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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/6560-8.txt b/6560-8.txt new file mode 100644 index 0000000..74b9c25 --- /dev/null +++ b/6560-8.txt @@ -0,0 +1,11792 @@ +The Project Gutenberg EBook of Deductive Logic, by St. George Stock + +Copyright laws are changing all over the world. Be sure to check the +copyright laws for your country before downloading or redistributing +this or any other Project Gutenberg eBook. + +This header should be the first thing seen when viewing this Project +Gutenberg file. Please do not remove it. Do not change or edit the +header without written permission. + +Please read the "legal small print," and other information about the +eBook and Project Gutenberg at the bottom of this file. Included is +important information about your specific rights and restrictions in +how the file may be used. You can also find out about how to make a +donation to Project Gutenberg, and how to get involved. + + +**Welcome To The World of Free Plain Vanilla Electronic Texts** + +**eBooks Readable By Both Humans and By Computers, Since 1971** + +*****These eBooks Were Prepared By Thousands of Volunteers!***** + + +Title: Deductive Logic + +Author: St. George Stock + +Release Date: September, 2004 [EBook #6560] +[Yes, we are more than one year ahead of schedule] +[This file was first posted on December 28, 2002] + +Edition: 10 + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THE PROJECT GUTENBERG EBOOK DEDUCTIVE LOGIC *** + + + + +Produced by Arno Peters, David Moynihan, Charles Franks +and the Online Distributed Proofreading Team. + + + + + +[Transcriber's Note: In this plain-text rendering, + .'. means therefore + [alpha], [beta], ..., [Alpha], [Beta], ... for Greek symbols] + + + + + +DEDUCTIVE LOGIC + +BY + +ST. GEORGE STOCK, M.A. + +PEMBROKE COLLEGE, OXFORD + + + + +PREFACE. + +One critic, who was kind enough to look at this book in manuscript, +recommended me to abandon the design of Publishing it, on the ground +that my logic was too like all other logics; another suggested to me +to cut out a considerable amount of new matter. The latter advice I +have followed; the former has encouraged me to hope that I shall not +be considered guilty of wanton innovation. The few novelties which I +have ventured to retain will, I trust, be regarded as legitimate +extensions of received lines of teaching. + +My object has been to produce a work which should be as thoroughly +representative of the present state of the logic of the Oxford Schools +as any of the text-books of the past. The qualities which I have aimed +at before all others have been clearness and consistency. For the task +which I have taken upon myself I may claim one qualification--that of +experience; since more than seventeen years have now elapsed since I +took my first pupil in logic for the Honour School of Moderations, and +during that time I have been pretty continuously engaged in studying +and teaching the subject. + +In acknowledging my obligations to previous writers I must begin with +Archbishop Whately, whose writings first gave me an interest in the +subject. The works of Mill and Hamilton have of course been freely +drawn upon. I have not followed either of those two great writers +exclusively, but have endeavoured to assimilate what seemed best in +both. To Professor Fowler I am under a special debt. I had not the +privilege of personal teaching from him in logic,--as I had in some +other subjects; but his book fell into my hands at an early period in +my mental training, and was so thoroughly studied as to have become a +permanent part of the furniture of my mind. Much the same may be said +of my relation to the late Professor Jevons's Elementary Lessons in +Logic. Two other books, which I feel bound to mention with special +emphasis, are Hansel's edition of Aldrich and McCosh's Laws of +Discursive Thought. If there be added to the foregoing Watts's Logic, +Thomson's Outlines of the Laws of Thought, Bain's Deductive Logic, +Jevons's Studies in Deductive Logic and Principles of Science, +Bradley's Principles of Logic, Abbott's Elements of Logic, Walker's +edition of Murray, Ray's Text-book of Deductive Logic, and +Weatherley's Rudiments of Logic, I think the list will be exhausted of +modern works from which I am conscious of having borrowed. But, not to +forget the sun, while thanking the manufacturers of lamps and candles, +I should add that I have studied the works of Aristotle according to +the measure of my time and ability. + +This work has had the great advantage of having been revised, while +still in manuscript, by Mr. Alfred Robinson, Fellow of New College, to +whom I cannot sufficiently express my obligation. I have availed +myself to the full of the series of criticisms which he was kind +enough to send me. As some additions have been made since then, he +cannot be held in anyway responsible for the faults which less kindly +critics may detect. + +For the examples at the end I am mainly indebted to others, and to a +large extent to my ingenious friend, the Rev. W. J. Priest of Merton +College. + +My thanks are due also to my friend and former pupil, Mr. Gilbert +Grindle, Scholar of Corpus, who has been at the pains to compose an +index, and to revise the proofs as they passed through the press. + +And last, but not least, I must set on record my gratitude to +Commander R. A. Stock, R.N., one of Her Majesty's Knights of Windsor, +without whose brotherly aid this work might never have been written, +and would certainly not have assumed exactly its present shape. + +OXFORD, + +_October_ 22, 1888. + + + + +CONTENTS. + +PREFACE. + +INTRODUCTION, §§ 1-56. + +PART I. Of Terms, §§ 57-171. + + CHAP. I. Of the Term as distinguished from other words, §§ 57-76. + + II. Of the Division of Things, §§ 77-85. + + III. Of the Divisions of Terms, §§ 86-165. + + IV. Of the Law of Inverse Variation of Extension and Intension, §§ + 166-171. + +PART II. Of Propositions, §§ 172-185. + + CHAP. I. Of the Proposition as distinguished from other Sentences, + §§ 172-185. + + II. Of the Copula, §§ 186-201. + + III. Of the Divisions of Propositions, §§ 202-273. + + IV. Of the Distribution of Terms, §§ 274-294. + + V. Of the Quantification of the Predicate, §§ 295-312. + + VI. Of the Heads of Predicables, §§ 313-346. + + VII. Of Definition, §§ 347-384. + + VIII. Of Division, §§ 385-425. + +PART III. Of Inferences, §§ 426-884. + + CHAP. I. Of Inferences in general, §§ 426-441. + + II. Of Deductive Inferences, §§ 442-448. + + III. Of Opposition, §§ 449-478. + + IV. Of Conversion, §§ 479-495. + + V. Of Permutation, §§ 496-502. + + VI. Of Compound Forms of Immediate Inference, §§ 503-532. + + VII. Of Other Forms of Immediate Inference, §§ 533-539. + + VIII. Of Mediate Inferences or Syllogisms, §§ 540-557. + + IX. Of Mood and Figure, §§ 558-568. + + X. Of the Canon of Reasoning, §§ 569-581. + + XI. Of the General Rules of Syllogism, §§ 582-598. + + XII. Of the Determination of the Legitimate Moods of Syllogism, §§ + 599-605. + + XIII. Of the Special Rules of the Four Figures, §§ 606-620. + + XIV. Of the Determination of the Moods that are valid in the Four + Figures, §§ 621-632. + + XV. Of the Special Canons of the Four Figures, §§ 633-647. + + XVI. Of the Special Uses of the Four Figures, §§ 648-655. + + XVII. Of the Syllogism with Three Figures, §§ 656-666. + + XVIII. Of Reduction, §§ 667-700. + + XIX. Of Immediate Inference as applied to Complex Propositions, §§ + 701-730. + + XX. Of Complex Syllogisms, §§ 731-743. + + XXI. Of the Reduction of the Partly Conjunctive Syllogism, §§ + 744-752. + + XXII. Of the Partly Conjunctive Syllogism regarded as all Immediate + Inference, §§ 753-759. + + XXIII. Of the Disjunctive Syllogism, §§ 760-765. + + XXIV. Of the Reduction of the Disjunctive Syllogism, §§ 766-769. + + XXV. Of the Disjunctive Syllogism regarded as an Immediate + Inference, §§ 770-777. + + XXVI. Of the Mixed Form of Complex Syllogism, §§ 778-795. + + XXVII. Of the Reduction of the Dilemma, §§ 796-797. + + XXVIII. Of the Dilemma regarded as an Immediate Inference, §§ + 798,799. + + XXIX. Of Trains of Reasoning, §§ 800-826. + + XXX. Of Fallacies, §§ 827-884. + +EXERCISES. + +INDEX. + + + + +INTRODUCTION. + +§ 1. LOGIC is divided into two branches, namely-- + + (1) Inductive, + + (2) Deductive. + +§ 2. The problem of inductive logic is to determine the actual truth +or falsity of propositions: the problem of deductive logic is to +determine their relative truth or falsity, that is to say, given such +and such propositions as true, what others will follow from them. + +§ 3. Hence in the natural order of treatment inductive logic precedes +deductive, since it is induction which supplies us with the general +truths, from which we reason down in our deductive inferences. + +§ 4. It is not, however, with logic as a whole that we are here +concerned, but only with deductive logic, which may be defined as The +Science of the Formal Laws of Thought. + +§ 5. In order fully to understand this definition we must know exactly +what is meant by 'thought,' by a 'law of thought,' by the term +'formal,' and by 'science.' + +§ 6. Thought, as here used, is confined to the faculty of +comparison. All thought involves comparison, that is to say, a +recognition of likeness or unlikeness. + +§ 7. The laws of thought are the conditions of correct thinking. The +term 'law,' however, is so ambiguous that it will be well to determine +more precisely in what sense it is here used. + +§ 8. We talk of the 'laws of the land' and of the 'laws of nature,' +and it is evident that we mean very different things by these +expressions. By a law in the political sense is meant a command +imposed by a superior upon an inferior and sanctioned by a penalty for +disobedience. But by the 'laws of nature' are meant merely certain +uniformities among natural phenomena; for instance, the 'law of +gravitation' means that every particle of matter does invariably +attract every other particle of matter in the universe. + +§ 9. The word 'law' is transferred by a metaphor from one of these +senses to the other. The effect of such a command as that described +above is to produce a certain amount of uniformity in the conduct of +men, and so, where we observe uniformity in nature, we assume that it +is the result of such a command, whereas the only thing really known +to us is the fact of uniformity itself. + +§ 10. Now in which of these two senses are we using the term 'laws of +thought'? The laws of the land, it is plain, are often violated, +whereas the laws of nature never can be so [Footnote: There is a sense +in which people frequently speak of the laws of nature being violated, +as when one says that intemperance or celibacy is a violation of the +laws of nature, but here by 'nature' is meant an ideal perfection in +the conditions of existence.]. Can the laws of thought be violated in +like manner with the laws of the land? Or are they inviolable like the +laws of nature? + +§ 11. In appearance they can be, and manifestly often are violated-for +how else could error be possible? But in reality they can not. No man +ever accepts a contradiction when it presents itself to the mind as +such: but when reasoning is at all complicated what does really +involve a contradiction is not seen to do so; and this sort of error +is further assisted by the infinite perplexities of language. + +§ 12. The laws of thought then in their ultimate expression are +certain uniformities which invariably hold among mental phenomena, and +so far they resemble the laws of nature: but in their complex +applications they may be violated owing to error, as the laws of the +land may be violated by crime. + +§ 13. We have now to determine the meaning of the expression 'formal +laws of thought.' + +§ 14. The distinction between form and matter is one which pervades +all nature. We are familiar with it in the case of concrete things. A +cup, for instance, with precisely the same form, may be composed of +very different matter-gold, silver, pewter, horn or what not? + +§ 15. Similarly in every act of thought we may distinguish two +things-- + + (1) the object thought about, + + (2) the way in which the mind thinks of it. + +The first is called the Matter; the second the Form of Thought. + +§ 16. Now Formal, which is another name for Deductive Logic, is +concerned only with the way in which the mind thinks, and has nothing +to do with the particular objects thought about. + +§ 17. Since the form may be the same, whilst the matter is different, +we may say that formal logic is concerned with the essential and +necessary elements of thought as opposed to such as are accidental and +contingent. By 'contingent' is meant what holds true in some cases, +but not in others. For instance, in the particular case of equilateral +triangles it is true to say, not only that 'all equilateral triangles +are equiangular,' but also that 'all equiangular triangles are +equilateral.' But the evidence for these two propositions is +independent. The one is not a formal consequence of the other. If it +were, we should be able to apply the same inference to all matter, and +assert generally that if all A is B, all B is A, which it is notorious +that we cannot do. + +§ 18. It remains now for the full elucidation of our definition to +determine what is meant by 'science.' + +§ 19. The question has often been discussed whether logic is a science +or an art. The answer to it must depend upon the meaning we assign to +these terms. + +§ 20. Broadly speaking, there is the same difference between Science +and Art as there is between knowing and doing. + + Science is systematized knowledge; + Art is systematized action. + Science is acquired by study; + Art is acquired by practice. + +§ 21. Now logic is manifestly a branch of knowledge, and does not +necessarily confer any practical skill. It is only the right use of +its rules in thinking which can make men think better. It is +therefore, in the broad sense of the terms, wholly a science and not +at all an art. + +§ 22. But this word 'art,' like most others, is ambiguous, and is +often used, not for skill displayed in practice, but for the knowledge +necessary thereto. This meaning is better conveyed by the term +'practical science.' + +§ 23. Science is either speculative or practical. In the first case we +study merely that we may know; in the latter that we may do. + + Anatomy is a speculative science; + Surgery is a practical science. + +In the first case we study the human frame in order that we may +understand its structure; in the second that we may assist its +needs. Whether logic is a speculative or a practical science depends +entirely upon the way in which it is treated. If we study the laws of +thought merely that we may know what they are, we are making it a +speculative science; if we study the same laws with a view to deducing +rules for the guidance of thought, we are making it a practical +science. + +§ 24. Hence logic may be declared to be both the science and the art +of thinking. It is the art of thinking in the same sense in which +grammar is the art of speaking. Grammar is not in itself the right +use of words, but a knowledge of it enables men to use words +correctly. In the same way a knowledge of logic enables men to think +correctly, or at least to avoid incorrect thoughts. As an art logic +may be called the navigation of the sea of thought. + +§ 25. The laws of thought are all reducible to the three following +axioms, which are known as The Three Fundamental Laws of Thought. + + (1) The Law of Identity-- + + Whatever is, is; + + or, in a more precise form, + + Every A is A. + + (2) The Law of Contradiction-- + + Nothing can both be and not be; + Nothing can be A and not A. + + (3) The Law of Excluded Middle-- + + Everything must either be or not be; + Everything is either A or not A. + +§ 26. Each of these principles is independent and self-evident. + +§ 27. If it were possible for the law of identity to be violated, no +violation of the law of contradiction would necessarily ensue: for a +thing might then be something else, without being itself at the same +time, which latter is what the law of contradiction militates +against. Neither would the law of excluded middle be infringed. For, +on the supposition, a thing would be something else, whereas all that +the law of excluded middle demands is that it should either be itself +or not. A would in this case adopt the alternative of being not A. + +§ 28. Again, the violation of the law of contradiction does not +involve any violation of the law of identity: for a thing might in +that case be still itself, so that the law of identity would be +observed, even though, owing to the law of contradiction not holding, +it were not itself at the same time. Neither would the law of excluded +middle be infringed. For a thing would, on the supposition, be both +itself and not itself, which is the very reverse of being neither. + +§ 29. Lastly, the law of excluded middle might be violated without a +violation of the law of contradiction: for we should then have a thing +which was neither A nor not A, but not a thing which was both at the +same time. Neither would the law of identity be infringed. For we +should in this case have a thing which neither was nor was not, so +that the conditions of the law of identity could not exist to be +broken. That law postulates that whatever is, is: here we have a thing +which never was to begin with. + +§ 30. These principles are of so simple a character that the +discussion of them is apt to be regarded as puerile. Especially is +this the case with regard to the law of identity. This principle in +fact is one of those things which are more honoured in the breach than +in the observance. Suppose for a moment that this law did not +hold--then what would become of all our reasoning? Where would be the +use of establishing conclusions about things, if they were liable to +evade us by a Protean change of identity? + +§ 31. The remaining two laws supplement each other in the following +way. The law of contradiction enables us to affirm of two exhaustive +and mutually exclusive alternatives, that it is impossible for both to +be true; the law of excluded middle entitles us to add, that it is +equally impossible for both to be false. Or, to put the same thing in +a different form, the law of contradiction lays down that one of two +such alternatives must be false; the law of excluded middle adds that +one must be true. + +§32. There are three processes of thought + + (1) Conception. + + (2) Judgement. + + (3) Inference or Reasoning. + +§ 33. Conception, which is otherwise known as Simple Apprehension, is +the act of forming in the mind the idea of anything, e.g. when we form +in the mind the idea of a cup, we are performing the process of +conception. + +§ 34. Judgement, in the sense in which it is here used [Footnote: +Sometimes the term 'judgement' is extended to the comparison of +nameless sense-impressions, which underlies the formation of +concepts. But this amounts to identifying judgement with thought in +general.] may be resolved into putting two ideas together in the +mind, and pronouncing as to their agreement or disagreement, e.g. we +have in our minds the idea of a cup and the idea of a thing made of +porcelain, and we combine them in the judgement--'This cup is made of +porcelain.' + +§ 35. Inference, or Reasoning, is the passage of the mind from one or +more judgements to another, e.g. from the two judgements 'Whatever is +made of porcelain is brittle,' and 'This cup is made of porcelain,' we +elicit a third judgement, 'This cup is brittle.' + +§ 36. Corresponding to these three processes there are three products +of thought, viz. + + (1) The Concept. + + (2) The Judgement. + + (3) The Inference. + +§ 37. Since our language has a tendency to confuse the distinction +between processes and products, [Footnote: E.g. We have to speak quite +indiscriminately of Sensation, Imagination, Reflexion, Sight, Thought, +Division, Definition, and so on, whether we mean in any case a process +or a product.] it is the more necessary to keep them distinct in +thought. Strictly we ought to speak of conceiving, judging and +inferring on the one hand, and, on the other, of the concept, the +judgement and the inference. + +The direct object of logic is the study of the products rather than of +the processes of thought. But, at the same time, in studying the +products we are studying the processes in the only way in which it is +possible to do so. For the human mind cannot be both actor and +spectator at once; we must wait until a thought is formed in our minds +before we can examine it. Thought must be already dead in order to be +dissected: there is no vivisection of consciousness. Thus we can never +know more of the processes of thought than what is revealed to us in +their products. + +§ 38. When the three products of thought are expressed in language, +they are called respectively + + (1) The Term. + + (2) The Proposition. + + (3) The Inference. + +§ 39. Such is the ambiguity of language that we have already used the +term 'inference' in three different senses--first, for the act or +process of inferring; secondly, for the result of that act as it +exists in the mind; and, thirdly, for the same thing as expressed in +language. Later on we shall have to notice a further ambiguity in its +use. + +§ 40. It has been declared that thought in general is the faculty of +comparison, and we have now seen that there are three products of +thought. It follows that each of these products of thought must be the +result of a comparison of some kind or other. + + The concept is the result of comparing attributes. + The judgement is the result of comparing concepts. + The inference is the result of comparing judgements. + +§ 41. In what follows we shall, for convenience, adopt the phraseology +which regards the products of thought as clothed in language in +preference to that which regards the same products as they exist in +the mind of the individual. For although the object of logic is to +examine thought pure and simple, it is obviously impossible to discuss +it except as clothed in language. Accordingly the three statements +above made may be expressed as follows-- + + The term is the result of comparing attributes. + The proposition is the result of comparing terms. + The inference is the result of comparing propositions. + +§ 42. There is an advantage attending the change of language in the +fact that the word 'concept' is not an adequate expression for the +first of the three products of thought, whereas the word 'term' is. By +a concept is meant a general notion, or the idea of a class, which +corresponds only to a common term. Now not only are common terms the +results of comparison, but singular terms, or the names of +individuals, are so too. + +§ 43. The earliest result of thought is the recognition of an +individual object as such, that is to say as distinguished and marked +off from the mass of its surroundings. No doubt the first impression +produced Upon the nascent intelligence of an infant is that of a +confused whole. It requires much exercise of thought to distinguish +this whole into its parts. The completeness of the recognition of an +individual object is announced by attaching a name to it. Hence even +an individual name, or singular term, implies thought or +comparison. Before the _child_ can attach a meaning to the word +'_mother_,' which to it is a singular term, it must have +distinguished between the set of impressions produced in it by one +object from those which are produced in it by others. Thus, when +Vergil says + + Incipe, parve puer, risu cognoscere matrem, + +he is exhorting the beatific infant to the exercise of the faculty of +comparison. + +§ 44. That a common term implies comparison does not need to be +insisted upon. It is because things resemble each other in certain of +their attributes that we call them by a common name, and this +resemblance could not be ascertained except by comparison, at some +time and by some one. Thus a common term, or concept, is the +compressed result of an indefinite number of comparisons, which lie +wrapped up in it like so many fossils, witnessing to prior ages of +thought. + +§ 45. In the next product of thought, namely, the proposition, we have +the result of a single act of comparison between two terms; and this +is why the proposition is called the unit of thought, as being the +simplest and most direct result of comparison. + +§ 46. In the third product of thought, namely, the inference, we have +a comparison of propositions either directly or by means of a +third. This will be explained later on. For the present we return to +the first product of thought. + +§ 47. The nature of singular terms has not given rise to much dispute; +but the nature of common terms has been the great battle-ground of +logicians. What corresponds to a singular term is easy to determine, +for the thing of which it is a name is there to point to: but the +meaning of a common term, like 'man' or 'horse,' is not so obvious as +people are apt to think on first hearing of the question. + +§ 48. A common term or class-name was known to medićval logicians +under the title of a Universal; and it was on the question 'What is a +Universal 7' that they split into the three schools of Realists, +Nominalists, and Conceptualists. Here are the answers of the three +schools to this question in their most exaggerated form-- + +§ 49. Universals, said the Realists, are substances having an +independent existence in nature. + +§ 50. Universals, said the Nominalists, are a mere matter of words, +the members of what we call a class having nothing in common but the +name. + +§ 51. Universals, said the Conceptualists, exist in the mind alone, +They are the conceptions under which the mind regards external +objects. + +§ 52. The origin of pure Realism is due to Plato and his doctrine of +'ideas'; for Idealism, in this sense, is not opposed to Realism, but +identical with it. Plato seems to have imagined that, as there was a +really existing thing corresponding to a singular term, such as +Socrates, so there must be a really existing thing corresponding to +the common term 'man.' But when once the existence of these general +objects is admitted, they swamp all other existences. For individual +men are fleeting and transitory--subject to growth, decay and +death--whereas the idea of man is imperishable and eternal. It is only +by partaking in the nature of these ideas that individual objects +exist at all. + +§ 53. Pure Nominalism was the swing of the pendulum of thought to the +very opposite extreme; while Conceptualism was an attempt to hit the +happy mean between the two. + +§ 54. Roughly it may be said that the Realists sought for the answer +to the question 'What is a Universal?' in the matter of thought, the +Conceptualists in the form, and the Nominalists in the expression. + +§ 55. A full answer to the question 'What is a Universal?' will bring +in something of the three views above given, while avoiding the +exaggeration of each. A Universal is a number of things that are +called by the same name; but they would not be called by the same name +unless they fell under the same conception in the mind; nor would they +fall under the same conception in the mind unless there actually +existed similar attributes in the several members of a class, causing +us to regard them under the same conception and to give them the same +name. Universals therefore do exist in nature, and not merely in the +mind of man: but their existence is dependent upon individual objects, +instead of individual objects depending for their existence upon +them. Aristotle saw this very clearly, and marked the distinction +between the objects corresponding to the singular and to the common +term by calling the former Primary and the latter Secondary +Existences. Rosinante and Excalibur are primary, but 'horse' and +'sword' secondary existences. + +§ 56. We have seen that the three products of thought are each one +stage in advance of the other, the inference being built upon the +proposition, as the proposition is built upon the term. Logic +therefore naturally divides itself into three parts. + + The First Part of Logic deals with the Term; + The Second Part deals with the Proposition; + The Third Part deals with the Inference. + + + + + +PART I.--OF TERMS. + + + + +CHAPTER 1. + +_Of the Term as distinguished from other words._ + + +§ 57. The word 'term' means a boundary. + +§ 58. The subject and predicate are the two terms, or boundaries, of a +proposition. In a proposition we start from a subject and end in a +predicate (§§ 182-4), there being nothing intermediate between the two +except the act of pronouncing as to their agreement or disagreement, +which is registered externally under the sign of the copula. Thus the +subject is the 'terminus a quo,' and the predicate is the 'terminus ad +quem.' + +§ 59. Hence it appears that the term by its very name indicates that +it is arrived at by an analysis of the proposition. It is the +judgement or proposition that is the true unit of thought and +speech. The proposition as a whole is prior in conception to the terms +which are its parts: but the parts must come before the whole in the +synthetic order of treatment. + +§ 60. A term is the same thing as a name or noun. + +§ 61. A name is a word, or collection of words, which serves as a mark +to recall or transmit the idea of a thing, either in itself or through +some of its attributes. + +§ 62. Nouns, or names, are either Substantive or Adjective. + +A Noun Substantive is the name of a thing in itself, that is to say, +without reference to any special attribute. + +§ 63. A Noun Adjective is a name which we are entitled to add to a +thing, when we know it to possess a given attribute. + +§ 64. The Verb, as such, is not recognised by logic, but is resolved +into predicate and copula, that is to say, into a noun which is +affirmed or denied of another, plus the sign of that affirmation or +denial. 'The kettle boils' is logically equivalent to 'The kettle is +boiling,' though it is by no means necessary to express the +proposition in the latter shape. Here we see that 'boils' is +equivalent to the noun 'boiling' together with the copula 'is,' which +declares its agreement with the noun 'kettle.' 'Boiling' here is a +noun adjective, which we are entitled to add to 'kettle,' in virtue of +certain knowledge which we have about the latter. Being a verbal noun, +it is called in grammar a participle, rather than a mere +adjective. The word 'attributive' in logic embraces both the adjective +and participle of grammar. + +§ 65. In grammar every noun is a separate word: but to logic, which is +concerned with the thought rather than with the expression, it is +indifferent whether a noun, or term, consists of one word or many. The +latter are known as 'many-worded names.' In the following passage, +taken at random from Butler's Analogy--'These several observations, +concerning the active principle of virtue and obedience to God's +commands, are applicable to passive submission or resignation to his +will'--we find the subject consisting of fourteen words, and the +predicate of nine. It is the exception rather than the rule to find a +predicate which consists of a single word. Many-worded names in +English often consist of clauses introduced by the conjunction 'that,' +as 'That letters should be written in strict conformity with nature is +true': often also of a grammatical subject with one or more dependent +clauses attached to it, as + + 'He who fights and runs away, + Will live to fight another day.' + +§ 66. Every term then is not a word, since a term may consist of a +collection of words. Neither is every word a term. 'Over,' for +instance, and 'swiftly,' and, generally, what are called particles in +grammar, do not by themselves constitute terms, though they may be +employed along with other words to make up a term. + +§ 67. The notions with which thought deals involve many subtle +relations and require many nice modifications. Language has +instruments, more or less perfect, whereby such relations and +modifications may be expressed. But these subsidiary aids to +expression do not form a notion which can either have something +asserted of it or be asserted itself of something else. + +§ 68. Hence words are divided into three classes-- + + (1) Categorematic; + + (2) Syncategorematic; + + (3) Acategorematic. + +§ 69. A Categorematic word is one which can be used by itself as a +term. + +§ 70. A Syncategorematic word is one which can help to form a term. + +§ 71. An Acategorematic word is one which can neither form, nor help +to form, a term [Footnote: Comparatively few of the parts of speech +are categorematic. Nouns, whether substantive or adjective, including +of course pronouns and participles, are so, but only in their +nominative cases, except when an oblique case is so used as to be +equivalent to an attributive. Verbs also are categorematic, but only +in three of their moods, the Indicative, the Infinitive, and the +Potential. The Imperative and Optative moods clearly do not convey +assertions at all, while the Subjunctive can only figure as a +subordinate member of some assertion. We may notice, too, that the +relative pronoun, unlike the rest, is necessarily syncategorematic, +for the same reason as the subjunctive mood. Of the remaining parts of +speech the article, adverb, preposition, and conjunction can never be +anything but syncategorematic, while the interjection is +acategorematic, like the vocative case of nouns and the imperative and +optative moods of verbs, which do not enter at all into the form of +sentence known as the proposition.]. + +§ 72. Categorematic literally means 'predicable.' 'Horse,' 'swift,' +'galloping' are categorematic. Thus we can say, 'The horse is swift,' +or 'The horse is galloping.' Each of these words forms a term by +itself, but 'over' and 'swiftly' can only help to form a term, as in +the proposition, 'The horse is galloping swiftly over the plain.' + +§ 73. A term then may be said to be a categorematic word or collection +of words, that is to say, one which can be used by itself as a +predicate. + +§ 74. To entitle a word or collection of words to be called a term, it +is not necessary that it should be capable of standing by itself as a +subject. Many terms which can be used as predicates are incapable of +being used as subjects: but every term which can be used as a subject +(with the doubtful exception of proper names) can be used also as a +predicate. The attributives 'swift' and 'galloping' are terms, quite +as much as the subject 'horse,' but they cannot themselves be used as +subjects. + +§ 75. When an attributive appears to be used as a subject, it is owing +to a grammatical ellipse. Thus in Latin we say 'Boni sapientes sunt,' +and in English 'The good are wise,' because it is sufficiently +declared by the inflexional form in the one case, and by the usage of +the language in the other, that men are signified. It is an accident +of language how far adjectives can be used as subjects. They cease to +be logical attributives the moment they are so used. + +§ 76. There is a sense in which every word may become categorematic, +namely, when it is used simply as a word, to the neglect of its proper +meaning. Thus we can say--'"Swiftly" is an adverb.' 'Swiftly' in this +sense is really no more than the proper name for a particular +word. This sense is technically known as the 'suppositio materialis' +of a word. + + + + +CHAPTER II. + +_Of the Division of Things._ + + +§ 77. Before entering on the divisions of terms it is necessary to +advert for a moment to a division of the things whereof they are +names. + +§ 78. By a 'thing' is meant simply an object of thought--whatever one +can think about. + +§ 79. Things are either Substances or Attributes. Attributes may be +sub-divided into Qualities and Relations. + + Thing + _______________|_______________ + | | + Substance Attribute + _____________|____________ + | | + Quality Relation + +§ 80. A Substance is a thing which can be conceived to exist by +itself. All bodies are material substances. The soul, as a thinking +subject, is an immaterial substance. + +§ 81. An Attribute is a thing which depends for its existence upon a +substance, e.g. greenness, hardness, weight, which cannot be +conceived to exist apart from green, hard, and heavy substances. + +§ 82. A Quality is an attribute which does not require more than one +substance for its existence. The attributes just mentioned are +qualities. There might be greenness, hardness, and weight, if there +were only one green, hard and heavy substance in the universe. + +§ 83. A Relation is an attribute which requires two or more substances +for its existence, e.g. nearness, fatherhood, introduction. + +§ 84. When we say that a substance can be conceived to exist by +itself, what is meant is that it can be conceived to exist +independently of other substances. We do not mean that substances can +be conceived to exist independently of attributes, nor yet out of +relation to a mind perceiving them. Substances, so far as we can know +them, are only collections of attributes. When therefore we say that +substances can be conceived to exist by themselves, whereas attributes +are dependent for their existence upon substances, the real meaning of +the assertion reduces itself to this, that it is only certain +collections of attributes which can be conceived to exist +independently; whereas single attributes depend for their existence +upon others. The colour, smoothness or solidity of a table cannot be +conceived apart from the extension, whereas the whole cluster of +attributes which constitutes the table can be conceived to exist +altogether independently of other 'such clusters. We can imagine a +table to exist, if the whole material universe were annihilated, and +but one mind left to perceive it. Apart from mind, however, we cannot +imagine it: since what we call the attributes of a material substance +are no more than the various modes in which we find our minds +affected. + +§ 85. The above division of things belongs rather to the domain of +metaphysics than of logic: but it is the indispensable basis of the +division of terms, to which we now proceed. + + + + +CHAPTER III. + +_Of the Division of Terms._ + + +§ 86. The following scheme presents to the eye the chief divisions of +terms. + + Term + Division of terms according to their place in thought. + Subject-Term + Attributive + + according to the kind of thing signified. + Abstract + Concrete + + according to Quantity in Extension. + Singular + Common + + according to Quality. + Positive + Privative + Negative + + according to number of meanings. + Univocal + Equivocal + + according to number of things involved in the name. + Absolute + Relative + + according to number of quantities. + Connotative + Non-connotative + + + +_Subject-term and Attributive._ + + +§ 87. By a Subject-term is meant any term which is capable of standing +by itself as a subject, e.g. 'ribbon,' 'horse.' + +§ 88. Attributives can only be used as predicates, not as subjects, +e.g. 'cherry-coloured,' 'galloping.' These can only be used in +conjunction with other words (syncategorematically) to make up a +subject. Thus we can say 'A cherry-coloured ribbon is becoming,' or 'A +galloping horse is dangerous.' + +§ 89. Attributives are contrivances of language whereby we indicate +that a subject has a certain attribute. Thus, when we say 'This paper +is white,' we indicate that the subject 'paper' possesses the +attribute whiteness. Logic, however, also recognises as attributives +terms which signify the non-possession of attributes. 