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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: The Mathematical Analysis of Logic % +% Being an Essay Towards a Calculus of Deductive Reasoning % +% % +% Author: George Boole % +% % +% Release Date: July 28, 2011 [EBook #36884] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK THE MATHEMATICAL ANALYSIS OF LOGIC *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{36884} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% babel: Greek language capabilities. Required. %% +%% %% +%% ifthen: Logical conditionals. 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no need to cross-ref +\newcommand{\atag}{\rlap{\quad$(a)$}} +\newcommand{\aref}{$(a)$} + +\newcommand{\btag}{\rlap{\quad$(b)$}} +\newcommand{\bref}{$(b)$} + +% "Label" tag: Other tag-like labels on displayed equations; no cross-refs +\newcommand{\Ltag}[1]{% + \ifthenelse{\equal{#1}{I}}{% + \tag*{#1\,\qquad} % Pad "I" on the right + }{ + \tag*{#1\qquad} + } +} +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +%% PG BOILERPLATE %% +\PGBoilerPlate +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +Project Gutenberg's The Mathematical Analysis of Logic, by George Boole + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Mathematical Analysis of Logic + Being an Essay Towards a Calculus of Deductive Reasoning + +Author: George Boole + +Release Date: July 28, 2011 [EBook #36884] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE MATHEMATICAL ANALYSIS OF LOGIC *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang +\end{PGtext} +\end{minipage} +\vfill +\end{center} + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\PageSep{i} +\FrontMatter +\begin{center} +\bfseries\large THE MATHEMATICAL ANALYSIS +\vfil + +\Large OF LOGIC, +\vfil + +\normalsize +BEING AN ESSAY TOWARDS A CALCULUS \\ +OF DEDUCTIVE REASONING. +\vfil + +BY GEORGE BOOLE. +\vfil + +\begin{Quote} +>Epikoinwno~usi d`e p~asai a<i >epist~hmai >all'hlais kat`a t`a koin'a. \Typo{Koin'a}{Koin`a} d`e +l'egw, o>~is qr~wntai <ws >ek to'utwn >apodeikn'untes; >all'' o>u per`i <~wn deikn'uousin, +\Typo{o>ude}{o>ud`e} <`o deikn'uousi. \\ +\selectlanguage{english} +\null\hfill\textsc{Aristotle}, \textit{Anal.\ Post.}, lib.~\textsc{i}. cap.~\textsc{xi}. +\end{Quote} +\vfil\vfil + +CAMBRIDGE: \\ +MACMILLAN, BARCLAY, \& MACMILLAN; \\ +LONDON: GEORGE BELL. \\ +\tb[0.25in] \\ +1847 +\normalfont +\end{center} +\PageSep{ii} +\newpage +\normalfont +\null +\vfill +\begin{center} +\scriptsize +PRINTED IN ENGLAND BY \\ +HENDERSON \& SPALDING \\ +LONDON. W.I +\end{center} +\PageSep{1} +\MainMatter + + +\Chapter{Preface.} + +\First{In} presenting this Work to public notice, I deem it not +irrelevant to observe, that speculations similar to those which +it records have, at different periods, occupied my thoughts. +In the spring of the present year my attention was directed +to the question then moved between Sir W.~Hamilton and +Professor De~Morgan; and I was induced by the interest +which it inspired, to resume the almost-forgotten thread of +former inquiries. It appeared to me that, although Logic +might be viewed with reference to the idea of quantity,\footnote + {See \Pageref{42}.} +it +had also another and a deeper system of relations. If it was +lawful to regard it from \emph{without}, as connecting itself through +the medium of Number with the intuitions of Space and Time, +it was lawful also to regard it from \emph{within}, as based upon +facts of another order which have their abode in the constitution +of the Mind. The results of this view, and of the +inquiries which it suggested, are embodied in the following +Treatise. + +It is not generally permitted to an Author to prescribe +the mode in which his production shall be judged; but there +are two conditions which I may venture to require of those +who shall undertake to estimate the merits of this performance. +The first is, that no preconceived notion of the impossibility +of its objects shall be permitted to interfere with that candour +and impartiality which the investigation of Truth demands; +the second is, that their judgment of the system as a whole +shall not be founded either upon the examination of only +\PageSep{2} +a part of it, or upon the measure of its conformity with any +received system, considered as a standard of reference from +which appeal is denied. It is in the general theorems which +occupy the latter chapters of this work,---results to which there +is no existing counterpart,---that the claims of the method, as +a Calculus of Deductive Reasoning, are most fully set forth. + +What may be the final estimate of the value of the system, +I have neither the wish nor the right to anticipate. The +estimation of a theory is not simply determined by its truth\Add{.} +It also depends upon the importance of its subject, and the +extent of its applications; beyond which something must still +be left to the arbitrariness of human Opinion. If the utility +of the application of Mathematical forms to the science of +Logic were solely a question of Notation, I should be content +to rest the defence of this attempt upon a principle which has +been stated by an able living writer: ``Whenever the nature +of the subject permits the reasoning process to be without +danger carried on mechanically, the language should be constructed +on as mechanical principles as possible; while in the +contrary case it should be so constructed, that there shall be +the greatest possible obstacle to a mere mechanical use of it.''\footnote + {Mill's \textit{System of Logic, Ratiocinative and Inductive}, Vol.~\textsc{ii}. p.~292.} +In one respect, the science of Logic differs from all others; +the perfection of its method is chiefly valuable as an evidence +of the speculative truth of its principles. To supersede the +employment of common reason, or to subject it to the rigour +of technical forms, would be the last desire of one who knows +the value of that intellectual toil and warfare which imparts +to the mind an athletic vigour, and teaches it to contend +with difficulties and to rely upon itself in emergencies. +\Signature{\textsc{Lincoln}, \textit{Oct.}~29, 1847.} +\PageSep{3} + + +%[**TN: Macro prints heading "MATHEMATICAL ANALYSIS OF LOGIC."] +\Chapter{Introduction.} + +\First{They} who are acquainted with the present state of the theory +of Symbolical Algebra, are aware, that the validity of the +processes of analysis does not depend upon the interpretation +of the symbols which are employed, but solely upon the laws +of their combination. Every system of interpretation which +does not affect the truth of the relations supposed, is equally +admissible, and it is thus that the same process may, under +one scheme of interpretation, represent the solution of a question +on the properties of numbers, under another, that of +a geometrical problem, and under a third, that of a problem +of dynamics or optics. This principle is indeed of fundamental +importance; and it may with safety be affirmed, that the recent +advances of pure analysis have been much assisted by the +influence which it has exerted in directing the current of +investigation. + +But the full recognition of the consequences of this important +doctrine has been, in some measure, retarded by accidental +circumstances. It has happened in every known form of +analysis, that the elements to be determined have been conceived +as measurable by comparison with some fixed standard. +The predominant idea has been that of magnitude, or more +strictly, of numerical ratio. The expression of magnitude, or +\PageSep{4} +of operations upon magnitude, has been the express object +for which the symbols of Analysis have been invented, and +for which their laws have been investigated. Thus the abstractions +of the modern Analysis, not less than the ostensive +diagrams of the ancient Geometry, have encouraged the notion, +that Mathematics are essentially, as well as actually, the Science +of Magnitude. + +The consideration of that view which has already been stated, +as embodying the true principle of the Algebra of Symbols, +would, however, lead us to infer that this conclusion is by no +means necessary. If every existing interpretation is shewn to +involve the idea of magnitude, it is only by induction that we +can assert that no other interpretation is possible. And it may +be doubted whether our experience is sufficient to render such +an induction legitimate. The history of pure Analysis is, it may +be said, too recent to permit us to set limits to the extent of its +applications. Should we grant to the inference a high degree +of probability, we might still, and with reason, maintain the +sufficiency of the definition to which the principle already stated +would lead us. We might justly assign it as the definitive +character of a true Calculus, that it is a method resting upon +the employment of Symbols, whose laws of combination are +known and general, and whose results admit of a consistent +interpretation. That to the existing forms of Analysis a quantitative +interpretation is assigned, is the result of the circumstances +by which those forms were determined, and is not to +be construed into a universal condition of Analysis. It is upon +the foundation of this general principle, that I purpose to +establish the Calculus of Logic, and that I claim for it a place +among the acknowledged forms of Mathematical Analysis, regardless +that in its object and in its instruments it must at +present stand alone. + +That which renders Logic possible, is the existence in our +minds of general notions,---our ability to conceive of a class, +and to designate its individual members by a common name. +\PageSep{5} +\Pagelabel{5}% +The theory of Logic is thus intimately connected with that of +Language. A successful attempt to express logical propositions +by symbols, the laws of whose combinations should be founded +upon the laws of the mental processes which they represent, +would, so far, be a step toward a philosophical language. But +this is a view which we need not here follow into detail.\footnote + {This view is well expressed in one of Blanco White's Letters:---``Logic is + for the most part a collection of technical rules founded on classification. The + Syllogism is nothing but a result of the classification of things, which the mind + naturally and necessarily forms, in forming a language. All abstract terms are + classifications; or rather the labels of the classes which the mind has settled.''---\textit{Memoirs + of the Rev.\ Joseph Blanco White}, vol.~\textsc{ii}. p.~163. See also, for a very + lucid introduction, Dr.~Latham's \textit{First Outlines of Logic applied to Language}, + Becker's \textit{German Grammar,~\etc.} Extreme Nominalists make Logic entirely + dependent upon language. For the opposite view, see Cudworth's \textit{Eternal + and Immutable Morality}, Book~\textsc{iv}. Chap.~\textsc{iii}.} +Assuming the notion of a class, we are able, from any conceivable +collection of objects, to separate by a mental act, those +which belong to the given class, and to contemplate them apart +from the rest. Such, or a similar act of election, we may conceive +to be repeated. The group of individuals left under consideration +may be still further limited, by mentally selecting +those among them which belong to some other recognised class, +as well as to the one before contemplated. And this process +may be repeated with other elements of distinction, until we +arrive at an individual possessing all the distinctive characters +which we have taken into account, and a member, at the same +time, of every class which we have enumerated. It is in fact +a method similar to this which we employ whenever, in common +language, we accumulate descriptive epithets for the sake of +more precise definition. + +Now the several mental operations which in the above case +we have supposed to be performed, are subject to peculiar laws. +It is possible to assign relations among them, whether as respects +the repetition of a given operation or the succession of +different ones, or some other particular, which are never violated. +It is, for example, true that the result of two successive acts is +\PageSep{6} +unaffected by the order in which they are performed; and there +are at least two other laws which will be pointed out in the +proper place. These will perhaps to some appear so obvious as +to be ranked among necessary truths, and so little important +as to be undeserving of special notice. And probably they are +noticed for the first time in this Essay. Yet it may with confidence +be asserted, that if they were other than they are, the +entire mechanism of reasoning, nay the very laws and constitution +of the human intellect, would be vitally changed. A Logic +might indeed exist, but it would no longer be the Logic we +possess. + +Such are the elementary laws upon the existence of which, +and upon their capability of exact symbolical expression, the +method of the following Essay is founded; and it is presumed +that the object which it seeks to attain will be thought to +have been very fully accomplished. Every logical proposition, +whether categorical or hypothetical, will be found to be capable +of exact and rigorous expression, and not only will the laws of +conversion and of syllogism be thence deducible, but the resolution +of the most complex systems of propositions, the separation +of any proposed element, and the expression of its value in +terms of the remaining elements, with every subsidiary relation +involved. Every process will represent deduction, every +mathematical consequence will express a logical inference. The +generality of the method will even permit us to express arbitrary +operations of the intellect, and thus lead to the demonstration +of general theorems in logic analogous, in no slight +degree, to the general theorems of ordinary mathematics. No +inconsiderable part of the pleasure which we derive from the +application of analysis to the interpretation of external nature, +arises from the conceptions which it enables us to form of the +universality of the dominion of law. The general formulæ to +which we are conducted seem to give to that element a visible +presence, and the multitude of particular cases to which they +apply, demonstrate the extent of its sway. Even the symmetry +\PageSep{7} +of their analytical expression may in no fanciful sense be +deemed indicative of its harmony and its consistency. Now I +do not presume to say to what extent the same sources of +pleasure are opened in the following Essay. The measure of +that extent may be left to the estimate of those who shall think +the subject worthy of their study. But I may venture to +assert that such occasions of intellectual gratification are not +here wanting. The laws we have to examine are the laws of +one of the most important of our mental faculties. The mathematics +we have to construct are the mathematics of the human +intellect. Nor are the form and character of the method, apart +from all regard to its interpretation, undeserving of notice. +There is even a remarkable exemplification, in its general +theorems, of that species of excellence which consists in freedom +from exception. And this is observed where, in the corresponding +cases of the received mathematics, such a character +is by no means apparent. The few who think that there is that +in analysis which renders it deserving of attention for its own +sake, may find it worth while to study it under a form in which +every equation can be solved and every solution interpreted. +Nor will it lessen the interest of this study to reflect that every +peculiarity which they will notice in the form of the Calculus +represents a corresponding feature in the constitution of their +own minds. + +It would be premature to speak of the value which this +method may possess as an instrument of scientific investigation. +I speak here with reference to the theory of reasoning, and to +the principle of a true classification of the forms and cases of +Logic considered as a Science.\footnote + {``Strictly a Science''; also ``an Art.''---\textit{Whately's Elements of Logic.} Indeed + ought we not to regard all Art as applied Science; unless we are willing, with + ``the multitude,'' to consider Art as ``guessing and aiming well''?---\textit{Plato, + Philebus.}} +The aim of these investigations +was in the first instance confined to the expression of the +received logic, and to the forms of the Aristotelian arrangement, +\PageSep{8} +but it soon became apparent that restrictions were thus introduced, +which were purely arbitrary and had no foundation in +the nature of things. These were noted as they occurred, and +will be discussed in the proper place. When it became necessary +to consider the subject of hypothetical propositions (in which +comparatively less has been done), and still more, when an +interpretation was demanded for the general theorems of the +Calculus, it was found to be imperative to dismiss all regard for +precedent and authority, and to interrogate the method itself for +an expression of the just limits of its application. Still, however, +there was no special effort to arrive at novel results. But +among those which at the time of their discovery appeared to be +such, it may be proper to notice the following. + +A logical proposition is, according to the method of this Essay, +expressible by an equation the form of which determines the +rules of conversion and of transformation, to which the given +proposition is subject. Thus the law of what logicians term +simple conversion, is determined by the fact, that the corresponding +equations are symmetrical, that they are unaffected by +a mutual change of place, in those symbols which correspond +to the convertible classes. The received laws of conversion +were thus determined, and afterwards another system, which is +thought to be more elementary, and more general. See Chapter, +\ChapRef{5}{On the Conversion of Propositions}. + +The premises of a syllogism being expressed by equations, the +elimination of a common symbol between them leads to a third +equation which expresses the conclusion, this conclusion being +always the most general possible, whether Aristotelian or not. +Among the cases in which no inference was possible, it was +found, that there were two distinct forms of the final equation. +It was a considerable time before the explanation of this fact +was discovered, but it was at length seen to depend upon the +presence or absence of a true medium of comparison between +the premises. The distinction which is thought to be new +is illustrated in the Chapter, \ChapRef{6}{On Syllogisms}. +\PageSep{9} + +The nonexclusive character of the disjunctive conclusion of +a hypothetical syllogism, is very clearly pointed out in the +examples of this species of argument. + +The class of logical problems illustrated in the chapter, \ChapRef{9}{On +the Solution of Elective Equations}, is conceived to be new: and +it is believed that the method of that chapter affords the means +of a perfect analysis of any conceivable system of propositions, +an end toward which the rules for the conversion of a single +categorical proposition are but the first step. + +However, upon the originality of these or any of these views, +I am conscious that I possess too slight an acquaintance with the +literature of logical science, and especially with its older literature, +to permit me to speak with confidence. + +It may not be inappropriate, before concluding these observations, +to offer a few remarks upon the general question of the +use of symbolical language in the mathematics. Objections +have lately been very strongly urged against this practice, on +the ground, that by obviating the necessity of thought, and +substituting a reference to general formulæ in the room of +personal effort, it tends to weaken the reasoning faculties. + +Now the question of the use of symbols may be considered +in two distinct points of view. First, it may be considered with +reference to the progress of scientific discovery, and secondly, +with reference to its bearing upon the discipline of the intellect. + +And with respect to the first view, it may be observed that +as it is one fruit of an accomplished labour, that it sets us at +liberty to engage in more arduous toils, so it is a necessary +result of an advanced state of science, that we are permitted, +and even called upon, to proceed to higher problems, than those +which we before contemplated. The practical inference is +obvious. If through the advancing power of scientific methods, +we find that the pursuits on which we were once engaged, +afford no longer a sufficiently ample field for intellectual effort, +the remedy is, to proceed to higher inquiries, and, in new +tracks, to seek for difficulties yet unsubdued. And such is, +\PageSep{10} +indeed, the actual law of scientific progress. We must be +content, either to abandon the hope of further conquest, or to +employ such aids of symbolical language, as are proper to the +stage of progress, at which we have arrived. Nor need we fear +to commit ourselves to such a course. We have not yet arrived +so near to the boundaries of possible knowledge, as to suggest +the apprehension, that scope will fail for the exercise of the +inventive faculties. + +In discussing the second, and scarcely less momentous question +of the influence of the use of symbols upon the discipline +of the intellect, an important distinction ought to be made. It +is of most material consequence, whether those symbols are +used with a full understanding of their meaning, with a perfect +comprehension of that which renders their use lawful, and an +ability to expand the abbreviated forms of reasoning which they +induce, into their full syllogistic \Typo{devolopment}{development}; or whether they +are mere unsuggestive characters, the use of which is suffered +to rest upon authority. + +The answer which must be given to the question proposed, +will differ according as the one or the other of these suppositions +is admitted. In the former case an intellectual discipline of a +high order is provided, an exercise not only of reason, but of +the faculty of generalization. In the latter case there is no +mental discipline whatever. It were perhaps the best security +against the danger of an unreasoning reliance upon symbols, +on the one hand, and a neglect of their just claims on the other, +that each subject of applied mathematics should be treated in the +spirit of the methods which were known at the time when the +application was made, but in the best form which those methods +have assumed. The order of attainment in the individual mind +would thus bear some relation to the actual order of scientific +discovery, and the more abstract methods of the higher analysis +would be offered to such minds only, as were prepared to +receive them. + +The relation in which this Essay stands at once to Logic and +\PageSep{11} +to Mathematics, may further justify some notice of the question +which has lately been revived, as to the relative value of the two +studies in a liberal education. One of the chief objections which +have been urged against the study of Mathematics in general, is +but another form of that which has been already considered with +respect to the use of symbols in particular. And it need not here +be further dwelt upon, than to notice, that if it avails anything, +it applies with an equal force against the study of Logic. The +canonical forms of the Aristotelian syllogism are really symbolical; +only the symbols are less perfect of their kind than those +of mathematics. If they are employed to test the validity of an +argument, they as truly supersede the exercise of reason, as does +a reference to a formula of analysis. Whether men do, in the +present day, make this use of the Aristotelian canons, except as +a special illustration of the rules of Logic, may be doubted; yet +it cannot be questioned that when the authority of Aristotle was +dominant in the schools of Europe, such applications were habitually +made. And our argument only requires the admission, +that the case is possible. + +But the question before us has been argued upon higher +grounds. Regarding Logic as a branch of Philosophy, and defining +Philosophy as the ``science of a real existence,'' and ``the +research of causes,'' and assigning as its \emph{main} business the investigation +of the ``why, (\textgreek{t`o d'ioti}),'' while Mathematics display +only the ``that, (\textgreek{t`o <ot`i}),'' Sir W.~Hamilton has contended, +not simply, that the superiority rests with the study of Logic, +but that the study of Mathematics is at once dangerous and useless.\footnote + {\textit{Edinburgh Review}, vol.~\textsc{lxii}. p.~409, and \textit{Letter to A. De~Morgan, Esq.}} +The pursuits of the mathematician ``have not only not +trained him to that acute scent, to that delicate, almost instinctive, +tact which, in the twilight of probability, the search and +discrimination of its finer facts demand; they have gone to cloud +his vision, to indurate his touch, to all but the blazing light, the +iron chain of demonstration, and left him out of the narrow confines +of his science, to a passive \emph{credulity} in any premises, or to +\PageSep{12} +an absolute \emph{incredulity} in all.'' In support of these and of other +charges, both argument and copious authority are adduced.\footnote + {The arguments are in general better than the authorities. Many writers + quoted in condemnation of mathematics (Aristo, Seneca, Jerome, Augustine, + Cornelius Agrippa,~\etc.)\ have borne a no less explicit testimony against other + sciences, nor least of all, against that of logic. The treatise of the last named + writer \textit{De~Vanitate Scientiarum}, must surely have been referred to by mistake.---\textit{Vide} + cap.~\textsc{cii}.} +I shall not attempt a complete discussion of the topics which +are suggested by these remarks. My object is not controversy, +and the observations which follow are offered not in the spirit +of antagonism, but in the hope of contributing to the formation +of just views upon an important subject. Of Sir W.~Hamilton +it is impossible to speak otherwise than with that respect which +is due to genius and learning. + +Philosophy is then described as the \emph{science of a real existence} +\Pagelabel{12}% +and \emph{the research of causes}. And that no doubt may rest upon +the meaning of the word \emph{cause}, it is further said, that philosophy +``mainly investigates the \emph{why}.'' These definitions are common +among the ancient writers. Thus Seneca, one of Sir W.~Hamilton's +authorities, \textit{Epistle}~\textsc{lxxxviii}., ``The philosopher seeks +and knows the \emph{causes} of natural things, of which the mathematician +searches out and computes the numbers and the measures.'' +It may be remarked, in passing, that in whatever +degree the belief has prevailed, that the business of philosophy +is immediately with \emph{causes}; in the same degree has every +science whose object is the investigation of \emph{laws}, been lightly +esteemed. Thus the Epistle to which we have referred, bestows, +by contrast with Philosophy, a separate condemnation on Music +and Grammar, on Mathematics and Astronomy, although it is +that of Mathematics only that Sir W.~Hamilton has quoted. + +Now we might take our stand upon the conviction of many +thoughtful and reflective minds, that in the extent of the meaning +above stated, Philosophy is impossible. The business of +true Science, they conclude, is with laws and phenomena. The +nature of Being, the mode of the operation of Cause, the \emph{why}, +\PageSep{13} +they hold to be beyond the reach of our intelligence. But we +do not require the vantage-ground of this position; nor is it +doubted that whether the aim of Philosophy is attainable or not, +the desire which impels us to the attempt is an instinct of our +higher nature. Let it be granted that the problem which has +baffled the efforts of ages, is not a hopeless one; that the +``science of a real existence,'' and ``the research of causes,'' +``that kernel'' for which ``Philosophy is still militant,'' do +not transcend the limits of the human intellect. I am then +compelled to assert, that according to this view of the nature of +Philosophy, \emph{Logic forms no part of it}. On the principle of +a true classification, we ought no longer to associate Logic and +Metaphysics, but Logic and Mathematics. + +Should any one after what has been said, entertain a doubt +upon this point, I must refer him to the evidence which will be +afforded in the following Essay. He will there see Logic resting +like Geometry upon axiomatic truths, and its theorems constructed +upon that general doctrine of symbols, which constitutes +the foundation of the recognised Analysis. In the Logic +of Aristotle he will be led to view a collection of the formulæ +of the science, expressed by another, but, (it is thought) less +perfect scheme of symbols. I feel bound to contend for the +absolute exactness of this parallel. It is no escape from the conclusion +to which it points to assert, that Logic not only constructs +a science, but also inquires into the origin and the nature of its +own principles,---a distinction which is denied to Mathematics. +``It is wholly beyond the domain of mathematicians,'' it is said, +``to inquire into the origin and nature of their principles.''---% +\textit{Review}, page~415. But upon what ground can such a distinction +be maintained? What definition of the term Science will +be found sufficiently arbitrary to allow such differences? + +The application of this conclusion to the question before us is +clear and decisive. The mental discipline which is afforded by +the study of Logic, \emph{as an exact science}, is, in species, the same +as that afforded by the study of Analysis. +\PageSep{14} + +Is it then contended that either Logic or Mathematics can +supply a perfect discipline to the Intellect? The most careful +and unprejudiced examination of this question leads me to doubt +whether such a position can be maintained. The exclusive claims +of either must, I believe, be abandoned, nor can any others, partaking +of a like exclusive character, be admitted in their room. +It is an important observation, which has more than once been +made, that it is one thing to arrive at correct premises, and another +thing to deduce logical conclusions, and that the business of life +depends more upon the former than upon the latter. The study +of the exact sciences may teach us the one, and it may give us +some general preparation of knowledge and of practice for the +attainment of the other, but it is to the union of thought with +action, in the field of Practical Logic, the arena of Human Life, +that we are to look for its fuller and more perfect accomplishment. + +I desire here to express my conviction, that with the advance +of our knowledge of all true science, an ever-increasing +harmony will be found to prevail among its separate branches. +The view which leads to the rejection of one, ought, if consistent, +to lead to the rejection of others. And indeed many +of the authorities which have been quoted against the study +of Mathematics, are even more explicit in their condemnation of +Logic. ``Natural science,'' says the Chian Aristo, ``is above us, +Logical science does not concern us.'' When such conclusions +are founded (as they often are) upon a deep conviction of the +preeminent value and importance of the study of Morals, we +admit the premises, but must demur to the inference. For it +has been well said by an ancient writer, that it is the ``characteristic +of the liberal sciences, not that they conduct us to Virtue, +but that they prepare us for Virtue;'' and Melancthon's sentiment, +``abeunt studia in mores,'' has passed into a proverb. +Moreover, there is a common ground upon which all sincere +votaries of truth may meet, exchanging with each other the +language of Flamsteed's appeal to Newton, ``The works of the +Eternal Providence will be better understood through your +labors and mine.'' +\PageSep{15} + + +\Chapter{First Principles.} + +\First{Let} us employ the symbol~$1$, or unity, to represent the +Universe, and let us understand it as comprehending every +conceivable class of objects whether actually existing or not, +it being premised that the same individual may be found in +more than one class, inasmuch as it may possess more than one +quality in common with other individuals. Let us employ the +letters $X$,~$Y$,~$Z$, to represent the individual members of classes, +$X$~applying to every member of one class, as members of that +particular class, and $Y$~to every member of another class as +members of such class, and so on, according to the received language +of treatises on Logic. + +Further let us conceive a class of symbols $x$,~$y$,~$z$, possessed +of the following character. + +The symbol~$x$ operating upon any subject comprehending +individuals or classes, shall be supposed to select from that +subject all the~$X$s which it contains. In like manner the symbol~$y$, +operating upon any subject, shall be supposed to select from +it all individuals of the class~$Y$ which are comprised in it, and +so on. + +When no subject is expressed, we shall suppose~$1$ (the Universe) +to be the subject understood, so that we shall have +\[ +x = x\quad (1), +\] +the meaning of either term being the selection from the Universe +of all the~$X$s which it contains, and the result of the operation +\PageSep{16} +being in common language, the class~$X$, \ie~the class of which +each member is an~$X$. + +From these premises it will follow, that the product~$xy$ will +represent, in succession, the selection of the class~$Y$, and the +selection from the class~$Y$ of such individuals of the class~$X$ as +are contained in it, the result being the class whose members are +both $X$s~and~$Y$s. And in like manner the product~$xyz$ will +represent a compound operation of which the successive elements +are the selection of the class~$Z$, the selection from it of +such individuals of the class~$Y$ as are contained in it, and the +selection from the result thus obtained of all the individuals of +the class~$X$ which it contains, the final result being the class +common to $X$,~$Y$, and~$Z$. + +From the nature of the operation which the symbols $x$,~$y$,~$z$, +are conceived to represent, we shall designate them as elective +symbols. An expression in which they are involved will be +called an elective function, and an equation of which the members +are elective functions, will be termed an elective equation. + +It will not be necessary that we should here enter into the +analysis of that mental operation which we have represented by +the elective symbol. It is not an act of Abstraction according +to the common acceptation of that term, because we never lose +sight of the concrete, but it may probably be referred to an exercise +of the faculties of Comparison and Attention. Our present +concern is rather with the laws of combination and of succession, +by which its results are governed, and of these it will suffice to +notice the following. + +1st. The result of an act of election is independent of the +grouping or classification of the subject. + +Thus it is indifferent whether from a group of objects considered +as a whole, we select the class~$X$, or whether we divide +the group into two parts, select the~$X$s from them separately, +and then connect the results in one aggregate conception. + +We may express this law mathematically by the equation +\[ +x(u + v) = xu + xv, +\] +\PageSep{17} +$u + v$ representing the undivided subject, and $u$~and~$v$ the +component parts of it. + +2nd. It is indifferent in what order two successive acts of +election are performed. + +Whether from the class of animals we select sheep, and from +the sheep those which are horned, or whether from the class of +animals we select the horned, and from these such as are sheep, +the result is unaffected. In either case we arrive at the class +\emph{horned sheep}. + +The symbolical expression of this law is +\[ +xy = yx. +\] + +3rd. The result of a given act of election performed twice, +or any number of times in succession, is the result of the same +act performed once. + +If from a group of objects we select the~$X$s, we obtain a class +of which all the members are~$X$s. If we repeat the operation +on this class no further change will ensue: in selecting the~$X$s +we take the whole. Thus we have +\[ +xx = x, +\] +or +\[ +x^{2} = x; +\] +and supposing the same operation to be $n$~times performed, we +have +\[ +x^{n} = x, +\] +which is the mathematical expression of the law above stated.\footnote + {The office of the elective symbol~$x$, is to select individuals comprehended + in the class~$X$. Let the class~$X$ be supposed to embrace the universe; then, + whatever the class~$Y$ may be, we have + \[ + xy = y. + \] + The office which $x$~performs is now equivalent to the symbol~$+$, in one at + least of its interpretations, and the index law~\Eqref{(3)} gives + \[ + +^{n} = +, + \] + which is the known property of that symbol.} + +The laws we have established under the symbolical forms +\begin{align*} +x(u + v) &= xu + xv, +\Tag{(1)} \\ +xy &= yx, +\Tag{(2)} \\ +x^{n} &= x, +\Tag{(3)} +\end{align*} +\PageSep{18} +are sufficient for the basis of a Calculus. From the first of these, +it appears that elective symbols are \emph{distributive}, from the second +that they are \emph{commutative}; properties which they possess in +common with symbols of \emph{quantity}, and in virtue of which, all +the processes of common algebra are applicable to the present +system. The one and sufficient axiom involved in this application +is that equivalent operations performed upon equivalent +subjects produce equivalent results.\footnote + {It is generally asserted by writers on Logic, that all reasoning ultimately + depends on an application of the dictum of Aristotle, \textit{de omni et~nullo}. ``Whatever + is predicated universally of any class of things, may be predicated in like + manner of any thing comprehended in that class.'' But it is agreed that this + dictum is not immediately applicable in all cases, and that in a majority of + instances, a certain previous process of reduction is necessary. What are the + elements involved in that process of reduction? Clearly they are as much + a part of general reasoning as the dictum itself. + + Another mode of considering the subject resolves all reasoning into an application + of one or other of the following canons,~viz.\ + + 1. If two terms agree with one and the same third, they agree with each + other. + + 2. If one term agrees, and another disagrees, with one and the same third, + these two disagree with each other. + + But the application of these canons depends on mental acts equivalent to + those which are involved in the before-named process of reduction. We have to + select individuals from classes, to convert propositions,~\etc., before we can avail + ourselves of their guidance. Any account of the process of reasoning is insufficient, + which does not represent, as well the laws of the operation which the + mind performs in that process, as the primary truths which it recognises and + applies. + + It is presumed that the laws in question are adequately represented by the + fundamental equations of the present Calculus. The proof of this will be found + in its capability of expressing propositions, and of exhibiting in the results of + its processes, every result that may be arrived at by ordinary reasoning.