overrightarrow{R}ʻʻ\text{V}\subset\alpha\cup\iotaʻ\Lambda.\overleftarrow{R}ʻʻ\text{V}\subset(\beta\cup\iotaʻ\Lambda)\cup\iotaʻ\Lambda.\\ +[\text{*70·12}] \equiv.R\in\alpha\rightarrow(\beta\cup\iotaʻ\Lambda) &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·21.</b> \(\vdash.\alpha\rightarrow\beta=(\alpha-\iotaʻ\Lambda)\rightarrow\beta=\alpha\rightarrow(\beta-\iotaʻ\Lambda)=(\alpha-\iotaʻ\Lambda)\rightarrow(\beta-\iotaʻ\Lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·222}.\supset\vdash\colon\Lambda{\sim}\in\alpha.\supset.\alpha-\iotaʻ\Lambda=\alpha:\Lambda{\sim}\in\beta.\supset.\beta-\iotaʻ\Lambda=\beta &\qquad \text{(1)}\\ +\vdash.\text{*51·221}.\supset\vdash\colon\Lambda\in\alpha.\supset.(\alpha-\iotaʻ\Lambda)\cup\iotaʻ\Lambda=\alpha:\Lambda\in\beta.\supset.(\beta-\iotaʻ\Lambda)\cup\beta=\beta &\qquad \text{(2)}\\ +\vdash.\text{(1)}.\supset\\ +\vdash:\Lambda{\sim}\in\alpha.\supset.(\alpha-\iotaʻ\Lambda)\rightarrow\beta=\alpha\rightarrow\beta.(\alpha-\iotaʻ\Lambda)\rightarrow(\beta-\iotaʻ\Lambda)=\alpha\rightarrow(\beta-\iotaʻ\Lambda) &\qquad \text{(3)}\\ +\vdash.\text{(2).*70·2}.\supset\\ +\vdash:\Lambda\in\alpha.\supset.(\alpha-\iotaʻ\Lambda)\rightarrow\beta=\alpha\rightarrow\beta.(\alpha-\iotaʻ\Lambda)\rightarrow(\beta-\iotaʻ\Lambda)=\alpha\rightarrow(\beta-\iotaʻ\Lambda) &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).*4·83}.\supset\\ +\vdash.(\alpha-\iotaʻ\Lambda)\rightarrow\beta=\alpha\rightarrow\beta.(\alpha-\iotaʻ\Lambda)\rightarrow(\beta-\iotaʻ\Lambda)=\alpha\rightarrow(\beta-\iotaʻ\Lambda) &\qquad \text{(5)}\\ +\text{Similarly} \vdash.\alpha\rightarrow(\beta-\iotaʻ\Lambda)=\alpha\rightarrow\beta.(\alpha-\iotaʻ\Lambda)\rightarrow(\beta-\iotaʻ\Lambda)=(\alpha-\iotaʻ\Lambda)\rightarrow\beta &\qquad \text{(6)}\\ +\vdash.\text{(5).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·22.</b> \(\vdash.\beta\rightarrow\alpha=\text{Cnv}ʻʻ(\alpha\rightarrow\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·6.*31·13}.\supset\\ +\vdash\colon\ldotp Q\in\text{Cnv}ʻʻ(\alpha\rightarrow\beta).&\equiv:(\exists R).R\in\alpha\rightarrow\beta.Q=\text{Cnv}ʻR:\\ +[\text{*70·12}] &\equiv:(\exists R).\overrightarrow{R}ʻʻ\text{V}\subset\alpha\cup\iotaʻ\Lambda.\overleftarrow{R}ʻʻ\subset\beta\cup\iotaʻ\Lambda.Q=\text{Cnv}ʻR:\\ +[\text{*32·24·241}] &\equiv:(\exists R).(\text{gs}ʻ\text{Cnv}ʻR)ʻʻ\subset\alpha\cup\iotaʻ\Lambda.\\ +&(\text{gs}ʻ\text{Cnv}ʻR)ʻʻ\subset\beta\cup\iotaʻ\Lambda.Q=\text{Cnv}ʻR:\\ +[\text{*13·193}] &\equiv:(\exists R).(\text{gs}ʻQ)ʻʻ\subset\alpha\cup\iotaʻ\Lambda.\\ +&(\text{gs}ʻQ)ʻʻ\subset\beta\cup\iotaʻ\Lambda.Q=\text{Cnv}ʻR:\\ +[\text{*32·23·231.*10·35}]&\equiv:\overleftarrow{Q}ʻʻ\subset\alpha\cup\iotaʻ\Lambda.\overrightarrow{Q}ʻʻ\subset\beta\cup\iotaʻ\Lambda:(\exists R).Q=\text{Cnv}ʻR:\\ +[\text{*31·33.*10·24}] &\equiv:\overleftarrow{Q}ʻʻ\subset\alpha\cup\iotaʻ\Lambda.\overrightarrow{Q}ʻʻ\subset\beta\cup\iotaʻ\Lambda:\\ +[\text{*70·12}] &\equiv:Q\in\beta\rightarrow\alpha\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_442">[Pg 442]</span></p> + +<p class="nind"><b>*70·3.</b> \(\vdash.\alpha\subset\gamma.\beta\subset\delta.\supset.\alpha\rightarrow\beta\subset\gamma\rightarrow\delta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*70·1}.&\supset\vdash\colon\text{Hp}.R\in\alpha\rightarrow\beta.\supset.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\alpha\overleftarrow{R}ʻʻ\text{ᗡ}ʻRʻ\subset\beta.\alpha\subset\gamma.\beta\subset\delta.\\ +[\text{*22·44}] &\supset.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\gamma.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\delta.\\ +[\text{*70·1}] &\supset.R\in\gamma\rightarrow\delta &\qquad \text{(1)}\\ +\vdash.\text{(1).Exp.*10·11·21}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·31.</b> \(\vdash.(\alpha\rightarrow\beta)\cap(\gamma\rightarrow\delta)=(\alpha\cap\gamma)\rightarrow(\beta\cap\delta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*70·1}.\supset\vdash\colon &R\in(\alpha\rightarrow\beta)\cap(\gamma\rightarrow\delta).\equiv.\\ +&\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\gamma.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\delta.\\ +[\text{*22·45}] &\equiv.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha\cap\gamma.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta\cap\delta.\\ +[\text{*70·1}] &\equiv.R\in(\alpha\cap\gamma)\rightarrow(\beta\cap\delta):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·32.</b> \(\vdash.(\alpha\rightarrow\beta)\cup(\gamma\rightarrow\delta)\subset(\alpha\cup\gamma)\rightarrow(\beta\cup\delta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*70·1}.\supset\vdash\colon\ldotp &R\in(\alpha\rightarrow\beta)\cup(\gamma\rightarrow\delta).\equiv:\\ +&\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta.\lor.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\gamma.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\delta:\\ +[\text{*3·26·27·48}]&\supset:\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha.\lor.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\gamma:\overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta.\lor.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\delta:\\ +[\text{*22·65}] &\supset:\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha\cup\gamma.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta\cup\delta:\\ +[\text{*70·1}] &\supset:R\in(\alpha\cup\gamma)\rightarrow(\beta\cup\delta)\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·4.</b> \(\vdash.\alpha\rightarrow\text{Cls}=\hat{R}(\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*70·1}.\supset\vdash\colon R\in\alpha\rightarrow\text{Cls}.&\equiv.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha.\overleftarrow{R}ʻʻ\text{D}ʻR\subset\text{Cls}.\\ +[\text{*37·761}] &\equiv.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·41.</b> \(\vdash.\text{Cls}\rightarrow\beta=\hat{R}(\overleftarrow{R}ʻʻ\text{D}ʻR\subset\beta) \quad[\text{Proof as in *70·4}]\)</p> + +<p class="nind"><b><a id="*70·42">*70·42</a>.</b> \(\vdash.\alpha\rightarrow\beta=(\alpha\rightarrow\text{Cls})\cap(\text{Cls}\rightarrow\beta) \quad[\text{*70·4·41}]\)</p> + +<p class="nind"><b>*70·43.</b> \(\vdash\colon\ldotp R\in\alpha\rightarrow\text{Cls}.\equiv:y\in\text{ᗡ}ʻR.\supset_{y}.\overrightarrow{R}ʻy\in\alpha \quad[\text{As in *70·11}]\)</p> + +<p class="nind"><b>*70·431.</b> \(\vdash\colon\ldotp R\in\text{Cls}\rightarrow\beta.\equiv:x\in\text{D}ʻR.\supset_{x}.\overleftarrow{R}ʻx\in\beta \quad[\text{As in *70·11}]\)</p> + +<p class="nind"><b>*70·44.</b> \(\vdash:R\in\alpha\rightarrow\text{Cls}.\equiv.\overrightarrow{R}ʻʻ\text{V}\subset\alpha\cup\iotaʻ\Lambda \quad[\text{As in *70·12}]\)</p> + +<p class="nind"><b>*70·441.</b> \(\vdash:R\in\text{Cls}\rightarrow\beta.\equiv.\overleftarrow{R}ʻʻ\text{V}\subset\beta\cup\iotaʻ\Lambda \quad[\text{As in *70·12}]\)</p> + +<p class="nind"><b>*70·45.</b> \(\vdash:R\in\alpha\rightarrow\text{Cls}.\equiv.(y).\overrightarrow{R}ʻy\in\alpha\cup\iotaʻ\Lambda \quad[\text{As in *70·13}]\)</p> + +<p><span class="pagenum" id="Page_443">[Pg 443]</span></p> + +<p class="nind"><b>*70·451.</b> \(\vdash:R\in\text{Cls}\rightarrow\beta.\equiv.(x).\overleftarrow{R}ʻx\in\beta\cup\iotaʻ\Lambda \quad[\text{As in *70·13}]\)</p> + +<p class="nind"><b>*70·46.</b> \(\vdash\colon\ldotp R\in\alpha\rightarrow\text{Cls}.\equiv:(y):\overrightarrow{R}ʻy\in\alpha.\lor.\overrightarrow{R}ʻy=\Lambda \quad[\text{As in *70·14}]\)</p> + +<p class="nind"><b>*70·461.</b> \(\vdash\colon\ldotp R\in\text{Cls}\rightarrow\beta.\equiv:(x):\overleftarrow{R}ʻx\in\beta.\lor.\overleftarrow{R}ʻx=\lambda \quad[\text{As in *70·14}]\)</p> + +<p class="nind"><b>*70·47.</b> \(\vdash\colon\ldotp R\in\alpha\rightarrow\text{Cls}.\equiv:\exists!\overrightarrow{R}ʻy.\supset_{y}.\overrightarrow{R}ʻy\in\alpha \quad[\text{As in *70·15}]\)</p> + +<p class="nind"><b>*70·471.</b> \(\vdash\colon\ldotp R\in\text{Cls}\rightarrow\beta.\equiv:\exists!\overleftarrow{R}ʻx.\supset_{x}.\overleftarrow{R}ʻx\in\beta \quad[\text{As in *70·15}]\)</p> + +<p class="nind"><b>*70·48.</b> \(\vdash\colon R\in\alpha\rightarrow\text{Cls}.\equiv.\text{D}ʻ\overrightarrow{R}\subset\alpha\cup\iotaʻ\Lambda \quad[\text{As in *70·16}]\)</p> + +<p class="nind"><b>*70·481.</b> \(\vdash\colon R\in\text{Cls}\rightarrow\beta.\equiv.\text{D}ʻ\overrightarrow{R}\subset\beta\cup\iotaʻ\Lambda \quad[\text{As in *70·16}]\)</p> + +<p class="nind"><b><a id="*70·5">*70·5</a>.</b> \(\vdash.\text{Cls}\rightarrow\alpha=\text{Cnv}ʻʻ(\rightarrow\alpha\text{Cls}).\alpha\rightarrow\text{Cls}=\text{Cnv}ʻʻ(\text{Cls}\rightarrow\alpha) \quad[\text{*70·22}]\)</p> + +<p class="nind"><b>*70·51.</b> + \(\vdash\colon\ldotp \xi,\eta\in\alpha.\supset_{\xi,\eta}.\xi\cap\eta\in\alpha\cup\iotaʻ\Lambda:\supset:R,S\in\rightarrow\alpha\text{Cls}.\supset.R\dot{\cap}S\in\alpha\rightarrow\text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·3}.&\vdash\colon\ldotp\text{Hp}.\supset:\overrightarrow{R}ʻy\in\alpha.\overrightarrow{S}ʻy\in\alpha.\supset.\{\text{sg}ʻ(R\dot{\cap}S)\}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*32·3.*51·15.*24·34}.&\supset\\ +&\vdash:\overrightarrow{R}ʻy\in\alpha.\overrightarrow{S}ʻy\in\iotaʻ\Lambda.\supset.\{\text{sg}ʻ(R\dot{\cap}S)\}ʻy=\Lambda.\\ +[\text{*51·236}] &\supset.\{\text{sg}ʻ(R\dot{\cap}S)\}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·4}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:\overrightarrow{R}ʻy\in\alpha.\overrightarrow{S}ʻy\in\in\alpha\cup\iotaʻ\Lambda.\supset.\{\text{sg}ʻ(R\dot{\cap}S)\}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(3)}\\ +\vdash.\text{*32·3.*51·15.*24·34}.&\supset\\ +&\vdash:\overrightarrow{R}ʻy\in\iotaʻ\Lambda.\overrightarrow{S}ʻy\in\alpha\cup\iotaʻ\Lambda.\supset.\{\text{sg}ʻ(R\dot{\cap}S)\}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).*4·4}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:\overrightarrow{R}ʻy,\overrightarrow{S}ʻy\in\in\alpha\cup\iotaʻ\Lambda.\supset.\{sgʻ(R\dot{\cap}S)\}ʻy\in\alpha\cup\iotaʻ\Lambda:\\ +[\text{*10·11·21·27.*70·45}] &\supset:R,S\in\alpha\rightarrow\text{Cls}.\supset.(y).\{\text{sg}ʻ(R\dot{\cap}S)\}ʻy\in\alpha\cup\iotaʻ\Lambda.\\ +[\text{*70·45.*32·23}] &\supset.R\dot{\cap}S\in\alpha\rightarrow\text{Cls}\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·52.</b> + \[\begin{align}&\vdash\colon\ldotp\xi,\eta\in\beta.\supset_{\xi,\eta}.\xi\cap\eta\in\beta\cup\iotaʻ\Lambda:\supset:R,\,S\in\text{Cls}\rightarrow\beta.\supset.R\dot{\cap}S\in\text{Cls}\rightarrow\beta\\ +&[\text{Proof as in *70·51}]\end{align}\]</p> + +<p class="nind"><b>*70·53.</b> + \[\begin{align}\vdash\colon\ldotp\xi,\eta\in\alpha.\supset_{\xi,\eta}.\xi\cap\eta\in\alpha\cup\iotaʻ\Lambda:\xi,\eta\in&\beta.\supset_{\xi,\eta}.\xi\cap\eta\in\beta\cup\iotaʻ\Lambda:\supset:\\ +&R,S\in\alpha\rightarrow\beta.\supset.R\dot{\cap}S\in\alpha\rightarrow\beta\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*70·5·51}.\supset\vdash\colon\ldotp\text{Hp}.\supset:R,\,S\in\alpha\rightarrow&\text{Cls}.R,S\in\text{Cls}\rightarrow\beta.\supset.\\ +&R\dot{\cap}S\in\alpha\rightarrow\text{Cls}.R\dot{\cap}S\in\text{Cls}\rightarrow\beta \qquad \text{(1)}\\ +\vdash.\text{(1).*70·42}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_444">[Pg 444]</span></p> + +<p class="nind"><b>*70·54.</b> \(\vdash:\text{ᗡ}ʻR\cap\text{ᗡ}ʻS=\lambda.R,S\in\alpha\rightarrow\text{Cls}.\supset.R\unicode{x228d}S\in\alpha\rightarrow\text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·15.*22·33}.\supset\\ +\vdash\colon\ldotp\text{ᗡ}ʻR\cap\text{ᗡ}ʻS=\Lambda.\supset:(y):{\sim}\{y\in\text{ᗡ}ʻR.y\in\text{ᗡ}ʻS\}:\\ +[\text{*33·41}] &\supset:(y):{\sim}\{\exists!\overrightarrow{R}ʻy.\exists!\overrightarrow{S}ʻy\}:\\ +[\text{*4·51.*24·51}]&\supset:(y):\overrightarrow{R}ʻy=\Lambda.\lor.\overrightarrow{S}ʻy=\Lambda:\\ +[\text{*24·36}] &\supset:(y):\overrightarrow{R}ʻy\cup\overrightarrow{S}ʻy=\overrightarrow{S}ʻy.\lor.\overrightarrow{R}ʻy\cup \overrightarrow{S}ʻy=\overrightarrow{R}ʻy &\qquad \text{(1)}\\ +\vdash.*70·45.\supset\\ +\vdash\colon\ldotp R,\,S\in\alpha\rightarrow\text{Cls}.&\supset:(y).\overrightarrow{R}ʻy\in\alpha\cup\iotaʻ\Lambda:(y).\overrightarrow{S}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:(y).\overrightarrow{R}ʻy\cup\overrightarrow{S}ʻy\in\alpha\cup\iotaʻ\Lambda:\\ +[\text{*32·32}] &\supset:(y).\{\text{sg}ʻ(R\unicode{x228d}S)\}ʻy\in\alpha\cup\iotaʻ\Lambda:\\ +[\text{*70·45}] &\supset:R\unicode{x228d}S\in\alpha\rightarrow\text{Cls}\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·55.</b> \[\begin{align}&\vdash:\text{D}ʻR\cap\text{D}ʻS=\Lambda.R,S\in\text{Cls}\rightarrow\beta.\supset.R\unicode{x228d}S\in\text{Cls}\rightarrow\beta\\ +&[\text{Proof as in *70·54}]\end{align}\]</p> + +<p class="nind"><b>*70·56.</b> \[\begin{align}&\vdash:\text{D}ʻR\cap\text{D}ʻS=\Lambda.\text{ᗡ}ʻR\cap\text{ᗡ}ʻS=\Lambda.R,S\in\alpha\rightarrow\beta.\supset.R\unicode{x228d}S\in\alpha\rightarrow\beta\\ +&[\text{*70·54·55·42}]\end{align}\]</p> + +<p class="nind"><b>*70·57.</b> \(\vdash:CʻR\cap\text{C}ʻS=\Lambda.R,\,S\in\alpha\rightarrow\beta.\supset.R\unicode{x228d}S\in\alpha\rightarrow\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·161}.&\supset\vdash.\text{D}ʻR\cap\text{D}ʻS\subset CʻR\cap CʻS.\text{ᗡ}ʻR\cap\text{ᗡ}ʻS\subset CʻR\cap CʻS.\\ +[\text{*24·13}] &\supset\vdash:CʻR\cap CʻS=\Lambda.\supset.\text{D}ʻR\cap\text{D}ʻS=\Lambda.\text{ᗡ}ʻR\cap\text{ᗡ}ʻS=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*70·56}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*70·6">*70·6</a>.</b> \(\vdash:S\in\alpha\rightarrow\text{Cls}.Rʻʻʻ\alpha\supset\alpha\cup\iotaʻ\Lambda.\supset.R\mid S\in\alpha\rightarrow\text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·31}. &\supset\vdash.\{\text{sg}ʻ(R\mid S)\}ʻʻ\text{V}=(R_{{\in}}\mid \overrightarrow{S})ʻʻ\text{V}\\ +[\text{*37·33}] &=R_{{\in}}ʻʻ\overrightarrow{S}ʻʻV &\qquad \text{(1)}\\ +\vdash.\text{(1).*70·44}.&\supset\vdash:S\in\alpha\rightarrow\text{Cls}.\supset.\{\text{sg}ʻ(R\mid S)\}ʻʻ\text{V}\subset R_{{\in}}ʻʻ(\alpha\cup\iotaʻ\Lambda) + &\qquad \text{(2)}\\ +\vdash.\text{*37·22}. & \supset\vdash.R_{{\in}}ʻʻ(\alpha\cup\iotaʻ\Lambda)=R_{{\in}}ʻʻ\alpha\cup R_{{\in}}ʻʻ\iotaʻ\Lambda\\ +[\text{*53·31}] &=R_{{\in}}ʻʻ\alpha\cup\iotaʻR_{{\in}}ʻ\Lambda\\ +[\text{(*37·04).*37·11·29}] &=Rʻʻʻ\alpha\cup\iotaʻ\Lambda &\qquad \text{(3)}\\ +\vdash.\text{(3).*22·66}.&\supset\vdash:Rʻʻʻ\alpha\supset\alpha\cup\iotaʻ\Lambda.\supset.R_{{\in}}ʻʻ(\alpha\cup\iotaʻ\Lambda)\supset\alpha\cup\iotaʻ\Lambda\cup\iotaʻ\Lambda.\\ +[\text{*22·56}] &\supset.R_{{\in}}ʻʻ(\alpha\cup\iotaʻ\Lambda)\supset\alpha\cup\iotaʻ\Lambda &\qquad \text{(4)}\\ +\vdash.\text{(2).(4)}. &\supset\vdash\supset\vdash:\text{Hp}.\supset.\{\text{sg}ʻ(R\mid S)\}ʻʻ\text{V}\supset\alpha\cup\iotaʻ\Lambda.\\ +[\text{*70·44}] &\supset.R\mid S\in\alpha\rightarrow\text{Cls}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_445">[Pg 445]</span></p> + +<p class="nind"><b><a id="*70·61">*70·61</a>.</b> \(\vdash:R\in\text{Cls}\rightarrow\beta.\breve{S}ʻʻʻ\beta\subset\beta\cup\iotaʻ\Lambda.\supset.R\mid S\in\text{Cls}\rightarrow\beta \quad[\text{As in *70·6}]\)</p> + +<p class="nind"><b>*70·62.</b> \(\vdash:R\in\alpha\rightarrow\text{Cls}.\supset.R\upharpoonright\in\alpha\rightarrow\text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·64.Transp}.&\supset\vdash:y{\sim}\in\gamma.\supset.y{\sim}\in\text{ᗡ}ʻ(R\upharpoonright\gamma).\\ +[\text{*33·41.*24·51}] &\supset.\{\text{sg}ʻ(R\upharpoonright\gamma)\}ʻy=\Lambda.\\ +[\text{*51·236}] &\supset.\{\text{sg}ʻ(R\upharpoonright\gamma)\}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*35·101.*4·73}.&\supset\vdash\colon\ldotp y\in\gamma.\supset:x(R\upharpoonright\gamma)y.\equiv_{x}.xRy:\\ +[\text{*20·15.*32·13·23}] &\supset:\{\text{sg}ʻ(R\upharpoonright\gamma)\}ʻy=\overrightarrow{R}ʻy &\qquad \text{(2)}\\ +\vdash.\text{*70·45}. &\supset\vdash:\text{Hp}.\supset.\overrightarrow{R}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash\colon\ldotp\text{Hp}.\supset:y\in\gamma.\supset.\{\text{sg}ʻ(R\upharpoonright\gamma)\}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*4·83}.&\supset\vdash:\text{Hp}.\supset.\{\text{sg}ʻ(R\upharpoonright\gamma)\}ʻy\in\alpha\cup\iotaʻ\Lambda &\qquad \text{(5)}\\ +\vdash.\text{(5).*10·11·21.*70·45}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*70·63.</b> \(\vdash:R\in\text{Cls}\rightarrow\beta.\supset.\delta\upharpoonleft R\in\text{Cls}\rightarrow\beta \quad[\text{As in *70·62}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_446">[Pg 446]</span></p> +<h2 class="nobreak" id="*71">*71. ONE-MANY, MANY-ONE, AND ONE-ONE RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *71.</p> + +<p>In this number we shall be concerned with the more elementary +properties of one-many, many-one, and one-one relations. These +properties are very numerous and very important. The properties of +many-one relations (<i>i.e.</i> of relations belonging to the class +\(\text{Cls} \rightarrow 1\)) result from those of one-many relations +by means of <a href="#*70·5">*70·5</a>, whence it follows that many-one relations are +the converses of one-many relations. It is thus only necessary to +interchange \(R\) and \(\breve{R}\), \(\text{D}\) and \(\text{ᗡ}\), +\(\overrightarrow{R}\) and \(\overleftarrow{R}\) in order to obtain +a property of a many-one relation from a property of a one-many +relation. Or we may repeat the various steps of any proof, making the +above interchanges at every step, and the analogous proposition will +result. For this reason, in what follows, we shall omit all proofs of +properties of many-one relations, confining ourselves to proving the +analogous properties of one-many relations.</p> + +<p>In virtue of <a href="#*70·42">*70·42</a>, one-one relations (<i>i.e.</i> relations belonging +to the class \(1 \rightarrow 1\)) are the relations which are both +one-many and many-one; hence their properties result from combining the +properties of one-many and many-one relations. We shall omit the proofs +when they consist merely in such combinations.</p> + +<p>A one-many relation gives rise to a descriptive function which is +existent whenever its argument belongs to the converse domain of +the relation. That is, if \(R \in 1 \rightarrow \text{Cls}\), we +have \(\text{E}!Rʻy\) whenever \(y \in \text{ᗡ}ʻR\). Conversely, if +a descriptive function \(Rʻy\) exists for the argument \(y\), then +\(R\) is one-many so far as that argument is concerned, <i>i.e.</i> +\(\overrightarrow{R}ʻy \in 1\). Thus we find +\[ +R \in 1 \rightarrow \text{Cls} .\equiv. \text{E}‼Rʻʻ\text{ᗡ}ʻR\text{.} +\]</p> + +<p>The descriptive function \(Rʻy\) derived from a one-many relation +\(R\) has thus a definite value whenever \(y \in \text{ᗡ}ʻR\), and +not otherwise. Thus the class of arguments for which such a function +exists is the converse domain of the relation which gives rise to the +function, <i>i.e.</i> +\[ +R \in 1 \rightarrow \text{Cls} .\supset. \hat{y}\{\text{E}!Rʻy\} = \text{ᗡ}ʻR\text{,} +\] +and the converse implication also holds.</p> + +<p>It often happens that a relation which is not in general +one-many becomes so when its domain, converse domain, or field +is subjected to some limitation. For example, let \(R\) be the +relation of parent to child, \(\alpha\) the class of males, and<span class="pagenum" id="Page_447">[Pg 447]</span> +\(\beta\) the class of females. Then \(R\) is not one-many, but +\(\alpha\upharpoonleft R\) and \(\beta\upharpoonleft R\) are one-many, +and in fact (\(\alpha\upharpoonleft R)ʻy\) = the father of \(y\), +(\(\beta\upharpoonleft R)ʻy\) = the mother of \(y\). We shall often +have occasion to deal with relations obtained by limitations imposed +on \(\text{D}\) or \(\text{ᗡ}\); thus \(\alpha(\text{D}\upharpoonright\lambda)R.\equiv.R\) +belongs to the class \(\lambda\), and has +\(\alpha\) for its domain. The class \(\lambda\) may be so constituted +that only one relation \(R\) fulfils this condition; in that +case, \(\text{D}\upharpoonright \lambda \in \text{Cls}\rightarrow 1\). +Since \(\text{D} \in 1\rightarrow \text{Cls}\), we find +\(\text{D}\upharpoonright \lambda \in \text{Cls}\rightarrow 1.\equiv.\text{D}\upharpoonright \lambda \in 1\rightarrow 1\). +This type of condition, \(\text{D}\upharpoonright \lambda \in 1\rightarrow 1\) +or \(\text{ᗡ}\upharpoonright \lambda \in 1\rightarrow 1\) +or \(C\upharpoonright \lambda \in 1\rightarrow 1\), is one which +frequently occurs in subsequent work. Another condition which often +occurs is \(F\upharpoonright \lambda \in \text{Cls}\rightarrow 1\). +When this condition is realized, a term \(x\) which belongs to the +field of one relation of the class \(\lambda\) does not belong to the +field of any other relation of this class, <i>i.e.</i> the fields of +relations of this class are mutually exclusive.</p> + +<p>For purposes of realizing imaginatively the properties of one-many +relations, it is often convenient to picture their structure as in the +accompanying figure.</p> + +<figure class="figcenter width500" id="i_447" style="width: 471px;"> +<img src="images/i_447.jpg" width="471" height="500" alt="Diagram +showing vector transformations R and R' under rotations about the x, +y and z axes, illustrating changes in direction between reference +frames."> +</figure> + +<p class="nind"> +Here \(x\), \(y\), \(z\), ... form the domain of \(R\), and all the +points in the oval marked \(\overleftarrow{R}ʻx\) are such that \(x\) +has the relation \(R\) to each of them, with similar conditions +for \(y\) and \(z\). What characterizes \(R\) as a \(1\rightarrow\text{Cls}\) +is the absence of overlapping in the ovals. For if +\(\overleftarrow{R}ʻx\) and \(\overleftarrow{R}ʻy\) had a point in +common, this would be a relatum both to \(x\) and \(y\), and both +\(x\) and \(y\) would be referents to it; whereas in a \(1\rightarrow +\text{Cls}\), no term has more than one referent.</p> + +<p><span class="pagenum" id="Page_448">[Pg 448]</span></p> + +<p>The above figure illustrates a very important property of one-many +relations, namely +\[ +R \in 1 \rightarrow \text{Cls} . \equiv . R \mid \breve{R} = I\upharpoonright \text{D}ʻR. +\]</p> + +<p>In the above figure, \(I\upharpoonright \text{D}ʻR\) is the relation of +identity confined to \(x\), \(y\), \(z\),.... If \(R\) were not a +\(1 \rightarrow \text{Cls}\), we could sometimes go from \(x\) to some term +of \(\overleftarrow{R}ʻx \cap \overleftarrow{R}ʻy\) by the relation +\(R\), and thence back to \(y\) by the relation \(\breve{R}\). But when +\(R \in 1 \rightarrow \text{Cls}\), \(R\mid \breve{R}\) must bring us +back to the point from which we started.</p> + +<p>When \(R \in 1 \rightarrow 1\), each of the ovals +\(\overleftarrow{R}ʻx\), \(\overleftarrow{R}ʻy\), +\(\overleftarrow{R}ʻz\), ... in the above figure shrinks to a single +point, so that \(\overleftarrow{R}ʻx = \iotaʻ\breve{R}ʻx\). Thus +when \(R\) is given as a \(1\rightarrow \text{Cls}\), it will be +a \(1 \rightarrow 1\) if \(Rʻy = Rʻz . \supset_{y, z} . y = z\). +This proposition is constantly used, and so is the consequence that +\(R\upharpoonright \beta\) is a \(1 \rightarrow 1\) if \(y, z \in \beta . Rʻy = Rʻz . \supset_{y, z} . y = z\). +(These propositions are *71·54·55 below.)</p> + +<p>The hypothesis \(R \in 1 \rightarrow \text{Cls}\) is equivalent to the +hypothesis +\[ +xRz . yRz . \supset_{x, y, z} . x= y +\] +(cf. <a href="#*71·17">*71·17</a>, below), and the hypothesis \(R \in \text{Cls} \rightarrow 1\) is equivalent to +\[ +xRy . xRz . \supset_{x, y, z} . y = z. +\]</p> + +<p>These are for many purposes the most convenient hypotheses to use.</p> + +<p>The most useful propositions in the present number are the following. +(We omit here propositions concerning \(\text{Cls} \rightarrow 1\) or +\(1 \rightarrow 1\) which are mere analogues of propositions concerning +\(1 \rightarrow \text{Cls}\).)</p> + +<p class="nind"><b>*71·16.</b> \(\vdash : R \in 1 \rightarrow \text{Cls} . \equiv . \text{E} ‼ Rʻʻ\text{ᗡ}ʻR\)</p> + +<p>This gives the connection of one-many relations with descriptive +functions. We have also</p> + +<p class="nind"><b>*71·163.</b> \(\vdash \colon\ldotp R \in 1 \rightarrow \text{Cls} . \equiv : y \in \text{ᗡ}ʻR . \equiv_{y} . \text{E} ! Rʻy\)</p> + +<p>For many of the constant relations defined from time to time, such as +\(\text{Cnv}\) or \(\text{D}\), the following proposition is useful:</p> + +<p class="nind"><b>*71·166.</b> \(\vdash : (y) . \text{E} ! Rʻy . \supset . R \in 1 \rightarrow \text{Cls}\)</p> + +<p class="nind"><b><a id="*71·17">*71·17</a>.</b> \(\vdash \colon\ldotp R \in 1 \rightarrow \text{Cls} . \equiv : xRz. yRz. \supset_{x, y, z} . x = y\)</p> + +<p>This might have been taken as the definition of one-many relations, +if we had not wished to derive them from the more general notion of +\(\alpha \rightarrow \beta\). In proving that a relation is one-many, +*71·17 is more often employed than any other proposition.</p> + +<p class="nind"><b>*71·22.</b> \(\vdash : R \in 1 \rightarrow \text{Cls}. S \unicode{x2abd} R . \supset . S \in 1 \rightarrow \text{Cls}\)</p> + +<p class="nind"><b>*71·25.</b> \(\vdash . R,\,S \in 1 \rightarrow \text{Cls} . \supset . R\mid S \in 1 \rightarrow \text{Cls}\)</p> + +<p class="nind"><b>*71·36.</b> \(\vdash \colon\ldotp R \in 1 \rightarrow \text{Cls} . \supset : x = Rʻy . \equiv . xRy\)</p> + +<p class="nind"><b>*71·381.</b> \(\vdash : R \in \text{Cls} \rightarrow 1 . \supset . Rʻʻ(\alpha - \beta) = Rʻʻ\alpha - Rʻʻ\beta\)</p> + +<p><span class="pagenum" id="Page_449">[Pg 449]</span></p> + +<p>(This proposition is more useful than the corresponding property of +\(1\rightarrow \text{Cls}.\))</p> + +<p class="nind"><b>*71·55.</b> \(\vdash\colon\colon R\in 1\rightarrow \text{Cls}.\supset\colon\ldotp R\upharpoonright \beta\in 1\rightarrow 1.\equiv:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z\)</p> + +<p>This proposition is constantly used. For example, putting \(\text{ᗡ}\) +for \(R\), it gives +\[ +\vdash\colon\ldotp \text{ᗡ}\upharpoonright \beta\in 1\rightarrow 1.\equiv:P,\,Q\in \beta.\text{ᗡ}ʻP=\text{ᗡ}ʻQ.\supset_{P,Q}.P=Q. +\]</p> + +<p>Most of the relations used to establish correlations in arithmetic are +obtained from a one-many relation, such as \(\text{ᗡ}\), by imposing +some limitation on the converse domain which makes the relation one-one.</p> + +<p class="nind"><b>*71·571.</b> \(\vdash\colon\ldotp y\in \beta.\supset_{y}.\text{E}!Rʻy:\equiv.R\upharpoonright \beta\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR\)</p> + +<p>Here "\(y\in \beta.\supset_{y}.\text{E}!Rʻy\)" is +\(\text{E}‼Rʻʻ\beta\), which has already played a large part as a +hypothesis, <i>e.g.</i> in <a href="#*37·6">*37·6</a> ff.</p> + +<p class="nind"><b>*71·7.</b> \(\vdash\colon\ldotp Q\in 1\rightarrow \text{Cls}.\supset:xP\mid Qz.\equiv.xP(Qʻz)\)</p> + +<p>Thus for example we shall have \(x(P\mid\text{Cnv})R.\equiv.xP(\text{Cnv}ʻR)\).</p> + +<hr class="tb"> + +<p class="nind"><b>*71·01.</b> \(\vdash.1\rightarrow \text{Cls}=\breve{R}(\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset 1) \quad[\text{*70·4}]\)</p> + +<p class="nind"><b>*71·02.</b> \(\vdash.\text{Cls}\rightarrow 1=\breve{R}(\overleftarrow{R}ʻʻ\text{D}ʻR\subset 1) \quad[\text{*70·41}]\)</p> + +<p class="nind"><b>*71·03.</b> \(\vdash.1\rightarrow 1=\breve{R}(\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset 1.\overleftarrow{R}ʻʻ\text{D}ʻR\subset 1) \quad[\text{*20·2.(*70·01)}]\)</p> + +<p class="nind"><b>*71·04.</b> \(\vdash.1\rightarrow 1=(1\rightarrow \text{Cls})\cap (\text{Cls}\rightarrow 1) \quad[\text{*70·42}]\)</p> + +<p class="nind"><b>*71·1.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\equiv.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset 1 \quad[\text{*20·33.*71·01}]\)</p> + +<p class="nind"><b>*71·101.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\equiv.\overleftarrow{R}ʻʻ\text{D}ʻR\subset 1 \quad[\text{*20·33.*71·02}]\)</p> + +<p class="nind"><b>*71·102.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\subset 1.\overleftarrow{R}ʻʻ\text{D}ʻR\subset 1 \quad[\text{*20·33.*71·03}]\)</p> + +<p class="nind"><b>*71·103.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.R\in 1\rightarrow \text{Cls}.R\in \text{Cls}\rightarrow 1 \quad[\text{*22·33.*71·04}]\)</p> + +<p class="nind"><b>*71·11.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\equiv.\overrightarrow{R}ʻʻ\text{V}\subset 1\cup \iotaʻ\Lambda \quad[\text{*70·44}]\)</p> + +<p class="nind"><b>*71·111.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\equiv.\overleftarrow{R}ʻʻ\text{V}\subset 1\cup \iotaʻ\Lambda \quad[\text{*70·441}]\)</p> + +<p class="nind"><b>*71·112.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.\overrightarrow{R}ʻʻ\text{V}\subset 1\cup \iotaʻ\Lambda.\overleftarrow{R}ʻʻ\text{V}\subset 1\cup \iotaʻ\Lambda \quad[\text{*70·12}]\)</p> + +<p class="nind"><b>*71·12.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\equiv.(y).\overrightarrow{R}ʻy\in 1\cup \iotaʻ\Lambda \quad[\text{*70·45}]\)</p> + +<p class="nind"><b>*71·121.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\equiv.(x).\overleftarrow{R}ʻx\in 1\cup \iotaʻ\Lambda \quad[\text{*70·451}]\)</p> + +<p class="nind"><b>*71·122.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\equiv:(y).\overrightarrow{R}ʻy\in 1\cup \iotaʻ\Lambda:(x).\overleftarrow{R}ʻx\in 1\cup \iotaʻ\Lambda \quad[\text{*70·13}]\)</p> + +<p class="nind"><b>*71·13.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\equiv:(y):\overrightarrow{R}ʻy\in 1.\lor.\overrightarrow{R}ʻy=\Lambda \quad[\text{*70·46}]\)</p> + +<p class="nind"><b>*71·131.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\equiv:(x):\overleftarrow{R}ʻx\in 1.\lor.\overleftarrow{R}ʻx=\Lambda \quad[\text{*70·461}]\)</p> + +<p><span class="pagenum" id="Page_450">[Pg 450]</span></p> + +<p class="nind"><b>*71·132.</b> + \(\vdash\colon\colon R\in 1\rightarrow 1.\equiv\colon\ldotp (y):\overrightarrow{R}ʻy\in 1.\lor.\overrightarrow{R}ʻy=\Lambda\colon\ldotp (x):\overleftarrow{R}ʻx\in 1.\lor.\overleftarrow{R}ʻx=\Lambda +\quad[\text{*70·14}]\)</p> + +<p class="nind"><b>*71·14.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\equiv:\exists !\overrightarrow{R}ʻy.\supset_{y}.\overrightarrow{R}ʻy\in 1 \quad[\text{*70·47}]\)</p> + +<p class="nind"><b>*71·141.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\equiv:\exists !\overleftarrow{R}ʻx.\supset_{x}.\overleftarrow{R}ʻx\in 1 \quad[\text{*70·471}]\)</p> + +<p class="nind"><b>*71·142.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\equiv:\exists !\overrightarrow{R}ʻy.\supset_{y}.\overrightarrow{R}ʻy\in 1:\exists !\overleftarrow{R}ʻx.\supset_{x}.\overleftarrow{R}ʻx\in + 1 \quad[\text{*70·15}]\)</p> + +<p class="nind"><b>*71·15.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\equiv.\text{D}ʻ\overrightarrow{R}\subset 1\cup \iotaʻ\Lambda \quad[\text{*70·48}]\)</p> + +<p class="nind"><b>*71·151.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\equiv.\text{D}ʻ\overleftarrow{R}\subset 1\cup \iotaʻ\Lambda \quad[\text{*70·481}]\)</p> + +<p class="nind"><b>*71·152.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.\text{D}ʻ\overrightarrow{R}\subset 1\cup \iotaʻ\Lambda.\text{D}ʻ\overleftarrow{R}\subset 1\cup \iotaʻ\Lambda \quad[\text{*70·16}]\)</p> + +<p class="nind"><b>*71·16.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\equiv.\text{E}‼Rʻʻ\text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·702.*71·1}.\supset\\ +\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.&\equiv:y\in \text{ᗡ}ʻR.\supset_{y}.\overrightarrow{R}ʻy\in 1:\\ +[\text{*53·3}] &\equiv:y\in \text{ᗡ}ʻR.\supset_{y}.\text{E}!Rʻy:\\ +[\text{*37·104}] &\equiv:\text{E}‼Rʻʻ\text{ᗡ}ʻR\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is very important; it exhibits the connection of +descriptive functions with one-many relations.</p> + +<p class="nind"><b>*71·161.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\equiv.\text{E}‼\breve{R}ʻʻ\text{D}ʻR\)</p> + +<p class="nind"><b>*71·162.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.\text{E}‼Rʻʻ\text{ᗡ}ʻR.\text{E}‼\breve{R}ʻʻ\text{D}ʻR\)</p> + +<p class="nind"><b>*71·163.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\equiv:y\in \text{ᗡ}ʻR.\equiv_{y}.\text{E}!Rʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·43}. & \supset\vdash:\text{E}!Rʻy.\supset.y\in \text{ᗡ}ʻR:\\ +[\text{*4·73}] &\supset\vdash\colon\ldotp y\in \text{ᗡ}ʻR.\supset.\text{E}!Rʻy:\equiv:y\in \text{ᗡ}ʻR.\equiv.\text{E}!Rʻy\colon\ldotp \\ +[\text{*10·11·271.*37·104}]&\supset\vdash\colon\ldotp \text{E}‼Rʻʻ\text{ᗡ}ʻR.\equiv:y\in \text{ᗡ}ʻR.\equiv_{y}.\text{E}!Rʻy &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·16}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·164.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\equiv:x\in \text{D}ʻR.\equiv_{x}.\text{E}!\breve{R}ʻx\)</p> + +<p class="nind"><b>*71·165.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\equiv:y\in \text{ᗡ}ʻR.\equiv_{y}.\text{E}!Rʻy:x\in \text{D}ʻR.\equiv_{x}.\text{E}!\breve{R}ʻx\)</p> + +<p class="nind"><b>*71·166.</b> \(\vdash:(y).\text{E}!Rʻy.\supset.R\in 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*2·02.*10·1}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \text{ᗡ}ʻR.\supset.\text{E}!Rʻy\colon\ldotp \\ +[\text{*10·11·21.*37·104}]&\supset\vdash:\text{Hp}.\supset.\text{E}‼Rʻʻ\text{ᗡ}ʻR.\\ +[\text{*71·16}] &\supset.R\in 1\rightarrow \text{Cls}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·167.</b> \(\vdash:(x).\text{E}!\breve{R}ʻx.\supset.R\in \text{Cls}\rightarrow 1\)</p> + +<p><span class="pagenum" id="Page_451">[Pg 451]</span></p> + +<p class="nind"><b>*71·168.</b> \(\vdash \colon\ldotp (y) . \text{E} ! Rʻy : (x) . \text{E} ! \breve{R}ʻx : \supset . R \in 1\rightarrow 1\)</p> + +<p class="nind"><b>*71·17.</b> \(\vdash \colon\ldotp R \in 1\rightarrow \text{Cls} . \equiv : xRz . yRz . \supset_{x, y, z} . x = y\)</p> + +<p>This proposition is constantly used in the sequel.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*52·4} . \supset \vdash \colon\ldotp \overrightarrow{R}ʻz \in 1 \cup \iotaʻ\Lambda . &\equiv : x, y \in \overrightarrow{R}ʻz . \supset_{x, y} . x = y :\\ +[\text{*32·18}] &\equiv : xRz . yRz . \supset_{x, y} . x = y \colon\ldotp \\ +[\text{*10·11·271.*11·21}] \supset \vdash \colon\ldotp (z) . \overrightarrow{R}ʻz \in 1 \cup \iotaʻ\Lambda . &\equiv : xRz . yRz . \supset_{x, y, z} . x = y &\qquad \text{(1)}\\ +\vdash . \text{(1) . *71·12}. \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·171.</b> \(\vdash \colon\ldotp R \in \text{Cls}\rightarrow 1 . \equiv : xRy . xRz . \supset_{x, y, z} . y = z\)</p> + +<p class="nind"><b>*71·172.</b> \(\vdash \colon\ldotp R \in 1\rightarrow 1 . \equiv : xRz . yRz . \supset_{x, y, z} . x = y : xRy . xRz . \supset_{x, y, z} . y = z\)</p> + +<p class="nind"><b>*71·18.</b> \(\vdash \colon\ldotp R \in 1\rightarrow \text{Cls} . \equiv : \exists ! \overleftarrow{R}ʻx \cap \overleftarrow{R}ʻy . \supset_{x, y} . x = y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*32·181 . *22·33} . \supset\\ +\vdash \colon\ldotp \exists ! \overleftarrow{R}ʻx \cap \overleftarrow{R}ʻy . \supset_{x, y} . x = y : &\equiv : (\exists z) . xRz . yRz . \supset_{x, y} . x = y :\\ +[\text{*10·23}] &\equiv : xRz . yRz . \supset_{x, y, z} . x = y :\\ +[\text{*71·17}] &\equiv : R \in 1\rightarrow \text{Cls} \colon\ldotp \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*71·181">*71·181</a>.</b> \(\vdash \colon\ldotp R \in \text{Cls}\rightarrow 1 . \equiv : \exists ! \overrightarrow{R}ʻy \cap \overrightarrow{R}ʻz . \supset_{y, z} . y = z\)</p> + +<p class="nind"><b>*71·182.</b> + \(\vdash \colon\colon R \in 1\rightarrow 1 . \equiv \colon\ldotp \exists ! \overleftarrow{R}ʻx \cap \overleftarrow{R}ʻy . \lor . \exists ! \overrightarrow{R}ʻx \cap \overrightarrow{R}ʻy : \supset_{x, y} . x = y\)</p> + +<p class="nind"><b>*71·19.</b> \(\vdash : R \in 1\rightarrow \text{Cls} . \equiv . R\mid \breve{R} = I\upharpoonright \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*34·1 . *31·11} . \supset \vdash . x (R\mid \breve{R}) y . \equiv . (\exists z). xRz . yRz &&\qquad \text{(1)}\\ +\vdash . \text{*50·1 . *35·101} . \supset \vdash . x (I\upharpoonright \text{D}ʻR)y . \equiv . x = y . y \in \text{D}ʻR &&\qquad \text{(2)}\\ +\vdash . \text{(1) . (2) . *21·43} . \supset\\ +\qquad\qquad\qquad\vdash \colon\colon R\mid \breve{R} = I\upharpoonright \text{D}ʻR . \equiv \colon\ldotp (\exists z) . xRz . yRz . &\equiv_{x, y} : x = y . y \in \text{D}ʻR :\\ +[\text{*33·13.*10·35}] &\equiv_{x, y} : (\exists z) . x = y . yRz :\\ +[\text{*13·194}] &\equiv_{x, y} : (\exists z) . x = y . xRz . yRz :\\ +[\text{*10·35}] &\equiv_{x, y} : x = y : (\exists z) . xRz. yRz \colon\ldotp \\ +[\text{*4·71}] &\equiv \colon\ldotp (\exists z) . xRz. yRz . \supset_{x, y} . x = y \colon\ldotp \\ +[\text{*10·23}] &\equiv \colon\ldotp xRz . yRz . \supset_{x, y, z} . x = y \colon\ldotp \\ +[\text{*71·17}] &\equiv \colon\ldotp R \in 1\rightarrow \text{Cls} \colon\colon \supset \vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·191.</b> \(\vdash : R \in \text{Cls}\rightarrow 1 . \equiv . \breve{R}\mid R = I\upharpoonright \text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*71·192.</b> \(\vdash : R \in 1\rightarrow 1 . \equiv . R\mid \breve{R} = I\upharpoonright \text{D}ʻR . \breve{R}\mid R = I\upharpoonright \text{ᗡ}ʻR\)</p> + +<p><span class="pagenum" id="Page_452">[Pg 452]</span></p> + +<p class="nind"><b>*71·2.</b> \[\begin{align}\vdash.\text{Cls}\rightarrow &1=\text{Cnv}ʻʻ(1\rightarrow \text{Cls}).\\ +&1\rightarrow \text{Cls}=\text{Cnv}ʻʻ(\text{Cls}\rightarrow 1).1\rightarrow 1=\text{Cnv}ʻʻ(1\rightarrow 1) \quad[\text{*70·22}]\end{align}\]</p> + +<p class="nind"><b>*71·21.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\equiv.\breve{R}\in \text{Cls}\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·62.*31·13}.\supset\vdash:R\in 1\rightarrow \text{Cls}.&\supset.\text{Cnv}ʻR\in \text{Cnv}ʻʻ(1\rightarrow \text{Cls}).\\ +[\text{*31·12.*71·2}] &\supset.\breve{R}\in \text{Cls}\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{*37·62.*31·13}.\supset\vdash:\breve{R}\in \text{Cls}\rightarrow 1.&\supset.\text{Cnv}ʻ\breve{R}\in \text{Cnv}ʻʻ(\text{Cls}\rightarrow 1).\\ +[\text{*31·33.*71·2}] &\supset.R\in 1\rightarrow \text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·211.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\equiv.\breve{R}\in 1\rightarrow \text{Cls}\)</p> + +<p class="nind"><b>*71·212.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.\breve{R}\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*71·22.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.S\unicode{x2abd}R.\supset.S\in 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·1}.\supset\\ +\vdash\colon\ldotp S\unicode{x2abd}R.\supset:xSz.ySz.\supset_{x,y,z}.xRz.yRz &\qquad \text{(1)}\\ +\vdash.\text{*71·17}.\supset\\ +\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:xRz.yRz.\supset_{x,y,z}.x=y &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*11·37}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:xSz.ySz.\supset_{x,y,z}.x=y:\\ +[\text{*71·17}]\supset:S\in 1\rightarrow \text{Cls}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·221.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.S\unicode{x2abd}R.\supset.S\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*71·222.</b> \(\vdash:R\in 1\rightarrow 1.S\unicode{x2abd}R.\supset.S\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*71·223.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\text{Rl}ʻR\subset 1\rightarrow \text{Cls} \quad[\text{*71·22.*61·2}]\)</p> + +<p class="nind"><b>*71·224.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\text{Rl}ʻR\subset \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*71·225.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\text{Rl}ʻR\subset 1\rightarrow 1\)</p> + +<p class="nind"><b>*71·23.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.R\dot{\cap}S\in 1\rightarrow \text{Cls} \quad[\text{*71·22.*23·43}]\)</p> + +<p class="nind"><b>*71·231.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.R\dot{\cap}S\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*71·232.</b> \(\vdash:R\in 1\rightarrow 1.\supset.R\dot{\cap}S\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*71·233.</b> \(\vdash:R,\, S\in 1\rightarrow \text{Cls}.\supset.R\dot{\cap}\breve{S}\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71.23}.&\supset\vdash:\text{Hp}.\supset.R\dot{\cap}\breve{S}\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*71·21}.&\supset\vdash:\text{Hp}.\supset.\breve{S}\in \text{Cls}\rightarrow 1.\\ +[\text{*71·231}]&\supset.R\dot{\cap}\breve{S}\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*71·103}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_453">[Pg 453]</span></p> + +<p class="nind"><b>*71·234.</b> \(\vdash:R,\,S\in \text{Cls}\rightarrow 1.\supset.R\dot{\cap}\breve{S}\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*71·235.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.S\in \text{Cls}\rightarrow 1.\supset.R\dot{\cap}S\in 1\rightarrow 1\)</p> + +<p class="nind"><b><a id="*71·24">*71·24</a>.</b> \(\vdash:R,\,S\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\cap \text{ᗡ}ʻS=\Lambda.\supset.R\unicode{x228d}S\in 1\rightarrow \text{Cls} \quad[\text{*70·54}]\)</p> + +<p class="nind"><b>*71·241.</b> \(\vdash:R,\,S\in \text{Cls}\rightarrow 1.\text{D}ʻR\cap \text{D}ʻS=\Lambda.\supset.R\unicode{x228d}S\in \text{Cls}\rightarrow 1 \quad[\text{*70·55}]\)</p> + +<p class="nind"><b>*71·242.</b> \(\vdash:R,\,S\in 1\rightarrow 1.\text{D}ʻR\cap \text{D}ʻS=\Lambda.\text{ᗡ}ʻR\cap \text{ᗡ}ʻS=\Lambda.\supset.R\unicode{x228d}S\in 1\rightarrow 1 +\quad[\text{*70·56}]\)</p> + +<p class="nind"><b>*71·243.</b> \(\vdash:R,\,S\in 1\rightarrow 1.CʻR\cap CʻS=\Lambda.\supset.R\unicode{x228d}S\in 1\rightarrow 1 \quad[\text{*70·57}]\)</p> + +<p class="nind"><b>*71·244.</b> \(\vdash:R,\,S\in 1\rightarrow \text{Cls}.R\upharpoonright \text{ᗡ}ʻS\unicode{x2abd}S.\supset.R\unicode{x228d}S\in 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·34.*4·4}.\supset\\ +\vdash\colon\ldotp x(R\unicode{x228d}S)z.y(R\unicode{x228d}S)z.\equiv:xRz.yRz.\lor.xRz.ySz.\lor.xSz.yRz.\lor.xSz.ySz &\qquad \text{(1)}\\ +\vdash.\text{*71·17}.\supset\vdash\colon\ldotp R,\,S\in 1\rightarrow \text{Cls}.\supset:xRz.yRz.\supset.x=y:xSz.ySz.\supset.x=y &\qquad \text{(2)}\\ +\vdash.\text{*33·14.*4·7}.\supset\vdash:xRz.ySz.\supset.xRz.ySz.z\in \text{ᗡ}ʻS.\\ +[\text{*35·101}] \supset.x(R\upharpoonright \text{ᗡ}ʻS)z.ySz &\qquad \text{(3)}\\ +\vdash.\text{(3)}.\supset\vdash\colon\ldotp R\upharpoonright \text{ᗡ}ʻS\unicode{x2abd}S.\supset:xRz.ySz.\supset.xSz.ySz &\qquad \text{(4)}\\ +\vdash.\text{(4)}\, \frac{y,\,x}{x,\,y}.\supset\vdash\colon\ldotp R\upharpoonright \text{ᗡ}ʻS\unicode{x2abd}S.\supset:xSz.yRz.\supset.xSz.ySz &\qquad \text{(5)}\\ +\vdash.\text{(2).(4).(5)}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xRz.ySz.\supset.x=y:xSz.yRz.\supset.x=y &\qquad \text{(6)}\\ +\vdash.\text{(1).(2).(6).*4·77}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x(R\unicode{x228d}S)z.y(R\unicode{x228d}S)z.\supset.x=y &\qquad \text{(7)}\\ +\vdash.\text{(7).*10·11·21.*71·17}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·245.</b> \(\vdash:R,S\in \text{Cls}\rightarrow 1.(\text{D}ʻS)\upharpoonleft R\unicode{x2abd}S.\supset.R\unicode{x228d}S\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*71·25.</b> \(\vdash:R,S\in 1\rightarrow \text{Cls}.\supset.R\mid S\in 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·17}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:ySx.zSx.\supset.y=z:\\ +[\text{Fact}] &\supset:uRy.ySx.vRz.zSx.\supset.y=z.uRy.vRv.\\ +[\text{*13·13}] &\supset.uRy.vRy.\\ +[\text{*71·17}] &\supset.u=v &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·3·54}.&\supset\\ +\vdash\colon\colon \text{Hp}.&\supset\colon\ldotp (\exists y).uRy.ySx:(\exists z).vRz.zSx:\supset.u=v\colon\ldotp \\ +[\text{*34·1}]&\supset\colon\ldotp u(R\mid S)x.v(R\mid S)x.\supset.u=v &\qquad \text{(2)}\\ +\vdash.\text{(2).*71·17}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·251.</b> \(\vdash:R,\,S\in \text{Cls}\rightarrow 1.\supset.R\mid S\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*71·252.</b> \(\vdash:R,\,S\in 1\rightarrow 1.\supset.R\mid S\in 1\rightarrow 1\)</p> + +<p>*71·25 may also be deduced from <a href="#*70·6">*70·6</a>, as follows:</p> + +<p><span class="pagenum" id="Page_454">[Pg 454]</span></p> + +<p><i>Alternative Dem. of</i> *71·25.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·301.*71·12}.&\supset\vdash:R\in 1\rightarrow \text{Cls}.\supset.Rʻʻ\iotaʻx\in 1\cup \iotaʻ\Lambda:\\ +[\text{*52·1}] &\supset\vdash:R\in 1\rightarrow \text{Cls}.\alpha\in 1.\supset.Rʻʻ\alpha\in 1\cup \iotaʻ\Lambda:\\ +[\text{*37·61·11·103}] &\supset\vdash:R\in 1\rightarrow \text{Cls}.\supset.Rʻʻʻ1\subset 1\cup \iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*70·6}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Similarly *71·251 may be deduced from <a href="#*70·61">*70·61</a>.</p> + +<p class="nind"><b>*71·26.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.R\upharpoonright \gamma\in 1\rightarrow \text{Cls} \quad[\text{*70·62}]\)</p> + +<p class="nind"><b>*71·261.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\beta\upharpoonleft R\in \text{Cls}\rightarrow 1 \quad[\text{*70·63}]\)</p> + +<p class="nind"><b>*71·27.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\beta\upharpoonleft R\in 1\rightarrow \text{Cls} \quad[\text{*35·44.*71·22}]\)</p> + +<p class="nind"><b>*71·271.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.R\upharpoonright \gamma\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*71.28.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\beta\upharpoonleft R\upharpoonright \gamma\in 1\rightarrow \text{Cls} \quad[\text{*35·442.*71·22}]\)</p> + +<p class="nind"><b>*71·281.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\beta\upharpoonleft R\upharpoonright \gamma\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*71·29.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\beta\upharpoonleft R,\,R\upharpoonright \gamma,\beta\upharpoonleft R\upharpoonright \gamma\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*71·31.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.y\in \text{ᗡ}ʻR.\supset.(Rʻy)Ry \quad[\text{*30·32.*71·163}]\)</p> + +<p class="nind"><b>*71·311.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.x\in \text{D}ʻR.\supset.xR(\breve{R}ʻx)\)</p> + +<p class="nind"><b>*71·312.</b> \(\vdash:R\in 1\rightarrow 1.x\in \text{D}ʻR.y\in \text{ᗡ}ʻR.\supset.xR(\breve{R}ʻx).(Rʻy)Ry\)</p> + +<p class="nind"><b>*71·32.</b> \[\begin{align}&\vdash\colon\colon R\in 1\rightarrow \text{Cls}.y\in \text{ᗡ}ʻR.\supset\colon\ldotp \psi(Rʻy).\equiv:(\exists x).xRy.\psi x:\equiv:xRy.\supset_{x}.\psi + x\\ +&[\text{*30·33.*71·163}]\end{align}\]</p> + +<p class="nind"><b>*71·321.</b> \(\vdash\colon\colon R\in \text{Cls}\rightarrow 1.x\in \text{D}ʻR.\supset\colon\ldotp \psi(\breve{R}ʻx).\equiv:(\exists y).xRy.\psi y:\equiv:xRy.\supset_{y}.\psi y\)</p> + +<p class="nind"><b>*71·33.</b> \(\vdash\colon\colon R\in 1\rightarrow \text{Cls}.\supset\colon\ldotp \psi(Rʻy):\equiv:(\exists x).xRy.\psi x:\equiv:y\in \text{ᗡ}ʻR:xRy.\supset_{x}.\psi x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·32.*5·32}.\supset\\ +\vdash\colon\colon \text{Hp}.\supset\colon\ldotp y\in \text{ᗡ}ʻR.\psi(Rʻy).&\equiv:y\in \text{ᗡ}ʻR:(\exists x).xRy.\psi x:\\ +&\equiv:y\in \text{ᗡ}ʻR:xRy.\supset_{x}.\psi x &\qquad \text{(1)}\\ +\vdash.\text{*14·21}.\supset\vdash:\psi(Rʻy).&\supset.\text{E}!Rʻy.\\ +[\text{*33·43}] &\supset.y\in \text{ᗡ}ʻR:\\ +[\text{*4·71}] &\supset\vdash\colon\ldotp y\in \text{ᗡ}ʻR.\psi(Rʻy).\equiv.\psi(Rʻy) &\qquad \text{(2)}\\ +\vdash.\text{*10·5}.&\supset\vdash:(\exists x).xRy.\psi x.\supset.(\exists x).xRy.\\ +[\text{*33·131}] &\supset.y\in \text{ᗡ}ʻR:\\ +[\text{*4·71}] &\supset\vdash\colon\ldotp y\in \text{ᗡ}ʻR:(\exists x).xRy.\psi x:\equiv.(\exists x).xRy.\psi x &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_455">[Pg 455]</span></p> + +<p class="nind"><b>*71·331.</b> \[\begin{align}\vdash\colon\colon R\in \text{Cls}\rightarrow 1.\supset\colon\ldotp \psi(\breve{R}ʻx).\equiv:(\exists y).&xRy.\psi y:\equiv:\\ +&x\in \text{D}ʻR:xRy.\supset_{y}.\psi y\end{align}\]</p> + +<p class="nind"><b>*71·332.</b> + \[\begin{align}&\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:Rʻy\in \alpha.\equiv.\exists !\overrightarrow{R}ʻy\cap \alpha.\equiv.y\in \text{ᗡ}ʻR.\overrightarrow{R}ʻy\subset \alpha\\ +&\left[\text{*71·33}\, \frac{x\in \alpha}{\psi x}\right]\end{align}\]</p> + +<p class="nind"><b>*71·333.</b> + \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:\breve{R}ʻx\in \alpha.\equiv.\exists !\overleftarrow{R}ʻx\cap \alpha.\equiv.x\in \text{D}ʻR.\overleftarrow{R}ʻx\subset \alpha\)</p> + +<p class="nind"><b>*71·34.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.R=S.y\in \text{ᗡ}ʻR.\supset.Rʻy=Sʻy \quad[\text{*30·36.*71·163}]\)</p> + +<p>*71·341. \(\vdash:R\in \text{Cls}\rightarrow 1.R=S.x\in \text{D}ʻR.\supset.\breve{R}ʻx=\breve{S}ʻx\)</p> + +<p class="nind"><b>*71·35.</b> \(\vdash\colon\colon R\in 1\rightarrow \text{Cls}.\supset\colon\ldotp y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.\supset_{y}.Rʻy=Sʻy:\equiv.R=S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·18}. &\supset\vdash\colon\ldotp R=S.\supset:y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.\equiv.y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻR.\\ +[\text{*22·56}] &\equiv.y\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·34}.&\supset\vdash\colon\colon \text{Hp}.R=S.\supset:y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.\supset_{y}.Rʻy=Sʻy &\qquad \text{(2)}\\ +\vdash.\text{(2).*33·45}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·351.</b> \(\vdash\colon\colon R\in \text{Cls}\rightarrow 1.\supset\colon\ldotp x\in \text{D}ʻR\cup \text{D}ʻS.\supset_{x}.\breve{R}ʻx=\breve{S}ʻx:\equiv.R=S\)</p> + +<p class="nind"><b>*71·352.</b> \[\begin{align}\vdash\colon\colon R\in 1\rightarrow 1.\supset\colon\ldotp y\in \text{ᗡ}ʻR\cup \text{ᗡ}ʻS.&\supset_{y}.Rʻy=Sʻy:\equiv:R=S:\\ +&\equiv:x\in \text{D}ʻR\cup \text{D}ʻS.\supset_{x}.\breve{R}ʻx=\breve{S}ʻx\end{align}\]</p> + +<p class="nind"><b>*71·36.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:x=Rʻy.\equiv.xRy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·4.*71·163}.\supset\\ +\vdash\colon\ldotp \text{Hp}.y\in \text{ᗡ}ʻR.\supset:x=Rʻy.\equiv.xRy &\qquad \text{(1)}\\ +\vdash.\text{*71·163.Transp}.\supset\\ +\vdash\colon\ldotp \text{Hp}.y{\sim}\in \text{ᗡ}ʻR.\supset.{\sim}\text{E}!Rʻy.\\ +[\text{*14·21.Transp}] \supset.{\sim}(x=Rʻy) &\qquad \text{(2)}\\ +\vdash.\text{*33·14.Transp}.\supset\vdash:y{\sim}\in \text{ᗡ}ʻR.\supset.{\sim}(xRy) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*5·21}.\supset\\ +\vdash\colon\ldotp \text{Hp}.y{\sim}\in \text{ᗡ}ʻR.\supset:x=Rʻy.\equiv.xRy &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*4·83}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·361.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:y=\breve{R}ʻx.\equiv.xRy\)</p> + +<p class="nind"><b>*71·362.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:x=Rʻy.\equiv.xRy.\equiv.y=\breve{R}ʻx\)</p> + +<p class="nind"><b>*71·37.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:y\in \breve{R}ʻʻ\alpha.\equiv.Rʻy\in \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·33}.\supset\vdash\colon\ldotp \text{Hp}.\supset:Rʻy\in \alpha.&\equiv.(\exists x).xRy.x\in \alpha.\\ +[\text{*37·105}] &\equiv.y\in \breve{R}ʻʻ\alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·371.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:x\in Rʻʻ\alpha.\equiv.\breve{R}ʻx\in \alpha\)</p> + +<p><span class="pagenum" id="Page_456">[Pg 456]</span></p> + +<p class="nind"><b>*71·38.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\breve{R}ʻʻ(\alpha-\beta)=\breve{R}ʻʻ\alpha-\breve{R}ʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·37}.\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \breve{R}ʻʻ(\alpha-\beta).&\equiv.Rʻy\in \alpha-\beta.\\ +[\text{*22·32.*14·21}] & \equiv.Rʻy\in \alpha.{\sim}(Rʻy\in \beta).\\ +[\text{*71·37}] &\equiv.y\in \breve{R}ʻʻ\alpha.{\sim}(y\in \breve{R}ʻʻ\beta).\\ +[\text{*22·32}] &\equiv.y\in \breve{R}ʻʻ\alpha-\breve{R}ʻʻ\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·381.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.Rʻʻ(\alpha-\beta)=Rʻʻ\alpha-Rʻʻ\beta\)</p> + +<p class="nind"><b>*71·4.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.Rʻʻ\beta=\hat{x}\{(\exists y).y\in \beta.x=Rʻy\} \quad[\text{*37·1.*71·36}]\)</p> + +<p class="nind"><b>*71·401.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}ʻʻ\beta=\hat{y}\{(\exists x).x\in \beta.y=\breve{R}ʻx\}\)</p> + +<p class="nind"><b>*71·41.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\text{D}ʻR=\hat{x}{(\exists y).x=Rʻy} \quad[\text{*33·11.*71·36}]\)</p> + +<p class="nind"><b>*71·411.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\text{ᗡ}ʻR=\hat{y}\{(\exists x).y=\breve{R}ʻx\}\)</p> + +<p class="nind"><b>*71·42.</b> \[\begin{align}&\vdash\colon\colon R\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR.\supset\colon\ldotp Rʻʻ\beta\subset \alpha.\equiv:y\in \beta.\supset_{y}.Rʻy\in + \alpha\\ +&[\text{*37·61.*71·16}]\end{align}\]</p> + +<p class="nind"><b>*71·421.</b> \(\vdash\colon\colon R\in \text{Cls}\rightarrow 1.\alpha\subset \text{D}ʻR.\supset\colon\ldotp \breve{R}ʻʻ\alpha\subset \beta.\equiv:x\in \alpha.\supset_{x}.\breve{R}ʻx\in + \beta\)</p> + +<p class="nind"><b>*71·43.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.y\in \alpha\cap \text{ᗡ}ʻR.\supset.Rʻy\in Rʻʻ\alpha \quad[\text{*37·62.*71·16}]\)</p> + +<p class="nind"><b>*71·431.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.x\in \alpha\cap \text{D}ʻR.\supset.\breve{R}ʻx\in \breve{R}ʻʻ\alpha\)</p> + +<p class="nind"><b>*71·44.</b> \[\begin{align}&\vdash\colon\colon R\in 1\rightarrow \text{Cls}.\alpha\subset \text{ᗡ}ʻR.\supset\colon\ldotp x\in Rʻʻ\alpha.\supset_{x}.\psi + x:\equiv:y\in \alpha.\supset_{y}.\psi(Rʻy)\\ +&[\text{*37·63.*71·16}]\end{align}\]</p> + +<p class="nind"><b>*71·441.</b> \(\vdash\colon\colon R\in \text{Cls}\rightarrow 1.\alpha\subset \text{D}ʻR.\supset\colon\ldotp y\in \breve{R}ʻʻ\alpha.\supset_{y}.\psi + y:\equiv:x\in \alpha.\supset_{x}.\psi(\breve{R}ʻx)\)</p> + +<p class="nind"><b>*71·45.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:(\exists x).x\in Rʻʻ\alpha.\psi x.\equiv.(\exists y).y\in \alpha.\psi(Rʻy)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·64.*71·16}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:(\exists x).x\in Rʻʻ(\alpha\cap \text{ᗡ}ʻR).\psi x.\equiv.(\exists y).y\in \alpha\cap \text{ᗡ}ʻR.\psi(Rʻy) &\qquad \text{(1)}\\ +\vdash.\text{*37·26}.&\supset\vdash.Rʻʻ(\alpha\cap \text{ᗡ}ʻR)=Rʻʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{*14·21}. &\supset\vdash:y\in \alpha.\psi(Rʻy).\supset.\text{E}!Rʻy.\\ +[\text{*33·43}] &\supset.y\in \text{ᗡ}ʻR:\\ +[\text{*4·71.*22·33}]&\supset\vdash:y\in \alpha.\psi(Rʻy).\equiv.y\in \alpha\cap \text{ᗡ}ʻR.\psi(Rʻy):\\ +[\text{*10·11·281}] &\supset\vdash:(\exists y).y\in \alpha.\psi(Rʻy).\equiv.(\exists y).y\in \alpha\cap \text{ᗡ}ʻR.\psi(Rʻy) &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·451.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:(\exists y).y\in \breve{R}ʻʻ\alpha.\psi y.\equiv.(\exists x).x\in \alpha.\psi(\breve{R}ʻx)\)</p> + +<p><span class="pagenum" id="Page_457">[Pg 457]</span></p> + +<p class="nind"><b>*71·46.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset Rʻʻ\beta.\supset.\alpha=Rʻʻ(\breve{R}ʻʻ\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·26}.\supset\vdash:Rʻʻ\beta=Rʻʻ(\beta\cap \text{ᗡ}ʻR).Rʻʻ(\breve{R}ʻʻ\alpha\cap \beta)=Rʻʻ(\breve{R}ʻʻ\alpha\cap \beta\cap \text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash.\text{*37·65.*71·16}.\supset\\ +\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset Rʻʻ(\beta\cap \text{ᗡ}ʻR).\supset.\alpha=Rʻʻ(\breve{R}ʻʻ\alpha\cap \beta\cap \text{ᗡ}ʻR) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·461.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\beta\subset \breve{R}ʻʻ\alpha.\supset.\beta=\breve{R}ʻʻ(Rʻʻ\beta\cap \alpha)\)</p> + +<p class="nind"><b>*71·47.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:\alpha\subset Rʻʻ\beta.\equiv.(\exists \gamma).\gamma\subset \beta.\alpha=Rʻʻ\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·46.*10·24.*22·43}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha\subset Rʻʻ\beta.\supset.(\exists \gamma).\gamma\subset \beta.\alpha=Rʻʻ\gamma &\qquad \text{(1)}\\ +\vdash.\text{*37·2.*10·11·23}. &\supset\vdash:(\exists \gamma).\gamma\subset \beta.\alpha=Rʻʻ\gamma.\supset.\alpha\subset Rʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·471.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:\beta\subset \breve{R}ʻʻ\alpha.\equiv.(\exists \gamma).\gamma\subset \alpha.\beta=\breve{R}ʻʻ\gamma\)</p> + +<p class="nind"><b>*71·48.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\text{D}ʻR_{\in}=\text{Cl}ʻ\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·24.*60·2}. &\supset\vdash.\text{D}ʻR_{\in}\subset \text{Cl}ʻ\text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*37·25.*71·47.*60·2}.&\supset\vdash:\text{Hp}.\alpha\in \text{Cl}ʻ\text{D}ʻR.\supset.(\exists \gamma).\gamma\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\gamma.\\ +[\text{*10·5.*37·23}] &\supset.\alpha\in \text{D}ʻR_{\in}:\\ +[\text{Exp.*10·11·21}] &\supset\vdash:\text{Hp}.\supset.\text{Cl}ʻ\text{D}ʻR\subset \text{D}ʻR_{\in} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·481.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\text{D}ʻ(\breve{R})_{\in}=\text{Cl}ʻ\text{ᗡ}ʻR\)</p> + +<p>The following proposition is used in the theory of derivatives of a +series (*216·411).</p> + +<p class="nind"><b>*71·49.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset \text{ᗡ}ʻR.\supset.Rʻʻʻ\text{Cl}ʻ\alpha=\text{Cl}ʻRʻʻ\alpha.Rʻʻʻ\text{Cl ex}ʻ\alpha=\text{Cl ex}ʻRʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·47.*60·2}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\gamma\in \text{Cl}ʻRʻʻ\alpha.\equiv.(\exists \beta).\beta\subset \alpha.\gamma=Rʻʻ\beta.\\ +[\text{*37·103}] &\equiv.\gamma\in Rʻʻʻ\text{Cl}ʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*37·43}. & \supset\vdash\colon\ldotp \text{Hp}.\beta\in \text{Cl}ʻ\alpha.\supset:\exists !\beta.\equiv.\exists !Rʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·491.</b> + \(\vdash:R\in \text{Cls}\rightarrow 1.\alpha\subset \text{D}ʻR.\supset.\breve{R}ʻʻʻ\text{Cl}ʻ\alpha=\text{Cl}ʻ\breve{R}ʻʻ\alpha.\breve{R}ʻʻʻ\text{Cl ex}ʻ\alpha=\text{Cl ex}ʻ\breve{R}ʻʻ\alpha\)</p> + +<p>This proposition is used in the theory of derivatives of a series +(*216·4) and in the theory of ordinal numbers (*251·11).</p> + +<p class="nind"><b>*71·5.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:xRy.\equiv.x=\breve{\iota}ʻ\overrightarrow{R}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·36.*30·1}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xRy.&\equiv.x=(℩x)(xRy).\\ +[\text{*51·56.*32·13}] &\equiv.x=\breve{\iota}ʻ\overrightarrow{R}ʻy\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_458">[Pg 458]</span></p> + +<p class="nind"><b>*71·501.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:xRy.\equiv.y=\breve{\iota}ʻ\overleftarrow{R}ʻx\)</p> + +<p class="nind"><b>*71·51.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.y\in \text{ᗡ}ʻR.\supset.Rʻy=\breve{\iota}ʻ\overrightarrow{R}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·31.*71·163}.\supset\vdash:\text{Hp}.&\supset.\iotaʻRʻy=\overrightarrow{R}ʻy.\\ +[\text{*51·51}] &\supset.Rʻy=\breve{\iota}ʻ\overrightarrow{R}ʻy:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·511.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.x\in \text{D}ʻR.\supset.\breve{R}ʻx=\breve{\iota}ʻ\overleftarrow{R}ʻx\)</p> + +<p class="nind"><b>*71·52.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.Rʻʻ\alpha=\breve{\iota}ʻʻ\overrightarrow{R}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1}. \supset\vdash.\breve{\iota}ʻʻ\overrightarrow{R}ʻʻ\alpha &=\hat{x}\{(\exists \beta).\beta\in \overrightarrow{R}ʻʻ\alpha.x\breve{\iota}\beta\}\\ +[\text{*51·51}] &=\hat{x}\{(\exists \beta).\beta\in \overrightarrow{R}ʻʻ\alpha.x=\breve{\iota}ʻ\beta\}\\ +[\text{*37·7}] &=\hat{x}\{(\exists \beta,y).y\in \alpha.\beta=\overrightarrow{R}ʻy.x=\breve{\iota}ʻ\beta\}\\ +[\text{*11·23.*13·195}] &=\hat{x}\{(\exists y).y\in \alpha.x=\breve{\iota}ʻ\overrightarrow{R}ʻy\} &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·5}.\supset\vdash:\text{Hp}.\supset.\breve{\iota}ʻʻ\overrightarrow{R}ʻʻ\alpha&=\hat{x}\{(\exists y).y\in \alpha.xRy\}\\ +[\text{*37·1}] &=Rʻʻ\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·521.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}ʻʻ\alpha=\breve{\iota}ʻʻ\overleftarrow{R}ʻʻ\alpha\)</p> + +<p class="nind"><b>*71·53.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\breve{R}ʻx=\breve{R}ʻy.\supset.x=y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}.\supset\vdash:\text{Hp}.&\supset.\text{E}!\breve{R}ʻx.\text{E}!\breve{R}ʻy.\\ +[\text{*30·32}] &\supset.xR(\breve{R}ʻx).yR(\breve{R}ʻy).\\ +[\text{*14·16}] &\supset.xR(\breve{R}ʻy).yR(\breve{R}ʻy).\\ +[\text{*71·17}] &\supset.x=y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·531.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.Rʻy=Rʻz.\supset.y=z\)</p> + +<p class="nind"><b>*71·532.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:Rʻy=Rʻz.\supset.y=z:\breve{R}ʻx=\breve{R}ʻy.\supset.x=y\)</p> + +<p class="nind"><b>*71·54.</b> \(\vdash\colon\colon R\in 1\rightarrow \text{Cls}.\supset\colon\ldotp R\in 1\rightarrow 1.\equiv:Rʻy=Rʻz.\supset_{y,z}.y=z\)</p> + +<p>This proposition and the next (*71·55) are very often used.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·36}.\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists x).xRy.xRz.&\equiv_{y,z}.(\exists x).x=Rʻy.x=Rʻz.\\ +[\text{*14·205}] &\equiv_{y,z}.Rʻy=Rʻz &\qquad \text{(1)}\\ +\vdash.\text{(1)}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp Rʻy=Rʻz.\supset_{y,z}.y=z:&\equiv:(\exists x).xRy.xRz.\supset_{y,z}.y=z:\\ +[\text{*10·23}] &\equiv:xRy.xRz.\supset_{x,y,z}.y=z:\\ +[\text{*71·171}] &\equiv:R\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{*71·103.*4·73}.\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in \text{Cls}\rightarrow 1.&\equiv.R\in 1\rightarrow 1 &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_459">[Pg 459]</span></p> + +<p class="nind"><b>*71·55.</b> \(\vdash\colon\colon R\in 1\rightarrow \text{Cls}.\supset\colon\ldotp R\upharpoonright \beta\in 1\rightarrow 1.\equiv:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·26}. \supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp R\upharpoonright \beta\in 1\rightarrow \text{Cls}\colon\ldotp \\ +[\text{*71·54}] \qquad\qquad\qquad\supset\colon\ldotp R\upharpoonright \beta\in 1\rightarrow 1.&\equiv:(R\upharpoonright \beta)ʻy=(R\upharpoonright \beta)ʻz.\supset_{y,z}.y=z:\\ +[\text{*35·7}] &\equiv:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·56.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.y\in \text{ᗡ}ʻR.\supset:Rʻy=Rʻz.\equiv.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·532}. &\supset\vdash:\text{Hp}.Rʻy=Rʻz.\supset.y=z &\qquad \text{(1)}\\ +\vdash.\text{*71·165.*30·37}. &\supset\vdash:\text{Hp}.y=z.\supset.Rʻy=Rʻz &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·561.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.x\in \text{D}ʻR.\supset:\breve{R}ʻx=\breve{R}ʻy.\equiv.x=y\)</p> + +<p class="nind"><b>*71·57.</b> \(\vdash\colon\ldotp Rʻy=Rʻz.\equiv_{y,z}.y=z:\equiv:R\in 1\rightarrow 1:(y).\text{E}!Rʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·1}.&\supset\vdash\colon\ldotp Rʻy=Rʻz.\equiv_{y,z}.y=z:\supset:Rʻy=Rʻy.\equiv_{y}.y=y:\\ +[\text{*13·15}] &\supset:(y).Rʻy=Rʻy:\\ +[\text{*14·28}] &\supset:(y).\text{E}!Rʻy &\qquad \text{(1)}\\ +[\text{*71.166}] &\supset:R\in 1\rightarrow \text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{(2)}.&\supset\vdash\colon\ldotp \text{Hp(2)}.\supset:R\in 1\rightarrow \text{Cls}:Rʻy=Rʻz.\supset_{y,z}.y=z:\\ +[\text{*71·54}] &\supset:R\in 1\rightarrow 1 &\qquad \text{(3)}\\ +\vdash.\text{(1).(3).*71·56}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·571.</b> \(\vdash\colon\ldotp y\in \beta.\supset_{y}.\text{E}!Rʻy:\equiv.R\upharpoonright \beta\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·16}.\supset\vdash\colon\ldotp R\upharpoonright \beta\in 1\rightarrow \text{Cls}.&\equiv:y\in \text{ᗡ}ʻ(R\upharpoonright \beta).\supset_{y}.\text{E}!(R\upharpoonright + \beta)ʻy:\\ +[\text{*35·64·7}] &\equiv:y\in \beta\cap \text{ᗡ}ʻR.\supset_{y}.y\in \beta.\text{E}!Rʻy:\\ +[\text{*22·33.*5·3}] &\equiv:y\in \beta\cap \text{ᗡ}ʻR.\supset_{y}.\text{E}!Rʻy &\qquad \text{(1)}\\ +\vdash.\text{(1).*22·621}.\supset\\ +\vdash\colon\ldotp R\upharpoonright \beta\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR.&\equiv:y\in \beta\cap \text{ᗡ}ʻR.\supset_{y}.\text{E}!Rʻy:\beta\cap \text{ᗡ}ʻR=\beta:\\ +[\text{*13·193}] &\equiv:y\in \beta.\supset_{y}.\text{E}!Rʻy:\beta\cap \text{ᗡ}ʻR=\beta &\qquad \text{(2)}\\ +\vdash.\text{*33·43}.\supset\vdash\colon\ldotp y\in \beta.\supset_{y}.\text{E}!Rʻy:\supset.\beta\subset \text{ᗡ}ʻR.\\ +[\text{*22·621}] &\supset.\beta\cap \text{ᗡ}ʻR=\beta &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*4·71}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·572.</b> \[\begin{align}&\vdash\colon\ldotp y\in \beta\cap \text{ᗡ}ʻR.\supset_{y}.\text{E}!Rʻy:\equiv.R\upharpoonright \beta\in 1\rightarrow \text{Cls}\\ +&[\text{*71·571.*35·351.*22·43}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_460">[Pg 460]</span></p> + +<p class="nind"><b>*71·58.</b> \(\vdash\colon\colon y,\,z\in \beta.\supset_{y,z}:Rʻy=Rʻz.\equiv.y=z\colon\ldotp \supset.R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·1}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp y\in \beta.\supset_{y}:Rʻy=Rʻy.\equiv.y=y:\\ +[\text{*13·15.*14·28}] \supset_{y}:\text{E}!Rʻy\colon\ldotp \\ +[\text{*71·571}] \supset\colon\ldotp R\upharpoonright \beta\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*3·26.Imp.*11·11·32}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z:\\ +[\text{*35·7}] \supset:(R\upharpoonright \beta)ʻy=(R\upharpoonright \beta)ʻz.\supset_{y,z}.y=z:\\ +[\text{*71·54.(1)}] \supset:R\upharpoonright \beta\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·59.</b> \(\vdash\colon\colon y,z\in \beta.\supset_{y,z}:Rʻy=Rʻz.\equiv.y=z\colon\ldotp \equiv.R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·56}.\supset\vdash\colon\colon R\upharpoonright \beta\in 1\rightarrow 1.\supset\colon\ldotp y\in \text{ᗡ}ʻ(R\upharpoonright \beta).\supset:(R\upharpoonright \beta)ʻy=(R\upharpoonright \beta)ʻz.\equiv.y=z\colon\ldotp\\ +[\text{*35·64·7}]\supset\colon\ldotp y\in \beta\cap \text{ᗡ}ʻR.\supset:y,z\in \beta.Rʻy=Rʻz.\equiv.y=z &\qquad \text{(1)}\\ +\vdash.\text{(1).*22·621}.\supset\vdash\colon\colon R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\supset\colon\ldotp \\ +\qquad\qquad\qquad y\in \beta.\supset:y,\,z\in \beta.Rʻy=Rʻz.\equiv.y=z\colon\ldotp \\ +[\text{*4·73}]\supset\colon\ldotp y,\,z\in \beta.\supset:Rʻy=Rʻz.\equiv.y=z &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·3}.\supset\vdash\colon\colon R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\supset\colon\ldotp \\ +\qquad\qquad\qquad y,\,z\in \beta.\supset_{y,z}:Rʻy=Rʻz.\equiv.y=z &\qquad \text{(3)}\\ +\vdash.\text{(3).*71·58}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in the theory of selections (<a href="#*80·91">*80·91</a>).</p> + +<p class="nind"><b>*71·6.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.R=\dot{s}ʻ\hat{P}\{(\exists y).y\in \text{ᗡ}ʻR.P=(Rʻy)\downarrow y\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11.*13·195}.\supset\\ +\vdash:x[\dot{s}ʻ\hat{P}\{(\exists y).y\in \text{ᗡ}ʻR.P=(Rʻy)\downarrow y\}]z.\equiv.\\ +\qquad\qquad\qquad(\exists y).y\in \text{ᗡ}ʻR.x\{(Rʻy)\downarrow y\}z.\\ +[\text{*55·13}] \equiv.(\exists y).y\in \text{ᗡ}ʻR.x=Rʻy.z=y.\\ +[\text{*13·195}]\equiv.z\in \text{ᗡ}ʻR.x=Rʻz &&\qquad \text{(1)}\\ +\vdash.\text{*71·36.*33·43}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:z\in \text{ᗡ}ʻR.x=Rʻz.\equiv.xRz &&\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·61.</b> \(\vdash:T\in 1\rightarrow \text{Cls}.\supset.Qʻʻʻ\overrightarrow{T}ʻʻ(\text{ᗡ}ʻT\cap \alpha)=\overrightarrow{Q}ʻʻTʻʻ\alpha\)</p> + +<p><span class="pagenum" id="Page_461">[Pg 461]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·103·67·111.*32·12}.\supset\\ +\vdash:\beta\in Qʻʻʻ\overrightarrow{T}ʻʻ(\text{ᗡ}ʻT\cap \alpha).&\equiv.(\exists x).x\in \text{ᗡ}ʻT\cap \alpha.\beta=Qʻʻ\overrightarrow{T}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*53·31.*71·16}.&\supset\vdash:\text{Hp}.x\in \text{ᗡ}ʻT\cap \alpha.\supset.Qʻʻ\overrightarrow{T}ʻx=\overrightarrow{Q}ʻTʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\beta\in Qʻʻʻ\overrightarrow{T}ʻʻ(\text{ᗡ}ʻT\cap \alpha).&\equiv.(\exists x).x\in \text{ᗡ}ʻT\cap \alpha.\beta=\overrightarrow{Q}ʻTʻx.\\ +[\text{*37·67.*71·16}] &\equiv.\beta\in \overrightarrow {Q}ʻʻTʻʻ(\text{ᗡ}ʻT\cap \alpha).\\ +[\text{*37·26}] &\equiv.\beta\in \overrightarrow {Q}ʻʻTʻʻ\alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·611.</b> \(\vdash:T\in \text{Cls}\rightarrow 1.\supset.Qʻʻʻ\overleftarrow{T}ʻʻ(\text{D}ʻT\cap \alpha)=\overrightarrow{Q}ʻʻ\breve{T}ʻʻ\alpha\)</p> + +<p class="nind"><b>*71·612.</b> \(\vdash:T\in 1\rightarrow \text{Cls}.\supset.\breve{Q}ʻʻʻ\overrightarrow{T}ʻʻ(\text{ᗡ}ʻT\cap \alpha)=\overleftarrow{Q}ʻʻTʻʻ\alpha\)</p> + +<p class="nind"><b>*71·613.</b> \(\vdash:T\in \text{Cls}\rightarrow 1.\supset.\breve{Q}ʻʻʻ\overleftarrow{T}ʻʻ(\text{D}ʻT\cap \alpha)=\overleftarrow{Q}ʻʻ\breve{T}ʻʻ\alpha\)</p> + +<p>*71·613 is used in the theory of series (*206·6), and in the theory of +"similarity of position" (*272·131).</p> + +<p class="nind"><b>*71·7.</b> \(\vdash\colon\ldotp Q\in 1\rightarrow \text{Cls}.\supset:xP\mid Qz.\equiv.xP(Qʻz)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·36}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:yQz.\equiv.y=Qʻz:\\ +[\text{Fact}] &\supset:xPy.yQz.\equiv.xPy.y=Qʻz:\\ +[\text{*10·281}]&\supset:(\exists y).xPy.yQz.\equiv.(\exists y).xPy.y=Qʻz:\\ +[\text{*34·1.*13·195}] &\supset:xP\mid Qz.\equiv.xP(Qʻz)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*71·701.</b> \(\vdash\colon\ldotp Q\in \text{Cls}\rightarrow 1.\supset:xQ\mid Pz.\equiv.(\breve{Q}ʻx)Pz\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_462">[Pg 462]</span></p> +<h2 class="nobreak" id="*72">*72. MISCELLANEOUS PROPOSITIONS CONCERNING ONE-MANY, +MANY-ONE, AND ONE-ONE RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *72.</p> + +<p>In this number we shall prove various propositions involving \(1 +\rightarrow \text{Cls}\), \(\text{Cls} \rightarrow 1\), or \(1 +\rightarrow 1\), but not embodying fundamental properties of these +classes of relations.</p> + +<p>The present number begins with various propositions (<a href="#*72·1">*72·1</a>-<a href="#*72·191">·191</a>) +showing that various special relations are one-many or one-one. The +most useful of these are</p> + +<p class="nind"><b>*72·182.</b> \(\vdash. x \downarrow y \in 1 \rightarrow 1\)</p> + +<p class="nind"><b>*72·184.</b> \(\vdash .x\downarrow, \downarrow x \in 1 \rightarrow 1\)</p> + +<p>We have next a set of propositions concerning \(RʻSʻz\) when \(R\) and +\(S\) are one-many, or \(Rʻ\breve{R}ʻz\) when \(R\) is one-one, and +kindred matters. The most useful of these is</p> + +<p class="nind"><b>*72·241.</b> \(\vdash \colon\ldotp R \in 1 \rightarrow 1 .\supset: y \in \text{ᗡ}ʻR .\equiv. y = \breve{R}ʻRʻy\)</p> + +<p>We have next a set of propositions (<a href="#*72·3">*72·3</a>-<a href="#*72·341">·341</a>) concerning products and +sums of classes of relations; of these the one most used is</p> + +<p class="nind"><b>*72·32.</b> \(\vdash \colon\ldotp \lambda \subset 1 \rightarrow \text{Cls}: P,Q \in \lambda. \exists! \text{ᗡ}ʻP \cap \text{ᗡ}ʻQ .\supset_{P,Q}. + P = Q :\supset.\dot{s}ʻ\lambda \in 1 \rightarrow \text{Cls}\)</p> + +<p>which is an extension of <a href="#*71·24">*71·24</a>.</p> + +<p>We have next a set of propositions (<a href="#*72·4">*72·4</a>—<a href="#*72·481">·481</a>) giving various +relations of \(\breve{R}ʻʻ\alpha\) and \(\breve{R}ʻʻ\beta\) when +\(R\in 1 \rightarrow \text{Cls}\), or of \(Rʻʻ\alpha\) and \(Rʻʻ\beta\) +when \(R \in \text{Cls} \rightarrow 1\). The more useful propositions +of this set are those that have the hypothesis \(R \in \text{Cls}\rightarrow 1\); +these are occasionally useful in arithmetic. We have</p> + +<p class="nind"><b>*72·401.</b> \(\vdash: R \in \text{Cls} \rightarrow 1 .\supset. Rʻʻ\alpha \cap Rʻʻ\beta = Rʻʻ(\alpha \cap \beta)\)</p> + +<p class="nind"><b>*72·411.</b> \(\vdash: R \in \text{Cls} \rightarrow 1 . \alpha \cap \beta = \Lambda .\supset. Rʻʻ\alpha \cap Rʻʻ\beta = \Lambda\)</p> + +<p><span class="pagenum" id="Page_463">[Pg 463]</span></p> + +<p>For example, the relation of son to father is many-one. Let \(\alpha\) += Cabinet Ministers, \(\beta\) = fools; then assuming \(\alpha \cap\beta = \Lambda\), +it will follow that the sons of Cabinet Ministers and the sons of +(male) fools have no common member. If we make \(R\) the relation of +son to parent (which is not many-one), it no longer follows that the +sons of Cabinet Ministers and the sons of fools have no common member.</p> + +<p>We have</p> + +<p class="nind"><b>*72·451.</b> \(\vdash:R \in \text{Cls}\rightarrow 1.\supset.R_{{\in}}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR \in 1\rightarrow 1\)</p> + +<p>The effect of this proposition is that if \(\alpha\) and \(\beta\) are both contained in +\(\text{ᗡ}ʻR\), and \(Rʻʻ\alpha=Rʻʻ\beta\), then \(\alpha=\beta\) (using \(R_{{\in}}ʻ\alpha=Rʻʻ\alpha)\).</p> + +<p>We next have a set of propositions concerned with the relations +of \(R_{{\in}}\) and (\(\breve{R})_{{\in}}\), or, what +comes to the same thing, with the circumstances under which +\(\alpha=Rʻʻ\beta.\equiv.\beta=\breve{R}ʻʻ\alpha\) and under which +\(Rʻʻ\breve{R}ʻʻ\alpha=\alpha\). We have</p> + +<p class="nind"><b>*72·502.</b> \(\vdash:R \in 1\rightarrow \text{Cls}.\alpha\subset \text{D}ʻR.\supset.Rʻʻ\breve{R}ʻʻ\alpha=\alpha\)</p> + +<p>Thus for example the fathers of the children of wise fathers are the +class of wise fathers; but the fathers of the children of wise parents +are not all wise, and the parents of the children of wise parents are +not all wise—the first because "\(\alpha\subset \text{D}ʻR\)" fails, +the second because "\(R \in 1\rightarrow \text{Cls}\)" fails.</p> + +<p>We have also</p> + +<p class="nind"><b>*72·52.</b> \(\vdash\colon\ldotp R \in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta\subset \text{ᗡ}ʻR.\supset:\alpha=Rʻʻ\beta.\equiv.\beta=\breve{R}ʻʻ\alpha\)</p> + +<p>We have next a set of propositions (<a href="#*72·59">*72·59</a>—<a href="#*72·66">·66</a>) in which the +relative product \(R\mid \breve{R}\) occurs if \(R \in 1\rightarrow\text{Cls}\), +or \(\breve{R}\mid R\) if \(R \in \text{Cls}\rightarrow 1\). +The most useful propositions in this set are</p> + +<p class="nind"><b>*72·591.</b> \(\vdash:R \in \text{Cls}\rightarrow 1.\supset.S\mid \breve{R}\mid R=S\upharpoonright \text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*72·601.</b> \(\vdash:R \in \text{Cls}\rightarrow 1.\text{ᗡ}ʻS\subset \text{ᗡ}ʻR.\supset.S\mid \breve{R}\mid R=S\)</p> + +<p class="nind"><b>*72·66.</b> \(\vdash:S^{2}\unicode{x2abd}S.S=\breve{S}.\equiv.(\exists R).R \in \text{Cls}\rightarrow 1.S=R\mid \breve{R}\)</p> + +<p>This is the "principle of abstraction." It shows that every relation +which has the formal properties of equality, <i>i.e.</i> which is +transitive and symmetrical, is equal to the relative product of a +many-one relation into its converse; <i>i.e.</i> whenever the relation +\(S\) holds between \(x\) and \(y\), there is a term \(\alpha\) such +that \(xR\alpha.yR\alpha\), where \(R\) is a many-one relation; +and <a href="#*72·64">*72·64</a> shows that this term \(\alpha\) may be taken to be +\(\overleftarrow{S}ʻx\), which is equal to \(\overleftarrow{S}ʻy\). +This principle embodies a great part of the reasons for our definitions +of the various kinds of numbers; in seeking these definitions, we +always have, to begin with, some transitive symmetrical relation which +we regard as sameness of number; thus by *72·64, the desired properties +of the numbers of the kind in question are secured by taking the number +of an object to be the class of objects to which the said object has +the transitive symmetrical relation in question. It is in this way +that we are led to define cardinal numbers as classes of classes, and +ordinal numbers as classes of relations.</p> + +<p><span class="pagenum" id="Page_464">[Pg 464]</span></p> + +<p>The remaining propositions of this number are of less importance, with +the exception of</p> + +<p class="nind"><b>*72·92.</b> \(\vdash:R \in 1\rightarrow \text{Cls}.S\unicode{x2abd}R.\supset.S=R\upharpoonright \text{ᗡ}ʻS\)</p> + +<p>This proposition shows that every relation contained in a one-many +relation is obtainable by a limitation of the converse domain. Thus +<i>e.g.</i> every relation contained in that of father to son can be +specified by specifying the class of sons who are to be its converse +domain; for then all the fathers of these sons must be included to +provide referents. But if we take the relation of parent and child, +which is not one-many or many-one, a contained relation is not +determinate even when both its domain and its converse domain are +given; for the relation may relate some of the children in any one +family to the father and some to the mother, and so long as all the +children and both parents are each related to some one by the relation, +the domain and converse domain remain unchanged by permutations within +the family.</p> + +<hr class="tb"> + +<p class="nind"><b><a id="*72·1">*72·1</a>.</b> \(\vdash.\dot{\Lambda} \in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*25·105}. &\supset\vdash.{\sim}(x\dot{\Lambda}z.y\dot{\Lambda}z).\\ +[\text{*2·21}] &\supset\vdash:x\dot{\Lambda}z.y\dot{\Lambda}z.\supset.x=y:\\ +[\text{*11·11.*71·17}]&\supset\vdash.\dot{\Lambda} \in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\text{Similarly} &\vdash.\dot{\Lambda} \in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*71·103}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·11.</b> \(\vdash.\text{Cnv} \in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·13.*71·166}.&\supset\vdash.\text{Cnv} \in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·54.*31·32·12}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·12.</b> \(\vdash.\overrightarrow{R},\overleftarrow{R} \in 1\rightarrow \text{Cls} \quad[\text{*32·12·121.*71·166}]\)</p> + +<p class="nind"><b>*72·121.</b> \(\vdash.\text{sg},\,\text{gs} \in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·22·221.*71·166}.&\supset\vdash.\text{sg},\,\text{gs} \in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{(1).*32·14·15·21·211.*71·54}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·13.</b> \(\vdash.\text{D} \in 1\rightarrow \text{Cls} \quad[\text{*33·12.*71·166}]\)</p> + +<p class="nind"><b>*72·131.</b> \(\vdash.\text{ᗡ} \in 1\rightarrow \text{Cls} \quad[\text{*33·121.*71·166}]\)</p> + +<p class="nind"><b>*72·132.</b> \(\vdash:C \in 1\rightarrow \text{Cls} \quad[\text{*33·122.*71·166}]\)</p> + +<p class="nind"><b>*72·14.</b> \(\vdash.x\unicode{x2640},\unicode{x2640}x \in 1\rightarrow \text{Cls} \quad[\text{*38·12.*71·166}]\)</p> + +<p>This proposition applies to a great many of the relations we have to +deal with, for example \(\upharpoonleft P\), \(P\upharpoonright\), +\(P \unicode{x0294f}\), \(P\mid\), \(\mid P\), \(x\downarrow\), +\(\downarrow x\), etc.</p> + +<p class="nind"><b>*72·15.</b> \(\vdash.P_{{\in}}\in 1\rightarrow \text{Cls} \quad[\text{*37·111.*71·166}]\)</p> + +<p><span class="pagenum" id="Page_465">[Pg 465]</span></p> + +<p>In *72·16 below, \(p\) has the meaning defined in <a href="#*40·01">*40·01</a>, and does +not represent a variable proposition. Similarly s in *72·161 has the +meaning defined in <a href="#*40·02">*40·02</a>.</p> + +<p class="nind"><b>*72·16.</b> \(\vdash.p\in 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*20·2.(*40·01)}.&\supset\vdash.pʻ\kappa=\hat{x}(\alpha\in \kappa.\supset_{\alpha}.x\in \alpha).\\ +[\text{*14·21}] &\supset\vdash.\text{E}!pʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·166}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·161.</b> \(\vdash.s\in 1\rightarrow \text{Cls} \quad[\text{Proof as in *72·16}]\)</p> + +<p class="nind"><b>*72·162.</b> \(\vdash.\dot{p}\in 1\rightarrow \text{Cls} \quad[\text{Proof as in *72·16}]\)</p> + +<p class="nind"><b>*72·163.</b> \(\vdash.\dot{s}\in 1\rightarrow \text{Cls} \quad[\text{Proof as in *72·16}]\)</p> + +<p class="nind"><b>*72·17.</b> \(\vdash.I\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·22.(*51·01)}.&\supset\vdash.(x).\overrightarrow Iʻx\in 1.\\ +[\text{*71·12}] &\supset\vdash.I\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·21.*50.2}.&\supset\vdash.I\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·18.</b> \(\vdash.\iota\in 1\rightarrow 1 \quad[\text{*51·23.*71·57}]\)</p> + +<p class="nind"><b>*72·181.</b> \(\vdash.\breve{\iota}\in 1\rightarrow 1 \quad[\text{*72·18.*71·212}]\)</p> + +<p class="nind"><b>*72·182.</b> \(\vdash.x\downarrow y\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·13}. &\supset\vdash:z(x\downarrow y)w.\equiv.z=x.w=y: &\qquad \text{(1)}\\ +[\text{*3·47}] & \supset\vdash:z(x\downarrow y)w.z'(x\downarrow y)w.\supset.z=x.z'=x.\\ +[\text{*13·172}] &\supset.z=z' &\qquad \text{(2)}\\ +\vdash.\text{(1).*3·47}.&\supset\vdash:z(x\downarrow y)w.z(x\downarrow y)w'.\supset.w=y.w'=y.\\ +[\text{*13·172}] &\supset.w=w' &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*71·172}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·184.</b> \(\vdash.x\downarrow ,\downarrow x\in 1\rightarrow 1 \quad[\text{*55·2.*71·57}]\)</p> + +<p class="nind"><b>*72·185.</b> \(\vdash.(\downarrow x)_{\in}\in 1\rightarrow 1 \quad[\text{*55·262.*37·11.*72·15.*71·54}]\)</p> + +<p class="nind"><b>*72·19.</b> \(\vdash.\text{Cl}\in 1\rightarrow 1 \quad[\text{*60·55.*71·57}]\)</p> + +<p class="nind"><b><a id="*72·191">*72·191</a>.</b> \(\vdash.\text{Rl}\in 1\rightarrow 1 \quad[\text{*61·55.*71·57}]\)</p> + +<p class="nind"><b>*72·192.</b> \(\vdash.\text{Cl ex}\in 1\rightarrow 1 \quad[\text{*60·56.*71·57}]\)</p> + +<p class="nind"><b>*72·193.</b> \(\vdash.\text{Rl ex}\in 1\rightarrow 1 \quad[\text{*61·56.*71·57}]\)</p> + +<p><span class="pagenum" id="Page_466">[Pg 466]</span></p> + +<p class="nind"><b>*72·2.</b> \(\vdash\colon\ldotp R,\,S\in 1\rightarrow \text{Cls}.\supset:x=RʻSʻz.\equiv.x(R\mid S)z.\equiv.x=(R\mid S)ʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·36}. \supset\vdash\colon\ldotp \text{Hp}.\supset:x=RʻSʻz.&\equiv.xR(Sʻz).\\ +[\text{*71·7}] &\equiv.x(R\mid S)z &\qquad \text{(1)}\\ +\vdash.\text{*71·36·25}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x(R\mid S)z.&\equiv.x=(R\mid S)ʻz &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·201.</b> \(\vdash\colon\ldotp R,\,S\in \text{Cls}\rightarrow 1.\supset:z=\breve{S}ʻ\breve{R}ʻx.\equiv.x(R\mid S)z.\equiv.z=(\breve{S}\mid \breve{R})ʻx\)</p> + +<p class="nind"><b>*72·202.</b> \(\vdash\colon\ldotp R,\,S\in 1\rightarrow 1.\supset:x=RʻSʻz.\equiv.x(R\mid S)z.\equiv.z=\breve{S}ʻ\breve{R}ʻx \quad[\text{*72·2·201}]\)</p> + +<p class="nind"><b>*72·21.</b> \(\vdash\colon\ldotp R,\,S\in 1\rightarrow \text{Cls}.\supset:z\in \breve{S}ʻʻ\text{ᗡ}ʻR.\equiv.\text{E}!RʻSʻz.\equiv.\text{E}!(R\mid S)ʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·25·163}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:z\in \text{ᗡ}ʻ(R\mid S).\equiv.\text{E}!(R\mid S)ʻz &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·32}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:z\in \breve{S}ʻʻ\text{ᗡ}ʻR.\equiv.\text{E}!(R\mid S)ʻz &\qquad \text{(2)}\\ +\vdash.\text{*72·2.*10·11·21·281}.\supset\\ +&\vdash\colon\ldotp \text{Hp}.\supset:(\exists x).x=RʻSʻz.\equiv.(\exists x).x=(R\mid S)ʻz:\\ +[\text{*14·204}] &\supset:\text{E}!RʻSʻz.\equiv.\text{E}!(R\mid S)ʻz &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·211.</b> \(\vdash\colon\ldotp R,\,S\in \text{Cls}\rightarrow 1.\supset:x\in Rʻʻ\text{D}ʻS.\equiv.\text{E}!\breve{S}ʻ\breve{R}ʻx.\equiv.\text{E}!(\breve{S}\mid \breve{R})ʻx\)</p> + +<p class="nind"><b>*72·22.</b> \(\vdash:R,\,S\in 1\rightarrow \text{Cls}.z\in \breve{S}ʻʻ\text{ᗡ}ʻR.\supset.RʻSʻz=(R\mid S)ʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·21}.\supset\vdash:\text{Hp}.&\supset.\text{E}!RʻSʻz.\\ +[\text{*34·41}] &\supset.RʻSʻz=(R\mid S)ʻz:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·221.</b> \(\vdash:R,\,S\in \text{Cls}\rightarrow 1.x\in Rʻʻ\text{D}ʻS.\supset.\breve{S}ʻ\breve{R}ʻx=(\breve{S}\mid \breve{R})ʻx\)</p> + +<p class="nind"><b>*72·23.</b> \(\vdash:R,\,S\in 1\rightarrow \text{Cls}.\supset.RʻʻSʻʻ\gamma=\hat{x}\{(\exists z).z\in \gamma.x=RʻSʻ\gamma\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·33}. \supset\vdash.RʻʻSʻʻ\gamma&=(R\mid S)ʻʻ\gamma &\qquad \text{(1)}\\ +\vdash.\text{*71·25·4}.\supset\vdash:\text{Hp}.\supset.(R\mid S)ʻʻ\gamma&=\hat{x}\{(\exists z).z\in \gamma.x=(R\mid S)ʻ\gamma\}\\ +[\text{*72·2}] & =\hat{x}\{(\exists z).z\in \gamma.x=RʻSʻ\gamma\} &\qquad \text{(2)}\\ +\vdash.(1).(2).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·24.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:x\in \text{D}ʻR.\equiv.x=Rʻ\breve{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·202.*71·212}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x=Rʻ\breve{R}ʻx.&\equiv.x(R\mid \breve{R})x.\\ +[\text{*71·192}] & \equiv.x(I\upharpoonright \text{D}ʻR)x.\\ +[\text{*35·101.*50·1}] &\equiv.x=x.x\in \text{D}ʻR.\\ +[\text{*13·15.*4·73}] &\equiv.x\in \text{D}ʻR\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·241.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:y\in \text{ᗡ}ʻR.\equiv.y=\breve{R}ʻRʻy\)</p> + +<p><span class="pagenum" id="Page_467">[Pg 467]</span></p> + +<p class="nind"><b>*72·242.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:\phi(Rʻ\breve{R}ʻz).\equiv.z\in \text{D}ʻR.\phi z:\phi(\breve{R}ʻRʻz).\equiv.z\in \text{ᗡ}ʻR.\phi z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·501·51}. \supset\vdash:\phi(Rʻ\breve{R}ʻz).&\equiv.(\exists x).x=Rʻ\breve{R}ʻz.\phi x &\qquad \text{(1)}\\ +\vdash.\text{(1).*72·2}. \supset\vdash\colon\ldotp \text{Hp}.\supset:\phi(Rʻ\breve{R}ʻz).&\equiv.(\exists x).x(R\mid \breve{R})z.\phi x.\\ +[\text{*71·192}] &\equiv.(\exists x).x=z.z\in \text{D}ʻR.\phi x.\\ +[\text{*13·195}] &\equiv.z\in \text{D}ʻR.\phi z &\qquad \text{(2)}\\ +\vdash.\text{(2)}\,\frac{\breve{R}}{R}.\,\text{*71·212}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\phi(\breve{R}ʻRʻz).&\equiv.z\in \text{ᗡ}ʻR.\phi z &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·243.</b> \(\vdash\colon\colon R\in 1\rightarrow 1.\supset\colon\ldotp z\in \text{D}ʻR.\phi z.\equiv_{z}.\psi(\breve{R}ʻz):\equiv:\phi(Rʻw).\equiv_{w}.w\in + \text{ᗡ}ʻR.\psi w\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·242}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp z\in \text{D}ʻR.\phi z.\equiv_{z}.\psi(\breve{R}ʻz):\supset:\\ +\qquad\qquad\qquad\phi(Rʻ\breve{R}ʻz).\equiv_{z}.\psi(\breve{R}ʻz):\\ +[\text{Fact}] \supset:\phi(Rʻ\breve{R}ʻz).w=\breve{R}ʻz.\equiv_{z,w}.\psi(\breve{R}ʻz).w=\breve{R}ʻz:\\ +[\text{*14·15}] \supset:\phi(Rʻw).w=\breve{R}ʻz.\equiv_{z,w}.\psi w.w=\breve{R}ʻz:\\ +[\text{*10·281}] \supset:(\exists z).\phi(Rʻw).w=\breve{R}ʻz.\equiv_{w}.(\exists z).\psi w.w=\breve{R}ʻz:\\ +[\text{*71·411}] \supset:\phi(Rʻw).w\in \text{ᗡ}ʻR.\equiv_{w}.\psi w.w\in \text{ᗡ}ʻR:\\ +[\text{*14·21.*71·163}] \supset:\phi(Rʻw).\equiv_{w}.\psi w.w\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1)}\,\frac{\breve{R}}{R}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp w\in \text{ᗡ}ʻR.\psi w.\equiv_{w}.\psi(Rʻw):\supset:\psi(\breve{R}ʻz).\equiv_{z}.\phi + z.z\in \text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in *272·4·41, which are used in the +theory of "rational series," <i>i.e.</i> series ordinally similar to +the series of rationals.</p> + +<p class="nind"><b>*72·25.</b> \(\vdash\colon\ldotp R \rightarrow 1 : (y) \text{E}! \supset. (y) . = \breve{R}ʻRʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·165}.\supset\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:(y).\text{E}!Rʻy.&\equiv.(y).y\in \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*72·241}.\supset\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:(y).y\in \text{ᗡ}ʻR.&\equiv.(y).y=\breve{R}ʻRʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).Imp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The propositions \(\text{Cnv}ʻ\text{Cnv}ʻP=P\) and +\(\breve{\iota}ʻ\iotaʻx=x\), which have been previously proved, are +particular cases of the above; the former is a particular case because +\(\text{Cnv}=\text{Cnv}ʻ\text{Cnv}\).</p> + +<p class="nind"><b>*72·26.</b> \(\vdash:(y).\text{E}!Rʻy.\supset.R=\overrightarrow{\in \mid R}\)</p> + +<p><span class="pagenum" id="Page_468">[Pg 468]</span></p> + +<p>In this proposition, the conditions of significance require that the +domain of \(R\) should consist of classes. This proposition is used in +*72·27.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*37·31}. \supset\vdash .\overrightarrow{{\in}\mid R} &= {\in}_{{\in}}\mid \overrightarrow{R}\\ +[\text{*62·32}] & = s\mid \overrightarrow{R} &\qquad \text{(1)}\\ +\vdash. \text{*53·31}. \supset\vdash : \text{Hp} .\supset . (y). sʻ\overrightarrow{R}ʻy &= sʻ\iotaʻRʻy\\ +[\text{*53·02}] &= Rʻy.\\ +[\text{*34·42}] \supset.s\mid \overrightarrow{R} &= R &\qquad \text{(2)}\\ +\vdash. \text{(1). (2)}. \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·27.</b> \(\vdash.\text{D}=\overrightarrow{{\in}\mid \text{D}}.\text{ᗡ}=\overrightarrow{{\in}·\text{ᗡ}} \quad[\text{*72·26.*33·12·121}]\)</p> + +<p>*72·27 is used in *74·63·631 and again in *163·15.</p> + +<p class="nind"><b><a id="*72·3">*72·3</a>.</b> \(\vdash: \exists ! \lambda \cap (1 \rightarrow \text{Cls}).\supset . \dot{p}ʻ\lambda \in 1 \rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*41·12. Fact}. &\supset\vdash: R \in \lambda . R \in 1 \rightarrow \text{Cls}.\supset.\dot{p}ʻ\lambda\unicode{x2abd}R . R \in 1 \rightarrow \text{Cls}.\\ +[\text{*71·22}] &\supset.\dot{p}ʻ\lambda \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash. \text{(1).*10·11·23}.&\supset\vdash: (\exists R).R \in \lambda .R \in 1 \rightarrow \text{Cls} .\supset. \dot{p}ʻ\lambda \in 1 \rightarrow \text{Cls} &\qquad \text{(2)}\\ +\vdash . \text{(2). *22·33}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·301.</b> \(\vdash: \exists ! \lambda \cap (\text{Cls} \rightarrow 1) .\supset . \dot{p}ʻ\lambda \in \text{Cls} \rightarrow 1\)</p> + +<p class="nind"><b>*72·302.</b> \(\vdash: \exists ! \lambda \cap (1 \rightarrow 1) .\supset . \dot{p}ʻ\lambda \in 1 \rightarrow 1\)</p> + +<p class="nind"><b>*72·303.</b> + \(\vdash: \exists ! \lambda \cap (1 \rightarrow \text{Cls}). \exists ! \lambda \cap (\text{Cls} \rightarrow 1).\supset.\dot{p}ʻ\lambda \in 1 \rightarrow 1 \quad[\text{*72·3·301}]\)</p> + +<p class="nind"><b>*72·31.</b> \(\vdash: \dot{s}ʻ\lambda \in 1 \rightarrow \text{Cls} . \supset . \lambda \subset 1 \rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*41·13}. &\supset\vdash:\dot{s}ʻ\lambda \in 1 \rightarrow \text{Cls}.P \in \lambda.\supset.\dot{s}ʻ\lambda \in 1 \rightarrow \text{Cls}.P \unicode{x2abd} \dot{s}ʻ\lambda.\\ +[\text{*71·22}] &\supset . P \in 1 \rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash. \text{(1). Exp . *10·11·21}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·311.</b> \(\vdash: \dot{s}ʻ\lambda \in \text{Cls} \rightarrow 1 .\supset. \lambda \subset \text{Cls} \rightarrow 1\)</p> + +<p class="nind"><b>*72·312.</b> \(\vdash: \dot{s}ʻ\lambda \in 1 \rightarrow 1 .\supset. \lambda \subset 1 \rightarrow 1\)</p> + +<p class="nind"><b>*72*32.</b> \(\vdash\colon\ldotp \lambda \subset 1 \rightarrow \text{Cls}:P,\,Q \in \lambda.\exists !\text{ᗡ}ʻP \cap \text{ᗡ}ʻQ.\supset_{P,Q}.P=Q:\supset.\dot{s}ʻ\lambda + \in 1 \rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*41·11. *11·54}. \supset\vdash: x(\dot{s}ʻ\lambda)z . y(\dot{s}ʻ\lambda)z. \equiv .\\ +(\exists P,Q).P,\,Q \in \lambda . xPz . yQz.\\ +[\text{*33·14.*4·71}] &\equiv. (\exists P,Q).P,\,Q \in \lambda .xPz.yQz.z \in \text{ᗡ}ʻP \cap \text{ᗡ}ʻQ &\qquad \text{(1)}\\ +\vdash. \text{(1). *4·71}. \supset\vdash \colon\ldotp \text{Hp} . \supset : x (\dot{s}ʻ\lambda)z. y (\dot{s}ʻ\lambda)z.\equiv.\\ +&(\exists P,Q).P,\, Q \in\lambda .xPz.yQz.z \in \text{ᗡ}ʻP \cap \text{ᗡ}ʻQ .P = Q.\\ +[\text{*13·195}] &\supset. (\exists P). P \in \lambda. xPz. yPz.\\ +[\text{*71·17.Hp}] &\supset. x = y &\qquad \text{(2)}\\ +\vdash. \text{(2). *11·11·3 . *71·17}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_469">[Pg 469]</span></p> + +<p class="nind"><b>*72·321.</b> \[\begin{align}&\vdash\colon\ldotp \lambda\subset \text{Cls}\rightarrow 1:P,\,Q\in \lambda.\exists !\text{D}ʻP\cap \text{D}ʻQ.\supset_{P,Q}.P=Q:\supset.\dot{s}ʻ\lambda\in + \text{Cls}\rightarrow 1\\ +&[\text{Proof as in *72·32}]\end{align}\]</p> + +<p class="nind"><b>*72·322.</b> \[\begin{align}\vdash\colon\ldotp \lambda\subset 1\rightarrow 1:&P,\,Q\in \lambda.\exists !\text{ᗡ}ʻP\cap \text{ᗡ}ʻQ.\supset_{P,Q}.P=Q:\\ +&P,\,Q\in \lambda.\exists !\text{D}ʻP\cap \text{D}ʻQ.\supset_{P,Q}.P=Q:\supset.\dot{s}ʻ\lambda\in 1\rightarrow 1\\ +[\text{*72·32·321}]\end{align}\]</p> + +<p class="nind"><b>*72·323.</b> \(\vdash\colon\ldotp \lambda\subset 1\rightarrow 1:P,\,Q\in \lambda.\exists !CʻP\cap CʻQ.\supset_{P,Q}.P=Q:\supset.\dot{s}ʻ\lambda\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·161.*22·49}.&\supset\vdash.\text{ᗡ}ʻP\cap \text{ᗡ}ʻQ\subset CʻP\cap CʻQ.\text{D}ʻP\cap \text{D}ʻQ\subset CʻP\cap CʻQ.\\ +[\text{*24·58}] &\supset\vdash:\exists !\text{ᗡ}ʻP\cap \text{ᗡ}ʻQ.\supset.\exists !CʻP\cap CʻQ:\\ +&\quad\exists !\text{D}ʻP\cap \text{D}ʻQ.\supset.\exists !CʻP\cap CʻQ &\qquad \text{(1)}\\ +\vdash.\text{(1).Syll}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:P,\,Q\in \lambda.\exists !\text{ᗡ}ʻP\cap \text{ᗡ}ʻQ.\supset_{P,Q}.P=Q:\\ +&\qquad\qquad\quad P,\,Q\in \lambda.\exists !\text{D}ʻP\cap \text{D}ʻQ.\supset_{P,Q}.P=Q &\qquad \text{(2)}\\ +\vdash.\text{(2).*72·322}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·34.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\exists !\kappa.\supset.pʻ\breve{R}ʻʻʻ\kappa=\breve{R}ʻʻpʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·35}.&\supset\vdash\colon\ldotp y\in pʻ\breve{R}ʻʻʻ\kappa.\equiv:\beta\in \kappa.\supset_{\beta}.y\in \breve{R}ʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·37}. &\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp y\in pʻ\breve{R}ʻʻʻ\kappa.\equiv:\beta\in \kappa.\supset_{\beta}.Rʻy\in \beta &\qquad \text{(2)}\\ +\vdash.\text{*14·21}. &\supset\vdash\colon\ldotp \beta\in \kappa.\supset.Rʻy\in \beta:\supset:\beta\in \kappa.\supset.\text{E}!Rʻy\colon\ldotp \\ +[\text{*10·52}] &\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \beta\in \kappa.\supset_{\beta}.Rʻy\in \beta:\supset.\text{E}!Rʻy &\qquad \text{(3)}\\ +\vdash.\text{*14·28.*40·1}. &\supset\vdash\colon\colon \text{E}!Rʻy.\supset\colon\ldotp \beta\in \kappa.\supset_{\beta}.Rʻy\in \beta:\equiv.Rʻy\in pʻ\kappa\colon\colon \\ +[\text{(2).(3).*5·32.*14·21}] \supset\vdash\colon\ldotp \text{Hp}.\supset:y\in pʻ\breve{R}ʻʻʻ\kappa.&\equiv.Rʻy\in pʻ\kappa.\\ +[\text{*71·37}] & \equiv.y\in \breve{R}ʻʻpʻ\kappa\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*72·341">*72·341</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\exists !\kappa.\supset.pʻRʻʻʻ\kappa=Rʻʻpʻ\kappa\)</p> + +<p>This proposition should be compared with <a href="#*40·37">*40·37</a> and <a href="#*40·37">*40·38</a>.</p> + +<p class="nind"><b><a id="*72·4">*72·4</a>.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\breve{R}ʻʻ\alpha\cap \breve{R}ʻʻ\beta=\breve{R}ʻʻ(\alpha\cap \beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·37}.\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \breve{R}ʻʻ\alpha\cap \breve{R}ʻʻ\beta.&\equiv.Rʻy\in \alpha.Rʻy\in \beta.\\ +[\text{*22·33}] &\equiv. Rʻy\in \alpha\cap \beta.\\ +[\text{*71·37}] &\equiv.y\in \breve{R}ʻʻ(\alpha\cap \beta)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>When \(R\) is not a \(1\rightarrow \text{Cls}\), we only have in +general (cf. <a href="#*37·21">*37·21</a>) +\[ +\breve{R}ʻʻ(\alpha\cap \beta)\subset \breve{R}ʻʻ\alpha\cap \breve{R}ʻʻ\beta. +\]</p> + +<p class="nind"><b>*72·401.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.Rʻʻ\alpha\cap Rʻʻ\beta=Rʻʻ(\alpha\cap \beta)\)</p> + +<p class="nind"><b>*72·41.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\cap \beta=\Lambda.\supset.\breve{R}ʻʻ\alpha\cap \breve{R}ʻʻ\beta=\Lambda \quad[\text{*72·4.*37·29}]\)</p> + +<p class="nind"><b>*72·411.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\alpha\cap \beta=\Lambda.\supset.Rʻʻ\alpha\cap Rʻʻ\beta=\Lambda\)</p> + +<p><span class="pagenum" id="Page_470">[Pg 470]</span></p> + +<p class="nind"><b>*72·42.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\exists !\breve{R}ʻʻ\alpha\cap\breve{R}ʻʻ\beta.\supset.\exists !\alpha\cap\beta \quad[\text{*72·41.Transp}]\)</p> + +<p class="nind"><b>*72·421.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\exists !Rʻʻ\alpha\cap Rʻʻ\beta.\supset.\exists !\alpha\cap\beta\)</p> + +<p class="nind"><b>*72·43.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\breve{R}ʻʻ\alpha=\breve{R}ʻʻ\beta.\supset.\alpha\cap\text{D}ʻR=\beta\cap\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·37}.\supset\colon\ldotp \text{Hp}.&\supset:Rʻy\in \alpha.\equiv_{y}.Rʻy\in \beta:\\ +[\text{Fact}] &\supset:z=Rʻy.Rʻy\in \alpha.\equiv_{y}.z=Rʻy.Rʻy\in \beta:\\ +[\text{*14·15}] &\supset:z=Rʻy.z\in \alpha.\equiv_{y}.z=Rʻy.z\in \beta:\\ +[\text{*10·281}] &\supset:(\exists y).z=Rʻy.z\in \alpha.\equiv.(\exists y).z=Rʻy.z\in \beta:\\ +[\text{*71·41.*10·35}] &\supset:z\in \text{D}ʻR.z\in \alpha.\equiv.z\in \text{D}ʻR.z\in\beta:\\ +[\text{*22·33}] &\supset:z\in \text{D}ʻR\cap\alpha.\equiv.z\in \text{D}ʻR\cap\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·431.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.Rʻʻ\alpha=Rʻʻ\beta.\supset.\alpha\cap\text{ᗡ}ʻR=\beta\cap\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*72·44.</b> \[\begin{align}&\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset \text{D}ʻR.\beta\subset \text{D}ʻR.\breve{R}ʻʻ\alpha=\breve{R}ʻʻ\beta.\supset.\alpha=\beta\\ +&[\text{*72·43.*22·621}]\end{align}\]</p> + +<p class="nind"><b>*72·441.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\alpha\subset \text{ᗡ}ʻR.\beta\subset \text{ᗡ}ʻR.Rʻʻ\alpha=Rʻʻ\beta.\supset.\alpha=\beta\)</p> + +<p>*72·441 is used in the theory of cardinal exponentiation (*116·659).</p> + +<p class="nind"><b>*72·45.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*60·2}. \supset\vdash:\alpha\subset \text{D}ʻR.\beta\subset \text{D}ʻR.\equiv.\alpha,\beta\in \text{Cl}ʻ\text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*37·11}.\supset\vdash:\breve{R}ʻʻ\alpha=\breve{R}ʻʻ\beta.\equiv.(\breve{R})_{\in}ʻ\alpha=(\breve{R})_{\in}ʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*72·44}.\supset\\ +\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:\alpha,\beta\in \text{Cl}ʻ\text{D}ʻR.(\breve{R})_{\in}ʻ\alpha=(\breve{R})_{\in}ʻ\beta.\supset_{\alpha,\beta}.\alpha=\beta:\\ +[\text{*71·55.*72·15}]\supset:(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\in 1\rightarrow 1\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·451.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*72·46.</b> \[\begin{align}&\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:\breve{R}ʻʻ\alpha=\breve{R}ʻʻ\beta.\equiv.\alpha\cap\text{D}ʻR=\beta\cap\text{D}ʻR\\ +&[\text{*72·43.*37·263}]\end{align}\]</p> + +<p class="nind"><b>*72·461.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:Rʻʻ\alpha=Rʻʻ\beta.\equiv.\alpha\cap\text{ᗡ}ʻR=\beta\cap\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*72·47.