'Not-white' is +an attributive equally with 'white.' + +§ 90. An Attributive then may be defined as a term which signifies the +possession, or non-possession, of an attribute by a subject. + +§ 91. It must be carefully noticed that attributives are not names of +attributes, but names of the things which possess the attributes, in +virtue of our knowledge that they possess them. Thus 'white' is the +name of all the things which possess the attribute whiteness, and +'virtuous' is a name; not of the abstract quality, virtue, itself, but +of the men and actions which possess it. It is clear that a term can +only properly be said to be a name of those things whereof it can be +predicated. Now, we cannot intelligibly predicate an attributive of +the abstract quality, or qualities, the possession of which it +implies. We cannot, for instance, predicate the term 'learned' of the +abstract quality of learning: but we may predicate it of the +individuals, Varro and Vergil. Attributives, then, are to be regarded +as names, not of the attributes which they imply, but of the things in +which those attributes are found. + +§ 92. Attributives, however, are names of things in a less direct way +than that in which subject-terms may be the names of the same +things. Attributives are names of things only in predication, whereas +subject-terms are names of things in or out of predication. The terms +'horse' and 'Bucephalus' are names of certain things, in this case +animals, whether we make any statement about them or not: but the +terms 'swift' and 'fiery' only become names of the same things in +virtue of being predicable of them. When we say 'Horses are swift' or +'Bucephalus was fiery,' the terms 'swift' and 'fiery' become names +respectively of the same things as 'horse' and 'Bucephalus.' This +function of attributives as names in a secondary sense is exactly +expressed by the grammatical term 'noun adjective.' An attributive is +not directly the name of anything. It is a name added on in virtue of +the possession by a given thing of a certain attribute, or, in some +cases, the non-possession. + +§ 93. Although attributives cannot be used as subjects, there is +nothing to prevent a subject-term from being used as a predicate, and +so assuming for the time being the functions of an attributive. When +we say 'Socrates was a man,' we convey to the mind the idea of the +same attributes which are implied by the attributive 'human.' But +those terms only are called attributives which can never be used +except as predicates. + +§ 94. This division into Subject-terms and Attributives may be +regarded as a division of terms according to their place in +thought. Attributives, as we have seen, are essentially predicates, +and can only be thought of in relation to the subject, whereas the +subject is thought of for its own sake. + + + +_Abstract and Concrete Terms_. + + +§ 95. An Abstract Term is the name of an attribute, e.g. whiteness +[Footnote: Since things cannot be spoken of except by their names, +there is a constantly recurring source of confusion between the thing +itself and the name of it. Take for instance 'whiteness.' The +attribute whiteness is a thing, the word 'whiteness' is a term.], +multiplication, act, purpose, explosion. + +§ 96. A Concrete Term is the name of a substance, e.g. a man, this +chair, the soul, God. + +§ 97. Abstract terms are so called as being arrived at by a process of +Abstraction. What is meant by Abstraction will be clear from a single +instance. The mind, in contemplating a number of substances, may draw +off, or abstract, its attention from all their other characteristics, +and fix it only on some point, or points, which they have in +common. Thus, in contemplating a number of three-cornered objects, we +may draw away our attention from all their other qualities, and fix it +exclusively upon their three-corneredness, thus constituting the +abstract notion of 'triangle.' Abstraction may be performed equally +well in the case of a single object: but the mind would not originally +have known on what points to fix its attention except by a comparison +of individuals. + +§ 98. Abstraction too may be performed upon attributes as well as +substances. Thus, having by abstraction already arrived at the notion +of triangle, square, and so on, we may fix our attention upon what +these have in common, and so rise to the higher abstraction of +'figure.' As thought becomes more complex, we may have abstraction on +abstraction and attributes of attributes. But, however many steps may +intervene, attributes may always be traced back to substances at +last. For attributes of attributes can mean at bottom nothing but the +co-existence of attributes in, or in connection with, the same +substances. + +§ 99. We have said that abstract terms are so called, as being arrived +at by abstraction: but it must not be inferred from this statement +that all terms which are arrived at by abstraction are abstract. If +this were so, all names would be abstract except proper names of +individual substances. All common terms, including attributives, are +arrived at by abstraction, but they are not therefore abstract terms. +Those terms only are called abstract, which cannot be applied to +substances at all. The terms 'man' and 'human' are names of the same +substance of which Socrates is a name. Humanity is a name only of +certain attributes of that substance, namely those which are shared by +others. All names of concrete things then are concrete, whether they +denote them individually or according to classes, and whether directly +and in themselves, or indirectly, as possessing some given attribute. + +§ 100. By a 'concrete thing' is meant an individual Substance +conceived of with all its attributes about it. The term is not +confined to material substances. A spirit conceived of under personal +attributes is as concrete as plum-pudding. + +§ 101. Since things are divided exhaustively into substances and +attributes, it follows that any term which is not the name of a thing +capable of being conceived to exist by itself, must be an abstract +term. Individual substances can alone be conceived to exist by +themselves: all their qualities, actions, passions, and +inter-relations, all their states, and all events with regard to them, +presuppose the existence of these individual substances. All names +therefore of such things as those just enumerated are abstract +terms. The term 'action,' for instance, is an abstract term. For how +could there be action without an agent? The term 'act' also is equally +abstract for the same reason. The difference between 'action' and +'act' is not the difference between abstract and concrete, but the +difference between the name of a process and the name of the +corresponding product. Unless acts can be conceived to exist without +agents they are as abstract as the action from which they result. + +§ 102. Since every term must be either abstract or concrete, it may be +asked--Are attributives abstract or concrete? The answer of course +depends upon whether they are names of substances or names of +attributes. But attributives, it must be remembered, are never +directly names of anything, in the way that subject-terms are; they +are only names of things in virtue of being predicated of +them. Whether an attributive is abstract or concrete, depends on the +nature of the subject of which it is asserted or denied. When we say +'This man is noble,' the term 'noble' is concrete, as being the name +of a substance: but when we say 'This act is noble,' the term 'noble' +is abstract, as being the name of an attribute. + +§ 103. The division of terms into Abstract and Concrete is based upon +the kind of thing signified. It involves no reference to actual +existence. There are imaginary as well as real substances. Logically a +centaur is as much a substance as a horse. + + + +_Terms._ + + +§ 104. A Singular Term is a name which can be applied, in the same +sense, to one thing only, e.g. 'John,' 'Paris,' 'the capital of +France,' 'this pen.' + +§ 105. A Common Term is a name which can be applied, in the same +sense, to a class of things, e.g. 'man,' 'metropolis,' 'pen.' + +In order that a term may be applied in the same sense to a number of +things, it is evident that it must indicate attributes which are +common to all of them. The term 'John' is applicable to a number of +things, but not in the same sense, as it does not indicate attributes. + +§ 106. Common terms are formed, as we have seen already (§ 99), by +abstraction, i. e. by withdrawing the attention from the attributes in +which individuals differ, and concentrating it upon those which they +have in common. + +§ 107. A class need not necessarily consist of more than two +things. If the sun and moon were the only heavenly bodies in the +universe, the word 'heavenly body' would still be a common term, as +indicating the attributes which are possessed alike by each. + +§ 108. This being so, it follows that the division of terms into +singular and common is as exhaustive as the preceding ones, since a +singular term is the name of one thing and a common term of more than +one. It is indifferent whether the thing in question be a substance or +an attribute; nor does it matter how complex it may be, so long as it +is regarded by the mind as one. + +§ 109. Since every term must thus be either singular or common, the +members of the preceding divisions must find their place under one or +both heads of this one. Subject-terms may plainly fall under either +head of singular or common: but attributives are essentially common +terms. Such names as 'green,' 'gentle,' 'incongruous' are applicable, +strictly in the same sense, to all the things which possess the +attributes which they imply. + +§ 110. Are abstract terms then, it may be asked, singular or common? +To this question we reply--That depends upon how they are used. The +term 'virtue,' for instance, in one sense, namely, as signifying moral +excellence in general, without distinction of kind, is strictly a +singular term, as being the name of one attribute: but as applied to +different varieties of moral excellence--justice, generosity, +gentleness and so on--it is a common term, as being a name which is +applicable, in the same sense, to a class of attributes. Similarly the +term 'colour,' in a certain sense, signifies one unvarying attribute +possessed by bodies, namely, the power of affecting the eye, and in +this sense it is a singular term: but as applied to the various ways +in which the eye may be affected, it is evidently a common term, being +equally applicable to red, blue, green, and every other colour. As +soon as we begin to abstract from attributes, the higher notion +becomes a common term in reference to the lower. By a 'higher notion' +is meant one which is formed by a further process of abstraction. The +terms 'red,' 'blue,' 'green,' etc., are arrived at by abstraction from +physical objects; 'colour' is arrived at by abstraction from them, and +contains nothing, but what is common to all. It therefore applies in +the same sense to each, and is a common term in relation to them. + +§ 111. A practical test as to whether an abstract term, in any given +case, is being used as a singular or common term, is to try whether +the indefinite article or the sign of the plural can be attached to +it. The term 'number,' as the name of a single attribute of things, +admits of neither of these adjuncts: but to talk of 'a number' or 'the +numbers, two, three, four,' etc., at once marks it as a common +term. Similarly the term 'unity' denotes a single attribute, admitting +of no shades of distinction: but when a writer begins to speak of 'the +unities' he is evidently using the word for a class of things of some +kind or other, namely, certain dramatical proprieties of composition. + + + +Proper _Names_ and _Designations_. + + +§ 112. Singular terms may be subdivided into Proper Names and +Designations. + +§ 113. A Proper Name is a permanent singular term applicable to a +thing in itself; a Designation is a singular term devised for the +occasion, or applicable to a thing only in so far as it possesses some +attribute. + +§ 114. 'Homer' is a proper name; 'this man,' 'the author of the Iliad' +are designations. + +§ 115. The number of things, it is clear, is infinite. For, granting +that the physical universe consists of a definite number of +atoms--neither one more nor one less--still we are far from having +exhausted the possible number of things. All the manifold material +objects, which are made up by the various combinations of these atoms, +constitute separate objects of thought, or things, and the mind has +further an indefinite power of conjoining and dividing these objects, +so as to furnish itself with materials of thought, and also of fixing +its attention by abstraction upon attributes, so as to regard them as +things, apart from the substances to which they belong. + +§ 116. This being so, it is only a very small number of things, which +are constantly obtruding themselves upon the mind, that have singular +terms permanently set apart to denote them. Human beings, some +domestic animals, and divisions of time and place, have proper names +assigned to them in most languages, e.g. 'John,' 'Mary,' 'Grip,' +'January,' 'Easter,' 'Belgium,' 'Brussels,' 'the Thames,' 'Ben-Nevis.' +Besides these, all abstract terms, when used without reference to +lower notions, are of the nature of proper names, being permanently +set apart to denote certain special attributes, e.g. 'benevolence,' +'veracity,' 'imagination,' 'indigestibility, 'retrenchment.' + +§ 117. But the needs of language often require a singular term to +denote some thing which has not had a proper name assigned to it. This +is effected by taking a common term, and so limiting it as to make it +applicable, under the given circumstances, to one thing only. Such a +limitation may be effected in English by prefixing a demonstrative or +the definite article, or by appending a description, e.g. 'this pen,' +'the sofa,' 'the last rose of summer.' When a proper name is unknown, +or for some reason, unavailable, recourse may be had to a designation, +e.g. 'the honourable member who spoke last but one.' + + + +_Collective Terms_. + + +§ 118. The division of terms into singular and common being, like +those which have preceded it, fundamental and exhaustive, there is +evidently no room in it for a third class of Collective Terms. Nor is +there any distinct class of terms to which that name can be given. The +same term may be used collectively or distributively in different +relations. Thus the term 'library,' when used of the books which +compose a library, is collective; when used of various collections of +books, as the Bodleian, Queen's library, and so on, it is +distributive, which, in this case, is the same thing as being a common +term. + +§ 119, The distinction between the collective and distributive use of +a term is of importance, because the confusion of the two is a +favourite source of fallacy. When it is said 'The plays of Shakspeare +cannot be read in a day,' the proposition meets with a very different +measure of acceptance according as its subject is understood +collectively or distributively. The word 'all' is perfectly ambiguous +in this respect. It may mean all together or each separately--two +senses which are distinguished in Latin by 'totus' or 'cunctus,' for +the collective, and 'omnis' for the distributive use. + +§ 120. What is usually meant however when people speak of a collective +term is a particular kind of singular term. + +§ 121. From this point of view singular terms may be subdivided into +Individual and Collective, by an Individual Term being meant the name +of one object, by a Collective Term the name of several considered as +one. 'This key' is an individual term; 'my bunch of keys' is a +collective term. + +§ 122. A collective term is quite as much the name of one thing as an +individual term is, though the thing in question happens to be a +group. A group is one thing, if we choose to think of it as one. For +the mind, as we have already seen, has an unlimited power of forming +its own things, or objects of thought. Thus a particular peak in a +mountain chain is as much one thing as the chain itself, though, +physically speaking, it is inseparable from it, just as the chain +itself is inseparable from the earth's surface. In the same way a +necklace is as much one thing as the individual beads which compose +it. + +§ 123. We have just seen that a collective term is the name of a group +regarded as one thing: but every term which is the name of such a +group is not necessarily a collective term. 'London,' for instance, is +the name of a group of objects considered as one thing. But 'London' +is not a collective term, whereas 'flock,' 'regiment,' and 'senate' +are. Wherein then lies the difference? It lies in this--that flock, +regiment and senate are groups composed of objects which are, to a +certain extent, similar, whereas London is a group made up of the most +dissimilar objects--streets and squares and squalid slums, fine +carriages and dirty faces, and so on. In the case of a true collective +term all the members of the group will come under some one common +name. Thus all the members of the group, flock of sheep, come under +the common name 'sheep,' all the members of the group 'regiment' under +the common name, 'soldier,' and so on. + +§ 124. The subdivision of singular terms into individual and +collective need not be confined to the names of concrete things. An +abstract term like 'scarlet,' which is the name of one definite +attribute, may be reckoned 'individual,' while a term like 'human +nature,' which is the name of a whole group of attributes, would more +fitly be regarded as collective. + +§ 126. The main division of terms, which we have been discussing, into +singular and collective, is based upon their Quantity in +Extension. This phrase will be explained presently. + +§ 126. We come now to a threefold division of terms into Positive, +Privative and Negative. It is based upon an implied two-fold division +into positive and non-positive, the latter member being subdivided +into Privative and Negative. + + Term + _______________|_______________ + | | + Positive Non-Positive + _____________|____________ + | | + Privative Negative + +If this division be extended, as it sometimes is, to terms in general, +a positive term must be taken to mean only the definite, or +comparatively definite, member of an exhaustive division in accordance +with the law of excluded middle (§ 25). Thus 'Socrates' and 'man' are +positive, as opposed to 'not-Socrates' and 'not-man.' + +§ 127. The chief value of the division, however, and especially of the +distinction drawn between privative and negative terms, is in relation +to attributives. + +From this point of view we may define the three classes of terms as +follows: + +A Positive Term signifies the presence of an attribute, e.g.: 'wise,' +'full.' + +A Negative Term signifies merely the absence of an attribute, +e.g. 'not-wise,' 'not-full.' + +A Privative Term signifies the absence of an attribute in a subject +capable of possessing it, e.g. 'unwise,' 'empty'. [Footnote: A +privative term is usually defined to mean one which signifies the +absence of an attribute where it was once possessed, or might have +been expected to be present, e.g. 'blind.' The utility of the slight +extension of meaning here assigned to the expression will, it is +hoped, prove its justification.] + +§ 128. Thus a privative term stands midway in meaning between the +other two, being partly positive and partly negative--negative in so +far as it indicates the absence of a certain attribute, positive in so +far as it implies that the thing which is declared to lack that +attribute is of such a nature as to be capable of possessing it. A +purely negative term conveys to the mind no positive information at +all about the nature of the thing of which it is predicated, but +leaves us to seek for it among the universe of things which fail to +exhibit a given attribute. + +A privative term, on the other hand, restricts us within a definite +sphere. The term 'empty' restricts us within the sphere of things +which are capable of fulness, that is, if the term be taken in its +literal sense, things which possess extension in three dimensions. + +§ 129. A positive and a negative term, which have the same matter, +must exhaust the universe between them, e.g. 'white' and 'not-white,' +since, according to the law of excluded middle, everything must be +either one or the other. To say, however, that a thing is 'not-white' +is merely to say that the term 'white' is inapplicable to it. +'Not-white' may be predicated of things which do not possess extension +as well as of those which do. Such a pair of terms as 'white' and +'not-white,' in their relation to one another, are called +Contradictories. + +§ 130. Contrary terms must be distinguished from +contradictory. Contrary terms are those which are most opposed under +the same head. Thus 'white' and 'black' are contrary terms, being the +most opposed under the same head of colour. 'Virtuous' and 'vicious' +again are contraries, being the most opposed under the same head of +moral quality. + +§ 131. A positive and a privative term in the same matter will always +be contraries, e.g. 'wise' and 'unwise,' 'safe' and 'unsafe': but +contraries do not always assume the shape of positive and privative +terms, but may both be positive in form, e.g. 'wise' and 'foolish,' +'safe' and 'dangerous.' + +§ 132. Words which are positive in form are often privative in +meaning, and vice versâ. This is the case, for instance, with the word +'safe,' which connotes nothing more than the absence of danger. We +talk of a thing involving 'positive danger' and of its being +'positively unsafe' to do so and so. 'Unhappy,' on the other hand, +signifies the presence of actual misery. Similarly in Latin 'inutilis' +signifies not merely that there is no benefit to be derived from a +thing, but that it is _positively injurious_. All such questions, +however, are for the grammarian or lexicographer, and not for the +logician. For the latter it is sufficient to know that corresponding +to every term which signifies the presence of some attribute there may +be imagined another which indicates the absence of the same attribute, +where it might be possessed, and a third which indicates its absence, +whether it might be possessed or not. + +§ 133. Negative terms proper are formed by the prefix 'not-' or +'non-,' and are mere figments of logic. We do not in practice require +to speak of the whole universe of objects minus those which possess a +given attribute or collection of attributes. We have often occasion to +speak of things which might be wise and are not, but seldom, if ever, +of all things other than wise. + +§ 134. Every privative attributive has, or may have, a corresponding +abstract term, and the same is the case with negatives: for the +absence of an attribute, is itself an attribute. Corresponding to +'empty,' there is 'emptiness'; corresponding to 'not-full' there may +be imagined the term 'not-fulness.' + +§ 135. The contrary of a given term always involves the contradictory, +but it involves positive elements as well. Thus 'black' is +'not-white,' but it is something more besides. Terms which, without +being directly contrary, involve a latent contradiction, are called +Repugnant, e.g. 'red' and 'blue.' All terms whatever which signify +attributes that exclude one another may be called Incompatible. + +§ 136. The preceding division is based on what is known as the Quality +of terms, a positive term being said to differ in quality from a +non-positive one. + + + +_Univocal and Equivocal Terms_. + + +§ 137. A term is said to be Univocal, when it has one and the same +meaning wherever it occurs. A term which has more than one meaning is +called Equivocal. 'Jam-pot,' 'hydrogen' are examples of univocal +terms; 'pipe' and 'suit' of equivocal. + +§ 138. This division does not properly come within the scope of logic, +since it is a question of language, not of thought. From the +logician's point of view an equivocal term is two or more different +terms, for the definition in each sense would be different. + +§ 139. Sometimes a third member is added to the same division under +the head of Analogous Terms. The word 'sweet,' for instance, is +applied by analogy to things so different in their own nature as a +lump of sugar, a young lady, a tune, a poem, and so on. Again, because +the head is the highest part of man, the highest part of a stream is +called by analogy 'the head.' It is plainly inappropriate to make a +separate class of analogous terms. Rather, terms become equivocal by +being extended by analogy from one thing to another. + + + +_Absolute and Relative Terms_. + + +§ 140. An Absolute term is a name given to a thing without reference +to anything else. + +§ 141. A Relative term is a name given to a thing with direct +reference to some other thing. + +§ 142. 'Hodge' and 'man' are absolute terms. 'Husband' 'father,' +'shepherd' are relative terms. 'Husband' conveys a direct reference to +'wife,' 'father' to 'Child,' 'shepherd' to 'sheep.' Given one term of +a relation, the other is called the correlative, e.g. 'subject' is +the correlative of 'ruler,' and conversely 'ruler' of 'subject.' The +two terms are also spoken of as a pair of correlatives. + +§ 143. The distinction between relative and absolute applies to +attributives as well as subject-terms. 'Greater,' 'near, 'like,' are +instances of attributives which everyone would recognise as relative. + +§ 144. A relation, it will be remembered, is a kind of attribute, +differing from a quality in that it necessarily involves more +substances than one. Every relation is at bottom a fact, or series of +facts, in which two or more substances play a part. A relative term +connotes this fact or facts from the point of view of one of the +substances, its correlative from that of the other. Thus 'ruler' and +'subject' imply the same set of facts, looked at from opposite points +of view. The series of facts itself, regarded from either side, is +denoted by the corresponding abstract terms, 'rule 'and 'subjection.' + +§ 145. It is a nice question whether the abstract names of relations +should themselves be considered relative terms. Difficulties will +perhaps be avoided by confining the expression 'relative _term_' +to names of concrete things. 'Absolute,' it must be remembered, is a +mere negative of 'relative,' and covers everything to which the +definition of the latter does not strictly apply. Now it can hardly be +said that 'rule' is a name given to a certain abstract thing with +direct reference to some other thing, namely, subjection. Rather +'rule' and 'subjection' are two names for identically the same series +of facts, according to the side from which we look at them. 'Ruler' +and 'subject,' on the other hand, are names of two distinct +substances, but each involving a reference to the other. + +§ 146. This division then may be said to be based on the number of +things involved in the name. + + + +_Connotative and Non-Connotative Terms._ + + +§ 147. Before explaining this division, it is necessary to treat of +what is called the Quantity of Terms. + + + +_Quantity of Terms._ + + +§ 148. A term is possessed of quantity in two ways-- + + (1) In Extension; + + (2) In Intension. + +§ 149. The Extension of a term is the number of things to which it +applies. + +§ 150. The Intension of a term is the number of attributes which it +implies. + +§ 151. It will simplify matters to bear in mind that the intension of +a term is the same thing as its meaning. To take an example, the term +'man' applies to certain things, namely, all the members of the human +race that have been, are, or ever will be: this is its quantity in +extension. But the term 'man' has also a certain meaning, and implies +certain attributes--rationality, animality, and a definite bodily +shape: the sum of these attributes constitutes its quantity in +intension. + +§ 152. The distinction between the two kinds of quantity possessed by +a term is also conveyed by a variety of expressions which are here +appended. + +Extension = breadth = compass = application = denotation. + +Intension = depth = comprehension = implication = connotation. + +Of these various expressions, 'application' and 'implication' have the +advantage of most clearly conveying their own meaning. 'Extension' and +'intension,' however, are more usual; and neither 'implication' nor +'connotation' is quite exact as a synonym for 'intension.' (§ 164.) + +§ 153. We now return to the division of terms into connotative and +non-connotative. + +§ 154. A term is said to connote attributes, when it implies certain +attributes at the same time that it applies to certain things distinct +therefrom. [Footnote: Originally 'connotative' was used in the same +sense in which we have used 'attributive,' for a word which directly +signifies the presence of an attribute and indirectly applies to a +subject. In this, its original sense, it was the subject which was +said to be connoted, and not the attribute.] + +§ 155. A term which possesses both extension and intension, distinct +from one another, is connotative. + +§ 156. A term which possesses no intension (if that be possible) or in +which extension and intension coincide is non-connotative. + +§ 157. The subject-term, 'man,' and its corresponding attributive, +'human,' have both extension and intension, distinct from one +another. They are therefore connotative. But the abstract term, +'humanity,' denotes the very collection of attributes, which was +before connoted by the concrete terms, 'man' and 'human.' In this +case, therefore, extension and intension coincide, and the term is +non-connotative. + +§ 158. The above remark must be understood to be limited to abstract +terms in their singular sense. When employed as common terms, abstract +terms possess both extension and intension distinct from one +another. Thus the term 'colour' applies to red, blue, and yellow, and +at the same time implies (i.e. connotes), the power of affecting the +eye. + +§ 159. Since all terms are names of things, whether substances or +attributes, it is clear that all terms must possess extension, though +the extension of singular terms is the narrowest possible, as being +confined to one thing. + +§ 160. Are there then any terms which possess no intension? To ask +this, is to ask--Are there any terms which have absolutely no meaning? +It is often said that proper names are devoid of meaning, and the +remark is, in a certain sense, true. When we call a being by the name +'man,' we do so because that being possesses human attributes, but +when we call the same being by the name, 'John,' we do not mean to +indicate the presence of any Johannine attributes. We simply wish to +distinguish that being, in thought and language, from other beings of +the same kind. Roughly speaking, therefore, proper names are devoid of +meaning or intension. But no name can be entirely devoid of +meaning. For, even setting aside the fact, which is not universally +true, that proper names indicate the sex of the owner, the mere act of +giving a name to a thing implies at least that the thing exists, +whether in fact or thought; it implies what we may call 'thinghood': +so that every term must carry with it some small amount of intension. + +§ 161. From another point of view, however, proper names possess more +intension than any other terms. For when we know a person, his name +calls up to our minds all the individual attributes with which we are +familiar, and these must be far more numerous than the attributes +which are conveyed by any common term which can be applied to +him. Thus the name 'John' means more to a person who knows him than +'attorney,' 'conservative,' 'scamp,' of 'vestry-man,' or any other +term which may happen to apply to him. This, however, is the acquired +intension of a term, and must be distinguished from the original +intension. The name 'John' was never meant to indicate the attributes +which its owner has, as a matter of fact, developed. He would be John +all the same, if he were none of these. + +§ 162. Hitherto we have been speaking only of christening-names, but +it is evident that family names have a certain amount of connotation +from the first. For when we dub John with the additional appellation +of Smith, we do not give this second name as a mere individual mark, +but intend thereby to indicate a relationship to other persons. The +amount of connotation that can be conveyed by proper names is very +noticeable in the Latin language. Let us take for an example the full +name of a distinguished Roman--Publius Cornelius Scipio Ćmilianus +Africanus minor. Here it is only the prćnomen, Publius, that can be +said to be a mere individual mark, and even this distinctly indicates +the sex of the owner. The nomen proper, Cornelius, declares the wearer +of it to belong to the illustrious gens Cornelia. The cognomen, +Scipio, further specifies him as a member of a distinguished family in +that gens. The agnomen adoptivum indicates his transference by +adoption from one gens to another. The second agnomen recalls the +fact of his victory over the Carthaginians, while the addition of the +word 'minor' distinguishes him from the former wearer of the same +title. The name, instead of being devoid of meaning, is a chapter of +history in itself. Homeric epithets, such as 'The Cloud-compeller,' +'The Earth-shaker' are instances of intensive proper names. Many of +our own family names are obviously connotative in their origin, +implying either some personal peculiarity, e.g. Armstrong, Cruikshank, +Courteney; or the employment, trade or calling of the original bearer +of the name, Smith, Carpenter, Baker, Clark, Leach, Archer, and so on; +or else his abode, domain or nationality, as De Caen, De Montmorency, +French, Langley; or simply the fact of descent from some presumably +more noteworthy parent, as Jackson, Thomson, Fitzgerald, O'Connor, +Macdonald, Apjohn, Price, Davids, etc. The question, however, whether +a term is connotative or not, has to be decided, not by its origin, +but by its use. We have seen that there are some proper names which, +in a rough sense, may be said to possess no intension. + +§ 163. The other kind of singular terms, namely, designations (§ 113) +are obviously connotative. We cannot employ even the simplest of them +without conveying more or less information about the qualities of the +thing which they are used to denote. When, for instance, we say 'this +table,' 'this book,' we indicate the proximity to the speaker of the +object in question. Other designations have a higher degree of +intension, as when we say 'the present prime minister of England,' +'the honourable member who brought forward this motion to-night.' +Such terms have a good deal of significance in themselves, apart from +any knowledge we may happen to possess of the individuals they denote. + +§ 164. We have seen that, speaking quite strictly, there are no terms +which are non-connotative: but, for practical purposes, we may apply +the expression to proper names, on the ground that they possess no +intension, and to singular abstract terms on the ground that their +extension and intension coincide. In the latter case it is indifferent +whether we call the quantity extension or intension. Only we cannot +call it 'connotation,' because that implies two quantities distinct +from one another. A term must already denote a subject before it can +be said to connote its attributes. + +§ 165. The division of terms into connotative and non-connotative is +based on their possession of one quantity or two. + + + + +CHAPTER IV. + +_Of the Law of Inverse Variation of Extension and Intension._ + + +§ 166. In a series of terms which fall under one another, as the +extension decreases, the intension increases, and vice versâ. Take for +instance the following series-- + + Thing + | + Substance + | + Matter + | + Organism + | + Animal + | + Vertebrate + | + Mammal + | + Ruminant + | + Sheep + | + This sheep. + +Here the term at the top possesses the widest possible extension, +since it applies to everything. But at the same time it possesses the +least possible amount of intension, implying nothing more than mere +existence, whether in fact or thought. On the other hand, the term at +the bottom possesses the greatest amount of intension, since it +implies all the attributes of, an individual superadded to those of +the class to which it belongs: but its extension is the narrowest +possible, being limited to one thing. + +§ 167. At each step in the descent from the term at the top, which is +called the 'Summum genus,' to the individual, we decrease the +extension by increasing the intension. Thus by adding on to the bare +notion of a thing the idea of independent existence, we descend to the +term 'substance,' This process is known as Determination, or +Specialisation. + +§ 168. Again, by withdrawing our attention from the individual +characteristics of a particular sheep, and fixing it upon those which +are common to it with other animals of the same kind, we arrive at the +common term, 'sheep.' Here we have increased the extension by +decreasing the intension. This process is known as Generalisation. + +§ 169. Generalisation implies abstraction, but we may have abstraction +without generalisation. + +§ 170. The following example is useful, as illustrating to the eye how +a decrease of extension is accompanied by an increase of intension. At +each step of the descent here we visibly tack on a fresh +attribute. [Footnote: This example is borrowed from Professor Jevons.] + + Ship + | + Steam-ship + | + Screw steam-ship + | + Iron screw steam-ship + | + British iron screw steam-ship. + +Could we see the classes denoted by the names the pyramid would be +exactly inverted. + +§ 171. The law of inverse variation of extension and intension must of +course be confined to the inter-relations of a series of terms of +which each can be predicated of the other until we arrive at the +bottom of the scale. It is not meant to apply to the extension and +intension of the same term. The increase of population does not add to +the meaning of 'baby.' + + + + + + +PART II.--OF PROPOSITIONS. + + + + +CHAPTER I. + +_Of the Proposition as distinguished from Other Sentences_. + + +§ 172. As in considering the term, we found occasion to distinguish it +from words generally, so now, in considering the proposition, it will +be well to begin by distinguishing it from other sentences. + +§ 173. Every proposition is a sentence, but every sentence is not a +proposition. + +§ 174. The field of logic is far from being conterminous with that of +language. Language is the mirror of man's whole nature, whereas logic +deals with language only so far as it gives clothing to the products +of thought in the narrow sense which we have assigned to that term. +Language has materials of every sort lying strewn about, among which +the logician has to seek for his proper implements. + +§ 175. Sentences may be employed for a variety of purposes-- + + (1) To ask a question; + + (2) To give an order; + + (3) To express a feeling; + + (4) To make a statement. + +These various uses give rise respectively to + + (1) The Interrogative Sentence; + + (2) The Imperative Sentence; + + (3) The Exclamatory Sentence; + + (4) The Enunciative Sentence; Indicative Potential. + +It is with the last of these only that logic is concerned. + +§ 176. The proposition, therefore, corresponds to the Indicative and +Potential, or Conditional, sentences of grammar. For it must be borne +in mind that logic recognises no difference between a statement of +fact and a supposition. 'It may rain to-morrow' is as much a +proposition as 'It is raining now.' + +§ 177. Leaving the grammatical aspect of the proposition, we must now +consider it from the purely logical point of view. + +§ 178. A proposition is a judgement expressed in words; and a +judgement is a direct comparison between two concepts. + +§ 179. The same thing may be expressed more briefly by saying that a +proposition is a direct comparison between two terms. + +§ 180. We say 'direct comparison,' because the syllogism also may be +described as a comparison between two terms: but in the syllogism the +two terms are compared indirectly, or by means of a third term. + +§ 181. A proposition may be analysed into two terms and a Copula, +which is nothing more than the sign of agreement or disagreement +between them. + +§ 182. The two terms are called the Subject and the Predicate (§ 58). + +§ 183. The Subject is that of which something is stated. + +§ 184. The Predicate is that which is stated of the subject. + +§ 185. Hence the subject is thought of for its own sake, and the +predicate for the sake of the subject. + + + + +CHAPTER II. + +Of _the Copula_. + + +§ 186. There are two kinds of copula, one for affirmative and one for +negative statements. + +§ 187. Materially the copula is expressed by some part of the verb 'to +be,' with or without the negative, or else is wrapped up in some +inflexional form of a verb. + +§ 188. The material form of the copula is an accident of language, and +a matter of indifference to logic. 'The kettle boils' is as logical a +form of expression as 'The kettle is boiling.' For it must be +remembered that the word 'is' here is a mere sign of agreement between +the two terms, and conveys no notion of actual existence. We may use +it indeed with equal propriety to express non-existence, as when we +say 'An idol is nothing.' + +§ 189. When the verb 'to be' expresses existence in fact it is known +in grammar as 'the substantive verb.' In this use it is predicate as +well as copula, as when we say 'God is,' which may be analysed, if we +please, into 'God is existent.' + +§ 190. We have laid down above that there are two kinds of copula, +affirmative and negative: but some logicians have maintained that the +copula is always affirmative. + +§ 191. What then, it may be asked, on this view, is the meaning of +negative propositions! To which the answer is, that a negative +proposition asserts an agreement between the subject and a negative +term. When, for instance, we say 'The whale is not a fish,' this would +be interpreted to mean 'The whale is a not-fish.' + +§ 192. Undoubtedly any negative proposition may be exhibited in an +affirmative form, since, by the law of excluded middle, given a pair +of contradictory terms, wherever the one can be asserted, the other +can be denied, and vice versâ. We shall find later on that this +principle gives rise to one of the forms of immediate inference. The +only question then can be, which is the more natural and legitimate +form of expression. It seems simpler to suppose that we assert the +agreement of 'whale' with 'not-fish' by implication only, and that +what we directly do is to predicate a disagreement between 'whale' and +the positive attributes connoted by 'fish.' For since 'not-fish' must +apply to every conceivable object of thought except those which fall +under the positive term 'fish,' to say that a whale is a 'not-fish,' +is to say that we have still to search for 'whale' throughout the +whole universe of being, minus a limited portion; which is only a more +clumsy way of saying that it is not to be found in that portion. + +§ 193. Again, the term 'not-fish' must be understood either in its +intension or in its extension. If it be understood in its intension, +what it connotes is simply the absence of the positive qualities which +constitute a fish, a meaning which is equally conveyed by the negative +form of proposition. We gain nothing in simplicity by thus confounding +assertion with denial. If, on the other hand, it is to be taken in +extension, this involves the awkwardness of supposing that the +predicative power of a term resides in its extensive capacity. + +§ 194. We therefore recognise predication as being of two +kinds--affirmation and negation--corresponding to which there are two +forms of copula. + +§ 195. On the other hand, other logicians have maintained that there +are many kinds of copula, since the copula must vary according to the +various degrees of probability with which we can assert or deny a +predicate of a subject. This view is technically known as the doctrine +of + + + +_The Modality of the Copula._ + + +§ 196. It may plausibly be maintained that the division of +propositions into affirmative and negative is not an exhaustive one, +since the result of an act of judgement is not always to lead the mind +to a clear assertion or a clear denial, but to leave it in more or +less doubt as to whether the predicate applies to the subject or +not. Instead of saying simply A is B, or A is not B, we may be led to +one of the following forms of proposition-- + + A is possibly B. + A is probably B. + A is certainly B. + +The adverbial expression which thus appears to qualify the copula is +known as 'the mode.' + +§ 197. When we say 'The accused may be guilty' we have a proposition +of very different force from 'The accused is guilty,' and yet the +terms appear to be the same. Wherein then does the difference lie? 