} + +The third law~\Eqref{(3)} we shall denominate the index law. It is +peculiar to elective symbols, and will be found of great importance +in enabling us to reduce our results to forms meet for +interpretation. + +From the circumstance that the processes of algebra may be +applied to the present system, it is not to be inferred that the +interpretation of an elective equation will be unaffected by such +processes. The expression of a truth cannot be negatived by +\PageSep{19} +a legitimate operation, but it may be limited. The equation +$y = z$ implies that the classes $Y$~and~$Z$ are equivalent, member +for member. Multiply it by a factor~$x$, and we have +\[ +xy = xz, +\] +which expresses that the individuals which are common to the +classes $X$~and~$Y$ are also common to $X$~and~$Z$, and \textit{vice versâ}. +This is a perfectly legitimate inference, but the fact which it +declares is a less general one than was asserted in the original +proposition. +\PageSep{20} + + +\Chapter{Of Expression and Interpretation.} + +\begin{Abstract} +A Proposition is a sentence which either affirms or denies, as, All men are +mortal, No creature is independent. + +A Proposition has necessarily two terms, as \emph{men}, \emph{mortal}; the former of which, +or the one spoken of, is called the subject; the latter, or that which is affirmed +or denied of the subject, the predicate. These are connected together by the +copula~\emph{is}, or \emph{is not}, or by some other modification of the substantive verb. + +The substantive verb is the only verb recognised in Logic; all others are +resolvable by means of the verb \emph{to be} and a participle or adjective, \eg~``The +Romans conquered''; the word conquered is both copula and predicate, being +equivalent to ``were (copula) victorious'' (predicate). + +A Proposition must either be affirmative or negative, and must be also either +universal or particular. Thus we reckon in all, four kinds of pure categorical +Propositions. + +1st. Universal-affirmative, usually represented by~$A$, +\[ +\text{Ex. All $X$s are $Y$s.} +\] + +2nd. Universal-negative, usually represented by~$E$, +\[ +\text{Ex. No $X$s are $Y$s.} +\] + +3rd. Particular-affirmative, usually represented by~$I$, +\[ +\text{Ex. Some $X$s are $Y$s.} +\] + +4th. Particular-negative, usually represented by~$O$,\footnote + {The above is taken, with little variation, from the Treatises of Aldrich + and Whately.} +\[ +\text{Ex. Some $X$s are not $Y$s.} +\] +\end{Abstract} + +1. To express the class, not-$X$, that is, the class including +all individuals that are not~$X$s. + +The class~$X$ and the class not-$X$ together make the Universe. +But the Universe is~$1$, and the class~$X$ is determined by the +symbol~$x$, therefore the class not-$X$ will be determined by +the symbol~$1 - x$. +\PageSep{21} + +Hence the office of the symbol $1 - x$ attached to a given +subject will be, to select from it all the not-$X$s which it +contains. + +And in like manner, as the product~$xy$ expresses the entire +class whose members are both $X$s and~$Y$s, the symbol $y(1 - x)$ +will represent the class whose members are $Y$s but not~$X$s, +and the symbol $(1 - x)(1 - y)$ the entire class whose members +are neither $X$s~nor~$Y$s. + +2. To express the Proposition, All $X$s are~$Y$s. + +As all the~$X$s which exist are found in the class~$Y$, it is +obvious that to select out of the Universe all~$Y$s, and from +these to select all~$X$s, is the same as to select at once from the +Universe all~$X$s. + +Hence +\[ +xy = x, +\] +or +\[ +x(1 - y) = 0. +\Tag{(4)} +\] + +3. To express the Proposition, No $X$s are~$Y$s. + +To assert that no $X$s are~$Y$s, is the same as to assert that +there are no terms common to the classes $X$~and~$Y$. Now +all individuals common to those classes are represented by~$xy$. +Hence the Proposition that No~$X$s are~$Y$s, is represented by +the equation +\[ +xy = 0. +\Tag{(5)} +\] + +4. To express the Proposition, Some $X$s are~$Y$s. + +If some $X$s are~$Y$s, there are some terms common to the +classes $X$~and~$Y$. Let those terms constitute a separate class~$V$, +to which there shall correspond a separate elective symbol~$v$, +then +\[ +v = xy. +\Tag{(6)} +\] +And as $v$~includes all terms common to the classes $X$~and~$Y$, +we can indifferently interpret it, as Some~$X$s, or Some~$Y$s. +\PageSep{22} + +5. To express the Proposition, Some $X$s are not~$Y$s. + +In the last equation write $1 - y$ for~$y$, and we have +\[ +v = x(1 - y), +\Tag{(7)} +\] +the interpretation of~$v$ being indifferently Some~$X$s or Some +not-$Y$s. + +The above equations involve the complete theory of categorical +Propositions, and so far as respects the employment of +analysis for the deduction of logical inferences, nothing more +can be desired. But it may be satisfactory to notice some particular +forms deducible from the third and fourth equations, and +susceptible of similar application. + +If we multiply the equation~\Eqref{(6)} by~$x$, we have +\[ +vx = x^{2}y = xy\quad\text{by~\Eqref{(3)}.} +\] + +Comparing with~\Eqref{(6)}, we find +\[ +v = vx, +\] +or +\[ +v(1 - x) = 0. +\Tag{(8)} +\] + +And multiplying~\Eqref{(6)} by~$y$, and reducing in a similar manner, +we have +\[ +v = vy, +\] +or +\[ +v(1 - y) = 0. +\Tag{(9)} +\] + +Comparing \Eqref{(8)} and~\Eqref{(9)}, +\[ +vx = vy = v. +\Tag{(10)} +\] + +And further comparing \Eqref{(8)} and~\Eqref{(9)} with~\Eqref{(4)}, we have as the +equivalent of this system of equations the Propositions +\[ +\begin{aligned} +&\text{All $V$s are~$X$s} \\ +&\text{All $V$s are~$Y$s} +\end{aligned} +\Rbrace{2}. +\] + +The system~\Eqref{(10)} might be used to replace~\Eqref{(6)}, or the single +equation +\[ +vx = vy, +\Tag{(11)} +\] +might be used, assigning to~$vx$ the interpretation, Some~$X$s, and +to~$vy$ the interpretation, Some~$Y$s. But it will be observed that +\PageSep{23} +this system does not express quite so much as the single equation~\Eqref{(6)}, +from which it is derived. Both, indeed, express the +Proposition, Some~$X$s are~$Y$s, but the system~\Eqref{(10)} does not +imply that the class~$V$ includes \emph{all} the terms that are common +to $X$~and~$Y$. + +In like manner, from the equation~\Eqref{(7)} which expresses the +Proposition Some~$X$s are not~$Y$s, we may deduce the system +\[ +vx = v(1 - y) = v, +\Tag{(12)} +\] +in which the interpretation of~$v(1 - y)$ is Some not-$Y$s. Since +in this case $vy = 0$, we must of course be careful not to interpret~$vy$ +as Some~$Y$s. + +If we multiply the first equation of the system~\Eqref{(12)},~viz. +\[ +vx = v(1 - y), +\] +by~$y$, we have +\begin{align*} +vxy &= vy(1 - y); \\ +\therefore vxy &= 0, +\Tag{(13)} +\end{align*} +which is a form that will occasionally present itself. It is not +necessary to revert to the primitive equation in order to interpret +this, for the condition that $vx$~represents Some~$X$s, shews +us by virtue of~\Eqref{(5)}, that its import will be +\[ +\text{Some~$X$s are not~$Y$s,} +\] +the subject comprising \emph{all} the~$X$s that are found in the class~$V$. + +Universally in these cases, difference of form implies a difference +of interpretation with respect to the auxiliary symbol~$v$, +and each form is interpretable by itself. + +Further, these differences do not introduce into the Calculus +a needless perplexity. It will hereafter be seen that they give +a precision and a definiteness to its conclusions, which could not +otherwise be secured. + +Finally, we may remark that all the equations by which +particular truths are expressed, are deducible from any one +general equation, expressing any one general Proposition, from +which those particular Propositions are necessary deductions. +\PageSep{24} +This has been partially shewn already, but it is much more fully +exemplified in the following scheme. + +The general equation +\[ +x = y, +\] +implies that the classes $X$~and~$Y$ are equivalent, member for +member; that every individual belonging to the one, belongs +to the other also. Multiply the equation by~$x$, and we have +\begin{align*} +x^{2} &= xy; \\ +\therefore x &= xy, +\end{align*} +which implies, by~\Eqref{(4)}, that all~$X$s are~$Y$s. Multiply the same +equation by~$y$, and we have in like manner +\[ +y = xy; +\] +the import of which is, that all~$Y$s are~$X$s. Take either of these +equations, the latter for instance, and writing it under the form +\[ +(1 - x)y = 0, +\] +we may regard it as an equation in which~$y$, an unknown +quantity, is sought to be expressed in terms of~$x$. Now it +will be shewn when we come to treat of the Solution of Elective +Equations (and the result may here be verified by substitution) +that the most general solution of this equation is +\[ +y = vx, +\] +which implies that All~$Y$s are~$X$s, and that Some~$X$s are~$Y$s. +Multiply by~$x$, and we have +\[ +vy = vx, +\] +which indifferently implies that some~$Y$s are~$X$s and some~$X$s +are~$Y$s, being the particular form at which we before arrived. + +For convenience of reference the above and some other +results have been classified in the annexed Table, the first +column of which contains propositions, the second equations, +and the third the conditions of final interpretation. It is to +be observed, that the auxiliary equations which are given in +this column are not independent: they are implied either +in the equations of the second column, or in the condition for +\PageSep{25} +the interpretation of~$v$. But it has been thought better to write +them separately, for greater ease and convenience. And it is +further to be borne in mind, that although three different forms +are given for the expression of each of the \emph{particular} propositions, +everything is really included in the first form. +\begin{table}[hbt!] +\caption{TABLE.} +\footnotesize +\begin{alignat*}{3} +&\text{The class~$X$} &&x \\ +&\text{The class not-$X$} &&1 - x \\ +% +&\!\begin{aligned} +&\text{All~$X$s are~$Y$s} \\ +&\text{All~$Y$s are~$X$s} +\end{aligned}\Rbrace{2} && x = y \\ +% +&\text{All~$X$s are~$Y$s} && x(1 - y) = 0 \\ +&\text{No~$X$s are~$Y$s} && \PadTo[r]{x(1 - y) = 0}{xy = 0} \\ +% +&\!\begin{aligned} +&\text{All~$Y$s are~$X$s} \\ +&\text{Some~$X$s are~$Y$s} +\end{aligned}\Rbrace{2} && y = vx +&&\begin{aligned} +&vx = \text{Some~$X$s} \\ +&v(1 - x) = 0. +\end{aligned} \\[8pt] +% +&\!\begin{aligned} +&\text{No~$Y$s are~$X$s} \\ +&\text{Some not-$X$s are~$Y$s} +\end{aligned}\Rbrace{2} && y = v(1 - x) +&&\begin{aligned} +v(1 - x) &= \text{some not-$X$s} \\ +vx &= 0. +\end{aligned} \\[8pt] +% +&\text{Some~$X$s are~$Y$s} && +\Lbrace{3}\begin{aligned} +&v = xy \\ +\text{or } &vx = vy \\ +\text{or } &vx(1 - y) = 0 +\end{aligned}\quad && +\begin{aligned} +&v = \text{some~$X$s or some~$Y$s} \\ +&vx = \text{some~$X$s},\ vy = \text{some~$Y$s} \\ +&v(1 - x) = 0,\ v(1 - y) = 0. +\end{aligned} \\[8pt] +% +&\text{Some~$X$s are not~$Y$s} && +\Lbrace{3}\begin{aligned} +&v = x(1 - y) \\ +\text{or } &vx = v(1 - y) \\ +\text{or } &vxy = 0 +\end{aligned} && +\begin{aligned} +&v = \text{some~$X$s, or some not-$Y$s} \\ +&vx = \text{some~$X$s}, v(1 - y) = \text{some not-$Y$s} \\ +&v(1 - x) = 0,\ vy = 0. +\end{aligned} +\end{alignat*} +\end{table} +\PageSep{26} + + +\Chapter{Of the Conversion of Propositions.} + +\begin{Abstract} +A Proposition is said to be converted when its terms are transposed; when +nothing more is done, this is called simple conversion; \eg +\begin{align*} +&\text{No virtuous man is a tyrant, \emph{is converted into}} \\ +&\text{No tyrant is a virtuous man.} +\intertext{\indent +Logicians also recognise conversion \textit{per accidens}, or by limitation, \eg} +&\text{All birds are animals, \emph{is converted into}} \\ +&\text{Some animals are birds.} +\intertext{And conversion by \emph{contraposition} or \emph{negation}, as} +&\text{Every poet is a man of genius, \emph{converted into}} \\ +&\text{He who is not a man of genius is not a poet.} +\end{align*} + +In one of these three ways every Proposition may be illatively converted, viz.\ +$E$~and~$I$ simply, $A$~and~$O$ by negation, $A$~and~$E$ by limitation. +\end{Abstract} + +The primary canonical forms already determined for the +expression of Propositions, are +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s,} &x(1 - y) &= 0, +\Ltag{A} \\ +&\text{No~$X$s are~$Y$s,} &xy &= 0, +\Ltag{E} \\ +&\text{Some~$X$s are~$Y$s,} &v &= xy, +\Ltag{I} \\ +&\text{Some~$X$s are not~$Y$s,} &v &= x(1 - y). +\Ltag{O} +\end{alignat*} + +On examining these, we perceive that $E$~and~$I$ are symmetrical +with respect to $x$~and~$y$, so that $x$~being changed into~$y$, +and $y$~into~$x$, the equations remain unchanged. Hence $E$~and~$I$ +may be interpreted into +\begin{gather*} +\text{No~$Y$s are~$X$s,} \\ +\text{Some~$Y$s are~$X$s,} +\end{gather*} +respectively. Thus we have the known rule of the Logicians, +that particular affirmative and universal negative Propositions +admit of simple conversion. +\PageSep{27} + +The equations $A$~and~$O$ may be written in the forms +\begin{gather*} +(1 - y)\bigl\{1 - (1 - x)\bigr\} = 0, \\ +v = (1 - y)\bigl\{1 - (1 - x)\bigr\}. +\end{gather*} + +Now these are precisely the forms which we should have +obtained if we had in those equations changed $x$~into~$1 - y$, +and $y$~into~$1 - x$, which would have represented the changing +in the original Propositions of the~$X$s into not-$Y$s, and the~$Y$s +into not-$X$s, the resulting Propositions being +\begin{gather*} +\text{All not-$Y$s are not-$X$s,} \\ +\text{Some not-$Y$s are not not-$X$s.}\atag +\end{gather*} +Or we may, by simply inverting the order of the factors in the +second member of~$O$, and writing it in the form +\[ +v = (1 - y)x, +\] +interpret it by~$I$ into +\[ +\text{Some not-$Y$s are~$X$s,} +\] +which is really another form of~\aref. Hence follows the rule, +that universal affirmative and particular negative Propositions +admit of negative conversion, or, as it is also termed, conversion +by contraposition. + +The equations $A$~and~$E$, written in the forms +\begin{align*} +(1 - y) x &= 0, \\ +yx &= 0, +\end{align*} +give on solution the respective forms +\begin{align*} +x &= vy, \\ +x &= v(1 - y), +\end{align*} +the correctness of which may be shewn by substituting these +values of~$x$ in the equations to which they belong, and observing +that those equations are satisfied quite independently of the nature +of the symbol~$v$. The first solution may be interpreted into +\[ +\text{Some~$Y$s are~$X$s,} +\] +and the second into +\[ +\text{Some not-$Y$s are~$X$s.} +\] +\PageSep{28} +From which it appears that universal-affirmative, and universal-negative +Propositions are convertible by limitation, or, as it has +been termed, \textit{per accidens}. + +The above are the laws of Conversion recognized by Abp.~Whately. +Writers differ however as to the admissibility of +negative conversion. The question depends on whether we will +consent to use such terms as not-$X$, not-$Y$. Agreeing with +those who think that such terms ought to be admitted, even +although they change the \emph{kind} of the Proposition, I am constrained +to observe that the present classification of them is +faulty and defective. Thus the conversion of No~$X$s are~$Y$s, +into All~$Y$s are not-$X$s, though perfectly legitimate, is not recognised +in the above scheme. It may therefore be proper to +examine the subject somewhat more fully. + +Should we endeavour, from the system of equations we have +obtained, to deduce the laws not only of the conversion, but +also of the general transformation of propositions, we should be +led to recognise the following distinct elements, each connected +with a distinct mathematical process. + +1st. The negation of a term, \ie~the changing of~$X$ into not-$X$, +or not-$X$ into~$X$. + +2nd. The translation of a Proposition from one \emph{kind} to +another, as if we should change +\[ +\text{All~$X$s are~$Y$s into Some~$X$s are~$Y$s,} +\Ltag{$A$~into~$I$} +\] +which would be lawful; or +\[ +\text{All~$X$s are~$Y$s into No~$X$s are~$Y$\Typo{.}{s,}} +\Ltag{$A$~into~$E$} +\] +which would be unlawful. + +3rd. The simple conversion of a Proposition. + +The conditions in obedience to which these processes may +lawfully be performed, may be deduced from the equations by +which Propositions are expressed. + +We have +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s\Add{,}}\qquad & x(1 - y) &= 0, +\Ltag{A} \\ +&\text{No~$X$s are~$Y$s\Add{,}} & xy &= 0. +\Ltag{E} +\end{alignat*} +\PageSep{29} + +Write $E$ in the form +\[ +x\bigl\{1 - (1 - y)\bigr\} = 0, +\] +%[** TN: "A" italicized in the original] +and it is interpretable by~$A$ into +\[ +\text{All~$X$s are not-$Y$s,} +\] +so that we may change +\[ +\text{No~$X$s are~$Y$s into All~$X$s are not-$Y$s.} +\] + +In like manner $A$~interpreted by~$E$ gives +\[ +\text{No~$X$s are not-$Y$s,} +\] +so that we may change +\[ +\text{All~$X$s are~$Y$s into No~$X$s are not-$Y$s.} +\] + +From these cases we have the following Rule: A universal-affirmative +Proposition is convertible into a universal-negative, +and, \textit{vice versâ}, by negation of the predicate. + +Again, we have +\begin{alignat*}{2} +&\text{Some~$X$s are~$Y$s\Add{,}} & v &= xy, \\ +&\text{Some~$X$s are not~$Y$s\Add{,}}\qquad& v &= x(1 - y). +\end{alignat*} +These equations only differ from those last considered by the +presence of the term~$v$. The same reasoning therefore applies, +and we have the Rule--- + +A particular-affirmative proposition is convertible into a particular-negative, +and \textit{vice versâ}, by negation of the predicate. + +Assuming the universal Propositions +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s\Add{,}}\qquad & x(1 - y) &= 0, \\ +&\text{No~$X$s are~$Y$s\Add{,}} & xy &= 0. +\end{alignat*} +Multiplying by~$v$, we find +\begin{align*} +vx(1 - y) &= 0, \\ +vxy &= 0, +\end{align*} +which are interpretable into +\begin{align*} +&\text{Some~$X$s are~$Y$s,} +\Ltag{I} \\ +&\text{Some~$X$s are not~$Y$s.} +\Ltag{O} +\end{align*} +\PageSep{30} + +Hence a universal-affirmative is convertible into a particular-affirmative, +and a universal-negative into a particular-negative +without negation of subject or predicate. + +Combining the above with the already proved rule of simple +conversion, we arrive at the following system of independent +laws of transformation. + +1st. An affirmative Proposition may be changed into its corresponding +negative ($A$~into~$E$, or $I$~into~$O$), and \textit{\Typo{vice versa}{vice versâ}}, +by negation of the predicate. + +2nd. A universal Proposition may be changed into its corresponding +particular Proposition, ($A$~into~$I$, or $E$~into~$O$). + +3rd. In a particular-affirmative, or universal-negative Proposition, +the terms may be mutually converted. + +Wherein negation of a term is the changing of~$X$ into not-$X$, +and \textit{vice versâ}, and is not to be understood as affecting the \emph{kind} +of the Proposition. + +Every lawful transformation is reducible to the above rules. +Thus we have +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s,} \\ +&\text{No~$X$s are not-$Y$s} &&\text{by 1st rule,} \\ +&\text{No not-$Y$s are~$X$s} &&\text{by 3rd rule,} \\ +&\text{All not-$Y$s are not-$X$s } &&\text{by 1st rule,} +\end{alignat*} +which is an example of \emph{negative conversion}. Again, +\begin{alignat*}{2} +&\text{No~$X$s are~$Y$s,} \\ +&\text{No~$Y$s are~$X$s} &&\text{3rd rule,} \\ +&\text{All~$Y$s are not-$X$s}\quad &&\text{1st rule,} +\end{alignat*} +which is the case already deduced. +\PageSep{31} + + +\Chapter{Of Syllogisms.} + +\begin{Abstract} +A Syllogism consists of three Propositions, the last of which, called the +conclusion, is a logical consequence of the two former, called the premises; +\Typo{e.g.}{\eg} +\begin{alignat*}{2} +&\text{\emph{Premises,}} && +\Lbrace{2}\begin{aligned} +&\text{All~$Y$s are~$X$s.} \\ +&\text{All~$Z$s are~$Y$s.} +\end{aligned} \\ +&\text{\emph{Conclusion,}}\quad && +\text{All~$Z$s are~$X$s.} +\end{alignat*} + +Every syllogism has three and only three terms, whereof that which is +the subject of the conclusion is called the \emph{minor} term, the predicate of the +conclusion, the \emph{major} term, and the remaining term common to both premises, +the middle term. Thus, in \Typo{ths}{the} above formula, $Z$~is the minor term, $X$~the +major term, $Y$~the middle term. + +The figure of a syllogism consists in the situation of the middle term with +respect to the terms of the conclusion. The varieties of figure are exhibited +in the annexed scheme. +\[ +\begin{array}{*{3}{c<{\qquad}}c@{}} +\ColHead{1st Fig.} & \ColHead{2nd Fig.} & \ColHead{3rd Fig.} & \ColHead{4th Fig.} \\ +YX & XY & YX & XY \\ +ZY & ZY & YZ & YZ \\ +ZX & ZX & ZX & ZX +\end{array} +\] + +When we designate the three propositions of a syllogism by their usual +symbols ($A$, $E$, $I$, $O$), and in their actual order, we are said to determine +the mood of the syllogism. Thus the syllogism given above, by way of +illustration, belongs to the mood~$AAA$ in the first figure. + +The moods of all syllogisms commonly received as valid, are represented +by the vowels in the following mnemonic verses. + +Fig.~1.---bArbArA, cElArEnt, dArII, fErIO que prioris. + +Fig.~2.---cEsArE, cAmEstrEs, \Typo{fEstIno}{fEstInO}, bArOkO, secundæ. + +Fig.~3.---Tertia dArAptI, dIsAmIs, dAtIsI, fElAptOn, \\ +\PadTo{\text{\indent Fig.~3.---}}{}bOkArdO, fErIsO, habet: quarta insuper addit. + +Fig.~4.---brAmAntIp, cAmEnEs, dImArIs, \Typo{fEsapO}{fEsApO}, frEsIsOn. +\end{Abstract} + +\First{The} equation by which we express any Proposition concerning +the classes $X$~and~$Y$, is an equation between the +symbols $x$~and~$y$, and the equation by which we express any +\PageSep{32} +Proposition concerning the classes $Y$~and~$Z$, is an equation +between the symbols $y$~and~$z$. If from two such equations +we eliminate~$y$, the result, if it do not vanish, will be an +equation between $x$~and~$z$, and will be interpretable into a +Proposition concerning the classes $X$~and~$Z$. And it will then +constitute the third member, or Conclusion, of a Syllogism, +of which the two given Propositions are the premises. + +The result of the elimination of~$y$ from the equations +\[ +\begin{alignedat}{2} +ay &+ b &&= 0, \\ +a'y &+ b' &&= 0, +\end{alignedat} +\Tag{(14)} +\] +is the equation +\[ +ab' - a'b = 0. +\Tag{(15)} +\] + +Now the equations of Propositions being of the first order +with reference to each of the variables involved, all the cases +of elimination which we shall have to consider, will be reducible +to the above case, the constants $a$,~$b$, $a'$,~$b'$, being +replaced by functions of $x$,~$z$, and the auxiliary symbol~$v$. + +As to the choice of equations for the expression of our +premises, the only restriction is, that the equations must not +\emph{both} be of the form $ay = 0$, for in such cases elimination would +be impossible. When both equations are of this form, it is +necessary to solve one of them, and it is indifferent which +we choose for this purpose. If that which we select is of +the form $xy = 0$, its solution is +\[ +y = v(1 - x), +\Tag{(16)} +\] +if of the form $(1 - x)y = 0$, the solution will be +\[ +y = vx, +\Tag{(17)} +\] +and these are the only cases which can arise. The reason +of this exception will appear in the sequel. + +For the sake of uniformity we shall, in the expression of +particular propositions, confine ourselves to the forms +\begin{alignat*}{2} +vx &= vy, &&\text{Some~$X$s are~$Y$s,} \\ +vx &= v(1 - y),\quad&&\text{Some~$X$s are not~$Y$s\Typo{,}{.}} +\end{alignat*} +\PageSep{33} +These have a closer analogy with \Eqref{(16)}~and~\Eqref{(17)}, than the other +forms which might be used. + +Between the forms about to be developed, and the Aristotelian +canons, some points of difference will occasionally be observed, +of which it may be proper to forewarn the reader. + +To the right understanding of these it is proper to remark, +that the essential structure of a Syllogism is, in some measure, +arbitrary. Supposing the order of the premises to be fixed, +and the distinction of the major and the minor term to be +thereby determined, it is purely a matter of choice which of +the two shall have precedence in the Conclusion. Logicians +have settled this question in favour of the minor term, but +it is clear, that this is a convention. Had it been agreed +that the major term should have the first place in the conclusion, +a logical scheme might have been constructed, less +convenient in some cases than the existing one, but superior +in others. What it lost in \textit{barbara}, it would gain in \textit{bramantip}. +Convenience is \emph{perhaps} in favour of the adopted arrangement,\footnote + {The contrary view was maintained by Hobbes. The question is very + fairly discussed in Hallam's \textit{Introduction to the Literature of Europe}, vol.~\textsc{iii}. + p.~309. In the rhetorical use of Syllogism, the advantage appears to rest + with the rejected form.} +but it is to be remembered that it is \emph{merely} an arrangement. + +Now the method we shall exhibit, not having reference +to one scheme of arrangement more than to another, will +always give the more general conclusion, regard being paid +only to its abstract lawfulness, considered as a result of pure +reasoning. And therefore we shall sometimes have presented +to us the spectacle of conclusions, which a logician would +pronounce informal, but never of such as a reasoning being +would account false. + +The Aristotelian canons, however, beside restricting the \emph{order} +of the terms of a conclusion, limit their nature also;---and +this limitation is of more consequence than the former. We +may, by a change of figure, replace the particular conclusion +\PageSep{34} +of \textit{bramantip} by the general conclusion of~\textit{barbara}; but we +cannot thus reduce to rule such inferences, as +\[ +\text{Some not-$X$s are not~$Y$s.} +\] + +Yet there are cases in which such inferences may lawfully +be drawn, and in unrestricted argument they are of frequent +occurrence. Now if an inference of this, or of any other +kind, is lawful in itself, it will be exhibited in the results +of our method. + +We may by restricting the canon of interpretation confine +our expressed results within the limits of the scholastic logic; +but this would only be to restrict ourselves to the use of a part +of the conclusions to which our analysis entitles us. + +The classification we shall adopt will be purely mathematical, +and we shall afterwards consider the logical arrangement to +which it corresponds. It will be sufficient, for reference, to +name the premises and the Figure in which they are found. + +\textsc{Class} 1st.---Forms in which $v$~does not enter. + +Those which admit of an inference are $AA$,~$EA$, Fig.~1; +$AE$,~$EA$, Fig.~2; $AA$,~$AE$, Fig.~4. + +Ex. $AA$, Fig.~1, and, by mutation of premises (change of +order), $AA$,~Fig.~4. +\begin{alignat*}{4} +&\text{All~$Y$s are~$X$s,}\qquad& +y(1 - x) &= 0,\qquad&& \text{or }& (1 - x) y &= 0, \\ +&\text{All~$Z$s are~$Y$s,} & +z(1 - y) &= 0, &&\text{or }& zy - z &= 0. +\end{alignat*} + +Eliminating~$y$ by~\Eqref{(13)} we have +\begin{gather*} +z(1 - x) = 0, \\ +\therefore\ \text{All~$Z$s are~$X$s.} +\end{gather*} + +A convenient mode of effecting the elimination, is to write +the equation of the premises, so that $y$~shall appear only as +a factor of one member in the first equation, and only as +a factor of the opposite member in the second equation, and +then to multiply the equations, omitting the~$y$. This method +we shall adopt. +\PageSep{35} + +Ex. $AE$, Fig.~2, and, by mutation of premises, $EA$, Fig\Typo{,}{.}~2. +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{No~$Z$s are~$Y$s,} & zy &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +\text{or } & x &=& xy\Add{,} \\ + &zy &=& 0\Add{,} \\ +\cline{2-4} + &zx &=& 0\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{No~$Z$s are~$X$s.}} +\end{array} +\] + +The only case in which there is no inference is~$AA$, Fig.~2, +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{All~$Z$s are~$Y$s,} & z(1 - y) &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +& x &=& xy\Add{,} \\ +&zy &=& z\Add{,} \\ +\cline{2-4} +&xz &=& xz\Add{,} \\ +\multicolumn{4}{l}{\rlap{$\therefore\ 0 = 0$.}} +\end{array} +\] + +\textsc{Class} 2nd.---When $v$~is introduced by the solution of an +equation. + +The lawful cases directly or indirectly\footnote + {We say \emph{directly} or \emph{indirectly}, mutation or conversion of premises being + in some instances required. Thus, $AE$ (fig.~1) is resolvable by \Chg{Fesapo}{\textit{fesapo}} (fig.~4), + or by \Chg{Ferio}{\textit{ferio}} (fig.~1). Aristotle and his followers rejected the fourth figure + as only a modification of the first, but this being a mere question of form, + either scheme may be termed Aristotelian.} +determinable by the +Aristotelian Rules are~$AE$, Fig.~1; $AA$, $AE$, $EA$, Fig.~3; +$EA$, Fig.~4. + +The lawful cases not so determinable, are $EE$, Fig.~1; $EE$, +Fig.~2; $EE$, Fig.~3; $EE$, Fig.~4. + +Ex. $AE$, Fig.~1, and, by mutation of premises, $EA$, Fig.~4. +\[ +\begin{alignedat}[t]{2} +&\text{All~$Y$s are~$X$s,}\qquad& y(1 - x) &= 0, \\ +&\text{No~$Z$s are~$Y$s,} & zy &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&y &=& vx\Add{,}\atag \\ +&0 &=& zy\Add{,} \\ +\cline{2-4} +&0 &=& vzx\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{Some~$X$s are not~$Z$s.}} +\end{array} +\] + +The reason why we cannot interpret $vzx = 0$ into Some~$Z$s +are not-$X$s, is that by the very terms of the first equation~\aref\ +the interpretation of~$vx$ is fixed, as Some~$X$s; $v$~is regarded +as the representative of Some, only with reference to the +class~$X$. +\PageSep{36} + +For the reason of our employing a solution of one of the +primitive equations, see the remarks on \Eqref{(16)}~and~\Eqref{(17)}. Had +we solved the second equation instead of the first, we should +have had +\begin{gather*} +\begin{aligned} +(1 - x)y &= 0, \\ +v(1 - z) &= y,\atag \\ +v(1 - z)(1 - x) &= 0,\btag +\end{aligned} \\ +\therefore\ \text{Some not-$Z$s are~$X$s.} +\end{gather*} + +Here it is to be observed, that the second equation~\aref\ fixes +the meaning of~$v(1 - z)$, as Some not-$Z$s. The full meaning +of the result~\bref\ is, that all the not-$Z$s which are found in +the class~$Y$ are found in the class~$X$, and it is evident that +this could not have been expressed in any other way. + +Ex.~2. $AA$, Fig.~3. +\[ +\begin{alignedat}[t]{2} +&\text{All~$Y$s are~$X$s,}\qquad& y(1 - x) &= 0, \\ +&\text{All~$Y$s are~$Z$s,} & y(1 - z) &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&y &=& vx\Add{,} \\ +&0 &=& y(1 - z)\Add{,} \\ +\cline{2-4} +&0 &=& vx(1 - z)\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{Some~$X$s are~$Z$s.}} +\end{array} +\] + +Had we solved the second equation, we should have had +as our result, Some~$Z$s are~$X$s. The form of the final equation +particularizes what~$X$s or what~$Z$s are referred to, and this +remark is general. + +The following, $EE$, Fig.~1, and, by mutation, $EE$, Fig.~4, +is an example of a lawful case not determinable by the Aristotelian +Rules. +\[ +\begin{alignedat}[t]{2} +&\text{No~$Y$s are~$X$s,}\qquad& xy &= 0, \\ +&\text{No~$Z$s are~$Y$s,} & zy &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&0 &=& xy\Add{,} \\ +&y &=& v(1 - z)\Add{,} \\ +\cline{2-4} +&0 &=& v(1 - z)x\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{Some not-$Z$s are not~$X$s.}} +\end{array} +\] + +\textsc{Class} 3rd.---When $v$~is met with in one of the equations, +but not introduced by solution. +\PageSep{37} + +The lawful cases determinable \emph{directly} or \emph{indirectly} by the +Aristotelian Rules, are $AI$,~$EI$, Fig.~1; $AO$, $EI$, $OA$, $IE$, +Fig.~2; $AI$, $AO$, $EI$, $EO$, $IA$, $IE$, $OA$, $OE$, Fig.~3; $IA$, $IE$, +Fig.~4. + +Those not so determinable are~$OE$, Fig.~1; $EO$, Fig.~4. + +The cases in which no inference is possible, are $AO$, $EO$, +$IA$, $IE$, $OA$, Fig.~1; $AI$, $EO$, $IA$, $OE$, Fig.~2; $OA$, $OE$, +$AI$, $EI$, $AO$, Fig.~4. + +Ex.~1. $AI$, Fig.~1, and, by mutation, $IA$, Fig.~4. +\[ +\begin{aligned}[t] +&\text{All~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are~$Y$s,} +\end{aligned} +\begin{array}[t]{>{\qquad}rr@{\,}c@{\,}l@{}} +&y(1 - x) &=& 0\Add{,} \\ +&vz &=& vy\Add{,} \\ +\cline{2-4} +&vz(1 - x) &=& 0\Add{,} \\ +\therefore\ & +\multicolumn{3}{l}{\rlap{Some~$Z$s are~$X$s.}} +\end{array} +\] + +Ex.~2. $AO$, Fig.~2, and, by mutation, $OA$, Fig.~2. +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{Some~$Z$s are not~$Y$s,} & vz &= v(1 - y), +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&x &=& xy\Add{,} \\ +&vy &=& v(1 - z)\Add{,} \\ +\cline{2-4} +&vx &=& vx(1 - z)\Add{,} \\ +&vxz&=& 0\Add{,} \\ +\multicolumn{4}{r}{\llap{$\therefore\ \text{Some~$Z$s are not~$X$s.}$}} +\end{array} +\] + +The interpretation of~$vz$ as Some~$Z$s, is implied, it will be +observed, in the equation $vz = v(1 - y)$ considered as representing +the proposition Some~$Z$s are not~$Y$s. + +The cases not determinable by the Aristotelian Rules are +$OE$, Fig.~1, and, by mutation, $EO$, Fig.~4. +\[ +\begin{aligned}[t] +&\text{Some~$Y$s are not~$X$s,} \\ +&\text{No~$Z$s are~$Y$s,} +\end{aligned}\qquad +\begin{array}[t]{>{\qquad}rr@{\,}c@{\,}l@{}} +&vy &=& v(1 - x)\Add{,} \\ +& 0 &=& zy\Add{,} \\ +\cline{2-4} +& 0 &=& v(1 - x)z\Add{,} \\ +\multicolumn{4}{c}{\makebox[0pt][c]{$\therefore$\ Some not-$X$s are not~$Z$s.}} +\end{array} +\] + +The equation of the first premiss here permits us to interpret +$v(1 - x)$, but it does not enable us to interpret~$vz$. +\PageSep{38} + +Of cases in which no inference is possible, we take as +examples--- + +$AO$, Fig.~1, and, by mutation, $OA$, Fig.~4\Typo{,}{.} +\[ +\begin{alignedat}[t]{2} +&\text{All~$Y$s are~$X$s,}\qquad& y(1 - x) &= 0, \\ +&\text{Some~$Z$s are not~$Y$s,} & vz &= v(1 - y)\Add{,}\atag +\end{alignedat}\qquad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +y(1 - x) &=& 0\Add{,} \\ +v(1 - z) &=& vy\Add{,} \\ +\cline{1-3} +v(1 - z)(1 - x) &=& 0\Add{,}\btag \\ +0&=& 0\Add{,} +\end{array} +\] +since the auxiliary equation in this case is $v(1 - z) = 0$. + +Practically it is not necessary to perform this reduction, but +it is satisfactory to do so. The equation~\aref, it is seen, defines~$vz$ +as Some~$Z$s, but it does not define $v(1 - z)$, so that we might +stop at the result of elimination~\bref, and content ourselves with +saying, that it is not interpretable into a relation between the +classes $X$~and~$Z$. + +Take as a second example $AI$, Fig.~2, and, by mutation, +$IA$, Fig.~2\Typo{,}{.} +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{Some~$Z$s are~$Y$s,} & vz &= vy, +\end{alignedat}\qquad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +x &=& xy\Add{,} \\ +vy &=& vz\Add{,} \\ +\cline{1-3} +vx &=& vxz\Add{,} \\ +\llap{$v(1 - z)x$}&=& 0\Add{,} \\ +0&=& 0, +\end{array} +\] +the auxiliary equation in this case being $v(1 - z)= 0$. + +Indeed in every case in this class, in which no inference +is possible, the result of elimination is reducible to the form +$0 = 0$. Examples therefore need not be multiplied. + +\textsc{Class} 4th.---When $v$~enters into both equations. + +No inference is possible in any case, but there exists a distinction +among the unlawful cases which is peculiar to this +class. The two divisions are, + +1st. When the result of elimination is reducible by the +auxiliary equations to the form $0 = 0$. The cases are $II$, $OI$, +\PageSep{39} +Fig.~1; $II$, $OO$, Fig.~2; $II$, $IO$, $OI$, $OO$, Fig.~3; $II$, $IO$, +Fig.~4. + +2nd. When the result of elimination is not reducible by the +auxiliary equations to the form $0 = 0$. + +The cases are $IO$, $OO$, Fig.~1; $IO$, $OI$, Fig.~2; $OI$, $OO$, +Fig.~4. + +Let us take as an example of the former case,~$II$, Fig.~3. +\[ +\begin{alignedat}[t]{2} +&\text{Some~$X$s are~$Y$s,}\qquad& vx &= vy, \\ +&\text{Some~$Z$s are~$Y$s,} & v'z &= v'y, +\end{alignedat}\qquad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +vx &=& vy\Add{,} \\ +v'y &=& v'z\Add{,} \\ +\cline{1-3} +vv'x &=& vv'z\Add{.} +\end{array} +\] + +Now the auxiliary equations $v(1 - x) = 0$, $v'(1 - z) = 0$, +%[** TN: Next word anomalously displayed in the original] +give +\[ +vx = v,\quad v'z = v'. +\] +Substituting we have +\begin{align*} +vv' &= vv', \\ +\therefore 0 &= 0. +\end{align*} + +As an example of the latter case, let us take $IO$, Fig.~1\Typo{,}{.} +\[ +\begin{alignedat}[t]{2} +&\text{Some~$Y$s are~$X$s,} & vy &= vx, \\ +&\text{Some~$Z$s are not~$Y$s,}\qquad& v'z &= v'(1 - y), +\end{alignedat}\quad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +vy &=& vx\Add{,} \\ +v'(1 - z) &=& v'y\Add{,} \\ +\cline{1-3} +vv'(1 - z) &=& vv'x\Add{.} +\end{array} +\] + +Now the auxiliary equations being $v(1 - x) = 0$, $v'(1 - z) = 0$, +the above reduces to $vv' = 0$. It is to this form that all similar +cases are reducible. Its interpretation is, that the classes $v$ +and~$v'$ have no common member, as is indeed evident. + +The above classification is purely founded on mathematical +distinctions. We shall now inquire what is the logical division +to which it corresponds. + +The lawful cases of the first class comprehend all those in +which, from two universal premises, a universal conclusion +may be drawn. We see that they include the premises of +\textit{barbara} and \textit{celarent} in the first figure, of \textit{cesare} and \textit{camestres} +in the second, and of \textit{bramantip} and \textit{camenes} in the fourth. +\PageSep{40} +The premises of \textit{bramantip} are included, because they admit +of an universal conclusion, although not in the same figure. + +The lawful cases of the second class are those in which +a particular conclusion only is deducible from two universal +premises. + +The lawful cases of the third class are those in which a +conclusion is deducible from two premises, one of which is +universal and the other particular. + +The fourth class has no lawful cases. + +Among the cases in which no inference of any kind is possible, +we find six in the fourth class distinguishable from the +others by the circumstance, that the result of elimination does +not assume the form $0 = 0$. The cases are +{\small +\[ +\Lbrace{2}\begin{aligned} +&\text{Some~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{aligned}\Rbrace{2}\quad +% +\Lbrace{2}\begin{aligned} +&\text{Some~$Y$s are not~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{aligned}\Rbrace{2}\quad +% +\Lbrace{2}\begin{aligned} +&\text{Some~$X$s are~$Y$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{aligned}\Rbrace{2} +\] +}% +and the three others which are obtained by mutation of +premises. + +It might be presumed that some logical peculiarity would +be found to answer to the mathematical peculiarity which we +have noticed, and in fact there exists a very remarkable one. +If we examine each pair of premises in the above scheme, we +shall find that there \emph{is virtually} no middle term, \emph{\ie~no medium +of comparison}, in any of them. Thus, in the first example, +the individuals spoken of in the first premiss are asserted to +belong to the class~$Y$, but those spoken of in the second +premiss are \emph{virtually} asserted to belong to the class not-$Y$: +nor can we by any lawful transformation or conversion alter +this state of things. The comparison will still be made with +the class~$Y$ in one premiss, and with the class not-$Y$ in the +other. + +Now in every case beside the above six, there will be found +a middle term, either expressed or implied. I select two +of the most difficult cases. +\PageSep{41} + +In $AO$, Fig.~1, viz. +\begin{align*} +&\text{All~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{align*} +we have, by \emph{negative conversion} of the first premiss, +\begin{align*} +&\text{All not-$X$s are not-$Y$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{align*} +and the middle term is now seen to be not-$Y$. + +Again, in $EO$, Fig.