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:\breve{R}ʻʻ\alpha=\text{ᗡ}ʻR.\equiv.\text{D}ʻR\subset \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·25.*72·46}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:\breve{R}ʻʻ\alpha=\text{ᗡ}ʻR.&\equiv.\alpha\cap\text{D}ʻR=\text{D}ʻR\cap\text{D}ʻR.\\ +[\text{*22·5·621}] &\equiv.\text{D}ʻR\subset \alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·471.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:Rʻʻ\alpha=\text{D}ʻR.\equiv.\text{ᗡ}ʻR\subset \alpha\)</p> + +<p><span class="pagenum" id="Page_471">[Pg 471]</span></p> + +<p class="nind"><b>*72·48.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\alpha,\beta\in \text{Cl}ʻ\text{D}ʻR.\supset:\breve{R}ʻʻ\alpha=\breve{R}ʻʻ\beta.\equiv.\alpha=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*22·621.\supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha=\beta.&\equiv.\alpha \cap \text{D}ʻR=\beta \cap \text{D}ʻR.\\ +[\text{*72·46}] &\equiv.\breve{R}ʻʻ\alpha=\breve{R}ʻʻ\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*72·481">*72·481</a>.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\alpha,\beta\in \text{Cl}ʻ\text{ᗡ}ʻR.\supset:Rʻʻ\alpha=Rʻʻ\beta.\equiv.\alpha=\beta\)</p> + +<p class="nind"><b>*72·49.</b> \(\vdash\colon\ldotp Q\in 1\rightarrow \text{Cls}.\supset:\text{ᗡ}ʻ(P\mid Q)=\text{ᗡ}ʻQ.\equiv.\text{D}ʻQ\subset \text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·47}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\breve{Q}ʻʻ\text{ᗡ}ʻP=\text{ᗡ}ʻQ.\equiv.\text{D}ʻQ\subset \text{ᗡ}ʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·32}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·491.</b> \(\vdash\colon\ldotp P\in \text{Cls}\rightarrow 1.\supset:\text{D}ʻ(P\mid Q)=\text{D}ʻP.\equiv.\text{ᗡ}ʻP\subset \text{D}ʻQ\)</p> + +<p class="nind"><b>*72·492.</b> \[\begin{align}\vdash\colon\ldotp &P\in \text{Cls}\rightarrow 1.Q\in 1\rightarrow \text{Cls}.\supset:\\ +&\text{D}ʻ(P\mid Q)=\text{D}ʻP.\text{ᗡ}ʻ(P\mid Q)=\text{ᗡ}ʻQ.\equiv.\text{ᗡ}ʻP=\text{D}ʻQ \quad[\text{*72·49·491}]\end{align}\]</p> + +<p class="nind"><b>*72·5.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.Rʻʻ\breve{R}ʻʻ\alpha=\alpha \cap \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·33}. &\supset\vdash.Rʻʻ\breve{R}ʻʻ\alpha=(R\mid \breve{R})ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·19}.&\supset\vdash:\text{Hp}.\supset.Rʻʻ\breve{R}ʻʻ\alpha=(I\upharpoonright \text{D}ʻR)ʻʻ\alpha\\ +[\text{*50·59}] &\qquad\qquad\qquad=\alpha\cap\text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·501.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}ʻʻRʻʻ\alpha=\alpha \cap \text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*72·502.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset \text{D}ʻR.\supset.Rʻʻ\breve{R}ʻʻ\alpha=\alpha \quad[\text{*72·5.*22·621}]\)</p> + +<p class="nind"><b>*72·503.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\alpha\subset \text{ᗡ}ʻR.\supset.\breve{R}ʻʻRʻʻ\alpha=\alpha\)</p> + +<p class="nind"><b>*72·504.</b> \(\vdash:\lambda\subset \text{D}ʻR_{\in}.\supset.R_{\in}ʻʻ\breve{R}_{\in}ʻʻ\lambda=\lambda \quad[\text{*72·502·15}]\)</p> + +<p>Note that \(\breve{R}_{\in}\) means \(\text{Cnv}ʻR_{\in}\), not +(\(\breve{R})_{\in}\). *72·504 is used in the theory of segments of a +series (*211·64).</p> + +<p class="nind"><b>*72·51.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha.\supset.\alpha=Rʻʻ\beta \quad[\text{*72·502.*20·18}]\)</p> + +<p class="nind"><b>*72·511.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset.\beta=\breve{R}ʻʻ\alpha \quad[\text{*72·503.*20·18}]\)</p> + +<p class="nind"><b>*72·512.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\supset:y\in \beta.\equiv.Rʻy\in Rʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·37}. &\supset\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:y\in \breve{R}ʻʻRʻʻ\beta.\equiv.Rʻy\in Rʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*72·503}. &\supset\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\supset:y\in \breve{R}ʻʻRʻʻ\beta.\equiv.y\in \beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·513.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1:(y).\text{E}!Rʻy:\supset:y\in \beta.\equiv.Rʻy\in Rʻʻ\beta \quad[\text{*72·512.*33·431}]\)</p> + +<p class="nind"><b>*72·52.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta\subset \text{ᗡ}ʻR.\supset:\alpha=Rʻʻ\beta.\equiv.\beta=\breve{R}ʻʻ\alpha \quad[\text{*72·51·511}]\)</p> + +<p><span class="pagenum" id="Page_472">[Pg 472]</span></p> + +<p class="nind"><b>*72·53.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\equiv.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·52.*5·32}.\supset\\ +\vdash\colon\ldotp R\in 1\rightarrow 1.&\supset:\alpha\subset \text{D}ʻR.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\equiv.\alpha\subset \text{D}ʻR.\beta\subset \text{ᗡ}ʻR.\beta=\breve{R}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*37·15}.&\supset\vdash:\alpha=Rʻʻ\beta.\supset.\alpha\subset \text{D}ʻR:\\ +[\text{*4·71}] &\supset\vdash:\alpha\subset \text{D}ʻR.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\equiv.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{*37·16}.&\supset\vdash:\beta=\breve{R}ʻʻ\alpha.\supset.\beta\subset \text{ᗡ}ʻR:\\ +[\text{*4·71}] &\supset\vdash:\alpha\subset \text{D}ʻR.\beta\subset \text{ᗡ}ʻR.\beta=\breve{R}ʻʻ\alpha.\equiv.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·54.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\text{Cnv}ʻ(R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR)=(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·131}.\supset\\ +\vdash:\beta\{\text{Cnv}ʻ(R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR)\}\alpha.\equiv.\alpha(R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR)\beta.\\ +[\text{*37·101.*35·101.*60·2}] \equiv.\alpha=Rʻʻ\beta.\beta\subset \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*37·102.*35·101.*60·2}.\supset\\ +\vdash:\beta\{(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\}\alpha.\equiv.\beta=\breve{R}ʻʻ\alpha.\alpha\subset \text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*72·53}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·541.</b> \[\begin{align}&\vdash:R\in 1\rightarrow 1.S=\breve{R}.\supset.\text{Cnv}ʻ(R_{\in}\upharpoonright \text{D}ʻS_{\in})=S_{\in}\upharpoonright + \text{D}ʻR_{\in}\\ +&[\text{*71·48·481.*72·54}]\end{align}\]</p> + +<p class="nind"><b>*72·55.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\alpha\upharpoonleft R=R\upharpoonright \breve{R}ʻʻ\alpha=\alpha\upharpoonleft R\upharpoonright \breve{R}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1.*71·36}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x(\alpha\upharpoonleft R)y.&\equiv.x\in \alpha.x=Rʻy.\\ +[\text{*14·15}] &\equiv.Rʻy\in \alpha.x=Rʻy.\\ +[\text{*71·37}] &\equiv.y\in \breve{R}ʻʻ\alpha.x=Rʻy.\\ +[\text{*71·36.*35·101}] &\equiv.x(R\upharpoonright \breve{R}ʻʻ\alpha)y &\qquad \text{(1)}\\ +\vdash.\text{(1).*35·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·551.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.R\upharpoonright \beta=(Rʻʻ\beta)\upharpoonleft R=(Rʻʻ\beta)\upharpoonleft R\upharpoonright \beta\)</p> + +<p class="nind"><b>*72·57.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.\lambda=\breve{Q}ʻʻ\mu.\supset.\mu\cap \text{D}ʻQ=Qʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·42}. &\supset\vdash:\lambda=\breve{Q}ʻʻ\mu.\supset.(\lambda\upharpoonleft \breve{Q})ʻʻ\mu=\breve{Q}ʻʻ\mu &\qquad \text{(1)}\\ +\vdash.\text{*37·421}. &\supset\vdash:\lambda=\breve{Q}ʻʻ\mu.\supset.(Q\upharpoonright \lambda)ʻʻ\breve{Q}ʻʻ\mu=Qʻʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash:\lambda=\breve{Q}ʻʻ\mu.\supset.(Q\upharpoonright \lambda)ʻʻ(\lambda\upharpoonleft \breve{Q})ʻʻ\mu=Qʻʻ\lambda &\qquad \text{(3)}\\ +\vdash.\text{*72·5.*35·52}.&\supset\vdash:Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.\supset.(Q\upharpoonright \lambda)ʻʻ(\lambda\upharpoonleft \breve{Q})ʻʻ\mu=\mu\cap \text{D}ʻQ &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_473">[Pg 473]</span></p> + +<p class="nind"><b><a id="*72·59">*72·59</a>.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.S\mid R\mid \breve{R}=S\upharpoonright \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·19}.\supset\vdash:\text{Hp}.\supset.S\mid R\mid \breve{R}=S\mid (I\upharpoonright \text{D}ʻR)\\ +[\text{*50·6}] \qquad\qquad\qquad\qquad\qquad=S\upharpoonright \text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·591.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.S\mid \breve{R}\mid R=S\upharpoonright \text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*72·6.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻS\subset \text{D}ʻR.\supset.S\mid R\mid \breve{R}=S \quad[\text{*72·59.*35·452}]\)</p> + +<p class="nind"><b>*72·601.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\text{ᗡ}ʻS\subset \text{ᗡ}ʻR.\supset.S\mid \breve{R}\mid R=S\)</p> + +<p class="nind"><b>*72·61.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻSʻ\subset \text{D}ʻR.\supset.S\mid R\mid \breve{R}\mid \breve{S}=S\mid \breve{S} \quad[\text{*72·6.*34·27}]\)</p> + +<p class="nind"><b>*72·611.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\text{ᗡ}ʻS\subset \text{ᗡ}ʻR.\supset.S\mid \breve{R}\mid R\mid \breve{S}=S\mid \breve{S}\)</p> + +<p>The following propositions lead up to the "principle of abstraction" +(<a href="#*72·66">*72·66</a>), which, though not explicitly referred to in the sequel, has +a certain intrinsic interest, and generalizes a type of reasoning +frequently employed by us.</p> + +<p class="nind"><b><a id="*72·62">*72·62</a>.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.S=R\mid \breve{R}.\supset.S^{2}=S.S=\breve{S}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·21}. &\supset\vdash:S=R\mid \breve{R}.\supset.S^{2}=R\mid (\breve{R}\mid R\mid \breve{R}) &\qquad \text{(1)}\\ +\vdash.\text{*72·6.*33·21}.&\supset\vdash:R\in 1\rightarrow \text{Cls}.\supset.\breve{R}\mid R\mid \breve{R}=\breve{R} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash:\text{Hp}.\supset.S^{2}&=R\mid \breve{R}\\ +[\text{Hp}] & =S &\qquad \text{(3)}\\ +\vdash.\text{(3).*34·7}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·621.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:y(\breve{R}\mid R)z.\equiv.Rʻy=Rʻz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·33}.\supset\vdash\colon\ldotp \text{Hp}.\supset:Rʻy=Rʻz.&\equiv.(\exists x).xRy.x=Rʻz.\\ +[\text{*71·36}] &\equiv.(\exists x).xRy.xRz.\\ +[\text{*31·11}] &\equiv.(\exists x).y\breve{R}x.xRz.\\ +[\text{*34·1}] &\equiv.y(\breve{R}\mid R)z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·622.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:y(R\mid \breve{R})z.\equiv.\breve{R}ʻy=\breve{R}ʻz\)</p> + +<p class="nind"><b>*72·63.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.S=R\mid \breve{R}.\supset.S^{2}=S.S=\breve{S}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·21}. &\supset\vdash:S=R\mid \breve{R}.\supset.S^{2}=(R\mid \breve{R}\mid R)\mid \breve{R} &\qquad \text{(1)}\\ +\vdash.\text{*72·601}.&\supset\vdash:R\in \text{Cls}\rightarrow 1.\supset.R\mid \breve{R}\mid R=R &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash:\text{Hp}.\supset.S^{2}=R\mid \breve{R}\\ +[\text{Hp}] &=S &\qquad \text{(3)}\\ +\vdash.\text{(3).*34·7}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_474">[Pg 474]</span></p> + +<p class="nind"><b><a id="*72·64">*72·64</a>.</b> \(\vdash:S^{2}=S.S=\breve{S}.R=\text{Cnv}ʻ(\overleftarrow{S}\upharpoonright \text{D}ʻS).\supset.R\in \text{Cls}\rightarrow 1.S=R\mid \breve{R}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·12.*71·26}.&\supset\vdash.\overleftarrow{S}\upharpoonright \text{D}ʻS\in 1\rightarrow \text{Cls}.\\ +[\text{*71·21}]&\supset\vdash:\text{Hp}.\supset.R\in \text{Cls}\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{(1).*72·622}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:y(R\mid \breve{R})z.&\equiv.\breve{R}ʻy=\breve{R}ʻz.\\ +[\text{*31·34.Hp}]&\equiv.(\overleftarrow{S}\upharpoonright \text{D}ʻS)ʻy=(\overleftarrow{S}\upharpoonright \text{D}ʻS)ʻz.\\ +[\text{*35·7}] &\equiv.y,z\in \text{D}ʻS.\overleftarrow{S}ʻy=\overleftarrow{S}ʻz.\\ +[\text{*34·85}] &\equiv.z\in \text{D}ʻS.ySz &\qquad \text{(2)}\\ +\vdash.\text{*31·11}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:ySz.\supset.zSy.\\ +[\text{*33·14}] &\supset.z\in \text{D}ʻS:\\ +[\text{*4·71}] &\supset:ySz.\equiv.z\in \text{D}ʻS.ySz &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·65.</b> \(\vdash:S^{2}=S.S=\breve{S}.\equiv.(\exists R).R\in \text{Cls}\rightarrow 1.S=R\mid \breve{R} \quad[\text{*72·63·64}]\)</p> + +<p class="nind"><b><a id="*72·66">*72·66</a>.</b> \(\vdash:S^{2}\unicode{x2abd}S.S=\breve{S}.\equiv.(\exists R).R\in \text{Cls}\rightarrow 1.S=R\mid \breve{R} \quad[\text{*72·65.*34·81}]\)</p> + +<p class="nind"><b>*72·7.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\overleftarrow{R}\upharpoonright \text{D}ʻR\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·4.*22·5}.&\supset\vdash:y,\,z\in \text{D}ʻR.\overleftarrow{R}ʻy=\overleftarrow{R}ʻz.\supset.\exists !\overleftarrow{R}ʻy\cap \overleftarrow{R}ʻz &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·18}.&\supset\vdash:y,\,z\in \text{D}ʻR.\overleftarrow{R}ʻy=\overleftarrow{R}ʻz.\supset.y=z &\qquad \text{(2)}\\ +\vdash.\text{(2).*72·12.*71·55}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·71.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\overrightarrow{R}\upharpoonright \text{ᗡ}ʻR\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*72·72.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\overrightarrow{R}\upharpoonright \text{ᗡ}ʻR,\overleftarrow{R}\upharpoonright \text{D}ʻR\in 1\rightarrow 1\)</p> + +<p class="nind"><b>*72·8.</b> \(\vdash:\lambda\subset \text{D}ʻx\downarrow .\supset.\text{ᗡ}\upharpoonright \lambda\in 1\rightarrow 1 \quad[\text{*55·28·22.*71·58}]\)</p> + +<p>The above proposition is used in <a href="#*72·62">*72·62</a>.</p> + +<p class="nind"><b>*72·81.</b> \(\vdash:\lambda\subset \text{D}ʻ\downarrow x.\supset.\text{D}\upharpoonright \lambda\in 1\rightarrow 1 \quad[\text{*55·281·221.*71·58}]\)</p> + +<p class="nind"><b>*72·9.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.S\unicode{x2abd}R.\supset:\text{E}!Sʻy.\equiv.Rʻy=Sʻy.\equiv.y\in \text{ᗡ}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·22}. \supset\vdash\colon\ldotp \text{Hp}.&\supset:S\in 1\rightarrow \text{Cls}:\\ +[\text{*71·163}] &\supset:\text{E}!Sʻy.\equiv.y\in \text{ᗡ}ʻS &\qquad \text{(1)}\\ +\vdash.\text{*14·21}.&\supset\vdash:Rʻy=Sʻy.\supset.\text{E}!Sʻy &\qquad \text{(2)}\\ +\vdash.\text{*30·32.(1)}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \text{ᗡ}ʻS.\supset.(Sʻy)Sy.\\ +[\text{Hp}] &\supset.(Sʻy)Ry.\\ +[\text{*71·36}] &\supset.Sʻy=Rʻy &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_475">[Pg 475]</span></p> + +<p class="nind"><b>*72·91.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.S\unicode{x2abd}R.\supset.\text{ᗡ}ʻ(R\dot{-}S)=\text{ᗡ}ʻR-\text{ᗡ}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·131.*23·33·35}.\supset\\ +\vdash:y\in \text{ᗡ}ʻ(R\dot{-}S).\equiv.(\exists x).xRy.{\sim}(xSy) &&\qquad \text{(1)}\\ +\vdash.\text{(1).*71·36}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:y\in \text{ᗡ}ʻ(R\dot{-}S).&\equiv.(\exists x).x=Rʻy.{\sim}(x=Sʻy).\\ +[\text{*14·15.*5·32}] &\equiv.(\exists x).x=Rʻy.{\sim}(Rʻy=Sʻy).\\ +[\text{*10·35.*14·204.*72·9}]&\equiv.\text{E}!Rʻy.{\sim}(y\in \text{ᗡ}ʻS).\\ +[\text{*71·163}] &\equiv.y\in \text{ᗡ}ʻR-\text{ᗡ}ʻS\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·911.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.S\unicode{x2abd}R.\supset.\text{D}ʻ(R\dot{-}S)=\text{D}ʻR-\text{D}ʻS\)</p> + +<p class="nind"><b>*72·92.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.S\unicode{x2abd}R.\supset.S=R\upharpoonright \text{ᗡ}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*23·1.*33·14}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:xSy.\supset_{x,y}.xRy.y\in \text{ᗡ}ʻS.\\ +[\text{*35·101}] &\qquad\qquad\qquad\qquad\supset_{x,y}.x(R\upharpoonright \text{ᗡ}ʻS)y:\\ +[\text{*23·1}] &\supset:S\unicode{x2abd}R\upharpoonright \text{ᗡ}ʻS &\qquad \text{(1)}\\ +\vdash.\text{*35·101.*71·36}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x(R\upharpoonright \text{ᗡ}ʻS)y.&\equiv.x=Rʻy.y\in \text{ᗡ}ʻS.\\ +[\text{*72·9}] &\equiv.x=Rʻy.Rʻy=Sʻy.\\ +[\text{*14·142}] &\supset.x=Sʻy.\\ +[\text{*30·31}] &\supset.xSy &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·3}. &\supset\vdash:\text{Hp}.\supset.R\upharpoonright \text{ᗡ}ʻS\unicode{x2abd}S &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·921.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.S\unicode{x2abd}R.\supset.S=(\text{D}ʻS)\upharpoonleft R\)</p> + +<p class="nind"><b>*72·93.</b> \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}S.\equiv:y\in \text{ᗡ}ʻR.\supset_{y}.(Rʻy)Sy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21.*4·71}.\supset\vdash\colon\colon y\in \text{ᗡ}ʻR.\supset_{y}.(Rʻy)Sy:\equiv\colon\ldotp\\ +&y\in \text{ᗡ}ʻR.\supset_{y}.\text{E}!Rʻy.(Rʻy)Sy\colon\ldotp\\ +[\text{*14·25}] &\equiv\colon\ldotp y\in \text{ᗡ}ʻR.\supset_{y}:\text{E}!Rʻy:xRy.\supset_{x}.xSy\colon\ldotp\\ +[\text{*10·29.*11·62}]&\equiv\colon\ldotp y\in \text{ᗡ}ʻR.\supset_{y}.\text{E}!Rʻy:y\in \text{ᗡ}ʻR.xRy.\supset_{x,y}.xSy\colon\ldotp\\ +[\text{*71·16.*33·14}] &\equiv\colon\ldotp R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}S\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*72·931.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.R\unicode{x2abd}S.\equiv:x\in DʻR.\supset_{x}.xS(\breve{R}ʻx)\)</p> + +<p class="nind"><b>*72·94.</b> \(\vdash\colon\ldotp R,\,S\in 1\rightarrow \text{Cls}.\supset:\dot{\exists}!R\dot{\cap}S.\equiv.(\exists y).Rʻy=Sʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·36}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\dot{\exists}!R\dot{\cap}S.&\equiv.(\exists x,y).x=Rʻy.x=Sʻy.\\ +[\text{*14·205}] &\equiv.(\exists y).Rʻy=Sʻy\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_476">[Pg 476]</span></p> +<h2 class="nobreak" id="*73">*73. SIMILARITY OF CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *73.</p> + +<p>Two classes \(\alpha\) and \(\beta\) are said to be <i>similar</i> +when there is a one-one relation whose domain is \(\alpha\) and whose +converse domain is \(\beta\). We express "\(\alpha\) is similar to +\(\beta\)" by the notation "\(\alpha \mathop{\text{ sm }} \beta\)." +When two classes are similar, they have the same cardinal number of +terms: it is this fact which gives importance to the relation of +similarity.</p> + +<p>We have +\[ +\alpha \mathop{\text{ sm }} \beta .\equiv. (\exists R) . R \in 1 \rightarrow 1 . \alpha = \text{D}ʻR . \beta = \text{ᗡ}ʻR\text{.} +\]</p> + +<p>The relation of similarity is that of the domain of a \(1 \rightarrow 1\) +to the converse domain, <i>i.e.</i> it is the relative product of +\(\text{D} \upharpoonright (1 \rightarrow 1)\) and \((1 \rightarrow 1)\upharpoonleft \breve{\text{ᗡ}}\), +or, what comes to the same thing, it is the relative product of +\(\text{D} \upharpoonright 1 \rightarrow 1\) and \(\breve{\text{ᗡ}}\).</p> + +<p>Most of the properties of similarity result immediately from those of +one-one relations and offer no difficulty of any kind.</p> + +<p>When there are relations which correlate \(\alpha\)'s with \(\beta\)'s +so as to make \(\alpha\) similar to \(\beta\), we denote the class of +such relations by "\(\alpha \mathop{\overline{\text{ sm }}} \beta\)." +Thus we have +\[ +\begin{align} +&\alpha \mathop{\overline{\text{ sm }}} \beta = 1 \rightarrow 1 \cap \overleftarrow{\text{D}}ʻ\alpha \cap \overleftarrow{ᗡ}ʻ\beta \quad &\text{Df}\\ +\text{and} \quad &\text{ sm } = \hat{\alpha}\hat{\beta}\{\exists! \alpha \mathop{\overline{\text{ sm }}} \beta\} \quad &\text{Df} \\ +\end{align} +\]</p> + +<p>When, as in this case, we have a descriptive double function closely +connected with a relation, we shall make it a practice to distinguish +the descriptive double function by a bar.</p> + +<p>It is to be observed that "\(\text{ sm }\)," like \(\Lambda\) and +\(\text{V}\) and \(1\) and \(1 \rightarrow 1\), is ambiguous as to +type, and only acquires a definite meaning when the types of its domain +and converse domain are specified. The domain and the converse domain +may or may not be of the same type, <i>i.e.</i> "\(\text{ sm }\)" may +or may not be a homogeneous relation. This enables us to speak of two +classes of different types as having the same number of terms. We +shall return to this point in connection with cardinal numbers (cf. +especially *102—*106).</p> + +<p>The propositions of the present number are important, and are very +frequently referred to throughout cardinal arithmetic. In order to +prove that two classes \(\alpha\) and \(\beta\) have the same cardinal +number of terms, it is<span class="pagenum" id="Page_477">[Pg 477]</span> generally necessary, in the fundamental +arithmetical propositions with which we are concerned, actually to +construct a relation \(R\) such that \(R\in \alpha\,\overline{\text{sm}}\,\beta\). +Such relation will be called a <i>correlator</i> of +\(\alpha\) and \(\beta\). It will usually be obtained by taking some +relation \(S\) for which we have (\(y).\text{E}!Sʻy\), and limiting the +converse domain to \(\beta\), so that \(S\upharpoonright \beta\) is the +required correlator. Very frequently we shall have \(S\in 1\rightarrow\text{Cls}\), +not \(S\in 1\rightarrow 1\), but \(\beta\) will be such +that \(S\upharpoonright \beta\in 1\rightarrow 1\).</p> + +<p>Among the more important propositions of the present number are the +following:</p> + +<p class="nind"><b>*73·142.</b> \(\vdash:R\upharpoonright \beta\in \alpha\overline{\text{ sm }}\beta.\equiv.R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta\)</p> + +<p><i>I.e.</i> \(R\upharpoonright \beta\) is a correlator of \(\alpha\) +and \(\beta\) if (1) \(R\upharpoonright \beta\) is one-one, (2) +\(\beta\) is contained in the converse domain of \(R\), (3) \(\alpha\) +is the class of those terms which have the relation \(R\) to members of +\(\beta\).</p> + +<p class="nind"><b>*73·2.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\text{D}ʻR \text{ sm } \text{ᗡ}ʻR.\text{ᗡ}ʻR \text{ sm } \text{D}ʻR\)</p> + +<p>This results immediately from the definition.</p> + +<p class="nind"><b>*73·22.</b> \(\vdash:R\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\supset.Rʻʻ\beta \text{ sm } \beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta\)</p> + +<p class="nind"><b>*73·3.</b> \(\vdash.\alpha \text{ sm } \alpha.I\upharpoonright \alpha\in \alpha\overline{\text{ sm }}\alpha\)</p> + +<p class="nind"><b>*73·31</b>. \(\vdash:\alpha \text{ sm } \beta.\equiv.\beta \text{ sm } \alpha\)</p> + +<p class="nind"><b>*73·32.</b> \(\vdash:\alpha \text{ sm } \beta.\beta \text{ sm } \gamma.\supset.\alpha \text{ sm } \gamma\)</p> + +<p>The above three propositions show that similarity is reflexive, +symmetrical and transitive.</p> + +<p class="nind"><b>*73·36.</b> \(\vdash\colon\ldotp \alpha \text{ sm } \beta.\supset:\exists !\alpha.\equiv.\exists !\beta\)</p> + +<p class="nind"><b>*73·41.</b> \(\vdash.\iotaʻʻ\alpha\text{ sm }\alpha.\iota\upharpoonright \alpha\in (\iotaʻʻ\alpha)\overline{\text{ sm }}\alpha\)</p> + +<p>Thus every class \(\alpha\) is similar to a class \(\iotaʻʻ\alpha\) of +higher type, and consisting wholly of unit classes.</p> + +<p class="nind"><b>*73·45.</b> \(\vdash.1=\hat{\beta}(\beta \text{ sm } \iotaʻx)\)</p> + +<p>Thus 1 is the class of all classes similar to any unit class.</p> + +<p class="nind"><b>*73·48.</b> \(\vdash.0=\hat{\beta}(\beta \text{ sm } \Lambda)\)</p> + +<p>Thus 0 is the class of all classes similar to the null-class.</p> + +<p class="nind"><b>*73·611.</b> \(\vdash.\downarrow xʻʻ\alpha \text{ sm } \alpha.(\downarrow x)\upharpoonright \alpha\in (\downarrow xʻʻ\alpha)\overline{\text{ sm }}\alpha\)</p> + +<p>This proposition is very often useful. For arithmetical purposes, +we often wish to obtain mutually exclusive classes. Now whether or +not \(\alpha\) and \(\beta\) be mutually exclusive, \(\downarrow xʻʻ\alpha\) +and \(\downarrow yʻʻ\beta\) are mutually exclusive provided +\(x \neq y\). Thus by means of the above proposition we can always +construct mutually exclusive classes each similar to a given class, +<i>i.e.</i> each having some assigned number of members.</p> + +<p class="nind"><b>*73·71.</b> \(\vdash:\alpha \text{ sm }\beta.\gamma \text{ sm }\delta.\alpha\cap \gamma=\Lambda.\beta\cap \delta=\Lambda.\supset.(\alpha\cup \gamma)\text{ sm }(\beta\cup \delta)\)</p> + +<p>This proposition is fundamental in the theory of addition.</p> + +<p><span class="pagenum" id="Page_478">[Pg 478]</span></p> + +<p class="nind"><b>*73·88.</b> \(\vdash:\alpha \text{ sm } \gamma.\beta \text{ sm } \delta.\gamma\subset \beta.\delta\subset \alpha.\supset.\alpha \text{ sm } \beta\)</p> + +<p><i>I.e.</i> "if \(\alpha\) is similar to a part of \(\beta\), and +\(\beta\) is similar to a part of \(\alpha\), then \(\alpha\) is +similar to \(\beta\)." This is the Schröder-Bernstein theorem. The +proof given below is due to Zermelo.</p> + +<hr class="tb"> + +<p class="nind"><b>*73·01.</b> \(\alpha\overline{\text{ sm }}\beta=1\rightarrow 1\cap \overleftarrow{\text{D}}ʻ\alpha\cap \overleftarrow{\text{ᗡ}}ʻ\beta \quad{\text{Df}}\)</p> + +<p class="nind"><b>*73·02.</b> \(\text{ sm }=\hat{\alpha}\hat{\beta}(\exists !\alpha\overline{\text{ sm }}\beta) \quad{\text{Df}}\)</p> + +<p class="nind"><b>*73·03.</b> \(\vdash:R\in \alpha\overline{\text{ sm }}\beta.\equiv.R\in 1\rightarrow 1.\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR \quad[\text{*33·6·61.(*73·01)}]\)</p> + +<p class="nind"><b>*73·04.</b> \(\vdash:\alpha \text{ sm } \beta.\equiv.\exists !\alpha\overline{\text{ sm }}\beta \quad[\text{(*73·02)}]\)</p> + +<p class="nind"><b><a id="*73·1">*73·1</a>.</b> \(\vdash:\alpha \text{ sm } \beta.\equiv.(\exists R).R\in 1\rightarrow 1.\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR \quad[\text{*73·03·04}]\)</p> + +<p class="nind"><b>*73·11.</b> \(\vdash:\alpha \text{ sm } \beta.\equiv.(\exists R).R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·42.*37·25}.\supset\\ +\vdash:R\in 1\rightarrow 1.\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR.\supset.R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha:\\ +[\text{*10·11·28}]\supset\vdash:(\exists R).R\in 1\rightarrow 1.\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR.\supset.\\ +\qquad\qquad\qquad\quad (\exists R).R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha:\\ +[\text{*73·1}]\supset\vdash:\alpha\text{ sm }\beta.\supset.(\exists R).R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*71·29.*37·4.*35·62}.\supset\\ +\vdash:R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha.\supset.\alpha\upharpoonleft R\in 1\rightarrow 1.\alpha=\text{D}ʻ(\alpha\upharpoonleft R).\beta=\text{ᗡ}ʻ(\alpha\upharpoonleft R).\\ +[\text{*10·24}] \qquad\qquad\qquad\supset.(\exists S).S\in 1\rightarrow 1.\alpha=\text{D}ʻS.\beta=\text{ᗡ}ʻS.\\ +[\text{*73·1}] \qquad\qquad\qquad\quad\supset.\alpha\text{ sm }\beta &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23}.\supset\\ +\vdash:(\exists R).R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha.\supset.\alpha\text{ sm }\beta &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·12.</b> \[\begin{align}&\vdash:\alpha\text{ sm }\beta.\equiv.(\exists R).R\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta\\ +&\text{[Proof as in *73·11}]\end{align}\]</p> + +<p class="nind"><b>*73·13.</b> \(\vdash:\alpha\text{ sm }\beta.\equiv.(\exists R).R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta\)</p> + +<p><span class="pagenum" id="Page_479">[Pg 479]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·103·271}. &\supset\vdash:R\in 1\rightarrow 1.\supset.R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1:\\ +[\text{Fact}] &\supset\vdash:R\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset.\\ +\qquad\qquad\qquad &R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta:\\ +[\text{*10·11·28.*73·12}]&\supset\vdash:\alpha\text{ sm }\beta.\supset.\\ +\qquad\qquad\qquad &(\exists R).R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*71·26}.&\supset\vdash:R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\supset.R\upharpoonright \beta\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\\ +[\text{*71·103}] &\supset.R\upharpoonright \beta\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{*35·65.*37·401}.\supset\\ +& \vdash:\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset.\beta=\text{ᗡ}ʻ(R\upharpoonright \beta).\alpha=Dʻ(R\upharpoonright \beta) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash:R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset.\\ +\qquad\qquad\qquad &R\upharpoonright \beta\in 1\rightarrow 1.\alpha=Dʻ(R\upharpoonright \beta).\beta=\text{ᗡ}ʻ(R\upharpoonright \beta).\\ +[\text{*10·24.*73·1}] & \supset.\alpha\text{ sm }\beta &\qquad \text{(4)}\\ +\vdash.\text{(4).*10·11·23}.\supset\\ +\vdash:(\exists R).&R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset.\alpha\text{ sm }\beta &\qquad \text{(5)}\\ +\vdash.\text{(1).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·131.</b> + \[\begin{align}&\vdash:\alpha\text{ sm }\beta.\equiv.(\exists R).R\in \text{Cls}\rightarrow 1.\alpha \upharpoonleft R\in 1\rightarrow \text{Cls}.\alpha\subset DʻR.\beta=\breve{R}ʻʻ\alpha\\ +&[\text{Proof as in *73·13}]\end{align}\]</p> + +<p class="nind"><b>*73·14.</b> \[\begin{align}&\vdash\colon\ldotp \alpha\text{ sm }\beta.\equiv:(\exists R):R\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta:\\ +& y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·55.*5·32}.\supset\\ +\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in 1\rightarrow 1.\equiv:\\ +\qquad\qquad\qquad R\in 1\rightarrow \text{Cls}:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z &\qquad \text{(1)}\\ +\vdash.\text{*71·26}. \supset\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:R\upharpoonright \beta\in 1\rightarrow \text{Cls}:\\ +[\text{*4·73.*71·103}] \supset:R\upharpoonright \beta\in 1\rightarrow 1.\equiv.R\upharpoonright \beta\in \text{Cls}\rightarrow 1\colon\ldotp \\ +[\text{*5·32}] \supset\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in 1\rightarrow 1.\equiv.R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon\ldotp (\exists R).R\in 1\rightarrow \text{Cls}.R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\equiv:\\ +\qquad\qquad\qquad(\exists R):R\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta:\\ +\qquad\qquad\qquad y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z &\qquad \text{(3)}\\ +\vdash.\text{(3).*73·13}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The use of this proposition in proving similarity is very frequent.</p> + +<p class="nind"><b>*73·141.</b> \[\begin{align}\vdash\colon\ldotp \alpha\text{ sm }\beta.\equiv:(\exists R):R\in \text{Cls}\rightarrow 1.\alpha\subset &\text{D}ʻR.\beta=\breve{R}ʻʻ\alpha:\\ +& y,z\in \alpha.\breve{R}ʻy=\breve{R}ʻz.\supset_{y,z}.y=z\\ +[\text{Proof as in *73·14}]\end{align}\]</p> + +<p class="nind"><b>*73·142.</b> \(\vdash:R\upharpoonright \beta\in \alpha\overline{\text{ sm }}\beta.\equiv.R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·03}.\supset\\ +\vdash:R\upharpoonright \beta\in \alpha\overline{\text{ sm }}\beta.&\equiv.R\upharpoonright \beta\in 1\rightarrow 1.\alpha=\text{D}ʻ(R\upharpoonright \beta).\beta=\text{ᗡ}ʻ(R\upharpoonright \beta).\\ +[\text{*37·401.*35·64}] &\equiv.R\upharpoonright \beta\in 1\rightarrow 1.\alpha=Rʻʻ\beta.\beta=\beta \cap \text{ᗡ}ʻR.\\ +[\text{*22·621}] &\equiv.R\upharpoonright \beta\in 1\rightarrow 1.\alpha=Rʻʻ\beta.\beta\subset \text{ᗡ}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·15.</b> \(\vdash:\alpha\text{ sm }\beta.\equiv.(\exists R).R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·12.*71·29}.&\supset\vdash:\alpha\text{ sm }\beta.\supset.(\exists R).R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*73·142·04}.&\supset\vdash:(\exists R).R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset.\alpha\text{ sm }\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_480">[Pg 480]</span></p> + +<p class="nind"><b><a id="*73·2">*73·2</a>.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\text{D}ʻR\text{ sm }\text{ᗡ}ʻR.\text{ᗡ}ʻR\text{ sm }\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*20·2.*3·21}.\supset\\ +\vdash:R\in 1\rightarrow 1.&\supset.R\in 1\rightarrow 1.\text{D}ʻR=\text{D}ʻR.\text{ᗡ}ʻR=\text{ᗡ}ʻR.\\ +[\text{*10·24}] &\supset.(\exists S).S\in 1\rightarrow 1.\text{D}ʻR=\text{D}ʻS.\text{ᗡ}ʻR=\text{ᗡ}ʻS.\\ +[\text{*73·1}] &\supset.\text{D}ʻR\text{ sm }\text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·212}.&\supset\vdash:R\in 1\rightarrow 1.\supset.\text{D}ʻ\breve{R}\text{ sm }\text{ᗡ}ʻ\breve{R}\\ +[\text{*33·2·21}] &\supset.\text{ᗡ}ʻR\text{ sm }\text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions, down to <a href="#*73·241">*73·241</a>, are deduced from +preceding propositions of this number just as "\(\text{D}ʻR\text{ sm }\text{ᗡ}ʻR\)" +was deduced in <a href="#*73·2">*73·2</a> from <a href="#*73·1">*73·1</a>. The proofs are therefore merely +indicated by references to the previous propositions of this number +which are used.</p> + +<p class="nind"><b>*73·21.</b> + \(\vdash:R\in 1\rightarrow 1.\alpha\subset \text{D}ʻR.\supset.\alpha\text{ sm }\breve{R}ʻʻ\alpha.\alpha\upharpoonleft R\in \alpha\overline{\text{ sm }}(\breve{R}ʻʻ\alpha) \quad[\text{*73·11}]\)</p> + +<p class="nind"><b>*73·22.</b> \(\vdash:R\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\supset.Rʻʻ\beta\text{ sm }\beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta \quad[\text{*73·12}]\)</p> + +<p class="nind"><b>*73·23.</b> \[\begin{align}\vdash:R\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR.&R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\supset.\\ +&Rʻʻ\beta\text{ sm }\beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta \quad[\text{*73·13}]\end{align}\]</p> + +<p class="nind"><b>*73·231.</b> \[\begin{align}\vdash:R\in \text{Cls}\rightarrow 1.\alpha\subset DʻR.&\alpha\upharpoonleft R\in 1\rightarrow \text{Cls}.\supset.\\ +&\alpha\text{ sm }\breve{R}ʻʻ\alpha.\alpha\upharpoonleft R\in \alpha\overline{\text{ sm }}(\breve{R}ʻʻ\alpha) \quad[\text{*73·131}]\end{align}\]</p> + +<p class="nind"><b>*73·24.</b> \[\begin{align}\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR&:y,\,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z:\supset.\\ +&Rʻʻ\beta\text{ sm }\beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta \quad[\text{*73·14·142}]\end{align}\]</p> + +<p class="nind"><b><a id="*73·241">*73·241</a>.</b> \[\begin{align}\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\alpha\subset \text{D}ʻR:&y,\,z\in \alpha.\breve{R}ʻy=\breve{R}ʻz.\supset_{y,z}.y=z:\supset.\\ +&\alpha\text{ sm }\breve{R}ʻʻ\alpha.\alpha\upharpoonleft R\in \alpha\overline{\text{ sm }}\breve{R}ʻʻ\alpha \quad[\text{*73·141·03}]\end{align}\]</p> + +<p class="nind"><b><a id="*73·25">*73·25</a>.</b> \(\vdash\colon\ldotp (y).\text{E}!Rʻy:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z:\supset.Rʻʻ\beta\text{ sm }\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·166}.&\supset\vdash:\text{Hp}.\supset.R\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*33·431}.&\supset\vdash:\text{Hp}.\supset.\beta\subset \text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z:\\ +[\text{*73·24}]&\supset:Rʻʻ\beta\text{ sm }\beta\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition will be convenient in such cases as the following: Let +\(\beta\) be a class of relations whose domains are mutually exclusive, +<i>i.e.</i> such that no two members of \(\beta\) have domains which +have a member in common, and suppose we wish to prove that the class +of these domains is similar to \(\beta\).<span class="pagenum" id="Page_481">[Pg 481]</span> The class of domains is +\(\text{D}ʻʻ\beta\), and we have (\(P).\text{E}!\text{D}ʻP\). Hence +we have only to prove (putting \(\text{D}\) in place of the \(R\) of +<a href="#*73·25">*73·25</a>) +\[ +P,\,Q\in \beta.\text{D}ʻP=\text{D}ʻQ.\supset_{P,Q}.P=Q, +\] +which, in the case supposed, is proved immediately.</p> + +<p class="nind"><b>*73·26.</b> \(\vdash\colon\ldotp (y).\text{E}!Rʻy:R\in 1\rightarrow 1:\supset.Rʻʻ\beta \text{ sm }\beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·431}.\supset\vdash:\text{Hp}.&\supset.R\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\\ +[\text{*73·22}] &\supset.Rʻʻ\beta\text{ sm }\beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·27.</b> \[\begin{align}&\vdash\colon\ldotp Rʻy=Rʻz.\equiv_{y,z}.y=z:\supset.Rʻʻ\beta\text{ sm }\beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta\\ +&[\text{*73·26.*71·57}]\end{align}\]</p> + +<p class="nind"><b>*73·28.</b> \[\begin{align}\vdash\colon\colon y,\,z \in \beta.\supset_{y,z}:Rʻy=Rʻz.\equiv.y=&z\colon\ldotp \supset.\\ +&Rʻʻ\beta \text{ sm }\beta.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta \end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{aligned} +\vdash.\text{*71·58.*73·03.*37·421}.\supset\vdash:\text{Hp}.\supset.R\upharpoonright \beta\in (Rʻʻ\beta)\overline{\text{ sm }}\beta:\supset\vdash.\text{Prop} +\end{aligned} +\]</p> + +<p class="nind"><b>*73·3.</b> \(\vdash.\alpha\text{ sm }\alpha.I\upharpoonright \alpha\in \alpha\overline{\text{ sm }}\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·31.*24·11}. &\supset\vdash.\alpha\subset \text{ᗡ}ʻI &\qquad \text{(1)}\\ +\vdash.\text{(1).*72·17.*50·16}.&\supset\vdash.I\in 1\rightarrow 1.\alpha\subset \text{ᗡ}ʻI.Iʻʻ\alpha=\alpha &\qquad \text{(2)}\\ +\vdash.\text{(2).*73·142·04}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This is the <i>reflexive</i> property of similarity. The conditions of +significance require that \(\alpha\) should be a class of some type, +but impose no restriction as to the type of class.</p> + +<p class="nind"><b>*73·301.</b> \(\vdash:R\in \alpha\overline{\text{ sm }}\beta.\equiv.\breve{R}\in \beta\overline{\text{ sm }}\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·03.*71·212.*33·2·21}.\supset\\ +\vdash:R\in \alpha\overline{\text{ sm }}\beta.\equiv.\breve{R}\in 1\rightarrow 1.\text{D}ʻ\breve{R}=\beta.\text{ᗡ}ʻ\breve{R}=\alpha.\\ +[\text{*73·03}] \equiv.\breve{R}\in \beta\overline{\text{ sm }}\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·31.</b> \(\vdash:\alpha\text{ sm }\beta.\equiv.\beta\text{ sm }\alpha \quad[\text{*73·301·04.*31·52}]\)</p> + +<p>This proposition shows that similarity is a <i>symmetrical</i> relation.</p> + +<p class="nind"><b>*73·311.</b> \(\vdash:R\in \alpha\overline{\text{ sm }}\beta.S\in \beta\overline{\text{ sm }}\gamma.\supset.R\mid S\in \alpha\overline{\text{ sm }}\gamma\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·03.*71·252}.\supset\vdash:\text{Hp}.\supset.R\mid S\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{*73·03.*37·32}.\supset\vdash:\text{Hp}.\supset.\text{D}ʻ(R\mid S)=Rʻʻ\beta.\text{ᗡ}ʻ(R\mid S)=\breve{S}ʻʻ\beta.\\ +\qquad\qquad\qquad\qquad\qquad\quad\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR.\beta=\text{D}ʻS.\gamma=\text{ᗡ}ʻS.\\ +[\text{*37·25}] \qquad\qquad\qquad\qquad\supset.\text{D}ʻ(R\mid S)=\alpha.\text{ᗡ}ʻ(R\mid S)=\gamma &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*73·03}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_482">[Pg 482]</span></p> + +<p class="nind"><b>*73·32.</b> \(\vdash:\alpha\text{ sm }\beta.\beta\text{ sm }\gamma.\supset.\alpha\text{ sm }\gamma \quad[\text{*73·311·04}]\)</p> + +<p>This proposition shows that similarity is a <i>transitive</i> relation. +Thus we have now proved that similarity is reflexive, symmetrical, and +transitive.</p> + +<p class="nind"><b>*73·33.</b> \(\vdash.\text{Cnv}ʻ\text{ sm }=\text{ sm } \quad[\text{*73·31.*31·131}]\)</p> + +<p class="nind"><b>*73·34.</b> \(\vdash.\text{ sm }^{2}=\text{ sm }\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*34·55.*73·32}. \supset\vdash.\text{ sm }^{2}\unicode{x2abd}\text{ sm } &\qquad \text{(1)}\\ +&\vdash.\text{(1).*73·33.*34·8}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·35.</b> \(\vdash.\text{D}ʻ\text{ sm }=\text{ᗡ}ʻ\text{ sm }=\text{Cls}\)</p> + + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·3}.&\supset\vdash.\hat{z}(\phi!z)\text{ sm }\hat{z}(\phi!z).\\ +[\text{*20·18}] &\supset\vdash:\alpha=\hat{z}(\phi!z).\supset.\alpha\text{ sm }\alpha:\\ +[\text{*10·11·23}]&\supset\vdash:(\exists \phi).\alpha=\hat{z}(\phi!z).\supset.\alpha\text{ sm }\alpha.\\ +[\text{*33·14}] &\supset.\alpha\in \text{D}ʻ\text{ sm }.\alpha\in \text{ᗡ}ʻ\text{ sm }:\\ +[\text{*20·4}] &\supset\vdash:\alpha\in \text{Cls}.\supset.\alpha\in \text{D}ʻ\text{ sm }.\alpha\in \text{ᗡ}ʻ\text{ sm } &\qquad \text{(1)}\\ +\vdash.\text{*73·1.*10·5}.\supset\\ +&\vdash\colon\ldotp \alpha\text{ sm }\beta.\supset:(\exists R).\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR:\\ +[\text{*10·5.*33·11·111}] &\supset:(\exists R).\alpha=\hat{x}{(\exists y).xRy}:(\exists R).\beta=\hat{y}{(\exists x).xRy}:\\ +[\text{*20·41·18}] &\supset:\alpha\in \text{Cls}.\beta\in \text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23}.\supset\\ +&\vdash\colon\ldotp (\exists \beta).\alpha\text{ sm }\beta.\supset.\alpha\in \text{Cls}:(\exists \alpha).\alpha\text{ sm }\beta.\supset.\beta\in \text{Cls}\colon\ldotp \\ +[\text{*33·13·131}]&\supset\vdash\colon\ldotp \alpha\in \text{D}ʻ\text{ sm }.\supset.\alpha\in \text{Cls}:\beta\in \text{ᗡ}ʻ\text{ sm }.\supset.\beta\in \text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·36.</b> \(\vdash\colon\ldotp \alpha\text{ sm }\beta.\supset:\exists !\alpha.\equiv.\exists !\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·24}.&\supset\vdash\colon\ldotp \alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR.\supset:\exists !\alpha.\equiv.\exists !\beta\colon\ldotp \\ +[\text{*3·42}] &\supset\vdash\colon\ldotp R\in 1\rightarrow 1.\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR.\supset:\exists !\alpha.\equiv.\exists !\beta\colon\ldotp \\ +[\text{*10·11·23}]&\supset\vdash\colon\ldotp (\exists R).R\in 1\rightarrow 1.\alpha=\text{D}ʻR.\beta=\text{ᗡ}ʻR.\supset:\exists !\alpha.\equiv.\exists !\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*73·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·37.</b> \(\vdash\colon\ldotp \alpha\text{ sm }\beta.\supset:\gamma\text{ sm }\alpha.\equiv.\gamma\text{ sm }\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·32}.&\supset\vdash:\alpha\text{ sm }\beta.\gamma\text{ sm }\alpha.\supset.\gamma\text{ sm }\beta &\qquad \text{(1)}\\ +\vdash.\text{*73·31}.&\supset\vdash:\alpha\text{ sm }\beta.\gamma\text{ sm }\beta.\supset.\beta\text{ sm }\alpha.\gamma\text{ sm }\beta.\\ +[\text{*73·32}] &\supset.\gamma\text{ sm }\alpha &\qquad \text{(2)}\\ +\vdash.(1).(2).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·4.</b> \(\vdash.\text{Cnv}ʻʻ\lambda\text{ sm }\lambda.\text{Cnv}\upharpoonright \lambda\in (\text{Cnv}ʻʻ\lambda)\overline{\text{ sm }}\lambda \quad[\text{*73·26.*72·11.*31·13}]\)</p> + +<p><span class="pagenum" id="Page_483">[Pg 483]</span></p> + +<p class="nind"><b>*73·41.</b> \(\vdash.\iotaʻʻ\alpha\text{ sm }\alpha.\iota\upharpoonright \alpha\in (\iotaʻʻ\alpha)\overline{\text{ sm }}\alpha \quad[\text{*73·26.*72·18.*51·12}]\)</p> + +<p>This proposition is useful, because it gives a class +(\(\iotaʻʻ\alpha)\) similar to \(\alpha\) but of higher type. Thus if +\(\mu\) is a cardinal number, and it is known that in a certain type +there are classes having \(\mu\) terms, it follows that there will be +classes having \(\mu\) terms in the next higher type, and therefore in +the next type above that, and so on. No corresponding means exist for +lowering the type.</p> + +<p class="nind"><b>*73·42.</b> \(\vdash:\alpha\subset 1.\supset.\alpha\text{ sm }\breve{\iota}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*52·13}.\supset\vdash:\text{Hp}.\supset.\alpha\subset\text{D}ʻ\iota &\qquad \text{(1)}\\ +&\vdash.\text{(1).*73·21.*72·18}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition gives a means of lowering the type without altering +the cardinal number, provided our class \(\alpha\) is composed wholly of +unit classes; for \(\breve{\iota}ʻʻ\alpha\) is of the type next below +the type of \(\alpha\). But when \(\alpha\) is not composed wholly of +unit classes, this construction fails.</p> + +<p class="nind"><b>*73·43.</b> \(\vdash.\iotaʻx\text{ sm }\iotaʻy.x\downarrow y\in (\iotaʻx)\overline{\text{ sm }}(\iotaʻy) \quad[\text{*55·15.*72·182.*73·2}]\)</p> + +<p class="nind"><b>*73·44.</b> \(\vdash\colon\ldotp \alpha\in 1.\supset:\beta\text{ sm }\alpha.\equiv.\beta\in 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·43}.&\supset\vdash\colon\ldotp \alpha=\iotaʻy.\supset:\beta=\iotaʻx.\supset.\beta\text{ sm }\alpha\colon\ldotp \\ +[\text{*10·11·23}] &\supset\vdash\colon\ldotp (\exists y).\alpha=\iotaʻy.\supset:\beta=\iotaʻx.\supset.\beta\text{ sm }\alpha\colon\ldotp\\ +[\text{*10·11·21·23}]&\supset\vdash\colon\ldotp (\exists y).\alpha=\iotaʻy.\supset:(\exists x).\beta=\iotaʻx.\supset.\beta\text{ sm }\alpha\colon\ldotp\\ +[\text{*52·1}] &\supset\vdash\colon\ldotp \alpha\in 1.\supset:\beta\in 1.\supset.\beta\text{ sm }\alpha &\qquad \text{(1)}\\ +\vdash.\text{*37·25}. &\supset\vdash\colon\ldotp R\in 1\rightarrow 1.\text{D}ʻR=\iotaʻx.\supset.\text{ᗡ}ʻR=\breve{R}ʻʻ\iotaʻx\\ +[\text{*53·31.*71·165}] &=\iotaʻ\breve{R}ʻx.\\ +[\text{*52·22}] &\supset.\text{ᗡ}ʻR\in 1\colon\ldotp\\ +[\text{*20·18}] &\supset\vdash\colon\ldotp R\in 1\rightarrow 1.\text{D}ʻR=\iotaʻx.\text{ᗡ}ʻR=\beta.\supset.\beta\in 1\colon\ldotp\\ +[\text{*10·11·23.*73·1}] &\supset\vdash:\iotaʻx \text{ sm }\beta.\supset.\beta\in 1:\\ +[\text{*20·18}] &\supset\vdash\colon\ldotp \alpha=\iotaʻx.\supset:\alpha\text{ sm }\beta.\supset.\beta\in 1\colon\ldotp\\ +[\text{*10·11·23}] &\supset\vdash\colon\ldotp (\exists x).\alpha=\iotaʻx.\supset:\alpha\text{ sm }\beta.\supset.\beta\in 1\colon\ldotp\\ +[\text{*73·31.*52·1}] &\supset\vdash\colon\ldotp \alpha\in 1.\supset:\beta\text{ sm }\alpha.\supset.\beta\in 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·45.</b> \(\vdash.1=\hat{\beta}(\beta\text{ sm }\iotaʻx)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*52·22.*73·44}.\supset\vdash:\beta\text{ sm }\iotaʻx.\equiv.\beta\in 1 &\qquad \text{(1)}\\ +&\vdash.\text{(1).*20·33}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·46.</b> \(\vdash.\Lambda\text{ sm }\Lambda \quad[\text{*72·1.*33·29.*73·2}]\)</p> + +<p><span class="pagenum" id="Page_484">[Pg 484]</span></p> + +<p class="nind"><b>*73·47.</b> \(\vdash:\beta\text{ sm }\Lambda.\equiv.\beta=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·46}.&\supset\vdash:\beta=\Lambda.\supset.\beta\text{ sm }\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*73·12.*10·5}.\supset\\ +&\vdash:\beta\text{ sm }\Lambda.\supset.(\exists R).\beta=Rʻʻ\Lambda.\\ +[\text{*37·29}] &\supset.\beta=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·48.</b> \(\vdash.0=\hat{\beta}(\beta\text{ sm }\Lambda) \quad[\text{*73·46.*51·11.(*54·01)}]\)</p> + +<p>The following proposition is used in the theory of double similarity +(*111·111).</p> + +<p class="nind"><b>*73·5.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR\unicode{x2abd}\text{ sm }\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·101.*37·101.*60·2}.\supset\\ +\vdash\colon\ldotp R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR\unicode{x2abd}\text{ sm }.\equiv:\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset_{\alpha,\beta}.\alpha\text{ + sm }\beta &&\qquad \text{(1)}\\ +\vdash.\text{*73·22}.\text{Exp}.\supset\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset.\alpha\text{ sm }\beta:\\ +[\text{(1).*11·11·3}] &\supset:R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR\unicode{x2abd}\text{ sm } &\qquad \text{(2)}\\ +\vdash.\text{*30·18.*51·12}.\supset\\ +\vdash\colon\ldotp \beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta.\supset_{\alpha,\beta}.\alpha\text{ sm }\beta:\supset:\iotaʻy\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\iotaʻy.\supset_{\alpha}.\alpha\text{ + sm }\iotaʻy:\\ +[\text{*51·2.*53·301}] &\supset:y\in \text{ᗡ}ʻR.\alpha=\overrightarrow{R}ʻy.\supset_{\alpha}.\alpha\text{ sm }\iotaʻy:\\ +[\text{*20·53.*73·44}] &\supset:y\in \text{ᗡ}ʻR.\supset.\overrightarrow{R}ʻy\in 1:\\ +[\text{*10·11·21.*37·702.*71·1}] &\supset:R\in 1\rightarrow \text{Cls}: &\qquad \text{(3)}\\ +[\text{*72·51.*37·16}] &\supset:\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha.\supset_{\alpha,\beta}.\beta\subset \text{ᗡ}ʻR.\alpha=Rʻʻ\beta &\qquad \text{(4)}\\ +\vdash.\text{(4).*4·7.*11·37}.\supset\vdash\colon\ldotp \text{Hp(4)}.\supset:\alpha\subset \text{D}ʻR.\beta=\breve{R}ʻʻ\alpha.\supset_{\alpha,\beta}.\alpha\text{ sm }\beta:\\ +\left[\text{(3)}\, \frac{\breve{R}}{R}.\text{*71·211.*73·31}\right] &\supset:R\in \text{Cls}\rightarrow 1 &\qquad \text{(5)}\\ +\vdash.\text{(1).(3).(5).*71·103}.&\supset\vdash:R_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻR\unicode{x2abd}\text{ sm }.\supset.R\in 1\rightarrow 1 &\qquad \text{(6)}\\ +\vdash.\text{(2).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·501.</b> \(\vdash:R\in 1\rightarrow 1.\equiv.(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\unicode{x2abd}\text{ sm }\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·212}.\supset\vdash:R\in 1\rightarrow 1.&\equiv.\breve{R}\in 1\rightarrow 1.\\ +[\text{*73·5}] &\equiv.(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻ\breve{R}\unicode{x2abd}\text{ sm }.\\ +[\text{*33·21}] &\equiv.(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\unicode{x2abd}\text{ sm }:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_485">[Pg 485]</span></p> + +<p class="nind"><b>*73·51.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset \text{D}ʻR.\supset.\overleftarrow{R}ʻʻ\alpha\text{ sm }\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·7}. &\supset\vdash:\text{Hp}.\supset.\overleftarrow{R}\upharpoonright \text{D}ʻR\in 1\rightarrow 1.\\ +[\text{*35·431.*71·222}] &\supset.\overleftarrow{R}\upharpoonright \alpha\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{*33·431.*32·121}.&\supset\vdash.\alpha\subset \text{ᗡ}ʻ\overleftarrow{R} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*72·12}.&\supset\vdash:\text{Hp}.\supset.\overleftarrow{R}\in 1\rightarrow \text{Cls}.\overleftarrow{R} \upharpoonright \alpha\in 1\rightarrow 1.\alpha\subset \text{ᗡ}ʻ\overleftarrow{R}.\\ +[\text{*73·23}] &\supset.\overleftarrow{R}ʻʻ\alpha\text{ sm }\alpha:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·511.</b> \[\begin{align}&\vdash:R\in \text{Cls}\rightarrow 1.\alpha\subset \text{ᗡ}ʻR.\supset.\overrightarrow{R}ʻʻ\alpha\text{ sm }\alpha\\ +&\left[\text{*73·51} \frac{\breve{R}}{R}.\text{*71·211.*33·2.*32·241}\right]\end{align}\]</p> + +<p class="nind"><b>*73·52.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset \text{Cls}ʻ\text{D}ʻR.\supset.(\breve{R})_{\in}ʻʻ\alpha\text{ sm }\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·45}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\in 1\rightarrow 1:\\ +[\text{*71·55.*72·15}] &\supset:\xi,\eta\in \text{Cl}ʻ\text{D}ʻR.(\breve{R})_{\in}ʻ\xi=(\breve{R})_{\in}ʻ\eta.\supset_{\xi,\eta}.\xi=\eta:\\ +[\text{Hp}] &\supset:\xi,\,\eta\in \alpha.(\breve{R})_{\in}ʻ\xi=(\breve{R})_{\in}ʻ\eta.\supset_{\xi,\eta}.\xi=\eta:\\ +[\text{*73·25.*37·111}]&\supset:(\breve{R})_{\in}ʻʻ\alpha\text{ sm }\alpha\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·521.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\beta\subset \text{Cl}ʻ\text{ᗡ}ʻR.\supset.R_{\in}ʻʻ\beta\text{ sm }\beta \quad[\text{Proof as in *73·52}]\)</p> + +<p class="nind"><b>*73·53.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\alpha\subset \text{Cl}ʻ\text{D}ʻR.\supset.\breve{R}ʻʻʻ\alpha\text{ sm }\alpha \quad[\text{*73·52.(*37·04)}]\)</p> + +<p class="nind"><b>*73·531.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\beta\subset \text{Cl}ʻ\text{ᗡ}ʻR.\supset.Rʻʻʻ\beta\text{ sm }\beta \quad[\text{*73·521.(*37·04)}]\)</p> + +<p class="nind"><b>*73·61.</b> \(\vdash.x\downarrow ʻʻ\alpha\text{ sm }\alpha.(x\downarrow )\upharpoonright \alpha\in (x\downarrow ʻʻ\alpha)\overline{\text{ sm }}\alpha \quad[\text{*73·27.*55·2}]\)</p> + +<p class="nind"><b>*73·611.</b> \(\vdash.\downarrow xʻʻ\alpha\text{ sm }\alpha.(\downarrow x)\upharpoonright \alpha\in (\downarrow xʻʻ\alpha)\overline{\text{ sm }}\alpha \quad[\text{*73·27.*55·201}]\)</p> + +<p class="nind"><b>*73·62.</b> + \(\vdash:\lambda\subset \text{D}ʻx\downarrow .\supset.\text{ᗡ}ʻʻ\lambda\text{ sm }\lambda.\text{ᗡ}\upharpoonright \lambda\in (\text{ᗡ}ʻʻ\lambda)\overline{\text{ sm }}\lambda \quad[\text{*73·23.*72·131·8}]\)</p> + +<p class="nind"><b>*73·621.</b> + \(\vdash:\lambda\subset \text{D}ʻ\downarrow x.\supset.\text{D}ʻʻ\lambda\text{ sm }\lambda.\text{D}\upharpoonright \lambda\in (\text{D}ʻʻ\lambda)\overline{\text{ sm }}\lambda \quad[\text{*73·23.*72·13·81}]\)</p> + +<p class="nind"><b>*73·63.</b> + \(\vdash:S\in \alpha\overline{\text{ sm }}\beta.T\upharpoonright \alpha,T\upharpoonright \beta\in 1\rightarrow 1.\alpha\cup \beta\subset \text{ᗡ}ʻT.\supset.T\mid S\mid \breve{T}\in (Tʻʻ\alpha)\overline{\text{ sm }}(Tʻʻ\beta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·03.*35·452·453}.&\supset\vdash:\text{Hp}.\supset.T\mid S\mid \breve{T}=T\mid \alpha\upharpoonleft S\upharpoonright \beta\mid \breve{T}\\ +[\text{*35·354}] &=T\upharpoonright \alpha\mid S\mid \beta\upharpoonleft \breve{T}.\\ +[\text{*35·52.*71·252.*73·03}] & \supset.T\mid S\mid \breve{T}\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{*37·32}. &\supset\vdash.\text{D}ʻ(T\mid S\mid \breve{T})=TʻʻSʻʻ\text{ᗡ}ʻT &\qquad \text{(2)}\\ +\vdash.\text{(2).*37·27.*73·03}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻ(T\mid S\mid \breve{T})=Tʻʻ\alpha &\qquad \text{(3)}\\ +\text{Similarly} & \vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(T\mid S\mid \breve{T})=Tʻʻ\beta &\qquad \text{(4)}\\ +\vdash.\text{(1).(3).(4).*73·03}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_486">[Pg 486]</span></p> + +<p>The above proposition is used once in connection with cardinal addition +(*112·231), and once in connection with cardinal multiplication +(*114·561).</p> + +<p>The following proposition (*73·69) is a lemma for *73·7.</p> + +<p class="nind"><b>*73·69.</b> + \(\vdash:R\in \alpha\overline{\text{ sm }}\beta.\alpha\cap \gamma=\Lambda.\beta\cap \gamma=\Lambda.\supset.R\unicode{x228d}I\upharpoonright \gamma\in (\alpha\cup \gamma)\overline{\text{ sm }}(\beta\cup \gamma)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·26·261.*50·5·52}.\supset\\ +\vdash:\text{D}ʻR=\alpha.\text{ᗡ}ʻR=\beta.S=R\unicode{x228d}I\upharpoonright \gamma.\supset.\text{D}ʻS=\alpha\cup \gamma.\text{ᗡ}ʻS=\beta\cup \gamma &\qquad \text{(1)}\\ +\vdash.\text{*71·242.*50·5·52}.\supset\\ +\vdash:\text{Hp(1)}.R\in 1\rightarrow 1.\alpha\cap \gamma=\Lambda.\beta\cap \gamma=\Lambda.\supset.R\unicode{x228d}I\upharpoonright \gamma\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*73·03}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·7.</b> \(\vdash:\alpha\text{ sm }\beta.\alpha\cap \gamma=\Lambda.\beta\cap \gamma=\Lambda.\supset.(\alpha\cup \gamma)\text{ sm }(\beta\cup \gamma) \quad[\text{*73·69·04}]\)</p> + +<p class="nind"><b>*73·701.</b> + \(\vdash:R\in \alpha\overline{\text{ sm }}\beta.S\in \gamma\overline{\text{ sm }}\delta.\alpha\cap \gamma=\Lambda.\beta\cap \delta=\Lambda.\supset.R\unicode{x228d}S\in (\alpha\cup \gamma)\overline{\text{ sm }}(\beta\cup \delta)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·030}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻR\cap \text{D}ʻS=\Lambda.\text{ᗡ}ʻR\cap \text{ᗡ}ʻS=\Lambda.R,\,S\in 1\rightarrow 1.\\ +[\text{*71·242}] &\supset.R\unicode{x228d}S\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{*33·26·261.*73·03}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻ(R\unicode{x228d}S)=\alpha\cup \gamma.\text{ᗡ}ʻ(R\unicode{x228d}S)=\beta\cup \delta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*73·03}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*73·71">*73·71</a>.</b> + \(\vdash:\alpha\text{ sm }\beta.\gamma\text{ sm }\delta.\alpha\cap \gamma=\Lambda.\beta\cap \delta=\Lambda.\supset.(\alpha\cup \gamma)\text{ sm }(\beta\cup \delta) \quad[\text{*73·701·04}]\)</p> + +<p class="nind"><b>*73·72.</b> \(\vdash:\alpha\cup \iotaʻx \text{ sm }\beta\cup \iotaʻy.x{\sim}\in \alpha.y{\sim}\in \beta.\supset.\alpha\text{ sm }\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·1}.\supset\\ +\vdash:\text{Hp}.&\supset.(\exists R).R\in 1\rightarrow 1.\text{D}ʻR=\alpha\cup \iotaʻx.\text{ᗡ}ʻR=\beta\cup \iotaʻy.x{\sim}\in \alpha.y{\sim}\in \beta &\qquad \text{(1)}\\ +\vdash.\text{*71·381}.&\supset\vdash:R\in 1\rightarrow 1.x\in \text{D}ʻR.y\in \text{ᗡ}ʻR.\supset.Rʻʻ(\text{ᗡ}ʻR-\iotaʻ\breve{R}ʻx-\iotaʻy)\\ +&=Rʻʻ\text{ᗡ}ʻR-Rʻʻ\iotaʻ\breve{R}ʻx-Rʻʻ\iotaʻy\\ +[\text{*37·25.*53·31}] & =\text{D}ʻR-\iotaʻRʻ\breve{R}ʻx-\iotaʻRʻy\\ +[\text{*72·24}] & =\text{D}ʻR-\iotaʻx-\iotaʻRʻy.\\ +[\text{*73·22}] &\supset.(\text{D}ʻR-\iotaʻx-\iotaʻRʻy)\text{ sm }(\text{ᗡ}ʻR-\iotaʻy-\iotaʻ\breve{R}ʻx) &\qquad \text{(2)}\\ +\vdash.\text{*71·362.*22·5}.&\supset\vdash:\text{Hp}(2).x=Rʻy.\supset.\\ +&\text{D}ʻR-\iotaʻx-\iotaʻRʻy=\text{D}ʻR-\iotaʻx.\text{ᗡ}ʻR-\iotaʻy-\iotaʻ\breve{R}ʻx=\text{ᗡ}ʻR-\iotaʻy.\\ +[\text{(2)}] &\supset.(\text{D}ʻR-\iotaʻx)\text{ sm }(\text{ᗡ}ʻR-\iotaʻy) &\qquad \text{(3)}\\ +\vdash.\text{*22·92.*33·43}.&\supset\vdash:\text{Hp}(2).x \neq Rʻy.\supset.\\ +&(\text{D}ʻR-\iotaʻx-\iotaʻRʻy)\cup \iotaʻRʻy=\text{D}ʻR-\iotaʻx &\qquad \text{(4)}\\ +\vdash . \text{*71·362}. &\supset\vdash: \text{Hp(4)} .\supset. y \neq \breve{R}ʻx.\\ +[\text{*22·92.*33·44}] &\supset. (\text{ᗡ}ʻR-\iotaʻy-\iotaʻ\breve{R}ʻx)\cup \iotaʻ\breve{R}ʻx=\text{ᗡ}ʻR-\iotaʻy &\qquad \text{(5)}\\ +\vdash.\text{(4).(5).*73·71·43.(2)}.&\supset\vdash: \text{Hp}(4).\supset.(\text{D}ʻR-\iotaʻx)\text{ sm }(\text{ᗡ}ʻR-\iotaʻy) &\qquad \text{(6)}\\ +\vdash.\text{(3).(6)}.&\supset\vdash: \text{Hp}(2) .\supset. (\text{D}ʻR-\iotaʻx)\text{ sm }(\text{ᗡ}ʻR-\iotaʻy) &\qquad \text{(7)}\\ +\vdash. \text{*51·211·22}.&\supset\vdash: \text{D}ʻR= \alpha \cup \iotaʻx.\text{ᗡ}ʻR = \beta\cup \iotaʻy.x{\sim}\in \alpha.y{\sim}\in \beta.\\ +&\supset.\text{D}ʻR-\iotaʻx = \alpha. \text{ᗡ}ʻR-\iotaʻy = \beta &\qquad \text{(8)}\\ +\vdash.\text{(7).(8)}. &\supset\vdash: R \in 1 \rightarrow 1. \text{Hp(8)}. \supset . \alpha \text{ sm } \beta &\qquad \text{(9)}\\ +\vdash.\text{(1).(9)}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p>The following propositions give the proof of the Schröder-Bernstein +theorem, namely: If one class is similar to part of another, and the +other is similar to part of the one, then the two classes are similar. +The proof here given is due to Zermelo<a id="FNanchor_60" href="#Footnote_60" class="fnanchor">[60]</a>. An explanation of the +following proof is given in connection with another proof in the +summary of <a href="#*94">*94</a>.</p> + +<p class="nind"><b>*73·8.</b> + \[\begin{align}\vdash:(\text{ᗡ}ʻR\subset \beta.\beta\subset \text{D}ʻR.\kappa = \hat{\alpha}(\alpha\subset \text{D}ʻR.\beta-\text{ᗡ}ʻR&\subset \alpha.\breve{R}ʻʻ\alpha\subset \alpha).\supset.\\ +&\text{D}ʻR\in \kappa.pʻ\kappa \subset \text{D}ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*22·42·43·44}. & \supset\vdash: \text{Hp}.\supset. \text{D}ʻR \subset \text{D}ʻR.\beta-\text{ᗡ}ʻR \subset \text{D}ʻR &\qquad \text{(1)}\\ +\vdash. \text{*22·44.*37·25}. &\supset\vdash: \text{Hp}.\supset . \breve{R}ʻʻ\text{D}ʻR \subset \text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash: \text{Hp}. \supset . \text{D}ʻR\in \kappa &\qquad \text{(3)}\\ +\vdash.\text{(3).*40·12}. &\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·801.</b> \(\vdash: \text{Hp} *73·8 . \supset . \beta - \text{ᗡ}ʻR pʻ\kappa\)</p> + +<p>Here "Hp*73·8" means "the hypothesis of *73·8."</p> + +<p><i>Dem.</i></p> + +<p>\[ +\vdash. *20·33 .\supset\vdash\colon\ldotp \text{Hp}. \supset : \alpha\in \kappa .\supset_{\alpha}.\beta - \text{ᗡ}ʻR \subset \alpha \colon\ldotp \supset\vdash . \text{Prop} +\]</p> + +<p class="nind"><b>*73·802.</b> \(\vdash: \text{Hp*73·8} .\supset .\breve{R}ʻʻpʻ\kappa \subset pʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*20·33} . \supset\vdash\colon\ldotp \text{Hp} .\supset : \alpha\in \kappa .\supset_{\alpha} .\breve{R}ʻʻ\alpha \subset \alpha &\qquad \text{(1)}\\ +\vdash. \text{(1). *40·81}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*73·81">*73·81</a>.</b> \(\vdash: \text{Hp*73·8} .\supset.pʻ\kappa\in \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\vdash.\text{*73·8·801·802}.\supset\vdash:\text{Hp}.\supset.pʻ\kappa\subset \text{D}ʻR.\beta-\text{ᗡ}ʻR\subset pʻ\kappa.\breve{R}ʻʻpʻ\kappa\subset pʻ\kappa:\supset\vdash. \text{Prop} +\]</p> + +<p class="nind"><b><a id="*73·811">*73·811</a>.</b> \(\vdash : \text{Hp*73·8} . \supset . \breve{R}ʻʻpʻ\kappa \subset pʻ\kappa - (\beta -\text{ᗡ}ʻR)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*37·16} .&\supset\vdash. \breve{R}ʻʻpʻ\kappa &\subset \text{ᗡ}ʻR\\ +[\text{*22·8}] &\subset - (-\text{ᗡ}ʻR)\\ +[\text{*22·81·43}] &\subset - (\beta-\text{ᗡ}ʻR) &\qquad \text{(1)}\\ +\vdash. \text{(1).*73·802} . \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_488">[Pg 488]</span></p> + +<p class="nind"><b>*73·812.</b> \(\vdash:\text{Hp}*73·8.x{\sim}\in (\beta-\text{ᗡ}ʻR)\cup \breve{R}ʻʻpʻ\kappa.\supset.\breve{R}ʻʻ(pʻ\kappa-\iotaʻx)\subset pʻ\kappa-\iotaʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·87}. &\supset\vdash:\text{Hp}.\supset.x{\sim}\in \breve{R}ʻʻpʻ\kappa.\\ +[\text{*51·36}] &\supset.\breve{R}ʻʻpʻ\kappa\subset -\iotaʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*73·802}.&\supset\vdash:\text{Hp}.\supset.\breve{R}ʻʻpʻ\kappa\subset pʻ\kappa-\iotaʻx.\\ +[\text{*37·2}] &\supset.\breve{R}ʻʻ(pʻ\kappa-\iotaʻx)\subset pʻ\kappa-\iotaʻx:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·82.</b> \(\vdash:\text{Hp}*73·812.\supset.pʻ\kappa-\iotaʻx=pʻ\kappa.x{\sim}\in pʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·87.*51·36}.&\supset\vdash:\text{Hp}.\supset.\beta-\text{ᗡ}ʻR\subset -\iotaʻx.\\ +[\text{*73·801}] &\supset.\beta-\text{ᗡ}ʻR\subset pʻ\kappa-\iotaʻx &\qquad \text{(1)}\\ +\vdash.\text{*73·8}.&\supset\vdash:\text{Hp}.\supset.pʻ\kappa-\iotaʻx\subset \text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*73·812}.&\supset\vdash:\text{Hp}.\supset.pʻ\kappa-\iotaʻx\in \kappa.\\ +[\text{*40·12}] & \supset.pʻ\kappa\subset pʻ\kappa-\iotaʻx.\\ +[\text{*51·36.*22·43}] &\supset.x{\sim}\in pʻ\kappa.pʻ\kappa-\iotaʻx=pʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·821.</b> \(\vdash:\text{Hp}*73·8.x\in pʻ\kappa-(\beta-\text{ᗡ}ʻR).\supset.x\in \breve{R}ʻʻpʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·82.Transp}.\supset\vdash:\text{Hp}*73·8.x\in pʻ\kappa.\supset.x\in (\beta-\text{ᗡ}ʻR)\cup \breve{R}ʻʻpʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*5·6}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*73·83">*73·83</a>.</b> \(\vdash:\text{Hp}*73·8.\supset.pʻ\kappa-(\beta-\text{ᗡ}ʻR)=\breve{R}ʻʻpʻ\kappa.pʻ\kappa=(\beta-\text{ᗡ}ʻR)\cup \breve{R}ʻʻpʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*73·821. &\supset\vdash:\text{Hp}.\supset.pʻ\kappa-(\beta-\text{ᗡ}ʻR)\subset \breve{R}ʻʻpʻ\kappa &\qquad \text{(1)}\\ +\vdash.(1).*73·811.&\supset\vdash:\text{Hp}.\supset.pʻ\kappa-(\beta-\text{ᗡ}ʻR)=\breve{R}ʻʻpʻ\kappa &\qquad \text{(2)}\\ +\vdash.(2).*24·47.*73·801.&\supset\vdash:\text{Hp}.\supset.pʻ\kappa=(\beta-\text{ᗡ}ʻR)\cup \breve{R}ʻʻpʻ\kappa &\qquad \text{(3)}\\ +\vdash.(2).(3).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·84.</b> \(\vdash:\text{Hp*73·8}.\supset.\beta=pʻ\kappa\cup (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*22·92.\supset\vdash:\text{Hp}.\supset.\beta=(\beta-\text{ᗡ}ʻR)\cup \text{ᗡ}ʻR\\ +[*22·92.*37·16] =(\beta-\text{ᗡ}ʻR)\cup \breve{R}ʻʻpʻ\kappa\cup (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa)\\ +[*73·83] =pʻ\kappa\cup (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·841.</b> \(\vdash:\text{Hp*73·8}.R\in 1\rightarrow 1.\supset.\beta\text{ sm }\text{ᗡ}ʻR.\beta\text{ sm }\text{D}ʻR\)</p> + + +<p><span class="pagenum" id="Page_489">[Pg 489]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*73·8·21.&\supset\vdash:\text{Hp}.\supset.pʻ\kappa\text{ sm }\breve{R}ʻʻpʻ\kappa &\qquad \text{(1)}\\ +\vdash.*24·21.&\supset\vdash.\breve{R}ʻʻpʻ\kappa\cap (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa)=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*73·83.*24·492.*73·801}.&\supset\\ +\vdash:\text{Hp}.&\supset.pʻ\kappa-\breve{R}ʻʻpʻ\kappa=\beta-\text{ᗡ}ʻR.\\ +[\text{*24·21}]&\supset.pʻ\kappa\cap (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa)=\Lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*73·7}.&\supset\\ +\vdash:\text{Hp}.&\supset.pʻ\kappa\cup (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa)\text{ sm }\breve{R}ʻʻpʻ\kappa\cup (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa).\\ +[\text{*73·84}] &\supset.\beta\text{ sm }\breve{R}ʻʻpʻ\kappa\cup (\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa).\\ +[\text{*22·92.*37·16}]&\supset.\beta\text{ sm }\text{ᗡ}ʻR &\qquad \text{(4)}\\ +\vdash.\text{(4).*73·2}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*73·85.</b> \(\vdash:R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \beta.\beta\subset \text{D}ʻR.\supset.\beta\text{ sm }\text{ᗡ}ʻR.\beta\text{ sm }\text{D}ʻR \quad[\text{*73·841}]\)</p> + +<p class="nind"><b>*73·86.</b> \[\begin{align}\vdash:\text{ᗡ}ʻR\subset \text{D}ʻS.\text{ᗡ}ʻS&\subset \text{D}ʻR.\supset.\\ +&\text{D}ʻ(R\mid S)=\text{D}ʻR.\text{ᗡ}ʻ(R\mid S)\subset \text{ᗡ}ʻS.\text{ᗡ}ʻS\subset \text{D}ʻ(R\mid S)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·321}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻ(R\mid S)=\text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*34·36}. &\supset\vdash:\text{ᗡ}ʻ(R\mid S)\subset \text{ᗡ}ʻS &\qquad \text{(2)}\\ +\vdash.\text{(1)}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻS\subset \text{D}ʻ(R\mid S) &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*73·87">*73·87</a>.</b> \(\vdash:R,\,S\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻS.\text{ᗡ}ʻS\subset \text{D}ʻR.\supset.\text{D}ʻR\text{ sm }\text{D}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·252}.\supset\vdash:\text{Hp}.&\supset.R\mid S\in 1\rightarrow 1.\\ +[\text{*73·86·85}]&\supset.\text{ᗡ}ʻS\text{ sm }\text{D}ʻR.\\ +[\text{*73·2}] &\supset.\text{D}ʻS\text{ sm }\text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*73·88">*73·88</a>.</b> \(\vdash:\alpha\text{ sm }\gamma.\beta\text{ sm }\delta.\gamma\subset \beta.\delta\subset \alpha.\supset.\alpha\text{ sm }\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*73·1}.&\supset\vdash:\text{Hp}.\supset.(\exists R,\,S).R,\,S\in 1\rightarrow 1.\text{D}ʻR=\alpha.\text{ᗡ}ʻR=\gamma.\\ +&\qquad\qquad\qquad\qquad\qquad\text{D}ʻS=\beta.\text{ᗡ}ʻS=\delta.\gamma\subset \beta.\delta\subset \alpha.\\ +[\text{*73·87}] &\supset.(\exists R,\,S).\text{D}ʻR=\alpha.\text{D}ʻS=\beta.\text{D}ʻR\text{ sm }\text{D}ʻS.\\ +[\text{*13·22}] &\supset.\alpha\text{ sm }\beta:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This is the Schröder-Bernstein theorem.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_60" href="#FNanchor_60" class="label">[60]</a> +<i>Math. Annalen</i>, vol. <span class="allsmcap">LXV</span>. Heft 2, February +1908.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_490">[Pg 490]</span></p> +<h2 class="nobreak" id="*74">*74. ON ONE-MANY AND MANY-ONE RELATIONS +WITH LIMITED FIELDS.</h2> +</div> + + +<p><i>Summary of</i> *74.</p> + +<p>The purpose of the present number is to collect together various +propositions in which we have such hypotheses as +\[ +R \upharpoonright \lambda \in 1 \rightarrow \text{Cls}, \kappa \upharpoonleft R \in 1 \rightarrow \text{Cls}\text{, etc.} +\] +or in which such hypotheses are shown to be deducible from others. +Hypotheses of this kind occur very frequently, and it is important to +be able to deal with them easily. For the sake of completeness, we +shall here repeat propositions previously proved on this subject.</p> + +<p>The propositions of this number are mostly of the nature of +lemmas, to be used in the theory of selections (Part II, Section +D), and in cardinal and ordinal arithmetic. The most useful of +them are *74·772·773·774·775. These propositions are concerned +with circumstances under which \(Q \parallel \breve{R}\) or \(\mid\breve{R}\), +with or without some limitation of the converse domain, +is a one-one relation. The reason they are important is that the +correlators by means of which many of the fundamental theorems of +cardinal and ordinal arithmetic are proved are such relations as +\(Q\parallel \breve{R}\) (with the converse domain limited) for +suitable values of \(Q\) and \(R\). The above-mentioned propositions +are as follows:</p> + +<p class="nind"><b>*74·772.</b> \(\vdash \colon\ldotp (x). \text{E}!Qʻx:(y).\text{E}!Rʻy:Q,\,R \in \text{Cls} \rightarrow 1 :\supset. Q \parallel \breve{R} \in 1 \rightarrow 1\)</p> + +<p>The hypothesis of this proposition will be verified if we put, for +example, \(Q = R = \downarrow x\). Thus \((\downarrow x) \parallel +(\text{Cnv}ʻ\downarrow x) \in 1 \rightarrow 1\). This proposition is +used in *116·531, which is used in proving one of the formal laws of +exponentiation, namely \(\mu^{\varpi} \times \nu^{\varpi} = (\mu \times +\nu)^{\varpi}\).</p> + +<p class="nind"><b><a id="*74·773">*74·773</a>.</b> + \[\begin{align}\vdash: &Q \upharpoonright \alpha, R \upharpoonright \beta \in \text{Cls} \rightarrow 1 . \alpha \subset \text{ᗡ}ʻQ.\beta \subset \text{ᗡ}ʻR.sʻ\text{D}ʻʻ\lambda \subset \alpha.sʻ\text{ᗡ}ʻʻ\lambda \subset \beta .\supset.\\ +&(Q \parallel \breve{R}) \upharpoonright \lambda \in 1 \rightarrow 1 . (Q \parallel \breve{R}) \upharpoonright \lambda \in \{(Q \parallel \breve{R})ʻʻ\lambda\}\mathop{\overline{\text{ sm }}} \lambda +\end{align}\]</p> + +<p>This proposition is used in connection with both cardinal and ordinal +multiplication and exponentiation. If \(Q \upharpoonright \alpha\) +and \(R \upharpoonright \beta\) correlate \(\gamma\) with \(\alpha\) +and \(\delta\) with \(\beta\), then if we take for \(\lambda\) the +class of all ordinal couples that can be formed of an \(\alpha\) and +a \(\beta\), (\(Q \parallel \breve{R})ʻʻ\lambda\) will be the class +of all couples<span class="pagenum" id="Page_491">[Pg 491]</span> that can be formed of a \(\gamma\) and a \(\delta\). +Thus in virtue of the above proposition, if \(\gamma\) is similar to +\(\alpha\) and \(\delta\) is similar to \(\beta\), the class of ordinal +couples formed of a \(\gamma\) and a \(\delta\) is similar to the class +of ordinal couples formed of an \(\alpha\) and a \(\beta\). This result +is useful because we define the product of the number of members of +\(\alpha\) and the number of members of \(\beta\) as the number of +ordinal couples formed of an \(\alpha\) and a \(\beta\).</p> + +<p class="nind"><b>*74·774.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1:(y).\text{E}!Rʻy:\supset.\mid \breve{R}\in 1\rightarrow 1\)</p> + +<p>This proposition is useful when, for example, \(R\) is \(\downarrow x\).</p> + +<p class="nind"><b>*74·775.</b> + \[\begin{align}\vdash:Q\upharpoonright &sʻ\text{D}ʻʻ\lambda,R\upharpoonright sʻ\text{ᗡ}ʻʻ\lambda\in \text{Cls}\rightarrow 1.sʻ\text{D}ʻʻ\lambda\subset \text{ᗡ}ʻQ.sʻ\text{ᗡ}ʻʻ\lambda\subset \text{ᗡ}ʻR.\supset.\\ +&(Q\Arrowvert \breve{R})\upharpoonright \lambda\in 1\rightarrow 1.(Q\Arrowvert \breve{R})\upharpoonright \lambda\in \{(Q\Arrowvert \breve{R})ʻʻ\lambda\}\overline{\text{ sm }}\lambda\end{align}\]</p> + +<p>This is a particular case of *74·773, and has similar uses.</p> + +<hr class="tb"> + +<p class="nind"><b>*74·1.</b> + \(\vdash\colon\colon R\upharpoonright \beta\in 1\rightarrow \text{Cls}.\supset\colon\ldotp R\upharpoonright \beta\in 1\rightarrow 1.\equiv:y,\,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*71·55.\supset\vdash\colon\colon \text{Hp}.\\ +\supset\colon\ldotp (R\upharpoonright \beta)\upharpoonright \beta\in 1\rightarrow 1.\equiv:y,\,z\in \beta.(R\upharpoonright \beta)ʻy=(R\upharpoonright \beta)ʻz.\supset_{y,z}.y=z\colon\ldotp + \\ +[*35·31·7] \supset\colon\ldotp R\upharpoonright \beta\in 1\rightarrow 1.\equiv:y,z\in \beta.Rʻy=Rʻz.\supset_{y,z}.y=z\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·11.</b> \(\vdash\colon\ldotp R\upharpoonright \beta\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻR.\equiv:\text{E}‼Rʻʻ\beta \quad[\text{*71·571.(*37·05)}]\)</p> + +<p class="nind"><b>*74·12.</b> \[\begin{align}\vdash\colon\colon R\upharpoonright \beta\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻR.\equiv\colon\ldotp y,z\in \beta.\supset_{y,z}:Rʻy=Rʻz.&\equiv.y=z\\ +&[\text{*71·59}]\end{align}\]</p> + +<p class="nind"><b>*74·13.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.(\breve{R})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻR\in 1\rightarrow 1 \quad[\text{*72·45}]\)</p> + +<p class="nind"><b>*74·131.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.R_{\in}\upharpoonright Clʻ\text{ᗡ}ʻR\in 1\rightarrow 1 \quad[\text{*72·451}]\)</p> + +<p class="nind"><b>*74·14.</b> + \(\vdash:R\in 1\rightarrow \text{Cls}.\beta=\breve{R}ʻʻ\alpha.\supset.\alpha\upharpoonleft R=R\upharpoonright \beta=\alpha\upharpoonleft R\upharpoonright \beta \quad[\text{*72·55}]\)</p> + +<p class="nind"><b>*74·141.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\alpha=Rʻʻ\beta.\supset.\alpha\upharpoonleft R=R\upharpoonright \beta=\alpha\upharpoonleft R\upharpoonright \beta \quad[\text{*72·551}]\)</p> + +<p class="nind"><b>*74.15.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.\lambda=\breve{Q}ʻʻ\kappa.\supset.\kappa\cap \text{D}ʻQ=Qʻʻ\lambda \quad[\text{*72·57}]\)</p> + +<p class="nind"><b>*74·151.</b> \(\vdash:\kappa\upharpoonleft Q\in \text{Cls}\rightarrow 1.\kappa=Qʻʻ\lambda.\supset.\lambda\cap \text{ᗡ}ʻQ=\breve{Q}ʻʻ\kappa\)</p> + +<p class="nind"><b>*74·16.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.\kappa\subset \text{D}ʻQ.\lambda=\breve{Q}ʻʻ\kappa.\supset.\kappa=Qʻʻ\lambda \quad[\text{*74·15.*22·621}]\)</p> + +<p class="nind"><b>*74·161.</b> \(\vdash:\kappa\upharpoonleft Q\in \text{Cls}\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda.\supset.\lambda=\breve{Q}ʻʻ\kappa\)</p> + +<p class="nind"><b>*74·17.</b> \(\vdash:Q\upharpoonright \breve{Q}ʻʻ\kappa\in 1\rightarrow \text{Cls}.\kappa\subset \text{D}ʻQ.\supset.\kappa=Qʻʻ\breve{Q}ʻʻ\kappa \quad[\text{*74·16}]\)</p> + +<p class="nind"><b>*74·171.</b> \(\vdash:(Qʻʻ\lambda)\upharpoonleft Q\in \text{Cls}\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\supset.\lambda=\breve{Q}ʻʻQʻʻ\lambda\)</p> + +<p class="nind"><b>*74·2.</b> \(\vdash:\breve{Q}ʻʻ\alpha\subset \beta.\supset.\alpha\upharpoonleft Q=\alpha\upharpoonleft Q\upharpoonright \beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*37·4.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻ(\alpha\upharpoonleft Q)\subset \beta.\\ +[*35·454] &\supset.\alpha\upharpoonleft Q=\alpha\upharpoonleft Q\upharpoonright \beta:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_492">[Pg 492]</span></p> + +<p class="nind"><b>*74·201.</b> \(\vdash:Qʻʻ\beta\subset \alpha.\supset.Q\upharpoonright \beta=\alpha\upharpoonleft Q\upharpoonright \beta \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*74·21.</b> \(\vdash.\alpha\upharpoonleft Q=\alpha\upharpoonleft Q\upharpoonright \breve{Q}ʻʻ\alpha \quad[\text{*74·2}]\)</p> + +<p class="nind"><b>*74·211.</b> \(\vdash.Q\upharpoonright \beta=(Qʻʻ\beta)\upharpoonleft Q\upharpoonright \beta \quad[\text{*74·201}]\)</p> + +<p class="nind"><b>*74·22.</b> \(\vdash:\text{D}ʻQ\subset \alpha.\supset.Q=\alpha\upharpoonleft Q \quad[\text{*35·451}]\)</p> + +<p class="nind"><b>*74·221.</b> \(\vdash:\text{ᗡ}ʻQ\subset \beta.\supset.Q=Q\upharpoonright \beta \quad[\text{*35·452}]\)</p> + +<p class="nind"><b>*74·23.</b> + \(\vdash:\alpha=Qʻʻ\breve{Q}ʻʻ\alpha.\supset.\alpha\upharpoonleft Q=Q\upharpoonright \breve{Q}ʻʻ\alpha=\alpha\upharpoonleft Q\upharpoonright \breve{Q}ʻʻ\alpha \quad[\text{*74·21·211}]\)</p> + +<p class="nind"><b>*74·231.</b> \(\vdash:\beta=\breve{Q}ʻʻQʻʻ\beta.\supset.Q\upharpoonright \beta=(Qʻʻ\beta)\upharpoonleft Q=(Qʻʻ\beta)\upharpoonleft Q\upharpoonright \beta \quad[\text{*74·21·211}]\)</p> + +<p class="nind"><b>*74·24.</b> \(\vdash:\alpha=Qʻʻ\beta.\beta=\breve{Q}ʻʻ\alpha.\supset.\alpha\upharpoonleft Q=Q\upharpoonright \beta=\alpha\upharpoonleft Q\upharpoonright \beta \quad[\text{*74·23}]\)</p> + +<p class="nind"><b>*74·25.</b> + \[\begin{align}&\vdash:Q\upharpoonright \beta\in 1\rightarrow \text{Cls}.\alpha\subset \text{D}ʻQ.\beta=\breve{Q}ʻʻ\alpha.\supset.\alpha\upharpoonleft Q=Q\upharpoonright \beta=\alpha\upharpoonleft Q\upharpoonright \beta\\ +&[\text{*74·16·24}]\end{align}\]</p> + +<p class="nind"><b>*74·251.</b> + \[\begin{align}&\vdash:\alpha\upharpoonleft Q\in \text{Cls}\rightarrow 1.\beta\subset \text{ᗡ}ʻQ.\alpha=Qʻʻ\beta.\supset.\alpha\upharpoonleft Q=Q\upharpoonright \beta=\alpha\upharpoonleft Q\upharpoonright \beta\\ +&[\text{**74·161·24}]\end{align}\]</p> + +<p class="nind"><b>*74·26.</b> + \(\vdash:Q\upharpoonright \beta\in 1\rightarrow 1.\alpha\subset \text{D}ʻQ.\beta=\breve{Q}ʻʻ\alpha.\equiv.\alpha\upharpoonleft Q\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻQ.\alpha=Qʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·25}.\supset\vdash:Q\upharpoonright \beta\in 1\rightarrow 1.\alpha\subset \text{D}ʻQ.\beta=\breve{Q}ʻʻ\alpha.\supset.\alpha\upharpoonleft Q=Q\upharpoonright \beta.\\ +[\text{*13·12}] \supset.\alpha\upharpoonleft Q\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{*37·16}.\supset\vdash:\beta=\breve{Q}ʻʻ\alpha.\supset.\beta\subset \text{ᗡ}ʻQ &\qquad \text{(2)}\\ +\vdash.\text{*74·16}.\supset\vdash:Q\upharpoonright \beta\in 1\rightarrow 1.\alpha\subset \text{D}ʻQ.\beta=\breve{Q}ʻʻ\alpha.\supset.\alpha=Qʻʻ\beta &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\\ +\vdash:Q\upharpoonright \beta\in 1\rightarrow 1.\alpha\subset \text{D}ʻQ.\beta=\breve{Q}ʻʻ\alpha.\supset.\alpha\upharpoonleft Q\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻQ.\alpha=Qʻʻ\beta &\qquad \text{(4)}\\ +\text{Similarly}\\ +\vdash:\alpha\upharpoonleft Q\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻQ.\alpha=Qʻʻ\beta.\supset.Q\upharpoonright \beta\in 1\rightarrow 1.\alpha\subset \text{D}ʻQ.\beta=\breve{Q}ʻʻ\alpha &\qquad \text{(5)}\\ +\vdash.\text{(4).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·27.</b> \(\vdash:Q\upharpoonright \beta\in 1\rightarrow 1.\beta=\breve{Q}ʻʻQʻʻ\beta.\equiv.(Qʻʻ\beta)\upharpoonleft Q\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·26} \frac{Qʻʻ\beta}{\alpha}.\supset\\ +\vdash:Q\upharpoonright \beta\in 1\rightarrow 1.Qʻʻ\beta\subset \text{D}ʻQ.\beta&=\breve{Q}ʻʻQʻʻ\beta.\equiv.\\ +&(Qʻʻ\beta)\upharpoonleft Q\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻQ.Qʻʻ\beta=Qʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·15.*20·2}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_493">[Pg 493]</span></p> + +<p class="nind"><b>*74·271.</b> + \[\begin{align}&\vdash:\alpha\upharpoonleft Q\in 1\rightarrow 1.\alpha=Qʻʻ\breve{Q}ʻʻ\alpha.\equiv.Q\upharpoonright \breve{Q}ʻʻ\alpha\in 1\rightarrow 1.\alpha\subset \text{D}ʻQ\\ +&\left[\text{*74·26}\, \frac{\breve{Q}ʻʻ\alpha}{\beta}\right]\end{align}\]</p> + +<p class="nind"><b>*74·3.</b> \(\vdash\colon\ldotp Q\upharpoonright \beta\in 1\rightarrow \text{Cls}:(\exists \alpha).\beta=\breve{Q}ʻʻ\alpha:\supset.\breve{Q}ʻʻQʻʻ\beta=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·15}.\supset\vdash:Q\upharpoonright \beta\in 1\rightarrow \text{Cls}.\beta&=\breve{Q}ʻʻ\alpha.\supset.\breve{Q}ʻʻQʻʻ\beta=\breve{Q}ʻʻ(\alpha \cap \text{D}ʻQ)\\ +[\text{*37·261}] &=\breve{Q}ʻʻ\alpha\\ +[\text{Hp}] &=\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·23·35}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·301.</b> \(\vdash\colon\ldotp \alpha\upharpoonleft Q\in \text{Cls}\rightarrow 1:(\exists \beta).\alpha=Qʻʻ\beta:\supset.Qʻʻ\breve{Q}ʻʻ\alpha=\alpha \quad[\text{Similar proof}]\)</p> + +<p class="nind"><b>*74·31.</b> \[\begin{align}\vdash:Q\upharpoonright &\beta\in 1\rightarrow \text{Cls}.\beta\in \text{D}ʻ(\breve{Q})_{\in}.\supset.\\ +&\beta=\breve{Q}ʻʻQʻʻ\beta.\beta\subset \text{ᗡ}ʻQ.Q\upharpoonright \beta=(Qʻʻ\beta)\upharpoonleft Q.(Qʻʻ\beta)\upharpoonleft Q\in 1\rightarrow \text{Cls}\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·3.*37·23}.&\supset\vdash:\text{Hp}.\supset.\beta=\breve{Q}ʻʻQʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*37·23·16}.&\supset\vdash:\text{Hp}.\supset.\beta\subset \text{ᗡ}ʻQ &\qquad \text{(2)}\\ +\vdash.\text{(1).*74·231}. &\supset\vdash:\text{Hp}.\supset.Q\upharpoonright \beta=(Qʻʻ\beta)\upharpoonleft Q &\qquad \text{(3)}\\ +[\text{*13·12}] &\supset.(Qʻʻ\beta)\upharpoonleft Q\in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·311.</b> \[\begin{align}&\vdash:\alpha\upharpoonleft Q\in \text{Cls}\rightarrow 1.\alpha\in \text{D}ʻQ_{\in}.\supset.\\ +&\alpha=Qʻʻ\breve{Q}ʻʻ\alpha.\alpha\subset \text{D}ʻQ.\alpha\upharpoonleft Q=Q\upharpoonright \breve{Q}ʻʻ\alpha.Q\upharpoonright \breve{Q}ʻʻ\alpha\in \text{Cls}\rightarrow 1\\ +&[\text{Similar proof}]\end{align}\]</p> + +<p class="nind"><b>*74·32.</b> \(\vdash:\kappa\subset \text{ᗡ}ʻR.R\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\supset.\overrightarrow{R}\upharpoonright \kappa\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·41}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:y,\,z\in \kappa.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz.\supset.(\exists x).xRy.xRz.\\ +[\text{*35·101}] &\supset.(\exists x).x(R\upharpoonright \kappa)y.x(R\upharpoonright \kappa)z.\\ +[\text{*71·171.Hp}] &\supset.y=z &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·55}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·4.</b> \(\vdash:P\mid (Q\upharpoonright \lambda)=P\mid Q.\equiv.\breve{Q}ʻʻ\text{ᗡ}ʻP\subset \lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·23}.\supset\vdash:P\mid (Q\upharpoonright \lambda)=P\mid Q.&\equiv.(P\mid Q)\upharpoonright \lambda=P\mid Q.\\ +[\text{*35·66}] &\equiv.\text{ᗡ}ʻ(P\mid Q)\subset \lambda.\\ +[\text{*37·32}] &\equiv.\breve{Q}ʻʻ\text{ᗡ}ʻP\subset \lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_494">[Pg 494]</span></p> + +<p class="nind"><b>*74·41.</b> \(\vdash:\text{ᗡ}ʻP\cap \text{D}ʻQ\subset \kappa.\supset.P\mid \kappa\upharpoonleft Q=P\mid Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13·131.*10·23}.\supset\\ +\vdash\colon\ldotp \text{Hp}. &\equiv:xPy.yQz.\supset_{x,y,z}.y\in \kappa:\\ +[\text{*4·71}] &\equiv:xPy.yQz.\equiv_{x,y,z}.xPy.yQz.y\in \kappa:\\ +[\text{*10·281}] &\supset:(\exists y).xPy.yQz.\equiv_{x,z}.(\exists y).xPy.yQz.y\in \kappa:\\ +[\text{*34·1.*35·1}]&\supset:x(P\mid Q)z.\equiv_{x,z}.x(P\mid \kappa\upharpoonleft Q)z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·42.</b> \(\vdash:\text{ᗡ}ʻP\subset Qʻʻ\lambda.\supset.\text{D}ʻ(P\mid Q\upharpoonright \lambda)=\text{D}ʻP \quad[\text{*37·321·401}]\)</p> + +<p class="nind"><b>*74·43.</b> \(\vdash:Qʻʻ\lambda\subset \text{ᗡ}ʻP.\supset.\text{ᗡ}ʻ(P\mid Q\upharpoonright \lambda)=\text{ᗡ}ʻQ\cap \lambda \quad[\text{*37·322·401.*35·64}]\)</p> + +<p class="nind"><b>*74·44.</b> \[\begin{align}\vdash:\text{ᗡ}ʻP=Qʻʻ\lambda.\supset.\text{D}ʻ(P\mid Q\upharpoonright \lambda)=\text{D}ʻP.\text{ᗡ}ʻ&(P\mid Q\upharpoonright \lambda)=\text{ᗡ}ʻQ\cap \lambda\\ +&[\text{*74·42·43}]\end{align}\]</p> + +<p class="nind"><b>*74·5.</b> \(\vdash:\text{E}!(P\upharpoonright \beta)ʻy.\equiv.y\in \beta.\text{E}!Pʻy.\equiv.(P\upharpoonright \beta)ʻy=Pʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·7}. &\supset\vdash:x=(P\upharpoonright \beta)ʻy.\equiv.y\in \beta.x=Pʻy &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·281}.&\supset\vdash\colon\ldotp (\exists x).x=(P\upharpoonright \beta)ʻy.\equiv:y\in \beta:(\exists x).x=Pʻy\colon\ldotp \\ +[\text{*14·204}] & \supset\vdash\colon\ldotp \text{E}!(P\upharpoonright \beta)ʻy.\equiv.y\in \beta.\text{E}!Pʻy &\qquad \text{(2)}\\ +\vdash.\text{*35·7}. &\supset\vdash:(P\upharpoonright \beta)ʻy=Pʻy.\equiv.y\in \beta.Pʻy=Pʻy.\\ +[\text{*14·28}] &\qquad\qquad\qquad\qquad\quad\equiv.y\in \beta.\text{E}!Pʻy &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·51.</b> \(\vdash\colon\ldotp \overrightarrow{P}ʻy\subset \alpha.\supset:\text{E}!(\alpha\upharpoonleft P)ʻy.\equiv.\text{E}!Pʻy.\equiv.Pʻy=(\alpha\upharpoonleft P)ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18.*35·1}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:xPy.\equiv_{x}.x(\alpha\upharpoonleft P)y: &\qquad \text{(1)}\\ +[\text{*30·34}] &\qquad\qquad\supset:\text{E}!(\alpha\upharpoonleft P)ʻy.\equiv.\text{E}!Pʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).*30·341}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\text{E}!Pʻy.\equiv.Pʻy=(\alpha\upharpoonleft P)ʻy &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·511.</b> + \[\begin{align}&\vdash\colon\ldotp \overleftarrow{P}ʻx\subset \beta.\supset:\text{E}!(\breve{P}\upharpoonright \beta)ʻx.\equiv.\text{E}!\breve{P}ʻx.\equiv.\breve{P}ʻx=(\breve{P}\upharpoonright \beta)ʻx\\ +&[\text{Proof as in *74·51}]\end{align}\]</p> + +<p class="nind"><b>*74·52.</b> \(\vdash:(Sʻʻ\beta)\upharpoonleft S\in 1\rightarrow \text{Cls}.\beta\subset \text{ᗡ}ʻS.y\in \beta.\supset.\{(Sʻʻ\beta)\upharpoonleft S\}ʻy=Sʻy.\text{E}!Sʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·18}.\supset\vdash:\text{Hp}.&\supset.\overrightarrow{S}ʻy\subset Sʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{*37.1}. \supset\vdash:\text{Hp}.&\supset.(\exists x).xSy.x\in Sʻʻ\beta.\\ +[\text{*33·131}] &\supset.y\in \text{ᗡ}ʻ\{(Sʻʻ\beta)\upharpoonleft S\}.\\ +[\text{*71·16}] &\supset.\text{E}!\{(Sʻʻ\beta)\upharpoonleft S\}ʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*74·51}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_495">[Pg 495]</span></p> + +<p class="nind"><b>*74·521.</b> + \[\begin{align}&\vdash:S\upharpoonright \breve{S}ʻʻ\beta\in \text{Cls}\rightarrow 1.\beta\subset \text{D}ʻS.y\in \beta.\supset.\{(\breve{S}ʻʻ\beta)\upharpoonleft \breve{S}\}ʻy=\breve{S}ʻy.\text{E}!\breve{S}ʻy\\ +&\left[\text{*74·52}\, \frac{\breve{S}}{S}\right]\end{align}\]</p> + +<p class="nind"><b>*74·53.</b> \(\vdash:(Sʻʻ\beta)\upharpoonleft S\in 1\rightarrow 1.\beta\subset \text{ᗡ}ʻS.y\in \beta.\supset.\breve{S}ʻSʻy=y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·1.*33·131}.\supset\vdash:\text{Hp}.&\supset.y\in \text{ᗡ}ʻ{(Sʻʻ\beta)\upharpoonleft S}.\\ +[\text{*72·241.*35·51}]& \supset.(\breve{S}\upharpoonright Sʻʻ\beta)ʻ\{(Sʻʻ\beta)\upharpoonleft S\}ʻy=y (&\qquad \text{(1)}\\ +\vdash.\text{*74·52}.&\supset\vdash:\text{Hp}.\supset.\{(Sʻʻ\beta)\upharpoonleft S\}ʻy=Sʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash:\text{Hp}.\supset.(\breve{S}\upharpoonright Sʻʻ\beta)ʻSʻy=y.\\ +[\text{*35·7}] &\supset.\breve{S}ʻSʻy=y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·531.</b> \[\begin{align}&\vdash:S\upharpoonright \breve{S}ʻʻ\beta\in 1\rightarrow 1.\beta\subset \text{D}ʻS.y\in \beta.\supset.Sʻ\breve{S}ʻy=y\\ +&\left[\text{*74·53}\, \frac{\breve{S}}{S}\right]\end{align}\]</p> + +<p class="nind"><b>*74·6.</b> + \(\vdash\colon\ldotp T\in 1\rightarrow 1.\lambda\subset \text{Cl}ʻ\text{ᗡ}ʻT.\kappa\subset \text{Cl}ʻ\text{D}ʻT.\supset:\kappa=T_{\in}ʻʻ\lambda.\equiv.\lambda=(\breve{T})_{\in}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·421}.\supset\vdash:\text{Hp}.\supset.&T_{\in}ʻʻ\lambda=(T_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻT)ʻʻ\lambda.\\ +&(\breve{T})_{\in}ʻʻ\kappa=\{(\breve{T})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻT\}ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*72·451·52}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:\kappa=(T_{\in}\upharpoonright \text{Cl}ʻ\text{ᗡ}ʻT)ʻʻ\lambda.&\equiv.\lambda=\{\text{Cnv}ʻ(T_{\in}\upharpoonright\text{Cl}ʻ\text{ᗡ}ʻT)\}ʻʻ\kappa.\\ +[\text{*72·54}] & \equiv.\lambda=\{(\breve{T})_{\in}\upharpoonright \text{Cl}ʻ\text{D}ʻT\}ʻʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·61.</b> \(\vdash\colon\ldotp T\in 1\rightarrow 1.\supset:\lambda\subset \text{Cl}ʻ\text{ᗡ}ʻT.\kappa=Tʻʻʻ\lambda.\equiv.\kappa\subset \text{Cl}ʻ\text{D}ʻT.\lambda=\breve{T}ʻʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·6.*37·103}.\supset\vdash\colon\ldotp \text{Hp}.\supset:&\kappa\subset \text{Cl}ʻ\text{D}ʻT.\lambda\subset \text{Cl}ʻ\text{ᗡ}ʻT.\kappa=Tʻʻʻ\lambda.\equiv.\\ +& \kappa\subset \text{Cl}ʻ\text{D}ʻT.\lambda\subset \text{Cl}ʻ\text{ᗡ}ʻT.\lambda=\breve{T}ʻʻʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*37·15·16}.\supset\vdash:&\kappa=Tʻʻʻ\lambda.\supset.\kappa\subset \text{Cl}ʻ\text{D}ʻT:\lambda=\breve{T}ʻʻʻ\kappa.\supset.\lambda\subset \text{Cl}ʻ\text{ᗡ}ʻT &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·71}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·62.</b> \(\vdash\colon\ldotp y,z\in \beta.y \neq z.\supset_{y,z}.\overrightarrow{S}ʻy\cap \overrightarrow{S}ʻz=\Lambda:\equiv.S\upharpoonright \beta\in \text{Cls}\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{Transp}.\supset\vdash\colon\ldotp &y,\,z\in \beta.y \neq z.\supset_{y,z}.\overrightarrow{S}ʻy\cap \overrightarrow{S}ʻz=\Lambda:\equiv:\\ +&y,z\in \beta.\exists !\overrightarrow{S}ʻy\cap \overrightarrow{S}ʻz.\supset_{y,z}.y=z:\\ +[\text{*32·18}] &\equiv:y,\,z\in \beta.xSy.xSz.\supset_{x,y,z}.y=z:\\ +[\text{*35·101}] &\equiv:x(S\upharpoonright \beta)y.x(S\upharpoonright \beta)z.\supset_{x,y,z}.y=z:\\ +[\text{*71·171}] &\equiv:S\upharpoonright \beta\in \text{Cls}\rightarrow 1\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_496">[Pg 496]</span></p> + +<p class="nind"><b>*74·63.</b> \[\begin{align}&\vdash\colon\ldotp P,\,Q\in \lambda.P \neq Q.\supset_{P,Q}.\text{D}ʻP\cap + \text{D}ʻQ=\Lambda:\equiv.\in \mid \text{D}\upharpoonright \lambda\in \text{Cls}\rightarrow 1\\ +&[\text{*74·62.*72·27}]\end{align}\]</p> + +<p class="nind"><b>*74·631.</b> \[\begin{align}&\vdash\colon\ldotp P,\,Q\in \lambda.P \neq Q.\supset_{P,Q}.\text{ᗡ}ʻP\cap + \text{ᗡ}ʻQ=\Lambda:\equiv.\in \mid \text{ᗡ}\upharpoonright \lambda\in \text{Cls}\rightarrow 1\\ +&[\text{*74·62.*72·27}]\end{align}\]</p> + +<p class="nind"><b>*74·632.</b> \[\begin{align}&\vdash\colon\ldotp P,\,Q\in \lambda.P \neq Q.\supset_{P,Q}.CʻP\cap CʻQ=\Lambda:\equiv.F\upharpoonright \lambda\in \text{Cls}\rightarrow 1\\ +&[\text{*74·62.*33·5}]\end{align}\]</p> + +<p class="nind"><b>*74·7.</b> \(\vdash:Q\in 1\rightarrow \text{Cls}.P\mid Q=Pʻ\mid Q.\supset.P\upharpoonright \text{D}ʻQ=Pʻ\upharpoonright \text{D}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·27}.\supset\vdash:\text{Hp}.&\supset.P\mid Q\mid \breve{Q}=Pʻ\mid Q\mid \breve{Q}.\\ +[\text{*72·59}] &\supset.P\upharpoonright \text{D}ʻQ=P'\upharpoonright \text{D}ʻQ:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·701.</b> \(\vdash:Q\in \text{Cls}\rightarrow 1.Q\mid P=Q\mid P'.\supset.(\text{ᗡ}ʻQ)\upharpoonleft P=(\text{ᗡ}ʻQ)\upharpoonleft P'\)</p> + +<p class="nind"><b>*74·71.</b> \[\begin{align}&\vdash\colon\ldotp Q\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻP\subset \text{D}ʻQ.\text{ᗡ}ʻP'\subset \text{D}ʻQ.\supset:P\mid Q=P'\mid Q.\equiv.P=P'\\ +&[\text{*74·7.*35·66.*34·28}]\end{align}\]</p> + +<p class="nind"><b>*74·711.</b> \(\vdash\colon\ldotp Q\in \text{Cls}\rightarrow 1.\text{D}ʻP\subset \text{ᗡ}ʻQ.\text{D}ʻP'\subset \text{ᗡ}ʻQ.\supset:Q\mid P=Q\mid P'.\equiv.P=P'\)</p> + +<p class="nind"><b>*74·72.</b> \(\vdash\colon\ldotp Q\in 1\rightarrow \text{Cls}:P\in \lambda.\supset_{P}.\text{ᗡ}ʻP\subset + \text{D}ʻQ:\supset.(\mid Q)\upharpoonright \lambda\in (\mid Qʻʻ\lambda)\overline{\text{ sm }}\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·71}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp P,\,P'\in \lambda.\supset_{P,P'}:P\mid Q=P'\mid Q.\equiv.P=P' &\qquad \text{(1)}\\ +\vdash.\text{(1).*73·28}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·721.</b> \(\vdash\colon\ldotp Q\in \text{Cls}\rightarrow 1:P\in \lambda.\supset_{P}.\text{D}ʻP\subset + \text{ᗡ}ʻQ:\supset.(Q\mid )\upharpoonright \lambda\in (Q\mid ʻʻ\lambda)\overline{\text{ sm }}\lambda\)</p> + +<p class="nind"><b>*74·73.</b> + \[\begin{align}&\vdash:Q\in 1\rightarrow \text{Cls}.sʻ\text{ᗡ}ʻʻ\lambda\subset \text{D}ʻQ.\supset.(\mid Q)\upharpoonright \lambda\in (\mid Qʻʻ\lambda)\overline{\text{ sm }}\lambda\\ +&[\text{*74·72.*40·43}]\end{align}\]</p> + +<p class="nind"><b>*74·731.</b> \(\vdash:Q\in \text{Cls}\rightarrow 1.sʻ\text{D}ʻʻ\lambda\subset \text{ᗡ}ʻQ.\supset.(Q\mid )\upharpoonright \lambda\in (Q\mid ʻʻ\lambda)\overline{\text{ sm }}\lambda\)</p> + +<p class="nind"><b>*74·74.</b> + \[\begin{align}&\vdash:Q\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻ\dot{s}ʻ\lambda\subset \text{D}ʻQ.\supset.(\mid Q)\upharpoonright \lambda\in (\mid Qʻʻ\lambda)\overline{\text{ sm }}\lambda\\ +&[\text{*74·73.*41·44}]\end{align}\]</p> + +<p class="nind"><b>*74·741.</b> \(\vdash:Q\in \text{Cls}\rightarrow 1.\text{D}ʻ\dot{s}ʻ\lambda\subset \text{ᗡ}ʻQ.\supset.(Q\mid )\upharpoonright \lambda\in (Q\mid ʻʻ\lambda)\overline{\text{ sm }}\lambda\)</p> + +<p class="nind"><b>*74·75.</b> + \(\vdash:\alpha\upharpoonleft Q\in 1\rightarrow \text{Cls}.\alpha\subset \text{D}ʻQ.sʻ\text{ᗡ}ʻʻ\lambda\subset \alpha.\supset.(\mid Q)\upharpoonright \lambda\in (\mid Qʻʻ\lambda)\overline{\text{ sm }}\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·43}. \supset\vdash\colon\ldotp \text{Hp}.&\supset:P\in \lambda.\supset_{P}.\text{ᗡ}ʻP\subset \alpha.\\ +[\text{*43·481}] & \supset_{P}.\mid QʻP=\mid (\alpha\upharpoonleft Q)ʻP:\\ +[\text{*37·69}] &\supset:\mid Qʻʻ\lambda=\mid (\alpha\upharpoonleft Q)ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{*43·491}. & \supset\vdash:\text{Hp}.\supset.(\mid Q)\upharpoonright \lambda=\{\mid (\alpha\upharpoonleft Q)\}\upharpoonright \lambda &\qquad \text{(2)}\\ +\vdash.\text{*74·73.*35·62}.&\supset\vdash:\text{Hp}.\supset.\{\mid (\alpha\upharpoonleft Q)\}\upharpoonright \lambda\in \{\mid (\alpha\upharpoonleft Q)ʻʻ\lambda\}\overline{\text{ sm }}\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·751.</b> + \[\begin{align}&\vdash:Q\upharpoonright \alpha\in \text{Cls}\rightarrow 1.\alpha\subset \text{ᗡ}ʻQ.sʻ\text{D}ʻʻ\lambda\subset \alpha.\supset.(Q\mid )\upharpoonright \lambda\in (Q\mid ʻʻ\lambda)\overline{\text{ sm }}\lambda\\ +&[\text{Proof as in *74·75, using *74·731, *43·48·49}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_497">[Pg 497]</span></p> + +<p class="nind"><b>*74·76.</b> \[\begin{align}\vdash: Q \in \text{Cls} &\rightarrow 1.R\in 1 \rightarrow \text{Cls}. Q \mid P \mid R = Q \mid P'\mid R. \supset .\\ +&(\text{ᗡ}ʻQ)\upharpoonleft P\upharpoonright \text{D}ʻR = (\text{ᗡ}ʻQ)\upharpoonleft P'\upharpoonright \text{D}ʻR \quad[\text{*74·7·701}]\end{align}\]</p> + +<p class="nind"><b>*74·761.</b> + \[\begin{align}\vdash\colon\ldotp \text{Hp} *74·76.\text{D}ʻP \subset & \text{ᗡ}ʻQ.\text{ᗡ}ʻP \subset \text{D}ʻR. \text{D}ʻP' \subset \text{ᗡ}ʻQ.\text{ᗡ}ʻP'\subset \text{D}ʻR.\supset:\\ +& Q\mid P\mid R = Q\mid P'\mid R.\equiv. P = P' [\text{*74·71·711}]\end{align}\]</p> + +<p class="nind"><b>*74·77.</b> \[\begin{align}\vdash: Q,\,R \in 1 \rightarrow \text{Cls}. sʻ\text{D}ʻʻ\lambda &\subset \text{D}ʻQ. sʻ\text{ᗡ}ʻʻ\lambda \subset \text{D}ʻR. \supset .\\ +&(\breve{Q} \Arrowvert R)\upharpoonright \lambda\in 1 \rightarrow 1.(\breve{Q} \Arrowvert R)\upharpoonright \lambda \in \{(\breve{Q}\Arrowvert R)ʻʻ\lambda\} \overline{\text{ sm }} \lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*74·761}\, \frac{\breve{Q}}{Q} . *40·43. \supset\\ +\vdash\colon\colon \text{Hp} .\supset\colon\ldotp P,\,P' \in \lambda .&\supset:\breve{Q}\mid P\mid R = \breve{Q}\mid P'\mid R . \equiv . P = P' :\\ +[\text{*43·112}] &\supset:(\breve{Q} \Arrowvert R)ʻP = (\breve{Q}\Arrowvert R)ʻP'. \equiv . P = P' &\qquad \text{(1)}\\ +\vdash.\text{(1). *73·28} . \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·771.</b> \[\begin{align}\vdash: Q,\,R \in \text{Cls} \rightarrow 1. sʻ&\text{D}ʻʻ\lambda \subset \text{ᗡ}ʻQ.sʻ\text{ᗡ}ʻʻ\lambda \subset \text{ᗡ}ʻR.\supset.\\ +&(Q\Arrowvert \breve{R})\upharpoonright \lambda\in 1\rightarrow 1.(Q\Arrowvert \breve{R})\upharpoonright \lambda\in \{(Q\Arrowvert \breve{R})ʻʻ\lambda\}\overline{\text{ sm }}\lambda\\ +\left[\text{*74·77}\, \frac{\breve{Q},\breve{R}}{Q,\,R}\right]\end{align}\]</p> + +<p>*74·772 and its immediate successors are of very great use in cardinal +and ordinal arithmetic.</p> + +<p class="nind"><b>*74·772.</b> \[\begin{align}&\vdash\colon\ldotp (x). \text{E}! Qʻx : (y). \text{E}! Rʻy : Q,\,R \in \text{Cls} \rightarrow 1:\supset.Q\Arrowvert \breve{R}\in 1\rightarrow 1\\ +&[\text{*74·771. *33·431}]\end{align}\]</p> + +<p class="nind"><b>*74·773.</b> + \[\begin{align}\vdash: Q\upharpoonright \alpha, R\upharpoonright \beta \in \text{Cls} \rightarrow & 1.\alpha \subset \text{ᗡ}ʻQ.\beta \subset \text{ᗡ}ʻR.sʻ\text{D}ʻʻ\lambda \subset \alpha.sʻ\text{ᗡ}ʻʻ\lambda \subset \beta.\supset.\\ +&(Q\Arrowvert \breve{R})\upharpoonright \lambda \in 1 \rightarrow 1. (Q\Arrowvert \breve{R})\upharpoonright \lambda \in \{(Q\Arrowvert \breve{R})ʻʻ\lambda\} \overline{\text{ sm }} \lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*35·64}. &\supset\vdash : \text{Hp}. \supset . sʻ\text{D}ʻʻ\lambda \subset \text{ᗡ}ʻ(Q\upharpoonright \alpha).sʻ\text{ᗡ}ʻʻ\lambda \subset \text{ᗡ}ʻ(R \upharpoonright \beta) &\qquad \text{(1)}\\ +\vdash. \text{*43·51}. &\supset\vdash : \text{Hp}. \supset . \{(Q\upharpoonright \alpha) \Arrowvert (\beta\upharpoonleft \breve{R})\} \upharpoonright \lambda = (Q \Arrowvert \breve{R})\upharpoonright \lambda &\qquad \text{(2)}\\ +\vdash. \text{(1) . (2). *74·771}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·774.</b> \(\vdash\colon\ldotp R \in \text{Cls} \rightarrow 1 : (y). \text{E}! Rʻy : \supset . \mid \breve{R} \in 1 \rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*71·166}. & \supset\vdash: \text{Hp} . \supset . \breve{R} \in \text{Cls} \rightarrow 1 &\qquad \text{(1)}\\ +\vdash. \text{*33·431}. & \supset\vdash: \text{Hp} . \supset . (P).\text{ᗡ}ʻP \subset \text{D}ʻ\breve{R} &\qquad \text{(2)}\\ +\vdash. \text{(1).(2).*74·71}\, \frac{\breve{R}}{Q} .&\supset\vdash \colon\ldotp \text{Hp}.\supset:P\mid \breve{R}=P'\mid \breve{R}.\equiv_{P,P'}. P = P' &\qquad \text{(3)}\\ +\vdash. \text{(3). *71·57} . \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_498">[Pg 498]</span></p> + +<p class="nind"><b>*74·775.</b> + \[\begin{align}\vdash:Q\upharpoonright sʻ\text{D}ʻʻ&\lambda,R\upharpoonright sʻ\text{ᗡ}ʻʻ\lambda\in \text{Cls}\rightarrow 1.sʻ\text{D}ʻʻ\lambda\subset \text{ᗡ}ʻQ.sʻ\text{ᗡ}ʻʻ\lambda\subset \text{ᗡ}ʻR.\supset.\\ +&(Q\Arrowvert \breve{R})\upharpoonright \lambda\in 1\rightarrow 1.(Q\Arrowvert \breve{R})\upharpoonright \lambda\in {(Q\Arrowvert \breve{R})ʻʻ\lambda}\overline{\text{ sm }}\lambda \quad[\text{*74·773}]\end{align}\]</p> + +<p class="nind"><b>*74·8.</b> \(\vdash:R\upharpoonright (\beta\cup \gamma)\in 1\rightarrow \text{Cls}.\equiv.R\upharpoonright \beta,R\upharpoonright \gamma\in 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·572}.\supset\vdash:R\upharpoonright (\beta\cup \gamma)\in 1\rightarrow \text{Cls}.&\equiv:y\in \text{ᗡ}ʻR\cap (\beta\cup \gamma).\supset_{y}.\text{E}!Rʻy:\\ +[\text{*22·68.*10·41}] &\equiv:y\in \text{ᗡ}ʻR\cap \beta.\supset_{y}.\text{E}!Rʻy:y\in \text{ᗡ}ʻR\cap \gamma.\supset_{y}.\text{E}!Rʻy:\\ +[\text{*71·572}] &\equiv:R\upharpoonright \beta,R\upharpoonright \gamma\in 1\rightarrow \text{Cls}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·801.</b> \(\vdash:(\beta\cup \gamma)\upharpoonleft R\in \text{Cls}\rightarrow 1.\equiv.\beta\upharpoonleft R,\gamma\upharpoonleft R\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*74·81.</b> \(\vdash:R\upharpoonright sʻ\kappa\in 1\rightarrow \text{Cls}.\equiv.R\upharpoonright ʻʻ\kappa\subset 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·572}.\supset\vdash\colon\ldotp R\upharpoonright sʻ\kappa\in 1\rightarrow \text{Cls}.&\equiv:y\in \text{ᗡ}ʻR\cap sʻ\kappa.\supset_{y}.\text{E}!Rʻy:\\ +[\text{*40·11.*10·35·23}] & \equiv:\alpha\in \kappa.y\in \text{ᗡ}ʻR\cap \alpha.\supset_{\alpha,y}.\text{E}!Rʻy:\\ +[\text{*11·62.