'In +the copula' would seem to be the obvious reply. We seem therefore +driven to admit that there are as many different kinds of copula as +there are different degrees of assurance with which a statement may be +made. + +§ 198. But there is another way in which modal propositions may be +regarded. Instead of the mode being attached to the copula, it may be +considered as itself constituting the predicate, so that the above +propositions would be analysed thus-- + + That A is B, is possible. + That A is B, is probable. + That A is B, is certain. + +§ 199. The subject here is itself a proposition of which we predicate +various degrees of probability. In this way the division of +propositions into affirmative and negative is rendered exhaustive. For +wherever before we had a doubtful assertion, we have now an assertion +of doubtfulness. + +§ 200. If degrees of probability can thus be eliminated from the +copula, much more so can expressions of time, which may always be +regarded as forming part of the predicate. 'The sun will rise +to-morrow' may be analysed into 'The sun is going to rise to-morrow.' +In either case the tense belongs equally to the predicate. It is often +an awkward task so to analyse propositions relative to past or future +time as to bring out the copula under the form 'is' or 'is not': but +fortunately there is no necessity for so doing, since, as has been +said before (§ 188), the material form of the copula is a matter of +indifference to logic. Indeed in affirmative propositions the mere +juxtaposition of the subject and predicate is often sufficient to +indicate their agreement, e.g. 'Most haste, worst speed,' chalepha +tha kala. It is because all propositions are not affirmative that we +require a copula at all. Moreover the awkwardness of expression just +alluded to is a mere accident of language. In Latin we may say with +equal propriety 'Sol orietur cras' or 'Sol est oriturus cras'; while +past time may also be expressed in the analytic form in the case of +deponent verbs, as 'Caesar est in Galliam profectus'--'Caesar is gone +into Gaul.' + +§ 201. The copula then may always be regarded as pure, that is, as +indicating mere agreement or disagreement between the two terms of the +proposition. + + + + +CHAPTER III. + +_Of the Divisions of Propositions_. + + +§ 202. The most obvious and the most important division of +propositions is into true and false, but with this we are not +concerned. Formal logic can recognise no difference between true and +false propositions. The one is represented by the same symbols as the +other. + +§ 203. We may notice, however, in passing, that truth and falsehood +are attributes of propositions and of propositions only. For something +must be predicated, i.e. asserted or denied, before we can have +either truth or falsehood. Neither concepts or terms, on the one hand, +nor reasonings, on the other, can properly be said to be true or +false. In the mere notion of a Centaur or of a black swan there is +neither truth nor falsehood; it is not until we make some statement +about these things, such as that 'black swans are found in Australia,' +or 'I met a Centaur in the High Street yesterday,' that the question +of truth or falsehood comes in. In such expressions as a 'true friend' +or 'a false patriot' there is a tacit reference to propositions. We +mean persons of whom the terms 'friend' and 'patriot' are truly or +falsely predicated. Neither can we with any propriety talk of true or +false reasoning. Reasoning is either valid or invalid: it is only the +premisses of our reasonings, which are propositions, that can be true +or false. We may have a perfectly valid process of reasoning which +starts from a false assumption and lands us in a false conclusion. + +§ 204. All truth and falsehood then are contained in propositions; and +propositions are divided according to the Quality of the Matter into +true and false. But the consideration of the matter is outside the +sphere of formal or deductive Logic. It is the problem of inductive +logic to establish, if possible, a criterion of evidence whereby the +truth or falsehood of propositions may be judged (§ 2). + +§ 205. Another usual division of propositions is into Pure and Modal, +the latter being those in which the copula is modified by some degree +of probability. This division is excluded by the view which has just +been taken of the copula, as being always simply affirmative or simply +negative. + +§ 206. We are left then with the following divisions of +propositions-- + + Proposition + according to Form + Simple + + Complex + Conjunctive + Disjunctive + + Universal + Singular + General + + according to Matter + Verbal + Real + + according to Quantity + Universal + Singular + General + + Particular + Indefinite + (strictly) Particular + + according to Quality + Affirmative + Negative + + + +_Simple and Complex Propositions_. + + +§ 207. A Simple Proposition is one in which a predicate is directly +affirmed or denied of a subject, e.g. 'Rain is falling.' + +§ 208. A simple proposition is otherwise known as Categorical. + +§ 209. A Complex Proposition is one in which a statement is made +subject to some condition, e.g. 'If the wind drops, rain will fall.' + +§ 210. Hence the complex proposition is also known as Conditional. + +§ 211. Every complex proposition consists of two parts-- + + (1) Antecedent; + + (2) Consequent. + +§ 212. The Antecedent is the condition on which another statement is +made to depend. It precedes the other in the order of thought, but may +either precede or follow it in the order of language. Thus we may say +indifferently--'If the wind drops, we shall have rain' or 'We shall +have rain, if the wind drops.' + +§ 213. The Consequent is the statement which is made subject to some +condition. + +§ 214. The complex proposition assumes two forms, + + (1) If A is B, C is D. + +This is known as the Conjunctive or Hypothetical proposition. + + (2) Either A is B or C is D. + +This is known as the Disjunctive proposition. + +§ 215. The disjunctive proposition may also appear in +the form + + A is either B or C, + +which is equivalent to saying + + Either A is B or A is C; + +or again in the form + + Either A or B is C, + +which is equivalent to saying + + Either A is C or B is C. + +§ 216. As the double nomenclature may cause some confusion, a scheme +is appended. + + Proposition + ____________|_____________ + | | + Simple Complex + (Categorical) (Conditional) + ___________|__________ + | | + Conjunctive Disjunctive. + (Hypothetical) + +§ 217. The first set of names is preferable. 'Categorical' properly +means 'predicable' and 'hypothetical' is a mere synonym for +'conditional.' + +§ 218. Let us examine now what is the real nature of the statement +which is made in the complex form of proposition. When, for instance, +we say 'If the sky falls, we shall catch larks,' what is it that we +really mean to assert? Not that the sky will fall, and not that we +shall catch larks, but a certain connection between the two, namely, +that the truth of the antecedent involves the truth of the +consequent. This is why this form of proposition is called +'conjunctive,' because in it the truth of the consequent is conjoined +to the truth of the antecedent. + +§ 219. Again, when we say 'Jones is either a knave or a fool,' what is +really meant to be asserted is--'If you do not find Jones to be a +knave, you may be sure that he is a fool.' Here it is the falsity of +the antecedent which involves the truth of the consequent; and the +proposition is known as 'disjunctive,' because the truth of the +consequent is disjoined from the truth of the antecedent. + +§ 220. Complex propositions then turn out to be propositions about +propositions, that is, of which the subject and predicate are +themselves propositions. But the nature of a proposition never varies +in thought. Ultimately every proposition must assume the form 'A is, +or is not, B.' 'If the sky falls, we shall catch larks' may be +compressed into 'Sky-falling is lark-catching.' + +§ 221. Hence this division turns upon the form of expression, and may +be said to be founded on the simplicity or complexity of the terms +employed in a proposition. + +§ 222. In the complex proposition there appears to be more than one +subject or predicate or both, but in reality there is only a single +statement; and this statement refers, as we have Seen, to a certain +connection between two propositions. + +§ 223. If there were logically, and not merely grammatically, more +than one subject or predicate, there would be more than one +proposition. Thus when we say 'The Jews and Carthaginians were Semitic +peoples and spoke a Semitic language,' we have four propositions +compressed into a single sentence for the sake of brevity. + +§ 224. On the other hand when we say 'Either the Carthaginians were of +Semitic origin or argument from language is of no value in ethnology,' +we have two propositions only in appearance. + +§ 225. The complex proposition then must be distinguished from those +contrivances of language for abbreviating expression in which several +distinct statements are combined into a single sentence. + + + +_Verbal and Real Propositions_. + + +§ 226. A Verbal Proposition is one which states nothing more about the +subject than is contained in its definition, e.g. 'Man is an animal'; +'Men are rational beings.' + +§ 227. A Real Proposition states some fact not contained in the +definition of the subject, e.g. 'Some animals have four feet.' + +§ 228. It will be seen that the distinction between verbal and real +propositions assumes a knowledge of the precise meaning of terms, that +is to say, a knowledge of definitions. + +§ 229. To a person who does not know the meaning of terms a verbal +proposition will convey as much information as a real one. To say 'The +sun is in mid-heaven at noon,' though a merely verbal proposition, +will convey information to a person who is being taught to attach a +meaning to the word 'noon.' We use so many terms without knowing their +meaning, that a merely verbal proposition appears a revelation to many +minds. Thus there are people who are surprised to hear that the lion +is a cat, though in its definition 'lion' is referred to the class +'cat.' The reason of this is that we know material objects far better +in their extension than in their intension, that is to say, we know +what things a name applies to without knowing the attributes which +those things possess in common. + +§ 230. There is nothing in the mere look of a proposition to inform us +whether it is verbal or real; the difference is wholly relative to, +and constituted by, the definition of the subject. When we have +accepted as the definition of a triangle that it is 'a figure +contained by three sides,' the statement of the further fact that it +has three angles becomes a real proposition. Again the proposition +'Man is progressive' is a real proposition. For though his +progressiveness is a consequence of his rationality, still there is no +actual reference to progressiveness contained in the usually accepted +definition, 'Man is a rational animal.' + +§ 231. If we were to admit, under the term 'verbal proposition,' all +statements which, though not actually contained in the definition of +the subject, are implied by it, the whole body of necessary truth +would have to be pronounced merely verbal, and the most penetrating +conclusions of mathematicians set down as only another way of stating +the simplest axioms from which they started. For the propositions of +which necessary truth is composed are so linked together that, given +one, the rest can always follow. But necessary truth, which is arrived +at 'a priori,' that is, by the mind's own working, is quite as real as +contingent truth, which is arrived at 'a posteriori,' or by the +teachings of experience, in other words, through our own senses or +those of others. + +§ 232. The process by which real truth, which is other than deductive, +is arrived at 'a priori' is known as Intuition. E.g. The mind sees +that what has three sides cannot but have three angles. + +§ 233. Only such propositions then must be considered verbal as state +facts expressly mentioned in the definition. + +§ 234. Strictly speaking, the division of propositions into verbal and +real is extraneous to our subject: since it is not the province of +logic to acquaint us with the content of definitions. + +§ 235, The same distinction as between verbal and real proposition, is +conveyed by the expressions 'Analytical' and 'Synthetical,' or +'Explicative' and 'Ampliative' judgements. + +§ 236. A verbal proposition is called analytical, as breaking up the +subject into its component notions. + +§ 237. A real proposition is called synthetical, as attaching some new +notion to the subject. + +§ 238. Among the scholastic logicians verbal propositions were known +as 'Essential,' because what was stated in the definition was +considered to be of the essence of the subject, while real +propositions were known as 'Accidental.' + + + +_Universal AND PARTICULAR Propositions_. + + +§ 239. A Universal proposition is one in which it is evident from the +form that the predicate applies to the subject in its whole extent. + +§ 240. When the predicate does not apply to the subject in its whole +extent, or when it is not clear that it does so, the proposition is +called Particular. + +§ 241. To say that a predicate applies to a subject in its whole +extent, is to say that it is asserted or denied of all the things of +which the subject is a name. + +§ 242. 'All men are mortal' is a universal proposition. + +§ 243. 'Some men are black' is a particular proposition. So also is +'Men are fallible;' for here it is not clear from the form whether +'all' or only 'some' is meant. + +§ 244. The latter kind of proposition is known as Indefinite, and must +be distinguished from the particular proposition strictly so called, +in which the predicate applies to part only of the subject. + +§ 245. The division into universal and particular is founded on the +Quantity of propositions. + +§ 246. The quantity of a proposition is determined by the quantity in +extension of its subject. + +§ 247. Very often the matter of an indefinite proposition is such as +clearly to indicate to us its quantity. When, for instance, we say +'Metals are elements,' we are understood to be referring to all +metals; and the same thing holds true of scientific statements in +general. Formal logic, however, cannot take account of the matter of +propositions; and is therefore obliged to set down all indefinite +propositions as particular, since it is not evident from the form that +they are universal. + +§ 248. Particular propositions, therefore, are sub-divided into such +as are Indefinite and such as are Particular, in the strict sense of +the term. + +§ 249. We must now examine the sub-division of universal propositions +into Singular and General. + +§ 250. A Singular proposition is one which has a singular term for its +subject, e.g. 'Virtue is beautiful.' + +§ 251. A General proposition is one which has for its subject a common +term taken in its whole extent. + +§ 252. Now when we say 'John is a man' or 'This table is oblong,' the +proposition is quite as universal, in the sense of the predicate +applying to the whole of the subject, as when we say 'All men are +mortal.' For since a singular term applies only to one thing, we +cannot avoid using it in its whole extent, if we use it at all. + +§ 253. The most usual signs of generality in a proposition are the +words 'all,' 'every,' 'each,' in affirmative, and the words 'no,' +'none,' 'not one,' &c. in negative propositions. + +§ 254. The terminology of the division of propositions according to +quantity is unsatisfactory. Not only has the indefinite proposition to +be set down as particular, even when the sense manifestly declares it +to be universal; but the proposition which is expressed in a +particular form has also to be construed as indefinite, _so_ that +an unnatural meaning is imparted to the word 'some,' as used in +logic. If in common conversation we were to say 'Some cows chew the +cud,' the person whom we were addressing would doubtless imagine us to +suppose that there were some cows which did not possess this +attribute. But in logic the word 'some' is not held to express more +than 'some at least, if not all.' Hence we find not only that an +indefinite proposition may, as a matter of fact, be strictly +particular, but that a proposition which appears to be strictly +particular may be indefinite. So a proposition expressed in precisely +the same form 'Some A is B' may be either strictly particular, if some +be taken to exclude all, or indefinite, if the word 'some' does not +exclude the possibility of the statement being true of all. It is +evident that the term 'particular' has become distorted from its +original meaning. It would naturally lead us to infer that a statement +is limited to part of the subject, whereas, by its being opposed to +universal, in the sense in which that term has been defined, it can +only mean that we have nothing to show us whether part or the whole is +spoken of. + +§ 255. This awkwardness of expression is due to the indefinite +proposition having been displaced from its proper position. Formerly +propositions were divided under three heads-- + + (1) Universal, + + (2) Particular, + + (3) Indefinite. + +But logicians anxious for simplification asked, whether a predicate in +any given case must not either apply to the whole of the subject or +not? And whether, therefore, the third head of indefinite propositions +were not as superfluous as the so-called 'common gender' of nouns in +grammar? + +§ 256. It is quite true that, as a matter of fact, any given predicate +must either apply to the whole of the subject or not, so that in the +nature of things there is no middle course between universal and +particular. But the important point is that we may not know whether +the predicate applies to the whole of the subject or not. The primary +division then should be into propositions whose quantity is known and +propositions whose quantity is unknown. Those propositions whose +quantity is known may be sub-divided into 'definitely universal' and +'definitely particular,' while all those whose quantity is unknown are +classed together under the term 'indefinite.' Hence the proper +division is as follows-- + + Proposition + __________|____________ + | | + Definite Indefinite + _____|_______ + | | + Universal Particular. + +§ 257. Another very obvious defeat of terminology is that the word +'universal' is naturally opposed to 'singular,' whereas it is here so +used as to include it; while, on the other hand, there is no obvious +difference between universal and general, though in the division the +latter is distinguished from the former as species from genus. + + + +_Affirmative and Negative Propositions._ + + +§ 258. This division rests upon the Quality of propositions. + +§ 259. It is the quality of the form to be affirmative or negative: +the quality of the matter, as we saw before (§ 204), is to be true or +false. But since formal logic takes no account of the matter of +thought, when we speak of 'quality' we are understood to mean the +quality of the form. + +§ 260. By combining the division of propositions +according to quantity with the division according to quality, +we obtain four kinds of proposition, namely-- + + (1) Universal Affirmative (A). + + (2) Universal Negative (E). + + (3) Particular Affirmative (I). + + (4) Particular Negative (O). + +§ 261. This is an exhaustive classification of propositions, and any +proposition, no matter what its form may be, must fall under one or +other of these four heads. For every proposition must be either +universal or particular, in the sense that the subject must either be +known to be used in its whole extent or not; and any proposition, +whether universal or particular, must be either affirmative or +negative, for by denying modality to the copula we have excluded +everything intermediate between downright assertion and denial. This +classification therefore may be regarded as a Procrustes' bed, into +which every proposition is bound to fit at its proper peril. + +§ 262. These four kinds of propositions are represented respectively +by the symbols A, E, I, O. + +§ 263. The vowels A and I, which denote the two affirmatives, occur in +the Latin words 'affirmo' and 'aio;' E and O, which denote the two +negatives, occur in the Latin word 'nego.' + + + +_Extensive and Intensive Propositions._ + + +§ 264. It is important to notice the difference between Extensive and +Intensive propositions; but this is not a division of propositions, +but a distinction as to our way of regarding them. Propositions may be +read either in extension or intension. Thus when we say 'All cows are +ruminants,' we may mean that the class, cow, is contained in the +larger class, ruminant. This is reading the proposition in +extension. Or we may mean that the attribute of chewing the cud is +contained in, or accompanies, the attributes which make up our idea of +'cow.' This is reading the proposition in intension. What, as a matter +of fact, we do mean, is a mixture of the two, namely, that the class, +cow, has the attribute of chewing the cud. For in the ordinary and +natural form of proposition the subject is used in extension, and the +predicate in intension, that is to say, when we use a subject, we are +thinking of certain objects, whereas when we use a predicate, we +indicate the possession of certain attributes. The predicate, however, +need not always be used in intension, e.g. in the proposition 'His +name is John' the predicate is not intended to convey the idea of any +attributes at all. What is meant to be asserted is that the name of +the person in question is that particular name, John, and not +Zacharias or Abinadab or any other name that might be given him. + +§ 265. Let it be noticed that when a proposition is read in extension, +the predicate contains the subject, whereas, when it is read in +intension, the subject contains the predicate. + + + +_Exclusive Propositions._ + + +§ 266. An Exclusive Proposition is so called because in it all but a +given subject is excluded from participation in a given predicate, +e.g. 'The good alone are happy,' 'None but the brave deserve the +fair,' 'No one except yourself would have done this.' + +§ 267. By the above forms of expression the predicate is declared to +apply to a given subject and to that subject only. Hence an exclusive +proposition is really equivalent to two propositions, one affirmative +and one negative. The first of the above propositions, for instance, +means that some of the good are happy, and that no one else is so. It +does not necessarily mean that all the good are happy, but asserts +that among the good will be found all the happy. It is therefore +equivalent to saying that all the happy are good, only that it puts +prominently forward in addition what is otherwise a latent consequence +of that assertion, namely, that some at least of the good are happy. + +§ 268. Logically expressed the exclusive proposition when universal +assumes the form of an E proposition, with a negative term for its +subject + + No not-A is B. + +§ 269. Under the head of exclusive comes the strictly particular +proposition, 'Some A is B,' which implies at the same time that 'Some +A is not B.' Here 'some' is understood to mean 'some only,' which is +the meaning that it usually bears in common language. When, for +instance, we say 'Some of the gates into the park are closed at +nightfall,' we are understood to mean 'Some are left open.' + + + +_Exceptive Propositions._ + + +§ 270. An Exceptive Proposition is so called as affirming the +predicate of the whole of the subject, with the exception of a certain +part, e.g. 'All the jury, except two, condemned the prisoner.' + +§ 271. This form of proposition again involves two distinct +statements, one negative and one affirmative, being equivalent to 'Two +of the jury did not condemn the prisoner; and all the rest did.' + +§ 272. The exceptive proposition is merely an affirmative way of +stating the exclusive-- + + No not-A is B = All not-A is not-B. + + No one but the sage is sane = All except the sage are mad. + + + +_Tautologous or Identical Propositions_ + + +§ 273. A Tautologous or Identical proposition affirms the subject of +itself, e.g. 'A man's a man,' 'What I have written, I have written,' +'Whatever is, is.' The second of these instances amounts formally to +saying 'The thing that I have written is the thing that I have +written,' though of course the implication is that the writing will +not be altered. + + + + +CHAPTER IV. + +_Of the Distribution of Terms._ + + +§ 274. The treatment of this subject falls under the second part of +logic, since distribution is not an attribute of terms in themselves, +but one which they acquire in predication. + +§ 275. A term is said to be distributed when it is known to be used in +its whole extent, that is, with reference to all the things of which +it is a name. When it is not so used, or is not known to be so used, +it is called undistributed. + +§ 276. When we say 'All men are mortal,' the subject is distributed, +since it is apparent from the form of the expression that it is used +in its whole extent. But when we say 'Men are miserable' or 'Some men +are black,' the subject is undistributed. + +§ 277. There is the same ambiguity attaching to the term +'undistributed' which we found to underlie the use of the term +'particular.' 'Undistributed' is applied both to a term whose quantity +is undefined, and to one whose quantity is definitely limited to a +part of its possible extent. + +§ 278. This awkwardness arises from not inquiring first whether the +quantity of a term is determined or undetermined, and afterwards +proceeding to inquire, whether it is determined as a whole or part of +its possible extent. As it is, to say that a term is distributed, +involves two distinct statements-- + + (1) That its quantity is known; + + (2) That its quantity is the greatest possible. + +The term 'undistributed' serves sometimes to contradict one of these +statements and sometimes to contradict the other. + +§ 279. With regard to the quantity of the subject of a proposition no +difficulty can arise. The use of the words 'all' or 'some,' or of a +variety of equivalent expressions, mark the subject as being +distributed or undistributed respectively, while, if there be nothing +to mark the quantity, the subject is for that reason reckoned +undistributed. + +§ 280. With regard to the predicate more difficulty may arise. + +§ 281. It has been laid down already that, in the ordinary form of +proposition, the subject is used in extension and the predicate in +intension. Let us illustrate the meaning of this by an example. If +someone were to say 'Cows are ruminants,' you would have a right to +ask him whether he meant 'all cows' or only 'some.' You would not by +so doing be asking for fresh information, but merely for a more +distinct explanation of the statement already made. The subject being +used in extension naturally assumes the form of the whole or part of a +class. But, if you were to ask the same person 'Do you mean that cows +are all the ruminants that there are, or only some of them?' he would +have a right to complain of the question, and might fairly reply, 'I +did not mean either one or the other; I was not thinking of ruminants +as a class. I wished merely to assert an attribute of cows; in fact, I +meant no more than that cows chew the cud.' + +§ 282. Since therefore a predicate is not used in extension at all, it +cannot possibly be known whether it is used in its whole extent or +not. + +§ 283. It would appear then that every predicate is necessarily +undistributed; and this consequence does follow in the case of +affirmative propositions. + +§ 284. In a negative proposition, however, the predicate, though still +used in intension, must be regarded as distributed. This arises from +the nature of a negative proposition. For we must remember that in any +proposition, although the predicate be not meant in extension, it +always admits of being so read. Now we cannot exclude one class from +another without at the same time wholly excluding that other from the +former. To take an example, when we say 'No horses are ruminants,' the +meaning we really wish to convey is that no member of the class, +horse, has a particular attribute, namely, that of chewing the +cud. But the proposition admits of being read in another form, namely, +'That no member of the class, horse, is a member of the class, +ruminant.' For by excluding a class from the possession of a given +attribute, we inevitably exclude at the same time any class of things +which possess that attribute from the former class. + +§ 285. The difference between the use of a predicate in an affirmative +and in a negative proposition may be illustrated to the eye as +follows. To say 'All A is B' may mean either that A is included in B +or that A and B are exactly co-extensive. + +[Illustration] + +§ 286. As we cannot be sure which of these two relations of A to B is +meant, the predicate B has to be reckoned undistributed, since a term +is held to be distributed only when we know that it is used in its +whole extent. + +§ 287. To say 'No A is B,' however, is to say that A falls wholly +outside of B, which involves the consequence that B falls wholly +outside of A. + +[Illustration] + +§ 288. Let us now apply the same mode of illustration to the +particular forms of proposition. + +§ 289. If I be taken in the strictly particular sense, there are, from +the point of view of extension, two things which may be meant when we +say 'Some A is B'-- + + (1) That A and B are two classes which overlap one another, that is + to say, have some members in common, e.g. 'Some cats are black.' + + [Illustration] + + (2) That B is wholly contained in A, which is an inverted way of + saying that all B is A, e.g. 'Some animals are men.' + + [Illustration] + +§ 290. Since we cannot be sure which of these two is meant, the +predicate is again reckoned undistributed. + +§ 291. If on the other hand 1 be taken in an indefinite sense, so as +to admit the possibility of the universal being true, then the two +diagrams which have already been used for A must be extended to 1, in +addition to its own, together with the remarks which we made in +connection with them (§§ 285-6). + +§ 292. Again, when we say 'Some A is not B,' we mean that some, if not +the whole of A, is excluded from the possession of the attribute B. In +either case the things which possess the attribute B are wholly +excluded either from a particular part or from the whole of A. The +predicate therefore is distributed. + +[Illustration] + +From the above considerations we elicit the following-- + +§ 293. Four Rules for the Distribution of Terms. + + (1) All universal propositions distribute their subject. + + (2) No particular propositions distribute their subject, + + (3) All negative propositions distribute their predicate. + + (4) No affirmative propositions distribute their predicate. + + +§ 294. The question of the distribution or non-distribution of the +subject turns upon the quantity of the proposition, whether universal +or particular; the question of the distribution or non-distribution of +the predicate turns upon the quality of the proposition, whether +affirmative or negative. + + + + +CHAPTER V. + +_Of the Quantification of the Predicate._ + + +§ 295. The rules that have been given for the distribution of terms, +together with the fourfold division of propositions into A, E, 1, 0, +are based on the assumption that it is the distribution or +non-distribution of the subject only that needs to be taken into +account in estimating the quantity of a proposition. + +§ 296. But some logicians have maintained that the predicate, though +seldom quantified in expression, must always be quantified in +thought--in other words, that when we say, for instance, 'All A is B,' +we must mean either that 'All A is all B' or only that 'All A is some +B.' + +§ 297. If this were so, it is plain that the number of possible +propositions would be exactly doubled, and that, instead of four +forms, we should now have to recognise eight, which may be expressed +as follows-- + + 1. All A is all B. ([upsilon]). + + 2. All A is some B. ([Lambda]). + + 3. No A is any B. ([Epsilon]). + + 4. No A is some B. ([eta]). + + 5. Some A is all B. ([Upsilon]). + + 6. Some A is some B. ([Iota]). + + 7. Some A is not any B. ([Omega]). + + 8. Some A is not some B. ([omega]). + +§ 298. It is evident that it is the second of the above propositions +which represents the original A, in accordance with the rule that 'No +affirmative propositions distribute their predicate' (§ 293). + +§ 299. The third represents the original E, in accordance with the +rule that 'All negative propositions distribute their predicate.' + +§ 300. The sixth represents the original I, in accordance with the +rule that 'No affirmative propositions distribute their predicate.' + +§ 301. The seventh represents the original O, in accordance with the +rule that 'All negative propositions distribute their predicate.' + +§ 302. Four new symbols are required, if the quantity of the predicate +as well as that of the subject be taken into account in the +classification of propositions. These have been supplied, somewhat +fancifully, as follows-- + +§ 303. The first, 'All A is all B,' which distributes both subject and +predicate, has been called [upsilon], to mark its extreme +universality. + +§ 304. The fourth, 'No A is some B,' is contained in E, and has +therefore been denoted by the symbol [eta], to show its connection +with E. + +§ 305. The fifth, 'Some A is all B,' is the exact converse of the +second, 'All A is some B,' and has therefore been denoted by the +symbol [Upsilon], which resembles an inverted A. + +§ 306. The eighth is contained in O, as part in whole, and has +therefore had assigned to it the symbol [omega], + +§ 307. The attempt to take the predicate in extension, instead of, as +it should naturally be taken, in intension, leads to some curious +results. Let us take, for instance, the u proposition. Either the sign +of quantity 'all' must be understood as forming part of the predicate +or not. If it is not, then the u proposition 'All A is all B' seems +to contain within itself, not one proposition, but two, namely, 'All A +is B' and 'All B is A.' But if on the other hand 'all' is understood +to form part of the predicate, then u is not really a general but a +singular proposition. When we say, 'All men are rational animals,' we +have a true general proposition, because the predicate applies to the +subject distributively, and not collectively. What we mean is that +'rational animal' may be affirmed of every individual in the class, +man. But when we say 'All men are all rational animals,' the predicate +no longer applies to the subject distributively, but only +collectively. For it is obvious that 'all rational animals' cannot be +affirmed of every individual in the class, man. What the proposition +means is that the class, man, is co-extensive with the class, rational +animal. The same meaning may be expressed intensively by saying that +the one class has the attribute of co-extension with the other. + +§ 308. Under the head o u come all propositions in which both subject +and predicate are singular terms, e.g. 'Homer was the author of the +Iliad,' 'Virtue is the way to happiness.' + +§ 309. The proposition [eta] conveys very little information to the +mind. 'No A is some B' is compatible with the A proposition in the +same matter. 'No men are some animals' may be true, while at the same +time it is true that 'All men are animals.' No men, for instance, are +the particular animals known as kangaroos. + +§ 310. The [omega] proposition conveys still less information than the +[eta]. For [omega] is compatible, not only with A, but with +[upsilon]. Even though 'All men are all rational animals,' it is still +true that 'Some men are not some rational animals': for no given human +being is the same rational animal as any other. + +§ 311. Nay, even when the [upsilon] is an identical proposition, +[omega] will still hold in the same matter. 