~1, +\begin{align*} +&\text{No~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{align*} +a proved conversion of the first premiss (see \ChapRef{5}{Conversion of +Propositions}), gives +\begin{align*} +&\text{All~$X$s are not-$Y$s,} \\ +&\text{Some~$Z$s are not-$Y$s,} +\end{align*} +and the middle term, the true medium of comparison, is plainly +\Pagelabel{41}% +not-$Y$, although as the not-$Y$s in the one premiss \emph{may be} +different from those in the other, no conclusion can be drawn. + +The mathematical condition in question, therefore,---the irreducibility +of the final equation to the form $0 = 0$,---adequately +represents the logical condition of there being no middle term, +or common medium of comparison, in the given premises. + +I am not aware that the distinction occasioned by the +presence or absence of a middle term, in the strict sense here +understood, has been noticed by logicians before. The distinction, +though real and deserving attention, is indeed by +no means an obvious one, and it would have been unnoticed +in the present instance but for the peculiarity of its mathematical +expression. + +What appears to be novel in the above case is the proof +of the existence of combinations of premises in which there +\PageSep{42} +is absolutely no medium of comparison. When such a medium +of comparison, or true middle term, does exist, the condition +that its quantification in both premises together shall exceed +its quantification as a single whole, has been ably and +\Pagelabel{42}% +clearly shewn by Professor De~Morgan to be necessary to +lawful inference (\textit{Cambridge Memoirs}, Vol.~\textsc{viii}.\ Part~3). And +this is undoubtedly the true principle of the Syllogism, viewed +from the standing-point of Arithmetic. + +I have said that it would be possible to impose conditions +of interpretation which should restrict the results of this calculus +to the Aristotelian forms. Those conditions would be, + +1st. That we should agree not to interpret the forms $v(1 - x)$, +$v(1 - z)$. + +2ndly. That we should agree to reject every interpretation in +which the order of the terms should violate the Aristotelian rule. + +Or, instead of the second condition, it might be agreed that, +the conclusion being determined, the order of the premises +should, if necessary, be changed, so as to make the syllogism +formal. + +From the \emph{general} character of the system it is indeed plain, +that it may be made to represent any conceivable scheme of +logic, by imposing the conditions proper to the case contemplated. + +We have found it, in a certain class of cases, to be necessary +to replace the two equations expressive of universal Propositions, +by their solutions; and it may be proper to remark, +that it would have been allowable in all instances to have +done this,\footnote + {It may be satisfactory to illustrate this statement by an example. In + \textit{\Chg{Barbara}{barbara}}, we should have + \[ + \begin{aligned}[t] + &\text{All~$Y$s are~$X$s,} \\ + &\text{All~$Z$s are~$Y$s,} + \end{aligned}\qquad + \begin{array}[t]{>{\qquad}r@{\,}c@{\,}l@{}} + y &=& vx\Add{,} \\ + z &=& v'y\Add{,} \\ + \cline{1-3} + z &=& vv'x\Add{,} \\ + \multicolumn{3}{c}{\makebox[0pt][c]{$\therefore$\ All~$Z$s are~$X$s.}} + \end{array} + \] +%[** TN: Footnote continues] + Or, we may multiply the resulting equation by~$1 - x$, which gives + \[ + z(1 - x) = 0, + \] + whence the same conclusion, All~$Z$s are~$X$s. + + Some additional examples of the application of the system of equations in + the text to the demonstration of general theorems, may not be inappropriate. + + Let $y$ be the term to be eliminated, and let $x$ stand indifferently for either of + the other symbols, then each of the equations of the premises of any given + syllogism may be put in the form + \[ + ay + bx = 0, + \GrTag[a]{(\alpha)} + \] + if the premiss is affirmative, and in the form + \[ + ay + b(1 - x) = 0, + \GrTag[b]{(\beta)} + \] + if it is negative, $a$~and~$b$ being either constant, or of the form~$±v$. To prove + this in detail, let us examine each kind of proposition, making $y$~successively + subject and predicate. + \begin{alignat*}{2} + A,\ &\text{All~$Y$s are~$X$s,} & y - vx &= 0, + \GrTag[c]{(\gamma)} \\ + &\text{All~$X$s are~$Y$s,} & x - vy &= 0, + \GrTag[d]{(\delta)} \\ +% + E,\ &\text{No~$Y$s are~$X$s,} & xy &= 0, \\ + &\text{No~$X$s are~$Y$s,} & y - v(1 - x) &= 0, + \GrTag[e]{(\epsilon)} \\ +% + I,\ &\text{Some~$X$s are~$Y$s,} && \\ + &\text{Some~$Y$s are~$X$s,} &vx - vy &= 0, + \GrTag[f]{(\zeta)} \\ +% + O,\ &\text{Some~$Y$s are not~$X$s,}\qquad& vy - v(1 - x) &= 0, + \GrTag[g]{(\eta)} \\ + &\text{Some~$X$s are not~$Y$s,} & vx &= v(1 - y), \\ + && \therefore vy - v(1 - x) &= 0. + \GrTag[h]{(\theta)} + \end{alignat*} + + The affirmative equations \GrEq[c]{(\gamma)},~\GrEq[d]{(\delta)} and~\GrEq[f]{(\zeta)}, belong to~\GrEq[a]{(\alpha)}, and the negative + equations \GrEq[e]{(\epsilon)},~\GrEq[g]{(\eta)} and~\GrEq[h]{(\theta)}, to~\GrEq[b]{(\beta)}. It is seen that the two last negative equations + are alike, but there is a difference of interpretation. In the former + \[ + v(1 - x) = \text{Some not-$X$s,} + \] + in the latter, + \[ + v(1 - x) = 0. + \] + + The utility of the two general forms of reference, \GrEq[a]{(\alpha)}~and~\GrEq[b]{(\beta)}, will appear + from the following application. + + 1st. \emph{A conclusion drawn from two affirmative propositions} is itself affirmative. + + By \GrEq[a]{(\alpha)} we have for the given propositions, + \begin{alignat*}{2} + ay &+ bx &&= 0, \\ + a'y &+ b'z &&= 0, + \end{alignat*} +%[** TN: Footnote continues] + and eliminating + \[ + ab'z - a'bx = 0, + \] + which is of the form~\GrEq[a]{(\alpha)}. Hence, if there is a conclusion, it is affirmative. + + 2nd. \emph{A conclusion drawn from an affirmative and a negative proposition is +negative.} + + By \GrEq[a]{(\alpha)}~and~\GrEq[b]{(\beta)}, we have for the given propositions + \begin{align*} + ay + bx &= 0, \\ + a'y + b'(1 - z) &= 0, \\ + \therefore\ a'bx - ab'(1 - z) &= 0, + \end{align*} + which is of the form~\GrEq[b]{(\beta)}. Hence the conclusion, if there is one, is negative. + + 3rd. \emph{A conclusion drawn from two negative premises will involve a negation, + \(not-$X$, not-$Z$\) in both subject and predicate, and will therefore be inadmissible in + the Aristotelian system, though just in itself.} + + For the premises being + \begin{alignat*}{2} + ay &+ b (1 - x) &&= 0, \\ + a'y &+ b'(1 - z) &&= 0, + \end{alignat*} + the conclusion will be + \[ + ab'(1 - z) - a'b(1 - x) = 0, + \] + which is only interpretable into a proposition that has a negation in each term. + + 4th. \emph{Taking into account those syllogisms only, in which the conclusion is the + most general, that can be deduced from the premises,---if, in an Aristotelian + syllogism, the minor premises be changed in quality \(from affirmative to negative + or from negative to affirmative\), whether it be changed in quantity or not, no conclusion + will be deducible in the same figure.} + + An Aristotelian proposition does not admit a term of the form not-$Z$ in the + subject,---Now on changing the quantity of the minor proposition of a syllogism, + we transfer it from the general form + \begin{align*} + ay + bz &= 0, \\ + \intertext{to the general form} + a'y + b'(1 - z) &= 0, + \end{align*} + see \GrEq[a]{(\alpha)}~\emph{and}~\GrEq[b]{(\beta)}, or \textit{vice versâ}. And therefore, in the equation of the conclusion, + there will be a change from~$z$ to~$1 - z$, or \textit{vice versâ}. But this is equivalent to + the change of~$Z$ into not-$Z$, or not-$Z$ into~$Z$. Now the subject of the original + conclusion must have involved a~$Z$ and not a not-$Z$, therefore the subject of the + new conclusion will involve a not-$Z$, and the conclusion will not be admissible + in the Aristotelian forms, except by conversion, which would render necessary + a change of Figure. + + Now the conclusions of this calculus are always the most general that can be + drawn, and therefore the above demonstration must not be supposed to extend + to a syllogism, in which a particular conclusion is deduced, when a universal + one is possible. This is the case with \textit{bramantip} only, among the Aristotelian + forms, and therefore the transformation of \textit{bramantip} into \textit{camenes}, and \textit{vice versâ}, + is the case of restriction contemplated in the preliminary statement of the + theorem. + + 5th. \emph{If for the minor premiss of an Aristotelian syllogism, we substitute its contradictory, + no conclusion is deducible in the same figure.} + + It is here only necessary to examine the case of \textit{bramantip}, all the others + being determined by the last proposition. + + On changing the minor of \textit{bramantip} to its contradictory, we have $AO$, + Fig.~4, and this admits of no legitimate inference. + + Hence the theorem is true without exception. Many other general theorems + may in like manner be proved.} +%[** TN: End of 3.5-page footnote] +so that every case of the Syllogism, without exception, +\PageSep{43} +might have been treated by equations comprised in +the general forms +\Pagelabel{43}% +\begin{alignat*}{3} + y &= vx, &&\text{or} & y - vx &= 0, +\Ltag{A} \\ + y &= v(1 - x),\qquad&&\text{or}\quad & y + vx - v &= 0, +\Ltag{E} \\ +vy &= vx, &&& vy - vx &= 0, +\Ltag{I} \\ +vy &= v(1 - x), &&& vy + vx - v &= 0. +\Ltag{O} +\end{alignat*} +\PageSep{44} + +Perhaps the system we have actually employed is better, +as distinguishing the cases in which $v$~only \emph{may} be employed, +\PageSep{45} +from those in which it \emph{must}. But for the demonstration of +certain general properties of the Syllogism, the above system +is, from its simplicity, and from the mutual analogy of its +forms, very convenient. We shall apply it to the following +theorem.\footnote + {This elegant theorem was communicated by the Rev.\ Charles Graves, + Fellow and Professor of Mathematics in Trinity College, Dublin, to whom the + Author desires further to record his grateful acknowledgments for a very + judicious examination of the former portion of this work, and for some new + applications of the method. The following example of Reduction \textit{ad~impossibile} + is among the number: + \[ + \begin{array}{rl<{\quad}r@{\,}c@{\,}l@{}} + \text{Reducend Mood,} & + \text{All~$X$s are~$Y$s,} & + 1 - y &=& v'(1 - x)\Add{,} \\ + \PadTxt{Reducend Mood,}{\textit{\Chg{Baroko}{baroko}}} & + \text{Some~$Z$s are not~$Y$s\Add{,}} & + vz &=& v(1 - y)\Add{,} \\ + \cline{3-5} +% + &\text{Some~$Z$s are not~$X$s\Add{,}} & + vz &=& vv'(1 - x)\Add{,} \\ +% + \text{Reduct Mood,} & + \text{All~$X$s are~$Y$s\Add{,}} & + 1 - y &=& v'(1 - x)\Add{,} \\ + \PadTxt{Reduct Mood,}{\textit{\Chg{Barbara}{barbara}}} & + \text{All~$Z$s are~$X$s\Add{,}} & + z(1 - x) &=& 0\Add{,} \\ + \cline{2-5} + &\text{All~$Z$s are~$Y$s\Add{,}} & + z(1 - y) &=& 0. + \end{array} + \] + + The conclusion of the reduct mood is seen to be the contradictory of the + suppressed minor premiss. Whence,~\etc. It may just be remarked that the + mathematical test of contradictory propositions is, that on eliminating one + elective symbol between their equations, the other elective symbol vanishes. + The \emph{ostensive} reduction of \textit{\Chg{Baroko}{baroko}} and \textit{\Chg{Bokardo}{bokardo}} involves no difficulty. + + Professor Graves suggests the employment of the equation $x = vy$ for the + primary expression of the Proposition All~$X$s are~$Y$s, and remarks, that on + multiplying both members by~$1 - y$, we obtain $x(1 - y) = 0$, the equation from + which we set out in the text, and of which the previous one is a solution.} + +Given the three propositions of a Syllogism, prove that there +is but one order in which they can be legitimately arranged, +and determine that order. + +All the forms above given for the expression of propositions, +are particular cases of the general form, +\[ +a + bx + cy = 0. +\] +\PageSep{46} + +Assume then for the premises of the given syllogism, the +equations +\begin{alignat*}{3} +a &+ bx &&+ cy &&= 0, +\Tag{(18)} \\ +a' &+ b'z &&+ c'y &&= 0, +\Tag{(19)} +\end{alignat*} +then, eliminating~$y$, we shall have for the conclusion +\[ +ac' - a'c + bc'x - b'cz = 0. +\Tag{(20)} +\] + +Now taking this as one of our premises, and either of the +original equations, suppose~\Eqref{(18)}, as the other, if by elimination +of a common term~$x$, between them, we can obtain a result +equivalent to the remaining premiss~\Eqref{(19)}, it will appear that +there are more than one order in which the Propositions may +be lawfully written; but if otherwise, one arrangement only +is lawful. + +Effecting then the elimination, we have +\[ +bc(a' + b'z + c'y) = 0, +\Tag{(21)} +\] +which is equivalent to~\Eqref{(19)} multiplied by a factor~$bc$. Now on +examining the value of this factor in the equations $A$,~$E$, $I$,~$O$, +we find it in each case to be $v$~or~$-v$. But it is evident, +that if an equation expressing a given Proposition be multiplied +by an extraneous factor, derived from another equation, +its interpretation will either be limited or rendered +impossible. Thus there will either be no result at all, or the +result will be a \emph{limitation} of the remaining Proposition. + +If, however, one of the original equations were +\[ +x = y,\quad\text{or}\quad x - y = 0, +\] +the factor~$bc$ would be~$-1$, and would \emph{not} limit the interpretation +of the other premiss. Hence if the first member of +a syllogism should be understood to represent the double +proposition All~$X$s are~$Y$s, and All~$Y$s are~$X$s, it would be +indifferent in what order the remaining Propositions were +written. +\PageSep{47} + +A more general form of the above investigation would be, +to express the premises by the equations +\begin{alignat*}{4} +a &+ bx &&+ cy &&+ dxy &&= 0, +\Tag{(22)} \\ +a' &+ b'z &&+ c'y &&+ d'zy &&= 0. +\Tag{(23)} +\end{alignat*} + +After the double elimination of $y$~and~$x$ we should find +\[ +(bc - ad)(a' + b'z + c'y + d'zy) = 0; +\] +and it would be seen that the factor $bc - ad$ must in every +case either vanish or express a limitation of meaning. + +The determination of the order of the Propositions is sufficiently +obvious. +\PageSep{48} + + +\Chapter{Of Hypotheticals.} + +\begin{Abstract} +A hypothetical Proposition is defined to be \emph{two or more categoricals united by +a copula} (or conjunction), and the different kinds of hypothetical Propositions +are named from their respective conjunctions, viz.\ conditional (if), disjunctive +(either, or),~\etc. + +In conditionals, that categorical Proposition from which the other results +is called the \emph{antecedent}, that which results from it the \emph{consequent}. + +Of the conditional syllogism there are two, and only two formulæ. + +1st. The constructive, +\begin{gather*} +\text{If $A$~is~$B$, then $C$~is~$D$,} \\ +\text{But $A$~is~$B$, therefore $C$~is~$D$.} +\end{gather*} + +2nd. The Destructive, +\begin{gather*} +\text{If $A$~is~$B$, then $C$~is~$D$,} \\ +\text{But $C$~is not~$D$, therefore $A$~is not~$B$.} +\end{gather*} + +A dilemma is a complex conditional syllogism, with several antecedents +in the major, and a disjunctive minor. +\end{Abstract} + +\First{If} we examine either of the forms of conditional syllogism +above given, we shall see that the validity of the argument +does not depend upon any considerations which have reference +to the terms $A$,~$B$,~$C$,~$D$, considered as the representatives +of individuals or of classes. We may, in fact, represent the +Propositions $A$~is~$B$, $C$~is~$D$, by the arbitrary symbols $X$~and~$Y$ +respectively, and express our syllogisms in such forms as the +following: +\begin{gather*} +\text{If $X$ is true, then $Y$ is true,} \\ +\text{But $X$ is true, therefore $Y$ is true.} +\end{gather*} + +Thus, what we have to consider is not objects and classes +of objects, but the truths of Propositions, namely, of those +\PageSep{49} +elementary Propositions which are embodied in the terms of +our hypothetical premises. + +To the symbols $X$,~$Y$,~$Z$, representative of Propositions, we +may appropriate the elective symbols $x$,~$y$,~$z$, in the following +sense. + +The hypothetical Universe,~$1$, shall comprehend all conceivable +cases and conjunctures of circumstances. + +The elective symbol~$x$ attached to any subject expressive of +such cases shall select those cases in which the Proposition~$X$ +is true, and similarly for $Y$~and~$Z$. + +If we confine ourselves to the contemplation of a given proposition~$X$, +and hold in abeyance every other consideration, +then two cases only are conceivable, viz.\ first that the given +Proposition is true, and secondly that it is false.\footnote + {It was upon the obvious principle that a Proposition is either true or false, + that the Stoics, applying it to assertions respecting future events, endeavoured + to establish the doctrine of Fate. It has been replied to their argument, that it +%[** TN: Italicized entire Latin phrase; only "est" italicized in original] + involves ``an abuse of the word \emph{true}, the precise meaning of which is \textit{id quod + res est}. An assertion respecting the future is neither true nor false.''---\textit{Copleston + on Necessity and Predestination}, p.~36. Were the Stoic axiom, however, presented + under the form, It is either certain that a given event will take place, + or certain that it will not; the above reply would fail to meet the difficulty. + The proper answer would be, that no merely verbal definition can settle the + question, what is the actual course and constitution of Nature. When we + affirm that it is either certain that an event will take place, or certain that + it will not take place, we tacitly assume that the order of events is necessary, + that the Future is but an evolution of the Present; so that the state of things + which is, completely determines that which shall be. But this (at least as respects + the conduct of moral agents) is the very question at issue. Exhibited + under its proper form, the Stoic reasoning does not involve an abuse of terms, + but a \textit{petitio principii}. + + It should be added, that enlightened advocates of the doctrine of Necessity + in the present day, viewing the end as appointed only in and through the + means, justly repudiate those practical ill consequences which are the reproach + of Fatalism.} +As these +cases together make up the Universe of the Proposition, and +as the former is determined by the elective symbol~$x$, the latter +is determined by the symbol~$1 - x$. + +But if other considerations are admitted, each of these cases +will be resolvable into others, individually less extensive, the +\PageSep{50} +number of which will depend upon the number of foreign considerations +admitted. Thus if we associate the Propositions $X$ +and~$Y$, the total number of conceivable cases will be found as +exhibited in the following scheme. +\[ +\begin{array}[b]{*{2}{l@{\ }}>{\qquad}c@{}} +\multicolumn{2}{c}{\ColHead{Cases.}} & +\multicolumn{1}{>{\qquad}c}{\ColHead{Elective expressions.}} \\ +\text{1st}& \text{$X$ true, $Y$ true\Add{,}} & xy\Add{,} \\ +\text{2nd}& \text{$X$ true, $Y$ false\Add{,}}& x(1 - y)\Add{,} \\ +\text{3rd}& \text{$X$ false, $Y$ true\Add{,}} & (1 - x)y\Add{,} \\ +\text{4th}& \text{$X$ false, $Y$ false\Add{,}}& (1 - x)(1 - y)\Add{.} +\end{array} +\Tag{(24)} +\] + +If we add the elective expressions for the two first of the +above cases the sum is~$x$, which is the elective symbol appropriate +to the more general case of $X$~being true independently +of any consideration of~$Y$; and if we add the elective expressions +in the two last cases together, the result is~$1 - x$, which +is the elective expression appropriate to the more general case +of $X$~being false. + +Thus the extent of the hypothetical Universe does not at +all depend upon the number of circumstances which are taken +into account. And it is to be noted that however few or many +those circumstances may be, the sum of the elective expressions +representing every conceivable case will be unity. Thus let +us consider the three Propositions, $X$,~It rains, $Y$,~It hails, +$Z$,~It freezes. The possible cases are the following: +\[ +\begin{array}{*{2}{l@{\ }}l@{}} +&\multicolumn{1}{c}{\ColHead{Cases.}} & +\multicolumn{1}{c}{\ColHead{Elective expressions.}} \\ +\text{1st}& \text{It rains, hails, and freezes,} & xyz\Add{,} \\ +\text{2nd}& \text{It rains and hails, but does not freeze\Add{,}}& xy(1 - z)\Add{,} \\ +\text{3rd}& \text{It rains and freezes, but does not hail\Add{,}}& xz(1 - y)\Add{,} \\ +\text{4th}& \text{It freezes and hails, but does not rain\Add{,}}& yz(1 - x)\Add{,} \\ +\text{5th}& \text{It rains, but neither hails nor freezes\Add{,}}& x(1 - y)(1 - z)\Add{,} \\ +\text{6th}& \text{It hails, but neither rains nor freezes\Add{,}}& y(1 - x)(1 - z)\Add{,} \\ +\text{7th}& \text{It freezes, but neither hails nor rains\Add{,}}& z(1 - x)(1 - y)\Add{,} \\ +\text{8th}& \text{It neither rains, hails, nor freezes\Add{,}}& (1 - x)(1 - y)(1 - z)\Add{,} \\ +\cline{3-3} +&&\multicolumn{1}{c}{1 = \text{sum\Add{.}}} +\end{array} +\] +\PageSep{51} + + +\Section{Expression of Hypothetical Propositions.} + +To express that a given Proposition~$X$ is true. + +The symbol $1 - x$ selects those cases in which the Proposition~$X$ +is false. But if the Proposition is true, there are no +such cases in its hypothetical Universe, therefore +\begin{align*} +1 - x &= 0, \\ +\intertext{or} +x &= 1. +\Tag{(25)} +\end{align*} + +To express that a given Proposition~$X$ is false. + +The elective symbol~$x$ selects all those cases in which the +Proposition is true, and therefore if the Proposition is false, +\[ +x = 0. +\Tag{(26)} +\] + +And in every case, having determined the elective expression +appropriate to a given Proposition, we assert the truth of that +Proposition by equating the elective expression to unity, and +its falsehood by equating the same expression to~$0$. + +To express that two Propositions, $X$~and~$Y$, are simultaneously +true. + +The elective symbol appropriate to this case is~$xy$, therefore +the equation sought is +\[ +xy = 1. +\Tag{(27)} +\] + +To express that two Propositions, $X$~and~$Y$, are simultaneously +false. + +The condition will obviously be +\begin{align*} +(1 - x)(1 - y) &= 1, \\ +\intertext{or} +x + y - xy &= 0. +\Tag{(28)} +\end{align*} + +To express that either the Proposition~$X$ is true, or the +Proposition~$Y$ is true. + +To assert that either one or the other of two Propositions +is true, is to assert that it is not true, that they are both false. +Now the elective expression appropriate to their both being +false is~$(1 - x)(1 - y)$, therefore the equation required is +\begin{align*} +(1 - x)(1 - y) &= 0, \\ +\intertext{or} +x + y - xy &= 1. +\Tag{(29)} +\end{align*} +\PageSep{52} + +And, by indirect considerations of this kind, may every disjunctive +Proposition, however numerous its members, be expressed. +But the following general Rule will usually be +preferable. + +\begin{Rule} +Consider what are those distinct and mutually exclusive +cases of which it is implied in the statement of the given Proposition, +that some one of them is true, and equate the sum of their +elective expressions to unity. This will give the equation of the +given Proposition. +\end{Rule} + +For the sum of the elective expressions for all distinct conceivable +cases will be unity. Now all these cases being mutually +exclusive, and it being asserted in the given Proposition that +some one case out of a given set of them is true, it follows that +all which are not included in that set are false, and that their +elective expressions are severally equal to~$0$. Hence the sum +of the elective expressions for the remaining cases, viz.\ those +included in the given set, will be unity. Some one of those +cases will therefore be true, and as they are mutually exclusive, +it is impossible that more than one should be true. Whence +the Rule in question. + +And in the application of this Rule it is to be observed, that +if the cases contemplated in the given disjunctive Proposition +are not mutually exclusive, they must be resolved into an equivalent +series of cases which are mutually exclusive. + +Thus, if we take the Proposition of the preceding example, +viz.\ Either $X$~is true, or $Y$~is true, and assume that the two +members of this Proposition are not exclusive, insomuch that +in the enumeration of possible cases, we must reckon that of +the Propositions $X$~and~$Y$ being both true, then the mutually +exclusive cases which fill up the Universe of the Proposition, +with their elective expressions, are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~true and $Y$~false,}& x(1 - y), \\ +\text{2nd,}& \text{$Y$~true and $X$~false,}& y(1 - x), \\ +\text{3rd,}& \text{$X$~true and $Y$~true,} & xy, +\end{array} +\] +\PageSep{53} +and the sum of these elective expressions equated to unity gives +\[ +x + y - xy = 1\Typo{.}{,} +\Tag{(30)} +\] +as before. But if we suppose the members of the disjunctive +Proposition to be exclusive, then the only cases to be considered +are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~true, $Y$~false,}& x(1 - y), \\ +\text{2nd,}& \text{$Y$~true, $X$~false,}& y(1 - x), +\end{array} +\] +and the sum of these elective expressions equated to~$0$, gives +\[ +x - 2xy + y = 1. +\Tag{(31)} +\] + +The subjoined examples will further illustrate this method. + +To express the Proposition, Either $X$~is not true, or $Y$~is not +true, the members being exclusive. + +The mutually exclusive cases are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~not true, $Y$~true,}& y(1 - x), \\ +\text{2nd,}& \text{$Y$~not true, $X$~true,}& x(1 - y), +\end{array} +\] +and the sum of these equated to unity gives +\[ +x - 2xy + y = 1, +\Tag{(32)} +\] +which is the same as~\Eqref{(31)}, and in fact the Propositions which +they represent are equivalent. + +To express the Proposition, Either $X$~is not true, or $Y$~is not +true, the members not being exclusive. + +To the cases contemplated in the last Example, we must add +the following, viz. +\[ +\text{$X$~not true, $Y$~not true,}\qquad (1 - x)(1 - y). +\] + +The sum of the elective expressions gives +\begin{gather*} +x(1 - y) + y(1 - x) + (1 - x)(1 - y) = 1, \\ +\intertext{or} +xy = 0. +\Tag{(33)} +\end{gather*} + +To express the disjunctive Proposition, Either $X$~is true, or +$Y$~is true, or $Z$~is true, the members being exclusive. +\PageSep{54} + +Here the mutually exclusive cases are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~true, $Y$~false, $Z$~false,}& x(1 - y)(1 - z), \\ +\text{2nd,}& \text{$Y$~true, $Z$~false, $X$~false,}& y(1 - z)(1 - x), \\ +\text{3rd,}& \text{$Z$~true, $X$~false, $Y$~false,}& z(1 - x)(1 - y), +\end{array} +\] +and the sum of the elective expressions equated to~$1$, gives, +upon reduction, +\[ +x + y + z - 2(xy + yz + zx) + 3xyz = 1. +\Tag{(34)} +\] + +The expression of the same Proposition, when the members +are in no sense exclusive, will be +\[ +(1 - x)(1 - y)(1 - z) = 0. +\Tag{(35)} +\] + +And it is easy to see that our method will apply to the +expression of any similar Proposition, whose members are +subject to any specified amount and character of exclusion. + +To express the conditional Proposition, If $X$~is true, $Y$~is +true. + +Here it is implied that all the cases of $X$~being true, are +cases of $Y$~being true. The former cases being determined +by the elective symbol~$x$, and the latter by~$y$, we have, in +virtue of~\Eqref{(4)}, +\[ +x(1 - y) = 0. +\Tag{(36)} +\] + +To express the conditional Proposition, If $X$~be true, $Y$~is +not true. + +The equation is obviously +\[ +xy = 0; +\Tag{(37)} +\] +this is equivalent to~\Eqref{(33)}, and in fact the disjunctive Proposition, +Either $X$~is not true, or $Y$~is not true, and the conditional +Proposition, If $X$~is true, $Y$~is not true, are equivalent. + +To express that If $X$~is not true, $Y$~is not true. + +In~\Eqref{(36)} write $1 - x$ for~$x$, and $1 - y$ for~$y$, we have +\[ +(1 - x)y = 0. +\] +\PageSep{55} + +The results which we have obtained admit of verification +in many different ways. Let it suffice to take for more particular +examination the equation +\[ +x - 2xy + y = 1, +\Tag{(38)} +\] +which expresses the conditional Proposition, Either $X$~is true, +or $Y$~is true, the members being in this case exclusive. + +First, let the Proposition~$X$ be true, then $x = 1$, and substituting, +we have +\[ +1 - 2y + y = 1,\qquad +\therefore -y = 0,\quad\text{or}\quad y = 0, +\] +which implies that $Y$~is not true. + +Secondly, let $X$~be not true, then $x = 0$, and the equation +gives +\[ +y = 1, +\Tag{(39)} +\] +which implies that $Y$~is true. In like manner we may proceed +with the assumptions that $Y$~is true, or that $Y$~is false. + +Again, in virtue of the property $x^{2} = x$, $y^{2} = y$, we may write +the equation in the form +\[ +x^{2} - 2xy + y^{2} = 1, +\] +and extracting the square root, we have +\[ +x - y = ±1, +\Tag{(40)} +\] +and this represents the actual case; for, as when $X$~is true +or false, $Y$~is respectively false or true, we have +\begin{gather*} +x = 1\quad\text{or}\quad 0, \\ +y = 0\quad\text{or}\quad 1, \\ +\therefore x - y = 1\quad\text{or}\quad -1. +\end{gather*} + +There will be no difficulty in the analysis of other cases. + + +\Section{Examples of Hypothetical Syllogism.} + +The treatment of every form of hypothetical Syllogism will +consist in forming the equations of the premises, and eliminating +the symbol or symbols which are found in more than one of +them. The result will express the conclusion. +\PageSep{56} + +1st. Disjunctive Syllogism. +\begin{align*} +&\begin{array}{l<{\qquad}@{}c@{}} +\text{Either $X$~is true, or $Y$~is true (exclusive),} & +x + y - 2xy = 1\Add{,} \\ +\text{But $X$~is true,} & x = 1\Add{,} \\ +\cline{2-2} +\text{Therefore $Y$~is not true,} & \therefore y = 0\Add{.} +\end{array} \\ +&\begin{array}{l<{\quad}@{}c@{}} +\text{Either $X$~is true, or $Y$~is true (not exclusive),}& +x + y - xy = 1\Add{,} \\ +\text{But $X$~is not true,}& x = 0\Add{,} \\ +\cline{2-2} +\text{Therefore $Y$~is true,}& \therefore y = 1\Add{.} +\end{array} +\end{align*} + +2nd. Constructive Conditional Syllogism. +\[ +\begin{array}{l<{\qquad}@{}c@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{But $X$~is true,}& x = 1\Add{,} \\ +\text{Therefore $Y$~is true,}& \therefore 1 - y = 0\quad\text{or}\quad y = 1. +\end{array} +\] + +3rd. Destructive Conditional Syllogism. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{But $Y$~is not true,}& y = 0\Add{,} \\ +\text{Therefore $X$~is not true,}& \therefore x = 0\Add{.} +\end{array} +\] + +4th. Simple Constructive Dilemma, the minor premiss exclusive. +\begin{alignat*}{2} +&\text{If $X$~is true, $Y$~is true,}& x(1 - y) &= 0, +\Tag{(41)} \\ +&\text{If $Z$~is true, $Y$~is true,}& z(1 - y) &= 0, +\Tag{(42)} \\ +&\text{But Either $X$~is true, or $Z$~is true,}\quad& +x + z - 2xz &= 1. +\Tag{(43)} +\end{alignat*} + +From the equations \Eqref{(41)},~\Eqref{(42)},~\Eqref{(43)}, we have to eliminate +$x$~and~$z$. In whatever way we effect this, the result is +\[ +y = 1; +\] +whence it appears that the Proposition~$Y$ is true. + +5th. Complex Constructive Dilemma, the minor premiss not +exclusive. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0, \\ +\text{If $W$~is true, $Z$~is true,}& w(1 - z) = 0, \\ +\text{Either $X$~is true, or $W$~is true,}& x + w - xw = 1. +\end{array} +\] + +From these equations, eliminating~$x$, we have +\[ +y + z - yz = 1, +\] +\PageSep{57} +which expresses the Conclusion, Either $Y$~is true, or $Z$~is true, +the members being \Chg{non-exclusive}{nonexclusive}. + +6th. Complex Destructive Dilemma, the minor premiss exclusive. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{If $W$~is true, $Z$~is true,}& w(1 - z) = 0\Add{,} \\ +\text{Either $Y$~is not true, or $Z$~is not true,}& y + z - 2yz = 1. +\end{array} +\] + +From these equations we must eliminate $y$~and~$z$. The +result is +\[ +xw = 0, +\] +which expresses the Conclusion, Either $X$~is not true, or $Y$~is +not true, the members \emph{not being exclusive}. + +7th. Complex Destructive Dilemma, the minor premiss not +exclusive. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{If $W$~is true, $Z$~is true,}& w(1 - z) = 0\Add{,} \\ +\text{Either $Y$~is not true, or $Z$~is not true,}& yz = 0. +\end{array} +\] + +On elimination of $y$~and~$z$, we have +\[ +xw = 0, +\] +which indicates the same Conclusion as the previous example. + +It appears from these and similar cases, that whether the +members of the minor premiss of a Dilemma are exclusive +or not, the members of the (disjunctive) Conclusion are never +exclusive. This fact has perhaps escaped the notice of logicians. + +The above are the principal forms of hypothetical Syllogism +which logicians have recognised. It would be easy, however, +to extend the list, especially by the blending of the disjunctive +and the conditional character in the same Proposition, of which +the following is an example. +\[ +\begin{array}{l<{\qquad}@{}c@{}} +\multicolumn{2}{l}{% + \text{If $X$~is true, then either $Y$~is true, or $Z$~is true,}} \\ + & x(1 - y - z + yz) = 0\Add{,} \\ +\text{But $Y$~is not true,}& y = 0\Add{,} \\ +\text{Therefore If $X$~is true, $Z$~is true,}& \therefore x(1 - z) = 0. +\end{array} +\] +\PageSep{58} + +That which logicians term a \emph{Causal} Proposition is properly +a conditional Syllogism, the major premiss of which is suppressed. + +The assertion that the Proposition~$X$ is true, \emph{because} the +Proposition~$Y$ is true, is equivalent to the assertion, +\begin{align*} +&\text{The Proposition~$Y$ is true,} \\ +&\text{\emph{Therefore} the Proposition X is true;} +\end{align*} +and these are the minor premiss and conclusion of the conditional +Syllogism, +\begin{align*} +&\text{If $Y$~is true, $X$~is true,} \\ +&\text{But $Y$~is true,} \\ +&\text{Therefore $X$~is true.} +\end{align*} +And thus causal Propositions are seen to be included in the +applications of our general method. + +Note, that there is a family of disjunctive and conditional +Propositions, which do not, of right, belong to the class considered +in this Chapter. Such are those in which the force +of the disjunctive or conditional particle is expended upon the +predicate of the Proposition, as if, speaking of the inhabitants +of a particular island, we should say, that they are all \emph{either +Europeans or Asiatics}; meaning, that it is true of each individual, +that he is either a European or an Asiatic. If we +appropriate the elective symbol~$x$ to the inhabitants, $y$~to +Europeans, and $z$~to Asiatics, then the equation of the above +Proposition is +\[ +x = xy + xz,\quad\text{or}\quad x(1 - y - z) = 0;\atag +\] +to which we might add the condition $yz = 0$, since no Europeans +are Asiatics. The nature of the symbols $x$,~$y$,~$z$, indicates that +the Proposition belongs to those which we have before designated +as \emph{Categorical}. Very different from the above is the +Proposition, Either all the inhabitants are Europeans, or they +are all Asiatics. Here the disjunctive particle separates Propositions. +The case is that contemplated in~\Eqref{(31)} of the present +Chapter; and the symbols by which it is expressed, +\PageSep{59} +although subject to the same laws as those of~\aref, have a totally +different interpretation.\footnote + {Some writers, among whom is Dr.\ Latham (\textit{First Outlines}), regard it as + the exclusive office of a conjunction to connect \emph{Propositions}, not \emph{words}. In this + view I am not able to agree. The Proposition, Every animal is \emph{either} rational + \emph{or} irrational, cannot be resolved into, \emph{Either} every animal is rational, \emph{or} every + animal is irrational. The former belongs to pure categoricals, the latter to + hypotheticals. In \emph{singular} Propositions, such conversions would seem to be + allowable. This animal is \emph{either} rational \emph{or} irrational, is equivalent to, \emph{Either} + this animal is rational, \emph{or} it is irrational. This peculiarity of \emph{singular} Propositions + would almost justify our ranking them, though truly universals, in + a separate class, as Ramus and his followers did.} + +The distinction is real and important. Every Proposition +which language can express may be represented by elective +symbols, and the laws of combination of those symbols are in +all cases the same; but in one class of instances the symbols +have reference to collections of objects, in the other, to the +truths of constituent Propositions. +\PageSep{60} + + +\Chapter{Properties of Elective Functions.} + +\First{Since} elective symbols combine according to the laws of +quantity, we may, by Maclaurin's theorem, expand a given +function~$\phi(x)$, in ascending powers of~$x$, known cases of failure +excepted. Thus we have +\[ +\phi(x) = \phi(0) + \phi'(0)x + \frac{\phi''(0)}{1·2}x^{2} + \etc. +\Tag{(44)} +\] + +Now $x^{2} = x$, $x^{3} = x$,~\etc., whence +\[ +\phi(x) = \phi(0) + x\bigl\{\phi'(0) + \frac{\phi''(0)}{1·2} + \etc.\bigr\}. +\Tag{(45)} +\] + +Now if in~\Eqref{(44)} we make $x = 1$, we have +\[ +\phi(1) = \phi(0) + \phi'(0) + \frac{\phi''(0)}{1·2} + \etc., +\] +whence +\[ +\phi'(0) + \frac{\phi''(0)}{1·2} + \frac{\phi'''(0)}{1·2·3} + \etc. + = \phi(1) - \phi(0). +\] + +Substitute this value for the coefficient of~$x$ in the second +member of~\Eqref{(45)}, and we have\footnote + {Although this and the following theorems have only been proved for those + forms of functions which are expansible by Maclaurin's theorem, they may be + regarded as true for all forms whatever; this will appear from the applications. + The reason seems to be that, as it is only through the one form of expansion + that elective functions become interpretable, no conflicting interpretation is + possible. + + The development of~$\phi(x)$ may also be determined thus. By the known formula + for expansion in factorials, + \[ + \phi(x) = \phi(0) + \Delta\phi(0)x + + \frac{\Delta^{2}\phi(0)}{1·2}x(x - 1) + \etc. + \] +%[** TN: Footnote continues] + Now $x$~being an elective symbol, $x(x - 1) = 0$, so that all the terms after the + second, vanish. Also $\Delta\phi(0) = \phi(1) - \phi(0)$, whence + \[ + \phi\bigl\{x = \phi(0)\bigr\} + \bigl\{\phi(1) - \phi(0)\bigr\}x. + \] + + The mathematician may be interested in the remark, that this is not the + only case in which an expansion stops at the second term. The expansions of + the compound operative functions $\phi\left(\dfrac{d}{dx} + x^{-1}\right)$ and $\phi\left\{x + \left(\dfrac{d}{dx}\right)^{-1}\right\}$ are, + respectively, + \[ + \phi\left(\frac{d}{dx}\right) + \phi'\left(\frac{d}{dx}\right)x^{-1}, + \] + and + \[ + \phi(x) + \phi'(x)\left(\frac{d}{dx}\right)^{-1}. + \] + + See \textit{Cambridge Mathematical Journal}, Vol.~\textsc{iv}. p.~219.} +\[ +\phi(x) = \phi(0) + \bigl\{\phi(1) - \phi(0)\bigr\}x, +\Tag{(46)} +\] +\PageSep{61} +which we shall also employ under the form +\[ +\phi(x) = \phi(1)x + \phi(0)(1 - x). +\Tag{(47)} +\] + +Every function of~$x$, in which integer powers of that symbol +are alone involved, is by this theorem reducible to the first +order. The quantities $\phi(0)$,~$\phi(1)$, we shall call the moduli +of the function~$\phi(x)$. They are of great importance in the +theory of elective functions, as will appear from the succeeding +Propositions. + +\Prop{1.} Any two functions $\phi(x)$,~$\psi(x)$, are equivalent, +whose corresponding moduli are equal. + +This is a plain consequence of the last Proposition. For since +\begin{align*} +\phi(x) &= \phi(0) + \bigl\{\phi(1) - \phi(0)\bigr\}x, \\ +\psi(x) &= \psi(0) + \bigl\{\psi(1) - \psi(0)\bigr\}x, +\end{align*} +it is evident that if $\phi(0) = \psi(0)$, $\phi(1) = \psi(1)$, the two +expansions will be equivalent, and therefore the functions which +they represent will be equivalent also. + +The converse of this Proposition is equally true, viz. + +If two functions are equivalent, their corresponding moduli +are equal. + +Among the most important applications of the above theorem, +we may notice the following. + +Suppose it required to determine for what forms of the +function~$\phi(x)$, the following equation is satisfied, viz. +\[ +\bigl\{\phi(x)\bigr\}^{n} = \phi(x). +\] +\PageSep{62} +Here we at once obtain for the expression of the conditions +in question, +\[ +\bigl\{\phi(0)\bigr\}^{n} = \phi(0)\Typo{.