*71·572}] &\equiv:\alpha\in \kappa.\supset_{\alpha}.R\upharpoonright \alpha\in 1\rightarrow \text{Cls}:\\ +[\text{*37·61}] &\equiv:R\upharpoonright ʻʻ\kappa\subset 1\rightarrow \text{Cls}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·811.</b> \(\vdash:(sʻ\kappa)\upharpoonleft R\in \text{Cls}\rightarrow 1.\equiv.\upharpoonleft Rʻʻ\kappa\subset \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*74·82.</b> + \(\vdash:(\beta\cup \gamma)\upharpoonleft R\in 1\rightarrow \text{Cls}.\equiv.\beta\upharpoonleft R,\gamma\upharpoonleft R\in 1\rightarrow \text{Cls}.\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1.*71·17}.\supset\\ +\vdash\colon\colon (\beta\cup \gamma)\upharpoonleft R\in 1\rightarrow \text{Cls}.&\equiv\colon\ldotp x,y\in \beta\cup \gamma.xRz.yRz.\supset_{x,y,z}.x=y\colon\ldotp\\ +[\text{*13·12}] & \supset\colon\ldotp x\in \beta.y\in \gamma.xRz.yRz.\supset_{x,y,z}.x\in \gamma\colon\ldotp \\ +[\text{Transp}] &\supset\colon\ldotp x\in \beta-\gamma.xRz.\supset_{x,y,z}.{\sim}(y\in \gamma.yRz)\colon\ldotp\\ +[\text{*10·21·252}] &\supset\colon\ldotp x\in \beta-\gamma.xRz.\supset_{x,z}.{\sim}(\exists y).y\in \gamma.yRz\colon\ldotp\\ +[\text{*10·28.*37·105}] &\supset\colon\ldotp z\in \breve{R}ʻʻ(\beta-\gamma).\supset_{z}.z{\sim}\in \breve{R}ʻʻ\gamma\colon\ldotp\\ +[\text{*24·39}] & \supset\colon\ldotp \breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·22}.\supset\\ +\vdash:(\beta\cup \gamma)\upharpoonleft R\in 1\rightarrow \text{Cls}.&\supset.\beta\upharpoonleft R,\gamma\upharpoonleft R\in 1\rightarrow \text{Cls}.\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*71·22}.&\supset\vdash:\beta\upharpoonleft R\in 1\rightarrow \text{Cls}.\supset.(\beta-\gamma)\upharpoonleft R\in 1\rightarrow \text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{*37·4}. &\supset\vdash:\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda.\supset.\text{ᗡ}ʻ(\beta-\gamma)\upharpoonleft R\cap \text{ᗡ}ʻ(\gamma\upharpoonleft R)=\Lambda &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).*71·24}.&\supset\vdash:\beta\upharpoonleft R,\gamma\upharpoonleft R\in 1\rightarrow \text{Cls}.\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda.\supset.\\ +& (\beta-\gamma)\upharpoonleft R\unicode{x228d}\gamma\upharpoonleft R\in 1\rightarrow \text{Cls}.\\ +[\text{*35·41}] & \supset.(\beta\cup \gamma)\upharpoonleft R\in 1\rightarrow \text{Cls} &\qquad \text{(5)}\\ +\vdash.\text{(2).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·821.</b> \[\begin{align}\vdash:R\upharpoonright (\beta\cup \gamma)\in \text{Cls}\rightarrow 1.&\equiv.\\ +& R\upharpoonright \beta,R\upharpoonright \gamma\in \text{Cls}\rightarrow 1.Rʻʻ(\beta-\gamma)\cap Rʻʻ\gamma=\Lambda\end{align}\]</p> + +<p><span class="pagenum" id="Page_499">[Pg 499]</span></p> + +<p class="nind"><b>*74·822.</b> + \[\begin{align}&\vdash:(\beta\cup \gamma)\upharpoonleft R\in 1\rightarrow 1.\equiv.\beta\upharpoonleft R,\,\gamma\upharpoonleft R\in 1\rightarrow 1.\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda\\ +&[\text{*74·82·801}]\end{align}\]</p> + +<p class="nind"><b>*74·823.</b> + \[\begin{align}&\vdash:R\upharpoonright (\beta\cup \gamma)\in 1\rightarrow 1.\equiv.R\upharpoonright \beta,\,R\upharpoonright \gamma\in 1\rightarrow 1.Rʻʻ(\beta-\gamma)\cap Rʻʻ\gamma=\Lambda\\ +&[\text{*74·8·821}]\end{align}\]</p> + +<p class="nind"><b>*74·83.</b> + \[\begin{align}&\vdash\colon\ldotp \breve{R}ʻʻ\beta\cap \breve{R}ʻʻ\gamma=\Lambda.\supset:(\beta\cup \gamma)\upharpoonleft R\in 1\rightarrow \text{Cls}.\equiv.\beta\upharpoonleft R,\gamma\upharpoonleft R\in 1\rightarrow \text{Cls}\\ +&[\text{*74·82}]\end{align}\]</p> + +<p class="nind"><b>*74·831.</b> + \(\vdash\colon\ldotp Rʻʻ\beta\cap Rʻʻ\gamma=\Lambda.\supset:R\upharpoonright (\beta\cup \gamma)\in \text{Cls}\rightarrow 1.\equiv.R\upharpoonright \beta,R\upharpoonright \gamma\in \text{Cls}\rightarrow 1\)</p> + +<p class="nind"><b>*74·832.</b> + \[\begin{align}&\vdash\colon\ldotp \breve{R}ʻʻ\beta\cap \breve{R}ʻʻ\gamma=\Lambda.\supset:(\beta\cup \gamma)\upharpoonleft R\in 1\rightarrow 1.\equiv.\beta\upharpoonleft R,\gamma\upharpoonleft R\in 1\rightarrow 1\\ +&[\text{*74·83·801}]\end{align}\]</p> + +<p class="nind"><b>*74·833.</b> + \[\begin{align}&\vdash\colon\ldotp Rʻʻ\beta\cap Rʻʻ\gamma=\Lambda.\supset:R\upharpoonright (\beta\cup \gamma)\in 1\rightarrow 1.\equiv.R\upharpoonright \beta,R\upharpoonright \gamma\in 1\rightarrow 1\\ +&[\text{*74·8·831}]\end{align}\]</p> + +<p class="nind"><b>*74·84.</b> \[\begin{align}\vdash\colon\ldotp (sʻ\kappa)\upharpoonleft R\in 1&\rightarrow \text{Cls}.\equiv:\\ +&\upharpoonleft Rʻʻ\kappa\subset 1\rightarrow \text{Cls}:\beta,\gamma\in \kappa.\supset_{\beta,\gamma}.\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·13.*35·43}.&\supset\vdash:\beta\in \kappa.\supset.\beta\upharpoonleft R\unicode{x2abd}(sʻ\kappa)\upharpoonleft R:\\ +[\text{*71·22}] & \supset\vdash\colon\ldotp (sʻ\kappa)\upharpoonleft R\in 1\rightarrow \text{Cls}.\supset:\beta\in \kappa.\supset.\beta\upharpoonleft R\in 1\rightarrow \text{Cls}:\\ +[\text{*37·61}] & \supset:\upharpoonleft Rʻʻ\kappa\subset 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*72·41.*37·421}.&\supset\vdash\colon\ldotp (sʻ\kappa)\upharpoonleft R\in 1\rightarrow \text{Cls}.\supset:\\ +&\beta,\gamma\in \kappa.\supset_{\beta,\gamma}.\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*37·105.*24·39}.\supset\\ +\vdash\colon\ldotp &\beta,\gamma\in \kappa.\supset_{\beta,\gamma}.\breve{R}ʻʻ(\beta-\gamma)\cap \breve{R}ʻʻ\gamma=\Lambda:\equiv:\\ +&\beta,\gamma\in \kappa.x\in \beta-\gamma.xRz.\supset_{\beta,\gamma}.{\sim}(\exists y).y\in \gamma.yRz:\\ +[\text{Transp}] \supset:&\beta,\gamma\in \kappa.x\in \beta.y\in \gamma.xRz.yRz.\supset_{\beta,\gamma}.x\in \gamma.\\ +[\text{*4·7}] & \supset_{\beta,\gamma}.x,y\in \gamma.xRz.yRz.\\ +[\text{*35·1}] &\supset_{\beta,\gamma}.x(\gamma\upharpoonleft R)z.y(\gamma\upharpoonleft R)z &\qquad \text{(3)}\\ +\vdash.\text{(3).*71·17}.\supset\vdash\colon\ldotp &\beta,\gamma\in \kappa.\supset_{\beta,\gamma}.\breve{R}ʻʻ(\beta-\gamma)\cap + \breve{R}ʻʻ\gamma=\Lambda:\upharpoonleft Rʻʻ\kappa\in 1\rightarrow \text{Cls}:\supset:\\ +& \beta,\gamma\in \kappa.x\in \beta.y\in \gamma.xRz.yRz.\supset_{\beta,\gamma,x,y,z}.x=y:\\ +[\text{*10·23.*40·11.*37·1}]&\supset:x\{(sʻ\kappa)\upharpoonleft R\}z.y\{(sʻ\kappa)\upharpoonleft R\}z.\supset_{x,y,z}.x=y:\\ +[\text{*71·17}] & \supset:sʻ\kappa\upharpoonleft R\in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*74·841.</b> \[\begin{align}\vdash\colon\ldotp R\upharpoonright sʻ\kappa\in \text{Cls}&\rightarrow 1.\equiv:\\ +&R\upharpoonright ʻʻ\kappa\subset \text{Cls}\rightarrow 1:\beta,\gamma\in \kappa.\supset_{\beta,\gamma}.Rʻʻ(\beta-\gamma)\cap Rʻʻ\gamma=\Lambda\end{align}\]</p> + +<p class="nind"><b>*74·842.</b> \[\begin{align}\vdash&\colon\ldotp (sʻ\kappa)\upharpoonleft R\in 1\rightarrow 1.\equiv:\\ +&\upharpoonleft Rʻʻ\kappa\subset 1\rightarrow 1:\beta,\gamma\in \kappa.\supset_{\beta,\gamma}.\breve{R}ʻʻ(\beta-\gamma)\cap + \breve{R}ʻʻ\gamma=\Lambda \quad[\text{*74·84·811}]\end{align}\]</p> + +<p class="nind"><b>*74·843.</b> \[\begin{align}\vdash&\colon\ldotp R\upharpoonright sʻ\kappa\in 1\rightarrow 1.\equiv:\\ +&R\upharpoonright ʻʻ\kappa\subset 1\rightarrow 1:\beta,\gamma\in \kappa.\supset_{\beta,\gamma}.Rʻʻ(\beta-\gamma)\cap +Rʻʻ\gamma=\Lambda \quad[\text{*74·81·841}]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_500">[Pg 500]</span></p> +<h2 class="nobreak" id="SECTION_D_b">SECTION D.<br> +<br> +SELECTIONS.</h2> +</div> + + +<p><i>Summary of Section D.</i></p> + +<p>The subject to be considered in this section is important chiefly +in connection with multiplication, both cardinal and ordinal. In +order to get a definition of multiplication which is not confined +to the case where the number of factors is finite, we have to seek +a construction by which, from a given class of classes, \(\kappa\) +say, we construct another class which, when \(\kappa\) is finite, +has that number of terms which, in the usual elementary sense, is +the product of the numbers of terms in the various classes which are +members of \(\kappa\), and which, whether \(\kappa\) is finite or not, +obeys as many as possible of the formal laws of multiplication. The +usual elementary sense of multiplication is derived from addition; +that is to say, \(\mu \times \nu\) is to be the number of terms in +\(sʻ\kappa\), where \(\kappa\) is a class of \(\mu\) mutually exclusive +classes each having \(\nu\) members, or vice versa. This sense can be +extended to any finite number of factors, but not to an infinite number +of factors; hence for a number of factors which may be infinite we +require a different definition, and this is derived from the theory of +<i>selections</i>.</p> + +<p>Selections are of two kinds, selections from classes of classes, and +selections from relations. The latter is the more general notion, from +which the former is derived. But as the former is an easier notion, we +will begin by explaining selections from classes of classes.</p> + +<p>Given a class of classes \(\kappa\), a class \(\mu\) is called a +<i>selected class</i> of \(\kappa\) when \(\mu\) is formed by choosing +one term out of each member of \(\kappa\). For example, if \(\kappa\) +consists of two members, \(\alpha\) and \(\beta\), and if \(x \in\alpha\) +and \(y \in \beta\), then \(\iotaʻx \cup \iotaʻy\) is a +selected class of \(\kappa\). If every constituency elects a local man, +Parliament is a selected class of the constituencies. If \(\kappa\) is +a class of mutually exclusive classes, <i>i.e.</i> a class no two of +whose members have any member in common, then a selected class consists +of only one term from each member of \(\kappa\); <i>i.e.</i> \(\mu\) is +a selected class if +\[ +\mu \subset sʻ\kappa : \alpha \in \kappa .\supset_{\alpha}. \mu \cap \alpha \in 1. +\] +But if \(\kappa\) is not a class of mutually exclusive classes, this +does not hold necessarily; for a term \(x\) which is a member of +both \(\alpha\) and \(\beta\) (where \(\alpha, \beta \in \kappa\)) +may be chosen as the representative of \(\alpha\), while some other +term may be<span class="pagenum" id="Page_501">[Pg 501]</span> chosen as the representative of \(\beta\), so that two +members of \(\beta\) may belong to the selected class. Again, if +\(\kappa\) is a class of mutually exclusive classes, the relation of +the representative to its class must be one-one, because, since no +term belongs to two classes which are members of \(\kappa\), no term +can be the representative of two classes. But when \(\kappa\) is not +a class of mutually exclusive classes, a term which belongs to two +classes \(\alpha\) and \(\beta\) may be chosen as the representative of +both. Thus the relation of the representative to its class may be only +one-many, not one-one.</p> + +<p>The relation of the representative to its class may be called a +<i>selective relation</i>. A selective relation of \(\kappa\) is +one which selects, from every class \(\alpha\) which is a member of +\(\kappa\), a certain member \(x\) as the <i>representative</i> of +\(\alpha\); that is, we have, if \(R\) is the selective relation, +\[ +\alpha\in \kappa.\supset_{\alpha}.Rʻ\alpha\in \alpha:\text{ᗡ}ʻR=\kappa. +\] +This condition is equivalent to +\[ +R\in 1\rightarrow\text{Cls}.R\unicode{x2abd}\in .\text{ᗡ}ʻR=\kappa. +\]</p> + +<p>If \(R\) is a selective relation, \(\text{D}ʻR\) is a selected class; +and if \(\mu\) is a selected class, there is a selective relation \(R\) +such that \(\mu=\text{D}ʻR\). Thus the study of selections from classes +of classes is wholly contained in the study of selective relations.</p> + +<p>The class of selective relations from a class \(\kappa\) is called +\({\in}_{\Delta}ʻ\kappa\). Thus +\[ +\begin{align} +R\in {\in}_{\Delta}ʻ\kappa.&\equiv.R\in 1\rightarrow\text{Cls}.R\unicode{x2abd}\in .\text{ᗡ}ʻR=\kappa,\\ +\text{and}\qquad\quad {\in}_{\Delta}ʻ\kappa&=(1\rightarrow\text{Cls})\cap \text{Rl}ʻ\in \cap \overleftarrow{\text{ᗡ}}ʻ\kappa. +\end{align} +\] +Then \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is the class of selected +classes.</p> + +<p>It will be seen that, if \(\alpha\in \kappa\), \(Rʻ\alpha\) may be any +member of \(\alpha\), and we get a different \(R\) for each different +member of \(\alpha\). Thus if we keep the representatives of all +the other members of \(\kappa\) unchanged, the number of selective +relations to be obtained by varying the representative of \(\alpha\) +is the number of members of \(\alpha\). Hence the number of selective +relations altogether may be fitly defined as the product of the +numbers of terms possessed by the various members of \(\kappa\). In +case \(\kappa\) is finite, this agrees with the usual definition of +multiplication; and whether \(\kappa\) is finite or infinite, the +product so defined obeys all the formal laws of multiplication.</p> + +<p>To illustrate the notion of selective relations, let us take a very +simple case, the case where \(\kappa\) consists of two classes +\(\alpha\) and \(\beta\), each of which has two members. Let \(x\) and +\(y\) be the members of \(\alpha\), \(z\) and \(w\) the members of +\(\beta\). We assume \(\alpha \neq \beta\), \(x \neq y\), \(z \neq w\). +Then the selective relations of \(\kappa\) are the following: +\[ +x\downarrow\alpha\unicode{x228d} z\downarrow\beta,\\ +x\downarrow\alpha\unicode{x228d} w\downarrow\beta,\\ +y\downarrow\alpha\unicode{x228d} z\downarrow\beta,\\ +y\downarrow\alpha\unicode{x228d} w\downarrow\beta. +\]<span class="pagenum" id="Page_502">[Pg 502]</span> +Thus they are four in number, <i>i.e.</i> the number of members of +\({\in}_{\Delta}ʻ\kappa\) is the product of the number of members of +\(\alpha\) and the number of members of \(\beta\). A similar process +would show that our definition of the product agrees with the usual +definition in any case in which all the numbers concerned are finite.</p> + +<p>Selections from <i>relations</i> are an obvious generalization of +selections from classes of classes. We had above +\[ +{\in}_{\Delta}ʻ\kappa = (1 \rightarrow \text{Cls}) \cap \text{Rl}ʻ{\in} \cap \overleftarrow{\text{ᗡ}}ʻ\kappa\text{.} +\] +We put, generally, +\[ +P_{\Delta}ʻ\kappa = (1 \rightarrow \text{Cls}) \cap \text{Rl}ʻP \cap \overleftarrow{\text{ᗡ}}ʻ\kappa\text{,} +\] +which we derive from the definition +\[ +P_{\Delta}=\hat{\lambda}\hat{\kappa}\{\lambda = (1 \rightarrow \text{Cls}) \cap \text{Rl}ʻP \cap \overleftarrow{\text{ᗡ}}ʻ\kappa \} \quad \text{Df}\text{.} +\] +This is the fundamental definition in the subject of selections. We +have, in virtue of this definition, +\[ +\vdash: R \in P_{\Delta}ʻ\kappa .\equiv. R \in 1 \rightarrow \text{Cls} . R \unicode{x2abd} P . \text{ᗡ}ʻR = \kappa \text{.} +\] +When \(\kappa = \text{ᗡ}ʻP\), we may call \(P_{\Delta}ʻ\kappa\) the +class of selections from \(P\). Thus generally, \(P_{\Delta}ʻ\kappa\) +is the class of selections from \(P \upharpoonright \kappa\) +provided \(\kappa \subset \text{ᗡ}ʻP\); and if this condition is not +fulfilled, \(P_{\Delta}ʻ\kappa = \Lambda\). We may call the class +\(P_{\Delta}ʻ\kappa\) the class of "\(P\)-selections from \(\kappa\)." +The class of "\(\in\)-selections from \(\kappa\)" will be what we +previously called the class of "selective relations of \(\kappa\)." +It will be observed that we have +\[ +R \in P_{\Delta}ʻ\kappa . y \in \kappa .\supset. Rʻy \in \overrightarrow{P}ʻy\text{.} +\] +Thus if \(\overrightarrow{P}ʻʻ\kappa\) is a class of mutually exclusive +classes, \(\text{D}ʻR\) selects one from each of these classes, and is +therefore a selective class of \(\overrightarrow{P}ʻʻ\kappa\); hence in +this case +\[ +\text{D}ʻʻP_{\Delta}ʻ\kappa = \text{D}ʻʻ{\in}_{\Delta}ʻ\overrightarrow{P}ʻʻ\kappa\text{.} +\]</p> + +<p>In Cardinal Arithmetic, \({\in}_{\Delta}ʻ\kappa\) is the important +notion, and the more general notion \(P_{\Delta}ʻ\kappa\) is seldom +required. In Ordinal Arithmetic, \(F_{\Delta}ʻ\kappa\) is the important +notion. It will be seen that +\[ +R \in F_{\Delta}ʻ\kappa .\equiv. R \in 1 \rightarrow \text{Cls} . R \unicode{x2abd} F . \text{ᗡ}ʻR = \kappa\text{.} +\] +Thus \(F_{\Delta}ʻ\kappa\) is only significant when \(\kappa\) is a +class of relations; in this case we have +\[ +R \in F_{\Delta}ʻ\kappa . Q \in \kappa .\supset. RʻQ \in CʻQ\text{.} +\] +<span class="pagenum" id="Page_503">[Pg 503]</span>Thus \(R\) chooses a representative member of the field of every +member of \(\kappa\). The most important case is when \(\kappa\) is of +the form \(CʻP\), where \(P\) is a serial relation whose field consists +of serial relations. Then \(F_{\Delta}ʻCʻP\) becomes the field of a +relation which may be defined as the ordinal product of the relations +composing \(CʻP\); in this way we get an infinite ordinal product +analogous to the infinite cardinal product. This will be explained at a +later stage (*172).</p> + +<p>Although it is chiefly \({\in}_{\Delta}ʻ\kappa\) and +\(F_{\Delta}ʻ\kappa\) that will be required in the sequel, we shall +treat \(P_{\Delta}ʻ\kappa\) generally, because this introduces +little extra complication, and most of the theorems which hold for +\({\in}_{\Delta}ʻ\kappa\) or \(F_{\Delta}ʻ\kappa\) have exact analogues +for \(P_{\Delta}ʻ\kappa\).</p> + +<p>\(P_{\Delta}ʻ\kappa\), as above defined, is the class of one-many +relations contained in \(P\) and having \(\kappa\) for their converse +domain. We know of no proof that there always are such relations when +\(\kappa\subset \text{ᗡ}ʻP\). In fact, the proposition +\[ +\kappa\subset \text{ᗡ}ʻP.\supset_{P,\kappa}.\exists !P_{\Delta}ʻ\kappa +\] +is equivalent to the "multiplicative axiom," <i>i.e.</i> to the axiom +that, given any class of mutually exclusive classes, none of which is +null, there is at least one class formed of one member from each of +these classes. (This equivalence is proved in <a href="#*88·36">*88·36</a>, below.) It is +also equivalent to Zermelo's axiom<a id="FNanchor_61" href="#Footnote_61" class="fnanchor">[61]</a>, which is +\[ +(\alpha).\exists !{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha; +\] +hence also it is equivalent to the proposition that every class can be +well-ordered. In the absence of evidence as to the truth or falsehood +of these various propositions, we shall not assume their truth, but +shall explicitly introduce them as hypotheses wherever they are +relevant.</p> + +<p>In the present section, we shall begin (<a href="#*80">*80</a>) by considering such +properties of \(P_{\Delta}ʻ\kappa\) as do not depend upon any +hypothesis as to \(P\). We shall then (<a href="#*81">*81</a>) proceed to consider +such further properties of \(P_{\Delta}ʻ\kappa\) as result from +the hypothesis \(P\upharpoonright \kappa\in \text{Cls}\rightarrow 1\). +This hypothesis is important, because it is verified in many +of the applications we wish to make, and because it leads to +important properties of \(P_{\Delta}ʻ\kappa\) which are not true in +general when \(P\) is not subject to any hypothesis. These special +properties are mostly due to the fact that when \(P\upharpoonright\kappa\) +is a many-one relation, \(P_{\Delta}ʻ\kappa\) consists of +one-one relations (not merely of one-many relations, as it does in +the general case). This is proved in <a href="#*81·1">*81·1</a>. We then (<a href="#*82">*82</a>) proceed +to consider the case of relative products, <i>i.e.</i> (\(P\mid Q)_{\Delta}ʻ\lambda\). +It will appear that, with a suitable hypothesis, +(\(P\mid Q)_{\Delta}ʻ\lambda=\mid QʻʻP_{\Delta}ʻQʻʻ\lambda\) and +\(\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda=DʻʻP_{\Delta}ʻQʻʻ\lambda\). +In the following number (<a href="#*83">*83</a>) we apply the results of <a href="#*80">*80</a> to the +particular case where \(P\) is replaced by \({\in}\), which is +the important case for cardinal arithmetic. In <a href="#*84">*84</a> we apply the +propositions of <a href="#*81">*81</a> to the case where \(P\) is replaced by \({\in}\), +and where, therefore, we have the hypothesis \({\in}\upharpoonright \kappa\in \text{Cls}\rightarrow 1\). +This hypothesis is equivalent to the hypothesis that no two members of +\(\kappa\) have any members in common, <i>i.e.</i> that +\[ +\alpha,\beta \in \kappa.\alpha \neq \beta.\supset_{\alpha,\beta}.\alpha\cap \beta=\Lambda. +\]<span class="pagenum" id="Page_504">[Pg 504]</span> +When \(\kappa\) fulfils this hypothesis, it is a class of +mutually exclusive classes. For classes of mutually exclusive +classes we adopt the notation "\(\text{Cls}^{2}\, \text{excl}\)." +It is shown in <a href="#*84·14">*84·14</a> that a \(\text{Cls}^{2}\, \text{excl}\) +is one for which we have \({\in}\upharpoonright \kappa \in\text{Cls}\rightarrow 1\). +When \(\kappa\) is a \(\text{Cls}^{2}\text{excl}\), \(\text{D}\upharpoonright P_{\Delta}ʻ\kappa\) +is a one-one relation, and \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\,\text{ sm }\, {\in}_{\Delta}ʻ\kappa\). +Also in this case \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) consists of +all classes formed of one member from each member of \(\kappa\), +<i>i.e.</i> all classes \(\mu\) such that +\[ +\mu\subset sʻ\kappa:\alpha \in \kappa.\supset_{\alpha}.\mu\cap \alpha \in 1. +\] +In <a href="#*85">*85</a>, we prove various important propositions, of which the chief is +a form of the associative law<a id="FNanchor_62" href="#Footnote_62" class="fnanchor">[62]</a>, namely +\[ +\vdash:\kappa \in \text{Cls}^{2} \text{excl}.\supset.{\in}_{\Delta}ʻsʻ\kappa\, \text{ sm }\, {\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa. +\] +Finally, in <a href="#*88">*88</a>, we consider the question of the existence of +selections. This cannot in general be proved when \(\kappa\) is an +infinite class. The assumption that \({\in}_{\Delta}ʻ\kappa\) is never +null unless one member of \(\kappa\) is null is equivalent to various +other assumptions, for example to the assumption that every class can +be well-ordered. One of these equivalent assumptions is called the +"multiplicative axiom." This axiom is equivalent to the assumption that +an arithmetical product cannot be zero unless one of its factors is +zero, and is regarded by some mathematicians as a self-evident truth. +This can be proved when the number of factors is finite, <i>i.e.</i> +when \(\kappa\) is a finite class, but not when the number of factors +is infinite. We have not assumed its truth in the general case where +it cannot be proved, but have included it in the hypotheses of all +propositions which depend upon it.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_61" href="#FNanchor_61" class="label">[61]</a> +See his "Beweis, dass jede Menge wohlgeordnet werden +kann," <i>Math. Annalen</i>, Vol. <span class="allsmcap">LIX</span>. pp. 514-516.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_62" href="#FNanchor_62" class="label">[62]</a> +Cf. notes to *42·1·11.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_505">[Pg 505]</span></p> +<h2 class="nobreak" id="*80">*80. ELEMENTARY PROPERTIES OF SELECTIONS.</h2> +</div> + + +<p><i>Summary of</i> *80.</p> + +<p>In this number, we shall give such properties of \(P_{\Delta}\) as +follow most directly from the definition, without any restrictive +hypothesis as to \(P\).</p> + +<p>If \(R \in P_{\Delta}ʻ\kappa\), \(R\) selects one member of +\(\overrightarrow{P}ʻy\), whenever \(y \in \kappa\), as the +selected referent of \(y\). For, since \(R \in 1 \rightarrow\text{Cls} . \text{ᗡ}ʻR = \kappa\), +we have \(y \in \kappa .\supset.\text{E}!Rʻy\); and since +\(R \unicode{x2abd} P\), we have \(y \in \kappa .\supset.(Rʻy)Py\), +<i>i.e.</i> \(y \in \kappa .\supset. Rʻy \in \overrightarrow{P}ʻy\). +Calling \(Rʻy\) the <i>selected referent</i> of \(y\), it +is evident that we may replace \(Rʻy\) by any other member +of \(\overrightarrow{P}ʻy\), and still have a member of +\(P_{\Delta}ʻ\kappa\). (This is proved in <a href="#*80·4">*80·4</a>.) Thus if +\(P_{\Delta}ʻ\kappa\) has any members at all, we can get as many +members as there are members of \(\overrightarrow{P}ʻy\) by merely +altering the selected referent of \(y\), leaving the other selected +referents unchanged.</p> + +<p>In the present section, we first prove various simple properties of +\(P_{\Delta}ʻ\kappa\). Most of these are almost immediate consequences +of</p> + +<p class="nind"><b>*80·14.</b> \(\vdash: R \in P_{\Delta}ʻ\kappa .\equiv. R \in 1 \rightarrow \text{Cls}. R \unicode{x2abd} P.\text{ᗡ}ʻR = \kappa\)</p> + +<p>The most useful of them are</p> + +<p class="nind"><b><a id="*80·2">*80·2</a>.</b> \(\vdash: \exists! P_{\Delta}ʻ\kappa .\supset. \kappa \subset \text{ᗡ}ʻP\)</p> + +<p class="nind"><b>*80·291.</b> \(\vdash: R \in P_{\Delta}ʻ\kappa .\supset. R \unicode{x2abd} P \upharpoonright \kappa\)</p> + +<p class="nind"><b>*80·3.</b> \(\vdash: R \in P_{\Delta}ʻ\kappa . y\in \kappa .\supset. \text{E}!Rʻy\)</p> + +<p class="nind"><b>*80·33.</b> \(\vdash: R \in P_{\Delta}ʻ\kappa .\supset. \text{D}ʻR \subset Pʻʻ\kappa\)</p> + +<p>We then have various propositions (<a href="#*80·4">*80·4</a>—<a href="#*80·46">·46</a>) concerned with \(x +\downarrow y\) when \(x P y\). Of these the most important are the +following:</p> + +<p class="nind"><b>*80·41.</b> \(\vdash: R \in P_{\Delta}ʻ\kappa . y \in \kappa . x'Py .\supset. [\{R \dot{-} (Rʻy) \downarrow y\} \unicode{x228d} x' \downarrow y] \in P_{\Delta}ʻ\kappa\)</p> + +<p><i>I.e.</i> given a selective relation \(R\), the selected referent of +\(y\) (where \(y \in \text{ᗡ}ʻP\)) may be replaced by any other term +having the relation \(P\) to \(y\), and we shall still have a selective +relation.</p> + +<p class="nind"><b>*80·45.</b> \(\vdash. P_{\Delta}ʻ\iotaʻy = \downarrow yʻʻ\overrightarrow{P}ʻy\)</p> + +<p>We then have a set of propositions (<a href="#*80·5">*80·5</a>—<a href="#*80·54">·54</a>) connecting +(\(P \unicode{x228d} Q)_{\Delta}ʻ(\kappa \cup \lambda)\) with +\(P_{\Delta}ʻ\kappa\) and \(Q_{\Delta}ʻ\lambda\). These are chiefly +useful as leading to the next set</p> + +<p><span class="pagenum" id="Page_506">[Pg 506]</span></p> + +<p class="nind"> +(*80·6—·69), connecting \(P_{\Delta}ʻ(\kappa\cup \lambda)\) with +\(P_{\Delta}ʻ\kappa\) and \(P_{\Delta}ʻ\lambda\). The most useful of +these are the following:</p> + +<p class="nind"><b>*80·6.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.\lambda\subset \kappa.\supset.R\upharpoonright \lambda\in P_{\Delta}ʻ\lambda\)</p> + +<p class="nind"><b>*80·65.</b> \(\vdash:\kappa\cap \lambda=\Lambda.R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.\supset.R\unicode{x228d}S\in P_{\Delta}ʻ(\kappa\cup \lambda)\)</p> + +<p class="nind"><b>*80·66.</b> \(\vdash\colon\ldotp \kappa\cap \lambda=\Lambda.\supset:M\in P_{\Delta}ʻ(\kappa\cup \lambda).\equiv.(\exists R,S).R\in P_{\Delta}ʻ\kappa.S\in + P_{\Delta}ʻ\lambda.M=R\unicode{x228d}S\)</p> + +<p>We have next a set of propositions (<a href="#*80·7">*80·7</a>—<a href="#*80·78">·78</a>) dealing with the +relations of \(M\) and \(M\dot{-}R\) when (<i>e.g.</i>) \(M\in +P_{\Delta}ʻ(\kappa\cup \lambda)\) and \(R\in P_{\Delta}ʻ\kappa\). These +propositions are seldom used, but they would be useful in considering +division.</p> + +<p>We next have a set of propositions (<a href="#*80·8">*80·8</a>—<a href="#*80·84">·84</a>) dealing with the +relations of \(P_{\Delta}ʻ\alpha\) and \(P_{\Delta}ʻ\beta\). The most +useful are</p> + +<p class="nind"><b>*80·81.</b> \(\vdash:\exists !P_{\Delta}ʻ\alpha.P_{\Delta}ʻ\alpha=P_{\Delta}ʻ\beta.\supset.\alpha=\beta\)</p> + +<p class="nind"><b>*80·82.</b> \(\vdash:\alpha \neq \beta.\supset.P_{\Delta}ʻ\alpha\cap P_{\Delta}ʻ\beta=\Lambda\)</p> + +<p>Finally, we have four propositions (<a href="#*80·9">*80·9</a>—<a href="#*80·93">·93</a>) on +\(P_{\Delta}ʻ(\iotaʻy\cup \iotaʻz)\) and one on \(P_{\Delta}ʻ(\beta\cup\iotaʻz)\). +The most useful of these is</p> + +<p class="nind"><b>*80·9.</b> \(\vdash\colon\ldotp y \neq z.\supset:M\in P_{\Delta}ʻ(\iotaʻy\cup \iotaʻz).\equiv.(\exists u,v).uPy.vPz.M=u\downarrow y\unicode{x228d}v\downarrow z\)</p> + +<hr class="tb"> + +<p class="nind"><b>*80·01.</b> \(P_{\Delta}=\hat{\lambda}\hat{\kappa}\{\lambda=(1\rightarrow \text{Cls})\cap \text{Rl}ʻP\cap \overleftarrow{\text{ᗡ}}ʻ\kappa\} \quad\text{Df}\)</p> + +<p class="nind"><b>*80·1.</b> \(\vdash:\lambda P_{\Delta}\kappa.\equiv.\lambda=(1\rightarrow \text{Cls})\cap \text{Rl}ʻP\cap \overleftarrow{\text{ᗡ}}ʻ\kappa \quad[\text{*21·3.(*80·01)}]\)</p> + +<p class="nind"><b>*80·11.</b> \(\vdash.P_{\Delta}ʻ\kappa=(1\rightarrow \text{Cls})\cap \text{Rl}ʻP\cap \overleftarrow{\text{ᗡ}}ʻ\kappa \quad[\text{*80·1.*30·3}]\)</p> + +<p class="nind"><b>*80·12.</b> \(\vdash.\text{E}!P_{\Delta}ʻ\kappa \quad[\text{*80·11.*14·21}]\)</p> + +<p class="nind"><b>*80·13.</b> \(\vdash:\lambda P_{\Delta}ʻ\kappa.\equiv.\lambda=P_{\Delta}ʻ\kappa \quad[\text{*80·12.*30·4}]\)</p> + +<p class="nind"><b>*80·14</b>. \[\begin{align}&\vdash:R\in P_{\Delta}ʻ\kappa.\equiv.R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}P.\text{ᗡ}ʻR=\kappa\\ +&[\text{*80·11.*20·43.*22·33.*61·2.*33·61}]\end{align}\]</p> + +<p class="nind"><b>*80·15.</b> \(\vdash:P\unicode{x2abd}Q.\supset.P_{\Delta}ʻ\kappa\subset Q_{\Delta}ʻ\kappa \quad[\text{*80·14}]\)</p> + +<p class="nind"><b>*80·16.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.R\unicode{x2abd}Q.\supset.R\in Q_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.&\supset\vdash:R\in P_{\Delta}ʻ\kappa.\supset.R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR=\kappa:\\ +[\text{Fact}] &\supset\vdash:R\in P_{\Delta}ʻ\kappa.R\unicode{x2abd}Q.\supset.R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR=\kappa.R\unicode{x2abd}Q.\\ +[\text{*80·14}] & \supset.R\in Q_{\Delta}ʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·17.</b> \(\vdash:Q\unicode{x2abd}P.\supset.Q_{\Delta}ʻ\kappa=P_{\Delta}ʻ\kappa\cap \text{Rl}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·15}. \supset\vdash:\text{Hp}.\supset.Q_{\Delta}ʻ\kappa\subset P_{\Delta}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*80·11}. \supset\vdash.Q_{\Delta}ʻ\kappa\subset \text{Rl}ʻQ &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash:\text{Hp}.\supset.Q_{\Delta}ʻ\kappa\subset P_{\Delta}ʻ\kappa\cap \text{Rl}ʻQ &\qquad \text{(3)}\\ +\vdash.\text{*80·16}. \supset\vdash.P_{\Delta}ʻ\kappa\cap \text{Rl}ʻQ\subset Q_{\Delta}ʻ\kappa &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_507">[Pg 507]</span></p> + +<p>This proposition is used in the theory of ordinal multiplication +(*172·162).</p> + +<p class="nind"><b>*80·2.</b> \(\vdash:\exists !P_{\Delta}ʻ\kappa.\supset.\kappa\subset \text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:R\in P_{\Delta}ʻ\kappa.&\supset.R\unicode{x2abd}P.\text{ᗡ}ʻR=\kappa.\\ +[\text{*33·264}] & \supset.\text{ᗡ}ʻR\subset \text{ᗡ}ʻP.\text{ᗡ}ʻR=\kappa.\\ +[\text{*13·13}] &\supset.\kappa\subset \text{ᗡ}ʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·23}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·21.</b> \(\vdash:{\sim}(\kappa\subset \text{ᗡ}ʻP).\supset.P_{\Delta}ʻ\kappa=\Lambda \quad[\text{*80·2.Transp}]\)</p> + +<p class="nind"><b>*80·22.</b> \(\vdash:P\upharpoonright \kappa=Q\upharpoonright \kappa.\supset.P_{\Delta}ʻ\kappa=Q_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·14}.&\supset\vdash\colon\colon \text{ᗡ}ʻR=\kappa.\supset\colon\ldotp xRy.\supset.y\in \kappa\colon\ldotp \\ +[\text{*5·44}] &\supset\colon\ldotp xRy.\supset.xPy:\equiv:xRy.\supset.xPy.y\in \kappa:\\ +[\text{*35·101}] &\equiv:xRy.\supset.x(P\upharpoonright \kappa)y &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·3·33}.\supset\\ +\vdash\colon\ldotp \text{ᗡ}ʻR=\kappa.\supset:R\unicode{x2abd}P.&\equiv.R\unicode{x2abd}P\upharpoonright \kappa &\qquad \text{(2)}\\ +\vdash.\text{(2)} \frac{Q}{P}.\supset\vdash\colon\ldotp \text{ᗡ}ʻR=\kappa.\supset:R\unicode{x2abd}Q.&\equiv.R\unicode{x2abd}Q\upharpoonright \kappa &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*13·12}.&\supset\vdash\colon\ldotp \text{ᗡ}ʻR=\kappa.P\upharpoonright \kappa=Q\upharpoonright \kappa.\supset:R\unicode{x2abd}P.\equiv.R\unicode{x2abd}Q &\qquad \text{(4)}\\ +\vdash.\text{(4).Comm.*5·32}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:R\unicode{x2abd}P.\text{ᗡ}ʻR=\kappa.\equiv.R\unicode{x2abd}Q.\text{ᗡ}ʻR=\kappa:\\ +[\text{*80·14}]\supset:R\in P_{\Delta}ʻ\kappa.&\equiv.R\in Q_{\Delta}ʻ\kappa\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·23.</b> \(\vdash.P_{\Delta}ʻ\kappa=(P\upharpoonright \kappa)_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·31.*22·5}.\supset\vdash.P\upharpoonright \kappa=(P\upharpoonright \kappa)\upharpoonright \kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*80·22}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·24.</b> \(\vdash:\kappa\subset \text{ᗡ}ʻP.Q=P\upharpoonright \kappa.\supset.P_{\Delta}ʻ\kappa=Q_{\Delta}ʻ\text{ᗡ}ʻQ \quad[\text{*35·65.*80·23}]\)</p> + +<p class="nind"><b>*80·25.</b> \(\vdash:\exists !P_{\Delta}ʻ\kappa.Q=P\upharpoonright \kappa.\supset.P_{\Delta}ʻ\kappa=Q_{\Delta}ʻ\text{ᗡ}ʻQ \quad[\text{*80·2·24}]\)</p> + +<p class="nind"><b><a id="*80·26">*80·26</a>.</b> \(\vdash.P_{\Delta}ʻ\Lambda=\iotaʻ\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:R\in P_{\Delta}ʻ\Lambda.&\equiv.R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}P.\text{ᗡ}ʻR=\Lambda.\\ +[\text{*33·241}] & \equiv.R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}P.R=\dot{\Lambda}.\\ +[\text{*13·193}] &\equiv.\dot{\Lambda}\in 1\rightarrow \text{Cls}.\dot{\Lambda}\unicode{x2abd}P.R=\dot{\Lambda}.\\ +[\text{*72·1.*25·12}] &\equiv.R=\dot{\Lambda}.\\ +[\text{*51·15}] &\equiv.R\in \iotaʻ\dot{\Lambda}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>Note that \(P_{\Delta}ʻ\Lambda\) is a unit class, not the null-class. It is owing to this +fact (as will appear later) that, if \(\mu\) is any cardinal, \(\mu^{0}=1\). See the note to +<a href="#*83·15">*83·15</a>.</p> + +<p><span class="pagenum" id="Page_508">[Pg 508]</span></p> + +<p class="nind"><b>*80·27.</b> \(\vdash:\exists !\kappa.\supset.\dot{\Lambda}_{\Delta}ʻ\kappa=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:R\in \dot{\Lambda}_{\Delta}ʻ\kappa.&\supset.R\unicode{x2abd}\dot{\Lambda}.\text{ᗡ}ʻR=\kappa.\\ +[\text{*25·13}] &\supset.R=\dot{\Lambda}.\text{ᗡ}ʻR=\kappa.\\ +[\text{*33·241}] &\supset.\kappa=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp.*10·11·21}.\supset\\ +\vdash:\exists !\kappa.\supset.(R).R{\sim}\in \dot{\Lambda}_{\Delta}ʻ\kappa.\\ +[\text{*24·15}] &\supset.\dot{\Lambda}_{\Delta}ʻ\kappa=\Lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·28.</b> \(\vdash:\exists !\kappa.\supset.\dot{\Lambda}{\sim}\in P_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash\colon\ldotp \exists !\kappa.&\supset:R\in P_{\Delta}ʻ\kappa.\supset_{R}.\exists !\text{ᗡ}ʻR:\\ +[\text{*33·241}] & \supset:R\in P_{\Delta}ʻ\kappa.\supset_{R}.\dot{\exists}!R:\\ +[\text{*25·63}] &\supset:\dot{\Lambda}{\sim}\in P_{\Delta}ʻ\kappa\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·29.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.\supset.R=R\upharpoonright \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻR=\kappa.\\ +[\text{*35·452}] & \supset.R=R\upharpoonright \kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·291.</b> \(\vdash\colon\ldotp R\in P_{\Delta}ʻ\kappa.\supset.R\unicode{x2abd}P\upharpoonright \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*33·14}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:xRy.&\supset_{x,y}.xPy.y\in \kappa.\\ +[\text{*35·101}] & \supset_{x,y}.x(P\upharpoonright \kappa)y\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·3.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.y\in \kappa.\supset.\text{E}!Rʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:\text{Hp}.&\supset.R\in 1\rightarrow \text{Cls}.y\in \text{ᗡ}ʻR.\\ +[\text{*71·163}] & \supset.\text{E}!Rʻy:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·31.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.y\in \kappa.\supset.Rʻy\in \overrightarrow{P}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:\text{Hp}.&\supset.R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}P.y\in \text{ᗡ}ʻR.\\ +[\text{*71·31}] & \supset.R\unicode{x2abd}P.(Rʻy)Ry.\\ +[\text{*23·441}] &\supset.(Rʻy)Py.\\ +[\text{*32·18}] &\supset.Rʻy\in \overrightarrow{P}ʻy:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·32.</b> \(\vdash\colon\ldotp R\in P_{\Delta}ʻ\kappa.\supset:y\in \kappa.\equiv.\text{E}!Rʻy.\equiv.Rʻy\in \overrightarrow{P}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\text{ᗡ}ʻR=\kappa:\\ +[\text{*33·43}] &\supset:\text{E}!Rʻy.\supset.y\in \kappa &\qquad \text{(1)}\\ +\vdash.\text{*14·21}.&\supset\vdash:Rʻy\in \overrightarrow{P}ʻy.\supset.\text{E}!Rʻy:\\ +[\text{(1)}] & \supset\vdash\colon\ldotp \text{Hp}.\supset:Rʻy\in \overrightarrow{P}ʻy.\supset.y\in \kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*80·3·31}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_509">[Pg 509]</span></p> + +<p class="nind"><b>*80·33.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.\supset.\text{D}ʻR\subset Pʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*37·25}.\supset\vdash:\text{Hp}.&\supset.\text{D}ʻR=Rʻʻ\kappa.R\unicode{x2abd}P.\\ +[\text{*37·201}] & \supset.\text{D}ʻR\subset Pʻʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·34.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.\supset.\text{E}‼Rʻʻ\kappa.Rʻʻ\kappa=\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*80.14.\supset\vdash:\text{Hp}.&\supset.R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR=\kappa.\\ +[*71·16.*37·25] &\supset.\text{E}‼Rʻʻ\kappa.Rʻʻ\kappa=\text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·35.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.\supset.\text{D}ʻR=\hat{x}\{(\exists y).y\in \kappa.x=Rʻy\} \quad[\text{*37·6.*80·34}]\)</p> + +<p class="nind"><b>*80·36.</b> \(\vdash:R,\,S\in P_{\Delta}ʻ\kappa.\supset.R\upharpoonright \alpha\unicode{x228d}S\upharpoonright -\alpha\in P_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·26}. & \supset\vdash:\text{Hp}.\supset.R\upharpoonright \alpha,S\upharpoonright -\alpha\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*35·64}. & \supset\vdash.\text{ᗡ}ʻ(R\upharpoonright \alpha)\cap \text{ᗡ}ʻ(S\upharpoonright -\alpha)=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*71·24}.&\supset\vdash:\text{Hp}.\supset.R\upharpoonright \alpha\unicode{x228d}S\upharpoonright -\alpha\in 1\rightarrow \text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{*35·64.*80·14}. \supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻ(R\upharpoonright \alpha)=\kappa\cap \alpha.\text{ᗡ}ʻ(S\upharpoonright -\alpha)=\kappa-\alpha.\\ +[\text{*24·41}] &\supset.\text{ᗡ}ʻ(R\upharpoonright \alpha\unicode{x228d}S\upharpoonright -\alpha)=\kappa &\qquad \text{(4)}\\ +\vdash.\text{*35·441.*80·14}.&\supset\vdash:\text{Hp}.\supset.R\upharpoonright \alpha\unicode{x2abd}P.S\upharpoonright -\alpha\unicode{x2abd}P.\\ +[\text{*23·59}]& \supset.R\upharpoonright \alpha\unicode{x228d}S\upharpoonright -\alpha\unicode{x2abd}P &\qquad \text{(5)}\\ +\vdash.\text{(3).(4).(5).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is used in dealing with greater and less among +cardinals (*117·68).</p> + +<p class="nind"><b><a id="*80·4">*80·4</a>.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.y\in \kappa.xRy.x'Py.\supset.\{(R\dot{-}x\downarrow y)\unicode{x228d}x'\downarrow y\}\in P_{\Delta}ʻ\kappa\)</p> + +<p>This proposition is important. It shows that, if \(R\in P_{\Delta}ʻ\kappa\) +and \(x\) is the selected referent of \(y\) (<i>i.e.</i> is +\(Rʻy)\), then \(x\) may be replaced by any other member of +\(\overrightarrow{P}ʻy\) without our ceasing to have a member of +\(P_{\Delta}ʻ\kappa\).</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·3}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:x\downarrow y\unicode{x2abd}R:\\ +[\text{*72·91}] \supset:\text{ᗡ}ʻ(R\dot{-}x\downarrow y)&=\text{ᗡ}ʻR-\text{ᗡ}ʻ(x\downarrow y)\\ +[\text{*80·14.*55·15}] & =\kappa-\iotaʻy &\qquad \text{(1)}\\ +\vdash.\text{(1).*33·261}. \supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ\{(R\dot{-}x\downarrow y)\unicode{x228d}x'\downarrow y\}&=(\kappa-\iotaʻy)\cup \text{ᗡ}ʻx'\downarrow y\\ +[\text{*55·15}] &=(\kappa-\iotaʻy)\cup \iotaʻy\\ +[\text{*51·221}] &=\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).*55·15}. \supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(R\dot{-}x\downarrow y)\cap \text{ᗡ}ʻ(x'\downarrow y)&=(\kappa-\iotaʻy)\cap \iotaʻy\\ +[\text{*24·21}] & =\Lambda.\\ +[\text{*71·24.*80·14}] & \supset.(R\dot{-}x\downarrow y)\unicode{x228d}x'\downarrow y\in 1\rightarrow \text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{*80·14.*55·3}.&\supset\vdash:\text{Hp}.\supset.R\dot{-}x\downarrow y\unicode{x2abd}P.x'\downarrow y\unicode{x2abd}P.\\ +[\text{*23·59}] & \supset.(R\dot{-}x\downarrow y)\unicode{x228d}x'\downarrow y\unicode{x2abd}P &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_510">[Pg 510]</span></p> + +<p class="nind"><b>*80·41.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.y\in \kappa.x'Py.\supset.[\{R\dot{-}(Rʻy)\downarrow y\}\unicode{x228d}x'\downarrow y]\in P_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·3.*30·32}.\supset\vdash:\text{Hp}.\supset.(Rʻy)Ry &\qquad \text{(1)}\\ +\vdash.\text{(1).*80·4}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·42.</b> \(\vdash:\exists !P_{\Delta}ʻ\kappa.\supset.\dot{s}ʻP_{\Delta}ʻ\kappa=P\upharpoonright \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·11}.&\supset\vdash:x(\dot{s}ʻP_{\Delta}ʻ\kappa)y.\equiv.(\exists R).R\in P_{\Delta}ʻ\kappa.xRy.\\ +[\text{*80·14}] & \supset.xPy.y\in \kappa.\\ +[\text{*35·101}] & \supset.x(P\upharpoonright \kappa)y &\qquad \text{(1)}\\ +\vdash.\text{*80·41.*35·101}.\supset\\ +\vdash:R\in P_{\Delta}ʻ\kappa.x(P\upharpoonright \kappa)y.&\supset.[\{R\dot{-}(Rʻy)\downarrow y\}\unicode{x228d}x\downarrow y]\in P_{\Delta}ʻ\kappa.\\ +[\text{*55·132}]&\supset.[\{R\dot{-}(Rʻy)\downarrow y\}\unicode{x228d}x\downarrow y]\in P_{\Delta}ʻ\kappa.x[\{R\dot{-}(Rʻy)\downarrow y\}\unicode{x228d}x\downarrow y]y.\\ +[\text{*41·141}]&\supset.x(\dot{s}ʻP_{\Delta}ʻ\kappa)y &\qquad \text{(2)}\\ +\vdash.\text{(2).Exp.*11·11·3}.&\supset\vdash:R\in P_{\Delta}ʻ\kappa.\supset.P\upharpoonright \kappa\unicode{x2abd}\dot{s}ʻP_{\Delta}ʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·23}. & \supset\vdash:\exists !P_{\Delta}ʻ\kappa.\supset.P\upharpoonright \kappa\unicode{x2abd}\dot{s}ʻP_{\Delta}ʻ\kappa &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·43.</b> \(\vdash:xPy.\equiv.x\downarrow y\in P_{\Delta}ʻ\iotaʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·182.*55·15}.&\supset\vdash.x\downarrow y\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻx\downarrow y=\iotaʻy &\qquad \text{(1)}\\ +\vdash.\text{*55·3}.& \supset\vdash:xPy.\equiv.x\downarrow y\unicode{x2abd}P &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·73}. &\supset\vdash:xPy.\equiv.x\downarrow y\unicode{x2abd}P.x\downarrow y\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻ(x\downarrow y)=\iotaʻy.\\ +[\text{*80·14}] & \equiv.x\downarrow y\in P_{\Delta}ʻ\iotaʻy:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·44.</b> \(\vdash:R\in P_{\Delta}ʻ\iotaʻy.\supset.R=(Rʻy)\downarrow y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:\text{Hp}.&\supset.R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR=\iotaʻy.\\ +[\text{*37·25}] &\supset.R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR=\iotaʻy.\text{D}ʻR=Rʻʻ\iotaʻy\\ +[\text{*53·31.*71·163}] &\qquad\qquad\qquad\qquad=\iotaʻRʻy.\\ +[\text{*55·16}] &\supset.R=(Rʻy)\downarrow y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·45.</b> \(\vdash.P_{\Delta}ʻ\iotaʻy=\downarrow yʻʻ\overrightarrow{P}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*38·131}.\supset\vdash:R\in \downarrow yʻʻ\overrightarrow{P}ʻy.&\equiv.(\exists x).x\in \overrightarrow{P}ʻy.R=x\downarrow y.\\ +[\text{*32·18}] &\equiv.(\exists x).xPy.R=x\downarrow y.\\ +[\text{*80·43}] & \supset.R\in P_{\Delta}ʻ\iotaʻy &\qquad \text{(1)}\\ +\vdash.\text{*80·44·31}.&\supset\vdash:R\in P_{\Delta}ʻ\iotaʻy.\supset.R=(Rʻy)\downarrow y.Rʻy\in \overrightarrow{P}ʻy.\\ +[\text{*14·205}] &\supset.(\exists x).R=x\downarrow y.x\in \overrightarrow{P}ʻy.\\ +[\text{*38·131}] &\supset.R\in \downarrow yʻʻ\overrightarrow{P}ʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_511">[Pg 511]</span></p> + +<p class="nind"><b><a id="*80·46">*80·46</a>.</b> \(\vdash:\exists !P_{\Delta}ʻ\iotaʻy.\equiv.\exists !\overrightarrow{P}ʻy.\equiv.y\in \text{ᗡ}ʻP \quad[\text{*80·45.*37·45.*33·41}]\)</p> + +<p class="nind"><b><a id="*80·5">*80·5</a>.</b> \(\vdash:\kappa\cap \lambda=\Lambda.R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.\supset.R\unicode{x228d}S\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup + \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.&\supset\vdash:\text{Hp}.\supset.R,\,S\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR=\kappa.\text{ᗡ}ʻS=\lambda.R\unicode{x2abd}P.S\unicode{x2abd}Q.\\ +[\text{Hp.*33·261.*23·72}]&\supset.R,\,S\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\cap \text{ᗡ}ʻS=\Lambda.\text{ᗡ}ʻ(R\unicode{x228d}S)=\kappa\cup \lambda.\\ +&\qquad\qquad\qquad\qquad\qquad\qquad\qquad R\unicode{x228d}S\unicode{x2abd}P\unicode{x228d}Q.\\ +[\text{*71·24}] &\supset.R\unicode{x228d}S\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻ(R\unicode{x228d}S)=\kappa\cup \lambda.R\unicode{x228d}S\unicode{x2abd}P\unicode{x228d}Q.\\ +[\text{*80·14}] &\supset.R\unicode{x228d}S\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·51.</b> \(\vdash:\lambda\cap \text{ᗡ}ʻP=\Lambda.R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.\supset.R\unicode{x228d}S\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup + \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·24}.\supset\vdash:\text{Hp}.&\supset.\exists !P_{\Delta}ʻ\kappa.\\ +[\text{*80·2}] & \supset.\kappa\subset \text{ᗡ}ʻP.\\ +[\text{*22·48}] & \supset.\kappa\cap \lambda\subset \text{ᗡ}ʻP\cap \lambda.\\ +[\text{Hp.*24·13}] & \supset.\kappa\cap \lambda=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*80·5}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·511.</b> \[\begin{align}\vdash:\kappa\cap \text{ᗡ}ʻQ=\Lambda.\lambda\cap \text{ᗡ}ʻP=\Lambda.M\in &(P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).\supset.\\ +& M\upharpoonright \kappa=M\dot{\cap}P.M\upharpoonright \lambda=M\dot{\cap}Q\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*23·621}.\supset\vdash:\text{Hp}.&\supset.M=M\dot{\cap}(P\unicode{x228d}Q).\\ +[\text{*35·17}] & \supset.M\upharpoonright \kappa=M\dot{\cap}(P\unicode{x228d}Q)\upharpoonright \kappa\\ +[\text{*35·644}] &=M\dot{\cap}P\upharpoonright \kappa\\ +[\text{*35·642.*25·24}] & =M\dot{\cap}(P\upharpoonright \kappa\unicode{x228d}P\upharpoonright \lambda)\\ +[\text{*35·412·17}] & =M\upharpoonright (\kappa\cup \lambda)\dot{\cap}P\\ +[\text{*80·29}] & =M\dot{\cap}P &\qquad \text{(1)}\\ +\vdash.\text{(1)}. \frac{Q,\,P,\,\lambda,\,\kappa}{P,\,Q,\,\kappa,\,\lambda} . &\supset\vdash:\text{Hp}.\supset.M\upharpoonright \lambda=M\dot{\cap}Q &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·52.</b> \[\begin{align}\vdash:\kappa\cap \text{ᗡ}ʻQ=\Lambda.\lambda\cap \text{ᗡ}ʻP=\Lambda.M\in (P\unicode{x228d}Q)_{\Delta}ʻ&(\kappa\cup \lambda).\supset.\\ +& M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in Q_{\Delta}ʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*71·26}.&\supset\vdash:\text{Hp}.\supset.M\upharpoonright \kappa,M\upharpoonright \lambda\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*80·511}. &\supset\vdash:\text{Hp}.\supset.M\upharpoonright \kappa=M\dot{\cap}P.M\upharpoonright \lambda=M\dot{\cap}Q.\\ +[\text{*23·43}] & \supset.M\upharpoonright \kappa\unicode{x2abd}P.M\upharpoonright \lambda\unicode{x2abd}Q &\qquad \text{(2)}\\ +\vdash.\text{*80·14.*22·58}.&\supset\vdash:\text{Hp}.\supset.\kappa\subset \text{ᗡ}ʻM.\lambda\subset \text{ᗡ}ʻM.\\ +[\text{*35·65}] & \supset.\text{ᗡ}ʻM\upharpoonright \kappa=\kappa.\text{ᗡ}ʻM\upharpoonright \lambda=\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_512">[Pg 512]</span></p> + +<p class="nind"><b>*80·53.</b> \[\begin{align}\vdash\colon\ldotp \kappa &\cap \text{ᗡ}ʻQ=\Lambda.\lambda\cap \text{ᗡ}ʻP=\Lambda.\supset:\\ +& M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).\equiv.(\exists R,S).R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.M=R\unicode{x228d}S\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\text{*80·52}. &\supset\vdash:\text{Hp}.M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).\supset.M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright + \lambda\in Q_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\text{*80·29}. &\supset\vdash:\text{Hp(1)}.\supset.M=M\upharpoonright (\kappa\cup \lambda)\\ +[\text{*35·412}] & =M\upharpoonright \kappa\unicode{x228d}M\upharpoonright \lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp \text{Hp}.\supset.M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).\supset.\\ +&\qquad\qquad\qquad(\exists R,S).R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.M=R\unicode{x228d}S &\qquad \text{(3)}\\ +\vdash.\text{*80·51}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.M=R\unicode{x228d}S.\supset.\\ +&\qquad\qquad\qquad M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda):\\ +[\text{*11·11·3·35}] & \supset:(\exists R,S).R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.M=R\unicode{x228d}S.\supset.\\ +&\qquad\qquad\qquad M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda) &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*80·54">*80·54</a>.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\cap \text{ᗡ}ʻQ=\Lambda.\lambda\cap \text{ᗡ}ʻP=\Lambda.\supset:\\ +& R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.\equiv.(\exists M).M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup + \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·51}.&\supset\vdash:\text{Hp}.R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.\supset.R\unicode{x228d}S\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup + \lambda) &\qquad \text{(1)}\\ +\vdash.\text{*80·14}.&\supset\vdash:\text{Hp(1)}.\supset.\kappa\cap \text{ᗡ}ʻS=\Lambda.\lambda\cap \text{ᗡ}ʻR=\Lambda.\\ +[\text{*35·644}] &\supset.(R\unicode{x228d}S)\upharpoonright \kappa=R\upharpoonright \kappa.(R\unicode{x228d}S)\upharpoonright \lambda=S\upharpoonright \lambda.\\ +[\text{*80·29}] &\supset.(R\unicode{x228d}S)\upharpoonright \kappa=R.(R\unicode{x228d}S)\upharpoonright \lambda=S &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash:\text{Hp}.R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda.\supset.\\ +&\qquad\qquad\qquad R\unicode{x228d}S\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).(R\unicode{x228d}S)\upharpoonright \kappa=R.(R\unicode{x228d}S)\upharpoonright \lambda=S.\\ +[\text{*10·24}] &\supset.(\exists M).M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).M\upharpoonright \kappa=R.M\upharpoonright \lambda=S &\qquad \text{(3)}\\ +\vdash.\text{*80·52}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda.\supset.\\ +&\qquad\qquad\qquad R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda:\\ +[\text{*10·11·21·23}] & \supset:(\exists M).M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda.\supset.\\ +&\qquad\qquad\qquad R\in P_{\Delta}ʻ\kappa.S\in Q_{\Delta}ʻ\lambda &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·6.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.\lambda\subset \kappa.\supset.R\upharpoonright \lambda\in P_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*71·26}. &\supset\vdash:\text{Hp}.\supset.R\upharpoonright \lambda\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*80·14.*35·441}.&\supset\vdash:\text{Hp}.\supset.R\upharpoonright \lambda\unicode{x2abd}P &\qquad \text{(2)}\\ +\vdash.\text{*80·14.*35·65}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻR\upharpoonright \lambda=\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·61.</b> \(\vdash:M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in P_{\Delta}ʻ\lambda.\supset.M\upharpoonright (\kappa\cup \lambda)\in P_{\Delta}ʻ(\kappa\cup + \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·6}.&\supset\vdash:M\upharpoonright \lambda\in P_{\Delta}ʻ\lambda.\supset.M\upharpoonright (\lambda-\kappa)\in P_{\Delta}ʻ(\lambda-\kappa):\\ +[\text{Fact}] &\supset\vdash:\text{Hp}.\supset.M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright (\lambda-\kappa)\in P_{\Delta}ʻ(\lambda-\kappa).\\ +[\text{*80·5.*24·21}] &\supset.M\upharpoonright \kappa\unicode{x228d}M\upharpoonright (\lambda-\kappa)\in P_{\Delta}ʻ\{\kappa\cup (\lambda-\kappa)\}.\\ +[\text{*35·412.*22·91}]&\supset.M\upharpoonright (\kappa\cup \lambda)\in P_{\Delta}ʻ(\kappa\cup \lambda):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_513">[Pg 513]</span></p> + +<p class="nind"><b>*80·62.</b> \(\vdash:M\in P_{\Delta}ʻ(\kappa\cup \lambda).\supset.M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in P_{\Delta}ʻ\lambda + \quad[\text{*80·6.*22·58}]\)</p> + +<p class="nind"><b>*80·621.</b> \(\vdash:M\upharpoonright (\kappa\cup \lambda)\in P_{\Delta}ʻ(\kappa\cup \lambda).\supset.M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright + \lambda\in P_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·31}.&\supset\vdash.\{M\upharpoonright (\kappa\cup \lambda)\}\upharpoonright \kappa=M\upharpoonright \{(\kappa\cup \lambda)\cap \kappa\}\\ +[\text{*22·631}] &\qquad\qquad\qquad\quad=M\upharpoonright \kappa &\qquad \text{(1)}\\ +\text{Similarly} &\vdash.\{M\upharpoonright (\kappa\cup \lambda)\}\upharpoonright \lambda=M\upharpoonright \lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*80·62}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·63.</b> \(\vdash:M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in P_{\Delta}ʻ\lambda.\equiv.M\upharpoonright (\kappa\cup \lambda)\in P_{\Delta}ʻ(\kappa\cup + \lambda) \quad[\text{*80·61·621}]\)</p> + +<p class="nind"><b>*80·64.</b> \(\vdash\colon\ldotp \text{ᗡ}ʻM=\kappa\cup \lambda.\supset:M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in P_{\Delta}ʻ\lambda.\equiv.M\in + P_{\Delta}ʻ(\kappa\cup \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·452}.\supset\vdash:\text{Hp}.\supset.M=M\upharpoonright (\kappa\cup \lambda) &\qquad \text{(1)}\\ +\vdash.\text{(1).*80·63}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·65.</b> \[\begin{align}&\vdash:\kappa\cap \lambda=\Lambda.R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.\supset.R\unicode{x228d}S\in P_{\Delta}ʻ(\kappa\cup + \lambda)\\ +&\left[\text{*80·5} \frac{P}{Q}.\text{*23·56}\right]\end{align}\]</p> + +<p class="nind"><b>*80·651.</b> \(\vdash:R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.\supset.R\unicode{x228d}S\upharpoonright (\lambda-\kappa)\in P_{\Delta}ʻ(\kappa\cup \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·6}.\supset\vdash:\text{Hp}.&\supset.S\upharpoonright (\lambda-\kappa)\in P_{\Delta}ʻ(\lambda-\kappa).\\ +[\text{*80·65}] &\supset.R\unicode{x228d}S\upharpoonright (\lambda-\kappa)\in P_{\Delta}ʻ\{\kappa\cup (\lambda-\kappa)\}.\\ +[\text{*22·91}] &\supset.R\unicode{x228d}S\upharpoonright (\lambda-\kappa)\in P_{\Delta}ʻ(\kappa\cup \lambda):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·66.</b> \[\begin{align}\vdash\colon\ldotp \kappa\cap &\lambda=\Lambda.\supset:\\ +& M\in P_{\Delta}ʻ(\kappa\cup \lambda).\equiv.(\exists R,S).R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.M=R\unicode{x228d}S\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·62}. &\supset\vdash:M\in P_{\Delta}ʻ(\kappa\cup \lambda).\supset.M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in P_{\Delta}ʻ\lambda + &\qquad \text{(1)}\\ +\vdash.\text{*35·452}.&\supset\vdash:M\in P_{\Delta}ʻ(\kappa\cup \lambda).\supset.M=M\upharpoonright (\kappa\cup \lambda)\\ +[\text{*35·412}] &\qquad\qquad\qquad\qquad\qquad =M\upharpoonright \kappa\unicode{x228d}M\upharpoonright \lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash:M\in P_{\Delta}ʻ(\kappa\cup \lambda).\supset.M\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in P_{\Delta}ʻ\lambda.M=M\upharpoonright + \kappa\unicode{x228d}M\upharpoonright \lambda.\\ +[\text{*11·36}] &\supset.(\exists R,S).R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.M=R\unicode{x228d}S &\qquad \text{(3)}\\ +\vdash.\text{*80·65}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.M=R\unicode{x228d}S.\supset.M\in P_{\Delta}ʻ(\kappa\cup + \lambda)\\ +[\text{*11·11·3·35}] &\supset:(\exists R,S).R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.M=R\unicode{x228d}S.\supset.\\ +&\qquad\qquad\qquad\quad M\in P_{\Delta}ʻ(\kappa\cup \lambda) &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·661.</b> \(\vdash:\kappa\cap \lambda=\Lambda.R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.\supset.R=(R\unicode{x228d}S)\upharpoonright + \kappa.S=(R\unicode{x228d}S)\upharpoonright \lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}. & \supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻR=\kappa.\text{ᗡ}ʻS\cap \kappa=\Lambda. &\qquad \text{(1)}\\ +[\text{*35·452}] & \supset.R\upharpoonright \kappa=R &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*35·644}.&\supset\vdash:\text{Hp}.\supset.(R\unicode{x228d}S)\upharpoonright \kappa=R &\qquad \text{(3)}\\ +\text{Similarly} &\vdash:\text{Hp}.\supset.(R\unicode{x228d}S)\upharpoonright \lambda=S &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_514">[Pg 514]</span></p> + +<p class="nind"><b>*80·67.</b> \[\begin{align}\vdash\colon\ldotp \kappa\cap \lambda=\Lambda.\supset:R\in P_{\Delta}ʻ&\kappa.S\in P_{\Delta}ʻ\lambda.\equiv.\\ +&(\exists M).M\in P_{\Delta}ʻ(\kappa\cup \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·65·661}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda.\supset.\\ +&R\unicode{x228d}S\in P_{\Delta}ʻ(\kappa\cup \lambda).R=(R\unicode{x228d}S)\upharpoonright \kappa.S=(R\unicode{x228d}S)\upharpoonright \lambda.\\ +[\text{*10·24}] &\supset.(\exists M).M\in P_{\Delta}ʻ(\kappa\cup \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda &\qquad \text{(1)}\\ +\vdash.\text{*80·62}.&\supset\vdash:M\in P_{\Delta}ʻ(\kappa\cup \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda.\supset.R\in P_{\Delta}ʻ\kappa.S\in + P_{\Delta}ʻ\lambda:\\ +[\text{*10·11·23}]&\supset\vdash:(\exists M).M\in P_{\Delta}ʻ(\kappa\cup \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda.\supset.\\ +&\qquad\qquad\qquad\qquad\qquad\qquad R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·68.</b> \(\vdash:R\in P_{\Delta}ʻ(\kappa-\iotaʻy).y\in \kappa.xPy.\supset.R\unicode{x228d}x\downarrow y\in P_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·43}. &\supset\vdash:\text{Hp}.\supset.x\downarrow y\in P_{\Delta}ʻ\iotaʻy &\qquad \text{(1)}\\ +\vdash.\text{*24·21}. &\supset\vdash.(\kappa-\iotaʻy)\cap \iotaʻy=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*80·65}.\supset\vdash:\text{Hp}.&\supset.R\unicode{x228d}x\downarrow y\in P_{\Delta}ʻ\{(\kappa-\iotaʻy)\cup \iotaʻy\}.\\ +[\text{*51·221}] &\supset.R\unicode{x228d}x\downarrow y\in P_{\Delta}ʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·69.</b> \(\vdash:\exists !P_{\Delta}ʻ(\kappa\cup \lambda).\equiv.\exists !P_{\Delta}ʻ\kappa.\exists !P_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·62}. &\supset\vdash:\exists !P_{\Delta}ʻ(\kappa\cup \lambda).\supset.\exists !P_{\Delta}ʻ\kappa.\exists !P_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{*80·6}. & \supset\vdash:\exists !P_{\Delta}ʻ\lambda.\supset.\exists !P_{\Delta}ʻ(\lambda-\kappa):\\ +[\text{Fact}] & \supset\vdash:\exists !P_{\Delta}ʻ\kappa.\exists !P_{\Delta}ʻ\lambda.\supset.\exists !P_{\Delta}ʻ\kappa.\exists !P_{\Delta}ʻ(\lambda-\kappa) + &\qquad \text{(2)}\\ +\vdash.\text{*80·65}. &\supset\vdash:R\in P_{\Delta}ʻ\kappa.S\in P_{\Delta}ʻ(\lambda-\kappa).\supset.R\unicode{x228d}S\in P_{\Delta}ʻ(\kappa\cup \lambda):\\ +[\text{*10·11·23}]&\supset\vdash:\exists !P_{\Delta}ʻ\kappa.\exists !P_{\Delta}ʻ(\lambda-\kappa).\supset.\exists !P_{\Delta}ʻ(\kappa\cup \lambda) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash:\exists !P_{\Delta}ʻ\kappa.\exists !P_{\Delta}ʻ\lambda.\supset.\exists !P_{\Delta}ʻ(\kappa\cup \lambda) &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*80·7">*80·7</a>.</b> \[\begin{align}\vdash:\text{ᗡ}ʻP\cap \text{ᗡ}ʻQ=\Lambda.\kappa\subset \text{ᗡ}ʻP.\lambda\subset \text{ᗡ}ʻQ.&M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup + \lambda).\supset.\\ +& M\dot{-}Q\in P_{\Delta}ʻ\kappa.M\dot{-}P\in Q_{\Delta}ʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·33.*80·14}.\supset\vdash:\text{Hp}.&\supset.P\dot{\cap}Q=\dot{\Lambda}.M\unicode{x2abd}P\unicode{x228d}Q.\\ +[\text{*25·491}] &\supset.M\dot{-}Q=M\dot{\cap}P.M\dot{-}P=M\dot{\cap}Q &\qquad \text{(1)}\\ +\vdash.\text{*22·48.*24·13}.\supset\vdash:\text{Hp}.&\supset.\kappa\cap \text{ᗡ}ʻQ=\Lambda.\lambda\cap \text{ᗡ}ʻP=\Lambda.\\ +[\text{*80·511·52}] &\supset.M\dot{\cap}P\in P_{\Delta}ʻ\kappa.M\dot{\cap}Q\in Q_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·71.</b> \(\vdash:\text{ᗡ}ʻP\cap \text{ᗡ}ʻQ=\Lambda.M\dot{-}Q\in P_{\Delta}ʻ\kappa.M\dot{-}P\in Q_{\Delta}ʻ\lambda.\supset.M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup + \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·33}.\supset\vdash:\text{Hp}.&\supset.P\dot{\cap}Q=\dot{\Lambda}.\\ +[\text{*25·493}] & \supset.M=(M\dot{-}P)\unicode{x228d}(M\dot{-}Q) &\qquad \text{(1)}\\ +\vdash.\text{*80·2}. &\supset\vdash:\text{Hp}.\supset.\lambda\subset \text{ᗡ}ʻQ.\\ +[\text{*22·48.*24·13}]&\supset.\lambda\cap \text{ᗡ}ʻP=\Lambda.\\ +[\text{*80·51}] &\supset.(M\dot{-}Q)\unicode{x228d}(M\dot{-}P)\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_515">[Pg 515]</span></p> + +<p class="nind"><b>*80·72.</b> \[\begin{align}\vdash\colon\ldotp &\text{ᗡ}ʻP\cap \text{ᗡ}ʻQ=\Lambda.\kappa\subset \text{ᗡ}ʻP.\lambda\subset \text{ᗡ}ʻQ.\supset:\\ +& M\in (P\unicode{x228d}Q)_{\Delta}ʻ(\kappa\cup \lambda).\equiv.M\dot{-}Q\in P_{\Delta}ʻ\kappa.M\dot{-}P\in Q_{\Delta}ʻ\lambda \quad[\text{*80·7·71}]\end{align}\]</p> + +<p class="nind"><b>*80·73.</b> \(\vdash:Q=P\upharpoonright \kappa.R=P\upharpoonright \lambda.\supset.P_{\Delta}ʻ(\kappa\cup \lambda)=(Q\unicode{x228d}R)_{\Delta}ʻ(\kappa\cup \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·412}.\supset\vdash:\text{Hp}.&\supset.Q\unicode{x228d}R=P\upharpoonright (\kappa\cup \lambda).\\ +[\text{*80·23}] &\supset.(Q\unicode{x228d}R)_{\Delta}ʻ(\kappa\cup \lambda)=P_{\Delta}ʻ(\kappa\cup \lambda):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·731.</b> \(\vdash:Q=P\upharpoonright \kappa.R=P\upharpoonright \lambda.\kappa\cup \lambda\subset \text{ᗡ}ʻP.\supset.\kappa=\text{ᗡ}ʻQ.\lambda=\text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·59}.\supset\vdash:\text{Hp}.&\supset.\kappa\subset \text{ᗡ}ʻP.\lambda\subset \text{ᗡ}ʻP.\\ +[\text{*35·65}] & \supset.\kappa=\text{ᗡ}ʻQ.\lambda=\text{ᗡ}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·732.</b> \(\vdash:Q=P\upharpoonright \kappa.R=P\upharpoonright \lambda.\kappa\cap \lambda=\Lambda.\supset.\text{ᗡ}ʻQ\cap \text{ᗡ}ʻR=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·64}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻQ\subset \kappa.\text{ᗡ}ʻR\subset \lambda.\\ +[\text{*22·49}] &\supset.\text{ᗡ}ʻQ\cap \text{ᗡ}ʻR\subset \kappa\cap \lambda.\\ +[\text{*24·13}] &\supset.\text{ᗡ}ʻQ\cap \text{ᗡ}ʻR=\Lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·74.</b> \[\begin{align}\vdash:\kappa\cap \lambda=\Lambda.&M\in P_{\Delta}ʻ(\kappa\cup \lambda).\supset.\\ +& M\upharpoonright \kappa=M\upharpoonright -\lambda=M\dot{-}P\upharpoonright \lambda.M\upharpoonright \lambda=M\upharpoonright -\kappa=M\dot{-}P\upharpoonright \kappa\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·4}.\supset\vdash:\text{Hp}.\supset.M\upharpoonright \kappa&=M\upharpoonright \{(\kappa\cup \lambda)-\lambda\}\\ +[\text{*35·31}] & ={M\upharpoonright (\kappa\cup \lambda)}\upharpoonright -\lambda\\ +[\text{*80·29}] &=M\upharpoonright -\lambda &\qquad \text{(1)}\\ +\vdash.\text{*80·732}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(P\upharpoonright \kappa)\cap \text{ᗡ}ʻ(P\upharpoonright \lambda)=\Lambda.\\ +[\text{*33·33}] &\supset.P\upharpoonright \kappa\dot{\cap}P\upharpoonright \lambda=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash.\text{*80·291}. &\supset\vdash:\text{Hp}.\supset.M\unicode{x2abd}P\upharpoonright (\kappa\cup \lambda).\\ +[\text{*35·412}] &\supset.M\unicode{x2abd}P\upharpoonright \kappa\unicode{x228d}P\upharpoonright \lambda &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*25·491}.&\supset\vdash:\text{Hp}.\supset.M\dot{-}P\upharpoonright \lambda=M\dot{\cap}P\upharpoonright \kappa\\ +[\text{*35·17}] & =(M\dot{\cap}P)\upharpoonright \kappa\\ +[\text{*80·14.*23·621}] & =M\upharpoonright \kappa &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.&\supset\vdash:\text{Hp}.\supset.M\upharpoonright \kappa=M\upharpoonright -\lambda=M\dot{-}P\upharpoonright \lambda &\qquad \text{(5)}\\ +\text{Similarly} & \vdash:\text{Hp}.\supset.M\upharpoonright \lambda=M\upharpoonright -\kappa=M\dot{-}P\upharpoonright \kappa &\qquad \text{(6)}\\ +\vdash.\text{(5).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·75.</b> \[\begin{align}&\vdash:\kappa\cap \lambda=\Lambda.M\in P_{\Delta}ʻ(\kappa\cup \lambda).\supset.M\dot{-}P\upharpoonright \lambda\in P_{\Delta}ʻ\kappa.M\dot{-}P\upharpoonright + \kappa\in P_{\Delta}ʻ\lambda\\ +&[\text{*80·62·74}]\end{align}\]</p> + +<p class="nind"><b>*80·76.</b> \(\vdash:M\in P_{\Delta}ʻ\mu.R\in P_{\Delta}ʻ\kappa.R\unicode{x2abd}M.\supset.M\dot{-}R\in P_{\Delta}ʻ(\mu-\kappa)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻR=\kappa.\text{ᗡ}ʻM=\mu &\qquad \text{(1)}\\ +\vdash.\text{*80·14.*72·91}.\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(M\dot{-}R)&=\text{ᗡ}ʻM-\text{ᗡ}ʻR\\ +[\text{(1)}] &=\mu-\kappa &\qquad \text{(2)}\\ +\vdash.\text{*80·14.*71·22}.&\supset\vdash:\text{Hp}.\supset.M\dot{-}R\in 1\rightarrow \text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{*80·14.*23·47}.&\supset\vdash:\text{Hp}.\supset.M\dot{-}R\unicode{x2abd}P &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_516">[Pg 516]</span></p> + +<p class="nind"><b>*80·761.</b> \(\vdash:\kappa\cap \lambda=\Lambda.M\in P_{\Delta}ʻ(\kappa\cup \lambda).R\in P_{\Delta}ʻ\kappa.R\unicode{x2abd}M.\supset.M\dot{-}R\in + P_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·76}.&\supset\vdash:\text{Hp}.\supset.M\dot{-}R\in P_{\Delta}ʻ\{(\kappa\cup \lambda)-\kappa\} &\qquad \text{(1)}\\ +\vdash.\text{*24·4}. &\supset\vdash:\text{Hp}.\supset.(\kappa\cup \lambda)-\kappa=\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·77.</b> \(\vdash:M\in P_{\Delta}ʻ\mu.M\dot{-}R\in P_{\Delta}ʻ(\mu-\kappa).R\unicode{x2abd}M.\kappa\subset \mu.\supset.R\in P_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·76}. &\supset\vdash:\text{Hp}.\supset.M\dot{-}(M\dot{-}R)\in P_{\Delta}ʻ\{\mu-(\mu-\kappa)\} &\qquad \text{(1)}\\ +\vdash.\text{*25·411}.&\supset\vdash:\text{Hp}.\supset.M=R\unicode{x228d}(M\dot{-}R) &\qquad \text{(2)}\\ +\vdash.\text{*25·21}. &\supset\vdash.R\dot{\cap}(M\dot{-}R)=\dot{\Lambda} &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*25·4}.&\supset\vdash:\text{Hp}.\supset.M\dot{-}(M\dot{-}R)=R &\qquad \text{(4)}\\ +\vdash.\text{*24·411·21·4}. &\supset\vdash:\text{Hp}.\supset.\mu-(\mu-\kappa)=\kappa &\qquad \text{(5)}\\ +\vdash.\text{(1).(4).(5)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·771.</b> \(\vdash:\kappa\cap \lambda=\Lambda.M\in P_{\Delta}ʻ(\kappa\cup \lambda).M\dot{-}R\in P_{\Delta}ʻ\lambda.R\unicode{x2abd}M.\supset.R\in + P_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·4}.\supset\vdash:\text{Hp}.\supset.\lambda=(\kappa\cup \lambda)-\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*80·77}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*80·78">*80·78</a>.</b> \(\vdash:M\in P_{\Delta}ʻ\mu.xMy.\supset.M\dot{-}x\downarrow y\in P_{\Delta}ʻ(\mu-\iotaʻy)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·3}. &\supset\vdash:\text{Hp}.\supset.x\downarrow y\unicode{x2abd}M &\qquad \text{(1)}\\ +\vdash.\text{*80·14}.&\supset\vdash:\text{Hp}.\supset.xPy.\\ +[\text{*80·43}] & \supset.x\downarrow y\in P_{\Delta}ʻ\iotaʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*80·76}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*80·8">*80·8</a>.</b> \(\vdash:\exists !P_{\Delta}ʻ\kappa.\supset.\text{ᗡ}ʻ\dot{s}ʻP_{\Delta}ʻ\kappa=\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·42}.\supset\vdash:\text{Hp}.\supset.\dot{s}ʻP_{\Delta}ʻ\kappa=P\upharpoonright \kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*80·2.*35·65}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·81.</b> \(\vdash:\exists !P_{\Delta}ʻ\alpha.P_{\Delta}ʻ\alpha=P_{\Delta}ʻ\beta.\supset.\alpha=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻ\dot{s}ʻP_{\Delta}ʻ\alpha=\text{ᗡ}ʻ\dot{s}ʻP_{\Delta}ʻ\beta.\\ +[\text{*80·8}]&\supset.\alpha=\beta:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*80·82.</b> \(\vdash:\alpha \neq\beta.\supset.P_{\Delta}ʻ\alpha\cap P_{\Delta}ʻ\beta=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.&\supset\vdash:R\in P_{\Delta}ʻ\alpha.S\in P_{\Delta}ʻ\beta.\supset.\text{ᗡ}ʻR=\alpha.\text{ᗡ}ʻS=\beta:\\ +[\text{*13·13}] &\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in P_{\Delta}ʻ\alpha.S\in P_{\Delta}ʻ\beta.\supset.\text{ᗡ}ʻR \neq\text{ᗡ}ʻS.\\ +[\text{*30·37.*33·121.Transp}] &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\supset.R \neq S &\qquad \text{(1)}\\ +\vdash.\text{(1).*24·37}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_517">[Pg 517]</span></p> + +<p>The following proposition is used in *80·84 and in the theory of double +similarity (*111·3).</p> + +<p class="nind"><b>*80·83.</b> \(\vdash.(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·12.*71·166}.&\supset\vdash.P_{\Delta}\in 1\rightarrow \text{Cls}.\\ +[\text{*71·27}] & \supset\vdash.(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.*35·1.*51·15.\supset\\ +\vdash:\lambda\{(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\}\alpha.\lambda{(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}}\beta.\\ +&\equiv.\lambda \neq\Lambda.\lambda P_{\Delta}\alpha.\lambda P_{\Delta}\beta.\\ +[\text{*24·54.*80·13}] &\equiv.\exists !\lambda.\lambda=P_{\Delta}ʻ\alpha.\lambda=P_{\Delta}ʻ\beta.\\ +[\text{*80·81}] &\supset.\alpha=\beta &\qquad \text{(2)}\\ +\vdash.\text{(2).*71·171}.&\supset\vdash.(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\in \text{Cls}\rightarrow 1 &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*80·84">*80·84</a>.</b> \(\vdash:\Lambda{\sim}\in P_{\Delta}ʻʻ\kappa.\supset.P_{\Delta}ʻʻ\kappa\text{ sm }\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*51·36}. & \supset\vdash:\text{Hp}.\supset.P_{\Delta}ʻʻ\kappa\subset -\iotaʻ\Lambda. &\qquad \text{(1)}\\ +[\text{*37·42}] &\supset.P_{\Delta}ʻʻ\kappa=\{(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\}ʻʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{*80·12.*33·431}. &\supset\vdash.\kappa\subset \text{ᗡ}ʻP_{\Delta}.\\ +[\text{*37·51}] &\supset\vdash.\kappa\subset \breve{P}_{\Delta}ʻʻP_{\Delta}ʻʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(1).*37·2}. \supset\vdash:\text{Hp}.\supset.\breve{P}_{\Delta}ʻʻP_{\Delta}ʻʻ\kappa&\subset \breve{P}_{\Delta}ʻʻ(-\iotaʻ\Lambda)\\ +[\text{*37·4}] &\subset \text{ᗡ}ʻ\{(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\} &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.& \supset\vdash:\text{Hp}.\supset.\kappa\subset \text{ᗡ}ʻ\{(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\} &\qquad \text{(5)}\\ +\vdash.\text{(5).*80·83.*73·22}.&\supset\vdash:\text{Hp}.\supset.\{(-\iotaʻ\Lambda)\upharpoonleft P_{\Delta}\}ʻʻ\kappa\text{ sm }\kappa &\qquad \text{(6)}\\ +\vdash.\text{(2).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The three following propositions are useful both in cardinal and in +ordinal multiplication (*113 and *172).</p> + +<p class="nind"><b><a id="*80·9">*80·9</a>.</b> \(\vdash\colon\ldotp y \neq z.\supset:M\in P_{\Delta}ʻ(\iotaʻy\cup \iotaʻz).\equiv.(\exists u,v).uPy.vPz.M=u\downarrow y\unicode{x228d}v\downarrow z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·45·66}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:M\in P_{\Delta}ʻ(\iotaʻy\cup \iotaʻz).\equiv.\\ +&(\exists R,S).R\in \downarrow yʻʻ\overrightarrow{P}ʻy.S\in \downarrow zʻʻ\overrightarrow{P}ʻz.M=R\unicode{x228d}S.\\ +[\text{*38·131.*32·18}] &\equiv.(\exists u,v).uPy.vPz.M=u\downarrow y\unicode{x228d}v\downarrow z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*80·91">*80·91</a>.</b> \(\vdash:M\in P_{\Delta}ʻ(\iotaʻy\cup \iotaʻz).\supset.M=(Mʻy)\downarrow y\unicode{x228d}(Mʻz)\downarrow z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·6.*80·14}.\supset\\ +\vdash:\text{Hp}.\supset.M&=\dot{s}ʻ\breve{Q}\{(\exists w).w\in \iotaʻy\cup \iotaʻz.Q=(Mʻw)\downarrow w\}\\ +[\text{*51·235}] &=\dot{s}ʻ\breve{Q}\{Q=(Mʻy)\downarrow y.\lor.Q=(Mʻz)\downarrow z\}\\ +[\text{*51·232}] &=\dot{s}ʻ\{\iotaʻ(Mʻy)\downarrow y\cup \iotaʻ(Mʻz)\downarrow z\}\\ +[\text{*53·13}] &=(Mʻy)\downarrow y\unicode{x228d}(Mʻz)\downarrow z:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_518">[Pg 518]</span></p> + +<p>80·9·91 can be extended, by precisely similar proofs, to any finite +number of variables \(y\), \(z\), .... They will, on occasion, be +assumed for three or four variables, without fresh proofs.</p> + +<p class="nind"><b>*80·92.</b> \(\vdash:y \neq z.\supset.\text{D}ʻʻP_{\Delta}ʻ(\iotaʻy\cup \iotaʻz)=\hat{\xi}\{(\exists u,v).uPy.vPz.\xi=\iotaʻu\cup \iotaʻv\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*55·15.*33·26}. &\supset\vdash.\text{D}ʻ(u\downarrow y\unicode{x228d}v\downarrow z)=\iotaʻu\cup \iotaʻv &\qquad \text{(1)}\\ +\vdash.\text{(1).*80·9.*37·6}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\xi \in \text{D}ʻʻP_{\Delta}ʻ(\iotaʻy\cup \iotaʻz).\equiv.\\ +&(\exists u,v,M).uPy.vPz.M=u\downarrow y\unicode{x228d}v\downarrow z.\xi=\iotaʻu\cup \iotaʻv.\\ +[\text{*13·19}] &\equiv.(\exists u,v).uPy.vPz.\xi=\iotaʻu\cup \iotaʻv\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*80·93">*80·93</a>.</b> \(\vdash:\exists !P_{\Delta}ʻ(\iotaʻy\cup \iotaʻz).\equiv.y,z \in \text{ᗡ}ʻP \quad[\text{*80·46·69}]\)</p> + +<p class="nind"><b>*80·94.</b> \(\vdash:\exists !P_{\Delta}ʻ(\beta\cup \iotaʻz).\equiv.\exists !P_{\Delta}ʻ\beta.z\in \text{ᗡ}ʻP \quad[\text{*80·46·69}]\)</p> + +<p>From this proposition, together with <a href="#*80·26">*80·26</a> (which gives \(\exists !P_{\Delta}ʻ\Lambda)\), +we shall obtain an inductive proof that \(P_{\Delta}ʻ\beta\) exists +whenever \(\beta\) is a finite class contained in \(\text{ᗡ}ʻP\) (cf. +*120·611).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_519">[Pg 519]</span></p> +<h2 class="nobreak" id="*81">*81. SELECTIONS FROM MANY-ONE RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *81.</p> + +<p>When \(P\upharpoonright \kappa\) is a many-one relation, +\(P_{\Delta}ʻ\kappa\) has many important properties which do not +hold in the general case. In the first place, \(P_{\Delta}ʻ\kappa\) +consists wholly of one-one relations. In the second place, if +\(R \in P_{\Delta}ʻ\kappa\), \(\text{D}ʻR\) takes one term and +no more out of each member of \(\overrightarrow{P}ʻʻ\kappa\). +Again, if \(R \in P_{\Delta}ʻ\kappa\), \(R\) is determinate +when \(\text{D}ʻR\) is given; <i>i.e.</i> \(R,\,S \in P_{\Delta}ʻ\kappa.\text{D}ʻR=\text{D}ʻS.\supset.R=S\). +It follows that +\(\text{D}ʻʻP_{\Delta}ʻ\kappa\) is similar to \(P_{\Delta}ʻ\kappa\); +hence the number of members of \(P_{\Delta}ʻ\kappa\) is the number +of ways of choosing one member out of each class belonging to +\(\overrightarrow{P}ʻʻ\kappa\). It should be remembered that when +\(P\upharpoonright \kappa\) is many-one, \(\overrightarrow{P}ʻʻ\kappa\) +is a class of mutually exclusive classes, <i>i.e.</i> no two different +members of \(\overrightarrow{P}ʻʻ\kappa\) have any common member. This +follows immediately from <a href="#*71·181">*71·181</a>.</p> + +<p>As explained in the introduction to this section, the propositions of +this number are chiefly useful on account of their application to the +case of \({\in}\). This application is made in <a href="#*84">*84</a>. The most important +propositions in this number are:</p> + +<p class="nind"><b><a id="*81·1">*81·1</a>.</b> \(\vdash:P\upharpoonright \kappa \in \text{Cls}\rightarrow 1.\supset.P_{\Delta}ʻ\kappa\subset 1\rightarrow 1\)</p> + +<p class="nind"><b>*81·14.</b> \(\vdash:P\upharpoonright \kappa \in \text{Cls}\rightarrow 1.R \in P_{\Delta}ʻ\kappa.\supset.R=(\text{D}ʻR)\upharpoonleft + P\upharpoonright \kappa=P\dot{\cap}\text{D}ʻR\uparrow \kappa\)</p> + +<p>This proposition, by exhibiting \(R\) as a function of \(\text{D}ʻR\), +leads immediately to</p> + +<p class="nind"><b>*81·21.</b> \(\vdash:P\upharpoonright \kappa \in \text{Cls}\rightarrow 1.\supset.\text{D}\upharpoonright P_{\Delta}ʻ\kappa \in 1\rightarrow 1.\text{D}ʻʻP_{\Delta}ʻ\kappa\,\text{ + sm }\,P_{\Delta}ʻ\kappa\)</p> + +<p>This is the principal proposition of this number. The following also is +important:</p> + +<p class="nind"><b>*81·22.</b> \(\vdash:P\upharpoonright \kappa \in \text{Cls}\rightarrow 1.\supset.\text{D}ʻʻP_{\Delta}ʻ\kappa=\hat{\mu}\{y \in \kappa.\supset_{y}.\mu\cap + \overrightarrow{P}ʻy \in 1:\mu\subset Pʻʻ\kappa\}\)</p> + +<hr class="tb"> + +<p class="nind"><b>*81·1.</b> \(\vdash:P\upharpoonright \kappa \in \text{Cls}\rightarrow 1.\supset.P_{\Delta}ʻ\kappa\subset 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}. &\supset\vdash:R \in P_{\Delta}ʻ\kappa.\supset.R \in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*80·291}.&\supset\vdash\colon\ldotp R \in P_{\Delta}ʻ\kappa.\supset:R\unicode{x2abd}P\upharpoonright \kappa:\\ +[\text{*71·221}] &\supset:P\upharpoonright \kappa \in \text{Cls}\rightarrow 1.\supset.R \in \text{Cls}\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_520">[Pg 520]</span></p> + +<p class="nind"><b>*81·11.</b> \(\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R\in P_{\Delta}ʻ\kappa.x\in \text{D}ʻR.\supset.\text{E}!\breve{R}ʻx.x(P\upharpoonright \kappa)\breve{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·165.*81·1}.\supset\vdash:\text{Hp}.&\supset.\text{E}!\breve{R}ʻx. &\qquad \text{(1)}\\ +[\text{*30·32.*3·11}] &\supset.xR(\breve{R}ʻx).\\ +[\text{*80·291}] &\supset.x(P\upharpoonright \kappa)\breve{R}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·12.</b> \[\begin{align}\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R\in P_{\Delta}ʻ\kappa.x\in \text{D}ʻ&R.\supset.\\ +&\breve{R}ʻx=(℩y)(y\in \kappa.xPy)=(\kappa\upharpoonleft \breve{P})ʻx\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·361}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:x(P\upharpoonright \kappa)\breve{R}ʻx.\equiv.\breve{R}ʻx=\{\text{Cnv}ʻ(P\upharpoonright \kappa)\}ʻx:\\ +[\text{*81·11}] &\supset:\breve{R}ʻx=\{\text{Cnv}ʻ(P\upharpoonright \kappa)\}ʻx\\ +[\text{*35·52}] &\qquad\quad=(\kappa\upharpoonleft \breve{P})ʻx &\qquad \text{(1)}\\ +[\text{*35·1}] &\qquad\quad=({℩}y)(y\in \kappa.xPy) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·13.</b> \(\vdash\colon\ldotp P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R\in P_{\Delta}ʻ\kappa.\supset:xRy.\equiv.x\in \text{D}ʻR.xPy.y\in \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*81·12}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp x\in \text{D}ʻR.&\supset:y=\breve{R}ʻx.\equiv.y=(\kappa\upharpoonleft \breve{P})ʻx:\\ +[\text{*71·361}] \supset:xRy.&\equiv.x(P\upharpoonright \kappa)y.\\ +[\text{*35·101}] &\equiv.xPy.y\in \kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*5·32}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:x\in \text{D}ʻR.xRy.&\equiv.x\in \text{D}ʻR.xPy.y\in \kappa:\\ +[\text{*33·14.*4·71}] &\supset:xRy.\equiv.x\in \text{D}ʻR.xPy.y\in \kappa\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·14.</b> \[\begin{align}&\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R\in P_{\Delta}ʻ\kappa.\supset.R=(\text{D}ʻR)\upharpoonleft + P\upharpoonright \kappa=P\dot{\cap}\text{D}ʻR\uparrow \kappa\\ +&[\text{*81·13.*35·102·822}]\end{align}\]</p> + +<p>This proposition, by exhibiting \(R\) as a function of \(\text{D}ʻR\), +shows that a member of \(P_{\Delta}ʻ\kappa\) is determinate +when its domain is given, provided \(P\upharpoonright \kappa\in\text{Cls}\rightarrow 1\).</p> + +<p class="nind"><b>*81·15.</b> \(\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R\in P_{\Delta}ʻ\kappa.y\in \kappa.\supset.\iotaʻRʻy=\text{D}ʻR\cap \overrightarrow{P}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*81·13}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:xRy.\equiv_{x}.x\in \text{D}ʻR.xPy:\\ +[\text{*32·18}] &\supset:x\in \overrightarrow{R}ʻy.\equiv_{x}.x\in \text{D}ʻR.x\in \overrightarrow{P}ʻy:\\ +[\text{*20·43.*22·33}] &\supset:\overrightarrow{R}ʻy=\text{D}ʻR\cap \overrightarrow{P}ʻy:\\ +[\text{*53·31.*71·163.*80·14}]&\supset:\iotaʻRʻy=\text{D}ʻR\cap \overrightarrow{P}ʻy\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_521">[Pg 521]</span></p> + +<p class="nind"><b>*81·2.</b> \(\vdash\colon\ldotp P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R,\,S\in P_{\Delta}ʻ\kappa.\supset:\text{D}ʻR=\text{D}ʻS.\equiv.R=S\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37.*33·12}.&\supset\vdash:R=S.\supset.\text{D}ʻR=\text{D}ʻS &\qquad \text{(1)}\\ +\vdash.\text{*81·14.*13·12}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\text{D}ʻR=\text{D}ʻS.\supset.R=P\dot{\cap}\text{D}ʻS\uparrow \kappa\\ +[\text{*81·14}] &\qquad\qquad\qquad\quad =S &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·21.</b> \[\begin{align}&\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\supset.\text{D}\upharpoonright P_{\Delta}ʻ\kappa\in 1\rightarrow 1.\text{D}ʻʻP_{\Delta}ʻ\kappa\text{ + sm }P_{\Delta}ʻ\kappa\\ +&[\text{*81·2.*71·59.*73·28}]\end{align}\]</p> + +<p>This proposition is very important. The class +\(\text{D}ʻʻP_{\Delta}ʻ\kappa\), when \(P\upharpoonright \kappa\in +\text{Cls}\rightarrow 1\), is formed, as we shall prove later, by +making every possible selection of one term out of each member of +\(\overrightarrow{P}ʻʻ\kappa\), each such selection giving us one +member of \(\text{D}ʻʻP_{\Delta}ʻ\kappa\). The fact that, with the +above hypothesis, the class of classes \(\text{D}ʻʻP_{\Delta}ʻ\kappa\) +has the same number of terms as \(P_{\Delta}ʻ\kappa\) (which results +from the above proposition), is of great utility in the theory of +cardinal multiplication and exponentiation.</p> + +<p class="nind"><b>*81·211.</b> \(\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\supset.\text{D}ʻʻP_{\Delta}ʻ\kappa\subset \hat{\mu}\{y\in \kappa.\supset_{y}.\mu\cap + \overrightarrow{P}ʻy\in 1:\mu\subset Pʻʻ\kappa\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*81·15.*52·1}.&\supset\vdash\colon\ldotp \text{Hp}.R\in P_{\Delta}ʻ\kappa.\mu=\text{D}ʻR.\supset:y\in \kappa.\supset_{y}.\mu\cap + \overrightarrow{P}ʻy\in 1\colon\ldotp \\ +[\text{*10·11·23·35}] &\supset\vdash\colon\ldotp \text{Hp}:(\exists R).R\in P_{\Delta}ʻ\kappa.\mu=\text{D}ʻR:\supset:y\in \kappa.\supset_{y}.\mu\cap + \overrightarrow{P}ʻy\in 1\colon\ldotp \\ +[\text{*37·6.*33·12}] &\supset\vdash\colon\ldotp \text{Hp}.\mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa.\supset:y\in \kappa.\supset_{y}.\mu\cap + \overrightarrow{P}ʻy\in 1 &\qquad \text{(1)}\\ +\vdash.\text{*80·291.*33·263}.\supset\\ +&\vdash:R\in P_{\Delta}ʻ\kappa.\mu=\text{D}ʻR.\supset.\mu\subset \text{D}ʻ(P\upharpoonright \kappa).\\ +[\text{*37·401}] &\supset.\mu\subset Pʻʻ\kappa:\\ +[\text{*10·11·23·35}] &\supset\vdash:(\exists R).R\in P_{\Delta}ʻ\kappa.\mu=\text{D}ʻR.\supset.\mu\subset Pʻʻ\kappa:\\ +[\text{*37·6.*33·12}] &\supset\vdash:\mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa.\supset.\mu\subset Pʻʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·212.</b> \(\vdash\colon\ldotp y\in \kappa.\supset_{y}.\mu\cap \overrightarrow{P}ʻy\in 1:\mu\subset Pʻʻ\kappa:\supset.\mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa.\mu\upharpoonleft + P\upharpoonright \kappa\in P_{\Delta}ʻ\kappa\)</p> + +<p><span class="pagenum" id="Page_522">[Pg 522]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·442.*37·402}.\supset\\ +\vdash:R=\mu\upharpoonleft P\upharpoonright \kappa.&\supset.R\unicode{x2abd}P.\text{ᗡ}ʻR=\kappa\cap \breve{P}ʻʻ\mu.\text{D}ʻR=\mu\cap Pʻʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*52·16}. & \supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \kappa.\supset_{y}.\exists !\mu\cap \overrightarrow{P}ʻy.\\ +[\text{*37·46.*32·241}] & \supset_{y}.y\in \breve{P}ʻʻ\mu:\\ +[\text{*22·1}] &\supset:\kappa\subset \breve{P}ʻʻ\mu &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*22·621}.&\supset\vdash:\text{Hp}.R=\mu\upharpoonleft P\upharpoonright \kappa.\supset.R\unicode{x2abd}P.\text{ᗡ}ʻR=\kappa.\text{D}ʻR=\mu &\qquad \text{(3)}\\ +\vdash.\text{*32·18.*35·102}.&\supset\vdash\colon\ldotp \text{Hp}(3).\supset:y\in \kappa.\supset_{y}.\overrightarrow{R}ʻy=\mu\cap \overrightarrow{P}ʻy.\\ +[\text{Hp}] &\supset_{y}.\overrightarrow{R}ʻy\in 1:\\ +[\text{*37·702}] &\supset:\overrightarrow{R}ʻʻ\kappa\subset 1:\\ +[\text{(3).*71·1}] & \supset:R\in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).*80·14}.&\supset\vdash:\text{Hp}.\supset.\mu\upharpoonleft P\upharpoonright \kappa\in P_{\Delta}ʻ\kappa.\text{D}ʻ(\mu\upharpoonleft + P\upharpoonright \kappa)=\mu. &\qquad \text{(5)}\\ +[\text{*37·6}] & \supset.\mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(6)}\\ +\vdash.\text{(5).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·22.</b> \[\begin{align}&\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\supset.\text{D}ʻʻP_{\Delta}ʻ\kappa=\hat{\mu}\{y\in \kappa.\supset_{y}.\mu\cap + \overrightarrow{P}ʻy\in 1:\mu\subset Pʻʻ\kappa\}\\ +&[\text{*81·211·212}]\end{align}\]</p> + +<p class="nind"><b>*81·221.</b> \(\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\supset.P_{\Delta}ʻ\kappa=\upharpoonleft (P\upharpoonright \kappa)ʻʻ\text{D}ʻʻP_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*81·14.*37·62}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:R\in P_{\Delta}ʻ\kappa.&\supset_{R}.R=(\text{D}ʻR)\upharpoonleft P\upharpoonright \kappa.\text{D}ʻR\in \text{D}ʻʻP_{\Delta}ʻ\kappa.\\ +[\text{*10·24}] &\supset_{R}.(\exists \mu).R=\mu\upharpoonleft P\upharpoonright \kappa.\mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa.\\ +[\text{*38·131}] &\supset_{R}.R\in \upharpoonleft (P\upharpoonright \kappa)ʻʻ\text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*81·22·212}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa.\supset_{\mu}.\mu\upharpoonleft + P\upharpoonright \kappa\in P_{\Delta}ʻ\kappa:\\ +[\text{*37·61}] & \supset:\upharpoonleft (P\upharpoonright \kappa)ʻʻ\text{D}ʻʻP_{\Delta}ʻ\kappa\subset P_{\Delta}ʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·23.</b> \(\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R\in P_{\Delta}ʻ\kappa.y\in + \kappa.\supset.\text{D}ʻR-\overrightarrow{P}ʻy=\text{D}ʻR-\iotaʻ\overrightarrow{R}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·93}.\supset\vdash.\text{D}ʻR-\overrightarrow{P}ʻy=\text{D}ʻR-(\text{D}ʻR\cap \overrightarrow{P}ʻy) &\qquad \text{(1)}\\ +\vdash.\text{*81·15}.\supset\vdash:\text{Hp}.\supset.\text{D}ʻR-(\text{D}ʻR\cap \overrightarrow{P}ʻy)=\text{D}ʻR-\iotaʻRʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·24.</b> \(\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa.y\in + \kappa.\supset.\mu-\overrightarrow{P}ʻy\in \text{D}ʻʻP_{\Delta}ʻ(\kappa-\iotaʻy)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·78}.&\supset\vdash:R\in P_{\Delta}ʻ\kappa.y\in \kappa.\supset.R\dot{-}(Rʻy)\downarrow y\in P_{\Delta}ʻ(\kappa-\iotaʻy).\\ +[\text{*37·62.*33·12}] & \supset.\text{D}ʻ\{R\dot{-}(Rʻy)\downarrow y\}\in \text{D}ʻʻP_{\Delta}ʻ(\kappa-\iotaʻy) &\qquad \text{(1)}\\ +\vdash.\text{*81·1.*80·14}.\supset\\ +\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.R\in P_{\Delta}ʻ\kappa.y\in \kappa.\supset.R\in 1\rightarrow 1.y\in \text{ᗡ}ʻR.\\ +[\text{*72·911.*71·31.*55·3}] &\supset.\text{D}ʻ\{R\dot{-}(Rʻy)\downarrow y\}=\text{D}ʻR-\iotaʻRʻy\\ +[\text{*81·23}] &\qquad\qquad\qquad\qquad =\text{D}ʻR-\overrightarrow{P}ʻy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash:\text{Hp(2)}.\text{D}ʻR=\mu.\supset.\mu-\overrightarrow{P}ʻy\in \text{D}ʻʻP_{\Delta}ʻ(\kappa-\iotaʻy) &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·23·35.*37·6.*33·12}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_523">[Pg 523]</span></p> + +<p class="nind"><b>*81·25.</b> \(\vdash: y \in \kappa . xPy. \mu \in \text{D}ʻʻP_{\Delta}ʻ(\kappa-\iotaʻy).\supset.\mu \cup \iotaʻx \in \text{D}ʻʻP_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·68} .&\supset\vdash: y \in \kappa. xPy. R \in P_{\Delta}ʻ(\kappa-\iotaʻy).\supset.R \unicode{x228d} x\downarrow y \in P_{\Delta}ʻ\kappa .\\ +[\text{*37·62}] & \supset . \text{D}ʻ( R \unicode{x228d} x\downarrow y) \in \text{D}ʻʻP_{\Delta}ʻ\kappa.\\ +[\text{*33·26.*55·15}] &\supset . \text{D}ʻR \cup \iotaʻx \in \text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(1)}\\ +\vdash. \text{(1)}.&\supset\vdash: y \in \kappa . xPy. R \in P_{\Delta}ʻ(\kappa-\iotaʻy). \mu = \text{D}ʻR .\supset. \mu \cup \iotaʻx \in \text{D}ʻʻP_{\Delta}ʻ\kappa + &\qquad \text{(2)}\\ +\vdash. \text{(2). *10·11·23·35 . *37·6} . \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·26.</b> \[\begin{align}\vdash\colon\ldotp P\upharpoonright \kappa \in \text{Cls} \rightarrow 1. y \in \kappa .\mu &\cap \overrightarrow{P}ʻy \in 1 . \supset :\\ +&\mu - \overrightarrow{P}ʻy \in \text{D}ʻʻP_{\Delta}ʻ(\kappa-\iotaʻy).\equiv.\mu \in \text{D}ʻʻP_{\Delta}ʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*81·24} . &\supset\vdash\colon\ldotp \text{Hp} . \supset : \mu \in \text{D}ʻʻP_{\Delta}ʻ\kappa .\supset. \mu - \overrightarrow{P}ʻy \in \text{D}ʻʻP_{\Delta}ʻ(\kappa + - \iotaʻy) &\qquad \text{(1)}\\ +\vdash. \text{*81·25} . &\supset\vdash\colon\ldotp \text{Hp}. \supset : \mu \cap \overrightarrow{P}ʻy = \iotaʻx.\mu - \overrightarrow{P}ʻy \in \text{D}ʻʻP_{\Delta}ʻ(\kappa + - \iotaʻy). \supset .\\ +&\qquad\qquad\quad(\mu - \overrightarrow{P}ʻy) \cup \iotaʻx \in \text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(2)}\\ +\vdash. \text{*22·551}. &\supset\vdash: \mu \cap \overrightarrow{P}ʻy=\iotaʻx.\supset.(\mu-\overrightarrow{P}ʻy) \cup \iotaʻx = (\mu-\overrightarrow{P}ʻy) \cup (\mu \cap \overrightarrow{P}ʻy)\\ +[\text{*24·41}] &\qquad\qquad\quad = \mu &\qquad \text{(2)}\\ +\vdash. \text{*52·1}. &\supset\vdash: \text{Hp} . \supset . (\exists x) . \mu \cap \overrightarrow{P}ʻy = \iotaʻx &\qquad \text{(4)}\\ +\vdash. \text{(2).(3).(4)} .&\supset\vdash\colon\ldotp \text{Hp} .\supset : \mu - \overrightarrow{P}ʻy \in \text{D}ʻʻP_{\Delta}ʻ(\kappa + - \iotaʻy) .\supset . \mu\in \text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(5)}\\ +\vdash. \text{(1). (5)}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·3.</b> \(\vdash: P\upharpoonright \kappa\in \text{Cls}\rightarrow 1. \lambda=\overrightarrow{P}ʻʻ\kappa.\supset.\text{D}ʻʻP_{\Delta}ʻ\kappa=\hat{\mu}\{\alpha\in + \lambda.\supset_{\alpha}.\mu\cap \alpha\in 1:\mu\subset sʻ\lambda\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*37·706}. \supset\vdash\colon\ldotp y \in \kappa .\supset_{y} . \mu \cap \overrightarrow{P}ʻy \in 1 :\equiv: \alpha \in \overrightarrow{P}ʻʻ\kappa.\supset_{\alpha} + . \mu\cap \alpha \in 1 &\qquad \text{(1)}\\ +\vdash. \text{*40·5}. \supset\vdash: \mu\subset Pʻʻ\kappa .\equiv. \mu \subset sʻ\overrightarrow{P}ʻʻ\kappa &\qquad \text{(2)}\\ +\vdash. \text{(1) . (2). *81·22}. \supset\\ +\vdash: P\upharpoonright \kappa \in \text{Cls} \rightarrow 1.\supset.\text{D}ʻʻP_{\Delta}ʻ\kappa = \hat{\mu}\{\alpha \in \overrightarrow{P}ʻʻ\kappa .\supset_{\alpha}. + \mu\cap \alpha\in 1 :\mu\subset sʻ\overrightarrow{P}ʻʻ\kappa\} &\qquad \text{(3)}\\ +\vdash. \text{(3).*13·12}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*81·31.</b> \(\vdash : P\upharpoonright \kappa, Q\upharpoonright \kappa \in \text{Cls} \rightarrow 1. \overrightarrow{P}ʻʻ\kappa = \overrightarrow{Q}ʻʻ\kappa.\supset. \text{D}ʻʻP_{\Delta}ʻ\kappa + = \text{D}ʻʻQ_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*81·3}. \supset\vdash: \text{Hp} .\supset. \text{D}ʻʻP_{\Delta}ʻ\kappa&= \hat{\mu}\{\alpha \in \overrightarrow{Q}ʻʻ\kappa .\supset_{\alpha}.\mu + \cap \alpha\in 1 :\mu\subset sʻ\overrightarrow{Q}ʻʻ\kappa\}\\ +[\text{*81·3}] & = \text{D}ʻʻQ_{\Delta}ʻ\kappa : \supset\vdash . \text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_524">[Pg 524]</span></p> +<h2 class="nobreak" id="*82">*82. SELECTIONS FROM RELATIVE PRODUCTS.</h2> +</div> + + +<p><i>Summary of</i> *82.</p> + +<p>The propositions contained in this number are not much used except in +connection with the associative law for cardinal multiplication, but +they have a certain intrinsic interest. We prove in this number that, +with a suitable hypothesis, (\(P\mid Q)_{\Delta}ʻ\lambda\) results +from \(P_{\Delta}ʻQʻʻ\lambda\) by multiplying each member by \(Q\), +<i>i.e.</i></p> + +<p class="nind"><b>*82·272.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\in \text{D}ʻ(\breve{Q})_{\in}.\supset.(P\mid Q)_{\Delta}ʻ\lambda=\mid + QʻʻP_{\Delta}ʻQʻʻ\lambda\)</p> + +<p>Also under a suitable hypothesis the domains of (\(P\mid Q)_{\Delta}ʻ\lambda\) +are the domains of \(P_{\Delta}ʻQʻʻ\lambda\), <i>i.e.</i></p> + +<p class="nind"><b>*82·32.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\supset.\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda=\text{D}ʻʻP_{\Delta}ʻQʻʻ\lambda\)</p> + +<p>In the applications of propositions of the present number +in <a href="#*85">*85</a>, \(P\) and \(Q\) are replaced by \({\in}\) and +\(\overrightarrow{Q}\). By <a href="#*62·26">*62·26</a>, \(\in \mid \overrightarrow{Q}=Q\); +thus we obtain relations between \(Q_{\Delta}ʻ\lambda\) and +\({\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\).</p> + +<hr class="tb"> + +<p class="nind"><b>*82·2.</b> \(\vdash:M\in P_{\Delta}ʻ\kappa.N\in Q_{\Delta}ʻ\lambda.Qʻʻ\lambda\subset \kappa.\supset.M\mid N\in (P\mid Q)_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.&\supset\vdash:\text{Hp}.\supset.M,\,N\in 1\rightarrow \text{Cls}.\\ +[\text{*71·25}] &\supset.M\mid N\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*80·14}. &\supset\vdash:\text{Hp}.\supset.M\unicode{x2abd}P.N\unicode{x2abd}Q.\\ +[\text{*34·34}] &\supset.M\mid N\unicode{x2abd}P\mid Q &\qquad \text{(2)}\\ +\vdash.\text{*80·14}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻM=\kappa.\\ +[\text{*37·32}] &\supset.\text{ᗡ}ʻ(M\mid N)=\breve{N}ʻʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{*80·14}.&\supset\vdash:\text{Hp}.\supset.N\unicode{x2abd}Q.\text{ᗡ}ʻN=\lambda. &\qquad \text{(4)}\\ +[\text{*37·201·25}] &\supset.Nʻʻ\lambda\subset Qʻʻ\lambda.Nʻʻ\lambda=\text{D}ʻN.\\ +[\text{Hp}] & \supset.\text{D}ʻN\subset \kappa.\\ +[\text{*37·271}] & \supset.\breve{N}ʻʻ\kappa=\text{ᗡ}ʻN &\qquad \text{(5)}\\ +\vdash.\text{(3).(4).(5)}.&\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(M\mid N)=\lambda &&\qquad \text{(6)}\\ +\vdash.\text{(1).(2).(6).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_525">[Pg 525]</span></p> + +<p class="nind"><b>*82·21.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.\lambda\subset \text{ᗡ}ʻQ.\supset.Q_{\Delta}ʻ\lambda=\iotaʻQ\upharpoonright \lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·291·14}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in Q_{\Delta}ʻ\lambda.\supset.R\unicode{x2abd}Q\upharpoonright \lambda.\text{ᗡ}ʻR=\lambda.\\ +[\text{*72·92}] &\supset.R=(Q\upharpoonright \lambda)\upharpoonright \text{ᗡ}ʻR.\text{ᗡ}ʻR=\lambda.\\ +[\text{*35·31}] &\supset.R=Q\upharpoonright \lambda &\qquad \text{(1)}\\ +\vdash.\text{35·441·65}.&\supset\vdash:\text{Hp}.\supset.Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.Q\upharpoonright \lambda\unicode{x2abd}Q.\text{ᗡ}ʻ(Q\upharpoonright \lambda)=\lambda.\\ +[\text{*80·14}] &\supset.Q\upharpoonright \lambda\in Q_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*51·141}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·22.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.\lambda=\breve{Q}ʻʻ\kappa.M\in P_{\Delta}ʻ\kappa.\supset.M\mid Q\in (P\mid Q)_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*37·32}.&\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(M\mid Q)=\breve{Q}ʻʻ\kappa.\\ +[\text{Hp}] & \supset.\text{ᗡ}ʻ(M\mid Q)=\lambda &\qquad \text{(1)}\\ +[\text{*35·452·23}] &\supset.M\mid Q=M\mid (Q\upharpoonright \lambda).\\ +[\text{*71·25.*80·14}] &\supset.M\mid Q\in 1\rightarrow \text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{*34·34.*80·14}.&\supset\vdash:\text{Hp}.\supset.M\mid Q\unicode{x2abd}P\mid Q &&\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·221.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow \text{Cls}.\lambda\subset \text{ᗡ}ʻQ.M\in P_{\Delta}ʻQʻʻ\lambda.\supset.M\mid + Q\upharpoonright \lambda\in (P\mid Q)_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·25.*80·14}. &\supset\vdash:\text{Hp}.\supset.M\mid Q\upharpoonright \lambda\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*34·34.*80·14}. &\supset\vdash:\text{Hp}.\supset.M\mid Q\upharpoonright \lambda\unicode{x2abd}P\mid Q &\qquad \text{(2)}\\ +\vdash.\text{*37·32.*35·64.*80.14}.&\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(M\mid Q\upharpoonright \lambda)=\lambda\cap \breve{Q}ʻʻQʻʻ\lambda\\ +[\text{*37·51.*22·621}] &\qquad\qquad\qquad\qquad\qquad\quad =\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·23.</b> \(\vdash:Q\upharpoonright \lambda\in \text{Cls}\rightarrow 1.\kappa=Qʻʻ\lambda.R\in (P\mid Q)_{\Delta}ʻ\lambda.\supset.R\mid \breve{Q}\in P_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.&\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻR=\lambda. &\qquad \text{(1)}\\ +[\text{*35·48}] &\supset.R\mid \breve{Q}=R\mid (\lambda \upharpoonleft \breve{Q})\\ +[\text{*35·51}] &=R\mid \text{Cnv}ʻ(Q\upharpoonright \lambda). &\qquad \text{(2)}\\ +[\text{*71·25}] &\supset.R\mid \breve{Q}\in 1\rightarrow \text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{*37·32}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(R\mid \breve{Q})=Qʻʻ\text{ᗡ}ʻR\\ +[\text{(1)}] &\qquad\qquad\qquad\qquad\quad =Qʻʻ\lambda\\ +[\text{Hp}] &\qquad\qquad\qquad\qquad\quad=\kappa &\qquad \text{(4)}\\ +\vdash.\text{*80·291}.&\supset\vdash:\text{Hp}.\supset.R\unicode{x2abd}(P\mid Q)\upharpoonright \lambda.\\ +[\text{*35·23}] &\supset.R\unicode{x2abd}P\mid (Q\upharpoonright \lambda).\\ +[\text{*34·34}] &\supset.R\mid \text{Cnv}ʻ(Q\upharpoonright \lambda)\unicode{x2abd}P\mid Q\upharpoonright \lambda\mid \text{Cnv}ʻ(Q\upharpoonright \lambda).\\ +[\text{(2).*72·59}] & \supset.R\mid \breve{Q}\unicode{x2abd}P\upharpoonright \text{D}ʻ(Q\upharpoonright \lambda).\\ +[\text{*35·441}] &\supset.R\mid \breve{Q}\unicode{x2abd}P &\qquad \text{(5)}\\ +\vdash.\text{(3).(4).(5).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_526">[Pg 526]</span></p> + +<p class="nind"><b><a id="*82·231">*82·231</a>.</b> \(\vdash:Q\upharpoonright \lambda\in \text{Cls}\rightarrow 1.R\in (P\mid Q)_{\Delta}ʻ\lambda.\supset.R\mid \breve{Q}\in P_{\Delta}ʻQʻʻ\lambda.R=R\mid + \breve{Q}\mid Q\upharpoonright \lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.&\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻR=\lambda. &\qquad \text{(1)}\\ +[\text{*74·41}] &\supset.R\mid \breve{Q}=R\mid \lambda\upharpoonleft \breve{Q}\\ +[\text{*35·51}] &=R\mid \text{Cnv}ʻ(Q\upharpoonright \lambda).\\ +[\text{*34·27}] &\supset.R\mid \breve{Q}\mid Q\upharpoonright \lambda=R\mid \text{Cnv}ʻ(Q\upharpoonright \lambda)\mid Q\upharpoonright \lambda\\ +[\text{*72·591}] &=R\upharpoonright \text{ᗡ}ʻ(Q\upharpoonright \lambda) &\qquad \text{(2)}\\ +\vdash.\text{*80·2}. &\supset\vdash:\text{Hp}.\supset.\lambda\subset \text{ᗡ}ʻ(P\mid Q).\\ +[\text{*34·36}] &\supset.\lambda\subset \text{ᗡ}ʻQ.\\ +[\text{*35·65}] &\supset.\text{ᗡ}ʻQ\upharpoonright \lambda=\lambda.\\ +[\text{(1).*74·221}] & \supset.R\upharpoonright \text{ᗡ}ʻ(Q\upharpoonright \lambda)=R &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash:\text{Hp}.\supset.R=R\mid \breve{Q}\mid Q\upharpoonright \lambda &\qquad \text{(4)}\\ +\vdash.\text{(4).*82·23}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·24.</b> \[\begin{align}\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\kappa\subset \text{D}ʻQ.