'All rational animals are +all rational animals': but, for all that, 'Some rational animals are +not some others.' This last form of proposition therefore is almost +wholly devoid of meaning. + +§ 312. The chief advantage claimed for the quantification of the +predicate is that it reduces every affirmative proposition to an exact +equation between its subject and predicate. As a consequence every +proposition would admit of simple conversion, that is to say, of +having the subject and predicate transposed without any further change +in the proposition. The forms also of Reduction (a term which will be +explained later on) would be simplified; and generally the +introduction of the quantified predicate into logic might be attended +with certain mechanical advantages. The object of the logician, +however, is not to invent an ingenious system, but to arrive at a true +analysis of thought. Now, if it be admitted that in the ordinary form +of proposition the subject is used in extension and the predicate in +intension, the ground for the doctrine is at once cut away. For, if +the predicate be not used in its extensive capacity at all, we plainly +cannot be called upon to determine whether it is used in its whole +extent or not. + + + + +CHAPTER VI. + +_Of the Heads of Predicables_. + + +§ 313. A predicate is something which is stated of a subject. + +§ 314. A predicable is something which can be stated of a subject. + +§ 315. The Heads of Predicables are a classification of the various +things which can be stated of a subject, viewed in their relation to +it. + +§ 316. The treatment of this topic, therefore, as it involves the +relation of a predicate to a subject, manifestly falls under the +second part of logic, which deals with the proposition. It is +sometimes treated under the first part of logic, as though the heads +of predicables were a classification of universal notions, i.e. common +terms, in relation to one another, without reference to their place in +the proposition. + +§ 317. The heads of predicables are commonly reckoned +as five, namely, + + (1) Genus. + + (2) Species. + + (3) Difference. + + (4) Property. + + (5) Accident. + +§ 318. We will first define these terms in the sense in which they are +now used, and afterwards examine the principle on which the +classification is founded and the sense in which they were originally +intended. + + (1) A Genus is a larger class containing under it smaller + classes. Animal is a genus in relation to man and brute. + + (2) A Species is a smaller class contained under a larger one. Man + is a species in relation to animal. + + (3) Difference is the attribute, or attributes, which distinguish + one species from others contained under the same genus. Rationality + is the attribute which distinguishes the species, man, from the + species, brute. + + N.B. The genus and the difference together make up the Definition of + a class-name, or common term. + + (4) A Property is an attribute which is not contained in the + definition of a term, but which flows from it. + + A Generic Property is one which flows from the genus. + + A Specific Property is one which flows from the difference. + + It is a generic property of man that he is mortal, which is a + consequence of his animality. It is a specific property of man that + he is progressive, which is a consequence of his rationality. + + (5) An Accident is an attribute, which is neither contained in the + definition, nor flows from it. + +§ 319. Accidents are either Separable or Inseparable. + +A Separable Accident is one which belongs only to some members of a +class. + +An Inseparable Accident is one which belongs to all the members of a +class. + +Blackness is a separable accident of man, an inseparable accident of +coals. + +§ 320. The attributes which belong to anything may be distinguished +broadly under the two heads of essential and non-essential, or +accidental. By the essential attributes of anything are meant those +which are contained in, or which flow from, the definition. Now it may +be questioned whether there can, in the nature of things, be such a +thing as an inseparable accident. For if an attribute were found to +belong invariably to all the members of a class, we should suspect +that there was some causal connection between it and the attributes +which constitute the definition, that is, we should suspect the +attribute in question to be essential and not accidental. Nevertheless +the term 'inseparable accident' may be retained as a cloak for our +ignorance, whenever it is found that an attribute does, as a matter of +fact, belong to all the members of a class, without there being any +apparent reason why it should do so. It has been observed that animals +which have horns chew the cud. As no one can adduce any reason why +animals that have horns should chew the cud any more than animals +which have not, we may call the fact of chewing the cud an inseparable +accident of horned animals. + +§ 321. The distinction between separable and inseparable accidents is +sometimes extended from classes to individuals. + +An inseparable accident of an individual is one which belongs to him +at all times. A separable accident of an individual is one which +belongs to him at one time and not at another. + +§ 322. It is an inseparable accident of an individual that he was born +at a certain place and on a certain date. It is a separable accident +of an individual that he resides at a certain place and is of a +certain age. + +§ 323. There are some remarks which it may be well to make about the +above five terms before we pass on to investigate the principle upon +which the division is based. + +§ 324. In the first place, it must of course be borne in mind that +genus and species are relative terms. No class in itself can be either +a genus or a species; it only becomes so in reference to some other +class, as standing to it in the relation of containing or contained. + +§ 325. Again, the distinction between genus and difference on the one +hand and property on the other is wholly relative to an assumed +definition. When we say 'Man is an animal,' 'Man is rational,' 'Man is +progressive,' there is nothing in the nature of these statements +themselves to tell us that the predicate is genus, difference, or +property respectively. It is only by a tacit reference to the accepted +definition of man that this becomes evident to us, Similarly, we +cannot know beforehand that the fact of a triangle having three sides +is its difference, and the fact of its having three angles a +property. It is only when we assume the definition of a triangle as a +three-sided figure that the fact of its having three angles sinks into +a property. Had we chosen to define it, in accordance with its +etymological meaning, as a figure with three angles, its +three-sidedness would then have been a mere property, instead of being +the difference; for these two attributes are so connected together +that, whichever is postulated, the other will necessarily follow. + +§ 326. Lastly, it must be noticed that we have not really defined the +term 'accident,' not having stated what it is, but only what it is +not. It has in fact been reserved as a residual head to cover any +attribute which is neither a difference nor a property. + +§ 327. If the five heads of predicables above given were offered to us +as an exhaustive classification of the possible relations in which the +predicate can stand to the subject in a proposition, the first thing +that would strike us is that they do not cover the case in which the +predicate is a singular term. In such a proposition as 'This man is +John,' we have neither a predication of genus or species nor of +attribute: but merely the identification of one term with another, as +applying to the same object. Such criticism as this, however, would be +entirely erroneous, since no singular term was regarded as a +predicate. A predicable was another name for a universal, the common +term being called a predicable in one relation and a universal in +another-a predicable, extensively, in so far as it was applicable to +several different things, a universal, intensively, in so far as the +attributes indicated were implied in several other notions, as the +attributes indicated by 'animal' are implied in 'horse,' 'sheep,' +'goat,' &c. + +§ 328. It would be less irrelevant to point out how the classification +breaks down in relation to the singular term as subject. When, for +instance, we say 'Socrates is an animal,' 'Socrates is a man,' there +is nothing in the proposition to show us whether the predicate is a +genus or a species: for we have not here the relation of class to +class, which gives us genus or species according to their relative +extension, but the relation of a class to an individual. + +§ 329. Again, when we say + + (1) Some animals are men, + + (2) Some men are black, + +what is there to tell us that the predicate is to be regarded in the +one case as a species and in the other as an accident of the subject? +Nothing plainly but the assumption of a definition already known. + +§ 330. But if this assumption be granted, the classification seems to +admit of a more or less complete defense by logic. + +For, given any subject, we can predicate of it either a class or an +attribute. + +When the predicate is a class, the term predicated is called a Genus, +if the subject itself be a class, or a Species, if it be an +individual. + +When, on the other hand, the predicate is an attribute, the attribute +predicated may be either the very attribute which distinguishes the +subject from other members of the same class, in which case it is +called the Difference, or it may be some attribute connected with the +definition, i.e. Property, or not connected with it, i.e. Accident. + +§ 331. These results may be exhibited in the following scheme-- + + Predicate + ________________|_________________ + | | + Class Attribute + ______|_______ __________|________ + | | | | + (Subject a (Subject a (The (Not the + common singular distinguishing distinguishing + term) term) Attribute) attribute) + Genus Species Difference + |___________________ + | | + (Connected (Not connected + with the with the + definition) definition) + Property Accident + +§ 332. The distinction which underlies this division between +predicating a class and predicating an attribute (in quid or in quale) +is a perfectly intelligible one, corresponding as it does to the +grammatical distinction between the predicate being a noun substantive +or a noun adjective. Nevertheless it is a somewhat arbitrary one, +since, even when the predicate is a class-name, what we are concerned +to convey to the mind, is the fact that the subject possesses the +attributes which are connoted by that class-name. We have not here the +difference between extensive and intensive predication, since, as we +have already seen (§ 264), that is not a difference between one +proposition and another, but a distinction in our mode of interpreting +any and every proposition. Whatever proposition we like to take may be +read either in extension or in intension, according as we fix our +minds on the fact of inclusion in a class or the fact of the +possession of attributes. + +§ 333. It will be seen that the term 'species,' as it appears in the +scheme, has a wholly different meaning from the current acceptation in +which it was defined above. Species, in its now accepted meaning, +signifies the relation of a smaller class to a larger one: as it was +originally intended in the heads of predicables it signifies a class +in reference to individuals. + +§ 334. Another point which requires to be noticed with regard to this +five-fold list of heads of predicables, if its object be to classify +the relations of a predicate to a subject, is that it takes no account +of those forms of predication in which class and attribute are +combined. Under which of the five heads would the predicates in the +following propositions fall? + + (1) Man is a rational animal. + + (2) Man is a featherless biped. + +In the one case we have a combination of genus and difference; in the +other we have a genus combined with an accident. + +§ 335. The list of heads of predicables which we have been discussing +is not derived from Aristotle, but from the 'Introduction' of +Porphyry, a Greek commentator who lived more than six centuries later. + + + +_Aristotle's Heads of Predicables_. + + +§ 336. Aristotle himself, by adopting a different basis of division, +has allowed room in his classification for the mixed forms of +predication above alluded to. His list contains only four heads, +namely, + + Genus ([Greek: génos]) + + Definition ([Greek: ňrismós]) + + Proprium ([Greek: îdion]) + + Accident ([Greek: sumbebekós]) + +§ 337. Genus here is not distinguished from difference. Whether we +say 'Man is an animal' or 'Man is rational,' we are equally understood +to be predicating a genus. + +§ 338. There is no account taken of species, which, when predicated, +resolves itself either into genus or accident. When predicated of an +individual, it is regarded as a genus, e.g. 'Socrates is a man'; when +predicated of a class, it is regarded as an accident, e.g. 'Some +animals are men.' + +§ 339. Aristotle's classification may easily be seen to be +exhaustive. For every predicate must either be coextensive with its +subject or not, i.e. predicable of the same things. And if the two +terms coincide in extension, the predicate must either coincide also +in intension with the subject or not. + +A predicate which coincides both in extension and intension with its +subject is exactly what is meant by a definition. One which coincides +in extension without coinciding in intension, that is, which applies +to the same things without expressing the whole meaning, of the +subject, is what is known as a Proprium or Peculiar Property. + +If, on the other hand, the two terms are not co-extensive, the +predicate must either partially coincide in intension with the subject +or not. [Footnote: The case could not arise of a predicate which was +entirely coincided in intension with a subject with which it was not +co-extensive. For, if the extension of the predicate were greater than +that of the subject, its intension would be less, and if less, +greater, in accordance with the law of inverse variation of the two +quantities (§ 166).] This is equivalent to saying that it must either +state part of the definition of the subject or not. Now the definition +is made up of genus and difference, either of which may form the +predicate: but as the two are indistinguishable in relation to a +single subject, they are lumped together for the present purpose under +the one head, genus. When the predicate, not being co-extensive, is +not even partially co-intensive with its subject, it is called an +Accident. + +§ 340. Proprium, it will be seen, differs from property. A proprium +is an attribute which is possessed by all the members of a class, and +by them alone, e.g. 'Men are the only religious animals.' + +§ 341. Under the head of definition must be included all propositions +in which the predicate is a mere synonym of the subject, e.g. 'Naso is +Ovid,' 'A Hebrew is a Jew,' 'The skipper is the captain.' In such +propositions the predicate coincides in extension with the subject, +and may be considered to coincide in intension where the intension of +both subject and predicate is at zero, as in the case of two proper +names. + +§ 342. Designations and descriptions will fall under the head of +'proprium' or peculiar property, e.g. 'Lord Salisbury is the present +prime minister of England,' 'Man is a mammal with hands and without +hair.' For here, while the terms are coincident in extension, they are +far from being so in intension. + +§ 343. The term 'genus' must be understood to include not only genus +in the accepted sense, but difference and generic property as well. + +§ 344. These results may be exhibited in the following +scheme-- + + Predicate + ___________________|______________ + | | + Coextensive with not + the subject coextensive + ________|_________ _____|________ + | | | | + Co-intensive not partially not at all + with the subject cointensive cointensive [Greek: sumbubekós] + [Greek: ňrismós] [Greek: îdion] [Greek: génos] Accident + ______|_____ ______|_____________ |________________ + | | | | | | | | + Defini- Synonym Designa- Descrip- Peculiar Genus Differ- Generic + tion tion tion Property ence Property + +§ 345. Thus Aristotle's four heads of predicables may be split up, if +we please, into nine-- + + 1. Definition \ + > [Greek: ňrismós]. + 2. Synonym / + + 3. Designation \ + | + 4. Description > [Greek: îdion]. + | + 5. Peculiar Property/ + + 6. Genus \ + | + 7. Difference > [Greek: génos]. + | + 8. Generic Property/ + + 9. Accident--[Greek: sumbebekós]. + +§ 346. We now pass on to the two subjects of Definition and Division, +the discussion of which will complete our treatment of the second part +of logic. Definition and division correspond respectively to the two +kinds of quantity possessed by terms. + +Definition is unfolding the quantity of a term in intension. + +Division is unfolding the quantity of a term in extension. + + + + +CHAPTER VII. + +_Of Definition._ + + +§ 347. To define a term is to unfold its intension, i.e. to explain +its meaning. + +§ 348. From this it follows that any term which possesses no intension +cannot be defined. + +§ 349. Hence proper names do not admit of definition, except just in +so far as they do possess some slight degree of intension: Thus we can +define the term 'John' only so far as to say that 'John' is the name +of a male person. This is said with regard to the original intension +of proper names; their acquired intension will be considered later. + +§ 350. Again, since definition is unfolding the intension of a term, +it follows that those terms will not admit of being defined whose +intension is already so simple that it cannot be unfolded further. Of +this nature are names of simple attributes, such as greenness, +sweetness, pleasure, existence. We know what these things are, but we +cannot define them. To a man who has never enjoyed sight, no language +can convey an idea of the greenness of the grass or the blueness of +the sky; and if a person were unaware of the meaning of the term +'sweetness,' no form of words could convey to him an idea of it. We +might put a lump of sugar into his mouth, but that would not be a +logical definition. + +§ 351. Thus we see that, for a thing to admit of definition, the idea +of it must be complex. Simple ideas baffle definition, but at the same +time do not require it. In defining we lay out the simpler ideas +which are combined in our notion of something, and so explain that +complex notion. We have defined 'triangle,' when we analyse it into +'figure' and 'contained by three lines.' Similarly we have defined +'substance' when we analyse it into 'thing' and 'which can be +conceived to exist by itself.' + +§ 352. But when we get to 'thing' we have reached a limit. The Summum +Genus, or highest class under which all things fall, cannot be defined +any more than a simple attribute; and for the very good reason that it +connotes nothing but pure being, which is the simplest of all +attributes. To say that a thing is an 'object of thought' is not +really to define it, but to explain its etymology, and to reclaim a +philosophical term from its abuse by popular language, in which it is +limited to the concrete and the lifeless. Again, to define it +negatively and to say that a thing is 'that which is not nothing' does +not carry us any further than we were before. The law of contradiction +warrants us in saying as much as that. + +§ 353. Definition is confined to subject-terms, and does not properly +extend to attributives. For definition is of things through names, and +an attributive out of predication is not the name of anything. The +attributive is defined, so far as it can be, through the corresponding +abstract term. + +§ 354. Common terms, other than attributives, ought always to admit of +definition. For things are distributed by the mind into classes owing +to their possessing certain attributes in common, and the definition +of the class-name can be effected by detailing these attributes, or at +least a sufficient number of them. + +§ 355. It is different with singular terms. Singular terms, when +abstract, admit of definition, in so far as they are not names of +attributes so simple as to evade analysis. When singular terms are +concrete, we have to distinguish between the two cases of proper names +and designations. Designations are connotative singular terms. They +are formed by limiting a common term to the 'case in hand.' Whatever +definition therefore fits the common term will fit also the +designation which is formed from it, if we add the attributes implied +by the limitations. Thus whatever definition fits the common term +'prime minister' will fit also the singular term 'the present prime +minister of England' by the addition to it of the attributes of place +and time which are indicated by the expression. Such terms as this +have a definite amount of intension, which can therefore be seized +upon and expounded by a definition. + +§ 356. But proper names, having no original intension of their own, +cannot be defined at all; whereas, if we look upon them from the point +of view of their acquired intension, they defy definition by reason of +the very complexity of their meaning. We cannot say exactly what +'John' and 'Mary' mean, because those names, to us who know the +particular persons denoted by them, suggest all the most trifling +accidents of the individual as well as the essential attributes of the +genus. + +§ 357. Definition serves the practical purpose of enabling us mentally +to distinguish, or, as the name implies, 'mark off' the thing defined +from all other things whatsoever. This may seem at first an endless +task, but there is a short cut by which the goal may be reached. For, +if we distinguish the thing in hand from the things which it is most +like, we shall, 'a fortiori,' have distinguished it from things to +which it bears a less resemblance. + +§ 358. Hence the first thing to do in seeking for a definition is to +fix upon the class into which the thing to be defined most naturally +falls, and then to distinguish the thing in question from the other +members of that class. If we were asked to define a triangle, we +would not begin by distinguishing it from a hawser, but from a square +and other figures with which it is more possible to confound it. The +class into which a thing falls is called its Genus, and the attribute +or attributes which distinguish it from other members of that class +are called its Difference. + +§ 359. If definition thus consists in referring a thing to a class, we +see a further reason why the summum genus of all things cannot be +defined. + +§ 360. We have said that definition is useful in enabling us to +distinguish things from one another in our minds: but this must not be +regarded as the direct object of the process. For this object may be +accomplished without giving a definition at all, by means of what is +called a Description. By a description is meant an enumeration of +accidents with or without the mention of some class-name. It is as +applicable to proper names as to common terms. When we say 'John Smith +lives next door on the right-hand side and passes by to his office +every morning at nine o'clock,' we have, in all probability, +effectually distinguished John Smith from other people: but living +next, &c., cannot be part of the intension of John Smith, since John +Smith may change his residence or abandon his occupation without +ceasing to be called by his name. Indirectly then definition serves +the purpose of distinguishing things in the mind, but its direct +object is to unfold the intension of terms, and so impart precision to +our thoughts by setting plainly before us the meaning of the words we +are using. + +§ 361. But when we say that definition is unfolding the intension of +terms, it must not be imagined that we are bound in defining to unfold +completely the intension of terms. This would be a tedious, and often +an endless, task. A term may mean, or convey to the mind, a good many +more attributes than those which are stated in its definition. There +is no limit indeed to the meaning which a term may legitimately +convey, except the common attributes of the things denoted by it. Who +shall say, for instance, that a triangle means a figure with three +sides, and does not mean a figure with three angles, or the surface of +the perpendicular bisection of a cone? Or again, that man means a +rational, and does not mean a speaking, a religious, or an aesthetic +animal, or a biped with two eyes, a nose, and a mouth? The only +attributes of which it can safely be asserted that they can form no +part of the intension of a term are those which are not common to all +the things to which the name applies. Thus a particular complexion, +colour, height, creed, nationality cannot form any part of the +intension of the term 'man.' But among the attributes common to a +class we cannot distinguish between essential and unessential, except +by the aid of definition itself. Formal logic cannot recognise any +order of priority between the attributes common to all the members of +a class, such as to necessitate our recognising some as genera and +differentiae and relegating others to the place of properties or +inseparable accidents. + +§ 362. The art of giving a good definition is to seize upon the +salient characteristics of the thing defined and those wherefrom the +largest number of other attributes can be deduced as consequences. To +do this well requires a special knowledge of the thing in question, +and is not the province of the formal logician. + +§ 363. We have seen already, in treating of the Heads of Predicables +(§ 325), that the difference between genus and difference on the one +hand and property on the other is wholly relative to some assumed +definition. Now definitions are always to a certain extent arbitrary, +and will vary with the point of view from which we consider the thing +required to be defined. Thus 'man' is usually contrasted with 'brute,' +and from this point of view it is held a sufficient definition of him +to say that he is 'a rational animal,' But a theologian might be more +anxious to contrast man with supposed incorporeal intelligences, and +from this point of view man would be defined as an 'embodied spirit.' + +§ 364. In the two definitions just given it will be noticed that we +have really employed exactly the same attributes, only their place as +genus and difference has been reversed. It is man's rational, or +spiritual, nature which distinguishes him from the brutes: but this is +just what he is supposed to have in common with incorporeal +intelligences, from whom he is differentiated by his animal nature. + +[Illustration] + +This illustration is sufficient to show us that, while there is no +absolute definition of anything, in the sense of a fixed genus and +difference, there may at the same time be certain attributes which +permanently distinguish the members of a given class from those of all +other classes. + +§ 365. The above remarks will have made it clear that the intension of +a term is often much too wide to be conveyed by any definition; and +that what a definition generally does is to select certain attributes +from the whole intension, which are regarded as being more typical of +the thing than the remainder. No definition can be expected to exhaust +the whole intension of a term, and there will always be room for +varying definitions of the same thing, according to the different +points of view from which it is approached. + +§ 366. Names of attributes lend themselves to definition far more +easily than names of substances. The reason of this is that names of +attributes are primarily intensive in force, whereas substances are +known to us in extension before they become known to us in +intension. There is no difficulty in defining a term like 'triangle' +or 'monarchy,' because these terms were expressly invented to cover +certain attributes; but the case is different with such terms as +'dog,' 'tree,' 'plant,' 'metal,' and other names of concrete +things. We none of us have any difficulty in recognising a dog or +tree, when we see them, or in distinguishing them from other animals +or plants respectively. We are therefore led to imagine that we know +the meaning of these terms. It is not until we are called upon for a +definition that we discover how superficial our knowledge really is of +the common attributes possessed by the things which these names +denote. + +§ 367. It might be imagined that a common name would never be given to +things except in virtue of our knowledge of their common +attributes. But as a matter of fact, the common name was first given +from a confused notion of resemblance, and we had afterwards to detect +the common attributes, when sometimes the name had been so extended +from one thing to another like it, that there were hardly any definite +attributes possessed in common by the earlier and later members of the +class. + +§ 368. This is especially the case where the meaning of terms has been +extended by analogy, e.g. head, foot, arm, post, pole, pipe, &c. + +§ 369. But in the progress of thought we come to form terms in which +the intensive capacity is everything. Of this kind notably are +mathematical conceptions. Terms of this kind, as we said before, lend +themselves readily to definition. + +§ 370. We may lay down then roughly that words are easy or difficult +of definition according as their intensive or extensive capacity +predominates. + +§ 371. There is a marked distinction to be observed between the +classes made by the mind of man and the classes made by nature, which +are known as 'real kinds.' In the former there is generally little or +nothing in common except the particular attribute which is selected as +the ground of classification, as in the case of red and white things, +which are alike only in their redness or whiteness; or else their +attributes are all necessarily connected, as in the case of circle, +square and triangle. But the members of nature's classes agree in +innumerable attributes which have no discoverable connection with one +another, and which must therefore, provisionally at least, be regarded +as standing in the relation of inseparable accidents to any particular +attributes which we may select for the purposes of definition. There +is no assignable reason why a rational animal should have hair on its +head or a nose on its face, and yet man, as a matter of fact, has +both; and generally the particular bodily configuration of man can +only be regarded as an inseparable accident of his nature as a +rational animal. + +§ 372. 'Real kinds' belong to the class of words mentioned above in +which the extension predominates over the intension. We know well +enough the things denoted by them, while most of us have only a dim +idea of the points of resemblance between these things. Nature's +classes moreover shade off into one another by such imperceptible +degrees that it is often impossible to fix the boundary line between +one class and another. A still greater source of perplexity in dealing +with real kinds is that it is sometimes almost impossible to fix upon +any attribute which is common to every individual member of the class +without exception. All that we can do in such cases is to lay down a +type of the class in its perfect form, and judge of individual +instances by the degree of their approximation to it. Again, real +kinds being known to us primarily in extension, the intension which we +attach to the names is hable to be affected by the advance of +knowledge. In dealing therefore with such terms we must be content +with provisional definitions, which adequately express our knowledge +of the things denoted by them, at the time, though a further study of +their attributes may induce us subsequently to alter the +definition. Thus the old definition of animal as a sentient organism +has been rendered inadequate by the discovery that so many of the +phenomena of sensation can be exhibited by plants, + +§ 373. But terms in which intension is the predominant idea are more +capable of being defined once for all. Aristotle's definitions of +'wealth' and 'monarchy' are as applicable now as in his own day, and +no subsequent discoveries of the properties of figures will render +Euclid's definitions unavailable. + +§ 374. We may distinguish therefore between two kinds of definition, +namely, + + (1) Final. + + (2) Provisional. + +§ 375. A distinction is also observed between Real and Nominal +Definitions. Both of these explain the meaning of a term: but a real +definition further assumes the actual existence of the thing +defined. Thus the explanation of the term 'Centaur' would be a +nominal, that of 'horse' a real definition. + +It is useless to assert, as is often done, that a nominal definition +explains the meaning of a term and a real definition the nature of a +thing; for, as we have seen already, the meaning of a term is whatever +we know of the nature of a thing. + +§ 376. It now remains to lay down certain rules for correct +definition. + +§ 377. The first rule that is commonly given is that a definition +should state the essential attributes of the thing defined. But this +amounts merely to saying that a definition should be a definition; +since it is only by the aid of definition that we can distinguish +between essential and non-essential among the common attributes +exhibited by a class of things. The rule however may be retained as a +material test of the soundness of a definition, in the sense that he +who seeks to define anything should fix upon its most important +attributes. To define man as a mammiferous animal having two hands, or +as a featherless biped, we feel to be absurd and incongruous, since +there is no reference to the most salient characteristic of man, +namely, his rationality. Nevertheless we cannot quarrel with these +definitions on formal, but only on material grounds. Again, if anyone +chose to define logic as the art of thinking, all we could say is that +we differ from him in opinion, as we think logic is more properly to +be regarded as the science of the laws of thought. But here also it is +on material grounds that we dissent from the definition. + +§ 378. Confining ourselves therefore to the sphere with which we are +properly concerned, we lay down the following + + + +_Rules for Definition._ + + + (1) A definition must be co-extensive with the term defined. + + (2) A definition must not state attributes which imply one another. + + (3) A definition must not contain the name defined, either directly + or by implication. + + (4) A definition must be clearer than the term defined. + + (5) A definition must not be negative, if it can be affirmative. + +Briefly, a definition must be adequate (1), terse (2), clear (4); and +must not be tautologous (3), or, if it can be avoided, negative (5). + +§ 379. It is worth while to notice a slight ambiguity in the term +'definition' itself. Sometimes it is applied to the whole proposition +which expounds the meaning of the term; at other times it is confined +to the predicate of this proposition. Thus in stating the first four +rules we have used the term in the latter sense, and in stating the +fifth in the former. + +§ 380. We will now illustrate the force of the above rules by giving +examples of their violation. + + Rule 1. Violations. A triangle is a figure with three equal sides. + + A square is a four-sided figure having all its sides equal. + +In the first instance the definition is less extensive than the term +defined, since it applies only to equilateral triangles. This fault +may be amended by decreasing the intension, which we do by eliminating +the reference to the equality of the sides. + +In the second instance the definition is more extensive than the term +defined. We must accordingly increase the intension by adding a new +attribute 'and all its angles right angles.' + + Rule 2. Violation. A triangle is a figure with three sides and three + angles. + +One of the chief merits of a definition is to be terse, and this +definition is redundant, since what has three sides cannot but have +three angles. + + Rule 3. Violations. A citizen is a person both of whose parents were + citizens. + + Man is a human being. + + Rule 4. Violations. A net is a reticulated fabric, decussated at + regular intervals. + + Life is the definite combination of heterogeneous changes, both + simultaneous and successive, in correspondence with external + co-existences and sequences. + + Rule 5. Violations. A mineral is that which is neither animal nor + vegetable. + + Virtue is the absence of vice. + +§ 381. The object of definition being to explain what a thing is, this +object is evidently defeated, if we confine ourselves to saying what +it is not. But sometimes this is impossible to be avoided. For there +are many terms which, though positive in form, are privative in force. +These terms serve as a kind of residual heads under which to throw +everything within a given sphere, which does not exhibit certain +positive attributes. Of this unavoidably negative nature was the +definition which we give of 'accident,' which amounted merely to +saying that it was any attribute which was neither a difference nor a +property. + +§ 382. The violation of Rule 3, which guards against defining a thing +by itself, is technically known as 'circulus in definiendo,' or +defining in a circle. This rule is often apparently violated, without +being really so. Thus Euclid defines an acute-angled triangle as one +which has three acute angles. This seems a glaring violation of the +rule, but is perfectly correct in its context; for it has already been +explained what is meant by the terms 'triangle' and 'acute angle,' and +all that is now required is to distinguish the acute-angled triangle +from its cognate species. He might have said that an acute-angled +triangle is one which has neither a right angle nor an obtuse angle: +but rightly preferred to throw the same statement into a positive +form. + +§ 383. The violation of Rule 4 is known as 'ignotum per ignotius' or +'per aeque ignotum.' This rule also may seemingly be violated when it +is not really so. For a definition may be correct enough from a +special point of view, which, apart from that particular context, +would appear ridiculous. From the point of view of conic sections, it +is correct enough to define a triangle as that section of a cone which +is formed by a plane passing through the vertex perpendicularly to the +base, but this could not be expected to make things clearer to a +person who was inquiring for the first time into the meaning of the +word triangle. But a real violation of the fourth rule may arise, not +only from obscurity, but from the employment of ambiguous language or +metaphor. To say that 'temperance is a harmony of the soul' or that +'bread is the staff of life,' throws no real light upon the nature of +the definiend. + +§ 384. The material correctness of a definition is, as we have already +seen, a matter extraneous to formal logic. An acquaintance with the +attributes which terms imply involves material knowledge quite as much +as an acquaintance with the things they apply to; knowledge of the +intension and of the extension of terms is alike acquired by +experience. No names are such that their meaning is rendered evident +by the very constitution of our mental faculties; yet nothing short of +this would suffice to bring the material content of definition within +the province of formal logic. + + + + +CHAPTER VIII. + +_Of Division._ + + +§ 385. To divide a term is to unfold its extension, that is, to set +forth the things of which it is a name. + +§ 386. But as in definition we need not completely unfold the +intension of a term, so in division we must not completely unfold its +extension. + +§ 387. Completely to unfold the extension of a term would involve +stating all the individual objects to which the name applies, a thing +which would be impossible in the case of most common terms. When it is +done, it is called Enumeration. To reckon up all the months of the +year from January to December would be an enumeration, and not a +division, of the term 'month.' + +§ 388. Logical division always stops short at classes. It may be +defined as the statement of the various classes of things that can be +called by a common name. Technically we may say that it consists in +breaking up a genus into its component species. + +§ 389. Since division thus starts with a class and ends with classes, +it is clear that it is only common terms which admit of division, and +also that the members of the division must themselves be common terms. + +§ 390. An 'individual' is so called as not admitting of logical +division. We may divide the term 'cow' into classes, as Jersey, +Devonshire, &c., to which the name 'cow' will still be applicable, but +the parts of an individual cow are no longer called by the name of the +whole, but are known as beefsteaks, briskets, &c. + +§ 391. In dividing a term the first requisite is to fix upon some +point wherein certain members of the class differ from others. The +point thus selected is called the Fundamentum Divisionis or Basis of +the Division. + +§ 392. The basis of the division will of course differ according to +the purpose in hand, and the same term will admit of being divided on +a number of different principles. Thus we may divide the term 'man,' +on the basis of colour, into white, black, brown, red, and yellow; or, +on the basis of locality, into Europeans, Asiatics, Africans, +Americans, Australians, New Zealanders, and Polynesians; or again, on +a very different principle, into men of nervous, sanguine, bilious, +lymphatic and mixed temperaments. + +§ 393. The term required to be divided is known as the Totum Divisum +or Divided Whole. It might also be called the Dividend. + +§ 394. The classes into which the dividend is split up are called the +Membra Dividentia, or Dividing Members. + +§ 395. Only two rules need be given for division-- + + (1) The division must be conducted on a single basis. + + (2) The dividing members must be coextensive with the divided whole. + +§ 396. More briefly, we may put the same thing thus--There must be no +cross-division (1) and the division must be exhaustive (2). + +§ 397. The rule, which is commonly given, that each dividing member +must be a common term, is already provided for under our definition of +the process. + +§ 398. The rule that the dividend must be predicable of each of the +dividing members is contained in our second rule; since, if there were +any term of which the dividend were not predicable, it would be +impossible for the dividing members to be exactly coextensive with it. +It would not do, for instance, to introduce mules and donkeys into a +division of the term horse. + +§ 399. Another rule, which is sometimes given, namely, that the +constituent species must exclude one another, is a consequence of our +first; for, if the division be conducted on a single principle, the +constituent species must exclude one another. The converse, however, +does not hold true. We may have a division consisting of mutually +exclusive members, which yet involves a mixture of different bases, +e.g. if we were to divide triangle into scalene, isosceles and +equiangular. This happens because two distinct attributes may be found +in invariable conjunction. + +§ 400. There is no better test, however, of the soundness of a +division than to try whether the species overlap, that is to say, +whether there are any individuals that would fall into two or more of +the classes. When this is found to be the case, we may be sure that we +have mixed two or more different fundamenta divisionis. If man, for +instance, were to be divided into European, American, Aryan, and +Semitic, the species would overlap; for both Europe and America +contain inhabitants of Aryan and Semitic origin. We have here members +of a division based on locality mixed up with members of another +division, which is based on race as indicated by language. + +§ 401. The classes which are arrived at by an act of division may +themselves be divided into smaller classes. This further process is +called Subdivision. + +§ 402. Let it be noticed that Rule 1 applies only to a single act of +division. The moment that we begin to subdivide we not only may, but +must, adopt a new basis of division; since the old one has, 'ex +hypothesi,' been exhausted. Thus, having divided men according to the +colour of their skins, if we wish to subdivide any of the classes, we +must look out for some fresh attribute wherein some men of the same +complexion differ from others, e.g. we might divide black men into +woolly-haired blacks, such as the Negroes, and straight-haired blacks, +like the natives of Australia. + +§ 403. We will now take an instance of division and +subdivision, with a view to illustrating some of the +technical terms which are used in connection with the +process. Keeping closely to our proper subject, we will +select as an instance a division of the products of thought, +which it is the province of logic to investigate. + + Product of thought + _______________|____________________________ + | | | + Term Proposition Inference + ____|___ ______|_____ _____|______ + | | | | | | + Singular Common Universal Particular Immediate Mediate + ___|___ ___|___ + | | | | + A E I O + +Here we have first a threefold division of the products of thought +based on their comparative complexity. The first two of these, namely, +the term and the proposition, are then subdivided on the basis of +their respective quantities. In the case of inference the basis of the +division is again the degree of complexity. The subdivision of the +proposition is carried a step further than that of the others. Having +exhausted our old basis of quantity, we take a new attribute, namely, +quality, on which to found the next step of subdivision. + +§ 404. Now in such a scheme of division and subdivision as the +foregoing, the highest class taken is known as the Summum Genus. Thus +the summum genus is the same thing as the divided whole, viewed in a +different relation. The