}{,}\quad +\bigl\{\phi(1)\bigr\}^{n} = \phi(1). +\Tag{(48)} +\] + +Again, suppose it required to determine the conditions under +which the following equation is satisfied, viz. +\[ +\phi(x)\psi(x) = \chi(x)\Typo{,}{.} +\] + +The general theorem at once gives +\[ +\phi(0)\psi(0) = \chi(0)\Typo{.}{,}\quad +\phi(1)\psi(1) = \chi(1). +\Tag{(49)} +\] + +This result may also be proved by substituting for~$\phi(x)$, +$\psi(x)$, $\chi(x)$, their expanded forms, and equating the coefficients +of the resulting equation properly reduced. + +All the above theorems may be extended to functions of more +than one symbol. For, as different elective symbols combine +with each other according to the same laws as symbols of quantity, +we can first expand a given function with reference to any +particular symbol which it contains, and then expand the result +with reference to any other symbol, and so on in succession, the +order of the expansions being quite indifferent. + +Thus the given function being~$\phi(xy)$ we have +\[ +\phi(xy) = \phi(x0) + \bigl\{\phi(x1) - \phi(x0)\bigr\}y, +\] +and expanding the coefficients with reference to~$x$, and reducing +\begin{align*} +\phi(xy) = \phi(00) + &+ \bigl\{\phi(10) - \phi(00)\bigr\}x + + \bigl\{\phi(01) - \phi(00)\bigr\}y \\ + &+ \bigl\{\phi(11) - \phi(10) - \phi(01) + \phi(00)\bigr\}xy, +\Tag{(50)} +\end{align*} +to which we may give the elegant symmetrical form +\begin{align*} +%[** TN: Not aligned in the original] +\phi(xy) = \phi(00)(1 - x)(1 - y) &+ \phi(01)y(1 - x) \\ + &+ \phi(10)x(1 - y) + \phi(11)xy, +\Tag{(51)} +\end{align*} +wherein we shall, in accordance with the language already +employed, designate $\phi(00)$, $\phi(01)$, $\phi(10)$, $\phi(11)$, as the +moduli of the function~$\phi(xy)$. + +By inspection of the above general form, it will appear that +any functions of two variables are equivalent, whose corresponding +moduli are all equal. +\PageSep{63} + +Thus the conditions upon which depends the satisfaction of +the equation, +\[ +\bigl\{\phi(xy)\bigr\}^{n} = \phi(xy) +\] +are seen to be +\[ +\begin{alignedat}{2} +\bigl\{\phi(00)\bigr\}^{n} &= \phi(00),\qquad& +\bigl\{\phi(01)\bigr\}^{n} &= \phi(01), \\ +\bigl\{\phi(10)\bigr\}^{n} &= \phi(10), & +\bigl\{\phi(11)\bigr\}^{n} &= \phi(11). +\end{alignedat} +\Tag{(52)} +\] + +And the conditions upon which depends the satisfaction of +the equation +\[ +\phi(xy)\psi(xy) = \chi(xy), +\] +are +\[ +\begin{alignedat}{2} +\phi(00)\psi(00) &= \chi(00),\qquad& +\phi(01)\psi(01) &= \chi(01), \\ +\phi(10)\psi(10) &= \chi(10),\qquad& +\phi(11)\psi(11) &= \chi(11). +\end{alignedat} +\Tag{(53)} +\] + +It is very easy to assign by induction from \Eqref{(47)}~and~\Eqref{(51)}, the +general form of an expanded elective function. It is evident +that if the number of elective symbols is~$m$, the number of the +moduli will be~$2^{m}$, and that their separate values will be obtained +by interchanging in every possible way the values $1$~and~$0$ in the +places of the elective symbols of the given function. The several +terms of the expansion of which the moduli serve as coefficients, +will then be formed by writing for each~$1$ that recurs under the +functional sign, the elective symbol~$x$,~\etc., which it represents, +and for each~$0$ the corresponding~$1 - x$,~\etc., and regarding these +as factors, the product of which, multiplied by the modulus from +which they are obtained, constitutes a term of the expansion. + +Thus, if we represent the moduli of any elective function +$\phi(xy\dots)$ by $a_{1}$,~$a_{2}$, $\dots,~a_{r}$, the function itself, when expanded +and arranged with reference to the moduli, will assume the form +\[ +\phi(xy) = a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}, +\Tag{(54)} +\] +in which $t_{1}t_{2}\dots t_{r}$ are functions of $x$,~$y$,~$\dots$, resolved into factors +of the forms $x$,~$y$,~$\dots$ $1 - x$, $1 - y$,~$\dots$~\etc. These functions satisfy +individually the index relations +\[ +t_{1}^{n} = t_{1},\quad +t_{2}^{n} = t_{2},\quad \etc., +\Tag{(55)} +\] +and the further relations, +\[ +t_{1}t_{2} = 0\dots t_{1}t_{2} = 0,~\etc., +\Tag{(56)} +\] +\PageSep{64} +the product of any two of them vanishing. This will at once +be inferred from inspection of the particular forms \Eqref{(47)}~and~\Eqref{(51)}. +Thus in the latter we have for the values of $t_{1}$,~$t_{2}$,~\etc., the forms +\[ +xy,\quad +x(1 - y),\quad +(1 - x)y,\quad +(1 - x)(1 - y); +\] +and it is evident that these satisfy the index relation, and that +their products all vanish. We shall designate $t_{1}t_{2}\dots$ as the constituent +functions of~$\phi(xy)$, and we shall define the peculiarity +of the vanishing of the binary products, by saying that those +functions are \emph{exclusive}. And indeed the classes which they +represent are mutually exclusive. + +The sum of all the constituents of an expanded function is +unity. An elegant proof of this Proposition will be obtained +by expanding~$1$ as a function of any proposed elective symbols. +Thus if in~\Eqref{(51)} we assume $\phi(xy) = 1$, we have $\phi(11) = 1$, +$\phi(10) = 1$, $\phi(01) = 1$, $\phi(00) = 1$, and \Eqref{(51)}~gives +\[ +1 = xy + x(1 - y) + (1 - x)y + (1 - x)(1 - y). +\Tag{(57)} +\] + +It is obvious indeed, that however numerous the symbols +involved, all the moduli of unity are unity, whence the sum +of the constituents is unity. + +We are now prepared to enter upon the question of the +general interpretation of elective equations. For this purpose +we shall find the following Propositions of the greatest service. + +\Prop{2.} If the first member of the general equation +$\phi(xy\dots) = 0$, be expanded in a series of terms, each of which +is of the form~$at$, $a$~being a modulus of the given function, then +for every numerical modulus~$a$ which does not vanish, we shall +have the equation +\[ +at = 0, +\] +and the combined interpretations of these several equations will +express the full significance of the original equation. + +For, representing the equation under the form +\[ +a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r} = 0. +\Tag{(58)} +\] + +Multiplying by~$t_{1}$ we have, by~\Eqref{(56)}, +\[ +a_{1}t_{1} = 0, +\Tag{(59)} +\] +\PageSep{65} +whence if $a_{1}$~is a numerical constant which does not vanish, +\[ +t_{1} = 0, +\] +and similarly for all the moduli which do not vanish. And +inasmuch as from these constituent equations we can form the +given equation, their interpretations will together express its +entire significance. + +Thus if the given equation were +\[ +x - y = 0,\quad \text{$X$s~and~$Y$s are identical,} +\Tag{(60)} +\] +we should have $\phi(11) = 0$, $\phi(10) = 1$, $\phi(01) = -1$, $\phi(00) = 0$, +so that the expansion~\Eqref{(51)} would assume the form +\[ +x(1 - y) - y(1 - x) = 0, +\] +whence, by the above theorem, +\begin{alignat*}{2} +x(1 - y) &= 0,\qquad& \text{All~$X$s are~$Y$s,} \\ +y(1 - x) &= 0, & \text{All~$Y$s are~$X$s,} +\end{alignat*} +results which are together equivalent to~\Eqref{(60)}. + +It may happen that the simultaneous satisfaction of equations +thus deduced, may require that one or more of the elective +symbols should vanish. This would only imply the nonexistence +of a class: it may even happen that it may lead to a final +result of the form +\[ +1 = 0, +\] +which would indicate the nonexistence of the logical Universe. +Such cases will only arise when we attempt to unite contradictory +Propositions in a single equation. The manner in which +the difficulty seems to be evaded in the result is characteristic. + +It appears from this Proposition, that the differences in the +interpretation of elective functions depend solely upon the +number and position of the vanishing moduli. No change in +the value of a modulus, but one which causes it to vanish, +produces any change in the interpretation of the equation in +which it is found. If among the infinite number of different +values which we are thus permitted to give to the moduli which +do not vanish in a proposed equation, any one value should be +\PageSep{66} +preferred, it is unity, for when the moduli of a function are all +either $0$~or~$1$, the function itself satisfies the condition +\[ +\bigl\{\phi(xy\dots)\bigr\}^{n} = \phi(xy\dots), +\] +and this at once introduces symmetry into our Calculus, and +provides us with fixed standards for reference. + +\Prop{3.} If $w = \phi(xy\dots)$, $w$,~$x$,~$y$,~$\dots$ being elective symbols, +and if the second member be completely expanded and arranged +in a series of terms of the form~$at$, we shall be permitted +to equate separately to~$0$ every term in which the modulus~$a$ +does not satisfy the condition +\[ +a^{n} = a, +\] +and to leave for the value of~$w$ the sum of the remaining terms. + +As the nature of the demonstration of this Proposition is +quite unaffected by the number of the terms in the second +member, we will for simplicity confine ourselves to the supposition +of there being four, and suppose that the moduli of the +two first only, satisfy the index law. + +We have then +\[ +w = a_{1}t_{1} + a_{2}t_{2} + a_{3}t_{3} + a_{4}t_{4}, +\Tag{(61)} +\] +with the relations +\[ +a_{1}^{n} = a_{1},\quad +a_{2}^{n} = a_{2}, +\] +in addition to the two sets of relations connecting $t_{1}$,~$t_{2}$, $t_{3}$,~$t_{4}$, +in accordance with \Eqref{(55)}~and~\Eqref{(56)}. + +Squaring~\Eqref{(61)}, we have +\[ +w = a_{1}t_{1} + a_{2}t_{2} + a_{3}^{2}t_{3} + a_{4}^{2}t_{4}, +\] +and subtracting~\Eqref{(61)} from this, +\[ +(a_{3}^{2} - a_{3})t_{3} + (a_{4}^{2} - a_{4})t_{4} = 0; +\] +and it being an hypothesis, that the coefficients of these terms +do not vanish, we have, by \PropRef{2}, +\[ +t_{3} = 0,\quad +t_{4} = 0, +\Tag{(62)} +\] +whence \Eqref{(61)}~becomes +\[ +w = a_{1}t_{1} + a_{2}t_{2}. +\] +The utility of this Proposition will hereafter appear. +\PageSep{67} + +\Prop{4.} The functions $t_{1}t_{2}\dots t_{r}$ being mutually exclusive, we +shall always have +\[ +\psi(a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}) + = \psi(a_{1})t_{1} + \psi(a_{2})t_{2} \dots + \psi(a_{r})t_{r}, +\Tag{(63)} +\] +whatever may be the values of $a_{1}a_{2}\dots a_{r}$ or the form of~$\psi$. + +%[** TN: Paragraph not indented in the original] +Let the function $a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}$ be represented by~$\phi(xy\dots)$, +then the moduli $a_{1}a_{2}\dots a_{r}$ will be given by the expressions +\[ +\phi(11\dots),\quad +\phi(10\dots),\quad +(\dots)\ \phi(00\dots). +\] + +Also +\begin{align*} +&\phantom{{}={}}\psi(a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}) + = \psi\bigl\{\phi(xy\dots)\bigr\} \\ + &= \psi\bigl\{\phi(11\dots)\bigr\}xy\dots + + \psi\bigl\{\phi(10\dots)\bigr\}x(1 - y)\dots \\ + &\qquad+ \psi\bigl\{\phi(00\dots)\bigr\}(1 - x)(1 - y)\dots \\ + &= \psi(a_{1})xy\dots + \psi(a_{2})x(1 - y)\dots + \psi(a_{r})(1 - x)(1 - y)\dots \\ + &= \psi(a_{1})t_{1} + \psi(a_{2})t_{2}\dots + \psi(a_{r})t_{r}. +\Tag{(64)} +\end{align*} + +It would not be difficult to extend the list of interesting +properties, of which the above are examples. But those which +we have noticed are sufficient for our present requirements. +The following Proposition may serve as an illustration of their +utility. + +\Prop{5.} Whatever process of reasoning we apply to a single +given Proposition, the result will either be the same Proposition +or a limitation of it. + +Let us represent the equation of the given Proposition under +its most general form, +\[ +a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r} = 0, +\Tag{(65)} +\] +resolvable into as many equations of the form $t = 0$ as there are +moduli which do not vanish. + +Now the most general transformation of this equation is +\[ +\psi(a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}) = \psi(0), +\Tag{(66)} +\] +provided that we attribute to~$\psi$ a perfectly arbitrary character, +allowing it even to involve new elective symbols, having \emph{any +proposed relation} to the original ones. +\PageSep{68} + +The development of~\Eqref{(66)} gives, by the last Proposition, +\[ +\psi(a_{1})t_{1} + \psi(a_{2})t_{2}\dots + \psi(a_{r})t_{r} = \psi(0). +\] +To reduce this to the general form of reference, it is only necessary +to observe that since +\[ +t_{1} + t_{2} \dots + t_{r} = 1, +\] +we may write for~$\psi(0)$, +\[ +\psi(0)(t_{1} + t_{2} \dots + t_{r}), +\] +whence, on substitution and transposition, +\[ +\bigl\{\psi(a_{1}) - \psi(0)\bigr\}t_{1} + +\bigl\{\psi(a_{2}) - \psi(0)\bigr\}t_{2} \dots + +\bigl\{\psi(a_{r}) - \psi(0)\bigr\}t_{r} = 0. +\] + +From which it appears, that if $a$~be any modulus of the +original equation, the corresponding modulus of the transformed +equation will be +\[ +\psi(a) - \psi(0). +\] + +If $a = 0$, then $\psi(a) - \psi(0) = \psi(0) - \psi(0) = 0$, whence +there are no \emph{new terms} in the transformed equation, and therefore +there are no \emph{new Propositions} given by equating its constituent +members to~$0$. + +Again, since $\psi(a) - \psi(0)$ may vanish without $a$~vanishing, +terms may be wanting in the transformed equation which existed +in the primitive. Thus some of the constituent truths of the +original Proposition may entirely disappear from the interpretation +of the final result. + +Lastly, if $\psi(a) - \psi(0)$ do not vanish, it must either be +a numerical constant, or it must involve new elective symbols. +In the former case, the term in which it is found will give +\[ +t = 0, +\] +which is one of the constituents of the original equation: in the +latter case we shall have +\[ +\bigl\{\psi(a\Typo{}{)} - \psi(0)\bigr\}t = 0, +\] +in which $t$~has a limiting factor. The interpretation of this +equation, therefore, is a limitation of the interpretation of~\Eqref{(65)}. +\PageSep{69} + +The purport of the last investigation will be more apparent +to the mathematician than to the logician. As from any mathematical +equation an infinite number of others may be deduced, +it seemed to be necessary to shew that when the original +equation expresses a logical Proposition, every member of the +derived series, even when obtained by expansion under a functional +sign, admits of exact and consistent interpretation. +\PageSep{70} + + +\Chapter{Of the Solution of Elective Equations.} + +\First{In} whatever way an elective symbol, considered as unknown, +may be involved in a proposed equation, it is possible to assign +its complete value in terms of the remaining elective symbols +considered as known. It is to be observed of such equations, +that from the very nature of elective symbols, they are necessarily +linear, and that their solutions have a very close analogy +with those of linear differential equations, arbitrary elective +symbols in the one, occupying the place of arbitrary constants +in the other. The method of solution we shall in the first place +illustrate by particular examples, and, afterwards, apply to the +investigation of general theorems. + +Given $(1 - x)y = 0$, (All~$Y$s are~$X$s), to determine~$y$ in +terms of~$x$. + +As $y$~is a function of~$x$, we may assume $y = vx + v'(1 - x)$, +(such being the expression of an arbitrary function of~$x$), the +moduli $v$~and~$v'$ remaining to be determined. We have then +\[ +(1 - x)\bigl\{vx + v'(1 - x)\bigr\} = 0, +\] +or, on actual multiplication, +\[ +v'(1 - x) = 0\Typo{:}{;} +\] +that this may be generally true, without imposing any restriction +upon~$x$, we must assume $v' = 0$, and there being no condition to +limit~$v$, we have +\[ +y = vx. +\Tag{(67)} +\] + +This is the complete solution of the equation. The condition +that $y$~is an elective symbol requires that $v$~should be an elective +\PageSep{71} +symbol also (since it must satisfy the index law), its interpretation +in other respects being arbitrary. + +Similarly the solution of the equation, $xy = 0$, is +\[ +y = v(1 - x). +\Tag{(68)} +\] + +Given $(1 - x)zy = 0$, (All~$Y$s which are~$Z$s are~$X$s), to determine~$y$. + +As $y$~is a function of $x$~and~$z$, we may assume +\[ +y = v(1 - x) (1 - z) + v'(1 - x)z + v''x(1 - z) + v'''zx. +\] +And substituting, we get +\[ +v'(1 - x)z = 0, +\] +whence $v' = 0$. The complete solution is therefore +\[ +y = v(1 - x)(1 - z) + v''x(1 - z) + v'''xz, +\Tag{(69)} +\] +$v'$,~$v''$,~$v'''$, being arbitrary elective symbols, and the rigorous +interpretation of this result is, that Every~$Y$ is \emph{either} a not-$X$ +and not-$Z$, or an~$X$ and not-$Z$, or an~$X$ and~$Z$. + +It is deserving of note that the above equation may, in consequence +of its linear form, be solved by adding the two +particular solutions with reference to $x$~and~$z$; and replacing +the arbitrary constants which each involves by an arbitrary +function of the other symbol, the result is +\[ +y = x\phi(z) + (1 - z)\psi(x). +\Tag{(70)} +\] + +To shew that this solution is equivalent to the other, it is +only necessary to substitute for the arbitrary functions $\phi(z)$, +$\psi(x)$, their equivalents +\[ +wz + w'(1 - z)\quad\text{and}\quad w''x + w'''(1 - x), +\] +we get +\[ +y = wxz + (w + w'')x(1 - z) + w'''(1 - x)(1 - z). +\] + +In consequence of the perfectly arbitrary character of $w'$~and~$w''$, +we may replace their sum by a single symbol~$w$, whence +\[ +y = wxz + w'x(1 - z) + w'''(1 - x)(1 - z), +\] +which agrees with~\Eqref{(69)}. +\PageSep{72} + +The solution of the equation $wx(1 - y)z = 0$, expressed by +arbitrary functions, is +\[ +z = (1 - w) \phi(xy) + (1 - x)\psi(wy) + y\chi(wx). +\Tag{(71)} +\] + +These instances may serve to shew the analogy which exists +between the solutions of elective equations and those of the +corresponding order of linear differential equations. Thus the +expression of the integral of a partial differential equation, +either by arbitrary functions or by a series with arbitrary coefficients, +is in strict analogy with the case presented in the two +last examples. To pursue this comparison further would minister +to curiosity rather than to utility. We shall prefer to contemplate +the problem of the solution of elective equations under +its most general aspect, which is the object of the succeeding +investigations. + +To solve the general equation $\phi(xy) = 0$, with reference to~$y$. + +If we expand the given equation with reference to $x$~and~$y$, +we have +\[ +%[** TN: Equation broken across two lines in the original +\phi(00)(1 - x)(1 - y) + \phi(01)(1 - x)y + \phi(10)x(1 - y) + + \phi(11)xy = 0, +\Tag{(72)} +\] +the coefficients $\phi(00)$~\etc.\ being numerical constants. + +Now the general expression of~$y$, as a function of~$x$, is +\[ +y = vx + v'(1 - x), +\] +$v$~and~$v'$ being unknown symbols to be determined. Substituting +this value in~\Eqref{(72)}, we obtain a result which may be written in +the following form, +\[ +%[** TN: Equation broken across two lines in the original +\bigl[\phi(10) + \bigl\{\phi(11) - \phi(10)\bigr\}v\bigr]x + + \bigl[\phi(00) + \bigl\{\phi(00) - \phi(00)\bigr\} v'\bigr](1 - x) = 0; +\] +and in order that this equation may be satisfied without any +way restricting the generality of~$x$, we must have +\begin{alignat*}{2} +\phi(10) &+ \bigl\{\phi(11) - \phi(10)\bigr\}v &&= 0, \\ +\phi(00) &+ \bigl\{\phi(01) - \phi(00)\bigr\}v' &&= 0, +\end{alignat*} +\PageSep{73} +from which we deduce +\[ +v = \frac{\phi(10)}{\phi(10) - \phi(11)}\;,\qquad +v' = \frac{\phi(00)}{\phi(01) - \phi(00)}\;, +\] +wherefore +\[ +y = \frac{\phi(10)}{\phi(10) - \phi(11)}\, x + + \frac{\phi(00)}{\phi(00) - \phi(01)}\, (1 - x). +\Tag{(73)} +\] + +Had we expanded the original equation with respect to $y$~only, +we should have had +\[ +\phi(x0) + \bigl\{\phi(x1) - \phi(x0)\bigr\}y = 0; +\] +but it might have startled those who are unaccustomed to the +processes of Symbolical Algebra, had we from this equation +deduced +\[ +y = \frac{\phi(x0)}{\phi(x0) - \phi(x1)}\;, +\] +because of the apparently meaningless character of the second +member. Such a result would however have been perfectly +lawful, and the expansion of the second member would have +given us the solution above obtained. I shall in the following +example employ this method, and shall only remark that those +to whom it may appear doubtful, may verify its conclusions by +the previous method. + +To solve the general equation $\phi(xyz) = 0$, or in other words +to determine the value of~$z$ as a function of $x$~and~$y$. + +Expanding the given equation with reference to~$z$, we have +\begin{gather*} +\phi(xy0) + \bigl\{\phi(xy1) - \phi(xy0)\bigr\}\Chg{·}{}z = 0; \\ +\therefore z = \frac{\phi(xy0)}{\phi(xy0) - \phi(xy1)}\;, +\Tag{(74)} +\end{gather*} +and expanding the second member as a function of $x$~and~$y$ by +aid of the general theorem, we have +\begin{multline*} +z = \frac{\phi(110)}{\phi(110) - \phi(111)}\, xy + + \frac{\phi(100)}{\phi(100) - \phi(101)}\, x(1 - y) \\ + + \frac{\phi(010)}{\phi(010) - \phi(011)}\, (1 - x)y + + \frac{\phi(000)}{\phi(000) - \phi(001)}\, (1 - x)(1 - y), +\Tag{(75)} +\end{multline*} +\PageSep{74} +and this is the complete solution required. By the same +method we may resolve an equation involving any proposed +number of elective symbols. + +In the interpretation of any general solution of this nature, +the following cases may present themselves. + +The values of the moduli $\phi(00)$, $\phi(01)$,~\etc.\ being constant, +one or more of the coefficients of the solution may assume +the form $\frac{0}{0}$~or~$\frac{1}{0}$. In the former case, the indefinite symbol~$\frac{0}{0}$ +must be replaced by an arbitrary elective symbol~$v$. In the +latter case, the term, which is multiplied by a factor~$\frac{1}{0}$ (or by +any numerical constant except~$1$), must be separately equated +to~$0$, and will indicate the existence of a subsidiary Proposition. +This is evident from~\Eqref{(62)}. + +Ex. Given $x(1 - y) = 0$, All~$X$s are~$Y$s, to determine~$y$ as +a function of~$x$. + +Let $\phi(xy) = x(1 - y)$, then $\phi(10) = 1$, $\phi(11) = 0$, $\phi(01) = 0$, +$\phi(00) = 0$; whence, by~\Eqref{(73)}, +\begin{align*} +y &= \frac{1}{1 - 0}\, x + \frac{0}{0 - 0}\, (1 - x) \\ + &= x + \tfrac{0}{0}(1 - x) \\ + &= x + v(1 - x), +\Tag{(76)} +\end{align*} +$v$~being an arbitrary elective symbol. The interpretation of this +result is that the class~$Y$ consists of the entire class~$X$ with an +indefinite remainder of not-$X$s. This remainder is indefinite in +the highest sense, \ie~it may vary from~$0$ up to the entire class +of not-$X$s. + +Ex. Given $x(1 - z) + z = y$, (the class~$Y$ consists of the +entire class~$Z$, with such not-$Z$s as are~$X$s), to find~$Z$. + +Here $\phi(xyz) = x(1 - z) - y + z$, whence we have the following +set of values for the moduli, +\begin{alignat*}{4} +\phi(110) &= 0,\quad& \phi(111) &= 0,\quad& \phi(100) &= 1,\quad& \phi(101) &= 1, \\ +\phi(010) &=-1,\quad& \phi(011) &= 0,\quad& \phi(000) &= 0,\quad& \phi(001) &= 1, +\end{alignat*} +and substituting these in the general formula~\Eqref{(75)}, we have +\[ +z = \tfrac{0}{0}xy + \tfrac{1}{0}x(1 - y) + (1 - x)y, +\Tag{(77)} +\] +\PageSep{75} +the infinite coefficient of the second term indicates the equation +\[ +x(1 - y) = 0,\quad\text{All~$X$s are~$Y$s;} +\] +and the indeterminate coefficient of the first term being replaced +by~$v$, an arbitrary elective symbol, we have +\[ +z = (1 - x)y + vxy, +\] +the interpretation of which is, that the class~$Z$ consists of all the~$Y$s +which are not~$X$s, and an \emph{indefinite} remainder of~$Y$s which +are~$X$s. Of course this indefinite remainder may vanish. The +two results we have obtained are logical inferences (not very +obvious ones) from the original Propositions, and they give us +all the information which it contains respecting the class~$Z$, and +its constituent elements. + +Ex. Given $x = y(1 - z) + z(1 - y)$. The class~$X$ consists of +all~$Y$s which are not-$Z$s, and all~$Z$s which are not-$Y$s: required +the class~$Z$. + +We have +\begin{alignat*}{4} +\phi(xyz) &= \rlap{$x - y(1 - z) - z(1 - y)$,} \\ +\phi(110) &= 0,\quad& \phi(111) &= 1,\quad& +\phi(100) &= 1,\quad& \phi(101) &= 0, \\ +% +\phi(010) &= -1,\quad& \phi(011) &= 0, & +\phi(000) &= 0, & \phi(001) &= -1; +\end{alignat*} +whence, by substituting in~\Eqref{(75)}, +\[ +z = x(1 - y) + y(1 - x), +\Tag{(78)} +\] +the interpretation of which is, the class~$Z$ consists of all~$X$s +which are not~$Y$s, and of all~$Y$s which are not~$X$s; an inference +strictly logical. + +Ex. Given $y\bigl\{1 - z(1 - x)\bigr\} = 0$, All~$Y$s are~$Z$s and not-$X$s. + +Proceeding as before to form the moduli, we have, on substitution +in the general formulæ, +\[ +z = \tfrac{1}{0}xy + + \tfrac{0}{0}x(1 - y) + + y(1 - x) + + \tfrac{0}{0}(1 - x)(1 - y), +\] +or +\begin{align*} +%[** TN: Unaligned in the original] +z &= y(1 - x) + vx(1 - y) + v'(1 - x)(1 - y) \\ + &= y(1 - x) + (1 - y)\phi(x), +\Tag{(79)} +\end{align*} +with the relation +\[ +xy = 0\Typo{:}{;} +\] +from these it appears that No~$Y$s are~$X$s, and that the class~$Z$ +\PageSep{76} +consists of all~$Y$s which are not~$X$s, and of an indefinite remainder +of not-$Y$s. + +This method, in combination with Lagrange's method of +indeterminate multipliers, may be very elegantly applied to the +treatment of simultaneous equations. Our limits only permit us +to offer a single example, but the subject is well deserving of +further investigation. + +Given the equations $x(1 - z) = 0$, $z(1 - y) = 0$, All~$X$s are~$Z$s, +All~$Z$s are~$Y$s, to determine the complete value of~$z$ with +any subsidiary relations connecting $x$~and~$y$. + +Adding the second equation multiplied by an indeterminate +constant~$\lambda$, to the first, we have +\[ +x(1 - z) + \lambda z(1 - y) = 0, +\] +whence determining the moduli, and substituting in~\Eqref{(75)}, +\[ +z = xy + \frac{1}{1 - \lambda}\, x(1 - y) + \tfrac{0}{0}(1 - x)y, +\Tag{(80)} +\] +from which we derive +\[ +z = xy + v(1 - x)y, +\] +with the subsidiary relation +\[ +x(1 - y) = 0\Typo{:}{;} +\] +the former of these expresses that the class~$Z$ consists of all~$X$s +that are~$Y$s, with an indefinite remainder of not-$X$s that are~$Y$s; +the latter, that All~$X$s are~$Y$s, being in fact the conclusion +of the syllogism of which the two given Propositions are the +premises. + +By assigning an appropriate meaning to our symbols, all the +equations we have discussed would admit of interpretation in +hypothetical, but it may suffice to have considered them as +examples of categoricals. + +That peculiarity of elective symbols, in virtue of which every +elective equation is reducible to a system of equations $t_{1} = 0$, +$t_{2} = 0$,~\etc., so constituted, that all the binary products $t_{1}t_{2}$, $t_{1}t_{3}$, +\etc., vanish, represents a general doctrine in Logic with reference +to the ultimate analysis of Propositions, of which it +may be desirable to offer some illustration. +\PageSep{77} + +Any of these constituents $t_{1}$,~$t_{2}$,~\etc.\ consists only of factors +of the forms $x$,~$y$,~$\dots$ $1 - w$,~$1 - z$,~\etc. In categoricals it therefore +represents a compound class, \ie~a class defined by the +presence of certain qualities, and by the absence of certain +other qualities. + +Each constituent equation $t_{1} = 0$,~\etc.\ expresses a denial of the +existence of some class so defined, and the different classes are +mutually exclusive. + +\begin{Rule}[] +Thus all categorical Propositions are resolvable into a denial of +the existence of certain compound classes, no member of one such +class being a member of another. +\end{Rule} + +The Proposition, All~$X$s are~$Y$s, expressed by the equation +$x(1 - y) = 0$, is resolved into a denial of the existence of a +class whose members are~$X$s and not-$Y$s. + +The Proposition Some~$X$s are~$Y$s, expressed by $v = xy$, is +resolvable as follows. On expansion, +\begin{gather*} +v - xy = vx(1 - y) + vy(1 - x) + v(1 - x)(1 - y) - xy(1 - v); \\ +\therefore +vx(1 - y) = 0,\quad +vy(1 - x) = 0,\quad +v(1 - x)(1 - y) = 0,\quad +(1 - v)xy = 0. +\end{gather*} + +The three first imply that there is no class whose members +belong to a certain unknown Some, and are~1st, $X$s~and not~$Y$s; +2nd, $Y$s~and not~$X$s; 3rd, not-$X$s and not-$Y$s. The fourth +implies that there is no class whose members are $X$s~and~$Y$s +without belonging to this unknown Some. + +From the same analysis it appears that \begin{Rule}[]all hypothetical Propositions +may be resolved into denials of the coexistence of the truth +or falsity of certain assertions. +\end{Rule} + +Thus the Proposition, If $X$~is true, $Y$~is true, is resolvable +by its equation $x(1 - y) = 0$, into a denial that the truth of~$X$ +and the falsity of~$Y$ coexist. + +And the Proposition Either $X$~is true, or $Y$~is true, members +exclusive, is resolvable into a denial, first, that $X$~and~$Y$ are +both true; secondly, that $X$~and~$Y$ are both false. + +But it may be asked, is not something more than a system of +negations necessary to the constitution of an affirmative Proposition? +is not a positive element required? Undoubtedly +\PageSep{78} +there is need of one; and this positive element is supplied +in categoricals by the assumption (which may be regarded as +a prerequisite of reasoning in such cases) that there \emph{is} a Universe +of conceptions, and that each individual it contains either +belongs to a proposed class or does not belong to it; in hypotheticals, +by the assumption (equally prerequisite) that there +is a Universe of conceivable cases, and that any given Proposition +is either true or false. Indeed the question of the +existence of conceptions (\textgreek{e>i >'esti}) is preliminary to any statement +%[** TN: Should be \textgreek{t'i >esti}? Not sufficiently certain to change.] +of their qualities or relations (\textgreek{t'i >'esti}).---\textit{Aristotle, Anal.\ Post.}\ +lib.~\textsc{ii}.\ cap.~2. + +It would appear from the above, that Propositions may be +regarded as resting at once upon a positive and upon a negative +foundation. Nor is such a view either foreign to the spirit +of Deductive Reasoning or inappropriate to its Method; the +latter ever proceeding by limitations, while the former contemplates +the particular as derived from the general. + + +%[** TN: Equation numbering restarts] +\Section{Demonstration of the Method of Indeterminate Multipliers, as +applied to Simultaneous Elective Equations.} + +To avoid needless complexity, it will be sufficient to consider +the case of three equations involving three elective symbols, +those equations being the most general of the kind. It will +be seen that the case is marked by every feature affecting +the character of the demonstration, which would present itself +in the discussion of the more general problem in which the +number of equations and the number of variables are both +unlimited. + +Let the given equations be +\[ +\phi(xyz) = 0,\quad +\psi(xyz) = 0,\quad +\chi(xyz) = 0. +\Tag[app]{(1)} +\] + +Multiplying the second and third of these by the arbitrary +constants $h$~and~$k$, and adding to the first, we have +\[ +\phi(xyz) + h\psi(xyz) + k\chi(xyz) = 0; +\Tag[app]{(2)} +\] +\PageSep{79} +and we are to shew, that in solving this equation with reference +to any variable~$z$ by the general theorem~\Eqref{(75)}, we shall obtain +not only the general value of~$z$ independent of $h$~and~$k$, but +also any subsidiary relations which may exist between $x$~and~$y$ +independently of~$z$. + +%[xref] +If we represent the general equation~\Eqref[app]{(2)} under the form +$F(xyz) = 0$, its solution may by~\Eqref{(75)} be written in the form +\[ +z = \frac{xy}{1 - \dfrac{F(111)}{F(110)}} + + \frac{x(1 - y)}{1 - \dfrac{F(101)}{F(100)}} + + \frac{y(1 - x)}{1 - \dfrac{F(011)}{F(010)}} + + \frac{(1 - x)(1 - y)}{1 - \dfrac{F(001)}{F(000)}}; +\] +and we have seen, that any one of these four terms is to be +equated to~$0$, whose modulus, which we may represent by~$M$, +does not satisfy the condition $M^{n} = M$, or, which is here the +same thing, whose modulus has any other value than $0$~or~$1$. + +Consider the modulus (suppose~$M_{1}$) of the first term, viz. +$\dfrac{1}{1 - \dfrac{F(111)}{F(110)}}$, and giving to the symbol~$F$ its full meaning, +we have +\[ +M_{1} = \frac{1}{1 - \dfrac{\phi(111) + h\psi(111) + k\chi(111)} + {\phi(110) + h\psi(110) + k\chi(110)}}. +\] + +It is evident that the condition $M_{1}^{n} = M_{1}$ cannot be satisfied +unless the right-hand member be independent of $h$~and~$k$; and +in order that this may be the case, we must have the function +$\dfrac{\phi(111) + h\psi(111) + k\chi(111)} + {\phi(110) + h\psi(110) + k\chi(110)}$ independent of $h$~and~$k$. + +Assume then +\[ +\frac{\phi(111) + h\psi(111) + k\chi(111)} + {\phi(110) + h\psi(110) + k\chi(110)} = c, +\] +$c$~being independent of $h$~and~$k$; we have, on clearing of fractions +and equating coefficients, +\[ +\phi(111) = c\phi(110),\quad +\psi(111) = c\psi(110),\quad +\chi(111) = c\chi(110); +\] +whence, eliminating~$c$, +\[ +\frac{\phi(111)}{\phi(110)} + = \frac{\psi(111)}{\psi(110)} + = \frac{\chi(111)}{\chi(110)}, +\] +\PageSep{80} +being equivalent to the triple system +\[ +\left.\begin{alignedat}{3} +&\phi(111)\psi(110) &&- \phi(110)\psi(111) &&= 0\Add{,} \\ +&\psi(111)\chi(110) &&- \psi(110)\chi(111) &&= 0\Add{,} \\ +&\chi(111)\phi(110) &&- \chi(110)\Typo{\psi}{\phi}(111) &&= 0\Add{;} +\end{alignedat} +\right\} +\Tag[app]{(3)} +\] +and it appears that if any one of these equations is not satisfied, +the modulus~$M_{1}$ will not satisfy the condition $M_{1}^{n} = M_{1}$, whence +the first term of the value of~$z$ must be equated to~$0$, and +we shall have +\[ +xy = 0, +\] +a relation between $x$~and~$y$ independent of~$z$. + +Now if we expand in terms of~$z$ each pair of the primitive +equations~\Eqref[app]{(1)}, we shall have +\begin{alignat*}{3} +&\phi(xy0) &&+ \bigl\{\phi(xy1) - \phi(xy0)\bigr\}z &&= 0, \\ +&\psi(xy0) &&+ \bigl\{\psi(xy1) - \psi(xy0)\bigr\}z &&= 0, \\ +&\chi(xy0) &&+ \bigl\{\chi(xy1) - \chi(xy0)\bigr\}z &&= 0, +\end{alignat*} +and successively eliminating~$z$ between each pair of these equations, +we have +\begin{alignat*}{3} +&\phi(xy1)\psi(xy0) &&- \phi(xy0)\psi(xy1) &&= 0, \\ +&\psi(xy1)\chi(xy0) &&- \psi(xy0)\chi(xy1) &&= 0, \\ +&\chi(xy1)\phi(xy0) &&- \chi(xy0)\phi(xy1) &&= 0, +\end{alignat*} +which express all the relations between $x$~and~$y$ that are formed +by the elimination of~$z$. Expanding these, and writing in full +the first term, we have +\begin{alignat*}{3} +&\bigl\{\phi(111)\psi(110) &&- \phi(110)\psi(111)\bigr\}xy &&+ \etc. = 0, \\ +&\bigl\{\psi(111)\chi(110) &&- \psi(110)\chi(111)\bigr\}xy &&+ \etc. = 0, \\ +&\bigl\{\chi(111)\phi(110) &&- \chi(110)\phi(111)\bigr\}xy &&+ \etc. = 0\Typo{:}{;} +\end{alignat*} +and it appears from \PropRef{2}.\ that if the coefficient of~$xy$ in any +of these equations does not vanish, we shall have the equation +\[ +xy = 0; +\] +but the coefficients in question are the same as the first members +of the system~\Eqref[app]{(3)}, and the two sets of conditions exactly agree. +Thus, as respects the first term of the expansion, the method of +indeterminate coefficients leads to the same result as ordinary +elimination; and it is obvious that from their similarity of form, +the same reasoning will apply to all the other terms. +\PageSep{81} + +Suppose, in the second place, that the conditions~\Eqref[app]{(3)} are satisfied +so that $M_{1}$~is independent of $h$~and~$k$. It will then indifferently +assume the equivalent forms +\[ +M_{1} = \frac{1}{1 - \dfrac{\phi(111)}{\phi(110)}} + = \frac{1}{1 - \dfrac{\psi(111)}{\psi(110)}} + = \frac{1}{1 - \dfrac{\chi(111)}{\chi(110)}}\Add{.} +\] + +These are the exact forms of the first modulus in the expanded +values of~$z$, deduced from the solution of the three +primitive equations singly. If this common value of~$M_{1}$ is $1$ +or $\frac{0}{0} = v$, the term will be retained in~$z$; if any other constant +value (except~$0$), we have a relation $xy = 0$, not given by elimination, +but deducible from the primitive equations singly, and +similarly for all the other terms. Thus in every case the expression +of the subsidiary relations is a necessary accompaniment +of the process of solution. + +It is evident, upon consideration, that a similar proof will +apply to the discussion of a system indefinite as to the number +both of its symbols and of its equations. +%[** TN: No page break in the original] + + +\Chapter{Postscript.} + +\First{Some} additional explanations and references which have +occurred to me during the printing of this work are subjoined. + +The remarks on the connexion between Logic and Language, +\Pageref{5}, are scarcely sufficiently explicit. Both the one and the +other I hold to depend very materially upon our ability to form +general notions by the faculty of abstraction. Language is an +instrument of Logic, but not an indispensable instrument. + +To the remarks on Cause, \Pageref{12}, I desire to add the following: +Considering Cause as an invariable antecedent in Nature, (which +is Brown's view), whether associated or not with the idea of +Power, as suggested by Sir~John Herschel, the knowledge of its +existence is a knowledge which is properly expressed by the word +\emph{that} (\textgreek{t`o <ot`i}), not by \emph{why} (\textgreek{t`o di<ot`i}). It is very remarkable that +the two greatest authorities in Logic, modern and ancient, agreeing +in the latter interpretation, differ most widely in its application +to Mathematics. Sir W.~Hamilton says that Mathematics +\PageSep{82} +exhibit only the \emph{that} (\textgreek{t`o <ot`i}): Aristotle says, The \emph{why} belongs +to mathematicians, for they have the demonstrations of Causes. +\textit{Anal.\ Post.}\ lib.~\textsc{i}., cap.~\textsc{xiv}. It must be added that Aristotle's +view is consistent with the sense (albeit an erroneous one) +which in various parts of his writings he virtually assigns to the +word Cause, viz.\ an antecedent in Logic, a sense according to +which the premises might be said to be the cause of the conclusion. +This view appears to me to give even to his physical +inquiries much of their peculiar character. + +Upon reconsideration, I think that the view on \Pageref{41}, as to the +presence or absence of a medium of comparison, would readily +follow from Professor De~Morgan's doctrine, and I therefore +relinquish all claim to a discovery. The mode in which it +appears in this treatise is, however, remarkable. + +I have seen reason to change the opinion expressed in +\Pagerefs{42}{43}. The system of equations there given for the expression +of Propositions in Syllogism is \emph{always} preferable to the one +before employed---first, in generality---secondly, in facility of +interpretation. + +In virtue of the principle, that a Proposition is either true or +false, every elective symbol employed in the expression of +hypotheticals admits only of the values $0$~and~$1$, which are the +only quantitative forms of an elective symbol. It is in fact +possible, setting out from the theory of Probabilities (which is +purely quantitative), to arrive at a system of methods and processes +for the treatment of hypotheticals exactly similar to those +which have been given. The two systems of elective symbols +and of quantity osculate, if I may use the expression, in the +points $0$~and~$1$. It seems to me to be implied by this, that +unconditional truth (categoricals) and probable truth meet together +in the constitution of contingent truth\Typo{;}{} (hypotheticals). +The general doctrine of elective symbols and all the more characteristic +applications are quite independent of any quantitative +origin. +\vfil +\begin{center} +\small +THE END. +\end{center} +\vfil\vfil +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\PGLicense +\begin{PGtext} +End of Project Gutenberg's The Mathematical Analysis of Logic, by George Boole + +*** END OF THIS PROJECT GUTENBERG EBOOK THE MATHEMATICAL ANALYSIS OF LOGIC *** + +***** This file should be named 36884-pdf.pdf or 36884-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/8/8/36884/ + +Produced by Andrew D. 