\lambda=\breve{Q}ʻʻ&\kappa.R\in (P\mid Q)_{\Delta}ʻ\lambda.\supset.\\ +&\kappa=Qʻʻ\lambda.R\mid \breve{Q}\in P_{\Delta}ʻ\kappa.R=R\mid \breve{Q}\mid Q\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·16}.\supset\vdash:\text{Hp}.&\supset.\kappa=Qʻʻ\lambda. &\qquad \text{(1)}\\ +[\text{*82·23}] & \supset.R\mid \breve{Q}\in P_{\Delta}ʻ\kappa.&\qquad \text{(2)}\\ +[\text{*80·14}] &\supset.\text{ᗡ}ʻ(R\mid \breve{Q})=\kappa.\\ +[\text{Hp}] &\supset.\breve{Q}ʻʻ\text{ᗡ}ʻ(R\mid \breve{Q})=\lambda.\\ +[\text{*74·4}] &\supset.R\mid \breve{Q}\mid Q\upharpoonright \lambda=R\mid \breve{Q}\mid Q.\\ +[\text{*82·231}] &\supset.R=R\mid \breve{Q}\mid Q &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·241.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\in \text{D}ʻ(\breve{Q})_{\in}.R\in (P\mid Q)_{\Delta}ʻ\lambda.\supset.R=R\mid \breve{Q}\mid Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*74·31}.\supset\vdash:\text{Hp}.\supset.\lambda&=\breve{Q}ʻʻQʻʻ\lambda\\ +[\text{*80·14}] &=\breve{Q}ʻʻQʻʻ\text{ᗡ}ʻR\\ +[\text{*37·32}] &=\breve{Q}ʻʻ\text{ᗡ}ʻ(R\mid \breve{Q}).\\ +[\text{*74·4}] &\supset.R\mid \breve{Q}\mid Q\upharpoonright \lambda=R\mid \breve{Q}\mid Q &\qquad \text{(1)}\\ +\vdash.\text{(1).*82·231}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·25.</b> \[\begin{align}\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\kappa\subset &\text{D}ʻQ.\lambda=\breve{Q}ʻʻ\kappa.R\in (P\mid Q)_{\Delta}ʻ\lambda.\supset.\\ +& (\exists M).M\in P_{\Delta}ʻ\kappa.R=M\mid Q \quad[\text{*82·24.*10·24}]\end{align}\]</p> + +<p class="nind"><b>*82·251.</b> \[\begin{align}&\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.R\in (P\mid Q)_{\Delta}ʻ\lambda.\supset.(\exists M).M\in P_{\Delta}ʻQʻʻ\lambda.R=M\mid + Q\upharpoonright \lambda\\ +&[\text{*82·231.*10·24}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_527">[Pg 527]</span></p> + +<p class="nind"><b>*82·26.</b> \[\begin{align}\vdash\colon\ldotp Q\upharpoonright \lambda&\in 1\rightarrow 1.\kappa\subset \text{D}ʻQ.\lambda=\breve{Q}ʻʻ\kappa.\supset:\\ +&R\in (P\mid Q)_{\Delta}ʻ\lambda.\equiv.(\exists M).M\in P_{\Delta}ʻ\kappa.R=M\mid Q \quad[\text{*82·22·25}]\end{align}\]</p> + +<p class="nind"><b>*82·261.</b> \[\begin{align}\vdash\colon\ldotp Q\upharpoonright &\lambda\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\supset:\\ +&R\in (P\mid Q)_{\Delta}ʻ\lambda.\equiv.(\exists M).M\in P_{\Delta}ʻQʻʻ\lambda.R=M\mid Q\upharpoonright \lambda\\ +[\text{*82·221·251}]\end{align}\]</p> + +<p class="nind"><b>*82·27.</b> \[\begin{align}&\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\kappa\subset \text{D}ʻQ.\lambda=\breve{Q}ʻʻ\kappa.\supset.(P\mid Q)_{\Delta}ʻ\lambda=\mid + QʻʻP_{\Delta}ʻ\kappa\\ +&[\text{*82·26.*43·121.*37·6}]\end{align}\]</p> + +<p class="nind"><b>*82·271.</b> \[\begin{align}&\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\supset.(P\mid Q)_{\Delta}ʻ\lambda=\mid + (Q\upharpoonright \lambda)ʻʻP_{\Delta}ʻQʻʻ\lambda\\ +&[\text{*82·261.*43·121.*37·6}]\end{align}\]</p> + +<p class="nind"><b>*82·272.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\in \text{D}ʻ(\breve{Q})_{\in}.\supset.(P\mid Q)_{\Delta}ʻ\lambda=\mid + QʻʻP_{\Delta}ʻQʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·23}.\supset\vdash:\text{Hp}.\supset.(\exists \mu).\lambda=\breve{Q}ʻʻ\mu.\\ +[\text{*37·261}] \qquad\qquad \supset.(\exists \mu).\lambda=\breve{Q}ʻʻ(\mu\cap \text{D}ʻQ).\\ +[\text{*22·43}] \qquad\qquad\supset.(\exists \kappa).\lambda=\breve{Q}ʻʻ\kappa.\kappa\subset \text{D}ʻQ &\qquad \text{(1)}\\ +\vdash.\text{*82·27.*74·16}.\supset\\ +\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\kappa\subset \text{D}ʻQ.\lambda=\breve{Q}ʻʻ\kappa.\supset.(P\mid Q)_{\Delta}ʻ\lambda=\mid QʻʻP_{\Delta}ʻQʻʻ\lambda + &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*10·11·23·35}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·28.</b> \[\begin{align}\vdash\colon\ldotp \kappa\upharpoonleft Q\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻ&Q.\kappa=Qʻʻ\lambda.\supset:\\ +&R\in (P\mid Q)_{\Delta}ʻ\lambda.\equiv.(\exists M).M\in P_{\Delta}ʻ\kappa.R=M\mid Q\\ +[\text{*82·26.*74·26}]\end{align}\]</p> + +<p class="nind"><b>*82·29.</b> \[\begin{align}&\vdash:\kappa\upharpoonleft Q\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda.\supset.(P\mid Q)_{\Delta}ʻ\lambda=\mid + QʻʻP_{\Delta}ʻ\kappa\\ +&[\text{*82·27.*74·26}]\end{align}\]</p> + +<p class="nind"><b>*82·291.</b> \[\begin{align}&\vdash:\kappa\upharpoonleft Q\in 1\rightarrow 1.\kappa\in \text{D}ʻQ_{\in}.\supset.(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\mid + QʻʻP_{\Delta}ʻ\kappa\\ +&[\text{Proof as in *82·272}]\end{align}\]</p> + +<p class="nind"><b>*82·3.</b> \(\vdash:M\in P_{\Delta}ʻQʻʻ\lambda.\supset.\text{D}ʻ(M\mid Q\upharpoonright \lambda)=\text{D}ʻM\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻM=Qʻʻ\lambda.\\ +[\text{*74·42}] &\supset.\text{D}ʻ(M\mid Q\upharpoonright \lambda)=\text{D}ʻM:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·31.</b> \(\vdash:R\in (P\mid Q)_{\Delta}ʻ\lambda.\supset.\text{D}ʻ(R\mid \breve{Q})=\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14·2}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻR=\lambda.\lambda\subset \text{ᗡ}ʻ(P\mid Q).\\ +[\text{*34·36}] &\supset.\text{ᗡ}ʻR\subset \text{ᗡ}ʻQ.\\ +[\text{*37·321}] &\supset.\text{D}ʻ(R\mid \breve{Q})=\text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_528">[Pg 528]</span></p> + +<p class="nind"><b>*82·32.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\supset.\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda=\text{D}ʻʻP_{\Delta}ʻQʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*82·271}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda=\text{D}ʻʻ\mid (Q\upharpoonright \lambda)ʻʻP_{\Delta}ʻQʻʻ\lambda:\\ +[\text{*37·67}] &\supset:\alpha\in \text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda.\equiv.(\exists M).M\in P_{\Delta}ʻQʻʻ\lambda.\alpha=\text{D}ʻ(M\mid Q\upharpoonright \lambda).\\ +[\text{*82·3}] &\supset.(\exists M).M\in P_{\Delta}ʻQʻʻ\lambda.\alpha=\text{D}ʻM.\\ +[\text{*37·6}] &\qquad\qquad\qquad\supset.\alpha\in \text{D}ʻʻP_{\Delta}ʻQʻʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{*82·3·221}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:M\in P_{\Delta}ʻQʻʻ\lambda.\supset.\text{D}ʻM=\text{D}ʻ(M\mid Q\upharpoonright \lambda).\\ +&\qquad\qquad\qquad\qquad M\mid (Q\upharpoonright \lambda)\in (P\mid Q)_{\Delta}ʻ\lambda.\\ +[\text{*37·62}] &\supset.\text{D}ʻM\in \text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda:\\ +[\text{*37·61}] &\supset:\text{D}ʻʻP_{\Delta}ʻQʻʻ\lambda\subset \text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·33.</b> \(\vdash:\kappa\upharpoonleft Q\in 1\rightarrow 1.\kappa\in \text{D}ʻQ_{\in}.\supset.\text{D}ʻʻ(P\mid + Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\text{D}ʻʻP_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·23·26}.\supset\vdash:\kappa\in \text{D}ʻQ_{\in}.\supset.(\exists \lambda).\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{*74·26}.\supset\\ +\vdash:\kappa\upharpoonleft Q\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda.\supset.Q\upharpoonright \lambda\in 1\rightarrow 1.\kappa\subset \text{D}ʻQ.\lambda=\breve{Q}ʻʻ\kappa. &\qquad \text{(2)}\\ +[\text{*82·32}] \supset.\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\lambda=\text{D}ʻʻP_{\Delta}ʻQʻʻ\lambda.\\ +[\text{(2).Hp(2)}]\supset.\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·23·35}.\supset\\ +\vdash\colon\ldotp \kappa\upharpoonleft Q\in 1\rightarrow 1:(\exists \lambda).\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda:\supset.\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions (*82·4·41·411·42) are lemmas for <a href="#*82·43">*83·43</a>, +which is used in the proof of *114·5, in the theory of cardinal +multiplication.</p> + +<p class="nind"><b>*82·4.</b> \(\vdash:T\in 1\rightarrow \text{Cls}.Pʻʻ\lambda\subset \text{ᗡ}ʻT.\supset.T\mid ʻʻP_{\Delta}ʻ\lambda\subset (T\mid P)_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*71·25}. &\supset\vdash:\text{Hp}.R\in P_{\Delta}ʻ\lambda.\supset.T\mid R\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*80·14.*34·34}. &\supset\vdash:\text{Hp}.R\in P_{\Delta}ʻ\lambda.\supset.T\mid R\unicode{x2abd}T\mid P &\qquad \text{(2)}\\ +\vdash.\text{*80·33}. &\supset\vdash:\text{Hp}.R\in P_{\Delta}ʻ\lambda.\supset.\text{D}ʻR\subset \text{ᗡ}ʻT.\\ +[\text{*37·322}] &\qquad\qquad\qquad\qquad\supset.\text{ᗡ}ʻ(T\mid R)=\text{ᗡ}ʻR.\\ +[\text{*80·14}] &\qquad\qquad\qquad\qquad\supset.\text{ᗡ}ʻ(T\mid R)=\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*80·14}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in P_{\Delta}ʻ\lambda.\supset.T\mid R\in (T\mid P)_{\Delta}ʻ\lambda\colon\ldotp + \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·41.</b> \(\vdash:T\in \text{Cls}\rightarrow 1.M\in (T\mid P)_{\Delta}ʻ\lambda.\supset.\breve{T}\mid M\in P_{\Delta}ʻ\lambda.M=T\mid \breve{T}\mid M\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*71·25}.&\supset\vdash:\text{Hp}.\supset.\breve{T}\mid M\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*80·14.*34·34}.&\supset\vdash:\text{Hp}.\supset.\breve{T}\mid M\unicode{x2abd}\breve{T}\mid T\mid P.\\ +[\text{*72·591.*34·2}] &\qquad\qquad\qquad\quad\unicode{x2abd}P &\qquad \text{(2)}\\ +\vdash.\text{*80·14.*34·36}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻM\subset \text{D}ʻT.\\ +[\text{*37·322}] &\qquad\qquad\supset.\text{ᗡ}ʻ(\breve{T}\mid M)=\text{ᗡ}ʻM.\\ +[\text{*80·14}] &\qquad\qquad\supset.\text{ᗡ}ʻ(\breve{T}\mid M)=\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_529">[Pg 529]</span></p> + +<p class="nind"><b>*82·411.</b> \(\vdash:T\in \text{Cls}\rightarrow 1.\supset.(T\mid P)_{\Delta}ʻ\lambda\subset T\mid ʻʻP_{\Delta}ʻ\lambda \quad[\text{*82·41}]\)</p> + +<p class="nind"><b>*82·42.</b> \(\vdash:T\in 1\rightarrow 1.Pʻʻ\lambda\subset \text{ᗡ}ʻT.\supset.(T\mid P)_{\Delta}ʻ\lambda=T\mid ʻʻP_{\Delta}ʻ\lambda \quad[\text{*82·4·411}]\)</p> + +<p class="nind"><b><a id="*82·43">*82·43</a>.</b> \[\begin{align}\vdash:T,\,Q\upharpoonright \lambda\in 1\rightarrow 1.Pʻʻ\lambda\subset \text{ᗡ}ʻT.\lambda\subset &\text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda.\supset.\\ +&(T\mid P\upharpoonright \lambda\mid \breve{Q})_{\Delta}ʻ\kappa=(T\Arrowvert \breve{Q})ʻʻP_{\Delta}ʻ\lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*82·27} \frac{\lambda,\kappa}{\kappa,\lambda} .\supset\vdash:Q\in 1\rightarrow 1.\lambda\subset DʻQ.\kappa=\breve{Q}ʻʻ\lambda.\supset.(P\mid Q)_{\Delta}ʻ\kappa=\mid + QʻʻP_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{(1)} \frac{\lambda\upharpoonleft \breve{Q}}{Q}. \supset\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\subset Dʻ(\lambda\upharpoonleft \breve{Q}).\kappa=(Q\upharpoonright \lambda)ʻʻ\lambda.\supset.\\ +\qquad\qquad\qquad\qquad(P\mid \lambda\upharpoonleft \breve{Q})_{\Delta}ʻ\kappa=(\lambda\upharpoonleft \breve{Q})ʻʻP_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).*35·61·354.*37·412.*43·481.*80·14}.\supset\\ +\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda.\supset.(P\upharpoonright \lambda\mid \breve{Q})_{\Delta}ʻ\kappa=\mid \breve{Q}ʻʻP_{\Delta}ʻ\lambda + &\qquad \text{(3)}\\ +\vdash.\text{(3)} \frac{T\mid P}{P}. \supset\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda.\supset.\\ +\qquad\qquad\qquad\qquad(T\mid P\upharpoonright \lambda\mid \breve{Q})_{\Delta}ʻ\kappa=\mid \breve{Q}ʻʻ(T\mid P)_{\Delta}ʻ\lambda &\qquad \text{(4)}\\ +\vdash.\text{(4).*82·42}.\supset\vdash:\text{Hp}.\supset.(T\mid P\upharpoonright \lambda\mid \breve{Q})_{\Delta}ʻ\kappa=\mid \breve{Q}ʻʻT\mid ʻʻP_{\Delta}ʻ\lambda\\ +[\text{*43·202.*37·33}] =(T\Arrowvert \breve{Q})ʻʻP_{\Delta}ʻ\lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·45.</b> \(\vdash:Q\upharpoonright \lambda\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\supset.(P\mid Q)_{\Delta}ʻ\lambda\text{ sm }P_{\Delta}ʻQʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*37·15}.&\supset\vdash:R\in P_{\Delta}ʻQʻʻ\lambda.\supset_{R}.\text{ᗡ}ʻR=Qʻʻ\lambda.Qʻʻ\lambda\subset \text{D}ʻQ.\\ +[\text{*14·15}] &\supset_{R}.\text{ᗡ}ʻR\subset \text{D}ʻQ:\\ +[\text{*74·72}] &\supset\vdash:\text{Hp}.\supset.\mid (Q\upharpoonright \lambda)ʻʻP_{\Delta}ʻQʻʻ\lambda\text{ sm }P_{\Delta}ʻQʻʻ\lambda.\\ +[\text{*82·271}] & \supset.(P\mid Q)_{\Delta}ʻ\lambda\text{ sm }P_{\Delta}ʻQʻʻ\lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*82·5.</b> \[\begin{align}\vdash:P\upharpoonright Qʻʻ\lambda\in \text{Cls}\rightarrow 1.Q\upharpoonright \lambda&\in 1\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\supset.\\ +&(P\mid Q)_{\Delta}ʻ\lambda\text{ sm }\text{D}ʻʻP_{\Delta}ʻQʻʻ\lambda \quad[\text{*82·45.*81·21}]\end{align}\]</p> + +<p class="nind"><b>*82·51.</b> \[\begin{align}\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\kappa\upharpoonleft Q\in 1&\rightarrow 1.\lambda\subset \text{ᗡ}ʻQ.\kappa=Qʻʻ\lambda.\supset.\\ +&(P\mid Q)_{\Delta}ʻ\lambda\text{ sm }\text{D}ʻʻP_{\Delta}ʻ\kappa \quad[\text{*82·5.*74·251}]\end{align}\]</p> + +<p class="nind"><b>*82·52.</b> \(\vdash:P\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\kappa\upharpoonleft Q\in 1\rightarrow 1.\kappa\in \text{D}ʻQ_{\in}.\supset.(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa\text{ + sm }\text{D}ʻʻP_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·23}. &\supset\vdash:\text{Hp}.\supset.(\exists \mu).\kappa=Qʻʻ\mu &\qquad \text{(1)}\\ +\vdash.\text{*37·26.*22·43}.\supset\\ +&\vdash:\kappa=Qʻʻ\mu.\lambda=\mu\cap \text{ᗡ}ʻQ.\supset.\kappa=Qʻʻ\lambda.\lambda\subset \text{ᗡ}ʻQ &\qquad \text{(2)}\\ +\vdash.\text{*74·161}. &\supset\vdash:\text{Hp}.\kappa=Qʻʻ\lambda.\lambda\subset \text{ᗡ}ʻQ.\supset.\lambda=\breve{Q}ʻʻ\kappa.\\ +[\text{*82·51}] &\supset.(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa\text{ sm }\text{D}ʻʻP_{\Delta}ʻ\kappa:\\ +[\text{*10·11·23·35}]&\supset\vdash\colon\ldotp \text{Hp}:(\exists \lambda).\kappa=Qʻʻ\lambda.\lambda\subset \text{ᗡ}ʻQ:\supset.(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa\text{ + sm }\text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash:\text{Hp}.\supset.(\exists \lambda).\kappa=Qʻʻ\lambda.\lambda\subset \text{ᗡ}ʻQ &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_530">[Pg 530]</span></p> + +<p class="nind"><b>*82·53.</b> + \[\begin{align}\vdash:&P\upharpoonright \kappa,R\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\kappa\upharpoonleft Q\in 1\rightarrow 1.\kappa\in \text{D}ʻQ_\in .\overrightarrow{P}ʻʻ\kappa=\overrightarrow{R}ʻʻ\kappa.\supset.\\ +&(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa\,\text{ sm }\,(R\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa.\\ +&\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\text{D}ʻʻ(R\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\\ +&\qquad\qquad\qquad\qquad\hat{\mu}\{\alpha\in \overrightarrow{P}ʻʻ\kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\mu\subset Pʻʻ\kappa\}\\ +&\qquad\qquad\qquad\qquad\qquad=\text{D}ʻʻP_{\Delta}ʻ\kappa=\text{D}ʻʻR_{\Delta}ʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*82·52}.&\supset\vdash:\text{Hp}.\supset.(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa\,\text{ sm }\,\text{D}ʻʻP_{\Delta}ʻ\kappa.\\ +[\text{*81·31}] &\supset.(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa\,\text{ sm }\,DʻʻR_{\Delta}ʻ\kappa.\\ +[\text{*82·52.*73·32}]&\supset.(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa\,\text{ sm }\,(R\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*82·33}.\supset\vdash:\text{Hp}.&\supset.\text{D}ʻʻ(P\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\text{D}ʻʻP_{\Delta}ʻ\kappa &\qquad \text{(2)}\\ +[\text{*81·31}] & =\text{D}ʻʻR_{\Delta}ʻ\kappa &\qquad \text{(3)}\\ +[\text{*81·3.*40·5}] &=\hat{\mu}\{\alpha\in \overrightarrow{P}ʻʻ\kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\mu\subset Pʻʻ\kappa\} &\qquad \text{(4)}\\ +\vdash.\text{*82·33}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻʻ(R\mid Q)_{\Delta}ʻ\breve{Q}ʻʻ\kappa=\text{D}ʻʻR_{\Delta}ʻ\kappa &\qquad \text{(5)}\\ +\vdash.\text{(1).(2).(3).(4).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_531">[Pg 531]</span></p> +<h2 class="nobreak" id="*83">*83. SELECTIONS FROM CLASSES OF CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *83.</p> + +<p>In this number, the general propositions which have been proved for +\(P_{\Delta}ʻ\kappa\) are to be applied to the important special case +where \(P\) is \(\in\). In this case, we have selections from classes +of classes: if \(R \in \in_{\Delta}ʻ\kappa\), \(R\) picks out a +<i>representative</i> \(Rʻ\alpha\) from each class \(\alpha\) which is +a member of \(\kappa\); <i>i.e.</i> we have +\[ +\alpha \in \kappa .\supset_{\alpha}. Rʻ\alpha \in \alpha. +\]</p> + +<p>The propositions of this number result from those of previous numbers +either immediately, by the substitution of \(\in\) for \(P\), or by the +use of propositions of *62, notably \(\overrightarrow{{\in}}ʻ\alpha = \alpha\) +(<a href="#*62·2">*62·2</a>), and \({\in}ʻʻ\kappa = sʻ\kappa\) (<a href="#*62·3">*62·3</a>).</p> + +<p>The propositions of the present number follow, in the main, the same +course as those of <a href="#*80">*80</a>, with \(\in\) substituted for \(P\) (except +that the special forms of propositions before <a href="#*80·2">*80·2</a> are not given). +We have first a set of propositions resulting immediately from early +propositions of *80. Of these the most used are:</p> + +<p class="nind"><b>*83·11.</b> \(\vdash: \Lambda \in \kappa .\supset. {\in}_{\Delta}ʻ\kappa = \Lambda\)</p> + +<p>This leads to the proposition that an arithmetical product is null if +one of its factors is null. (We cannot prove the converse universally +without assuming the multiplicative axiom.)</p> + +<p class="nind"><b>*83·15.</b> \(\vdash. {\in}_{\Delta}ʻ\Lambda = \iotaʻ\dot{\Lambda}\)</p> + +<p>Thus \({\in}_{\Delta}ʻ\Lambda\) is a unit class. This is the source of +the proposition \(\mu^{0} = 1\), where \(\mu\) is a cardinal (cf. note to +*83·15).</p> + +<p class="nind"><b>*83·2.</b> \(\vdash \colon\ldotp R \in {\in}_{\Delta}ʻ\kappa .\supset: \alpha \in \kappa .\equiv. \text{E}!Rʻ\alpha .\equiv. Rʻ\alpha \in \alpha\)</p> + +<p>Here \(Rʻ\alpha\) is the "representative" of \(\alpha\).</p> + +<p class="nind"><b>*83·21.</b> \(\vdash: R \in {\in}_{\Delta}ʻ\kappa .\supset. \text{D}ʻR \subset sʻ\kappa\)</p> + +<p>We have next a set of propositions (<a href="#*83·4">*83·4</a>—<a href="#*83·44">·44</a>) on selections from unit +classes and classes of unit classes. We have</p> + +<p><span class="pagenum" id="Page_532">[Pg 532]</span></p> + +<p class="nind"><b>*83·41.</b> \(\vdash. {\in}_{\Delta}ʻ\iotaʻ\alpha \mathop{\text{ sm }} \alpha\)</p> + +<p>This leads to the proposition that a product of one factor is equal to +that factor.</p> + +<p class="nind"><b>*83·43.</b> \(\vdash:\kappa\subset 1.\supset.{\in}_{\Delta}ʻ\kappa=\iotaʻ(\breve{\iota}\upharpoonright \kappa)=\iotaʻ({\in}\upharpoonright \kappa)\)</p> + +<p>This leads to</p> + +<p class="nind"><b>*83·44.</b> \(\vdash:\kappa\subset 1.\supset.{\in}_{\Delta}ʻ\kappa \in 1\)</p> + +<p>whence it follows that a product of factors, each of which is one, is +one. This holds even if the number of factors is infinite or zero.</p> + +<p>We have next a set of propositions (<a href="#*83·5">*83·5</a>—<a href="#*83·58">*·58</a>) on changing the +representative of a class, and on selections from a class of classes +some of which are unit classes. These propositions are seldom referred +to in the sequel.</p> + +<p>We have next (<a href="#*83·6">*83·6</a>—<a href="#*83·74">·74</a>) a set of propositions on +the domains of selections, <i>i.e.</i> on the class +\(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\). We have</p> + +<p class="nind"><b>*83·66.</b> \(\vdash:\exists !{\in}_{\Delta}ʻ\kappa.\supset.sʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa=sʻ\kappa\)</p> + +<p>(The hypothesis here cannot be dispensed with unless we assume the +multiplicative axiom.)</p> + +<p class="nind"><b>*83·7.</b> \(\vdash.\text{D}ʻʻ{\in}_{\Delta}ʻ\iotaʻ\alpha=\iotaʻʻ\alpha\)</p> + +<p class="nind"><b><a id="*83·71">*83·71</a>.</b> \(\vdash.\text{D}ʻʻ{\in}_{\Delta}ʻ\iotaʻʻ\alpha=\iotaʻ\alpha.\text{D}ʻ\alpha\upharpoonleft \breve{\iota}=\alpha\)</p> + +<p>We have next two propositions (*83·8·81) on the types of +\({\in}_{\Delta}ʻ\kappa\) and \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\). The +type of \(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\) is the same as that of +\(\kappa\) (<a href="#*83·81">*83·81</a>).</p> + +<p>The last set of propositions in this number (<a href="#*83·9">*83·9</a>—<a href="#*83·904">·904</a>) deals with +the existence of selections. We have</p> + +<p class="nind"><b>*83·9.</b> \(\vdash.\exists !{\in}_{\Delta}ʻ\Lambda\)</p> + +<p class="nind"><b>*83·901.</b> \(\vdash:\exists !{\in}_{\Delta}ʻ\iotaʻ\alpha.\equiv.\exists !\alpha\)</p> + +<p class="nind"><b>*83·904.</b> \(\vdash:\exists !{\in}_{\Delta}ʻ(\kappa\cup \iotaʻ\beta).\equiv.\exists !{\in}_{\Delta}ʻ\kappa.\exists !\beta\)</p> + +<p>From these propositions we shall deduce by mathematical induction that +whenever \(\kappa\) is a finite class, \({\in}_{\Delta}ʻ\kappa\) exists +unless \(\Lambda \in\kappa\) (cf. *120·62). Thus a product consisting +of a finite number of factors (which may themselves be either finite or +infinite) can only vanish if one of the factors vanishes.</p> + +<hr class="tb"> + +<p class="nind"><b>*83·1.</b> \(\vdash:\exists !{\in}_{\Delta}ʻ\kappa.\supset.\Lambda{\sim}{\in}\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·2}.&\supset\vdash:\text{Hp}.\supset.\kappa\subset \text{ᗡ}ʻ{\in}.\\ +[\text{*62·231}] &\supset.\Lambda{\sim}{\in}\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·11.</b> \(\vdash:\Lambda \in \kappa.\supset.{\in}_{\Delta}ʻ\kappa=\Lambda \quad[\text{*83·1.Transp}]\)</p> + +<p><span class="pagenum" id="Page_533">[Pg 533]</span></p> + +<p class="nind"><b>*83·12.</b> \(\vdash.{\in}_{\Delta}ʻ\kappa=({\in}\upharpoonright \kappa)_{\Delta}ʻ\kappa \quad[\text{*80·23}]\)</p> + +<p class="nind"><b>*83·13.</b> \(\vdash:\Lambda{\sim}\in \kappa.Q=\in \upharpoonright \kappa.\supset.{\in}_{\Delta}ʻ\kappa=Q_{\Delta}ʻ\text{ᗡ}ʻQ \quad[\text{*80·24.*62·231}]\)</p> + +<p class="nind"><b>*83·14.</b> \(\vdash:\exists !{\in}_{\Delta}ʻ\kappa.Q=\in \upharpoonright \kappa.\supset.{\in}_{\Delta}ʻ\kappa=Q_{\Delta}ʻ\text{ᗡ}ʻQ \quad[\text{*83·1·13}]\)</p> + +<p class="nind"><b><a id="*83·15">*83·15</a>.</b> \(\vdash.{\in}_{\Delta}ʻ\Lambda=\iotaʻ\dot{\Lambda} \quad[\text{*80·26}]\)</p> + +<p>In virtue of this proposition, the product of 0 cardinal numbers is +1—a proposition of which a particular case, namely \(\mu^{0}=1\), is +familiar. This arithmetical proposition results from the above as +follows. We shall define the product of the numbers of members of +\(\kappa\) as the number of members of \({\in}_{\Delta}ʻ\kappa\). +Thus when \(\kappa=\Lambda\), the number of members of +\({\in}_{\Delta}ʻ\kappa\) is a product of 0 factors. Now by the +above proposition, \({\in}_{\Delta}ʻ\Lambda\) has one member, namely +\(\dot{\Lambda}\). Hence a product of 0 factors is 1.</p> + +<p class="nind"><b>*83·16.</b> \(\vdash:\exists !\kappa.\supset.\dot{\Lambda}{\sim}\in {\in}_{\Delta}ʻ\kappa \quad[\text{*80·28}]\)</p> + +<p class="nind"><b>*83·2.</b> \(\vdash\colon\ldotp R\in {\in}_{\Delta}ʻ\kappa.\supset:\alpha\in \kappa.\equiv.\text{E}!Rʻ\alpha.\equiv.Rʻ\alpha\in \alpha \quad[\text{*80·32.*62·2}]\)</p> + +<p class="nind"><b>*83·21.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\supset.\text{D}ʻR\subset sʻ\kappa \quad[\text{*80·33.*62·3}]\)</p> + +<p class="nind"><b>*83·22.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\supset.\text{E}‼Rʻʻ\kappa.Rʻʻ\kappa=\text{D}ʻR \quad[\text{*80·34}]\)</p> + +<p class="nind"><b>*83·23.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\supset.\text{D}ʻR=\hat{x}\{(\exists \alpha).\alpha\in \kappa.x=Rʻ\alpha\} \quad[\text{*80·35}]\)</p> + +<p class="nind"><b>*83·24.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\alpha\in \kappa.x\in \alpha.\supset.[\{R\dot{-}(Rʻ\alpha)\downarrow \alpha\}\unicode{x228d}x\downarrow \alpha]\in {\in}_{\Delta}ʻ\kappa + \quad[\text{*80·41}]\)</p> + +<p class="nind"><b>*83·25.</b> \(\vdash:\exists !{\in}_{\Delta}ʻ\kappa.\supset.\dot{s}ʻ{\in}_{\Delta}ʻ\kappa=\in \upharpoonright \kappa \quad[\text{*80·42}]\)</p> + +<p class="nind"><b>*83·26.</b> \(\vdash:Q=\in \upharpoonright \kappa.\exists !Q_{\Delta}ʻ\kappa.\supset.\dot{s}ʻQ_{\Delta}ʻ\kappa=Q \quad[\text{*83·12·25}]\)</p> + +<p class="nind"><b><a id="*83·27">*83·27</a>.</b> \(\vdash\colon\ldotp R\unicode{x2abd}\in .R\in 1\rightarrow \text{Cls}.\equiv:\alpha\in \text{ᗡ}ʻR.\supset_{\alpha}.Rʻ\alpha\in \alpha \quad[\text{*62·45.*71·16}]\)</p> + +<p class="nind"><b>*83·271.</b> \(\vdash\colon\ldotp R\in {\in}_{\Delta}ʻ\text{ᗡ}ʻR.\equiv:\alpha\in \text{ᗡ}ʻR.\supset_{\alpha}.Rʻ\alpha\in \alpha \quad[\text{*83·27.*80·14}]\)</p> + +<p class="nind"><b><a id="*83·28">*83·28</a>.</b> \[\begin{align}&\vdash\colon\ldotp R\in {\in}_{\Delta}ʻ\kappa.\equiv:\alpha\in \kappa.\supset_{\alpha}.Rʻ\alpha\in \alpha:\text{ᗡ}ʻR=\kappa\\ +&[\text{*83·27.*80·14.*14·15}]\end{align}\]</p> + +<p class="nind"><b>*83·29.</b> \(\vdash\colon\ldotp R\in {\in}_{\Delta}ʻ\kappa.\equiv:\alpha\in \kappa.\equiv_{\alpha}.Rʻ\alpha\in \alpha:\text{ᗡ}ʻR=\kappa \quad[\text{*83·2·28}]\)</p> + +<p class="nind"><b>*83·3.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\cap \lambda=\Lambda.\supset:M\in {\in}_{\Delta}ʻ(\kappa\cup \lambda).\equiv.\\ +&(\exists R,\,S).R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.M=R\unicode{x228d}S \quad[\text{*80·66}]\end{align}\]</p> + +<p class="nind"><b>*83·31.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\cap \lambda=\Lambda.\supset:R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.\equiv.\\ +&(\exists M).M\in {\in}_{\Delta}ʻ(\kappa\cup \lambda).R=M\upharpoonright \kappa.S=M\upharpoonright \lambda \quad[\text{*80·67}]\end{align}\]</p> + +<p class="nind"><b><a id="*83·4">*83·4</a>.</b> \(\vdash.{\in}_{\Delta}ʻ\iotaʻ\alpha=\downarrow \alphaʻʻ\alpha \quad[\text{*80·45.*62·2}]\)</p> + +<p class="nind"><b>*83·41.</b> \(\vdash.{\in}_{\Delta}ʻ\iotaʻ\alpha\text{ sm }\alpha \quad[\text{*83·4.*73·611}]\)</p> + +<p><span class="pagenum" id="Page_534">[Pg 534]</span></p> + +<p>This proposition shows that a cardinal product of one factor +is equal to that one factor. For the number of members of +\({\in}_{\Delta}ʻ\iotaʻ\alpha\) is the product of the numbers of +members of members of \(\iotaʻ\alpha\), <i>i.e.</i> it is a product +whose only factor is the number of members of \(\alpha\). By the +above proposition, this product is equal to the number of members of +\(\alpha\).</p> + +<p class="nind"><b>*83·42.</b> \(\vdash.{\in}_{\Delta}ʻ\iotaʻʻ\alpha=\iotaʻ(\alpha\upharpoonleft \breve{\iota})=\iotaʻ(\breve{\iota}\upharpoonright \iotaʻʻ\alpha)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·12}. &\supset\vdash.{\in}_{\Delta}ʻ\iotaʻʻ\alpha=(\in \upharpoonright \iotaʻʻ\alpha)_{\Delta}ʻ\iotaʻʻ\alpha\\ +[\text{*62·56}] &\qquad\qquad\quad=(\breve{\iota}\upharpoonright \iotaʻʻ\alpha)_{\Delta}ʻ\iotaʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*72·181.*71·26}.&\supset\vdash.\breve{\iota}\upharpoonright \iotaʻʻ\alpha\in 1\rightarrow \text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{*37·15.*33·21}. &\supset\vdash.\iotaʻʻ\alpha\subset \text{ᗡ}ʻ\breve{\iota}.\\ +[\text{*35·65}] &\supset\vdash.\iotaʻʻ\alpha=\text{ᗡ}ʻ(\breve{\iota}\upharpoonright ʻʻ\alpha) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*82·21}.&\supset\vdash.(\breve{\iota}\upharpoonright \iotaʻʻ\alpha)_{\Delta}ʻ\iotaʻʻ\alpha=\iotaʻ\{(\breve{\iota}\upharpoonright + \iotaʻʻ\alpha)\upharpoonright \iotaʻʻ\alpha\}\\ +[\text{*35·31}] &\qquad\qquad\qquad =\iotaʻ(\breve{\iota}\upharpoonright \iotaʻʻ\alpha) &\qquad \text{(4)}\\ +[\text{*62·56}] &\qquad\qquad\qquad =\iotaʻ(\alpha\upharpoonleft \breve{\iota}) &\qquad \text{(5)}\\ +\vdash.\text{(1).(4).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that a cardinal product whose factors +are all 1 is 1. For \(\iotaʻʻ\alpha\) is a class whose members +are all unit classes, and thus the number of members of +\({\in}_{\Delta}ʻ\iotaʻʻ\alpha\) is the product of a number of 1's; and +by the above proposition, \({\in}_{\Delta}ʻ\iotaʻʻ\alpha\) is a unit +class, its sole member being \(\alpha\upharpoonleft \breve{\iota}\). +This result is rendered more explicit by *83·43·44.</p> + +<p class="nind"><b>*83·43.</b> \(\vdash:\kappa\subset 1.\supset.{\in}_{\Delta}ʻ\kappa=\iotaʻ(\breve{\iota}\upharpoonright \kappa)=\iotaʻ(\in \upharpoonright \kappa)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·42}.&\supset\vdash:\kappa=\iotaʻʻ\alpha.\supset.{\in}_{\Delta}ʻ\kappa=\iotaʻ(\breve{\iota}\upharpoonright \kappa) &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·23}.\supset\\ +& \vdash:(\exists \alpha).\kappa=\iotaʻʻ\alpha.\supset.{\in}_{\Delta}ʻ\kappa=\iotaʻ(\breve{\iota}\upharpoonright \kappa):\\ +[\text{*52·31}] &\supset\vdash:\kappa\subset 1.\supset.{\in}_{\Delta}ʻ\kappa=\iotaʻ(\breve{\iota}\upharpoonright \kappa)\\ +[\text{*62·55}] &\qquad\qquad\qquad\qquad =\iotaʻ(\in \upharpoonright \kappa):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*83·44">*83·44</a>.</b> \(\vdash:\kappa\subset 1.\supset.{\in}_{\Delta}ʻ\kappa\in 1 \quad[\text{*83·43.*52·22}]\)</p> + +<p class="nind"><b><a id="*83·5">*83·5</a>.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\alpha{\sim}\in \kappa.x\in \alpha.\supset.R\unicode{x228d}x\downarrow \alpha\in {\in}_{\Delta}ʻ(\kappa\cup \iotaʻ\alpha)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·43}. \supset\vdash:\text{Hp}.\supset.x\downarrow \alpha\in {\in}_{\Delta}ʻ\iotaʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*51·211}.\supset\vdash:\text{Hp}.\supset.\kappa\cap \iotaʻ\alpha=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*80·65}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>It follows from this proposition that if \(\kappa\) is a class of +classes for which there are selections, and if one member (not null) +be added to \(\kappa\), there are still selections from the resulting +class of classes.</p> + +<p class="nind"><b>*83·51.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\alpha\in \kappa.\supset.R\dot{-}(Rʻ\alpha)\downarrow \alpha\in {\in}_{\Delta}ʻ(\kappa-\iotaʻ\alpha) \quad[\text{*80·78}]\)</p> + +<p><span class="pagenum" id="Page_535">[Pg 535]</span></p> + +<p class="nind"><b>*83·52.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\alpha\in \kappa.x\in \alpha.\supset.{R\dot{-}(Rʻ\alpha)\downarrow \alpha}\unicode{x228d}x\downarrow \alpha\in {\in}_{\Delta}ʻ\kappa + \quad[\text{*80·41}]\)</p> + +<p class="nind"><b>*83·54.</b> \(\vdash:\kappa\cap \lambda=\Lambda.\lambda\subset 1.R\in {\in}_{\Delta}ʻ\kappa.\supset.R\unicode{x228d}\breve{\iota}\upharpoonright \lambda\in {\in}_{\Delta}ʻ(\kappa\cup + \lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·65}.\supset\vdash\colon\ldotp \text{Hp}.\supset:S\in {\in}_{\Delta}ʻ\lambda.\supset.R\unicode{x228d}S\in {\in}_{\Delta}ʻ(\kappa\cup \lambda) &\qquad \text{(1)}\\ +\vdash.\text{*83·43}.\supset\vdash:\text{Hp}.\supset.\breve{\iota}\upharpoonright \lambda\in {\in}_{\Delta}ʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·55.</b> \(\vdash:\kappa\cap \lambda=\Lambda.\lambda\subset 1.S\in {\in}_{\Delta}ʻ(\kappa\cup + \lambda).\supset.S\dot{-}\breve{\iota}\upharpoonright \lambda\in {\in}_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·66}. &\supset\vdash:\text{Hp}.\supset.(\exists M,\,N).M\in {\in}_{\Delta}ʻ\kappa.N\in \in _{\Delta}ʻ\lambda.S=M\unicode{x228d}N.\\ +[\text{*83·43.*51·15}] &\supset.(\exists M).M\in {\in}_{\Delta}ʻ\kappa.S=M\unicode{x228d}\breve{\iota}\upharpoonright \lambda &\qquad \text{(1)}\\ +\vdash.\text{*80·14.*35·64}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:M\in {\in}_{\Delta}ʻ\kappa.\supset.\text{ᗡ}ʻM\cap \text{ᗡ}ʻ(\breve{\iota}\upharpoonright \lambda)=\Lambda.\\ +[\text{*33·33}] &\supset.M\dot{\cap}\breve{\iota}\upharpoonright \lambda=\dot{\Lambda}.\\ +[\text{*25·4}] &\supset.(M\unicode{x228d}\breve{\iota}\upharpoonright \lambda)\dot{-}\breve{\iota}\upharpoonright \lambda=M.\\ +[\text{*13·12}] &\supset:M\in {\in}_{\Delta}ʻ\kappa.S=M\unicode{x228d}\breve{\iota}\upharpoonright \lambda.\supset.S\dot{-}\breve{\iota}\upharpoonright \lambda\in {\in}_{\Delta}ʻ\kappa + &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·21·23}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:(\exists M).M\in {\in}_{\Delta}ʻ\kappa.S=M\unicode{x228d}\breve{\iota}\upharpoonright + \lambda.\supset.S\dot{-}\breve{\iota}\upharpoonright \lambda\in {\in}_{\Delta}ʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·56.</b> \(\vdash:\kappa\cap \lambda=\Lambda.\lambda\subset 1.\supset.{\in}_{\Delta}ʻ(\kappa\cup \lambda)=\hat{M}\{(\exists R).R\in {\in}_{\Delta}ʻ\kappa.M=R\unicode{x228d}\breve{\iota}\upharpoonright + \lambda\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·66}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +M\in {\in}_{\Delta}ʻ(\kappa\cup \lambda).&\equiv.(\exists R,\,S).R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.M=R\unicode{x228d}S.\\ +[\text{*83·43}] &\equiv.(\exists R).R\in {\in}_{\Delta}ʻ\kappa.M=R\unicode{x228d}\breve{\iota}\upharpoonright \lambda\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in the theory of cardinal +multiplication (*114·41).</p> + +<p class="nind"><b>*83·57.</b> \(\vdash:\kappa\cap \lambda=\Lambda.\lambda\subset 1.\supset.{\in}_{\Delta}ʻ(\kappa\cup \lambda)\text{ sm }{\in}_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·56.*38·131}. &\supset\vdash:\text{Hp}.\supset.{\in}_{\Delta}ʻ(\kappa\cup \lambda)=(\unicode{x228d}\breve{\iota}\upharpoonright \lambda)ʻʻ{\in}_{\Delta}ʻ\kappa + &\qquad \text{(1)}\\ +\vdash.\text{*80·14.*35·64}. &\supset\vdash:\text{Hp}.R\in {\in}_{\Delta}ʻ\kappa.\supset.\text{ᗡ}ʻR\cap \text{ᗡ}ʻ(\breve{\iota}\upharpoonright \lambda)=\Lambda.\\ +[\text{*33·33}] & \supset.R\dot{\cap}\breve{\iota}\upharpoonright \lambda=\dot{\Lambda}.\\ +[\text{*25·4}] &\supset.R=(R\unicode{x228d}\breve{\iota}\upharpoonright \lambda)\dot{-}\breve{\iota}\upharpoonright \lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).*23·481.*13·172}.&\supset\vdash:\text{Hp}.R,\,S\in {\in}_{\Delta}ʻ\kappa.R\unicode{x228d}\breve{\iota}\upharpoonright + \lambda=S\unicode{x228d}\breve{\iota}\upharpoonright \lambda.\supset.R=S:\\ +[\text{Exp.*11·11·3.*38·11}] &\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +& R,\,S\in {\in}_{\Delta}ʻ\kappa.(\unicode{x228d}\breve{\iota}\upharpoonright \lambda)ʻR=(\unicode{x228d}\breve{\iota}\upharpoonright \lambda)ʻS.\supset_{R,S}.R=S:\\ +[\text{*38·12.*73·25}] & \supset:(\unicode{x228d}\breve{\iota}\upharpoonright \lambda)ʻʻ{\in}_{\Delta}ʻ\kappa\text{ sm }{\in}_{\Delta}ʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_536">[Pg 536]</span></p> + +<p class="nind"><b><a id="*83·58">*83·58</a>.</b> \(\vdash.{\in}_{\Delta}ʻ\kappa \text{ sm } {\in}_{\Delta}ʻ(\kappa-1)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·41·21.*22·43}.\supset\vdash.\kappa=(\kappa-1)\cup (\kappa\cap 1).(\kappa-1)\cap (\kappa\cap 1)=\Lambda.\kappa\cap 1\subset 1 &\qquad \text{(1)}\\ +\vdash.\text{(1).*83·57}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that in a product any number of factors each +equal to 1 may be omitted without altering the value of the product.</p> + +<p>The following propositions, down to <a href="#*83·74">*83·74</a>, are concerned with the +domains of selective relations, <i>i.e.</i> with the selected classes.</p> + +<p class="nind"><b><a id="*83·6">*83·6</a>.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\alpha\in \kappa.\supset.\exists !\alpha\cap \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·2}.\supset\vdash:\text{Hp}.&\supset.Rʻ\alpha\in \alpha.\\ +[\text{*33·43}]&\supset.Rʻ\alpha\in \alpha\cap \text{D}ʻR.\\ +[\text{*10·24}]&\supset.\exists !\alpha\cap \text{D}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·61.</b> \(\vdash:R\in {\in}_{\Delta}ʻ\kappa.\alpha\in \kappa.\alpha\cap sʻ(\kappa-\iotaʻ\alpha)=\Lambda.\supset.\alpha\cap \text{D}ʻR=\iotaʻRʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·27}.\supset\vdash\colon\ldotp \alpha\cap sʻ(\kappa-\iotaʻ\alpha)=\Lambda.&\equiv:\beta\in \kappa-\iotaʻ\alpha.\supset_{\beta}.\alpha\cap \beta=\Lambda:\\ +[\text{Transp.*51·15}] &\equiv:\beta\in \kappa.\exists !\alpha\cap \beta.\supset_{\beta}.\beta=\alpha &\qquad \text{(1)}\\ +\vdash.\text{*83·23}.\supset\vdash\colon\ldotp \text{Hp}. \supset:x\in \text{D}ʻR.&\equiv.(\exists \beta).\beta\in \kappa.x=Rʻ\beta.\\ +[\text{*10·35.*14·15}] \supset:x\in \alpha\cap \text{D}ʻR.&\equiv.(\exists \beta).\beta\in \kappa.x=Rʻ\beta.Rʻ\beta\in \alpha.\\ +[\text{*83·2}] &\equiv.(\exists \beta).\beta\in \kappa.x=Rʻ\beta.Rʻ\beta\in \alpha\cap \beta.\\ +[\text{(1).*4·71}] &\equiv.(\exists \beta).\beta\in \kappa.x=Rʻ\beta.Rʻ\beta\in \alpha\cap \beta.\alpha=\beta.\\ +[\text{*13·195.*22·5}] &\equiv.\alpha\in \kappa.x=Rʻ\alpha.Rʻ\alpha\in \alpha.\\ +[\text{Hp.*4·73.*83·2}] &\equiv.x=Rʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(2).*51·15}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·62.</b> \(\vdash:\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\supset.\mu\subset sʻ\kappa \quad[\text{*83·21.*37·63}]\)</p> + +<p class="nind"><b><a id="*83·63">*83·63</a>.</b> \(\vdash:sʻ\kappa\cap sʻ\lambda=\Lambda.\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).\supset.\mu\cap sʻ\kappa\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\mu\cap + sʻ\lambda\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·62}. &\supset\vdash:M\in {\in}_{\Delta}ʻ(\kappa\cup \lambda).\supset.M\upharpoonright \kappa\in {\in}_{\Delta}ʻ\kappa.M\upharpoonright \lambda\in {\in}_{\Delta}ʻ\lambda. + &\qquad \text{(1)}\\ +[\text{*83·21}] &\supset.\text{D}ʻM\upharpoonright \kappa\subset sʻ\kappa.\text{D}ʻM\upharpoonright \lambda\subset sʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).*24·494}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:M\in {\in}_{\Delta}ʻ(\kappa\cup \lambda).\supset.\\ +&\text{D}ʻM\upharpoonright \kappa=(\text{D}ʻM\upharpoonright \kappa\cup \text{D}ʻM\upharpoonright \lambda)-sʻ\lambda.\text{D}ʻM\upharpoonright \lambda=(\text{D}ʻM\upharpoonright \kappa\cup \text{D}ʻM\upharpoonright \lambda)-sʻ\kappa.\\ +[\text{*33·26.*35·412.*80·29}] &\supset.\text{D}ʻM\upharpoonright \kappa=\text{D}ʻM-sʻ\lambda.\text{D}ʻM\upharpoonright \lambda=\text{D}ʻM-sʻ\kappa.\\ +[\text{*24·491}] &\supset.\text{D}ʻM\upharpoonright \kappa=\text{D}ʻM\cap sʻ\kappa.\text{D}ʻM\upharpoonright \lambda=\text{D}ʻM\cap sʻ\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(3).*37·6}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +& M\in {\in}_{\Delta}ʻ(\kappa\cup \lambda).\supset.\text{D}ʻM\cap sʻ\kappa\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\text{D}ʻM\cap + sʻ\lambda\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda:\\ +[\text{*37·63}]&\supset:\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).\supset.\mu\cap sʻ\kappa\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\mu\cap + sʻ\lambda\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_537">[Pg 537]</span></p> + +<p class="nind"><b>*83·64.</b> \[\begin{align}\vdash\colon\ldotp \kappa\cap &\lambda=\Lambda.\supset:\\ +&\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).\equiv.(\exists \rho,\sigma).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda.\mu=\rho\cup + \sigma\end{align}\]</p> + +<p>Observe that the hypothesis required here is \(\kappa\cap\lambda=\Lambda\), +not \(sʻ\kappa\cap sʻ\lambda=\Lambda\) as in <a href="#*83·63">*83·63</a>.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·66}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:M\in {\in}_{\Delta}ʻ(\kappa\cup \lambda).\mu=\text{D}ʻM.\equiv.\\ +&(\exists R,\,S).R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.M = R\unicode{x228d}S.\mu=\text{D}ʻM.\\ +[\text{*13·193.*33·26}]&\equiv.(\exists R,S).R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.M=R\unicode{x228d}S.\mu=\text{D}ʻR\cup \text{D}ʻS &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21·281.*37·6}.\supset\\ +\vdash\colon\colon \text{Hp}.&\supset\colon\ldotp \mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).\equiv:\\ +&(\exists M,R,S).R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.M=R\unicode{x228d}S.\mu=\text{D}ʻR\cup \text{D}ʻS:\\ +[\text{*10·35}] &\equiv:(\exists R,S):R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.\mu=\text{D}ʻR\cup \text{D}ʻS:(\exists M).M=R\unicode{x228d}S:\\ +[\text{*21·2}] &\equiv:(\exists R,S).R\in {\in}_{\Delta}ʻ\kappa.S\in {\in}_{\Delta}ʻ\lambda.\mu=\text{D}ʻR\cup \text{D}ʻS:\\ +[\text{*13·22}] &\equiv:(\exists R,S,\rho,\sigma).R\in {\in}_{\Delta}ʻ\kappa.\rho=\text{D}ʻR.S\in {\in}_{\Delta}ʻ\lambda.\sigma=\text{D}ʻS.\mu=\rho\cup \sigma:\\ +[\text{*11·24·54}] &\equiv:(\exists \rho,\sigma):(\exists R).R\in {\in}_{\Delta}ʻ\kappa.\rho=\text{D}ʻR:(\exists S).S\in {\in}_{\Delta}ʻ\lambda.\sigma=\text{D}ʻS.\\ +&\mu=\rho\cup \sigma:\\ +[\text{*37·6.*10·35}] &\equiv:(\exists \rho,\sigma).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda.\mu=\rho\cup + \sigma\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is used in connection with cardinal +multiplication (*115·14).</p> + +<p class="nind"><b>*83·641.</b> \[\begin{align}\vdash\colon\ldotp sʻ\kappa&\cap sʻ\lambda=\Lambda.\supset:\\ +&\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).\equiv.(\exists \rho,\sigma).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda.\mu=\rho\cup + \sigma\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·25}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\kappa\cap \lambda=\Lambda\cap \text{Cls}.\lor.\kappa\cap \lambda=\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*83·64}.&\supset\vdash\colon\ldotp \kappa\cap \lambda=\Lambda\cap \text{Cls}.\supset:\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).\equiv.\\ +&(\exists \rho,\sigma).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda.\mu=\rho\cup \sigma &\qquad \text{(2)}\\ +\vdash.\text{*51·16}.&\supset\vdash\colon\ldotp \kappa\cap \lambda=\iotaʻ\Lambda.\supset:\Lambda\in \kappa.\Lambda\in \lambda:\\ +[\text{*83·11}] &\supset:{\in}_{\Delta}ʻ\kappa=\Lambda.{\in}_{\Delta}ʻ\lambda=\Lambda.{\in}_{\Delta}ʻ(\kappa\cup \lambda)=\Lambda:\\ +[\text{*37·29}] &\supset:\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa=\Lambda.\text{D}ʻʻ{\in}_{\Delta}ʻ\lambda=\Lambda.\text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda)=\Lambda:\\ +[\text{*24·15}] &\supset:\mu{\sim}\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda):(\rho).\rho{\sim}\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa:\\ +[\text{*11·55.Transp.*10·252}] &\supset:\mu{\sim}\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda):\\ +&{\sim}(\exists \rho,\sigma).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda.\mu=\rho\cup \sigma:\\ +[\text{*5·21}] &\supset:\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).\equiv.\\ +&(\exists \rho,\sigma).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda.\mu=\rho\cup \sigma &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_538">[Pg 538]</span></p> + +<p class="nind"><b>*83·65.</b> \[\begin{align}\vdash:sʻ\kappa\cap sʻ\lambda=\Lambda.\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).&\supset.\\ +&\mu-sʻ\kappa\in \text{D}ʻʻ{\in}_{\Delta}ʻ\lambda.\mu-sʻ\lambda\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·62} \supset\vdash:\text{Hp}.&\supset.\mu\subset sʻ(\kappa\cup \lambda).\\ +[\text{*40·171}] &\supset.\mu\subset sʻ\kappa\cup sʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*24·491}.&\supset\vdash:\text{Hp}.\supset.\mu-sʻ\kappa=\mu\cap sʻ\lambda.\mu-sʻ\lambda=\mu\cap sʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(2).*83·63}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·66.</b> \(\vdash:\exists !{\in}_{\Delta}ʻ\kappa.\supset.sʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa=sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·43}.\supset\vdash.sʻ\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa&=\text{D}ʻ\dot{s}ʻ{\in}_{\Delta}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*83·25}.\supset\vdash:\text{Hp}.\supset.\text{D}ʻ\dot{s}ʻ{\in}_{\Delta}ʻ\kappa&=\text{D}ʻ\in \upharpoonright \kappa\\ +[\text{*62·43}] & =sʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·7.</b> \(\vdash.\text{D}ʻʻ{\in}_{\Delta}ʻ\iotaʻ\alpha=\iotaʻʻ\alpha \quad[\text{*83·4.*55·261}]\)</p> + +<p class="nind"><b>*83·71.</b> \(\vdash.\text{D}ʻʻ{\in}_{\Delta}ʻ\iotaʻʻ\alpha=\iotaʻ\alpha.\text{D}ʻ\alpha\upharpoonleft \breve{\iota}=\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·42}.\supset\vdash.\text{D}ʻʻ{\in}_{\Delta}ʻ\iotaʻʻ\alpha&=\text{D}ʻʻ\iotaʻ(\alpha\upharpoonleft \breve{\iota})\\ +[\text{*53·31}] & =\iotaʻ\text{D}ʻ(\alpha\upharpoonleft \breve{\iota}) &\qquad \text{(1)}\\ +[\text{*35·61}] & =\iotaʻ(\alpha\cap \text{D}ʻ\breve{\iota})\\ +[\text{*33·2}] & =\iotaʻ(\alpha\cap \text{ᗡ}ʻ\iota)\\ +[\text{*51·17.*24·26}] & =\iotaʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·72.</b> \(\vdash:\kappa\subset 1.\supset.\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa=\iotaʻsʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·43}.\supset\vdash:\text{Hp}.\supset.\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa&=\text{D}ʻʻ\iotaʻ(\in \upharpoonright \kappa)\\ +[\text{*53·31}] &=\iotaʻ\text{D}ʻ(\in \upharpoonright \kappa)\\ +[\text{*62·43}] & =\iotaʻsʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>*83·73·731 are lemmas for <a href="#*83·74">*83·74</a>.</p> + +<p class="nind"><b>*83·73.</b> \[\begin{align}\vdash:\kappa\cap \lambda=\Lambda.\lambda&\subset 1.\supset.\\ +&\text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda)=\hat{\sigma}\{(\exists \rho).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma=\rho\cup sʻ\lambda\}\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·56.*37·6}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +\sigma\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa\cup \lambda).&\equiv.(\exists R,S).R\in {\in}_{\Delta}ʻ\kappa.S=R\unicode{x228d}\breve{\iota}\upharpoonright + \lambda.\sigma=\text{D}ʻS.\\ +[\text{*13·193}] & \equiv.(\exists R,S).R\in {\in}_{\Delta}ʻ\kappa.S=R\unicode{x228d}\breve{\iota}\upharpoonright + \lambda.\sigma=\text{D}ʻ(R\unicode{x228d}\breve{\iota}\upharpoonright \lambda).\\ +[\text{*62·43·55}] &\equiv.(\exists R,S).R\in {\in}_{\Delta}ʻ\kappa.S=R\unicode{x228d}\breve{\iota}\upharpoonright \lambda.\sigma=\text{D}ʻR\cup sʻ\lambda.\\ +[\text{*10·35.*21·2}] &\equiv.(\exists R).R\in {\in}_{\Delta}ʻ\kappa.\sigma=\text{D}ʻR\cup sʻ\lambda.\\ +[\text{*37·64}] & \equiv.(\exists \rho).\rho\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\sigma=\rho\cup sʻ\lambda\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_539">[Pg 539]</span></p> + +<p class="nind"><b><a id="*83·731">*83·731</a>.</b> \(\vdash\colon\ldotp \lambda\subset 1.\supset:sʻ\kappa\cap sʻ\lambda=\Lambda.\supset.\kappa\cap \lambda=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*53·25.*51·16}.&\supset\vdash\colon\ldotp sʻ\kappa\cap sʻ\lambda=\Lambda.\supset:\kappa\cap \lambda=\Lambda.\lor.\Lambda\in\lambda &\qquad \text{(1)}\\ +\vdash.\text{*52·16}. &\supset\vdash\colon\ldotp \lambda\subset 1.\supset:\alpha\in\lambda.\supset_{\alpha}.\exists !\alpha:\\ +[\text{*24·63}] &\supset:\Lambda{\sim}\in\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*83·74">*83·74</a>.</b> \(\vdash:sʻ\kappa\cap sʻ\lambda=\Lambda.\lambda\subset 1.\supset.\text{D}ʻʻ\in_{\Delta}ʻ(\kappa\cup \lambda)\text{ sm }\text{D}ʻʻ\in_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·73·731.*38·131}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻʻ\in_{\Delta}ʻ(\kappa\cup \lambda)=(\cup sʻ\lambda)ʻʻ\text{D}ʻʻ\in_{\Delta}ʻ\kappa + &\qquad \text{(1)}\\ +\vdash.\text{*83·62.*24·13}.\supset\\ +\vdash\colon\colon \text{Hp}.&\supset\colon\ldotp \mu,\,\nu\in\in_{\Delta}ʻ\kappa.\supset:\mu\cap sʻ\lambda=\Lambda.\nu\cap sʻ\lambda=\Lambda:\\ +[\text{*24·481}] &\supset:\mu\cup sʻ\lambda=u\cup sʻ\lambda.\equiv.\mu=\nu:\\ +[\text{*38·11}] &\supset:(\cup sʻ\lambda)ʻ\mu=(\cup sʻ\lambda)ʻu.\equiv.\mu=\nu &\qquad \text{(2)}\\ +\vdash.\text{(2).*73·28}.&\supset\vdash:\text{Hp}.\supset.(\cup sʻ\lambda)ʻʻ\text{D}ʻʻ\in_{\Delta}ʻ\kappa\text{ sm }\text{D}ʻʻ\in_{\Delta}ʻ\kappa &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*83·8.</b> \(\vdash.\in_{\Delta}ʻ\kappa\subset .t_{10}ʻ\kappa.\in_{\Delta}ʻ\kappa\in tʻt_{10}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14.*83·21.*35·83}.&\supset\vdash:R\in\in_{\Delta}ʻ\kappa.\supset.R\unicode{x2abd}sʻ\kappa\uparrow \kappa.\\ +[\text{*63·105.(*63·03)}] &\supset.R\unicode{x2abd}t_{1}ʻ\kappa\uparrow t_{0}ʻ\kappa.\\ +[\text{*64·201}] &\supset.R\in tʻ(t_{1}ʻ\kappa\uparrow t_{0}ʻ\kappa).\\ +[\text{(*64·021)}] &\supset.R\in t_{10}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*63·371}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*83·81">*83·81</a>.</b> \(\vdash.\text{D}ʻʻ\in_{\Delta}ʻ\kappa\subset t_{0}ʻ\kappa.Dʻʻ\in_{\Delta}ʻ\kappa\in tʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +&\vdash.\text{*83·62}.&\supset\vdash:\mu\in \text{D}ʻʻ\in_{\Delta}ʻ\kappa.\supset.\mu\subset sʻ\kappa.\\ +&[\text{*63·105.(*63·03)}] &\supset.\mu\subset t_{1}ʻ\kappa.\\ +&[\text{*63·51}] &\supset.\mu\in t_{0}ʻ\kappa &\qquad \text{(1)}\\ +&\vdash.\text{(1).*63·371}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*83·9">*83·9</a>.</b> \(\vdash.\exists !\in_{\Delta}ʻ\Lambda \quad[\text{*83·15}]\)</p> + +<p class="nind"><b>*83·901.</b> \(\vdash:\exists !\in_{\Delta}ʻ\iotaʻ\alpha.\equiv.\exists !\alpha \quad[\text{*80·46.*62·2}]\)</p> + +<p class="nind"><b>*83·902.</b> \(\vdash:\exists !\in_{\Delta}ʻ(\kappa\cup \lambda).\equiv.\exists !\in_{\Delta}ʻ\kappa.\exists !\in_{\Delta}ʻ\lambda \quad[\text{*80·69}]\)</p> + +<p class="nind"><b>*83·903.</b> \(\vdash:\exists !\in_{\Delta}ʻ(\iotaʻ\alpha\cup \iotaʻ\beta).\equiv.\exists !\alpha.\exists !\beta \quad[\text{*83·901·902}]\)</p> + +<p class="nind"><b><a id="*83·904">*83·904</a>.</b> \(\vdash:\exists !\in_{\Delta}ʻ(\kappa\cup \iotaʻ\beta).\equiv.\exists !\in_{\Delta}ʻ\kappa.\exists !\beta \quad[\text{*83·901·902}]\)</p> + +<p>*83·9·904 leads to an inductive proof (to be given later) of +\(\exists !\in_{\Delta}ʻ\kappa\) whenever \(\kappa\) is a finite +class of classes none of which is \(\Lambda\).</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_540">[Pg 540]</span></p> +<h2 class="nobreak" id="*84">*84. CLASSES OF MUTUALLY EXCLUSIVE CLASSES.</h2> +</div> + + +<p><i>Summary of</i> *84.</p> + +<p>A class \(\kappa\) of mutually exclusive classes is one such that, +if \(\alpha\) and \(\beta\) are two different members of \(\kappa\), +\(\alpha\) and \(\beta\) have no common members; <i>i.e.</i> it is +a class composed of non-overlapping classes. Classes of mutually +exclusive classes have many important properties. They are important +in cardinal arithmetic, among other reasons, because if \(\kappa\) +is a class of mutually exclusive classes, the cardinal number of +\(sʻ\kappa\) is the sum of the cardinal numbers of the members of +\(\kappa\). Also if \(\kappa\) is a class of mutually exclusive +classes, the number of selected classes of \(\kappa\) (<i>i.e.</i> +\(\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\)) is the same as the number of +selective relations (<i>i.e.</i> \({\in}_{\Delta}ʻ\kappa\))</p> + +<p>"\(\kappa\) is a class of mutually exclusive classes" is written +"\(\kappa \in \text{Cls}^2 ~ \text{excl}\)."</p> + +<p>An important case is when no member of \(\kappa\) is null; in this case +we write +\[ +\kappa \in \text{Cls} ~ \text{ex}^2 ~ \text{excl}\text{.} +\]</p> + +<p>For a \(\text{Cls}^2 ~ \text{excl}\) which is contained in a class of +classes \(\gamma\), we write +\[ +\text{Cl} ~ \text{excl}ʻ\gamma\text{,} +\] +on the analogy of the notation \(\text{Cl}ʻ\gamma\).</p> + +<p>The definitions are as follows:</p> + +<p class="nind"><b>*84·01.</b> \(\text{Cls}^2 ~ \text{excl} = \hat{\kappa}(\alpha, \beta \in \kappa . \alpha \neq \beta .\supset_{\alpha, \beta}. + \alpha \cap \beta = \Lambda) \quad \text{Df}\)</p> + +<p class="nind"><b>*84·02.</b> \(\text{Cl} ~ \text{excl}ʻ\gamma = \text{Cls}^2 ~ \text{excl} \cap \text{Cl}ʻ\gamma \quad \text{Df}\)</p> + +<p class="nind"><b>*84·03.</b> \(\text{Cls} ~ \text{ex}^2 ~ \text{excl} = \text{Cls}^2 ~ \text{excl} - \overleftarrow{{\in}}ʻ\Lambda \quad \text{Df}\)</p> + +<p>The propositions of this number begin (<a href="#*84·1">*84·1</a>—<a href="#*84·14">·14</a>) with various +equivalent forms for the definitions. Of these the most useful are:</p> + +<p class="nind"><b>*84·11.</b> \(\vdash \colon\ldotp \kappa \in \text{Cls}^{2} ~ \text{excl} .\equiv: \alpha, \beta \in \kappa . \exists! \alpha \cap \beta .\supset_{\alpha,\beta}.\alpha + = \beta\)</p> + +<p class="nind"><b>*84·13.</b> \(\vdash : \kappa \in \text{Cls} ~ \text{ex}^2 ~ \text{excl} .\equiv. \kappa \in \text{Cls}^2 ~ \text{excl} . \Lambda \mathrel{{\sim}{\in}} \kappa\)</p> + +<p class="nind"><b><a id="*84·14">*84·14</a>.</b> \(\vdash : \kappa \in \text{Cls}^2 ~ \text{excl} .\equiv. {\in} \upharpoonright \kappa \in \text{Cls} \rightarrow 1\)</p> + +<p>The last of these is specially important, because it renders the +propositions of <a href="#*81">*81</a> applicable to \({\in}_{\Delta}ʻ\kappa\) when +<span class="pagenum" id="Page_541">[Pg 541]</span>\(\kappa \in \text{Cls}^2 ~ \text{excl}\).</p> + +<p>We have next (<a href="#*84·2">*84·2</a>—<a href="#*84·28">·28</a>) a set of propositions dealing with various +special cases, such as \(\Lambda\) and 1. The most useful of these are</p> + +<p class="nind"><b>*84·23.</b> \(\vdash.\iotaʻ\alpha\in \text{Cls}^2\, \text{excl}\)</p> + +<p class="nind"><b>*84·241.</b> \(\vdash.\iotaʻʻ\alpha\in \text{Cls}^2\, \text{excl}\)</p> + +<p class="nind"><b>*84·25.</b> \(\vdash:\kappa\in \text{Cls}^2\, \text{excl}.\lambda\subset \kappa.\supset.\lambda\in \text{Cls}^{2}\, \text{excl}\)</p> + +<p>We next have a set of propositions (<a href="#*84·3">*84·3</a>—<a href="#*84·37">·37</a>) which are immediate +consequences of propositions in <a href="#*81">*81</a>, by means of <a href="#*84·14">*84·14</a>. The most +useful of these is</p> + +<p class="nind"><b>*84·3.</b> \(\vdash:\kappa\in \text{Cls}^{2}\, \text{excl}.\supset.\in _{\Delta}ʻ\kappa\subset 1\rightarrow 1\)</p> + +<p>We next have a set of propositions (<a href="#*84·4">*84·4</a>—<a href="#*84·43">·43</a>) dealing with the +domains of selections from a \(\text{Cls}^{2}\, \text{excl}\). These are +for the most part still immediate consequences of propositions in *81, +in virtue of *84·14. The most useful are</p> + +<p class="nind"><b>*84·41.</b> \(\vdash:\kappa\in \text{Cls}^{2}\, \text{excl}.\supset.\text{D}\upharpoonright \in _{\Delta}ʻ\kappa\in 1\rightarrow 1.\text{D}ʻʻ\in _{\Delta}ʻ\kappa\text{ + sm }\in _{\Delta}ʻ\kappa\)</p> + +<p class="nind"><b><a id="*84·412">*84·412</a>.</b> \(\vdash:\kappa\in \text{Cls}^{2}\, \text{excl}.\supset.\text{D}ʻʻ\in_{\Delta}ʻ\kappa=\hat{\mu}\{\alpha\in \kappa.\supset_{\alpha}.\mu + \cap \alpha\in 1:\mu \subset sʻ\kappa\}\)</p> + +<p class="nind"><b><a id="*84·43">*84·43</a>.</b> \(\vdash\colon\ldotp \alpha,\,\beta\in \text{Cls}^{2}\, \text{excl}.sʻ\alpha=sʻ\beta.\supset:\alpha\subset \text{D}ʻʻ\in _{\Delta}ʻ\beta.\equiv.\beta\subset + \text{D}ʻʻ\in _{\Delta}ʻ\alpha\)</p> + +<p>This proposition applies to such cases as the relations of rows and +columns. Imagine any set of terms arranged in rows and columns so as to +form a rectangle. Then each column is a selection from the rows, and +each row is a selection from the columns. This is a particular case of +the above proposition.</p> + +<p>We next have a set of propositions on \(\overrightarrow{R}ʻʻ\kappa\), +\(Rʻʻʻ\kappa\), and \(P_{\Delta}ʻʻ\kappa\) (<a href="#*84·5">*84·5</a>—<a href="#*84·55">·55</a>). The most +important of these are</p> + +<p class="nind"><b>*84·51.</b> \(\vdash:R\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\supset.\overrightarrow{R}ʻʻ\kappa\in \text{Cls}^{2}\, \text{excl}\)</p> + +<p class="nind"><b>*84·53.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\kappa\in \text{Cls}^{2}\, \text{excl}.\supset.Rʻʻʻ\kappa\in \text{Cls}^{2}\, \text{excl}\)</p> + +<p>Finally we have a set of propositions (<a href="#*84·59">*84·59</a>—<a href="#*84·62">·62</a>) showing +circumstances under which \(\kappa\cup \lambda\) is a \(\text{Cls}^{2}\,\text{excl}\). +The only one of these which is used subsequently is</p> + +<p class="nind"><b>*84·62.</b> \(\vdash\colon\ldotp \alpha \neq \beta.\supset:\iotaʻ\alpha\cup \iotaʻ\beta\in \text{Cls}^{2}\, \text{excl}.\equiv.\alpha\cap \beta=\Lambda\)</p> + +<hr class="tb"> + +<p class="nind"><b>*84·01.</b> \(\text{Cls}^2\, \text{excl}=\hat{\kappa}(\alpha,\beta\in \kappa.\alpha \neq \beta.\supset_{\alpha,\beta}.\alpha\cap \beta=\Lambda) \quad\text{Df}\)</p> + +<p class="nind"><b>*84·02.</b> \(\text{Cl excl}ʻ\gamma=\text{Cls}^2\, \text{excl}\cap \text{Cl}ʻ\gamma \quad\text{Df}\)</p> + +<p class="nind"><b>*84·03.</b> \(\text{Cls}^2\, \text{excl}=\text{Cls}^2\, \text{excl}-\overleftarrow{\in}ʻ\Lambda \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*84·1">*84·1</a>.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^2\, \text{excl}.\equiv:\alpha,\beta\in \kappa.\alpha \neq \beta.\supset_{\alpha,\beta}.\alpha\cap + \beta=\Lambda\\ +&[\text{*20·3.(*84·01)}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_542">[Pg 542]</span></p> + +<p class="nind"><b>*84·11.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^2\,\text{excl}.\equiv:\alpha,\,\beta\in + \kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta\\ +&[\text{*84·1.Transp}]\end{align}\]</p> + +<p class="nind"><b>*84·12.</b> \[\begin{align}\vdash\colon\ldotp \kappa\in \text{Cl excl}ʻ\gamma.&\equiv:\alpha,\beta\in \kappa.\alpha \neq \beta.\supset_{\alpha,\beta}.\alpha\cap + \beta=\Lambda:\kappa\subset \gamma:\equiv:\\ +&\kappa\in \text{Cls}^2\,\text{excl}.\kappa\subset \gamma [\text{*20·3.(*84·02).*22·33.*84·1}]\end{align}\]</p> + +<p class="nind"><b>*84·121.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cl excl}ʻ\gamma.\equiv:\alpha,\beta\in \kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta:\kappa\subset + \gamma\\ +&[\text{*20·3.(*84·02).*22·33.*84·11}]\end{align}\]</p> + +<p class="nind"><b>*84·13.</b> \(\vdash:\kappa\in \text{Cls}^2\,\text{excl}.\equiv.\kappa\in \text{Cls}^2\,\text{excl}.\Lambda{\sim}\in \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·33·35.(*84·03)}.\supset\\ +\vdash:\kappa\in \text{Cls}^2\,\text{excl}.&\equiv.\kappa\in \text{Cls}^2\,\text{excl}.\kappa{\sim}\in \overleftarrow{\in}ʻ\Lambda.\\ +[\text{*62·21}] &\equiv.\kappa\in \text{Cls}^2\,\text{excl}.\Lambda{\sim}\in \kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·131.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^2\,\text{excl}.\equiv:\alpha,\,\beta\in \kappa.\alpha \neq\beta.\supset_{\alpha,\beta}.\alpha\cap + \beta=\Lambda:\Lambda{\sim}\in \kappa\\ +&[\text{*84·13·1}]\end{align}\]</p> + +<p class="nind"><b>*84·132.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^2\,\text{excl}.\equiv:\alpha,\,\beta\in \kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta:\Lambda{\sim}\in + \kappa\\ +&[\text{*84·13·11}]\end{align}\]</p> + +<p class="nind"><b>*84·133.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^2\,\text{excl}.\equiv:\alpha,\,\beta\in \kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta:\alpha\in + \kappa.\supset_{\alpha}.\exists !\alpha\\ +&[\text{*84·132.*24·63}]\end{align}\]</p> + +<p class="nind"><b>*84·134.</b> \(\vdash\colon\colon \kappa\in \text{Cls}^2\,\text{excl}.\equiv\colon\ldotp \alpha,\,\beta\in \kappa.\supset_{\alpha,\beta}:\exists + !\alpha.\exists !\beta:\exists !\alpha\cap \beta.\supset.\alpha=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*11·59}.&\supset\vdash\colon\ldotp \alpha\in \kappa.\supset_{\alpha}.\exists !\alpha:\equiv:\alpha,\,\beta\in \kappa.\supset_{\alpha,\beta}.\exists + !\alpha.\exists !\beta\colon\ldotp &\qquad \text{(1)}\\ +\vdash.\text{*4·87.*11·33}.&\supset\vdash\colon\colon \alpha,\beta\in \kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta:\equiv\colon\ldotp \\ +&\alpha,\beta\in \kappa.\supset_{\alpha,\beta}:\exists !\alpha\cap \beta.\supset.\alpha=\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*84·133}.&\supset\vdash\colon\colon \kappa\in \text{Cls}^2\,\text{excl}.\equiv\colon\ldotp \\ +&\alpha,\,\beta\in \kappa.\supset_{\alpha,\beta}.\exists !\alpha.\exists !\beta\colon\ldotp \alpha,\,\beta\in \kappa.\supset_{\alpha,\beta}:\exists + !\alpha\cap \beta.\supset.\alpha=\beta\colon\ldotp \\ +[\text{*11·391}]\equiv\colon\ldotp \alpha,\beta\in \kappa.&\supset_{\alpha,\beta}:\exists + !\alpha.\exists !\beta:\exists !\alpha\cap \beta.\supset.\alpha=\beta\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·135.</b> \(\vdash\colon\colon \kappa\in \text{Cls}^2\,\text{excl}.\equiv\colon\ldotp \alpha,\beta\in \kappa.\supset_{\alpha,\beta}:\exists + !\alpha\cap \beta.\equiv.\alpha=\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*84·133.*22·5.*13·191}.\supset\\ +\vdash\colon\colon \kappa\in \text{Cls}^2\,\text{excl}.&\equiv\colon\ldotp \alpha,\beta\in \kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta:\\ +&\qquad\qquad\qquad\alpha,\beta\in \kappa.\alpha=\beta.\supset_{\alpha.\beta}.\exists !\alpha\cap \beta\colon\ldotp \\ +[\text{*11·31}] &\equiv\colon\ldotp (\alpha,\,\beta)\colon\ldotp \alpha,\,\beta\in \kappa.\exists !\alpha\cap \beta.\supset.\alpha=\beta:\\ +&\qquad\qquad\qquad\alpha,\beta\in \kappa.\alpha=\beta.\supset.\exists !\alpha\cap \beta\colon\ldotp \\ +[\text{*4·87.Comp.*11·33}]&\equiv\colon\ldotp (\alpha,\,\beta)\colon\ldotp \alpha,\,\beta\in \kappa.\supset:\exists !\alpha\cap \beta.\equiv.\alpha=\beta\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_543">[Pg 543]</span></p> + +<p class="nind"><b>*84·14.</b> \(\vdash:\kappa\in \text{Cls}^2\,\text{excl}.\equiv.\in \upharpoonright \kappa\in \text{Cls}\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·23.*84·11}.&\supset\vdash\colon\ldotp \kappa\in \text{Cls}^2\,\text{excl}.\equiv:\alpha,\beta\in + \kappa.x\in \alpha.x\in \beta.\supset_{x,\alpha,\beta}.\alpha=\beta:\\ +[\text{*35·101}] &\equiv:x(\in \upharpoonright \kappa)\alpha.x(\in \upharpoonright \kappa)\beta.\supset_{x,\alpha,\beta}.\alpha=\beta:\\ +[\text{*71·171}] &\equiv:\in \upharpoonright \kappa\in \text{Cls}\rightarrow 1\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition is important, since it enables us to apply the +propositions of <a href="#*81">*81</a> to \({\in}_{\Delta}ʻ\kappa\) when \(\kappa\in \text{Cls}^{2}\,\text{excl}\).</p> + +<p class="nind"><b><a id="*84·2">*84·2</a>.</b> \(\vdash.\Lambda\cap \text{Cls}\in \text{Cls}\,\text{ex}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·105.*11·57}.&\supset\vdash.(\alpha,\,\beta).\alpha,\,\beta{\sim}\in \Lambda\cap \text{Cls}.\\ +[\text{*11·25·63}] &\supset\vdash\colon\ldotp \alpha,\beta\in \Lambda\cap \text{Cls}.\supset_{\alpha,\beta}:\exists !\alpha\cap \beta.\equiv.\alpha=\beta\colon\ldotp \\ +[\text{*84·135}] &\supset\vdash.\Lambda\cap \text{Cls}\in \text{Cls}\,\text{ex}^{2}\,\text{excl} +\end{array} +\]</p> + +<p class="nind"><b>*84·21.</b> \(\vdash.1(\text{Cls})\subset \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Note.</i> \(1\,(\text{Cls})\) is the class of all unit classes +whose members are classes; this results from <a href="#*65·02">*65·02</a>. Thus "\(\alpha\in 1\,(\text{Cls})\)" +is equivalent to "\(\alpha\) consists of one class."</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·33.(*65·02)}.&\supset\vdash\colon\ldotp \alpha\in 1(\text{Cls}).\equiv:\alpha\in 1.\alpha\subset \text{Cls}:\\ +[\text{*52·16}] &\supset:\beta,\gamma\in \alpha.\supset_{\beta,\gamma}.\beta=\gamma:\\ +[\text{*3·41}] &\supset:\beta,\gamma\in \alpha.\exists !\beta\cap \gamma.\supset_{\beta,\gamma}.\beta=\gamma:\\ +[\text{*84·11}] &\supset:\alpha\in \text{Cls}^{2}\,\text{excl}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·22.</b> \(\vdash.1\in \text{Cls ex}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·46}.\supset\vdash\colon\ldotp \alpha,\,\beta\in 1.\supset:\exists !\alpha\cap \beta.\equiv.\alpha=\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·135}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·23.</b> \(\vdash.\iotaʻ\alpha\in \text{Cls}^{2}\,\text{excl} \quad[\text{*84·21.*52·22}]\)</p> + +<p class="nind"><b>*84·24.</b> \(\vdash:\exists !\alpha.\supset.\iotaʻ\alpha\in \text{Cls}\,\text{ex}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*13·191}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\beta=\alpha.\supset_{\beta}.\exists !\beta:\\ +[\text{*51·15}] &\supset:\beta\in \iotaʻ\alpha.\supset_{\beta}.\exists !\beta:\\ +[\text{*24·63}] &\supset:\Lambda{\sim}\in \iotaʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·23·13}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·241.</b> \(\vdash.\iotaʻʻ\alpha\in \text{Cls}\,\text{ex}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·3}.\supset\vdash\colon\ldotp \beta,\gamma\in \iotaʻʻ\alpha.&\supset_{\beta,\gamma}:\beta,\gamma\in 1:\\ +[\text{*52·46}] & \supset_{\beta,\gamma}:\exists !\beta\cap \gamma.\equiv.\beta=\gamma &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·135}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_544">[Pg 544]</span></p> + +<p class="nind"><b>*84·242.</b> \(\vdash:\kappa\subset 1.\supset.\kappa\in \text{Cls ex}^{2}\,\text{excl} \quad[\text{*52·46.*84·135}]\)</p> + +<p class="nind"><b>*84·25.</b> \(\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\lambda\subset \kappa.\supset.\lambda\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·1.*11·59}.&\supset\vdash\colon\ldotp \lambda\subset \kappa.\supset:\alpha,\,\beta\in \lambda.\supset_{\alpha,\beta}.\alpha,\,\beta\in \kappa:\\ +[\text{*11·38}] &\supset:\alpha,\beta\in \lambda.\alpha \neq \beta.\supset_{\alpha,\beta}.\alpha,\,\beta\in \kappa.\alpha \neq \beta &\qquad \text{(1)}\\ +\vdash.\text{*84·1}. &\supset\vdash\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{excl}.\supset:\alpha,\,\beta\in \kappa.\alpha \neq \beta.\supset_{\alpha,\beta}.\alpha + \cap \beta=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*11·37}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha,\beta\in \lambda.\alpha\neq \beta.\supset_{\alpha,\beta}.\alpha \cap \beta=\Lambda:\\ +[\text{*84·1}]&\supset:\lambda\in \text{Cls}^{2}\,\text{excl}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·26.</b> \(\vdash:\kappa\in \text{Cls ex}^{2}\,\text{excl}.\lambda\subset \kappa.\supset.\lambda\in \text{Cls ex}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*84·13·25}. &\supset\vdash:\text{Hp}.\supset.\lambda\in \text{Cls}^{2}\,\text{excl} &\qquad \text{(1)}\\ +\vdash.\text{*22·1.*10·1}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\Lambda\in \lambda.\supset.\Lambda\in \kappa:\\ +[\text{Transp}] &\supset:\Lambda{\sim}\in \kappa.\supset.\Lambda{\sim}\in \lambda &\qquad \text{(2)}\\ +\vdash.\text{*84·13} &\supset\vdash:\text{Hp}.\supset.\Lambda{\sim}\in \kappa &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash:\text{Hp}.\supset.\Lambda{\sim}\in \lambda &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*84·13}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*84·28">*84·28</a>.</b> \(\vdash:\kappa\in \text{Cl excl}ʻ\gamma.\lambda\subset \kappa.\gamma\subset \delta.\supset.\lambda\in \text{Cl excl}ʻ\delta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*84·12·25}.&\supset\vdash:\text{Hp}.\supset.\lambda\in \text{Cls}^{2}\,\text{excl} &\qquad \text{(1)}\\ +\vdash.\text{*84·12}.&\supset\vdash:\text{Hp}.\supset.\kappa\subset \gamma.\lambda\subset \kappa.\gamma\subset \delta.\\ +[\text{*22·44}] &\supset.\lambda\subset \delta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*84·12}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with selections from +a \(\text{Cls}^{2}\, \text{excl}\). In virtue of <a href="#*84·14">*84·14</a>, the +propositions of <a href="#*81">*81</a> which have the hypothesis \(R\upharpoonright\kappa\in \text{Cls}\rightarrow 1\) +become applicable when \(R\) is \({\in}\) and \(\kappa\) is a +\(\text{Cls}^{2}\, \text{excl}\). Thus \({\in}_{\Delta}ʻ\kappa\) has +many important properties when \(\kappa\) is a \(\text{Cls}^{2}\,\text{excl}\) +which it does not have in the general case.</p> + +<p class="nind"><b><a id="*84·3">*84·3</a>.</b> \(\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\supset.{\in}_{\Delta}ʻ\kappa\subset 1\rightarrow 1 \quad[\text{*84·14.*81·1}]\)</p> + +<p class="nind"><b>*84·31.</b> \(\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.R\in \in _{\Delta}ʻ\kappa.x\in \text{D}ʻR.\supset.\text{E}!\breve{R}ʻx \quad[\text{*84·14.*81·11}]\)</p> + +<p class="nind"><b>*84·32.</b> \[\begin{align}&\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.R\in {\in} _{\Delta}ʻ\kappa.x\in \text{D}ʻR.\supset.x\in \breve{R}ʻx.\breve{R}ʻx\in \kappa\\ +&[\text{*84·14.*81·11.*35·101}]\end{align}\]</p> + +<p class="nind"><b>*84·33.</b> \[\begin{align}&\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.R\in {\in}_{\Delta}ʻ\kappa.x\in + \text{D}ʻR.\supset.\breve{R}ʻx=(℩\alpha)(\alpha\in \kappa.x\in \alpha)=(\kappa\upharpoonleft \breve{\in})ʻx\\ +&[\text{*84·14.*81·12}]\end{align}\]</p> + +<p class="nind"><b>*84·34.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{excl}.R\in {\in}_{\Delta}ʻ\kappa.\supset:xR\alpha.\equiv.x\in + \alpha.x\in \text{D}ʻR.\alpha\in \kappa\\ +&[\text{*81·13.*84·14}]\end{align}\]</p> + +<p class="nind"><b>*84·341.</b> \[\begin{align}&\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.R\in {\in}_{\Delta}ʻ\kappa.\supset.R=\text{D}ʻR\upharpoonleft + \in \upharpoonright \kappa=\in \dot{\cap}\text{D}ʻR\uparrow \kappa\\ +&[\text{*81·14.*84·14}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_545">[Pg 545]</span></p> + +<p class="nind"><b>*84·342.</b> \[\begin{align}&\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}. R\in {\in}_{\Delta}ʻ\kappa.\alpha\in \kappa.\supset.\iotaʻRʻ\alpha=\alpha\cap \text{D}ʻR\\ +&[\text{*81·15.*84·14.*62·2}]\end{align}\]</p> + +<p class="nind"><b>*84·35.</b> \(\vdash\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{excl}.\supset:R\in {\in}_{\Delta}ʻ\kappa.\equiv.R\in + 1\rightarrow 1.R\unicode{x2abd}\in \upharpoonright \kappa.\text{ᗡ}ʻR=\text{ᗡ}ʻ\in \upharpoonright \kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*84·13}.&\supset\vdash:\text{Hp}.\supset.\Lambda{\sim}\in \kappa.\\ +[\text{*62·42}] &\supset.\text{ᗡ}ʻ\in \upharpoonright \kappa=\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*71·103.*80·14}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:R\in 1\rightarrow 1.R\unicode{x2abd}\in \upharpoonright \kappa.\text{ᗡ}ʻR=\text{ᗡ}ʻ\in \upharpoonright \kappa.\supset.R\in {\in}_{\Delta}ʻ\kappa + &\qquad \text{(2)}\\ +\vdash.\text{(1).*80·14}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:R\in {\in}_{\Delta}ʻ\kappa.\supset.\text{ᗡ}ʻR=\text{ᗡ}ʻ\in \upharpoonright \kappa &\qquad \text{(3)}\\ +\vdash.\text{(3).*80·291.*84·3}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:R\in {\in}_{\Delta}ʻ\kappa.\supset.R\in + 1\rightarrow 1.R\unicode{x2abd}\in \upharpoonright \kappa.\text{ᗡ}ʻR=\text{ᗡ}ʻ\in \upharpoonright \kappa &\qquad \text{(4)}\\ +\vdash.\text{(2).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*84·37">*84·37</a>.</b> \(\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\exists !{\in}_{\Delta}ʻ\kappa.\supset.\kappa\in \text{Cls ex}^{2}\,\text{excl} + \quad[\text{*83·1.*84·13}]\)</p> + +<p class="nind"><b><a id="*84·4">*84·4</a>.</b> \(\vdash\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{excl}.R,\,S\in {\in}_{\Delta}ʻ\kappa.\supset:\text{D}ʻR=\text{D}ʻS.\equiv.R=S \quad[\text{*81·2.*84·14}]\)</p> + +<p class="nind"><b>*84·41.</b> \(\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\supset.\text{D}\upharpoonright {\in}_{\Delta}ʻ\kappa\in 1\rightarrow 1.\text{D}ʻʻ\in _{\Delta}ʻ\kappa\text{ + sm }{\in}_{\Delta}ʻ\kappa \quad[\text{*81·21.*84·14}]\)</p> + +<p>This is an important proposition, since it shows that, when \(\kappa\) +is a \(\text{Cls}^{2}\,\text{excl}\), the number of classes that can be +selected from \(\kappa\) is the product of the numbers of the various +classes that are members of \(\kappa\).</p> + +<p class="nind"><b>*84·411.</b> \(\vdash\colon\ldotp \alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\mu\subset sʻ\kappa:\supset.\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa + \quad[\text{*81·212.*62·2·3}]\)</p> + +<p class="nind"><b>*84·412.</b> \[\begin{align}&\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\supset.\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa=\hat{\mu}\{\alpha\in \kappa.\supset_{\alpha}.\mu\cap + \alpha\in 1:\mu\subset sʻ\kappa\}\\ +&[\text{*81·22.*84·14.*62·2·3}]\end{align}\]</p> + +<p>This proposition gives what might be taken as the definition of the +class of selected classes, namely +\[ +\hat{\mu}\{\alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\mu\subset sʻ\kappa\}. +\]</p> + +<p>We might, starting with this as our definition, deal with the class +of selected classes without first considering selective relations. +The disadvantages of this method would be, first, that it requires +that \(\kappa\) should be a \(\text{Cls}^{2}\,\text{excl}\) if it is +to give the results desired in arithmetic; secondly, that it is much +more cumbrous technically than the method which proceeds by selective +relations; thirdly, that it does not enable us to deal with selection +from a class of classes as a particular case of selection from a +relation (namely from \({\in} \upharpoonright \kappa)\), and therefore +does not yield theorems of such generality as those obtained by the +method adopted above.</p> + +<p class="nind"><b>*84·42.</b> \[\begin{align}&\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\alpha\in \kappa.\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa.\supset.\mu-\alpha\in + \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa-\iotaʻ\alpha)\\ +&[\text{*81·24.*84·14.*62·2}]\end{align}\]</p> + +<p class="nind"><b>*84·421.</b> \[\begin{align}&\vdash:\alpha\in \kappa.x\in \alpha.\mu\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa-\iotaʻ\alpha).\supset.\mu\cup + \iotaʻx\in \text{D}ʻʻ{\in}_{\Delta}ʻ\kappa\\ +&[\text{*81·25}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_546">[Pg 546]</span></p> + +<p class="nind"><b>*84·422.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{excl}.\alpha\in \kappa.\mu\cap \alpha\in 1.\supset:\mu-\alpha\in \text{D}ʻʻ{\in}_{\Delta}ʻ(\kappa-\iotaʻ\alpha).\equiv.\mu\in + \text{D}ʻʻ\in _{\Delta}ʻ\alpha\\ +&[\text{*81·26.*84·14.*62·2}]\end{align}\]</p> + +<p class="nind"><b>*84·43.</b> \(\vdash\colon\ldotp \alpha,\,\beta\in \text{Cls}^{2}\,\text{excl}.sʻ\alpha=sʻ\beta.\supset:\alpha\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\beta.\equiv.\beta\subset + \text{D}ʻʻ{\in}_{\Delta}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*84·412}.\supset\vdash\colon\colon\ldotp \text{Hp}.\supset\colon\colon \\ +\alpha\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\beta.&\equiv\colon\ldotp \xi\in \alpha.\supset_{\xi}:\eta\in \beta.\supset_{\eta}.\xi\cap + \eta\in 1:\xi\subset sʻ\beta\colon\ldotp \\ +[\text{*40·13.Hp}]&\equiv\colon\ldotp \xi\in \alpha.\supset_{\xi}:\eta\in \beta.\supset_{\eta}.\xi\cap \eta\in 1\colon\ldotp \\ +[\text{*10·542·21}]&\equiv\colon\ldotp \eta\in \beta.\supset_{\eta}:\xi\in \alpha.\supset_{\xi}.\xi\cap \eta\in 1\colon\ldotp \\ +[\text{*40·13.Hp}] &\equiv\colon\ldotp \eta\in \beta.\supset_{\eta}:\xi\in \alpha.\supset_{\xi}.\xi\cap \eta\in 1:\eta\subset sʻ\alpha\colon\ldotp \\ +[\text{*84·412}] &\equiv\colon\ldotp \eta\in \beta.\supset_{\eta}:\eta\in \text{D}ʻʻ{\in}_{\Delta}ʻ\alpha\colon\colon\ldotp\supset\vdash.\text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b><a id="*84·5">*84·5</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·181}.& \supset\vdash\colon\ldotp \text{Hp}.\supset:\exists !\overrightarrow{R}ʻx\cap \overrightarrow{R}ʻy.\supset_{x,y}.x=y.\\ +[\text{*30·37}] & \supset_{x,y}.\overrightarrow{R}ʻx=\overrightarrow{R}ʻy &\qquad \text{(1)}\\ +\vdash.\text{*33·41.*11·59}.&\supset\vdash:x,\,y\in \text{ᗡ}ʻR.\supset_{x,y}.\exists !\overrightarrow{R}ʻx.\exists !\overrightarrow{R}ʻy &&\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp x,\,y\in \text{ᗡ}ʻR.\supset_{x,y}:\\ +&\qquad\qquad\qquad\exists !\overrightarrow{R}ʻx.\exists !\overrightarrow{R}ʻy:\exists !\overrightarrow{R}ʻx\cap \overrightarrow{R}ʻy.\supset.\overrightarrow{R}ʻx=\overrightarrow{R}ʻy\colon\ldotp \\ +[\text{*37·63}] &\supset\colon\ldotp \alpha,\beta\in \overrightarrow{R}ʻʻ\text{ᗡ}ʻR.\supset_{\alpha,\beta}:\exists + !\alpha.\exists !\beta:\exists !\alpha\cap \beta.\supset.\alpha=\beta\colon\ldotp \\ +[\text{*84·134}] &\supset\colon\ldotp \overrightarrow{R}ʻʻ\text{ᗡ}ʻR\in \text{Cls}^{2}\,\text{excl}\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>It might be supposed that the converse of the above +would also hold. But this is not the case; for although +\(\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\in\text{Cls}^{2}\,\text{excl}\) +secures that \(\overrightarrow{R}ʻx\) and \(\overrightarrow{R}ʻy\) +cannot overlap when they are unequal, yet we may have +\(\overrightarrow{R}ʻx=\overrightarrow{R}ʻy\) without having +\(x = y\), so that if \(\overrightarrow{R}ʻx = \alpha= \overrightarrow{R}ʻy\), +we shall have \(z\in \alpha.\supset.zRx.zRy\), +whence, if \(\exists !\alpha.x \neq y\), it follows that +\(R\) is not a \(\text{Cls}\,\rightarrow 1\) even if +\(\overrightarrow{R}ʻʻ\text{ᗡ}ʻR\in \text{Cls}^{2}\,\text{excl}\).</p> + +<p class="nind"><b>*84·51.</b> \(\vdash:R\upharpoonright \kappa\in Cls\rightarrow 1.\supset.\overrightarrow{R}ʻʻ\kappa\in\text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·171.*35·101}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:xRy.y\in \kappa.xRz.z\in \kappa.\supset_{x,y,z}.y=z.\\ +[\text{*30·37}] \qquad\qquad\qquad\qquad\qquad\quad\supset_{x,y,z}.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz:\\ +[\text{*32·18}]\supset:y,z\in \kappa.x\in \overrightarrow{R}ʻy\cap \overrightarrow{R}ʻz.\supset_{x,y,z}.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz:\\ +[\text{*10·23}]\supset:y,z\in \kappa.\exists !\overrightarrow{R}ʻy\cap \overrightarrow{R}ʻz.\supset_{y,z}.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz:\\ +[\text{*37·63}]\supset:\alpha,\beta\in \overrightarrow{R}ʻʻ\kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta:\\ +[\text{*84·11}]\supset:\overrightarrow{R}ʻʻ\kappa\in \text{Cls}^{2}\,\text{excl}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_547">[Pg 547]</span></p> + +<p class="nind"><b>*84·52.</b> \(\vdash:R\upharpoonright \kappa\in \text{Cls}\rightarrow 1.\kappa\subset \text{ᗡ}ʻR.\supset.\overrightarrow{R}ʻʻ\kappa\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·2}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha\in \overrightarrow{R}ʻʻ\kappa.\supset.\alpha\in \overrightarrow{R}ʻʻ\text{ᗡ}ʻR.\\ +[\text{*37·77}] &\supset.\exists !\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·51·13.*24·63}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·521.</b> \(\vdash:\overrightarrow{R}\upharpoonright \beta\in 1\rightarrow 1.\overrightarrow{R}ʻʻ\beta\in \text{Cls}^{2}\,\text{excl}.\supset.R\upharpoonright + \beta\in \text{Cls}\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·55.*84·11}.\supset\\ +\vdash\colon\ldotp \overrightarrow{R}\upharpoonright \beta\in 1\rightarrow 1.\overrightarrow{R}ʻʻ\beta\in \text{Cls}^{2}\,\text{excl}.\supset.&y,\,z\in + \beta.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz.\supset_{y,z}.y=z:\\ +&y,\,z\in \beta.\exists !\overrightarrow{R}ʻy\cap \overrightarrow{R}ʻz.\supset_{y,z}.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz:\\ +[\text{*11·37}] &\supset:y,\,z\in \beta.\exists !\overrightarrow{R}ʻy\cap \overrightarrow{R}ʻz.\supset_{y,z}.y=z:\\ +[\text{*74·62.Transp}] &\supset:R\upharpoonright \beta\in \text{Cls}\rightarrow 1\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is a lemma for *84·522, which is used in an +important proposition on relations of mutually exclusive relations +(*163·17).</p> + +<p class="nind"><b>*84·522.</b> + \(\vdash\colon\ldotp \beta\subset \text{ᗡ}ʻR.\supset:R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\equiv.\overrightarrow{R}\upharpoonright \beta\in 1\rightarrow 1.\overrightarrow{R}ʻʻ\beta\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·31}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:y,\,z\in \beta.\supset.\exists !\overrightarrow{R}ʻy.\exists !\overrightarrow{R}ʻz:\\ +[\text{*22·5}] &\supset:y,\,z\in \beta.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz.\supset.\exists !\overrightarrow{R}ʻy\cap \overrightarrow{R}ʻz:\\ +[\text{*74·62}] &\supset:R\upharpoonright \beta\in \text{Cls}\rightarrow 1.y,\,z\in \beta.\overrightarrow{R}ʻy=\overrightarrow{R}ʻz.\supset.y=z:\\ +[\text{*71·55}] &\supset:R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\supset.\overrightarrow{R}\upharpoonright \beta\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·51}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:R\upharpoonright \beta\in \text{Cls}\rightarrow 1.\supset.\overrightarrow{R}\upharpoonright \beta\in 1\rightarrow 1.\overrightarrow{R}ʻʻ\beta\in \text{Cls}^{2}\,\text{excl} &\qquad \text{(2)}\\ +\vdash.\text{(2).*84·521}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*84·53.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\kappa\in \text{Cls}^{2}\,\text{excl}.\supset.Rʻʻʻ\kappa\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·421}.\supset\\ +\vdash:R\in \text{Cls}\rightarrow 1.\alpha,\beta\in \kappa.\exists !Rʻʻ\alpha\cap Rʻʻ\beta.\supset.\exists !\alpha\cap \beta &\qquad \text{(1)}\\ +\vdash.\text{(1).Syll}.\supset\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1:\alpha,\beta\in \kappa.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta:\supset:\\ +\alpha,\,\beta\in \kappa.\exists !Rʻʻ\alpha\cap Rʻʻ\beta.\supset_{\alpha,\beta}.\alpha=\beta.\\ +[\text{*30·37.*37·11·111}] \supset_{\alpha,\beta}.Rʻʻ\alpha=Rʻʻ\beta:\\ +[\text{*37·63.(*37·04)}]\supset:\rho,\,\sigma\in Rʻʻʻ\kappa.\exists !\rho\cap \sigma.\supset_{\rho,\sigma}.\rho=\sigma &\qquad \text{(2)}\\ +\vdash.\text{(2).*84·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_548">[Pg 548]</span></p> + +<p class="nind"><b>*84·54.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\kappa\in \text{Cls}^{2}\,\text{excl}.\supset.\breve{R}ʻʻʻ\kappa\in \text{Cls}^{2}\,\text{excl} + \quad\left[\text{*84·53} \frac{\breve{R}}{R}\right]\)</p> + +<p class="nind"><b><a id="*84·55">*84·55</a>.</b> \(\vdash.P_{\Delta}ʻʻ\kappa\in \text{Cls}^{2}\,\text{excl} \quad[\text{*80·82}]\)</p> + +<p class="nind"><b><a id="*84·59">*84·59</a>.</b> \(\vdash:\kappa\cup \lambda\in \text{Cls}^{2}\,\text{excl}.\equiv.\kappa,\,\lambda\in \text{Cls}^{2}\,\text{excl}.sʻ(\kappa-\lambda)\cap sʻ\lambda=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*84·14}.\supset\vdash:\kappa\cup \lambda\in \text{Cls}^{2}\,\text{excl}.&\equiv.\in \upharpoonright (\kappa\cup \lambda)\in \text{Cls}\rightarrow 1.\\ +[\text{*74·821}] &\equiv.\in \upharpoonright \kappa,\in \upharpoonright \lambda\in \text{Cls}\rightarrow 1.{\in}ʻʻ(\kappa-\lambda)\cap {\in}ʻʻ\lambda=\Lambda.\\ +[\text{*84·14.*62·3}] &\equiv.\kappa,\lambda\in \text{Cls}^{2}\,\text{excl}.sʻ(\kappa-\lambda)\cap sʻ\lambda=\Lambda +\end{array} +\]</p> + +<p class="nind"><b>*84·6.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\cap \lambda=\Lambda.\supset:\kappa\cup \lambda\in \text{Cls}^{2}\,\text{excl}.\equiv.\kappa,\,\lambda\in \text{Cls}^{2}\,\text{excl}.sʻ\kappa\cap + sʻ\lambda=\Lambda\\ +&[\text{*84·59.*24·313}]\end{align}\]</p> + +<p class="nind"><b>*84·61.</b> \[\begin{align}&\vdash\colon\ldotp \beta{\sim}\in \kappa.\supset:\kappa\cup \iotaʻ\beta\in \text{Cls}^{2}\,\text{excl}.\equiv.\kappa\in \text{Cls}^{2}\,\text{excl}.\beta\cap + sʻ\kappa=\Lambda\\ +&[\text{*51·211.*53·02.*84·23·6}]\end{align}\]</p> + +<p class="nind"><b><a id="*84·62">*84·62</a>.</b> \[\begin{align}&\vdash\colon\ldotp \alpha \neq \beta.\supset:\iotaʻ\alpha\cup \iotaʻ\beta\in \text{Cls}^{2}\,\text{excl}.\equiv.\alpha\cap \beta=\Lambda\\ +&[\text{*84·61.*51·15.*53·02.*84·23}]\end{align}\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_549">[Pg 549]</span></p> +<h2 class="nobreak" id="*85">*85. MISCELLANEOUS PROPOSITIONS.</h2> +</div> + + +<p><i>Summary of</i> *85.</p> + +<p>In this number certain important propositions are proved, and the other +propositions of this number are mainly lemmas. The most important +propositions are the following:</p> + +<p><a href="#*85·1">*85·1</a> and <a href="#*85·14">*85·14</a>, which show that if \(Q \upharpoonright\lambda\) +is a \(\text{Cls} \rightarrow 1\), then the +domains of \(Q_{\Delta}ʻ\lambda\) are the same as the +domains of \({\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\), +and \(Q_{\Delta}ʻ\lambda\) is similar to +\({\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\), thus reducing the +problem of selections from many-one relations to that of selections +from classes of classes.</p> + +<p><a href="#*85·27">*85·27</a> and <a href="#*85·43">*85·43</a>, which show that if \(\kappa \in \text{Cls}^{2} ~ \text{excl}\), +\(P_{\Delta}ʻsʻ\kappa\) consists of the relational sums +of the domains of \({\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\) and is similar +to \({\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\); <i>i.e.</i> the class of +\(P\)-selections from \(sʻ\kappa\) is similar to the class obtained +as follows: take the members of \(\kappa\) one by one, and form the +\(P\)-selections of each; we thus obtain a class of classes, each class +being of the form \(P_{\Delta}ʻ\alpha\), where \(\alpha \in \kappa\); +we then make a selection from this class of classes; this selection is +a member of \({\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\); the number of such +selections is the same as the number of \(P_{\Delta}ʻsʻ\kappa\).</p> + +<p><a href="#*85·28">*85·28</a> and <a href="#*85·44">*85·44</a>, which are special cases of *85·27 and *85·43, +but more useful than these. *85·44 is the source of the associative +law in cardinal multiplication; it states that, if \(\kappa\) is a +\(\text{Cls}^{2} ~ \text{excl}\), \({\in}_{\Delta}ʻsʻ\kappa\) has the +same number of members as \({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\). +(On associative laws in general, see the notes to *42·1·11.) +That is to say, if we form the class of selective relations +(\({\in}_{\Delta}ʻ\alpha)\) for every \(\alpha\) which is a member +of \(\kappa\), and then form the class of selective relations +for \({\in}_{\Delta}ʻʻ\kappa\), we get the same number of terms +as if we proceeded to form the class of selective relations for +\({\in}_{\Delta}ʻsʻ\kappa\). The way in which this proposition yields +the associative law of multiplication may be explained as follows. +We shall define the product of the numbers of members of \(\alpha\) +as the number of \({\in}_{\Delta}ʻ\alpha\). Thus <i>e.g.</i> if the +numbers of the members of \(\alpha\) are \(\mu_{\alpha 1}, \mu_{\alpha 2}, \mu_{\alpha 3}\), +the number of \({\in}_{\Delta}ʻ\alpha\) is +\(\mu_{\alpha 1} × \mu_{\alpha 2} × \mu_{\alpha 3}\). Suppose the +other members of \(\kappa\) are \(\beta\) and \(\gamma\), and that +\(\beta\) and \(\gamma\) again have three members each. Then the +number of \({\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\) is the product of +the numbers of \({\in}_{\Delta}ʻ\alpha\), \({\in}_{\Delta}ʻ\beta\), +\({\in}_{\Delta}ʻ\gamma\), <i>i.e.</i> it is the product of +\(\mu_{\alpha 1} \times \mu_{\alpha 2} \times \mu_{\alpha 3}\), +\(\mu_{\beta 1} \times \mu_{\beta 2} \times \mu_{\beta 3}\) and +<span class="pagenum" id="Page_550">[Pg 550]</span>\(\mu_{\gamma 1} \times \mu_{\gamma 2} \times \mu_{\gamma 3}\).</p> + +<p>But the numbers of the members of \(sʻ\kappa\) are +\[ +\mu_{\alpha 1}, \mu_{\alpha 2}, \mu_{\alpha 3}, \mu_{\beta 1}, \mu_{\beta 2}, \mu_{\beta 3}, \mu_{\gamma 1}, \mu_{\gamma 2}, + \mu_{\gamma 3}. +\]</p> + +<p>Thus the number of \({\in}_{\Delta}ʻsʻ\kappa\) is +\[ +\mu_{\alpha 1}×\mu_{\alpha 2}×\mu_{\alpha 3}×\mu_{\beta 1}×\mu_{\beta 2}×\mu_{\beta 3}×\mu_{\gamma 1}×\mu_{\gamma 2}×\mu_{\gamma 3}. +\]</p> + +<p>Hence <a href="#*85·44">*85·44</a> enables us to conclude that +\[ +\begin{align} +(\mu_{\alpha 1}×\mu_{\alpha 2}×\mu_{\alpha 3})×(\mu_{\beta 1}×\mu_{\beta 2}×\mu_{\beta 3})×(\mu_{\gamma 1}×\mu_{\gamma 2}×\mu_{\gamma 3}) \\ +=\mu_{\alpha 1}×\mu_{\alpha 2}×\mu_{\alpha 3}×\mu_{\beta 1}×\mu_{\beta 2}×\mu_{\beta 3}×\mu_{\gamma 1}×\mu_{\gamma 2}×\mu_{\gamma 3},\\ +\end{align} +\] +which is a case of the associative law. In fact *85·44 gives us this +law in its general form, when the number of brackets, and of factors in +each bracket, may be infinite or finite indifferently.</p> + +<p>Another important pair of propositions is *85·53·54. These enable us to +reduce the problem of selections for <i>any</i> relation to the problem +of selections from a class of classes. The method is as follows: Given +any term \(x\), form the class of ordered couples of which \(x\) is +relatum while the referent is a term having the relation \(P\) to +\(x\). Call this class of couples \(P \unicode{x21a7} x\). Form this +class for every \(x\) which is a member of \(\alpha\); we thus obtain a +class of classes, namely \(P \unicode{x21a7}ʻʻ\alpha\). Then the number +of selections from this class of classes is the same as the number of +\(P_{\Delta}ʻ\alpha\).</p> + +<p>We have one other important pair of propositions in this number, +namely *85·61·63. These show that what is called "Zermelo's axiom" is +equivalent to what is called the "multiplicative axiom." Zermelo's +axiom<a id="FNanchor_63" href="#Footnote_63" class="fnanchor">[63]</a> is to the effect that if \(\alpha\) is any class, +\({\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha\) is never null, <i>i.e.</i> +\((\alpha).\exists! {\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha\). The +"multiplicative axiom" is to the effect that if \(\kappa \in \text{Cls ex}^{2} ~ \text{excl}\), +there is at least one class formed by taking one representative +from each member of \(\kappa\), which is equivalent to +\[ +\kappa \in \text{Cls ex}^{2} ~ \text{excl} .\supset_{\kappa}.\exists! {\in}_{\Delta}ʻ\kappa. +\]</p> + +<p>In <a href="#*85·63">*85·63</a>, these two axioms are shown to be equivalent. From Zermelo's +theorem<a id="FNanchor_64" href="#Footnote_64" class="fnanchor">[64]</a> it follows that both are equivalent to the assumption that +every class can be well-ordered. This will be proved later (*258).</p> + +<p>The above-mentioned propositions, stated symbolically, are as follows:</p> + +<p class="nind"><b><a id="*85·1">*85·1</a>.</b> \(\vdash: Q \upharpoonright \lambda \in \text{Cls} \rightarrow 1 .\supset. \text{D}ʻʻQ_{\Delta}ʻ\lambda = \text{D}ʻʻ{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\)</p> + +<p class="nind"><b><a id="*85·14">*85·14</a>.</b> \(\vdash: Q \upharpoonright \lambda \in \text{Cls} \rightarrow 1 .\supset. Q_{\Delta}ʻ\lambda \mathop{\text{ sm }} {\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\)</p> + +<p class="nind"><b>*85·27.</b> \(\vdash: \kappa \in \text{Cls}^{2} ~ \text{excl} .\supset. P_{\Delta}ʻsʻ\kappa=\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\)</p> + +<p class="nind"><b>*85·28.</b> \(\vdash: \kappa \in \text{Cls}^{2} ~ \text{excl} .\supset. {\in}_{\Delta}ʻsʻ\kappa = \dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\)</p> + +<p><span class="pagenum" id="Page_551">[Pg 551]</span></p> + +<p class="nind"><b>*85·43.</b> \(\vdash: \kappa \in \text{Cls}^2 ~ \text{excl} .\supset. P_{\Delta}ʻsʻ\kappa \mathop{\text{ sm }} {\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\)</p> + +<p class="nind"><b>*85·44.</b> \(\vdash\colon\kappa\in\text{Cls}^{2}\text{excl}.\supset.{\in}_{\Delta}ʻsʻ\kappa\text{ sm }{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa\)</p> + +<p>The following propositions depend upon the definition</p> + +<p class="nind"><b>*85·5.</b> \(P \unicode{x21A7} y=\downarrow yʻʻ\overrightarrow{P}ʻy \quad\text{Df}\)</p> + +<p><i>I.e.</i> \(P \unicode{x21A7} y\) is the class of all couples whose +relatum is y while the referent has the relation \(P\) to \(y\). We +then have</p> + +<p class="nind"><b>*85·53.</b> \(\vdash.P_{\Delta}ʻ\alpha=\breve{s}ʻʻDʻʻ{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha\)</p> + +<p>giving a construction for \(P_{\Delta}ʻ\alpha\) by means of +\({\in}_{\Delta}\), and</p> + +<p class="nind"><b>*85·54.</b> \(\vdash.P_{\Delta}ʻ\alpha\text{ sm }{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha\)</p> + +<p>which reduces the question of the existence of \(P\)-selections to that +of the existence of \({\in}\)-selections.</p> + +<p class="nind"><b>*85·61.</b> + \(\vdash.{\in}\unicode{x21A7}ʻʻ\kappa\in\text{Cls}^{2}\text{excl}.{\in}_{\Delta}ʻ\kappa=\breve{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}\unicode{x21A7}ʻʻ\kappa.{\in}_{\Delta}ʻ\kappa\text{ + sm }{\in}_{\Delta}ʻ{\in}\unicode{x21A7}ʻʻ\kappa\)</p> + +<p>This proposition gives a construction for any \({\in}\)-selection in +terms of an \({\in}\)-selection from a \(\text{Cls}^{2}\,\text{excl}\), +and reduces the question of the existence of the former to that of +the existence of the latter. A particularly important case is when +\(\kappa=\text{Cl ex}ʻ\alpha\). This is considered in</p> + +<p class="nind"><b>*85·63.</b> \(\vdash\colon{\in}\unicode{x21A7}ʻʻ\text{Cl ex}ʻ\alpha\in\text{Cls ex}^{2} \text{excl}:\exists!{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha.\equiv.\exists!{\in}_{\Delta}ʻ{\in}\unicode{x21A7}ʻʻ\text{Cl ex}ʻ\alpha\)</p> + +<hr class="tb"> + +<p class="nind"><b>*85·1.</b> \(\vdash\colon Q\upharpoonright \lambda\in\text{Cls}\rightarrow 1.\supset.DʻʻQ_{\Delta}ʻ\lambda=Dʻʻ{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*81·3}. \supset \vdash :\text{Hp}.&\supset .\text{D}ʻʻQ_{\Delta }ʻ \lambda =\hat{\mu}\{\alpha \in \overrightarrow{Q}ʻʻ\lambda .\supset _{\alpha}.\mu + \cap\alpha \in 1:\mu \subset sʻ \overrightarrow{Q}ʻʻ \lambda \} &\qquad \text{(1)}\\ +\vdash.\text{*84·51}.\supset \vdash:\text{Hp}.&\supset .\overrightarrow{Q}ʻʻ \lambda \in \text{Cls}^{2}\text{excl}.\\ +[\text{*84·412}] &\supset .\text{D}ʻʻ {\in}_{\Delta}ʻ \overrightarrow{Q}ʻʻ\lambda =\hat{\mu}\{\alpha \in \overrightarrow{Q}ʻʻ \lambda .\supset _{\alpha }.\mu + \cap\alpha \in 1:\mu \subset sʻ \overrightarrow{Q}ʻʻ \lambda\} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·11.</b> + \(\vdash\colon\overrightarrow{Q}\upharpoonright\lambda\in 1\rightarrow 1.\supset.\text{D}ʻʻ(P\mid\overrightarrow{Q})_{\Delta}ʻ\lambda=\text{D}ʻʻP_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·431.*32·12}.&\supset \vdash :\text{Hp}.\supset .\lambda \subset ᗡʻ \overrightarrow{Q} &\qquad \text{(1)}\\ +\vdash.\text{(1).*82·32}. &\supset \vdash:\text{Hp}.\supset .\text{D}ʻʻ(P\mid\overrightarrow{Q})_{\Delta }ʻ \lambda =\text{D}ʻʻ P_{\Delta }ʻ + \overrightarrow{Q}ʻʻ \lambda :\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·111.</b> \(\vdash\colon M\in{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda.\supset.\text{D}ʻ(M\mid\overrightarrow{Q}\upharpoonright\lambda)=DʻM \quad[\text{*82·3}]\)</p> + +<p class="nind"><b>*85·112.</b> \(\vdash\colon M\in{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda.\supset.M\mid\overrightarrow{Q}\upharpoonright \lambda\in Q_{\Delta}ʻ\lambda + \quad\left[\text{*82·221} \frac{{\in},\,\overrightarrow{Q}}{P,\,Q}.\text{*62·26}\right]\)</p> + +<p class="nind"><b>*85·12.</b> \(\vdash\colon\overrightarrow{Q}\upharpoonright \lambda\in 1\rightarrow 1.\supset.DʻʻQ_{\Delta}ʻ\lambda=\text{D}ʻʻ{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*62·26}.&\supset \vdash .\text{D}ʻʻ Q_{\Delta}ʻ \lambda =\text{D}ʻʻ ({\in}\mid\overrightarrow{Q} )_{\Delta}ʻ\lambda &\qquad \text{(1)}\\ +\vdash .\text{*82·32}.&\supset \vdash :\text{Hp}.\supset .\text{D}ʻʻ ({\in} \mid\overrightarrow{Q})_{\Delta }ʻ \lambda =\text{D}ʻʻ {\in}_{\Delta}ʻ + \overrightarrow{Q}ʻʻ \lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_552">[Pg 552]</span></p> + +<p>This proposition is used in connection with ordinal multiplication +(*173·14).</p> + +<p class="nind"><b>*85·13.</b> + \(\vdash:\overrightarrow{Q}\upharpoonright\lambda\in 1\rightarrow 1.R\in Q_{\Delta}ʻ\lambda.\supset.R\mid\text{Cnv}ʻ\overrightarrow{Q}\in{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*62·26}.&\supset\vdash\colon\text{Hp}.\supset.\overrightarrow{Q}\upharpoonright\lambda\in 1\rightarrow 1.R\in({\in}\mid \overrightarrow{Q})_{\Delta}ʻ\lambda.\\ +[\text{*82·231}] &\supset.R\mid\text{Cnv}ʻ\overrightarrow{Q}\in{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition, the hypothesis required +as to \(\overrightarrow{Q}\) by <a href="#*82·231">*82·231</a> is only +\(\overrightarrow{Q}\upharpoonright \lambda \in \text{Cls}\rightarrow 1\); +but since \(\overrightarrow{Q}\in 1\rightarrow \text{Cls}\), +\(\overrightarrow{Q}\upharpoonright \lambda \in \text{Cls}\rightarrow 1.\equiv .\overrightarrow{Q}\upharpoonright \lambda \in 1\rightarrow 1\).</p> + +<p>The above proposition is used in connection with "families" (<a href="#*97·31">*97·31</a>).</p> + +<p class="nind"><b>*85·14.</b> \(\vdash:Q\upharpoonright\lambda\in\text{Cls}\rightarrow 1.\supset.Q_{\Delta}ʻ\lambda\,\text{ sm }\,{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*81·21}.&\supset\vdash\colon\text{Hp}.\supset.Q_{\Delta}ʻ\lambda\,\text{ sm }\,\text{D}ʻʻQ_{\Delta}ʻ\lambda.\\ +[\text{*85·1}] &\supset.Q_{\Delta}ʻ\lambda\,\text{ sm }\,\text{D}ʻʻ{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda &\qquad \text{(1)}\\ +\vdash.\text{*84·51}.&\supset\vdash\colon\text{Hp}.\supset.\overrightarrow{Q}ʻʻ\lambda\in\text{Cls}^{2}\,\text{excl}.\\ +[\text{*84·41}] &\supset.\text{D}ʻʻ{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda\,\text{ sm }\,{\in}_{\Delta}ʻ\overrightarrow{Q}ʻʻ\lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>*85·21·22 are lemmas for <a href="#*85·24">*85·24</a>, which, with <a href="#*85·26">*85·26</a>, is required for +<a href="#*85·27">*85·27</a>.</p> + +<p class="nind"><b>*85·21.</b> \(\vdash:\alpha\in\kappa.M\in {P}_{\Delta}ʻsʻ\kappa.\supset.M\upharpoonright\alpha\in P_{\Delta}ʻ\alpha \quad[\text{*80·6.*40·13}]\)</p> + +<p class="nind"><b><a id="*85·22">*85·22</a>.</b> \(\vdash:M\in {P}_{\Delta}ʻsʻ\kappa.\supset.M\upharpoonright \mid\kappa \upharpoonleft \breve{P}_{\Delta}\in {\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\dot{s}ʻ\text{D}ʻ(M\upharpoonright + \mid\kappa \upharpoonleft \breve{P}_{\Delta})=M\)</p> + +<p>Here \(M\upharpoonright \mid\kappa \upharpoonleft \breve{P}_{\Delta}\in {\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\) +can also be written \(\{(M\upharpoonright )\mid(\kappa \upharpoonleft \breve{P}_{\Delta})\}\in ({\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa)\). +The brackets are omitted because no other meaning is possible.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*85·21}. &\supset\vdash\colon\ldotp\text{Hp}.\supset:\alpha\in\kappa.\supset_{\alpha}.\exists!P_{\Delta}ʻ\alpha:\\ +[\text{*80·81}] &\supset:\alpha,\beta\in\kappa.P_{\Delta}ʻ\alpha=P_{\Delta}ʻ\beta.\supset_{\alpha,\beta}.\alpha=\beta:\\ +[\text{*80·12.*71·166·55}] &\supset:P_{\Delta}\upharpoonright \kappa\in 1\rightarrow 1:\\ +[\text{*35·52}] &\supset:\kappa\upharpoonleft\breve{P}_{\Delta}\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{(1).*72·14.*71·25}.&\supset\vdash:\text{Hp}.\supset.M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta}\in 1\rightarrow\text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{*34·1.*30·4}. &\supset\vdash:R\{M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta}\}\lambda.\equiv.\\ +&(\exists \alpha).R=M\upharpoonright \alpha.\alpha\in\kappa.\lambda=P_{\Delta}ʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(3).*85·21}. &\supset\vdash:\text{Hp}.\supset.M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta}\unicode{x2abd}{\in} &\qquad \text{(4)}\\ +\vdash.\text{*37·322.*33·431}. &\supset\vdash.\text{ᗡ}ʻ(M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta})=\text{ᗡ}ʻ(\kappa\upharpoonleft\breve{P}_{\Delta})\\ +[\text{*37·4}] &=P_{\Delta}ʻʻ\kappa &\qquad \text{(5)}\\ +\vdash.