term which is called the divided whole with +reference to a single act of division, is called the summum genus +whenever subdivision has taken place. + +§ 405. The classes at which the division stops, that is, any which are +not subdivided, are known as the Infimae Species. + +§ 406. All classes intermediate between the summum genus and the +infimae species are called Subaltern Genera or Subaltern Species, +according to the way they are looked at, being genera in relation to +the classes below them and species in relation to the classes above +them. + +§ 407. Any classes which fall immediately under the same genus are +called Cognate Species, e.g. singular and common terms are cognate +species of term. + +§ 408. The classes under which any lower class successively falls are +called Cognate Genera. The relation of cognate species to one another +is like that of children of the same parents, whereas cognate genera +resemble a line of ancestry. + +§ 409. The Specific Difference of anything is the attribute or +attributes which distinguish it from its cognate species. Thus the +specific difference of a universal proposition is that the predicate +is known to apply to the whole of the subject. A specific difference +is said to constitute the species. + +§ 410. The specific difference of a higher class becomes a Generic +Difference with respect to the class below it. A generic difference +then may be said to be the distinguishing attribute of the whole class +to which a given species belongs. The generic difference is common to +species that are cognate to one another, whereas the specific +difference is peculiar to each. It is the generic difference of an A +proposition that it is universal, the specific difference that it is +affirmative. + +§ 411. The same distinction is observed between the specific and +generic properties of a thing. A Specific Property is an attribute +which flows from the difference of a thing itself; a Generic Property +is an attribute which flows from the difference of the genus to which +the thing belongs. It is a specific property of an E proposition that +its predicate is distributed, a generic property that its contrary +cannot be true along with it (§ 465); for this last characteristic +flows from the nature of the universal proposition generally. + +§ 412. It now remains to say a few words as to the place in logic of +the process of division. Since the attributes in which members of the +same class differ from one another cannot possibly be indicated by +their common name, they must be sought for by the aid of experience; +or, to put the same thing in other words, since all the infimae +species are alike contained under the summum genus, their distinctive +attributes can be no more than separable accidents when viewed in +relation to the summum genus. Hence division, being always founded on +the possession or non-possession of accidental attributes, seems to +lie wholly outside the sphere of formal logic. This however is not +quite the case, for, in virtue of the Law of Excluded Middle, there is +always open to us, independently of experience, a hypothetical +division by dichotomy. By dichotomy is meant a division into two +classes by a pair of contradictory terms, e.g. a division of the +class, man, into white and not-white. Now we cannot know, +independently of experience, that any members of the class, man, +possess whiteness; but we may be quite sure, independently of all +experience, that men are either white or not. Hence division by +dichotomy comes strictly within the province of formal logic. Only it +must be noticed that both sides of the division must be hypothetical. +For experience alone can tell us, on the one hand, that there are any +men that are white, and on the other, that there are any but white +men. + +§ 413. What we call a division on a single basis is in reality the +compressed result of a scheme of division and subdivision by +dichotomy, in which a fresh principle has been introduced at every +step. Thus when we divide men, on the basis of colour, into white, +black, brown, red and yellow, we may be held to have first divided men +into white and not-white, and then to have subdivided the men that are +not-white into black and not-black, and so on. From the strictly +formal point of view this division can only be represented as +follows-- + + Men + ___________________|_____ + | | + White (if any) Not-white (if any) + _________________|_____ + | | + Black (if any) Not-black (if any) + __________________|____ + | | + Brown (if any) Not-brown (if any) + ____________________|____ + | | + Red (if any) Not-red (if any). + +§ 414. Formal correctness requires that the last term in such a series +should be negative. We have here to keep the term 'not-red' open, to +cover any blue or green men that might turn up. It is only experience +that enables us to substitute the positive term 'yellow' for +'not-red,' since we know as a matter of fact that there are no men but +those of the five colours given in the original division. + +§ 415. Any correct logical division always admits of being arrived at +by the longer process of division and subdivision by dichotomy. For +instance, the term quadrilateral, or four-sided rectilinear figure, is +correctly divided into square, oblong, rhombus, rhomboid and +trapezium. The steps of which this division consists are as follows-- + + Quadrilateral + __________|_________ + | | + Parallelogram Trapezium + _____|_____________________ + | | + Rectangle Non-rectangle + ___|___ _____|_____ + | | | | + Square Oblong Rhombus Rhomboid. + +§ 416. In reckoning up the infimae species in such a scheme, we must +of course be careful not to include any class which has been already +subdivided; but no harm is done by mixing an undivided class, like +trapezium, with the subdivisions of its cognate species. + +§ 417. The two processes of definition and division are intimately +connected with one another. Every definition suggests a division by +dichotomy, and every division supplies us at once with a complete +definition of all its members. + +§ 418. Definition itself, so far as concerns its content, is, as we +have already seen, extraneous to formal logic: but when once we have +elicited a genus and difference out of the material elements of +thought, we are enabled, without any further appeal to experience, to +base thereon a division by dichotomy. Thus when man has been defined +as a rational animal, we have at once suggested to us a division of +animal into rational and irrational. + +§ 419. Again, the addition of the attributes, rational and irrational +respectively, to the common genus, animal, ipso facto supplies us with +definitions of the species, man and brute. Similarly, when we +subdivided rectangle into square and oblong on the basis of the +equality or inequality of the adjacent sides, we were by so doing +supplied with a definition both of square and oblong--'A square is a +rectangle having all its sides equal,' and 'An oblong is a rectangle +which has only its opposite sides equal.' + +§ 420. The definition of a square just given amounts to the same thing +as Euclid's definition, but it complies with a rule which has value as +a matter of method, namely, that the definition should state the +Proximate Genus of the thing defined. + +§ 421. Since definition and division are concerned with the intension +and extension of terms, they are commonly treated of under the first +part of logic: but as the treatment of the subject implies a +familiarity with the Heads of Predicables, which in their turn imply +the proposition, it seems more desirable to deal with them under the +second. + +§ 422. We have already had occasion to distinguish division from +Enumeration. The latter is the statement of the individual things to +which a name applies. In enumeration, as in division, the wider term +is predicable of each of the narrower ones. + +§ 423. Partition is the mapping out of a physical whole into its +component parts, as when we say that a tree consists of roots, stem, +and branches. In a partition the name of the whole is not predicable +of each of the parts. + +§ 424. Distinction is the separation from one another of the various +meanings of an equivocal term. The term distinguished is predicable +indeed of each of the members, but of each in a different sense. An +equivocal term is in fact not one but several terms, as would quickly +appear, if we were to use definitions in place of names. + +§ 425. We have seen that a logical whole is a genus viewed in relation +to its underlying species. From this must be distinguished a +metaphysical whole, which is a substance viewed in relation to its +attributes, or a class regarded in the same way. Logically, man is a +part of the class, animal; metaphysically, animal is contained in +man. Thus a logical whole is a whole in extension, while a +metaphysical whole is a whole in intension. From the former point of +view species is contained in genus; from the latter genus is contained +in species. + + + + + +PART III.--OF INFERENCES. + + + + +CHAPTER I. + +_Of Inferences in General_. + + +§ 426. To infer is to arrive at some truth, not by direct experience, +but as a consequence of some truth or truths already known. If we see +a charred circle on the grass, we infer that somebody has been +lighting a fire there, though we have not seen anyone do it. This +conclusion is arrived at in consequence of our previous experience of +the effects of fire. + +§ 427. The term Inference is used both for a process and for a product +of thought. + +As a process inference may be defined as the passage of the mind from +one or more propositions to another. + +As a product of thought inference may be loosely declared to be the +result of comparing propositions. + +§ 428. Every inference consists of two parts-- + + (1) the truth or truths already known; + + (2) the truth which we arrive at therefrom. + +The former is called the Antecedent, the latter the Consequent. But +this use of the terms 'antecedent' and 'consequent' must be carefully +distinguished from the use to which they were put previously, to +denote the two parts of a complex proposition. + +§ 429. Strictly speaking, the term inference, as applied to a product +of thought, includes both the antecedent and consequent: but it is +often used for the consequent to the exclusion of the +antecedent. Thus, when we have stated our premisses, we say quite +naturally, 'And the inference I draw is so and so.' + +§ 430. Inferences are either Inductive or Deductive. In induction we +proceed from the less to the more general; in deduction from the more +to the less general, or, at all events, to a truth of not greater +generality than the one from which we started. In the former we work +up to general principles; in the latter we work down from them, and +elicit the particulars which they contain. + +§ 431. Hence induction is a real process from the known to the +unknown, whereas deduction is no more than the application of +previously existing knowledge; or, to put the same thing more +technically, in an inductive inference the consequent is not contained +in the antecedent, in a deductive inference it is. + +§ 432. When, after observing that gold, silver, lead, and other +metals, are capable of being reduced to a liquid state by the +application of heat, the mind leaps to the conclusion that the same +will hold true of some other metal, as platinum, or of all metals, we +have then an inductive inference, in which the conclusion, or +consequent, is a new proposition, which was not contained in those +that went before. We are led to this conclusion, not by reason, but by +an instinct which teaches us to expect like results, under like +circumstances. Experience can tell us only of the past: but we allow +it to affect our notions of the future through a blind belief that +'the thing that hath been, it is that which shall be; and that which +is done is that which shall be done; and there is no new thing under +the sun.' Take away this conviction, and the bridge is cut which +connects the known with the unknown, the past with the future. The +commonest acts of daily life would fail to be performed, were it not +for this assumption, which is itself no product of the reason. Thus +man's intellect, like his faculties generally, rests upon a basis of +instinct. He walks by faith, not by sight. + +§ 433. It is a mistake to talk of inductive reasoning, as though it +were a distinct species from deductive. The fact is that inductive +inferences are either wholly instinctive, and so unsusceptible of +logical vindication, or else they may be exhibited under the form of +deductive inferences. We cannot be justified in inferring that +platinum will be melted by heat, except where we have equal reason for +asserting the same thing of copper or any other metal. In fact we are +justified in drawing an individual inference only when we can lay down +the universal proposition, 'Every metal can be melted by heat.' But +the moment this universal proposition is stated, the truth of the +proposition in the individual instance flows from it by way of +deductive inference. Take away the universal, and we have no logical +warrant for arguing from one individual case to another. We do so, as +was said before, only in virtue of that vague instinct which leads us +to anticipate like results from like appearances. + +§ 434. Inductive inferences are wholly extraneous to the science of +formal logic, which deals only with formal, or necessary, inferences, +that is to say with deductive inferences, whether immediate or +mediate. These are called formal, because the truth of the consequent +is apparent from the mere form of the antecedent, whatever be the +nature of the matter, that is, whatever be the terms employed in the +proposition or pair of propositions which constitutes the +antecedent. In deductive inference we never do more than vary the form +of the truth from which we started. When from the proposition 'Brutus +was the founder of the Roman Republic,' we elicit the consequence 'The +founder of the Roman Republic was Brutus,' we certainly have nothing +more in the consequent than was already contained in the antecedent; +yet all deductive inferences may be reduced to identities as palpable +as this, the only difference being that in more complicated cases the +consequent is contained in the antecedent along with a number of other +things, whereas in this case the consequent is absolutely all that the +antecedent contains. + +§ 435. On the other hand, it is of the very essence of induction that +there should be a process from the known to the unknown. Widely +different as these two operations of the mind are, they are yet both +included under the definition which we have given of inference, as the +passage of the mind from one or more propositions to another. It is +necessary to point this out, because some logicians maintain that all +inference must be from the known to the unknown, whereas others +confine it to 'the carrying out into the last proposition of what was +virtually contained in the antecedent judgements.' + +§ 436. Another point of difference that has to be noticed between +induction and deduction is that no inductive inference can ever attain +more than a high degree of probability, whereas a deductive inference +is certain, but its certainty is purely hypothetical. + +§ 437. Without touching now on the metaphysical difficulty as to how +we pass at all from the known to the unknown, let us grant that there +is no fact better attested by experience than this--'That where the +circumstances are precisely alike, like results follow.' But then we +never can be absolutely sure that the circumstances in any two cases +are precisely alike. All the experience of all past ages in favour of +the daily rising of the sun is not enough to render us theoretically +certain that the sun will rise tomorrow We shall act indeed with a +perfect reliance upon the assumption of the coming day-break; but, for +all that, the time may arrive when the conditions of the universe +shall have changed, and the sun will rise no more. + +§ 438. On the other hand a deductive inference has all the certainty +that can be imparted to it by the laws of thought, or, in other words, +by the structure of our mental faculties; but this certainty is purely +hypothetical. We may feel assured that if the premisses are true, the +conclusion is true also. But for the truth of our premisses we have to +fall back upon induction or upon intuition. It is not the province of +deductive logic to discuss the material truth or falsity of the +propositions upon which our reasonings are based. This task is left to +inductive logic, the aim of which is to establish, if possible, a test +of material truth and falsity. + +§ 439. Thus while deduction is concerned only with the relative truth +or falsity of propositions, induction is concerned with their actual +truth or falsity. For this reason deductive logic has been termed the +logic of consistency, not of truth. + +§ 440. It is not quite accurate to say that in deduction we proceed +from the more to the less general, still less to say, as is often +said, that we proceed from the universal to the particular. For it may +happen that the consequent is of precisely the same amount of +generality as the antecedent. This is so, not only in most forms of +immediate inference, but also in a syllogism which consists of +singular propositions only, e.g. + + The tallest man in Oxford is under eight feet. + So and so is the tallest man in Oxford. + .'. So and so is under eight feet. + +This form of inference has been named Traduction; but there is no +essential difference between its laws and those of deduction. + +§ 441. Subjoined is a classification of inferences, which will serve +as a map of the country we are now about to explore. + + Inference + ________________________|__________ + | | + Inductive Deductive + _________________|_______________ + | | + Immediate Mediate + ___________|__________ ______|______ + | | | | + Simple Compound Simple Complex + ______|________________ | ______|_____________|_ + | | | | | | | + Opposition Conversion Permutation | Conjunctive Disjunctive Dilemma + | + _________|________ + | | + Conversion Conversion + by by + Negation position + + + + +CHAPTER II. + +_Of Deductive Inferences._ + + +$ 442. Deductive inferences are of two kinds--Immediate and Mediate. + +§ 443. An immediate inference is so called because it is effected +without the intervention of a middle term, which is required in +mediate inference. + +§ 444. But the distinction between the two might be conveyed with at +least equal aptness in this way-- + +An immediate inference is the comparison of two propositions directly. + +A mediate inference is the comparison of two propositions by means of +a third. + +§ 445. In that sense of the term inference in which it is confined to +the consequent, it may be said that-- + +An immediate inference is one derived from a single proposition. + +A mediate inference is one derived from two propositions conjointly. + +§ 446. There are never more than two propositions in the antecedent of +a deductive inference. Wherever we have a conclusion following from +more than two propositions, there will be found to be more than one +inference. + +§ 447. There are three simple forms of immediate inference, namely +Opposition, Conversion and Permutation. + +§ 448. Besides these there are certain compound forms, in which +permutation is combined with conversion. + + + + +CHAPTER III. + +_Of Opposition._ + + +§ 449. Opposition is an immediate inference grounded on the relation +between propositions which have the same terms, but differ in quantity +or in quality or in both. + +§ 450. In order that there should be any formal opposition between two +propositions, it is necessary that their terms should be the +same. There can be no opposition between two such propositions as +these-- + + (1) All angels have wings. + + (2) No cows are carnivorous. + +§ 451. If we are given a pair of terms, say A for subject and B for +predicate, and allowed to affix such quantity and quality as we +please, we can of course make up the four kinds of proposition +recognised by logic, namely, + + A. All A is B. + + E. No A is B. + + I. Some A is B. + + O. Some A is not B. + +§ 452. Now the problem of opposition is this: Given the truth or +falsity of any one of the four propositions A, E, I, O, what can be +ascertained with regard to the truth or falsity of the rest, the +matter of them being supposed to be the same? + +§ 453. The relations to one another of these four propositions +are usually exhibited in the following scheme-- + + A . . . . Contrary . . . . E + . . . . + . . . . + . . . . + . . . . + . . . . + . . . . + Subaltern Contradictory Subaltern + . . . . + . . . . + . . . . + . . . . + . . . . + . . . . + I . . . Sub-contrary . . . O + +§ 454. Contrary Opposition is between two universals which differ in +quality. + +§ 455. Sub-contrary Opposition is between two particulars which differ +in quality. + +§ 456. Subaltern Opposition is between two propositions which differ +only in quantity. + +§ 457. Contradictory Opposition is between two propositions which +differ both in quantity and in quality. + +§ 458. Subaltern Opposition is also known as Subalternation, and of +the two propositions involved the universal is called the Subalternant +and the particular the Subalternate. Both together are called +Subalterns, and similarly in the other forms of opposition the two +propositions involved are known respectively as Contraries, +Sub-contraries and Contradictories. + +§ 459. For the sake of convenience some relations are classed under +the head of opposition in which there is, strictly speaking, no +opposition at all between the two propositions involved. + +§ 460. Between sub-contraries there is an apparent, but not a real +opposition, since what is affirmed of one part of a term may often +with truth be denied of another. Thus there is no incompatibility +between the two statements. + + (1) Some islands are inhabited. + + (2) Some islands are not inhabited. + +§ 461. In the case of subaltern opposition the truth of the universal +not only may, but must, be compatible with that of the particular. + +§ 462. Immediate Inference by Relation would be a more appropriate +name than Opposition; and Relation might then be subdivided into +Compatible and Incompatible Relation. By 'compatible' is here meant +that there is no conflict between the _truth_ of the two +propositions. Subaltern and sub-contrary opposition would thus fall +under the head of compatible relation; contrary and contradictory +relation under that of incompatible relation. + + Relation + ______________|_____________ + | | + Compatible Incompatible + ______|_____ _____|_______ + | | | | + Subaltern Sub-contrary Contrary Contradictory. + +§ 463. It should be noticed that the inference in the case of +opposition is from the truth or falsity of one of the opposed +propositions to the truth or falsity of the other. + +§ 464. We will now lay down the accepted laws of inference with regard +to the various kinds of opposition. + +§ 465. Contrary propositions may both be false, but cannot both be +true. Hence if one be true, the other is false, but not vice versâ. + +§ 466. Sub-contrary propositions may both be true, but cannot both be +false. Hence if one be false, the other is true, but not vice versâ. + +§ 467. In the case of subaltern propositions, if the universal be +true, the particular is true; and if the particular be false, the +universal is false; but from the truth of the particular or the +falsity of the universal no conclusion can be drawn. + +§ 468. Contradictory propositions cannot be either true or false +together. Hence if one be true, the other is false, and vice versâ. + +§ 469. By applying these laws of inference we obtain the following +results-- + + If A be true, E is false, O false, I true. + + If A be false, E is unknown, O true, I unknown. + + If E be true, O is true, I false, A false. + + If E be false, O is unknown, I true, A unknown. + + If O be true, I is unknown, A false, E unknown. + + If O be false, I is true, A true, E false. + + If I be true, A is unknown, E false, O unknown. + + If I be false, A is false, E true, O true. + +§ 470. It will be seen from the above that we derive more information +from deriving a particular than from denying a universal. Should this +seem surprising, the paradox will immediately disappear, if we reflect +that to deny a universal is merely to assert the contradictory +particular, whereas to deny a particular is to assert the +contradictory universal. It is no wonder that we should obtain more +information from asserting a universal than from asserting a +particular. + +§ 471. We have laid down above the received doctrine with regard to +opposition: but it is necessary to point out a flaw in it. + +When we say that of two sub-contrary propositions, if one be false, +the other is true, we are not taking the propositions I and O in their +now accepted logical meaning as indefinite (§ 254), but rather in +their popular sense as 'strict particular' propositions. For if I and +O were taken as indefinite propositions, meaning 'some, if not all,' +the truth of I would not exclude the possibility of the truth of A, +and, similarly, the truth of O would not exclude the possibility of +the truth of E. Now A and E may both be false. Therefore I and O, +being possibly equivalent to them, may both be false also. In that +case the doctrine of contradiction breaks down as well. For I and O +may, on this showing, be false, without their contradictories E and A +being thereby rendered true. This illustrates the awkwardness, which +we have previously had occasion to allude to, which ensures from +dividing propositions primarily into universal and particular, instead +of first dividing them into definite and indefinite, and particular (§ +256). + +§ 472. To be suddenly thrown back upon the strictly particular view of +I and O in the special case of opposition, after having been +accustomed to regard them as indefinite propositions, is a manifest +inconvenience. But the received doctrine of opposition does not even +adhere consistently to this view. For if I and O be taken as strictly +particular propositions, which exclude the possibility of the +universal of the same quality being true along with them, we ought not +merely to say that I and O may both be true, but that if one be true +the other must also be true. For I being true, A is false, and +therefore O is true; and we may argue similarly from the truth of O to +the truth of I, through the falsity of E. Or--to put the Same thing in +a less abstract form--since the strictly particular proposition means +'some, but not all,' it follows that the truth of one sub-contrary +necessarily carries with it the truth of the other, If we lay down +that some islands only are inhabited, it evidently follows, or rather +is stated simultaneously, that there are some islands also which are +not inhabited. For the strictly particular form of proposition 'Some A +only is B' is of the nature of an exclusive proposition, and is really +equivalent to two propositions, one affirmative and one negative. + +§ 473. It is evident from the above considerations that the doctrine +of opposition requires to be amended in one or other of two +ways. Either we must face the consequences which follow from regarding +I and O as indefinite, and lay down that sub-contraries may both be +false, accepting the awkward corollary of the collapse of the doctrine +of contradiction; or we must be consistent with ourselves in regarding +I and O, for the particular purposes of opposition, as being strictly +particular, and lay down that it is always possible to argue from the +truth of one sub-contrary to the truth of the other. The latter is +undoubtedly the better course, as the admission of I and O as +indefinite in this connection confuses the theory of opposition +altogether. + +§ 474. Of the several forms of opposition contradictory opposition is +logically the strongest. For this three reasons may be given-- + + (1) Contradictory opposites differ both in quantity and in quality, + whereas others differ only in one or the other. + + (2) Contradictory opposites are incompatible both as to truth and + falsity, whereas in other cases it is only the truth _or_ + falsity of the two that is incompatible. + + (3) Contradictory opposition is the safest form to adopt in + argument. For the contradictory opposite refutes the adversary's + proposition as effectually as the contrary, and is not so hable to a + counter-refutation. + +§ 475. At first sight indeed contrary opposition appears stronger, +because it gives a more sweeping denial to the adversary's +assertion. If, for instance, some person with whom we were arguing +were to lay down that 'All poets are bad logicians,' we might be +tempted in the heat of controversy to maintain against him the +contrary proposition 'No poets are bad logicians.' This would +certainly be a more emphatic contradiction, but, logically considered, +it would not be as sound a one as the less obtrusive contradictory, +'Some poets are not bad logicians,' which it would be very difficult +to refute. + +§ 476. The phrase 'diametrically opposed to one another' seems to be +one of the many expressions which have crept into common language from +the technical usage of logic. The propositions A and O and E and I +respectively are diametrically opposed to one another in the sense +that the straight lines connecting them constitute the diagonals of +the parallelogram in the scheme of opposition. + +§ 477. It must be noticed that in the case of a singular proposition +there is only one mode of contradiction possible. Since the quantity +of such a proposition is at the minimum, the contrary and +contradictory are necessarily merged into one. There is no way of +denying the proposition 'This house is haunted,' save by maintaining +the proposition which differs from it only in quality, namely, 'This +house is not haunted.' + +478. A kind of generality might indeed he imparted even to a singular +proposition by expressing it in the form 'A is always B.' Thus we may +say, 'This man is always idle'--a proposition which admits of being +contradicted under the form 'This man is sometimes not idle.' + + + + +CHAPTER IV. + +_Of Conversion._ + + +§ 479. Conversion is an immediate inference grounded On the +transposition of the subject and predicate of a proposition. + +§ 480. In this form of inference the antecedent is technically known +as the Convertend, i.e. the proposition to be converted, and the +consequent as the Converse, i.e. the proposition which has been +converted. + +§ 481. In a loose sense of the term we may be said to have converted a +proposition when we have merely transposed the subject and predicate, +when, for instance, we turn the proposition 'All A is B' into 'All B +is A' or 'Some A is not B' into 'Some B is not A.' But these +propositions plainly do not follow from the former ones, and it is +only with conversion as a form of inference--with Illative Conversion +as it is called--that Logic is concerned. + +§ 482. For conversion as a form of inference two rules have been laid +down-- + + (1) No term must be distributed in the converse which was not + distributed in the convertend. + + (2) The quality of the converse must be the same as that of the + convertend. + +§ 483. The first of these rules is founded on the nature of things. A +violation of it involves the fallacy of arguing from part of a term to +the whole. + +§ 484. The second rule is merely a conventional one. We may make a +valid inference in defiance of it: but such an inference will be seen +presently to involve something more than mere conversion. + +§ 485. There are two kinds of conversion-- + + (1) Simple. + + (2) Per Accidens or by Limitation. + +§ 486. We are said to have simply converted a proposition when the +quantity remains the same as before. + +§ 487. We are said to have converted a proposition per accidens, or by +limitation, when the rules for the distribution of terms necessitate a +reduction in the original quantity of the proposition. + +§ 488. + + A can only be converted per accidens. + + E and I can be converted simply. + + O cannot be converted at all. + +§ 489. The reason why A can only be converted per accidens is that, +being affirmative, its predicate is undistributed (§ 293). Since 'All +A is B' does not mean more than 'All A is some B,' its proper converse +is 'Some B is A.' For, if we endeavoured to elicit the inference, 'All +B is A,' we should be distributing the term B in the converse, which +was not distributed in the convertend. Hence we should be involved in +the fallacy of arguing from the part to the whole. Because 'All +doctors are men' it by no means follows that 'All men are doctors.' + +§ 499. E and I admit of simple conversion, because the quantity of the +subject and predicate is alike in each, both subject and predicate +being distributed in E and undistributed in I. + + + / No A is B. + E < + \ .'. No B is A. + + / Some A is B. + I < + \ .'. Some B is A. + +§ 491. The reason why O cannot be converted at all is that its subject +is undistributed and that the proposition is negative. Now, when the +proposition is converted, what was the subject becomes the predicate, +and, as the proposition must still be negative, the former subject +would now be distributed, since every negative proposition distributes +its predicate. Hence we should necessarily have a term distributed in +the converse which was not distributed in the convertend. From 'Some +men are not doctors,' it plainly does not follow that 'Some doctors +are not men'; and, generally from 'Some A is not B' it cannot be +inferred that 'Some B is not A,' since the proposition 'Some A is not +B' admits of the interpretation that B is wholly contained in A. + +[Illustration] + +§ 492. It may often happen as a matter of fact that in some given +matter a proposition of the form 'All B is A' is true simultaneously +with 'All A is B.' Thus it is as true to say that 'All equiangular +triangles are equilateral' as that 'All equilateral triangles are +equiangular.' Nevertheless we are not logically warranted in inferring +the one from the other. Each has to be established on its separate +evidence. + +§ 493. On the theory of the quantified predicate the difference +between simple conversion and conversion by limitation disappears. For +the quantity of a proposition is then no longer determined solely by +reference to the quantity of its subject. 'All A is some B' is of no +greater quantity than 'Some B is all A,' if both subject and predicate +have an equal claim to be considered. + +§ 494. Some propositions occur in ordinary language in which the +quantity of the predicate is determined. This is especially the case +when the subject is a singular term. Such propositions admit of +conversion by a mere transposition of their subject and predicate, +even though they fall under the form of the A proposition, e.g. + + Virtue is the condition of happiness. + .'. The condition of happiness is virtue. + +And again, + + Virtue is a condition of happiness. + .'. A condition of happiness is virtue. + +In the one case the quantity of the predicate is determined by the +form of the expression as distributed, in the other as undistributed. + +§ 495. Conversion offers a good illustration of the principle on which +we have before insisted, namely, that in the ordinary form of +proposition the subject is used in extension and the predicate in +intension. For when by conversion we change the predicate into the +subject, we are often obliged to attach a noun substantive to the +predicate, in order that it may be taken in extension, instead of, as +before, in intension, e.g. + + Some mothers are unkind. + .'. Some unkind persons are mothers. + +Again, + + Virtue is conducive to happiness. + .'. One of the things which are conducive to happiness is virtue. + + + + +CHAPTER V. + +_Of Permutation._ + + +§ 496. Permutation [Footnote: Called by some writers Obversion.] is an +immediate inference grounded on a change of quality in a proposition +and a change of the predicate into its contradictory-term. + +§ 497. In less technical language we may say that permutation is +expressing negatively what was expressed affirmatively and vice versâ. + +§ 498. Permutation is equally applicable to all the four +forms of proposition. + + (A) All A is B. + .'. No A is not-B (E). + + (E) No A is B. + .'. All A is not-B (A). + + (I) Some A is B. + .'. Some A is not not-B (O). + + (O) Some A is not B. + .'. Some A is not-B (I). + +§ 499, Or, to take concrete examples-- + + (A) All men are fallible. + .'. No men are not-fallible (E). + + (E) No men are perfect. + .'. All men are not-perfect (A). + + (I) Some poets are logical. + .'. Some poets are not not-logical (O). + + (O) Some islands are not inhabited. + .'. Some islands are not-inhabited (I). + +§ 500. The validity of permutation rests on the principle of excluded +middle, namely--That one or other of a pair of contradictory terms +must be applicable to a given subject, so that, when one may be +predicated affirmatively, the other may be predicated negatively, and +vice versâ (§ 31). + +§ 501. Merely to alter the quality of a proposition would of course +affect its meaning; but when the predicate is at the same time changed +into its contradictory term, the original meaning of the proposition +is retained, whilst the form alone is altered. Hence we may lay down +the following practical rule for permutation-- + +Change the quality of the proposition and change the predicate into +its contradictory term. + +§ 502. The law of excluded middle holds only with regard to +contradictories. It is not true of a pair of positive and privative +terms, that one or other of them must be applicable to any given +subject. For the subject may happen to fall wholly outside the sphere +to which such a pair of terms is limited. But since the fact of a term +being applied is a sufficient indication of its applicability, and +since within a given sphere positive and privative terms are as +mutually destructive as contradictories, we may in all cases +substitute the privative for the negative term in immediate inference +by permutation, which will bring the inferred proposition more into +conformity with the ordinary usage of language. Thus the concrete +instances given above will appear as follows-- + + (A) All men are fallible. + .'. No men are infallible (E). + + (E) No men are perfect. + .'. All men are imperfect (A). + + (I) Some poets are logical. + .'. Some poets are not illogical (O). + + (O) Some islands are not inhabited. + .'. Some islands are uninhabited (I). + + + + +CHAPTER VI. + +_Of Compound Forms of Immediate Inference._ + + +§ 503. Having now treated of the three simple forms of immediate +inference, we go on to speak of the compound forms, and first of + + + +_Conversion by Negation._ + + +§ 504. When A and O have been permuted, they become respectively E and +I, and, in this form, admit of simple conversion. We have here two +steps of inference: but the process may be performed at a single +stroke, and is then known as Conversion by Negation. Thus from 'All A +is B' we may infer 'No not-B is A,' and again from 'Some A is not B' +we may infer 'Some not-B is A.' The nature of these inferences will be +seen better in concrete examples. + +§ 505. + + (A) All poets are imaginative. + .'. No unimaginative persons are poets (E). + + (O) Some parsons are not clerical. + .'. Some unclerical persons are parsons (I). + +§ 506. The above inferences, when analysed, will be found to resolve +themselves into two steps, namely, + + (1) Permutation. + + (2) Simple Conversion. + + (A) All A is B. + .'. No A is not-B (by permutation). + .'. No not-B is A (by simple conversion). + + (O) Some A is not B. + .'. Some A is not-B (by permutation). + .'. Some not-B is A (by simple conversion). + +§ 507. The term conversion by negation has been arbitrarily limited to +the exact inferential procedure of permutation followed by simple +conversion. Hence it necessarily applies only to A and 0 propositions, +since these when permuted become E and 1, which admit of simple +conversion; whereas E and 1 themselves are permuted into A and 0, +which do not. There seems to be no good reason, however, why the term +'conversion by negation' should be thus restricted in its meaning; +instead of being extended to the combination of permutation with +conversion, no matter in what order the two processes may be +performed. If this is not done, inferences quite as legitimate as +those which pass under the title of conversion by negation are left +without a name. + +§ 508. From E and 1 inferences may be elicited as follows-- + + (E) No A is B. + .'. All B is not-A (A). + + (I) Some A is B. + .'. Some B is not not-A (O). + + (E) No good actions are unbecoming. + .'. All unbecoming actions are not-good (A). + + (I) Some poetical persons are logicians. + .'. Some logicians are not unpoetical (O). + +Or, taking a privative term for our subject, + + Some unpractical persons are statesmen. + .'. Some statesmen are not practical. + +§ 509. When the inferences just given are analysed, it will be found +that the process of simple conversion precedes that of permutation. + +§ 510. In the case of the E proposition a compound inference can be +drawn even in the original order of the processes, + + No A is B. + .'. Some not-B is A. + + No one who employs bribery is honest. + .'. Some dishonest men employ bribery. + +The inference here, it must be remembered, does not refer to matter of +fact, but means that one of the possible forms of dishonesty among men +is that of employing bribery. + +§ 511. If we analyse the preceding, we find that the second step is +conversion by limitation. + + No A is B. + .'. All A is not-B (by permutation). + .'