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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: The Mathematical Analysis of Logic % +% Being an Essay Towards a Calculus of Deductive Reasoning % +% % +% Author: George Boole % +% % +% Release Date: July 28, 2011 [EBook #36884] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK THE MATHEMATICAL ANALYSIS OF LOGIC *** +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{36884} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% babel: Greek language capabilities. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: The Mathematical Analysis of Logic + Being an Essay Towards a Calculus of Deductive Reasoning + +Author: George Boole + +Release Date: July 28, 2011 [EBook #36884] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE MATHEMATICAL ANALYSIS OF LOGIC *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang +\end{PGtext} +\end{minipage} +\vfill +\end{center} + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\PageSep{i} +\FrontMatter +\begin{center} +\bfseries\large THE MATHEMATICAL ANALYSIS +\vfil + +\Large OF LOGIC, +\vfil + +\normalsize +BEING AN ESSAY TOWARDS A CALCULUS \\ +OF DEDUCTIVE REASONING. +\vfil + +BY GEORGE BOOLE. +\vfil + +\begin{Quote} +>Epikoinwno~usi d`e p~asai a<i >epist~hmai >all'hlais kat`a t`a koin'a. \Typo{Koin'a}{Koin`a} d`e +l'egw, o>~is qr~wntai <ws >ek to'utwn >apodeikn'untes; >all'' o>u per`i <~wn deikn'uousin, +\Typo{o>ude}{o>ud`e} <`o deikn'uousi. \\ +\selectlanguage{english} +\null\hfill\textsc{Aristotle}, \textit{Anal.\ Post.}, lib.~\textsc{i}. cap.~\textsc{xi}. +\end{Quote} +\vfil\vfil + +CAMBRIDGE: \\ +MACMILLAN, BARCLAY, \& MACMILLAN; \\ +LONDON: GEORGE BELL. \\ +\tb[0.25in] \\ +1847 +\normalfont +\end{center} +\PageSep{ii} +\newpage +\normalfont +\null +\vfill +\begin{center} +\scriptsize +PRINTED IN ENGLAND BY \\ +HENDERSON \& SPALDING \\ +LONDON. W.I +\end{center} +\PageSep{1} +\MainMatter + + +\Chapter{Preface.} + +\First{In} presenting this Work to public notice, I deem it not +irrelevant to observe, that speculations similar to those which +it records have, at different periods, occupied my thoughts. +In the spring of the present year my attention was directed +to the question then moved between Sir W.~Hamilton and +Professor De~Morgan; and I was induced by the interest +which it inspired, to resume the almost-forgotten thread of +former inquiries. It appeared to me that, although Logic +might be viewed with reference to the idea of quantity,\footnote + {See \Pageref{42}.} +it +had also another and a deeper system of relations. If it was +lawful to regard it from \emph{without}, as connecting itself through +the medium of Number with the intuitions of Space and Time, +it was lawful also to regard it from \emph{within}, as based upon +facts of another order which have their abode in the constitution +of the Mind. The results of this view, and of the +inquiries which it suggested, are embodied in the following +Treatise. + +It is not generally permitted to an Author to prescribe +the mode in which his production shall be judged; but there +are two conditions which I may venture to require of those +who shall undertake to estimate the merits of this performance. +The first is, that no preconceived notion of the impossibility +of its objects shall be permitted to interfere with that candour +and impartiality which the investigation of Truth demands; +the second is, that their judgment of the system as a whole +shall not be founded either upon the examination of only +\PageSep{2} +a part of it, or upon the measure of its conformity with any +received system, considered as a standard of reference from +which appeal is denied. It is in the general theorems which +occupy the latter chapters of this work,---results to which there +is no existing counterpart,---that the claims of the method, as +a Calculus of Deductive Reasoning, are most fully set forth. + +What may be the final estimate of the value of the system, +I have neither the wish nor the right to anticipate. The +estimation of a theory is not simply determined by its truth\Add{.} +It also depends upon the importance of its subject, and the +extent of its applications; beyond which something must still +be left to the arbitrariness of human Opinion. If the utility +of the application of Mathematical forms to the science of +Logic were solely a question of Notation, I should be content +to rest the defence of this attempt upon a principle which has +been stated by an able living writer: ``Whenever the nature +of the subject permits the reasoning process to be without +danger carried on mechanically, the language should be constructed +on as mechanical principles as possible; while in the +contrary case it should be so constructed, that there shall be +the greatest possible obstacle to a mere mechanical use of it.''\footnote + {Mill's \textit{System of Logic, Ratiocinative and Inductive}, Vol.~\textsc{ii}. p.~292.} +In one respect, the science of Logic differs from all others; +the perfection of its method is chiefly valuable as an evidence +of the speculative truth of its principles. To supersede the +employment of common reason, or to subject it to the rigour +of technical forms, would be the last desire of one who knows +the value of that intellectual toil and warfare which imparts +to the mind an athletic vigour, and teaches it to contend +with difficulties and to rely upon itself in emergencies. +\Signature{\textsc{Lincoln}, \textit{Oct.}~29, 1847.} +\PageSep{3} + + +%[**TN: Macro prints heading "MATHEMATICAL ANALYSIS OF LOGIC."] +\Chapter{Introduction.} + +\First{They} who are acquainted with the present state of the theory +of Symbolical Algebra, are aware, that the validity of the +processes of analysis does not depend upon the interpretation +of the symbols which are employed, but solely upon the laws +of their combination. Every system of interpretation which +does not affect the truth of the relations supposed, is equally +admissible, and it is thus that the same process may, under +one scheme of interpretation, represent the solution of a question +on the properties of numbers, under another, that of +a geometrical problem, and under a third, that of a problem +of dynamics or optics. This principle is indeed of fundamental +importance; and it may with safety be affirmed, that the recent +advances of pure analysis have been much assisted by the +influence which it has exerted in directing the current of +investigation. + +But the full recognition of the consequences of this important +doctrine has been, in some measure, retarded by accidental +circumstances. It has happened in every known form of +analysis, that the elements to be determined have been conceived +as measurable by comparison with some fixed standard. +The predominant idea has been that of magnitude, or more +strictly, of numerical ratio. The expression of magnitude, or +\PageSep{4} +of operations upon magnitude, has been the express object +for which the symbols of Analysis have been invented, and +for which their laws have been investigated. Thus the abstractions +of the modern Analysis, not less than the ostensive +diagrams of the ancient Geometry, have encouraged the notion, +that Mathematics are essentially, as well as actually, the Science +of Magnitude. + +The consideration of that view which has already been stated, +as embodying the true principle of the Algebra of Symbols, +would, however, lead us to infer that this conclusion is by no +means necessary. If every existing interpretation is shewn to +involve the idea of magnitude, it is only by induction that we +can assert that no other interpretation is possible. And it may +be doubted whether our experience is sufficient to render such +an induction legitimate. The history of pure Analysis is, it may +be said, too recent to permit us to set limits to the extent of its +applications. Should we grant to the inference a high degree +of probability, we might still, and with reason, maintain the +sufficiency of the definition to which the principle already stated +would lead us. We might justly assign it as the definitive +character of a true Calculus, that it is a method resting upon +the employment of Symbols, whose laws of combination are +known and general, and whose results admit of a consistent +interpretation. That to the existing forms of Analysis a quantitative +interpretation is assigned, is the result of the circumstances +by which those forms were determined, and is not to +be construed into a universal condition of Analysis. It is upon +the foundation of this general principle, that I purpose to +establish the Calculus of Logic, and that I claim for it a place +among the acknowledged forms of Mathematical Analysis, regardless +that in its object and in its instruments it must at +present stand alone. + +That which renders Logic possible, is the existence in our +minds of general notions,---our ability to conceive of a class, +and to designate its individual members by a common name. +\PageSep{5} +\Pagelabel{5}% +The theory of Logic is thus intimately connected with that of +Language. A successful attempt to express logical propositions +by symbols, the laws of whose combinations should be founded +upon the laws of the mental processes which they represent, +would, so far, be a step toward a philosophical language. But +this is a view which we need not here follow into detail.\footnote + {This view is well expressed in one of Blanco White's Letters:---``Logic is + for the most part a collection of technical rules founded on classification. The + Syllogism is nothing but a result of the classification of things, which the mind + naturally and necessarily forms, in forming a language. All abstract terms are + classifications; or rather the labels of the classes which the mind has settled.''---\textit{Memoirs + of the Rev.\ Joseph Blanco White}, vol.~\textsc{ii}. p.~163. See also, for a very + lucid introduction, Dr.~Latham's \textit{First Outlines of Logic applied to Language}, + Becker's \textit{German Grammar,~\etc.} Extreme Nominalists make Logic entirely + dependent upon language. For the opposite view, see Cudworth's \textit{Eternal + and Immutable Morality}, Book~\textsc{iv}. Chap.~\textsc{iii}.} +Assuming the notion of a class, we are able, from any conceivable +collection of objects, to separate by a mental act, those +which belong to the given class, and to contemplate them apart +from the rest. Such, or a similar act of election, we may conceive +to be repeated. The group of individuals left under consideration +may be still further limited, by mentally selecting +those among them which belong to some other recognised class, +as well as to the one before contemplated. And this process +may be repeated with other elements of distinction, until we +arrive at an individual possessing all the distinctive characters +which we have taken into account, and a member, at the same +time, of every class which we have enumerated. It is in fact +a method similar to this which we employ whenever, in common +language, we accumulate descriptive epithets for the sake of +more precise definition. + +Now the several mental operations which in the above case +we have supposed to be performed, are subject to peculiar laws. +It is possible to assign relations among them, whether as respects +the repetition of a given operation or the succession of +different ones, or some other particular, which are never violated. +It is, for example, true that the result of two successive acts is +\PageSep{6} +unaffected by the order in which they are performed; and there +are at least two other laws which will be pointed out in the +proper place. These will perhaps to some appear so obvious as +to be ranked among necessary truths, and so little important +as to be undeserving of special notice. And probably they are +noticed for the first time in this Essay. Yet it may with confidence +be asserted, that if they were other than they are, the +entire mechanism of reasoning, nay the very laws and constitution +of the human intellect, would be vitally changed. A Logic +might indeed exist, but it would no longer be the Logic we +possess. + +Such are the elementary laws upon the existence of which, +and upon their capability of exact symbolical expression, the +method of the following Essay is founded; and it is presumed +that the object which it seeks to attain will be thought to +have been very fully accomplished. Every logical proposition, +whether categorical or hypothetical, will be found to be capable +of exact and rigorous expression, and not only will the laws of +conversion and of syllogism be thence deducible, but the resolution +of the most complex systems of propositions, the separation +of any proposed element, and the expression of its value in +terms of the remaining elements, with every subsidiary relation +involved. Every process will represent deduction, every +mathematical consequence will express a logical inference. The +generality of the method will even permit us to express arbitrary +operations of the intellect, and thus lead to the demonstration +of general theorems in logic analogous, in no slight +degree, to the general theorems of ordinary mathematics. No +inconsiderable part of the pleasure which we derive from the +application of analysis to the interpretation of external nature, +arises from the conceptions which it enables us to form of the +universality of the dominion of law. The general formulć to +which we are conducted seem to give to that element a visible +presence, and the multitude of particular cases to which they +apply, demonstrate the extent of its sway. Even the symmetry +\PageSep{7} +of their analytical expression may in no fanciful sense be +deemed indicative of its harmony and its consistency. Now I +do not presume to say to what extent the same sources of +pleasure are opened in the following Essay. The measure of +that extent may be left to the estimate of those who shall think +the subject worthy of their study. But I may venture to +assert that such occasions of intellectual gratification are not +here wanting. The laws we have to examine are the laws of +one of the most important of our mental faculties. The mathematics +we have to construct are the mathematics of the human +intellect. Nor are the form and character of the method, apart +from all regard to its interpretation, undeserving of notice. +There is even a remarkable exemplification, in its general +theorems, of that species of excellence which consists in freedom +from exception. And this is observed where, in the corresponding +cases of the received mathematics, such a character +is by no means apparent. The few who think that there is that +in analysis which renders it deserving of attention for its own +sake, may find it worth while to study it under a form in which +every equation can be solved and every solution interpreted. +Nor will it lessen the interest of this study to reflect that every +peculiarity which they will notice in the form of the Calculus +represents a corresponding feature in the constitution of their +own minds. + +It would be premature to speak of the value which this +method may possess as an instrument of scientific investigation. +I speak here with reference to the theory of reasoning, and to +the principle of a true classification of the forms and cases of +Logic considered as a Science.\footnote + {``Strictly a Science''; also ``an Art.''---\textit{Whately's Elements of Logic.} Indeed + ought we not to regard all Art as applied Science; unless we are willing, with + ``the multitude,'' to consider Art as ``guessing and aiming well''?---\textit{Plato, + Philebus.}} +The aim of these investigations +was in the first instance confined to the expression of the +received logic, and to the forms of the Aristotelian arrangement, +\PageSep{8} +but it soon became apparent that restrictions were thus introduced, +which were purely arbitrary and had no foundation in +the nature of things. These were noted as they occurred, and +will be discussed in the proper place. When it became necessary +to consider the subject of hypothetical propositions (in which +comparatively less has been done), and still more, when an +interpretation was demanded for the general theorems of the +Calculus, it was found to be imperative to dismiss all regard for +precedent and authority, and to interrogate the method itself for +an expression of the just limits of its application. Still, however, +there was no special effort to arrive at novel results. But +among those which at the time of their discovery appeared to be +such, it may be proper to notice the following. + +A logical proposition is, according to the method of this Essay, +expressible by an equation the form of which determines the +rules of conversion and of transformation, to which the given +proposition is subject. Thus the law of what logicians term +simple conversion, is determined by the fact, that the corresponding +equations are symmetrical, that they are unaffected by +a mutual change of place, in those symbols which correspond +to the convertible classes. The received laws of conversion +were thus determined, and afterwards another system, which is +thought to be more elementary, and more general. See Chapter, +\ChapRef{5}{On the Conversion of Propositions}. + +The premises of a syllogism being expressed by equations, the +elimination of a common symbol between them leads to a third +equation which expresses the conclusion, this conclusion being +always the most general possible, whether Aristotelian or not. +Among the cases in which no inference was possible, it was +found, that there were two distinct forms of the final equation. +It was a considerable time before the explanation of this fact +was discovered, but it was at length seen to depend upon the +presence or absence of a true medium of comparison between +the premises. The distinction which is thought to be new +is illustrated in the Chapter, \ChapRef{6}{On Syllogisms}. +\PageSep{9} + +The nonexclusive character of the disjunctive conclusion of +a hypothetical syllogism, is very clearly pointed out in the +examples of this species of argument. + +The class of logical problems illustrated in the chapter, \ChapRef{9}{On +the Solution of Elective Equations}, is conceived to be new: and +it is believed that the method of that chapter affords the means +of a perfect analysis of any conceivable system of propositions, +an end toward which the rules for the conversion of a single +categorical proposition are but the first step. + +However, upon the originality of these or any of these views, +I am conscious that I possess too slight an acquaintance with the +literature of logical science, and especially with its older literature, +to permit me to speak with confidence. + +It may not be inappropriate, before concluding these observations, +to offer a few remarks upon the general question of the +use of symbolical language in the mathematics. Objections +have lately been very strongly urged against this practice, on +the ground, that by obviating the necessity of thought, and +substituting a reference to general formulć in the room of +personal effort, it tends to weaken the reasoning faculties. + +Now the question of the use of symbols may be considered +in two distinct points of view. First, it may be considered with +reference to the progress of scientific discovery, and secondly, +with reference to its bearing upon the discipline of the intellect. + +And with respect to the first view, it may be observed that +as it is one fruit of an accomplished labour, that it sets us at +liberty to engage in more arduous toils, so it is a necessary +result of an advanced state of science, that we are permitted, +and even called upon, to proceed to higher problems, than those +which we before contemplated. The practical inference is +obvious. If through the advancing power of scientific methods, +we find that the pursuits on which we were once engaged, +afford no longer a sufficiently ample field for intellectual effort, +the remedy is, to proceed to higher inquiries, and, in new +tracks, to seek for difficulties yet unsubdued. And such is, +\PageSep{10} +indeed, the actual law of scientific progress. We must be +content, either to abandon the hope of further conquest, or to +employ such aids of symbolical language, as are proper to the +stage of progress, at which we have arrived. Nor need we fear +to commit ourselves to such a course. We have not yet arrived +so near to the boundaries of possible knowledge, as to suggest +the apprehension, that scope will fail for the exercise of the +inventive faculties. + +In discussing the second, and scarcely less momentous question +of the influence of the use of symbols upon the discipline +of the intellect, an important distinction ought to be made. It +is of most material consequence, whether those symbols are +used with a full understanding of their meaning, with a perfect +comprehension of that which renders their use lawful, and an +ability to expand the abbreviated forms of reasoning which they +induce, into their full syllogistic \Typo{devolopment}{development}; or whether they +are mere unsuggestive characters, the use of which is suffered +to rest upon authority. + +The answer which must be given to the question proposed, +will differ according as the one or the other of these suppositions +is admitted. In the former case an intellectual discipline of a +high order is provided, an exercise not only of reason, but of +the faculty of generalization. In the latter case there is no +mental discipline whatever. It were perhaps the best security +against the danger of an unreasoning reliance upon symbols, +on the one hand, and a neglect of their just claims on the other, +that each subject of applied mathematics should be treated in the +spirit of the methods which were known at the time when the +application was made, but in the best form which those methods +have assumed. The order of attainment in the individual mind +would thus bear some relation to the actual order of scientific +discovery, and the more abstract methods of the higher analysis +would be offered to such minds only, as were prepared to +receive them. + +The relation in which this Essay stands at once to Logic and +\PageSep{11} +to Mathematics, may further justify some notice of the question +which has lately been revived, as to the relative value of the two +studies in a liberal education. One of the chief objections which +have been urged against the study of Mathematics in general, is +but another form of that which has been already considered with +respect to the use of symbols in particular. And it need not here +be further dwelt upon, than to notice, that if it avails anything, +it applies with an equal force against the study of Logic. The +canonical forms of the Aristotelian syllogism are really symbolical; +only the symbols are less perfect of their kind than those +of mathematics. If they are employed to test the validity of an +argument, they as truly supersede the exercise of reason, as does +a reference to a formula of analysis. Whether men do, in the +present day, make this use of the Aristotelian canons, except as +a special illustration of the rules of Logic, may be doubted; yet +it cannot be questioned that when the authority of Aristotle was +dominant in the schools of Europe, such applications were habitually +made. And our argument only requires the admission, +that the case is possible. + +But the question before us has been argued upon higher +grounds. Regarding Logic as a branch of Philosophy, and defining +Philosophy as the ``science of a real existence,'' and ``the +research of causes,'' and assigning as its \emph{main} business the investigation +of the ``why, (\textgreek{t`o d'ioti}),'' while Mathematics display +only the ``that, (\textgreek{t`o <ot`i}),'' Sir W.~Hamilton has contended, +not simply, that the superiority rests with the study of Logic, +but that the study of Mathematics is at once dangerous and useless.\footnote + {\textit{Edinburgh Review}, vol.~\textsc{lxii}. p.~409, and \textit{Letter to A. De~Morgan, Esq.}} +The pursuits of the mathematician ``have not only not +trained him to that acute scent, to that delicate, almost instinctive, +tact which, in the twilight of probability, the search and +discrimination of its finer facts demand; they have gone to cloud +his vision, to indurate his touch, to all but the blazing light, the +iron chain of demonstration, and left him out of the narrow confines +of his science, to a passive \emph{credulity} in any premises, or to +\PageSep{12} +an absolute \emph{incredulity} in all.'' In support of these and of other +charges, both argument and copious authority are adduced.\footnote + {The arguments are in general better than the authorities. Many writers + quoted in condemnation of mathematics (Aristo, Seneca, Jerome, Augustine, + Cornelius Agrippa,~\etc.)\ have borne a no less explicit testimony against other + sciences, nor least of all, against that of logic. The treatise of the last named + writer \textit{De~Vanitate Scientiarum}, must surely have been referred to by mistake.---\textit{Vide} + cap.~\textsc{cii}.} +I shall not attempt a complete discussion of the topics which +are suggested by these remarks. My object is not controversy, +and the observations which follow are offered not in the spirit +of antagonism, but in the hope of contributing to the formation +of just views upon an important subject. Of Sir W.~Hamilton +it is impossible to speak otherwise than with that respect which +is due to genius and learning. + +Philosophy is then described as the \emph{science of a real existence} +\Pagelabel{12}% +and \emph{the research of causes}. And that no doubt may rest upon +the meaning of the word \emph{cause}, it is further said, that philosophy +``mainly investigates the \emph{why}.'' These definitions are common +among the ancient writers. Thus Seneca, one of Sir W.~Hamilton's +authorities, \textit{Epistle}~\textsc{lxxxviii}., ``The philosopher seeks +and knows the \emph{causes} of natural things, of which the mathematician +searches out and computes the numbers and the measures.'' +It may be remarked, in passing, that in whatever +degree the belief has prevailed, that the business of philosophy +is immediately with \emph{causes}; in the same degree has every +science whose object is the investigation of \emph{laws}, been lightly +esteemed. Thus the Epistle to which we have referred, bestows, +by contrast with Philosophy, a separate condemnation on Music +and Grammar, on Mathematics and Astronomy, although it is +that of Mathematics only that Sir W.~Hamilton has quoted. + +Now we might take our stand upon the conviction of many +thoughtful and reflective minds, that in the extent of the meaning +above stated, Philosophy is impossible. The business of +true Science, they conclude, is with laws and phenomena. The +nature of Being, the mode of the operation of Cause, the \emph{why}, +\PageSep{13} +they hold to be beyond the reach of our intelligence. But we +do not require the vantage-ground of this position; nor is it +doubted that whether the aim of Philosophy is attainable or not, +the desire which impels us to the attempt is an instinct of our +higher nature. Let it be granted that the problem which has +baffled the efforts of ages, is not a hopeless one; that the +``science of a real existence,'' and ``the research of causes,'' +``that kernel'' for which ``Philosophy is still militant,'' do +not transcend the limits of the human intellect. I am then +compelled to assert, that according to this view of the nature of +Philosophy, \emph{Logic forms no part of it}. On the principle of +a true classification, we ought no longer to associate Logic and +Metaphysics, but Logic and Mathematics. + +Should any one after what has been said, entertain a doubt +upon this point, I must refer him to the evidence which will be +afforded in the following Essay. He will there see Logic resting +like Geometry upon axiomatic truths, and its theorems constructed +upon that general doctrine of symbols, which constitutes +the foundation of the recognised Analysis. In the Logic +of Aristotle he will be led to view a collection of the formulć +of the science, expressed by another, but, (it is thought) less +perfect scheme of symbols. I feel bound to contend for the +absolute exactness of this parallel. It is no escape from the conclusion +to which it points to assert, that Logic not only constructs +a science, but also inquires into the origin and the nature of its +own principles,---a distinction which is denied to Mathematics. +``It is wholly beyond the domain of mathematicians,'' it is said, +``to inquire into the origin and nature of their principles.''---% +\textit{Review}, page~415. But upon what ground can such a distinction +be maintained? What definition of the term Science will +be found sufficiently arbitrary to allow such differences? + +The application of this conclusion to the question before us is +clear and decisive. The mental discipline which is afforded by +the study of Logic, \emph{as an exact science}, is, in species, the same +as that afforded by the study of Analysis. +\PageSep{14} + +Is it then contended that either Logic or Mathematics can +supply a perfect discipline to the Intellect? The most careful +and unprejudiced examination of this question leads me to doubt +whether such a position can be maintained. The exclusive claims +of either must, I believe, be abandoned, nor can any others, partaking +of a like exclusive character, be admitted in their room. +It is an important observation, which has more than once been +made, that it is one thing to arrive at correct premises, and another +thing to deduce logical conclusions, and that the business of life +depends more upon the former than upon the latter. The study +of the exact sciences may teach us the one, and it may give us +some general preparation of knowledge and of practice for the +attainment of the other, but it is to the union of thought with +action, in the field of Practical Logic, the arena of Human Life, +that we are to look for its fuller and more perfect accomplishment. + +I desire here to express my conviction, that with the advance +of our knowledge of all true science, an ever-increasing +harmony will be found to prevail among its separate branches. +The view which leads to the rejection of one, ought, if consistent, +to lead to the rejection of others. And indeed many +of the authorities which have been quoted against the study +of Mathematics, are even more explicit in their condemnation of +Logic. ``Natural science,'' says the Chian Aristo, ``is above us, +Logical science does not concern us.'' When such conclusions +are founded (as they often are) upon a deep conviction of the +preeminent value and importance of the study of Morals, we +admit the premises, but must demur to the inference. For it +has been well said by an ancient writer, that it is the ``characteristic +of the liberal sciences, not that they conduct us to Virtue, +but that they prepare us for Virtue;'' and Melancthon's sentiment, +``abeunt studia in mores,'' has passed into a proverb. +Moreover, there is a common ground upon which all sincere +votaries of truth may meet, exchanging with each other the +language of Flamsteed's appeal to Newton, ``The works of the +Eternal Providence will be better understood through your +labors and mine.'' +\PageSep{15} + + +\Chapter{First Principles.} + +\First{Let} us employ the symbol~$1$, or unity, to represent the +Universe, and let us understand it as comprehending every +conceivable class of objects whether actually existing or not, +it being premised that the same individual may be found in +more than one class, inasmuch as it may possess more than one +quality in common with other individuals. Let us employ the +letters $X$,~$Y$,~$Z$, to represent the individual members of classes, +$X$~applying to every member of one class, as members of that +particular class, and $Y$~to every member of another class as +members of such class, and so on, according to the received language +of treatises on Logic. + +Further let us conceive a class of symbols $x$,~$y$,~$z$, possessed +of the following character. + +The symbol~$x$ operating upon any subject comprehending +individuals or classes, shall be supposed to select from that +subject all the~$X$s which it contains. In like manner the symbol~$y$, +operating upon any subject, shall be supposed to select from +it all individuals of the class~$Y$ which are comprised in it, and +so on. + +When no subject is expressed, we shall suppose~$1$ (the Universe) +to be the subject understood, so that we shall have +\[ +x = x\quad (1), +\] +the meaning of either term being the selection from the Universe +of all the~$X$s which it contains, and the result of the operation +\PageSep{16} +being in common language, the class~$X$, \ie~the class of which +each member is an~$X$. + +From these premises it will follow, that the product~$xy$ will +represent, in succession, the selection of the class~$Y$, and the +selection from the class~$Y$ of such individuals of the class~$X$ as +are contained in it, the result being the class whose members are +both $X$s~and~$Y$s. And in like manner the product~$xyz$ will +represent a compound operation of which the successive elements +are the selection of the class~$Z$, the selection from it of +such individuals of the class~$Y$ as are contained in it, and the +selection from the result thus obtained of all the individuals of +the class~$X$ which it contains, the final result being the class +common to $X$,~$Y$, and~$Z$. + +From the nature of the operation which the symbols $x$,~$y$,~$z$, +are conceived to represent, we shall designate them as elective +symbols. An expression in which they are involved will be +called an elective function, and an equation of which the members +are elective functions, will be termed an elective equation. + +It will not be necessary that we should here enter into the +analysis of that mental operation which we have represented by +the elective symbol. It is not an act of Abstraction according +to the common acceptation of that term, because we never lose +sight of the concrete, but it may probably be referred to an exercise +of the faculties of Comparison and Attention. Our present +concern is rather with the laws of combination and of succession, +by which its results are governed, and of these it will suffice to +notice the following. + +1st. The result of an act of election is independent of the +grouping or classification of the subject. + +Thus it is indifferent whether from a group of objects considered +as a whole, we select the class~$X$, or whether we divide +the group into two parts, select the~$X$s from them separately, +and then connect the results in one aggregate conception. + +We may express this law mathematically by the equation +\[ +x(u + v) = xu + xv, +\] +\PageSep{17} +$u + v$ representing the undivided subject, and $u$~and~$v$ the +component parts of it. + +2nd. It is indifferent in what order two successive acts of +election are performed. + +Whether from the class of animals we select sheep, and from +the sheep those which are horned, or whether from the class of +animals we select the horned, and from these such as are sheep, +the result is unaffected. In either case we arrive at the class +\emph{horned sheep}. + +The symbolical expression of this law is +\[ +xy = yx. +\] + +3rd. The result of a given act of election performed twice, +or any number of times in succession, is the result of the same +act performed once. + +If from a group of objects we select the~$X$s, we obtain a class +of which all the members are~$X$s. If we repeat the operation +on this class no further change will ensue: in selecting the~$X$s +we take the whole. Thus we have +\[ +xx = x, +\] +or +\[ +x^{2} = x; +\] +and supposing the same operation to be $n$~times performed, we +have +\[ +x^{n} = x, +\] +which is the mathematical expression of the law above stated.\footnote + {The office of the elective symbol~$x$, is to select individuals comprehended + in the class~$X$. Let the class~$X$ be supposed to embrace the universe; then, + whatever the class~$Y$ may be, we have + \[ + xy = y. + \] + The office which $x$~performs is now equivalent to the symbol~$+$, in one at + least of its interpretations, and the index law~\Eqref{(3)} gives + \[ + +^{n} = +, + \] + which is the known property of that symbol.} + +The laws we have established under the symbolical forms +\begin{align*} +x(u + v) &= xu + xv, +\Tag{(1)} \\ +xy &= yx, +\Tag{(2)} \\ +x^{n} &= x, +\Tag{(3)} +\end{align*} +\PageSep{18} +are sufficient for the basis of a Calculus. From the first of these, +it appears that elective symbols are \emph{distributive}, from the second +that they are \emph{commutative}; properties which they possess in +common with symbols of \emph{quantity}, and in virtue of which, all +the processes of common algebra are applicable to the present +system. The one and sufficient axiom involved in this application +is that equivalent operations performed upon equivalent +subjects produce equivalent results.\footnote + {It is generally asserted by writers on Logic, that all reasoning ultimately + depends on an application of the dictum of Aristotle, \textit{de omni et~nullo}. ``Whatever + is predicated universally of any class of things, may be predicated in like + manner of any thing comprehended in that class.'' But it is agreed that this + dictum is not immediately applicable in all cases, and that in a majority of + instances, a certain previous process of reduction is necessary. What are the + elements involved in that process of reduction? Clearly they are as much + a part of general reasoning as the dictum itself. + + Another mode of considering the subject resolves all reasoning into an application + of one or other of the following canons,~viz.\ + + 1. If two terms agree with one and the same third, they agree with each + other. + + 2. If one term agrees, and another disagrees, with one and the same third, + these two disagree with each other. + + But the application of these canons depends on mental acts equivalent to + those which are involved in the before-named process of reduction. We have to + select individuals from classes, to convert propositions,~\etc., before we can avail + ourselves of their guidance. Any account of the process of reasoning is insufficient, + which does not represent, as well the laws of the operation which the + mind performs in that process, as the primary truths which it recognises and + applies. + + It is presumed that the laws in question are adequately represented by the + fundamental equations of the present Calculus. The proof of this will be found + in its capability of expressing propositions, and of exhibiting in the results of + its processes, every result that may be arrived at by ordinary reasoning.} + +The third law~\Eqref{(3)} we shall denominate the index law. It is +peculiar to elective symbols, and will be found of great importance +in enabling us to reduce our results to forms meet for +interpretation. + +From the circumstance that the processes of algebra may be +applied to the present system, it is not to be inferred that the +interpretation of an elective equation will be unaffected by such +processes. The expression of a truth cannot be negatived by +\PageSep{19} +a legitimate operation, but it may be limited. The equation +$y = z$ implies that the classes $Y$~and~$Z$ are equivalent, member +for member. Multiply it by a factor~$x$, and we have +\[ +xy = xz, +\] +which expresses that the individuals which are common to the +classes $X$~and~$Y$ are also common to $X$~and~$Z$, and \textit{vice versâ}. +This is a perfectly legitimate inference, but the fact which it +declares is a less general one than was asserted in the original +proposition. +\PageSep{20} + + +\Chapter{Of Expression and Interpretation.