\text{(2).(4).(5).*80·14}.&\supset\vdash:\text{Hp}.\supset.\{M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta}\}\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa + &\qquad \text{(6)}\\ +\vdash.\text{*37·32.*35·62}. &\supset\vdash.\text{D}ʻ(M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta})=M\upharpoonright ʻʻ\kappa.\\ +[\text{*41·35}] &\supset\vdash.\dot{s}ʻ\text{D}ʻ(M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta})=M\upharpoonright sʻ\kappa &\qquad \text{(7)}\\ +\vdash.\text{(7).*80·29}. &\supset\vdash:\text{Hp}.\supset.\dot{s}ʻDʻ(M\upharpoonright\mid\kappa\upharpoonleft\breve{P}_{\Delta})=M &\qquad \text{(8)}\\ +\vdash.\text{(6).(8)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_553">[Pg 553]</span></p> + +<p class="nind"><b><a id="*85·24">*85·24</a>.</b> \(\vdash.P_{\Delta}ʻsʻ\kappa\subset\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*85·22}.\supset\vdash\colon M\in P_{\Delta}ʻsʻ\kappa.&\supset.(\exists X).X\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.M=\dot{s}ʻ\text{D}ʻX.\\ +[\text{*37·67}] &\supset.M\in\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are lemmas for <a href="#*85·26">*85·26</a>.</p> + +<p class="nind"><b>*85·241.</b> \(\vdash:X\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\alpha\in\kappa.\supset.XʻP_{\Delta}ʻ\alpha\in P_{\Delta}ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·2}.&\supset\vdash\colon\ldotp X\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\supset:\lambda\in P_{\Delta}ʻʻ\kappa.\supset_\lambda.Xʻ\lambda\in\lambda:\\ +[\text{*37·63}] &\supset:\alpha\in\kappa.\supset_\alpha.XʻP_{\Delta}ʻ\alpha\in P_{\Delta}ʻ\alpha\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·243.</b> \(\vdash:\kappa\in\text{Cls}^{2}\,\text{excl}.X\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\supset.\dot{s}ʻ\text{D}ʻX\in 1\rightarrow\text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·21}. &\supset\vdash\colon\text{Hp}.\supset.\text{D}ʻX\subset sʻP_{\Delta}ʻʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*40·151.*80·11}.&\supset\vdash.sʻP_{\Delta}ʻʻ\kappa\subset 1\rightarrow\text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash\colon\text{Hp}.\supset.\text{D}ʻX\subset 1\rightarrow\text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{*80·35.*11·45·55}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:M,N\in\text{D}ʻX.\exists!\text{ᗡ}ʻM\cap\text{ᗡ}ʻN.\supset.\\ +&(\exists \alpha,\beta).\alpha,\beta\in\kappa.M=XʻP_{\Delta}ʻ\alpha.N=XʻP_{\Delta}ʻ\beta.\exists!\text{ᗡ}ʻM\cap\text{ᗡ}ʻN.\\ +[\text{*85·241.*80·14}]&\supset.(\exists \alpha,\beta).\alpha,\beta\in\kappa.M=XʻP_{\Delta}ʻ\alpha.N=XʻP_{\Delta}ʻ\beta.\\ +&\exists!\text{ᗡ}ʻM\cap\text{ᗡ}ʻN.\alpha=\text{ᗡ}ʻM.\beta=\text{ᗡ}ʻN.\\ +[\text{*13·22}] &\supset.\text{ᗡ}ʻM,\text{ᗡ}ʻN\in\kappa.M=XʻP_{\Delta}ʻ\text{ᗡ}ʻM.N=XʻP_{\Delta}ʻ\text{ᗡ}ʻN.\\ +&\exists!\text{ᗡ}ʻM\cap\text{ᗡ}ʻN.\\ +[\text{*84·11}] &\supset.\text{ᗡ}ʻM=\text{ᗡ}ʻN.M=XʻP_{\Delta}ʻ\text{ᗡ}ʻM.N=XʻP_{\Delta}ʻ\text{ᗡ}ʻN.\\ +[\text{*30·37}] &\supset.M=N &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).*72·32}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·244.</b> \(\vdash:X\in {\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\supset.\dot{s}ʻ\text{D}ʻX\unicode{x2abd}P\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·21.*40·4}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:R\in\text{D}ʻX.\supset_R.(\exists \alpha).\alpha\in\kappa.R\in P_{\Delta}ʻ\alpha.\\ +[\text{*80·14}] &\supset_R.R\unicode{x2abd}P:\\ +[\text{*41·151}] &\supset:\dot{s}ʻ\text{D}ʻX\unicode{x2abd}P\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·245.</b> \(\vdash:X\in {\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\supset.\text{ᗡ}ʻ\dot{s}ʻ\text{D}ʻX=sʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*85·241.*80·14}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:\alpha\in\kappa.\supset_\alpha.\text{ᗡ}ʻXʻP_{\Delta}ʻ\alpha=\alpha:\\ +[\text{*50·17}] &\supset:\text{ᗡ}ʻʻXʻʻP_{\Delta}ʻʻ\kappa=\kappa:\\ +[\text{*80·34}] &\supset:\text{ᗡ}ʻʻ\text{D}ʻX=\kappa:\\ +[\text{*41·44}] &\supset:\text{ᗡ}ʻ\dot{s}\text{D}ʻX=sʻ\kappa\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·25.</b> \[\begin{align}\vdash:\kappa\in\text{Cls}^{2}\,\text{excl}.X\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\supset.\dot{s}ʻ\text{D}ʻX\in P_{\Delta}ʻsʻ\kappa\\ +\quad[\text{*85·243·244·245.*80·14}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_554">[Pg 554]</span></p> + +<p class="nind"><b><a id="*85·26">*85·26</a>.</b> \(\vdash\colon\kappa\in\text{Cls}^{2}\,\text{excl}.\supset.\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\subset P_{\Delta}ʻsʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*85·25}.&\supset\vdash\colon\ldotp\text{Hp}.\supset:X\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\supset_{X}.\dot{s}ʻ\text{D}ʻX\in P_{\Delta}ʻsʻ\kappa:\\ +[\text{*37·61·33}] &\supset:\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\subset P_{\Delta}ʻsʻ\kappa\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*85·27">*85·27</a>.</b> \(\vdash\colon\kappa\in\text{Cls}^{2}\,\text{excl}.\supset.P_{\Delta}ʻsʻ\kappa=\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa + \quad[\text{*85·24·26}]\)</p> + +<p class="nind"><b><a id="*85·28">*85·28</a>.</b> \(\vdash\colon\kappa\in\text{Cls}^{2}\,\text{excl}.\supset.{\in}_{\Delta}ʻsʻ\kappa=\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa + \quad\left[\text{*85·27}\, \frac{{\in}}{P}\right]\)</p> + +<p>The following proposition is a lemma for <a href="#*85·31">*85·31</a>.</p> + +<p class="nind"><b>*85·3.</b> \(\vdash\colon M\in P_{\Delta}ʻ\alpha.z\in\alpha.\supset.Mʻz\unicode{x2abd}\dot{s}ʻ\text{D}ʻM.Mʻz\unicode{x2abd}\dot{s}ʻ\overrightarrow{P}ʻz\)</p> + +<p>The conditions of significance here and in *85·31·32·33·34 require +\(\text{D}ʻP\subset\text{Rel}\).</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·32.*33·43}.&\supset\vdash\colon\text{Hp}.\supset.Mʻz\in\text{D}ʻM.Mʻz\in\overrightarrow{P}ʻz.\\ +[\text{*41·13}] &\supset.Mʻz\unicode{x2abd}\dot{s}ʻ\text{D}ʻM.Mʻz\unicode{x2abd}\dot{s}ʻ\overrightarrow{P}ʻz:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions, down to <a href="#*85·42">*85·42</a> inclusive, deal +with circumstances under which we can infer \(M=N\) from +\(\dot{s}ʻ\text{D}ʻM=\dot{s}ʻ\text{D}ʻN\). *85·32·33·34 are not subsequently used; +the remainder are used in proving <a href="#*85·43">*85·43</a>.</p> + +<p class="nind"><b><a id="*85·31">*85·31</a>.</b> \[\begin{align}\vdash\colon\ldotp &z,w\in\alpha.z\neq w.\supset_{z,w}.\dot{s}ʻ\overrightarrow{P}ʻz\dot{\cap}\dot{s}ʻ\overrightarrow{P}ʻw=\dot{\Lambda}:\supset:\\ +&M,N\in P_{\Delta}ʻ\alpha.\dot{s}ʻ\text{D}ʻM=\dot{s}ʻ\text{D}ʻN.\supset.M=N\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*25·54}.&\supset\vdash\colon\text{Hp}.z,w\in\alpha.\dot{\exists}!\dot{s}ʻ\overrightarrow{P}ʻz\dot{\cap}\dot{s}ʻ\overrightarrow{P}ʻw.\supset_{z,w}.z=w:\\ +[\text{*11·35}] &\supset\vdash\colon\text{Hp}.z,w\in\alpha.u(\dot{s}ʻ\overrightarrow{P}ʻz)v.u(\dot{s}ʻ\overrightarrow{P}ʻw)v.\supset_{z,w,u,v}.z=w &\qquad \text{(1)}\\ +\vdash.\text{*85·3}. &\supset\vdash\colon\colon\text{Hp}.z\in\alpha.M,N\in P_{\Delta}ʻ\alpha.\dot{s}ʻ\text{D}ʻM=\dot{s}ʻDʻN.\supset\colon\ldotp\\ +&u(Mʻz)v.\supset:z\in\alpha.u(\dot{s}ʻ\overrightarrow{P}ʻz)v.u(\dot{s}ʻ\text{D}ʻN)v:\\ +[\text{*80·35}] &\supset:z\in\alpha.u(\dot{s}ʻ\overrightarrow{P}ʻz)v:(\exists w).w\in\alpha.u(Nʻw)v:\\ +[\text{*85·3.*10·35}]&\supset:(\exists w).z,w\in\alpha.u(\dot{s}ʻ\overrightarrow{P}ʻz)v.u(\dot{s}ʻ\overrightarrow{P}ʻw)v.u(Nʻw)v:\\ +[\text{(1).*10·28}] &\supset:(\exists w).z=w.u(Nʻw)v:\\ +[\text{*13·195}] & \supset:u(Nʻz)v &\qquad \text{(2)}\\ +\vdash.\text{(2). Exp.*10·11·21}.&\supset\vdash\colon\ldotp\text{Hp(2)}.\supset:z\in\alpha.\supset_{z}.Mʻz\unicode{x2abd}Nʻz &\qquad \text{(3)}\\ +\text{Similarly} &\supset\vdash\colon\ldotp\text{Hp(2)}.\supset:z\in\alpha.\supset_{z}.Nʻz\unicode{x2abd}Mʻz &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}. &\supset\vdash\colon\ldotp\text{Hp(2)}.\supset:z\in\alpha.\supset_{z}.Mʻz=Nʻz:\\ +[\text{*33·45.*80·14}] &\supset:M=N\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_555">[Pg 555]</span></p> + +<p class="nind"><b><a id="*85·32">*85·32</a>.</b> \[\begin{align}\vdash\colon\ldotp &z,w\in\alpha .z\neq w.\supset_{z,w}.sʻ Cʻʻ\overrightarrow{P}ʻ z\cap sʻ Cʻʻ\overrightarrow{P}ʻ w=\Lambda:\supset:\\ +&M,N\in P_{\Delta}ʻ\alpha .\dot{s}ʻ \text{D}ʻ M=\dot{s}ʻ \text{D}ʻ N.\supset .M=N\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*41·45}.\supset\\ +\vdash\colon\ldotp\text{Hp}.\supset:z,w\in\alpha.z\neq w.&\supset_{z,w}.Cʻ\dot{s}ʻ\overrightarrow{P}ʻz\cap Cʻ\dot{s}ʻ\overrightarrow{P}ʻw=\Lambda.\\ +[\text{*33·34}] &\supset_{z,w}.\dot{s}ʻ\overrightarrow{P}ʻz\dot{\cap}\dot{s}ʻ\overrightarrow{P}ʻw=\dot{\Lambda} \qquad\qquad \text{(1)}\\ +\vdash.\text{(1).*85·31}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·33.</b> \[\begin{align}\vdash\colon\ldotp &z,w\in\alpha .z\neq w.\supset_{z,w}.sʻ \text{D}ʻʻ\overrightarrow{P}ʻ z\cap sʻ \text{D}ʻʻ\overrightarrow{P}ʻ w=\Lambda :\supset:\\ +&M,N\in P_{\Delta}ʻ\alpha .\dot{s}\text{D}ʻ M=\dot{s}ʻ \text{D}ʻ N.\supset .M=N \quad[\text{*41·43.*33·32.*85·31}]\end{align}\]</p> + +<p>The proof proceeds exactly as in <a href="#*85·32">*85·32</a>.</p> + +<p class="nind"><b>*85·34.</b> \[\begin{align}\vdash\colon\ldotp &z,w\in\alpha.z \neq w.\supset_{z,w}.sʻ\text{ᗡ}ʻʻ\overrightarrow{P}ʻz\cap sʻ\text{ᗡ}ʻʻ\overrightarrow{P}ʻw=\Lambda:\supset:\\ +&M,N\in P_{\Delta}ʻ\alpha.\dot{s}ʻDʻM=\dot{s}ʻ\text{D}ʻN.\supset.M=N \quad[\text{*41·44.*33·33.*85·31}]\end{align}\]</p> + +<p>The following propositions, *85·4·41·42, are lemmas for *85·43·44, +which latter are of fundamental importance, since they are the source +of the associative law in cardinal arithmetic.</p> + +<p class="nind"><b>*85·4.</b> \[\begin{align}\vdash\colon\ldotp &\lambda,\mu\in\kappa.\lambda\neq\mu.\supset_{\lambda,\mu}.\dot{s}ʻ\lambda\dot{\cap}\dot{s}ʻ\mu=\dot{\Lambda}:\supset:\\ +&M,N\in{\in}_{\Delta}ʻ\kappa.\dot{s}ʻDʻM=\dot{s}ʻDʻN.\supset.M=N \quad\left[\text{*85·31}\, \frac{{\in}}{P}.\text{*62·2}\right]\end{align}\]</p> + +<p class="nind"><b>*85·41.</b> + \(\vdash\colon\ldotp \kappa\in\text{Cls}^{2}\,\text{excl}.\supset:\alpha,\beta\in\kappa.\alpha\neq\beta.\supset.\dot{s}ʻP_{\Delta}ʻ\alpha\dot{\cap}\dot{s}ʻP_{\Delta}ʻ\beta=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·14}.&\supset\vdash\colon x(\dot{s}ʻP_{\Delta}ʻ\alpha)y.x(\dot{s}ʻP_{\Delta}ʻ\beta)y.\supset_{x,y}.y\in\alpha.y\in\beta.\\ +[\text{*22·33.*10·24}] &\supset_{x,y}.\exists!.\alpha\cap\beta:\\ +[\text{Transp}] &\supset\vdash\colon\alpha\cap\beta=\Lambda.\supset.\dot{s}ʻP_{\Delta}ʻ\alpha\dot{\cap}\dot{s}ʻP_{\Delta}ʻ\beta=\dot{\Lambda} \qquad\qquad \text{(1)}\\ +\vdash.\text{(1).*84·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*85·42">*85·42</a>.</b> \(\vdash\colon \kappa\in\text{Cls}^{2}\,\text{excl}.M,N\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\dot{s}ʻ\text{D}ʻM=\dot{s}ʻ\text{D}ʻN.\supset.M=N\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37.Transp}.&\supset\vdash\colon P_{\Delta}ʻ\alpha\neq P_{\Delta}ʻ\beta.\supset_{\alpha,\beta}.\alpha\neq\beta:\\ +[\text{Fact}] &\supset\vdash\colon \kappa\in\text{Cls}^{2}\,\text{excl}.\alpha,\beta\in\kappa.P_{\Delta}ʻ\alpha\neq P_{\Delta}ʻ\beta.\supset_{\alpha,\beta}.\\ +&\qquad\kappa\in\text{Cls}^{2}\,\text{excl}.\alpha,\beta\in\kappa.\alpha\neq\beta.\\ +[\text{*85·41}] &\supset_{\alpha,\beta}.\dot{s}ʻP_{\Delta}ʻ\alpha\dot{\cap}\dot{s}ʻP_{\Delta}ʻ\beta=\dot{\Lambda}:\\ +[\text{*37·63}]&\supset\vdash\colon \kappa\in\text{Cls}^{2}\,\text{excl}.\lambda,\mu\in P_{\Delta}ʻʻ\kappa.\lambda\neq\mu.\supset_{\lambda,\mu}.\dot{s}ʻ\lambda\dot{\cap}\dot{s}ʻ\mu=\dot{\Lambda} + \qquad\qquad \text{(1)}\\ +\vdash.\text{(1).*85·4}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_556">[Pg 556]</span></p> + +<p class="nind"><b><a id="*85·43">*85·43</a>.</b> \(\vdash\colon\kappa\in\text{Cls}^{2}\,\text{excl}.\supset.P_{\Delta}ʻsʻ\kappa\,\text{ sm }\,{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·41}.&\supset\vdash.(M).\dot{s}ʻ\text{D}ʻM=(\dot{s}\mid\text{D})ʻM.\\ +[\text{*13·12}] \supset\vdash\colon\ldotp &M,\,N\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.\dot{s}ʻ\text{D}ʻM=\dot{s}ʻ\text{D}ʻN.\supset_{M,N}.M=N:\supset:\\ +&M,\,N\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.(\dot{s}\mid\text{D})ʻM=(\dot{s}\mid\text{D})ʻN.\supset_{M,N}.M=N &\qquad \text{(1)}\\ +\vdash.\text{(1).*85·42}.\supset\\ +\vdash\colon\ldotp\kappa\in\text{Cls}^{2}\,\text{excl}.&\supset:M,N\in{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa.(\dot{s}\mid\text{D})ʻM=(\dot{s}\mid\text{D})ʻN.\supset_{M,N}.M=N:\\ +[\text{*73·25}] &\supset:(\dot{s}\mid\text{D})ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\,\text{ sm }\,{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa:\\ +[\text{*37·33}] &\supset:\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\,\text{ sm }\,{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa:\\ +[\text{*85·27}] &\supset:P_{\Delta}ʻsʻ\kappa\,\text{ sm }\,{\in}_{\Delta}ʻP_{\Delta}ʻʻ\kappa\colon\ldotp\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*85·44">*85·44</a>.</b> \(\vdash\colon\kappa\in\text{Cls}^{2}\,\text{excl}.\supset.{\in}_{\Delta}ʻsʻ\kappa\,\text{ sm }\,{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa + \quad\left[\text{*85·43}\,\frac{{\in}}{P}\right]\)</p> + +<p>The following proposition is used in connection with cardinal +multiplication (*114·301).</p> + +<p class="nind"><b>*85·45.</b> \(\vdash\colon\kappa\cap\lambda=\Lambda.\supset.{\in}_{\Delta}ʻ\kappa\cup\lambda\,\text{ + sm }\,{\in}_{\Delta}ʻ(\iotaʻ{\in}_{\Delta}ʻ\kappa\cup\iotaʻ{\in}_{\Delta}ʻ\lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*85·44}.\supset\\ +\vdash\colon \iotaʻ\kappa\cup\iotaʻ\lambda\in\text{Cls}^{2}\,\text{excl}.&\supset.{\in}_{\Delta}ʻsʻ(\iotaʻ\kappa\cup\iotaʻ\lambda)\,\text{ sm }\,{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ(\iotaʻ\kappa\cup\iotaʻ\lambda) + &\qquad \text{(1)}\\ +\vdash.\text{*24·57}. \supset\vdash\colon\ldotp\text{Hp}.&\supset:\kappa\neq\lambda.\lor.\kappa=\Lambda.\lambda=\Lambda:\\ +[\text{*84·62·23}] &\supset:\iotaʻ\kappa\cup\iotaʻ\lambda\in\text{Cls}^{2}\,\text{excl} &\qquad \text{(2)}\\ +\vdash.\text{*53·11·32}.&\supset\vdash.sʻ(\iotaʻ\kappa\cup\iotaʻ\lambda)=\kappa\cup\lambda.{\in}_{\Delta}ʻʻ(\iotaʻ\kappa\cup\iotaʻ\lambda)=(\iotaʻ{\in}_{\Delta}ʻ\kappa\cup\iotaʻ{\in}_{\Delta}ʻ\lambda &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The purpose of the following propositions, down to <a href="#*85·55">*85·55</a>, is to +show how to get from a class of classes a class of selections +having the same number of terms as \(P_{\Delta}ʻ\kappa\). For this +purpose we introduce a new notation, representing a rather important +analysis of the couples contained in a given relation. A couple +\(x\downarrow y\) is contained in a relation \(P\) when \(xPy\); thus +if, keeping \(y\) fixed, we form the class of couples \(\downarrow yʻʻ\overrightarrow{P}ʻy\), +all these couples are contained in \(P\). We put</p> + +<p class="nind"><b>*85·5.</b> \(P\unicode{x21A7}y=\downarrow yʻʻ\overrightarrow{P}ʻy \quad\text{Df}\)</p> + +<p><span class="pagenum" id="Page_557">[Pg 557]</span></p> + +<p>Then \(P\unicode{x21A7}ʻʻ\text{ᗡ}ʻP\in\text{Cls ex}^{2}\,\text{excl}\). +Also \(sʻP\unicode{x21A7}ʻʻ\text{ᗡ}ʻP\) is the +class of all couples contained in \(P\), and +\(\dot{s}ʻsʻP\unicode{x21A7}ʻʻ\text{ᗡ}ʻP=P\). We shall now prove that +\(P_{\Delta}ʻ\alpha=\dot{s}ʻʻDʻʻ{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha\), +so that every member of \(P_{\Delta}ʻ\alpha\) can be derived from a +member of \({\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha\), and the problem +of the existence of \(P_{\Delta}ʻ\alpha\) is reduced to that of the +existence of selections from a class of mutually exclusive existent +classes.</p> + +<p class="nind"><b>*85·51.</b> \(\vdash .P_{\Delta}ʻ\iotaʻx=\downarrow xʻʻ\overrightarrow{P}ʻ x=P \unicode{x21A7} \quad[\text{*80·45.(*85·5)}]\)</p> + +<p class="nind"><b>*85·52.</b> \(\vdash .P_{\Delta}ʻʻ\iotaʻʻ\alpha =P\unicode{x21A7}ʻʻ\alpha \quad[\text{*37·35.*85·51}]\)</p> + +<p class="nind"><b>*85·53.</b> \(\vdash .P_{\Delta}ʻ\alpha =\dot{s}ʻʻ Dʻʻ{\in}_{\Delta}ʻ P\unicode{x21A7}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*84·241.*53·22}.&\supset\vdash.\iotaʻʻ\alpha\in\text{Cls}^{2}\,\text{excl}.sʻ\iotaʻʻ\alpha=\alpha.\\ +[\text{*85·27}] \supset\vdash.P_{\Delta}ʻ\alpha &= \dot{s}ʻʻDʻʻ{\in}_{\Delta}ʻP_{\Delta}ʻʻ\iotaʻʻ\alpha\\ +[\text{*85·52}] &= \dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·54.</b> \(\vdash .P_{\Delta}ʻ\alpha \text{ sm } {\in}_{\Delta}ʻ P\unicode{x21A7}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{ll} +\vdash. \text{*84·241.*53·22} .&\supset\vdash.\iotaʻʻ\alpha\in\text{Cls}^{2}\,\text{excl}.sʻ\iotaʻʻ\alpha=\alpha.\\ +[*85·43] &\supset\vdash. P_{\Delta}ʻ\alpha\,\text{ sm }\,{\in}_{\Delta}ʻP_{\Delta}ʻʻ\iotaʻʻ\alpha.\\ +[*85·52] &\supset\vdash. P_{\Delta}ʻ\alpha\,\text{ sm }\,{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha.\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p>The following proposition is frequently useful.</p> + +<p class="nind"><b><a id="*85·55">*85·55</a>.</b> \(\vdash . P_{\Delta}ʻ\alpha\,\text{ sm }\,\text{D}ʻʻ{\in}_{\Delta}ʻ P\unicode{x21A7}ʻʻ\alpha .P\unicode{x21A7}ʻʻ\alpha\in\text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*85·51.*80·14}.&\supset\vdash:R\in P\unicode{x21A7}x.\supset.\text{ᗡ}ʻR=\iotaʻx:R\in P\unicode{x21A7}y.\supset.\text{ᗡ}ʻR=\iotaʻy:\\ +[\text{*3·47}] &\supset\vdash:R\in P\unicode{x21A7}x\cap P\unicode{x21A7}y.\supset.\text{ᗡ}ʻR=\iotaʻx.\text{ᗡ}ʻR=\iotaʻy.\\ +[\text{*13·171.*51·23}]& \supset.x=y.\\ +[\text{*30·37}] &\supset.P\unicode{x21A7}x = P\unicode{x21A7}y:\\ +[\text{*10·11·23}] &\supset\vdash:\exists! P\unicode{x21A7}x\cap P\unicode{x21A7}y.\supset.P\unicode{x21A7}x=P\unicode{x21A7}y:\\ +[\text{*3·42.*11·11}] &\supset\vdash: x,y\in\alpha.\exists!P\unicode{x21A7}x\cap P\unicode{x21A7}y.\supset_{x,y}.P\unicode{x21A7}x=P\unicode{x21A7}y:\\ +[\text{*37·63}] &\supset\vdash: \lambda,\mu \in P\unicode{x21A7}ʻʻ\alpha. \exists!\lambda\cap\mu .\supset_{\lambda,\mu}.\lambda=\mu:\\ +[\text{*84·11}] &\supset\vdash.P\unicode{x21A7}ʻʻ\alpha\in\text{Cls}^{2}\,\text{excl}. &\qquad \text{(1)}\\ +[\text{*84·41}] &\supset\vdash.\text{D}ʻʻ{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha\,\text{ sm }\,{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha.\\ +[\text{*85·54}] &\supset\vdash. P_{\Delta}ʻ\alpha\,\text{ sm }\,\text{D}ʻʻ{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·56.</b> \(\vdash :P\upharpoonright\alpha\in\text{Cls}\rightarrow 1.\supset .{\in}_{\Delta}ʻ\overrightarrow{P}ʻʻ\alpha\,\text{ sm }\,{\in}_{\Delta}ʻ + P\unicode{x21A7}ʻʻ\alpha \quad[\text{*85·14·54}]\)</p> + +<p class="nind"><b>*85·6.</b> \(\vdash . {\in}_{\Delta}ʻʻ\iotaʻʻ\kappa = \hat{\mu}\{(\exists\beta).\beta\in\kappa .\mu = \downarrow \betaʻʻ\beta\}={\in}\unicode{x21A7}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·67}.\supset\vdash.{\in}_{\Delta}ʻʻ\iotaʻʻ\kappa&= \hat{\mu}\{(\exists \beta).\beta\in\kappa.\mu = {\in}_{\Delta}ʻ\iotaʻ\beta\}\\ +[\text{*83·4}] &= \hat{\mu}\{(\exists \beta).\beta\in\kappa.\mu = \downarrow\betaʻʻ\beta\} &\qquad \text{(1)}\\ +\vdash.\text{(1).*85·52}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is frequently employed.</p> + +<p><span class="pagenum" id="Page_558">[Pg 558]</span></p> + +<p class="nind"><b>*85·601.</b> + \(\vdash .\in \unicode{x21A7} \alpha =\downarrow \alphaʻʻ \alpha .\in \unicode{x21A7} \alpha \text{ sm }\alpha .\in \unicode{x21A7}ʻʻ \kappa \text{ sm }\kappa .\in \unicode{x21A7} \in 1\rightarrow 1.\text{E}!\in \unicode{x21A7}ʻ\alpha \)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*85·51.*62·2}. &\supset \vdash .\in \unicode{x21A7} \alpha =\downarrow \alphaʻʻ \alpha &\qquad \text{(1)}\\ +[\text{*73·611}] &\supset \vdash .\in \unicode{x21A7} \alpha\,\text{ sm }\,\alpha &\qquad \text{(2)}\\ +\vdash .\text{*38·12}.&\supset \vdash .\text{E}!\in \unicode{x21A7}ʻ \alpha &\qquad \text{(3)}\\ +[\text{*71·166}] &\supset \vdash .\in \unicode{x21A7} \in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ +\vdash .\text{(2).*73·47}. \supset \vdash :\alpha =\Lambda .\in \unicode{x21A7} \alpha =\in \unicode{x21A7} \beta .&\supset .\in \unicode{x21A7} \beta =\Lambda .\\ +[\text{*73·47.(2)}] &\supset .\beta =\Lambda &\qquad \text{(5)}\\ +\vdash .\text{(1).*38·131}. &\supset \vdash :x\in \alpha .\in \unicode{x21A7} \alpha =\in \unicode{x21A7} \beta .\supset .x\downarrow \alpha \in \downarrow \beta ʻʻ \beta .\\ +[\text{*38·131}] & \supset .(\exists y).x\downarrow \alpha =y\downarrow \beta .\\ +[\text{*55·202}] & \supset .\alpha =\beta &\qquad \text{(6)}\\ +\vdash .\text{(6).*10·11·23·35}.&\supset \vdash :\exists !\alpha .\in \unicode{x21A7} \alpha =\in \unicode{x21A7} \beta .\supset .\alpha =\beta &\qquad \text{(7)}\\ +\vdash .\text{(5).(7)}.& \supset \vdash :\in \unicode{x21A7} \alpha =\in \unicode{x21A7} \beta .\supset .\alpha =\beta &\qquad \text{(8)}\\ +\vdash .\text{(4).(8).*71·54}.&\supset \vdash .\in \unicode{x21A7} \in 1\rightarrow 1 &\qquad \text{(9)}\\ +\vdash .\text{(9).(3).*73·26}. & \supset \vdash .\in \unicode{x21A7}ʻʻ \kappa\,\text{ sm }\,\kappa &\qquad \text{(10)}\\ +\vdash .\text{(1).(2).(3).(9).(10)}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·61.</b> \(\vdash .\in \unicode{x21A7}ʻʻ \kappa \in \text{Cls}^{2}\text{excl}.{\in}_{\Delta}ʻ \kappa =\dot{s}ʻʻ \text{D}ʻʻ {\in}_{\Delta}ʻ + \in \unicode{x21A7} ʻʻ \kappa .{\in}_{\Delta}ʻ \kappa \text{ sm }{\in}_{\Delta}ʻ \in \unicode{x21A7}ʻʻ \kappa +\quad\left[\text{*85·53·54·55}\,\frac{{\in}}{P}\right]\)</p> + +<p class="nind"><b>*85·62.</b> \(\vdash :\exists !{\in}_{\Delta}ʻ \kappa .\equiv .\exists !{\in}_{\Delta}ʻ \in \unicode{x21A7} ʻʻ \kappa \quad[\text{*85·61.*73·36}]\)</p> + +<p class="nind"><b><a id="*85·63">*85·63</a>.</b> \(\vdash :\in \unicode{x21A7}ʻʻ \text{Cl ex}ʻ \alpha \in \text{Cls ex}^{2}\text{excl}:\exists !{\in}_{\Delta}ʻ + \text{Cl}\,\text{ex}ʻ \alpha .\equiv .\exists !{\in}_{\Delta}ʻ \in \unicode{x21A7}ʻʻ \text{Cl ex}ʻ \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*85·6.*60·21}.\supset\\ +&\vdash :\lambda \in {\in} \unicode{x21A7}ʻʻ \text{Cl ex}ʻ \alpha .\equiv .(\exists \beta ).\beta \subset\alpha .\exists !\beta .\lambda =\downarrow \betaʻʻ \beta &\qquad \text{(1)}\\ +\vdash .\text{*73·611·36}.&\supset \vdash :\exists !\beta .\lambda =\downarrow \beta ʻʻ\beta .\supset .\exists !\lambda :\\ +[\text{*3·42}] &\supset \vdash :\beta \subset\alpha .\exists !\beta .\lambda =\downarrow \betaʻʻ \beta .\supset .\exists !\lambda :\\ +[\text{*10·11·23}] &\supset \vdash :(\exists \beta ).\beta \subset\alpha .\exists !\beta .\lambda =\downarrow \beta ʻʻ \beta .\supset .\exists !\lambda &\qquad \text{(2)}\\ +\vdash .\text{(1).(2)}. &\supset \vdash :\lambda \in (\in \unicode{x21A7}ʻʻ \text{Cl ex}ʻ \alpha ).\supset .\exists !\lambda :\\ +[\text{*10·11.*24·63}]&\supset \vdash .\Lambda \sim \in ({\in} \unicode{x21A7}ʻʻ \text{Cl ex}ʻ \alpha ) &\qquad \text{(3)}\\ +\vdash .\text{(3).*85·61.*84·13}.&\supset \vdash .{\in} \unicode{x21A7}ʻʻ\text{Cl ex}ʻ \alpha \in \text{Cls}\,\text{ex}^{2}\text{excl} &\qquad \text{(4)}\\ +\vdash .\text{(4).*85·62}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><i>Note.</i> (\(\alpha).\exists !{\in}_{\Delta}ʻ \text{Cl}\,\text{ex}ʻ\alpha\) +is "Zermelo's axiom." The above proposition shows that this is +true if +\[ +\kappa \in \text{Cls}\,\text{ex}^{2}\,\text{excl}.\supset _{\kappa }.\exists !{\in}_{\Delta}ʻ \kappa, +\]<span class="pagenum" id="Page_559">[Pg 559]</span> +which again is true if +\[ +\kappa \in \text{Cls}\,\text{ex}^{2}\,\text{excl}.\supset:(\exists \mu ):\alpha \in \kappa .\supset_{\alpha}.\mu \cap \alpha \in 1 +\] +in virtue of <a href="#*84·412">*84·412</a>. The last of these is the "multiplicative axiom," +which is thus shown to imply "Zermelo's axiom."</p> + +<p>The following propositions lead up to <a href="#*85·72">*85·72</a>, which is used in the +theory of double similarity (*111·3).</p> + +<p class="nind"><b>*85·7.</b> \[\begin{align}\vdash \colon\ldotp \beta \in \lambda .\supset _{\beta }.Rʻ \beta \subset \beta :M\in\, &{\in}_{\Delta}ʻ Rʻʻ \lambda :\supset.\\ +&M\mid R\upharpoonright \lambda \in {\in}_{\Delta}ʻ \lambda .\text{D}ʻ (M\mid R\upharpoonright \lambda )=\text{D}ʻ M\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*14·21}.\supset \vdash \colon\ldotp \text{Hp}.&\supset :\beta \in \lambda .\supset _{\beta }.\text{E}!Rʻ \beta :\\ +[\text{*74·11}] &\supset :R\upharpoonright \lambda \in 1\rightarrow \text{Cls}.\lambda \subset \text{ᗡ}ʻ R &&\qquad \text{(1)}\\ +[\text{*80·14.*71·25}] &\supset :M\mid R\upharpoonright \lambda \in 1\rightarrow \text{Cls} &&\qquad \text{(2)}\\ +\vdash .\text{(1).*71·7.*35·7}.&\supset \vdash \colon\ldotp \text{Hp}.\supset :x(M\mid R\upharpoonright \lambda )\beta .\supset .\beta \in \lambda .xM(Rʻ \beta ).\\ +[\text{*80·14}.\,\text{Hp}] &\supset .\beta \in \lambda .x\in Rʻ \beta .\\ +[\text{Hp}] &\supset .x\in \beta &&\qquad \text{(3)}\\ +\vdash .\text{*80·14.*74·44}.&\supset\\ +& \vdash :\text{Hp}.\supset .\text{D}ʻ (M\mid R\upharpoonright \lambda )=\text{D}ʻ M.\text{ᗡ}ʻ (M\mid R\upharpoonright \lambda )&=\lambda \cap \text{ᗡ}ʻ R\\ +[\text{(1)}] &&=\lambda &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4).*80·14}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*85·701.</b> \(\vdash \colon\ldotp \beta \in \lambda .\supset _{\beta }.Rʻ \beta \subset \beta :\supset .\text{D}ʻʻ {\in}_{\Delta}ʻ + Rʻʻ \lambda \subset \text{D}ʻʻ {\in}_{\Delta}ʻ \lambda \quad[\text{*85·7}]\)</p> + +<p class="nind"><b>*85·702.</b> \[\begin{align}&\vdash \colon\ldotp \beta \in \lambda .\supset _{\beta }.Rʻ\text{Cl}ʻ \beta \in\text{Cl}ʻ\beta :\supset .\text{D}ʻʻ {\in}_{\Delta}ʻ + Rʻʻ\text{Cl}ʻʻ \lambda \subset \text{D}ʻʻ {\in}_{\Delta}ʻ \lambda\\ +&\quad\left[\text{*85·701}\frac{R\mid \text{Cl}}{R}\right]\end{align}\]</p> + +<p class="nind"><b>*85·71.</b> \(\vdash :R\in {\in}_{\Delta}ʻ\text{Cl}ʻʻ\lambda .\supset .\text{D}ʻʻ {\in}_{\Delta}ʻ\text{D}ʻ R\subset \text{D}ʻʻ {\in}_{\Delta}ʻ + \lambda \quad[\text{*85·702.*83·2}]\)</p> + +<p>This proposition asserts that if we can select one sub-class out of +each member of \(\lambda\) (where \(\lambda\) is a class of classes), +then selections from the sub-classes so obtained are selections from +\(\lambda\).</p> + +<p class="nind"><b><a id="*85·72">*85·72</a>.</b> \[\begin{align}\vdash \colon\ldotp (Sʻʻ \beta )\upharpoonleft S\in 1\rightarrow 1:\beta \in \lambda .\supset _{\beta }.Rʻ \beta &\subset Sʻ \beta :\supset.\\ +&\text{D}ʻʻ {\in}_{\Delta}ʻ Rʻʻ \lambda \subset \text{D}ʻʻ {\in}_{\Delta}ʻ Sʻʻ \lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash .\text{*14·21.*33·43}.&\supset \vdash \colon\ldotp \text{Hp}.\supset :\beta \in \lambda .\supset .\beta \in \text{ᗡ}ʻ S &\qquad \text{(1)}\\ +\vdash .\text{*85·701}\, \frac{R\mid\breve{S},Sʻʻ \lambda}{R,\lambda} .\supset\\ +\vdash \colon\ldotp \gamma \in Sʻʻ \lambda .&\supset _{\gamma }.(R\mid \breve{S})ʻ \gamma \subset \gamma :\supset .\text{D}ʻʻ {\in}_{\Delta}ʻ + Rʻʻ \breve{S}ʻʻ Sʻʻ \lambda \subset \text{D}ʻʻ {\in}_{\Delta}ʻ Sʻʻ\lambda &\qquad \text{(2)}\\ +\vdash .\text{*37·63.*14·21}.\supset\\ +\vdash\colon\colon \text{Hp}.\supset \colon\ldotp \gamma \in Sʻʻ \lambda .\supset _{\gamma }.(R\mid + \breve{S})ʻ \gamma \subset \gamma :&\equiv :\beta \in \lambda .\supset _{\beta }.(R\mid\breve{S})ʻ Sʻ \beta \subset Sʻ \beta :\\ +[\text{*74·53.(1)}] &\equiv :\beta \in \lambda .\supset _{\beta }.Rʻ \beta \subset Sʻ \beta &\qquad \text{(3)}\\ +\vdash .\text{*74·171}.\supset \vdash: \text{Hp}.&\supset .\breve{S}ʻʻ Sʻʻ \lambda =\lambda &\qquad \text{(4)}\\ +\vdash .\text{(2).(3).(4)}.\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_560">[Pg 560]</span></p> + +<p>The following proposition is a lemma employed in the theory of double +similarity (*111·313).</p> + +<p class="nind"><b>*85·81.</b> + \[\begin{align}\vdash\colon\ldotp \lambda\in \text{Cls}^{2}\text{excl}:\beta\in\lambda.\supset_{\beta}.sʻ\text{ᗡ}ʻʻTʻ&\beta\subset\beta:R\in{\in}_{\Delta}ʻTʻʻ\lambda:\supset:\\ +&\beta\in\lambda.\supset_{\beta}.(\breve{s}ʻ\text{D}ʻR)\upharpoonright \beta=RʻTʻ\beta\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:\beta\in\lambda.\supset.\exists!Tʻ\beta: &\qquad \text{(1)}\\ +[\text{*83·2.*37·6}] &\supset:\beta\in\lambda.\supset.RʻTʻ\beta\in Tʻ\beta. &\qquad \text{(2)}\\ +[\text{*35·452.Hp}] &\supset.RʻTʻ\beta=(RʻTʻ\beta)\upharpoonright \beta &\qquad \text{(3)}\\ +\vdash.\text{(1).*83·22}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\beta\in\lambda.\supset.\exists!RʻTʻ\beta.\\ +[\text{*33·43.*41·13}] &\supset.RʻTʻ\beta\unicode{x2abd}\breve{s}ʻDʻR.\\ +[\text{*35·461}] &\supset.(RʻTʻ\beta)\upharpoonright \beta \upharpoonright (\breve{s}ʻDʻR)\upharpoonright \beta.\\ +[\text{(3)}] &\supset.RʻTʻ\beta\unicode{x2abd}(\breve{s}ʻ\text{D}ʻR)\upharpoonright \beta &\qquad \text{(4)}\\ +\vdash.\text{(1).*37·6.*83·23}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:DʻR=\breve{M}\{(\exists \gamma).\gamma\in\lambda.M=RʻTʻ\gamma\}:\\ +[\text{*41·11.*13·195}] &\supset:x(\breve{s}ʻ\text{D}ʻR)y.\equiv.(\exists \gamma).\gamma\in\lambda.x(RʻTʻ\gamma)y:\\ +[\text{*35·101}] &\supset:x{(\breve{s}ʻ\text{D}ʻR)\upharpoonright \beta}y.\equiv.(\exists \gamma).\gamma\in\lambda.x(RʻTʻ\gamma)y.y\in\beta &\qquad \text{(5)}\\ +\vdash.\text{(2).*33·14}.&\supset\vdash\colon\ldotp \text{Hp}.\gamma\in\lambda.\supset:x(RʻTʻ\gamma)y.\supset.y\in\text{ᗡ}ʻRʻTʻ\gamma.RʻTʻ\gamma\in Tʻ\gamma.\\ +[\text{*40·4}] &\supset.y\in sʻ\text{ᗡ}ʻʻTʻ\gamma.\\ +[\text{Hp}] &\supset.y\in\gamma &\qquad \text{(6)}\\ +\vdash.\text{(5).(6)}. &\supset\vdash\colon\ldotp \text{Hp}.\supset\colon\ldotp\beta\in\lambda.\supset:\\ +&x{(\breve{s}ʻ\text{D}ʻR)\upharpoonright \beta}y.\equiv.(\exists \gamma).\beta,\gamma\in\lambda.x(RʻTʻ\gamma)y.y\in\beta.y\in\gamma.\\ +[\text{*84·11.Hp}] &\supset.(\exists \gamma).\beta ,\gamma\in\lambda.x(RʻTʻ\gamma)y.\beta=\gamma.\\ +[\text{*13·195}] &\supset.x(RʻTʻ\beta)y &\qquad \text{(7)}\\ +\vdash.\text{(4).(7)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_63" href="#FNanchor_63" class="label">[63]</a> +See <i>Math. Annalen</i>, Vol. <span class="allsmcap">LIX</span>.</p> + +</div> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_64" href="#FNanchor_64" class="label">[64]</a> +<i>loc. cit.</i></p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_561">[Pg 561]</span></p> +<h2 class="nobreak" id="*88">*88. CONDITIONS FOR THE EXISTENCE OF SELECTIONS.</h2> +</div> + + +<p><i>Summary of</i> *88.</p> + + +<p>The existence of selections cannot, so far as is known at present, be +proved in general. That is, we cannot prove any of the following: +\[ +\begin{align} +&(P,\kappa):\kappa\subset\text{ᗡ}ʻP.\supset.\exists!P_{\Delta}ʻ\kappa\\ +&(P,\kappa):P\in\text{Cls}\rightarrow 1.\kappa\subset\text{ᗡ}ʻP.\supset.\exists!P_{\Delta}ʻ\kappa\\ +&(P).\exists!P_{\Delta}ʻ\text{ᗡ}ʻ\\ +&(\kappa):\Lambda{\sim}\in\kappa.\supset.\exists!{\in}_{\Delta}ʻ\kappa\\ +&(\kappa):\kappa\in \text{Cls ex}^{2}\text{excl}.\supset.\exists!{\in}_{\Delta}ʻ\kappa\\ +&(\alpha).\exists!{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha\\ +&(\kappa)\colon\ldotp \kappa\in \text{Cls ex}^{2}\text{excl}.\supset:(\exists \mu):\alpha\in\kappa.\supset_\alpha.\mu\cap\alpha\in 1 +\end{align} +\]</p> + +<p>These various propositions can be shown to be all equivalent <i>inter +se</i>; and in virtue of Zermelo's theorem (cf. *258), they are +equivalent to the proposition "every class can be well-ordered." In +the present number we have to prove the above equivalences, as well +as certain propositions giving the existence of selections in various +particular cases.</p> + +<p>The most apparently obvious of the above propositions is the last, +namely: "If \(\kappa\) is a class of mutually exclusive classes, no +one of which is null, there is at least one class \(\mu\) which takes +one and only one member from each member of \(\kappa\)." This we shall +define as the "multiplicative axiom."</p> + +<p>We will call \(P\) a <i>multipliable</i> relation (denoted by "Rel +Mult") if \(P_{\Delta}ʻ\text{ᗡ}ʻP\) exists, or, what is equivalent, if +\(\kappa\subset\text{ᗡ}ʻP.\supset_\kappa.\exists!P_{\Delta}ʻ\kappa\) +Thus we put +\[ +\text{Rel Mult}=\breve{P}\{\exists!P_{\Delta}ʻ\text{ᗡ}ʻP\} \quad\text{Df}. +\]</p> + +<p>We will call \(\kappa\) a <i>multipliable</i> class of classes if +\({\in}_{\Delta}ʻ\kappa\) exists, <i>i.e.</i> we put +\[ +\text{Cls}^{2} \text{Mult}=\breve{\kappa}\{\exists!{\in}_{\Delta}ʻ\kappa\} \quad\text{Df}. +\]</p> + +<p>The multiplicative axiom will be denoted by "\(\text{Mult ax}\)." Thus we put +\[ +\text{Mult ax}.=\colon\ldotp \kappa\in\text{Cls ex}^{2} \text{excl}.\supset_\kappa:(\exists \mu):\alpha\in \kappa.\supset_\alpha.\mu\cap\alpha\in 1 \quad\text{Df}. +\]</p> + +<p>In the present number, we shall first give various equivalent forms +of the assumption that \(P\) is a multipliable relation (<a href="#*88·1">*88·1</a>—<a href="#*88·15">·15</a>); +we shall then do the same for multipliable classes of classes +(<a href="#*88·2">*88·2</a>—<a href="#*88·26">·26</a>); next we shall give various equivalent forms of the +multiplicative axiom (<a href="#*88·3">*88·3</a>—<a href="#*88·39">·39</a>).<span class="pagenum" id="Page_562">[Pg 562]</span> (Some important equivalent forms +cannot be given at this stage, as they depend upon definitions not +yet given, such as the definitions of cardinal multiplication and +of well-ordered series. Cf. *114·26 and *258·37.) Finally we shall +give propositions showing that various special classes of classes +are multipliable. Most of these propositions will not be used in +the sequel, but they illustrate the nature of the difficulties +involved in proving that a class of classes is multipliable, and +some of them show that mere size does not prevent a class from +being multipliable. For example, <a href="#*88·48">*88·48</a> shows that, given any class +of classes \(\kappa\), if each member \(\alpha\) is replaced by +\(\iotaʻʻ\alpha\cup\iotaʻ\alpha\), the result is a multipliable class +of classes; but the only effect of this change is to increase the +number of members of each member of our class of classes by one.</p> + +<p>The chief propositions in this number which are afterwards referred to +are the following:</p> + +<p class="nind"><b>*88·22.</b> \(\vdash :\kappa\in \text{Cls}^{2}\,\text{Mult}.\lambda \subset \kappa.\supset .\lambda \in \text{Cls}^{2}\,\text{Mult}\)</p> + +<p class="nind"><b>*88·32.</b> \(\vdash \colon\ldotp \text{Mult}\,\text{ax}.\equiv :\kappa\in \text{Cls}\,\text{ex}^{2}\,\text{excl}.\supset_{\kappa}.\exists !{\in}_{\Delta}ʻ \kappa\)</p> + +<p class="nind"><b>*88·33.</b> \(\vdash :\text{Mult}\,\text{ax}.\equiv .(\alpha ).\exists !{\in}_{\Delta}ʻ \text{Cl}\,\text{ex}ʻ \alpha\)</p> + +<p class="nind"><b>*88·361.</b> \(\vdash \colon\ldotp \text{Mult}\,\text{ax}.\equiv :\kappa\subset \text{ᗡ}ʻ R.\equiv_{R,\kappa}.\exists !R_{\Delta}ʻ \kappa\)</p> + +<p class="nind"><b>*88·37.</b> \(\vdash \colon\ldotp \text{Mult}\,\text{ax}.\equiv :\Lambda \in \kappa.\supset_{\kappa}.\exists !{\in}_{\Delta}ʻ \kappa\)</p> + +<p>The above is usually the most convenient form of the multiplicative +axiom.</p> + +<p class="nind"><b>*88·372.</b> \(\vdash \colon\ldotp \text{Mult}\,\text{ax}.\equiv:\Lambda \in \kappa.\equiv_{\kappa}.{\in}_{\Delta}ʻ\kappa=\Lambda \)</p> + +<p>This proposition is used in *114, to prove that the multiplicative +axiom is equivalent to the proposition that a cardinal product vanishes +when, and only when, one of its factors vanishes.</p> + +<hr class="tb"> + +<p class="nind"><b>*88·01.</b> \(\text{Rel}\,\text{Mult}=\hat{P}\{\exists!P_{\Delta}ʻ \text{ᗡ}ʻ P\} \quad\text{Df}\)</p> + +<p class="nind"><b>*88·02.</b> \(\text{Cls}^{2}\,\text{Mult}=\hat{\kappa} \{\exists!{\in}_{\Delta}ʻ \kappa\} \quad\text{Df}\)</p> + +<p class="nind"><b>*88·03.</b> \(\text{Mult ax}.=\colon\ldotp \kappa\in \text{Cls}\,\text{ex}^{2}\,\text{excl}.\supset_{\kappa}:(\exists \mu ):\alpha \in \kappa.\supset_{\alpha }.\mu + \cap \alpha \in 1 \quad\text{Df}\)</p> + +<p class="nind"><b><a id="*88·1">*88·1</a>.</b> \(\vdash:P\in \text{Rel}\,\text{Mult}.\equiv .\exists !P_{\Delta}ʻ\text{ᗡ}ʻP \quad[\text{*20·3.(*88·01)}]\)</p> + +<p class="nind"><b>*88·11.</b> \(\vdash:P\in \text{Rel}\,\text{Mult}.\lambda \subset \text{ᗡ}ʻ P.\supset .\exists !P_{\Delta}ʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·6}. &\supset\vdash:R\in P_{\Delta}ʻ\text{ᗡ}ʻP.\lambda \subset \text{ᗡ}ʻ P.\supset .R\upharpoonright\lambda \in P_{\Delta}ʻ\lambda .\\ +[\text{*10·24}] &\supset.\exists!P_{\Delta}ʻ\lambda:\\ +[\text{*10·11·23·35}]&\supset \vdash :\exists !P_{\Delta}ʻ\text{ᗡ}ʻP.\lambda \subset \text{ᗡ}ʻ P.\supset .\exists !P_{\Delta}ʻ \lambda \qquad\qquad \text{(1)}\\ +\vdash.\text{(1).*88·1}.&\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_563">[Pg 563]</span></p> + +<p class="nind"><b>*88·12</b> \(\vdash\colon\ldotp P\in \text{Rel Mult}\ldotp \equiv :\lambda \subset \text{ᗡ}ʻP\ldotp \supset _{\lambda }\ldotp \exists !P_{\Delta }ʻ\lambda \)</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*88·11.Exp.*10·11·21}. \supset\\ +\vdash\colon\ldotp P\in \text{Rel Mult}\ldotp \supset :\lambda \subset \text{ᗡ}ʻP\ldotp \supset _{\lambda }\ldotp \exists !P_{\Delta }ʻ\lambda &\qquad \text{(1)}\\ +\vdash. \text{*10·1.*22·42}. \supset\\ +\vdash\colon\ldotp\lambda \subset \text{ᗡ}ʻP\ldotp \supset _{\lambda }.\supset. \exists !P_{\Delta }ʻ\lambda :\supset \ldotp \exists !P_{\Delta }ʻ\text{ᗡ}ʻP\ldotp\\ +[\text{*88·1}] \supset \ldotp P\in \text{Rel Mult} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. \supset \vdash\ldotp \text{Prop} +\vdash.\text{(1).*88·2}. \supset \vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*88·13</b> \(\vdash:P\in \text{Rel Mult}\ldotp \equiv \ldotp \exists !{\in}_{\Delta }ʻP\unicode{x21A7} ʻʻ\text{ᗡ}ʻP\quad [\text{*85·54.*73·36.*88·1}]\)</p> + +<p class="nind"><b>*88·14</b> \(\vdash\colon\ldotp \kappa \subset \text{ᗡ}ʻP\ldotp \supset :P\upharpoonright \kappa \in \text{Rel Mult}\ldotp \equiv \ldotp \exists !P_{\Delta }ʻ\kappa \)</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*80·23}. &\supset \vdash:\exists !P_{\Delta }ʻ\kappa \ldotp \equiv \ldotp \exists !(P\upharpoonright \kappa )_{\Delta }ʻ\kappa &\qquad \text{(1)}\\ +\vdash. \text{*35·65}. &\supset \vdash:\kappa \subset \text{ᗡ}ʻP\ldotp \supset \ldotp \text{ᗡ}ʻ(P\upharpoonright \kappa )=\kappa &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. \supset \vdash\colon\ldotp \text{Hp}\ldotp \supset :\exists !P_{\Delta }ʻ\kappa \ldotp &\equiv \ldotp \exists !(P\upharpoonright \kappa )_{\Delta }ʻ\text{ᗡ}ʻ(P\upharpoonright + \kappa )\ldotp\\ +[\text{*88·1}] &\equiv \ldotp P\upharpoonright \kappa \in \text{Rel Mult}\colon\ldotp\supset \vdash\ldotp \text{Prop} +\vdash.\text{(1).*88·2}. \supset \vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*88·15">*88·15</a>.</b> \(\vdash\colon\ldotp \text{ᗡ}ʻP=\text{V}\ldotp \supset :P\upharpoonright \kappa \in \text{Rel Mult}\ldotp \equiv \ldotp \exists !P_{\Delta }ʻ\kappa + \quad [\text{*88·14.*24·11}]\)</p> + +<p class="nind"><b><a id="*88·2">*88·2</a>.</b> \(\vdash:\kappa \in \text{Cls}^{2}\text{Mult}\ldotp \equiv \ldotp \exists !{\in}_{\Delta }ʻ\kappa \quad [\text{*20·3.(*88·02)}]\)</p> + +<p class="nind"><b>*88·21.</b> \(\vdash:P\in\text{Rel Mult}\ldotp \equiv \ldotp P\unicode{x21A7} ʻʻ\text{ᗡ}ʻP\in \text{Cls}^{2}\text{Mult} \quad [\text{*88·13·2}]\)</p> + +<p class="nind"><b>*88·22.</b> \(\vdash:\kappa \in \text{Cls}^{2}\text{Mult}\ldotp \lambda \subset \kappa \ldotp \supset \ldotp \lambda \in\text{Cls}^{2}\text{Mult}\)</p> + +<p><i>Dem</i>. +\[ +\begin{array}{l} +\vdash. \text{*80·6}. &\supset \vdash:R\in {\in}_{\Delta }ʻ\kappa \ldotp \lambda \subset \kappa \ldotp \supset \ldotp R\upharpoonright \lambda \in {\in}_{\Delta }ʻ\lambda + \ldotp\\ +[\text{*10·24}] &\supset \ldotp \exists !{\in}_{\Delta }ʻ\lambda :\\ +[\text{*10·11·23·35}]&\supset \vdash:\exists !{\in}_{\Delta }ʻ\kappa \ldotp \lambda \subset \kappa \ldotp \supset \ldotp \exists !{\in}_{\Delta }ʻ\lambda + &\qquad \text{(1)}\\ +\vdash.\text{(1).*88·2}. \supset \vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*88·23</b> \(\vdash:\kappa \in \text{Cls}^{2}\text{Mult}\ldotp \supset \ldotp \text{Cl}ʻ\kappa \subset \text{Cls}^{2}\text{Mult} \quad [\text{*88·22.*60·2}]\)</p> + +<p class="nind"><b>*88·24</b> \(\vdash\colon\ldotp P\in \text{Cls}\rightarrow 1\ldotp \supset :P\in \text{Rel Mult}\ldotp \equiv \ldotp \overrightarrow{P}ʻʻ\text{ᗡ}ʻP\in \text{Cls}^{2}\text{Mult}\)</p> + +<p><i>Dem</i>. +\[ +\begin{aligned} +&\vdash. \text{*85·14.*73·36}. \supset \vdash\colon\ldotp\text{Hp}\ldotp \supset :\exists !P_{\Delta }ʻ\text{ᗡ}ʻP\ldotp \equiv \ldotp \exists !{\in}_{\Delta }ʻ\overrightarrow{P}ʻʻ\text{ᗡ}ʻP + \quad (1)\\ +&\vdash. \text{(1).*88·1·2}.\supset \vdash\ldotp \text{Prop} +\end{aligned} +\]</p> + +<p class="nind"><b>*88·25</b> + \(\vdash\colon\ldotp P\upharpoonright \kappa \in \text{Cls}\rightarrow 1\ldotp \kappa \subset \text{ᗡ}ʻP\ldotp \supset :P\upharpoonright \kappa \in \text{Rel Mult}\ldotp \equiv \ldotp \overrightarrow{P}ʻʻ\kappa \in \text{Cls}^{2}\text{Mult}\)</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*85·14.*73·36}. \supset\\ +\vdash\colon\ldotp\text{Hp}\ldotp \quad &\supset :\exists !P_{\Delta }ʻ\kappa \ldotp \equiv \ldotp \exists !{\in}_{\Delta }ʻ\overrightarrow{P}ʻʻ\kappa :\\ +[\text{*88·14·2}]&\supset :P\upharpoonright \kappa \in \text{Rel Mult}\ldotp \equiv \ldotp \overrightarrow{P}ʻʻ\kappa \in \text{Cls}^{2}\text{Mult}\colon\ldotp\supset + \vdash\ldotp \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_564">[Pg 564]</span></p> + +<p class="nind"><b><a id="*88·26">*88·26</a>.</b> \(\vdash\colon\colon \kappa\in \text{Cls}^{2}\,\text{excl}.\supset\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{Mult}.\equiv:(\exists + \mu):\alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*88·2.*37·45}.\supset\vdash:\kappa\in \text{Cls}^{2}\,\text{Mult}.\equiv.\exists !\text{D}ʻʻ{\in}_{\Delta}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·412}.\supset\\ +\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{Mult}.\equiv:(\exists \mu):\alpha\in \kappa.\supset_{\alpha}.\mu\cap + \alpha\in 1:\mu\subset sʻ\kappa: &\qquad \text{(2)}\\ +[\text{*10·5}] \supset:(\exists \mu):\alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1 &\qquad \text{(3)}\\ +\vdash.\text{*40·13.*22·621}.\supset\vdash:\alpha\in \kappa.\supset_{\alpha}.sʻ\kappa\cap \alpha=\alpha.\\ +[\text{*22·481}] \supset_{\alpha}.\mu\cap sʻ\kappa\cap \alpha=\mu\cap \alpha:\\ +[\text{*2·77.*10·27}] \supset\vdash\colon\ldotp \alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\supset:\alpha\in \kappa.\supset_{\alpha}.\mu\cap + sʻ\kappa\cap \alpha\in 1 &\qquad \text{(4)}\\ +\vdash.\text{(4).*22·43}.\supset\vdash\colon\ldotp \alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\supset:\\ +\alpha\in \kappa.\supset_{\alpha}.\mu\cap sʻ\kappa\cap \alpha\in 1:\mu\cap sʻ\kappa\subset sʻ\kappa:\\ +[\text{*10·24}] \supset:(\exists \nu):\alpha\in \kappa.\supset_{\alpha}.\nu \cap \alpha\in 1:\nu\subset sʻ\kappa &\qquad \text{(5)}\\ +\vdash.\text{(5).*10·11·23}.\supset\\ +\vdash\colon\ldotp (\exists \mu):\alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\supset:(\exists \nu):\alpha\in \kappa.\supset_{\alpha}.\nu\cap + \alpha\in 1:\nu\subset sʻ\kappa &\qquad \text{(6)}\\ +\vdash.\text{(6).(2)}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp (\exists \mu):\alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1:\supset.\kappa\in \text{Cls}^{2}\,\text{Mult} + &\qquad \text{(7)}\\ +\vdash.\text{(3).(7)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*88·3">*88·3</a>.</b> \[\begin{align}&\vdash\colon\colon \text{Mult ax}.\equiv\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{excl}.\supset_{\kappa}:(\exists + \mu):\alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1\\ +&[\text{*4·2.(*88·03)}]\end{align}\]</p> + +<p class="nind"><b>*88·31.</b> \(\vdash:\text{Mult ax}.\equiv.\text{Cls}^{2}\,\text{excl}\subset\text{Cls}^{2}\,\text{Mult}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*88·26.*5·74}.\supset\vdash\colon\colon &\kappa\in \text{Cls}^{2}\,\text{excl}.\supset_{\kappa}.\kappa\in \text{Cls}^{2}\,\text{Mult}:\equiv\colon\ldotp \\ +&\kappa\in \text{Cls}^{2}\,\text{excl}.\supset_{\kappa}:(\exists \mu):\alpha\in \kappa.\supset_{\alpha}.\mu\cap \alpha\in 1\colon\ldotp \\ +[\text{*88·3}] &\equiv\colon\ldotp \text{Mult ax}\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*88·32.</b> \(\vdash\colon\ldotp \text{Mult ax}.\equiv:\kappa\in \text{Cls}^{2}\,\text{excl}.\supset_{\kappa}.\exists !{\in}_{\Delta}ʻ\kappa \quad[\text{*88·31·2}]\)</p> + +<p class="nind"><b>*88·33.</b> \(\vdash:\text{Mult ax}.\equiv.(\alpha).\exists !{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha\)</p> + +<p>Note that (\(\alpha).\exists !{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha\) is Zermelo's axiom.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*88·32.*85·63}.&\supset\vdash:\text{Mult ax}.\supset.\exists !{\in}_{\Delta}ʻ\in \unicode{x21A7}ʻʻ\text{Cl ex}ʻ\alpha.\\ +[\text{*85·63}] &\supset.\exists !{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*60·57}. &\supset\vdash.\kappa\subset\text{Cl}ʻsʻ\kappa.\\ +[\text{*60·24}] &\supset\vdash.\kappa-\iotaʻ\Lambda\subset\text{Cl ex}ʻsʻ\kappa.\\ +[\text{*84·13}] &\supset\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\supset.\kappa\subset\text{Cl ex}ʻsʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(2).*80·6}.&\supset\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.R\in {\in}_{\Delta}ʻ\text{Cl ex}ʻsʻ\kappa.\supset.R\upharpoonright \kappa\in \in _{\Delta}ʻ\kappa + &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·28·35}.&\supset\vdash:\kappa\in \text{Cls}^{2}\,\text{excl}.\exists !{\in}_{\Delta}ʻ\text{Cl ex}ʻsʻ\kappa.\supset_{\kappa}.\exists + !\in _{\Delta}ʻ\kappa:\\ +[\text{*10·1}]\supset\vdash\colon\ldotp (\alpha).\exists !{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha.&\supset:\kappa\in \text{Cls}^{2}\,\text{excl}.\supset_{\kappa}.\exists + !{\in}_{\Delta}ʻ\kappa:\\ +[\text{*88·32}] &\supset:\text{Mult ax} &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_565">[Pg 565]</span></p> + +<p class="nind"><b>*88·34.</b> \(\vdash:\text{Mult ax}.\equiv.\text{Cls}\rightarrow 1\subset \text{Rel Mult}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*84·5.*88·32}.&\supset\vdash\colon\ldotp \text{Mult ax}.\supset:R\in \text{Cls}\rightarrow 1.\supset.\exists !{\in}_{\Delta}ʻ\overrightarrow{R}ʻʻ\text{ᗡ}ʻR.\\ +[\text{*85·14.*73·36}] &\supset.\exists !R_{\Delta}ʻ\text{ᗡ}ʻR.\\ +[\text{*88·1}] &\supset.R\in \text{Rel Mult} &\qquad \text{(1)}\\ +\vdash.\text{*84·14}.&\supset\vdash\colon\ldotp \text{Cls}\rightarrow 1\subset \text{Rel Mult}.\supset:\\ +& \kappa\in \text{Cls}^{2}\,\text{excl}.\supset.\in \upharpoonright \kappa\in \text{Rel Mult}.\\ +[\text{*88·1}] &\supset.\exists !(\in \upharpoonright \kappa)_{\Delta}ʻ\text{ᗡ}ʻ\in \upharpoonright \kappa.\\ +[\text{*84·13.*62·42}] &\supset.\exists !(\in \upharpoonright \kappa)_{\Delta}ʻ\kappa.\\ +[\text{*80·23}] &\supset.\exists !{\in}_{\Delta}ʻ\kappa &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·21.*88·32}.&\supset\vdash:\text{Cls}\rightarrow 1\subset \text{Rel Mult}.\supset.\text{Mult ax} &&\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*88·35.</b> \(\vdash:\text{Mult ax}.\equiv.(R).R\in \text{Rel Mult}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·45.*55·121.(*85·5)}.&\supset\vdash:\exists !P\unicode{x21A7}x.\equiv.\exists !\overrightarrow{P}ʻx.\\ +[\text{*33·41}] &\equiv.x\in \text{ᗡ}ʻP &\qquad \text{(1)}\\ +\vdash.\text{(1)*10·11.*37·63}.&\supset\vdash:\alpha\in P\unicode{x21A7}ʻʻ\text{ᗡ}ʻP.\supset_{\alpha}.\exists !\alpha:\\ +[\text{*24·63}] &\supset\vdash.\Lambda{\sim}\in P\unicode{x21A7}ʻʻ\text{ᗡ}ʻP &\qquad \text{(2)}\\ +\vdash.\text{(2).*84·13.*85·55}.&\supset\vdash.P\unicode{x21A7}ʻʻ\text{ᗡ}ʻP\in \text{Cls}^{2}\,\text{excl}.\\ +[\text{*88·32}] &\supset\vdash:\text{Mult ax}.\supset.\exists !{\in}_{\Delta}ʻP\unicode{x21A7}ʻʻ\text{ᗡ}ʻP.\\ +[\text{*85·54.*73·36}] &\supset.\exists !P_{\Delta}ʻ\text{ᗡ}ʻP.\\ +[\text{*88·1}] &\supset.P\in \text{Rel Mult} &\qquad \text{(3)}\\ +\vdash.\text{*10·1.*88·1}.&\supset\vdash:(R).R\in \text{Rel Mult}.\supset.\exists !(\in \upharpoonright \text{Cl ex}ʻ\alpha)_{\Delta}ʻ\text{ᗡ}ʻ(\in \upharpoonright \text{Cl ex}ʻ\alpha).\\ +[\text{*62·42}] &\supset.\exists !(\in \upharpoonright \text{Cl ex}ʻ\alpha)_{\Delta}ʻ\text{Cl ex}ʻ\alpha.\\ +[\text{*80·23}] &\supset.\exists !{\in}_{\Delta}ʻ\text{Cl ex}ʻ\alpha &\qquad \text{(4)}\\ +\vdash.\text{(4).*10·11·21.*88·33}.&\supset\vdash:(R).R\in \text{Rel Mult}.\supset.\text{Mult ax} &\qquad \text{(5)}\\ +\vdash.\text{(3).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*88·36">*88·36</a>.</b> \(\vdash\colon\ldotp \text{Mult ax}.\equiv:\kappa\subset \text{ᗡ}ʻR.\supset_{R,\kappa}.\exists !R_{\Delta}ʻ\kappa \quad[\text{*88·35·12}]\)</p> + +<p class="nind"><b>*88·361.</b> \(\vdash\colon\ldotp \text{Mult ax}.\equiv:\kappa\subset \text{ᗡ}ʻR.\equiv_{R,\kappa}.\exists !R_{\Delta}ʻ\kappa \quad[\text{*88·36.*80·2}]\)</p> + +<p class="nind"><b>*88·37.</b> \(\vdash\colon\ldotp \text{Mult ax}.\equiv:\Lambda{\sim}\in \kappa.\supset_{\kappa}.\exists !{\in}_{\Delta}ʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*88·36.*62·231}.&\supset\vdash\colon\ldotp \text{Mult ax}.\supset:\Lambda{\sim}\in \kappa.\supset_{\kappa}.\exists !{\in}_{\Delta}ʻ\kappa &\qquad \text{(1)}\\ +\vdash.\text{*84·13.*88·32}.&\supset\vdash\colon\ldotp \Lambda{\sim}\in \kappa.\supset_{\kappa}.\exists !{\in}_{\Delta}ʻ\kappa:\supset.\text{Mult ax} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_566">[Pg 566]</span></p> + +<p class="nind"><b>*88·371.</b> \(\vdash\colon\ldotp \text{Mult ax}.\equiv:\Lambda{\sim}\in \kappa.\equiv_{\kappa}.\exists !{\in}_{\Delta}ʻ\kappa \quad[\text{*88·37.*83·1}]\)</p> + +<p class="nind"><b>*88·372.</b> \(\vdash\colon\ldotp \text{Mult ax}.\equiv:\Lambda\in \kappa.\equiv_{\kappa}.{\in}_{\Delta}ʻ\kappa=\Lambda \quad[\text{*88·371.Transp}]\)</p> + +<p>This proposition shows that the multiplicative axiom is equivalent to +the assumption that a cardinal product is zero when, and only when, one +of its factors is zero.</p> + +<p class="nind"><b>*88·373.</b> \(\vdash:\text{Mult ax}.\equiv.\text{Cl}ʻ(\text{Cls}-\iotaʻ\Lambda)\subset \text{Cls}^{2}\,\text{Mult}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·63.*53·5}.\supset\vdash\colon\ldotp \Lambda{\sim}\in \kappa.&\equiv:\alpha\in \kappa.\supset_{\alpha}.\alpha\in \text{Cls}-\iotaʻ\Lambda:\\ +[\text{*22·1}] &\equiv:\kappa\subset \text{Cls}-\iotaʻ\Lambda:\\ +[\text{*60·2}] &\equiv:\kappa\in \text{Cl}ʻ(\text{Cls}-\iotaʻ\Lambda) &\qquad \text{(1)}\\ +\vdash.\text{(1).*88·37}. \supset\vdash\colon\ldotp \text{Mult ax}.&\equiv:\kappa\in \text{Cl}ʻ(\text{Cls}-\iotaʻ\Lambda).\supset_{\kappa}.\exists !\in _{\Delta}ʻ\kappa:\\ +[\text{*88·2}] &\equiv:\text{Cl}ʻ(\text{Cls}-\iotaʻ\Lambda)\subset \text{Cls}^{2}\,\text{Mult}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*88·38.</b> \(\vdash:\text{Mult ax}.\equiv.\text{Cls}-\iotaʻ\Lambda\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*88·23·373}]\)</p> + +<p class="nind"><b><a id="*88·39">*88·39</a>.</b> \(\vdash:\text{Mult ax}.\equiv.(\exists R).R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}\in .\text{D}ʻR=\text{V}.\text{ᗡ}ʻR=\text{Cls}-\iotaʻ\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*88·38·2.*80·14}.\supset\\ +\vdash:\text{Mult ax}.&\equiv.(\exists R).R\in 1\rightarrow \text{Cls}.R\unicode{x2abd}\in .\text{ᗡ}ʻR=\text{Cls}-\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*51·161.*53·5}.&\supset\vdash:\text{ᗡ}ʻR=\text{Cls}-\iotaʻ\Lambda.\supset.\iotaʻx\in \text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash.\text{*23·621}.&\supset\vdash:R\unicode{x2abd}\in .\supset.R=R\dot{\cap}\in &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash:R\unicode{x2abd}\in .\text{ᗡ}ʻR=\text{Cls}-\iotaʻ\Lambda.\supset.\iotaʻx\in \text{ᗡ}ʻ(R\dot{\cap}\in).\\ +[\text{*33·131}] &\supset.(\exists y).yR\iotaʻx.y\in \iotaʻx.\\ +[\text{*51·15}] &\supset.(\exists y).yR\iotaʻx.y=x.\\ +[\text{*13·195}] &\supset.xR\iotaʻx.\\ +[\text{*33·14}] &\supset.x\in \text{D}ʻR &\qquad \text{(4)}\\ +\vdash.\text{(4).*10·11·21.*24·14}.&\supset\vdash:R\unicode{x2abd}\in .\text{ᗡ}ʻR=\text{Cls}-\iotaʻ\Lambda.\supset.\text{D}ʻR=\text{V} &\qquad \text{(5)}\\ +\vdash.\text{(1).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned with certain cases in which a +construction exists by which the existence of selections can be proved.</p> + +<p class="nind"><b>*88·4.</b> \(\vdash.\kappa\upharpoonleft \breve{\text{Cl}}\in {\in}_{\Delta}ʻ\text{Cl}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·19.*71·27}.&\supset\vdash.\kappa\upharpoonleft \breve{\text{Cl}}\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*35·52·101}. &\supset\vdash:\alpha(\kappa\upharpoonleft \breve{\text{Cl}})\lambda.\equiv.\alpha\in \kappa.\lambda=\text{Cl}ʻ\alpha.\\ +[\text{*60·34}] & \supset.\alpha\in \lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11}.&\supset\vdash.\kappa\upharpoonleft \breve{\text{Cl}}\unicode{x2abd}\in &\qquad \text{(3)}\\ +\vdash.\text{*35·52}.& \supset\vdash.\text{ᗡ}ʻ(\kappa\upharpoonleft \breve{\text{Cl}})=\text{D}ʻ(\text{Cl}\upharpoonright \kappa)\\ +[\text{*37·401}] & =\text{Cl}ʻʻ\kappa &\qquad \text{(4)}\\ +\vdash.\text{(1).(3).(4).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_567">[Pg 567]</span></p> + +<p class="nind"><b>*88·41.</b> \(\vdash.\text{Cl}ʻʻ\kappa\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*88·4·2}]\)</p> + +<p class="nind"><b>*88·411.</b> \(\vdash.\kappa\in \text{D}ʻʻ{\in}_{\Delta}ʻ\text{Cl}ʻʻ\kappa\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·52}. \supset\vdash.\text{D}ʻ(\kappa\upharpoonleft \breve{\text{Cl}})&=\text{ᗡ}ʻ(\text{Cl}\upharpoonright \kappa)\\ +[\text{*35·65.*33·431}] &=\kappa &\qquad \text{(1)}\\ +\vdash.\text{(1).*88·4}.&\supset\vdash.(\exists R).R\in {\in}_{\Delta}ʻ\text{Cl}ʻʻ\kappa.\text{D}ʻR=\kappa.\\ +[\text{*37·6}] &\supset\vdash.\kappa\in \text{D}ʻʻ{\in}_{\Delta}ʻ\text{Cl}ʻʻ\kappa.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*88·42.</b> \(\vdash:\kappa\in \text{Cls}^{2}\,\text{Mult}.\exists !\alpha.\equiv.\kappa\cup \iotaʻ\alpha\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*83·904.*88·2}]\)</p> + +<p>In virtue of this proposition, as will be proved later, every finite +class of existent classes is a \(\text{Cls}^{2}\,\text{Mult}\). For we +have \(\dot{\Lambda}\in {\in}_{\Delta}ʻ\Lambda\); and, by the above, a +\(\text{Cls}^{2}\,\text{Mult}\) remains a \(\text{Cls}^{2}\,\text{Mult}\) +when one existent class is added as an additional member; hence the +result follows by induction.</p> + +<p class="nind"><b>*88·43.</b> \(\vdash:sʻ\kappa\in \text{Cls}^{2}\,\text{Mult}.\supset.{\in}_{\Delta}ʻʻ\kappa\in \text{Cls}^{2}\,\text{Mult}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*88·2}.\supset\vdash:\text{Hp}.&\supset.\exists !{\in}_{\Delta}ʻsʻ\kappa.\\ +[\text{*85·24}] &\supset.\exists !\dot{s}ʻʻ\text{D}ʻʻ{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa.\\ +[\text{*37·45}] &\supset.\exists !{\in}_{\Delta}ʻ{\in}_{\Delta}ʻʻ\kappa.\\ +[\text{*88·2}] &\supset.{\in}_{\Delta}ʻʻ\kappa\in \text{Cls}^{2}\,\text{Mult}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*88·431.</b> \[\begin{align}&\vdash\colon\ldotp \kappa\in \text{Cls}^{2}\,\text{excl}.\supset:{\in}_{\Delta}ʻʻ\kappa\in \text{Cls}^{2}\,\text{Mult}.\equiv.sʻ\kappa\in + \text{Cls}^{2}\,\text{Mult}\\ +&[\text{*88·2.*85·28.*37·45}]\end{align}\]</p> + +<p class="nind"><b>*88·44.</b> \(\vdash:\text{Cl ex}ʻsʻ\kappa\in \text{Cls}^{2}\,\text{Mult}.\supset.\kappa-\iotaʻ\Lambda\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*60·57.*88·22}]\)</p> + +<p class="nind"><b>*88·441.</b> \(\vdash:\Lambda{\sim}\in \kappa.\text{Cl ex}ʻsʻ\kappa\in \text{Cls}^{2}\,\text{Mult}.\supset.\kappa\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*88·44}]\)</p> + +<p class="nind"><b><a id="*88·45">*88·45</a>.</b> \(\vdash:\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda.P=\hat{x}\hat{\alpha}\{x\in \text{ᗡ}ʻR.\alpha=\overrightarrow{R}ʻx\cup \iotaʻx\}.\supset.P\in {\in}_{\Delta}ʻ\text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·3}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xP\alpha.&\equiv_{x,\alpha}.x\in \text{ᗡ}ʻR.\alpha=\overrightarrow{R}ʻx\cup \iotaʻx. &\qquad \text{(1)}\\ +[\text{*51·16}] &\supset_{x,\alpha}.x\in \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).*33·15.*51·2}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:xP\alpha.&\supset_{x}.\alpha=\overrightarrow{R}ʻx\cup \iotaʻx.\overrightarrow{R}ʻx\subset \text{D}ʻR.\iotaʻx\subset \text{ᗡ}ʻR.\\ +[\text{*24·494}] & \supset_{x}.\alpha-\text{D}ʻR=\iotaʻx &\qquad \text{(3)}\\ +\vdash.\text{(3).*11·59}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xP\alpha.yP\alpha.&\supset_{x,y}.\alpha-\text{D}ʻR=\iotaʻx.\alpha-\text{D}ʻR=\iotaʻy.\\ +[\text{*20·23.*51·23}] &\supset_{x,y}.x=y:\\ +[\text{*71·17}] & \supset:P\in 1\rightarrow \text{Cls} &\qquad \text{(4)}\\ +\vdash.\text{(2).(4).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_568">[Pg 568]</span></p> + +<p class="nind"><b><a id="*88·46">*88·46</a>.</b> \[\begin{align}\vdash:\text{D}ʻR\cap \text{ᗡ}ʻR=\Lambda.\lambda=\hat{\alpha}\{(\exists x).x\in \text{ᗡ}ʻR.\alpha=\overrightarrow{R}ʻx\cup &\iotaʻx\}.\supset.\\ +&\lambda\in \text{Cls}^{2}\,\text{Mult}\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·3.*10·281.*33·131}.&\supset\vdash\colon\ldotp P=\hat{x}\hat{\alpha}{x\in \text{ᗡ}ʻR.\alpha=\overrightarrow{R}ʻx\cup \iotaʻx}.\supset:\\ +&\alpha\in \text{ᗡ}ʻP.\equiv_{\alpha}.(\exists x).x\in \text{ᗡ}ʻR.\alpha=\overrightarrow{R}ʻx\cup \iotaʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*88·45}.&\supset\vdash:\text{Hp}.\supset.\hat{x}\hat{\alpha}{x\in \text{ᗡ}ʻR.\alpha=\overrightarrow{R}ʻx\cup \iotaʻx}\in {\in}_{\Delta}ʻ\lambda.\\ +[\text{*10·24}] & \supset.\exists !{\in}_{\Delta}ʻ\lambda.\\ +[\text{*88·2}] &\supset.\lambda\in \text{Cls}^{2}\,\text{Mult}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*88·47">*88·47</a>.</b> \(\vdash:P=\hat{\alpha}\hat{\beta}\{\alpha\in \kappa.\beta=\iotaʻʻ\alpha\cup \iotaʻ\alpha\}.\supset.P\in {\in}_{\Delta}ʻ\text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*21·3}. \supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha P\beta.&\equiv_{\alpha,\beta}.\alpha\in \kappa.\beta=\iotaʻʻ\alpha\cup \iotaʻ\alpha. &\qquad \text{(1)}\\ +[\text{*51·16}] &\supset_{\alpha,\beta}.\alpha\in \beta &\qquad \text{(2)}\\ +\vdash.\text{(1).*11·59}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha P\beta.\gamma P\beta.&\supset_{\alpha,\beta,\gamma}.\beta=\iotaʻʻ\alpha\cup + \iotaʻ\alpha.\beta=\iotaʻʻ\gamma\cup \iotaʻ\gamma.\\ +[\text{*40·171.*53·22·02}] &\supset_{\alpha,\beta,\gamma}.sʻ\beta=\alpha.sʻ\beta=\gamma.\\ +[\text{*20·23}] & \supset_{\alpha,\beta,\gamma}.\alpha=\gamma\\ +[\text{*71·17}] &\supset:P\in 1\rightarrow \text{Cls} &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*88·48">*88·48</a>.</b> \(\vdash.\hat{\beta}\{(\exists \alpha).\alpha\in \kappa.\beta=\iotaʻʻ\alpha\cup \iotaʻ\alpha\}\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*88·47}]\)</p> + +<p>The proof proceeds as in <a href="#*88·46">*88·46</a>.</p> + +<p class="nind"><b>*88·5.</b> \(\vdash.\Lambda\cap \text{Cls}\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*83·9.*88·2}]\)</p> + +<p class="nind"><b>*88·51.</b> \(\vdash:\exists !\alpha.\supset.\iotaʻ\alpha\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*83·901.*88·2}]\)</p> + +<p class="nind"><b>*88·52.</b> \(\vdash.\iotaʻʻ\alpha\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*83·42}]\)</p> + +<p class="nind"><b>*88·53.</b> \(\vdash:\kappa\subset 1.\supset.\kappa\in \text{Cls}^{2}\,\text{Mult} \quad[\text{*83·44}]\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_569">[Pg 569]</span></p> +<h2 class="nobreak" id="SECTION_E_b">SECTION E.<br> +<br> +INDUCTIVE RELATIONS.</h2> +</div> + + +<p><i>Summary of Section E.</i></p> + +<p>The subjects to be treated in this section are certain general +ideas of which a particular instance is afforded by mathematical +induction. Mathematical induction is, in fact, the application to the +number-series of a conception which is applicable to all relations, +and is often very important. The conception in question is that +which we shall call the <i>ancestral relation</i> with respect to +a given relation. If \(R\) is the given relation, we denote the +corresponding ancestral relation by "\(R_{\unicode{x2217}}\)"; the +name is chosen because, if \(R\) is the relation of parent and +child, \(R_{\unicode{x2217}}\) will be the relation of ancestor and +descendant—where, for convenience of language, we include \(x\) among +his own ancestors if \(x\) is a parent or a child of anything.</p> + +<p>It would commonly be said that \(a\) has to \(z\) the relation of +ancestor to descendant if there are a certain number of intermediate +people \(b\), \(c\), \(d\), ... such that in the series \(a\), \(b\), +\(c\), \(d\), ... \(z\) each term has to the next the relation of +parent and child. But this is not an adequate definition, because the +dots in +\[ +\unicode{x201c}a, b, c, d, \ldots z \unicode{x201d} +\] +represent an unanalysed idea. We may then try to amend this definition +by saying that there is a finite class \(\alpha\) of intermediate terms +such that one member (\(b\)) of \(\alpha\) is a child of \(a\), one +(\(y\)) is a parent of \(z\), every member of \(\alpha\) except \(b\) +is a child of one (and only one) member of \(\alpha\), and every member +of \(\alpha\) except \(y\) is a parent of one (and only one) member of +\(\alpha\). This definition is open to several objections. In the first +place, it is very complicated; in the second place, there will, in +regard to a general relation, be difficulty in securing the uniqueness +of the member of \(\alpha\) which is to be a parent (or a child) of a +given member of \(\alpha\); in the third place (and this is the really +fatal objection) the proposed definition states that \(\alpha\) is +to be a <i>finite</i> class, and we shall find that finitude, in the +relevant sense, is only defined by means of the very conception of the +ancestral relation which we are here engaged in defining. In fact, if +\(N\) denotes the relation of \(\nu\) to \(\nu + 1\), where \(\nu\) is +a cardinal number, then a finite cardinal (in the sense we require) is +<span class="pagenum" id="Page_570">[Pg 570]</span>one to which 0 has the relation \(N_{\unicode{x2217}}\), <i>i.e.</i> +one of which 0 is an ancestor with respect to the relation +\[ +\hat{\nu}\hat{\mu}\,(\mu=\nu+1). +\] +Hence we must not use the notion of finitude in defining the ancestral +relation. In fact, the ancestral relation is defined as follows.</p> + +<p>Let us call \(\mu\) a <i>hereditary class with respect</i> to \(R\) if +\(\breve{R}ʻʻ\mu\subset\mu\), <i>i.e.</i> if successors of \(\mu\)'s +(with respect to \(R\)) are \(\mu\)'s. Thus, for example, if \(\mu\) is +the class of persons named Smith, \(\mu\) is hereditary with respect to +the relation of father to son. If \(\mu\) is the Peerage, \(\mu\) is +hereditary with respect to the relation of father to surviving eldest +son. If \(\mu\) is numbers greater than 100, \(\mu\) is hereditary with +respect to the relation of \(\nu\) to \(\nu+1\); and so on. If now +\(a\) is an ancestor of \(z\), and \(\mu\) is a hereditary class to +which \(a\) belongs, then \(z\) also belongs to this class. Conversely, +if \(z\) belongs to every hereditary class to which \(a\) belongs, +then (in the sense in which a is one of his own ancestors if \(a\) is +anybody's parent or child) \(a\) must be an ancestor of \(z\). For to +have \(a\) for one's ancestor is a hereditary property which belongs +to \(a\), and therefore, by hypothesis, to \(z\). Hence \(a\) is an +ancestor of \(z\) when, and only when, \(a\) belongs to the field of +the relation in question and \(z\) belongs to every hereditary class to +which \(a\) belongs. This property may be used to define the ancestral +relation; <i>i.e.</i> since we have +\[ +aR_{\unicode{x2217}}z.\equiv:a\in CʻR:\breve{R}ʻʻ\mu\subset \mu.a\in \mu.\supset_{\mu}.z\in \mu +\] +we put +\[ +R_{\unicode{x2217}}=\hat{a}\hat{z}\{a\in CʻR:\breve{R}ʻʻ\mu\subset \mu.a\in \mu.\supset_{\mu}.z\in \mu\} \quad\text{Df}. +\] +We then have +\[ +\vdash:a\in CʻR.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻa=\hat{z}\{\breve{R}ʻʻ\mu\subset \mu.a\in \mu.\supset_{\mu}.z\in \mu\}. +\] +Here \(\overleftarrow{R}_{\unicode{x2217}}ʻa\) may be called "the +descendants of \(a\)." It is the class of terms of which \(a\) is an +ancestor.</p> + +<p>To make plain the relation of the above to mathematical induction, put +0 for \(a\), and \(\hat{\alpha}\hat{\beta}\,(\beta=\alpha+1)\) for +\(R\). Then, since 1=0+1, we have \(0\in CʻR\). Again +\[ +\breve{R}ʻʻ\mu\subset \mu.\equiv:\alpha\in \mu.\supset_{\alpha}.\alpha+1\in \mu. +\] +Thus we find +\[ +\overleftarrow{R}_{\unicode{x2217}}ʻ0=\hat{\beta}\{\alpha\in \mu.\supset_{\alpha}.\alpha+1\in \mu:0\in \mu:\supset_{\mu}.\beta\in \mu\}. +\] +Thus if \(\beta\) is a descendant of 0, \(\beta\) belongs to every +class to which 0 belongs and to which \(\alpha+1\) belongs whenever +\(\alpha\) belongs. Hence mathematical induction, starting from 0, +will prove properties of \(\beta\). In elementary mathematics it is +customary to speak as if this held of <i>all</i> integers, <i>i.e.</i> +as if \(\overleftarrow{R}_{\unicode{x2217}}ʻ0\) (as above defined) +included all integers; but in fact only <i>finite</i> integers (in<span class="pagenum" id="Page_571">[Pg 571]</span> one +of the two senses which the word <i>finite</i> may have) belong to the +class \(\overleftarrow{R}_{\unicode{x2217}}ʻ0\), and they belong to it +<i>by definition</i>, being defined as the class +\[ +\hat{\beta}\{\alpha\in \mu.\supset_{\alpha}.\alpha+1\in \mu:0\in \mu:\supset_{\mu}.\beta\in \mu\}, +\] +<i>i.e.</i> as \(\overleftarrow{R}_{\unicode{x2217}}ʻ0\) in the above +sense. To infinite numbers, inductive proofs of this kind starting from +0 cannot be applied.</p> + +<p>The study of \(R_{\unicode{x2217}}\) will occupy <a href="#*90">*90</a>. The +relation \(R_{\unicode{x2217}}\) holds between \(x\) and \(y\) if +\(x(I\upharpoonright C'R)y\) or \(xRy\) or \(xR^{2}y\) or etc. The +study of this "etc." occupies <a href="#*91">*91</a>, "on the powers of a relation." We +may, for many technical purposes, regard \(I\upharpoonright CʻR\) as +the 0th power of \(R\); the other powers are \(R\), \(R^{2}\), etc. If +\(S\) is a power of \(R\), so is \(S\mid R\). Now \(S\mid R\) is \(\mid RʻS\), +according to the definition in <a href="#*38">*38</a>. Thus if we have +\[ +R\in \mu:S\in \mu.\supset_{S}.S\mid R\in \mu:\supset_{\mu}.P\in \mu, +\] +\(P\) must be a power of \(R\), because the class of powers of \(R\) is +a value of \(\mu\) which satisfies the hypothesis +\[ +R\in \mu:S\in \mu.\supset_{S}.S\mid R\in \mu. +\] +Conversely, if \(P\) is a power of \(R\), then \(P\) is reached by +repetitions of the process of turning \(S\) into \(S\mid R\), starting +this process with \(R\). Hence if \(P\) is a power of \(R\), we shall +have +\[ +R\in \mu:S\in \mu.\supset_{S}.S\mid R\in \mu:\supset_{\mu}.P\in \mu. +\]</p> + +<p>Consequently, if we denote the class of powers of \(R\) by +\(\text{Pot}ʻR\), we have +\[ +P\in \text{Pot}ʻR.\equiv\colon\ldotp R\in \mu:S\in \mu.\supset_{S}.S\mid R\in \mu:\supset_{\mu}.P\in \mu. +\] +We might use this as the definition of \(\text{Pot}ʻR\); but we can +get a somewhat simpler form. For the above is shown, without much +difficulty, to be equivalent to +\[ +P\in \text{Pot}ʻR.\equiv.P(\mid R)_{\unicode{x2217}}R, +\] +that is, \(P\) belongs to the ancestry of \(R\) with respect to \(\mid R\), +in other words, \(P\) is reached from \(R\) by proceeding along +the series +\[ +R,\quad \mid RʻR,\quad \mid Rʻ\mid RʻR, \quad\text{etc}. +\] +which is the same as the series +\[ +R,\quad R^{2},\quad R^{3}, \quad\text{etc}. +\] +The relation \((\mid R)_{\unicode{x2217}}\) is important on its own +account. We put +\[ +R_{\text{ts}}=(\mid R)_{\unicode{x2217}} \quad\text{Df}, +\] +and then we put +\[ +\text{Pot}ʻR=\overrightarrow{R}_{\text{ts}}ʻR \quad\text{Df}. +\]</p> + +<p>We often want to include \(I\upharpoonright CʻR\) among the powers +of \(R\); the class consisting of \(\text{Pot}ʻR\) together with +\(I\upharpoonright CʻR\) we call \(\text{Potid}ʻR\). The definition is +\[ +\text{Potid}ʻR=\overrightarrow{R}_{\text{ts}}ʻ(I\upharpoonright CʻR), +\]<span class="pagenum" id="Page_572">[Pg 572]</span> +whence we easily prove +\[ +\text{Potid}ʻR=\text{Pot}ʻR\cup \iotaʻ(I\upharpoonright CʻR). +\] +The relation of being related by some power of \(R\) (other than +\(I\upharpoonright CʻR\)) is a very important one. We denote it by +\(R_{\text{po}}\), and put +\[ +R_{\text{po}}=\dot{s}ʻ\text{Pot}ʻR \quad\text{Df}. +\] +Thus when \(xR_{\text{po}}y\), we have one of \(xRy\), \(xR^{2}y\), +\(xR^{3}y\), etc. It is easy to prove that +\[ +R_{\unicode{x2217}}=R_{\text{po}}\unicode{x228d}I\upharpoonright CʻR. +\] +In a series in which every term (except the first, if there is a first) +has an immediate predecessor, and every term (except the last, if there +is a last) has an immediate successor, if \(R\) is the relation of a +term to its immediate successor, \(R_{\text{po}}\) is the relation of +any earlier term to any later one.</p> + +<p>The next number (<a href="#*92">*92</a>) concerns itself with some special properties of +the powers of one-many, many-one and one-one relations.</p> + +<p>The next number (<a href="#*93">*93</a>) analyses the field of a relation into successive +<i>generations</i>; <i>e.g.</i> if the relation is that of parent +and child, the first generation will consist of Adam and Eve, the +second of their children, the third of their grandchildren, and so +on, taking always the longest route from Adam and Eve when there +have been intermarriages between generations. That is, taking any +relation \(P\), the first generation is \(\text{D}ʻP-\text{ᗡ}ʻP\), +the second is \(\text{ᗡ}ʻP-\text{ᗡ}ʻ(P^{2})\), the third is +\(\text{ᗡ}ʻ(P^{2})-\text{ᗡ}ʻ(P^{3})\), and so on. Generally, if \(T\) is a +power of \(P\) (including \(I\upharpoonright CʻP\)), the corresponding +generation is +\[ +\begin{array}{l} +&\text{ᗡ}ʻT-\text{ᗡ}ʻ(T\mid P),\\ +\textit{i.e.} &\text{ᗡ}ʻT-\breve{P}ʻʻ\text{ᗡ}ʻT. +\end{array} +\]</p> + +<p>In order to express this more conveniently, we introduce a new +symbol \(\text{min}_{P}\), which is required also on other grounds, +especially in series. "\(\text{min}_{P}\)" may be read "minimum +with respect to \(P\)." We regard "\(xPy\)" as "\(x\) precedes +\(y\)"; then in a class \(\alpha\), the "minima of \(\alpha\)" will +be those members of \(\alpha\) which belong to \(CʻP\) and are not +preceded by any other members of \(\alpha\), <i>i.e.</i> \(\alpha\cap CʻP-\breve{P}ʻʻ\alpha\). +We put therefore +\[ +\begin{array}{l} +& x\, \text{min}_{P}\,\alpha.\equiv.x\in \alpha\cap CʻP-\breve{P}ʻʻ\alpha,\\ +\textit{i.e.} & \text{min}_{P}=\hat{x}\hat{\alpha}(x\in \alpha\cap CʻP-\breve{P}ʻʻ\alpha) \quad\text{Df}. +\end{array} +\] +Hence we have +\[ +\overrightarrow{\text{min}}_{P}ʻ\alpha=\alpha\cap CʻP-\breve{P}ʻʻ\alpha, +\] +<i>i.e.</i> \(\overrightarrow{\text{min}}_{P}ʻ\alpha\) consists of +those members of \(\alpha\cap CʻP\) which are not preceded by any other +members of \(\alpha\). (If \(\alpha\) has a single first term, this +term is \(\text{min}_{P}ʻ\alpha\).) Thus we have, when \(T\) is a power of +\(P\), +\[ +\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\text{ᗡ}ʻT-\breve{P}ʻʻ\text{ᗡ}ʻT. +\]<span class="pagenum" id="Page_573">[Pg 573]</span> +Thus \(\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT\), where \(T\) is any +power of \(P\) (including \(I\upharpoonright CʻP\)) is the generation +of \(P\) corresponding to \(T\); thus the whole class of generations is +\(\overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\text{Potid}ʻP\). Hence we +put +\[ +\text{gen}ʻP=\overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\text{Potid}ʻP \quad\text{Df}, +\] +where "\(\text{gen}\)" stands for "generation."</p> + +<p>The notation "\(\text{min}_{P}\)" will not be much used until we come +to series, but then it will be constantly used. At present, we shall +only give such properties of \(\text{min}_{P}\) as are necessary for +our immediate purposes, but in Part V (on series) we shall devote a +number (*205) to its properties.</p> + +<p>In this number we also introduce the notation "\(xBP\)" for "\(x\in \text{D}ʻP-\text{ᗡ}ʻP\)." +"\(xBP\)" may be read "\(x\) begins \(P\)." If there is a single +beginning of \(P\), this is \(BʻP\); otherwise the class of beginnings +is \(\overrightarrow{B}ʻP\), which =\(\text{D}ʻP-\text{ᗡ}ʻP\). Thus +if \(P\) is the relation of father and son, \(BʻP\) = Adam; if \(P\) +is the relation of parent and child, \(\overrightarrow{B}ʻP\) = Adam +and Eve. \(Bʻ\breve{P}\) will be the end of \(P\), if there is one; +generally, \(\overrightarrow{B}ʻ\breve{P}\) will be the class of ends, +<i>i.e.</i> \(\text{ᗡ}ʻP-\text{D}ʻP\). The first generation of \(P\) +is \(\overrightarrow{B}ʻP\). If \(P\in 1\rightarrow \text{Cls}\), any +generation of \(P\) is \(\breve{T}ʻʻ\overrightarrow{B}ʻP\), where \(T\) +is the corresponding power of \(P\).</p> + +<p>The field of a relation consists, in general, not only of the +generations of \(P\), but also of another part, the part in which, +however far we go backwards, we never reach a beginning. This part is +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\). The two parts \(sʻ\text{gen}ʻP\) and +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\) are mutually exclusive, and together +exhaust \(CʻP\).</p> + +<p>The two next numbers, <a href="#*94">*94</a> and <a href="#*95">*95</a>, are hardly ever relevant in +subsequent propositions, and may therefore be omitted by any reader +who is not interested in their subject-matter. *94 deals with powers +of relative products. It is only used in the following number (*95), +on "equi-factor relations." The matter to be dealt with in this number +may be explained as follows. In dealing with correlations and similar +topics, we often wish to consider the series of relations +\[ +R,\quad P\mid R\mid Q,\quad P^{2}\mid R\mid Q^{2},\quad P^{3}\mid R\mid Q^{3}, \quad\text{etc}. +\] +Now we have not yet at our command a definition of \(P^{\nu}\), where +\(\nu\) is any finite number; thus we cannot define a general term +of this series as \(P^{\nu}\mid R\mid Q^{\nu}\). We need therefore a +different method of definition. We have +\[ +\begin{aligned} +P\mid R\mid Q=(P\Arrowvert Q)ʻR,\\ +P^{2}\mid R\mid Q^{2}=(P\Arrowvert Q)^{2}ʻR, +\end{aligned} +\] +and so on. Thus if \(T\) is any power of (\(P\mid)\mid (\mid Q)\), a +general term of our series is \(TʻR\). For convenience of notation, we +put +\[ +P\unicode{x2217} Q=\text{sg}ʻ(P\Arrowvert Q)_\unicode{x2217} \quad\text{Df}. +\]<span class="pagenum" id="Page_574">[Pg 574]</span> +Then our series consists of (\(P\unicode{x2217}Q)ʻR\). The sum of all +relations of this class is considered in this number.</p> + +<p>The principal propositions proved in <a href="#*94">*94</a> and <a href="#*95">*95</a> are two which have the +same hypothesis as the Schröder-Bernstein theorem, namely +\[ +R,\,S\in 1\rightarrow 1.\text{ᗡ}ʻS\subset \text{D}ʻR.\text{ᗡ}ʻR\subset \text{D}ʻS. +\] +These two propositions state that, with the above hypothesis,</p> + +<p>\[ +\begin{array}{l} +&sʻ\text{gen}ʻ(R\mid S)\text{ sm } sʻ{\text{gen}}ʻ(S\mid R)\\ +\text{and} &pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\text{ sm } pʻ\text{ᗡ}ʻʻ(S\mid R). +\end{array} +\] +The two combined reconstitute the Schröder-Bernstein theorem, since +\[ +\begin{array}{l} +& sʻ\text{gen}ʻ(R\mid S)\cup pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)=\text{D}ʻR\\ +\text{and} & sʻ\text{gen}ʻ(S\mid R)\cup pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)=\text{D}ʻS. +\end{array} +\] +Thus they present, so to speak, an itemized account of the equality +proved by the Schröder-Bernstein theorem.</p> + +<p><a href="#*96">*96</a>, on the posterity of a term, is concerned with the properties +of \(\overleftarrow{R}_{\unicode{x2217}}ʻx\), chiefly when +\(R\in \text{Cls}\rightarrow 1\). In this case, in general, +\(\overleftarrow{R}_{\unicode{x2217}}ʻx\) consists of two parts, +first an open series and then a cyclic series. Either of these may +vanish, or may reduce to a single term. If we call the two parts +\(\beta\) and \(\gamma\), the whole of \(\beta\) precedes the whole +of \(\gamma\), and \(\beta\upharpoonleft R\), \(\gamma\upharpoonleft R\in 1\rightarrow 1\). +Thus if either \(\beta\) or \(\gamma\) vanishes, +\(\overleftarrow{R}_{\unicode{x2217}}ʻx\upharpoonleft R\in 1\rightarrow 1\). +If \(\gamma\) vanishes, the series never returns into itself, +that is, \(\overleftarrow{R}_{\unicode{x2217}}ʻx\upharpoonleft R_{\text{po}}\unicode{x2abd}J\). +If \(\gamma\) exists, there is a definite power of \(R\), say +\(T\), such that \(y\in \gamma.\supset_{y}.yTy\). If \(\beta\) and +\(\gamma\) both exist, there is one term, namely the successor +of the last term of \(\beta\), which has just two immediate +predecessors, one in \(\beta\) and one in \(\gamma\); every other +term of \(\overleftarrow{R}_{\text{po}}ʻx\) has only one immediate +predecessor in \(\overleftarrow{R}_{\unicode{x2217}}ʻx\). Thus +\(\overleftarrow{R}_{\unicode{x2217}}ʻx\) is shaped like a \(Q\), with +\(x\) at the tip of the tail.</p> + +<p><span class="pagenum" id="Page_575">[Pg 575]</span></p> + +<p><a href="#*97">*97</a> deals with the analysis of the field of a relation into families. +Taking any member \(x\) of \(CʻR\), the family of \(x\) with +respect to \(R\) is \(\overrightarrow{R}_{\unicode{x2217}}ʻx\cup \overleftarrow{R}_{\unicode{x2217}}ʻx\), +which we write \(\overleftrightarrow{R}_{\unicode{x2217}}ʻx\). Thus the +class of families is \(\overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\). +Those families which contain a member of \(\overrightarrow{B}ʻR\) +are \(\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\). +If we regard \(\breve{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\) +as arranged in a rectangle, in which the +generations are the successive rows, then +\(\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\) +will be the columns. Thus the relation of \(\text{gen}ʻR\) to +\(\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\) may be +regarded as a generalized form of the relation of rows and columns. +Under a suitable hypothesis, each row is a selection from the columns, +and each column a selection from the rows. This is expressed in the +following proposition: +\[ +\begin{align} +\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ&\breve{R}\in \text{gen}ʻR\cup \iotaʻ\Lambda.\supset.\\ +&\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset \text{D}ʻʻ{\in}_{\Delta}ʻ(\text{gen}ʻR-\iotaʻ\Lambda).\text{gen}ʻR-\iotaʻ\Lambda\subset + \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR +\end{align} +\] +whence we derive existence-theorems for selections in the cases +concerned.</p> + +<p>The importance of the ideas dealt with in the present section is very +great. These ideas dominate the treatment of finite and infinite, +the theory of progressions and \(\aleph_{0}\), and the transition +from series generated by one-one or many-one relations of consecutive +terms to series generated by transitive relations of <i>before</i> +and <i>after</i>. Wherever, in short, mathematical induction is used +the ideas treated in this section are required. The portions of our +subsequent work in which this section is most referred to are the two +sections on finite and infinite cardinals and ordinals (Part III, +Section C and Part V, Section E). In the general theory of cardinals, +<i>i.e.</i> in Part III, Sections A and B, before the distinction of +finite and infinite has been introduced, the present section will be +seldom if ever referred to<a id="FNanchor_65" href="#Footnote_65" class="fnanchor">[65]</a>.</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_65" href="#FNanchor_65" class="label">[65]</a> +The present section is based on the work of Frege, who +first defined the ancestral relation. See his <i>Begriffsschrift</i> +(Halle, 1879), Part <span class="allsmcap">III</span>., pp. 55-87. Cf. also his +<i>Grundgesetze der Arithmetik</i>, Vol. <span class="allsmcap">I</span>. (Jena, 1893), §§ +45, 46 (pp. 59, 60). In this work the ancestral relation is used to +prove the properties of finite cardinals and \(\aleph_{0}\).</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_576">[Pg 576]</span></p> +<h2 class="nobreak" id="*90">*90. ON THE ANCESTRAL RELATION.</h2> +</div> + + +<p><i>Summary of</i> *90.</p> + +<p>If \(R\) is any relation, "\(x R_{\unicode{x2217}} y\)" is to mean +"\(x\) is an ancestor of \(y\) with respect to \(R\)," where a term +counts as its own ancestor provided it belongs to the field of \(R\). +The definition of \(R_{\unicode{x2217}}\) is as follows:</p> + +<p><b>90·01.</b> \(R_{\unicode{x2217}} = \hat{x}\hat{y}\{x \in CʻR:\breve{R}ʻʻ\mu \subset \mu . x \in \mu .\supset_{\mu}. y \in \mu\} +\quad \text{Df}\)</p> + +<p>That is, \(x R_{\unicode{x2217}} y\) is to hold when \(x\) belongs to +the field of \(R\), and \(y\) belongs to every hereditary class to +which \(x\) belongs; a hereditary class being a class \(\mu\) such that +\(\breve{R}ʻʻ\mu \subset \mu\), <i>i.e.</i> such that all successors of +\(\mu\)'s are \(\mu\)'s.</p> + +<p class="nind"><b><a id="*90·02">*90·02</a>.</b> \(\breve{R}_{\unicode{x2217}} = \text{Cnv}ʻR_{\unicode{x2217}} \quad \text{Df}\)</p> + +<p>This definition serves merely to decide the ambiguity +between \((\breve{R})_{\unicode{x2217}}\) and +\(\text{Cnv}ʻR_{\unicode{x2217}}\), either of which might be meant of +\(\breve{R}_{\unicode{x2217}}\). It will be shown, however, that the +two are equal (<a href="#*90·132">*90·132</a>).</p> + +<p>The most important propositions of this number are the following:</p> + +<p><b>90·112.</b> \(\vdash\colon\ldotp x R_{\unicode{x2217}} y: {\phi}z . z R w .\supset_{z,w}.{\phi}w:{\phi}x :\supset. {\phi}y\)</p> + +<p><i>I.e.</i> if \(x R_{\unicode{x2217}} y\) and if \({\phi}\hat{z}\) is +a hereditary property belonging to \(x\), then it belongs to \(y\).</p> + +<p class="nind"><b>*90·12.</b> \(\vdash: x \in CʻR .\equiv. x R_{\unicode{x2217}} x\)</p> + +<p><i>I.e.</i> \(R_{\unicode{x2217}}\) is reflexive throughout the field +of \(R\), but not elsewhere.</p> + +<p class="nind"><b>*90·14</b> \(\vdash. \text{D}ʻR_{\unicode{x2217}} = \text{ᗡ}ʻR_{\unicode{x2217}} = CʻR_{\unicode{x2217}} = CʻR\)</p> + +<p class="nind"><b>*90·15.</b> \(\vdash. I \upharpoonright CʻR \unicode{x2abd} R_{\unicode{x2217}}\)</p> + +<p class="nind"><b>*90·151.</b> \(\vdash. R \unicode{x2abd} R_{\unicode{x2217}}\)</p> + +<p class="nind"><b>*90·16.</b> \(\vdash. R_{\unicode{x2217}} \mid R \unicode{x2abd} R_{\unicode{x2217}}\)</p> + +<p class="nind"><b>*90·163.</b> \(\vdash. \breve{R}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx \subset \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>I.e.</i> \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) is a hereditary +class.</p> + +<p><span class="pagenum" id="Page_577">[Pg 577]</span></p> + +<p class="nind"><b>*90·17.</b> \(\vdash. R_{\unicode{x2217}}^{2} = R_{\unicode{x2217}}\)</p> + +<p class="nind"><b>*90·21.</b> \(\vdash:\alpha\subset CʻR. \equiv .\alpha\subset \breve{R}_{\unicode{x2217}}ʻʻ\alpha. \equiv .\alpha\subset R_{\unicode{x2217}}ʻʻ\alpha\)</p> + +<p class="nind"><b>*90·22.</b> \(\vdash:\breve{R}ʻʻ\alpha\subset \alpha. \equiv .\breve{R}_{\unicode{x2217}}ʻʻ\alpha\subset \alpha\)</p> + +<p><i>I.e.</i> the classes that are hereditary with respect to +\(R\) are the same as those that are hereditary with respect to +\(R_{\unicode{x2217}}\).</p> + +<p class="nind"><b>*90·31.</b> \(\vdash.R_{\unicode{x2217}} = I\upharpoonright CʻR\unicode{x228d}R_{\unicode{x2217}}\mid R\)</p> + +<p class="nind"><b>*90·32.</b> \(\vdash.R\mid R_{\unicode{x2217}} = R\unicode{x228d}R\mid R_{\unicode{x2217}}\mid R = R_{\unicode{x2217}}\mid R\)</p> + +<p class="nind"><b>*90·33.</b> \(\vdash.R_{\unicode{x2217}}ʻʻ\alpha = (\alpha\cap CʻR)\cup R_{\unicode{x2217}}ʻʻRʻʻ\alpha = (\alpha\cap CʻR)\cup RʻʻR_{\unicode{x2217}}ʻʻ\alpha\)</p> + +<p class="nind"><b>*90·4.</b> \(\vdash·(R_{\unicode{x2217}})_{\unicode{x2217}} = R_{\unicode{x2217}}\)</p> + +<hr class="tb"> + +<p class="nind"><b>*90·01.</b> \(R_{\unicode{x2217}} = \hat{x}\hat{y}\{x\in CʻR:\breve{R}ʻʻ\mu\subset \mu.x\in \mu.\supset_{\mu}.y\in \mu\} \quad\text{Df}\)</p> + +<p class="nind"><b>*90·02.</b> \(\breve{R}_{\unicode{x2217}} = \text{Cnv}ʻR_{\unicode{x2217}} \quad\text{Df}\)</p> + +<p class="nind"><b>*90·1.</b> \(\vdash\colon\ldotp xR_{\unicode{x2217}}y. \equiv :x\in CʻR:\breve{R}ʻʻ\mu\subset \mu.x\in \mu.\supset_{\mu}.y\in \mu \quad[\text{*21·3.(*90·01)}]\)</p> + +<p class="nind"><b>*90·101.</b> \(\vdash:\breve{R}ʻʻ\mu\subset \mu. \equiv .Rʻʻ-\mu\subset -\mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·171}.\supset\vdash\colon\ldotp \breve{R}ʻʻ\mu\subset \mu. &\equiv :x\in \mu.xRy.\supset_{x,y}.y\in \mu:\\ +[\text{Transp}] &\equiv :y\in - \mu.xRy.\supset_{x,y}.x\in - \mu:\\ +[\text{*37·17}] &\equiv :Rʻʻ - \mu\subset - \mu\colon\ldotp \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p>*90·102 is a lemma for <a href="#*90·11">*90·11</a>.</p> + +<p class="nind"><b><a id="*90·102">*90·102</a>.</b> \(\vdash\colon\ldotp \breve{R}ʻʻ\mu\subset \mu.x\in \mu.\supset_{\mu}.y\in \mu: \equiv :Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·101}.\supset\\ +\vdash\colon\ldotp \breve{R}ʻʻ\mu\subset \mu.x\in \mu.\supset.y\in \mu: &\equiv :Rʻʻ - \mu\subset - \mu.x\in \mu.\supset.y\in \mu:\\ +[\text{Transp}] &\equiv :Rʻʻ - \mu\subset - \mu.y\in - \mu.\supset.x\in - \mu &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·271}.\supset\\ +\vdash:\breve{R}ʻʻ\mu\subset \mu.x\in \mu.\supset_\mu.y\in \mu: &\equiv :Rʻʻ - \mu\subset - \mu.y\in - \mu.\supset_{\mu}.x\in - \mu:\\ +[\text{*22·94}] &\equiv :Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu\colon\ldotp \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*90·11">*90·11</a>.</b> \(\vdash\colon\ldotp xR_{\unicode{x2217}}y. \equiv :x\in CʻR:Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu \quad[\text{*90·1·102}]\)</p> + +<p class="nind"><b>*90·111.</b> \[\begin{align}&\vdash\colon\colon xR_{\unicode{x2217}}y. \equiv \colon\ldotp x\in CʻR\colon\ldotp z\in \mu.zRw.\supset_{z,w}.w\in \mu:x\in \mu:\supset_{\mu}.y\in + \mu\\ +&[\text{*90·1.*37·171}]\end{align}\]</p> + +<p class="nind"><b><a id="*90·112">*90·112</a>.</b> \(\vdash\colon\ldotp xR_{\unicode{x2217}}y:\phi z.zRw.\supset_{z,w}.\phi w:\phi x:\supset.\phi y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·111} \frac{\hat{z}(\phi z)}{\mu} .\supset\\ +\vdash\colon\colon xR_{\unicode{x2217}}y.&\supset\colon\ldotp z\in \hat{z}(\phi z).zRw.\supset_{z,w}.w\in + \hat{z}(\phi z):x\in \hat{z}(\phi z):\supset:y\in \hat{z}(\phi z)\colon\ldotp\\ +[\text{*20·3}] &\supset\colon\ldotp \phi z.zRw.\supset_{z,w}.\phi w:\phi x:\supset.\phi y &\qquad \text{(1)}\\ +\vdash.\text{(1).Imp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_578">[Pg 578]</span></p> + +<p class="nind"><b>*90·12.</b> \(\vdash:x\in CʻR.\equiv.xR_{\unicode{x2217}}x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·1}. &\supset\vdash:xR_{\unicode{x2217}}x.\supset.x\in CʻR &\qquad \text{(1)}\\ +\vdash.\text{*3·27.*10·11}.&\supset\vdash:\breve{R}ʻʻ\mu\subset \mu.x\in \mu.\supset_{\mu}.x\in \mu:\\ +[\text{*3·21}] &\supset\vdash\colon\ldotp x\in CʻR.\supset:x\in CʻR:\breve{R}ʻʻ\mu\subset \mu.x\in \mu.\supset_{\mu}.x\in \mu:\\ +[\text{*90·1}] &\qquad\qquad\qquad\supset:xR_{\unicode{x2217}}x &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·13.</b> \(\vdash:xR_{\unicode{x2217}}y.\supset.x,y\in CʻR.xR_{\unicode{x2217}}x.yR_{\unicode{x2217}}y\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·16.*33·161}.&\supset\vdash.\breve{R}ʻʻCʻR\subset CʻR &\qquad \text{(1)}\\ +\vdash.\text{*90·1}. &\supset\vdash:xR_{\unicode{x2217}}y.\supset.x\in CʻR &\qquad \text{(2)}\\ +\vdash.\text{*90·1} \frac{CʻR}{\mu}. &\supset\vdash\colon\ldotp xR_{\unicode{x2217}}y.\supset:\breve{R}ʻʻCʻR\subset CʻR.x\in CʻR.\supset.y\in CʻR:\\ +[\text{(1).(2)}] &\qquad\qquad\quad\supset:y\in CʻR &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*90·12}.&\supset\vdash:xR_{\unicode{x2217}}y.\supset.xR_{\unicode{x2217}}x.yR_{\unicode{x2217}}y &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following proposition is a lemma for <a href="#*90·132">*90·132</a>.</p> + +<p class="nind"><b>*90·131.</b> \(\vdash\colon\ldotp xR_{\unicode{x2217}}y.\equiv:y\in CʻR:Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·11·13}.\supset\\ +\vdash\colon\ldotp xR_{\unicode{x2217}}y.&\supset:y\in CʻR:Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu &\qquad \text{(1)}\\ +\vdash.\text{*37·15.*33·161}.&\supset\vdash.RʻʻCʻR\subset CʻR &\qquad \text{(2)}\\ +\vdash.\text{*10·1}.&\supset\vdash\colon\ldotp y\in CʻR:Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu:\supset:\\ +&y\in CʻR:RʻʻCʻR\subset CʻR.y\in CʻR.\supset.x\in CʻR:\\ +[\text{*5·33}] &\supset:RʻʻCʻR\subset CʻR.\supset.x\in CʻR:\\ +[\text{(2)}] &\supset:x\in CʻR &\qquad \text{(3)}\\ +\vdash.\text{(3).*5·3}.&\supset\vdash\colon\ldotp y\in CʻR:Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu:\supset:\\ +&x\in CʻR:Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu:\\ +[\text{*90·11}] &\supset:xR_{\unicode{x2217}}y &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*90·132">*90·132</a>.</b> \(\vdash.(\breve{R})_{\unicode{x2217}}=\breve{R}_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·33.*33·22.*90·1}.\supset\\ +\vdash\colon\ldotp y(\breve{R})_{\unicode{x2217}}x.&\equiv:y\in CʻR:Rʻʻ\mu\subset \mu.y\in \mu.\supset_{\mu}.x\in \mu:\\ +[\text{*90·131}] &\equiv:xR_{\unicode{x2217}}y:\\ +[\text{*31·11}] &\equiv:y\breve{R}_{\unicode{x2217}}x\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_579">[Pg 579]</span></p> + +<p>In accordance with our general convention as regards suffixes, +and with the definition <a href="#*90·02">*90·02</a>, \(\breve{R}_{\unicode{x2217}}\) +means \(\text{Cnv}ʻR_{\unicode{x2217}}\), not +(\(\breve{R})_{\unicode{x2217}}\).</p> + +<p class="nind"><b>*90·14.</b> \(\vdash.\text{D}ʻR_\unicode{x2217} = \text{ᗡ}ʻR_\unicode{x2217} = CʻR_\unicode{x2217} = CʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash·\text{*90·12.*33·14·17}.&\supset\vdash:x\in CʻR.\supset.x\in \text{D}ʻR_\unicode{x2217}.x\in \text{ᗡ}ʻR_\unicode{x2217}.x\in CʻR_\unicode{x2217} &\qquad \text{(1)}\\ +\vdash.\text{*33·13}. &\supset\vdash:x\in \text{D}ʻR_\unicode{x2217}. \equiv .(\exists y).xR_\unicode{x2217}y.\\ +[\text{*90·13}] &\qquad\qquad\qquad\supset.x\in CʻR &\qquad \text{(2)}\\ +\text{Similarly} &\vdash:x\in \text{ᗡ}ʻR_\unicode{x2217}.\supset.x\in CʻR &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*33·16}. &\supset\vdash:x\in CʻR_\unicode{x2217}.\supset.x\in CʻR &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·141.</b> \(\vdash:\dot{\exists}!R_\unicode{x2217}. \equiv .\dot{\exists}!R \quad[\text{*90·14.*33·24}]\)</p> + +<p class="nind"><b>*90·15.</b> \(\vdash.I\upharpoonright CʻR\unicode{x2abd}R_\unicode{x2217}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash·\text{*50·1.*35·101}.\supset\vdash:x(I\upharpoonright CʻR)y. &\equiv .x = y.y\in CʻR.\\ +[\text{*90·12}] &\equiv .x = y.yR_\unicode{x2217}y.\\ +[\text{*13·13}] &\supset.xR_\unicode{x2217}y:\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p>Note that \(I\upharpoonright CʻR\) may be conveniently regarded as +the 0th power of \(R\). By *50·64·65, when multiplied by \(R\) it +gives \(R\); also it is contained in \(R\mid \breve{R}\), \(R^{2}\mid \breve{R}^{2}\), +etc. I has properties, as regards relational multiplication, +analogous to those of 1 in ordinary multiplication; thus to regard +\(I\upharpoonright CʻR\) as the 0th power of \(R\) is analogous to +regarding 1 as the 0th power of \(n\), where \(n\) is a number.</p> + +<p class="nind"><b>*90·151.</b> \(\vdash.R\unicode{x2abd}R_\unicode{x2217}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*11·1}.\supset&\vdash\colon\colon z\in \mu.zRw.\supset_{z,w}.w\in \mu:\supset\colon\ldotp x\in \mu.xRy.\supset.y\in \mu\colon\ldotp \\ +[\text{Exp.Comm}] &\supset\colon\ldotp xRy.\supset:x\in \mu.\supset.y\in \mu &\qquad \text{(1)}\\ +\vdash.\text{(1).Comm.Imp}.\supset\\ +&\vdash\colon\colon xRy.\supset\colon\ldotp z\in \mu.zRw.\supset_{z,w}.w\in \mu:x\in \mu:\supset.y\in \mu &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·21}.\supset\\ +&\vdash\colon\colon xRy.\supset\colon\ldotp z\in \mu.zRw.\supset_{z,w}.w\in \mu:x\in \mu:\supset_{\mu}.y\in \mu\colon\ldotp \\ +[\text{*90·111.*33·17}] &\supset\colon\ldotp xR_\unicode{x2217}y\colon\colon \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·16.</b> \(\vdash.R_\unicode{x2217}\mid R\unicode{x2abd}R_\unicode{x2217}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*11·1}.\supset\vdash\colon\ldotp z\in \mu.zRw.\supset_{z,w}.w\in \mu:\supset:y\in \mu.yRv.\supset.v\in \mu &\qquad \text{(1)}\\ +\vdash.\text{*90·111.*10·1.Fact}.\supset\\ +\vdash\colon\colon xR_\unicode{x2217}y.yRv.\supset\colon\ldotp z\in \mu.zRw.\supset_{z,w}.w\in \mu:x\in \mu:\supset.y\in \mu.yRv &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\\ +\vdash\colon\colon xR_\unicode{x2217}y.yRv.\supset\colon\ldotp z\in \mu.zRw.\supset_{z,w}.w\in \mu:x\in \mu:\supset.v\in \mu &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·21.*90·111}.\supset\\ +\vdash:xR_\unicode{x2217}y.yRv.\supset.xR_\unicode{x2217}v &\qquad \text{(4)}\\ +\vdash.\text{(4).*10·11·23.*34·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_580">[Pg 580]</span></p> + +<p class="nind"><b>*90·161</b> \(\vdash :S\unicode{x2abd} R_{\unicode{x2217}}\ldotp \supset \ldotp S\mid R\unicode{x2abd} R_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·34}. \supset \vdash :\text{Hp}\ldotp \supset \ldotp S\mid R\unicode{x2abd} R_{\unicode{x2217}}\mid R &\qquad \text{(1)}\\ +\vdash.\text{(1).*90·16}.\supset \vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·162</b> \(\vdash \ldotp R^{2}\unicode{x2abd} R_{\unicode{x2217}} \quad [\text{*90·151·161}]\)</p> + +<p class="nind"><b>*90·163</b> \(\vdash \ldotp \breve{R}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx\subset \overleftarrow{R}_{\unicode{x2217}}ʻx \quad[\text{*37·301.·32·19.*90·16}]\)</p> + +<p>This proposition is important, since it proves that +\(\overleftarrow{R}_{\unicode{x2217}}ʻx\) is a hereditary class.</p> + +<p class="nind"><b>*90·164</b> \(\vdash \ldotp \breve{R}ʻʻ\breve{R}_{\unicode{x2217}}ʻʻ\alpha \subset \breve{R}_{\unicode{x2217}}ʻʻ\alpha \quad[\text{*37·33·201.*90·16}]\)</p> + +<p>This proposition shows that \(\breve{R}_{\unicode{x2217}}ʻʻ\alpha \) is +a hereditary class.</p> + +<p class="nind"><b>*90·17</b> \(\vdash \ldotp R_{\unicode{x2217}}^{2}=R_{\unicode{x2217}}\)</p> + +<p>Note that \(R_{\unicode{x2217}}^{2}\) means (\(R_{\unicode{x2217}})^{2},\text{ not }(R^{2})_{\unicode{x2217}}\).</p> + +<p><i>Dem</i>.</p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·13}. \supset \vdash :xR_{\unicode{x2217}}y\ldotp &\supset \ldotp xR_{\unicode{x2217}}y\ldotp yR_{\unicode{x2217}}y\ldotp\\ +[\text{*34·5.*10·24}] &\supset \ldotp xR_{\unicode{x2217}}^{2}y &\quad\text{(1)}\\ +\vdash.\text{*90·163·1} \frac{\overleftarrow{R}ʻx}{\mu}. &\supset \vdash \colon\ldotp yR_{\unicode{x2217}}z\ldotp \supset :y\epsilon \overleftarrow{R}_{\unicode{x2217}}ʻx\ldotp + \supset \ldotp z\epsilon \overleftarrow{R}_{\unicode{x2217}}ʻx:\\ +[\text{*32·181}] &\supset :xR_{\unicode{x2217}}y\ldotp \supset \ldotp xR_{\unicode{x2217}}z&\quad \text{(2)}\\ +\vdash.\text{(2).Imp}.&\supset \vdash :xR_{\unicode{x2217}}y\ldotp yR_{\unicode{x2217} }z\ldotp \supset \ldotp xR_{\unicode{x2217}}z:\\ +[\text{*11·11.*34·55}] &\supset \vdash :R_{\unicode{x2217}}^{2}\unicode{x2abd} R_{\unicode{x2217}}&\quad\text{(3)}\\ +\vdash.\text{(1).(3)}. \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·171</b> \(\vdash \ldotp \breve{R}_{\unicode{x2217}}ʻʻ\breve{R}_{\unicode{x2217} }ʻʻ\alpha =\breve{R}_{\unicode{x2217}}ʻʻ\alpha \quad[\text{*90·17.*37·33}]\)</p> + +<p class="nind"><b>*90·172</b> \(\vdash \ldotp R\mid R_{\unicode{x2217}}\unicode{x2abd} R_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·151}. &\supset \vdash \ldotp R\mid R_{\unicode{x2217} }\unicode{x2abd} R_{\unicode{x2217}}^{2} &\qquad \text{(1)}\\ +\vdash.\text{(1).*90·17}. &\supset \vdash \ldotp \text{Prop}\\ +\end{array} +\]</p> + +<p class="nind"><b>*90·18</b> \(\vdash :P\unicode{x2abd} Q\ldotp \supset \ldotp P_{\unicode{x2217} }\unicode{x2abd} Q_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·265}. \supset \vdash \colon\ldotp \text{Hp}\ldotp &\supset :x\in CʻP\ldotp \supset \ldotp x\in CʻQ &\quad \text{(1)}\\ +\vdash.\text{*37·201}. &\supset \vdash \colon\colon \text{Hp}\ldotp \supset \colon\ldotp \breve{P}ʻʻ\mu \subset \breve{Q}ʻʻ\mu \colon\ldotp\\ +[\text{*22·44}] &\supset \colon\ldotp \breve{Q}ʻʻ\mu \subset \mu \ldotp \supset \ldotp \breve{P}ʻʻ\mu \subset \mu \colon\ldotp\\ +[\text{Fact}] &\supset \colon\ldotp \breve{Q}ʻʻ\mu \subset \mu \ldotp x\in \mu \ldotp \supset \ldotp \breve{P}ʻʻ\mu \subset \mu \ldotp x\in \mu \colon\ldotp\\ +[\text{Syll}] &\supset \colon\ldotp \breve{P}ʻʻ\mu \subset \mu \ldotp x\in \mu \ldotp \supset \ldotp y\in \mu :\supset :\breve{Q}ʻʻ\mu \subset \mu \ldotp x\in \mu \ldotp \supset \ldotp y\in \mu &\quad \text{(2)}\\ +\vdash.\text{(2).*10·11·21·27}. \supset\\ +\vdash \colon\colon \text{Hp}. &\supset \colon\ldotp \breve{P}ʻʻ\mu \subset \mu \ldotp x\in \mu \ldotp \supset _{\mu }\ldotp + y\in \mu :\supset :\breve{Q}ʻʻ\mu \subset \mu \ldotp x\in \mu \ldotp \supset _{\mu }\ldotp y\in \mu &\quad \text{(3)}\\ +\vdash.\text{(1).(3).*90·1}.&\supset \vdash \colon\ldotp \text{Hp}\ldotp \supset :xP_{\unicode{x2217}}y\ldotp \supset \ldotp xQ_{\unicode{x2217}}y\colon\ldotp + \supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_581">[Pg 581]</span></p> + +<p class="nind"><b>*90·21.</b> \(\vdash:\alpha\subset CʻR . \equiv .\alpha\subset \breve{R}_\unicode{x2217}ʻʻ\alpha. \equiv .\alpha\subset R_\unicode{x2217}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·7}.&\supset\vdash\colon\ldotp \alpha\subset CʻR.\supset:x\in \alpha.\supset.x\in \alpha.x\in CʻR.\\ +[\text{*90·12}] &\supset.x\in \alpha.xR_\unicode{x2217}x.\\ +[\text{*10·24.*37·1·105}] &\supset.x\in \breve{R}_\unicode{x2217}ʻʻ\alpha.x\in R_\unicode{x2217}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*37·16}.&\supset\vdash:\alpha\subset \breve{R}_\unicode{x2217}ʻʻ\alpha.\supset.\alpha\subset \text{ᗡ}ʻR_\unicode{x2217}.\\ +[\text{*90.14}] &\supset.\alpha\subset CʻR &\qquad \text{(2)}\\ +\vdash.\text{*37·15.*90·14}.&\supset\vdash:\alpha\subset R_\unicode{x2217}ʻʻ\alpha.\supset.\alpha\subset CʻR &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·22.</b> \(\vdash:\breve{R}ʻʻ\alpha\subset \alpha. \equiv .\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·1}.&\supset\vdash\colon\ldotp xR_\unicode{x2217}y.\supset_{x,y}:\breve{R}ʻʻ\alpha\subset \alpha.x\in \alpha.\supset.y\in \alpha\colon\ldotp\\ +[\text{Comm}] &\supset\vdash\colon\ldotp \breve{R}ʻʻ\alpha\subset \alpha.\supset:xR_\unicode{x2217}y.x\in \alpha.\supset_{x,y}.y\in \alpha:\\ +[\text{*37·171}] &\supset:\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \alpha &\qquad \text{(1)}\\ +\vdash.\text{*90·151.*37·201}.&\supset\vdash.\breve{R}ʻʻ\alpha\subset \breve{R}_\unicode{x2217}ʻʻ\alpha.\\ +[\text{*22·44}] &\supset\vdash:\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \alpha.\supset.\breve{R}ʻʻ\alpha\subset \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·23.</b> \(\vdash:\alpha\subset CʻR.\breve{R}ʻʻ\alpha\subset \alpha. \equiv .