. Some not-B is A (by conversion per accidens). + +§ 512. From A again an inference can be drawn in the reverse order of +conversion per accidens followed by permutation-- + + All A is B. + .'. Some B is not not-A. + + All ingenuous persons are agreeable. + .'. Some agreeable persons are not disingenuous. + +§ 513. The intermediate link between the above two propositions is the +converse per accidens of the first--'Some B is A.' This inference, +however, coincides with that from 1 (§ 508), as the similar inference +from E (§ 510) coincides with that from 0 (§ 506). + +§ 514. All these inferences agree in the essential feature of +combining permutation with conversion, and should therefore be classed +under a common name. + +§ 515. Adopting then this slight extension of the term, we define +conversion by negation as--A form of conversion in which the converse +differs in quality from the convertend, and has the contradictory of +one of the original terms. + +§ 516. A still more complex form of immediate inference is known as + + + +_Conversion by Contraposition._ + + +This mode of inference assumes the following form-- + + All A is B. + .'. All not-B is not-A. + + All human beings are fallible. + .'. All infallible beings are not-human. + +§ 517. This will be found to resolve itself on analysis into three +steps of inference in the following order-- + + (1) Permutation. + + (2) Simple Conversion. + + (3) Permutation. + +§ 518. Let us verify this statement by performing the three steps. + + All A is B. + .'. No A is not-B (by permutation). + .'. No not-B is A (by simple conversion). + .'. All not-B is not-A (by permutation). + + All Englishmen are Aryans. + .'. No Englishmen are non-Aryans. + .'. No non-Aryans are Englishmen. + .'. All non-Aryans are non-Englishmen. + +§ 519. Conversion by contraposition may be complicated in appearance +by the occurrence of a negative term in the subject or predicate or +both, e.g. + + All not-A is B. + .'. All not-B is A. + +Again, + + All A is not-B. + .'. All B is not-A. + +Lastly, + + All not-A is not-B. + .'. All B is A. + +§ 520. The following practical rule will be found of use for the right +performing of the process-- + + Transpose the subject and predicate, and substitute for each its + contradictory term. + +§ 521. As concrete illustrations of the above forms of inference we +may take the following-- + + All the men on this board that are not white are red. + .'. All the men On this board that are not red are white. + +Again, + + All compulsory labour is inefficient. + .'. All efficient labour is free (=non-compulsory). + +Lastly, + + All inexpedient acts are unjust. + .'. All just acts are expedient. + +§ 522. Conversion by contraposition may be said to +rest on the following principle-- + + If one class be wholly contained in another, whatever is external to + the containing class is external also to the class contained. + + [Illustration] + +§ 523. The same principle may be expressed intensively as follows:-- + + If an attribute belongs to the whole of a subject, whatever fails to + exhibit that attribute does not come under the subject. + +§ 524. This statement contemplates conversion by contraposition only +in reference to the A proposition, to which the process has hitherto +been confined. Logicians seem to have overlooked the fact that +conversion by contraposition is as applicable to the O as to the A +proposition, though, when expressed in symbols, it presents a more +clumsy appearance. + + Some A is not B. + .'. Some not-B is not not-A. + + Some wholesome things are not pleasant. + .'. Some unpleasant things are not unwholesome. + +§ 525. The above admits of analysis in exactly the same way as the +same process when applied to the A proposition. + + Some A is not B. + .'. Some A is not-B (by permutation). + .'. Some not-B is A (by simple conversion). + .'. Some not-B is not not-A (by permutation). + +The result, as in the case of the A proposition, is the converse by +negation of the original proposition permuted. + +§ 526. Contraposition may also be applied to the E proposition by the +use of conversion per accidens in the place of simple conversion. But, +owing to the limitation of quantity thus effected, the result arrived +at is the same as in the case of the O proposition. Thus from 'No +wholesome things are pleasant' we could draw the same inference as +before. Here is the process in symbols, when expanded. + + No A is B. + .'. All A is not-B (by permutation). + .'. Some not-B is A (by conversion per accidens). + .'. Some not-B is not not-A (by permutation). + + +§ 527. In its unanalysed form conversion by contraposition may be +defined generally as--A form of conversion in which both subject and +predicate are replaced by their contradictories. + +§ 528. Conversion by contraposition differs in several respects from +conversion by negation. + + (1) In conversion by negation the converse differs in quality from + the convertend: whereas in conversion by contraposition the quality + of the two is the same. + + (2) In conversion by negation we employ the contradictory either of + the subject or predicate, but in conversion by contraposition we + employ the contradictory of both. + + (3) Conversion by negation involves only two steps of immediate + inference: conversion by contraposition three. + +§ 529. Conversion by contraposition cannot be applied to the ordinary +E proposition except by limitation (§ 526). + +From 'No A is B' we cannot infer 'No not-B is not-A.' For, if we +could, the contradictory of the latter, namely, 'Some not-B is not-A' +would be false. But it is manifest that this is not necessarily +false. For when one term is excluded from another, there must be +numerous individuals which fall under neither of them, unless it +should so happen that one of the terms is the direct contradictory of +the other, which is clearly not conveyed by the form of the expression +'No A is B. 'No A is not-A' stands alone among E propositions in +admitting of full conversion by contraposition, and the form of that +is the same after it as before. + +§ 530. Nor can conversion by contraposition be applied at all to I. + +[Illustration] + +From 'Some A is B' we cannot infer that 'Some not-B is not-A.' For +though the proposition holds true as a matter of fact, when A and B +are in part mutually exclusive, yet this is not conveyed by the form +of the expression. It may so happen that B is wholly contained under +A, while A itself contains everything. In this case it will be true +that 'No not-B is not-A,' which contradicts the attempted +inference. Thus from the proposition 'Some things are substances' it +cannot be inferred that 'Some not-substances are not-things,' for in +this case the contradictory is true that 'No not-substances are +not-things'; and unless an inference is valid in every case, it is not +formally valid at all. + +§ 531. It should be noticed that in the case of the [nu] proposition +immediate inferences are possible by mere contraposition without +conversion. + + All A is all B. + .'. All not-A is not-B. + +For example, if all the equilateral triangles are all the equiangular, +we know at once that all non-equilateral triangles are also +non-equiangular. + +§ 532. The principle upon which this last kind of inference rests is +that when two terms are co-extensive, whatever is excluded from the +one is excluded also from the other. + + + + +CHAPTER VII. + +_Of other Forms of Immediate Inference._ + + +§ 533. Having treated of the main forms of immediate inference, +whether simple or compound, we will now close this subject with a +brief allusion to some other forms which have been recognised by +logicians. + +§ 534. Every statement of a relation may furnish us with ail immediate +inference in which the same fact is presented from the opposite +side. Thus from 'John hit James' we infer 'James was hit by John'; +from 'Dick is the grandson of Tom' we infer 'Tom is the grandfather of +Dick'; from 'Bicester is north-east of Oxford' we infer 'Oxford is +south-west of Bicester'; from 'So and so visited the Academy the day +after he arrived in London' we infer 'So and so arrived in London the +day before he visited the Academy'; from 'A is greater than B' we +infer 'B is less than A'; and so on without limit. Such inferences as +these are material, not formal. No law can be laid down for them +except the universal postulate, that + + 'Whatever is true in one form of words is true in every other form + of words which conveys the same meaning.' + +§ 535. There is a sort of inference which goes under the title of +Immediate Inference by Added Determinants, in which from some +proposition already made another is inferred, in which the same +attribute is attached both to the subject and the predicate, e.g., + + A horse is a quadruped. + .'. A white horse is a white quadruped. + +§ 536. Such inferences are very deceptive. The attributes added must +be definite qualities, like whiteness, and must in no way involve a +comparison. From 'A horse is a quadruped' it may seem at first sight +to follow that 'A swift horse is a swift quadruped.' But we need not +go far to discover how little formal validity there is about such an +inference. From 'A horse is a quadruped' it by no means follows that +'A slow horse is a slow quadruped'; for even a slow horse is swift +compared with most quadrupeds. All that really follows here is that +'A slow horse is a quadruped which is slow for a horse.' Similarly, +from 'A Bushman is a man' it does not follow that 'A tall Bushman is a +tall man,' but only that 'A tall Bushman is a man who is tall for a +Bushman'; and so on generally. + +§ 537. Very similar to the preceding is the process known as Immediate +Inference by Complex Conception, e.g. + + A horse is a quadruped. + .'. The head of a horse is the head of a quadruped. + +§ 538. This inference, like that by added determinants, from which it +differs in name rather than in nature, may be explained on the +principle of Substitution. Starting from the identical proposition, +'The head of a quadruped is the head of a quadruped,' and being given +that 'A horse is a quadruped,' so that whatever is true of 'quadruped' +generally we know to be true of 'horse,' we are entitled to substitute +the narrower for the wider term, and in this manner we arrive at the +proposition, + + The head of a horse is the head of a quadruped. + +§ 539. Such an inference is valid enough, if the same caution be +observed as in the case of added determinants, that is, if no +difference be allowed to intervene in the relation of the fresh +conception to the generic and the specific terms. + + + + +CHAPTER VIII. + +_Of Mediate Inferences or Syllogisms._ + + +§ 540. A Mediate Inference, or Syllogism, consists of two +propositions, which are called the Premisses, and a third proposition +known as the Conclusion, which flows from the two conjointly. + +§ 541. In every syllogism two terms are compared with one another by +means of a third, which is called the Middle Term. In the premisses +each of the two terms is compared separately with the middle term; and +in the conclusion they are compared with one another. + +§ 542. Hence every syllogism consists of three terms, one of which +occurs twice in the premisses and does not appear at all in the +conclusion. This term is called the Middle Term. The predicate of the +conclusion is called the Major Term and its subject the Minor Term. + +§ 543. The major and minor terms are called the Extremes, as opposed +to the Mean or Middle Term. + +§ 544. The premiss in which the major term is compared with the middle +is called the Major Premiss. + +§ 545. The other premiss, in which the minor term is compared with the +middle, is called the Minor Premiss. + +§ 546. The order in which the premisses occur in a syllogism is +indifferent, but it is usual, for convenience, to place the major +premiss first. + +§ 547. The following will serve as a typical instance of a syllogism-- + + Middle term Major term \ + Major Premiss. All mammals are warm-blooded | Antecedent + > or + Minor term Middle term | Premisses + Minor Premiss. All whales are mammals / + + Minor term Major term \ Consequent or + .'. All whales are warm-blooded > Conclusion. + +§ 548. The reason why the names 'major, 'middle' and 'minor' terms +were originally employed is that in an affirmative syllogism such as +the above, which was regarded as the perfect type of syllogism, these +names express the relative quantity in extension of the three terms. + +[Illustration] + +§ 549. It must be noticed however that, though the middle term cannot +be of larger extent than the major nor of smaller extent than the +minor, if the latter be distributed, there is nothing to prevent all +three, or any two of them, from being coextensive. + +§ 550. Further, when the minor term is undistributed, we either have a +case of the intersection of two classes, from which it cannot be told +which of them is the larger, or the minor term is actually larger than +the middle, when it stands to it in the relation of genus to species, +as in the following syllogism-- + + All Negroes have woolly hair. + Some Africans are Negroes. + .'. Some Africans have woolly hair. + + [Illustration] + +§ 551. Hence the names are not applied with strict accuracy even in +the case of the affirmative syllogism; and when the syllogism is +negative, they are not applicable at all: since in negative +propositions we have no means of comparing the relative extension of +the terms employed. Had we said in the major premiss of our typical +syllogism, 'No mammals are cold-blooded,' and drawn the conclusion 'No +whales are cold-blooded,' we could not have compared the relative +extent of the terms 'mammal' and 'cold-blooded,' since one has been +simply excluded from the other. + + [Illustration] + +§ 552. So far we have rather described than defined the syllogism. All +the products of thought, it will be remembered, are the results of +comparison. The syllogism, which is one of them, may be so regarded in +two ways-- + + (1) As the comparison of two propositions by means of a third. + + (2) As the comparison of two terms by means of a third or middle + term. + +§ 553. The two propositions which are compared with one another are +the major premiss and the conclusion, which are brought into +connection by means of the minor premiss. Thus in the syllogism above +given we compare the conclusion 'All whales are warm-blooded' with the +major premiss 'All mammals are warm-blooded,' and find that the former +is contained under the latter, as soon as we become acquainted with +the intermediate proposition 'All whales are mammals.' + +§ 554. The two terms which are compared with one another are of course +the major and minor. + +§ 555. The syllogism is merely a form into which our deductive +inferences may be thrown for the sake of exhibiting their +conclusiveness. It is not the form which they naturally assume in +speech or writing. Practically the conclusion is generally stated +first and the premisses introduced by some causative particle as +'because,' 'since,' 'for,' &c. We start with our conclusion, and then +give the reason for it by supplying the premisses. + +§ 556. The conclusion, as thus stated first, was called by logicians +the Problema or Quaestio, being regarded as a problem or question, to +which a solution or answer was to be found by supplying the premisses. + +§ 557. In common discourse and writing the syllogism is usually stated +defectively, one of the premisses or, in some cases, the conclusion +itself being omitted. Thus instead of arguing at full length + + All men are fallible, + The Pope is a man, + .'. The Pope is fallible, + +we content ourselves with saying 'The Pope is fallible, for he is a +man,' or 'The Pope is fallible, because all men are so'; or perhaps we +should merely say 'All men are fallible, and the Pope is a man,' +leaving it to the sagacity of our hearers to supply the desired +conclusion. A syllogism, as thus elliptically stated, is commonly, +though incorrectly, called an Enthymeme. When the major premiss is +omitted, it is called an Enthymeme of the First Order; when the minor +is omitted, an Enthymeme of the Second Order; and when the conclusion +is omitted an Enthymeme of the Third Order. + + + + +CHAPTER IX. + +_Of Mood and Figure._ + + +§ 558. Syllogisms may differ in two ways-- + + (1) in Mood; + + (2) in Figure. + +§ 559. Mood depends upon the kind of propositions employed. Thus a +syllogism consisting of three universal affirmatives, AAA, would be +said to differ in mood from one consisting of such propositions as EIO +or any other combination that might be made. The syllogism previously +given to prove the fallibility of the Pope belongs to the mood +AAA. Had we drawn only a particular conclusion, 'Some Popes are +fallible,' it would have fallen into the mood AAI. + +§ 560. Figure depends upon the arrangement of the terms in the +propositions. Thus a difference of figure is internal to a difference +of mood, that is to say, the same mood can be in any figure. + +§ 561. We will now show how many possible varieties there are of mood +and figure, irrespective of their logical validity. + +§ 562. And first as to mood. + +Since every syllogism consists of three propositions, and each of +these propositions may be either A, E, I, or O, it is clear that there +will be as many possible moods as there can be combinations of four +things, taken three together, with no restrictions as to +repetition. It will be seen that there are just sixty-four of such +combinations. For A may be followed either by itself or by E, I, or +O. Let us suppose it to be followed by itself. Then this pair of +premisses, AA, may have for its conclusion either A, E, I, or O, thus +giving four combinations which commence with AA. In like manner there +will be four commencing with AE, four with AI, and four with AO, +giving a total of sixteen combinations which commence with +A. Similarly there will be sixteen commencing with E, sixteen with I, +sixteen with O--in all sixty-four. It is very few, however, of these +possible combinations that will be found legitimate, when tested by +the rules of syllogism. + +§ 563. Next as to figure. + +There are four possible varieties of figure in a syllogism, as may be +seen by considering the positions that can be occupied by the middle +term in the premisses. For as there are only two terms in each +premiss, the position occupied by the middle term necessarily +determines that of the others. It is clear that the middle term must +either occupy the same position in both premisses or not, that is, it +must either be subject in both or predicate in both, or else subject +in one and predicate in the other. Now, if we are not acquainted with +the conclusion of our syllogism, we do not know which is the major and +which the minor term, and have therefore no means of distinguishing +between one premiss and another; consequently we must Stop here, and +say that there are only three different arrangements possible. But, if +the Conclusion also be assumed as known, then we are able to +distinguish one premiss as the major and the other as the minor; and +so we can go further, and lay down that, if the middle term does not +hold the same position in both premisses, it must either be subject in +the major and predicate in the minor, or else predicate in the major +and subject in the minor. + +§ 564. Hence there result + + _The Four Figures._ + +When the middle term is subject in the major and predicate in the +minor, we are said to have the First Figure. + +When the middle term is predicate in both premisses, we are said to +have the Second Figure. + +When the middle term is subject in both premisses, we are said to have +the Third Figure. + +When the middle term is predicate in the major premiss and subject in +the minor, we are said to have the Fourth Figure. + +§ 565. Let A be the major term; B the middle. C the minor. + + Figure I. Figure II. Figure III. Figure IV. + B--A A--B B--A A--B + C--B C--B B--C B--C + C--A C--A C--A C--A + +All these figures are legitimate, though the fourth is comparatively +valueless. + +§ 566. It will be well to explain by an instance the meaning of the +assertion previously made, that a difference of figure is internal to +a difference of mood. We will take the mood EIO, and by varying the +position of the terms, construct a syllogism in it in each of the four +figures. + + I. + E No wicked man is happy. + I Some prosperous men are wicked. + O .'. Some prosperous men are not happy. + + II. + E No happy man is wicked. + I Some prosperous men are wicked. + O .'. Some prosperous men are not happy. + + III. + E No wicked man is happy. + I Some wicked men are prosperous. + O .'. Some prosperous men are not happy. + + IV. + E No happy man is wicked. + I Some wicked men are prosperous. + O .'. Some prosperous men are not happy. + +§ 567. In the mood we have selected, owing to the peculiar nature of +the premisses, both of which admit of simple conversion, it happens +that the resulting syllogisms are all valid. But in the great majority +of moods no syllogism would be valid at all, and in many moods a +syllogism would be valid in one figure and invalid in another. As yet +however we are only concerned with the conceivable combinations, apart +from the question of their legitimacy. + +§ 568. Now since there are four different figures and sixty-four +different moods, we obtain in all 256 possible ways of arranging three +terms in three propositions, that is, 256 possible forms of syllogism. + + + + +CHAPTER X. + +_Of the Canon of Reasoning._ + + +& 569. The first figure was regarded by logicians as the only perfect +type of syllogism, because the validity of moods in this figure may be +tested directly by their complying, or failing to comply, with a +certain axiom, the truth of which is self-evident. This axiom is known +as the Dictum de Omni et Nullo. It may be expressed as follows-- + + Whatever may be affirmed or denied of a whole class may be affirmed + or denied of everything contained in that class. + +§ 570. This mode of stating the axiom contemplates predication as +being made in extension, whereas it is more naturally to be regarded +as being made in intension. + +§ 571. The same principle may be expressed intensively as follows-- + + Whatever has certain attributes has also the attributes which + invariably accompany them .[Footnote: Nota notae est nota rei + ipsius. 'Whatever has any mark has that which it is a mark of.' + Mill, vol. i, p. 201,] + +§ 572. By Aristotle himself the principle was expressed in a neutral +form thus-- + + 'Whatever is stated of the predicate will be stated also of the + subject [Footnote: [Greek: osa katŕ toű kategorouménou légetai pánta kaě + katŕ toű hypokeiménou rhaetésetai]. Cat. 3, § I].' + +This way of putting it, however, is too loose. + +§ 573. The principle precisely stated is as follows-- + + Whatever may be affirmed or denied universally of the predicate of + an affirmative proposition, may be affirmed or denied also of the + subject. + +§ 574. Thus, given an affirmative proposition 'Whales are mammals,' if +we can affirm anything universally of the predicate 'mammals,' as, for +instance, that 'All mammals are warm-blooded,' we shall be able to +affirm the same of the subject 'whales'; and, if we can deny anything +universally of the predicate, as that 'No mammals are oviparous,' we +shall be able to deny the same of the subject. + +§ 575. In whatever way the supposed canon of reasoning may be stated, +it has the defect of applying only to a single figure, namely, the +first. The characteristic of the reasoning in that figure is that some +general rule is maintained to hold good in a particular case. The +major premiss lays down some general principle, whether affirmative or +negative; the minor premiss asserts that a particular case falls under +this principle; and the conclusion applies the general principle to +the particular case. But though all syllogistic reasoning may be +tortured into conformity with this type, some of it finds expression +more naturally in other ways. + +§ 576. Modern logicians therefore prefer to abandon the Dictum de Omni +et Nullo in any shape, and to substitute for it the following three +axioms, which apply to all figures alike. + + + +_Three Axioms of Mediale Inference._ + + + (1) If two terms agree with the same third term, they agree with one + another. + + (2) If one term agrees, and another disagrees, with the same third + term, they disagree with one another. + + (3) If two terms disagree with the same third term, they may or may + not agree with one another. + +§ 577. The first of these axioms is the principle of all affirmative, +the second of all negative, syllogisms; the third points out the +conditions under which no conclusion can be drawn. If there is any +agreement at all between the two terms and the third, as in the cases +contemplated in the first and second axioms, then we have a conclusion +of some kind: if it is otherwise, we have none. + +§ 578. It must be understood with regard to these axioms that, when we +speak of terms agreeing or disagreeing with the same third term, we +mean that they agree or disagree with the same part of it. + +§ 579. Hence in applying these axioms it is necessary to bear in mind +the rules for the distinction of terms. Thus from + + All B is A, + No C is B, + +the only inference which can be drawn is that Some A is not C (which +alters the figure from the first to the fourth). For it was only part +of A which was known to agree with B. On the theory of the quantified +predicate we could draw the inference No C is some A. + +§ 580. It is of course possible for terms to agree with different +parts of the same third term, and yet to have no connection with one +another. Thus + + All birds fly. + All bats fly. + +But we do not infer therefrom that bats are birds or vice versâ. + +§ 581. On the other hand, had we said,-- + + All birds lay eggs, + No bats lay eggs, + +we might confidently have drawn the conclusion + + No bats are birds + +For the term 'bats,' being excluded from the whole of the term 'lay +eggs,' is thereby necessarily excluded from that part of it which +coincides with 'birds.' + +[Illustration] + + + + +CHAPTER XI. + +_Of the Generad Rules of Syllogism._ + + +§ 582. We now proceed to lay down certain general rules to which all +valid syllogisms must conform. These are divided into primary and +derivative. + +I. _Primary_. + + (1) A syllogism must consist of three propositions only. + + (2) A syllogism must consist of three terms only. + + (3) The middle term must be distributed at least once in the + premisses. + + (4) No term must be distributed in the conclusion which was not + distributed in the premisses. + + (5) Two negative premisses prove nothing. + + (6) If one premiss be negative, the conclusion must be negative. + + (7) If the conclusion be negative, one of the premisses must be + negative: but if the conclusion be affirmative, both premisses must + be affirmative. + +II. _Derivative_. + + (8) Two particular premisses prove nothing. + + (9) If one premiss be particular, the conclusion must be particular. + +§ 583. The first two of these rules are involved in the definition of +the syllogism with which we started. We said it might be regarded +either as the comparison of two propositions by means of a third or as +the comparison of two terms by means of a third. To violate either of +these rules therefore would be inconsistent with the fundamental +conception of the syllogism. The first of our two definitions indeed +(§ 552) applies directly only to the syllogisms in the first figure; +but since all syllogisms may be expressed, as we shall presently see, +in the first figure, it applies indirectly to all. When any process +of mediate inference appears to have more than two premisses, it will +always be found that there is more than one syllogism. If there are +less than three propositions, as in the fallacy of 'begging the +question,' in which the conclusion simply reiterates one of the +premisses, there is no syllogism at all. + +With regard to the second rule, it is plain that any attempted +syllogism which has more than three terms cannot conform to the +conditions of any of the axioms of mediate inference. + +§ 584. The next two rules guard against the two fallacies which are +fatal to most syllogisms whose constitution is unsound. + +§ 585. The violation of Rule 3 is known as the Fallacy of +Undistributed Middle. The reason for this rule is not far to seek. +For if the middle term is not used in either premiss in its whole +extent, we may be referring to one part of it in one premiss and to +quite another part of it in another, so that there will be really no +middle term at all. From such premisses as these-- + + All pigs are omnivorous, + All men are omnivorous, + +it is plain that nothing follows. Or again, take these premisses-- + + Some men are fallible, + All Popes are men. + +Here it is possible that 'All Popes' may agree with precisely that +part of the term 'man,' of which it is not known whether it agrees +with 'fallible' or not. + +§ 586. The violation of Rule 4 is known as the Fallacy of Illicit +Process. If the major term is distributed in the conclusion, not +having been distributed in the premiss, we have what is called Illicit +Process of the Major; if the same is the case with the minor term, we +have Illicit Process of the Minor. + +§ 587. The reason for this rule is that if a term be used in its whole +extent in the conclusion, which was not so used in the premiss in +which it occurred, we would be arguing from the part to the whole. It +is the same sort of fallacy which we found to underlie the simple +conversion of an A proposition. + +§ 588. Take for instance the following-- + + All learned men go mad. + John is not a learned man. + .'. John will not go mad. + +In the conclusion 'John' is excluded from the whole class of persons +who go mad, whereas in the premisses, granting that all learned men go +mad, it has not been said that they are all the men who do so. We have +here an illicit process of the major term. + +§ 589. Or again take the following-- + + All Radicals are covetous. + All Radicals are poor. + .'. All poor men are covetous. + +The conclusion here is certainly not warranted by our premisses. For +in them we spoke only of some poor men, since the predicate of an +affirmative proposition is undistributed. + +§ 590. Rule 5 is simply another way of stating the third axiom of +mediate inference. To know that two terms disagree with the same third +term gives us no ground for any inference as to whether they agree or +disagree with one another, e.g. + + Ruminants are not oviparous. + Sheep are not oviparous. + +For ought that can be inferred from the premisses, sheep may or may +not be ruminants. + +§ 591. This rule may sometimes be violated in appearance, though not +in reality. For instance, the following is perfectly legitimate +reasoning. + + No remedy for corruption is effectual that does not render it + useless. + Nothing but the ballot renders corruption useless. + .'. Nothing but the ballot is an effectual remedy for corruption. + +But on looking into this we find that there are four terms-- + + No not-A is B. + No not-C is A. + .'. No not-C is B. + +The violation of Rule 5 is here rendered possible by the additional +violation of Rule 2. In order to have the middle term the same in both +premisses we are obliged to make the minor affirmative, thus + + No not-A is B. + All not-C is not-A. + .'. No not-C is B. + + No remedy that fails to render corruption useless is effectual. + All but the ballot fails to render corruption useless. + .'. Nothing but the ballot is effectual. + +§ 592. Rule 6 declares that, if one premiss be negative, the +conclusion must be negative. Now in compliance with Rule 5, if one +premiss be negative, the other must be affirmative. We have therefore +the case contemplated in the second axiom, namely, of one term +agreeing and the other disagreeing with the same third term; and we +know that this can only give ground for a judgement of disagreement +between the two terms themselves--in other words, to a negative +conclusion. + +§ 593. Rule 7 declares that, if the conclusion be negative, one of the +premisses must be negative; but, if the conclusion be affirmative, +both premisses must be affirmative. It is plain from the axioms that a +judgement of disagreement can only be elicited from a judgement of +agreement combined with a judgement of disagreement, and that a +judgement of agreement can result only from two prior judgements of +agreement. + +§ 594. The seven rules already treated of are evident by their own +light, being of the nature of definitions and axioms: but the two +remaining rules, which deal with particular premisses, admit of being +proved from their predecessors. + +§ 595. Proof of Rule 8.--_That two particular premisses prove +nothing_. + +We know by Rule 5 that both premisses cannot be negative. Hence they +must be either both affirmative, II, or one affirmative and one +negative, IO or OI. + +Now II premisses do not distribute any term at all, and therefore the +middle term cannot be distributed, which would violate Rule 3. + +Again in IO or OI premisses there is only one term distributed, +namely, the predicate of the O proposition. But Rule 3 requires that +this one term should be the middle term. Therefore the major term must +be undistributed in the major premiss. But since one of the premisses +is negative, the conclusion must be negative, by Rule 6. And every +negative proposition distributes its predicate. Therefore the major +term must be distributed where it occurs as predicate of the +conclusion. But it was not distributed in the major premiss. Therefore +in drawing any conclusion we violate Rule 4 by an illicit process of +the major term. + +§ 596. Proof of Rule 9.--_That_, _if_ one _premiss be +particular_, _the conclusion must be particular_. + +Two negative premisses being excluded by Rule 5, and two particular by +Rule 8, the only pairs of premisses we can have are-- + + AI, AO, EI. + +Of course the particular premiss may precede the universal, but the +order of the premisses will not affect the reasoning. + +AI premisses between them distribute one term only. This must be the +middle term by Rule 3. Therefore the conclusion must be particular, as +its subject cannot be distributed, + +AO and EI premisses each distribute two terms, one of which must be +the middle term by Rule 3: so that there is only one term left which +may be distributed in the conclusion. But the conclusion must be +negative by Rule 4. Therefore its predicate must be distributed. +Hence its subject cannot be so. Therefore the conclusion must be +particular. + +§ 597. Rules 6 and 9 are often lumped together in a single +expression--'The conclusion must follow the weaker part,' negative +being considered weaker than affirmative, and particular than +universal. + +§ 598. The most important rules of syllogism are summed up in the +following mnemonic lines, which appear to have been perfected, though +not invented, by a medićval logician known as Petrus Hispanus, who was +afterwards raised to the Papal Chair under the title of Pope John XXI, +and who died in 1277-- + + Distribuas medium, nec quartus terminus adsit; + Utraque nec praemissa negans, nec particularis; + Sectetur partem conclusio deteriorem, + Et non distribuat, nisi cum praemissa, negetve. + + + + +CHAPTER XII. + +_Of the Determination of the Legitimate Moods of Syllogism._ + + +§ 599. It will be remembered that there were found to be 64 possible +moods, each of which might occur in any of the four figures, giving us +altogether 256 possible varieties of syllogism. The task now before us +is to determine how many of these combinations of mood and figure are +legitimate. + +§ 600. By the application of the preceding rules we are enabled to +reduce the 64 possible moods to 11 valid ones. This may be done by a +longer or a shorter method. The longer method, which is perhaps easier +of comprehension, is to write down the 64 possible moods, and then +strike out such as violate any of the rules of syllogism. + + AAA -AEA- -AIA- -AOA- + -AAE- AEE -AIE- -AOE- + AAI -AEI- AII -AOI- + -AAO- AEO -AIO- AOO + + -EAA- -EEA- -EIA- -EOA- + EAE -EEE- -EIE- -EOE- + -EAI- -EEI- -EII- -EOI- + EAO -EEO- EIO -EOO- + +[Illustration] + +§ 601. The batches which are crossed are those in which the premisses +can yield no conclusion at all, owing to their violating Rule 6 or 9; +in the rest the premises are legitimate, but a wrong conclusion is +drawn from each of them as are translineated. + +§ 602. IEO stands alone, as violating Rule 4. This may require a +little explanation. + +Since the conclusion is negative, the major term, which is its +predicate, must be distributed. But the major premiss, being 1, does +not distribute either subject or predicate. Hence IEO must always +involve an illicit process of the major. + +§ 603. The II moods which have been left valid, after being tested by +the syllogistic rules, are as follows-- + + AAA. AAI. AEE. AEO. AII. AOO. + EAE. EAO. EIO. + IAI. + OAO. + +§ 604. We will now arrive at the same result by a shorter and more +scientific method. This method consists in first determining what +pairs of premisses are valid in accordance with Rules 6 and g, and +then examining what conclusions may be legitimately inferred from them +in accordance with the other rules of syllogism. + +§ 605. The major premiss may be either A, E, I or O. If it is A, the +minor also may be either A, E, I or O. If it is E, the minor can only +be A or I. If it is I, the minor can only be A or E. If it is O, the +minor can only be A. Hence there result 9 valid pairs of premisses. + + AA. AE. AI. AO. + EA. EI. + IA. IE. + OA. + +Three of these pairs, namely AA, AE, EA, yield two conclusions apiece, +one universal and one particular, which do not violate any of the +rules of syllogism; one of them, IE, yields no conclusion at all; the +remaining five have their conclusion limited to a single proposition, +on the principle that the conclusion must follow the weaker part. +Hence we arrive at the same result as before, of II legitimate moods-- + + AAA. AAI. AEE. AEO. EAE. EAO. + AII. AOO. EIO. IAI. OAO. + + + + + +CHAPTER XIII. + +_Of the Special Rules of the Four Figures_. + + +§ 606. Our next task must be to determine how far the 11 moods which +we arrived at in the last chapter are valid in the four figures. But +before this can be done, we must lay down the + + + +_Special Rules of the Four Figures_. + + + FIGURE 1. + + Rule 1, The minor premiss must be affirmative. + + Rule 2. The major premiss must be universal. + + + FIGURE II. + + Rule 1. One or other premiss must be negative. + + Rule 2. The conclusion must be negative. + + Rule 3. The major premiss must be universal. + + + FIGURE III. + + Rule 1. The minor premiss must be affirmative. + + Rule 2. The conclusion must be particular. + + + FIGURE IV. + + Rule 1. When the major premiss is affirmative, the minor must be + universal. + + Rule 2. When the minor premiss is particular, the major must be + negative. + + Rule 3, When the minor premiss is affirmative, the conclusion must + be particular. + + Rule 4. When the conclusion is negative, the major premiss must be + universal. + + Rule 5. The conclusion cannot be a universal affirmative. + + Rule 6. Neither of the premisses can be a particular negative. + + +§ 607. The special rules of the first figure are merely a reassertion +in another form of the Dictum de Omni et Nullo. For if the major +premiss were particular, we should not have anything affirmed or +denied of a whole class; and if the minor premiss were negative, we +should not have anything declared to be contained in that class. +Nevertheless these rules, like the rest, admit of being proved from +the position of the terms in the figure, combined with the rules for +the distribution of terms (§ 293). + + + +_Proof of the Special Rules of the Four Figures._ + + +FIGURE 1. + +§ 608. Proof of Rule 1.--_The minor premiss must be affirmative_. + + B--A + C--B + C--A + +If possible, let the minor premiss be negative. Then the major must be +affirmative (by Rule 5), [Footnote: This refers to the General Rules +of Syllogism.] and the conclusion must be negative (by Rule 6). But +the major being affirmative, its predicate is undistributed; and the +conclusion being negative, its predicate is distributed. Now the major +term is in this figure predicate both in the major premiss and in the +conclusion. Hence there results illicit process of the major +term. Therefore the minor premiss must be affirmative. + +§ 609. Proof of Rule 2.--_The major premiss must be universal._ + +Since the minor premiss is affirmative, the middle term, which is its +predicate, is undistributed there. Therefore it must be distributed in +the major premiss, where it is subject. Therefore the major premiss +must be universal. + + +FIGURE II. + +§ 610. Proof of Rule 1,--_One or other premiss must be negative_. + + A--B + C--B + C--A + +The middle term being predicate in both premisses, one or other must +be negative; else there would be undistributed middle. + +§ 611. Proof of Rule 2.--_The conclusion must be negative._ + +Since one of the premisses is negative, it follows that the conclusion +also must be so (by Rule 6). + +§ 612. Proof of Rule 3.--_The major premiss must be universal._ + +The conclusion being negative, the major term will there be +distributed. But the major term is subject in the major +premiss. Therefore the major premiss must be universal (by Rule 4). + + +FIGURE III. + +§ 613. Proof of Rule 1.--_The minor premiss must be affirmative._ + + B--A + B--C + C--A + +The proof of this rule is the same as in the first figure, the two +figures being alike so far as the major term is concerned. + +§ 614. Proof of Rule 2.