} + +\begin{Abstract} +A Proposition is a sentence which either affirms or denies, as, All men are +mortal, No creature is independent. + +A Proposition has necessarily two terms, as \emph{men}, \emph{mortal}; the former of which, +or the one spoken of, is called the subject; the latter, or that which is affirmed +or denied of the subject, the predicate. These are connected together by the +copula~\emph{is}, or \emph{is not}, or by some other modification of the substantive verb. + +The substantive verb is the only verb recognised in Logic; all others are +resolvable by means of the verb \emph{to be} and a participle or adjective, \eg~``The +Romans conquered''; the word conquered is both copula and predicate, being +equivalent to ``were (copula) victorious'' (predicate). + +A Proposition must either be affirmative or negative, and must be also either +universal or particular. Thus we reckon in all, four kinds of pure categorical +Propositions. + +1st. Universal-affirmative, usually represented by~$A$, +\[ +\text{Ex. All $X$s are $Y$s.} +\] + +2nd. Universal-negative, usually represented by~$E$, +\[ +\text{Ex. No $X$s are $Y$s.} +\] + +3rd. Particular-affirmative, usually represented by~$I$, +\[ +\text{Ex. Some $X$s are $Y$s.} +\] + +4th. Particular-negative, usually represented by~$O$,\footnote + {The above is taken, with little variation, from the Treatises of Aldrich + and Whately.} +\[ +\text{Ex. Some $X$s are not $Y$s.} +\] +\end{Abstract} + +1. To express the class, not-$X$, that is, the class including +all individuals that are not~$X$s. + +The class~$X$ and the class not-$X$ together make the Universe. +But the Universe is~$1$, and the class~$X$ is determined by the +symbol~$x$, therefore the class not-$X$ will be determined by +the symbol~$1 - x$. +\PageSep{21} + +Hence the office of the symbol $1 - x$ attached to a given +subject will be, to select from it all the not-$X$s which it +contains. + +And in like manner, as the product~$xy$ expresses the entire +class whose members are both $X$s and~$Y$s, the symbol $y(1 - x)$ +will represent the class whose members are $Y$s but not~$X$s, +and the symbol $(1 - x)(1 - y)$ the entire class whose members +are neither $X$s~nor~$Y$s. + +2. To express the Proposition, All $X$s are~$Y$s. + +As all the~$X$s which exist are found in the class~$Y$, it is +obvious that to select out of the Universe all~$Y$s, and from +these to select all~$X$s, is the same as to select at once from the +Universe all~$X$s. + +Hence +\[ +xy = x, +\] +or +\[ +x(1 - y) = 0. +\Tag{(4)} +\] + +3. To express the Proposition, No $X$s are~$Y$s. + +To assert that no $X$s are~$Y$s, is the same as to assert that +there are no terms common to the classes $X$~and~$Y$. Now +all individuals common to those classes are represented by~$xy$. +Hence the Proposition that No~$X$s are~$Y$s, is represented by +the equation +\[ +xy = 0. +\Tag{(5)} +\] + +4. To express the Proposition, Some $X$s are~$Y$s. + +If some $X$s are~$Y$s, there are some terms common to the +classes $X$~and~$Y$. Let those terms constitute a separate class~$V$, +to which there shall correspond a separate elective symbol~$v$, +then +\[ +v = xy. +\Tag{(6)} +\] +And as $v$~includes all terms common to the classes $X$~and~$Y$, +we can indifferently interpret it, as Some~$X$s, or Some~$Y$s. +\PageSep{22} + +5. To express the Proposition, Some $X$s are not~$Y$s. + +In the last equation write $1 - y$ for~$y$, and we have +\[ +v = x(1 - y), +\Tag{(7)} +\] +the interpretation of~$v$ being indifferently Some~$X$s or Some +not-$Y$s. + +The above equations involve the complete theory of categorical +Propositions, and so far as respects the employment of +analysis for the deduction of logical inferences, nothing more +can be desired. But it may be satisfactory to notice some particular +forms deducible from the third and fourth equations, and +susceptible of similar application. + +If we multiply the equation~\Eqref{(6)} by~$x$, we have +\[ +vx = x^{2}y = xy\quad\text{by~\Eqref{(3)}.} +\] + +Comparing with~\Eqref{(6)}, we find +\[ +v = vx, +\] +or +\[ +v(1 - x) = 0. +\Tag{(8)} +\] + +And multiplying~\Eqref{(6)} by~$y$, and reducing in a similar manner, +we have +\[ +v = vy, +\] +or +\[ +v(1 - y) = 0. +\Tag{(9)} +\] + +Comparing \Eqref{(8)} and~\Eqref{(9)}, +\[ +vx = vy = v. +\Tag{(10)} +\] + +And further comparing \Eqref{(8)} and~\Eqref{(9)} with~\Eqref{(4)}, we have as the +equivalent of this system of equations the Propositions +\[ +\begin{aligned} +&\text{All $V$s are~$X$s} \\ +&\text{All $V$s are~$Y$s} +\end{aligned} +\Rbrace{2}. +\] + +The system~\Eqref{(10)} might be used to replace~\Eqref{(6)}, or the single +equation +\[ +vx = vy, +\Tag{(11)} +\] +might be used, assigning to~$vx$ the interpretation, Some~$X$s, and +to~$vy$ the interpretation, Some~$Y$s. But it will be observed that +\PageSep{23} +this system does not express quite so much as the single equation~\Eqref{(6)}, +from which it is derived. Both, indeed, express the +Proposition, Some~$X$s are~$Y$s, but the system~\Eqref{(10)} does not +imply that the class~$V$ includes \emph{all} the terms that are common +to $X$~and~$Y$. + +In like manner, from the equation~\Eqref{(7)} which expresses the +Proposition Some~$X$s are not~$Y$s, we may deduce the system +\[ +vx = v(1 - y) = v, +\Tag{(12)} +\] +in which the interpretation of~$v(1 - y)$ is Some not-$Y$s. Since +in this case $vy = 0$, we must of course be careful not to interpret~$vy$ +as Some~$Y$s. + +If we multiply the first equation of the system~\Eqref{(12)},~viz. +\[ +vx = v(1 - y), +\] +by~$y$, we have +\begin{align*} +vxy &= vy(1 - y); \\ +\therefore vxy &= 0, +\Tag{(13)} +\end{align*} +which is a form that will occasionally present itself. It is not +necessary to revert to the primitive equation in order to interpret +this, for the condition that $vx$~represents Some~$X$s, shews +us by virtue of~\Eqref{(5)}, that its import will be +\[ +\text{Some~$X$s are not~$Y$s,} +\] +the subject comprising \emph{all} the~$X$s that are found in the class~$V$. + +Universally in these cases, difference of form implies a difference +of interpretation with respect to the auxiliary symbol~$v$, +and each form is interpretable by itself. + +Further, these differences do not introduce into the Calculus +a needless perplexity. It will hereafter be seen that they give +a precision and a definiteness to its conclusions, which could not +otherwise be secured. + +Finally, we may remark that all the equations by which +particular truths are expressed, are deducible from any one +general equation, expressing any one general Proposition, from +which those particular Propositions are necessary deductions. +\PageSep{24} +This has been partially shewn already, but it is much more fully +exemplified in the following scheme. + +The general equation +\[ +x = y, +\] +implies that the classes $X$~and~$Y$ are equivalent, member for +member; that every individual belonging to the one, belongs +to the other also. Multiply the equation by~$x$, and we have +\begin{align*} +x^{2} &= xy; \\ +\therefore x &= xy, +\end{align*} +which implies, by~\Eqref{(4)}, that all~$X$s are~$Y$s. Multiply the same +equation by~$y$, and we have in like manner +\[ +y = xy; +\] +the import of which is, that all~$Y$s are~$X$s. Take either of these +equations, the latter for instance, and writing it under the form +\[ +(1 - x)y = 0, +\] +we may regard it as an equation in which~$y$, an unknown +quantity, is sought to be expressed in terms of~$x$. Now it +will be shewn when we come to treat of the Solution of Elective +Equations (and the result may here be verified by substitution) +that the most general solution of this equation is +\[ +y = vx, +\] +which implies that All~$Y$s are~$X$s, and that Some~$X$s are~$Y$s. +Multiply by~$x$, and we have +\[ +vy = vx, +\] +which indifferently implies that some~$Y$s are~$X$s and some~$X$s +are~$Y$s, being the particular form at which we before arrived. + +For convenience of reference the above and some other +results have been classified in the annexed Table, the first +column of which contains propositions, the second equations, +and the third the conditions of final interpretation. It is to +be observed, that the auxiliary equations which are given in +this column are not independent: they are implied either +in the equations of the second column, or in the condition for +\PageSep{25} +the interpretation of~$v$. But it has been thought better to write +them separately, for greater ease and convenience. And it is +further to be borne in mind, that although three different forms +are given for the expression of each of the \emph{particular} propositions, +everything is really included in the first form. +\begin{table}[hbt!] +\caption{TABLE.} +\footnotesize +\begin{alignat*}{3} +&\text{The class~$X$} &&x \\ +&\text{The class not-$X$} &&1 - x \\ +% +&\!\begin{aligned} +&\text{All~$X$s are~$Y$s} \\ +&\text{All~$Y$s are~$X$s} +\end{aligned}\Rbrace{2} && x = y \\ +% +&\text{All~$X$s are~$Y$s} && x(1 - y) = 0 \\ +&\text{No~$X$s are~$Y$s} && \PadTo[r]{x(1 - y) = 0}{xy = 0} \\ +% +&\!\begin{aligned} +&\text{All~$Y$s are~$X$s} \\ +&\text{Some~$X$s are~$Y$s} +\end{aligned}\Rbrace{2} && y = vx +&&\begin{aligned} +&vx = \text{Some~$X$s} \\ +&v(1 - x) = 0. +\end{aligned} \\[8pt] +% +&\!\begin{aligned} +&\text{No~$Y$s are~$X$s} \\ +&\text{Some not-$X$s are~$Y$s} +\end{aligned}\Rbrace{2} && y = v(1 - x) +&&\begin{aligned} +v(1 - x) &= \text{some not-$X$s} \\ +vx &= 0. +\end{aligned} \\[8pt] +% +&\text{Some~$X$s are~$Y$s} && +\Lbrace{3}\begin{aligned} +&v = xy \\ +\text{or } &vx = vy \\ +\text{or } &vx(1 - y) = 0 +\end{aligned}\quad && +\begin{aligned} +&v = \text{some~$X$s or some~$Y$s} \\ +&vx = \text{some~$X$s},\ vy = \text{some~$Y$s} \\ +&v(1 - x) = 0,\ v(1 - y) = 0. +\end{aligned} \\[8pt] +% +&\text{Some~$X$s are not~$Y$s} && +\Lbrace{3}\begin{aligned} +&v = x(1 - y) \\ +\text{or } &vx = v(1 - y) \\ +\text{or } &vxy = 0 +\end{aligned} && +\begin{aligned} +&v = \text{some~$X$s, or some not-$Y$s} \\ +&vx = \text{some~$X$s}, v(1 - y) = \text{some not-$Y$s} \\ +&v(1 - x) = 0,\ vy = 0. +\end{aligned} +\end{alignat*} +\end{table} +\PageSep{26} + + +\Chapter{Of the Conversion of Propositions.} + +\begin{Abstract} +A Proposition is said to be converted when its terms are transposed; when +nothing more is done, this is called simple conversion; \eg +\begin{align*} +&\text{No virtuous man is a tyrant, \emph{is converted into}} \\ +&\text{No tyrant is a virtuous man.} +\intertext{\indent +Logicians also recognise conversion \textit{per accidens}, or by limitation, \eg} +&\text{All birds are animals, \emph{is converted into}} \\ +&\text{Some animals are birds.} +\intertext{And conversion by \emph{contraposition} or \emph{negation}, as} +&\text{Every poet is a man of genius, \emph{converted into}} \\ +&\text{He who is not a man of genius is not a poet.} +\end{align*} + +In one of these three ways every Proposition may be illatively converted, viz.\ +$E$~and~$I$ simply, $A$~and~$O$ by negation, $A$~and~$E$ by limitation. +\end{Abstract} + +The primary canonical forms already determined for the +expression of Propositions, are +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s,} &x(1 - y) &= 0, +\Ltag{A} \\ +&\text{No~$X$s are~$Y$s,} &xy &= 0, +\Ltag{E} \\ +&\text{Some~$X$s are~$Y$s,} &v &= xy, +\Ltag{I} \\ +&\text{Some~$X$s are not~$Y$s,} &v &= x(1 - y). +\Ltag{O} +\end{alignat*} + +On examining these, we perceive that $E$~and~$I$ are symmetrical +with respect to $x$~and~$y$, so that $x$~being changed into~$y$, +and $y$~into~$x$, the equations remain unchanged. Hence $E$~and~$I$ +may be interpreted into +\begin{gather*} +\text{No~$Y$s are~$X$s,} \\ +\text{Some~$Y$s are~$X$s,} +\end{gather*} +respectively. Thus we have the known rule of the Logicians, +that particular affirmative and universal negative Propositions +admit of simple conversion. +\PageSep{27} + +The equations $A$~and~$O$ may be written in the forms +\begin{gather*} +(1 - y)\bigl\{1 - (1 - x)\bigr\} = 0, \\ +v = (1 - y)\bigl\{1 - (1 - x)\bigr\}. +\end{gather*} + +Now these are precisely the forms which we should have +obtained if we had in those equations changed $x$~into~$1 - y$, +and $y$~into~$1 - x$, which would have represented the changing +in the original Propositions of the~$X$s into not-$Y$s, and the~$Y$s +into not-$X$s, the resulting Propositions being +\begin{gather*} +\text{All not-$Y$s are not-$X$s,} \\ +\text{Some not-$Y$s are not not-$X$s.}\atag +\end{gather*} +Or we may, by simply inverting the order of the factors in the +second member of~$O$, and writing it in the form +\[ +v = (1 - y)x, +\] +interpret it by~$I$ into +\[ +\text{Some not-$Y$s are~$X$s,} +\] +which is really another form of~\aref. Hence follows the rule, +that universal affirmative and particular negative Propositions +admit of negative conversion, or, as it is also termed, conversion +by contraposition. + +The equations $A$~and~$E$, written in the forms +\begin{align*} +(1 - y) x &= 0, \\ +yx &= 0, +\end{align*} +give on solution the respective forms +\begin{align*} +x &= vy, \\ +x &= v(1 - y), +\end{align*} +the correctness of which may be shewn by substituting these +values of~$x$ in the equations to which they belong, and observing +that those equations are satisfied quite independently of the nature +of the symbol~$v$. The first solution may be interpreted into +\[ +\text{Some~$Y$s are~$X$s,} +\] +and the second into +\[ +\text{Some not-$Y$s are~$X$s.} +\] +\PageSep{28} +From which it appears that universal-affirmative, and universal-negative +Propositions are convertible by limitation, or, as it has +been termed, \textit{per accidens}. + +The above are the laws of Conversion recognized by Abp.~Whately. +Writers differ however as to the admissibility of +negative conversion. The question depends on whether we will +consent to use such terms as not-$X$, not-$Y$. Agreeing with +those who think that such terms ought to be admitted, even +although they change the \emph{kind} of the Proposition, I am constrained +to observe that the present classification of them is +faulty and defective. Thus the conversion of No~$X$s are~$Y$s, +into All~$Y$s are not-$X$s, though perfectly legitimate, is not recognised +in the above scheme. It may therefore be proper to +examine the subject somewhat more fully. + +Should we endeavour, from the system of equations we have +obtained, to deduce the laws not only of the conversion, but +also of the general transformation of propositions, we should be +led to recognise the following distinct elements, each connected +with a distinct mathematical process. + +1st. The negation of a term, \ie~the changing of~$X$ into not-$X$, +or not-$X$ into~$X$. + +2nd. The translation of a Proposition from one \emph{kind} to +another, as if we should change +\[ +\text{All~$X$s are~$Y$s into Some~$X$s are~$Y$s,} +\Ltag{$A$~into~$I$} +\] +which would be lawful; or +\[ +\text{All~$X$s are~$Y$s into No~$X$s are~$Y$\Typo{.}{s,}} +\Ltag{$A$~into~$E$} +\] +which would be unlawful. + +3rd. The simple conversion of a Proposition. + +The conditions in obedience to which these processes may +lawfully be performed, may be deduced from the equations by +which Propositions are expressed. + +We have +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s\Add{,}}\qquad & x(1 - y) &= 0, +\Ltag{A} \\ +&\text{No~$X$s are~$Y$s\Add{,}} & xy &= 0. +\Ltag{E} +\end{alignat*} +\PageSep{29} + +Write $E$ in the form +\[ +x\bigl\{1 - (1 - y)\bigr\} = 0, +\] +%[** TN: "A" italicized in the original] +and it is interpretable by~$A$ into +\[ +\text{All~$X$s are not-$Y$s,} +\] +so that we may change +\[ +\text{No~$X$s are~$Y$s into All~$X$s are not-$Y$s.} +\] + +In like manner $A$~interpreted by~$E$ gives +\[ +\text{No~$X$s are not-$Y$s,} +\] +so that we may change +\[ +\text{All~$X$s are~$Y$s into No~$X$s are not-$Y$s.} +\] + +From these cases we have the following Rule: A universal-affirmative +Proposition is convertible into a universal-negative, +and, \textit{vice versâ}, by negation of the predicate. + +Again, we have +\begin{alignat*}{2} +&\text{Some~$X$s are~$Y$s\Add{,}} & v &= xy, \\ +&\text{Some~$X$s are not~$Y$s\Add{,}}\qquad& v &= x(1 - y). +\end{alignat*} +These equations only differ from those last considered by the +presence of the term~$v$. The same reasoning therefore applies, +and we have the Rule--- + +A particular-affirmative proposition is convertible into a particular-negative, +and \textit{vice versâ}, by negation of the predicate. + +Assuming the universal Propositions +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s\Add{,}}\qquad & x(1 - y) &= 0, \\ +&\text{No~$X$s are~$Y$s\Add{,}} & xy &= 0. +\end{alignat*} +Multiplying by~$v$, we find +\begin{align*} +vx(1 - y) &= 0, \\ +vxy &= 0, +\end{align*} +which are interpretable into +\begin{align*} +&\text{Some~$X$s are~$Y$s,} +\Ltag{I} \\ +&\text{Some~$X$s are not~$Y$s.} +\Ltag{O} +\end{align*} +\PageSep{30} + +Hence a universal-affirmative is convertible into a particular-affirmative, +and a universal-negative into a particular-negative +without negation of subject or predicate. + +Combining the above with the already proved rule of simple +conversion, we arrive at the following system of independent +laws of transformation. + +1st. An affirmative Proposition may be changed into its corresponding +negative ($A$~into~$E$, or $I$~into~$O$), and \textit{\Typo{vice versa}{vice versâ}}, +by negation of the predicate. + +2nd. A universal Proposition may be changed into its corresponding +particular Proposition, ($A$~into~$I$, or $E$~into~$O$). + +3rd. In a particular-affirmative, or universal-negative Proposition, +the terms may be mutually converted. + +Wherein negation of a term is the changing of~$X$ into not-$X$, +and \textit{vice versâ}, and is not to be understood as affecting the \emph{kind} +of the Proposition. + +Every lawful transformation is reducible to the above rules. +Thus we have +\begin{alignat*}{2} +&\text{All~$X$s are~$Y$s,} \\ +&\text{No~$X$s are not-$Y$s} &&\text{by 1st rule,} \\ +&\text{No not-$Y$s are~$X$s} &&\text{by 3rd rule,} \\ +&\text{All not-$Y$s are not-$X$s } &&\text{by 1st rule,} +\end{alignat*} +which is an example of \emph{negative conversion}. Again, +\begin{alignat*}{2} +&\text{No~$X$s are~$Y$s,} \\ +&\text{No~$Y$s are~$X$s} &&\text{3rd rule,} \\ +&\text{All~$Y$s are not-$X$s}\quad &&\text{1st rule,} +\end{alignat*} +which is the case already deduced. +\PageSep{31} + + +\Chapter{Of Syllogisms.} + +\begin{Abstract} +A Syllogism consists of three Propositions, the last of which, called the +conclusion, is a logical consequence of the two former, called the premises; +\Typo{e.g.}{\eg} +\begin{alignat*}{2} +&\text{\emph{Premises,}} && +\Lbrace{2}\begin{aligned} +&\text{All~$Y$s are~$X$s.} \\ +&\text{All~$Z$s are~$Y$s.} +\end{aligned} \\ +&\text{\emph{Conclusion,}}\quad && +\text{All~$Z$s are~$X$s.} +\end{alignat*} + +Every syllogism has three and only three terms, whereof that which is +the subject of the conclusion is called the \emph{minor} term, the predicate of the +conclusion, the \emph{major} term, and the remaining term common to both premises, +the middle term. Thus, in \Typo{ths}{the} above formula, $Z$~is the minor term, $X$~the +major term, $Y$~the middle term. + +The figure of a syllogism consists in the situation of the middle term with +respect to the terms of the conclusion. The varieties of figure are exhibited +in the annexed scheme. +\[ +\begin{array}{*{3}{c<{\qquad}}c@{}} +\ColHead{1st Fig.} & \ColHead{2nd Fig.} & \ColHead{3rd Fig.} & \ColHead{4th Fig.} \\ +YX & XY & YX & XY \\ +ZY & ZY & YZ & YZ \\ +ZX & ZX & ZX & ZX +\end{array} +\] + +When we designate the three propositions of a syllogism by their usual +symbols ($A$, $E$, $I$, $O$), and in their actual order, we are said to determine +the mood of the syllogism. Thus the syllogism given above, by way of +illustration, belongs to the mood~$AAA$ in the first figure. + +The moods of all syllogisms commonly received as valid, are represented +by the vowels in the following mnemonic verses. + +Fig.~1.---bArbArA, cElArEnt, dArII, fErIO que prioris. + +Fig.~2.---cEsArE, cAmEstrEs, \Typo{fEstIno}{fEstInO}, bArOkO, secundć. + +Fig.~3.---Tertia dArAptI, dIsAmIs, dAtIsI, fElAptOn, \\ +\PadTo{\text{\indent Fig.~3.---}}{}bOkArdO, fErIsO, habet: quarta insuper addit. + +Fig.~4.---brAmAntIp, cAmEnEs, dImArIs, \Typo{fEsapO}{fEsApO}, frEsIsOn. +\end{Abstract} + +\First{The} equation by which we express any Proposition concerning +the classes $X$~and~$Y$, is an equation between the +symbols $x$~and~$y$, and the equation by which we express any +\PageSep{32} +Proposition concerning the classes $Y$~and~$Z$, is an equation +between the symbols $y$~and~$z$. If from two such equations +we eliminate~$y$, the result, if it do not vanish, will be an +equation between $x$~and~$z$, and will be interpretable into a +Proposition concerning the classes $X$~and~$Z$. And it will then +constitute the third member, or Conclusion, of a Syllogism, +of which the two given Propositions are the premises. + +The result of the elimination of~$y$ from the equations +\[ +\begin{alignedat}{2} +ay &+ b &&= 0, \\ +a'y &+ b' &&= 0, +\end{alignedat} +\Tag{(14)} +\] +is the equation +\[ +ab' - a'b = 0. +\Tag{(15)} +\] + +Now the equations of Propositions being of the first order +with reference to each of the variables involved, all the cases +of elimination which we shall have to consider, will be reducible +to the above case, the constants $a$,~$b$, $a'$,~$b'$, being +replaced by functions of $x$,~$z$, and the auxiliary symbol~$v$. + +As to the choice of equations for the expression of our +premises, the only restriction is, that the equations must not +\emph{both} be of the form $ay = 0$, for in such cases elimination would +be impossible. When both equations are of this form, it is +necessary to solve one of them, and it is indifferent which +we choose for this purpose. If that which we select is of +the form $xy = 0$, its solution is +\[ +y = v(1 - x), +\Tag{(16)} +\] +if of the form $(1 - x)y = 0$, the solution will be +\[ +y = vx, +\Tag{(17)} +\] +and these are the only cases which can arise. The reason +of this exception will appear in the sequel. + +For the sake of uniformity we shall, in the expression of +particular propositions, confine ourselves to the forms +\begin{alignat*}{2} +vx &= vy, &&\text{Some~$X$s are~$Y$s,} \\ +vx &= v(1 - y),\quad&&\text{Some~$X$s are not~$Y$s\Typo{,}{.}} +\end{alignat*} +\PageSep{33} +These have a closer analogy with \Eqref{(16)}~and~\Eqref{(17)}, than the other +forms which might be used. + +Between the forms about to be developed, and the Aristotelian +canons, some points of difference will occasionally be observed, +of which it may be proper to forewarn the reader. + +To the right understanding of these it is proper to remark, +that the essential structure of a Syllogism is, in some measure, +arbitrary. Supposing the order of the premises to be fixed, +and the distinction of the major and the minor term to be +thereby determined, it is purely a matter of choice which of +the two shall have precedence in the Conclusion. Logicians +have settled this question in favour of the minor term, but +it is clear, that this is a convention. Had it been agreed +that the major term should have the first place in the conclusion, +a logical scheme might have been constructed, less +convenient in some cases than the existing one, but superior +in others. What it lost in \textit{barbara}, it would gain in \textit{bramantip}. +Convenience is \emph{perhaps} in favour of the adopted arrangement,\footnote + {The contrary view was maintained by Hobbes. The question is very + fairly discussed in Hallam's \textit{Introduction to the Literature of Europe}, vol.~\textsc{iii}. + p.~309. In the rhetorical use of Syllogism, the advantage appears to rest + with the rejected form.} +but it is to be remembered that it is \emph{merely} an arrangement. + +Now the method we shall exhibit, not having reference +to one scheme of arrangement more than to another, will +always give the more general conclusion, regard being paid +only to its abstract lawfulness, considered as a result of pure +reasoning. And therefore we shall sometimes have presented +to us the spectacle of conclusions, which a logician would +pronounce informal, but never of such as a reasoning being +would account false. + +The Aristotelian canons, however, beside restricting the \emph{order} +of the terms of a conclusion, limit their nature also;---and +this limitation is of more consequence than the former. We +may, by a change of figure, replace the particular conclusion +\PageSep{34} +of \textit{bramantip} by the general conclusion of~\textit{barbara}; but we +cannot thus reduce to rule such inferences, as +\[ +\text{Some not-$X$s are not~$Y$s.} +\] + +Yet there are cases in which such inferences may lawfully +be drawn, and in unrestricted argument they are of frequent +occurrence. Now if an inference of this, or of any other +kind, is lawful in itself, it will be exhibited in the results +of our method. + +We may by restricting the canon of interpretation confine +our expressed results within the limits of the scholastic logic; +but this would only be to restrict ourselves to the use of a part +of the conclusions to which our analysis entitles us. + +The classification we shall adopt will be purely mathematical, +and we shall afterwards consider the logical arrangement to +which it corresponds. It will be sufficient, for reference, to +name the premises and the Figure in which they are found. + +\textsc{Class} 1st.---Forms in which $v$~does not enter. + +Those which admit of an inference are $AA$,~$EA$, Fig.~1; +$AE$,~$EA$, Fig.~2; $AA$,~$AE$, Fig.~4. + +Ex. $AA$, Fig.~1, and, by mutation of premises (change of +order), $AA$,~Fig.~4. +\begin{alignat*}{4} +&\text{All~$Y$s are~$X$s,}\qquad& +y(1 - x) &= 0,\qquad&& \text{or }& (1 - x) y &= 0, \\ +&\text{All~$Z$s are~$Y$s,} & +z(1 - y) &= 0, &&\text{or }& zy - z &= 0. +\end{alignat*} + +Eliminating~$y$ by~\Eqref{(13)} we have +\begin{gather*} +z(1 - x) = 0, \\ +\therefore\ \text{All~$Z$s are~$X$s.} +\end{gather*} + +A convenient mode of effecting the elimination, is to write +the equation of the premises, so that $y$~shall appear only as +a factor of one member in the first equation, and only as +a factor of the opposite member in the second equation, and +then to multiply the equations, omitting the~$y$. This method +we shall adopt. +\PageSep{35} + +Ex. $AE$, Fig.~2, and, by mutation of premises, $EA$, Fig\Typo{,}{.}~2. +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{No~$Z$s are~$Y$s,} & zy &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +\text{or } & x &=& xy\Add{,} \\ + &zy &=& 0\Add{,} \\ +\cline{2-4} + &zx &=& 0\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{No~$Z$s are~$X$s.}} +\end{array} +\] + +The only case in which there is no inference is~$AA$, Fig.~2, +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{All~$Z$s are~$Y$s,} & z(1 - y) &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +& x &=& xy\Add{,} \\ +&zy &=& z\Add{,} \\ +\cline{2-4} +&xz &=& xz\Add{,} \\ +\multicolumn{4}{l}{\rlap{$\therefore\ 0 = 0$.}} +\end{array} +\] + +\textsc{Class} 2nd.---When $v$~is introduced by the solution of an +equation. + +The lawful cases directly or indirectly\footnote + {We say \emph{directly} or \emph{indirectly}, mutation or conversion of premises being + in some instances required. Thus, $AE$ (fig.~1) is resolvable by \Chg{Fesapo}{\textit{fesapo}} (fig.~4), + or by \Chg{Ferio}{\textit{ferio}} (fig.~1). Aristotle and his followers rejected the fourth figure + as only a modification of the first, but this being a mere question of form, + either scheme may be termed Aristotelian.} +determinable by the +Aristotelian Rules are~$AE$, Fig.~1; $AA$, $AE$, $EA$, Fig.~3; +$EA$, Fig.~4. + +The lawful cases not so determinable, are $EE$, Fig.~1; $EE$, +Fig.~2; $EE$, Fig.~3; $EE$, Fig.~4. + +Ex. $AE$, Fig.~1, and, by mutation of premises, $EA$, Fig.~4. +\[ +\begin{alignedat}[t]{2} +&\text{All~$Y$s are~$X$s,}\qquad& y(1 - x) &= 0, \\ +&\text{No~$Z$s are~$Y$s,} & zy &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&y &=& vx\Add{,}\atag \\ +&0 &=& zy\Add{,} \\ +\cline{2-4} +&0 &=& vzx\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{Some~$X$s are not~$Z$s.}} +\end{array} +\] + +The reason why we cannot interpret $vzx = 0$ into Some~$Z$s +are not-$X$s, is that by the very terms of the first equation~\aref\ +the interpretation of~$vx$ is fixed, as Some~$X$s; $v$~is regarded +as the representative of Some, only with reference to the +class~$X$. +\PageSep{36} + +For the reason of our employing a solution of one of the +primitive equations, see the remarks on \Eqref{(16)}~and~\Eqref{(17)}. Had +we solved the second equation instead of the first, we should +have had +\begin{gather*} +\begin{aligned} +(1 - x)y &= 0, \\ +v(1 - z) &= y,\atag \\ +v(1 - z)(1 - x) &= 0,\btag +\end{aligned} \\ +\therefore\ \text{Some not-$Z$s are~$X$s.} +\end{gather*} + +Here it is to be observed, that the second equation~\aref\ fixes +the meaning of~$v(1 - z)$, as Some not-$Z$s. The full meaning +of the result~\bref\ is, that all the not-$Z$s which are found in +the class~$Y$ are found in the class~$X$, and it is evident that +this could not have been expressed in any other way. + +Ex.~2. $AA$, Fig.~3. +\[ +\begin{alignedat}[t]{2} +&\text{All~$Y$s are~$X$s,}\qquad& y(1 - x) &= 0, \\ +&\text{All~$Y$s are~$Z$s,} & y(1 - z) &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&y &=& vx\Add{,} \\ +&0 &=& y(1 - z)\Add{,} \\ +\cline{2-4} +&0 &=& vx(1 - z)\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{Some~$X$s are~$Z$s.}} +\end{array} +\] + +Had we solved the second equation, we should have had +as our result, Some~$Z$s are~$X$s. The form of the final equation +particularizes what~$X$s or what~$Z$s are referred to, and this +remark is general. + +The following, $EE$, Fig.~1, and, by mutation, $EE$, Fig.~4, +is an example of a lawful case not determinable by the Aristotelian +Rules. +\[ +\begin{alignedat}[t]{2} +&\text{No~$Y$s are~$X$s,}\qquad& xy &= 0, \\ +&\text{No~$Z$s are~$Y$s,} & zy &= 0, +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&0 &=& xy\Add{,} \\ +&y &=& v(1 - z)\Add{,} \\ +\cline{2-4} +&0 &=& v(1 - z)x\Add{,} \\ +\multicolumn{4}{l}{\therefore\ \rlap{Some not-$Z$s are not~$X$s.}} +\end{array} +\] + +\textsc{Class} 3rd.---When $v$~is met with in one of the equations, +but not introduced by solution. +\PageSep{37} + +The lawful cases determinable \emph{directly} or \emph{indirectly} by the +Aristotelian Rules, are $AI$,~$EI$, Fig.~1; $AO$, $EI$, $OA$, $IE$, +Fig.~2; $AI$, $AO$, $EI$, $EO$, $IA$, $IE$, $OA$, $OE$, Fig.~3; $IA$, $IE$, +Fig.~4. + +Those not so determinable are~$OE$, Fig.~1; $EO$, Fig.~4. + +The cases in which no inference is possible, are $AO$, $EO$, +$IA$, $IE$, $OA$, Fig.~1; $AI$, $EO$, $IA$, $OE$, Fig.~2; $OA$, $OE$, +$AI$, $EI$, $AO$, Fig.~4. + +Ex.~1. $AI$, Fig.~1, and, by mutation, $IA$, Fig.~4. +\[ +\begin{aligned}[t] +&\text{All~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are~$Y$s,} +\end{aligned} +\begin{array}[t]{>{\qquad}rr@{\,}c@{\,}l@{}} +&y(1 - x) &=& 0\Add{,} \\ +&vz &=& vy\Add{,} \\ +\cline{2-4} +&vz(1 - x) &=& 0\Add{,} \\ +\therefore\ & +\multicolumn{3}{l}{\rlap{Some~$Z$s are~$X$s.}} +\end{array} +\] + +Ex.~2. $AO$, Fig.~2, and, by mutation, $OA$, Fig.~2. +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{Some~$Z$s are not~$Y$s,} & vz &= v(1 - y), +\end{alignedat}\quad +\begin{array}[t]{rr@{\,}c@{\,}l@{}} +&x &=& xy\Add{,} \\ +&vy &=& v(1 - z)\Add{,} \\ +\cline{2-4} +&vx &=& vx(1 - z)\Add{,} \\ +&vxz&=& 0\Add{,} \\ +\multicolumn{4}{r}{\llap{$\therefore\ \text{Some~$Z$s are not~$X$s.}$}} +\end{array} +\] + +The interpretation of~$vz$ as Some~$Z$s, is implied, it will be +observed, in the equation $vz = v(1 - y)$ considered as representing +the proposition Some~$Z$s are not~$Y$s. + +The cases not determinable by the Aristotelian Rules are +$OE$, Fig.~1, and, by mutation, $EO$, Fig.~4. +\[ +\begin{aligned}[t] +&\text{Some~$Y$s are not~$X$s,} \\ +&\text{No~$Z$s are~$Y$s,} +\end{aligned}\qquad +\begin{array}[t]{>{\qquad}rr@{\,}c@{\,}l@{}} +&vy &=& v(1 - x)\Add{,} \\ +& 0 &=& zy\Add{,} \\ +\cline{2-4} +& 0 &=& v(1 - x)z\Add{,} \\ +\multicolumn{4}{c}{\makebox[0pt][c]{$\therefore$\ Some not-$X$s are not~$Z$s.}} +\end{array} +\] + +The equation of the first premiss here permits us to interpret +$v(1 - x)$, but it does not enable us to interpret~$vz$. +\PageSep{38} + +Of cases in which no inference is possible, we take as +examples--- + +$AO$, Fig.~1, and, by mutation, $OA$, Fig.~4\Typo{,}{.} +\[ +\begin{alignedat}[t]{2} +&\text{All~$Y$s are~$X$s,}\qquad& y(1 - x) &= 0, \\ +&\text{Some~$Z$s are not~$Y$s,} & vz &= v(1 - y)\Add{,}\atag +\end{alignedat}\qquad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +y(1 - x) &=& 0\Add{,} \\ +v(1 - z) &=& vy\Add{,} \\ +\cline{1-3} +v(1 - z)(1 - x) &=& 0\Add{,}\btag \\ +0&=& 0\Add{,} +\end{array} +\] +since the auxiliary equation in this case is $v(1 - z) = 0$. + +Practically it is not necessary to perform this reduction, but +it is satisfactory to do so. The equation~\aref, it is seen, defines~$vz$ +as Some~$Z$s, but it does not define $v(1 - z)$, so that we might +stop at the result of elimination~\bref, and content ourselves with +saying, that it is not interpretable into a relation between the +classes $X$~and~$Z$. + +Take as a second example $AI$, Fig.~2, and, by mutation, +$IA$, Fig.~2\Typo{,}{.} +\[ +\begin{alignedat}[t]{2} +&\text{All~$X$s are~$Y$s,}\qquad& x(1 - y) &= 0, \\ +&\text{Some~$Z$s are~$Y$s,} & vz &= vy, +\end{alignedat}\qquad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +x &=& xy\Add{,} \\ +vy &=& vz\Add{,} \\ +\cline{1-3} +vx &=& vxz\Add{,} \\ +\llap{$v(1 - z)x$}&=& 0\Add{,} \\ +0&=& 0, +\end{array} +\] +the auxiliary equation in this case being $v(1 - z)= 0$. + +Indeed in every case in this class, in which no inference +is possible, the result of elimination is reducible to the form +$0 = 0$. Examples therefore need not be multiplied. + +\textsc{Class} 4th.---When $v$~enters into both equations. + +No inference is possible in any case, but there exists a distinction +among the unlawful cases which is peculiar to this +class. The two divisions are, + +1st. When the result of elimination is reducible by the +auxiliary equations to the form $0 = 0$. The cases are $II$, $OI$, +\PageSep{39} +Fig.~1; $II$, $OO$, Fig.~2; $II$, $IO$, $OI$, $OO$, Fig.~3; $II$, $IO$, +Fig.~4. + +2nd. When the result of elimination is not reducible by the +auxiliary equations to the form $0 = 0$. + +The cases are $IO$, $OO$, Fig.~1; $IO$, $OI$, Fig.~2; $OI$, $OO$, +Fig.~4. + +Let us take as an example of the former case,~$II$, Fig.~3. +\[ +\begin{alignedat}[t]{2} +&\text{Some~$X$s are~$Y$s,}\qquad& vx &= vy, \\ +&\text{Some~$Z$s are~$Y$s,} & v'z &= v'y, +\end{alignedat}\qquad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +vx &=& vy\Add{,} \\ +v'y &=& v'z\Add{,} \\ +\cline{1-3} +vv'x &=& vv'z\Add{.} +\end{array} +\] + +Now the auxiliary equations $v(1 - x) = 0$, $v'(1 - z) = 0$, +%[** TN: Next word anomalously displayed in the original] +give +\[ +vx = v,\quad v'z = v'. +\] +Substituting we have +\begin{align*} +vv' &= vv', \\ +\therefore 0 &= 0. +\end{align*} + +As an example of the latter case, let us take $IO$, Fig.~1\Typo{,}{.} +\[ +\begin{alignedat}[t]{2} +&\text{Some~$Y$s are~$X$s,} & vy &= vx, \\ +&\text{Some~$Z$s are not~$Y$s,}\qquad& v'z &= v'(1 - y), +\end{alignedat}\quad +\begin{array}[t]{r@{\,}c@{\,}l@{}} +vy &=& vx\Add{,} \\ +v'(1 - z) &=& v'y\Add{,} \\ +\cline{1-3} +vv'(1 - z) &=& vv'x\Add{.} +\end{array} +\] + +Now the auxiliary equations being $v(1 - x) = 0$, $v'(1 - z) = 0$, +the above reduces to $vv' = 0$. It is to this form that all similar +cases are reducible. Its interpretation is, that the classes $v$ +and~$v'$ have no common member, as is indeed evident. + +The above classification is purely founded on mathematical +distinctions. We shall now inquire what is the logical division +to which it corresponds. + +The lawful cases of the first class comprehend all those in +which, from two universal premises, a universal conclusion +may be drawn. We see that they include the premises of +\textit{barbara} and \textit{celarent} in the first figure, of \textit{cesare} and \textit{camestres} +in the second, and of \textit{bramantip} and \textit{camenes} in the fourth. +\PageSep{40} +The premises of \textit{bramantip} are included, because they admit +of an universal conclusion, although not in the same figure. + +The lawful cases of the second class are those in which +a particular conclusion only is deducible from two universal +premises. + +The lawful cases of the third class are those in which a +conclusion is deducible from two premises, one of which is +universal and the other particular. + +The fourth class has no lawful cases. + +Among the cases in which no inference of any kind is possible, +we find six in the fourth class distinguishable from the +others by the circumstance, that the result of elimination does +not assume the form $0 = 0$. The cases are +{\small +\[ +\Lbrace{2}\begin{aligned} +&\text{Some~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{aligned}\Rbrace{2}\quad +% +\Lbrace{2}\begin{aligned} +&\text{Some~$Y$s are not~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{aligned}\Rbrace{2}\quad +% +\Lbrace{2}\begin{aligned} +&\text{Some~$X$s are~$Y$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{aligned}\Rbrace{2} +\] +}% +and the three others which are obtained by mutation of +premises. + +It might be presumed that some logical peculiarity would +be found to answer to the mathematical peculiarity which we +have noticed, and in fact there exists a very remarkable one. +If we examine each pair of premises in the above scheme, we +shall find that there \emph{is virtually} no middle term, \emph{\ie~no medium +of comparison}, in any of them. Thus, in the first example, +the individuals spoken of in the first premiss are asserted to +belong to the class~$Y$, but those spoken of in the second +premiss are \emph{virtually} asserted to belong to the class not-$Y$: +nor can we by any lawful transformation or conversion alter +this state of things. The comparison will still be made with +the class~$Y$ in one premiss, and with the class not-$Y$ in the +other. + +Now in every case beside the above six, there will be found +a middle term, either expressed or implied. I select two +of the most difficult cases. +\PageSep{41} + +In $AO$, Fig.~1, viz. +\begin{align*} +&\text{All~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{align*} +we have, by \emph{negative conversion} of the first premiss, +\begin{align*} +&\text{All not-$X$s are not-$Y$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{align*} +and the middle term is now seen to be not-$Y$. + +Again, in $EO$, Fig.