\alpha = \breve{R}_\unicode{x2217}ʻʻ\alpha \quad[\text{*90·21·22}]\)</p> + +<p>*90·23 is useful in the theory of sections of a series (*211). A +section of the series generated by \(R\) is defined as a class +\(\alpha\) satisfying +\[ +\alpha\subset CʻR.Rʻʻ\alpha\subset \alpha. +\]</p> + +<p class="nind"><b>*90·24.</b> \(\vdash:\breve{R}ʻʻ\mu\subset \mu.\alpha\subset \mu.\supset.\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·2}. &\supset\vdash:\text{Hp}.\supset.\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \breve{R}_\unicode{x2217}ʻʻ\mu &\qquad \text{(1)}\\ +\vdash.\text{*90·22}. &\supset\vdash:\text{Hp}.\supset.\breve{R}_\unicode{x2217}ʻʻ\mu\subset \mu &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p>This proposition shows that if \(\mu\) is a hereditary class which +contains \(\alpha\), then \(\mu\) contains all the descendants of +\(\alpha\)'s.</p> + +<p class="nind"><b>*90·25.</b> \(\vdash:\alpha\subset CʻR.\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \mu.\supset.\alpha\subset \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·21}.\supset\vdash:\text{Hp}.&\supset.\alpha\subset \breve{R}_\unicode{x2217}ʻʻ\alpha.\\ +[\text{Hp}] &\supset.\alpha\subset \mu:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_582">[Pg 582]</span></p> + +<p class="nind"><b>*90·26.</b> \(\vdash\colon\ldotp \alpha\subset CʻR.\breve{R}ʻʻ\mu\subset \mu.\supset:\alpha\subset \mu. \equiv .\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·24}. &\supset\vdash\colon\ldotp Hp.\supset:\alpha\subset \mu.\supset.\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \mu &\qquad \text{(1)}\\ +\vdash.\text{*90·25}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \mu.\supset.\alpha\subset \mu &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·27.</b> \(\vdash\colon\ldotp \alpha\subset CʻR.\supset:\alpha\cup \breve{R}ʻʻ\mu\subset \mu. \equiv .\breve{R}_\unicode{x2217}ʻʻ\alpha\cup \breve{R}ʻʻ\mu\subset \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·26.Exp.*5·32}.\supset\\ +\vdash\colon\ldotp \alpha\subset CʻR.&\supset:\breve{R}ʻʻ\mu\subset \mu.\alpha\subset \mu. \equiv .\breve{R}ʻʻ\mu\subset \mu.\breve{R}_\unicode{x2217}ʻʻ\alpha\subset \mu:\\ +[\text{*22·59}] &\supset:\alpha\cup \breve{R}ʻʻ\mu\subset \mu. \equiv .\breve{R}_\unicode{x2217}ʻʻ\alpha\cup \breve{R}ʻʻ\mu\subset \mu\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·31.</b> \(\vdash.R_\unicode{x2217} = I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·15·16}.&\supset\vdash.I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R\unicode{x2abd}R_\unicode{x2217} &\qquad \text{(1)}\\ +[\text{Fact}] &\supset\vdash:x(I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R)z.zRw.\supset.xR_\unicode{x2217}z.zRw.\\ +[\text{*10·24.*34·1}] &\supset.x(R_\unicode{x2217}\mid R)w.\\ +[\text{*23·58}] &\supset.x(I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R)w &\qquad \text{(2)}\\ +\vdash.\text{*90·13.*50·3}.&\supset\vdash:xR_\unicode{x2217}y.\supset.xIx.x\in CʻR.\\ +[\text{*35·101}] &\supset.x(I\upharpoonright CʻR)x.\\ +[\text{*23·58}] &\supset.x(I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R)x.\\ +[\text{*4·7}] &\supset.xR_\unicode{x2217}y.x(I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R)x &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*90·112}\, \frac{x(I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R)z}{\phi z} .\supset\\ +&\vdash:xR_\unicode{x2217}y.\supset.x(I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R)y &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the last line of the above proof, the process is as +follows. Writing \(\phi z\) for \(x(I\upharpoonright CʻR\unicode{x228d}R_\unicode{x2217}\mid R)z\), +(2) becomes \(\phi z.zRw.\supset.\phi w\), while (3) becomes +\(xR_\unicode{x2217}y.\supset.xR_\unicode{x2217}y.\phi x\). Hence, by +(2) and (3), +\[ +xR_\unicode{x2217}y.\supset:xR_\unicode{x2217}y:\phi z.zRw.\supset_{z,w}.\phi w:\phi x. +\] +Hence, by <a href="#*90·112">*90·112</a>, \(xR_\unicode{x2217}y.\supset.\phi y\), which is the +proposition to be proved.</p> + +<p class="nind"><b>*90·311.</b> \(\vdash.R_\unicode{x2217} = I\upharpoonright CʻR\unicode{x228d}R\mid R_\unicode{x2217}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·31} \frac{\breve{R}}{R}.\text{*90·132}.\supset\\ +\vdash.\breve{R}_\unicode{x2217} &= I\upharpoonright Cʻ\breve{R}\unicode{x228d}\breve{R}_\unicode{x2217}\mid \breve{R}\\ +[\text{*33·22.*34·2}] &= I\upharpoonright CʻR\unicode{x228d}\text{Cnv}ʻ(R\mid R_\unicode{x2217})\\ +[\text{*50·5·51}] &= \text{Cnv}ʻ(I\upharpoonright CʻR)\unicode{x228d}\text{Cnv}ʻ(R\mid R_\unicode{x2217})\\ +[\text{*31·15}] &= \text{Cnv}ʻ(I\upharpoonright CʻR\unicode{x228d}R\mid R_\unicode{x2217}) &\qquad \text{(1)}\\ +\vdash.\text{(1).*31·32}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_583">[Pg 583]</span></p> + +<p class="nind"><b>*90·32.</b> \(\vdash.R\mid R_{\unicode{x2217}}=R\unicode{x228d}R\mid R_{\unicode{x2217}}\mid R=R_{\unicode{x2217}}\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·31}.\supset\vdash.R\mid R_{\unicode{x2217}}&=R\mid I\upharpoonright CʻR\unicode{x228d}R\mid R_{\unicode{x2217}}\mid R\\ +[\text{*50·64}] &=R\unicode{x228d}R\mid R_{\unicode{x2217}}\mid R &\qquad \text{(1)}\\ +[\text{*50·65}] &=(I\upharpoonright CʻR)\mid R\unicode{x228d}R\mid R_{\unicode{x2217}}\mid R\\ +[\text{*90·311.*34·26}] &=R_{\unicode{x2217}}\mid R &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·33.</b> \(\vdash.R_{\unicode{x2217}}ʻʻ\alpha=(\alpha\cap CʻR)\cup R_{\unicode{x2217}}ʻʻRʻʻ\alpha=(\alpha\cap CʻR)\cup RʻʻR_{\unicode{x2217}}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·31.*37·221}.\supset\\ +\vdash.R_{\unicode{x2217}}ʻʻ\alpha=(I\upharpoonright CʻR)ʻʻ\alpha\cup (R_{\unicode{x2217}}\mid R)ʻʻ\alpha\\ +[\text{*37·412·33}] = Iʻʻ(CʻR\cap \alpha)\cup R_{\unicode{x2217}}ʻʻRʻʻ\alpha\\ +[\text{*50·16}] = (CʻR\cap \alpha)\cup R_{\unicode{x2217}}ʻʻRʻʻ\alpha &\qquad \text{(1)}\\ +\text{Similarly, by *90·311},\\ +\vdash.R_{\unicode{x2217}}ʻʻ\alpha=(CʻR\cap \alpha)\cup RʻʻR_{\unicode{x2217}}ʻʻ\alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·331.</b> \[\begin{align}&\vdash.\breve{R}_{\unicode{x2217}}ʻʻ\alpha=(\alpha\cap CʻR)\cup \breve{R}_{\unicode{x2217}}ʻʻ\breve{R}ʻʻ\alpha=(\alpha\cap + CʻR)\cup \breve{R}ʻʻ\breve{R}_{\unicode{x2217}}ʻʻ\alpha\\ +&[\text{Proof as in *90·33}]\end{align}\]</p> + +<p class="nind"><b>*90·34.</b> \[\begin{align}&\vdash:\alpha\subset CʻR.\supset.R_{\unicode{x2217}}ʻʻ\alpha=\alpha\cup R_{\unicode{x2217}}ʻʻRʻʻ\alpha=\alpha\cup + RʻʻR_{\unicode{x2217}}ʻʻ\alpha\\ +&[\text{*90·33.*22·621}]\end{align}\]</p> + +<p class="nind"><b>*90·341.</b> \[\begin{align}&\vdash:\alpha\subset CʻR.\supset.\breve{R}_{\unicode{x2217}}ʻʻ\alpha=\alpha\cup \breve{R}_{\unicode{x2217}}ʻʻ\breve{R}ʻʻ\alpha=\alpha\cup + \breve{R}ʻʻ\breve{R}_{\unicode{x2217}}ʻʻ\alpha\\ +&[\text{*90·331.*22·621}]\end{align}\]</p> + +<p class="nind"><b>*90·35.</b> \(\vdash\colon\ldotp xR\mid R_{\unicode{x2217}}z.\supset:\breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset_{\mu}.z\in \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·181}.\supset\vdash\colon\ldotp xRy.&\supset:y\in \overleftarrow{R}ʻx:\\ +[\text{*22·46}] &\supset:\overleftarrow{R}ʻx\subset \mu.\supset.y\in \mu:\\ +[\text{Fact}] &\supset:\breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset.\breve{R}ʻʻ\mu\subset \mu.y\in \mu &\qquad \text{(1)}\\ +\vdash.\text{*90·1}. &\supset\vdash\colon\ldotp yR_{\unicode{x2217}}z.\supset:\breve{R}ʻʻ\mu\subset \mu.y\in \mu.\supset.z\in \mu &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp xRy.yR_{\unicode{x2217}}z.\supset:\breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset.z\in \mu\colon\ldotp\\ +[\text{*10·11·23.*34·1}]&\supset\vdash\colon\ldotp xR\mid R_{\unicode{x2217}}z.\supset:\breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset.z\in \mu &\qquad \text{(3)}\\ +\vdash.\text{(3).*10·11·21}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_584">[Pg 584]</span></p> + +<p class="nind"><b>*90·351.</b> \(\vdash\colon\ldotp \breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset_{\mu}.z\in \mu:\supset.xR\mid R_{\unicode{x2217}}z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·172.Fact}.\supset\vdash:xR\mid R_{\unicode{x2217}}z.zRw.&\supset.xR_{\unicode{x2217}}z.zRw.\\ +[\text{*34·1}] &\supset.xR_{\unicode{x2217}}\mid Rw.\\ +[\text{*90·32}] & \supset.xR\mid R_{\unicode{x2217}}w &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·171}.&\supset\vdash.\breve{R}ʻʻ\hat{z}(xR\mid R_{\unicode{x2217}}z)\subset \hat{z}(xR\mid R_{\unicode{x2217}}z) &\qquad \text{(2)}\\ +\vdash.\text{*90·32}. & \supset\vdash:xRy.\supset.xR\mid R_{\unicode{x2217}}y:\\ +[\text{*32·181.*20·3}] &\supset\vdash:y\in \overleftarrow{R}ʻx.\supset.y\in \hat{z}(xR\mid R_{\unicode{x2217}}z):\\ +[\text{*10·11.*22·1}] &\supset\vdash.\overleftarrow{R}ʻx\subset \hat{z}(xR\mid R_{\unicode{x2217}}z) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*10·1}.\supset\\ +\vdash\colon\ldotp \breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset_{\mu}.z\in \mu:&\supset.z\in \hat{z}(xR\mid R_{\unicode{x2217}}z).\\ +[\text{*20·3}] &\supset.xR\mid R_{\unicode{x2217}}z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·36.</b> \(\vdash\colon\ldotp xR\mid R_{\unicode{x2217}}z.\equiv:\breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset_{\mu}.z\in + \mu \quad[\text{*90·35·351}]\)</p> + +<p class="nind"><b>*90·4.</b> \(\vdash.(R_{\unicode{x2217}})_{\unicode{x2217}}=R_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·151·18}.\supset\vdash.R_{\unicode{x2217}}\unicode{x2abd}(R_{\unicode{x2217}})_{\unicode{x2217}} &&\qquad \text{(1)}\\ +\vdash.\text{*90·112}\, \frac{R_{\unicode{x2217}},xR_{\unicode{x2217}}z}{R,\,\,\phi z} .\supset\\ +\vdash\colon\ldotp x(R_{\unicode{x2217}})_{\unicode{x2217}}y:xR_{\unicode{x2217}}z.zR_{\unicode{x2217}}w.\supset_{z,w}.xR_{\unicode{x2217}}w:xR_{\unicode{x2217}}x:\supset.xR_{\unicode{x2217}}y &&\qquad \text{(2)}\\ +\vdash.\text{*90·13}.\supset\vdash:x(R_{\unicode{x2217}})_{\unicode{x2217}}y.&\supset.x\in CʻR_{\unicode{x2217}}.\\ +[\text{*90·14}] & \supset.x\in CʻR.\\ +[\text{*90·12}] &\supset.xR_{\unicode{x2217}}x &\qquad \text{(3)}\\ +\vdash.\text{*90·17}.&\supset\vdash:xR_{\unicode{x2217}}z.zR_{\unicode{x2217}}w.\supset_{z,w}.xR_{\unicode{x2217}}w &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4)}.&\supset\vdash:x(R_{\unicode{x2217}})_{\unicode{x2217}}y.\supset.xR_{\unicode{x2217}}y &\qquad \text{(5)}\\ +\vdash.\text{(1).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·41.</b> \(\vdash.CʻP_{\unicode{x2217}}\unicode{x0294f}\alpha=\alpha\cap CʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·41}. & \supset\vdash.CʻP_{\unicode{x2217}}\unicode{x0294f}\alpha=\alpha\cap (P_{\unicode{x2217}}ʻʻ\alpha\cup \breve{P}_{\unicode{x2217}}ʻʻ\alpha) + &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·15·16.*90·14}.& \supset\vdash.CʻP_{\unicode{x2217}}\unicode{x0294f}\alpha \subset \alpha \cap CʻP &\qquad \text{(2)}\\ +\vdash.\text{*90·33·331}.&\supset\vdash.\alpha\cap CʻP\subset P_{\unicode{x2217}}ʻʻ\alpha\cap \breve{P}_{\unicode{x2217}}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(3).(1)}. &\supset\vdash.\alpha\cap CʻP\subset CʻP_{\unicode{x2217}}\unicode{x0294f}\alpha &\qquad \text{(4)}\\ +\vdash.\text{(2).(4)}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*90·42.</b> \(\vdash.(Q_{\unicode{x2217}}\unicode{x0294f}\alpha)_{\unicode{x2217}}=Q_{\unicode{x2217}}\unicode{x0294f}\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·18}.&\supset\vdash.(Q_{\unicode{x2217}}\unicode{x0294f}\alpha)_{\unicode{x2217}}\unicode{x2abd}(Q_{\unicode{x2217}})_{\unicode{x2217}}\\ +[\text{*90·4}] &\qquad\qquad\qquad\unicode{x2abd}Q_{\unicode{x2217}} &\qquad \text{(1)}\\ +\vdash.\text{*90·13}. &\supset\vdash:x(Q_{\unicode{x2217}}\unicode{x0294f}\alpha)_{\unicode{x2217}}y.\supset.x,y\in CʻQ_{\unicode{x2217}}\unicode{x0294f}\alpha.\\ +[\text{*90·41}] &\supset.x,\,y\in \alpha &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.(Q_{\unicode{x2217}}\unicode{x0294f}\alpha)_{\unicode{x2217}}\unicode{x2abd}Q_{\unicode{x2217}}\unicode{x0294f}\alpha + &\qquad \text{(3)}\\ +\vdash.\text{(3).*90·151}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_585">[Pg 585]</span></p> +<h2 class="nobreak" id="*91">*91. ON POWERS OF A RELATION.</h2> +</div> + + +<p><i>Summary of</i> *91.</p> + +<p>In the present number, we consider the class of relations +\[ +R, R^{2}, R^{3}, \ldots +\]</p> + +<p>Each of these has to its predecessor the relation \(\mid R\); we have +\[ +R^{2} = \mid RʻR, R^{3} = \mid RʻR^{2}, \text{etc.} +\] +Thus every term of the series has to \(R\) the relation (\(\mid +R)_{\unicode{x2217}}\); hence the powers of \(R\) may be defined +as those relations which have to \(R\) the relation (\(\mid +R)_{\unicode{x2217}}\). The series of powers starting with \(I +\upharpoonright CʻR\) instead of with \(R\) is similarly composed of +those relations which have to \(I \upharpoonright CʻR\) the relation +\((\mid R)_{\unicode{x2217}}\). (This class consists of the previous +class together with \(I \upharpoonright CʻR\).) To say that the +relation \(R_{\unicode{x2217}}\) holds between \(x\) and \(y\) turns +out to be equivalent to saying that one of the relations +\[ +I \upharpoonright CʻR,\quad R,\quad R^{2},\quad R^{3},\, \ldots +\] +holds between \(x\) and \(y\); and to say that the relation +\(R \mid R_{\unicode{x2217}}\) holds between \(x\) and \(y\) +turns out to be equivalent to saying that one of the relations +\[ +R,\quad R^{2},\quad R^{3}, \ldots +\] +holds between \(x\) and \(y\). Thus we might have begun by defining +powers of \(R\), and proceeded to define \(R_{\unicode{x2217}}\) as +their sum.</p> + +<p>For notational convenience we put +\[ +R_{\text{ts}} = (\mid R)_{\unicode{x2217}} \quad \text{Df}\text{.} +\] +Then the definition of powers of \(R\) excluding \(I \upharpoonright CʻR\) is +\[ +\text{Pot}ʻR = \overrightarrow{R}_{\text{ts}}ʻR \quad \text{Df}\text{,} +\] +and the definition of powers of \(R\) including \(I \upharpoonright CʻR\) is +\[ +\text{Potid}ʻR = \overrightarrow{R}_{\text{ts}}ʻ(I \upharpoonright CʻR) \quad \text{Df}\text{.} +\] +(Here the letters "id" are added to suggest that identity is to be +added to \(\text{Pot}ʻR\).)</p> + +<p>We put also +\[ +R_{\text{po}}=\dot{s}ʻ\text{Pot}ʻR \quad \text{Df}\text{.} +\]</p> + +<p><span class="pagenum" id="Page_586">[Pg 586]</span></p> + +<p>Many of the propositions in this number are very often used. Among the +more important propositions are the following:</p> + +<p class="nind"><b>*91·17.</b> \(\vdash\colon\ldotp P\in \text{Potid}ʻR:\phi S.\supset_{S}.\phi(S\mid R):\phi(I\upharpoonright CʻR):\supset.\phi P\)</p> + +<p class="nind"><b>*91·171.</b> \(\vdash\colon\ldotp P\in \text{Pot}ʻR:\phi S.\supset_{S}.\phi(S\mid R):\phi R:\supset.\phi P\)</p> + +<p class="nind"><b>*91·373.</b> \(\vdash\colon\ldotp P\in \text{Pot}ʻR.\supset_{P}.\phi P:\equiv:\phi R:S\in \text{Pot}ʻR.\phi S.\supset_{S}.\phi(S\mid R)\)</p> + +<p>These are formulae of induction. The first two state that if the +property \(\phi\) is hereditary with respect to \(\mid R\), then if +\(\phi\) belongs to \(I\upharpoonright CʻR\) it belongs to any member +of \(\text{Potid}ʻR\), while if \(\phi\) belongs to \(R\) it belongs +to any member of \(\text{Pot}ʻR\). The third gives a form of induction +which is sometimes more powerful than the second. It states that if +\(\phi\) is hereditary provided its argument is a power of \(R\), and +if \(\phi R\), then every power of \(R\) satisfies \(\phi\), and vice +versa.</p> + +<p class="nind"><b>*91·23.</b> \(\vdash.\text{Potid}ʻR=\iotaʻ(I\upharpoonright CʻR)\cup \text{Pot}ʻR\)</p> + +<p class="nind"><b>*91·24.</b> \(\vdash.\text{Pot}ʻR=\mid Rʻʻ\text{Potid}ʻR\)</p> + +<p>These two propositions are very useful as giving relations of +\(\text{Pot}ʻR\) and \(\text{Potid}ʻR\).</p> + +<p class="nind"><b>*91·27.</b> \(\vdash:P\in \text{Potid}ʻR.\supset.CʻP\subset CʻR\)</p> + +<p class="nind"><b>*91·271.</b> \(\vdash:P\in \text{Pot}ʻR.\supset.\text{D}ʻP\subset DʻR.\text{ᗡ}ʻP\subset \text{ᗡ}ʻR\)</p> + +<p>We do not have in general \(P\in \text{Pot}ʻR.\supset.\text{D}ʻP=\text{D}ʻR.\text{ᗡ}ʻP=\text{ᗡ}ʻR\). +If \(R\) is the sort of relation which generates a series (<i>i.e.</i> +is either itself serial, or such that \(R_{\text{po}}\) is serial), +the above would characterize a series without a first or last term. To +illustrate the matter, consider a series of four terms, \(x\), \(y\), +\(z\), \(w\), and let \(R\) be the relation of immediately preceding +in this series. Thus \(R\) holds between \(x\) and \(y\), \(y\) and +\(z\), \(z\) and \(w\). Then \(R^{2}\) holds between \(x\) and \(z\), +\(y\) and \(w\); thus \(z\), which belongs to \(\text{D}ʻR\), does not +belong to \(\text{D}ʻR^{2}\). \(R^{3}\) holds only between \(x\) and \(w\); +thus neither \(y\) nor \(z\) belongs to \(\text{D}ʻR^{3}\). All powers +of \(R\) beyond the third are null. On the other hand, if we take a +cyclic relation, such as that of left-hand neighbour at a dinner-table, +we shall always have \(\text{D}ʻP=\text{D}ʻR.\text{ᗡ}ʻP=\text{ᗡ}ʻR\), +whatever power of \(R\, P\) may be.</p> + +<p class="nind"><b>*91·282.</b> \(\vdash:P\in \text{Pot}ʻR.\supset.P\mid R\in \text{Pot}ʻR\)</p> + +<p>This proposition shows that \(\text{Pot}ʻR\) is a hereditary class with +respect to \(\mid R\).</p> + +<p class="nind"><b>*91·34.</b> \(\vdash:P,\,Q\in \text{Potid}ʻR.\supset.P\mid Q=Q\mid P\)</p> + +<p>This proposition states that the relative product is commutative when +each factor is \(I\upharpoonright CʻR\) or a power of \(R\).</p> + +<p>We come next to propositions concerning \(R_{\text{po}}\). We have</p> + +<p class="nind"><b>*91·502.</b> \(\vdash.R\unicode{x2abd}R_{\text{po}}\)</p> + +<p><span class="pagenum" id="Page_587">[Pg 587]</span></p> + +<p class="nind"><b>*91·504.</b> \(\vdash.\text{D}ʻR_{\text{po}}=\text{D}ʻR.\text{ᗡ}ʻR_{\text{po}}=\text{ᗡ}ʻR.CʻR_{\text{po}}=CʻR\)</p> + +<p class="nind"><b>*91·511.</b> \(\vdash.R_{\text{po}}\mid R\unicode{x2abd}R_{\text{po}}\)</p> + +<p class="nind"><b>*91·52.</b> \(\vdash.R_{\text{po}}=R_{\unicode{x2217}}\mid R=R\mid R_{\unicode{x2217}}\)</p> + +<p class="nind"><b>*91·54.</b> \(\vdash.R_{\unicode{x2217}}=I\upharpoonright CʻR\unicode{x228d}R_{\text{po}}\)</p> + +<p>*91·52·54 are fundamental in the theory of inductive relations.</p> + +<p class="nind"><b>*91·542.</b> \(\vdash:xR_{\unicode{x2217}}y.x \neq y.\equiv.xR_{\text{po}}y.x \neq y\)</p> + +<p>This proposition is particularly useful when (as often happens) +we have \(R_{\text{po}}\unicode{x2abd}J\). In that case, it gives +\(R_{\text{po}}=R_{\unicode{x2217}}\dot{\cap}J\).</p> + +<p class="nind"><b>*91·55.</b> \(\vdash.R_{\unicode{x2217}}=\dot{s}ʻ\text{Potid}ʻR\)</p> + +<p class="nind"><b>*91·56.</b> \(\vdash.R_{\text{po}}^{2}\unicode{x2abd}R_{\text{po}}\)</p> + +<p>Thus \(R_{\text{po}}\) is always transitive, which is one of the three +characteristics of serial relations (cf. *204). We shall find that +\(R_{\text{po}}\) is often serial when \(R\) is not so.</p> + +<p class="nind"><b>*91·574.</b> \(\vdash.R_{\unicode{x2217}}\mid R_{\text{po}}=R_{\text{po}}\mid R_{\unicode{x2217}}=R_{\text{po}}=R\mid + R_{\unicode{x2217}}=R_{\unicode{x2217}}\mid R\)</p> + +<p class="nind"><b>*91·602.</b> \(\vdash.(R_{\text{po}})_{\unicode{x2217}}=R_{\unicode{x2217}}\)</p> + +<hr class="tb"> + +<p class="nind"><b>*91·01.</b> \(R_{\text{st}}=(R\mid)_{\unicode{x2217}} \quad\text{Df}\)</p> + +<p class="nind"><b>*91·02.</b> \(R_{\text{ts}}=(\mid R)_{\unicode{x2217}} \quad\text{Df}\)</p> + +<p class="nind"><b>*91·03.</b> \(\text{Pot}ʻR=\overrightarrow{R}_{\text{ts}}ʻR \quad\text{Df}\)</p> + +<p class="nind"><b>*91·04.</b> \(\text{Potid}ʻR=\overrightarrow{R}_{\text{ts}}ʻ(I\upharpoonright CʻR) \quad\text{Df}\)</p> + +<p class="nind"><b>*91·05.</b> \(R_{\text{po}}=\dot{s}ʻ\text{Pot}ʻR \quad\text{Df}\)</p> + +<p>The first two of the above definitions are introduced merely for +notational convenience. The other three represent ideas of great +importance. The last is especially useful when a series is given as the +field of a one-one relation between consecutive terms—as, <i>e.g.</i>, +when the series of natural numbers is given as the field of the +relation of \(n\) to \(n + 1\). Then \(R_{\text{po}}\) is the relation +of any earlier term to any later term—<i>e.g.</i>, in the above case +of the natural numbers, the relation of a less integer to a greater.</p> + +<p class="nind"><b>*91·1.</b> \(\vdash\colon\colon PR_{\text{st}}Q.\equiv\colon\ldotp S\in \mu.\supset_{S}.R\mid S\in \mu:Q\in \mu:\supset_{\mu}.P\in \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*4·2.(*91·01)}.\supset\\ +\vdash\colon\colon PR_{\text{st}}Q.&\equiv\colon\ldotp P(R\mid )_{\unicode{x2217}}Q\colon\ldotp \\ +[\text{*90·11}] & \equiv\colon\ldotp P\in Cʻ(R\mid ):(R\mid )ʻʻ\mu\subset \mu.Q\in \mu.\supset_{\mu}.P\in \mu\colon\ldotp \\ +[\text{*43·3.*33·161}]&\equiv\colon\ldotp (R\mid )ʻʻ\mu\subset \mu.Q\in \mu.\supset_{\mu}.P\in \mu\colon\ldotp \\ +[\text{*37·61}] &\equiv\colon\ldotp S\in \mu.\supset_{S}.R\mid ʻS\in \mu:Q\in \mu:\supset_{\mu}.P\in \mu\colon\ldotp \\ +[\text{*43·11}] &\equiv\colon\ldotp S\in \mu.\supset_{S}.R\mid S\in \mu:Q\in \mu:\supset_{\mu}.P\in \mu\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_588">[Pg 588]</span></p> + +<p class="nind"><b><a id="*91·11">*91·11</a>.</b> \(\vdash\colon\colon PR_{\text{ts}}Q.\equiv\colon\ldotp S\in \mu.\supset_{S}.S\mid R\in \mu:Q\in \mu:\supset_{\mu}.P\in \mu\)</p> + +<p class="nind"><b>*91·12.</b> \(\vdash:P\in \text{Pot}ʻR.\equiv.PR_{\text{ts}}R \quad[\text{*32·18.(*91·03)}]\)</p> + +<p class="nind"><b>*91·13.</b> \[\begin{align}&\vdash\colon\colon P\in \text{Pot}ʻR.\equiv\colon\ldotp S\in \mu.\supset_{S}.S\mid R\in \mu:R\in \mu:\supset_{\mu}.P\in \mu\\ +&[\text{*91·11·12}]\end{align}\]</p> + +<p class="nind"><b>*91·14.</b> \(\vdash:P\in \text{Potid}ʻR.\equiv.PR_{\text{ts}}(I\upharpoonright CʻR) \quad[\text{*32·18.(*91·04)}]\)</p> + +<p class="nind"><b>*91·15.</b> \[\begin{align}&\vdash\colon\colon P\in \text{Potid}ʻR.\equiv\colon\ldotp S\in \mu.\supset_{S}.S\mid R\in \mu:I\upharpoonright CʻR\in \mu:\supset_{\mu}.P\in + \mu\\ +&[\text{*91·11·14}]\end{align}\]</p> + +<p class="nind"><b>*91·16.</b> \[\begin{align}&\vdash\colon\colon xR_{\text{po}}y.\equiv\colon\ldotp (\exists P)\colon\ldotp S\in \mu.\supset_{S}.S\mid R\in \mu:R\in \mu:\supset_{\mu}.P\in + \mu\colon\ldotp xPy\\ +&[\text{*41·11.(*91·05).*91·13}]\end{align}\]</p> + +<p class="nind"><b>*91·17.</b> \[\begin{align}&\vdash\colon\ldotp P\in \text{Potid}ʻR:\phi S.\supset_{S}.\phi(S\mid R):\phi(I\upharpoonright CʻR):\supset.\phi P\\ +&\left[\text{*91·15} \frac{\hat{S}(\phi S)}{\mu}\right]\end{align}\]</p> + +<p class="nind"><b>*91·171.</b> \[\begin{align}&\vdash\colon\ldotp P\in \text{Pot}ʻR:\phi S.\supset_{S}.\phi(S\mid R):\phi R:\supset.\phi P\\ +&\left[*91·13 \frac{\hat{S}(\phi S)}{\mu}\right]\end{align}\]</p> + +<p>These propositions are of great importance, because they enable us to +prove that a property \(\phi\) belongs to every power of \(R\) if it +belongs to \(R\) (or \(I\upharpoonright CʻR\)) and also belongs to +\(S\mid R\) whenever it belongs to \(S\).</p> + +<p class="nind"><b>*91·2.</b> \(\vdash:QR_{\text{ts}}P.\supset.(Q\mid R)R_{\text{ts}}P\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·101.(*91·02)}.\supset\vdash:\text{Hp}.&\supset.(Q\mid R)(\mid R)Q.Q(\mid R)_{\unicode{x2217}}P.\\ +[\text{*90·172}] & \supset.(Q\mid R)(\mid R)_{\unicode{x2217}}P.\\ +[\text{Id.(*91·02)}] &\supset.(Q\mid R)R_{\text{ts}}P:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·201.</b> \(\vdash:QR_{\text{st}}P.\supset.(R\mid Q)R_{\text{st}}P \quad[\text{Proof as in *91·2}]\)</p> + +<p class="nind"><b>*91·204.</b> \(\vdash:P\{R_{\text{ts}}\mid (\mid R)\}Q.\equiv.PR_{\text{ts}}(Q\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·1}.\supset\vdash:P\{R_{\text{ts}}\mid (\mid R)\}Q.&\equiv.(\exists T).PR_{\text{ts}}T.T(\mid R)Q.\\ +[\text{*43·101}] &\equiv.(\exists T).PR_{\text{ts}}T.T=Q\mid R.\\ +[\text{*13·195}] &\equiv.PR_{\text{ts}}(Q\mid R):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·205.</b> \(\vdash:P\{R_{\text{st}}\mid (R\mid )\}Q.\equiv.PR_{\text{st}}(R\mid Q)\)</p> + +<p class="nind"><b>*91·21.</b> \(\vdash.R_{\text{ts}}=I\unicode{x228d}R_{\text{ts}}\mid (\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·31.(*91·02)}.\supset\vdash.R_{\text{ts}}&=I\upharpoonright Cʻ(\mid R)\unicode{x228d}R_{ts}\mid (\mid R)\\ +[\text{*43·311}] &=I\unicode{x228d}R_{\text{ts}}\mid (\mid R).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_589">[Pg 589]</span></p> + +<p class="nind"><b>*91·211</b> \(\vdash \ldotp R_{\text{st}}=I\unicode{x228d} R_{\text{st}}\mid (R\mid)\)</p> + +<p class="nind"><b>*91·212</b> \(\vdash \colon\ldotp PR_{\text{ts}}Q\ldotp \equiv :P=Q\ldotp \lor \ldotp PR_{\text{ts}}(Q\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·21.*50·1}. \supset \vdash \colon\ldotp PR_{\text{ts}}Q\ldotp &\equiv :P=Q\ldotp \lor \ldotp P\{R_{\text{ts}}\mid (\mid R)\}Q:\\ +[\text{*91·204}] &\equiv :P=Q\ldotp \lor \ldotp PR_{\text{ts}}(Q\mid R)\colon\ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·213</b> \(\vdash \colon\ldotp PR_{\text{st}}Q\ldotp \equiv :P=Q\ldotp \lor \ldotp PR_{\text{st}}(R\mid Q)\)</p> + +<p class="nind"><b>*91·22</b> \(\vdash \ldotp \overrightarrow{R}_{\text{ts}}ʻQ=\iota ʻQ\cup \overrightarrow{R}_{\text{ts}}ʻ(Q\mid R)\quad[\text{*91·212.*32·18.*51·15}]\)</p> + +<p class="nind"><b>*91·221</b> \(\vdash \ldotp \overrightarrow{R}_{\text{st}}ʻQ=\iota ʻQ\cup \overrightarrow{R}_{\text{st}}ʻ(R\mid Q)\)</p> + +<p class="nind"><b>*91·23</b> \(\vdash \ldotp \text{Potid}ʻR=\iota ʻ(I\upharpoonright CʻR)\cup \text{Pot}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*91·2.(*91·04)}. \supset \vdash \ldotp \text{Potid}ʻR&=\iota ʻ(I\upharpoonright CʻR)\cup \overrightarrow{R}_{\text{ts}}ʻ\{(I\upharpoonright CʻR)\mid R\}\\ +[\text{*50·65.(*91·03)}] &=\iota ʻ(I\upharpoonright CʻR)\cup \text{Pot}ʻR\ldotp \supset \vdash \ldotp \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·231</b> \(\vdash \ldotp \overrightarrow{R}_{\text{ts}}ʻI=\iota ʻI\cup \text{Pot}ʻR \quad[\text{*91·22.(*91·03).*50·4}]\)</p> + +<p class="nind"><b>*91·24</b> \(\vdash \ldotp \text{Pot}ʻR=\mid Rʻʻ\text{Potid}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·12}. \supset \vdash :P\in \text{Pot}ʻR\ldotp &\equiv \ldotp PR_{ts}R\ldotp\\ +[\text{*50·65}] &\equiv \ldotp PR_{\text{ts}}(I\upharpoonright CʻR\mid R)\ldotp\\ +[\text{*91·204}] &\equiv \ldotp P\{R_{\text{ts}}\mid (\mid R)\}(I\upharpoonright CʻR)\ldotp\\ +[\text{*90·32.(*91·02)}] &\equiv \ldotp P\{(\mid R)\mid R_{\text{ts}}\}(I\upharpoonright CʻR)\ldotp\\ +[\text{*37·3}] &\equiv \ldotp P\in \mid Rʻʻ\overrightarrow{R}_{ts}ʻ(I\upharpoonright CʻR)\ldotp\\ +[\text{*4·2.(*91·04)}] &\equiv \ldotp P\in \mid Rʻʻ\text{Potid}ʻR:\supset \vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·241</b> \(\vdash :TR_{\text{ts}}P\ldotp \supset \ldotp (Q\mid T)R_{\text{ts}}(Q\mid P)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·212}. &\supset \vdash \ldotp (Q\mid P)R_{\text{ts}}(Q\mid P) &\qquad\text{(1)}\\ +\vdash.\text{*91·2}. &\supset \vdash :(Q\mid S)R_{\text{ts}}(Q\mid P)\ldotp \supset \ldotp (Q\mid S\mid R)R_{\text{ts}}(Q\mid P) &\qquad\text{(2)}\\ +\vdash.\text{(1).(2).*91·11}\, \frac{\hat{S}\{(Q\mid S)R_{ts}(Q\mid P)\}}{\mu} \ldotp \supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p>The last line of the above proof is obtained as follows: writing +\(\mu\) for \(\hat{S}{(Q\mid S)R_{\text{ts}}(Q\mid P)},\) (1) becomes +\[ +\begin{aligned} +P&\in \mu &\qquad\text{(1)},\\ +\text{while (2) becomes}\qquad\qquad\qquad S\in \mu \ldotp \supset &\ldotp S\mid R\in \mu &\qquad\text{(2)}. +\end{aligned} +\]</p> + +<p>But by <a href="#*91·11">*91·11</a>, writing \(T\) for the \(P\) of *91·11, and \(P\) for the +\(Q\), we have +\[ +TR_{\text{ts}}P\ldotp \supset \colon\ldotp S\in \mu \ldotp \supset _{S}\ldotp S\mid R\in \mu :P\in \mu :\supset \ldotp T\in\mu \ldotp +\]</p> + +<p>Hence, by (1) and (2), \(TR_{\text{ts}}P.\supset.T\in \mu\), <i>i.e</i>. +\[ +TR_{\text{ts}}P.\supset.(Q\mid T)R_{\text{ts}}(Q\mid P), +\] +which is the proposition to be proved.</p> + +<p><span class="pagenum" id="Page_590">[Pg 590]</span></p> + +<p class="nind"><b>*91·242.</b> \(\vdash:SR_{\text{ts}}(Q\mid P).\supset.S\in Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·22.*43·11}.&\supset\vdash.Q\mid P\in Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻP &\qquad \text{(1)}\\ +\vdash.\text{*37·1.*43·1}.\supset\\ +\vdash:S\in Q\mid ʻʻR_{\text{ts}}ʻP.&\equiv.(\exists T).T\in \overrightarrow{R}_{\text{ts}}ʻP.S=Q\mid T.\\ +[\text{*91·2}] &\supset.(\exists T).T\mid R\in \overrightarrow{R}_{\text{ts}}ʻP.S\mid R=Q\mid T\mid R.\\ +[\text{*37·1.*43·1}] &\supset.S\mid R\in Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻP &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·11}\, \frac{Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻP}{\mu} .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·25.</b> \(\vdash.\overrightarrow{R}_{\text{ts}}ʻ(Q\mid P)=Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·242}. &\supset\vdash.\overrightarrow{R}_{\text{ts}}ʻ(Q\mid P)\subset Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻP &\qquad \text{(1)}\\ +\vdash.\text{*91·241}. &\supset\vdash:T\in \overrightarrow{R}_{\text{ts}}ʻP.S=Q\mid T.\supset.S\in \overrightarrow{R}_{\text{ts}}ʻ(Q\mid P):\\ +[\text{*10·11·23}]&\supset\vdash:(\exists T).T\in \overrightarrow{R}_{\text{ts}}ʻP.S=Q\mid T.\supset.S\in \overrightarrow{R}_{\text{ts}}ʻ(Q\mid P):\\ +[\text{*37·1.*43·1}]&\supset\vdash:S\in Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻP.\supset.S\in \overrightarrow{R}_{\text{ts}}ʻ(Q\mid P) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·251.</b> \(\vdash.\overrightarrow{R}_{\text{st}}ʻ(Q\mid P)=\mid Pʻʻ\overrightarrow{R}_{\text{st}}ʻQ \quad[\text{Proof as in *91·25}]\)</p> + +<p class="nind"><b>*91·26.</b> \(\vdash.\overrightarrow{R}_{\text{ts}}ʻQ=Q\mid ʻʻ\overrightarrow{R}_{\text{ts}}ʻI \quad\left[\text{*91·25}\, \frac{I}{P}\right]\)</p> + +<p class="nind"><b>*91·261.</b> \(\vdash.\overrightarrow{R}_{\text{st}}ʻQ=\mid Qʻʻ\overrightarrow{R}_{\text{st}}ʻI \quad\left[\text{*91·251}\, \frac{I,\,Q}{Q,\,P}\right]\)</p> + +<p class="nind"><b>*91·262.</b> \[\begin{align}&\vdash:\text{ᗡ}ʻQ\subset CʻR.\supset.\overrightarrow{R}_{\text{ts}}ʻQ=Q\mid ʻʻ\text{Potid}ʻR\\ +&\left[\text{*91·25}\, \frac{I\upharpoonright CʻR}{P} .\text{*50·62.(*91·04)}\right]\end{align}\]</p> + +<p class="nind"><b>*91·263.</b> \(\vdash.\overrightarrow{R}_{\text{ts}}ʻ(Q\mid R)=Q\mid ʻʻ\text{Pot}ʻR \quad\left[\text{*91·25}\, \frac{R}{P}.\,\text{(*91·03)}\right]\)</p> + +<p class="nind"><b>*91·264.</b> \(\vdash.\text{Pot}ʻR=\iotaʻR\cup R\mid ʻʻ\text{Pot}ʻR \quad\left[\text{*91·22·263}\, \frac{R}{Q}\right]\)</p> + +<p class="nind"><b>*91·27.</b> \(\vdash:P\in \text{Potid}ʻR.\supset.CʻP\subset CʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·5·52}.&\supset\vdash.Cʻ(I\upharpoonright CʻR)=CʻR.\\ +[\text{*22·42}] &\supset\vdash.Cʻ(I\upharpoonright CʻR)\subset CʻR &\qquad \text{(1)}\\ +\vdash.\text{*34·38}. &\supset\vdash:CʻS\subset CʻR.\supset.Cʻ(S\mid R)\subset CʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·17}\, \frac{CʻS\subset CʻR}{\phi S} .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_591">[Pg 591]</span></p> + +<p class="nind"><b><a id="*91·271">*91·271</a>.</b> \(\vdash:P\in \text{Pot}ʻR.\supset.\text{D}ʻP\subset \text{D}ʻR.\text{ᗡ}ʻP\subset \text{ᗡ}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·42}.\supset\vdash.\text{D}ʻR\subset \text{D}ʻR.\text{ᗡ}ʻR\subset \text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*34·36}.\supset\vdash:\text{D}ʻS\subset \text{D}ʻR.\supset.\text{D}ʻ(S\mid R)\subset \text{D}ʻR.\text{ᗡ}ʻ(S\mid R)\subset \text{ᗡ}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·17}\, \frac{\text{D}ʻS\subset \text{D}ʻR.\text{ᗡ}ʻS\subset \text{ᗡ}ʻR}{\phi S} .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·28.</b> \(\vdash:P\in \text{Potid}ʻR.\supset.P\mid R\in \text{Pot}ʻR \quad[\text{*91·24}]\)</p> + +<p class="nind"><b>*91·281.</b> \(\vdash:\text{Pot}ʻR\subset \text{Potid}ʻR.\mid Rʻʻ\text{Potid}ʻR\subset \text{Potid}ʻR \quad[\text{*91·23·24}]\)</p> + +<p class="nind"><b>*91·282.</b> \(\vdash:P\in \text{Pot}ʻR.\supset.P\mid R\in \text{Pot}ʻR \quad[\text{*91·28·281}]\)</p> + +<p class="nind"><b>*91·283.</b> \(\vdash:\mid Rʻʻ\text{Pot}ʻR\subset \text{Pot}ʻR \quad[\text{*91·282}]\)</p> + +<p>The following propositions show that the relative product of two powers +of \(R\) is commutative, <i>i.e.</i> (cf. <a href="#*91·34">*91·34</a>) +\[ +P,\,Q\in \text{Potid}ʻR.\supset.P\mid Q=Q\mid P. +\]</p> + +<p>We also have (cf. <a href="#*91·341">*91·341</a>) +\[ +P,\,Q\in \text{Potid}ʻR.\supset.P\mid Q\in \text{Potid}ʻR. +\]</p> + +<p>It is these propositions (as will appear in the sequel) which are the +source of the commutative law for the addition of finite ordinals. +Ordinals in general are not commutative, just as relative products +in general are not commutative; but owing to the fact that relative +products whose factors are powers of a given relation are commutative, +<i>finite</i> ordinals are commutative.</p> + +<p class="nind"><b>*91·3.</b> \(\vdash:P\in \text{Potid}ʻR.\supset.R\mid P=P\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·64·65}.\supset\vdash.R\mid I\upharpoonright CʻR=I\upharpoonright CʻR\mid R &\qquad \text{(1)}\\ +\vdash.\text{*34·21}. \supset\vdash.R\mid (S\mid R)=(R\mid S)\mid R &\qquad \text{(2)}\\ +\vdash.\text{*34·27}. \supset\vdash:R\mid S=S\mid R.\supset.(R\mid S)\mid R=(S\mid R)\mid R.\\ +[\text{(2)}] \qquad\qquad\qquad\qquad\qquad\supset.R\mid (S\mid R)=(S\mid R)\mid R &\qquad \text{(3)}\\ +\vdash.\text{*91·17}\, \frac{R\mid S=S\mid R}{\phi S} .\supset\\ +\vdash\colon\ldotp P\in \text{Potid}ʻR:R\mid S=S\mid R.\supset_S.R\mid (S\mid R)=(S\mid R)\mid R:R\mid I\upharpoonright CʻR=I\upharpoonright CʻR\mid R:\\ +\qquad\qquad\qquad\qquad\qquad\qquad\supset.R\mid P=P\mid R &\qquad \text{(4)}\\ +\vdash.\text{(1).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·301.</b> \(\vdash:P\in \overrightarrow{R}_{\text{st}}ʻ(I\upharpoonright CʻR).\supset.R\mid P=P\mid R \quad[\text{Proof as in *91·3}]\)</p> + +<p class="nind"><b>*91·302.</b> \(\vdash.\mid Rʻʻ\text{Potid}ʻR=R\mid ʻʻ\text{Potid}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·3.*13·182}.\supset\vdash\colon\ldotp P\in \text{Potid}ʻR.&\supset:S=R\mid P.\equiv.S=P\mid R:\\ +[\text{*43·1·101}] &\supset:S(R\mid )P.\equiv.S(\mid R)P &\qquad \text{(1)}\\ +\vdash.\text{(1).*5·32}.&\supset\vdash:P\in \text{Potid}ʻR.S(R\mid )P.\equiv.P\in \text{Potid}ʻR.S(\mid R)P:\\ +[\text{*10·11·281}]\supset\vdash:&(\exists P).P\in \text{Potid}ʻR.S(R\mid )P.\equiv.\\ +&(\exists P).P\in \text{Potid}ʻR.S(\mid R)P:\\ +[\text{*37·1}] & \supset\vdash:S\in R\mid ʻʻ\text{Potid}ʻR.\equiv.S\in \mid Rʻʻ\text{Potid}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_592">[Pg 592]</span></p> + +<p class="nind"><b>*91·303.</b> \(\vdash.\mid Rʻʻ\overrightarrow{R}_{\text{st}}ʻ(I\upharpoonright CʻR)=R\mid ʻʻ\overrightarrow{R}_{\text{st}}ʻ(I\upharpoonright + CʻR) \quad[\text{Proof as in *91·302}]\)</p> + +<p class="nind"><b>*91·304.</b> \(\vdash.\mid Rʻʻ\text{Pot}ʻR=R\mid ʻʻ\text{Pot}ʻR \quad[\text{Proof as in *91·302}]\)</p> + +<p class="nind"><b>*91·31.</b> \(\vdash.\text{Pot}ʻR=R\mid ʻʻ\text{Potid}ʻR \quad[\text{*91·24·301}]\)</p> + +<p class="nind"><b>*91·33.</b> \(\vdash.\text{Potid}ʻR=\overrightarrow{R}_{\text{st}}ʻ(I\upharpoonright CʻR)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·23}.&\supset\vdash.I\upharpoonright CʻR\in \text{Potid}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*91·3}. \supset\vdash:P\in \text{Potid}ʻR.&\supset.R\mid P=P\mid R.\\ +[*91·281] & \supset.R\mid P\in \text{Potid}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·1}\, \frac{\text{Potid}ʻR}{\mu} .&\supset\vdash:PR_{\text{st}}(I\upharpoonright CʻR).\supset.P\in \text{Potid}ʻR &\qquad \text{(3)}\\ +\vdash.\text{*91·301}.\supset\vdash:PR_{\text{st}}(I\upharpoonright CʻR).&\supset.P\mid R=R\mid P.\\ +[\text{*91·201}] & \supset.(P\mid R)R_{\text{st}}(I\upharpoonright CʻR) &\qquad \text{(4)}\\ +\vdash.\text{*91·213}.&\supset\vdash.(I\upharpoonright CʻR)R_{\text{st}}(I\upharpoonright CʻR) &\qquad \text{(5)}\\ +\vdash.\text{(4).(5).*91·17}.&\supset\vdash:P\in \text{Potid}ʻR.\supset.PR_{\text{st}}(I\upharpoonright CʻR) &\qquad \text{(6)}\\ +\vdash.\text{(3).(6)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·331.</b> \(\vdash.\text{Pot}ʻR=\overrightarrow{R}_{\text{st}}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·24·33}.\supset\vdash.\text{Pot}ʻR&=\mid Rʻʻ\overrightarrow{R}_{\text{st}}ʻ(I\upharpoonright CʻR)\\ +[\text{*91·251.*50·65}] & =\overrightarrow{R}_{\text{st}}ʻR.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*91·34">*91·34</a>.</b> \(\vdash:P,\,Q\in \text{Potid}ʻR.\supset.P\mid Q=Q\mid P\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\text{*50·62.*91·27}.&\supset\vdash:P\in \text{Potid}ʻR.\supset.P\mid (I\upharpoonright CʻR)=P\\ +[\text{*50·63.*91·27}] &\qquad\qquad\qquad\qquad\qquad =(I\upharpoonright CʻR)\mid P &\qquad \text{(1)}\\ +\vdash.\text{*34·27}. &\supset\vdash:P\in \text{Potid}ʻR.P\mid S=S\mid P.\supset.P\mid S\mid R=S\mid P\mid R\\ +[\text{*91·3}] &\qquad\qquad\qquad\qquad\qquad =S\mid R\mid P &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·17}\, \frac{P\mid S=S\mid P}{\phi S} .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This is the commutative law for the relative product of two powers of +\(R\).</p> + +<p class="nind"><b><a id="*91·341">*91·341</a>.</b> \(\vdash:P,\,Q\in \text{Potid}ʻR.\supset.P\mid Q\in \text{Potid}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·62.*91·27}.&\supset\vdash:P\in \text{Potid}ʻR.\supset.P\mid (I\upharpoonright CʻR)=P.\\ +[\text{*13·12}] & \supset.P\mid (I\upharpoonright CʻR)\in \text{Potid}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*91·281}. &\supset\vdash:P\mid S\in \text{Potid}ʻR.\supset.P\mid S\mid R\in \text{Potid}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·17}\, \frac{P\mid S\in \text{Potid}ʻR}{\phi S} .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_593">[Pg 593]</span></p> + +<p class="nind"><b>*91·342.</b> \(\vdash:P\in \text{Potid}ʻR.Q\in \text{Pot}ʻR.\supset.P\mid Q\in \text{Pot}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·28}. &\supset\vdash:P\in \text{Potid}ʻR.\supset.P\mid R\in \text{Pot}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*91·282}.&\supset\vdash:P\mid Q\in \text{Pot}ʻR.\supset.P\mid Q\mid R\in \text{Pot}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·171}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·343.</b> \(\vdash:P,\,Q\in \text{Pot}ʻR.\supset.P\mid Q\in \text{Pot}ʻR \quad[\text{*91·342·23}]\)</p> + +<p class="nind"><b>*91·35.</b> \(\vdash.I\upharpoonright CʻR\in \text{Potid}ʻR \quad[\text{*91·23}]\)</p> + +<p class="nind"><b>*91·351.</b> \(\vdash.R\in \text{Pot}ʻR \quad[\text{*91·264}]\)</p> + +<p class="nind"><b>*91·352.</b> \(\vdash.R^{2}\in \text{Pot}ʻR \quad[\text{*91·282·351}]\)</p> + +<p class="nind"><b>*91·36.</b> \(\vdash:P\in \text{Pot}ʻR.\supset.P\mid R,R\mid P\in \text{Pot}ʻR \quad[\text{*91·343·351}]\)</p> + +<p class="nind"><b>*91·37.</b> \(\vdash\colon\ldotp \text{Potid}ʻR\subset \mu.\equiv:I\upharpoonright CʻR\in \mu:S\in \text{Potid}ʻR.S\in \mu.\supset_{S}.S\mid R\in \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·281·35}.\supset\\ +\vdash\colon\ldotp I\upharpoonright CʻR\in \mu:S\in \text{Potid}ʻR.S\in \mu.\supset_{S}.S\mid R\in \mu:\equiv:\\ +I\upharpoonright CʻR\in \text{Potid}ʻR.I\upharpoonright CʻR\in \mu:S\in \text{Potid}ʻR.S\in \mu.\supset_{S}.S\mid R\in \text{Potid}ʻR.S\mid R\in \mu:\\ +[\text{*91·17}]\supset:P\in \text{Potid}ʻR.\supset.P\in \mu &\qquad \text{(1)}\\ +\vdash.\text{*91·35}. \supset\vdash:\text{Potid}ʻR\subset \mu.\supset.I\upharpoonright CʻR\in \mu &\qquad \text{(2)}\\ +\vdash.\text{*91·281}.\supset\vdash\colon\ldotp \text{Potid}ʻR\subset \mu.\supset:S\in \text{Potid}ʻR.\supset_{S}.S\mid R\in \mu:\\ +[\text{*3·41}] \qquad\qquad\qquad\qquad\supset:S\in \text{Potid}ʻR.S\in \mu.\supset_{S}.S\mid R\in \mu &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·371.</b> \[\begin{align}\vdash\colon\ldotp P\in \text{Potid}ʻ&R.\supset_{P}.\phi P:\equiv:\\ +&\phi(I\upharpoonright CʻR):S\in \text{Potid}ʻR.\phi S.\supset_{S}.\phi(S\mid R) \quad[\text{*91·37}]\end{align}\]</p> + +<p class="nind"><b>*91·372.</b> \[\begin{align}&\vdash\colon\ldotp \text{Pot}ʻR\subset \mu.\equiv:R\in \mu:S\in \text{Pot}ʻR.S\in \mu.\supset_{S}.S\mid R\in \mu\\ +&[\text{Proof as in *91·37}]\end{align}\]</p> + +<p class="nind"><b>*91·373.</b> \[\begin{align}&\vdash\colon\ldotp P\in \text{Pot}ʻR.\supset_{P}.\phi P:\equiv:\phi R:S\in \text{Pot}ʻR.\phi S.\supset_{S}.\phi(S\mid R)\\ +&[\text{*91·372}]\end{align}\]</p> + +<p class="nind"><b>*91·41.</b> \(\vdash.\overrightarrow{R}_{\text{ts}}ʻ(P\mid R)=P\mid ʻʻ\text{Pot}ʻR \quad\left[\text{*91·25}\, \frac{P,\,R}{Q,\,P}.\text{(*91·03)}\right]\)</p> + +<p class="nind"><b>*91·411.</b> \(\vdash.\overrightarrow{R}_{\text{st}}ʻ(R\mid P)=\mid Pʻʻ\text{Pot}ʻR \quad\left[\text{*91·251}\, \frac{R}{Q}.\text{*91·331}\right]\)</p> + +<p class="nind"><b>*91·42.</b> \(\vdash.\overrightarrow{R}_{\text{ts}}ʻP=\iotaʻP\cup P\mid ʻʻ\text{Pot}ʻR \quad[\text{*91·22·41}]\)</p> + +<p class="nind"><b>*91·421.</b> \(\vdash.\overrightarrow{R}_{\text{st}}ʻP=\iotaʻP\cup \mid Pʻʻ\text{Pot}ʻR \quad[\text{*91·221·411}]\)</p> + +<p class="nind"><b>*91·43.</b> \(\vdash:P\in \text{Pot}ʻR.QR_{\text{ts}}P.\supset.Q\in \text{Pot}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·42}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:Q=P.\lor.Q\in P\mid ʻʻ\text{Pot}ʻR:\\ +[\text{*37·1.*43·101}] &\supset:Q=P.\lor.(\exists T).T\in \text{Pot}ʻR.Q=P\mid T:\\ +[\text{*13·12.*91·343}] &\supset:Q\in \text{Pot}ʻR\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_594">[Pg 594]</span></p> + +<p class="nind"><b>*91·431.</b> \(\vdash:P\in \text{Potid}ʻR.QR_{\text{ts}}P.\supset.Q\in \text{Potid}ʻR \quad[\text{Proof as in *91·43}]\)</p> + +<p class="nind"><b>*91·44.</b> \(\vdash\colon\ldotp P,\,Q\in \text{Potid}ʻR.\supset:QR_{\text{ts}}P.\lor .PR_{\text{ts}}Q\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·14}. &\supset\vdash:P\in \text{Potid}ʻR.\supset.PR_{\text{ts}}(I\upharpoonright CʻR) &\qquad \text{(1)}\\ +\vdash.\text{*91·2}. &\supset\vdash:QR_{\text{ts}}P.\supset.(Q\mid R)R_{\text{ts}}P &\qquad \text{(2)}\\ +\vdash.\text{*91·212}.&\supset\vdash\colon\ldotp PR_{\text{ts}}Q.\supset:P=Q.\lor .PR_{\text{ts}}(Q\mid R) &\qquad \text{(3)}\\ +\vdash.\text{*91·212}.&\supset\vdash:P=Q.\supset.QR_{\text{ts}}P.\\ +[\text{*91·2}] &\supset.(Q\mid R)R_{\text{ts}}P &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.&\supset\vdash\colon\ldotp PR_{\text{ts}}Q.\supset:(Q\mid R)R_{\text{ts}}P.\lor .PR_{\text{ts}}(Q\mid R) &\qquad \text{(5)}\\ +\vdash.\text{(2).(5)}.&\supset\vdash\colon\ldotp QR_{\text{ts}}P.\lor .PR_{\text{ts}}Q:\supset:(Q\mid R)R_{\text{ts}}P.\lor .PR_{\text{ts}}(Q\mid + R) &\qquad \text{(6)}\\ +\vdash.\text{(1).(6).*91·17}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·45.</b> \(\vdash\colon\ldotp P,\,Q\in \text{Potid}ʻR.\supset:(\exists T):T\in \text{Potid}ʻR:Q=P\mid T.\lor .P=Q\mid T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·262·27}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\overrightarrow{R}_{\text{ts}}ʻP=P\mid ʻʻ\text{Potid}ʻR.\overrightarrow{R}_{\text{ts}}ʻQ=Q\mid + ʻʻ\text{Potid}ʻR:\\ +[\text{*37·1.*43·1}] &\supset:QR_{\text{ts}}P.\equiv.(\exists T).T\in \text{Potid}ʻR.Q=P\mid T:\\ +& PR_{\text{ts}}Q.\equiv.(\exists T).T\in \text{Potid}ʻR.P=Q\mid T &\qquad \text{(1)}\\ +\vdash.\text{(1).*91·44.*10·42}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·46.</b> \[\begin{align}&\vdash\colon\ldotp P,\,Q\in \text{Potid}ʻR.\supset:(\exists T):T\in \text{Potid}ʻR:Q=T\mid P.\lor .P=T\mid Q\\ +&[\text{*91·45·34}]\end{align}\]</p> + +<p>The remainder of this number is concerned with \(R_{\text{po}}\) and +its relations to \(R_{\unicode{x2217}}\).</p> + +<p class="nind"><b>*91·502.</b> \(\vdash.R\unicode{x2abd}R_{\text{po}} \quad[\text{*91·351.(*91·05).*41·13}]\)</p> + +<p class="nind"><b>*91·503.</b> \(\vdash.R^{2}\unicode{x2abd}R_{\text{po}} \quad[\text{*91·352.(*91·05).*41·13}]\)</p> + +<p class="nind"><b>*91·504.</b> \(\vdash.\text{D}ʻR_{\text{po}}=\text{D}ʻR.\text{ᗡ}ʻR_{\text{po}}=\text{ᗡ}ʻR.CʻR_{\text{po}}=CʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·502}. & \supset\vdash.\text{D}ʻR\subset \text{D}ʻR_{\text{po}} &\qquad \text{(1)}\\ +\vdash.\text{*91·271.*40·43}.&\supset\vdash.sʻ\text{D}ʻʻ\text{Pot}ʻR\subset \text{D}ʻR.\\ +[\text{*41·43}] &\supset\vdash.\text{D}ʻR_{\text{po}}\subset \text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash.\text{D}ʻR=\text{D}ʻR_{\text{po}} &\qquad \text{(3)}\\ +\text{Similarly} & \vdash.\text{ᗡ}ʻR=\text{ᗡ}ʻR_{\text{po}}.CʻR=CʻR_{\text{po}} &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions are concerned mainly with the relations of +\(R_{\text{po}}\) and \(R_{\unicode{x2217}}\). These relations are +embodied in the propositions +\[ +\begin{array}{l} +&R_{\text{po}}=R_{\unicode{x2217}}\mid R=R\mid R_{\unicode{x2217}} &\quad(\text{*91·52)}\\ +&R_{\unicode{x2217}}=I\upharpoonright CʻR\unicode{x228d}R_{\text{po}} &\quad(\text{*91·54)}\\ +\text{and} &R_{\unicode{x2217}}=\dot{s}ʻ\text{Potid}ʻR &\quad(\text{*91·55)} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_595">[Pg 595]</span></p> + +<p class="nind"><b>*91·51.</b> \(\vdash.R_{\text{po}}\mid R=R\mid R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·421.(*91·05)}.\supset\vdash.R_{\text{po}}\mid R&=\dot{s}ʻ\mid Rʻʻ\text{Pot}ʻR\\ +[\text{*91·304}] & =\dot{s}ʻR\mid ʻʻ\text{Pot}ʻR\\ +[\text{*43·42.(*91·05)}] & =R\mid R_{\text{po}}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·511.</b> \(\vdash.R_{\text{po}}\mid R\unicode{x2abd}R_{\text{po}} \quad[\text{*43·421.*91·283.*41·161}]\)</p> + +<p class="nind"><b>*91·512.</b> \(\vdash.R_{\text{po}}\unicode{x2abd}R_{\unicode{x2217}}\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·32}.&\supset\vdash.R\unicode{x2abd}R_{\unicode{x2217}}\mid R &\qquad \text{(1)}\\ +\vdash.\text{*90·16}.&\supset\vdash:S\unicode{x2abd}R_{\unicode{x2217}}\mid R.\supset.S\unicode{x2abd}R_{\unicode{x2217}}.\\ +[\text{*34·34}] &\supset.S\mid R\unicode{x2abd}R_{\unicode{x2217}}\mid R &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·171}\, \frac{S\unicode{x2abd}R_{\unicode{x2217}}\mid R}{\phi S}.&\supset\vdash:P\in \text{Pot}ʻR.\supset.P\unicode{x2abd}R_{\unicode{x2217}}\mid R:\\ +[\text{*41·151.(*91·05)}] &\supset\vdash.R_{\text{po}}\unicode{x2abd}R_{\unicode{x2217}}\mid R.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·513.</b> \(\vdash.R_{\unicode{x2217}}\unicode{x2abd}\dot{s}ʻ\text{Potid}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·112}\, \frac{x(\dot{s}ʻ\text{Potid}ʻR)z}{\phi z} .\supset\\ +\vdash\colon\ldotp xR_{\unicode{x2217}}y:x(\dot{s}ʻ\text{Potid}ʻR)z.zRw.\supset_{z,w}.x(\dot{s}ʻ\text{Potid}ʻR)w:\\ +& x(\dot{s}ʻ\text{Potid}ʻR)x:\supset.x(\dot{s}ʻ\text{Potid}ʻR)y &\qquad \text{(1)}\\ +\vdash.\text{*43·421}. & \supset\vdash.(\dot{s}ʻ\text{Potid}ʻR)\mid R=\dot{s}ʻ\mid Rʻʻ\text{Potid}ʻR\\ +[\text{*91·281.*41·161}] \unicode{x2abd}\dot{s}ʻ\text{Potid}ʻR.\\ +[\text{*34·1.*10·23}]\supset\vdash:x(\dot{s}ʻ\text{Potid}ʻR)z.zRw.&\supset_{z,w}.x(\dot{s}ʻ\text{Potid}ʻR)w &\qquad \text{(2)}\\ +\vdash.\text{*90·13}. & \supset\vdash:xR_{\unicode{x2217}}y.\supset.x\in CʻR.\\ +[\text{*50·3.*35·101}] &\supset.x(I\upharpoonright CʻR)x.\\ +[\text{*91·35.*41·13}] &\supset.x(\dot{s}ʻ\text{Potid}ʻR)x &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*4·71·73}.&\supset\vdash:\text{Hp(1)}.\equiv.xR_{\unicode{x2217}}y &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.&\supset\vdash:xR_{\unicode{x2217}}y.\supset.x(\dot{s}ʻ\text{Potid}ʻR)y:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·514.</b> \(\vdash.R_{\unicode{x2217}}\mid R\unicode{x2abd}R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·513}.\supset\vdash.R_{\unicode{x2217}}\mid R&\unicode{x2abd}(\dot{s}ʻ\text{Potid}ʻR)\mid R\\ +[\text{*43·421}] &\unicode{x2abd}(\dot{s}ʻ\mid Rʻʻ\text{Potid}ʻR\\ +[\text{*91·24}] &\unicode{x2abd}\dot{s}ʻ\text{Pot}ʻR\\ +[\text{(*91·05}] &\unicode{x2abd}R_{\text{po}}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_596">[Pg 596]</span></p> + +<p class="nind"><b>*91·52.</b> \(\vdash.R_{\text{po}}=R_{\unicode{x2217}}\mid R=R\mid R_{\unicode{x2217}} \quad[\text{*91·512·514.*90·32}]\)</p> + +<p class="nind"><b>*91·521.</b> \(\vdash : P \in \text{Potid}ʻR. \equiv . \breve{P} \in \text{Potid}ʻ\breve{R}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*91·15}\, \frac{\text{Cnv}ʻʻ\mu}{\mu} . &\supset\vdash \colon\colon \breve{P} \in \text{Potid}ʻ\breve{R} . \supset \colon\ldotp \\ +&I\upharpoonright CʻR \in \text{Cnv}ʻʻ\mu:S\in \text{Cnv}ʻʻ\mu.\supset_{S}.S\mid \breve{R}\in \text{Cnv}ʻʻ\mu:\supset.\breve{P}\in \text{Cnv}ʻʻ\mu &\qquad \text{(1)}\\ +\vdash . \text{*72·513.11}. &\supset\vdash : \breve{P}\in \text{Cnv}ʻʻ\mu . \equiv . P\in \mu &\qquad \text{(2)}\\ +\vdash . \text{(2). *50·5·51}. &\supset\vdash : I\upharpoonright CʻR \in \text{Cnv}ʻʻ\mu . \equiv . I\upharpoonright CʻR \in \mu &\qquad \text{(3)}\\ +\vdash . \text{*31·51}. & \supset\vdash \colon\ldotp S \in \text{Cnv}ʻʻ\mu . \supset_{S}. S\mid \breve{R} \in \text{Cnv}ʻʻ\mu :\equiv:\\ +&\breve{S}\in \text{Cnv}ʻʻ\mu. \supset_{S}. \breve{S}\mid \breve{R} \in \text{Cnv}ʻʻ\mu :\\ +[\text{(2).*34·2}] &\equiv : S \in \mu. \supset_{S} . R \mid S \in \mu &\qquad \text{(4)}\\ +\vdash . \text{(1) . (2). (3). (4)}. \supset\\ +\vdash \colon\colon \breve{P} \in \text{Potid}ʻ\breve{R} .&\supset\colon\ldotp I\upharpoonright CʻR\in \mu: S\in \mu .\supset_{S} + . S\mid R\in \mu:\supset.P\in \mu &\qquad \text{(5)}\\ +\vdash . \text{(5). *10·11·21 .*91·15}. \supset\\ +\vdash : \breve{P} \in \text{Potid}ʻ\breve{R}. &\supset . P \in \text{Potid}ʻR &\qquad \text{(6)}\\ +\vdash .\text{(6)}\,\frac{\breve{P},\breve{R}}{P,\,R}.\text{*31·33}.&\supset\vdash:P\in \text{Potid}ʻR.\supset.\breve{P}\in \text{Potid}ʻ\breve{R}&\qquad \text{(7)}\\ +\vdash . \text{(6) . (7)}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·522.</b> \(\vdash: P \in \text{Pot}ʻR .\equiv. \breve{P} \in \text{Pot}ʻ\breve{R} \quad[\text{Proof as in *91·521}]\)</p> + +<p class="nind"><b>*91·53.</b> \(\vdash. \breve{R}_{\text{po}} = (\breve{R})_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*91·52}. \supset\vdash. \breve{R}_{\text{po}} &= \breve{R}\mid \breve{R}_{\unicode{x2217}}\\ +[\text{*90·132}] & = \breve{R}\mid (\breve{R})_{\unicode{x2217}}\\ +[\text{*91·52}] & = (\breve{R})_{\text{po}}. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·54.</b> \(\vdash . R_{\unicode{x2217}} = I \upharpoonright CʻR \unicode{x228d} R_{\text{po}} \quad[\text{*90·31.*91·52}]\)</p> + +<p class="nind"><b>*91·541.</b> \(\vdash . R_{\unicode{x2217}}\dot{\cap}J=R_{\text{po}}\dot{\cap}J \quad[\text{*25·401.(*50·02).*35·441. *91·54}]\)</p> + +<p class="nind"><b>*91·542.</b> \(\vdash : xR_{\unicode{x2217}}y.x \neq y.\equiv. xR_{\text{po}}y.x \neq y \quad[\text{*91·541 .*50·11}]\)</p> + +<p class="nind"><b>*91·543.</b> \(\vdash . R_{\unicode{x2217}}ʻʻ\beta = (\beta \cap CʻR) \cup R_{\text{po}}ʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*91·54.*37·221}.\supset\vdash.R_{\unicode{x2217}}ʻʻ\beta &= (I\upharpoonright CʻR)ʻʻ\beta \cup R_{\text{po}}ʻʻ\beta\\ +[\text{*50·59}] &= (\beta \cap CʻR) \cup R_{\text{po}}ʻʻ\beta.\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·544.</b> \(\vdash . \breve{R}_{\unicode{x2217}}ʻʻ\beta = (\beta \cap CʻR) \cup \breve{R}_{\text{po}}ʻʻ\beta\)</p> + +<p class="nind"><b>*91·545.</b> \(\vdash : \beta \subset CʻR.\supset. R_{\unicode{x2217}}ʻʻ\beta = \beta \cup R_{\text{po}}ʻʻ\beta \quad[\text{*91·543.*22·621}]\)</p> + +<p><span class="pagenum" id="Page_597">[Pg 597]</span></p> + +<p class="nind"><b>*91·546.</b> \(\vdash : \beta \subset CʻR. \supset . \breve{R}_{\unicode{x2217}}ʻʻ\beta = \beta \cup \breve{R}_{\text{po}}ʻʻ\beta\)</p> + +<p class="nind"><b><a id="*91·55">*91·55</a>.</b> \(\vdash.R_{\unicode{x2217}}=\dot{s}ʻ\text{Potid}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·23}.\supset\vdash.\dot{s}ʻ\text{Potid}ʻR&=\dot{s}ʻ\{\iotaʻ(I\upharpoonright CʻR)\cup \text{Pot}ʻR\}\\ +[\text{*53·17.(*91·05)}] & =I\upharpoonright CʻR\unicode{x228d}R_{\text{po}}\\ +[\text{*91·54}] & =R_{\unicode{x2217}}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·56.</b> \(\vdash.R_{\text{po}}^{2}\unicode{x2abd}R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·52}.\supset\vdash.R_{\text{po}}^{2}&=R_{\unicode{x2217}}\mid R\mid R_{\unicode{x2217}}\mid R\\ +[\text{*90·16}] & \unicode{x2abd}R_{\unicode{x2217}}\mid R_{\unicode{x2217}}\mid R\\ +[\text{*90·17}] & \unicode{x2abd}R_{\unicode{x2217}}\mid R\\ +[\text{*91·52}] & \unicode{x2abd}R_{\text{po}}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·561.</b> \(\vdash\colon\ldotp S\unicode{x2abd}R_{\text{po}}.T\unicode{x2abd}R_{\text{po}}.\supset.S\mid T\unicode{x2abd}R_{\text{po}} + \quad[\text{*34·34.*91·56}]\)</p> + +<p class="nind"><b>*91·562.</b> \(\vdash:S\unicode{x2abd}R_{\text{po}}.\supset.S\mid R\unicode{x2abd}R_{\text{po}}.R\mid S\unicode{x2abd}R_{\text{po}} \quad[\text{*91·561·502}]\)</p> + +<p class="nind"><b>*91·57.</b> \(\vdash.R_{\text{po}}=R\unicode{x228d}R_{\text{po}}\mid R=R\unicode{x228d}R\mid R_{\text{po}} \quad[\text{*90·32.*91·52}]\)</p> + +<p class="nind"><b>*91·571.</b> \(\vdash.R_{\text{po}}\mid R=R\mid R_{\text{po}} \quad[\text{*91·52}]\)</p> + +<p class="nind"><b>*91·572.</b> \(\vdash.R_{\text{po}}\dot{-}(R_{\text{po}}\mid R)\unicode{x2abd}R \quad[\text{*91·57.*22·9·43}]\)</p> + +<p class="nind"><b>*91·573.</b> \(\vdash.R_{\text{po}}\dot{-}(R\mid R_{\text{po}})\unicode{x2abd}R \quad[\text{*91·571·572}]\)</p> + +<p class="nind"><b>*91·574.</b> \(\vdash.R_{\unicode{x2217}}\mid R_{\text{po}}=R_{\text{po}}\mid R_{\unicode{x2217}}=R_{\text{po}}=R\mid + R_{\unicode{x2217}}=R_{\unicode{x2217}}\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·52}.\supset\vdash.R_{\unicode{x2217}}\mid R_{\text{po}}&=R_{\unicode{x2217}}\mid R_{\unicode{x2217}}\mid R\\ +[\text{*90·17}] &=R_{\unicode{x2217}}\mid R &\qquad \text{(1)}\\ +\vdash.\text{*91·52}.\supset\vdash.R_{\text{po}}\mid R_{\unicode{x2217}}&=R\mid R_{\unicode{x2217}}\mid R_{\unicode{x2217}}\\ +[\text{*90·17}] & =R\mid R_{\unicode{x2217}} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·52}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·575.</b> \(\vdash.R_{\text{po}}^{2}=R\mid R_{\text{po}}=R_{\text{po}}\mid R=R^{2}\mid R_{\unicode{x2217}}=R_{\unicode{x2217}}\mid + R^{2}=R\mid R_{\unicode{x2217}}\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·574·52}.\supset\vdash.R_{\text{po}}^{2}=R\mid R_{\text{po}}=R_{\text{po}}\mid R &\qquad \text{(1)}\\ +\vdash.\text{(1).*91·52}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·58.</b> \(\vdash:P\in \text{Potid}ʻR.\supset.P\unicode{x2abd}R_{\unicode{x2217}} \quad[\text{*91·55.*41·13}]\)</p> + +<p class="nind"><b>*91·581.</b> \(\vdash:P\in \text{Pot}ʻR.\supset.P\unicode{x2abd}R_{\text{po}} \quad[\text{*41·13.(*91·05)}]\)</p> + +<p class="nind"><b>*91·59.</b> \(\vdash:R\unicode{x2abd}S.\supset.R_{\text{po}}\unicode{x2abd}S_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·18}.\supset\vdash:\text{Hp}.&\supset.R_{\unicode{x2217}}\unicode{x2abd}S_{\unicode{x2217}}.\\ +[\text{*34·34}] & \supset.R_{\unicode{x2217}}\mid R\unicode{x2abd}S_{\unicode{x2217}}\mid S.\\ +[\text{*91·52}] & \supset.R_{\text{po}}\unicode{x2abd}S_{\text{po}}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_598">[Pg 598]</span></p> + +<p class="nind"><b>*91·6.</b> \(\vdash:Q\in \text{Pot}ʻR.\supset.\text{Pot}ʻQ\subset \text{Pot}ʻR.Q_{\text{po}}\unicode{x2abd}R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·171}\, \frac{Q,\,S\in \text{Pot}ʻR}{R,\,\,\phi S} .\supset\\ +\vdash\colon\ldotp P\in \text{Pot}ʻQ:S\in \text{Pot}ʻR.&\supset_{S}.S\mid Q\in \text{Pot}ʻR:Q\in \text{Pot}ʻR:\supset.P\in \text{Pot}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*91·343}. &\supset\vdash\colon\ldotp Q\in \text{Pot}ʻR.\supset:S\in \text{Pot}ʻR.\supset_{S}\mid Q\in \text{Pot}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. & \supset\vdash:P\in \text{Pot}ʻQ.Q\in \text{Pot}ʻR.\supset.P\in \text{Pot}ʻR:\\ +[\text{Exp.*10·11·21}]&\supset\vdash:Q\in \text{Pot}ʻR.\supset.\text{Pot}ʻQ\subset \text{Pot}ʻR. &\qquad \text{(3)}\\ +[\text{*41·161}] & \supset.Q_{\text{po}}\unicode{x2abd}R_{\text{po}} &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·601.</b> \(\vdash.(R_{\text{po}})_{\text{po}}=R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·502}.&\supset\vdash.R_{\text{po}}\unicode{x2abd}(R_{\text{po}})_{\text{po}} &\qquad \text{(1)}\\ +\vdash.\text{*91·171}\, \frac{R_{\text{po}},\,S\unicode{x2abd}R_{\text{po}}}{R,\,\phi S} .\supset\\ +\vdash\colon\ldotp P\in \text{Pot}ʻR_{\text{po}}:S\unicode{x2abd}R_{\text{po}}.&\supset_{S}.S\mid + R_{\text{po}}\unicode{x2abd}R_{\text{po}}:R_{\text{po}}\unicode{x2abd}R_{\text{po}}:\supset.P\unicode{x2abd}R_{\text{po}} &\qquad \text{(2)}\\ +\vdash.\text{*34·34.*91·56}. &\supset\vdash:S\unicode{x2abd}R_{\text{po}}.\supset_{S}.S\mid R_{\text{po}}\unicode{x2abd}R_{\text{po}} &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*23·42}.&\supset\vdash:P\in \text{Pot}ʻR_{\text{po}}.\supset.P\unicode{x2abd}R_{\text{po}}:\\ +[\text{*41·151}] &\supset\vdash.(R_{\text{po}})_{\text{po}}\unicode{x2abd}R_{\text{po}} &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·602.</b> \(\vdash.(R_{\text{po}})_{\unicode{x2217}}=R_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·54}.\supset\vdash.(R_{\text{po}})_{\unicode{x2217}}&=I\upharpoonright CʻR_{\text{po}}\unicode{x228d}(R_{\text{po}})_{\text{po}}\\ +[\text{*91·504·601}] & =I\upharpoonright CʻR\unicode{x228d}R_{\text{po}}\\ +[\text{*91·54}] & =R_{\unicode{x2217}}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·603.</b> \(\vdash.(R_{\unicode{x2217}})_{\text{po}}=R_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·52}.\supset\vdash.(R_{\unicode{x2217}})_{\text{po}}&=(R_{\unicode{x2217}})_{\unicode{x2217}}\mid R_{\unicode{x2217}}\\ +[\text{*90·4}] & =R_{\unicode{x2217}}\mid R_{\unicode{x2217}}\\ +[\text{*90·17}] & =R_{\unicode{x2217}}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·62.</b> \(\vdash\colon\ldotp xR_{\text{po}}y.\equiv:\breve{R}ʻʻ\mu\subset \mu.\overleftarrow{R}ʻx\subset \mu.\supset_{\mu}.y\in \mu \quad[\text{*91·52.*90·36}]\)</p> + +<p><span class="pagenum" id="Page_599">[Pg 599]</span></p> + +<p>This formula should be compared with <a href="#*90·11">*90·11</a>, in which an +analogous formula is given for \(R_{\unicode{x2217}}\). It will +be observed that here we do not require to add \(x\in CʻR\), +for if \(\overleftarrow{R}ʻx=\Lambda\), the above formula leads +to \(xR_{\text{po}}y.\supset.y\in \Lambda\), <i>i.e.</i> to +\({\sim}(xR_{\text{po}}y)\). Hence \(xR_{\text{po}}y.\supset.\exists!\overleftarrow{R}ʻx\), +<i>i.e.</i> \(xR_{\text{po}}y.\supset.x\in\text{D}ʻR\). +It will be observed that \(xR_{\text{po}}y\) holds whenever \(y\) +belongs to every hereditary class which contains the immediate +successors of \(x\), whereas \(xR_{\unicode{x2217}}y\) holds whenever +\(y\) belongs to every hereditary class to which \(x\) itself belongs.</p> + +<p class="nind"><b>*91·7.</b> \(\vdash.R_{\text{po}}ʻʻ\text{ᗡ}ʻR=\text{D}ʻR.\breve{R}_{\text{po}}ʻʻ\text{D}ʻR=\text{ᗡ}ʻR \quad[\text{*91·504.*37·25}]\)</p> + +<p class="nind"><b>*91·71.</b> \(\vdash:Rʻʻ\mu\subset \mu.\equiv.R_{\text{po}}ʻʻ\mu\subset \mu.\equiv.R_{\unicode{x2217}}ʻʻ\mu\subset \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·22·132}.\supset\vdash:Rʻʻ\mu\subset \mu.&\equiv.R_{\unicode{x2217}}ʻʻ\mu\subset \mu. &\qquad \text{(1)}\\ +[\text{*91·602}] & \equiv.(R_{\text{po}})_{\unicode{x2217}}ʻʻ\mu\subset \mu.\\ +\left[\text{(1)}\, \frac{R_{\text{po}}}{R}\right] & \equiv.R_{\text{po}}ʻʻ\mu\subset \mu &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·711.</b> \(\vdash:Rʻʻ\mu\subset \mu.\supset.R_{\text{po}}ʻʻ\mu=Rʻʻ\mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·71·52.*37·2}.&\supset\vdash:\text{Hp}.\supset.R_{\text{po}}ʻʻ\mu\subset Rʻʻ\mu &\qquad \text{(1)}\\ +\vdash.\text{*91·502}. & \supset\vdash.Rʻʻ\mu\subset R_{\text{po}}ʻʻ\mu &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in the theory of minimum points in a +series (*205·68).</p> + +<p class="nind"><b>*91·72.</b> \(\vdash.Rʻʻ(\alpha\cup R_{\text{po}}ʻʻ\alpha)=R_{\text{po}}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·22·33}.\supset\vdash.Rʻʻ(\alpha\cup R_{\text{po}}ʻʻ\alpha)&=Rʻʻ\alpha\cup (R\mid R_{\text{po}})ʻʻ\alpha\\ +[\text{*37·221}] & =(R\unicode{x228d}R\mid R_{\text{po}})ʻʻ\alpha\\ +[\text{*91·57}] & =R_{\text{po}}ʻʻ\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·721.</b> \(\vdash.\breve{R}ʻʻ(\alpha\cup \breve{R}_{\text{po}}ʻʻ\alpha)=\breve{R}_{\text{po}}ʻʻ\alpha + \quad\left[\text{*91·72}\, \frac{\breve{R}}{R}.\text{*91·53}\right]\)</p> + +<p class="nind"><b><a id="*91·73">*91·73</a>.</b> \(\vdash\colon\ldotp P,\,Q\in \text{Potid}ʻR.P \neq Q.\supset:(\exists T):T\in \text{Pot}ʻR:Q=P\mid T.\lor.P=Q\mid T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·45}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:(\exists T):T\in \text{Potid}ʻR:Q=P\mid T.P\mid T \neq P.\lor.P=Q\mid T.Q\mid T \neq Q &\qquad \text{(1)}\\ +\vdash.\text{*91·504.*50·62}.&\supset\vdash:P\in \text{Potid}ʻR.\supset.P\mid I\upharpoonright CʻR=P:\\ +[\text{Transp}] &\supset\vdash:P,\,T\in \text{Potid}ʻR.P\mid T \neq P.\supset.T \neq I\upharpoonright CʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:(\exists T):T\in \text{Potid}ʻR.T \neq I\upharpoonright CʻR:Q=P\mid T.\lor.P=Q\mid T &\qquad \text{(3)}\\ +\vdash.\text{*91·23}.&\supset\vdash:T\in \text{Potid}ʻR.T \neq I\upharpoonright CʻR.\supset.T\in \text{Pot}ʻR &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*91·731">*91·731</a>.</b> \[\begin{align}&\vdash\colon\ldotp P,\,Q\in \text{Potid}ʻR.P \neq Q.\supset:(\exists T):T\in \text{Pot}ʻR:Q=T\mid P.\lor.P=T\mid Q\\ +&[\text{*91·73·34}]\end{align}\]</p> + +<p>By means of <a href="#*91·73">*91·73</a> or <a href="#*91·731">*91·731</a>, the powers of \(R\) can often be +arranged in a series, the rule of arrangement being that \(P\) comes +earlier than \(Q\) if<span class="pagenum" id="Page_600">[Pg 600]</span> \(Q = P\mid T\), and later in the converse case. +But we shall only get an open series from this arrangement if +\(P\in\text{Potid}ʻR.T\in \text{Pot}ʻR. \supset_{P,T}.P\mid T \neq P\); +otherwise the powers from a certain point onwards form a cyclic series.</p> + +<p class="nind"><b>*91·732.</b> \[\begin{align}\vdash\colon\ldotp P,Q\in \text{Potid}ʻR.&P \neq Q.\supset:\\ +&(\exists S):S\in \text{Potid}ʻR:Q = S\mid R\mid P.\lor.P = S\mid R\mid Q\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·731·24}.\supset\\ +\vdash\colon\ldotp \text{Hp}. &\supset:(\exists S,T):S\in \text{Potid}ʻR.T=S\mid R:Q = T\mid P.\lor.P = T\mid Q:\\ +[\text{*13·195}]&\supset:(\exists S):S\in \text{Potid}ʻR:Q = S\mid R\mid P.\lor.P = S\mid R\mid Q\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*91·74.</b> \(\vdash.\breve{R}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx = \overleftarrow{R}_{\text{po}}ʻx.Rʻʻ\overrightarrow{R}_{\unicode{x2217}}ʻx + = \overrightarrow{R}_{\text{po}}ʻx \quad[\text{*91·52.*37·302}]\)</p> + +<p class="nind"><b>*91·75.</b> \(\vdash.R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} = R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\text{po}} + = R_{\text{po}}\unicode{x228d}\breve{R}_{\unicode{x2217}} = R_{\text{po}}\unicode{x228d}I\upharpoonright CʻR\unicode{x228d}\breve{R}_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·5·51}. &\supset\vdash.\text{Cnv}ʻ(I\upharpoonright CʻR) = I\upharpoonright CʻR.\\ +[\text{*91·54}] &\supset\vdash.\breve{R}_{\unicode{x2217}} = I\upharpoonright CʻR\unicode{x228d}\breve{R}_{\text{po}}. &\qquad \text{(1)}\\ +[\text{*91·54.*23·56}] \supset\vdash.R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} &= R_{\text{po}}\unicode{x228d}I\upharpoonright + CʻR\unicode{x228d}\breve{R}_{\text{po}} &\qquad \text{(2)}\\ +[\text{*91·54}] & = R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\text{po}} &\qquad \text{(3)}\\ +[\text{(1)}] &= R_{\text{po}} \unicode{x228d} \breve{R}_{\unicode{x2217}} &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_601">[Pg 601]</span></p> +<h2 class="nobreak" id="*92">*92. POWERS OF ONE-MANY AND MANY-ONE RELATIONS.</h2> +</div> + + +<p><i>Summary of</i> *92.</p> + +<p>If \(R \in \text{Cls} \rightarrow 1\), it follows that, starting from +a given term \(x\), there is only one series of terms \(x_{1}, x_{2}, x_{3}, \ldots\) +such that +\[ +x R x_{1} . x_{1} R x_{2} . x_{2} R x_{3} . \ldots \text{.} +\]</p> + +<p>Thus for example the relation of son to father is a \(\text{Cls}\rightarrow 1\); +and starting from a given man, the series of ancestors +in the direct male line (which is the above series \(x_{1}\), +\(x_{2}\), \(x_{3}\),...) is unique and determinate. A result of this +property of many-one relations is that if, starting from a term \(y\), +we go backwards a certain number of steps to a term \(x\), and then +forward a greater number of steps to a term \(z\), we must pass through +\(y\) in going from \(x\) to \(z\); while if the number of steps from +\(x\) to \(z\) is less than that from \(x\) to \(y\), \(z\) must lie +on the road from \(x\) to \(y\). These facts are expressed by the +proposition: +\[ +R \in \text{Cls} \rightarrow 1 .\supset. \breve{R}_{\unicode{x2217}} \mid R_{\unicode{x2217}} \unicode{x2abd} R_{\unicode{x2217}} + \unicode{x228d} \breve{R}_{\unicode{x2217}}\text{.} +\]</p> + +<p>In the present number, we have to establish various propositions of +this kind.</p> + +<p>We prove in this number various propositions which are used in the +discussion of "families" in <a href="#*96">*96</a> and <a href="#*97">*97</a>, and some which are used in +the theory of finite and infinite. But on the whole the propositions +of this number are not much used. The most important of them are the +following:</p> + +<p class="nind"><b>*92·11.</b> \(\vdash: R \in 1 \rightarrow \text{Cls} .\supset. R_{\text{po}} \mid \breve{R} \unicode{x2abd} R_{\unicode{x2217}} . R_{\text{po}} + \mid \breve{R} = R_{\unicode{x2217}} \upharpoonright \text{D}ʻR\)</p> + +<p>with a similar proposition (<a href="#*92·111">*92·111</a>) for \(\text{Cls} \rightarrow 1\).</p> + +<p class="nind"><b>*92·132.</b> \(\vdash: R \in 1 \rightarrow \text{Cls} . Q,\,T \in \text{Potid}ʻR .\supset. Q \mid T \mid \breve{Q} \unicode{x2abd} T\)</p> + +<p>with a similar proposition (<a href="#*92·133">*92·133</a>) for \(\text{Cls} \rightarrow 1\).</p> + +<p class="nind"><b>*92·14.</b> \(\vdash: \text{ᗡ}ʻR \subset \text{D}ʻR . Q \in \text{Pot}ʻR .\supset. \text{D}ʻQ = \text{D}ʻR\)</p> + +<p>On this proposition, compare the remarks on <a href="#*91·271">*91·271</a> in the introduction +to <a href="#*91">*91</a>. If \(R\) is a serial relation, \(\text{ᗡ}ʻR \subset +\text{D}ʻR\) is the condition that the series may have no last term.</p> + +<p class="nind"><b>*92·31.</b> \(\vdash: R \in 1 \rightarrow \text{Cls} .\supset. R_{\unicode{x2217}} \mid \breve{R}_{\unicode{x2217}} = R_{\unicode{x2217}} + \unicode{x228d} \breve{R}_{\unicode{x2217}}\)</p> + +<p class="nind"><b>*92·311.</b> \(\vdash: R \in \text{Cls} \rightarrow 1 .\supset. \breve{R}_{\unicode{x2217}} \mid R_{\unicode{x2217}} = R_{\unicode{x2217}} + \unicode{x228d} \breve{R}_{\unicode{x2217}}\)</p> + +<p><span class="pagenum" id="Page_602">[Pg 602]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*92·1.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\text{Potid}ʻR\subset 1\rightarrow \text{Cls}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·17.*71·26}.&\supset\vdash.I\upharpoonright CʻR\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*71·25}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:S\in 1\rightarrow \text{Cls}.\supset.SʻR\in 1\rightarrow \text{Cls} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·17}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·101.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\text{Potid}ʻR\subset \text{Cls}\rightarrow 1 \quad[\text{Proof as in *92·1}]\)</p> + +<p class="nind"><b>*92·102.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\text{Potid}ʻR\subset 1\rightarrow 1 \quad[\text{Proof as in *92·1}]\)</p> + +<p class="nind"><b>*92·11.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.R_{\text{po}}\mid \breve{R}\unicode{x2abd}R_{\unicode{x2217}}.R_{\text{po}}\mid + \breve{R} = R_{\unicode{x2217}}\upharpoonright \text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·52}. &\supset\vdash.R_{\text{po}}\mid \breve{R} = R_{\unicode{x2217}}\mid R\mid \breve{R} &\qquad \text{(1)}\\ +\vdash.\text{*71·19}. &\supset\vdash:\text{Hp}.\supset.R\mid \breve{R} = I\upharpoonright \text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*50·6}.&\supset\vdash:\text{Hp}.\supset.R_{\text{po}}\mid \breve{R} = R_{\unicode{x2217}}\upharpoonright \text{D}ʻR &\qquad \text{(3)}\\ +\vdash.\text{(3).*35·441}. \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*92·111">*92·111</a>.</b> \[\begin{align}&\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}\mid R_{\text{po}}\unicode{x2abd}R_{\unicode{x2217}}.\breve{R}\mid R_{\text{po}} + = (\text{ᗡ}ʻR)\upharpoonleft R_{\unicode{x2217}}\\ +&[\text{Proof as in *92·11}]\end{align}\]</p> + +<p class="nind"><b>*92·112.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.R\mid R_{\text{po}}\mid \breve{R} = R_{\text{po}}\upharpoonright \text{D}ʻR \quad[\text{*92·11.*91·52}]\)</p> + +<p class="nind"><b>*92·113.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}\mid R_{\text{po}}\mid R = (\text{ᗡ}ʻR)\upharpoonleft R_{\text{po}} \quad[\text{*92·111.*91·52}]\)</p> + +<p class="nind"><b>*92·12.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \text{D}ʻR.\supset.R_{\text{po}}\mid \breve{R} = R_{\unicode{x2217}} \quad[\text{*92·11.*35·66}]\)</p> + +<p class="nind"><b>*92·121.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\text{D}ʻR\subset \text{ᗡ}ʻR.\supset.\breve{R}\mid R_{\text{po}} = R_{\unicode{x2217}} \quad[\text{*92·111.*35·63}]\)</p> + +<p class="nind"><b>*92·13.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.Q,\,T\in \text{Potid}ʻR.\supset.T\mid Q\mid \breve{Q} = T\upharpoonright \text{D}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·1}.\supset\vdash:\text{Hp}.&\supset.Q\in 1\rightarrow \text{Cls}.\\ +[\text{*71·19}] &\supset.Q\mid \breve{Q} = I\upharpoonright \text{D}ʻQ.\\ +[\text{*50·6}] &\supset.T\mid Q\mid \breve{Q} = T\upharpoonright \text{D}ʻQ:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·131.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.Q,\,T\in \text{Potid}ʻR.\supset.\breve{Q}\mid Q\mid T= (\text{ᗡ}ʻQ)\upharpoonleft T\)</p> + +<p>In this number, when proofs have been given for \(R\in 1\rightarrow\text{Cls}\), +we shall omit the proofs of corresponding propositions +for \(R\in \text{Cls}\rightarrow 1\), as these are always exactly +analogous to the proofs for \(R\in 1\rightarrow \text{Cls}\).</p> + +<p class="nind"><b>*92·132.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.Q,\,T\in \text{Potid}ʻR.\supset.Q\mid T\mid \breve{Q}\unicode{x2abd}T \quad[\text{*92·13.*91·34}]\)</p> + +<p class="nind"><b><a id="*92·133">*92·133</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.Q,\,T\in \text{Potid}ʻR.\supset.\breve{Q}\mid T\mid Q\unicode{x2abd}T\)</p> + +<p><span class="pagenum" id="Page_603">[Pg 603]</span></p> + +<p class="nind"><b>*92·14.</b> \(\vdash:\text{ᗡ}ʻR\subset \text{D}ʻR.Q\in \text{Pot}ʻR.\supset.\text{D}ʻQ=\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·271}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\text{ᗡ}ʻQ\subset \text{D}ʻR:\\ +[\text{*37·321}]&\supset:\text{D}ʻ(Q\mid R)=\text{D}ʻQ:\\ +[\text{*13·182}]&\supset:\text{D}ʻQ=\text{D}ʻR.\supset.\text{D}ʻ(Q\mid R)=\text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*13·15}.&\supset\vdash.\text{D}ʻR=\text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·171}\, \frac{\text{D}ʻS=\text{D}ʻR}{\phi S} .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·141.</b> \(\vdash:\text{D}ʻR\subset \text{ᗡ}ʻR.Q\in \text{Pot}ʻR.\supset.\text{ᗡ}ʻQ=\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*92·142.</b> \(\vdash:\text{ᗡ}ʻR\subset \text{D}ʻR.Q\in \text{Potid}ʻR.\supset.\text{D}ʻQ=\text{D}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·5·52}.&\supset\vdash:Q=I\upharpoonright CʻR.\supset.\text{D}ʻQ=CʻR &\qquad \text{(1)}\\ +\vdash.\text{*33·181}. &\supset\vdash:\text{Hp}.\supset.CʻR=\text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash:\text{Hp}.Q=I\upharpoonright CʻR.\supset.\text{D}ʻQ=\text{D}ʻR &\qquad \text{(3)}\\ +\vdash.\text{*91·23}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:Q=I\upharpoonright CʻR.\lor.Q\in \text{Pot}ʻR &\qquad \text{(4)}\\ +\vdash.\text{(3).(4).*92·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·143.</b> \(\vdash:\text{D}ʻR\subset \text{ᗡ}ʻR.Q\in \text{Potid}ʻR.\supset.\text{ᗡ}ʻQ=\text{ᗡ}ʻR\)</p> + +<p class="nind"><b>*92·144.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \text{D}ʻR.Q\in \text{Potid}ʻR.\supset.\text{ᗡ}ʻQ\subset \text{D}ʻR.\text{ᗡ}ʻQ\subset \text{D}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·271}.& \supset\vdash:\text{Hp}.Q\in \text{Pot}ʻR.\supset.\text{ᗡ}ʻQ\subset \text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*50·5·52}.&\supset\vdash:Q=I\upharpoonright CʻR.\supset.\text{ᗡ}ʻQ=CʻR &\qquad \text{(2)}\\ +\vdash.\text{*33·181}. &\supset\vdash:\text{Hp}.\supset.CʻR=\text{D}ʻR &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*23·42}.&\supset\vdash:\text{Hp}.Q=I\upharpoonright CʻR.\supset.\text{ᗡ}ʻQ\subset \text{D}ʻR &\qquad \text{(4)}\\ +\vdash.\text{*91·23}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:Q=I\upharpoonright CʻR.\lor.Q\in \text{Pot}ʻR &\qquad \text{(5)}\\ +\vdash.\text{(1).(4).(5).*92·142}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·145.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\text{D}ʻR\subset \text{ᗡ}ʻR.Q\in \text{Potid}ʻR.\supset.\text{D}ʻQ\subset \text{ᗡ}ʻR.\text{D}ʻQ\subset \text{ᗡ}ʻQ\)</p> + +<p class="nind"><b>*92·146.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \text{D}ʻR.Q,\,T\in \text{Potid}ʻR.\supset.T\upharpoonright \text{D}ʻQ=T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·142·144}.\supset\vdash:\text{Hp}.&\supset.\text{D}ʻQ=\text{D}ʻR.\text{ᗡ}ʻT\subset \text{D}ʻR.\\ +[\text{*13·13}] & \supset.\text{ᗡ}ʻT\subset \text{D}ʻQ.\\ +[\text{*35·66}] & \supset.T\upharpoonright \text{D}ʻQ=T:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·147.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\text{D}ʻR\subset \text{ᗡ}ʻR.Q,\,T\in \text{Potid}ʻR.\supset.(\text{ᗡ}ʻQ)\upharpoonleft T=T\)</p> + +<p class="nind"><b>*92·15.</b> \[\begin{align}&\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \text{D}ʻR.Q,\,T\in \text{Potid}ʻR.\supset.T\mid Q\mid \breve{Q}=T\\ +&[\text{*92·13·146}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_604">[Pg 604]</span></p> + +<p class="nind"><b>*92·151.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\text{D}ʻR\subset \text{ᗡ}ʻR.Q,T\in \text{Potid}ʻR.\supset.\breve{Q}\mid Q\mid T=T\)</p> + +<p class="nind"><b>*92·152.</b> \[\begin{align}&\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \text{D}ʻR.Q,\,T\in \text{Potid}ʻR.\supset.Q\mid T\mid \breve{Q}=T\\ +&[\text{*92·15.*91·34}]\end{align}\]</p> + +<p class="nind"><b>*92·153.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\text{D}ʻR\subset \text{ᗡ}ʻR.Q,\,T\in \text{Potid}ʻR.\supset.\breve{Q}\mid T\mid Q=T\)</p> + +<p class="nind"><b>*92·16.</b> \[\begin{align}\vdash\colon\ldotp R\in &1\rightarrow \text{Cls}.P,\,Q\in \text{Potid}ʻR.\supset:\\ +&(\exists T):T\in \text{Potid}ʻR:P\mid \breve{Q}=T\upharpoonright \text{D}ʻQ.\lor.P\mid \breve{Q}=\text{Cnv}ʻ(T\upharpoonright \text{D}ʻP)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·46}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists T):T\in \text{Potid}ʻR:Q=T\mid P.\lor.P=T\mid Q &\qquad \text{(1)}\\ +\vdash.\text{*92·13}.&\supset\vdash:\text{Hp}.T\in \text{Potid}ʻR.P=T\mid Q.\supset.P\mid \breve{Q}=T\upharpoonright \text{D}ʻQ &\qquad \text{(2)}\\ +\vdash.\text{*92·13}.&\supset\vdash:\text{Hp}.T\in \text{Potid}ʻR.Q=T\mid P.\supset.Q\mid \breve{P}=T\upharpoonright \text{D}ʻP.\\ +[\text{*34·2}] &\supset.P\mid \breve{Q}=\text{Cnv}ʻ(T\upharpoonright \text{D}ʻP) &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·161.</b> \[\begin{align}\vdash\colon\ldotp &R\in \text{Cls}\rightarrow 1.P,\,Q\in \text{Potid}ʻR.\supset:\\ +&(\exists T):T\in \text{Potid}ʻR:\breve{Q}\mid P=(\text{ᗡ}ʻQ)\upharpoonleft T.\lor.\breve{Q}\mid P=\text{Cnv}ʻ\{(\text{ᗡ}ʻP)\upharpoonleft T\}\end{align}\]</p> + +<p class="nind"><b>*92·17.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.P,\,Q\in \text{Potid}ʻR.\supset.(\exists T).T\in \text{Potid}ʻR.P\mid \breve{Q}\unicode{x2abd}T\unicode{x228d}\breve{T}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·441}.&\supset\vdash:P\mid \breve{Q}=T\upharpoonright \text{D}ʻQ.\supset.P\mid \breve{Q}\unicode{x2abd}T.\\ +[\text{*23·58}] &\supset.P\mid \breve{Q}\unicode{x2abd}T\unicode{x228d}\breve{T} &\qquad \text{(1)}\\ +\vdash.\text{*35·52·44}.&\supset\vdash:P\mid \breve{Q}=\text{Cnv}ʻ(T\upharpoonright \text{D}ʻP).\supset.P\mid \breve{Q}\unicode{x2abd}\breve{T}.\\ +[\text{*23·58}] &\supset.P\mid \breve{Q}\unicode{x2abd}T\unicode{x228d}\breve{T} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*92·16}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·171.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.P,Q\in \text{Potid}ʻR.\supset.(\exists T).T\in \text{Potid}ʻR.\breve{Q}\mid P\unicode{x2abd}T\unicode{x228d}\breve{T}\)</p> + +<p class="nind"><b>*92·18.</b> \[\begin{align}\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \text{D}ʻR.P,Q\in \text{Potid}ʻ&R.\supset.\\ +&P\mid \breve{Q}\in \text{Potid}ʻR\cup \text{Potid}ʻ\breve{R}\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·16·146}.\supset\\ +\vdash\colon\ldotp \text{Hp}. & \supset:(\exists T):T\in \text{Potid}ʻR:P\mid \breve{Q}=T.\lor.P\mid \breve{Q}=\breve{T}:\\ +[\text{*10·42}] &\supset:(\exists T).T\in \text{Potid}ʻR.P\mid \breve{Q}=T.\lor.(\exists T).T\in \text{Potid}ʻR.P\mid \breve{Q}=\breve{T}:\\ +[\text{*91·521}]&\supset:(\exists T).T\in \text{Potid}ʻR.P\mid \breve{Q}=T.\lor.(\exists T).T\in \text{Potid}ʻ\breve{R}.P\mid \breve{Q}=T:\\ +[\text{*13·195}]&\supset:P\mid \breve{Q}\in \text{Potid}ʻR.\lor.P\mid \breve{Q}\in \text{Potid}ʻ\breve{R}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·181.</b> \[\begin{align}\vdash:R\in \text{Cls}\rightarrow 1.\text{D}ʻR\subset \text{ᗡ}ʻR.P,Q\in \text{Potid}ʻ&R.\supset.\\ +&\breve{Q}\mid P\in \text{Potid}ʻR\cup \text{Potid}ʻ\breve{R}\end{align}\]</p> + +<p><span class="pagenum" id="Page_605">[Pg 605]</span></p> + +<p class="nind"><b>*92·19.</b> \[\begin{align}\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \text{D}ʻR.P,&Q\in \text{Potid}ʻR.\supset:\\ +&P\mid \breve{Q}\in \text{Potid}ʻR.\lor.Q\mid \breve{P}\in \text{Potid}ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·18}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:P\mid \breve{Q}\in \text{Potid}ʻR.\lor.P\mid \breve{Q}\in \text{Potid}ʻ\breve{R} &\qquad \text{(1)}\\ +\vdash.\text{*91·521.*34·2}.&\supset\vdash:P\mid \breve{Q}\in \text{Potid}ʻ\breve{R}.\equiv.Q\mid \breve{P}\in \text{Potid}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·191.</b> \[\begin{align}\vdash:R\in \text{Cls}\rightarrow 1.\text{D}ʻR\subset \text{ᗡ}ʻR.P\mid Q &\in \text{Potid}ʻR.\supset:\\ +&\breve{P}\mid Q\in \text{Potid}ʻR.\lor.\breve{Q}\mid P\in \text{Potid}ʻR\end{align}\]</p> + +<p class="nind"><b>*92·3.</b> + \(\vdash:R\in 1\rightarrow \text{Cls}.P,\,Q\in \text{Potid}ʻR.\supset.P\mid \breve{Q}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·58}.&\supset\vdash:T\in \text{Potid}ʻR.\supset.T\unicode{x228d}\breve{T}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}:\\ +[\text{*23·44}] &\supset\vdash:T\in \text{Potid}ʻR.P\mid \breve{Q}\unicode{x2abd}T\unicode{x228d}\breve{T}.\supset.P\mid \breve{Q}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}:\\ +[\text{*10·11·23}]&\supset\vdash:(\exists T).T\in \text{Potid}ʻR.P\mid \breve{Q}\unicode{x2abd}T\unicode{x228d}\breve{T}.\supset.P\mid \breve{Q}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} &\qquad \text{(1)}\\ +\vdash.\text{(1).*92·17}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·301.</b> + \(\vdash:R\in \text{Cls}\rightarrow 1.P,\,Q\in \text{Potid}ʻR.\supset.\breve{P}\mid Q\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\)</p> + +<p class="nind"><b><a id="*92·31">*92·31</a>.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.R_{\unicode{x2217}}\mid + \breve{R}_{\unicode{x2217}}=R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·14.*50·64}. & \supset\vdash.R_{\unicode{x2217}}=R_{\unicode{x2217}}\mid I\upharpoonright CʻR &\qquad \text{(1)}\\ +\vdash.\text{*90·15·132.*33·22}.&\supset\vdash.I\upharpoonright CʻR\unicode{x2abd}\breve{R}_{\unicode{x2217}}.\\ +[\text{*34·34}] &\supset\vdash.R_{\unicode{x2217}}\mid I\upharpoonright CʻR\unicode{x2abd}R_{\unicode{x2217}}\mid \breve{R}_{\unicode{x2217}}.\\ +[\text{(1)}] &\supset\vdash.R_{\unicode{x2217}}\unicode{x2abd}R_{\unicode{x2217}}\mid \breve{R}_{\unicode{x2217}} &\qquad \text{(2)}\\ +\text{Similarly} & \vdash.\breve{R}_{\unicode{x2217}}\unicode{x2abd}R_{\unicode{x2217}}\mid \breve{R}_{\unicode{x2217}} &\qquad \text{(3)}\\ +\vdash.\text{*91·55.*90·132}. & \supset\vdash.R_{\unicode{x2217}}\mid \breve{R}_{\unicode{x2217}}=\dot{s}ʻ\text{Potid}ʻR\mid \dot{s}ʻ\text{Potid}ʻ\breve{R}\\ +[\text{*41·51}] & =\dot{s}ʻ\hat{T}\{(\exists P,Q).P\in \text{Potid}ʻR.Q\in \text{Potid}ʻ\breve{R}.T=P\mid Q\}\\ +[\text{*91·521}] & =\dot{s}ʻ\hat{T}\{(\exists P,Q).P,\,Q\in \text{Potid}ʻR.T=P\mid \breve{Q}\} &\qquad \text{(4)}\\ +\vdash.\text{*92·3}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists P,Q).P,Q\in \text{Potid}ʻR.T=P\mid \breve{Q}.\supset_{T}.T\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}:\\ +[\text{*41·151}] &\supset:\dot{s}ʻ\hat{T}\{(\exists P,Q).P,\,Q\in \text{Potid}ʻR.T=P\mid \breve{Q}\}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} + &\qquad \text{(5)}\\ +\vdash.\text{(4).(5)}.&\supset\vdash:\text{Hp}.\supset.R_{\unicode{x2217}}\mid + \breve{R}_{\unicode{x2217}}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} &\qquad \text{(6)}\\ +\vdash.\text{(2).(3).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·311.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}_{\unicode{x2217}}\mid + R_{\unicode{x2217}}=R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\)</p> + +<p><span class="pagenum" id="Page_606">[Pg 606]</span></p> + +<p class="nind"><b>*92·312.</b> \(\vdash:R\in 1\rightarrow 1.\supset.R_{\unicode{x2217}}\mid \breve{R}_{\unicode{x2217}}=\breve{R}_{\unicode{x2217}}\mid + R_{\unicode{x2217}}=R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} \quad[\text{*92·31·311}]\)</p> + +<p class="nind"><b>*92·32.</b> \(\vdash:R\in 1\rightarrow 1.\supset.(R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}})\mid + (R\unicode{x228d}\breve{R})\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·25·26}. &\supset\vdash.(R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}})\mid (R\unicode{x228d}\breve{R})=R_{\unicode{x2217}}\mid + R\unicode{x228d}R_{\unicode{x2217}}\mid \breve{R}\unicode{x228d}\breve{R}_{\unicode{x2217}}\mid R\unicode{x228d}\breve{R}_{\unicode{x2217}}\mid + \breve{R} &\qquad \text{(1)}\\ +\vdash.\text{*90·16·132}.&\supset\vdash.R_{\unicode{x2217}}\mid R\unicode{x2abd}R_{\unicode{x2217}}.\breve{R}_{\unicode{x2217}}\mid + \breve{R}\unicode{x2abd}\breve{R}_{\unicode{x2217}} &\qquad \text{(2)}\\ +\vdash.\text{*90·151}. &\supset\vdash.R_{\unicode{x2217}}\mid \breve{R}\unicode{x2abd}R_{\unicode{x2217}}\mid \breve{R}_{\unicode{x2217}}.\breve{R}_{\unicode{x2217}}\mid + R\unicode{x2abd}\breve{R}_{\unicode{x2217}}\mid R_{\unicode{x2217}} &\qquad \text{(3)}\\ +\vdash.\text{(3).*92·312}.&\supset\vdash:\text{Hp}.\supset.R_{\unicode{x2217}}\mid + \breve{R}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}.\breve{R}_{\unicode{x2217}}\mid R\unicode{x2abd}R_{\unicode{x2217}}\cup + \breve{R}_{\unicode{x2217}} &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(4)}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·33.</b> \(\vdash:R\in 1\rightarrow 1.\supset.(R\unicode{x228d}\breve{R})_{\unicode{x2217}}=R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·18}. &\supset\vdash.R_{\unicode{x2217}}\unicode{x2abd}(R\unicode{x228d}\breve{R})_{\unicode{x2217}}.\breve{R}_{\unicode{x2217}}\unicode{x2abd}(R\unicode{x228d}\breve{R})_{\unicode{x2217}}.\\ +[\text{*23·59}] &\supset\vdash.R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\unicode{x2abd}(R\unicode{x228d}\breve{R})_{\unicode{x2217}} + &\qquad \text{(1)}\\ +\vdash.\text{*33·272}. & \supset\vdash.I\upharpoonright Cʻ(R\unicode{x228d}\breve{R})=I\upharpoonright CʻR.\\ +[\text{*90·15.*23·58}]&\supset\vdash.I\upharpoonright Cʻ(R\unicode{x228d}\breve{R})\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} + &\qquad \text{(2)}\\ +\vdash.\text{*92·32.*34·34}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:S\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}.\supset.S\mid + (R\unicode{x228d}\breve{R})\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*91·17}\, \frac{R\unicode{x228d}\breve{R},S\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}}{R,\,\,\phi S} .\supset\\ +&\vdash\colon\ldotp \text{Hp}.\supset:P\in \text{Potid}ʻ(R\unicode{x228d}\breve{R}).\supset_{P}.P\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}:\\ +[\text{*41·151}] &\supset:\dot{s}ʻ\text{Potid}ʻ(R\unicode{x228d}\breve{R})\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}:\\ +[\text{*91·55}] & \supset:(R\unicode{x228d}\breve{R})_{\unicode{x2217}}\unicode{x2abd}R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}} + &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*92·34.</b> \(\vdash:R\in 1\rightarrow 1.\supset.(R\unicode{x228d}\breve{R})_{\text{po}}=R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·33.*91·52}.\supset\\ +\vdash:\text{Hp}.\supset.\\ +(R\unicode{x228d}\breve{R})_{\text{po}}&=(R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}})\mid (R\unicode{x228d}\breve{R})\\ +[\text{*34·25·26}] & =R_{\unicode{x2217}}\mid R\unicode{x228d}\breve{R}_{\unicode{x2217}}\mid R\unicode{x228d}R_{\unicode{x2217}}\mid + \breve{R}\unicode{x228d}\breve{R}_{\unicode{x2217}}\mid \breve{R}\\ +[\text{*91·52·54·57}] & =R_{\text{po}}\unicode{x228d}(I\upharpoonright CʻR\unicode{x228d}\breve{R}\unicode{x228d}\breve{R}_{\unicode{x2217}}\mid + \breve{R})\mid R\unicode{x228d}(I\upharpoonright CʻR\unicode{x228d}R\unicode{x228d}R_{\unicode{x2217}}\mid R)\mid \breve{R}\unicode{x228d}\breve{R}_{\text{po}}\\ +[\text{*50·65.*71·192.*72·59·591}]\\ +& =R_{\text{po}}\unicode{x228d}R\unicode{x228d}I\upharpoonright \text{ᗡ}ʻR\unicode{x228d}\breve{R}_{\unicode{x2217}}\upharpoonright + \text{ᗡ}ʻR\unicode{x228d}\breve{R}\unicode{x228d}I\upharpoonright \text{D}ʻR\unicode{x228d}R_{\unicode{x2217}}\upharpoonright \text{D}ʻR\unicode{x228d}\breve{R}_{\text{po}}\\ +[\text{*35·412.*91·502}]\\ +&=R_{\text{po}}\unicode{x228d}I\upharpoonright CʻR\unicode{x228d}\breve{R}_{\unicode{x2217}}\upharpoonright \text{ᗡ}ʻR\unicode{x228d}R_{\unicode{x2217}}\upharpoonright + \text{D}ʻR\unicode{x228d}\breve{R}_{\text{po}}\\ +[\text{*91·75}] &=R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}\upharpoonright + \text{ᗡ}ʻR\unicode{x228d}R_{\unicode{x2217}}\upharpoonright \text{D}ʻR\\ +[\text{*35·441}]& =R_{\unicode{x2217}}\unicode{x228d}\breve{R}_{\unicode{x2217}}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_607">[Pg 607]</span></p> +<h2 class="nobreak" id="*93">*93. INDUCTIVE ANALYSIS OF THE FIELD OF A RELATION.</h2> +</div> + + +<p><i>Summary of</i> *93.</p> + +<p>For this number, we introduce three new notations, of which the first +two will be used constantly, especially in the theory of series, while +the third will be seldom used except in the present section. The two +which are constantly used are +\[ +xBP, ~ \text{meaning} ~ x \in \text{D}ʻP - \text{ᗡ}ʻP +\] +\[ +\text{and} \quad x \mathop{\text{min}_{P}} \alpha, ~ \text{meaning} ~ x \in \alpha \cap CʻP-\breve{P}ʻʻ\alpha\text{,} +\] +<i>i.e.</i> \(x\) is a member of \(\alpha\) and of \(CʻP\), and no member of \(\alpha\) precedes \(x\) in \(CʻP\).</p> + +<p>The letter \(B\) may be regarded as standing for "begins." Thus if we +take any member \(y\) of \(CʻP\), and proceed backwards and forwards as +far as possible by \(P\)-steps, we obtain a series which may be called +the "family" of \(y\): this series, if it has a first term, has one +which is a member of \(\text{D}ʻP-\text{ᗡ}ʻP\); thus the members of +\(\text{D}ʻP-\text{ᗡ}ʻP\) are the beginners of families. For example, +if \(P\) is the relation of a peer to his heir, "\(xBP\)" will mean +"\(x\) is a peer who is not the heir of a peer"; thus \(x\) is the +first of his family. If \(P\) is the relation of parent and child, +"\(xBP\)" will be satisfied only by Adam and Eve; and so for other +relations.</p> + +<p>The definition of \(B\) is +\[ +B = \hat{x}\hat{P}(x \in \text{D}ʻP - \text{ᗡ}ʻP) \quad \text{Df}\text{.} +\] +Hence \(\overrightarrow{B}ʻP = \text{D}ʻP - \text{ᗡ}ʻP\). If \(P\) is +the generating relation of a series which has a first term, that first +term is \(BʻP\); if there is a last term it is \(Bʻ\breve{P}\).</p> + +<p>If \(\alpha\) is any class, we may call a term \(x\) a <i>minimum</i> +of \(\alpha\) with respect to \(P\) if it is a member of \(\alpha\) and +of \(CʻP\), but does not follow any member of \(\alpha\), <i>i.e.</i> +is not a member of \(\breve{P}ʻʻ\alpha\). We denote this relation of +\(x\) to \(\alpha\) by "\(\text{min}_{P}\)"; thus we have +\[ +x \mathop{\text{min}_{P}} \alpha .\equiv. x \in \alpha \cap CʻP - \breve{P}ʻʻ\alpha\text{,} +\] +and the definition of \(\text{min}_{P}\) is +\[ +\text{min}_{P} = \hat{x}\hat{\alpha}(x \in \alpha \cap CʻP - \breve{P}ʻʻ\alpha) \quad \text{Df}\text{.} +\]</p> + +<p><span class="pagenum" id="Page_608">[Pg 608]</span></p> + +<p>We shall also, when convenient, write "\(\text{min}(P)\)" in place of +"\(\text{min}_{P}\)."</p> + +<p>We have \(\overrightarrow{\text{min}}_{P}ʻ\alpha = \alpha \cap CʻP - \breve{P}ʻʻ\alpha\).</p> + +<p>If \(P\) is serial, \(\overrightarrow{\text{min}}_{P}ʻ\alpha\) reduces +to a single term if it is not null; thus if a class \(\alpha\) has a +first term, this term is \(\text{min}_{P}ʻ\alpha\). We also put +\[ +\text{max}_{P} = \text{min}(\breve{P}) \quad \text{Df}\text{,} +\] +and then \(\text{max}_{P}ʻ\alpha\), if it exists, is the last +term of \(\alpha\) in the \(P\)-series. Thus if \(\alpha\) +is the class of peers, and \(P\) is the relation of father +to son, \(\overrightarrow{\text{min}_{P}}ʻ\alpha\) consists +of those peers who are the first of their line, while +\(\overrightarrow{\text{max}_{P}}ʻ\alpha\) consists of those peers who +are the last of their line. If \(\alpha\) is a class of numbers, and +\(P\) is the relation of less to greater, \(\text{min}_{P}ʻ\alpha\) +is the smallest member of \(\alpha\) (if it exists), and +\(\text{max}_{P}ʻ\alpha\) is the largest (if it exists).</p> + +<p>\(B\) and "\(\text{max}_{P}\)" and "\(\text{min}_{P}\)" will be used constantly in connection with +series, where the two latter will be considered in detail, but the present number +is more specially concerned with a less general idea, namely that of <i>generations</i>. +Take, <i>e.g.</i>, the relation of parent and child; let us call it \(P\). Then +the first generation consists of those who are parents but not children, +<i>i.e.</i> \(\overrightarrow{B}ʻP\); the second consists of those who are children but not grandchildren, +<i>i.e.</i> \(\text{ᗡ}ʻP - \text{ᗡ}ʻP^{2}\), <i>i.e.</i> \(\text{ᗡ}ʻP - \breve{P}ʻʻ\text{ᗡ}ʻP\), <i>i.e.</i> \(\overrightarrow{\text{min}_{P}}ʻ\text{ᗡ}ʻP\); + the third consists of those +who are grandchildren but not great-grandchildren, <i>i.e.</i> \(\text{ᗡ}ʻP^{2} - \text{ᗡ}ʻP^{3}\), <i>i.e.</i> +\(\text{ᗡ}ʻP^{2} - \breve{P}ʻʻ\text{ᗡ}ʻP^{2}\), <i>i.e.</i> +\(\overrightarrow{\text{min}_{P}}ʻ\text{ᗡ}ʻP^{2}\); and so on. Also we +have +\[ +\overrightarrow{B}ʻP = \overrightarrow{\text{min}_{P}}ʻ\text{ᗡ}ʻ(I \upharpoonright CʻP)\text{;} +\] +hence the generations of \(P\) are +\(\overrightarrow{\text{min}_{P}}ʻʻ\text{ᗡ}ʻʻ\text{Potid}ʻP\). Thus we +put +\[ +\text{gen}ʻP = \overrightarrow{\text{min}_{P}}ʻʻ\text{ᗡ}ʻʻ\text{Potid}ʻP \quad \text{Df}\text{,} +\] +where "\(\text{gen}\)" stands for "generation."</p> + +<p>When \(P\) is a one-many relation, such as that of father and son, +every generation is of the form \(\breve{T}ʻʻ\overrightarrow{B}ʻP\), +where \(T\) is a power of \(P\) (including \(I \upharpoonright CʻP\)). +When \(P\) is not a one-many relation, this is not in general the case.</p> + +<p>The generations of \(P\) do not in general exhaust the field of +\(P\). For \(x\) will only belong to a generation of \(P\) if \(x\) +can be reached by successive \(P\)-steps starting from a member of +\(\overrightarrow{B}ʻP\). If some of the families constituting the +field of \(P\) have no beginning, the members of these families will +not belong to any generation of \(P\). Such terms together constitute +the class +\[ +\begin{align} +&pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP,\\ +\text{or} \quad &pʻ\text{ᗡ}ʻʻ\text{Potid}ʻP,\\ +\end{align} +\] +which is the same class.</p> + +<p><span class="pagenum" id="Page_609">[Pg 609]</span></p> + +<p>Thus the field of \(P\) may be divided into two mutually exclusive +portions, \(sʻ\text{gen}ʻP\) and \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\).</p> + +<p>The present number begins with some elementary properties of \(B\) +and \(\text{min}_{P}\) and \(\text{max}_{P}\). We then (<a href="#*93·2">*93·2</a>—<a href="#*93·275">·275</a>) +consider such properties of generations as do not demand any hypothesis +as to \(P\). We prove</p> + +<p class="nind"><b>*93·25.</b> \(\vdash.\text{gen}ʻP\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p class="nind"><b>*93·261.</b> \(\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=pʻ\text{ᗡ}ʻʻ\text{Potid}ʻP.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\subset \text{ᗡ}ʻP\)</p> + +<p>and we prove (*93·274·275) that \(sʻ\text{gen}ʻP\) and +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\) are mutually exclusive, and together +constitute \(CʻP\). We then proceed to a set of propositions +(<a href="#*93·3">*93·3</a>—<a href="#*93·41">·41</a>) demanding that P should be one-many or many-one or +one-one. We prove</p> + +<p class="nind"><b>*93·32.</b> \(\vdash\colon\ldotp P\in 1\rightarrow \text{Cls}.\supset:\alpha\in \text{gen}ʻP.\equiv.(\exists T).T\in \text{Potid}ʻP.\alpha=\breve{T}ʻʻ\overrightarrow{B}ʻP\)</p> + +<p class="nind"><b>*93·36.</b> \(\vdash:P\in 1\rightarrow \text{Cls}.\supset.sʻ\text{gen}ʻP=\breve{P}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻP\)</p> + +<p class="nind"><b>*93·381.</b> \(\vdash\colon\ldotp P\in \text{Cls}\rightarrow 1.\supset:x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ\breve{P}.\equiv.\overleftarrow{P}_{\unicode{x2217}}ʻx\subset + \text{D}ʻP.x\in CʻP\)</p> + +<p>and various other properties of \(\text{gen}ʻP\) and +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\) when \(P\in 1\rightarrow \text{Cls}\).</p> + +<p>The propositions of this number are used throughout the rest of this +section; they are also used in the cardinal theory of finite and +infinite. The early propositions, down to <a href="#*93·12">*93·12</a> inclusive, are also +used in the theory of series.</p> + +<hr class="tb"> + +<p class="nind"><b>*93·01.</b> \(B=\hat{x}\hat{P}(x\in \text{D}ʻP-\text{ᗡ}ʻP) \quad\text{Df}\)</p> + +<p class="nind"><b>*93·02.</b> \(\text{min}_{P}=\text{min}(P)=\hat{x}\hat{\alpha}(x\in \alpha\cap CʻP-\breve{P}ʻʻ\alpha) \quad\text{Df}\)</p> + +<p class="nind"><b>*93·021.</b> \(\text{max}_{P}=\text{max}(P)=\text{min}(\breve{P}) \quad\text{Df}\)</p> + +<p class="nind"><b>*93·03.</b> \(\text{gen}ʻP=\overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\text{Potid}ʻP \quad\text{Df}\)</p> + +<p class="nind"><b>*93·1.</b> \(\vdash:xBP.\equiv.x\in \text{D}ʻP-\text{ᗡ}ʻP \quad[\text{*21·3.(*93·01)}]\)</p> + +<p class="nind"><b>*93·101.</b> \(\vdash.\overrightarrow{B}ʻP=\text{D}ʻP-\text{ᗡ}ʻP \quad[\text{*93·1.*32·18}]\)</p> + +<p class="nind"><b>*93·102.</b> \[\begin{align}&\vdash:x=BʻP.\equiv.x=\breve{\iota}ʻ(\text{D}ʻP-\text{ᗡ}ʻP).\equiv.\text{D}ʻP-\text{ᗡ}ʻP\in 1.x\in \text{D}ʻP-\text{ᗡ}ʻP\\ +&[\text{*93·101.*53·4}]\end{align}\]</p> + +<p class="nind"><b>*93·103.</b> \(\vdash.\overrightarrow{B}ʻP=CʻP-\text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·9.*33·16}.&\supset\vdash.CʻP-\text{ᗡ}ʻP=\text{D}ʻP-\text{ᗡ}ʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*93·101}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_610">[Pg 610]</span></p> + +<p class="nind"><b>*93·104.</b> \(\vdash:xBR.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻx=\iotaʻx.\overrightarrow{R}_{\text{po}}ʻx=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·1}.&\supset\vdash:\text{Hp}.\supset.x\in CʻR.\\ +[\text{*90·12}] &\supset.x\in \overrightarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*91·504}. &\supset\vdash:\exists !\overrightarrow{R}_{\text{po}}ʻx.\supset.x\in \text{ᗡ}ʻR:\\ +[\text{Transp.*93·1}] &\supset\vdash:xBR.\supset.\overrightarrow{R}_{\text{po}}ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*91·542}. &\supset\vdash:yR_{\unicode{x2217}}x.y \neq x.\supset.yR_{\text{po}}x:\\ +[\text{*32·18}] &\supset\vdash\colon\ldotp yR_{\unicode{x2217}}x.\supset:y=x.\lor.y\in \overrightarrow{R}_{\text{po}}ʻx &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:yR_{\unicode{x2217}}x.\supset.y=x &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}. & \supset\vdash:\text{Hp}.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻx=\iotaʻx &\qquad \text{(5)}\\ +\vdash.\text{(2).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·11.</b> \(\vdash:x\, \text{min}_{P}\,\alpha.\equiv.x\in \alpha\cap CʻP-\breve{P}ʻʻ\alpha \quad[\text{(*93·02)}]\)</p> + +<p class="nind"><b>*93·111.</b> \(\vdash.\overrightarrow{\text{min}}_{P}ʻ\alpha=\alpha\cap CʻP-\breve{P}ʻʻ\alpha \quad[\text{*93·11.*32·18}]\)</p> + +<p class="nind"><b>*93·112.</b> \(\vdash.\overrightarrow{B}ʻP=\overrightarrow{\text{min}}_{P}ʻ\text{D}ʻP=\overrightarrow{\text{min}}_{P}ʻCʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·111}.\supset\vdash.\overrightarrow{\text{min}}_{P}ʻ\text{D}ʻP&=\text{D}ʻP-\breve{P}ʻʻ\text{D}ʻP\\ +[\text{*37·25}] & =\text{D}ʻP-\text{ᗡ}ʻP\\ +[\text{*93·101}] & =\overrightarrow{B}ʻP &\qquad \text{(1)}\\ +\text{Similarly} \qquad\qquad\qquad\quad \vdash.\overrightarrow{\text{min}}_{P}ʻCʻP&=\overrightarrow{B}ʻP &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·113.</b> \(\vdash.\overrightarrow{\text{min}}_{P}ʻ\alpha\subset \alpha\cap CʻP \quad[\text{*93·111}]\)</p> + +<p class="nind"><b>*93·114.</b> \(\vdash.\text{max}_{P}=\text{min}(\breve{P}) \quad[\text{(*93·021)}]\)</p> + +<p class="nind"><b>*93·115.</b> \(\vdash:x\, \text{max}_{P}\,\alpha.\equiv.x\in \alpha\cap CʻP-Pʻʻ\alpha \quad[\text{*93·11·114}]\)</p> + +<p class="nind"><b>*93·116.</b> \(\vdash.\overrightarrow{\text{max}}_{P}ʻ\alpha=\alpha\cap CʻP-Pʻʻ\alpha \quad[\text{*93·115.*32·18}]\)</p> + +<p class="nind"><b>*93·117.</b> \(\vdash.\overrightarrow{B}ʻ\breve{P}=\overrightarrow{\text{max}}_{P}ʻ\text{ᗡ}ʻP=\overrightarrow{\text{max}}_{P}ʻCʻP \quad[\text{*93·112·114}]\)</p> + +<p class="nind"><b>*93·118.</b> \(\vdash.\overrightarrow{\text{max}}_{P}ʻ\alpha\subset \alpha\cap CʻP \quad[\text{*93·116}]\)</p> + +<p class="nind"><b><a id="*93·12">*92·12</a>.</b> \(\vdash.\overrightarrow{B}ʻ\breve{P}=\text{ᗡ}ʻP-\text{D}ʻP=CʻP-\text{D}ʻP \quad[\text{*93·101·103.*33·2·21·22}]\)</p> + +<p class="nind"><b>*93·13.</b> \(\vdash.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻ(I\upharpoonright CʻP)=\overrightarrow{B}ʻP \quad[\text{*50·5·52.