--_The conclusion must be particular_. + +The minor premiss being affirmative, the minor term, which is its +predicate, will be undistributed there. Hence it must be undistributed +in the conclusion (by Rule 4). Therefore the conclusion must be +particular. + + +FIGURE IV. + +§ 615. Proof of Rule I.--_When the major premiss is affirmative, +the minor must be universal_. + +If the minor were particular, there would be undistributed +middle. [Footnote: Shorter proofs are employed in this figure, as the +student is by this time familiar with the method of procedure.] + +§ 616. Proof of Rule 2.--_When the minor premiss is particular, the +major must be negative._ + + A--B + B--C + C--A + +This rule is the converse of the preceding, and depends upon the same +principle. + +§ 617. Proof of Rule 3.--_When the minor premiss is affirmative, the +conclusion must be particular._ + +If the conclusion were universal, there would be illicit process of +the minor. + +§ 618. Proof of Rule 4.--_When the conclusion is negative, the major +premiss must_ be universal. + +If the major premiss were particular, there would be illicit process +of the major. + +§ 619. Proof of Rule 5.--_The conclusion CANNOT be A UNIVERSAL +affirmative_. + +The conclusion being affirmative, the premisses must be so too (by +Rule 7). Therefore the minor term is undistributed in the minor +premiss, where it is predicate. Hence it cannot be distributed in the +conclusion (by Rule 4). Therefore the affirmative conclusion must be +particular. + +§ 620. Proof of Rule 6.--_Neither of the premisses can lie a, +PARTICULAR NEGATIVE_. + +If the major premiss were a particular negative, the conclusion would +be negative. Therefore the major term would be distributed in the +conclusion. But the major premiss being particular, the major term +could not be distributed there. Therefore we should have an illicit +process of the major term. + +If the minor premiss were a particular negative, then, since the major +must be affirmative (by Rule 5), we should have undistributed middle. + + + + +CHAPTER XIV + +_Of the Determination of the Moods that are valid in the Four +Figures._ + + +§ 621. By applying the special rules just given we shall be able to +determine how many of the eleven legitimate moods are valid in the +four figures. + +$622. These eleven legitimate moods were found to be + + AAA. AAI. AEE. AEO. AII. AOO. EAE. + EAO. EIO. IAI. OAO. + + +FIGURE 1. + +§ 623. The rule that the major premiss must be universal excludes the +last two moods, IAI, OAO. The rule that the minor premiss must be +affirmative excludes three more, namely, AEE, AEO, AOO. + +Thus we are left with six moods which are valid in the first figure, +namely, + + AAA. EAE. AII. EIO. AAI. EAO. + + +FIGURE II. + +§ 624. The rule that one premiss must be negative excludes four moods, +namely, AAA, AAI, AII, IAI. The rule that the major must be universal +excludes OAO. Thus we are left with six moods which are valid in the +second figure, namely, + + EAE. AEE. EIO. AOO. EAO. AEO. + + +FIGURE III. + +§ 625. The rule that the conclusion must be particular confines us to +eight moods, two of which, namely AEE and AOO, are excluded by the +rule that the minor premiss must be affirmative. + +Thus we are left with six moods which are valid in the third figure, +namely, + + AAI. IAI. AII. EAO. OAO. EIO. + + +FIGURE IV. + +§ 626. The first of the eleven moods, AAA, is excluded by the rule +that the conclusion cannot be a universal affirmative. + +Two more moods, namely AOO and OAO, are excluded by the rule that +neither of the premisses can be a particular negative. + +AII violates the rule that when the major premiss is affirmative, the +minor must be universal. + +EAE violates the rule that, when the minor premiss is affirmative, the +conclusion must be particular. Thus we are left with six moods which +are valid in the fourth figure, namely, + + AAI. AEE. IAI. EAO. EIO. AEO. + +§ 627. Thus the 256 possible forms of syllogism have been reduced to +two dozen legitimate combinations of mood and figure, six moods being +valid in each of the four figures. + + FIGURE I. AAA. EAE. AII. EIO. (AAI. EAO.) + + FIGURE II. EAE. AEE. EIO. AGO. (EAO. AEO.) + + FIGURE III. AAI. IAI. AII. EAO. OAO. EIO. + + FIGURE IV. AAI. AEE. IAI. EAO. EIO. (AEO.) + +§ 628. The five moods enclosed in brackets, though valid, are +useless. For the conclusion drawn is less than is warranted by the +premisses. These are called Subaltern Moods, because their conclusions +might be inferred by subalternation from the universal conclusions +which can justly be drawn from the same premisses. Thus AAI is +subaltern to AAA, EAO to EAE, and so on with the rest. + +§ 629. The remaining 19 combinations of mood and figure, which are +loosely called 'moods,' though in strictness they should be called +'figured moods,' are generally spoken of under the names supplied by +the following mnemonics-- + + Barbara, Celarent, Darii, Ferioque prioris; + Cesare, Camestres, Festino, Baroko secundć; + Tertia Darapti, Disamis, Datisi, Felapton, + Bokardo, Ferison habet; Quarta insuper addit + Bramantip, Camenes, Dimaris, Fesapo, Fresison: + Quinque Subalterni, totidem Generalibus orti, + Nomen habent nullum, nee, si bene colligis, usum. + +§ 630. The vowels in these lines indicate the letters of the mood. All +the special rules of the four figures can be gathered from an +inspection of them. The following points should be specially noted. + +The first figure proves any kind of conclusion, and is the only one +which can prove A. + +The second figure proves only negatives. + +The third figure proves only particulars. + +The fourth figure proves any conclusion except A. + +§ 631. The first figure is called the Perfect, and the rest the +Imperfect figures. The claim of the first to be regarded as the +perfect figure may be rested on these grounds-- + + 1. It alone conforms directly to the Dictum de Omni et Nullo. + + 2. It suffices to prove every kind of conclusion, and is the only + figure in which a universal affirmative proposition can be + established. + + 3. It is only in a mood of this figure that the major, middle and + minor terms are to be found standing in their relative order of + extension. + +§ 632. The reason why a universal affirmative, which is of course +infinitely the most important form of proposition, can only be proved +in the first figure may be seen as follows. + +_Proof that A can only be established in figure I._ + +An A conclusion necessitates both premisses being A propositions (by +Rule 7). But the minor term is distributed in the conclusion, as being +the subject of an A proposition, and must therefore be distributed in +the minor premiss, in order to which it must be the subject. Therefore +the middle term must be the predicate and is consequently +undistributed. In order therefore that the middle term may be +distributed, it must be subject in the major premiss, since that also +is an A proposition. But when the middle term is subject in the major +and predicate in the minor premiss, we have what is called the first +figure. + + + + +CHAPTER XV. + +_Of the Special Canons of the Four Figures._ + + +§ 633. So far we have given only a negative test of legitimacy, having +shown what moods are not invalidated by running counter to any of the +special rules of the four figures. We will now lay down special canons +for the four figures, conformity to which will serve as a positive +test of the validity of a given mood in a given figure. The special +canon of the first figure--will of course be practically equivalent to +the Dictum de Omni et Nullo. All of them will be expressed in terms of +extension, for the sake of perspicuity. + + + +_Special Canons of the Four Figures._ + + +FIGURE 1. + +§ 634. CANON. If one term wholly includes or excludes another, which +wholly or partly includes a third, the first term wholly or partly +includes or excludes the third. + +Here four cases arise-- + + [Illustration] + + (1) Total inclusion (Barbara). + + All B is A. + All C is B. + .'. All C is A. + + + [Illustration] + + (2) Partial inclusion (Darii). + + All B is A. + Some C is B. + .'. Some C is A. + + + [Illustration] + + (3) Total exclusion (Celarent). + + No B is A. + All C is B. + .'. No C is A. + + + [Illustration] + + (4) Partial exclusion (Ferio). + + No B is A. + Some C is B. + .'. Some C is not A. + + FIGURE II. + +§ 635. CANON. If one term is excluded from another, which wholly or +partly includes a third, or is included in another from which a third +is wholly or partly excluded, the first is excluded from the whole or +part of the third. + +Here we have four cases, all of exclusion-- + + (1) Total exclusion on the ground of inclusion in an excluded term + (Cesare). + + [Illustration] + + No A is B. + All C is B. + .'. No C is A. + + (2) Partial exclusion on the ground of a similar partial inclusion + (Festino). + + [Illustration] + + No A is B. + Some C is B. + .'. Some C is not A. + + (3) Total exclusion on the ground of exclusion from an including + term (Camestres). + + [Illustration] + + All A is B. + No C is B. + .'. No C is A. + + (4) Partial exclusion on the ground of a similar partial exclusion + (Baroko). + + [Illustration] + + All A is B. + Some C is not B. + .'. Some C is not A. + + FIGURE III. + +§ 636. CANON. If two terms include another term in common, or if the +first includes the whole and the second a part of the same term, or +vice versâ, the first of these two terms partly includes the second; +and if the first is excluded from the whole of a term which is wholly +or in part included in the second, or is excluded from part of a term +which is wholly included in the second, the first is excluded from +part of the second. + +Here it is evident from the statement that six cases arise-- + + (1) Total inclusion of the same term in two others + (Darapti). + + [Illustration] + + All B is A. + All B is C. + .'. some C is A. + + (2) Total inclusion in the first and partial inclusion + in the second (Datisi). + + [Illustration] + + All B is A. + Some B is C. + .'. some C is A. + + (3) Partial inclusion in the first and total inclusion in + the second (Disamis). + + [Illustration] + + Some B is A. + All B is C. + .'. some C is A. + + (4) Total exclusion of the first from a term which is + wholly included in the second (Felapton). + + [Illustration] + + No B is A. + All B is C. + .'. some C is not A. + + (5) Total exclusion of the first from a term which is + partly included in the second (Ferison). + + [Illustration] + + No B is A. + Some B is C. + .'. some C is not A. + + (6) Exclusion of the first from part of a term which + is wholly included in the second (Bokardo). + + [Illustration] + + Some B is not A. + All B is C. + .'. Some C is not A. + + FIGURE IV. + +§ 637. CANON. If one term is wholly or partly included in another +which is wholly included in or excluded from a third, the third term +wholly or partly includes the first, or, in the case of total +inclusion, is wholly excluded from it; and if a term is excluded from +another which is wholly or partly included in a third, the third is +partly excluded from the first. + +Here we have five cases-- + + (1) Of the inclusion of a whole term (Bramsntip). + + [Illustration] + All A is B. + All B is C. + .'. Some C is (all) A. + + (2) Of the inclusion of part of a term (DIMARIS). + + [Illustration] + + Some A is B. + All B is C. + .'. Some C is (some) A, + + (3) Of the exclusion of a whole term (Camenes). + + [Illustration] + + All A is B. + No B is C. + .'. No C is A. + + (4) Partial exclusion on the ground of including + the whole of an excluded term (Fesapo). + + [Illustration] + + No A is B. + All B is C. + .'. Some C is not A. + + (5) Partial exclusion on the ground of including + part of an excluded term (Fresison). + + [Illustration] + + No A is B. + Some B is C. + .'. Some C is not A. + +§ 638. It is evident from the diagrams that in the subaltern moods the +conclusion is not drawn directly from the premisses, but is an +immediate inference from the natural conclusion. Take for instance AAI +in the first figure. The natural conclusion from these premisses is +that the minor term C is wholly contained in the major term A. But +instead of drawing this conclusion we go on to infer that something +which is contained in C, namely some C, is contained in A. + + [Illustration] + + All B is A. + All C is B. + .'. all C is A. + .'. some C is A. + +Similarly in EAO in figure 1, instead of arguing that the whole of C +is excluded from A, we draw a conclusion which really involves a +further inference, namely that part of C is excluded from A. + + [Illustration] + + No B is A. + All C is B. + .'. no C is A. + .'. some C is not A. + +§ 639. The reason why the canons have been expressed in so cumbrous a +form is to render the validity of all the moods in each figure at once +apparent from the statement. For purposes of general convenience they +admit of a much more compendious mode of expression. + +§ 640. The canon of the first figure is known as the Dictum de Omni et +Nullo-- + + What is true (distributively) of a whole term is true of all that it + includes. + +§ 641. The canon of the second figure is known as the Dictum de +Diverse-- + + If one term is contained in, and another excluded from a third term, + they are mutually excluded. + +§ 642. The canon of the third figure is known as the Dictum de Exemplo +et de Excepto-- + + Two terms which contain a common part partly agree, or, if one + contains a part which the other does not, they partly differ. + +§ 643. The canon of the fourth figure has had no name assigned to it, +and does not seem to admit of any simple expression. Another mode of +formulating it is as follows:-- + + Whatever is affirmed of a whole term may have partially affirmed of + it whatever is included in that term (Bramantip, Dimaris), and + partially denied of it whatever is excluded (Fesapo); whatever is + affirmed of part of a term may have partially denied of it whatever + is wholly excluded from that term (Fresison); and whatever is denied + of a whole term may have wholly denied of it whatever is wholly + included in that term (Camenes). + +§ 644. From the point of view of intension the canons of the first +three figures may be expressed as follows. + +§ 645. Canon of the first figure. Dictum de Omni et Nullo-- + + An attribute of an attribute of anything is an attribute of the + thing itself. + +§ 646. Canon of the second figure. Dictum de Diverso-- + + If a subject has an attribute which a class has not, or vice versa, + the subject does not belong to the class. + +§ 647. Canon of the third figure. + + 1. Dictum de Exemplo-- + + If a certain attribute can be affirmed of any portion of the + members of a class, it is not incompatible with the distinctive + attributes of that class. + + 2. Dictum de Excepto-- + + If a certain attribute can be denied of any portion of the members + of a class, it is not inseparable from the distinctive attributes + of that class. + + + + +CHAPTER XVI. + +_Of the Special Uses of the Four Figures._ + + +§ 648. The first figure is useful for proving the properties of a +thing. + +§ 649. The second figure is useful for proving distinctions between +things. + +§ 650. The third figure is useful for proving instances or exceptions. + +§ 651. The fourth figure is useful for proving the species of a genus. + +FIGURE 1. + +§ 652. + + B is or is not A. + C is B. + .'. C is or is not A. + +We prove that C has or has not the property A by predicating of it B, +which we know to possess or not to possess that property. + + Luminous objects are material. + Comets are luminous. + .'. Comets are material. + + No moths are butterflies. + The Death's head is a moth. + .'. The Death's head is not a butterfly. + +FIGURE II. + +§ 653. + + A is B. A is not B. + C is not B. C is B. + .'. C is not A. .'. C is not A. + +We establish the distinction between C and A by showing that A has an +attribute which C is devoid of, or is devoid of an attribute which C +has. + + All fishes are cold-blooded. + A whale is not cold-blooded. + .'. A whale is not a fish. + + No fishes give milk. + A whale gives milk. + .'. A whale is not a fish. + +FIGURE III. + +§ 654. + + B is A. B is not A. + B is C. B is C. + .'. Some C is A. .'. Some C is not A. + +We produce instances of C being A by showing that C and A meet, at all +events partially, in B. Thus if we wish to produce an instance of the +compatibility of great learning with original powers of thought, we +might say + + Sir William Hamilton was an original thinker. + Sir William Hamilton was a man of great learning. + .'. Some men of great learning are original thinkers. + +Or we might urge an exception to the supposed rule about Scotchmen +being deficient in humour under the same figure, thus-- + + Sir Walter Scott was not deficient in humour. + Sir Walter Scott was a Scotchman. + .'. Some Scotchmen are not deficient in humour. + +FIGURE IV. + +§ 655. + + All A is B, No A is B. + All B is C. All B is C. + .'. Some C is A .'.Some C is not A. + +We show here that A is or is not a species of C by showing that A +falls, or does not fall, under the class B, which itself falls under +C. Thus-- + + All whales are mammals. + All mammals are warm-blooded. + .'. Some warm-blooded animals are whales. + No whales are fishes. + All fishes are cold-blooded. + .'. Some cold-blooded animals are not whales. + + + + +CHAPTER XVII. + +_Of the Syllogism with three figures._ + +§ 656. It will be remembered that in beginning to treat of figure (§ +565) we pointed out that there were either four or three ligures +possible according as the conclusion was assumed to be known or +not. For, if the conclusion be not known, we cannot distinguish +between the major and the minor term, nor, consequently, between one +premiss and another. On this view the first and the fourth figures are +the same, being that arrangement of the syllogism in which the middle +term occupies a different position in one premiss from what it does in +the other. We will now proceed to constitute the legitimate moods and +figures of the syllogism irrespective of the conclusion. + +§ 657. When the conclusion is set out of sight, the number of possible +moods is the same as the number of combinations that can be made of +the four things, A, E, I, O, taken two together, without restriction +as to repetition. These are the following 16:-- + + AA EA IA OA + AE -EE- IE -OE- + AI EI -II- -OI- + AO -EO- -IO- -OO- + +of which seven may be neglected as violating the general rules of the +syllogism, thus leaving us with nine valid moods-- + + AA. AE. AI. AO. EA. EI. IA. IE. OA. + +§ 658. We will now put these nine moods successively into the three +figures. By so doing it will become apparent how far they are valid in +each. + +§ 659. Let it be premised that + + when the extreme in the premiss that stands first is predicate in + the conclusion, we are said to have a Direct Mood; + + when the extreme in the premiss that stands second is predicate in + the conclusion, we are said to have an Indirect Mood. + +§ 660. FIGURE 1. + + _Mood AA._ + All B is A. + All C is B. + .'. All C is A, or Some A is C, (Barbara & Bramantip). + + _Mood AE._ + All B is A. + No C is B. + .'. Illicit Process, or Some A is not C, (Fesapo). + + _Mood AI._ + All B is A. + Some C is B. + .'. Some C is A, or Some A is C. (Darii & Disamis). + + _Mood AO._ + All B is A. + Some C is not B. + .'. Illicit Process, (Ferio). + + _Mood EA._ + No B is A. + All C is B. + .'. No C is A, or No A is C, (Celarent & Camenes). + + _Mood EI._ + No B is A. + Some C is B. + .'. Some C is not A, or Illicit Process. + + _Mood IA._ + Some B is A. + All C is B. + .'. Undistributed Middle. + + _Mood IE._ + Some B is C. Some B is not A. + No A is B. All C is B. + .'. Illicit Process, or Some C is not A, (Fresison). + + _Mood OA._ + Some B is not A. + All C is B. + .'. Undistributed Middle. + +§ 661. Thus we are left with six valid moods, which yield four direct +conclusions and five indirect ones, corresponding to the four moods of +the original first figure and the five moods of the original fourth, +which appear now as indirect moods of the first figure. + +§ 662. But why, it maybe asked, should not the moods of the first +figure equally well be regarded as indirect moods of the fourth? For +this reason-that all the moods of the fourth figure can be elicited +out of premisses in which the terms stand in the order of the first, +whereas the converse is not the case. If, while retaining the quantity +and quality of the above premisses, i. e. the mood, we were in each +case to transpose the terms, we should find that we were left with +five valid moods instead of six, since AI in the reverse order of the +terms involves undistributed middle; and, though we should have +Celarent indirect to Camenes, and Darii to Dimaris, we should never +arrive at the conclusion of Barbara or have anything exactly +equivalent to Ferio. In place of Barbara, Bramantip would yield as an +indirect mood only the subaltern AAI in the first figure. Both Fesapo +and Fresison would result in an illicit process, if we attempted to +extract the conclusion of Ferio from them as an indirect mood. The +nearest approach we could make to Ferio would be the mood EAO in the +first figure, which may be elicited indirectly from the premisses of +CAMENES, being subaltern to CELARENT. For these reasons the moods of +the fourth figure are rightly to be regarded as indirect moods of the +first, and not vice versâ. + +$663. FIGURE II. + + + _Mood AA._ + All A is B. + All C is B. + .'. Undistributed Middle. + + _Mood AE._ + All A is B. + No C is B. + .'. No C is A, or No A is C, (Camestres & Cesare). + + _Mood AI._ + All A is B. + Some C is B. + .'. Undistributed Middle. + + _Mood AO._ + All A is B. + Some C is not B. + .'. Some C is not A, (Baroko), or Illicit Process. + + _Mood EA._ + No A is B. + All C is B. + .'. No C is A, or No A is C, (Cesare & Carnestres). + + + _Mood EI_ + No A is B. + Some C is B. + .'. Some C is not A, (Festino), or Illicit Process. + + _Mood IA._ + Some A is B. + All C is B. + .'. Undistributed Middle. + + _Mood IE._ + Some A is B. + No C is B. + .'. Illicit Process, or Some A is not C, (Festino). + + _Mood OA._ + Some A is not B. + All C is B. + .'. Illicit Process, or Some A is not C, (Baroko). + +§ 664. Here again we have six valid moods, which yield four direct +conclusions corresponding to Cesare, CARNESTRES, FESTINO and +BAROKO. The same four are repeated in the indirect moods. + +§ 665. FIGURE III. + + _Mood AA._ + All B is A. + All B is C. + .'. Some C is A, or Some A is C, (Darapti). + + _Mood AE._ + All B is A. + No B is C. + .'. Illicit Process, or Some A is not C, (Felapton). + + _Mood AI._ + All B is A, + Some B is C. + .'. Some C is A, or Some A is C, (Datisi & Disamis). + + _Mood AO._ + All B is A. + Some B is not C. + .'. Illicit Process, Or Some A is not C, (Bokardo). + + _Mood EA._ + No B is A. + All B is C. + .'. Some C is not A, (Felapton), or Illicit Process. + + _Mood EI._ + No B is A. + Some B is C. + .'. Some C is not A, (Ferison), or Illicit Process. + + _Mood IA._ + Some B is A. + All B is C. + .'. Some C is A, Or Some A is C, (Disamis & Datisi). + + _Mood IE._ + Some B is A. + No B is C. + .'. Illicit Process, or Some A is not C, (Ferison). + + _Mood QA._ + Some B is not A. + All B is C. + .'. Some C is not A, (Bokardo), or Illicit Process. + +§ 666. In this figure every mood is valid, either directly or +indirectly. We have six direct moods, answering to Darapti, Disamis, +Datisi, Felapton, Bokardo and Ferison, which are simply repeated by +the indirect moods, except in the case of Darapti, which yields a +conclusion not provided for in the mnemonic lines. Darapti, though +going under one name, has as much right to be considered two moods as +Disamis and Datisi. + + + + +CHAPTER XVIII. + +_Of Reduction._ + + +§ 667. We revert now to the standpoint of the old logicians, who +regarded the Dictum de Omni et Nullo as the principle of all +syllogistic reasoning. From this point of view the essence of mediate +inference consists in showing that a special case, or class of cases, +comes under a general rule. But a great deal of our ordinary reasoning +does not conform to this type. It was therefore judged necessary to +show that it might by a little manipulation be brought into conformity +with it. This process is called Reduction. + +§ 668. Reduction is of two kinds-- + + (1) Direct or Ostensive. + + (2) Indirect or Ad Impossibile. + +§ 669. The problem of direct, or ostensive, reduction is this-- + + Given any mood in one of the imperfect figures (II, III and IV) how + to alter the form of the premisses so as to arrive at the same + conclusion in the perfect figure, or at one from which it can be + immediately inferred. The alteration of the premisses is effected by + means of immediate inference and, where necessary, of transposition. + +§ 670. The problem of indirect reduction, or reductio (per +deductionem) ad impossibile, is this--Given any mood in one of the +imperfect figures, to show by means of a syllogism in the perfect +figure that its conclusion cannot be false. + +§ 671. The object of reduction is to extend the sanction of the Dictum +de Omni et Nullo to the imperfect figures, which do not obviously +conform to it. + +§ 672. The mood required to be reduced is called the Reducend; that to +which it conforms, when reduced, is called the Reduct. + + + +_Direct or Ostensive Reduction._ + + +§ 673. In the ordinary form of direct reduction, the only kind of +immediate inference employed is conversion, either simple or by +limitation; but the aid of permutation and of conversion by negation +and by contraposition may also be resorted to. + +§ 674. There are two moods, Baroko and Bokardo, which cannot be +reduced ostensively except by the employment of some of the means last +mentioned. Accordingly, before the introduction of permutation into +the scheme of logic, it was necessary to have recourse to some other +expedient, in order to demonstrate the validity of these two +moods. Indirect reduction was therefore devised with a special view to +the requirements of Baroko and Bokardo: but the method, as will be +seen, is equally applicable to all the moods of the imperfect figures. + +§ 675. The mnemonic lines, 'Barbara, Celarent, etc., provide complete +directions for the ostensive reduction of all the moods of the second, +third, and fourth figures to the first, with the exception of Baroko +and Bokardo. The application of them is a mere mechanical trick, which +will best be learned by seeing the process performed. + +§ 676. Let it be understood that the initial consonant of each name of +a figured mood indicates that the reduct will be that mood which +begins with the same letter. Thus the B of Bramantip indicates that +Bramantip, when reduced, will become Barbara. + +§ 677. Where m appears in the name of a reducend, me shall have to +take as major that premiss which before was minor, and vice versa-in +other words, to transpose the premisses, m stands for mutatio or +metathesis. + +§ 678. s, when it follows one of the premisses of a reducend, +indicates that the premiss in question must be simply converted; when +it follows the conclusion, as in Disamis, it indicates that the +conclusion arrived at in the first figure is not identical in form +with the original conclusion, but capable of being inferred from it by +simple conversion. Hence s in the middle of a name indicates something +to be done to the original premiss, while s at the end indicates +something to be done to the new conclusion. + +§ 679. P indicates conversion per accidens, and what has just been +said of s applies, mutatis mutandis, to p. + +§ 680. k may be taken for the present to indicate that Baroko and +Bokardo cannot be reduced ostensively. + +§ 681. FIGURE II. + + Cesare. \ / Celarent. + No A is B. \ = / No B is A. + All C is B. / \ All C is B. + .'. No C is A. / \ .'. No C is A. + + Camestres. \ / Celarent. + All A is B. \ = / No B is C. + No C is B. / \ All A is B. + .'. No C is A. / \ .'. No A is C. + .'. No C is A. + + Festino. Ferio. + No A is B. \ / No B is A. + Some C is B. | = | Some C is B. + .'. Some C is not A./ \ .'. Some C is not A. + [Baroko] + +§ 682. FIGURE III. + + Darapti. \ / Darii. + All B is A. \ = / All B is A. + All B is C. / \ Some C is B. + .'. Some C is A. / \ Some C is A. + + Disamis. \ / Darii. + Some B is A. \ = / All B is C. + All B is C. / \ Some A is B. + .'. Some C is A. / \ .'. Some A is C. + .'. Some C is A. + + Datisi. \ / Darii. + All B is A. \ = / All B is A. + Some B is C. / \ Some C is B. + .'. Some C is A. / \ .'. Some C is A. + + Felapton. \ / Ferio. + No B is A. \ = / No B is A. + All B is C. / \ Some C is B. + .'. Some C is not-A. / \ .'. Some C is not-A. + + [Bokardo]. + + Ferison. \ / Ferio. + No B is A. \ = / No B is A. + Some B is C. / \ Some C is B + .'. Some C is not A. / \ .'. Some C is not A. + +§ 683. FIGURE IV. + + Bramantip. \ / Barbara. + All A is B. \ = / All B is C. + All B is C. / \ All A is B. + .. Some C is A. / \ .. All A is C. + .'. Some C is A. + + Camenes Celarent + All A is B \ / No B is C. + No B is C. | = | All A is B. + .. No C is A./ \ .'. No A is C. + .'. No C is A. + + Dimaris. Darii. + Some A is B. \ / All B is C. + All B is C. | = | Some A is B. + .'. Some C is A./ \ .'. Some A is C. + .'. Some C is A. + + Fesapo. Ferio. + No A is B. \ / No B is A. + All B is C. | = | Some C is B. + .'. Some C is not A./ \ .'. Some C is not A. + + Fresison. Ferio. + No A is B. \ / No B is A. + Some B is C. | = | Some C is B. + .'. Some C is not A./ \ .'. Some C is not A. + +§ 684. The reason why Baroko and Bokardo cannot be reduced ostensively +by the aid of mere conversion becomes plain on an inspection of +them. In both it is necessary, if we are to obtain the first figure, +that the position of the middle term should be changed in one +premiss. But the premisses of both consist of A and 0 propositions, of +which A admits only of conversion by limitation, the effect of which +would be to produce two particular premisses, while 0 does not admit +of conversion at all, + +It is clear then that the 0 proposition must cease to be 0 before we +can get any further. Here permutation comes to our aid; while +conversion by negation enables us to convert the A proposition, +without loss of quantity, and to elicit the precise conclusion we +require out of the reduct of Boltardo. + + (Baroko) Fanoao. Ferio. + All A is B. \ / No not-B is A. + Some C is not-B. | = | Some C is not-B. + .'. Some C is not-A./ \ .'. Some C is not-A. + + (Bokardo) Donamon. Darii. + Some B is not-A. \ / All B is C. + All B is C. | = | Some not-A is B + .'. Some C is not-A./ \ .'. Some not-A is C. + .'. Some C is not-A. + +§ 685. In the new symbols, Fanoao and Donamon, [pi] has been +adopted as a symbol for permutation; n signifies conversion by +negation. In Donamon the first n stands for a process which resolves +itself into permutation followed by simple conversion, the second for +one which resolves itself into simple conversion followed by +permutation, according to the extended meaning which we have given to +the term 'conversion by negation.' If it be thought desirable to +distinguish these two processes, the ugly symbol Do[pi]samos[pi] may +be adopted in place of Donamon. + +§ 686. The foregoing method, which may be called Reduction by +Negation, is no less applicable to the other moods of the second +figure than to Baroko. The symbols which result from providing for its +application would make the second of the mnemonic lines run thus-- + + Benare[pi], Cane[pi]e, Denilo[pi], Fano[pi]o secundae. + +§ 687. The only other combination of mood and figure in which it will +be found available is Camenes, whose name it changes to Canene. + +§ 688. + + (Cesare) Benarea. Barbara. + No A is B. \ / All B is not-A. + All C is B. | = | All C is B. + .'. No C is A. / \ .'. All C is not-A. + .'. No C is A. + + (Camestres) Cane[pi]e. Celarent. + All A is B. \ / No not-B is A. + No C is B. | = | All C is not-B. + .'. No C is A. / \ .'. No C is A. + + (Festino) Denilo[pi]. Darii. + No A is B. \ / All B is not-A. + Some C is B. | = | Some C is B. + .'. Some C is not A./ \ .'. Some C is not-A. + .'. Some C is not A. + + (Camenes) Canene. Celarent. + All A is B. \ / No not-B is A. + No B is C. | = | All C is not-B. + .'. No C is A. / \ .'. No C is A. + +§ 689. The following will serve as a concrete instance of Cane[pi]e +reduced to the first figure. + + All things of which we have a perfect idea are perceptions. + A substance is not a perception. + .'. A substance is not a thing of which we have a perfect idea. + +When brought into Celarent this becomes-- + + No not-perception is a thing of which we have a perfect idea. + A substance is a not-perception. + .'. No substance is a thing of which we have a perfect idea. + +§ 690. We may also bring it, if we please, into Barbara, by permuting +the major premiss once more, so as to obtain the contrapositive of the +original-- + + All not-perceptions are things of which we have an imperfect idea. + All substances are not-perceptions. + .'. All substances are things of which we have an imperfect idea. + + + +_Indirect Reduction._ + + +§ 691. We will apply this method to Baroko. + + All A is B. All fishes are oviparous. + Some C is not B. Some marine animals are not oviparous. + .'. Some C is not A. .'. Some marine animals are not fishes. + +§ 692. The reasoning in such a syllogism is evidently conclusive: but +it does not conform, as it stands, to the first figure, nor +(permutation apart) can its premisses be twisted into conformity with +it. But though we cannot prove the conclusion true in the first +figure, we can employ that figure to prove that it cannot be false, by +showing that the supposition of its falsity would involve a +contradiction of one of the original premisses, which are true ex +hypothesi. + +§ 693. If possible, let the conclusion 'Some C is not A' be +false. Then its contradictory 'All C is A' must be true. Combining +this as minor with the original major, we obtain premisses in the +first figure, + + All A is B, All fishes are oviparous, + All C is A, All marine animals are fishes, + +which lead to the conclusion + + All C is B, All marine animals are oviparous. + +But this conclusion conflicts with the original minor, 'Some C is not +B,' being its contradictory. But the original minor is ex hypothesi +true. Therefore the new conclusion is false. Therefore it must either +be wrongly drawn or else one or both of its premisses must be false. +But it is not wrongly drawn; since it is drawn in the first figure, to +which the Dictum de Omni et Nullo applies. Therefore the fault must +lie in the premisses. But the major premiss, being the same with that +of the original syllogism, is ex hypothesi true. Therefore the minor +premiss, 'All C is A,' is false. But this being false, its +contradictory must be true. Now its contradictory is the original +conclusion, 'Some C is not A,' which is therefore proved to be true, +since it cannot be false. + +§ 694. It is convenient to represent the two syllogisms in +juxtaposition thus-- + + Baroko. Barbara. + All A is B. All A is B. + Some C is not B. \/ All C is A. + .'. Some C is not A. /\ All C is B. + +§ 695. The lines indicate the propositions which conflict with one +another. The initial consonant of the names Baroko and Eokardo +indicates that the indirect reduct will be Barbara. The k indicates +that the O proposition, which it follows, is to be dropped out in the +new syllogism, and its place supplied by the contradictory of the old +conclusion. + +§ 696. In Bokardo the two syllogisms will stand thus-- + + Bokardo. Barbara. + Some B is not A. \ / All C is A. + All B is C. X All B is C. + .'. Some C is not A./ \ .'. All B is A. + +§ 697. The method of indirect reduction, though invented with a +special view to Baroko and Bokardo, is applicable to all the moods of +the imperfect figures. The following modification of the mnemonic +lines contains directions for performing the process in every +case:--Barbara, Celarent, Darii, Ferioque prioris; Felake, Dareke, +Celiko, Baroko secundae; Tertia Cakaci, Cikari, Fakini, Bekaco, +Bokardo, Dekilon habet; quarta insuper addit Cakapi, Daseke, Cikasi, +Cepako, Cesďkon. + +§ 698. The c which appears in two moods of the third figure, Cakaci +and Bekaco, signifies that the new conclusion is the contrary, instead +of, as usual, the contradictory of the discarded premiss. + +§ 699. The letters s and p, which appear only in the fourth figure, +signify that the new conclusion does not conflict directly with the +discarded premiss, but with its converse, either simple or per +accidens, as the case may be. + +§ 700. l, n and r are meaningless, as in the original lines. + + + + +CHAPTER XIX. + +_Of Immediate Inference as applied to Complex Propositions._ + + +§ 701. So far we have treated of inference, or reasoning, whether +mediate or immediate, solely as applied to simple propositions. But it +will be remembered that we divided propositions into simple and +complex. I t becomes incumbent upon us therefore to consider the laws +of inference as applied to complex propositions. Inasmuch however as +every complex proposition is reducible to a simple one, it is evident +that the same laws of inference must apply to both. + +§ 702. We must first make good this initial statement as to the +essential identity underlying the difference of form between simple +and complex propositions. + +§ 703. Complex propositions are either Conjunctive or Disjunctive (§ +214). + +§ 704. Conjunctive propositions may assume any of the four forms, A, +E, I, O, as follows-- + + (A) If A is B, C is always D. + (E) If A is B, C is never D. + (I) If A is B, C is sometimes D. + (O) If A is B, C is sometimes not D. + +§ 705. These admit of being read in the form of simple propositions, +thus-- + + (A) If A is B, C is always D = All cases of A being B are cases of C + being D. (Every AB is a CD.) + + (E) If A is B, C is never D = No cases of A being B are cases of C + being D. (No AB is a CD.) + + (I) If A is B, C is sometimes D = Some cases of A being B are cases + of C being D. (Some AB's are CD's.) + + (O) If A is B, C is sometimes not D = Some cases of A being B are + not cases of C being D. (Some AB's are not CD's.) + +§ 706. Or, to take concrete examples, + + (A) If kings are ambitious, their subjects always suffer. + = All cases of ambitious kings are cases of subjects suffering. + + (E) If the wind is in the south, the river never freezes. + = No cases of wind in the south are cases of the river freezing. + + (I) If a man plays recklessly, the luck sometimes goes against him. + = Some cases of reckless playing are cases of going against one. + + (O) If a novel has merit, the public sometimes do not buy it. + = Some cases of novels with merit are not cases of the public buying. + +§ 707. We have seen already that the disjunctive differs from the +conjunctive proposition in this, that in the conjunctive the truth +of the antecedent involves the truth of the consequent, whereas in the +disjunctive the falsity of the antecedent involves the truth of the +consequent. The disjunctive proposition therefore + + Either A is B or C is D + +may be reduced to a conjunctive + + If A is not B, C is D, + +and so to a simple proposition with a negative term for subject. + + All cases of A not being B are cases of C being D. + (Every not-AB is a CD.) + +§ 708. It is true that the disjunctive proposition, more than any +other form, except U, seems to convey two statements in one +breath. Yet it ought not, any more than the E proposition, to be +regarded as conveying both with equal directness. The proposition 'No +A is B' is not considered to assert directly, but only implicitly, +that 'No B is A.' In the same way the form 'Either A is B or C is D' +ought to be interpreted as meaning directly no more than this, 'If A +is not B, C is D.' It asserts indeed by implication also that 'If C is +not D, A is B.' But this is an immediate inference, being, as we shall +presently see, the contrapositive of the original. When we say 'So and +so is either a knave or a fool,' what we are directly asserting is +that, if he be not found to be a knave, he will be found to be a +fool. By implication we make the further statement that, if he be not +cleared of folly, he will stand condemned of knavery. This inference +is so immediate that it seems indistinguishable from the former +proposition: but since the two members of a complex proposition play +the part of subject and predicate, to say that the two statements are +identical would amount to asserting that the same proposition can have +two subjects and two predicates. From this point of view it becomes +clear that there is no difference but one of expression between the +disjunctive and the conjunctive proposition. The disjunctive is +merely a peculiar way of stating a conjunctive proposition with a +negative antecedent. + +§ 709. Conversion of Complex Propositions. + + A / If A is B, C is always D. + \ .'. If C is D, A is sometimes B. + + E / If A is B, C is never D. + \ .'. If C is D, A is never B. + + I / If A is S, C is sometimes D. + \ .'