~1, +\begin{align*} +&\text{No~$Y$s are~$X$s,} \\ +&\text{Some~$Z$s are not~$Y$s,} +\end{align*} +a proved conversion of the first premiss (see \ChapRef{5}{Conversion of +Propositions}), gives +\begin{align*} +&\text{All~$X$s are not-$Y$s,} \\ +&\text{Some~$Z$s are not-$Y$s,} +\end{align*} +and the middle term, the true medium of comparison, is plainly +\Pagelabel{41}% +not-$Y$, although as the not-$Y$s in the one premiss \emph{may be} +different from those in the other, no conclusion can be drawn. + +The mathematical condition in question, therefore,---the irreducibility +of the final equation to the form $0 = 0$,---adequately +represents the logical condition of there being no middle term, +or common medium of comparison, in the given premises. + +I am not aware that the distinction occasioned by the +presence or absence of a middle term, in the strict sense here +understood, has been noticed by logicians before. The distinction, +though real and deserving attention, is indeed by +no means an obvious one, and it would have been unnoticed +in the present instance but for the peculiarity of its mathematical +expression. + +What appears to be novel in the above case is the proof +of the existence of combinations of premises in which there +\PageSep{42} +is absolutely no medium of comparison. When such a medium +of comparison, or true middle term, does exist, the condition +that its quantification in both premises together shall exceed +its quantification as a single whole, has been ably and +\Pagelabel{42}% +clearly shewn by Professor De~Morgan to be necessary to +lawful inference (\textit{Cambridge Memoirs}, Vol.~\textsc{viii}.\ Part~3). And +this is undoubtedly the true principle of the Syllogism, viewed +from the standing-point of Arithmetic. + +I have said that it would be possible to impose conditions +of interpretation which should restrict the results of this calculus +to the Aristotelian forms. Those conditions would be, + +1st. That we should agree not to interpret the forms $v(1 - x)$, +$v(1 - z)$. + +2ndly. That we should agree to reject every interpretation in +which the order of the terms should violate the Aristotelian rule. + +Or, instead of the second condition, it might be agreed that, +the conclusion being determined, the order of the premises +should, if necessary, be changed, so as to make the syllogism +formal. + +From the \emph{general} character of the system it is indeed plain, +that it may be made to represent any conceivable scheme of +logic, by imposing the conditions proper to the case contemplated. + +We have found it, in a certain class of cases, to be necessary +to replace the two equations expressive of universal Propositions, +by their solutions; and it may be proper to remark, +that it would have been allowable in all instances to have +done this,\footnote + {It may be satisfactory to illustrate this statement by an example. In + \textit{\Chg{Barbara}{barbara}}, we should have + \[ + \begin{aligned}[t] + &\text{All~$Y$s are~$X$s,} \\ + &\text{All~$Z$s are~$Y$s,} + \end{aligned}\qquad + \begin{array}[t]{>{\qquad}r@{\,}c@{\,}l@{}} + y &=& vx\Add{,} \\ + z &=& v'y\Add{,} \\ + \cline{1-3} + z &=& vv'x\Add{,} \\ + \multicolumn{3}{c}{\makebox[0pt][c]{$\therefore$\ All~$Z$s are~$X$s.}} + \end{array} + \] +%[** TN: Footnote continues] + Or, we may multiply the resulting equation by~$1 - x$, which gives + \[ + z(1 - x) = 0, + \] + whence the same conclusion, All~$Z$s are~$X$s. + + Some additional examples of the application of the system of equations in + the text to the demonstration of general theorems, may not be inappropriate. + + Let $y$ be the term to be eliminated, and let $x$ stand indifferently for either of + the other symbols, then each of the equations of the premises of any given + syllogism may be put in the form + \[ + ay + bx = 0, + \GrTag[a]{(\alpha)} + \] + if the premiss is affirmative, and in the form + \[ + ay + b(1 - x) = 0, + \GrTag[b]{(\beta)} + \] + if it is negative, $a$~and~$b$ being either constant, or of the form~$±v$. To prove + this in detail, let us examine each kind of proposition, making $y$~successively + subject and predicate. + \begin{alignat*}{2} + A,\ &\text{All~$Y$s are~$X$s,} & y - vx &= 0, + \GrTag[c]{(\gamma)} \\ + &\text{All~$X$s are~$Y$s,} & x - vy &= 0, + \GrTag[d]{(\delta)} \\ +% + E,\ &\text{No~$Y$s are~$X$s,} & xy &= 0, \\ + &\text{No~$X$s are~$Y$s,} & y - v(1 - x) &= 0, + \GrTag[e]{(\epsilon)} \\ +% + I,\ &\text{Some~$X$s are~$Y$s,} && \\ + &\text{Some~$Y$s are~$X$s,} &vx - vy &= 0, + \GrTag[f]{(\zeta)} \\ +% + O,\ &\text{Some~$Y$s are not~$X$s,}\qquad& vy - v(1 - x) &= 0, + \GrTag[g]{(\eta)} \\ + &\text{Some~$X$s are not~$Y$s,} & vx &= v(1 - y), \\ + && \therefore vy - v(1 - x) &= 0. + \GrTag[h]{(\theta)} + \end{alignat*} + + The affirmative equations \GrEq[c]{(\gamma)},~\GrEq[d]{(\delta)} and~\GrEq[f]{(\zeta)}, belong to~\GrEq[a]{(\alpha)}, and the negative + equations \GrEq[e]{(\epsilon)},~\GrEq[g]{(\eta)} and~\GrEq[h]{(\theta)}, to~\GrEq[b]{(\beta)}. It is seen that the two last negative equations + are alike, but there is a difference of interpretation. In the former + \[ + v(1 - x) = \text{Some not-$X$s,} + \] + in the latter, + \[ + v(1 - x) = 0. + \] + + The utility of the two general forms of reference, \GrEq[a]{(\alpha)}~and~\GrEq[b]{(\beta)}, will appear + from the following application. + + 1st. \emph{A conclusion drawn from two affirmative propositions} is itself affirmative. + + By \GrEq[a]{(\alpha)} we have for the given propositions, + \begin{alignat*}{2} + ay &+ bx &&= 0, \\ + a'y &+ b'z &&= 0, + \end{alignat*} +%[** TN: Footnote continues] + and eliminating + \[ + ab'z - a'bx = 0, + \] + which is of the form~\GrEq[a]{(\alpha)}. Hence, if there is a conclusion, it is affirmative. + + 2nd. \emph{A conclusion drawn from an affirmative and a negative proposition is +negative.} + + By \GrEq[a]{(\alpha)}~and~\GrEq[b]{(\beta)}, we have for the given propositions + \begin{align*} + ay + bx &= 0, \\ + a'y + b'(1 - z) &= 0, \\ + \therefore\ a'bx - ab'(1 - z) &= 0, + \end{align*} + which is of the form~\GrEq[b]{(\beta)}. Hence the conclusion, if there is one, is negative. + + 3rd. \emph{A conclusion drawn from two negative premises will involve a negation, + \(not-$X$, not-$Z$\) in both subject and predicate, and will therefore be inadmissible in + the Aristotelian system, though just in itself.} + + For the premises being + \begin{alignat*}{2} + ay &+ b (1 - x) &&= 0, \\ + a'y &+ b'(1 - z) &&= 0, + \end{alignat*} + the conclusion will be + \[ + ab'(1 - z) - a'b(1 - x) = 0, + \] + which is only interpretable into a proposition that has a negation in each term. + + 4th. \emph{Taking into account those syllogisms only, in which the conclusion is the + most general, that can be deduced from the premises,---if, in an Aristotelian + syllogism, the minor premises be changed in quality \(from affirmative to negative + or from negative to affirmative\), whether it be changed in quantity or not, no conclusion + will be deducible in the same figure.} + + An Aristotelian proposition does not admit a term of the form not-$Z$ in the + subject,---Now on changing the quantity of the minor proposition of a syllogism, + we transfer it from the general form + \begin{align*} + ay + bz &= 0, \\ + \intertext{to the general form} + a'y + b'(1 - z) &= 0, + \end{align*} + see \GrEq[a]{(\alpha)}~\emph{and}~\GrEq[b]{(\beta)}, or \textit{vice versâ}. And therefore, in the equation of the conclusion, + there will be a change from~$z$ to~$1 - z$, or \textit{vice versâ}. But this is equivalent to + the change of~$Z$ into not-$Z$, or not-$Z$ into~$Z$. Now the subject of the original + conclusion must have involved a~$Z$ and not a not-$Z$, therefore the subject of the + new conclusion will involve a not-$Z$, and the conclusion will not be admissible + in the Aristotelian forms, except by conversion, which would render necessary + a change of Figure. + + Now the conclusions of this calculus are always the most general that can be + drawn, and therefore the above demonstration must not be supposed to extend + to a syllogism, in which a particular conclusion is deduced, when a universal + one is possible. This is the case with \textit{bramantip} only, among the Aristotelian + forms, and therefore the transformation of \textit{bramantip} into \textit{camenes}, and \textit{vice versâ}, + is the case of restriction contemplated in the preliminary statement of the + theorem. + + 5th. \emph{If for the minor premiss of an Aristotelian syllogism, we substitute its contradictory, + no conclusion is deducible in the same figure.} + + It is here only necessary to examine the case of \textit{bramantip}, all the others + being determined by the last proposition. + + On changing the minor of \textit{bramantip} to its contradictory, we have $AO$, + Fig.~4, and this admits of no legitimate inference. + + Hence the theorem is true without exception. Many other general theorems + may in like manner be proved.} +%[** TN: End of 3.5-page footnote] +so that every case of the Syllogism, without exception, +\PageSep{43} +might have been treated by equations comprised in +the general forms +\Pagelabel{43}% +\begin{alignat*}{3} + y &= vx, &&\text{or} & y - vx &= 0, +\Ltag{A} \\ + y &= v(1 - x),\qquad&&\text{or}\quad & y + vx - v &= 0, +\Ltag{E} \\ +vy &= vx, &&& vy - vx &= 0, +\Ltag{I} \\ +vy &= v(1 - x), &&& vy + vx - v &= 0. +\Ltag{O} +\end{alignat*} +\PageSep{44} + +Perhaps the system we have actually employed is better, +as distinguishing the cases in which $v$~only \emph{may} be employed, +\PageSep{45} +from those in which it \emph{must}. But for the demonstration of +certain general properties of the Syllogism, the above system +is, from its simplicity, and from the mutual analogy of its +forms, very convenient. We shall apply it to the following +theorem.\footnote + {This elegant theorem was communicated by the Rev.\ Charles Graves, + Fellow and Professor of Mathematics in Trinity College, Dublin, to whom the + Author desires further to record his grateful acknowledgments for a very + judicious examination of the former portion of this work, and for some new + applications of the method. The following example of Reduction \textit{ad~impossibile} + is among the number: + \[ + \begin{array}{rl<{\quad}r@{\,}c@{\,}l@{}} + \text{Reducend Mood,} & + \text{All~$X$s are~$Y$s,} & + 1 - y &=& v'(1 - x)\Add{,} \\ + \PadTxt{Reducend Mood,}{\textit{\Chg{Baroko}{baroko}}} & + \text{Some~$Z$s are not~$Y$s\Add{,}} & + vz &=& v(1 - y)\Add{,} \\ + \cline{3-5} +% + &\text{Some~$Z$s are not~$X$s\Add{,}} & + vz &=& vv'(1 - x)\Add{,} \\ +% + \text{Reduct Mood,} & + \text{All~$X$s are~$Y$s\Add{,}} & + 1 - y &=& v'(1 - x)\Add{,} \\ + \PadTxt{Reduct Mood,}{\textit{\Chg{Barbara}{barbara}}} & + \text{All~$Z$s are~$X$s\Add{,}} & + z(1 - x) &=& 0\Add{,} \\ + \cline{2-5} + &\text{All~$Z$s are~$Y$s\Add{,}} & + z(1 - y) &=& 0. + \end{array} + \] + + The conclusion of the reduct mood is seen to be the contradictory of the + suppressed minor premiss. Whence,~\etc. It may just be remarked that the + mathematical test of contradictory propositions is, that on eliminating one + elective symbol between their equations, the other elective symbol vanishes. + The \emph{ostensive} reduction of \textit{\Chg{Baroko}{baroko}} and \textit{\Chg{Bokardo}{bokardo}} involves no difficulty. + + Professor Graves suggests the employment of the equation $x = vy$ for the + primary expression of the Proposition All~$X$s are~$Y$s, and remarks, that on + multiplying both members by~$1 - y$, we obtain $x(1 - y) = 0$, the equation from + which we set out in the text, and of which the previous one is a solution.} + +Given the three propositions of a Syllogism, prove that there +is but one order in which they can be legitimately arranged, +and determine that order. + +All the forms above given for the expression of propositions, +are particular cases of the general form, +\[ +a + bx + cy = 0. +\] +\PageSep{46} + +Assume then for the premises of the given syllogism, the +equations +\begin{alignat*}{3} +a &+ bx &&+ cy &&= 0, +\Tag{(18)} \\ +a' &+ b'z &&+ c'y &&= 0, +\Tag{(19)} +\end{alignat*} +then, eliminating~$y$, we shall have for the conclusion +\[ +ac' - a'c + bc'x - b'cz = 0. +\Tag{(20)} +\] + +Now taking this as one of our premises, and either of the +original equations, suppose~\Eqref{(18)}, as the other, if by elimination +of a common term~$x$, between them, we can obtain a result +equivalent to the remaining premiss~\Eqref{(19)}, it will appear that +there are more than one order in which the Propositions may +be lawfully written; but if otherwise, one arrangement only +is lawful. + +Effecting then the elimination, we have +\[ +bc(a' + b'z + c'y) = 0, +\Tag{(21)} +\] +which is equivalent to~\Eqref{(19)} multiplied by a factor~$bc$. Now on +examining the value of this factor in the equations $A$,~$E$, $I$,~$O$, +we find it in each case to be $v$~or~$-v$. But it is evident, +that if an equation expressing a given Proposition be multiplied +by an extraneous factor, derived from another equation, +its interpretation will either be limited or rendered +impossible. Thus there will either be no result at all, or the +result will be a \emph{limitation} of the remaining Proposition. + +If, however, one of the original equations were +\[ +x = y,\quad\text{or}\quad x - y = 0, +\] +the factor~$bc$ would be~$-1$, and would \emph{not} limit the interpretation +of the other premiss. Hence if the first member of +a syllogism should be understood to represent the double +proposition All~$X$s are~$Y$s, and All~$Y$s are~$X$s, it would be +indifferent in what order the remaining Propositions were +written. +\PageSep{47} + +A more general form of the above investigation would be, +to express the premises by the equations +\begin{alignat*}{4} +a &+ bx &&+ cy &&+ dxy &&= 0, +\Tag{(22)} \\ +a' &+ b'z &&+ c'y &&+ d'zy &&= 0. +\Tag{(23)} +\end{alignat*} + +After the double elimination of $y$~and~$x$ we should find +\[ +(bc - ad)(a' + b'z + c'y + d'zy) = 0; +\] +and it would be seen that the factor $bc - ad$ must in every +case either vanish or express a limitation of meaning. + +The determination of the order of the Propositions is sufficiently +obvious. +\PageSep{48} + + +\Chapter{Of Hypotheticals.} + +\begin{Abstract} +A hypothetical Proposition is defined to be \emph{two or more categoricals united by +a copula} (or conjunction), and the different kinds of hypothetical Propositions +are named from their respective conjunctions, viz.\ conditional (if), disjunctive +(either, or),~\etc. + +In conditionals, that categorical Proposition from which the other results +is called the \emph{antecedent}, that which results from it the \emph{consequent}. + +Of the conditional syllogism there are two, and only two formulć. + +1st. The constructive, +\begin{gather*} +\text{If $A$~is~$B$, then $C$~is~$D$,} \\ +\text{But $A$~is~$B$, therefore $C$~is~$D$.} +\end{gather*} + +2nd. The Destructive, +\begin{gather*} +\text{If $A$~is~$B$, then $C$~is~$D$,} \\ +\text{But $C$~is not~$D$, therefore $A$~is not~$B$.} +\end{gather*} + +A dilemma is a complex conditional syllogism, with several antecedents +in the major, and a disjunctive minor. +\end{Abstract} + +\First{If} we examine either of the forms of conditional syllogism +above given, we shall see that the validity of the argument +does not depend upon any considerations which have reference +to the terms $A$,~$B$,~$C$,~$D$, considered as the representatives +of individuals or of classes. We may, in fact, represent the +Propositions $A$~is~$B$, $C$~is~$D$, by the arbitrary symbols $X$~and~$Y$ +respectively, and express our syllogisms in such forms as the +following: +\begin{gather*} +\text{If $X$ is true, then $Y$ is true,} \\ +\text{But $X$ is true, therefore $Y$ is true.} +\end{gather*} + +Thus, what we have to consider is not objects and classes +of objects, but the truths of Propositions, namely, of those +\PageSep{49} +elementary Propositions which are embodied in the terms of +our hypothetical premises. + +To the symbols $X$,~$Y$,~$Z$, representative of Propositions, we +may appropriate the elective symbols $x$,~$y$,~$z$, in the following +sense. + +The hypothetical Universe,~$1$, shall comprehend all conceivable +cases and conjunctures of circumstances. + +The elective symbol~$x$ attached to any subject expressive of +such cases shall select those cases in which the Proposition~$X$ +is true, and similarly for $Y$~and~$Z$. + +If we confine ourselves to the contemplation of a given proposition~$X$, +and hold in abeyance every other consideration, +then two cases only are conceivable, viz.\ first that the given +Proposition is true, and secondly that it is false.\footnote + {It was upon the obvious principle that a Proposition is either true or false, + that the Stoics, applying it to assertions respecting future events, endeavoured + to establish the doctrine of Fate. It has been replied to their argument, that it +%[** TN: Italicized entire Latin phrase; only "est" italicized in original] + involves ``an abuse of the word \emph{true}, the precise meaning of which is \textit{id quod + res est}. An assertion respecting the future is neither true nor false.''---\textit{Copleston + on Necessity and Predestination}, p.~36. Were the Stoic axiom, however, presented + under the form, It is either certain that a given event will take place, + or certain that it will not; the above reply would fail to meet the difficulty. + The proper answer would be, that no merely verbal definition can settle the + question, what is the actual course and constitution of Nature. When we + affirm that it is either certain that an event will take place, or certain that + it will not take place, we tacitly assume that the order of events is necessary, + that the Future is but an evolution of the Present; so that the state of things + which is, completely determines that which shall be. But this (at least as respects + the conduct of moral agents) is the very question at issue. Exhibited + under its proper form, the Stoic reasoning does not involve an abuse of terms, + but a \textit{petitio principii}. + + It should be added, that enlightened advocates of the doctrine of Necessity + in the present day, viewing the end as appointed only in and through the + means, justly repudiate those practical ill consequences which are the reproach + of Fatalism.} +As these +cases together make up the Universe of the Proposition, and +as the former is determined by the elective symbol~$x$, the latter +is determined by the symbol~$1 - x$. + +But if other considerations are admitted, each of these cases +will be resolvable into others, individually less extensive, the +\PageSep{50} +number of which will depend upon the number of foreign considerations +admitted. Thus if we associate the Propositions $X$ +and~$Y$, the total number of conceivable cases will be found as +exhibited in the following scheme. +\[ +\begin{array}[b]{*{2}{l@{\ }}>{\qquad}c@{}} +\multicolumn{2}{c}{\ColHead{Cases.}} & +\multicolumn{1}{>{\qquad}c}{\ColHead{Elective expressions.}} \\ +\text{1st}& \text{$X$ true, $Y$ true\Add{,}} & xy\Add{,} \\ +\text{2nd}& \text{$X$ true, $Y$ false\Add{,}}& x(1 - y)\Add{,} \\ +\text{3rd}& \text{$X$ false, $Y$ true\Add{,}} & (1 - x)y\Add{,} \\ +\text{4th}& \text{$X$ false, $Y$ false\Add{,}}& (1 - x)(1 - y)\Add{.} +\end{array} +\Tag{(24)} +\] + +If we add the elective expressions for the two first of the +above cases the sum is~$x$, which is the elective symbol appropriate +to the more general case of $X$~being true independently +of any consideration of~$Y$; and if we add the elective expressions +in the two last cases together, the result is~$1 - x$, which +is the elective expression appropriate to the more general case +of $X$~being false. + +Thus the extent of the hypothetical Universe does not at +all depend upon the number of circumstances which are taken +into account. And it is to be noted that however few or many +those circumstances may be, the sum of the elective expressions +representing every conceivable case will be unity. Thus let +us consider the three Propositions, $X$,~It rains, $Y$,~It hails, +$Z$,~It freezes. The possible cases are the following: +\[ +\begin{array}{*{2}{l@{\ }}l@{}} +&\multicolumn{1}{c}{\ColHead{Cases.}} & +\multicolumn{1}{c}{\ColHead{Elective expressions.}} \\ +\text{1st}& \text{It rains, hails, and freezes,} & xyz\Add{,} \\ +\text{2nd}& \text{It rains and hails, but does not freeze\Add{,}}& xy(1 - z)\Add{,} \\ +\text{3rd}& \text{It rains and freezes, but does not hail\Add{,}}& xz(1 - y)\Add{,} \\ +\text{4th}& \text{It freezes and hails, but does not rain\Add{,}}& yz(1 - x)\Add{,} \\ +\text{5th}& \text{It rains, but neither hails nor freezes\Add{,}}& x(1 - y)(1 - z)\Add{,} \\ +\text{6th}& \text{It hails, but neither rains nor freezes\Add{,}}& y(1 - x)(1 - z)\Add{,} \\ +\text{7th}& \text{It freezes, but neither hails nor rains\Add{,}}& z(1 - x)(1 - y)\Add{,} \\ +\text{8th}& \text{It neither rains, hails, nor freezes\Add{,}}& (1 - x)(1 - y)(1 - z)\Add{,} \\ +\cline{3-3} +&&\multicolumn{1}{c}{1 = \text{sum\Add{.}}} +\end{array} +\] +\PageSep{51} + + +\Section{Expression of Hypothetical Propositions.} + +To express that a given Proposition~$X$ is true. + +The symbol $1 - x$ selects those cases in which the Proposition~$X$ +is false. But if the Proposition is true, there are no +such cases in its hypothetical Universe, therefore +\begin{align*} +1 - x &= 0, \\ +\intertext{or} +x &= 1. +\Tag{(25)} +\end{align*} + +To express that a given Proposition~$X$ is false. + +The elective symbol~$x$ selects all those cases in which the +Proposition is true, and therefore if the Proposition is false, +\[ +x = 0. +\Tag{(26)} +\] + +And in every case, having determined the elective expression +appropriate to a given Proposition, we assert the truth of that +Proposition by equating the elective expression to unity, and +its falsehood by equating the same expression to~$0$. + +To express that two Propositions, $X$~and~$Y$, are simultaneously +true. + +The elective symbol appropriate to this case is~$xy$, therefore +the equation sought is +\[ +xy = 1. +\Tag{(27)} +\] + +To express that two Propositions, $X$~and~$Y$, are simultaneously +false. + +The condition will obviously be +\begin{align*} +(1 - x)(1 - y) &= 1, \\ +\intertext{or} +x + y - xy &= 0. +\Tag{(28)} +\end{align*} + +To express that either the Proposition~$X$ is true, or the +Proposition~$Y$ is true. + +To assert that either one or the other of two Propositions +is true, is to assert that it is not true, that they are both false. +Now the elective expression appropriate to their both being +false is~$(1 - x)(1 - y)$, therefore the equation required is +\begin{align*} +(1 - x)(1 - y) &= 0, \\ +\intertext{or} +x + y - xy &= 1. +\Tag{(29)} +\end{align*} +\PageSep{52} + +And, by indirect considerations of this kind, may every disjunctive +Proposition, however numerous its members, be expressed. +But the following general Rule will usually be +preferable. + +\begin{Rule} +Consider what are those distinct and mutually exclusive +cases of which it is implied in the statement of the given Proposition, +that some one of them is true, and equate the sum of their +elective expressions to unity. This will give the equation of the +given Proposition. +\end{Rule} + +For the sum of the elective expressions for all distinct conceivable +cases will be unity. Now all these cases being mutually +exclusive, and it being asserted in the given Proposition that +some one case out of a given set of them is true, it follows that +all which are not included in that set are false, and that their +elective expressions are severally equal to~$0$. Hence the sum +of the elective expressions for the remaining cases, viz.\ those +included in the given set, will be unity. Some one of those +cases will therefore be true, and as they are mutually exclusive, +it is impossible that more than one should be true. Whence +the Rule in question. + +And in the application of this Rule it is to be observed, that +if the cases contemplated in the given disjunctive Proposition +are not mutually exclusive, they must be resolved into an equivalent +series of cases which are mutually exclusive. + +Thus, if we take the Proposition of the preceding example, +viz.\ Either $X$~is true, or $Y$~is true, and assume that the two +members of this Proposition are not exclusive, insomuch that +in the enumeration of possible cases, we must reckon that of +the Propositions $X$~and~$Y$ being both true, then the mutually +exclusive cases which fill up the Universe of the Proposition, +with their elective expressions, are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~true and $Y$~false,}& x(1 - y), \\ +\text{2nd,}& \text{$Y$~true and $X$~false,}& y(1 - x), \\ +\text{3rd,}& \text{$X$~true and $Y$~true,} & xy, +\end{array} +\] +\PageSep{53} +and the sum of these elective expressions equated to unity gives +\[ +x + y - xy = 1\Typo{.}{,} +\Tag{(30)} +\] +as before. But if we suppose the members of the disjunctive +Proposition to be exclusive, then the only cases to be considered +are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~true, $Y$~false,}& x(1 - y), \\ +\text{2nd,}& \text{$Y$~true, $X$~false,}& y(1 - x), +\end{array} +\] +and the sum of these elective expressions equated to~$0$, gives +\[ +x - 2xy + y = 1. +\Tag{(31)} +\] + +The subjoined examples will further illustrate this method. + +To express the Proposition, Either $X$~is not true, or $Y$~is not +true, the members being exclusive. + +The mutually exclusive cases are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~not true, $Y$~true,}& y(1 - x), \\ +\text{2nd,}& \text{$Y$~not true, $X$~true,}& x(1 - y), +\end{array} +\] +and the sum of these equated to unity gives +\[ +x - 2xy + y = 1, +\Tag{(32)} +\] +which is the same as~\Eqref{(31)}, and in fact the Propositions which +they represent are equivalent. + +To express the Proposition, Either $X$~is not true, or $Y$~is not +true, the members not being exclusive. + +To the cases contemplated in the last Example, we must add +the following, viz. +\[ +\text{$X$~not true, $Y$~not true,}\qquad (1 - x)(1 - y). +\] + +The sum of the elective expressions gives +\begin{gather*} +x(1 - y) + y(1 - x) + (1 - x)(1 - y) = 1, \\ +\intertext{or} +xy = 0. +\Tag{(33)} +\end{gather*} + +To express the disjunctive Proposition, Either $X$~is true, or +$Y$~is true, or $Z$~is true, the members being exclusive. +\PageSep{54} + +Here the mutually exclusive cases are +\[ +\begin{array}{l@{\ }l<{\qquad}c@{}} +\text{1st,}& \text{$X$~true, $Y$~false, $Z$~false,}& x(1 - y)(1 - z), \\ +\text{2nd,}& \text{$Y$~true, $Z$~false, $X$~false,}& y(1 - z)(1 - x), \\ +\text{3rd,}& \text{$Z$~true, $X$~false, $Y$~false,}& z(1 - x)(1 - y), +\end{array} +\] +and the sum of the elective expressions equated to~$1$, gives, +upon reduction, +\[ +x + y + z - 2(xy + yz + zx) + 3xyz = 1. +\Tag{(34)} +\] + +The expression of the same Proposition, when the members +are in no sense exclusive, will be +\[ +(1 - x)(1 - y)(1 - z) = 0. +\Tag{(35)} +\] + +And it is easy to see that our method will apply to the +expression of any similar Proposition, whose members are +subject to any specified amount and character of exclusion. + +To express the conditional Proposition, If $X$~is true, $Y$~is +true. + +Here it is implied that all the cases of $X$~being true, are +cases of $Y$~being true. The former cases being determined +by the elective symbol~$x$, and the latter by~$y$, we have, in +virtue of~\Eqref{(4)}, +\[ +x(1 - y) = 0. +\Tag{(36)} +\] + +To express the conditional Proposition, If $X$~be true, $Y$~is +not true. + +The equation is obviously +\[ +xy = 0; +\Tag{(37)} +\] +this is equivalent to~\Eqref{(33)}, and in fact the disjunctive Proposition, +Either $X$~is not true, or $Y$~is not true, and the conditional +Proposition, If $X$~is true, $Y$~is not true, are equivalent. + +To express that If $X$~is not true, $Y$~is not true. + +In~\Eqref{(36)} write $1 - x$ for~$x$, and $1 - y$ for~$y$, we have +\[ +(1 - x)y = 0. +\] +\PageSep{55} + +The results which we have obtained admit of verification +in many different ways. Let it suffice to take for more particular +examination the equation +\[ +x - 2xy + y = 1, +\Tag{(38)} +\] +which expresses the conditional Proposition, Either $X$~is true, +or $Y$~is true, the members being in this case exclusive. + +First, let the Proposition~$X$ be true, then $x = 1$, and substituting, +we have +\[ +1 - 2y + y = 1,\qquad +\therefore -y = 0,\quad\text{or}\quad y = 0, +\] +which implies that $Y$~is not true. + +Secondly, let $X$~be not true, then $x = 0$, and the equation +gives +\[ +y = 1, +\Tag{(39)} +\] +which implies that $Y$~is true. In like manner we may proceed +with the assumptions that $Y$~is true, or that $Y$~is false. + +Again, in virtue of the property $x^{2} = x$, $y^{2} = y$, we may write +the equation in the form +\[ +x^{2} - 2xy + y^{2} = 1, +\] +and extracting the square root, we have +\[ +x - y = ±1, +\Tag{(40)} +\] +and this represents the actual case; for, as when $X$~is true +or false, $Y$~is respectively false or true, we have +\begin{gather*} +x = 1\quad\text{or}\quad 0, \\ +y = 0\quad\text{or}\quad 1, \\ +\therefore x - y = 1\quad\text{or}\quad -1. +\end{gather*} + +There will be no difficulty in the analysis of other cases. + + +\Section{Examples of Hypothetical Syllogism.} + +The treatment of every form of hypothetical Syllogism will +consist in forming the equations of the premises, and eliminating +the symbol or symbols which are found in more than one of +them. The result will express the conclusion. +\PageSep{56} + +1st. Disjunctive Syllogism. +\begin{align*} +&\begin{array}{l<{\qquad}@{}c@{}} +\text{Either $X$~is true, or $Y$~is true (exclusive),} & +x + y - 2xy = 1\Add{,} \\ +\text{But $X$~is true,} & x = 1\Add{,} \\ +\cline{2-2} +\text{Therefore $Y$~is not true,} & \therefore y = 0\Add{.} +\end{array} \\ +&\begin{array}{l<{\quad}@{}c@{}} +\text{Either $X$~is true, or $Y$~is true (not exclusive),}& +x + y - xy = 1\Add{,} \\ +\text{But $X$~is not true,}& x = 0\Add{,} \\ +\cline{2-2} +\text{Therefore $Y$~is true,}& \therefore y = 1\Add{.} +\end{array} +\end{align*} + +2nd. Constructive Conditional Syllogism. +\[ +\begin{array}{l<{\qquad}@{}c@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{But $X$~is true,}& x = 1\Add{,} \\ +\text{Therefore $Y$~is true,}& \therefore 1 - y = 0\quad\text{or}\quad y = 1. +\end{array} +\] + +3rd. Destructive Conditional Syllogism. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{But $Y$~is not true,}& y = 0\Add{,} \\ +\text{Therefore $X$~is not true,}& \therefore x = 0\Add{.} +\end{array} +\] + +4th. Simple Constructive Dilemma, the minor premiss exclusive. +\begin{alignat*}{2} +&\text{If $X$~is true, $Y$~is true,}& x(1 - y) &= 0, +\Tag{(41)} \\ +&\text{If $Z$~is true, $Y$~is true,}& z(1 - y) &= 0, +\Tag{(42)} \\ +&\text{But Either $X$~is true, or $Z$~is true,}\quad& +x + z - 2xz &= 1. +\Tag{(43)} +\end{alignat*} + +From the equations \Eqref{(41)},~\Eqref{(42)},~\Eqref{(43)}, we have to eliminate +$x$~and~$z$. In whatever way we effect this, the result is +\[ +y = 1; +\] +whence it appears that the Proposition~$Y$ is true. + +5th. Complex Constructive Dilemma, the minor premiss not +exclusive. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0, \\ +\text{If $W$~is true, $Z$~is true,}& w(1 - z) = 0, \\ +\text{Either $X$~is true, or $W$~is true,}& x + w - xw = 1. +\end{array} +\] + +From these equations, eliminating~$x$, we have +\[ +y + z - yz = 1, +\] +\PageSep{57} +which expresses the Conclusion, Either $Y$~is true, or $Z$~is true, +the members being \Chg{non-exclusive}{nonexclusive}. + +6th. Complex Destructive Dilemma, the minor premiss exclusive. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{If $W$~is true, $Z$~is true,}& w(1 - z) = 0\Add{,} \\ +\text{Either $Y$~is not true, or $Z$~is not true,}& y + z - 2yz = 1. +\end{array} +\] + +From these equations we must eliminate $y$~and~$z$. The +result is +\[ +xw = 0, +\] +which expresses the Conclusion, Either $X$~is not true, or $Y$~is +not true, the members \emph{not being exclusive}. + +7th. Complex Destructive Dilemma, the minor premiss not +exclusive. +\[ +\begin{array}{l<{\qquad}@{}r@{}} +\text{If $X$~is true, $Y$~is true,}& x(1 - y) = 0\Add{,} \\ +\text{If $W$~is true, $Z$~is true,}& w(1 - z) = 0\Add{,} \\ +\text{Either $Y$~is not true, or $Z$~is not true,}& yz = 0. +\end{array} +\] + +On elimination of $y$~and~$z$, we have +\[ +xw = 0, +\] +which indicates the same Conclusion as the previous example. + +It appears from these and similar cases, that whether the +members of the minor premiss of a Dilemma are exclusive +or not, the members of the (disjunctive) Conclusion are never +exclusive. This fact has perhaps escaped the notice of logicians. + +The above are the principal forms of hypothetical Syllogism +which logicians have recognised. It would be easy, however, +to extend the list, especially by the blending of the disjunctive +and the conditional character in the same Proposition, of which +the following is an example. +\[ +\begin{array}{l<{\qquad}@{}c@{}} +\multicolumn{2}{l}{% + \text{If $X$~is true, then either $Y$~is true, or $Z$~is true,}} \\ + & x(1 - y - z + yz) = 0\Add{,} \\ +\text{But $Y$~is not true,}& y = 0\Add{,} \\ +\text{Therefore If $X$~is true, $Z$~is true,}& \therefore x(1 - z) = 0. +\end{array} +\] +\PageSep{58} + +That which logicians term a \emph{Causal} Proposition is properly +a conditional Syllogism, the major premiss of which is suppressed. + +The assertion that the Proposition~$X$ is true, \emph{because} the +Proposition~$Y$ is true, is equivalent to the assertion, +\begin{align*} +&\text{The Proposition~$Y$ is true,} \\ +&\text{\emph{Therefore} the Proposition X is true;} +\end{align*} +and these are the minor premiss and conclusion of the conditional +Syllogism, +\begin{align*} +&\text{If $Y$~is true, $X$~is true,} \\ +&\text{But $Y$~is true,} \\ +&\text{Therefore $X$~is true.} +\end{align*} +And thus causal Propositions are seen to be included in the +applications of our general method. + +Note, that there is a family of disjunctive and conditional +Propositions, which do not, of right, belong to the class considered +in this Chapter. Such are those in which the force +of the disjunctive or conditional particle is expended upon the +predicate of the Proposition, as if, speaking of the inhabitants +of a particular island, we should say, that they are all \emph{either +Europeans or Asiatics}; meaning, that it is true of each individual, +that he is either a European or an Asiatic. If we +appropriate the elective symbol~$x$ to the inhabitants, $y$~to +Europeans, and $z$~to Asiatics, then the equation of the above +Proposition is +\[ +x = xy + xz,\quad\text{or}\quad x(1 - y - z) = 0;\atag +\] +to which we might add the condition $yz = 0$, since no Europeans +are Asiatics. The nature of the symbols $x$,~$y$,~$z$, indicates that +the Proposition belongs to those which we have before designated +as \emph{Categorical}. Very different from the above is the +Proposition, Either all the inhabitants are Europeans, or they +are all Asiatics. Here the disjunctive particle separates Propositions. +The case is that contemplated in~\Eqref{(31)} of the present +Chapter; and the symbols by which it is expressed, +\PageSep{59} +although subject to the same laws as those of~\aref, have a totally +different interpretation.\footnote + {Some writers, among whom is Dr.\ Latham (\textit{First Outlines}), regard it as + the exclusive office of a conjunction to connect \emph{Propositions}, not \emph{words}. In this + view I am not able to agree. The Proposition, Every animal is \emph{either} rational + \emph{or} irrational, cannot be resolved into, \emph{Either} every animal is rational, \emph{or} every + animal is irrational. The former belongs to pure categoricals, the latter to + hypotheticals. In \emph{singular} Propositions, such conversions would seem to be + allowable. This animal is \emph{either} rational \emph{or} irrational, is equivalent to, \emph{Either} + this animal is rational, \emph{or} it is irrational. This peculiarity of \emph{singular} Propositions + would almost justify our ranking them, though truly universals, in + a separate class, as Ramus and his followers did.} + +The distinction is real and important. Every Proposition +which language can express may be represented by elective +symbols, and the laws of combination of those symbols are in +all cases the same; but in one class of instances the symbols +have reference to collections of objects, in the other, to the +truths of constituent Propositions. +\PageSep{60} + + +\Chapter{Properties of Elective Functions.} + +\First{Since} elective symbols combine according to the laws of +quantity, we may, by Maclaurin's theorem, expand a given +function~$\phi(x)$, in ascending powers of~$x$, known cases of failure +excepted. Thus we have +\[ +\phi(x) = \phi(0) + \phi'(0)x + \frac{\phi''(0)}{1·2}x^{2} + \etc. +\Tag{(44)} +\] + +Now $x^{2} = x$, $x^{3} = x$,~\etc., whence +\[ +\phi(x) = \phi(0) + x\bigl\{\phi'(0) + \frac{\phi''(0)}{1·2} + \etc.\bigr\}. +\Tag{(45)} +\] + +Now if in~\Eqref{(44)} we make $x = 1$, we have +\[ +\phi(1) = \phi(0) + \phi'(0) + \frac{\phi''(0)}{1·2} + \etc., +\] +whence +\[ +\phi'(0) + \frac{\phi''(0)}{1·2} + \frac{\phi'''(0)}{1·2·3} + \etc. + = \phi(1) - \phi(0). +\] + +Substitute this value for the coefficient of~$x$ in the second +member of~\Eqref{(45)}, and we have\footnote + {Although this and the following theorems have only been proved for those + forms of functions which are expansible by Maclaurin's theorem, they may be + regarded as true for all forms whatever; this will appear from the applications. + The reason seems to be that, as it is only through the one form of expansion + that elective functions become interpretable, no conflicting interpretation is + possible. + + The development of~$\phi(x)$ may also be determined thus. By the known formula + for expansion in factorials, + \[ + \phi(x) = \phi(0) + \Delta\phi(0)x + + \frac{\Delta^{2}\phi(0)}{1·2}x(x - 1) + \etc. + \] +%[** TN: Footnote continues] + Now $x$~being an elective symbol, $x(x - 1) = 0$, so that all the terms after the + second, vanish. Also $\Delta\phi(0) = \phi(1) - \phi(0)$, whence + \[ + \phi\bigl\{x = \phi(0)\bigr\} + \bigl\{\phi(1) - \phi(0)\bigr\}x. + \] + + The mathematician may be interested in the remark, that this is not the + only case in which an expansion stops at the second term. The expansions of + the compound operative functions $\phi\left(\dfrac{d}{dx} + x^{-1}\right)$ and $\phi\left\{x + \left(\dfrac{d}{dx}\right)^{-1}\right\}$ are, + respectively, + \[ + \phi\left(\frac{d}{dx}\right) + \phi'\left(\frac{d}{dx}\right)x^{-1}, + \] + and + \[ + \phi(x) + \phi'(x)\left(\frac{d}{dx}\right)^{-1}. + \] + + See \textit{Cambridge Mathematical Journal}, Vol.~\textsc{iv}. p.~219.} +\[ +\phi(x) = \phi(0) + \bigl\{\phi(1) - \phi(0)\bigr\}x, +\Tag{(46)} +\] +\PageSep{61} +which we shall also employ under the form +\[ +\phi(x) = \phi(1)x + \phi(0)(1 - x). +\Tag{(47)} +\] + +Every function of~$x$, in which integer powers of that symbol +are alone involved, is by this theorem reducible to the first +order. The quantities $\phi(0)$,~$\phi(1)$, we shall call the moduli +of the function~$\phi(x)$. They are of great importance in the +theory of elective functions, as will appear from the succeeding +Propositions. + +\Prop{1.} Any two functions $\phi(x)$,~$\psi(x)$, are equivalent, +whose corresponding moduli are equal. + +This is a plain consequence of the last Proposition. For since +\begin{align*} +\phi(x) &= \phi(0) + \bigl\{\phi(1) - \phi(0)\bigr\}x, \\ +\psi(x) &= \psi(0) + \bigl\{\psi(1) - \psi(0)\bigr\}x, +\end{align*} +it is evident that if $\phi(0) = \psi(0)$, $\phi(1) = \psi(1)$, the two +expansions will be equivalent, and therefore the functions which +they represent will be equivalent also. + +The converse of this Proposition is equally true, viz. + +If two functions are equivalent, their corresponding moduli +are equal. + +Among the most important applications of the above theorem, +we may notice the following. + +Suppose it required to determine for what forms of the +function~$\phi(x)$, the following equation is satisfied, viz. +\[ +\bigl\{\phi(x)\bigr\}^{n} = \phi(x). +\] +\PageSep{62} +Here we at once obtain for the expression of the conditions +in question, +\[ +\bigl\{\phi(0)\bigr\}^{n} = \phi(0)\Typo{.