*93·112}]\)</p> + +<p class="nind"><b>*93·131.</b> \(\vdash.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻP=\text{ᗡ}ʻP-\text{ᗡ}ʻP^{2}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·111}.\supset\vdash.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻP&=\text{ᗡ}ʻP-\breve{P}ʻʻ\text{ᗡ}ʻP\\ +[\text{*37·36}] &=\text{ᗡ}ʻP-\text{ᗡ}ʻP^{2}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_611">[Pg 611]</span></p> + +<p class="nind"><b>*93·132.</b> \(\vdash.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\text{ᗡ}ʻT-\text{ᗡ}ʻ(T\mid P)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·111}.\supset\vdash.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT&=\text{ᗡ}ʻT-\breve{P}ʻʻ\text{ᗡ}ʻT\\ +[\text{*37·32}] &=\text{ᗡ}ʻT-\text{ᗡ}ʻ(T\mid P).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*93·2">*92·2</a>.</b> \[\begin{align}&\vdash:\alpha\in \text{gen}ʻP.\equiv.(\exists T).T\in \text{Potid}ʻP.\alpha=\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT\\ +&[\text{*37·67.(*93·03)}]\end{align}\]</p> + +<p class="nind"><b>*93·21.</b> \[\begin{align}&\vdash:\alpha\in \text{gen}ʻP.\equiv.(\exists T).T\in \text{Potid}ʻP.\alpha=\text{ᗡ}ʻT-\text{ᗡ}ʻ(T\mid P)\\ +&[\text{*93·2·132}]\end{align}\]</p> + +<p class="nind"><b>*93·22.</b> \(\vdash.\overrightarrow{B}ʻP\in \text{gen}ʻP \quad[\text{*93·2·13.*91·35}]\)</p> + +<p class="nind"><b>*93·221.</b> \(\vdash.\text{ᗡ}ʻP-\text{ᗡ}ʻP^{2}\in \text{gen}ʻP \quad[\text{*93·2·131.*91·351·23}]\)</p> + +<p class="nind"><b>*93·23.</b> \(\vdash.\text{gen}ʻP=\iotaʻ\overrightarrow{B}ʻP\cup \overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·23.*37·22}.\supset\\ +\vdash.\text{gen}ʻP&=\overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\iotaʻ(I\upharpoonright CʻP)\cup \overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\\ +[\text{*53·31}]&=\iotaʻ\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻ(I\upharpoonright CʻP)\cup \overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\\ +[\text{*93·13}]&=\iotaʻ\overrightarrow{B}ʻP\cup \overrightarrow{\text{min}}_{P}ʻʻ\text{ᗡ}ʻʻ\text{Pot}ʻP +\end{array} +\]</p> + +<p class="nind"><b>*93·231.</b> \(\vdash\colon\ldotp S,\,T\in \text{Potid}ʻP.S \neq T.\supset:\text{ᗡ}ʻS\subset \breve{P}ʻʻ\text{ᗡ}ʻT.\lor.\text{ᗡ}ʻT\subset \breve{P}ʻʻ\text{ᗡ}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·732}.\supset\\ +\vdash\colon\ldotp \text{Hp}. \supset:(\exists M):S&=M\mid P\mid T.\lor.T=M\mid P\mid S:\\ +[\text{*91·3}]\supset:(\exists M):S&=M\mid T\mid P.\lor.T=M\mid S\mid P &\qquad \text{(1)}\\ +\vdash.\text{*34·36}.&\supset\vdash:S=M\mid T\mid P.\supset.\text{ᗡ}ʻS\subset \text{ᗡ}ʻ(T\mid P).\\ +[\text{*37·32}] &\supset.\text{ᗡ}ʻS\subset \breve{P}ʻʻ\text{ᗡ}ʻT &\qquad \text{(2)}\\ +\text{Similarly} &\vdash:T=M\mid S\mid P.\supset.\text{ᗡ}ʻT\subset \breve{P}ʻʻ\text{ᗡ}ʻS &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·24.</b> \(\vdash:S,\,T\in \text{Potid}ʻP.S \neq T.\supset.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻS\cap \overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*24·3}. \supset\vdash:\text{ᗡ}ʻS\subset \breve{P}ʻʻ\text{ᗡ}ʻT.&\supset.\text{ᗡ}ʻS-\breve{P}ʻʻ\text{ᗡ}ʻT=\Lambda.\\ +[\text{*24·34}] & \supset.\text{ᗡ}ʻS\cap \text{ᗡ}ʻT-\breve{P}ʻʻ\text{ᗡ}ʻT=\Lambda.\\ +[\text{*24·34}] &\supset.(\text{ᗡ}ʻS-\breve{P}ʻʻ\text{ᗡ}ʻS)\cap (\text{ᗡ}ʻT-\breve{P}ʻʻ\text{ᗡ}ʻT)=\Lambda.\\ +[\text{*93·111}] & \supset.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻS\cap \overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1)}\, \frac{T,\,S}{S,\,T} .\supset\vdash:\text{ᗡ}ʻT\subset \breve{P}ʻʻ\text{ᗡ}ʻS.&\supset.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻS\cap \overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\Lambda + &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*93·231}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_612">[Pg 612]</span></p> + +<p class="nind"><b>*93·25.</b> \(\vdash.\text{gen}ʻP\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*30·37.Transp}.\supset\\ +\vdash\colon\ldotp S,\,T\in &\text{Potid}ʻP.\alpha=\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻS.\beta=\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT.\alpha \neq \beta.\supset:\\ +& S,\,T\in \text{Potid}ʻP.S \neq T:\\ +[\text{*93·24}]&\supset:\alpha\cap \beta=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·35·54}.\supset\\ +\vdash\colon\ldotp (\exists S).S\in &\text{Potid}ʻP.\alpha=\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻS:(\exists + T).T\in \text{Potid}ʻP.\beta=\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT:\\ +&\alpha \neq \beta:\supset.\alpha\cap \beta=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).*93·2}.&\supset\vdash:\alpha,\beta\in \text{gen}ʻP.\alpha \neq \beta.\supset.\alpha\cap \beta=\Lambda &\qquad \text{(3)}\\ +\vdash.\text{(3).*84·1}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·26.</b> \(\vdash:S,\,T\in \text{Potid}ʻP.T\in \mid Sʻʻ\text{Pot}ʻP.\supset.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻS\cap + \overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·24}.\supset\vdash:\text{Hp}.&\supset.T\in \mid Sʻʻ\mid Pʻʻ\text{Potid}ʻP.\\ +[\text{*43·101.*37·67}]&\supset.(\exists M).T=M\mid P\mid S.\\ +[\text{*91·3}] & \supset.(\exists M).T=M\mid S\mid P.\\ +[\text{*34·36.*37·32}] &\supset.\text{ᗡ}ʻT\subset \breve{P}ʻʻ\text{ᗡ}ʻS.\\ +[\text{*24·3}] & \supset.\text{ᗡ}ʻT-\breve{P}ʻʻ\text{ᗡ}ʻS=\Lambda.\\ +[\text{*24·34}] &\supset.(\text{ᗡ}ʻS-\breve{P}ʻʻ\text{ᗡ}ʻS)\cap (\text{ᗡ}ʻT-\breve{P}ʻʻ\text{ᗡ}ʻT)=\Lambda.\\ +[\text{*93·111.*91·27}]&\supset.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻS\cap \overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\Lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·261.</b> \(\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=pʻ\text{ᗡ}ʻʻ\text{Potid}ʻP.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\subset \text{ᗡ}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·23}. \supset\vdash.\text{ᗡ}ʻʻ\text{Potid}ʻP&=\text{ᗡ}ʻʻ\text{Pot}ʻP\cup \iotaʻ\text{ᗡ}ʻ(I\upharpoonright CʻP)\\ +[\text{*50·5·52}] & =\text{ᗡ}ʻʻ\text{Pot}ʻP\cup \iotaʻCʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*53·14}. &\supset\vdash.pʻ\text{ᗡ}ʻʻ\text{Potid}ʻP=pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\cap CʻP &\qquad \text{(2)}\\ +\vdash.\text{*40·12.*91·351}. &\supset\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\subset \text{ᗡ}ʻP &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*22·621}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·27.</b> \(\vdash\colon\ldotp x\in CʻP.\supset:x{\sim}\in sʻ\text{gen}ʻP.\equiv.x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p><span class="pagenum" id="Page_613">[Pg 613]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·11.*10·51}.\supset\\ +\vdash\colon\ldotp x{\sim}\in sʻ\text{gen}ʻP.&\equiv:\alpha\in \text{gen}ʻP.\supset_{\alpha}.x{\sim}\in \alpha:\\ +[\text{*93·21}] &\equiv:T\in \text{Potid}ʻP.\supset_{T}.x{\sim}\in \text{ᗡ}ʻT-\text{ᗡ}ʻ(T\mid P):\\ +[\text{*4·53.*5·6}] &\equiv:T\in \text{Potid}ʻP.x\in \text{ᗡ}ʻT.\supset_{T}.x\in \text{ᗡ}ʻ(T\mid P) &\qquad \text{(1)}\\ +\vdash.\text{*50·5·52}.&\supset\vdash:x\in CʻP.\supset.x\in \text{ᗡ}ʻ(I\upharpoonright CʻP) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\colon x\in CʻP.\supset\colon\ldotp x~\in sʻ\text{gen}ʻP.\equiv:\\ +&x\in \text{ᗡ}ʻ(I\upharpoonright CʻP):T\in \text{Potid}ʻP.x\in \text{ᗡ}ʻT.\supset_{T}.x\in \text{ᗡ}ʻ(T\mid P):\\ +[\text{*91·371}]&\equiv:T\in \text{Potid}ʻP.\supset_{T}.x\in \text{ᗡ}ʻT:\\ +[\text{*40·41}]&\equiv:x\in pʻ\text{ᗡ}ʻʻ\text{Potid}ʻP:\\ +[\text{*93·261}]&\equiv:x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·271.</b> \(\vdash.CʻP-sʻ\text{gen}ʻP=pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*5·32.*93·27}.\supset\vdash:x\in CʻP-sʻ\text{gen}ʻP.&\equiv.x\in CʻP.x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP.\\ +[\text{*93·261.*4·71}] &\equiv.x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·272.</b> \(\vdash.sʻ\text{gen}ʻP\subset CʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·2·113}.\supset\vdash:\alpha\in \text{gen}ʻP.&\supset.(\exists T).T\in \text{Potid}ʻP.\alpha\subset \text{ᗡ}ʻT.\\ +[\text{*91·27}] &\supset.\alpha\subset CʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*40·151}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*93·273">*93·273</a>.</b> \(\vdash.CʻP-pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=sʻ\text{gen}ʻP \quad[\text{*93·271·272.*24·492}]\)</p> + +<p class="nind"><b>*93·274.</b> \(\vdash.CʻP=sʻ\text{gen}ʻP\cup pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP \quad[\text{*24·411.*93·271·272}]\)</p> + +<p class="nind"><b><a id="*93·275">*93·275</a>.</b> \(\vdash.sʻ\text{gen}ʻP\cap pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=\Lambda \quad[\text{*93·271.*24·21}]\)</p> + +<p class="nind"><b><a id="*93·3">*93·3</a>.</b> \(\vdash:P\in 1\rightarrow \text{Cls}.T\in \text{Potid}ʻP.\supset.\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\breve{T}ʻʻ\overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·38.*93·101}. \supset\vdash:\text{Hp}.\supset.\breve{T}ʻʻ\overrightarrow{B}ʻP&=\breve{T}ʻʻ\text{D}ʻP-\breve{T}ʻʻ\text{ᗡ}ʻP\\ +[\text{*37·25}] & =\breve{T}ʻʻ\text{D}ʻP-\breve{T}ʻʻ\breve{P}ʻʻ\text{D}ʻP\\ +[\text{*37·33.*91·3}] &=\breve{T}ʻʻ\text{D}ʻP-\breve{P}ʻʻ\breve{T}ʻʻ\text{D}ʻP\\ +[\text{*93·111.*91·27}] & =\overrightarrow{\text{min}}_{P}ʻ\breve{T}ʻʻ\text{D}ʻP &\qquad \text{(1)}\\ +\vdash.\text{*91·271.*37·271}.\supset\vdash:T\in \text{Pot}ʻP.\supset.\breve{T}ʻʻ\text{D}ʻP&=\text{ᗡ}ʻT &\qquad \text{(2)}\\ +\vdash.\text{*50·5·51·59}. \supset\vdash:T=I\upharpoonright CʻP.\supset.\breve{T}ʻʻ\text{D}ʻP&=\text{D}ʻP.\\ +[\text{*93·112}] \supset.\overrightarrow{\text{min}}_{P}ʻ\breve{T}ʻʻ\text{D}ʻP&=\overrightarrow{B}ʻP\\ +[\text{*93·13}] & =\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*91·23}.\supset\vdash:T\in \text{Potid}ʻP.\supset.\overrightarrow{\text{min}}_{P}ʻ\breve{T}ʻʻ\text{D}ʻP&=\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT + &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·31.</b> \(\vdash:P\in 1\rightarrow \text{Cls}.\supset.\breve{P}ʻʻ\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT=\overrightarrow{\text{min}}_{P}ʻ(T\mid P)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·38.*93·111.*37·265}.\supset\\ +\vdash:\text{Hp}.\supset.\breve{P}ʻʻ\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻT&=\breve{P}ʻʻ\text{ᗡ}ʻT-\breve{P}ʻʻ\breve{P}ʻʻ\text{ᗡ}ʻT\\ +[\text{*37·32}]&=\text{ᗡ}ʻ(T\mid P)-\breve{P}ʻʻ\text{ᗡ}ʻ(T\mid P)\\ +[\text{*93·111.*34·36}] & =\overrightarrow{\text{min}}_{P}ʻ\text{ᗡ}ʻ(T\mid P):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_614">[Pg 614]</span></p> + +<p class="nind"><b>*93·32.</b> + \[\begin{align}&\vdash\vdash\colon\ldotp P\in 1\rightarrow \text{Cls}.\supset:\alpha\in \text{gen}ʻP.\equiv.(\exists T).T\in \text{Potid}ʻP.\alpha=\breve{T}ʻʻ\overrightarrow{B}ʻP\\ +&[\text{*93·2·3}]\end{align}\]</p> + +<p class="nind"><b>*93·33.</b> \[\begin{align}&\vdash\vdash:P\in 1\rightarrow \text{Cls}.\alpha\in \text{gen}ʻP.\supset.\breve{P}ʻʻ\alpha\in \text{gen}ʻP\\ +&[\text{*93·2·31.*91·28·281}]\end{align}\]</p> + +<p class="nind"><b>*93·34.</b> \(\vdash:P\in 1\rightarrow \text{Cls}.\supset.\breve{P}ʻʻ\overrightarrow{B}ʻP\in \text{gen}ʻP \quad[\text{*93·22·33}]\)</p> + +<p class="nind"><b>*93·35.</b> \(\vdash:P\in 1\rightarrow \text{Cls}.\alpha\in \text{gen}ʻP.T\in \text{Potid}ʻP.\supset.\breve{T}ʻʻ\alpha\in \text{gen}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·341.*37·33.*34·2}.\supset\\ +\vdash:S,\,T\in \text{Potid}ʻP.\alpha=\breve{S}ʻʻ\overrightarrow{B}ʻP.\supset.S\mid T\in \text{Potid}ʻP.\breve{T}ʻʻ\alpha=\{\text{Cnv}ʻ(S\mid T)\}ʻʻ\overrightarrow{B}ʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*93·32}.\supset\vdash:\text{Hp(1)}.P\in 1\rightarrow \text{Cls}.\supset.\breve{T}ʻʻ\alpha\in \text{gen}ʻP &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23·35.*93·32}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·36.</b> \(\vdash:P\in 1\rightarrow \text{Cls}.\supset.sʻ\text{gen}ʻP=\breve{P}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·32}.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \\ +y\in sʻ\text{gen}ʻP.&\equiv:(\exists T).T\in \text{Potid}ʻP.y\in \breve{T}ʻʻ\overrightarrow{B}ʻP:\\ +[\text{*37·105}] &\equiv:(\exists T,x).T\in \text{Potid}ʻP.x\in \overrightarrow{B}ʻP.xTy:\\ +[\text{*11·55}] &\equiv:(\exists x):x\in \overrightarrow{B}ʻP:(\exists T).T\in \text{Potid}ʻP.xTy:\\ +[\text{*41·11}] &\equiv:(\exists x).x\in \overrightarrow{B}ʻP.x(\dot{s}ʻ\text{Potid}ʻP)y:\\ +[\text{*91·55}] &\equiv:(\exists x).x\in \overrightarrow{B}ʻP.xR_{\unicode{x2217}}y:\\ +[\text{*37·105}] &\equiv:y\in \breve{P}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻP\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·37.</b> \(\vdash:P\in 1\rightarrow \text{Cls}.\supset.CʻP=\breve{P}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻP\cup pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP \quad[\text{*93·274·36}]\)</p> + +<p class="nind"><b>*93·38.</b> \(\vdash\colon\ldotp P\in 1\rightarrow \text{Cls}.\supset:x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP.\equiv.\overrightarrow{P}_{\unicode{x2217}}ʻx\subset \text{ᗡ}ʻP.x\in CʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·271·36}.\supset\\ +\vdash\colon\colon \text{Hp}.\supset\colon\ldotp x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP.&\equiv:x\in CʻP.x{\sim}\in \breve{P}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻP:\\ +[\text{*37·105.*10·51}] &\equiv:x\in CʻP:yP_{\unicode{x2217}}x.\supset_{y}.y{\sim}\in \overrightarrow{B}ʻP:\\ +[\text{*93·101.*22·84·8}] &\equiv:x\in CʻP:yP_{\unicode{x2217}}x.\supset_{y}.y\in \text{ᗡ}ʻP\cup -\text{D}ʻP:\\ +[\text{*90·13.*33·16}] &\equiv:x\in CʻP:yP_{\unicode{x2217}}x.\supset_{y}.\\ +&\qquad\qquad\qquad y\in (\text{ᗡ}ʻP\cup -\text{D}ʻP)\cap (\text{ᗡ}ʻP\cup \text{D}ʻP):\\ +[\text{*22·69.*24·21}] &\equiv:x\in CʻP:yP_{\unicode{x2217}}x.\supset_{y}.y\in \text{ᗡ}ʻP\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_615">[Pg 615]</span></p> + +<p class="nind"><b>*93·381.</b> \(\vdash\colon\ldotp P \in \text{Cls} \rightarrow 1.\supset:x \in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ\breve{P}.\equiv.\overleftarrow{P}_{\unicode{x2217}}ʻx\subset + \text{D}ʻP. x\in CʻP\)</p> + +<p class="nind"><b>*93·382.</b> \[\begin{align}\vdash\colon\ldotp &P \in 1 \rightarrow 1.\supset : x \in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP \cap pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ\breve{P}. \equiv .\\ +&\overrightarrow{P}_{\unicode{x2217}}ʻx \cup \overleftarrow{P}_{\unicode{x2217}}ʻx + \subset \text{D}ʻP \cap \text{ᗡ}ʻP. x \in CʻP \quad[\text{*93·38·381·261. *90·31·311}]\end{align}\]</p> + +<p class="nind"><b>*93·4.</b> + \(\vdash: P \in 1 \rightarrow \text{Cls}.\text{ᗡ}ʻP \subset \text{D}ʻP. \exists !\overrightarrow{B}ʻP. T \in \text{Potid}ʻP.\supset.\exists !\overrightarrow{\text{min}}_{p}ʻ\text{ᗡ}ʻT\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash . \text{*93·13} . \supset\vdash : \text{Hp} .&\supset.\exists ! \overrightarrow{\text{min}}_{p}ʻ\text{ᗡ}ʻ(I\upharpoonright CʻP) &\qquad \text{(1)}\\ +\vdash . \text{*93·113 .*33*181}. \supset\vdash\colon\ldotp \text{Hp}. &\supset : \overrightarrow{\text{min}}_{p}ʻ\text{ᗡ}ʻT \subset \text{D}ʻP :\\ +[\text{*37·431}] &\supset: \exists ! \overrightarrow{\text{min}}_{p}ʻ\text{ᗡ}ʻT .\supset.\exists ! \breve{P}ʻʻ\overrightarrow{\text{min}}_{p}ʻ\text{ᗡ}ʻT.\\ +[\text{*93·31}] & \supset. \exists !\overrightarrow{\text{min}}_{p}ʻ\text{ᗡ}ʻ (T \mid P) &\qquad \text{(2)}\\ +\vdash . \text{(1). (2). *91·17}. \supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*93·41">*93·41</a>.</b> \[\begin{align}&\vdash : P \in 1 \rightarrow \text{Cls}.\text{ᗡ}ʻP \subset \text{D}ʻP. \exists ! \overrightarrow{B}ʻP . \supset . \text{gen}ʻP \in \text{Cls ex}^{2}\, + \text{excl}\\ +&[\text{*93·2·4·25 .*84·13 .*24·63}]\end{align}\]</p> + +<p class="nind"><b>*93·412.</b> \(\vdash . \breve{P}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP \subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*93·261}. \supset\vdash . \breve{P}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP &= \breve{P}ʻʻpʻ\text{ᗡ}ʻʻ\text{Potid}ʻP\\ +[\text{*40·37}] &\subset pʻ\breve{P}ʻʻʻ\text{ᗡ}ʻʻ\text{Potid}ʻP\\ +[\text{*43·411}] &\subset pʻ\text{ᗡ}ʻʻ\mid Pʻʻ\text{Potid}ʻP\\ +[\text{*91·24}] &\subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP. \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·42.</b> \(\vdash : P \in 1 \rightarrow \text{Cls} . \supset . \breve{P}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP = pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*93·261}. \supset\vdash . \breve{P}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP &= \breve{P}ʻʻpʻ\text{ᗡ}ʻʻ\text{Potid}ʻP &\qquad \text{(1)}\\ +\vdash. \text{(1) . *72·34. *91·35 . *10·24}. \supset\\ +\vdash: \text{Hp}. \supset . \breve{P}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻP&=pʻ\breve{P}ʻʻʻ\text{ᗡ}ʻʻ\text{Potid}ʻP\\ +[\text{*43·411}] &=pʻ\text{ᗡ}ʻʻ\mid Pʻʻ\text{Potid}ʻP\\ +[\text{*91·24}] & =pʻ\text{ᗡ}ʻʻ \text{Pot}ʻ P : \supset\vdash . \text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*93·431.</b> \(\vdash . pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP = pʻ\text{ᗡ}ʻʻ \mid Pʻʻ\text{Pot}ʻP\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*91·264·304}. &\supset\vdash. \text{Pot}ʻP = \iotaʻP \cup \mid Pʻʻ\text{Pot}ʻP.\\ +[\text{*53·14}] &\supset\vdash .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP = \text{ᗡ}ʻP \cap pʻ\text{ᗡ}ʻʻ\mid Pʻʻ\text{Pot}ʻP.\\ +[\text{*91·271·283.*40·151·23}] &\supset\vdash .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=pʻ\text{ᗡ}ʻʻ\mid Pʻʻ\text{Pot}ʻP .\supset\vdash. \text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_616">[Pg 616]</span></p> + +<p>The following propositions, not being needed in subsequent +propositions, are here inserted without proof, merely for the sake of +their intrinsic interest.</p> + +<p class="nind"><b>*93·5.</b> \(\vdash:T\in \text{Potid}ʻP.\supset.\overrightarrow{P}_{\text{ts}}ʻT=\overrightarrow{P}_{\text{st}}ʻT=T\mid ʻʻ\text{Potid}ʻP=\mid Tʻʻ\text{Potid}ʻP\)</p> + +<p class="nind"><b>*93·51.</b> \(\vdash:T\in \text{Pot}ʻP.\supset.\text{Pot}ʻT\subset \overrightarrow{P}_{\text{ts}}ʻT\subset \text{Pot}ʻP\)</p> + +<p class="nind"><b>*93·52.</b> \(\vdash:T\in \text{Pot}ʻP.\supset.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻT=pʻ\text{ᗡ}ʻʻ\overrightarrow{P}_{\text{ts}}ʻT=pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p class="nind"><b>*93·53.</b> \(\vdash:S,\,T\in \text{Pot}ʻP.xSx.\supset.(\exists y).y(S\mid T)x\)</p> + +<p class="nind"><b>*93·54.</b> \(\vdash:S\in \text{Pot}ʻP.xSx.\supset.x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p class="nind"><b>*93·55.</b> \(\vdash.Cʻ(P_{\text{po}}\dot{\cap}I)\subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + +<p class="nind"><b>*93·56.</b> \(\vdash:\dot{\exists}!(P_{\text{po}}\dot{\cap}I).\supset.\exists !pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP\)</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_617">[Pg 617]</span></p> +<h2 class="nobreak" id="*94">*94. ON POWERS OF RELATIVE PRODUCTS.</h2> +</div> + + +<p><i>Summary of</i> *94.</p> + +<p>In this number we shall be chiefly concerned with propositions +connecting powers of \(R \mid S\) with powers of \(S \mid R\). If \(P\) +is a power of \(R \mid S\), \(S \mid P \mid R\) will be a power of \(S\mid R\). +If \(P\) is a power of \(R \mid S\), it is a product of the +form +\[ +(R \mid S) \mid (R \mid S) \mid \ldots \mid (R \mid S). +\]</p> + +<p>If we transfer the initial \(R\) to the end, we get a power of \(S \mid R\). +Thus there is a power of \(S \mid R\), say \(T\), such that +\[ +P \mid R = R \mid T\text{.} +\]</p> + +<p>If \(R \in 1 \rightarrow \text{Cls} . \text{ᗡ}ʻ(R \mid S) \subset\text{D}ʻR\), we find +\[ +R \mid (S \mid R) \mid (S \mid R) \mid \ldots \mid (S \mid R) \mid \breve{R} = (R \mid S) \mid (R \mid S) \ldots (R \mid S) +\] +by rearranging and observing that \(R \mid \breve{R} = I \upharpoonright \text{D}ʻR\). Thus +\[ +R \in 1 \rightarrow \text{Cls} . \text{ᗡ}ʻ(R \mid S) \subset \text{D}ʻR . P \in \text{Pot}ʻR \mid S .\supset. (\exists T) . T \in \text{Pot}ʻS \mid R . P = R \mid T \mid \breve{R}\text{.} +\]</p> + +<p>Expressions of the form \(R \mid T \mid \breve{R}\) are constantly +needed. They will be specially dealt with in *150, and will occur +constantly in the sequel.</p> + +<p>The above connections of \(\text{Pot}ʻ(R \mid S)\) and \(\text{Pot}ʻ(S \mid R)\) are embodied in the following propositions:</p> + +<p class="nind"><b>*94·14.</b> \(\vdash. \mid Rʻʻ\text{Pot}ʻ(R \mid S) = R \mid ʻʻ\text{Pot}ʻ(S \mid R)\)</p> + +<p class="nind"><b>*94·21.</b> \(\vdash. \text{Pot}ʻ(S \mid R) = (S \parallel R)ʻʻ\{\text{Pot}ʻ(R \mid S) \cup \iotaʻI\}\)</p> + +<p class="nind"><b>*94·31.</b> \(\vdash: R \in 1 \rightarrow \text{Cls} . \text{ᗡ}ʻ(R \mid S) \subset \text{D}ʻR .\supset. \text{Pot}ʻ(R \mid S) = (R \parallel \breve{R})ʻʻ\text{Pot}ʻ(S \mid R)\)</p> + +<p>From <a href="#*94·4">*94·4</a> to <a href="#*94·54">*94·54</a>, the propositions are all concerned with \(pʻ\text{ᗡ}ʻʻ(R\mid S)\) +and \(pʻ\text{ᗡ}ʻʻ(S\mid R)\). We prove</p> + +<p class="nind"><b>*94·5.</b> \(\vdash. pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S \mid R) = pʻ\text{ᗡ}ʻʻR \mid ʻʻ\text{Pot}ʻ(S \mid R)\)</p> + +<p class="nind"><b>*94·51.</b> \(\vdash: R \in 1 \rightarrow \text{Cls} .\supset. pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S \mid R) = \breve{R}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R \mid S)\)</p> + +<p><span class="pagenum" id="Page_618">[Pg 618]</span></p> + +<p>Finally we prove (*94·53·54) that if either \(R\) is one-one +and \(\text{ᗡ}ʻ(R \mid S) \subset \text{D}ʻR\), or \(S\) +is one-one and \(\text{ᗡ}ʻ(S \mid R) \subset \text{D}ʻS\), +then \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R \mid S)\) is similar to +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S \mid R)\).</p> + +<p>The only proposition of this number which is ever subsequently +referred to is the last, <a href="#*94·64">*94·64</a>, which, owing to the fact that the +Schröder-Bernstein theorem has been already proved (<a href="#*73·88">*73·88</a>), is only +used in <a href="#*95·23">*95·23</a>. But *95·23 itself is never referred to again. The +reader may therefore omit the reading of the propositions of this +number (as also of <a href="#*95">*95</a>) without detriment to the understanding of what +follows; he should, however, read the summaries.</p> + +<p>The chief importance of the propositions in the present number is +when \(R\) and \(S\) fulfil the hypothesis of the Schröder-Bernstein +theorem, <i>i.e.</i> +\[ +R,\,S \in 1 \rightarrow 1 . \text{ᗡ}ʻR \subset \text{D}ʻS . \text{ᗡ}ʻS \subset \text{D}ʻR. +\]</p> + +<p>In this case, \(R \mid S\) gives what we may call a "reflexion" of +\(\text{D}ʻR\) into part of itself; this part may be again reflected by +\(R\mid S\) into a part of itself, and so on. The terms in \(\text{D}ʻR\) +which are eliminated sooner or later by this process of reflexion +constitute \(sʻ\text{gen}ʻ(R \mid S)\), since any one reflexion +eliminates terms which constitute one generation of \(R \mid S\). +The terms not eliminated by any number of reflexions constitute +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R \mid S)\). These two sets of terms together +constitute \(\text{D}ʻ(R \mid S)\), <i>i.e.</i> \(\text{D}ʻR\). In +this number and *95 we shall prove that, with the Schröder-Bernstein +hypothesis, +\[ +sʻ\text{gen}ʻ(R \mid S) \mathop{\text{ sm }} sʻ\text{gen}ʻ(S \mid R).pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R \mid S) \mathop{\text{ sm }} pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S \mid R)\text{.} +\]</p> + +<p>These two propositions together yield a proof of the Schröder-Bernstein +theorem, in virtue of *93·274·275. This proof is essentially the same +as Bernsteinʻs published originally by Borel<a id="FNanchor_66" href="#Footnote_66" class="fnanchor">[66]</a>.</p> + +<p>The nature of the two proofs of the Schröder-Bernstein theorem, namely +Zermelo's (that given in <a href="#*73">*73</a>) and Bernstein's (that to be given in this +number and <a href="#*95">*95</a>) will be best apprehended by means of figures.</p> + +<figure class="figcenter width500" id="i_618" style="width: 525px;"> +<img src="images/i_618.jpg" width="525" height="500" alt="Geometrical +diagram which represents mathematical relationships between +transformations."> +</figure> + +<p>In Zermelo's proof, we first prove that if \(R\) is one-one, and +\(\beta\) is a class contained in \(\text{D}ʻR\) and containing +\(\text{ᗡ}ʻR\), then \(\beta\) is similar both to \(\text{D}ʻR\) +and to \(\text{ᗡ}ʻR\). In the figure, the points of the outer +rectangle form \(\text{D}ʻR\), those of the<span class="pagenum" id="Page_619">[Pg 619]</span> inner rectangle form +\(\text{ᗡ}ʻR\), and those of the outer oval form \(\beta\). Thus the +shaded portion of the figure is \(\beta-\text{ᗡ}ʻR\). We now define +a class of classes \(\kappa\) by the following characteristics: +\(\alpha\) is a member of \(\kappa\) if (1) \(\alpha\) is contained in +\(\text{D}ʻR\), (2) \(\alpha\) contains the whole of the shaded area, +(3) \(\breve{R}ʻʻ\alpha\subset \alpha\), <i>i.e.</i> if \(x\) is a +member of \(\alpha\), so is any term to which \(x\) has the relation +\(R\). Our proposition is obtained by considering \(pʻ\kappa\), +<i>i.e.</i> the area common to all the members of \(\kappa\). We +prove (<a href="#*73·81">*73·81</a>) that \(pʻ\kappa\in \kappa\) and (<a href="#*73·811">*73·811</a>) that +\(\breve{R}ʻʻpʻ\kappa\) does not contain any of the shaded area. In +the figure, \(\breve{R}ʻʻpʻ\kappa\) is the smaller oval. We then +prove (<a href="#*73·83">*73·83</a>) that \(pʻ\kappa\) consists entirely of the shaded +portion and the smaller oval. Hence \(\beta\) (the larger oval) +consists of two mutually exclusive parts, namely \(pʻ\kappa\) and +\(\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa\), the latter being that part of the +inner rectangle which lies outside the inner oval. Assuming now that +\(R\) is one-one, \(pʻ\kappa\) is similar to \(\breve{R}ʻʻpʻ\kappa\); +hence, adding \(\text{ᗡ}ʻR-\breve{R}ʻʻpʻ\kappa\), it follows that +\(\beta\) is similar to \(\text{ᗡ}ʻR\), and therefore to \(\text{D}ʻR\).</p> + +<p>In order to obtain hence the Schröder-Bernstein theorem, it is +only necessary to replace \(R\) by \(R\mid S\) and \(\beta\) by +\(\text{ᗡ}ʻS\), and to assume further that \(S\) is a one-one whose +domain contains \(\text{ᗡ}ʻR\). Then \(\text{D}ʻR=\text{D}ʻ(R\mid S)\), +and we obtain (<a href="#*73·87">*73·87</a>) \(\text{ᗡ}ʻS \text{ sm } \text{D}ʻS\), and +therefore \(\text{D}ʻS \text{ sm } \text{D}ʻR\), which was to be proved.</p> + +<figure class="figcenter width500" id="i_619" style="width: 940px;"> +<img src="images/i_619.jpg" width="940" height="500" alt="A schematic +diagram showing two parallel systems each containing nested regions."> +</figure> + +<p>In Bernstein's proof, we have the two relations \(R\) and \(S\) +from the beginning. In the left-hand part of the figure, the +outer rectangle is \(\text{D}ʻR\), which = \(\text{D}ʻ(R\mid +S)\), the oval is \(\text{ᗡ}ʻS\), and the second rectangle is +\(\text{ᗡ}ʻ(R\mid S)\). Thus the points of the outer but not the +second rectangle form the first generation of \(R\mid S\). Within +\(\text{ᗡ}ʻ(R\mid S)\) we can form a third rectangle, which will +be \(\breve{S}ʻʻ\breve{R}ʻʻ\text{ᗡ}ʻ(R\mid S)\), <i>i.e.</i> +\(\text{ᗡ}ʻ(R\mid S)^{2}\). The points belonging to the second +rectangle but not to the third form the second generation of +\(R\mid S\). We can proceed in this way to continually smaller +rectangles. The points which sooner or later are left outside +some rectangle form \(sʻ\text{gen}ʻ(R\mid S)\); those which are +common to all the rectangles form \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\). +A similar analysis,<span class="pagenum" id="Page_620">[Pg 620]</span> exhibited in the right-hand part of the +figure, may be applied to \(\text{D}ʻS\), which is thus divided +into \(sʻ\text{gen}ʻ(S\mid R)\) and \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)\). +We prove in this number (<a href="#*94·53">*94·53</a>) that, with a hypothesis +which is part of the hypothesis of the Schröder-Bernstein +theorem, \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S) \text{ sm }pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)\); +in the next number (<a href="#*95·71">*95·71</a>) we prove that with the hypothesis of the +Schröder-Bernstein theorem, \(sʻ\text{gen}ʻ(R\mid S) \text{ sm } sʻ\text{gen}ʻ(S\mid R)\). +Hence by addition, \(\text{D}ʻR \text{ sm }\text{D}ʻS\).</p> + +<hr class="tb"> + +<p class="nind"><b>*94·12.</b> \(\vdash:P\in \text{Pot}ʻ(R\mid S).\supset.(\exists T).T\in \text{Pot}ʻ(S\mid R).P\mid R=R\mid T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·21}. &\supset\vdash.(R\mid S)\mid R=R\mid (S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{*91·36.*34·27}.&\supset\vdash:T\in \text{Pot}ʻ(S\mid R).P\mid R=R\mid T.\supset.\\ +&T\mid S\mid R\in \text{Pot}ʻ(S\mid R).P\mid R\mid S\mid R=R\mid T\mid S\mid R.\\ +[\text{*10·24}] &\supset.(\exists T).T\in \text{Pot}ʻ(S\mid R).P\mid R\mid S\mid R=R\mid T &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23}.&\supset\vdash:(\exists T).T\in \text{Pot}ʻ(S\mid R).P\mid R=R\mid T.\supset.\\ +&(\exists T).T\in \text{Pot}ʻ(S\mid R).P\mid R\mid S\mid R=R\mid T &\qquad \text{(3)}\\ +\vdash.\text{(1).(3).*91·171}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·13.</b> \[\begin{align}&\vdash:T\in \text{Pot}ʻ(S\mid R).\supset.(\exists P).P\in \text{Pot}ʻ(R\mid S).P\mid R=R\mid T\\ +&[\text{Proof as in *94·12}]\end{align}\]</p> + +<p class="nind"><b>*94·14.</b> \(\vdash.\mid Rʻʻ\text{Pot}ʻ(R\mid S)=R\mid ʻʻ\text{Pot}ʻ(S\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*94·12.*43·111·1.*37·1}.&\supset\vdash:P\in \text{Pot}ʻ(R\mid S).\supset.\mid RʻP\in R\mid ʻʻ\text{Pot}ʻ(S\mid R):\\ +[\text{*37·61}]&\supset\vdash.\mid Rʻʻ\text{Pot}ʻ(R\mid S)\subset R\mid ʻʻ\text{Pot}ʻ(S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{*94·13.*43·11·101.*37·1}.\supset\\ +&\vdash:T\in \text{Pot}ʻ(S\mid R).\supset.R\mid ʻT\in \mid Rʻʻ\text{Pot}ʻ(R\mid S):\\ +[\text{*37·61}]&\supset\vdash.R\mid ʻʻ\text{Pot}ʻ(S\mid R)\subset \mid Rʻʻ\text{Pot}ʻ(R\mid S) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·2.</b> \(\vdash:P\in \text{Pot}ʻ(R\mid S)\cup \iotaʻI.\supset.S\mid P\mid R\in \text{Pot}ʻ(S\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·21}.&\supset\vdash.S\mid (R\mid S)\mid R=(S\mid R)^{2}.\\ +[\text{*91·352}]&\supset\vdash.S\mid (R\mid S)\mid R\in \text{Pot}ʻ(S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{*34·21.*91·282}.\supset\\ +\vdash:S\mid P\mid R\in \text{Pot}ʻ(S\mid R).\supset.S\mid (P\mid R\mid S)&\mid R=(S\mid P\mid R)\mid S\mid R.\\ +&(S\mid P\mid R)\mid S\mid R\in \text{Pot}ʻ(S\mid R) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·171}\, \frac{S\mid P\mid R\in \text{Pot}ʻ(S\mid R)}{\phi S} .\supset\\ +\vdash:P\in \text{Pot}ʻ(R\mid S).&\supset.S\mid P\mid R\in \text{Pot}ʻ(S\mid R) &\qquad \text{(3)}\\ +\vdash.\text{*50·4.*91·351}.&\supset\vdash.S\mid I\mid R\in \text{Pot}ʻ(S\mid R) &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_621">[Pg 621]</span></p> + +<p class="nind"><b>*94·201.</b> \(\vdash:T\in \text{Pot}ʻ(S\mid R).\supset.(\exists P).P\in \text{Pot}ʻ(R\mid S)\cup \iotaʻI.T=S\mid P\mid R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·4.*51·16}. &\supset\vdash.S\mid R=S\mid I\mid R.I\in \text{Pot}ʻ(R\mid S)\cup \iotaʻI.\\ +[\text{*10·24}] & \supset\vdash.(\exists P).P\in \text{Pot}ʻ(R\mid S)\cup \iotaʻI.S\mid R=S\mid P\mid R &\qquad \text{(1)}\\ +\vdash.\text{*91·282.*34·21}.&\supset\vdash:P\in \text{Pot}ʻ(R\mid S).T=S\mid P\mid R.\supset.\\ +&\qquad\qquad\qquad P\mid R\mid S\in \text{Pot}ʻ(R\mid S).T\mid S\mid R=S\mid (P\mid R\mid S)\mid R.\\ +[\text{*10·24}] & \supset.(\exists Q).Q\in \text{Pot}ʻ(R\mid S).T\mid S\mid R=S\mid Q\mid R &\qquad \text{(2)}\\ +\vdash.\text{*50·4.*34·21}.\supset\\ +\vdash:P=I.T=S\mid P\mid R.\supset.T\mid S\mid R=S\mid (R\mid S)\mid R.\\ +[\text{*91·351}] &\supset.(\exists Q).Q\in \text{Pot}ʻ(R\mid S).T\mid S\mid R=S\mid Q\mid R &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*10·11·23}.&\supset\vdash:(\exists P).P\in \text{Pot}ʻ(R\mid S)\cup \iotaʻI.T=S\mid P\mid R.\supset.\\ +&\qquad\qquad\qquad (\exists Q).Q\in \text{Pot}ʻ(R\mid S).T\mid S\mid R=S\mid Q\mid R.\\ +[\text{*22·58}] &\supset.(\exists Q).Q\in \text{Pot}ʻ(R\mid S)\cup \iotaʻI.T\mid S\mid R=S\mid Q\mid R &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*91·171}\, \frac{S\mid R,(\exists P).P\in \text{Pot}ʻ(R\mid S)\cup \iotaʻI.T=S\mid P\mid R}{P,\quad\phi T} .\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·21.</b> \(\vdash.\text{Pot}ʻ(S\mid R)=(S\Arrowvert R)ʻʻ\{\text{Pot}ʻ(R\mid S)\cup \iotaʻI\}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*94·2.*43·112.*37·61}. &\supset\vdash.(S\Arrowvert R)ʻʻ\{\text{Pot}ʻ(R\mid S)\cup \iotaʻI\}\subset \text{Pot}ʻ(S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{*94·201.*43·102.*37·1}.&\supset\vdash.\text{Pot}ʻ(S\mid R)\subset (S\Arrowvert R)ʻʻ\{\text{Pot}ʻ(R\mid S)\cup \iotaʻI\} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·22.</b> \[\begin{align}\vdash\colon\ldotp \text{ᗡ}ʻR\subset \text{D}ʻS.\lor.\text{D}ʻS\subset \text{ᗡ}ʻR&:\supset.\\ +&\text{Pot}ʻ(S\mid R)=(S\Arrowvert R)ʻʻ\text{Potid}ʻ(R\mid S)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*94·21.*43·112.*50·4.*53·31}.\supset\\ +\vdash.\text{Pot}ʻ(S\mid R)=(S\Arrowvert R)ʻʻ\text{Pot}ʻ(R\mid S)\cup \iotaʻ(S\mid R) &&\qquad \text{(1)}\\ +\vdash.\text{*37·321}.\supset\vdash:\text{ᗡ}ʻR\subset \text{D}ʻS.&\supset.\text{D}ʻR=\text{D}ʻ(R\mid S).\\ +[\text{*33·161}] &\supset.\text{D}ʻR\subset Cʻ(R\mid S).\\ +[\text{*50·63}] &\supset.I\upharpoonright Cʻ(R\mid S)\mid R=R.\\ +[\text{*34·28}] &\supset.S\mid I\upharpoonright Cʻ(R\mid S)\mid R=S\mid R.\\ +[\text{*43·112}] & \supset.(S\Arrowvert R)ʻI\upharpoonright Cʻ(R\mid S)=S\mid R &\qquad \text{(2)}\\ +\text{Similarly} &\vdash:\text{D}ʻS\subset \text{ᗡ}ʻR.\supset.(S\Arrowvert R)ʻI\upharpoonright Cʻ(R\mid S)=S\mid R &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\\ +\vdash:\text{Hp}.&\supset.\text{Pot}ʻ(S\mid R)=(S\Arrowvert R)ʻʻ\text{Pot}ʻ(R\mid S)\cup \iotaʻ(S\Arrowvert R)ʻI\upharpoonright Cʻ(R\mid S)\\ +[\text{*91·23}] & =(S\Arrowvert R)ʻʻ\text{Potid}ʻ(R\mid S):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_622">[Pg 622]</span></p> + +<p class="nind"><b>*94·3.</b> \[\begin{align}\vdash\colon\ldotp R\in 1\rightarrow &\text{Cls}.\text{ᗡ}ʻ(R\mid S)\subset \text{D}ʻR.\supset:\\ +& P\in \text{Pot}ʻ(R\mid S).\equiv.(\exists T).T\in \text{Pot}ʻ(S\mid R).P=R\mid T\mid \breve{R}\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*94·12}. &\supset\vdash:P\in \text{Pot}ʻ(R\mid S).\supset.(\exists T).T\in \text{Pot}ʻ(S\mid R).P\mid R\mid \breve{R}=R\mid T\mid \breve{R} &\qquad \text{(1)}\\ +\vdash.\text{*91·271}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:P\in \text{Pot}ʻ(R\mid S).\supset.\text{ᗡ}ʻP\subset \text{D}ʻR.\\ +[\text{*72·6}] & \supset.P\mid R\mid \breve{R}=P &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash\colon\ldotp \text{Hp}.\supset:P\in \text{Pot}ʻ&(R\mid S).\supset.\\ +&(\exists T).T\in \text{Pot}ʻ(S\mid R).P=R\mid T\mid \breve{R} &\qquad \text{(3)}\\ +\vdash.\text{*94·13}. \supset\vdash:T\in \text{Pot}ʻ(S\mid R).\supset.\\ +&(\exists P).P\in \text{Pot}ʻ(R\mid S).P\mid R\mid \breve{R}=R\mid T\mid \breve{R} &\qquad \text{(4)}\\ +\vdash.\text{(2).(4)}. \supset\vdash\colon\ldotp \text{Hp}.\supset:T\in \text{Pot}ʻ(S\mid R).\supset.\\ +&(\exists P).P\in \text{Pot}ʻ(R\mid S).P=R\mid T\mid \breve{R}.\\ +[\text{*13·195}] &\supset.R\mid T\mid \breve{R}\in \text{Pot}ʻ(R\mid S):\\ +[\text{*13·12}] &\supset:T\in \text{Pot}ʻ(S\mid R).P=R\mid T\mid \breve{R}.\supset.P\in \text{Pot}ʻ(R\mid S) &\qquad \text{(5)}\\ +\vdash.\text{(5).*10·11·21·23}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:(\exists T).T\in \text{Pot}ʻ(S\mid R).P=R\mid T\mid \breve{R}.\supset.P\in \text{Pot}ʻ(R\mid S) &\qquad \text{(6)}\\ +\vdash.\text{(3).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·31.</b> \[\begin{align}&\vdash:R\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻ(R\mid S)\subset \text{D}ʻR.\supset.\text{Pot}ʻ(R\mid S)=(R\parallel\breve{R})ʻʻ\text{Pot}ʻ(S\mid R)\\ +&[\text{*94·3}]\end{align}\]</p> + +<p>The following series of propositions lead up to the proof that when +\(R\in 1\rightarrow 1.\text{ᗡ}ʻ(R\mid S)\subset \text{D}ʻR\), or +\(S\in 1\rightarrow 1.\text{ᗡ}ʻ(S\mid R)\subset \text{D}ʻS\), we have +\[ +pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\text{ sm } pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R). +\]</p> + +<p class="nind"><b><a id="*94·4">*94·4</a>.</b> \[\begin{align}\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)&=pʻ\text{ᗡ}ʻʻ\mid Sʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)\\ +&=pʻ\breve{S}ʻʻʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)\\ +& =pʻ\breve{S}ʻʻʻ\breve{R}ʻʻʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·431}.\supset\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)&=pʻ\text{ᗡ}ʻʻ\mid (R\mid S)ʻʻ\text{Pot}ʻ(R\mid S)\\ +[\text{*43·201.*37·33}] & =pʻ\text{ᗡ}ʻʻ\mid Sʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S) &\qquad \text{(1)}\\ +[\text{*43·411}] & =pʻ\breve{S}ʻʻʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S) &\qquad \text{(2)}\\ +[\text{*43·411}] & =pʻ\breve{S}ʻʻʻ\breve{R}ʻʻʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S) &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_623">[Pg 623]</span></p> + +<p class="nind"><b>*94·401.</b> \(\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)=pʻ\text{ᗡ}ʻʻR\mid ʻʻS\mid ʻʻ\text{Pot}ʻ(R\mid S)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·431.*91·304}.\supset\\ +\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)&=pʻ\text{ᗡ}ʻʻ(R\mid S)\mid ʻʻ\text{Pot}ʻ(R\mid S)\\ +[\text{*43·2.*37·33}] & =pʻ\text{ᗡ}ʻʻR\mid ʻʻS\mid ʻʻ\text{Pot}ʻ(R\mid S).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*94·402">*94·402</a>.</b> \(\vdash.pʻ\text{ᗡ}ʻʻR\mid ʻʻ\lambda\subset pʻ\text{ᗡ}ʻʻ\lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·11.*34·36}.&\supset\vdash.(P).\text{ᗡ}ʻR\mid ʻP\subset \text{ᗡ}ʻP &\qquad \text{(1)}\\ +\vdash.\text{(1).*40·451}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·41.</b> \[\begin{align}\vdash:S\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻ(S\mid R)&\subset \text{D}ʻS.\supset.\\ +& Sʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)=pʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*40·12.*91·351}.\supset\vdash.pʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)&\subset \text{ᗡ}ʻ\mid Rʻ(R\mid S)\\ +[\text{*43·111}] & \subset \text{ᗡ}ʻ(R\mid S\mid R)\\ +[\text{*34·36}] &\subset \text{ᗡ}ʻ(S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{(1)}.\supset\vdash:\text{Hp}.&\supset.pʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)\subset \text{D}ʻS.\\ +[\text{*72·502}] \supset.pʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)&=Sʻʻ\breve{S}ʻʻpʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)\\ +[\text{*72·34}] &=Sʻʻpʻ\breve{S}ʻʻʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)\\ +[\text{*94·4}] & =Sʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·42.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.\breve{R}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)=pʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·34}.\supset\vdash:\text{Hp}.\supset.\breve{R}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)&=pʻ\breve{R}ʻʻʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\\ +[\text{*43·411}] & =pʻ\text{ᗡ}ʻʻ\mid Rʻʻ\text{Pot}ʻ(R\mid S):\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·43.</b> \[\begin{align}\vdash:R,S\in 1\rightarrow &\text{Cls}.\text{ᗡ}ʻ(S\mid R)\subset \text{D}ʻS.\supset.\\ +& Sʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)=\breve{R}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S) \quad[\text{*94·41·42}]\end{align}\]</p> + +<p class="nind"><b>*94·441.</b> \[\begin{align}\vdash:S\in 1\rightarrow &\text{Cls}.\text{ᗡ}ʻ(S\mid R)\subset \text{D}ʻS.\supset.\\ +& Sʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)=pʻ\text{ᗡ}ʻʻR\mid ʻʻ\text{Pot}ʻ(S\mid R) \quad[\text{*94·14·41}]\end{align}\]</p> + +<p class="nind"><b>*94·442.</b> \[\begin{align}&\vdash:R\in 1\rightarrow \text{Cls}.\supset.\breve{R}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)=pʻ\text{ᗡ}ʻʻR\mid ʻʻ\text{Pot}ʻ(S\mid R)\\ +&[\text{*94·14·42}]\end{align}\]</p> + +<p class="nind"><b>*94·5.</b> \(\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)=pʻ\text{ᗡ}ʻʻR\mid ʻʻ\text{Pot}ʻ(S\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*94·402}.&\supset\vdash.pʻ\text{ᗡ}ʻʻR\mid ʻʻ\text{Pot}ʻ(S\mid R)\subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{*94·402}.&\supset\vdash.pʻ\text{ᗡ}ʻʻS\mid ʻʻR\mid ʻʻ\text{Pot}ʻ(S\mid R)\subset pʻ\text{ᗡ}ʻʻR\mid ʻʻ\text{Pot}ʻ(S\mid R).\\ +[\text{*94·401}] &\supset\vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)\subset pʻ\text{ᗡ}ʻʻR\mid ʻʻ\text{Pot}ʻ(S\mid R) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_624">[Pg 624]</span></p> + +<p class="nind"><b>*94·51.</b> \[\begin{align}&\vdash:R\in 1\rightarrow \text{Cls}.\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)=\breve{R}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\\ +&[\text{*94·5·442}]\end{align}\]</p> + +<p class="nind"><b>*94·52.</b> \[\begin{align}\vdash:S\in 1\rightarrow \text{Cls}.&\text{ᗡ}ʻ(S\mid R)\subset \text{D}ʻS.\supset .\\ +& pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)=Sʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S) \quad[\text{*94·5·441}]\end{align}\]</p> + +<p class="nind"><b><a id="*94·53">*94·53</a>.</b> \[\begin{align}\vdash:R\in 1\rightarrow 1.\text{ᗡ}ʻ(R\mid S)\subset \text{D}ʻ&R.\supset .\\ +& pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\text{ sm } pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·261}.&\supset \vdash.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\subset \text{ᗡ}ʻ(R\mid S) &\qquad \text{(1)}\\ +\vdash.\text{(1)}.& \supset \vdash:\text{Hp}.\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\subset \text{D}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(2).*94·51.*73·21}.\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*94·54">*94·54</a>.</b> \[\begin{align}&\vdash:S\in 1\rightarrow 1.\text{ᗡ}ʻ(S\mid R)\subset \text{D}ʻS.\supset .pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(R\mid S)\text{ sm } pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ(S\mid R)\\ +&\left[\text{*94·53}\, \frac{S,R}{R,S}\right]\\ +&[\text{Or, *94·52.*93·261.*73·22}]\end{align}\]</p> + +<p class="nind"><b>*94·6.</b> \(\vdash\colon\ldotp R\mid S=S\mid R.\supset :M\in \text{Pot}ʻR.N\in \text{Pot}ʻS.\supset .M\mid N=N\mid M\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·27·28}.&\supset \vdash:\text{Hp}.M\mid S=S\mid M.\supset .M\mid R\mid S=S\mid M\mid R &\qquad \text{(1)}\\ +\vdash.\text{(1).*91·171} \frac{M\mid S=S\mid M}{\phi M} .\supset \\ +\vdash\colon\ldotp\text{Hp}.M\in \text{Pot}ʻR.&\supset :M\mid S=S\mid M: &\qquad \text{(2)}\\ +\left[\text{(2)}\, \frac{S,\,M,\,N}{R,\,S,\,M}\right] &\supset :N\in \text{Pot}ʻS.\supset .M\mid N=N\mid M\colon\ldotp\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·61.</b> \[\begin{align}\vdash\colon\ldotp R\mid S=S\mid R.\supset :&M\in \text{Pot}ʻR.\supset .M\mid S_{\text{po}}=S_{\text{po}}\mid M:\\ +& N\in \text{Pot}ʻS.\supset .N\mid R_{\text{po}}=R_{\text{po}}\mid N\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·42}.&\supset \vdash.M\mid S_{\text{po}}=\dot{s}ʻM\mid ʻʻ\text{Pot}ʻS &\qquad \text{(1)}\\ +\vdash.\text{(1).*94·6}.\supset \vdash:\text{Hp}.M\in \text{Pot}ʻR.\supset .M\mid S_{\text{po}}&=\dot{s}ʻ\mid Mʻʻ\text{Pot}ʻS\\ +[\text{*43·421}] & =S_{\text{po}}\mid M &\qquad \text{(2)}\\ +\vdash.\text{(2)}\, \frac{S,\,R}{R,\,S}. &\supset \vdash:\text{Hp}.N\in \text{Pot}ʻS.\supset .N\mid R_{\text{po}}=R_{\text{po}}\mid N &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*94·62.</b> \(\vdash:R\mid S=S\mid R.\supset .R_{\text{po}}\mid S_{\text{po}}=S_{\text{po}}\mid R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·42.*94·61}.\supset \vdash:\text{Hp}.\supset .R_{\text{po}}\mid S_{\text{po}}&=\dot{s}ʻ\mid R_{\text{po}}ʻʻ\text{Pot}ʻS\\ +[\text{*43·421}] & =S_{\text{po}}\mid R_{\text{po}}:\supset \vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_625">[Pg 625]</span></p> + +<p class="nind"><b>*94·63.</b> \(\vdash : R\mid S = S\mid R. \supset .(R\mid S)_{\text{po}} \unicode{x2abd} R_{\text{po}}\mid S_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*91·5020}. &\supset\vdash . R\mid S \unicode{x2abd} R_{\text{po}} S_{\text{po}} &\qquad \text{(1)}\\ +\vdash. \text{*94·61}. \supset\vdash : \text{Hp}.M \unicode{x2abd} R_{\text{po}}\mid S_{\text{po}} . \supset . M\mid R\mid S &\unicode{x2abd} R_{\text{po}}\mid + R\mid S_{\text{po}}\mid S\\ +[\text{*91·511}] &\unicode{x2abd} R_{\text{po}}\mid S_{\text{po}} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·171}. \supset\vdash \colon\ldotp \text{Hp}. &\supset : M \in \text{Pot}ʻ(R\mid S).\supset. M \unicode{x2abd} R_{\text{po}}\mid S_{\text{po}}:\\ +[\text{*41·151}] & \supset : (R\mid S)_{\text{po}} \unicode{x2abd} R_{\text{po}}\mid S_{\text{po}} \colon\ldotp \supset\vdash .\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*94·64">*94·64</a>.</b> \(\vdash: R\mid S = S\mid R.\supset.(R\mid S)_{\unicode{x2217}} \unicode{x2abd} R_{\unicode{x2217}}\mid S_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·36}. & \supset\vdash . \text{D}ʻ(R\mid S) \subset \text{D}ʻR. \text{ᗡ}ʻ(S\mid R) \subset \text{ᗡ}ʻR.\\ +[\text{*33·16}] & \supset\vdash : \text{Hp}. \supset . Cʻ(R\mid S) \subset CʻR &\qquad \text{(1)}\\ +\text{Similarly} &\vdash: \text{Hp}. \supset . Cʻ(R\mid S) \subset CʻS &\qquad \text{(2)}\\ +\vdash. \text{(1).(2).*50·6.*35·31}. &\supset\vdash : \text{Hp}.\supset. I \upharpoonright Cʻ(R\mid S) \unicode{x2abd} I\upharpoonright CʻR\mid I\upharpoonright CʻS &\qquad \text{(3)}\\ +\vdash. \text{(3).*94·63.*91·54}. &\supset\vdash . \text{Prop} +\end{array} +\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_66" href="#FNanchor_66" class="label">[66]</a> +<i>Leçons sur la théorie des fonctions</i> (Paris, 1898), +Note I (pp. 102-7).</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_626">[Pg 626]</span></p> +<h2 class="nobreak" id="*95">*95. ON THE EQUI-FACTOR RELATION.</h2> +</div> + + +<p><i>Summary of</i> *95.</p> + +<p>The purpose of this number may be explained as follows. Consider the +series of relations +\[ +R, P \mid R \mid Q,\, P^{2} \mid R \mid Q^{2},\, P^{3} \mid R \mid Q^{3}, \ldots\text{;} +\] +it is required to find a means of defining this series without the use +of numbers. If we used numbers, and had the definition given later +(*301) of \(P^{\nu}\) where \(\nu\) is any finite integer, the general +term of the series would be \(P^{\nu} \mid R \mid Q^{\nu}\). But we have +not yet defined numbers, and we therefore desire some means, not +involving numbers, of expressing what is intended when we say that, +in a given term of the series, the same power of \(P\) and of \(Q\) +is to be involved. This we do as follows. Using the definition of \(P\parallel Q\) +in <a href="#*43">*43</a>, we have +\[ +P \mid R \mid Q = (P \parallel Q)ʻR . P^{2} \mid R \mid Q^{2} = (P \parallel Q)^{2}ʻR . P^{3} \mid R \mid Q^{3} = (P \parallel Q)^{3}ʻR \ldots\text{.} +\]</p> + +<p>Thus the general term of our series is got by taking any power \(S\) of +(\(P \parallel Q)\), and forming \(SʻR\). The whole of the terms of the +series are therefore constituted by the terms which have to \(R\) the +relation (\(P \parallel Q)_{\unicode{x2217}}\); <i>i.e.</i> they are +\(\{\text{sg}ʻ(P \parallel Q)_{\unicode{x2217}}\}ʻR\). For convenience +of notation we put<a id="FNanchor_67" href="#Footnote_67" class="fnanchor">[67]</a> +\[ +(P\unicode{x2217}Q) = \text{sg}ʻ(P \parallel Q)_{\unicode{x2217}} \quad \text{Dft} ~ \text{[*95]} +\]</p> + +<p>Thus the class of relations we wish to consider is +(\(P\unicode{x2217}Q)ʻR\).</p> + +<p>To illustrate the nature of (\(P\unicode{x2217}Q)ʻR\), suppose \(R\) +is the relation "first cousin," while \(P\) is the relation of child +to parent and \(Q\) is the relation of parent to child. Then \(P \mid +R \mid Q\) is the relation "second cousin," \(P^{2} \mid R \mid Q^{2}\) is +the relation "third cousin," and so on. Thus (\(P*Q)ʻR\) is the class +of all relations of cousinship which do not involve a difference of +generation; and "\(x\{\dot{s}ʻ(P\unicode{x2217}Q)ʻR\}y\)" will mean +"\(x\) is a cousin of \(y\) in the same generation."</p> + +<p>Most of the propositions in this number are inserted because they are +required in the proof of <a href="#*95·52">*95·52</a>, which states that, under suitable +circumstances, \(\dot{s}ʻ(P\unicode{x2217}Q)ʻR \in 1 \rightarrow 1\). +This proposition itself is proved mainly because it is required in the +proof of <a href="#*95·63">*95·63</a>, which states that, if \(P\), \(Q\) are one-one's each +of which has its converse domain contained in its domain, and if the<span class="pagenum" id="Page_627">[Pg 627]</span> +first generation of \(P\) is similar to the first generation of \(Q\), +then the sum of the generations of \(P\) is similar to the sum of the +generations of \(Q\). This leads immediately to a proposition (<a href="#*95·71">*95·71</a>) +which is half of the Schröder-Bernstein theorem (the other half being +<a href="#*94·53">*94·53</a> or <a href="#*94·54">*94·54</a>), namely: "If \(R\) and \(S\) are one-one's each of +which has its converse domain contained in the domain of the other, +then the sum of the generations of \(R\mid S\) is similar to the sum of +the generations of \(S\mid R\)."</p> + +<hr class="tb"> + +<p class="nind"><b>*95·01.</b> (\(P\unicode{x2217}Q)=\text{sg}ʻ\{(P\Arrowvert Q)_{\unicode{x2217}}\} \quad\text{Dft}\, [\text{*95}]\)</p> + +<p class="nind"><b>*95·1.</b> \(\vdash\colon\colon M\in (P\unicode{x2217}Q)ʻR.\equiv\colon\ldotp R\in \mu:N\in \mu.\supset_{N}.P\mid N\mid Q\in \mu:\supset_{\mu}.M\in \mu\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18.(*95·01)}.\supset\\ +\vdash\colon\colon M\in (P\unicode{x2217}Q)ʻR.&\equiv\colon\ldotp M(P\Arrowvert Q)_{\unicode{x2217}}R\colon\ldotp \\ +[\text{*90·111}] & \equiv\colon\ldotp M\in Cʻ(P\Arrowvert Q)\colon\ldotp N\in \mu.T(P\Arrowvert Q)N.\supset_{N,T}.T\in \mu:\\ +&\qquad\qquad\qquad\qquad R\in \mu:\supset_{\mu}.M\in \mu\colon\ldotp \\ +[\text{*43·302·102}]&\equiv\colon\ldotp N\in \mu.T=P\mid N\mid Q.\supset_{T,N}.T\in \mu:R\in \mu:\supset_{\mu}.M\in \mu\colon\ldotp \\ +[\text{*13·191}] & \equiv\colon\ldotp N\in \mu.\supset_{N}.P\mid N\mid Q\in \mu:R\in \mu:\supset_{\mu}.M\in \mu\colon\colon \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*95·11">*95·11</a>.</b> \(\vdash\colon\ldotp \phi R:\phi N.\supset_{N}.\phi(P\mid N\mid Q):\supset:M\in (P\unicode{x2217}Q)ʻR.\supset_{M}.\phi M\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·1}\, \frac{\hat{N}(\phi N)}{\mu} .\supset\\ +\vdash\colon\colon M\in (P\unicode{x2217}Q)ʻR.\supset\colon\ldotp \phi R:\phi N.\supset_{N}.\phi(P\mid N\mid Q):\supset.\phi M &\qquad \text{(1)}\\ +\vdash.\text{(1).Comm.*10·11·21}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·12.</b> \(\vdash\colon\ldotp M\in (P\unicode{x2217}Q)ʻR.\supset_{M}.\phi(P\mid M\mid Q):\supset:N\in (P\unicode{x2217}Q)ʻR-\iotaʻR.\supset_{N}.\phi N\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*43·112}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\equiv:M\in (P\unicode{x2217}Q)ʻR.\supset_{M}.\phi\{(P\Arrowvert Q)ʻM\}:\\ +[\text{*37·63}]\equiv:N\in (P\Arrowvert Q)ʻʻ(P\unicode{x2217}Q)ʻR.\supset_{N}.\phi N &\qquad \text{(1)}\\ +\vdash.\text{*90·311}\, \frac{P\Arrowvert Q}{R} .\supset\\ +\vdash:N\in (P\unicode{x2217}Q)ʻR-\iotaʻR.\supset.N\in (P\Arrowvert Q)ʻʻ(P\unicode{x2217}Q)ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·13.</b> \(\vdash.R\in (P\unicode{x2217}Q)ʻR \quad[\text{*95·1}]\)</p> + +<p class="nind"><b>*95·131.</b> \(\vdash.P\mid R\mid Q\in (P\unicode{x2217}Q)ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·151}\, \frac{P\Arrowvert Q}{R} .\supset\vdash:S(P\Arrowvert Q)R.\supset.S(P\Arrowvert Q)_{\unicode{x2217}}R &\qquad \text{(1)}\\ +\vdash.\text{(1).*43·102.(*95·01)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_628">[Pg 628]</span></p> + +<p class="nind"><b>*95·132.</b> \[\begin{align}&\vdash:M\in (P \unicode{x2217} Q)ʻR.\supset.P\mid M\mid Q\in (P \unicode{x2217} Q)ʻR\\ +&\left[\text{*90·172}\, \frac{P\Arrowvert Q}{R} .\text{*43·102}\right]\end{align}\]</p> + +<p class="nind"><b>*95·14.</b> \(\vdash\colon\ldotp \phi R:N\in (P \unicode{x2217} Q)ʻR.\phi N.\supset_{N}.\phi(P\mid N\mid Q):\supset:M\in (P \unicode{x2217} Q)ʻR.\supset_{M}.\phi M\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·13·132}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +\phi R.R\in (P \unicode{x2217} Q)ʻR:N\in (P \unicode{x2217} Q)ʻR.\phi N.\supset_{N}.P\mid N\mid Q\in (P \unicode{x2217} Q)ʻR.\phi(P\mid N\mid Q):\\ +[\text{*95·11}]\supset:M\in (P \unicode{x2217} Q)ʻR.\supset_{M}.M\in (P \unicode{x2217} Q)ʻR.\phi M\colon\ldotp \supset\vdash.\text{Prop} +\end{array}\]</p> + +<p>The use of <a href="#*95·11">*95·11</a> in the last line of the above proof proceeds by +substituting \(M\in (P \unicode{x2217} Q)ʻR.\phi M\) for \(\phi M\).</p> + +<p class="nind"><b>*95·21.</b> \(\vdash:M\in (P \unicode{x2217} Q)ʻR.\supset.(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Pot}ʻQ\cup \iotaʻI.M=S\mid R\mid T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·4}.&\supset\vdash.R=I\mid R\mid I.\\ +[\text{*51·16}]&\supset\vdash.(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Pot}ʻQ\cup \iotaʻI.R=S\mid R\mid T &\qquad \text{(1)}\\ +\vdash.\text{*91·36·351.*50·4.*34·27·28}.\supset\\ +\vdash:S\in \text{Pot}ʻ&P\cup \iotaʻI.T\in \text{Pot}ʻQ\cup \iotaʻI.M=S\mid R\mid T.\supset.\\ +&P\mid S\in \text{Pot}ʻP\cup \iotaʻI.T\mid Q\in \text{Pot}ʻQ\cup \iotaʻI.P\mid M\mid Q=(P\mid S)\mid R\mid (T\mid Q).\\ +[\text{*11·36}]&\supset.(\exists S',T').S'\in \text{Pot}ʻP\cup \iotaʻI.T'\in \text{Pot}ʻQ\cup \iotaʻI.P\mid M\mid Q=S'\mid R\mid T' &\qquad \text{(2)}\\ +\vdash.\text{(2).*11·11·35}.\supset\\ +\vdash:&(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Pot}ʻQ\cup \iotaʻI.M=S\mid R\mid T.\supset.\\ +&(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Pot}ʻQ\cup \iotaʻI.P\mid M\mid Q=S\mid R\mid T &\qquad \text{(3)}\\ +\vdash.\text{(1).(3).*95·11}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·211.</b> \[\begin{align}\vdash:\text{ᗡ}ʻR\subset CʻQ.M &\in (P \unicode{x2217} Q)ʻR.\supset.\\ +&(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Potid}ʻQ.M=S\mid R\mid T\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·62·4}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:S\mid R\mid I\upharpoonright CʻQ=S\mid R\mid I:\\ +[\text{*51·239.*91·23}] &\supset:(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Potid}ʻQ.M=S\mid R\mid T.\equiv.\\ +&(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Pot}ʻQ\cup \iotaʻI.M=S\mid R\mid T:\\ +[\text{*95·21}] &\supset:(\exists S,T).S\in \text{Pot}ʻP\cup \iotaʻI.T\in \text{Potid}ʻQ.M=S\mid R\mid T\colon\ldotp \\ +&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·212.</b> \[\begin{align}\vdash:\text{D}ʻR &\subset CʻP.M\in (P \unicode{x2217} Q)ʻR.\supset.\\ +&(\exists S,T).S\in \text{Potid}ʻP.T\in \text{Pot}ʻQ\cup \iotaʻI.M=S\mid R\mid T\\ +&[\text{Proof as in *95·211}]\end{align}\]</p> + +<p class="nind"><b>*95·22.</b> \[\begin{align}\vdash:\text{D}ʻR &\subset CʻP.\text{ᗡ}ʻR\subset CʻQ.M\in (P \unicode{x2217} Q)ʻR.\supset.\\ +&(\exists S,T).S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.M=S\mid R\mid T\\ +[\text{Proof as in *95·211}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_629">[Pg 629]</span></p> + +<p class="nind"><b>*95·221.</b> \(\vdash:T\in \text{Pot}ʻQ.\supset.(\exists S).S\in \text{Pot}ʻP.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·131.*91·351}.&\supset\vdash.(\exists S).S\in \text{Pot}ʻP.S\mid R\mid Q\in (P \unicode{x2217} Q)ʻR &\qquad \text{(1)}\\ +\vdash.\text{*95·132}.\supset\\ +\vdash:S\in \text{Pot}ʻP.T\in \text{Pot}ʻQ.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR.&\supset.P\mid S\mid R\mid T\mid Q\in (P \unicode{x2217} Q)ʻR.\\ +[\text{*91·36}] & \supset.(\exists S').S'\in \text{Pot}ʻP.S'\mid R\mid T\mid Q\in (P \unicode{x2217} Q)ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·373}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·222.</b> \[\begin{align}&\vdash:S\in \text{Pot}ʻP.\supset.(\exists T).T\in \text{Pot}ʻQ.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR\\ +&[\text{Proof as in *95·221}]\end{align}\]</p> + +<p class="nind"><b><a id="*95·23">*95·23</a>.</b> \(\vdash:M\in (P \unicode{x2217} Q)ʻR.\supset.M(P_{\text{st}}\mid Q_{\text{ts}})R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18.(*95·01)}.\supset\vdash:\text{Hp}.&\supset.M\{(P\Arrowvert Q)_{\unicode{x2217}}\}R.\\ +[\text{*43·202}] &\supset.M\{(P\mid )\mid (\mid Q)\}_{\unicode{x2217}}R.\\ +[\text{*43·202.*94·64}] & \supset.M\{(P\mid )_{\unicode{x2217}}\mid (\mid Q)_{\unicode{x2217}}\}R.\\ +[\text{(*91·01·02)}] &\supset.M(P_{\text{st}}\mid Q_{\text{ts}})R.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·24.</b> \(\vdash:M\in (P \unicode{x2217} Q)ʻR.\supset.M(Q_{\text{ts}}\mid P_{\text{st}})R \quad[\text{Proof as in *95·23}]\)</p> + +<p class="nind"><b>*95·3.</b> \(\vdash\colon\ldotp \dot{\exists}!R.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\text{ᗡ}ʻR\subset \text{D}ʻQ.\supset:T\in \text{Potid}ʻQ.\supset.\dot{\exists}!R\mid T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*50·62}. & \supset\vdash:\text{Hp}.\supset.R\mid I\upharpoonright CʻQ=R.\\ +[\text{*13·12}] & \supset.\dot{\exists}!R\mid (I\upharpoonright CʻQ) &\qquad \text{(1)}\\ +\vdash.\text{*91·27.*33·181}.&\supset\vdash\colon\ldotp \text{Hp}.T\in \text{Potid}ʻQ.\supset:\text{ᗡ}ʻT\subset \text{D}ʻQ:\\ +[\text{*34·35}] &\supset:\dot{\exists}!T.\supset.\dot{\exists}!(T\mid Q) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·371}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·301.</b> \[\begin{align}&\vdash\colon\ldotp \dot{\exists}!R.\text{D}ʻP\subset \text{ᗡ}ʻP.\text{D}ʻR\subset \text{ᗡ}ʻP.\supset:S\in \text{Potid}ʻP.\supset.\dot{\exists}!S\mid R\\ +&[\text{Proof as in *95·3}]\end{align}\]</p> + +<p class="nind"><b>*95·302.</b> \(\vdash\colon\ldotp \text{ᗡ}ʻQ\subset \text{D}ʻQ.\text{ᗡ}ʻR\subset \text{D}ʻQ.\supset:T\in \text{Potid}ʻQ.\supset.\text{ᗡ}ʻ(R\mid T)\subset \text{D}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·271.*34·36}.\supset\vdash:T\in \text{Potid}ʻQ.\supset.\text{ᗡ}ʻ(R\mid T)\subset \text{ᗡ}ʻQ &\qquad \text{(1)}\\ +\vdash.\text{(1).*22·44}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·303.</b> \[\begin{align}&\vdash\colon\ldotp \text{D}ʻR\subset \text{ᗡ}ʻP.\text{D}ʻP\subset \text{ᗡ}ʻP.\supset:S\in \text{Potid}ʻP.\supset.\text{D}ʻ(S\mid R)\subset \text{ᗡ}ʻP\\ +&[\text{Proof as in *95·302}]\end{align}\]</p> + +<p class="nind"><b>*95·304.</b> \[\begin{align}&\vdash\colon\ldotp \text{ᗡ}ʻQ\subset \text{D}ʻQ.\text{ᗡ}ʻR\subset \text{D}ʻQ.\text{D}ʻP\subset \text{ᗡ}ʻP.\text{D}ʻR\subset \text{ᗡ}ʻP.\supset:\\ +&S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\supset.\text{D}ʻ(S\mid R\mid T)\subset \text{ᗡ}ʻP.\text{ᗡ}ʻ(S\mid R\mid T)\subset \text{D}ʻQ\\ +&[\text{*95·302·303.*34·36}]\end{align}\]</p> + +<p class="nind"><b>*95·305.</b> \[\begin{align}&\vdash\colon\ldotp \text{Hp}*95·304.\supset:M\in (P \unicode{x2217} Q)ʻR.\supset.\text{D}ʻM\subset \text{ᗡ}ʻP.\text{ᗡ}ʻM\subset \text{D}ʻQ\\ +&[\text{*95·304·22}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_630">[Pg 630]</span></p> + +<p class="nind"><b><a id="*95·31">*95·31</a>.</b> \(\vdash\colon\ldotp \text{Hp*95·304}.\dot{\exists}!R.\supset:S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\supset.\dot{\exists}!S\mid R\mid T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·142·143}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\supset.\\ +&\text{D}ʻR\subset \text{ᗡ}ʻS.\text{ᗡ}ʻR\subset \text{D}ʻT.\\ +[\text{*34·361}] &\supset.\dot{\exists}!S\mid R\mid T\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·32.</b> \(\vdash\colon\ldotp \text{Hp*95·31}.\supset:M\in (P \unicode{x2217} Q)ʻR.\supset.\dot{\exists}!M \quad[\text{*95·31·22}]\)</p> + +<p class="nind"><b>*95·33.</b> \(\vdash:\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ.\supset.\text{ᗡ}ʻ(S\mid R\mid T)\subset \breve{T}ʻʻ\overrightarrow{B}ʻQ\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·36}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻ(S\mid R)\subset \overrightarrow{B}ʻQ.\\ +[\text{*37·32·2}] &\supset.\text{ᗡ}ʻ(S\mid R\mid T)\subset \breve{T}ʻʻ\overrightarrow{B}ʻQ:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·34.</b> + \[\begin{align}&\vdash:\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ.M\in (P \unicode{x2217} Q)ʻR.\supset.(\exists T).T\in \text{Potid}ʻQ.\text{ᗡ}ʻM\subset \breve{T}ʻʻ\overrightarrow{B}ʻQ\\ +&[\text{*95·33·211}]\end{align}\]</p> + +<p class="nind"><b>*95·35.</b> + \[\begin{align}&\vdash:Q\in 1\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ.M\in (P \unicode{x2217} Q)ʻR.\supset.(\exists \alpha).\alpha\in \text{gen}ʻQ.\text{ᗡ}ʻM\subset \alpha\\ +&[\text{*95·34.*93·32}]\end{align}\]</p> + +<p class="nind"><b>*95·351.</b> \[\begin{align}\vdash\colon\ldotp Q\in 1&\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ.\supset:\\ +&T,\,T'\in \text{Potid}ʻQ.\exists !\text{ᗡ}ʻ(S\mid R\mid T)\cap \text{ᗡ}ʻ(S'\mid R\mid T').\supset.T=T'\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·33}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +T,\,T'\in \text{Potid}ʻQ.\exists !\text{ᗡ}ʻ(S\mid R\mid T)\cap \text{ᗡ}ʻ(S'\mid R\mid T').&\supset.\exists !\breve{T}ʻʻ\overrightarrow{B}ʻQ\cap \breve{T}'ʻʻ\overrightarrow{B}ʻQ.\\ +[\text{*93·3}] &\supset.\exists !\overrightarrow{\text{min}}_{Q}ʻ\text{ᗡ}ʻT\cap \overrightarrow{\text{min}}_{Q}ʻ\text{ᗡ}ʻT'.\\ +[\text{*93·24.Transp}] &\supset.T=T'\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·352.</b> \[\begin{align}\vdash\colon\ldotp P\in \text{Cls}&\rightarrow 1.\text{D}ʻR\subset \overrightarrow{B}ʻ\breve{P}.\supset:\\ +&S,\,S'\in \text{Potid}ʻP.\exists !\text{D}ʻ(S\mid R\mid T)\cap \text{D}ʻ(S'\mid R\mid T').\supset.S=S'\\ +&[\text{Proof as in *95·351}]\end{align}\]</p> + +<p class="nind"><b>*95·36.</b> \[\begin{align}\vdash\colon\ldotp Q\in 1&\rightarrow \text{Cls}.\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ.\dot{\exists}!R.\text{D}ʻR\subset \text{ᗡ}ʻP.\\ +&\text{D}ʻP\subset \text{ᗡ}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\supset:\\ +&S,\,S'\in \text{Potid}ʻP.T,T'\in \text{Potid}ʻQ.S\mid R\mid T=S'\mid R\mid T'.\supset.T=T'\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·31.*93·101}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +&S,\,S'\in \text{Potid}ʻP.T,\,T'\in \text{Potid}ʻQ.S\mid R\mid T=S'\mid R\mid T'.\supset.\\ +&\dot{\exists}!S\mid R\mid T.S\mid R\mid T=S'\mid R\mid T'.\\ +[\text{*22·5.*33·24}] &\supset.\exists !\text{ᗡ}ʻ(S\mid R\mid T)\cap \text{ᗡ}ʻ(S'\mid R\mid T').\\ +[\text{*95·351}] &\supset.T=T'\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_631">[Pg 631]</span></p> + +<p class="nind"><b>*95·361.</b> \[\begin{align}\vdash\colon\ldotp P\in \text{Cls}\rightarrow 1.\text{D}ʻR&\subset \overrightarrow{B}ʻ\breve{P}.\dot{\exists}!R.\text{D}ʻP\subset \text{ᗡ}ʻP.\\ +&\text{ᗡ}ʻR\subset \text{D}ʻQ.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\supset:\\ +&S,\,S'\in \text{Potid}ʻP.T,\,T'\in \text{Potid}ʻQ.S\mid R\mid T=S'\mid R\mid T'.\supset.S=S'\\ +&[\text{Proof as in *95·36}]\end{align}\]</p> + +<p class="nind"><b>*95·37.</b> + \[\begin{align}\vdash\colon\ldotp P\in \text{Cls}\rightarrow 1.Q\in 1\rightarrow \text{Cls}.\text{D}ʻR&\subset \overrightarrow{B}ʻ\breve{P}.\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ.\\ +&\text{D}ʻP\subset \text{ᗡ}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\supset:\\ +&S,\,S'\in \text{Potid}ʻP.T,T'\in \text{Potid}ʻQ.S\mid R\mid T=S'\mid R\mid T'.\supset.S=S'.T=T'\\ +&[\text{*95·36·361}]\end{align}\]</p> + +<p class="nind"><b>*95·38.</b> \(\vdash\colon\ldotp \exists !\overrightarrow{B}ʻQ\cap \text{ᗡ}ʻR.\supset:T\in \text{Pot}ʻQ.\supset.R\mid T \neq R\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·271}.\supset\vdash:T\in \text{Pot}ʻQ.&\supset.\text{ᗡ}ʻ(R\mid T)\subset \text{ᗡ}ʻQ.\\ +[\text{*93·101}] &\supset.\text{ᗡ}ʻ(R\mid T)\cap \overrightarrow{B}ʻQ=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*24·54}.\supset\vdash:\text{Hp}. &\supset.{\sim}\{\text{ᗡ}ʻR\cap \overrightarrow{B}ʻQ=\Lambda\} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*13·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·381.</b> \[\begin{align}&\vdash\colon\ldotp \exists !\overrightarrow{B}ʻ\breve{P}\cap \text{D}ʻR.\supset:S\in \text{Pot}ʻP.\supset.S\mid R \neq R\\ +&[\text{Proof as in *95·38}]\end{align}\]</p> + +<p class="nind"><b>*95·382.</b> \[\begin{align}\vdash\colon\ldotp \exists !\overrightarrow{B}ʻ\breve{P}&\cap \text{D}ʻR.\lor.\exists !\overrightarrow{B}ʻQ\cap \text{ᗡ}ʻR:\supset:\\ +&S\in \text{Pot}ʻP.T\in \text{Pot}ʻQ.\supset.S\mid R\mid T \neq R\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·271.*93·101}.&\supset\vdash:T\in \text{Pot}ʻQ.\supset.\text{ᗡ}ʻ(S\mid R\mid T)\cap \overrightarrow{B}ʻQ=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*24·54}.& \supset\vdash:\exists !\overrightarrow{B}ʻQ\cap \text{ᗡ}ʻR.\supset.{\sim}{\text{ᗡ}ʻR\cap \overrightarrow{B}ʻQ=\Lambda} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*13·14}.&\supset\vdash\colon\ldotp \exists !\overrightarrow{B}ʻQ\cap \text{ᗡ}ʻR.\supset:T\in \text{Pot}ʻQ.\supset.S\mid R\mid T \neq R &\qquad \text{(3)}\\ +\vdash.\text{*91·271.*93·101}.&\supset\vdash:S\in \text{Pot}ʻP.\supset.\text{D}ʻ(S\mid R\mid T)\cap \overrightarrow{B}ʻ\breve{P}=\Lambda &\qquad \text{(4)}\\ +\vdash.\text{*24·54}. & \supset\vdash:\exists !\overrightarrow{B}ʻ\breve{P}\cap \text{D}ʻR.\supset.{\sim}\{\text{D}ʻR\cap \overrightarrow{B}ʻ\breve{P}=\Lambda\} &\qquad \text{(5)}\\ +\vdash.\text{(4).(5).*13·14}. &\supset\vdash\colon\ldotp \exists !\overrightarrow{B}ʻ\breve{P}\cap \text{D}ʻR.\supset:S\in \text{Pot}ʻP.\supset.S\mid R\mid T \neq R &\qquad \text{(6)}\\ +\vdash.\text{(3).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·383.</b> \[\begin{align}&\vdash\colon\ldotp \dot{\exists}!R:\text{D}ʻR\subset \overrightarrow{B}ʻ\breve{P}.\lor.\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ:\supset:\\ +&S\in \text{Pot}ʻP.T\in \text{Pot}ʻQ.\supset.S\mid R\mid T \neq R \quad[\text{*95·382.*33·24.*22·621}]\end{align}\]</p> + +<p class="nind"><b>*95·4.</b> \[\begin{align}\vdash:M\in (P \unicode{x2217} Q)ʻR.S\in \text{Pot}ʻP.T\in \text{Pot}ʻQ.S\mid R\mid T\in &(P \unicode{x2217} Q)ʻR.\supset.\\ +&S\mid M\mid T\in (P \unicode{x2217} Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{Simp}.&\supset\vdash:\text{Hp}.\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR &\qquad \text{(1)}\\ +\vdash.\text{*91·34.*95·132}.\supset\\ +\vdash:\text{Hp}.S\mid M\mid T\in (P \unicode{x2217} Q)ʻR.&\supset.S\mid P\mid M\mid Q\mid T=P\mid S\mid M\mid T\mid Q.\\ +&P\mid S\mid M\mid T\mid Q\in (P \unicode{x2217} Q)ʻR.\\ +[\text{*13·13}] & \supset.S\mid (P\mid M\mid Q)\mid T\in (P \unicode{x2217} Q)ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*95·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_632">[Pg 632]</span></p> + +<p class="nind"><b>*95·41.</b> \[\begin{align}&\vdash\colon\ldotp P\in \text{Cls}\rightarrow 1.Q\in 1\rightarrow \text{Cls}.\text{D}ʻP\subset \text{ᗡ}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\supset:\\ +& S,\,S'\in \text{Potid}ʻP.T,\,T'\in \text{Potid}ʻQ.\supset.\breve{S}\mid S\mid S'\mid N\mid T'\mid T\mid \breve{T} = S'\mid N\mid T'\\ +&[\text{*92·15·151}]\end{align}\]</p> + +<p class="nind"><b>*95·411.</b> \[\begin{align}&\vdash\colon\ldotp \text{Hp*95·41}.\supset:\\ +& S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.M\in (P \unicode{x2217} Q)ʻR.\supset.M = \breve{S}\mid S\mid M\mid T\mid \breve{T}\\ +&[\text{*95·41·22}]\end{align}\]</p> + +<p class="nind"><b>*95·42.</b> \(\vdash\colon\ldotp \text{Hp*95·41}.\supset:M\in (P \unicode{x2217} Q)ʻR - \iotaʻR.\supset.\breve{P}\mid M\mid \breve{Q}\in (P \unicode{x2217} Q)ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·411. *91·351·281}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:M\in (P \unicode{x2217} Q)ʻR.\supset.\breve{P}\mid (P\mid M\mid Q)\mid \breve{Q}\in (P \unicode{x2217} Q)ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*95·12}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·43.</b> \[\begin{align}\vdash\colon\ldotp \text{Hp*95·41}.\text{Hp*95·382}.&\supset:S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\\ +& P\mid S\mid R\mid T\mid Q\in (P \unicode{x2217} Q)ʻR.\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·42·382.*91·28·3}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\\ +& P\mid S\mid R\mid T\mid Q\in (P \unicode{x2217} Q)ʻR.\supset.\breve{P}\mid P\mid S\mid R\mid T\mid Q\mid \breve{Q}\in (P \unicode{x2217} Q)ʻR &\qquad \text{(1)}\\ +\vdash.\text{*95·41}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\supset.\\ +&\qquad\qquad\qquad\qquad\breve{P}\mid P\mid S\mid R\mid T\mid Q\mid \breve{Q} = S\mid R\mid T &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·431.</b> \[\begin{align}\vdash:\text{Hp*95·43}.S\in &\text{Potid}ʻP.T\in \text{Potid}ʻQ.M\in (P \unicode{x2217} Q)ʻR.\\ +& P\mid S\mid M\mid T\mid Q\in (P \unicode{x2217} Q)ʻR.\supset.S\mid M\mid T\in (P \unicode{x2217} Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·22}.\supset\vdash:\text{Hp}.&\supset.(\exists S',T').S'\in \text{Potid}ʻP.T'\in \text{Potid}ʻQ.M = S'\mid R\mid T'.\\ +&\qquad\qquad\qquad\qquad P\mid S\mid M\mid T\mid Q\in (P \unicode{x2217} Q)ʻR.\\ +[\text{*91·341}] &\supset.(\exists S',T').S'\in \text{Potid}ʻP.T'\in \text{Potid}ʻQ.M = S'\mid R\mid T'.\\ +&S\mid S'\in \text{Potid}ʻP.T'\mid T\in \text{Potid}ʻQ.P\mid S\mid S'\mid R\mid T'\mid T\mid Q\in (P \unicode{x2217} Q)ʻR.\\ +[\text{*95·43}] &\supset.(\exists S',T').S'\in \text{Potid}ʻP.T'\in \text{Potid}ʻQ.M = S'\mid R\mid T'.\\ +&\qquad\qquad\qquad\qquad S\mid S'\mid R\mid T'\mid T\in (P \unicode{x2217} Q)ʻR.\\ +[\text{*13·195}] &\supset.S\mid M\mid T\in (P \unicode{x2217} Q)ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·44.</b> \[\begin{align}\vdash\colon\ldotp \text{Hp*95·43}.& S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\supset:\\ +& M\in (P \unicode{x2217} Q)ʻR.S\mid M\mid T\in (P \unicode{x2217} Q)ʻR.\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR\end{align}\]</p> + +<p><span class="pagenum" id="Page_633">[Pg 633]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{Id}.&\supset\vdash\colon\colon \phi M.\equiv_{M}:S\mid + M\mid T\in (P \unicode{x2217} Q)ʻR.\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR\colon\ldotp \supset.\phi R &\qquad \text{(1)}\\ +\vdash.\text{*95·431.*91·3}.\supset\\ +\vdash\colon\colon\ldotp \text{Hp}.&\supset\colon\colon S\mid P\mid M\mid Q\mid T\in (P \unicode{x2217} Q)ʻR.\supset\colon\ldotp S\mid M\mid T\in (P \unicode{x2217} Q)ʻR\colon\ldotp\\ +[\text{*2·27}] &\supset\colon\ldotp S\mid M\mid T\in (P \unicode{x2217} Q)ʻR.\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR:\supset.\\ +&\qquad\qquad\qquad\qquad S\mid R\mid T\in (P \unicode{x2217} Q)ʻR &\qquad \text{(2)}\\ +\vdash.\text{(2).Comm}.\supset\\ +\vdash\colon\colon \text{Hp}.&\supset\colon\ldotp S\mid M\mid T\in (P \unicode{x2217} Q)ʻR.\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR:\supset:\\ +& S\mid (P\mid M\mid Q)\mid T\in (P \unicode{x2217} Q)ʻR.\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR &\qquad \text{(3)}\\ +\vdash.\text{(3)}.&\supset\colon\ldotp \text{Hp}.\text{Hp(1)}.\supset:\phi M.\supset.\phi(P\mid M\mid Q) &\qquad \text{(4)}\\ +\vdash.\text{(1).(4).*95·14}.&\supset\vdash:\text{Hp}.\text{Hp(1)}.M\in (P \unicode{x2217} Q)ʻR.\supset.\phi M:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·45.</b> \[\begin{align}&\vdash\colon\ldotp \text{Hp*95·43}.S,\,S'\in \text{Potid}ʻP.T,\,T'\in \text{Potid}ʻQ.\\ +& S\mid S'\mid R\mid T'\mid T\in (P \unicode{x2217} Q)ʻR.\supset:S\mid R\mid T\in (P \unicode{x2217} Q)ʻR.\equiv.S'\mid R\mid T'\in (P \unicode{x2217} Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·44}.\supset\vdash:\text{Hp}.S'\mid R\mid T'\in (P \unicode{x2217} Q)ʻR.&\supset.S\mid R\mid T\in (P \unicode{x2217} Q)ʻR &\qquad \text{(1)}\\ +\vdash.\text{*91·34}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:S'\mid S\mid R\mid T\mid T'\in (P \unicode{x2217} Q)ʻR:\\ +[\text{*95·44}] & \supset:S\mid R\mid T\in (P \unicode{x2217} Q)ʻR.\supset.S'\mid R\mid T'\in (P \unicode{x2217} Q)ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·46.</b> \[\begin{align}\vdash\colon\ldotp \text{Hp}*95·41.\dot{\exists}!R.\text{D}ʻR\subset \overrightarrow{B}ʻ\breve{P}.\text{ᗡ}ʻ&R\subset \overrightarrow{B}ʻ\breve{Q}.\supset:\\ +&T\in \text{Pot}ʻQ.\supset.R\mid T{\sim}\in (P \unicode{x2217} Q)ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·38}.\supset\vdash\colon\ldotp \text{Hp}.T\in \text{Pot}ʻQ.&\supset:R\mid T \neq R:\\ +[\text{*95·42}] &\supset:R\mid T\in (P \unicode{x2217} Q)ʻR.\supset.\breve{P}\mid R\mid T\mid \breve{Q}\in (P \unicode{x2217} Q)ʻR.\\ +[\text{*95·32}] &\supset.\dot{\exists}!\breve{P}\mid R\mid T\mid \breve{Q}.\\ +[\text{*34·31}] &\supset.\dot{\exists}!\breve{P}\mid R.\\ +[\text{*34·3}] & \supset.\exists !\text{D}ʻP\cap \text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*93·101}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻP\cap \text{D}ʻR=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).(1).Transp}.&\supset\vdash:\text{Hp}.T\in \text{Pot}ʻQ.\supset.R\mid T{\sim}\in (P \unicode{x2217} Q)ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·47.</b> \[\begin{align}\vdash:\text{Hp*95·46}.S\in \text{Potid}ʻP.T,&T'\in \text{Potid}ʻQ.\\ +& S\mid R\mid T,S\mid R\mid T'\in (P \unicode{x2217} Q)ʻR.\supset.T=T'\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·45}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists U):U\in \text{Potid}ʻQ:T=U\mid T'.\lor.T'=U\mid T &\qquad \text{(1)}\\ +\vdash.\text{*50·62.*91·35}.&\supset\vdash:\text{Hp}.\supset.S=S\mid I\upharpoonright CʻP.I\upharpoonright CʻP\in \text{Potid}ʻP &\qquad \text{(2)}\\ +\vdash.\text{*95·45.*33·24.*22·621.(2)}.\supset\\ +\vdash:\text{Hp}.U\in \text{Potid}ʻQ.T=U\mid T'.&\supset.I\upharpoonright CʻP\mid R\mid U\in (P \unicode{x2217} Q)ʻR.U\in \text{Potid}ʻQ.\\ +[\text{*50·63}] &\supset.R\mid U\in (P \unicode{x2217} Q)ʻR.U\in \text{Potid}ʻQ.\\ +[\text{*95·46.Transp}] &\supset.U{\sim}\in \text{Pot}ʻQ.U\in \text{Potid}ʻQ.\\ +[\text{*91·23}] &\supset.U=I\upharpoonright CʻQ.\\ +[\text{*91·27.*50·63}] &\supset.U\mid T'=T'.\\ +[\text{*13·12}] & \supset.T=T' &\qquad \text{(3)}\\ +\text{Similarly}\qquad\qquad\qquad \vdash:\text{Hp}.U\in \text{Potid}ʻQ.T'&=U\mid T.\supset.T=T' &\qquad \text{(4)}\\ +\vdash.\text{(1).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_634">[Pg 634]</span></p> + +<p class="nind"><b>*95·471.</b> \[\begin{align}\vdash:\text{Hp*95·46}.S,\,S'\in &\text{Potid}ʻP.T\in \text{Potid}ʻQ.\\ +& S\mid R\mid T,S'\mid R\mid T\in (P \unicode{x2217} Q)ʻR.\supset.S=S'\\ +[\text{Proof as in *95·47}]\end{align}\]</p> + +<p class="nind"><b>*95·51.</b> \(\vdash:\text{Hp*95·46}.M,\,M'\in (P \unicode{x2217} Q)ʻR.\exists !\text{ᗡ}ʻM\cap \text{ᗡ}ʻM'.\supset.M=M'\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·22}.\supset\vdash:\text{Hp}.\supset.(\exists S,S',T,T').&S,\,S'\in \text{Potid}ʻP.T,\,T'\in \text{Potid}ʻQ.\\ +& M=S\mid R\mid T.M'=S'\mid R\mid T'.\\ +& S\mid R\mid T,S'\mid R\mid T'\in (P \unicode{x2217} Q)ʻR.\\ +& \exists !\text{ᗡ}ʻ(S\mid R\mid T)\cap \text{ᗡ}ʻ(S'\mid R\mid T').\\ +[\text{*95·351}]\supset.(\exists S,S',T).&S,\,S'\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.M=S\mid R\mid T.M'=S'\mid R\mid T.\\ +& S\mid R\mid T,\,S'\mid R\mid T\in (P \unicode{x2217} Q)ʻR.\\ +[\text{*95·471}]&\supset.(\exists S,T).S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.M=S\mid R\mid T.M'=S\mid R\mid T.\\ +[\text{*13·172}]&\supset.M=M':\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·511.</b> \[\begin{align}&\vdash:\text{Hp*95·46}.M,\,M'\in (P \unicode{x2217} Q)ʻR.\exists !\text{D}ʻM\cap \text{D}ʻM'.\supset.M=M'\\ +&[\text{Proof as in *95·51}]\end{align}\]</p> + +<p class="nind"><b><a id="*95·52">*95·52</a>.</b> + \[\begin{align}\vdash:P,\,Q,\,R\in 1\rightarrow 1.\text{D}ʻP\subset \text{ᗡ}ʻP.\text{ᗡ}ʻQ&\subset \text{D}ʻQ.\text{D}ʻR\subset \overrightarrow{B}ʻ\breve{P}.\text{ᗡ}ʻR\subset \overrightarrow{B}ʻQ.\supset.\\ +&\dot{s}ʻ(P \unicode{x2217} Q)ʻR\in 1\rightarrow 1\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·21.*34·32}.\supset\vdash:R=\dot{\Lambda}.&\supset.(P \unicode{x2217} Q)ʻR\subset \iotaʻ\dot{\Lambda}.\\ +[\text{*53·04}] &\supset.\dot{s}ʻ(P \unicode{x2217} Q)ʻR=\dot{\Lambda}.\\ +[\text{*72·1}] & \supset.\dot{s}ʻ(P \unicode{x2217} Q)ʻR\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{*92·102.*95·21.*71·252}.&\supset\vdash:\text{Hp}.M\in (P \unicode{x2217} Q)ʻR.\supset.M\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{*41·11}.&\supset\vdash:x\{\dot{s}ʻ(P \unicode{x2217} Q)ʻR\}y.x\{\dot{s}ʻ(P \unicode{x2217} Q)ʻR\}z.\supset.\\ +& (\exists M,M').M,\,M'\in (P \unicode{x2217} Q)ʻR.xMy.xM'z.\\ +[\text{*33·14}] & \supset.(\exists M,M').M,\,M'\in (P \unicode{x2217} Q)ʻR.xMy.xM'z.\exists !\text{D}ʻM\cap \text{D}ʻM' &\qquad \text{(3)}\\ +\vdash.\text{(3).*95·511}.\supset\\ +\vdash:\text{Hp}.\dot{\exists}!R.\text{Hp(3)}.&\supset.(\exists M).M\in (P \unicode{x2217} Q)ʻR.xMy.xMz.\\ +[\text{(2)}] &\supset.y=z &\qquad \text{(4)}\\ +\text{Similarly}\\ +\vdash:\text{Hp}.\dot{\exists}!R.x\{\dot{s}ʻ(P \unicode{x2217} Q)ʻR\}z.y\{\dot{s}ʻ(P \unicode{x2217} Q)ʻR\}z.&\supset.x=y &\qquad \text{(5)}\\ +\vdash.\text{(4).(5).*71·172}.&\supset\vdash:\text{Hp}.\dot{\exists}!R.\supset.\dot{s}ʻ(P \unicode{x2217} Q)ʻR\in 1\rightarrow 1 &\qquad \text{(6)}\\ +\vdash.\text{(1).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·6.</b> \[\begin{align}\vdash:\text{D}ʻR\subset \text{ᗡ}ʻP.\text{D}ʻP\subset \text{ᗡ}ʻP.\text{ᗡ}ʻR=&\overrightarrow{B}ʻQ.Q\in 1\rightarrow \text{Cls}.\supset.\\ +&\text{ᗡ}ʻʻ(P \unicode{x2217} Q)ʻR=\text{gen}ʻQ\end{align}\]</p> + +<p><span class="pagenum" id="Page_635">[Pg 635]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·143}.\supset\vdash:\text{Hp}.S\in \text{Potid}ʻP.&\supset.\text{ᗡ}ʻS=\text{ᗡ}ʻP.\\ +[\text{Hp}] &\supset.\text{D}ʻR\subset \text{ᗡ}ʻS.\\ +[\text{*37·322}] & \supset.\text{ᗡ}ʻ(S\mid R)=\text{ᗡ}ʻR.\\ +[\text{*37·32}] & \supset.\text{ᗡ}ʻ(S\mid R\mid T)=\breve{T}ʻʻ\text{ᗡ}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1)}.&\supset\vdash:\text{Hp}.S\in \text{Potid}ʻP.T\in \text{Potid}ʻQ.\supset.\text{ᗡ}ʻ(S\mid R\mid T)=\breve{T}ʻʻ\overrightarrow{B}ʻQ. &\qquad \text{(2)}\\ +[\text{*93·32}] &\supset.\text{ᗡ}ʻ(S\mid R\mid T)\in \text{gen}ʻQ &\qquad \text{(3)}\\ +\vdash.\text{(3).*95·22}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻʻ(P \unicode{x2217} Q)ʻR\subset \text{gen}ʻQ &\qquad \text{(4)}\\ +\vdash.\text{(2).*95·221.*93·32}.&\supset\vdash:\text{Hp}.\supset.\text{gen}ʻQ\subset \text{ᗡ}ʻʻ(P \unicode{x2217} Q)ʻR &\qquad \text{(5)}\\ +\vdash.\text{(4).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·601.</b> \[\begin{align}\vdash:\text{ᗡ}ʻR\subset \text{D}ʻQ.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\text{D}ʻR=\overrightarrow{B}ʻ\breve{P}.P\in &\text{Cls}\rightarrow 1.\supset.\\ +&\text{D}ʻʻ(P \unicode{x2217} Q)ʻR=\text{gen}ʻ\breve{P}\\ +&[\text{Proof as in *95·6}]\end{align}\]</p> + +<p class="nind"><b>*95·61.</b> + \[\begin{align}&\vdash:P,Q,R\in 1\rightarrow 1.\text{D}ʻP\subset \text{ᗡ}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\text{D}ʻR=\overrightarrow{B}ʻ\breve{P}.\text{ᗡ}ʻR=\overrightarrow{B}ʻQ.\supset.\\ +&\dot{s}ʻ(P \unicode{x2217} Q)ʻR\in 1\rightarrow 1.\text{D}ʻ\dot{s}ʻ(P \unicode{x2217} Q)ʻR=sʻ\text{gen}ʻ\breve{P}.\text{ᗡ}ʻ\dot{s}ʻ(P \unicode{x2217} Q)ʻR=sʻ\text{gen}ʻQ\\ +&[\text{*95·52·6·601.*41·43·44}]\end{align}\]</p> + +<p class="nind"><b>*95·62.</b> \(\vdash:\text{Hp*95·61}.\supset.sʻ\text{gen}ʻP \text{ sm } sʻ\text{gen}ʻQ \quad[\text{*95·61.*73·2}]\)</p> + +<p class="nind"><b><a id="*95·63">*95·63</a>.</b> \[\begin{align}\vdash:P,\,Q\in 1\rightarrow 1.\text{ᗡ}ʻP\subset \text{D}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.&\overrightarrow{B}ʻP \text{ sm }\overrightarrow{B}ʻQ.\supset.\\ +&sʻ\text{gen}ʻP \text{ sm } sʻ\text{gen}ʻQ\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*95·62}\, \frac{\breve{P}}{P} .&\supset\vdash:P,\,Q,\,R\in 1\rightarrow 1.\text{ᗡ}ʻP\subset \text{D}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\\ +&\text{D}ʻR=\overrightarrow{B}ʻP.\text{ᗡ}ʻR=\overrightarrow{B}ʻQ.\supset.sʻ\text{gen}ʻP \text{ sm } sʻ\text{gen}ʻQ &\qquad \text{(1)}\\ +\vdash.(1).*10·11·23·35.*73·1.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*95·64.</b> \[\begin{align}\vdash:P,\,Q\in 1&\rightarrow 1.\text{ᗡ}ʻP\subset \text{D}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\overrightarrow{B}ʻP \text{ sm }\overrightarrow{B}ʻQ.\\ +&pʻ\text{ᗡ}ʻʻ\text{Pot}ʻP=\Lambda.pʻ\text{ᗡ}ʻʻ\text{Pot}ʻQ=\Lambda.\supset.\text{D}ʻP \text{ sm } \text{D}ʻQ\\ +&[\text{*95·63.*93·274.*33·181}]\end{align}\]</p> + +<p class="nind"><b><a id="*95·65">*95·65</a>.</b> \[\begin{align}\vdash:&P,Q\in 1\rightarrow 1.\text{ᗡ}ʻP\subset \text{D}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ.\overrightarrow{B}ʻP \text{ sm }\overrightarrow{B}ʻQ.\\ +&CʻP=\breve{P}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻP.CʻQ=\breve{Q}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻQ.\supset.CʻP \text{ sm } CʻQ\\ +&[\text{*95·63.*93·36}]\end{align}\]</p> + +<p>The following example may illustrate the scope of <a href="#*95·65">*95·65</a>. Let \(R\), +\(S\) be the generating relations of two well-ordered series, neither +of which has a last term. Put \(P=R\dot{-}R^{2}.Q=S\dot{-}S^{2}\). Then +\(P\) is the relation of immediately preceding in the \(R\)-series, and +\(Q\) is the relation of immediately preceding in the \(S\)-series. We +shall have +\[ +P,\,Q\in 1\rightarrow 1.\text{ᗡ}ʻP\subset \text{D}ʻP.\text{ᗡ}ʻQ\subset \text{D}ʻQ. +\]<span class="pagenum" id="Page_636">[Pg 636]</span> +Also, except in certain exceptional cases, \(\overrightarrow{B}ʻP\), +\(\overrightarrow{B}ʻQ\) are the first derivatives of the two series +(including the first terms of the two series). +\[ +\unicode{x201c}CʻP=\breve{P}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻP\unicode{x201d} +\] +states that, starting from any term of the series and going backwards, a finite +number of steps will bring us to a member of the first derivative, which is +true. Hence, by <a href="#*95·65">*95·65</a>, neglecting certain exceptional cases, we arrive at +the result that if the first derivatives of two well-ordered series have the +same cardinal number of terms, then the series themselves have the same +cardinal number of terms. This proposition can of course be proved otherwise; +the above is merely mentioned as an illustration of the results of +*95·65.</p> + +<p class="nind"><b>*95·7.</b> \(\vdash:R,\,S\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻS.\text{ᗡ}ʻS\subset \text{D}ʻR.\supset.\overrightarrow{B}ʻ(R\mid S)\text{ sm }\overrightarrow{B}ʻ(S\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·101.*24·412.*37·16·321}.\supset\\ +\vdash:\text{Hp}.&\supset.\overrightarrow{B}ʻ(R\mid S)=(\text{D}ʻR-\text{ᗡ}ʻS)\cup (\text{ᗡ}ʻS-\breve{S}ʻʻ\text{ᗡ}ʻR).\\ +&\overrightarrow{B}ʻ(S\mid R)=(\text{D}ʻS-\text{ᗡ}ʻR)\cup (\text{ᗡ}ʻR-\breve{R}ʻʻ\text{ᗡ}ʻS) &\qquad \text{(1)}\\ +\vdash.\text{*71·38.*37·32}. & \supset\vdash:\text{Hp}.\supset.\breve{R}ʻʻ(\text{D}ʻR-\text{ᗡ}ʻS)=\text{ᗡ}ʻR-\breve{R}ʻʻ\text{ᗡ}ʻS &\qquad \text{(2)}\\ +\vdash.\text{*71·381.*37·32}. &\supset\vdash:\text{Hp}.\supset.Sʻʻ(\text{ᗡ}ʻS-\breve{S}ʻʻ\text{ᗡ}ʻR)=\text{D}ʻS-Sʻʻ\breve{S}ʻʻ\text{ᗡ}ʻR\\ +[\text{*72·502}] &\qquad\qquad\qquad =\text{D}ʻS-\text{ᗡ}ʻR &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).*73·21·22}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻR-\text{ᗡ}ʻS \text{ sm } \text{ᗡ}ʻR-\breve{R}ʻʻ\text{ᗡ}ʻS.\\ +&\qquad\qquad\qquad\text{ᗡ}ʻS-\breve{S}ʻʻ\text{ᗡ}ʻR \text{ sm } \text{ᗡ}ʻS-\text{ᗡ}ʻR &\qquad \text{(4)}\\ +\vdash.\text{*24·21}.& \supset\vdash:\text{Hp}.\supset.(\text{D}ʻR-\text{ᗡ}ʻS)\cap (\text{ᗡ}ʻS-\breve{S}ʻʻ\text{ᗡ}ʻR)=\Lambda.\\ +&\qquad\qquad\qquad(\text{ᗡ}ʻR-\breve{R}ʻʻ\text{ᗡ}ʻS)\cap (\text{ᗡ}ʻS-\text{ᗡ}ʻR)=\Lambda &\qquad \text{(5)}\\ +\vdash.\text{(1).(4).(5).*73·71}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*95·71">*95·71</a>.</b> \(\vdash:R,\,S\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻS.\text{ᗡ}ʻS\subset \text{D}ʻR.\supset.sʻ\text{gen}ʻ(R\mid S)\text{ sm } sʻ\text{gen}ʻ(S\mid R)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*34·36.*37·321}. &\supset\vdash:\text{Hp}.\supset.\text{ᗡ}ʻ(R\mid S)\subset \text{D}ʻ(R\mid S).\text{ᗡ}ʻ(S\mid R)\subset \text{D}ʻ(S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{*71·252}.& \supset\vdash:\text{Hp}.\supset.R\mid S,\,S\mid R\in 1\rightarrow 1 &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*95·7·63}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This proposition and <a href="#*94·53">*94·53</a> or <a href="#*94·54">*94·54</a> together reconstitute the +Schröder-Bernstein theorem (<a href="#*73·88">*73·88</a>). For, in virtue of *93·274·275 and +<a href="#*73·71">*73·71</a>, they together give +\[ +R,\,S\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻS.\text{ᗡ}ʻS\subset \text{D}ʻR.\supset.Cʻ(R\mid S)\text{ sm } Cʻ(S\mid R), +\] +and with this hypothesis +\[ +Cʻ(R\mid S)=\text{D}ʻR.Cʻ(S\mid R)=\text{D}ʻS. +\]</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_67" href="#FNanchor_67" class="label">[67]</a> +This notation is used in the present number only. In +*257, we shall introduce a different and wholly unconnected meaning for +(\(P\unicode{x2217}Q)\). A temporary definition is indicated by the +letters "\(\text{Dft}\)" followed by a reference in square brackets to +the number or numbers in which the definition is used.</p> + +</div> +</div> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_637">[Pg 637]</span></p> +<h2 class="nobreak" id="*96">*96. ON THE POSTERITY OF A TERM.</h2> +</div> + + +<p><i>Summary of</i> *96.</p> + +<p>By the "posterity" of a term with respect to a relation \(R\) +we mean the class \(\overleftarrow{R}_{\unicode{x2217}}ʻx\). +In the present number, we shall be chiefly concerned with +the relation (\(\overleftarrow{R}_{\unicode{x2217}}ʻx) +\upharpoonleft R\), <i>i.e.</i> the relation \(R\) confined +to the posterity of \(x\). We shall also be concerned with +(\(\overleftarrow{R}_{\unicode{x2217}}ʻx) \upharpoonleft R_{\unicode{x2217}}\) +and (\(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\), +which, as is proved in <a href="#*96·13">*96·13</a>, are respectively +\[ +\{(\overleftarrow{R}_{\unicode{x2217}}ʻx) \upharpoonleft R\}_{\unicode{x2217}} ~\text{and}~ \{(\overleftarrow{R}_{\unicode{x2217}}ʻx) \upharpoonleft R\}_{\text{po}}. +\]</p> + +<p>The most interesting case is when \(R \in \text{Cls} \rightarrow +1\). In this case, \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) is in +general shaped like a \(Q\), with \(x\) at the tip of the tail; that +is, \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) may be divided into two +parts, the first an open series, the second a closed series. If \(y\) +is the junction of the two, we shall have +\[ +x R_{\unicode{x2217}} z .z R_{\text{po}} y .\supset. {\sim}(z R_{\text{po}} z)\text{,} +\] +\[ +y R_{\unicode{x2217}} z .\supset. z R_{\text{po}}z\text{;} +\] +\[ +\text{in fact,} \quad (\exists P): P \in \text{Pot}ʻR : y R_{\unicode{x2217}} z .\supset_{z}. z P z\text{.} +\]</p> + +<p>We have also, when \(R \in \text{Cls} \rightarrow 1\), +\[ +y,\,z \in \overleftarrow{R}_{\unicode{x2217}}ʻx .\supset: y R_{\unicode{x2217}} z .\lor. z R_{\unicode{x2217}} y\text{.} +\]</p> + +<p>It thus appears that \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) +is divided into two parts, the first consisting of those terms +\(z\) for which \({\sim}(z R_{\text{po}} z)\), the second of those +for which \(z R_{\text{po}} z\). The first wholly precedes the +second; the first exists if \({\sim}(x R_{\text{po}} x)\), the +second if \(\dot{\exists}!\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}} \dot{\cap} I\}\). +Every term in \(\overleftarrow{R}_{\text{po}}ʻx\) has one and only +one immediate predecessor, except the term (if it exists) at the +junction of the tail and circle of the \(Q\); this term has just +two immediate predecessors, one in the tail and one in the circle. +But if either the tail or the circle is null, then every term in +\(\overleftarrow{R}_{\text{po}}ʻx\) has only one immediate predecessor, +and therefore +\[ +(\overleftarrow{R}_{\unicode{x2217}}ʻx) \upharpoonleft R \in 1 \rightarrow 1\text{.} +\]<span class="pagenum" id="Page_638">[Pg 638]</span> +\[ +\text{Put} \quad (I_{R}ʻx = \overleftarrow{R}_{\unicode{x2217}}ʻx \cap \hat{z}(z R_{\text{po}} z) \quad \text{Dft} +\] +\[ +J_{R}ʻx = \overleftarrow{R}_{\unicode{x2217}}ʻx \cup \hat{z}\{{\sim}(z R_{\text{po}} z)\} \quad \text{Dft} +\] +(these definitions being only to apply within <a href="#*96">*96</a>). Then \(J_{R}ʻx\) is +the open part of the series \(\overleftarrow{R}_{\unicode{x2217}}ʻx\), +and \(I_{R}ʻx\) is the circular part. The open part wholly precedes the +circular part, provided \(R \in \text{Cls} \rightarrow 1\); <i>i.e.</i> +\[ +R \in \text{Cls} \rightarrow 1 .\supset. J_{R}ʻx \supset pʻ\overrightarrow{R}_{\text{po}}ʻʻI_{R}ʻx\text{.} +\]</p> + +<p>If \(J_{R}ʻx\) and \(I_{R}ʻx\) both exist, \(J_{R}ʻx\) has a last +term, say \(y\). The successor of this term, \(\breve{R}ʻy\), is the +only term in \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) which has two +immediate predecessors in \(\overleftarrow{R}_{\unicode{x2217}}ʻx\), +namely \(y\) and \(\breve{\iota}ʻ(I_{R}ʻx \cap +\overrightarrow{R}ʻ\breve{R}ʻy)\).</p> + +<p>The most important applications of the propositions of the +present number are in the theory of finite and infinite, both +cardinal and ordinal. When \(R\) is many-one, then if \(I_{R}ʻx\) +exists, or, more generally, if \(J_{R}ʻx\) has a last term, +\(\overleftarrow{R}_{\unicode{x2217}}ʻx\) is a finite class, +<i>i.e.</i> what we shall call a "\(\text{Cls} ~ \text{induct}\)" (cf. +*120). That is, we have +\[ +\vdash: R \in \text{Cls} \rightarrow 1 . \text{E}! ~ \text{max}_{R}ʻJ_{R}ʻx .\supset. \overleftarrow{R}_{\unicode{x2217}}ʻx \in \text{Cls} ~ \text{induct}\text{.} +\]</p> + +<p>If \(J_{R}ʻx\) exists, but has no last term, +\(\overleftarrow{R}_{\unicode{x2217}}ʻx\) is a <i>progression</i> (cf. +*122) when its terms are arranged in the order generated by \(R\). +That is, giving to \(\aleph_{0}\) and \(\omega\) the meanings given by +Cantor (cf. *123 and *263), and using "\(\text{Prog}\)" for the class +of one-one relations which generate progressions, we have +\[ +\begin{align} +\vdash: R \in \text{Cls} \rightarrow 1&.{\sim}E! ~ \text{max}_{R}ʻJ_{R}ʻx.\exists! J_{R}ʻx .\supset.\\ +&\overleftarrow{R}_{\unicode{x2217}}ʻx \in \aleph_{0} . (\overleftarrow{R}_{\unicode{x2217}}ʻx) \upharpoonleft R \in \text{Prog} . (\overleftarrow{R}_{\unicode{x2217}}ʻx) + \upharpoonleft R_{\text{po}} \in \omega.\\ +\end{align} +\]</p> + +<p>Another very important proposition in the proof of which the +present number is useful is *121·47, which proves that if \(R\) +is either one-many or many-one, and \(a\) and \(z\) are any two +terms whatever, then \(\overleftarrow{R}_{\unicode{x2217}}ʻa \cap\overrightarrow{R}_{\unicode{x2217}}ʻz\) +(which we call the "interval" from \(a\) to \(z\)) is always a finite +class. The proof that progressions are well-ordered series depends upon +the propositions of this number, since it uses *122·23, which depends +upon <a href="#*96·52">*96·52</a>.</p> + +<p>The present number begins with a series of propositions (ending with +<a href="#*96·16">*96·16</a>) on \(\alpha \upharpoonleft R_{\text{po}}\) and +\(\alpha\upharpoonleft R_{\unicode{x2217}}\), both in general and when +\(\alpha = \overleftarrow{R}_{\unicode{x2217}}ʻx\). We then proceed to a few +propositions (<a href="#*96·2">*96·2</a>-<a href="#*96·25">·25</a>) on (\(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\) +when \(R \in 1 \rightarrow \text{Cls}\); with +the exception of <a href="#*96·24">*96·24</a>, these propositions are all used in the +cardinal theory of finite and infinite. They are, however, less +important than the subsequent propositions, which are concerned with +<span class="pagenum" id="Page_639">[Pg 639]</span>\(\overleftarrow{R}_{\unicode{x2217}}ʻx\) when \(R \in \text{Cls}\rightarrow 1\).</p> + +<p>If \(R\) is a many-one relation, and x is a member of +\(\text{D}ʻR\), the relation \(R\) in general arranges +\(\overleftarrow{R}_{\unicode{x2217}}ʻx\) (<i>i.e.</i> the posterity +of \(x\)) in a figure such as is here given. The relation \(R\) holds +between each dot and the next, starting from \(x\), and travelling +round the circle in the sense indicated by the arrow. The dots from +\(x\) to \(y\) constitute \(J_{R}ʻx\), and the dots in the circle +constitute \(I_{R}ʻx\). \(y\) is the last term of \(J_{R}ʻx\), +<i>i.e.</i> \(\text{max}_{R}'J_{R}ʻx\); \(w\) is \(\breve{R}ʻy\), and +\(z\) is \(\breve{\iota}ʻ(\overrightarrow{R}ʻw\cap I_{R}ʻx)\), or, +what comes to the same thing, \(\{(I_{R}ʻx)\upharpoonleft R\}ʻw\). +\(w\) is the only term which has more than one immediate predecessor +in \(\overleftarrow{R}_{\unicode{x2217}}ʻx\); \(w\) always exists if +neither \(J_{R}ʻx\) nor \(I_{R}ʻx\) is null, and conversely, if \(w\) +exists, neither \(J_{R}ʻx\) nor \(I_{R}ʻx\) is null. The proof of these +propositions is long; the following are useful stages in the proof.</p> + +<p>If \(xRx\), the whole posterity of \(x\) is \(x\) itself (<a href="#*96·33">*96·33</a>); if +\(xRy\) and \(yRx\), \(x\) and \(y\) constitute the whole posterity of +\(x\) (<a href="#*96·331">*96·331</a>), and so on. The successors of members of \(I_{R}ʻx\) +belong to \(I_{R}ʻx\) (<a href="#*96·341">*96·341</a>), and the predecessors of members of +\(J_Rʻx\), if they belong to \(\overleftarrow{R}_{\unicode{x2217}}ʻx\), +belong to \(J_{R}ʻx\) (<a href="#*96·351">*96·351</a>). (It should be observed that, +since \(R\) is only assumed to be many-one, not one-one, +every member of \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) +may have any number of predecessors which do not belong to +\(\overleftarrow{R}_{\unicode{x2217}}ʻx)\). We have a series +of propositions, beginning with <a href="#*96·4">*96·4</a>, which deal with the +hypothesis \(yRw.zRw\). We prove (<a href="#*96·42">*96·42</a>) that if \(yRw.zRw\) and +\(yR_{\text{po}}z\), then \(zR_{\text{po}}z\), <i>i.e.</i> \(z\) +belongs to \(I_{R}ʻx\). We prove (<a href="#*96·431">*96·431</a>) that \(J_{R}ʻx\) wholly +precedes \(I_{R}ʻx\); that (\(J_{R}ʻx)\upharpoonleft R\) and +(\(I_{R}ʻx)\upharpoonleft R\) are both one-one (<a href="#*96·45">*96·45</a>), so that if +\(yRw.zRw.y\neq z\), one of \(y\) and \(z\) must belong to \(J_{R}ʻx\) +and the other to \(I_Rʻx\) (<a href="#*96·441">*96·441</a>). Hence it follows (<a href="#*96·453">*96·453</a>) +that if either \(xR_{\text{po}}x\) (in which case \(J_{R}ʻx={po})\) +or (\(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\unicode{x2abd}J\) +(in which case \(I_{R}ʻx=\Lambda)\), +then (\(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\) is +a one-one relation. (This proposition is used twice in the cardinal +theory of finite and infinite, namely in *121·43 and *122·17.) Hence we +arrive at the proposition (<a href="#*96·47">*96·47</a>) that if two different members \(y\) +and \(z\) of \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) both immediately +precede a term \(w\), then one of \(y\) and \(z\) (say \(y\)) is the +last term of \(J_{R}ʻx\), \(w\) is its immediate successor and \(z\) is +the immediate predecessor of \(w\) in \(I_{R}ʻx\), <i>i.e.</i> we have +\[ +y=\text{max}_{R}ʻJ_{R}ʻx.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.z=\{(I_{R}ʻx)\upharpoonleft R\}ʻ\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx. +\] +Thus \(y\), \(z\), \(w\) are unique if they exist. We prove next +(<a href="#*96·475">*96·475</a>) that \(y\), \(z\), \(w\) exist when, and only when, neither +\(I_{R}ʻx\) nor \(J_{R}ʻx\) is null.</p> + +<p>It follows from the above propositions that if \(R\) is one-one, either +\(I_{R}ʻx\) or \(J_{R}ʻx\) must be null (<a href="#*96·491">*96·491</a>), <i>i.e.</i> the +posterity of a term is either an open series or a cycle, and cannot +have the \(Q\)-shape.</p> + +<p><span class="pagenum" id="Page_640">[Pg 640]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*96·01.</b> \(I_{R}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx\cap \hat{z}(zR_{\text{po}}z) \quad\text{Dft}\, [\text{*96}]\)</p> + +<p class="nind"><b>*96·02.</b> \(J_{R}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx-I_{R}ʻx \quad\text{Dft}\, [\text{*96}]\)</p> + +<p class="nind"><b>*96·1.</b> \(\vdash:z\in I_{R}ʻx.\equiv.xR_{\unicode{x2217}}z.zR_{\text{po}}z \quad[\text{*20·3.*32·181.(*96·01)}]\)</p> + +<p class="nind"><b>*96·101.</b> \(\vdash:z\in J_{R}ʻx.\equiv.xR_{\unicode{x2217}}z.{\sim}(zR_{\text{po}}z) \quad[\text{*96·1.*22·93.(*96·02)}]\)</p> + +<p class="nind"><b>*96·102.</b> \(\vdash.\overleftarrow{R}_{\unicode{x2217}}ʻx=J_{R}ʻx\cup I_{R}ʻx.J_{R}ʻx\cap I_{R}ʻx=\Lambda \quad[\text{*24·41·21.(*96·01·02)}]\)</p> + +<p class="nind"><b>*96·103.</b> \(\vdash.(J_{R}ʻx)\upharpoonleft R_{\text{po}}\unicode{x2abd}J\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·101}.\supset\vdash\colon\ldotp y\{(J_{R}ʻx)\upharpoonleft + R_{\text{po}}\}z.&\equiv:xR_{\unicode{x2217}}y.{\sim}(yR_{\text{po}}y).yR_{\text{po}}z:\\ +[\text{*13·14}] &\supset:y \neq z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·104.</b> \(\vdash:I_{R}ʻx=\Lambda.\equiv.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R_{\text{po}}\unicode{x2abd}J.\equiv.J_{R}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·1}.\supset\vdash\colon\ldotp I_{R}ʻx=\Lambda.&\equiv:xR_{\unicode{x2217}}y.\supset_{y}.{\sim}(yR_{\text{po}}y):\\ +[\text{*13·196}] &\equiv:xR_{\unicode{x2217}}y.yR_{\text{po}}z.\supset_{y,z}.y \neq z:\\ +[\text{*35·1}] &\equiv:(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\unicode{x2abd}J &\qquad \text{(1)}\\ +\vdash.\text{(1).*96·102}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·11.</b> \(\vdash.(\alpha\upharpoonleft R)_{\text{po}}\unicode{x2abd}\alpha\upharpoonleft R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·502.*35·46}.&\supset\vdash.\alpha\upharpoonleft R\unicode{x2abd}\alpha\upharpoonleft R_{\text{po}} &\qquad \text{(1)}\\ +\vdash.\text{*35·1}.\supset\\ +\vdash\colon\ldotp P\unicode{x2abd}\alpha\upharpoonleft R_{\text{po}}.&\supset:xPy.y(\alpha\upharpoonleft R)z.\supset.x\in \alpha.xR_{\text{po}}y.yRz.\\ +[\text{*91·511.*35·1}] & \supset.x(\alpha\upharpoonleft R_{\text{po}})z.\\ +[\text{*34·1}] & \supset:P\mid (\alpha\upharpoonleft R)\unicode{x2abd}\alpha\upharpoonleft R_{\text{po}} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·171}.&\supset\vdash:P\in \text{Pot}ʻ(\alpha\upharpoonleft R).\supset.P\unicode{x2abd}\alpha\upharpoonleft R_{\text{po}}:\\ +[\text{*41·151}] & \supset\vdash.(\alpha\upharpoonleft R)_{\text{po}}\unicode{x2abd}\alpha\upharpoonleft R_{\text{po}}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·111">*96·111</a>.</b> \(\vdash:\breve{R}ʻʻ\alpha\subset \alpha.\supset.(\alpha\upharpoonleft R)_{\text{po}}=\alpha\upharpoonleft R_{\text{po}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·502}.&\supset\vdash.\alpha\upharpoonleft R\unicode{x2abd}(\alpha\upharpoonleft R)_{\text{po}} &\qquad \text{(1)}\\ +\vdash.\text{*90·22.