}{,}\quad +\bigl\{\phi(1)\bigr\}^{n} = \phi(1). +\Tag{(48)} +\] + +Again, suppose it required to determine the conditions under +which the following equation is satisfied, viz. +\[ +\phi(x)\psi(x) = \chi(x)\Typo{,}{.} +\] + +The general theorem at once gives +\[ +\phi(0)\psi(0) = \chi(0)\Typo{.}{,}\quad +\phi(1)\psi(1) = \chi(1). +\Tag{(49)} +\] + +This result may also be proved by substituting for~$\phi(x)$, +$\psi(x)$, $\chi(x)$, their expanded forms, and equating the coefficients +of the resulting equation properly reduced. + +All the above theorems may be extended to functions of more +than one symbol. For, as different elective symbols combine +with each other according to the same laws as symbols of quantity, +we can first expand a given function with reference to any +particular symbol which it contains, and then expand the result +with reference to any other symbol, and so on in succession, the +order of the expansions being quite indifferent. + +Thus the given function being~$\phi(xy)$ we have +\[ +\phi(xy) = \phi(x0) + \bigl\{\phi(x1) - \phi(x0)\bigr\}y, +\] +and expanding the coefficients with reference to~$x$, and reducing +\begin{align*} +\phi(xy) = \phi(00) + &+ \bigl\{\phi(10) - \phi(00)\bigr\}x + + \bigl\{\phi(01) - \phi(00)\bigr\}y \\ + &+ \bigl\{\phi(11) - \phi(10) - \phi(01) + \phi(00)\bigr\}xy, +\Tag{(50)} +\end{align*} +to which we may give the elegant symmetrical form +\begin{align*} +%[** TN: Not aligned in the original] +\phi(xy) = \phi(00)(1 - x)(1 - y) &+ \phi(01)y(1 - x) \\ + &+ \phi(10)x(1 - y) + \phi(11)xy, +\Tag{(51)} +\end{align*} +wherein we shall, in accordance with the language already +employed, designate $\phi(00)$, $\phi(01)$, $\phi(10)$, $\phi(11)$, as the +moduli of the function~$\phi(xy)$. + +By inspection of the above general form, it will appear that +any functions of two variables are equivalent, whose corresponding +moduli are all equal. +\PageSep{63} + +Thus the conditions upon which depends the satisfaction of +the equation, +\[ +\bigl\{\phi(xy)\bigr\}^{n} = \phi(xy) +\] +are seen to be +\[ +\begin{alignedat}{2} +\bigl\{\phi(00)\bigr\}^{n} &= \phi(00),\qquad& +\bigl\{\phi(01)\bigr\}^{n} &= \phi(01), \\ +\bigl\{\phi(10)\bigr\}^{n} &= \phi(10), & +\bigl\{\phi(11)\bigr\}^{n} &= \phi(11). +\end{alignedat} +\Tag{(52)} +\] + +And the conditions upon which depends the satisfaction of +the equation +\[ +\phi(xy)\psi(xy) = \chi(xy), +\] +are +\[ +\begin{alignedat}{2} +\phi(00)\psi(00) &= \chi(00),\qquad& +\phi(01)\psi(01) &= \chi(01), \\ +\phi(10)\psi(10) &= \chi(10),\qquad& +\phi(11)\psi(11) &= \chi(11). +\end{alignedat} +\Tag{(53)} +\] + +It is very easy to assign by induction from \Eqref{(47)}~and~\Eqref{(51)}, the +general form of an expanded elective function. It is evident +that if the number of elective symbols is~$m$, the number of the +moduli will be~$2^{m}$, and that their separate values will be obtained +by interchanging in every possible way the values $1$~and~$0$ in the +places of the elective symbols of the given function. The several +terms of the expansion of which the moduli serve as coefficients, +will then be formed by writing for each~$1$ that recurs under the +functional sign, the elective symbol~$x$,~\etc., which it represents, +and for each~$0$ the corresponding~$1 - x$,~\etc., and regarding these +as factors, the product of which, multiplied by the modulus from +which they are obtained, constitutes a term of the expansion. + +Thus, if we represent the moduli of any elective function +$\phi(xy\dots)$ by $a_{1}$,~$a_{2}$, $\dots,~a_{r}$, the function itself, when expanded +and arranged with reference to the moduli, will assume the form +\[ +\phi(xy) = a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}, +\Tag{(54)} +\] +in which $t_{1}t_{2}\dots t_{r}$ are functions of $x$,~$y$,~$\dots$, resolved into factors +of the forms $x$,~$y$,~$\dots$ $1 - x$, $1 - y$,~$\dots$~\etc. These functions satisfy +individually the index relations +\[ +t_{1}^{n} = t_{1},\quad +t_{2}^{n} = t_{2},\quad \etc., +\Tag{(55)} +\] +and the further relations, +\[ +t_{1}t_{2} = 0\dots t_{1}t_{2} = 0,~\etc., +\Tag{(56)} +\] +\PageSep{64} +the product of any two of them vanishing. This will at once +be inferred from inspection of the particular forms \Eqref{(47)}~and~\Eqref{(51)}. +Thus in the latter we have for the values of $t_{1}$,~$t_{2}$,~\etc., the forms +\[ +xy,\quad +x(1 - y),\quad +(1 - x)y,\quad +(1 - x)(1 - y); +\] +and it is evident that these satisfy the index relation, and that +their products all vanish. We shall designate $t_{1}t_{2}\dots$ as the constituent +functions of~$\phi(xy)$, and we shall define the peculiarity +of the vanishing of the binary products, by saying that those +functions are \emph{exclusive}. And indeed the classes which they +represent are mutually exclusive. + +The sum of all the constituents of an expanded function is +unity. An elegant proof of this Proposition will be obtained +by expanding~$1$ as a function of any proposed elective symbols. +Thus if in~\Eqref{(51)} we assume $\phi(xy) = 1$, we have $\phi(11) = 1$, +$\phi(10) = 1$, $\phi(01) = 1$, $\phi(00) = 1$, and \Eqref{(51)}~gives +\[ +1 = xy + x(1 - y) + (1 - x)y + (1 - x)(1 - y). +\Tag{(57)} +\] + +It is obvious indeed, that however numerous the symbols +involved, all the moduli of unity are unity, whence the sum +of the constituents is unity. + +We are now prepared to enter upon the question of the +general interpretation of elective equations. For this purpose +we shall find the following Propositions of the greatest service. + +\Prop{2.} If the first member of the general equation +$\phi(xy\dots) = 0$, be expanded in a series of terms, each of which +is of the form~$at$, $a$~being a modulus of the given function, then +for every numerical modulus~$a$ which does not vanish, we shall +have the equation +\[ +at = 0, +\] +and the combined interpretations of these several equations will +express the full significance of the original equation. + +For, representing the equation under the form +\[ +a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r} = 0. +\Tag{(58)} +\] + +Multiplying by~$t_{1}$ we have, by~\Eqref{(56)}, +\[ +a_{1}t_{1} = 0, +\Tag{(59)} +\] +\PageSep{65} +whence if $a_{1}$~is a numerical constant which does not vanish, +\[ +t_{1} = 0, +\] +and similarly for all the moduli which do not vanish. And +inasmuch as from these constituent equations we can form the +given equation, their interpretations will together express its +entire significance. + +Thus if the given equation were +\[ +x - y = 0,\quad \text{$X$s~and~$Y$s are identical,} +\Tag{(60)} +\] +we should have $\phi(11) = 0$, $\phi(10) = 1$, $\phi(01) = -1$, $\phi(00) = 0$, +so that the expansion~\Eqref{(51)} would assume the form +\[ +x(1 - y) - y(1 - x) = 0, +\] +whence, by the above theorem, +\begin{alignat*}{2} +x(1 - y) &= 0,\qquad& \text{All~$X$s are~$Y$s,} \\ +y(1 - x) &= 0, & \text{All~$Y$s are~$X$s,} +\end{alignat*} +results which are together equivalent to~\Eqref{(60)}. + +It may happen that the simultaneous satisfaction of equations +thus deduced, may require that one or more of the elective +symbols should vanish. This would only imply the nonexistence +of a class: it may even happen that it may lead to a final +result of the form +\[ +1 = 0, +\] +which would indicate the nonexistence of the logical Universe. +Such cases will only arise when we attempt to unite contradictory +Propositions in a single equation. The manner in which +the difficulty seems to be evaded in the result is characteristic. + +It appears from this Proposition, that the differences in the +interpretation of elective functions depend solely upon the +number and position of the vanishing moduli. No change in +the value of a modulus, but one which causes it to vanish, +produces any change in the interpretation of the equation in +which it is found. If among the infinite number of different +values which we are thus permitted to give to the moduli which +do not vanish in a proposed equation, any one value should be +\PageSep{66} +preferred, it is unity, for when the moduli of a function are all +either $0$~or~$1$, the function itself satisfies the condition +\[ +\bigl\{\phi(xy\dots)\bigr\}^{n} = \phi(xy\dots), +\] +and this at once introduces symmetry into our Calculus, and +provides us with fixed standards for reference. + +\Prop{3.} If $w = \phi(xy\dots)$, $w$,~$x$,~$y$,~$\dots$ being elective symbols, +and if the second member be completely expanded and arranged +in a series of terms of the form~$at$, we shall be permitted +to equate separately to~$0$ every term in which the modulus~$a$ +does not satisfy the condition +\[ +a^{n} = a, +\] +and to leave for the value of~$w$ the sum of the remaining terms. + +As the nature of the demonstration of this Proposition is +quite unaffected by the number of the terms in the second +member, we will for simplicity confine ourselves to the supposition +of there being four, and suppose that the moduli of the +two first only, satisfy the index law. + +We have then +\[ +w = a_{1}t_{1} + a_{2}t_{2} + a_{3}t_{3} + a_{4}t_{4}, +\Tag{(61)} +\] +with the relations +\[ +a_{1}^{n} = a_{1},\quad +a_{2}^{n} = a_{2}, +\] +in addition to the two sets of relations connecting $t_{1}$,~$t_{2}$, $t_{3}$,~$t_{4}$, +in accordance with \Eqref{(55)}~and~\Eqref{(56)}. + +Squaring~\Eqref{(61)}, we have +\[ +w = a_{1}t_{1} + a_{2}t_{2} + a_{3}^{2}t_{3} + a_{4}^{2}t_{4}, +\] +and subtracting~\Eqref{(61)} from this, +\[ +(a_{3}^{2} - a_{3})t_{3} + (a_{4}^{2} - a_{4})t_{4} = 0; +\] +and it being an hypothesis, that the coefficients of these terms +do not vanish, we have, by \PropRef{2}, +\[ +t_{3} = 0,\quad +t_{4} = 0, +\Tag{(62)} +\] +whence \Eqref{(61)}~becomes +\[ +w = a_{1}t_{1} + a_{2}t_{2}. +\] +The utility of this Proposition will hereafter appear. +\PageSep{67} + +\Prop{4.} The functions $t_{1}t_{2}\dots t_{r}$ being mutually exclusive, we +shall always have +\[ +\psi(a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}) + = \psi(a_{1})t_{1} + \psi(a_{2})t_{2} \dots + \psi(a_{r})t_{r}, +\Tag{(63)} +\] +whatever may be the values of $a_{1}a_{2}\dots a_{r}$ or the form of~$\psi$. + +%[** TN: Paragraph not indented in the original] +Let the function $a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}$ be represented by~$\phi(xy\dots)$, +then the moduli $a_{1}a_{2}\dots a_{r}$ will be given by the expressions +\[ +\phi(11\dots),\quad +\phi(10\dots),\quad +(\dots)\ \phi(00\dots). +\] + +Also +\begin{align*} +&\phantom{{}={}}\psi(a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}) + = \psi\bigl\{\phi(xy\dots)\bigr\} \\ + &= \psi\bigl\{\phi(11\dots)\bigr\}xy\dots + + \psi\bigl\{\phi(10\dots)\bigr\}x(1 - y)\dots \\ + &\qquad+ \psi\bigl\{\phi(00\dots)\bigr\}(1 - x)(1 - y)\dots \\ + &= \psi(a_{1})xy\dots + \psi(a_{2})x(1 - y)\dots + \psi(a_{r})(1 - x)(1 - y)\dots \\ + &= \psi(a_{1})t_{1} + \psi(a_{2})t_{2}\dots + \psi(a_{r})t_{r}. +\Tag{(64)} +\end{align*} + +It would not be difficult to extend the list of interesting +properties, of which the above are examples. But those which +we have noticed are sufficient for our present requirements. +The following Proposition may serve as an illustration of their +utility. + +\Prop{5.} Whatever process of reasoning we apply to a single +given Proposition, the result will either be the same Proposition +or a limitation of it. + +Let us represent the equation of the given Proposition under +its most general form, +\[ +a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r} = 0, +\Tag{(65)} +\] +resolvable into as many equations of the form $t = 0$ as there are +moduli which do not vanish. + +Now the most general transformation of this equation is +\[ +\psi(a_{1}t_{1} + a_{2}t_{2} \dots + a_{r}t_{r}) = \psi(0), +\Tag{(66)} +\] +provided that we attribute to~$\psi$ a perfectly arbitrary character, +allowing it even to involve new elective symbols, having \emph{any +proposed relation} to the original ones. +\PageSep{68} + +The development of~\Eqref{(66)} gives, by the last Proposition, +\[ +\psi(a_{1})t_{1} + \psi(a_{2})t_{2}\dots + \psi(a_{r})t_{r} = \psi(0). +\] +To reduce this to the general form of reference, it is only necessary +to observe that since +\[ +t_{1} + t_{2} \dots + t_{r} = 1, +\] +we may write for~$\psi(0)$, +\[ +\psi(0)(t_{1} + t_{2} \dots + t_{r}), +\] +whence, on substitution and transposition, +\[ +\bigl\{\psi(a_{1}) - \psi(0)\bigr\}t_{1} + +\bigl\{\psi(a_{2}) - \psi(0)\bigr\}t_{2} \dots + +\bigl\{\psi(a_{r}) - \psi(0)\bigr\}t_{r} = 0. +\] + +From which it appears, that if $a$~be any modulus of the +original equation, the corresponding modulus of the transformed +equation will be +\[ +\psi(a) - \psi(0). +\] + +If $a = 0$, then $\psi(a) - \psi(0) = \psi(0) - \psi(0) = 0$, whence +there are no \emph{new terms} in the transformed equation, and therefore +there are no \emph{new Propositions} given by equating its constituent +members to~$0$. + +Again, since $\psi(a) - \psi(0)$ may vanish without $a$~vanishing, +terms may be wanting in the transformed equation which existed +in the primitive. Thus some of the constituent truths of the +original Proposition may entirely disappear from the interpretation +of the final result. + +Lastly, if $\psi(a) - \psi(0)$ do not vanish, it must either be +a numerical constant, or it must involve new elective symbols. +In the former case, the term in which it is found will give +\[ +t = 0, +\] +which is one of the constituents of the original equation: in the +latter case we shall have +\[ +\bigl\{\psi(a\Typo{}{)} - \psi(0)\bigr\}t = 0, +\] +in which $t$~has a limiting factor. The interpretation of this +equation, therefore, is a limitation of the interpretation of~\Eqref{(65)}. +\PageSep{69} + +The purport of the last investigation will be more apparent +to the mathematician than to the logician. As from any mathematical +equation an infinite number of others may be deduced, +it seemed to be necessary to shew that when the original +equation expresses a logical Proposition, every member of the +derived series, even when obtained by expansion under a functional +sign, admits of exact and consistent interpretation. +\PageSep{70} + + +\Chapter{Of the Solution of Elective Equations.} + +\First{In} whatever way an elective symbol, considered as unknown, +may be involved in a proposed equation, it is possible to assign +its complete value in terms of the remaining elective symbols +considered as known. It is to be observed of such equations, +that from the very nature of elective symbols, they are necessarily +linear, and that their solutions have a very close analogy +with those of linear differential equations, arbitrary elective +symbols in the one, occupying the place of arbitrary constants +in the other. The method of solution we shall in the first place +illustrate by particular examples, and, afterwards, apply to the +investigation of general theorems. + +Given $(1 - x)y = 0$, (All~$Y$s are~$X$s), to determine~$y$ in +terms of~$x$. + +As $y$~is a function of~$x$, we may assume $y = vx + v'(1 - x)$, +(such being the expression of an arbitrary function of~$x$), the +moduli $v$~and~$v'$ remaining to be determined. We have then +\[ +(1 - x)\bigl\{vx + v'(1 - x)\bigr\} = 0, +\] +or, on actual multiplication, +\[ +v'(1 - x) = 0\Typo{:}{;} +\] +that this may be generally true, without imposing any restriction +upon~$x$, we must assume $v' = 0$, and there being no condition to +limit~$v$, we have +\[ +y = vx. +\Tag{(67)} +\] + +This is the complete solution of the equation. The condition +that $y$~is an elective symbol requires that $v$~should be an elective +\PageSep{71} +symbol also (since it must satisfy the index law), its interpretation +in other respects being arbitrary. + +Similarly the solution of the equation, $xy = 0$, is +\[ +y = v(1 - x). +\Tag{(68)} +\] + +Given $(1 - x)zy = 0$, (All~$Y$s which are~$Z$s are~$X$s), to determine~$y$. + +As $y$~is a function of $x$~and~$z$, we may assume +\[ +y = v(1 - x) (1 - z) + v'(1 - x)z + v''x(1 - z) + v'''zx. +\] +And substituting, we get +\[ +v'(1 - x)z = 0, +\] +whence $v' = 0$. The complete solution is therefore +\[ +y = v(1 - x)(1 - z) + v''x(1 - z) + v'''xz, +\Tag{(69)} +\] +$v'$,~$v''$,~$v'''$, being arbitrary elective symbols, and the rigorous +interpretation of this result is, that Every~$Y$ is \emph{either} a not-$X$ +and not-$Z$, or an~$X$ and not-$Z$, or an~$X$ and~$Z$. + +It is deserving of note that the above equation may, in consequence +of its linear form, be solved by adding the two +particular solutions with reference to $x$~and~$z$; and replacing +the arbitrary constants which each involves by an arbitrary +function of the other symbol, the result is +\[ +y = x\phi(z) + (1 - z)\psi(x). +\Tag{(70)} +\] + +To shew that this solution is equivalent to the other, it is +only necessary to substitute for the arbitrary functions $\phi(z)$, +$\psi(x)$, their equivalents +\[ +wz + w'(1 - z)\quad\text{and}\quad w''x + w'''(1 - x), +\] +we get +\[ +y = wxz + (w + w'')x(1 - z) + w'''(1 - x)(1 - z). +\] + +In consequence of the perfectly arbitrary character of $w'$~and~$w''$, +we may replace their sum by a single symbol~$w$, whence +\[ +y = wxz + w'x(1 - z) + w'''(1 - x)(1 - z), +\] +which agrees with~\Eqref{(69)}. +\PageSep{72} + +The solution of the equation $wx(1 - y)z = 0$, expressed by +arbitrary functions, is +\[ +z = (1 - w) \phi(xy) + (1 - x)\psi(wy) + y\chi(wx). +\Tag{(71)} +\] + +These instances may serve to shew the analogy which exists +between the solutions of elective equations and those of the +corresponding order of linear differential equations. Thus the +expression of the integral of a partial differential equation, +either by arbitrary functions or by a series with arbitrary coefficients, +is in strict analogy with the case presented in the two +last examples. To pursue this comparison further would minister +to curiosity rather than to utility. We shall prefer to contemplate +the problem of the solution of elective equations under +its most general aspect, which is the object of the succeeding +investigations. + +To solve the general equation $\phi(xy) = 0$, with reference to~$y$. + +If we expand the given equation with reference to $x$~and~$y$, +we have +\[ +%[** TN: Equation broken across two lines in the original +\phi(00)(1 - x)(1 - y) + \phi(01)(1 - x)y + \phi(10)x(1 - y) + + \phi(11)xy = 0, +\Tag{(72)} +\] +the coefficients $\phi(00)$~\etc.\ being numerical constants. + +Now the general expression of~$y$, as a function of~$x$, is +\[ +y = vx + v'(1 - x), +\] +$v$~and~$v'$ being unknown symbols to be determined. Substituting +this value in~\Eqref{(72)}, we obtain a result which may be written in +the following form, +\[ +%[** TN: Equation broken across two lines in the original +\bigl[\phi(10) + \bigl\{\phi(11) - \phi(10)\bigr\}v\bigr]x + + \bigl[\phi(00) + \bigl\{\phi(00) - \phi(00)\bigr\} v'\bigr](1 - x) = 0; +\] +and in order that this equation may be satisfied without any +way restricting the generality of~$x$, we must have +\begin{alignat*}{2} +\phi(10) &+ \bigl\{\phi(11) - \phi(10)\bigr\}v &&= 0, \\ +\phi(00) &+ \bigl\{\phi(01) - \phi(00)\bigr\}v' &&= 0, +\end{alignat*} +\PageSep{73} +from which we deduce +\[ +v = \frac{\phi(10)}{\phi(10) - \phi(11)}\;,\qquad +v' = \frac{\phi(00)}{\phi(01) - \phi(00)}\;, +\] +wherefore +\[ +y = \frac{\phi(10)}{\phi(10) - \phi(11)}\, x + + \frac{\phi(00)}{\phi(00) - \phi(01)}\, (1 - x). +\Tag{(73)} +\] + +Had we expanded the original equation with respect to $y$~only, +we should have had +\[ +\phi(x0) + \bigl\{\phi(x1) - \phi(x0)\bigr\}y = 0; +\] +but it might have startled those who are unaccustomed to the +processes of Symbolical Algebra, had we from this equation +deduced +\[ +y = \frac{\phi(x0)}{\phi(x0) - \phi(x1)}\;, +\] +because of the apparently meaningless character of the second +member. Such a result would however have been perfectly +lawful, and the expansion of the second member would have +given us the solution above obtained. I shall in the following +example employ this method, and shall only remark that those +to whom it may appear doubtful, may verify its conclusions by +the previous method. + +To solve the general equation $\phi(xyz) = 0$, or in other words +to determine the value of~$z$ as a function of $x$~and~$y$. + +Expanding the given equation with reference to~$z$, we have +\begin{gather*} +\phi(xy0) + \bigl\{\phi(xy1) - \phi(xy0)\bigr\}\Chg{·}{}z = 0; \\ +\therefore z = \frac{\phi(xy0)}{\phi(xy0) - \phi(xy1)}\;, +\Tag{(74)} +\end{gather*} +and expanding the second member as a function of $x$~and~$y$ by +aid of the general theorem, we have +\begin{multline*} +z = \frac{\phi(110)}{\phi(110) - \phi(111)}\, xy + + \frac{\phi(100)}{\phi(100) - \phi(101)}\, x(1 - y) \\ + + \frac{\phi(010)}{\phi(010) - \phi(011)}\, (1 - x)y + + \frac{\phi(000)}{\phi(000) - \phi(001)}\, (1 - x)(1 - y), +\Tag{(75)} +\end{multline*} +\PageSep{74} +and this is the complete solution required. By the same +method we may resolve an equation involving any proposed +number of elective symbols. + +In the interpretation of any general solution of this nature, +the following cases may present themselves. + +The values of the moduli $\phi(00)$, $\phi(01)$,~\etc.\ being constant, +one or more of the coefficients of the solution may assume +the form $\frac{0}{0}$~or~$\frac{1}{0}$. In the former case, the indefinite symbol~$\frac{0}{0}$ +must be replaced by an arbitrary elective symbol~$v$. In the +latter case, the term, which is multiplied by a factor~$\frac{1}{0}$ (or by +any numerical constant except~$1$), must be separately equated +to~$0$, and will indicate the existence of a subsidiary Proposition. +This is evident from~\Eqref{(62)}. + +Ex. Given $x(1 - y) = 0$, All~$X$s are~$Y$s, to determine~$y$ as +a function of~$x$. + +Let $\phi(xy) = x(1 - y)$, then $\phi(10) = 1$, $\phi(11) = 0$, $\phi(01) = 0$, +$\phi(00) = 0$; whence, by~\Eqref{(73)}, +\begin{align*} +y &= \frac{1}{1 - 0}\, x + \frac{0}{0 - 0}\, (1 - x) \\ + &= x + \tfrac{0}{0}(1 - x) \\ + &= x + v(1 - x), +\Tag{(76)} +\end{align*} +$v$~being an arbitrary elective symbol. The interpretation of this +result is that the class~$Y$ consists of the entire class~$X$ with an +indefinite remainder of not-$X$s. This remainder is indefinite in +the highest sense, \ie~it may vary from~$0$ up to the entire class +of not-$X$s. + +Ex. Given $x(1 - z) + z = y$, (the class~$Y$ consists of the +entire class~$Z$, with such not-$Z$s as are~$X$s), to find~$Z$. + +Here $\phi(xyz) = x(1 - z) - y + z$, whence we have the following +set of values for the moduli, +\begin{alignat*}{4} +\phi(110) &= 0,\quad& \phi(111) &= 0,\quad& \phi(100) &= 1,\quad& \phi(101) &= 1, \\ +\phi(010) &=-1,\quad& \phi(011) &= 0,\quad& \phi(000) &= 0,\quad& \phi(001) &= 1, +\end{alignat*} +and substituting these in the general formula~\Eqref{(75)}, we have +\[ +z = \tfrac{0}{0}xy + \tfrac{1}{0}x(1 - y) + (1 - x)y, +\Tag{(77)} +\] +\PageSep{75} +the infinite coefficient of the second term indicates the equation +\[ +x(1 - y) = 0,\quad\text{All~$X$s are~$Y$s;} +\] +and the indeterminate coefficient of the first term being replaced +by~$v$, an arbitrary elective symbol, we have +\[ +z = (1 - x)y + vxy, +\] +the interpretation of which is, that the class~$Z$ consists of all the~$Y$s +which are not~$X$s, and an \emph{indefinite} remainder of~$Y$s which +are~$X$s. Of course this indefinite remainder may vanish. The +two results we have obtained are logical inferences (not very +obvious ones) from the original Propositions, and they give us +all the information which it contains respecting the class~$Z$, and +its constituent elements. + +Ex. Given $x = y(1 - z) + z(1 - y)$. The class~$X$ consists of +all~$Y$s which are not-$Z$s, and all~$Z$s which are not-$Y$s: required +the class~$Z$. + +We have +\begin{alignat*}{4} +\phi(xyz) &= \rlap{$x - y(1 - z) - z(1 - y)$,} \\ +\phi(110) &= 0,\quad& \phi(111) &= 1,\quad& +\phi(100) &= 1,\quad& \phi(101) &= 0, \\ +% +\phi(010) &= -1,\quad& \phi(011) &= 0, & +\phi(000) &= 0, & \phi(001) &= -1; +\end{alignat*} +whence, by substituting in~\Eqref{(75)}, +\[ +z = x(1 - y) + y(1 - x), +\Tag{(78)} +\] +the interpretation of which is, the class~$Z$ consists of all~$X$s +which are not~$Y$s, and of all~$Y$s which are not~$X$s; an inference +strictly logical. + +Ex. Given $y\bigl\{1 - z(1 - x)\bigr\} = 0$, All~$Y$s are~$Z$s and not-$X$s. + +Proceeding as before to form the moduli, we have, on substitution +in the general formulć, +\[ +z = \tfrac{1}{0}xy + + \tfrac{0}{0}x(1 - y) + + y(1 - x) + + \tfrac{0}{0}(1 - x)(1 - y), +\] +or +\begin{align*} +%[** TN: Unaligned in the original] +z &= y(1 - x) + vx(1 - y) + v'(1 - x)(1 - y) \\ + &= y(1 - x) + (1 - y)\phi(x), +\Tag{(79)} +\end{align*} +with the relation +\[ +xy = 0\Typo{:}{;} +\] +from these it appears that No~$Y$s are~$X$s, and that the class~$Z$ +\PageSep{76} +consists of all~$Y$s which are not~$X$s, and of an indefinite remainder +of not-$Y$s. + +This method, in combination with Lagrange's method of +indeterminate multipliers, may be very elegantly applied to the +treatment of simultaneous equations. Our limits only permit us +to offer a single example, but the subject is well deserving of +further investigation. + +Given the equations $x(1 - z) = 0$, $z(1 - y) = 0$, All~$X$s are~$Z$s, +All~$Z$s are~$Y$s, to determine the complete value of~$z$ with +any subsidiary relations connecting $x$~and~$y$. + +Adding the second equation multiplied by an indeterminate +constant~$\lambda$, to the first, we have +\[ +x(1 - z) + \lambda z(1 - y) = 0, +\] +whence determining the moduli, and substituting in~\Eqref{(75)}, +\[ +z = xy + \frac{1}{1 - \lambda}\, x(1 - y) + \tfrac{0}{0}(1 - x)y, +\Tag{(80)} +\] +from which we derive +\[ +z = xy + v(1 - x)y, +\] +with the subsidiary relation +\[ +x(1 - y) = 0\Typo{:}{;} +\] +the former of these expresses that the class~$Z$ consists of all~$X$s +that are~$Y$s, with an indefinite remainder of not-$X$s that are~$Y$s; +the latter, that All~$X$s are~$Y$s, being in fact the conclusion +of the syllogism of which the two given Propositions are the +premises. + +By assigning an appropriate meaning to our symbols, all the +equations we have discussed would admit of interpretation in +hypothetical, but it may suffice to have considered them as +examples of categoricals. + +That peculiarity of elective symbols, in virtue of which every +elective equation is reducible to a system of equations $t_{1} = 0$, +$t_{2} = 0$,~\etc., so constituted, that all the binary products $t_{1}t_{2}$, $t_{1}t_{3}$, +\etc., vanish, represents a general doctrine in Logic with reference +to the ultimate analysis of Propositions, of which it +may be desirable to offer some illustration. +\PageSep{77} + +Any of these constituents $t_{1}$,~$t_{2}$,~\etc.\ consists only of factors +of the forms $x$,~$y$,~$\dots$ $1 - w$,~$1 - z$,~\etc. In categoricals it therefore +represents a compound class, \ie~a class defined by the +presence of certain qualities, and by the absence of certain +other qualities. + +Each constituent equation $t_{1} = 0$,~\etc.\ expresses a denial of the +existence of some class so defined, and the different classes are +mutually exclusive. + +\begin{Rule}[] +Thus all categorical Propositions are resolvable into a denial of +the existence of certain compound classes, no member of one such +class being a member of another. +\end{Rule} + +The Proposition, All~$X$s are~$Y$s, expressed by the equation +$x(1 - y) = 0$, is resolved into a denial of the existence of a +class whose members are~$X$s and not-$Y$s. + +The Proposition Some~$X$s are~$Y$s, expressed by $v = xy$, is +resolvable as follows. On expansion, +\begin{gather*} +v - xy = vx(1 - y) + vy(1 - x) + v(1 - x)(1 - y) - xy(1 - v); \\ +\therefore +vx(1 - y) = 0,\quad +vy(1 - x) = 0,\quad +v(1 - x)(1 - y) = 0,\quad +(1 - v)xy = 0. +\end{gather*} + +The three first imply that there is no class whose members +belong to a certain unknown Some, and are~1st, $X$s~and not~$Y$s; +2nd, $Y$s~and not~$X$s; 3rd, not-$X$s and not-$Y$s. The fourth +implies that there is no class whose members are $X$s~and~$Y$s +without belonging to this unknown Some. + +From the same analysis it appears that \begin{Rule}[]all hypothetical Propositions +may be resolved into denials of the coexistence of the truth +or falsity of certain assertions. +\end{Rule} + +Thus the Proposition, If $X$~is true, $Y$~is true, is resolvable +by its equation $x(1 - y) = 0$, into a denial that the truth of~$X$ +and the falsity of~$Y$ coexist. + +And the Proposition Either $X$~is true, or $Y$~is true, members +exclusive, is resolvable into a denial, first, that $X$~and~$Y$ are +both true; secondly, that $X$~and~$Y$ are both false. + +But it may be asked, is not something more than a system of +negations necessary to the constitution of an affirmative Proposition? +is not a positive element required? Undoubtedly +\PageSep{78} +there is need of one; and this positive element is supplied +in categoricals by the assumption (which may be regarded as +a prerequisite of reasoning in such cases) that there \emph{is} a Universe +of conceptions, and that each individual it contains either +belongs to a proposed class or does not belong to it; in hypotheticals, +by the assumption (equally prerequisite) that there +is a Universe of conceivable cases, and that any given Proposition +is either true or false. Indeed the question of the +existence of conceptions (\textgreek{e>i >'esti}) is preliminary to any statement +%[** TN: Should be \textgreek{t'i >esti}? Not sufficiently certain to change.] +of their qualities or relations (\textgreek{t'i >'esti}).---\textit{Aristotle, Anal.\ Post.}\ +lib.~\textsc{ii}.\ cap.~2. + +It would appear from the above, that Propositions may be +regarded as resting at once upon a positive and upon a negative +foundation. Nor is such a view either foreign to the spirit +of Deductive Reasoning or inappropriate to its Method; the +latter ever proceeding by limitations, while the former contemplates +the particular as derived from the general. + + +%[** TN: Equation numbering restarts] +\Section{Demonstration of the Method of Indeterminate Multipliers, as +applied to Simultaneous Elective Equations.} + +To avoid needless complexity, it will be sufficient to consider +the case of three equations involving three elective symbols, +those equations being the most general of the kind. It will +be seen that the case is marked by every feature affecting +the character of the demonstration, which would present itself +in the discussion of the more general problem in which the +number of equations and the number of variables are both +unlimited. + +Let the given equations be +\[ +\phi(xyz) = 0,\quad +\psi(xyz) = 0,\quad +\chi(xyz) = 0. +\Tag[app]{(1)} +\] + +Multiplying the second and third of these by the arbitrary +constants $h$~and~$k$, and adding to the first, we have +\[ +\phi(xyz) + h\psi(xyz) + k\chi(xyz) = 0; +\Tag[app]{(2)} +\] +\PageSep{79} +and we are to shew, that in solving this equation with reference +to any variable~$z$ by the general theorem~\Eqref{(75)}, we shall obtain +not only the general value of~$z$ independent of $h$~and~$k$, but +also any subsidiary relations which may exist between $x$~and~$y$ +independently of~$z$. + +%[xref] +If we represent the general equation~\Eqref[app]{(2)} under the form +$F(xyz) = 0$, its solution may by~\Eqref{(75)} be written in the form +\[ +z = \frac{xy}{1 - \dfrac{F(111)}{F(110)}} + + \frac{x(1 - y)}{1 - \dfrac{F(101)}{F(100)}} + + \frac{y(1 - x)}{1 - \dfrac{F(011)}{F(010)}} + + \frac{(1 - x)(1 - y)}{1 - \dfrac{F(001)}{F(000)}}; +\] +and we have seen, that any one of these four terms is to be +equated to~$0$, whose modulus, which we may represent by~$M$, +does not satisfy the condition $M^{n} = M$, or, which is here the +same thing, whose modulus has any other value than $0$~or~$1$. + +Consider the modulus (suppose~$M_{1}$) of the first term, viz. +$\dfrac{1}{1 - \dfrac{F(111)}{F(110)}}$, and giving to the symbol~$F$ its full meaning, +we have +\[ +M_{1} = \frac{1}{1 - \dfrac{\phi(111) + h\psi(111) + k\chi(111)} + {\phi(110) + h\psi(110) + k\chi(110)}}. +\] + +It is evident that the condition $M_{1}^{n} = M_{1}$ cannot be satisfied +unless the right-hand member be independent of $h$~and~$k$; and +in order that this may be the case, we must have the function +$\dfrac{\phi(111) + h\psi(111) + k\chi(111)} + {\phi(110) + h\psi(110) + k\chi(110)}$ independent of $h$~and~$k$. + +Assume then +\[ +\frac{\phi(111) + h\psi(111) + k\chi(111)} + {\phi(110) + h\psi(110) + k\chi(110)} = c, +\] +$c$~being independent of $h$~and~$k$; we have, on clearing of fractions +and equating coefficients, +\[ +\phi(111) = c\phi(110),\quad +\psi(111) = c\psi(110),\quad +\chi(111) = c\chi(110); +\] +whence, eliminating~$c$, +\[ +\frac{\phi(111)}{\phi(110)} + = \frac{\psi(111)}{\psi(110)} + = \frac{\chi(111)}{\chi(110)}, +\] +\PageSep{80} +being equivalent to the triple system +\[ +\left.\begin{alignedat}{3} +&\phi(111)\psi(110) &&- \phi(110)\psi(111) &&= 0\Add{,} \\ +&\psi(111)\chi(110) &&- \psi(110)\chi(111) &&= 0\Add{,} \\ +&\chi(111)\phi(110) &&- \chi(110)\Typo{\psi}{\phi}(111) &&= 0\Add{;} +\end{alignedat} +\right\} +\Tag[app]{(3)} +\] +and it appears that if any one of these equations is not satisfied, +the modulus~$M_{1}$ will not satisfy the condition $M_{1}^{n} = M_{1}$, whence +the first term of the value of~$z$ must be equated to~$0$, and +we shall have +\[ +xy = 0, +\] +a relation between $x$~and~$y$ independent of~$z$. + +Now if we expand in terms of~$z$ each pair of the primitive +equations~\Eqref[app]{(1)}, we shall have +\begin{alignat*}{3} +&\phi(xy0) &&+ \bigl\{\phi(xy1) - \phi(xy0)\bigr\}z &&= 0, \\ +&\psi(xy0) &&+ \bigl\{\psi(xy1) - \psi(xy0)\bigr\}z &&= 0, \\ +&\chi(xy0) &&+ \bigl\{\chi(xy1) - \chi(xy0)\bigr\}z &&= 0, +\end{alignat*} +and successively eliminating~$z$ between each pair of these equations, +we have +\begin{alignat*}{3} +&\phi(xy1)\psi(xy0) &&- \phi(xy0)\psi(xy1) &&= 0, \\ +&\psi(xy1)\chi(xy0) &&- \psi(xy0)\chi(xy1) &&= 0, \\ +&\chi(xy1)\phi(xy0) &&- \chi(xy0)\phi(xy1) &&= 0, +\end{alignat*} +which express all the relations between $x$~and~$y$ that are formed +by the elimination of~$z$. Expanding these, and writing in full +the first term, we have +\begin{alignat*}{3} +&\bigl\{\phi(111)\psi(110) &&- \phi(110)\psi(111)\bigr\}xy &&+ \etc. = 0, \\ +&\bigl\{\psi(111)\chi(110) &&- \psi(110)\chi(111)\bigr\}xy &&+ \etc. = 0, \\ +&\bigl\{\chi(111)\phi(110) &&- \chi(110)\phi(111)\bigr\}xy &&+ \etc. = 0\Typo{:}{;} +\end{alignat*} +and it appears from \PropRef{2}.\ that if the coefficient of~$xy$ in any +of these equations does not vanish, we shall have the equation +\[ +xy = 0; +\] +but the coefficients in question are the same as the first members +of the system~\Eqref[app]{(3)}, and the two sets of conditions exactly agree. +Thus, as respects the first term of the expansion, the method of +indeterminate coefficients leads to the same result as ordinary +elimination; and it is obvious that from their similarity of form, +the same reasoning will apply to all the other terms. +\PageSep{81} + +Suppose, in the second place, that the conditions~\Eqref[app]{(3)} are satisfied +so that $M_{1}$~is independent of $h$~and~$k$. It will then indifferently +assume the equivalent forms +\[ +M_{1} = \frac{1}{1 - \dfrac{\phi(111)}{\phi(110)}} + = \frac{1}{1 - \dfrac{\psi(111)}{\psi(110)}} + = \frac{1}{1 - \dfrac{\chi(111)}{\chi(110)}}\Add{.} +\] + +These are the exact forms of the first modulus in the expanded +values of~$z$, deduced from the solution of the three +primitive equations singly. If this common value of~$M_{1}$ is $1$ +or $\frac{0}{0} = v$, the term will be retained in~$z$; if any other constant +value (except~$0$), we have a relation $xy = 0$, not given by elimination, +but deducible from the primitive equations singly, and +similarly for all the other terms. Thus in every case the expression +of the subsidiary relations is a necessary accompaniment +of the process of solution. + +It is evident, upon consideration, that a similar proof will +apply to the discussion of a system indefinite as to the number +both of its symbols and of its equations. +%[** TN: No page break in the original] + + +\Chapter{Postscript.} + +\First{Some} additional explanations and references which have +occurred to me during the printing of this work are subjoined. + +The remarks on the connexion between Logic and Language, +\Pageref{5}, are scarcely sufficiently explicit. Both the one and the +other I hold to depend very materially upon our ability to form +general notions by the faculty of abstraction. Language is an +instrument of Logic, but not an indispensable instrument. + +To the remarks on Cause, \Pageref{12}, I desire to add the following: +Considering Cause as an invariable antecedent in Nature, (which +is Brown's view), whether associated or not with the idea of +Power, as suggested by Sir~John Herschel, the knowledge of its +existence is a knowledge which is properly expressed by the word +\emph{that} (\textgreek{t`o <ot`i}), not by \emph{why} (\textgreek{t`o di<ot`i}). It is very remarkable that +the two greatest authorities in Logic, modern and ancient, agreeing +in the latter interpretation, differ most widely in its application +to Mathematics. Sir W.~Hamilton says that Mathematics +\PageSep{82} +exhibit only the \emph{that} (\textgreek{t`o <ot`i}): Aristotle says, The \emph{why} belongs +to mathematicians, for they have the demonstrations of Causes. +\textit{Anal.\ Post.}\ lib.~\textsc{i}., cap.~\textsc{xiv}. It must be added that Aristotle's +view is consistent with the sense (albeit an erroneous one) +which in various parts of his writings he virtually assigns to the +word Cause, viz.\ an antecedent in Logic, a sense according to +which the premises might be said to be the cause of the conclusion. +This view appears to me to give even to his physical +inquiries much of their peculiar character. + +Upon reconsideration, I think that the view on \Pageref{41}, as to the +presence or absence of a medium of comparison, would readily +follow from Professor De~Morgan's doctrine, and I therefore +relinquish all claim to a discovery. The mode in which it +appears in this treatise is, however, remarkable. + +I have seen reason to change the opinion expressed in +\Pagerefs{42}{43}. The system of equations there given for the expression +of Propositions in Syllogism is \emph{always} preferable to the one +before employed---first, in generality---secondly, in facility of +interpretation. + +In virtue of the principle, that a Proposition is either true or +false, every elective symbol employed in the expression of +hypotheticals admits only of the values $0$~and~$1$, which are the +only quantitative forms of an elective symbol. It is in fact +possible, setting out from the theory of Probabilities (which is +purely quantitative), to arrive at a system of methods and processes +for the treatment of hypotheticals exactly similar to those +which have been given. The two systems of elective symbols +and of quantity osculate, if I may use the expression, in the +points $0$~and~$1$. It seems to me to be implied by this, that +unconditional truth (categoricals) and probable truth meet together +in the constitution of contingent truth\Typo{;}{} (hypotheticals). +The general doctrine of elective symbols and all the more characteristic +applications are quite independent of any quantitative +origin. +\vfil +\begin{center} +\small +THE END. +\end{center} +\vfil\vfil +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\PGLicense +\begin{PGtext} +End of Project Gutenberg's The Mathematical Analysis of Logic, by George Boole + +*** END OF THIS PROJECT GUTENBERG EBOOK THE MATHEMATICAL ANALYSIS OF LOGIC *** + +***** This file should be named 36884-pdf.pdf or 36884-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/8/8/36884/ + +Produced by Andrew D. 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