*91·54}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:P\in \text{Pot}ʻR.x\in \alpha.xPy.\supset.y\in \alpha:\\ +[\text{*35·1.Fact}]&\supset:P\in \text{Pot}ʻR.x(\alpha\upharpoonleft P)y.yRz.\supset.y(\alpha\upharpoonleft R)z:\\ +[\text{*91·511}] &\supset:P\in \text{Pot}ʻR.\alpha\upharpoonleft P\unicode{x2abd}(\alpha\upharpoonleft R)_{\text{po}}.\supset.(\alpha\upharpoonleft + P)\mid R\unicode{x2abd}(\alpha\upharpoonleft R)_{\text{po}} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·373}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:P\in \text{Pot}ʻR.\supset.\alpha\upharpoonleft P\unicode{x2abd}(\alpha\upharpoonleft R)_{\text{po}}:\\ +[\text{*41·52}] & \supset:\alpha\upharpoonleft R_{\text{po}}\unicode{x2abd}(\alpha\upharpoonleft R)_{\text{po}}:\\ +[\text{*96·11}] & \supset:\alpha\upharpoonleft R_{\text{po}}=(\alpha\upharpoonleft R)_{\text{po}}\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_641">[Pg 641]</span></p> + +<p class="nind"><b>*96·112.</b> \(\vdash:\alpha\subset \text{D}ʻR.\breve{R}ʻʻ\alpha\subset \alpha.\supset.(\alpha\upharpoonleft R)_{\unicode{x2217}}=\alpha\upharpoonleft R_{\unicode{x2217}}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·62.*37·4}. &\supset\vdash:\text{Hp}.\supset.Cʻ(\alpha\upharpoonleft R)=\alpha\cup \breve{R}ʻʻ\alpha\\ +[\text{*22·62}] &\qquad\qquad\qquad\qquad\qquad =\alpha.\\ +[\text{*50·5}] & \supset.I\upharpoonright Cʻ(\alpha\upharpoonleft R)=\alpha\upharpoonleft I &\qquad \text{(1)}\\ +\vdash.\text{*50·53}. & \supset\vdash.\alpha\upharpoonleft I\upharpoonright CʻR=(\alpha\cap CʻR)\upharpoonleft I &\qquad \text{(2)}\\ +\vdash.\text{(2).*22·621}. & \supset\vdash:\text{Hp}.\supset.\alpha\upharpoonleft I\upharpoonright CʻR=\alpha\upharpoonleft I &\qquad \text{(3)}\\ +\vdash.\text{*91·54}.& \supset\vdash:(\alpha\upharpoonleft R)_{\unicode{x2217}}=(\alpha\upharpoonleft R)_{\text{po}}\unicode{x228d}I\upharpoonright + Cʻ(\alpha\upharpoonleft R) &\qquad \text{(4)}\\ +\vdash.\text{*91·54.*35·42}.&\supset\vdash:\alpha\upharpoonleft R_{\unicode{x2217}}=\alpha\upharpoonleft R_{\text{po}}\unicode{x228d}\alpha\upharpoonleft + I\upharpoonright CʻR &\qquad \text{(5)}\\ +\vdash.\text{(1).(3).(4).(5).*96·111}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·121.</b> \(\vdash:Rʻʻ\alpha\subset \alpha.\supset.(R\upharpoonright \alpha)_{\text{po}}=R_{\text{po}}\upharpoonright \alpha \quad[\text{Proof as in *96·111}]\)</p> + +<p class="nind"><b>*96·122.</b> \(\vdash:\alpha\subset \text{ᗡ}ʻR.Rʻʻ\alpha\subset \alpha.\supset.(R\upharpoonright \alpha)_{\unicode{x2217}}=R_{\unicode{x2217}}\upharpoonright + \alpha \quad[\text{Proof as in *96·112}]\)</p> + +<p class="nind"><b><a id="*96·13">*96·13</a>.</b> \(\vdash.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}=\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\}_{\text{po}} + \quad[\text{*96·111.*90·163}]\)</p> + +<p class="nind"><b>*96·131.</b> \(\vdash.x\in \text{D}ʻR.\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\unicode{x2217}}=\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R\}_{\unicode{x2217}} \quad[\text{*96·112.*90·163}]\)</p> + +<p class="nind"><b>*96·14.</b> \(\vdash:x\in CʻR.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx=\iotaʻx\cup \overleftarrow{R}_{\text{po}}ʻx \quad[\text{*91·54.*32·33}]\)</p> + +<p class="nind"><b>*96·141.</b> \(\vdash.Cʻ(\alpha\upharpoonleft R_{\unicode{x2217}})=\breve{R}_{\unicode{x2217}}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·61.*37·4.*90·14}.\supset\vdash.Cʻ(\alpha\upharpoonleft R_{\unicode{x2217}})&=(\alpha\cap CʻR)\cup \breve{R}_{\unicode{x2217}}ʻʻ\alpha\\ +[\text{*90·331}] & =\breve{R}_{\unicode{x2217}}ʻʻ\alpha.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·142.</b> \(\vdash.Cʻ(\alpha\upharpoonleft R_{\text{po}})=(\alpha\cap \text{D}ʻR)\cup \breve{R}_{\text{po}}ʻʻ\alpha \quad[\text{*35·61.*37·4.*91·504}]\)</p> + +<p class="nind"><b>*96·143.</b> \(\vdash.Cʻ(\alpha\upharpoonleft R_{\text{po}})=\breve{R}_{\unicode{x2217}}ʻʻ(\alpha\cap \text{D}ʻR)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·261.*91·504}.\supset\vdash.\breve{R}_{\text{po}}ʻʻ\alpha=\breve{R}_{\text{po}}ʻʻ(\alpha\cap \text{D}ʻR) &\qquad \text{(1)}\\ +\vdash.\text{(1).*91·546.*96·142}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·144.</b> \(\vdash:\alpha\cap \text{ᗡ}ʻR\subset \breve{R}_{\unicode{x2217}}ʻʻ(\alpha\cap + \text{D}ʻR).\supset.Cʻ(\alpha\upharpoonleft R_{\text{po}})=\breve{R}_{\unicode{x2217}}ʻʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*22·62}.\supset\vdash:\text{Hp}.&\supset.\breve{R}_{\unicode{x2217}}ʻʻ(\alpha\cap \text{D}ʻR)=(\alpha\cap \text{ᗡ}ʻR)\cup \breve{R}_{\unicode{x2217}}ʻʻ(\alpha\cap + \text{D}ʻR)\\ +[\text{*91·546}] & =(\alpha\cap \text{ᗡ}ʻR)\cup (\alpha\cap \text{D}ʻR)\cup \breve{R}_{\text{po}}ʻʻ(\alpha\cap \text{D}ʻR)\\ +[\text{*37·261.*91·504}] &=(\alpha\cap CʻR)\cup \breve{R}_{\text{po}}ʻʻ\alpha\\ +[\text{*91·544}] & =\breve{R}_{\unicode{x2217}}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*96·143}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_642">[Pg 642]</span></p> + +<p class="nind"><b>*96·15.</b> \(\vdash:\text{D}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\}=\overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \text{D}ʻR.\text{ᗡ}ʻ{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R}=\overleftarrow{R}_{\text{po}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·61}.\supset\vdash.\text{D}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\}&=\overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*37·4}. \supset\vdash.\text{ᗡ}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\}&=\breve{R}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx\\ +[\text{*91·74}] & =\overleftarrow{R}_{\text{po}}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·151.</b> \(\vdash:x\in \text{D}ʻR.\supset.Cʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\}=\overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·14}.&\supset\vdash:\text{Hp}.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx\cap \text{D}ʻR=\iotaʻx\cup (\overleftarrow{R}_{\text{po}}ʻx\cap \text{D}ʻR).\\ +[\text{*22·63}] &\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx\cap \text{D}ʻR)\cup \overleftarrow{R}_{\text{po}}ʻx=\iotaʻx\cup \overleftarrow{R}_{\text{po}}ʻx\\ +[\text{*96·14}] &\qquad\qquad\qquad\qquad\qquad\quad =\overleftarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*96·15}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·152.</b> \(\vdash.\breve{R}_{\unicode{x2217}}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx \quad[\text{*90·17}]\)</p> + +<p class="nind"><b>*96·153.</b> + \(\vdash.\breve{R}_{\unicode{x2217}}ʻʻ\overleftarrow{R}_{\text{po}}ʻx=\breve{R}_{\text{po}}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\text{po}}ʻx + \quad[\text{*91·574}]\)</p> + +<p class="nind"><b>*96·154.</b> \(\vdash.Cʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\unicode{x2217}}\}=\overleftarrow{R}_{\unicode{x2217}}ʻx + \quad[\text{*96·141·152}]\)</p> + +<p class="nind"><b>*96·155.</b> \(\vdash.\text{D}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\}=\overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \text{D}ʻR.\text{ᗡ}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\}=\overleftarrow{R}_{\text{po}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·61.*91·504}.\supset\vdash.\text{D}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\}&=\overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \text{D}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*37·4}. \supset\vdash.\text{ᗡ}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R_{\text{po}}\}&=\breve{R}_{\text{po}}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx\\ +[\text{*96·153}] & =\overleftarrow{R}_{\text{po}}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·156.</b> \(\vdash.Cʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\}=(\iotaʻx\cap \text{D}ʻR)\cup \overleftarrow{R}_{\text{po}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·155}.\supset\\ +\vdash.Cʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\}&=(\overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \text{D}ʻR)\cup \overleftarrow{R}_{\text{po}}ʻx\\ +[\text{*91·54}] & =(\iotaʻx\cap CʻR\cap \text{D}ʻR)\cup (\overleftarrow{R}_{\text{po}}ʻx\cap \text{D}ʻR)\cup \overleftarrow{R}_{\text{po}}ʻx\\ +[\text{*22·62.*33·161}] &=(\iotaʻx\cap \text{D}ʻR)\cup \overleftarrow{R}_{\text{po}}ʻx.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·157.</b> \(\vdash:x\in \text{D}ʻR.\supset.Cʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\}=\overleftarrow{R}_{\unicode{x2217}}ʻx + \quad[\text{*96·156·14}]\)</p> + +<p class="nind"><b>*96·158.</b> \(\vdash:x{\sim}\in \text{D}ʻR.\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}=\dot{\Lambda}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·504}.\supset\vdash:\text{Hp}.&\supset.x{\sim}\in \text{D}ʻR_{\text{po}}.\\ +[\text{*33·4}] &\supset.\overleftarrow{R}_{\text{po}}ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*96·155}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_643">[Pg 643]</span></p> + +<p class="nind"><b>*96·159.</b> \(\vdash:\dot{\exists}!(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}.\supset.Cʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R_{\text{po}}\}=\overleftarrow{R}_{\unicode{x2217}}ʻx \quad[\text{*96·157·158}]\)</p> + +<p class="nind"><b><a id="*96·16">*96·16</a>.</b> \(\vdash.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R=R \unicode{x0294f} \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*35·1}.\supset\vdash:y\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\}z.&\equiv.y\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yRz.\\ +[\text{*90·16.*4·71}] & \equiv.y\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yRz.z\in \overleftarrow{R}_{\unicode{x2217}}ʻx.\\ +[\text{*36·13}] &\equiv.y(R \unicode{x0294f} \overleftarrow{R}_{\unicode{x2217}}ʻx)z:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·2">*96·2</a>.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R=R\upharpoonright \overleftarrow{R}_{\text{po}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·55}.\supset\vdash:\text{Hp}.\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R&=R\upharpoonright \breve{R}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx\\ +[\text{*91·74}] & =R\upharpoonright \overleftarrow{R}_{\text{po}}ʻx:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·21.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.xBR.\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R=R\upharpoonright \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·14}. &\supset\vdash:\text{Hp}.\supset.R\upharpoonright \overleftarrow{R}_{\unicode{x2217}}ʻx=R\upharpoonright + \iotaʻx\unicode{x228d}R\upharpoonright \overleftarrow{R}_{\text{po}}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*35·64.*93·1}.\supset\vdash:\text{Hp}.&\supset.\text{ᗡ}ʻ(R\upharpoonright \iotaʻx)=\Lambda.\\ +[\text{*33·241}] &\supset.R\upharpoonright \iotaʻx=\dot{\Lambda} &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. &\supset\vdash:\text{Hp}.\supset.R\upharpoonright \overleftarrow{R}_{\unicode{x2217}}ʻx=R\upharpoonright \overleftarrow{R}_{\text{po}}ʻx\\ +[\text{*96·2}] &\qquad\qquad\qquad\qquad\quad =(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·22.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.{\sim}(xRx).\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\unicode{x2abd}J\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11}. \supset\vdash:xQy.yRy.&\supset.xQy.yRy.y\breve{Q}x.\\ +[\text{*10·24.*34·1}] & \supset.xQ\mid R\mid \breve{Q}x &\qquad \text{(1)}\\ +\vdash.\text{(1).*92·132}.\supset\vdash:R\in 1\rightarrow \text{Cls}.&\supset:Q\in \text{Potid}ʻR.xQy.yRy.\supset.xRx:\\ +[\text{*10·11·21·23·35.*91·55}]& \supset:xR_{\unicode{x2217}}y.yRy.\supset.xRx:\\ +[\text{Transp}] &\supset:{\sim}(xRx).xR_{\unicode{x2217}}y.\supset.{\sim}(yRy):\\ +[\text{*13·196}] & \supset:{\sim}(xRx).xR_{\unicode{x2217}}y.yRz.\supset.y \neq z:\\ +[\text{*32·181.*35·1}] & \supset:{\sim}(xRx).\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\unicode{x2abd}J\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·23.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.xBR.\supset.I_{R}ʻx=\Lambda.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\unicode{x2abd}J\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11}.\supset\vdash:xQy.yTy.&\supset.xQy.yTy.y\breve{Q}x.\\ +[\text{*34·1}] &\supset.xQ\mid T\mid \breve{Q}x &\qquad \text{(1)}\\ +\vdash.\text{(1).*92·132}.\supset\\ +\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.&\supset:Q,\,T\in \text{Potid}ʻR.xQy.yTy.\supset.xTx:\\ +[\text{*91·271}] & \supset:Q\in \text{Potid}ʻR.T\in \text{Pot}ʻR.xQy.yTy.\supset.x\in \text{ᗡ}ʻR:\\ +[\text{*11·11·3·35·54.*91·55.(*91·05)}]&\supset:y\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yR_{\text{po}}y.\supset.x\in \text{ᗡ}ʻR:\\ +[\text{Transp.*93·1}] &\supset:xBR.\supset.{\sim}(y\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yR_{\text{po}}y):\\ +[\text{*96·1.*10·11·21}] &\supset:xBR.\supset.I_{R}ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(2).*96·104}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_644">[Pg 644]</span></p> + +<p class="nind"><b><a id="*96·24">*96·24</a>.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.CʻR=\breve{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\supset.R_{\text{po}}\unicode{x2abd}J\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·105}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:y\in CʻR.\supset.(\exists x).x\in \overrightarrow{B}ʻR.xR_{\unicode{x2217}}y:\\ +[\text{*91·504}] &\supset:yR_{\text{po}}z.\supset.(\exists x).x\in \overrightarrow{B}ʻR.xR_{\unicode{x2217}}y:\\ +[\text{*4·7.*32·18·181}] &\supset:yR_{\text{po}}z.\supset.(\exists x).xBR.y\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yR_{\text{po}}z.\\ +[\text{*96·23}] &\qquad\quad\quad \supset.yJz\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·25">*96·25</a>.</b> + \(\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.xBR.xR_{\unicode{x2217}}y:yR_{\unicode{x2217}}z.\lor.zR_{\unicode{x2217}}y:\supset.xR_{\unicode{x2217}}z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·17}. \supset\vdash:xR_{\unicode{x2217}}y.yR_{\unicode{x2217}}z.\supset.xR_{\unicode{x2217}}z &\qquad \text{(1)}\\ +\vdash.\text{*92·31.*91·75}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:xR_{\unicode{x2217}}y.zR_{\unicode{x2217}}y.\supset:xR_{\unicode{x2217}}z.\lor.zR_{\text{po}}x &\qquad \text{(2)}\\ +\vdash.\text{*91·504.*93·1}.\supset\vdash:xBR.\supset.~(zR_{\text{po}}x) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xR_{\unicode{x2217}}y.zR_{\unicode{x2217}}y.\supset.xR_{\unicode{x2217}}z &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The following propositions lead up to <a href="#*96·32">*96·32</a>, <i>i.e.</i> +\[ +\vdash:R\in 1\rightarrow 1.xR_{\unicode{x2217}}y.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻx\cup \overleftarrow{R}_{\unicode{x2217}}ʻx=\overrightarrow{R}_{\unicode{x2217}}ʻy\cup + \overleftarrow{R}_{\unicode{x2217}}ʻy, +\] +which is a proposition used in the following number (<a href="#*97">*97</a>).</p> + +<p>*96·3·301·302·303 are also frequently used elsewhere.</p> + +<p class="nind"><b>*96·3.</b> \(\vdash:xR_{\unicode{x2217}}y.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻy\subset \overleftarrow{R}_{\unicode{x2217}}ʻx \quad[\text{*90·17}]\)</p> + +<p class="nind"><b>*96·301.</b> \(\vdash:xR_{\unicode{x2217}}y.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻx\subset \overrightarrow{R}_{\unicode{x2217}}ʻy \quad[\text{*90·17}]\)</p> + +<p class="nind"><b>*96·302.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.xR_{\unicode{x2217}}y.xR_{\unicode{x2217}}z.\supset:yR_{\unicode{x2217}}z.\lor.zR_{\unicode{x2217}}y + \quad[\text{*92·311}]\)</p> + +<p class="nind"><b>*96·303.</b> \[\begin{align}&\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.xR_{\unicode{x2217}}y.xR_{\unicode{x2217}}z.y + \neq z.\supset:yR_{\text{po}}z.\lor.zR_{\text{po}}y\\ +&[\text{*96·302.*91·542}]\end{align}\]</p> + +<p class="nind"><b>*95·31.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.xR_{\unicode{x2217}}y.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx\subset \overrightarrow{R}_{\unicode{x2217}}ʻy\cup + \overleftarrow{R}_{\unicode{x2217}}ʻy \quad[\text{*96·302}]\)</p> + +<p class="nind"><b>*96·311.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.xR_{\unicode{x2217}}y.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻy\subset \overrightarrow{R}_{\unicode{x2217}}ʻx\cup + \overleftarrow{R}_{\unicode{x2217}}ʻx \quad[\text{*92·31}]\)</p> + +<p class="nind"><b><a id="*96·32">*96·32</a>.</b> \(\vdash:R\in 1\rightarrow 1.xR_{\unicode{x2217}}y.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻx\cup + \overleftarrow{R}_{\unicode{x2217}}ʻx=\overrightarrow{R}_{\unicode{x2217}}ʻy\cup \overleftarrow{R}_{\unicode{x2217}}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·301·31}.&\supset\vdash:R\in \text{Cls}\rightarrow 1.xR_{\unicode{x2217}}y.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻx\cup \overleftarrow{R}_{\unicode{x2217}}ʻx\subset + \overrightarrow{R}_{\unicode{x2217}}ʻy\cup \overleftarrow{R}_{\unicode{x2217}}ʻy &\qquad \text{(1)}\\ +\vdash.\text{*96·3·311}. &\supset\vdash:R\in 1\rightarrow \text{Cls}.xR_{\unicode{x2217}}y.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻy\cup \overleftarrow{R}_{\unicode{x2217}}ʻy\subset + \overrightarrow{R}_{\unicode{x2217}}ʻx\cup \overleftarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_645">[Pg 645]</span></p> + +<p class="nind"><b><a id="*96·33">*96·33</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.xRx.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx=\iotaʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·171}.\supset\vdash\colon\ldotp \text{Hp}.\supset:z=x.zRw.\supset_{z,w}.w=x &\qquad \text{(1)}\\ +\vdash.\text{(1).*13·15.*90·112}\, \frac{z=x}{\phi z} .\supset\vdash:xR_{\unicode{x2217}}y.\supset.y=x &\qquad \text{(2)}\\ +\vdash.\text{*90·12}.\supset\vdash:\text{Hp}.\supset.xR_{\unicode{x2217}}x &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xR_{\unicode{x2217}}y.\equiv.y=x\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·331">*96·331</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.xRy.yRx.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx=\iotaʻx\cup \iotaʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·151·162}. & \supset\vdash:\text{Hp}.\supset.\iotaʻx\cup \iotaʻy\subset \overleftarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*71·171}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:z=x.zRw.\supset_{z,w}.w=y.\\ +[\text{*51·232}] & \supset_{z,w}.w\in \iotaʻx\cup \iotaʻy &\qquad \text{(2)}\\ +\vdash.\text{*71·171}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:z=y.zRw.\supset_{z,w}.w=x.\\ +[\text{*51·232}] &\supset_{z,w}.w\in \iotaʻx\cup \iotaʻy &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:z\in \iotaʻx\cup \iotaʻy.zRw.\supset_{z,w}.w\in \iotaʻx\cup \iotaʻy &\qquad \text{(4)}\\ +\vdash.\text{*51·16}. &\supset\vdash.x\in \iotaʻx\cup \iotaʻy &\qquad \text{(5)}\\ +\vdash.\text{(4).(5).*90·112}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:xR_{\unicode{x2217}}z.\supset.z\in \iotaʻx\cup \iotaʻy &\qquad \text{(6)}\\ +\vdash.\text{(1).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>This process of proof can obviously be extended to any finite cycle of +terms.</p> + +<p class="nind"><b>*96·34.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}_{\text{po}}ʻʻ\hat{z}(zR_{\text{po}}z)\subset \hat{z}(zR_{\text{po}}z)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11.*34·1}.&\supset\vdash:zR_{\text{po}}z.zRw.\supset.w\breve{R}\mid R_{\text{po}}\mid Rw &\qquad \text{(1)}\\ +\vdash.\text{(1).*92·113}. \supset\vdash\colon\ldotp \text{Hp}.&\supset:zR_{\text{po}}z.zRw.\supset.wR_{\text{po}}w:\\ +[\text{*20·3}] &\supset:z\in \hat{z}(zR_{\text{po}}z).zRw.\supset.w\in \hat{z}(zR_{\text{po}}z):\\ +[\text{*37·171}] & \supset:\breve{R}ʻʻ\hat{z}(zR_{\text{po}}z)\subset \hat{z}(zR_{\text{po}}z):\\ +[\text{*91·71·53}] &\supset:\breve{R}_{\text{po}}ʻʻ\hat{z}(zR_{\text{po}}z)\subset \hat{z}(zR_{\text{po}}z)\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·341">*96·341</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}_{\text{po}}ʻʻI_{R}ʻx\subset I_{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·21.(*96·01)}.\supset\vdash.\breve{R}_{\text{po}}ʻʻI_{R}ʻx&\subset \breve{R}_{\text{po}}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \breve{R}_{\text{po}}ʻʻ\hat{z}(zR_{\text{po}}z)\\ +[\text{*90·163.*91·602}] &\subset \overleftarrow{R}_{\unicode{x2217}}ʻx\cap \breve{R}_{\text{po}}ʻʻ\hat{z}(zR_{\text{po}}z) &\qquad \text{(1)}\\ +\vdash.\text{(1).*96·34}. \supset\vdash:\text{Hp}.\supset.\breve{R}_{\text{po}}ʻʻI_{R}ʻx&\subset \overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \hat{z}(zR_{\text{po}}z)\\ +[\text{(*96·01)}] & \subset I_{R}ʻx:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_646">[Pg 646]</span></p> + +<p class="nind"><b>*96·342.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.\breve{R}_{\unicode{x2217}}ʻʻI_{R}ʻx\subset I_{R}ʻx \quad[\text{*96·341.*91·71}]\)</p> + +<p class="nind"><b>*96·35.</b> \[\begin{align}&\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:{\sim}(wR_{\text{po}}w).zR_{\text{po}}w.\supset.{\sim}(zR_{\text{po}}z)\\ +&[\text{*96·34.Transp}]\end{align}\]</p> + +<p class="nind"><b><a id="*96·351">*96·351</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.R_{\text{po}}ʻʻJ_{R}ʻx\cap \overleftarrow{R}_{\unicode{x2217}}ʻx\subset J_{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·35.Fact.*96·101}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:w\in J_{R}ʻx.zR_{\text{po}}w.z\in \overleftarrow{R}_{\unicode{x2217}}ʻx.\supset.z\in J_{R}ʻx\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·352.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.R_{\unicode{x2217}}ʻʻJ_{R}ʻx\cap \overleftarrow{R}_{\unicode{x2217}}ʻx\subset J_{R}ʻx + \quad[\text{*91·543.*96·351}]\)</p> + +<p>The following propositions are lemmas for *96·45·47.</p> + +<p class="nind"><b><a id="*96·4">*96·4</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.S,\,T\in \text{Pot}ʻR.ySy.yTz.\supset.zSz\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11}.\supset\vdash:\text{Hp}.&\supset.z\breve{T}\mid S\mid Tz.\\ +[\text{*92·133}] &\supset.zSz:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·401.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.S,\,T\in \text{Pot}ʻR.ySy.yTz.yRw.zRw.\supset.wSw.wTw\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11}. \supset\vdash:\text{Hp}.&\supset.w\breve{R}z.z\breve{T}y.ySy.yTz.zRw.\\ +[\text{*34·1·2}] &\supset.w\{\text{Cnv}ʻ(T\mid R)\mid S\mid (T\mid R)\}w &\qquad \text{(1)}\\ +\vdash.\text{*91·282}. & \supset\vdash:\text{Hp}.\supset.T\mid R\in \text{Pot}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*92·133}.&\supset\vdash:\text{Hp}.\supset.wSw &\qquad \text{(3)}\\ +\vdash.\text{*31·11}. \supset\vdash:\text{Hp}.&\supset.w\breve{R}y.yTz.zRw.\\ +[\text{*34·1}] &\supset.w\breve{R}\mid T\mid Rw.\\ +[\text{*91·351.*92·133}] &\supset.wTw &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·402.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.T\in \text{Pot}ʻR.yRy.yTz.yRw.zRw.\supset.y=w.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·171}. & \supset\vdash:\text{Hp}.\supset.y=w &\qquad \text{(1)}\\ +\vdash.\text{*96·4.*91·351}.\supset\vdash:\text{Hp}.&\supset.zRz.\\ +[\text{*71·171}] &\supset.z=w &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·403.</b> \[\begin{align}\vdash:R\in \text{Cls}\rightarrow 1.S,\,T\in \text{Pot}ʻR.yS\mid Ry.yTz.yRw.&zRw.\supset.\\ +&wSy.wSz.y=z\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11}. \supset\vdash:\text{Hp}.&\supset.w\breve{R}\mid S\mid Ry.\\ +[\text{*92·133}] &\supset.wSy &\qquad \text{(1)}\\ +\vdash.\text{*96·4.*91·343}. \supset\vdash:\text{Hp}.&\supset.zS\mid Rz.\\ +[\text{*31·11}] & \supset.w\breve{R}\mid S\mid Rz.\\ +[\text{*92·133}] &\supset.wSz &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*92·101.*71·171}.&\supset\vdash:\text{Hp}.\supset.y=z &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_647">[Pg 647]</span></p> + +<p class="nind"><b>*96·41.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.S,\,T\in \text{Pot}ʻR.ySy.yTz.yRw.zRw.\supset.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·264·304}.&\supset\vdash.\text{Pot}ʻR=\iotaʻR\cup \mid Rʻʻ\text{Pot}ʻR.\\ +[\text{*51·236}] &\supset\vdash\colon\ldotp S\in \text{Pot}ʻR.\equiv:S=R.\lor.(\exists S').S'\in \text{Pot}ʻR.S=S'\mid R &\qquad \text{(1)}\\ +\vdash.\text{*96·402}.\supset\\ +\vdash\colon\ldotp S=R.&\supset:R\in \text{Cls}\rightarrow 1.T\in \text{Pot}ʻR.ySy.yTz.yRw.zRw.\supset.y=z &\qquad \text{(2)}\\ +\vdash.\text{*96·403}.\supset\\ +\vdash\colon\ldotp (\exists S').S'\in &\text{Pot}ʻR.S=S'\mid R.\supset:\\ +&R\in \text{Cls}\rightarrow 1.T\in \text{Pot}ʻR.ySy.yTz.yRw.zRw.\supset.y=z &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash\colon\ldotp S\in \text{Pot}ʻR.\supset:\\ +&R\in \text{Cls}\rightarrow 1.T\in \text{Pot}ʻR.ySy.yTz.yRw.zRw.\supset.y=z\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·42">*96·42</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.yRw.zRw.yR_{\text{po}}z.\supset.zR_{\text{po}}z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*31·11}.\supset\vdash:\text{Hp}.&\supset.w\breve{R}y.yR_{\text{po}}z.\\ +[\text{*92·11·1}] &\supset.wR_{\unicode{x2217}}z.\\ +[\text{Hp.*34·1}] &\supset.zR\mid R_{\unicode{x2217}}z.\\ +[\text{*91·52}] &\supset.zR_{\text{po}}z:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·421.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.y,\,z\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yRw.zRw.y + \neq z.\supset:yR_{\text{po}}y.\lor.zR_{\text{po}}z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·303}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:yR_{\text{po}}z.\lor.zR_{\text{po}}y &\qquad \text{(1)}\\ +\vdash.\text{*96·42}. &\supset\vdash:\text{Hp}.yR_{\text{po}}z.\supset.zR_{\text{po}}z &\qquad \text{(2)}\\ +\vdash.\text{*96·42}. &\supset\vdash:\text{Hp}.zR_{\text{po}}y.\supset.yR_{\text{po}}y &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·431">*96·431</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.y\in J_{R}ʻx.z\in I_{R}ʻx.\supset.yR_{\text{po}}z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·102}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:y \neq z:\\ +[\text{*96·303}] &\supset:yR_{\text{po}}z.\lor.zR_{\text{po}}y &\qquad \text{(1)}\\ +\vdash.\text{*96·341}. \supset\vdash\colon\ldotp \text{Hp}.&\supset:zR_{\text{po}}y.\supset.y\in I_{R}ʻx:\\ +[\text{Transp.*96·102}] &\supset:y\in J_{R}ʻx.\supset.{\sim}(zR_{\text{po}}y) &\qquad \text{(2)}\\ +\vdash.\text{(2)}. &\supset\vdash:\text{Hp}.\supset.{\sim}(zR_{\text{po}}y) &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·432.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.y,z\in I_{R}ʻx.yRw.zRw.\supset.y=z\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·1}. &\supset\vdash:\text{Hp}.\supset.(\exists S,T).S,\,T\in \text{Pot}ʻR.ySy.zTz &\qquad \text{(1)}\\ +\vdash.\text{*96·303}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:y=z:\lor:(\exists U):U\in \text{Pot}ʻR:yUz.\lor.zUy &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\\ +&\qquad y=z:\lor:(\exists S,T,U):S,\,T,\,U\in \text{Pot}ʻR.ySy.zTz:yUz.\lor.zUy &\qquad \text{(3)}\\ +\vdash.\text{*96·41}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists S,U).S,\,U\in \text{Pot}ʻR.ySy.yUz.\supset.y=z &\qquad \text{(4)}\\ +\vdash.\text{*96·41}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists T,U).T,\,U\in \text{Pot}ʻR.zTz.zUy.\supset.y=z &\qquad \text{(5)}\\ +\vdash.\text{(4).(5)}.&\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp \\ +&\qquad (\exists S,T,U):S,\,T,\,U\in \text{Pot}ʻR.ySy.zTz:yUz.\lor.zUy:\supset.y=z &\qquad \text{(6)}\\ +\vdash.\text{(3).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_648">[Pg 648]</span></p> + +<p class="nind"><b>*96·44.</b> \[\begin{align}&\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.y,z\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yRw.zRw.y \neq z.\supset:y\in I_{R}ʻx.\lor.z\in + I_{R}ʻx\\ +&[\text{*96·421·1}]\end{align}\]</p> + +<p class="nind"><b><a id="*96·441">*96·441</a>.</b> \[\begin{align}\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.y,z\in &\overleftarrow{R}_{\unicode{x2217}}ʻx.yRw.zRw.y \neq z.\supset:\\ +&w\in I_{R}ʻx:y\in J_{R}ʻx.z\in I_{R}ʻx.\lor.y\in I_{R}ʻx.z\in J_{R}ʻx\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·432.Transp.(*96·02)}.\supset\\ +\vdash\colon\ldotp \text{Hp}.\supset:z\in I_{R}ʻx.&\supset.y\in J_{R}ʻx:y\in I_{R}ʻx.\supset.z\in J_{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*96·44}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in J_{R}ʻx.z\in I_{R}ʻx.\lor.y\in I_{R}ʻx.z\in J_{R}ʻx &\qquad \text{(2)}\\ +\vdash.\text{*91·502.*96·341}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:z\in I_{R}ʻx.\supset.w\in I_{R}ʻx:y\in I_{R}ʻx.\supset.w\in I_{R}ʻx:\\ +[\text{*96·44}] & \supset:w\in I_{R}ʻx &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·442.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.y,\,z\in J_{R}ʻx.yRw.zRw.\supset.y=z \quad[\text{*96·44.Transp}]\)</p> + +<p>The following proposition (*96·45) is important.</p> + +<p class="nind"><b><a id="*96·45">*96·45</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\supset.(J_{R}ʻx)\upharpoonleft R,(I_{R}ʻx)\upharpoonleft R\in 1\rightarrow 1 \quad[\text{*96·442·432}]\)</p> + +<p class="nind"><b>*96·451.</b> \[\begin{align}&\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1:J_{R}ʻx=\Lambda.\lor.I_{R}ʻx=\Lambda:\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R\in 1\rightarrow 1\\ +&[\text{*96·45·102}]\end{align}\]</p> + +<p class="nind"><b>*96·452.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:\exists !J_{R}ʻx.\equiv.x\in J_{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*10·24}. &\supset\vdash:x\in J_{R}ʻx.\supset.\exists !J_{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*96·342}. \supset\vdash:\text{Hp}.x\in I_{R}ʻx.&\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx\subset I_{R}ʻx.\\ +[\text{*96·102}] & \supset.J_{R}ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{*96·101}. \supset\vdash:\exists !J_{R}ʻx.&\supset.\exists !\overleftarrow{R}_{\unicode{x2217}}ʻx.\\ +[\text{*90·13}] & \supset.xR_{\unicode{x2217}}x &\qquad \text{(3)}\\ +\vdash.\text{(3).(2).Transp}.\supset\vdash:\text{Hp}.\exists !J_{R}ʻx.&\supset.x\in \overleftarrow{R}_{\unicode{x2217}}ʻx-I_{R}ʻx.\\ +[\text{(*96·02)}] &\supset.x\in J_{R}ʻx &\qquad \text{(4)}\\ +\vdash.\text{(1).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·453">*96·453</a>.</b> \(\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1:xR_{\text{po}}x.\lor.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R_{\text{po}}\unicode{x2abd}J:\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·452.Transp}.&\supset\vdash:R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.\supset.J_{R}ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*96·104}.&\supset\vdash:R\in \text{Cls}\rightarrow 1.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R_{\text{po}}\unicode{x2abd}J.\supset.I_{R}ʻx=\Lambda + &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*96·451}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·46.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.y,y'\in J_{R}ʻx.\breve{R}ʻy,\breve{R}ʻy'\in I_{R}ʻx.\supset.y=y'\)</p> + +<p><span class="pagenum" id="Page_649">[Pg 649]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·111}.\supset\\ +\vdash:R\in \text{Cls}\rightarrow 1.y\in J_{R}ʻx.\breve{R}ʻy\in I_{R}ʻx.yR_{\text{po}}y'.&\supset.\breve{R}ʻy\in + I_{R}ʻx.\breve{R}ʻyR_{\unicode{x2217}}y'.\\ +[\text{*96·342}] & \supset.y'\in I_{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp}.&\supset\vdash:R\in \text{Cls}\rightarrow 1.y,\,y'\in J_{R}ʻx.\breve{R}ʻy\in I_{R}ʻx.\supset.{\sim}(yR_{\text{po}}y') + &\qquad \text{(2)}\\ +\vdash.\text{(2)}\, \frac{y',y}{y,y'} .\supset\\ +&\vdash:R\in \text{Cls}\rightarrow 1.y,\,y'\in J_{R}ʻx.\breve{R}ʻy'\in I_{R}ʻx.\supset.{\sim}(y'R_{\text{po}}y) &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.&\supset\vdash:\text{Hp}.\supset.{\sim}(yR_{\text{po}}y').{\sim}(y'R_{\text{po}}y).\\ +[\text{*96·303.Transp}]&\supset.y=y':\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·461.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.y\in J_{R}ʻx.\breve{R}ʻy\in I_{R}ʻx.\supset.y=\text{max}_{R}ʻJ_{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*14·21}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:\text{E}!\breve{R}ʻy:\\ +[\text{*30·13}] \supset:\breve{R}ʻy{\sim}\in J_{R}ʻx.&\equiv.{\sim}(\breve{R}ʻy\in J_{R}ʻx).\\ +[\text{*71·371.Transp}] & \equiv.y{\sim}\in RʻʻJ_{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*93·115.*96·102}.&\supset\vdash:\text{Hp}.\supset.y \text{max}_{R}(J_{R}ʻx) &\qquad \text{(2)}\\ +\vdash.\text{*96·431}.\supset\vdash\colon\ldotp \text{Hp}.y'\in J_{R}ʻx.&\supset:y'R_{\text{po}}\breve{R}ʻy:\\ +[\text{*91·504}] & \supset:y'\in \text{D}ʻR:\\ +[\text{*71·164}] & \supset:\text{E}!\breve{R}ʻy':\\ +[\text{*30·13}] \supset:\breve{R}ʻy'{\sim}\in J_{R}ʻx.&\equiv.{\sim}(\breve{R}ʻy'\in J_{R}ʻx).\\ +[\text{*71·371.Transp}] & \equiv.y'{\sim}\in RʻʻJ_{R}ʻx &\qquad \text{(3)}\\ +\vdash.\text{(3).*93·115}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:y'\text{max}_{R}(J_{R}ʻx).\supset.y'\in J_{R}ʻx.\breve{R}ʻy'{\sim}\in J_{R}ʻx.\\ +[\text{*96·102}] & \supset.y'\in J_{R}ʻx.\breve{R}ʻy'\in I_{R}ʻx.\\ +[\text{*96·46}] & \supset.y=y' &\qquad \text{(4)}\\ +\vdash.\text{(2).(4).*30·31}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·462">*96·462</a>.</b> \[\begin{align}\vdash:R &\in \text{Cls}\rightarrow 1.y\in J_{R}ʻx.z\in I_{R}ʻx.yRw.zRw.\supset.\\ +& y=\text{max}_{R}ʻJ_{R}ʻx.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.z=\{(I_{R}ʻx)\upharpoonleft + R\}ʻ\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·441·102.*71·361}.\supset\\ +\vdash:\text{Hp}.&\supset.w\in I_{R}ʻx.w=\breve{R}ʻy.\\ +[\text{*96·461}] &\supset.y=\text{max}_{R}ʻJ_{R}ʻx.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*96·45}.\supset\vdash:\text{Hp}.&\supset.z=\{(I_{R}ʻx)\upharpoonleft R\}ʻw &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_650">[Pg 650]</span></p> + +<p>The above proposition, since it exhibits \(y\), \(z\), \(w\) as +functions of \(x\) and \(R\), shows that there is at most one \(w\) +in \(\overleftarrow{R}_{\unicode{x2217}}ʻx\) having more than one +immediate predecessor, and that this one has exactly one immediate +predecessor in \(J_{R}ʻx\) and one in \(I_{R}ʻx\). (These results +require <a href="#*96·441">*96·441</a>, in addition to <a href="#*96·462">*96·462</a>.) Thus we arrive at the +following proposition:</p> + +<p class="nind"><b><a id="*96·47">*96·47</a>.</b> \[\begin{align}\vdash\colon\ldotp R\in \text{Cls}\rightarrow &1.y,z\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yRw.zRw.y + \neq z.\supset:w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx:\\ +&y = \text{max}_{R}ʻJ_{R}ʻx.z=\{(I_{R}ʻx)\upharpoonleft R\}ʻ\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\lor.\\ +&z = \text{max}_{R}ʻJ_{R}ʻx.y=\{(I_{R}ʻx)\upharpoonleft R\}ʻ\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx\\ +\quad[\text{*96·441·462}]\end{align}\]</p> + +<p>We still have to prove +\[ +R\in \text{Cls}\rightarrow 1.\exists !J_{R}ʻx.\exists !I_{R}ʻx.\supset.(\exists y,z,w).y,z\in \overleftarrow{R}_{\unicode{x2217}}ʻx.yRw.zRw.y \neq z, +\] +or, what comes to the same thing because of <a href="#*96·441">*96·441</a>, +\[ +R\in \text{Cls}\rightarrow 1.\exists !J_{R}ʻx.\exists !I_{R}ʻx.\supset.(\exists y,z,w).y\in J_{R}ʻx.z\in I_{R}ʻx.yRw.zRw. +\] +This is effected in the following propositions.</p> + +<p class="nind"><b>*96·472.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.\exists !J_{R}ʻx.\exists !I_{R}ʻx.\supset.(\exists y).y\in J_{R}ʻx.\breve{R}ʻy\in I_{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·1}.\supset\vdash\colon\ldotp x\in J_{R}ʻx.\breve{R}ʻʻJ_{R}ʻx\subset J_{R}ʻx.&\supset:xR_{\unicode{x2217}}y.\supset.y\in J_{R}ʻx:\\ +[\text{*96·104}] &\supset:I_{R}ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp.*96·452}.\supset\vdash:\text{Hp}.&\supset.\exists !\breve{R}ʻʻJ_{R}ʻx-J_{R}ʻx.\\ +[\text{*71·401}] &\supset.(\exists y,z).y\in J_{R}ʻx.z=\breve{R}ʻy.z{\sim}\in J_{R}ʻx.\\ +[\text{*13·195}] &\supset.(\exists y).y\in J_{R}ʻx.\breve{R}ʻy{\sim}\in J_{R}ʻx.\\ +[\text{*96·102}] &\supset.(\exists y).y\in J_{R}ʻx.\breve{R}ʻy\in I_{R}ʻx:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·473.</b> \[\begin{align}&\vdash:R\in \text{Cls}\rightarrow 1.\exists !J_{R}ʻx.\exists + !I_{R}ʻx.\supset.\text{E}!\text{max}_{R}ʻJ_{R}ʻx.\text{E}!\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx\\ +&[\text{*96·461·472}]\end{align}\]</p> + +<p class="nind"><b>*96·474.</b> \[\begin{align}\vdash:&R\in \text{Cls}\rightarrow 1.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\supset.\\ +&\text{E}!\{(I_{R}ʻx)\upharpoonleft R\}ʻw.\text{E}!\text{max}_{R}ʻJ_{R}ʻx.\{(J_{R}ʻx)\upharpoonleft R\}ʻw=\text{max}_{R}ʻJ_{R}ʻx\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·361}. \supset\vdash:\text{Hp}.&\supset.(\text{max}_{R}ʻJ_{R}ʻx)Rw. &\qquad \text{(1)}\\ +[\text{*14·21}] &\supset.\text{E}!\text{max}_{R}ʻJ_{R}ʻx. &\qquad \text{(2)}\\ +[\text{*93·11}] &\supset.\text{max}_{R}ʻJ_{R}ʻx\in J_{R}ʻx.\\ +[\text{(1).*96·45}] &\supset.\{(J_{R}ʻx)\upharpoonleft R\}ʻw=\text{max}_{R}ʻJ_{R}ʻx &\qquad \text{(3)}\\ +\vdash.\text{(2).*93·11}.&\supset\vdash:\text{Hp}.\supset.\text{max}_{R}ʻJ_{R}ʻx{\sim}\in RʻʻJ_{R}ʻx.\\ +[\text{*71·371.*30·13}] &\supset.\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx{\sim}\in J_{R}ʻx.\\ +[\text{Hp.*96·102}] &\supset.w\in I_{R}ʻx.\\ +[\text{*96·1.*91·52}] &\supset.wR_{po}w.wR_{\unicode{x2217}}\mid Rw.\\ +[\text{*34·1}] &\supset.(\exists z).wR_{po}w.wR_{\unicode{x2217}}z.zRw.\\ +[\text{*96·342}] &\supset.(\exists z).z\in I_{R}ʻx.zRw.\\ +[\text{*96·45}] &\supset.\text{E}!\{(I_{R}ʻx)\upharpoonleft R\}ʻw &\qquad \text{(4)}\\ +\vdash.\text{(2).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_651">[Pg 651]</span></p> + +<p class="nind"><b><a id="*96·475">*96·475</a>.</b> \[\begin{align}&\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.\supset:\text{E}!\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\equiv.\exists !J_{R}ʻx.\exists + !I_{R}ʻx\\ +&[\text{*96·473·474}]\end{align}\]</p> + +<p>This proposition and *96·45·47 embody the main results of this number.</p> + +<p class="nind"><b>*96·48.</b> \[\begin{align}\vdash\colon\ldotp R &\in \text{Cls}\rightarrow 1.S=(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R.w\in \overleftarrow{R}_{\text{po}}ʻx.\supset:\\ +&{\sim}(w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx).\equiv.\overrightarrow{S}ʻw\in 1:w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\equiv.\overrightarrow{S}ʻw\in 2\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·15.*33·41}.&\supset\vdash:\text{Hp}.\supset.\exists !\overrightarrow{S}ʻw &\qquad \text{(1)}\\ +\vdash.\text{*96·47}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:(\exists y,z).ySw.zSw.y \neq z.\supset.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx:\\ +[\text{(1).*52·41}] &\supset:\overrightarrow{S}ʻw{\sim}\in 1.\supset.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx &\qquad \text{(2)}\\ +\vdash.\text{*96·474·102}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\supset.\overrightarrow{S}ʻw{\sim}\in 1 &\qquad \text{(3)}\\ +\vdash.\text{(2).(3).Transp}.& \supset\vdash\colon\ldotp \text{Hp}.\supset:{\sim}(w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx).\equiv.\overrightarrow{S}ʻw\in 1 &\qquad \text{(4)}\\ +\vdash.\text{(2).*52·4.*54·101}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:\overrightarrow{S}ʻw\in 2.\supset.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx &\qquad \text{(5)}\\ +\vdash.\text{*96·474·102}.& \supset\vdash\colon\ldotp \text{Hp}.\supset:w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\supset.\\ +&\text{E}!\{(J_{R}ʻx)\upharpoonleft R\}ʻw.\text{E}!\{(I_{R}ʻx)\upharpoonleft R\}ʻw.\iotaʻ\{(J_{R}ʻx)\upharpoonleft R\}ʻw\cup \iotaʻ\{(I_{R}ʻx)\upharpoonleft + R\}ʻw=\overrightarrow{S}ʻw.\\ +[\text{*96·102.*54·101}]&\supset.\overrightarrow{S}ʻw\in 2 &\qquad \text{(6)}\\ +\vdash.\text{(5).(6)}.&\supset\colon\ldotp \text{Hp}.\supset:w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\equiv.\overrightarrow{S}ʻw\in 2 &\qquad \text{(7)}\\ +\vdash.\text{(4).(7)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>In the above proposition we write +"\({\sim}(w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx)\)" rather than +"\(w \neq \breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx\)," because the latter implies the +existence of \(\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx\).</p> + +<p class="nind"><b>*96·49.</b> \[\begin{align}\vdash\colon\colon R\in \text{Cls}\rightarrow 1.x\in \text{D}ʻ&R.\supset\colon\ldotp \\ +&(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R\in 1\rightarrow 1.\equiv:I_{R}ʻx=\Lambda.\lor.J_{R}ʻx=\Lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·48.Transp}.&\supset\vdash\colon\ldotp \text{Hp}.S=(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft R.\supset:\\ +& w\in \overleftarrow{R}_{\text{po}}ʻx.w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.\equiv.w\in \overleftarrow{R}_{\text{po}}ʻx.\overrightarrow{S}ʻw{\sim}\in 1:\\ +[\text{*96·15.*91·52}]\supset:w=\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.&\equiv.w\in \text{ᗡ}ʻS.\overrightarrow{S}ʻw{\sim}\in 1:\\ +[\text{*14·204}] \supset:\text{E}!\breve{R}ʻ\text{max}_{R}ʻJ_{R}ʻx.&\equiv.(\exists w).w\in \text{ᗡ}ʻS.\overrightarrow{S}ʻw{\sim}\in 1:\\ +[\text{*96·475.*71·1}]\supset:\exists !J_{R}ʻx.\exists !I_{R}ʻx.&\equiv.S{\sim}\in 1\rightarrow \text{Cls}.\\ +[\text{*71·261·103}] & \equiv.S{\sim}\in 1\rightarrow 1 &\qquad \text{(1)}\\ +\vdash.\text{(1).Transp}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·491">*96·491</a>.</b> \(\vdash\colon\ldotp R\in 1\rightarrow 1.\supset:I_{R}ʻx=\Lambda.\lor.J_{R}ʻx=\Lambda\)</p> + +<p><i>Dem.</i> +\[ +\begin{array}{l} +\vdash.\text{*96·49}. \supset\vdash\colon\ldotp \text{Hp}.x\in \text{D}ʻR.&\supset:I_{R}ʻx=\Lambda.\lor.J_{R}ʻx=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*91·54·504}.&\supset\vdash:\text{Hp}.x{\sim}\in \text{D}ʻR.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx=\iotaʻx\cap CʻR.{\sim}(xR_{\text{po}}x).\\ +[\text{*96·1}] &\supset.I_{R}ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_652">[Pg 652]</span></p> + +<p class="nind"><b>*96·492.</b> \[\begin{align}\vdash\colon\ldotp R\in 1\rightarrow 1.x\in \text{D}ʻR.&\supset:\\ +&{\sim}(xR_{\text{po}}x).\equiv.I_{R}ʻx=\Lambda:xR_{\text{po}}x.\equiv.J_{R}ʻx=\Lambda\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·1·101}.\supset\\ +\vdash:I_{R}ʻx=\Lambda.&\supset.{\sim}(xR_{\text{po}}x):x\in \text{D}ʻR.{\sim}(xR_{\text{po}}x).\supset.\exists !J_{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).*96·491}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:{\sim}(xR_{\text{po}}x).\equiv.I_{R}ʻx=\Lambda &\qquad \text{(2)}\\ +\text{Similarly} & \vdash\colon\ldotp \text{Hp}.\supset:xR_{\text{po}}x.\equiv.J_{R}ʻx=\Lambda &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The above proposition is used in *122·52.</p> + +<p>The following propositions, not being needed in the sequel, are merely +stated:</p> + +<p>\[ +\begin{array}{l} +\vdash:R\in \text{Cls}\rightarrow 1.\exists !J_{R}ʻx.\exists !I_{R}ʻx.\supset.I_{R}ʻx\cap \breve{R}ʻʻJ_{R}ʻx\in 1.J_{R}ʻx\cap RʻʻI_{R}ʻx\in 1\\ +\vdash:R\in \text{Cls}\rightarrow 1.\supset.(\exists S).S\in \text{Pot}ʻR.(I_{R}ʻx)\upharpoonleft R_{\text{po}}\unicode{x2abd}S\\ +\vdash:R\in \text{Cls}\rightarrow 1.J_{R}ʻx=\Lambda.\supset.(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft I\in \text{Pot}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}ʻx)\upharpoonleft + R\} +\end{array} +\]</p> + +<p class="nind"><b>*96·5.</b> \(\vdash:R\in 1\rightarrow 1.x\in \text{D}ʻR.\supset.\overrightarrow{R}_{\text{po}}ʻ\breve{R}ʻx=\overrightarrow{R}_{\unicode{x2217}}ʻx=\overrightarrow{R}_{\text{po}}ʻx\cup + \iotaʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*71·7}.\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \overrightarrow{R}_{\text{po}}ʻ\breve{R}ʻx.&\equiv.yR_{\text{po}}\mid \breve{R}x.\\ +[\text{*92·11}] & \equiv.yR_{\unicode{x2217}}x.x\in \text{D}ʻR.\\ +[\text{Hp.*4·71}]&\equiv.yR_{\unicode{x2217}}x:\\ +\left[\text{*32·18.*96·14}\, \frac{\breve{R}}{R}\right] \supset:\overrightarrow{R}_{\text{po}}ʻ\breve{R}ʻx=\overrightarrow{R}_{\unicode{x2217}}ʻx=\overrightarrow{R}_{\text{po}}ʻx\cup + \iotaʻx\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*96·501.</b> \(\vdash:R\in 1\rightarrow 1.x\in \text{ᗡ}ʻR.\supset.\overleftarrow{R}_{\text{po}}ʻRʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\text{po}}ʻx\cup + \iotaʻx\)</p> + +<p class="nind"><b>*96·502.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.xRy.\supset.\overrightarrow{R}_{\unicode{x2217}}ʻy=\overrightarrow{R}_{\unicode{x2217}}ʻx\cup \iotaʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{aligned} +\vdash.*90·31.\supset\vdash\colon\colon \text{Hp}.\supset\colon\ldotp zR_{\unicode{x2217}}y.\equiv:zR_{\unicode{x2217}}(Rʻy).\lor.z=y\colon\colon \supset\vdash.\text{Prop} +\end{aligned} +\]</p> + +<p class="nind"><b>*96·51.</b> \(\vdash:R\in 1\rightarrow 1.\alpha\subset \breve{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\alpha\subset \breve{R}_{\text{po}}ʻʻ\alpha.\supset.\alpha=\Lambda\)</p> + +<p><span class="pagenum" id="Page_653">[Pg 653]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·105}.\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \alpha.&\supset_{y}.(\exists x).x\in \alpha.xR_{\text{po}}y.\\ +[\text{*32·18}] & \supset_{y}.\exists !\alpha\cap \overrightarrow{R}_{\text{po}}ʻy:\\ +[\text{*14·18·21}] \supset:\breve{R}ʻx\in \alpha.&\supset.\exists !\alpha\cap \overrightarrow{R}_{\text{po}}ʻ\breve{R}ʻx.\\ +[\text{*96·5}] & \supset.\exists !\alpha\cap \overrightarrow{R}_{\unicode{x2217}}ʻx:\\ +[\text{Transp}] \supset:\alpha\cap \overrightarrow{R}_{\unicode{x2217}}ʻx=\Lambda.xRy.&\supset.y{\sim}\in \alpha.\\ +[\text{*51·211}] & \supset.\alpha\cap (\overrightarrow{R}_{\unicode{x2217}}ʻx\cup \iotaʻy)=\Lambda.\\ +[\text{*96·502}] & \supset.\alpha\cap \overrightarrow{R}_{\unicode{x2217}}ʻy=\Lambda &\qquad \text{(1)}\\ +\vdash.\text{*91·504}.&\supset\vdash\colon\ldotp \alpha\subset \breve{R}_{\text{po}}ʻʻ\alpha.\supset:\alpha\subset \text{ᗡ}ʻR:\\ +[\text{*93·104}] &\supset:x\in {\sim}\overrightarrow{B}ʻR.\supset.\alpha\cap \overrightarrow{R}_{\unicode{x2217}}ʻx=\Lambda &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*90·112}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:x\in \overrightarrow{B}ʻR.xR_{\unicode{x2217}}y.\supset.\alpha\cap + \overrightarrow{R}_{\unicode{x2217}}ʻy=\Lambda.\\ +[\text{*90·13}] & \supset.y{\sim}\in \alpha:\\ +[\text{*37·105}] &\supset:\breve{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\cap \alpha=\Lambda &\qquad \text{(3)}\\ +\vdash.\text{*22·621}. &\supset\vdash:\text{Hp}.\supset.\alpha=\breve{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\cap \alpha &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*96·52">*96·52</a>.</b> \(\vdash:R\in 1\rightarrow 1.\alpha\subset \breve{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\exists + !\alpha.\supset.\exists !\overrightarrow{\text{min}}(R_{\text{po}})ʻ\alpha\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*96·51.Transp}.&\supset\vdash:\text{Hp}.\supset.\exists !\alpha-\breve{R}_{\text{po}}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*93·111}. &\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>The proposition is used in *122·23.</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<p><span class="pagenum" id="Page_654">[Pg 654]</span></p> +<h2 class="nobreak" id="*97">*97. ANALYSIS OF THE FIELD OF A RELATION INTO FAMILIES.</h2> +</div> + + +<p><i>Summary of</i> *97.</p> + +<p>In this number, we consider not only the posterity +of a term, but the ancestry and posterity together, +<i>i.e.</i> \(\overrightarrow{R}_{\unicode{x2217}}ʻx \cup\overleftarrow{R}_{\unicode{x2217}}ʻx\). +We put +\[ +\overleftrightarrow{R}ʻx = \overrightarrow{R}ʻx \cup (\iotaʻx \cap CʻR) \cup \overleftarrow{R}ʻx \quad \text{Df}\text{.} +\]</p> + +<p>Thus the whole family of a term, <i>i.e.</i> its ancestry and posterity +together, is \(\overleftrightarrow{R}_{\unicode{x2217}}ʻx\). The most +important case here is when \(R \in 1 \rightarrow 1\); in this case +families are mutually exclusive, <i>i.e.</i> we have +\[ +\vdash: R \in 1 \rightarrow 1 .\supset. \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR \in \text{Cls} ~ \text{ex}^{2} ~ \text{excl}\text{.} +\]</p> + +<p>In case \(R \in 1 \rightarrow 1\) and \(y\) belongs to +a family which has a beginning, <i>i.e.</i> in case +\(\exists!\overleftrightarrow{R}_{\unicode{x2217}}ʻy \cap +\overrightarrow{B}ʻR\), the whole family of \(y\) consists of the +posterity of the beginning, <i>i.e.</i> we have +\[ +\vdash: R \in 1 \rightarrow 1 . xBR . x R_{\unicode{x2217}} y .\supset. \overleftrightarrow{R}_{\unicode{x2217}}ʻy = \overleftarrow{R}_{\unicode{x2217}}ʻx\text{,} +\] +whence</p> + +<p class="nind"><b>*97·21.</b> \(\vdash: R \in 1 \rightarrow 1 .\supset. \overleftrightarrow{R}_{\unicode{x2217}}ʻʻsʻ\text{gen}ʻR = \overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\)</p> + +<figure class="figcenter width500" id="i_654" style="width: 894px;"> +<img src="images/i_654.jpg" width="894" height="500" alt="A grid of +dots where an oval enclosed some dots showing field mapping."> +</figure> + + +<p>When \(R \in 1 \rightarrow 1\), the relation of \(\text{gen}ʻR\) +to \(\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\) +may be pictured as the relation of rows to columns. <i>E.g.</i> +let the field of \(R\) consist of the dots in the accompanying +rectangle, and let each dot have the relation \(R\) to the dot +below it. Then the top row is \(\overrightarrow{B}ʻR\), the second +<span class="pagenum" id="Page_655">[Pg 655]</span>row is \(\text{ᗡ}ʻR - \text{ᗡ}ʻR^{2}\), the third is \(\text{ᗡ}ʻR^{2} - \text{ᗡ}ʻR^{3}\), +and so on; thus the rows are the generations of \(R\). Again, +if \(x\) is any dot in the top row, the column beginning +with \(x\) is \(\overleftarrow{R}_{\unicode{x2217}}ʻx\), +and if \(y\) is any member of this column, the column is +\(\overleftrightarrow{R}_{\unicode{x2217}}ʻy\). Thus the columns are +the families of \(R\). It will be seen that in the case represented +by the above figure, every family consists of a selection from the +generations, and every generation consists of a selection from the +families, <i>i.e.</i> +\[ +\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR \subset \text{D}ʻʻ{\in}_{\Delta}ʻ\text{gen}ʻR + . \text{gen}ʻR \subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\text{.} +\]</p> + +<p>The circumstances under which this occurs will be considered in the +present number (<a href="#*97·3">*97·3</a>—<a href="#*97·47">·47</a>). The results are summed up in <a href="#*97·47">*97·47</a>.</p> + +<p>The remaining propositions (<a href="#*97·5">*97·5</a>—<a href="#*97·58">·58</a>) are concerned with +<i>circular</i> families of one-one relations. If \(R \in 1 \rightarrow 1\), +\(\overleftrightarrow{R}_{\unicode{x2217}}ʻx\) is a +circular family if \(x R_{\text{po}} x\). In that case, we have +\(x R_{\text{po}} y .\supset. y R_{\text{po}} x\); moreover there is a +definite power of \(R\), say \(P\), such that every member of the +family of \(x\) has the relation \(P\) to itself (<a href="#*97·54">*97·54</a>). (The same +will hold, of course, of all powers of \(P\).) The families of a +\(1 \rightarrow 1\) are all either circular or open, <i>i.e.</i> we have +(<a href="#*97·55">*97·55</a>) either \(y \in \overleftrightarrow{R}_{\unicode{x2217}}ʻx .\supset_{y}. y R_{\text{po}} y\), +or \(y \in \overleftrightarrow{R}_{\unicode{x2217}}ʻx .\supset_{y}. {\sim}(y R_{\text{po}} y)\). +The \(Q\)-shaped families considered in <a href="#*96">*96</a> are +not possible for a \(1 \rightarrow 1\), since in such families the +term at the junction of the tail and the circle has two predecessors. +The family of any member of \(sʻ\text{gen}ʻR\) must be open (<a href="#*97·57">*97·57</a>). +The family of a member of \(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR\) need not be +closed, but cannot have a beginning; if open, it forms a series of +type \(^{\unicode{x2217}}\omega\) or \({}^{\unicode{x2217}}\omega + \omega\), +according as it has or has not an end<a id="FNanchor_68" href="#Footnote_68" class="fnanchor">[68]</a>. +Finite open families are contained in \(sʻ\text{gen}ʻR \cap +sʻ\text{gen}ʻ\breve{R}\); families of type \(\omega\) are contained +in \(sʻ\text{gen}ʻR \cap pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ\breve{R}\); those of +type \({}^{\unicode{x2217}}\omega\), in \(sʻ\text{gen}ʻ\breve{R} \cap pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR\); +those of type \({}^{\unicode{x2217}}\omega + \omega\) +and circular families are contained in +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR \cap pʻ\text{ᗡ}ʻʻ\text{Pot}ʻ\breve{R}\). +Those of type \(^{\unicode{x2217}}\omega + \omega\) are distinguished +from circular families by the fact that in the former we do not have +\(x R_{\text{po}} x\), while in the latter we do have this.</p> + +<p>In addition to the propositions already mentioned, the most useful +propositions of the present number are the following:</p> + +<p class="nind"><b>*97·13.</b> \(\vdash. \overleftrightarrow{R}_{\unicode{x2217}}ʻx = \overrightarrow{R}_{\unicode{x2217}}ʻx \cup \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p class="nind"><b>*97·17.</b> \(\vdash. \overleftrightarrow{R}_{\unicode{x2217}}ʻx = \overleftrightarrow{R}_{\text{po}}ʻx = \overrightarrow{R}_{\unicode{x2217}}ʻx + \cup \overleftarrow{R}_{\text{po}}ʻx = \overrightarrow{R}_{\text{po}}ʻx \cup \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p class="nind"><b>*97·5.</b> \(\vdash: R \in \text{Cls} \rightarrow 1 . x R_{\text{po}} x . x R_{\text{po}} y .\supset. y R_{po} x\)</p> + +<p class="nind"><b>*97·501.</b> \(\vdash: R \in 1 \rightarrow \text{Cls} . x R_{\text{po}} x . y R_{\text{po}} x .\supset. x R_{\text{po}} y\)</p> + +<p><span class="pagenum" id="Page_656">[Pg 656]</span></p> + +<hr class="tb"> + +<p class="nind"><b>*97·01.</b> \(\overleftrightarrow{R}ʻx=\overrightarrow{R}ʻx\cup(\iotaʻx\cap CʻR)\cup\overleftarrow{R}ʻx \quad\text{Df}\)</p> + +<p>Observe that "\(\iotaʻx\cap CʻR\)" means that \(x\) is to be included +if it is a member of \(CʻR\), but not otherwise; for \(\iotaʻx\cap CʻR=\iotaʻx\) +if \(x\in CʻR\), and otherwise \(\iotaʻx\cap CʻR=\Lambda\).</p> + +<p class="nind"><b>*97·1.</b> \(\vdash\colon\ldotp y\in \overleftrightarrow{R}ʻx.\equiv:yRx.\lor.y=x.x\in CʻR.\lor.xRy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18·181.*51·15.(*97·01)}.\supset\\ +\vdash\colon\ldotp y\in \overleftrightarrow{R}ʻx.&\equiv:yRx.\lor.y=x.y\in CʻR.\lor.xRy:\\ +[\text{*13·193}]&\equiv:yRx.\lor.y=x.x\in CʻR.\lor.xRy\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·101.</b> \(\vdash:y\in \overleftrightarrow{R}ʻx.\equiv.x\in \overleftrightarrow{R}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*32·18·181.*51·15.(*97·01)}.\supset\\ +\vdash\colon\ldotp x\in \overleftrightarrow{R}ʻy.&\equiv:xRy.\lor.x=y.x\in CʻR.\lor.yRx:\\ +[\text{*97·1}]&\equiv:y\in \overleftrightarrow{R}ʻx\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·11.</b> \(\vdash.sʻ\overleftrightarrow{R}ʻʻCʻR=CʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·1.*40·11}.\supset\\ +\vdash\colon\ldotp y\in sʻ\overleftrightarrow{R}ʻʻCʻR.&\equiv:(\exists x).yRx.\lor.(\exists x).y=x.x\in CʻR.\lor.(\exists x).xRy:\\ +[\text{*33·13·131.*13·195}]&\equiv:y\in \text{D}ʻR.\lor.y\in CʻR.\lor.y\in \text{ᗡ}ʻR:\\ +[\text{*33·16}] & \equiv:y\in CʻR\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·111.</b> \(\vdash:x\in CʻR.\equiv.x\in \overleftrightarrow{R}ʻx.\equiv.\exists !\overleftrightarrow{R}ʻx\)</p> + +<p><i>\text{D}em.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·1}.\supset\vdash\colon\ldotp x\in \overleftrightarrow{R}ʻx.&\equiv:xRx.\lor.x\in CʻR:\\ +[\text{*33·17}] &\equiv:x\in CʻR &\qquad \text{(1)}\\ +\vdash.\text{*97·1}.\supset\vdash\colon\ldotp \exists !\overleftrightarrow{R}ʻx.&\equiv:(\exists y):yRx.\lor.xRy:\lor:(\exists y).x\in CʻR.y=x:\\ +[\text{*33·132.*13·19}] & \equiv:x\in CʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·12.</b> \(\vdash.\Lambda{\sim}\in \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·111.*37·63}.\supset\vdash:\alpha\in \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR.\supset_{\alpha}.\exists !\alpha &\qquad \text{(1)}\\ +\vdash.\text{(1).*24·63}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_657">[Pg 657]</span></p> + +<p class="nind"><b>*97·13.</b> \(\vdash.\overleftrightarrow{R}_{\unicode{x2217}}ʻx = \overrightarrow{R}_{\unicode{x2217}}ʻx \cup \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Note.</i> \(\overleftrightarrow{R}_{\unicode{x2217}}\) is to mean (\(\overleftrightarrow{R}_{\unicode{x2217}})\), not (\(\overleftrightarrow{R})_{\unicode{x2217}}\). + The latter is unmeaning, since +\(\overleftrightarrow{R}\) is never a homogeneous relation, and therefore its square and higher +powers are unmeaning.</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash. \text{*90·12} . &\supset\vdash:y = x.y\in CʻR.\supset.yR_{\unicode{x2217}}x:\\ +[\text{*51·15}] &\supset\vdash.\iotaʻx\cap CʻR\subset \overleftarrow{R}_{\unicode{x2217}}ʻx.\\ +[\text{*90·14}] &\supset\vdash.\iotaʻx\cap CʻR_{\unicode{x2217}}\subset \overleftarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(1)}\\ +\vdash.\text{(1).(*97·01)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·14.</b> \(\vdash:R\in 1\rightarrow 1.xR_{\unicode{x2217}}y.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftrightarrow{R}_{\unicode{x2217}}ʻy + \quad[\text{*96·32.*97·13}]\)</p> + +<p class="nind"><b>*97·15.</b> + \(\vdash:R\in 1\rightarrow 1.x\in \overleftrightarrow{R}_{\unicode{x2217}}ʻy.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftrightarrow{R}_{\unicode{x2217}}ʻy\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·13}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xR_{\unicode{x2217}}y.\lor.yR_{\unicode{x2217}}x &\qquad \text{(1)}\\ +\vdash.\text{(1).*97·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·16.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\in \text{Cls}^{2}\,\text{excl}\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·15}.\supset\vdash\colon\ldotp \text{Hp}.\supset:x\in \overleftrightarrow{R}_{\unicode{x2217}}ʻy.x\in + \overleftrightarrow{R}_{\unicode{x2217}}ʻz.&\supset_{x}.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftrightarrow{R}_{\unicode{x2217}}ʻy.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftrightarrow{R}_{\unicode{x2217}}ʻz.\\ +[\text{*13·171}] &\supset_{x}.\overleftrightarrow{R}_{\unicode{x2217}}ʻy=\overleftrightarrow{R}_{\unicode{x2217}}ʻz:\\ +[\text{*10·23}] &\supset:\exists !\overleftrightarrow{R}_{\unicode{x2217}}ʻy\cap + \overleftrightarrow{R}_{\unicode{x2217}}ʻz.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻy=\overleftrightarrow{R}_{\unicode{x2217}}ʻz &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·3.*37·63}.\supset\\ +\vdash\colon\ldotp \text{Hp}.&\supset:\alpha,\beta\in \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR.\exists !\alpha\cap \beta.\supset_{\alpha,\beta}.\alpha=\beta + &\qquad \text{(2)}\\ +\vdash.\text{(2).*97·12.*84·132}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·17.</b> \(\vdash.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftrightarrow{R}_{\text{po}}ʻx=\overrightarrow{R}_{\unicode{x2217}}ʻx\cup + \overleftarrow{R}_{\text{po}}ʻx=\overrightarrow{R}_{\text{po}}ʻx\cup \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·13.*91·54}.\supset\vdash.\overleftrightarrow{R}_{\unicode{x2217}}ʻx&=\overrightarrow{R}_{\text{po}}ʻx\cup (\iotaʻx\cap CʻR)\cup \overleftarrow{R}_{\text{po}}ʻx + &\qquad \text{(1)}\\ +[\text{*91·504.(*97·01)}] &=\overleftrightarrow{R}_{\text{po}}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).*91·54}. &\supset\vdash.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overrightarrow{R}_{\text{po}}ʻx\cup + \overleftarrow{R}_{\unicode{x2217}}ʻx=\overrightarrow{R}_{\unicode{x2217}}ʻx\cup \overleftarrow{R}_{\text{po}}ʻx &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·18.</b> \(\vdash.Cʻ(R\unicode{x0294f}\overleftrightarrow{R}ʻx)=\overleftrightarrow{R}ʻx\)</p> + + +<p><span class="pagenum" id="Page_658">[Pg 658]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·41}.&\supset\vdash.Cʻ(R\unicode{x0294f}\overleftrightarrow{R}ʻx)\subset \overleftrightarrow{R}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*97·1.*36·13}.\supset\\ +\vdash\colon\ldotp x\in CʻR.y\in \overrightarrow{R}ʻx\cup \overleftarrow{R}ʻx.&\supset:x(R\unicode{x0294f}\overleftrightarrow{R}ʻx)y.\lor.y(R\unicode{x0294f}\overleftrightarrow{R}ʻx)x:\\ +[\text{*33·17}] &\supset:x,y\in Cʻ(R\unicode{x0294f}\overleftrightarrow{R}ʻx) &\qquad \text{(2)}\\ +\vdash.\text{(2).*97·1}. & \supset\vdash:x\in CʻR.\supset.\overleftrightarrow{R}ʻx\subset Cʻ(R\unicode{x0294f}\overleftrightarrow{R}ʻx) &\qquad \text{(3)}\\ +\vdash.\text{*97·111.Transp}.&\supset\vdash:x{\sim}\in CʻR.\supset.\overleftrightarrow{R}ʻx\subset Cʻ(R\unicode{x0294f}\overleftrightarrow{R}ʻx) &\qquad \text{(4)}\\ +\vdash.\text{(1).(3).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·2.</b> \(\vdash:xBR.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·104.*97·13}.\supset\vdash:\text{Hp}.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\iotaʻx\cup \overleftarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*93·101.*90·12}.\supset\vdash:\text{Hp}.\supset.x\in \overleftarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·21.</b> \(\vdash:R\in 1\rightarrow 1.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻʻsʻ\text{gen}ʻR=\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·14·2}.\supset\vdash\colon\ldotp \text{Hp}.\supset:xBR.xR_{\unicode{x2217}}y.&\supset.\overleftarrow{R}_{\unicode{x2217}}ʻx=\overleftrightarrow{R}_{\unicode{x2217}}ʻy.\\ +[\text{*37·62}] & \supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻy\in \overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR:\\ +[\text{*93·36}] &\supset:y\in sʻ\text{gen}ʻR.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻy\in \overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR:\\ +[\text{*37·61}] & \supset:\overleftrightarrow{R}_{\unicode{x2217}}ʻʻsʻ\text{gen}ʻR\subset \overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR &\qquad \text{(1)}\\ +\vdash.\text{*97·2.*93·22}.&\supset\vdash.\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset \overleftrightarrow{R}_{\unicode{x2217}}ʻʻsʻ\text{gen}ʻR + &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·22.</b> \[\begin{align}&\vdash:R\in 1\rightarrow 1.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\cup + \overleftrightarrow{R}_{\unicode{x2217}}ʻʻpʻ\text{ᗡ}ʻʻPotʻR=\overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\\ +&[\text{*97·21.*93·37}]\end{align}\]</p> + +<p class="nind"><b>*97·23.</b> \(\vdash\colon\colon \overleftrightarrow{R}ʻʻCʻR\in 0\cup 1.\equiv\colon\ldotp x,\,y\in CʻR.\supset_{x,y}:x=y.\lor.xRy.\lor.yRx\)</p> + + +<p><span class="pagenum" id="Page_659">[Pg 659]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*52·4.(*54·01)}.\supset\\ +\vdash\colon\colon\ldotp \overleftrightarrow{R}ʻʻCʻR\in 0\cup 1.&\equiv\colon\colon \alpha,\beta\in \overleftrightarrow{R}ʻʻCʻR.\supset_{\alpha,\beta}.\alpha=\beta\colon\colon \\ +[\text{*37·63}] & \equiv\colon\colon x,\,y\in CʻR.\supset_{x,y}.\overleftrightarrow{R}ʻx=\overleftrightarrow{R}ʻy\colon\colon \\ +[\text{*97·1}] \equiv\colon\colon x,\,y\in CʻR.\supset_{x,y}\colon\ldotp &zRx.\lor.x\in CʻR.z=x.\lor.xRz.\equiv_{z}:\\ +&zRy.\lor.y\in CʻR.z=y.\lor.yRz\colon\ldotp \\ +[\text{*4·71}] \equiv\colon\colon x,\,y\in CʻR.\supset_{x,y}\colon\ldotp &zRx.\lor.z=x.\lor.xRz:\equiv_{z}:\\ +&zRy.\lor.z=y.\lor.yRz\colon\ldotp &\qquad \text{(1)}\\ +[\text{*10·1}] \supset\colon\ldotp x,\,y\in CʻR.\supset_{x,y}\colon\ldotp &xRx.\lor.x=x.\lor.xRx:\equiv:\\ +&xRy.\lor.x=y.\lor.yRx\colon\ldotp \\ +[\text{*13·15}] \supset_{x,y}\colon\ldotp &xRy.\lor.x=y.\lor.yRx &\qquad \text{(2)}\\ +\vdash.\text{*10·1}.&\supset\vdash\colon\colon x,y\in CʻR:z\in CʻR\colon\ldotp x,y\in CʻR.\supset_{x,y}:xRy.\lor.x=y.\lor.yRx\colon\ldotp \supset\colon\ldotp \\ +&\qquad\qquad\qquad xRz.\lor.x=z.\lor.zRx:yRz.\lor.y=z.\lor.zRy\colon\ldotp \\ +[\text{*5·1}] & \supset\colon\ldotp xRz.\lor.x=z.\lor.zRx:\equiv:yRz.\lor.y=z.\lor.zRy &\qquad \text{(3)}\\ +\vdash.\text{*33·132.Transp.*13·14}.\supset\\ +\vdash\colon\colon x,\,y\in CʻR:z{\sim}\in CʻR:&\supset\colon\ldotp {\sim}(xRz.\lor.zRx).x \neq z:{\sim}(yRz.\lor.zRy).y \neq z\colon\ldotp \\ +[\text{*5·21}] &\supset\colon\ldotp xRz.\lor.x=z.\lor.zRx:\equiv:\\ +&\qquad\qquad\qquad yRz.\lor.y=z.\lor.zRy &\qquad \text{(4)}\\ +\vdash.\text{(3).(4)}.&\supset\vdash\colon\colon\ldotp x,\,y\in CʻR.\supset_{x,y}:xRy.\lor.x=y.\lor.yRx\colon\ldotp \supset\colon\colon \\ +&\qquad x,\,y\in CʻR.\supset_{x,y}\colon\ldotp xRz.\lor.x=z.\lor.zRx:\equiv_{z}:yRz.\lor.y=z.\lor.zRy &\qquad \text{(5)}\\ +\vdash.\text{(1).(2).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·231.</b> \(\vdash\colon\ldotp \overleftrightarrow{R}ʻʻCʻR\in 0\cup 1.\equiv:x\in CʻR.\supset_{x}.CʻR=\overrightarrow{R}ʻx\cup \iotaʻx\cup \overleftarrow{R}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.*97·23.*32·18·181.*51·15.\supset\\ +\vdash\colon\ldotp \overleftrightarrow{R}ʻʻCʻR\in 0\cup 1.&\equiv:x\in CʻR.\supset.CʻR\subset \overrightarrow{R}ʻx\cup \iotaʻx\cup \overleftarrow{R}ʻx &\qquad \text{(1)}\\ +\vdash.*33·152.*51·2.&\supset\vdash:x\in CʻR.\supset.\overrightarrow{R}ʻx\cup \iotaʻx\cup \overleftarrow{R}ʻx\subset CʻR &\qquad \text{(2)}\\ +\vdash.(1).(2).\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·24.</b> \(\vdash\colon\ldotp \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\in 0\cup 1.\equiv:x\in CʻR.\supset_{x}.CʻR=\overrightarrow{R}_{\unicode{x2217}}ʻx\cup + \overleftarrow{R}_{\unicode{x2217}}ʻx\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·231.*90·14}.\supset\\ +\vdash\colon\ldotp \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\in 0\cup 1.&\equiv:x\in CʻR.\supset_{x}.CʻR=\overrightarrow{R}_{\unicode{x2217}}ʻx\cup + \iotaʻx\cup \overleftarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(1)}\\ +\vdash.\text{*90·12}. &\supset\vdash:x\in CʻR.\supset.\iotaʻx\subset \overrightarrow{R}_{\unicode{x2217}}ʻx &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*22·62}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·241.</b> \(\vdash\colon\colon \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\in + 0\cup 1.\equiv\colon\ldotp x,y\in CʻR.\supset_{x,y}:xR_{\unicode{x2217}}y.\lor.yR_{\unicode{x2217}}x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·24.*32·18·181}.\supset\\ +\vdash\colon\colon\ldotp\overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\in 0\cup 1.\equiv\colon\colon x\in CʻR.\supset_{x}\colon\ldotp + y\in CʻR.\equiv_{y}:xR_{\unicode{x2217}}y.\lor.yR_{\unicode{x2217}}x &\qquad \text{(1)}\\ +\vdash.\text{*90·13}.\supset\vdash\colon\ldotp xR_{\unicode{x2217}}y.\lor.yR_{\unicode{x2217}}x:\supset.y\in CʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*4·73}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·242.</b> \[\begin{align}\vdash\colon\colon \overleftrightarrow{R}_{\unicode{x2217}}ʻʻCʻR\in + 0\cup 1.&\equiv\colon\ldotp x,\,y\in CʻR.\supset_{x,y}:x=y.\lor.xR_{\text{po}}y.\lor.yR_{\text{po}}x\colon\ldotp \\ +&\equiv\colon\ldotp \overleftrightarrow{R}_{\text{po}}ʻʻCʻR\in 0\cup 1\\ +[\text{*91·542.*97·23.*91·504}]\end{align}\]</p> + +<p><span class="pagenum" id="Page_660">[Pg 660]</span></p> + +<p>The remaining propositions of this number (except <a href="#*97·5">*97·5</a> ff.) are +concerned with proving that, under certain hypotheses, +\[ +\begin{array}{l} +&\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\text{gen}ʻR,\,\, \textit{i.e.}\,\, \overleftrightarrow{R}_{\unicode{x2217}}ʻʻsʻ\text{gen}ʻR\subset + \text{D}ʻʻ{\in}_{\Delta}ʻ\text{gen}ʻR,\\ +\text{and} &\qquad\qquad\text{gen}ʻR-\iotaʻ\Lambda\subset \text{D}ʻʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR. +\end{array} +\]</p> + +<p>These propositions have the merit of proving the existence of +selections in the cases to which they apply.</p> + +<p class="nind"><b><a id="*97·3">*97·3</a>.</b> \(\vdash.\overleftarrow{R}_{\unicode{x2217}}\upharpoonright \overrightarrow{B}ʻR\in 1\rightarrow 1\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*90·12}.\supset\\ +\vdash\colon\ldotp x,\,y\in \overrightarrow{B}ʻR.\overleftarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻy.&\supset:y\in + \overleftarrow{R}_{\unicode{x2217}}ʻx:\\ +[\text{*91·54}] & \supset:y=x.\lor.xR_{\text{po}}y &\qquad \text{(1)}\\ +\vdash.\text{*91·504}. & \supset\vdash:xR_{\text{po}}y.\supset.y\in \text{ᗡ}ʻR:\\ +[\text{Transp.*93·101}]&\supset\vdash:y\in \overrightarrow{B}ʻR.\supset.{\sim}(xR_{\text{po}}y) &\qquad \text{(2)}\\ +\vdash.\text{(1).(2)}. & \supset\vdash:x,\,y\in \overrightarrow{B}ʻR.\overleftarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻy.\supset.x=y + &\qquad \text{(3)}\\ +\vdash.\text{(3).*71·55.*72·12}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·301.</b> \(\vdash.I\upharpoonright \overrightarrow{B}ʻR\in (\breve{R}_{\unicode{x2217}})_{\Delta}ʻ\overrightarrow{B}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*72·17}. &\supset\vdash.I\upharpoonright \overrightarrow{B}ʻR\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*90·15}. &\supset\vdash.I\upharpoonright \overrightarrow{B}ʻR\unicode{x2abd}\breve{R}_{\unicode{x2217}} &\qquad \text{(2)}\\ +\vdash.\text{*50·5·52}.&\supset\vdash.\text{ᗡ}ʻI\upharpoonright \overrightarrow{B}ʻR=\overrightarrow{B}ʻR &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3).*80·14}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*97·31">*97·31</a>.</b> \(\vdash.(\overrightarrow{B}ʻR)\upharpoonleft \text{Cnv}ʻ\overleftarrow{R}_{\unicode{x2217}}\in {\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\text{D}ʻ\{(\overrightarrow{B}ʻR)\upharpoonleft + \text{Cnv}ʻ\overleftarrow{R}_{\unicode{x2217}}\}=\overrightarrow{B}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·3.*85·13}\, \frac{\breve{R}_{\unicode{x2217}}}{Q}.\supset\\ +\vdash:S\in (\breve{R}_{\unicode{x2217}})_{\Delta}ʻ\overrightarrow{B}ʻR.&\supset.S\mid \text{Cnv}ʻ\overleftarrow{R}_{\unicode{x2217}}\in + {\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*97·301}. &\supset\vdash.I\upharpoonright \overrightarrow{B}ʻR\mid \text{Cnv}ʻ\overleftarrow{R}_{\unicode{x2217}}\in + {\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\\ +[\text{*50·61}] & \supset\vdash.(\overrightarrow{B}ʻR)\upharpoonleft \text{Cnv}ʻ\overleftarrow{R}_{\unicode{x2217}}\in {\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR + &\qquad \text{(2)}\\ +\vdash.\text{*35·62.*33·431}.&\supset\vdash.\text{D}ʻ{(\overrightarrow{B}ʻR)\upharpoonleft \text{Cnv}ʻ\overleftarrow{R}_{\unicode{x2217}}}=\overrightarrow{B}ʻR &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_661">[Pg 661]</span></p> + +<p class="nind"><b>*97·32.</b> \(\vdash.\overrightarrow{B}ʻR\in \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR \quad[\text{*97·31}]\)</p> + +<p class="nind"><b>*97·33.</b> \(\vdash:R\in 1\rightarrow 1.\alpha\subset sʻ\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta.\beta\subset + sʻ\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha=\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·15.Fact}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \beta.x\in \overleftrightarrow{R}_{\unicode{x2217}}ʻy.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻy=\overleftrightarrow{R}_{\unicode{x2217}}ʻx.y\in \beta.\\ +[\text{*37·62}] & \supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx\in \overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta &\qquad \text{(1)}\\ +\vdash.\text{(1).*10·11·21·23.*40·4}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:x\in sʻ\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta.\supset_{x}.\overleftrightarrow{R}_{\unicode{x2217}}ʻx\in \overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta:\\ +[\text{Hp.Syll}] &\supset:x\in \alpha.\supset_{x}.\overleftrightarrow{R}_{\unicode{x2217}}ʻx\in \overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta:\\ +[\text{*37·61}] & \supset:\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha\subset \overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{*40·4}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:y\in \beta.\supset.(\exists x).x\in \alpha.y\in \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\\ +[\text{*97·15}] & \supset.(\exists x).x\in \alpha.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftrightarrow{R}_{\unicode{x2217}}ʻy\\ +[\text{*37·62}] &\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻy\in \overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha &\qquad \text{(3)}\\ +\vdash.\text{(3).*37·61}.&\supset\vdash:\text{Hp}.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta\subset \overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha + &\qquad \text{(4)}\\ +\vdash.\text{(2).(4)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·34.</b> + \(\vdash:R\in 1\rightarrow 1.\beta\in \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha=\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*83·6·62}. & \supset\vdash\colon\ldotp \text{Hp}.\supset:x\in \alpha.\supset_{x}.\exists !\beta\cap \overleftrightarrow{R}_{\unicode{x2217}}ʻx:\beta\subset + sʻ\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\alpha &\qquad \text{(1)}\\ +\vdash.\text{*40·4.*97·101}. &\supset\vdash\colon\ldotp x\in \alpha.\supset_{x}.\exists !\beta\cap \overleftrightarrow{R}_{\unicode{x2217}}ʻx:\equiv.\alpha\subset + sʻ\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*97·33}.&\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·341</b>. + \[\begin{align}&\vdash:R \in 1\rightarrow 1.\beta\in \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻʻ\beta=\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\\ +&\left[\text{*97·34}\, \frac{\overrightarrow{B}ʻR}{\alpha} .\text{*97·2}\right]\end{align}\]</p> + +<p class="nind"><b>*97·35.</b> \[\begin{align}\vdash:R &\in \text{Cls}\rightarrow 1.T\in PotidʻR.\overrightarrow{B}ʻR\subset \text{D}ʻT.\supset.\\ +&\text{Cnv}ʻ{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright + \overrightarrow{B}ʻR)\mid T}\in {\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\text{ᗡ}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright + \overrightarrow{B}ʻR)\mid T\}=\breve{T}ʻʻ\overrightarrow{B}ʻR\end{align}\]</p> + +<p><span class="pagenum" id="Page_662">[Pg 662]</span></p> + + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·3.*92·101}.&\supset\vdash:\text{Hp}.\supset.\text{Cnv}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright + \overrightarrow{B}ʻR)\mid T\}\in 1\rightarrow \text{Cls} &\qquad \text{(1)}\\ +\vdash.\text{*35·101.*30·4}.&\supset\\ +&\vdash:\alpha\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright \overrightarrow{B}ʻR)\mid T\}y.\equiv.(\exists x).x\in \overrightarrow{B}ʻR.\alpha=\overleftarrow{R}_{\unicode{x2217}}ʻx.xTy + &\qquad \text{(2)}\\ +\vdash.\text{*91·58}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:xTy.\supset.y\in \overleftarrow{R}_{\unicode{x2217}}ʻx:\\ +[\text{*13·12}] &\supset:\alpha=\overleftarrow{R}_{\unicode{x2217}}ʻx.xTy.\supset.y\in \alpha &\qquad \text{(3)}\\ +\vdash.\text{(2).(3)}. &\supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright \overrightarrow{B}ʻR)\mid T\}y.\supset_{\alpha,y}.y\in + \alpha:\\ +[\text{*23·1.*31·131}] &\supset:\text{Cnv}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright \overrightarrow{B}ʻR)\mid T\}\unicode{x2abd}\in &\qquad \text{(4)}\\ +\vdash.\text{*37·321.*35·65}.&\supset\vdash:\text{Hp}.\supset.\text{D}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright + \overrightarrow{B}ʻR)\mid T\}=\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR &\qquad \text{(5)}\\ +\vdash.\text{(1).(4).(5).*80·14}.&\supset\vdash:\text{Hp}.\supset.\text{Cnv}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright + \overrightarrow{B}ʻR)\mid T\}\in {\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR. &\qquad \text{(6)}\\ +\vdash.\text{*35·65}. & \supset\vdash.\text{ᗡ}ʻ(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright \overrightarrow{B}ʻR)=\overrightarrow{B}ʻR.\\ +[\text{*37·32}] & \supset\vdash.\text{ᗡ}ʻ\{(\overleftarrow{R}_{\unicode{x2217}}\upharpoonright \overrightarrow{B}ʻR)\mid T\}=\breve{T}ʻʻ\overrightarrow{B}ʻR &\qquad \text{(7)}\\ +\vdash.\text{(6).(7)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·36.</b> + \[\begin{align}&\vdash:R\in \text{Cls}\rightarrow 1.T\in \text{Potid}ʻR.\overrightarrow{B}ʻR\subset \text{D}ʻT.\supset.\breve{T}ʻʻ\overrightarrow{B}ʻR\in \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\\ +&[\text{*97·35}]\end{align}\]</p> + +<p class="nind"><b>*97·37.</b> + \(\vdash:R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\supset.\text{gen}ʻR\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·14}.\supset\vdash\colon\ldotp \text{Hp}.\supset:T\in \text{Potid}ʻR.\supset.\overrightarrow{B}ʻR\subset \text{D}ʻT &\qquad \text{(1)}\\ +\vdash.\text{*93·32}.\supset\vdash\colon\ldotp \text{Hp}.\supset:\alpha\in \text{gen}ʻR.\equiv.(\exists T).T\in \text{Potid}ʻR.\alpha=\breve{T}ʻʻ\overrightarrow{B}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*97·36}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·38.</b> \(\vdash:R\in 1\rightarrow 1.\text{ᗡ}ʻR\subset \text{D}ʻR.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset + \text{D}ʻʻ{\in}_{\Delta}ʻ\text{gen}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·36.*40·52}.\supset\vdash:\text{Hp}.\supset.sʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR=sʻ\text{gen}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·43.*97·37.*93·25.*97·16·21}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·4.</b> \(\vdash:S\in \text{Pot}ʻR.\supset.\breve{S}ʻʻ\overrightarrow{B}ʻ\breve{R}=\Lambda\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·31}.&\supset\vdash:\text{Hp}.\supset.(\exists T).T\in \text{Potid}ʻR.S=R\mid T.\\ +[\text{*37·341}] &\supset.(\exists T).T\in \text{Potid}ʻR.\breve{S}ʻʻ\overrightarrow{B}ʻ\breve{R}=\breve{T}ʻʻ\breve{R}ʻʻ\overrightarrow{B}ʻ\breve{R}\\ +[\text{*37·261·29.*93·101}] &\qquad\qquad\qquad\qquad\qquad\qquad =\Lambda.\\ +[\text{*10·35}] &\supset.\breve{S}ʻʻ\overrightarrow{B}ʻ\breve{R}=\Lambda:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·401.</b> \(\vdash\colon\ldotp x\in \text{D}ʻR:S\in \text{Pot}ʻR.xSy.\supset_{S,y}.y\in \text{D}ʻR:\supset:S\in \text{Pot}ʻR.\supset_{S}.x\in \text{D}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*33·13}.\supset\vdash\colon\ldotp \text{Hp}.\supset:S\in \text{Pot}ʻR.xSy.&\supset_{S,y}.(\exists z).yRz.xSy.\\ +[\text{*34·1.*33·13}] & \supset_{S,y}.x\in \text{D}ʻ(S\mid R):\\ +[\text{*10·28.*33·13}] &\supset:S\in \text{Pot}ʻR.x\in \text{D}ʻS.\supset_{S}.x\in \text{D}ʻ(S\mid R) &\qquad \text{(1)}\\ +\vdash.\text{(1).*91·373}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·402.</b> \[\begin{align}\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1:(\exists S).S\in \text{Pot}ʻR.x{\sim}\in &\text{D}ʻS:\supset.\\ +&(\exists S).S\in \text{Pot}ʻR.\breve{S}ʻx\in \overrightarrow{B}ʻ\breve{R}\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·401.Transp}.\supset\vdash:\text{Hp}.&\supset.(\exists S,y).S\in \text{Pot}ʻR.xSy.y{\sim}\in \text{D}ʻR.\\ +[\text{*91·271.*33·14.*93·101}]&\supset.(\exists S,y).S\in \text{Pot}ʻR.xSy.y\in \overrightarrow{B}ʻ\breve{R}.\\ +[\text{*71·321}] & \supset.(\exists S).S\in \text{Pot}ʻR.\breve{S}ʻx\in \overrightarrow{B}ʻ\breve{R}:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_663">[Pg 663]</span></p> + +<p class="nind"><b>*97·403.</b> \[\begin{align}\vdash:R\in \text{Cls}\rightarrow 1.x\in \overrightarrow{B}ʻR.T\in \text{Pot}ʻR.&\overrightarrow{B}ʻ\breve{R} = \breve{T}ʻʻ\overrightarrow{B}ʻR.\supset.\\ +&(\exists S).S\in \text{Pot}ʻR.x{\sim}\in \text{D}ʻS\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·131}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:xTy.xTz.zRw.\supset.yRw.\\ +[\text{*33·14}] &\supset.y{\sim}\in \overrightarrow{B}ʻ\breve{R} &\qquad \text{(1)}\\ +\vdash.\text{(1).*11·11·3·35}.\supset\\ +\vdash\colon\colon \text{Hp}.&\supset\colon\ldotp xTy:(\exists z,w).xTz.zRw:\supset.y{\sim}\in \overrightarrow{B}ʻ\breve{R}\colon\ldotp \\ +[\text{*34·1.*33·13}] &\supset\colon\ldotp xTy.x\in \text{D}ʻ(T\mid R).\supset.y{\sim}\in \overrightarrow{B}ʻ\breve{R}\colon\ldotp \\ +[\text{Transp}] &\supset\colon\ldotp xTy.y\in \overrightarrow{B}ʻ\breve{R}.\supset.x{\sim}\in \text{D}ʻ(T\mid R) &\qquad \text{(2)}\\ +\vdash.\text{*10·24}.&\supset\vdash:\text{Hp}.x{\sim}\in \text{D}ʻT.\supset.(\exists S).S\in \text{Pot}ʻR.x{\sim}\in \text{D}ʻS &\qquad \text{(3)}\\ +\vdash·\text{*37·105}.&\supset\vdash:\text{Hp}.xTy.\supset.y\in \overrightarrow{B}ʻ\breve{R}.\\ +[\text{(2)}] &\supset.x{\sim}\in \text{D}ʻ(T\mid R) &\qquad \text{(4)}\\ +\vdash.\text{(4).*10·11·23·35.*33·13}.\supset\\ +\vdash:\text{Hp}.x\in \text{D}ʻT.&\supset.x{\sim}\in \text{D}ʻ(T\mid R).\\ +[\text{*91·282}] &\supset.(\exists S).S\in \text{Pot}ʻR.x{\sim}\in \text{D}ʻS &\qquad \text{(5)}\\ +\vdash.\text{(3).(5)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·41.</b> \[\begin{align}\vdash:R\in \text{Cls}\rightarrow 1.x\in {\sim}\overrightarrow{B}ʻR.T\in \text{Pot}ʻR.&\overrightarrow{B}ʻ\breve{R} = \breve{T}ʻʻ\overrightarrow{B}ʻR.\supset.\\ +&(\exists S).S\in \text{Pot}ʻR.\breve{S}ʻx\in \overrightarrow{B}ʻ\breve{R}\\ +&[\text{*97·402·403}]\end{align}\]</p> + +<p class="nind"><b>*97·42.</b> + \(\vdash:R\in 1\rightarrow 1.x\in \overrightarrow{B}ʻR.S,T\in \text{Pot}ʻR.\overrightarrow{B}ʻ\breve{R} = \breve{T}ʻʻ\overrightarrow{B}ʻR.\breve{S}ʻx\in \overrightarrow{B}ʻ\breve{R}.\supset.S = T\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*37·6}. \supset\vdash:\text{Hp}.&\supset.(\exists y).y\in \overrightarrow{B}ʻR.\breve{S}ʻx = \breve{T}ʻy &\qquad \text{(1)}\\ +\vdash.\text{*37·62.(1)}.\supset\vdash:\text{Hp}.&\supset.\breve{S}ʻx\in \breve{S}ʻʻ\overrightarrow{B}ʻR\cap \breve{T}ʻʻ\overrightarrow{B}ʻR.\\ +[\text{*93·3}] &\supset.\breve{S}ʻx\in \overrightarrow{\text{min}}_{R}ʻ\text{ᗡ}ʻS\cap \overrightarrow{\text{min}}_{R}ʻ\text{ᗡ}ʻT.\\ +[\text{*93·24.Transp}] &\supset.S = T:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·43.</b> \(\vdash:R\in 1\rightarrow 1.T\in \text{Pot}ʻR.\overrightarrow{B}ʻ\breve{R} = \breve{T}ʻʻ\overrightarrow{B}ʻR.\supset.\overrightarrow{B}ʻR\subset \text{D}ʻT\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·42}.\supset\\ +\vdash\colon\ldotp \text{Hp}.x\in \overrightarrow{B}ʻR.&\supset:S\in \text{Pot}ʻR.\breve{S}ʻx\in \overrightarrow{B}ʻ\breve{R}.\supset.\breve{T}ʻx\in \overrightarrow{B}ʻ\breve{R}:\\ +[\text{*10·11·21·23}] &\supset:(\exists S).S\in \text{Potid}ʻR.\breve{S}ʻx\in \overrightarrow{B}ʻ\breve{R}.\supset.\breve{T}ʻx\in \overrightarrow{B}ʻ\breve{R}:\\ +[\text{*97·41}] &\supset:\breve{T}ʻx\in \overrightarrow{B}ʻ\breve{R}:\\ +[\text{*14·21}] &\supset:\text{E}!\breve{T}ʻx:\\ +[\text{*33·44}] &\supset:x\in \text{D}ʻT\colon\ldotp \supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p><span class="pagenum" id="Page_664">[Pg 664]</span></p> + +<p class="nind"><b>*97·44.</b> + \(\vdash:R\in 1\rightarrow 1.S,\,T\in \text{Pot}ʻR.\overrightarrow{B}ʻ\breve{R}=\breve{T}ʻʻ\overrightarrow{B}ʻR.\exists !\breve{S}ʻʻ\overrightarrow{B}ʻR.\supset.\overrightarrow{B}ʻR\subset \text{D}ʻS\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*91·45}.\supset\vdash\colon\ldotp \text{Hp}.&\supset:(\exists U):U\in \text{Potid}ʻR:S=U\mid T.\lor.T=U\mid S &\qquad \text{(1)}\\ +\vdash.\text{*97·4}. \supset\vdash\colon\ldotp \text{Hp}.&\supset:U\in \text{Pot}ʻR.S=U\mid T.\supset.\breve{S}ʻʻ\overrightarrow{B}ʻR=\Lambda:\\ +[\text{*91·23}] & \supset:U\in \text{Potid}ʻR.S=U\mid T.\exists !\breve{S}ʻʻ\overrightarrow{B}ʻR.\supset.U=I\upharpoonright CʻR.\\ +[\text{*50·63.*91·271}] & \supset.S=T.\\ +[\text{*97·43}] & \supset.\overrightarrow{B}ʻR\subset \text{D}ʻS &\qquad \text{(2)}\\ +\vdash.\text{*91·34}.&\supset\vdash\colon\ldotp \text{Hp}.\supset:U\in \text{Potid}ʻR.T=U\mid S.\supset.T=S\mid U.\\ +[\text{*34·36}] & \supset.\text{D}ʻT\subset \text{D}ʻS.\\ +[\text{*97·43}] & \supset.\overrightarrow{B}ʻR\subset \text{D}ʻS &\qquad \text{(3)}\\ +\vdash.\text{(1).(2).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·45.</b> + \(\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}\in \text{gen}ʻR.\supset.\text{gen}ʻR-\iotaʻ\Lambda\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·44.*10·11·23·35.*93·32}.\supset\\ +\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}\in \text{gen}ʻR.S\in \text{Pot}ʻR.\exists !\breve{S}ʻʻ\overrightarrow{B}ʻR.\supset.\overrightarrow{B}ʻR\subset \text{D}ʻS.\\ +[\text{*97·36}] \supset.\breve{S}ʻʻ\overrightarrow{B}ʻR\in \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR &\qquad \text{(1)}\\ +\vdash.\text{(1).*13·12}.\supset\\ +\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}\in \text{gen}ʻR.S\in \text{Pot}ʻR.\alpha=\breve{S}ʻʻ\overrightarrow{B}ʻR.\exists !\alpha.\\ + \supset.\alpha\in \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR &\qquad \text{(2)}\\ +\vdash.\text{(2).*10·11·23·35.*93·32}.\supset\\ +\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}\in \text{gen}ʻR.\alpha\in \text{gen}ʻR.\exists !\alpha.\supset.\alpha\in \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR + &\qquad \text{(3)}\\ +\vdash.\text{(3).*53·52}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*97·46">*97·46</a>.</b> \(\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}\in \text{gen}ʻR.\supset.\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset + \text{D}ʻʻ{\in}_{\Delta}ʻ(\text{gen}ʻR-\iotaʻ\Lambda)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·36.*40·52}.\supset\vdash:\text{Hp}.\supset.sʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR&=sʻ\text{gen}ʻR\\ +[\text{*53·18}] & =sʻ(\text{gen}ʻR-\iotaʻ\Lambda) &\qquad \text{(1)}\\ +\vdash.\text{(1).*84·43.*97·45·16·21.*93·25}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*97·47">*97·47</a>.</b> \[\begin{align}\vdash:R &\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}\in \text{gen}ʻR\cup \iotaʻ\Lambda.\supset.\\ +&\mid \text{gen}ʻR-\iotaʻ\Lambda\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset + \text{D}ʻʻ{\in}_{\Delta}ʻ(\text{gen}ʻR-\iotaʻ\Lambda) +\end{align}\]</p> + + +<p><span class="pagenum" id="Page_665">[Pg 665]</span></p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·32}.&\supset\vdash:\overrightarrow{B}ʻR=\Lambda.\supset.\text{gen}ʻR=\iotaʻ\Lambda &\qquad \text{(1)}\\ +\vdash.\text{(1).*37·29}.&\supset\vdash:\overrightarrow{B}ʻR=\Lambda.\supset.\text{gen}ʻR-\iotaʻ\Lambda=\Lambda.\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR=\Lambda.\\ +[\text{*24·12}] &\supset.\text{gen}ʻR-\iotaʻ\Lambda\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\\ +&\qquad\qquad\qquad \overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset \text{D}ʻʻ{\in}_{\Delta}ʻ(\text{gen}ʻR-\iotaʻ\Lambda) &\qquad \text{(2)}\\ +\vdash.\text{*24·3.Fact}.\supset\\ +\vdash:R\in 1\rightarrow 1.\exists !\overrightarrow{B}ʻR.\overrightarrow{B}ʻ\breve{R}=\Lambda.&\supset.R\in 1\rightarrow 1.\exists !\overrightarrow{B}ʻR.\text{ᗡ}ʻR\subset \text{D}ʻR. &\qquad \text{(3)}\\ +[\text{*93·41}] & \supset.\Lambda{\sim}\in \text{gen}ʻR.\\ +[\text{*51·222}] & \supset.\text{gen}ʻR-\iotaʻ\Lambda=\text{gen}ʻR.\\ +[\text{(3).*97·37·38}] &\supset.\text{gen}ʻR-\iotaʻ\Lambda\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\\ +&\qquad\qquad\qquad \overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset \text{D}ʻʻ{\in}_{\Delta}ʻ(\text{gen}ʻR-\iotaʻ\Lambda) &\qquad \text{(4)}\\ +\vdash.\text{(2).(4)}.\supset\\ +\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}=\Lambda.&\supset.\text{gen}ʻR-\iotaʻ\Lambda\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\\ +&\qquad\qquad\qquad\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset \text{D}ʻʻ{\in}_{\Delta}ʻ(\text{gen}ʻR-\iotaʻ\Lambda) &\qquad \text{(5)}\\ +\vdash.\text{*97·45·46}.\supset\\ +\vdash:R\in 1\rightarrow 1.\overrightarrow{B}ʻ\breve{R}\in \text{gen}ʻR.&\supset.\text{gen}ʻR-\iotaʻ\Lambda\subset \text{D}ʻʻ{\in}_{\Delta}ʻ\overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR.\\ +&\qquad\qquad\qquad \overleftarrow{R}_{\unicode{x2217}}ʻʻ\overrightarrow{B}ʻR\subset \text{D}ʻʻ{\in}_{\Delta}ʻ(\text{gen}ʻR-\iotaʻ\Lambda) &\qquad \text{(6)}\\ +\vdash.\text{(5).(6)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b><a id="*97·5">*97·5</a>.</b> \(\vdash:R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.xR_{\text{po}}y.\supset.yR_{\text{po}}x\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*92·111}.&\supset\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.xRy.\supset:yR_{\unicode{x2217}}x:\\ +[\text{*91·54}] &\supset:y=x.\lor.yR_{\text{po}}x:\\ +[\text{Hp}] & \supset:yR_{\text{po}}x &\qquad \text{(1)}\\ +\vdash.\text{*10·1.*34·1}.&\supset\vdash\colon\ldotp R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.P\in \text{Pot}ʻR:\\ +& xPy.\supset_{y}.yR_{\text{po}}x:xP\mid Rz:\supset:(\exists y).yR_{\text{po}}x.yRz:\\ +[\text{*92·111}] & \supset:zR_{\unicode{x2217}}x:\\ +[\text{*91·54}] &\supset:z=x.\lor.zR_{\text{po}}x:\\ +[\text{Hp}] &\supset:zR_{\text{po}}x &\qquad \text{(2)}\\ +\vdash.\text{(1).(2).*91·171}.&\supset\vdash:R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.P\in \text{Pot}ʻR.xPy.\supset.yR_{\text{po}}x:\\ +[\text{(*91·05)}] & \supset\vdash:R\in \text{Cls}\rightarrow 1.xR_{\text{po}}x.xR_{\text{po}}y.\supset.yR_{\text{po}}x:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·501.</b> \(\vdash:R\in 1\rightarrow \text{Cls}.xR_{\text{po}}x.yR_{\text{po}}x.\supset.xR_{\text{po}}y \quad[\text{Proof as in *97·5}]\)</p> + +<p class="nind"><b>*97·51.</b> + \[\begin{align}&\vdash:R\in 1\rightarrow 1.xR_{\text{po}}x.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overrightarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx\cap \overrightarrow{R}_{\unicode{x2217}}ʻx\\ +&[\text{*97·5·501·17}]\end{align}\]</p> + +<p class="nind"><b>*97·52.</b> \(\vdash:R\in 1\rightarrow 1.xR_{\text{po}}x.xR_{\text{po}}y.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx=\overleftarrow{R}_{\unicode{x2217}}ʻx\cap + \overrightarrow{R}_{\unicode{x2217}}ʻy \quad[\text{*97·5·501·51·14}]\)</p> + +<p><span class="pagenum" id="Page_666">[Pg 666]</span></p> + +<p class="nind"><b>*97·53.</b> \(\vdash:R\in 1\rightarrow 1.P\in \text{Pot}ʻR.xPx.y\in \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\supset.yPy \quad[\text{*92·132·133}]\)</p> + +<p class="nind"><b><a id="*97·54">*97·54</a>.</b> \[\begin{align}&\vdash:R\in 1\rightarrow 1.xR_{\text{po}}x.\supset.(\exists P).P\in \text{Pot}ʻR.P\upharpoonright \overleftrightarrow{R}_{\unicode{x2217}}ʻx=I\upharpoonright + \overleftrightarrow{R}_{\unicode{x2217}}ʻx\\ +&[\text{*97·53}]\end{align}\]</p> + +<p class="nind"><b><a id="*97·55">*97·55</a>.</b> \(\vdash\colon\colon R\in 1\rightarrow 1.\supset\colon\ldotp y\in \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\supset_{y}.yR_{\text{po}}y:\lor:y\in + \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\supset_{y}.{\sim}(yR_{\text{po}}y)\)</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*97·53}.& \supset\vdash\colon\ldotp \text{Hp}.xR_{\text{po}}x.\supset:y\in \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\supset_{y}.yR_{\text{po}}y + &\qquad \text{(1)}\\ +\vdash.\text{(1)}\, \frac{y,\,x}{x,\,y}.\text{Transp}.&\supset\vdash:\text{Hp}.{\sim}(xR_{\text{po}}x).x\in + \overleftrightarrow{R}_{\unicode{x2217}}ʻy.\supset.{\sim}(yR_{\text{po}}y) &\qquad \text{(2)}\\ +\vdash.\text{(2).*97·101}. &\supset\vdash\colon\ldotp \text{Hp}.{\sim}(xR_{\text{po}}x).\supset:y\in + \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\supset_{y}.{\sim}(yR_{\text{po}}y) &\qquad \text{(3)}\\ +\vdash.\text{(1).(3)}.\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p class="nind"><b>*97·56.</b> + \[\begin{align}&\vdash\colon\ldotp R\in 1\rightarrow 1.x\in \overrightarrow{B}ʻR.\supset:y\in \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\supset_{y}.{\sim}(yR_{\text{po}}y)\\ +&[\text{*96·23·1.*97·55}]\end{align}\]</p> + +<p class="nind"><b><a id="*97·57">*97·57</a>.</b> + \[\begin{align}&\vdash\colon\ldotp R\in 1\rightarrow 1.x\in sʻ\text{gen}ʻR.\supset:y\in \overleftrightarrow{R}_{\unicode{x2217}}ʻx.\supset_{y}.{\sim}(yR_{\text{po}}y)\\ +&[\text{*97·21·56}]\end{align}\]</p> + +<p class="nind"><b><a id="*97·58">*97·58</a>.</b> \[\begin{align}\vdash\colon\ldotp R\in 1\rightarrow \text{Cls}.\supset:x\in sʻ\text{gen}ʻR.&\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx\subset sʻ\text{gen}ʻR:\\ +& x\in pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻx\subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR\end{align}\]</p> + +<p><i>Dem.</i></p> + +<p>\[ +\begin{array}{l} +\vdash.\text{*93·412}. & \supset\vdash.\breve{R}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻR\subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR &\qquad \text{(1)}\\ +[\text{*90·101.*93·273.*37·265}]&\supset\vdash.Rʻʻsʻ\text{gen}ʻR\subset sʻ\text{gen}ʻR &\qquad \text{(2)}\\ +\vdash.\text{*93·33.*40·8}. & \supset\vdash:R\in 1\rightarrow \text{Cls}.\supset.\breve{R}ʻʻsʻ\text{gen}ʻR\subset sʻ\text{gen}ʻR. &\qquad \text{(3)}\\ +[\text{*90·101.*93·271.*37·265}] &\supset.Rʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻR\subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR &\qquad \text{(4)}\\ +\vdash.\text{(1).(2).(3).(4).*90·22}.\supset\\ +\vdash:R\in 1\rightarrow \text{Cls}.\supset.\overleftrightarrow{R}_{\unicode{x2217}}ʻʻsʻ\text{gen}ʻR&\subset sʻ\text{gen}ʻR.\\ +&\overleftrightarrow{R}_{\unicode{x2217}}ʻʻpʻ\text{ᗡ}ʻʻ\text{Pot}ʻR\subset pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR:\supset\vdash.\text{Prop} +\end{array} +\]</p> + +<p>It follows from this proposition that every family is either wholly +contained in the generations of \(R\) or wholly contained in +\(pʻ\text{ᗡ}ʻʻ\text{Pot}ʻR\), which may be called the <i>residue</i> of +the field of \(R\).</p> + + +<div class="footnotes"><h3>FOOTNOTES:</h3> + +<div class="footnote"> + +<p class="nind"><a id="Footnote_68" href="#FNanchor_68" class="label">[68]</a> +Here the type "\(^{\unicode{x2217}}\omega\)" is the +type of converses of relations of type \(\omega\), <i>i.e.</i> the +type of the negative integers in order of magnitude, ending with +\(-1\), \(\omega\) being the type of the positive integers in order of +magnitude, and therefore \(^{\unicode{x2217}}\omega + \omega\) being +the type of negative and positive integers in order of magnitude.</p> + +</div> +</div> + + + + +<hr class="tb"> + + +<p class="nindc space-above2 space-below2"> +CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS</p> + + +<hr class="chap x-ebookmaker-drop"> + +<div class="chapter"> +<div class="transnote spa1"> +<p class="nindc"><b>TRANSCRIBER’S NOTES</b></p> + +<p>All items in the Errata, from all three volumes, have been +added and corrected accordingly. +</p> + +<p>The author's notation as ‘*2·37·38’ is an abbreviation for *2·37 +and *2·38. For this reason, these numbers were not considered in the +cross-references.</p> + +<p>The lemma *84.44 cited on page 326 was not described by the authors +in the corresponding section.</p> + +</div></div> + +</body> +</html> diff --git a/78050-src/README-math.txt b/78050-src/README-math.txt new file mode 100644 index 0000000..adabf5a --- /dev/null +++ b/78050-src/README-math.txt @@ -0,0 +1,96 @@ +MathJax HTML source file instructions +================================== +This project is a math heavy eBook. The source is a preliminary HTML file that +uses MathJax to define mathematical expressions, which is processed to generate +a final HTML file with SVG images. + +This source file is kept for the purpose of applying errata fixes. Although the +MathJax takes some learning, it is clearer than the generated final. This also +allows the SVG images to be regenerated with changes. + + +Tools +===== +See the ppmath GitHub repository: + https://github.com/DistributedProofreaders/ppmath. +Follow the instructions to install m2svg. + +Command line: + m2svg -i input.htm -o output.htm + +- The SVG files will be placed in a subdirectory of the working directory + called "images". + +- In the converted file, the maths expressions, delimited by the tags `\[` + and `\]` for *display* expressions or `\(` and `\)` for *inline* + expressions, are replaced by `<img>` links. + +- The "data-tex" attribute will contain the original maths expression. + + +Inline code example +=================== +For the expression \(\mathrm{AB}^{2} = \mathrm{AG} \times \mathrm{BD}\), the +input `\(\mathrm{AB}^{2} = \mathrm{AG} \times \mathrm{BD}\)` + +becomes + `<span class="nowrap"><img style="vertical-align: -0.186ex; width: 16.872ex; + height: 2.253ex;" src="images/4.svg" alt="" data-tex="\mathrm{AB}^{2} + = \mathrm{AG} \times \mathrm{BD}">,</span>` + +The file images/4.svg displays the desired expression. + + +Source files structure +====================== +(eBook 75107 is used as an example) + +- 75107/ + - README-math.txt (this file) + - 75107-h/ + - 75107-h.htm (final HTML file) + - images/ + - 75107-src/ + - 75107-src.htm (source HTML file with MathJax) + + +SVG fixup for ebookmaker +======================== +Now, the SVG files contain a "data-variant" attribute that causes errors. +It needs to be removed by downloading and running this utility: +https://github.com/user-attachments/files/25548572/remove_data_variant_attribute.py + +Command line: + python remove_data_variant_attribute.py images + +Hopefully, this step will be removed in the future. + + +Submission process +================== +- Generated final HTML and images should be submitted as normal. +- In addition, the source HTML will be included, and needs to be renamed + to #####-src.htm by the whitewasher or the Workflow app. +- This readme will need to be added by the whitewasher or the Workflow app. + - Having it with the eBook makes it obvious, and avoids issues with + procedures changing in the future. + + +Errata process +============== +(eBook 75107 is used as an example) + +1. Download the project files using Errata Workbench, and unzip. +2. Install m2svg if not already done. +3. Make desired changes to 75107-src.htm. +4. Execute command line `m2svg -i 75107-src.htm -o 75107-h.htm` + - The image files will be placed in a subdirectory of the working directory + called images. +5. Move 75107-h.htm to the 75107-h directory. +6. Move the contents of the images directory to the 75107-h/images directory. + - Rename the existing 75107-h/images directory to images-old. + - Move the new images directory to 75107-h. + - Check images-old, move any non-generated images (JPG, PNG, etc.). to + images. + - Remove images-old and any other temporary files. +7. Zip the project directory and upload to Errata Workbench. diff --git a/LICENSE.txt b/LICENSE.txt index 6c72794..b5dba15 100644 --- a/LICENSE.txt +++ b/LICENSE.txt @@ -7,5 +7,5 @@ the "Copyright How-To" at https://www.gutenberg.org. No investigation has been made concerning possible copyrights in jurisdictions other than the United States. Anyone seeking to utilize -this eBook outside of the United States should confirm copyright +this book outside of the United States should confirm copyright status under the laws that apply to them. @@ -1,2 +1,2 @@ -Project Gutenberg (https://www.gutenberg.org) public repository for eBook #78050 -(https://www.gutenberg.org/ebooks/78050) +Project Gutenberg (https://www.gutenberg.org) public repository for +book #78050 (https://www.gutenberg.org/ebooks/78050) |