. If C is D, A is sometimes B. + +§ 710. Exactly the same rules of conversion apply to conjunctive as to +simple propositions. + +§ 711. A can only be converted per accidens, as above. + +The original proposition + + 'If A is B, C is always D' + +is equivalent to the simple proposition + + 'All cases of A being B are cases of C being D.' + +This, when converted, becomes + + 'Some cases of C being D are cases of A being B,' + +which, when thrown back into the conjunctive form, becomes + + 'If C is D, A is sometimes B.' + +§ 712. This expression must not be misunderstood as though it +contained any reference to actual existence. The meaning might be +better conveyed by the form + + 'If C is D, A may be B.' + +But it is perhaps as well to retain the other, as it serves to +emphasize the fact that formal logic is concerned only with the +connection of ideas. + +§ 713. A concrete instance will render the point under discussion +clearer. The example we took before of an A proposition in the +conjunctive form-- + + 'If kings are ambitious, their subjects always suffer' + +may be converted into + + 'If subjects suffer, it may be that their kings are ambitious,' + +i.e. among the possible causes of suffering on the part of subjects is +to be found the ambition of their rulers, even if every actual case +should be referred to some other cause. It is in this sense only that +the inference is a necessary one. But then this is the only sense +which formal logic is competent to recognise. To judge of conformity +to fact is no part of its province. From 'Every AB is a CD' it follows +that ' Some CD's are AB's' with exactly the same necessity as that +with which 'Some B is A' follows from 'All A is B.' In the latter case +also neither proposition may at all conform to fact. From 'All +centaurs are animals' it follows necessarily that 'Some animals are +centaurs': but as a matter of fact this is not true at all. + +§ 714. The E and the I proposition may be converted simply, as above. + +§ 715. O cannot be converted at all. From the proposition + + 'If a man runs a race, he sometimes does not win it,' + +it certainly does not follow that + + 'If a man wins a race, he sometimes does not run it.' + +§ 716. There is a common but erroneous notion that all conditional +propositions are to be regarded as affirmative. Thus it has been +asserted that, even when we say that 'If the night becomes cloudy, +there will be no dew,' the proposition is not to be regarded as +negative, on the ground that what we affirm is a relation between the +cloudiness of night and the absence of dew. This is a possible, but +wholly unnecessary, mode of regarding the proposition. It is precisely +on a par with Hobbes's theory of the copula in a simple proposition +being always affirmative. It is true that it may always be so +represented at the cost of employing a negative term; and the same is +the case here. + +§ 717. There is no way of converting a disjunctive proposition except +by reducing it to the conjunctive form. + +§ 718. _Permutation of Complex Propositions_. + + (A) If A is B, C is always D. + .'. If A is B, C is never not-D. (E) + + (E) If A is B, C is never D. + .'. If A is B, C is always not-D. (A) + + (I) If A is B, C is sometimes D. + .'. If A is B, C is sometimes not not-D. (O) + + (O) If A is B, C is sometimes not D. + .'. If A is B, C is sometimes not-D. (I) + +§ 719. + + (A) If a mother loves her children, she is always kind to them. + .'. If a mother loves her children, she is never unkind to + them. (E) + + (E) If a man tells lies, his friends never trust him. + .'. If a man tells lies, his friends always distrust him. (A) + + (I) If strangers are confident, savage dogs are sometimes friendly. + .'. If strangers are confident, savage dogs are sometimes not + unfriendly. (O) + + (O) If a measure is good, its author is sometimes not popular. + .'. If a measure is good, its author is sometimes + unpopular. (I) + +§ 720. The disjunctive proposition may be permuted as it stands +without being reduced to the conjunctive form. + + Either A is B or C is D. + .'. Either A is B or C is not not-D. + + Either a sinner must repent or he will be damned. + .'. Either a sinner must repent or he will not be saved. + +§ 721. _Conversion by Negation of Complex Propositions._ + + (A) If A is B, C is always D. + .'. If C is not-D, A is never B. (E) + + (E) If A is B, C is never D. + .'. If C is D, A is always not-B. (A) + + (I) If A is B, C is sometimes D. + .'. If C is D, A is sometimes not not-B. (O) + + (O) If A is B, C is sometimes not D. + .'. If C is not-D, A is sometimes B. (I) + + (E per acc.) If A is B, C is never D. + .'. If C is not-D, A is sometimes B. (I) + + (A per ace.) If A is B, C is always D. + .'. If C is D, A is sometimes not not-D. (O) + +§ 722. + + (A) If a man is a smoker, he always drinks. + .'. If a man is a total abstainer, he never smokes. (E) + + (E) If a man merely does his duty, no one ever thanks him. + .'. If people thank a man, he has always done more than his + duty. (A) + + (I) If a statesman is patriotic, he sometimes adheres to a party. + .'. If a statesman adheres to a party, he is sometimes not + unpatriotic. (O) + + (O) If a book has merit, it sometimes does not sell. + .'. If a book fails to sell, it sometimes has merit. (I) + + (E per acc.) If the wind is high, rain never falls. + .'. If rain falls, the wind is sometimes high. (I) + + (A per acc.) If a thing is common, it is always cheap. + .'. If a thing is cheap, it is sometimes not uncommon. (O) + +§ 723. When applied to disjunctive propositions, the distinctive +features of conversion by negation are still discernible. In each of +the following forms of inference the converse differs in quality from +the convertend and has the contradictory of one of the original terms +(§ 515). + +§ 724. + + (A) Either A is B or C is always D. + .'. Either C is D or A is never not-B. (E) + + (E) Either A is B or C is never D. + .'. Either C is not-D or A is always B. (A) + + (I) Either A is B or C is sometimes D. + .'. Either C is not-D or A is sometimes not B. (O) + + (O) Either A is B or C is sometimes not D. + .'. Either C is D or A is sometimes not-B. (I) + +§ 725. + + (A) Either miracles are possible or every ancient historian is + untrustworthy. + .'. Either ancient historians are untrustworthy or miracles are + not impossible. (E) + + (E) Either the tide must turn or the vessel can not make the port. + .'. Either the vessel cannot make the port or the tide must + turn. (A) + + (1) Either he aims too high or the cartridges are sometimes bad. + .'. Either the cartridges are not bad or he sometimes does not + aim too high. (0) + + (O) Either care must be taken or telegrams will sometimes not be + correct. + .'. Either telegrams are correct or carelessness is sometimes + shown. (1) + +§ 726. In the above examples the converse of E looks as if it had +undergone no change but the mere transposition of the +alternative. This appearance arises from mentally reading the E as an +A proposition: but, if it were so taken, the result would be its +contrapositive, and not its converse by negation. + +§ 727. The converse of I is a little difficult to grasp. It becomes +easier if we reduce it to the equivalent conjunctive-- + + 'If the cartridges are bad, he sometimes does not aim too high.' + +Here, as elsewhere, 'sometimes' must not be taken to mean more than +'it may be that.' + +§ 728. _Conversion by Contraposition of Complex Propositions._ + +As applied to conjunctive propositions conversion by contraposition +assumes the following forms-- + + (A) If A is B, C is always D. + .'. If C is not-D, A is always not-B. + + (O) If A is B, C is sometimes not D. + .'. If C is not-D, A is sometimes not not-B. + + (A) If a man is honest, he is always truthful. + .'. If a man is untruthful, he is always dishonest. + + (O) If a man is hasty, he is sometimes not malevolent. + .'. If a man is benevolent, he is sometimes not unhasty. + +§ 729. As applied to disjunctive propositions conversion by +contraposition consists simply in transposing the two alternatives. + + (A) Either A is B or C is D. + .'. Either C is D or A is B. + +For, when reduced to the conjunctive shape, the reasoning would run +thus-- + + If A is not B, C is D. + .'. If C is not D, A is B. + +which is the same in form as + + All not-A is B. + .'. All not-B is A. + +Similarly in the case of the O proposition + + (O) Either A is B or C is sometimes not D. + .'. Either C is D or A is sometimes not B. + +§ 730. On comparing these results with the converse by negation of +each of the same propositions, A and 0, the reader will see that they +differ from them, as was to be expected, only in being permuted. The +validity of the inference may be tested, both here and in the case of +conversion by negation, by reducing the disjunctive proposition to the +conjunctive, and so to the simple form, then performing the process as +in simple propositions, and finally throwing the converse, when so +obtained, back into the disjunctive form. We will show in this manner +that the above is really the contrapositive of the 0 proposition. + + (O) Either A is B or C is sometimes not D. + + = If A is not B, C is sometimes not D. + + = Some cases of A not being B are not cases of C being D. (Some A is + not B.) + + = Some cases of C not being D are not cases of A being B. (Some + not-B is not not-A.) + + = If C is not D, A is sometimes not B. + + = Either C is D or A is sometimes not B. + + + + +CHAPTER XX. + +_Of Complex Syllogisms_. + + +§ 731. A Complex Syllogism is one which is composed, in whole or part, +of complex propositions. + +§ 732. Though there are only two kinds of complex proposition, there +are three varieties of complex syllogism. For we may have + + (1) a syllogism in which the only kind of complex proposition + employed is the conjunctive; + + (2) a syllogism in which the only kind of complex proposition + employed is the disjunctive; + + (3) a syllogism which has one premiss conjunctive and the other + disjunctive. + +The chief instance of the third kind is that known as the Dilemma. + + Syllogism + ___________________|_______________ + | | + Simple Complex + (Categorical) (Conditional) + _____________________|_______________ + | | | + Conjunctive Disjunctive Dilemma + (Hypothetical) + + + +_The Conjunctive Syllogism_. + + +§ 733. The Conjunctive Syllogism has one or both premisses conjunctive +propositions: but if only one is conjunctive, the other must be a +simple one. + +§ 734. Where both premisses are conjunctive, the conclusion will be of +the same character; where only one is conjunctive, the conclusion will +be a simple proposition. + +§ 735. Of these two kinds of conjunctive syllogisms we will first take +that which consists throughout of conjunctive propositions. + + + +_The Wholly Conjunctive Syllogism_. + + +§ 736. Wholly conjunctive syllogisms do not differ essentially from +simple ones, to which they are immediately reducible. They admit of +being constructed in every mood and figure, and the moods of the +imperfect figures may be brought into the first by following the +ordinary rules of reduction. For instance-- + + Cesare. Celarent. + + If A is B, C is never D. \ / If C is D, A is never B. + If E is F, C is always D. | = | If E is F, C is always D. + .'. If E is F, A is never B. / \ .'. If E is F, A is never B. + + If it is day, the stars never shine.\ /If the stars shine, it is never day. + If it is night, the stars always \=/ If it is night, the stars always + shine. / \ shine. + .'. If it is night, it is never day / \.'. If it is night, it is never day. + + Disamis. Darii. + If C is D, A is sometimes B. \ / If C is D, E is always F. + If C is D, E is always F. | = | If A is B, C is sometimes D. + If E is F, A is sometimes B. / \ .'. If A is B, E is sometimes F. + .'. If E is F, A is sometimes B. + + If she goes, I sometimes go. \ / If she goes, he always goes, + If she goes, he always goes. | = | If I go, she sometimes goes. + .'. If he goes, I sometimes go. / \ .'. If I go, he sometimes goes. + .'. If he goes, I sometimes go. + + + +_The Partly Conjunctive Syllogism._ + + +§ 737. It is this kind which is usually meant when the Conjunctive or +Hypothetical Syllogism is spoken of. + +§ 738. Of the two premisses, one conjunctive and one simple, the +conjunctive is considered to be the major, and the simple premiss the +minor. For the conjunctive premiss lays down a certain relation to +hold between two propositions as a matter of theory, which is applied +in the minor to a matter of fact. + +§ 739. Taking a conjunctive proposition as a major premiss, there are +four simple minors possible. For we may either assert or deny the +antecedent or the consequent of the conjunctive. + + Constructive Mood. Destructive Mood. + (1) If A is B, C is D. (2) If A is B, C is D. + A is B. C is not D. + .'. C is D. .'. A is not B. + + (3) If A is B, C is D. (4) If A is B, C is D. + A is not B. C is D. + No conclusion. No conclusion. + +§ 740. When we take as a minor 'A is not B ' (3), it is clear that we +can get no conclusion. For to say that C is D whenever A is B gives us +no right to deny that C can be D in the absence of that +condition. What we have predicated has been merely inclusion of the +case AB in the case CD. + +[Illustration] + +§ 741. Again, when we take as a minor, 'C is D' (4), we can get no +universal conclusion. For though A being B is declared to involve as a +consequence C being D, yet it is possible for C to be D under other +circumstances, or from other causes. Granting the truth of the +proposition 'If the sky falls, we shall catch larks,' it by no means +follows that there are no other conditions under which this result can +be attained. + +§ 742. From a consideration of the above four cases we elicit the +following + + + +_Canon of the Conjunctive Syllogism._ + + +To affirm the antecedent is to affirm the consequent, and to deny the +consequent is to deny the antecedent: but from denying the antecedent +or affirming the consequent no conclusion follows. + +§ 743. There is a case, however, in which we can legitimately deny the +antecedent and affirm the consequent of a conjunctive proposition, +namely, when the relation predicated between the antecedent and the +consequent is not that of inclusion but of coincidence--where in fact +the conjunctive proposition conforms to the type u. + +For example-- + + _Denial of the Antecedent_. + If you repent, then only are you forgiven. + You do not repent. + .'. You are not forgiven. + + _Affirmation of the Consequent_. + If you repent, then only are you forgiven. + You are forgiven. + .'. You repent. + + + + +CHAPTER XXI. + +_Of the Reduction of the Partly Conjunctive Syllogism._ + + +§ 744. Such syllogisms as those just treated of, if syllogisms they +are to be called, have a major and a middle term visible to the eye, +but appear to be destitute of a minor. The missing minor term is +however supposed to be latent in the transition from the conjunctive +to the simple form of proposition. When we say 'A is B,' we are taken +to mean, 'As a matter of fact, A is B' or 'The actual state of the +case is that A is B.' The insertion therefore of some such expression +as 'The case in hand,' or 'This case,' is, on this view, all that is +wanted to complete the form of the syllogism. When reduced in this +manner to the simple type of argument, it will be found that the +constructive conjunctive conforms to the first figure and the +destructive conjunctive to the second. + + _Constructive Mood_. _Barbara_. + + If A is B, C is D. \ / All cases of A being B are cases of + \ = / C being D. + A is B. / \ This is a case of A being B. + .'. C is D. / \ .'. This is a case of C being D. + + _Destructive Mood._ Camestres. + + If A is B, C is D. \ / All cases of A being B are cases of + \ = / C being D. + C is not D. / \ This is not a case of C being D. + .'. A is not B. / \ .'. This is not a case of A being B. + +§ 745. It is apparent from the position of the middle term that the +constructive conjunctive must fall into the first figure and the +destructive conjunctive into the second. There is no reason, however, +why they should be confined to the two moods, Barbara and +Carnestres. If the inference is universal, whether as general or +singular, the mood is Barbara or Carnestres; if it is particular, the +mood is Darii or Baroko. + + Barbara. Camestres. + If A is B, C is always D. \ If A is B, C is always D. \ + A is always B. \ C is never D. \ + .'. C is always D. \ .'. A is never B. \ + | | + If A is B, C is always D. / If A is B, C is always D. / + A is in this case B. / C is not in this case D. / + .'. C is in this case D. / .'. A is not in this case B. / + + Darii. Baroko. + + If A is B, C is always D. If A is B, C is never D. + A is sometimes B. C is sometimes not D. + .'. C is sometimes D. .'. A is sometimes not B. + +§ 746. The remaining moods of the first and second figure are obtained +by taking a negative proposition as the consequent in the major +premiss. + + Celarent. Ferio. + If A is B, C is never D. If A is B, C is never D. + A is always B. A is sometimes B. + .'. C is never D. .'. C is sometimes not D. + + _Cesare_. Festino. + If A is B, C is never D. If A is B, C is never D. + C is always D. C is sometimes D. + .'. A is never B. .'. A is sometimes not B. + +§ 747. As the partly conjunctive syllogism is thus reducible to the +simple form, it follows that violations of its laws must correspond +with violations of the laws of simple syllogism. By our throwing the +illicit moods into the simple form it will become apparent what +fallacies are involved in them. + + _Denial of Anteceded_. + + If A is B, C is D. \ / All cases of A being B are cases of C + \ = / being D. + A is not B. / \ This is not a case of A being B. + .'. C is not D. / \ .'. This is not a case of C being D. + +Here we see that the denial of the antecedent amounts to illicit +process of the major term. + +§ 7481 _Affirmation of Consequent_. + + If A is B, C is D. \ / All Cases of A being B are cases of C + | = | being D. + C is D. / \ This is a case of C being D. + +Here we see that the affirmation of the consequent amounts to +undistributed middle. + +§ 749. If we confine ourselves to the special rules of the four +figures, we see that denial of the antecedent involves a negative +minor in the first figure, and affirmation of the consequent two +affirmative premisses in the second. Or, if the consequent in the +major premiss were itself negative, the affirmation of it would amount +to the fallacy of two negative premisses. Thus-- + + If A is B, C is not D. \ / No cases of A being B are cases of C + | = | being D. + C is not D. / \ This is not a case of C being D. + +§ 750. The positive side of the canon of the conjunctive +syllogism--'To affirm the antecedent is to affirm the consequent,' +corresponds with the Dictum de Omni. For whereas something (viz. C +being D) is affirmed in the major of all conceivable cases of A being +B, the same is affirmed in the conclusion of something which is +included therein, namely, 'this case,' or 'some cases,' or even 'all +actual cases.' + +§ 751. The negative side--'to deny the consequent is to deny the +antecedent'--corresponds with the Dictum de Diverse (§ 643). For +whereas in the major all conceivable cases of A being B are included +in C being D, in the minor 'this case,' or 'some cases,' or even 'all +actual cases' of C being D, are excluded from the same notion. + +§ 752. The special characteristic of the partly conjunctive syllogism +lies in the transition from hypothesis to fact. We might lay down as +the appropriate axiom of this form of argument, that 'What is true in +the abstract is true--in the concrete,' or 'What is true in theory is +also true in fact,' a proposition which is apt to be neglected or +denied. But this does not vitally distinguish it from the ordinary +syllogism. For though in the latter we think rather of the transition +from a general truth to a particular application of it, yet at bottom +a general truth is nothing but a hypothesis resting upon a slender +basis of observed fact. The proposition 'A is B' may be expressed in +the form 'If A is, B is.' To say that 'All men are mortal' may be +interpreted to mean that 'If we find in any subject the attributes of +humanity, the attributes of mortality are sure to accompany them.' + + + + +CHAPTER XXII. + +_Of the Partly Conjunctive Syllogism regarded as an Immediate +Inference_. + + +§ 753. It is the assertion of fact in the minor premiss, where we have +the application of an abstract principle to a concrete instance, which +alone entitles the partly conjunctive syllogism to be regarded as a +syllogism at all. Apart from this the forms of semi-conjunctive +reasoning run at once into the moulds of immediate inference. + +§ 754. The constructive mood will then be read in this way-- + + If A is B, C is D, + .'. A being B, C is D. + +reducing itself to an instance of immediate inference by subaltern +opposition-- + + Every case of A being B, is a case of C being D. + .'. Some particular case of A being B is a case of C being D. + +§ 755. Again, the destructive conjunctive will read as follows-- + + If A is B, C is D, + .'. C not being D, A is not B. + +which is equivalent to + + All cases of A being B are cases of C being D. + .'. Whatever is not a case of C being D is not a case of A being B. + .'. Some particular case of C not being D is not a case of A being + B. + +But what is this but an immediate inference by contraposition, coming +under the formula + + All A is B, + .'. All not-B is not-A, + +and followed by Subalternation? + +§ 756. The fallacy of affirming the consequent becomes by this mode of +treatment an instance of the vice of immediate inference known as the +simple conversion of an A proposition. 'If A is B, C is D' is not +convertible with 'If C is D, A is B' any more than 'All A is B' is +convertible with 'All B is A.' + +§ 757. We may however argue in this way + + If A is B, C is D, + C is D, + .'. A may be B, + +which is equivalent to saying, + + When A is B, C is always D, + .'. When C is D, A is sometimes B, + +and falls under the legitimate form of conversion of A per accidens-- + + All cases of A being B are cases of C being D. + .'. Some cases of C being D are cases of A being B. + +§ 758. The fallacy of denying the antecedent assumes the following +form-- + + If A is B, C is D, + .'. If A is not B, C is not D, + +equivalent to-- + + All cases of A being B are cases of C being D. + .'. Whatever is not a case of A being B is not a case of C being D. + +This is the same as to argue-- + + All A is B, + .'. All not-A is not-B, + +an erroneous form of immediate inference for which there is no special +name, but which involves the vice of simple conversion of A, since +'All not-A is not-B' is the contrapositive, not of 'All A is B,' but +of its simple converse 'All B is A.' + +§ 759. The above-mentioned form of immediate inference, however +(namely, the employment of contraposition without conversion), is +valid in the case of the U proposition; and so also is simple +conversion. Accordingly we are able, as we have seen, in dealing with +a proposition of that form, both to deny the antecedent and to assert +the consequent with impunity-- + + If A is B, then only C is D, + .'. A not being B, C is not D; + +and again, C being D, A must be B. + + + + +CHAPTER XXIII. + +_Of the Disjunctive Syllogism_. + + +§ 760. Roughly speaking, a Disjunctive Syllogism results from the +combination of a disjunctive with a simple premiss. As in the +preceding form, the complex proposition is regarded as the major +premiss, since it lays down a hypothesis, which is applied to fact in +the minor. + +§ 761. The Disjunctive Syllogism may be exactly defined as follows-- + + A complex syllogism, which has for its major premiss a disjunctive + proposition, either the antecedent or consequent of which is in the + minor premiss simply affirmed or denied. + +§ 762. Thus there are four types of disjunctive syllogism possible. + +_Constructive Moods._ + + (1) Either A is B or C is D. (2) Either A is B or C is D. + A is not B. C is not D. + .'. C is D. .'. A is B. + + Either death is annihilation or we are immortal. + Death is not annihilation. + .'. We are immortal. + + Either the water is shallow or the boys will be drowned. + The boys are not drowned. + .'. The water is shallow. + +_Destructive Moods_. + + (3) Either A is B or C is D. (4) Either A is B or C is D. + A is B. C is D. + .'. C is not D. .'. A is not B. + +§ 763. Of these four, however, it is only the constructive moods that +are formally conclusive. The validity of the two destructive moods is +contingent upon the kind of alternatives selected. If these are such +as necessarily to exclude one another, the conclusion will hold, but +not otherwise. They are of course mutually exclusive whenever they +embody the result of a correct logical division, as 'Triangles are +either equilateral, isosceles or scalene.' Here, if we affirm one of +the members, we are justified in denying the rest. When the major thus +contains the dividing members of a genus, it may more fitly be +symbolized under the formula, 'A is either B or C.' But as this admits +of being read in the shape, 'Either A is B or A is C,' we retain the +wider expression which includes it. Any knowledge, however, which we +may have of the fact that the alternatives selected in the major are +incompatible must come to us from material sources; unless indeed we +have confined ourselves to a pair of contradictory terms (A is either +B or not-B). There can be nothing in the form of the expression to +indicate the incompatibility of the alternatives, since the same form +is employed when the alternatives are palpably compatible. When, for +instance, we say, 'A successful student must be either talented or +industrious,' we do not at all mean to assert the positive +incompatibility of talent and industry in a successful student, but +only the incompatibility of their negatives--in other words, that, if +both are absent, no student can be successful. Similarly, when it is +said, 'Either your play is bad or your luck is abominable,' there is +nothing in the form of the expression to preclude our conceiving that +both may be the case. + +§ 764. There is no limit to the number of members in the disjunctive +major. But if there are only two alternatives, the conclusion will be +a simple proposition; if there are more than two, the conclusion will +itself be a disjunctive. Thus-- + + Either A is B or C is D or E is F or G is H. + E is not F. + .'. Either A is B or C is D or G is H. + +§ 765. The Canon of the Disjunctive Syllogism may be laid down as +follows-- + + To deny one member is to affirm the rest, either simply or + disjunctively; but from affirming any member nothing follows. + + + + +CHAPTER XXIV. + +_Of the Reduction of the Disjunctive Syllogism._ + +§ 766. We have seen that in the disjunctive syllogism the two +constructive moods alone are formally valid. The first of these, +namely, the denial of the antecedent, will in all cases give a simple +syllogism in the first figure; the second of them, namely, the denial +of the consequent, will in all cases give a simple syllogism in the +second figure. + + _Denial of Antecedent_ = Barbara. + + Either A is B or C is D. + A is not B. + .'.C is D + + is equal to + + If A is not B, C is D. + A is not B. + .'. C is D. + + is equal to + + All cases of A not being B are cases of C being D. + This is a case of A not being B. + .'. This is a case of C being D. + + + _Denial of Consequent_ = Camestres. + + Either A is E or C is D. + C is not D. + .'. A is B. + + is equal to + + If A is not B, C is D. + C is not D. + .'. A is B. + + is equal to + + All cases of A not being B are cases of C being D. + This is not a case of C being D. + .'. This is not a case of A being B. + +§ 767. The other moods of the first and second figures can be obtained +by varying the quality of the antecedent and consequent in the major +premiss and reducing the quantity of the minor. + +§ 768. The invalid destructive moods correspond with the two invalid +types of the partly conjunctive syllogism, and have the same fallacies +of simple syllogism underlying them. Affirmation of the antecedent of +a disjunctive is equivalent to the semi-conjunctive fallacy of denying +the antecedent, and therefore involves the ordinary syllogistic +fallacy of illicit process of the major. + +Affirmation of the consequent of a disjunctive is equivalent to the +same fallacy in the semi-conjunctive form, and therefore involves the +ordinary syllogistic fallacy of undistributed middle. + + _Affirmation of Antecedent_ = _Illicit Major_. + + Either A is B or C is D. + A is B. + .'. C is not D. + + is equal to + + If A is not B, C is D. + A is B. + .'. C is not D. + + is equal to + + All cases of A not being B are cases of C being D. + This is not a case of A not being B. + .'. This is not a case of C not being D. + + + _Affirmation of Consequent_ = _Undistributed Middle_. + + Either A is B or C is D. + C is D. + + is equal to + + If A is not B, C is D. + C is D. + + is equal to + + All cases of A not being B are cases of C being D. + This is a case of C being D. + +§ 769. So far as regards the consequent, the two species of complex +reasoning hitherto discussed are identical both in appearance and +reality. The apparent difference of procedure in the case of the +antecedent, namely, that it is affirmed in the partly conjunctive, but +denied in the disjunctive syllogism, is due merely to the fact that in +the disjunctive proposition the truth of the consequent is involved in +the falsity of the antecedent, so that the antecedent being +necessarily negative, to deny it in appearance is in reality to assert +it. + + + + +CHAPTER XXV. + +_The Disjunctive Syllogism regarded as an Immediate Inference_. + + +§ 770. If no stress be laid on the transition from disjunctive +hypothesis to fact, the disjunctive syllogism will run with the same +facility as its predecessor into the moulds of immediate inference. + +§ 771. + + _Denial of Antecedent_. Subalternation. + + Either A is B or C is D, Every case of A not being B + is a case of C being D. + .'. A not being B, C is D. .'. Some case of A not being B + is a case of C being D. + +§ 772. + + _Denial of Consequent_. Conversion by Contraposition + + Subalternation. + + Either A is B or C is D. All cases of A not being B + are cases of C being D. + .'. C not being D, A is B .'. All cases of C not being D are + cases of A being B. + .'. Some case of C not being D is + a case of A being B. + +§ 773. Similarly the two invalid types of disjunctive syllogism will +be found to coincide with fallacies of immediate inference. + +§ 774. + + _Affirmation of Antecedent_. Contraposition without + Conversion. + + Either A is B or C is D. All cases of A not being B are + cases of C being D. + .'. A being B, C is not D .'. All cases of A being B are + cases of C not being D. + +§ 775. The affirmation of the antecedent thus comes under the +formula-- + + All not-A is B, + .'. All A is not-B, + +a form of inference which cannot hold except where A and B are known +to be incompatible. Who, for instance, would assent to this?-- + + All non-boating men play cricket. + .'. All boating men are non-cricketers. + +§ 776. + + _Affirmation of Consequent_. Simple Conversion of A. + + Either A is B or C is D. All cases of A not being B are + cases of C being D. + .'.C being D, A is not B. .'. All cases of C being D are + cases of A not being B. + +§ 777. We may however argue in this way-- + + Conversion of A per accidens. + Either A is B or C is D. All cases of A not being B + are cases of C being D. + .'. C being D, A is sometimes B. .'. Some cases of C being D are + cases of A not being B. + + The men who pass this examination must have either talent or industry. + .'. Granting that they are industrious, they may be without talent. + + + + +CHAPTER XXVI. + +_Of the Mixed Form of Complex Syllogism_. + + +§ 778. Under this head are included all syllogisms in which a +conjunctive is combined with a disjunctive premiss. The best known +form is + + + +_The Dilemma_. + + +§ 779. The Dilemma may be defined as-- + + A complex syllogism, having for its major premiss a conjunctive + proposition with more than one antecedent, or more than one + consequent, or both, which (antecedent or consequent) the minor + premiss disjunctively affirms or denies. + +§ 780. It will facilitate the comprehension of the dilemma, if the +following three points are borne in mind-- + + (1) that the dilemma conforms to the canon of the partly conjunctive + syllogism, and therefore a valid conclusion can be obtained only by + affirming the antecedent or denying the consequent; + + (2) that the minor premiss must be disjunctive; + + (3) that if only the antecedent be more than one, the conclusion + will be a simple proposition; but if both antecedent and consequent + be more than one, the conclusion will itself be disjunctive. + +§ 781. The dilemma, it will be seen, differs from the partly +conjunctive syllogism chiefly in the fact of having a disjunctive +affirmation of the antecedent or denial of the consequent in the +minor, instead of a simple one. It is this which constitutes the +essence of the dilemma, and which determines its possible +varieties. For if only the antecedent or only the consequent be more +than one, we must, in order to obtain a disjunctive minor, affirm the +antecedent or deny the consequent respectively; whereas, if there be +more than one of both, it is open to us to take either course. This +gives us four types of dilemma. + +§ 782. + + (1). _Simple Constructive._ + + If A is B or C is D, E is F. + Either A is B or C is D. + .'. E is F. + + (2). _Simple Destructive._ + + If A is B, C is D and E is F. + Either C is not D or E is not F. + .'. A is not B. + + (3). _Complex Constructive._ + + If A is B, C is D; and if E is F, G is H. + Either A is B or E is F. + .'. Either C is D or G is H. + + (4). _Complex Destructive_. + + If A is B, C is D; and if E is F, G is H. + Either C is not D or G is not H. + .'. Either A is not B or E is not F. + +§ 783. + + (1). _Simple Constructive_. + + If she sinks or if she swims, there will be an end of her. + She must either sink or swim. + .'. There will be an end of her. + + (2). _Simple Destructive_. + + If I go to Town, I must pay for my ticket and pay my hotel bill. + Either I cannot pay for my ticket or I cannot pay my hotel bill. + .'. I cannot go to Town. + + (3). _Complex Constructive_. + + If I stay in this room, I shall be burnt to death, and if I jump + out of the window, I shall break my neck. + I must either stay in the room or jump out of the window. + .'. I must either be burnt to death or break my neck. + + (4). _Complex Destructive_. + + If he were clever, he would see his mistake; and + if he were candid, he would acknowledge it. + Either he does not see his mistake or he will not acknowledge it. + .'. Either he is not clever or he is not candid. + +§ 784. It must be noticed that the simple destructive dilemma would +not admit of a disjunctive consequent. If we said, + + If A is B, either C is D or E is F, + Either C is not D or E is not F, + +we should not be denying the consequent. For 'E is not F' would make +it true that C is D, and 'C is not D' would make it true that E is F; +so that in either case we should have one of the alternatives true, +which is just what the disjunctive form 'Either C is D or E is F' +insists upon. + +§ 785. In the case of the complex constructive dilemma the several +members, instead of being distributively assigned to one another, may +be connected together as a whole--thus-- + + If either A is B or E is F, either C is D or G is H. + Either A is B or E is F. + .'. Either C is D or G is H. + +In this shape the likeness of the dilemma to the partly conjunctive +syllogism is more immediately recognisable. The major premiss in this +shape is vaguer than in the former. For each antecedent has now a +disjunctive choice of consequents, instead of being limited to +one. This vagueness, however, does not affect the conclusion. For, so +long as the conclusion is established, it does not matter from which +members of the major its own members flow. + +§ 786. It must be carefully noticed that we cannot treat the complex +destructive dilemma in the same way. + + If either A is B or E is F, either C is D or G is H. + Either C is not D or G is not H. + +Since the consequents are no longer connected individually with the +antecedents, a disjunctive denial of them leaves it still possible for +the antecedent as a whole to be true. For 'C is not D' makes it true +that G is H, and 'G is not H' makes it true that C is D. In either +case then one is true, which is all that was demanded by the +consequent of the major. Hence the consequent has not really been +denied. + +§ 787. For the sake of simplicity we have limited the examples to the +case of two antecedents or consequents. But we may have as many of +either as we please, so as to have a Trilemma, a Tetralemma, and so +on. + +TRILEMMA. + + If A is B, C is D; and if E is F, G is H; and if K is L, M is N. + Either A is B or E is F or K is L. + .'. Either C is D or G is H or K is L. + +§ 788. Having seen what the true dilemma is, we shall now examine some +forms of reasoning which resemble dilemmas without being so. + +§ 789. This, for instance, is not a dilemma-- + + If A is B or if E is F, C is D. + But A is B and E is F. + .'. C is D. + + If he observes the sabbath or if he refuses to eat pork, he is a + Jew. + But he both observes the sabbath and refuses to eat pork. + .'. He is a Jew. + +What we have here is a combination of two partly conjunctive +syllogisms with the same conclusion, which would have been established +by either of them singly. The proof is redundant. + +§ 790. Neither is the following a dilemma-- + + If A is B, C is D and E is F. + Neither C is D nor E is F. + .'. A is not B. + + If this triangle is equilateral, its sides and its angles will be + equal. + But neither its sides nor its angles are equal. + .'. It is not equilateral. + +This is another combination of two conjunctive syllogisms, both +pointing to the same conclusion. The proof is again redundant. In this +case we have the consequent denied in both, whereas in the former we +had the antecedent affirmed. It is only for convenience that such +arguments as these are thrown into the form of a single +syllogism. Their real distinctness may be seen from the fact that we +here deny each proposition separately, thus making two independent +statements--C is not D and E is not F. But in the true instance of the +simple destructive dilemma, what we deny is not the truth of the two +propositions contained in the consequent, but their compatibility; in +other words we make a disjunctive denial. + +§ 791. Nor yet is the following a dilemma-- + + If A is B, either C is D or E is F. + Neither C is D nor E is F. + .'. A is not B. + + If the barometer falls there will be either wind or rain. + There is neither wind nor rain. + .'. The barometer has not fallen. + +What we have here is simply a conjunctive major with the consequent +denied in the minor. In the consequent of the major it is asserted +that the two propositions, 'C is D' and 'E is F' cannot both be false; +and in the minor this is denied by the assertion that they are both +false. + +§ 792. A dilemma is said to be rebutted or retorted, when another +dilemma is made out proving an opposite conclusion. If the dilemma be +a sound one, and its premisses true, this is of course impossible, and +any appearance of contradiction that may present itself on first sight +must vanish on inspection. The most usual mode of rebutting a dilemma +is by transposing and denying the consequents in the major-- + + If A is B, C is D; and if E is F, G is H. + Either A is B or E is F. + .'. Either C is D or G is H. + +The same rebutted-- + + If A is B, G is not H; and if E is F, C is not D. + Either A is B or E is F. + .'. Either G is not H or C is not D. + = Either C is not D or G is not H. + +§ 793. Under this form comes the dilemma addressed by the Athenian +mother to her son--'Do not enter public life: for, if you say what is +just, men will hate you; and, if you say what is unjust, the gods will +hate you' to which the following retort was made--'I ought to enter +public life: for, if 1 say what is just, the gods will love me; and, +if 1 say what is unjust, men will love me.' But the two conclusions +here are quite compatible. A man must, on the given premisses, be both +hated and loved, whatever course he takes. So far indeed are two +propositions of the form + + Either C is D or G is H, + and Either C is not D or G is not H, + +from being incompatible, that they express precisely the same thing +when contradictory alternatives have been selected, e.g.-- + + Either a triangle is equilateral or non-equilateral. + Either a triangle is non-equilateral or equilateral. + +§ 794. Equally illusory is the famous instance of rebutting a dilemma +contained in the story of Protagoras and Euathlus +(Aul. Gell. Noct. Alt. v. 10), Euathlus was a pupil of Protagoras in +rhetoric. He paid half the fee demanded by his preceptor before +receiving lessons, and agreed to pay the remainder when he won his +first case. But as he never proceeded to practise at the bar, it +became evident that he meant to bilk his tutor. Accordingly Protagoras +himself instituted a law-suit against him, and in the preliminary +proceedings before the jurors propounded to him the following +dilemma--'Most foolish young man, whatever be the issue of this suit, +you must pay me what I claim: for, if the verdict be given in your +favour, you are bound by our bargain; and if it be given against you, +you are bound by the decision of the jurors.' The pupil, however, was +equal to the occasion, and rebutted the dilemma as follows. 'Most +sapient master, whatever be the issue of this suit, I shall not pay +you what you claim: for, if the verdict be given in my favour, I am +absolved by the decision of the jurors; and, if it be given against +me, I am absolved by our bargain.' The jurors are said to have been so +puzzled by the conflicting plausibility of the arguments that they +adjourned the case till the Greek Kalends. It is evident, however, +that a grave injustice was thus done to Protagoras. His dilemma was +really invincible. In the counter-dilemma of Euathlus we are meant to +infer that Protagoras would actually lose his fee, instead of merely +getting it in one way rather than another. In either case he would +both get and lose his fee, in the sense of getting it on one plea, and +not getting it on another: but in neither case would he actually lose +it. + +§ 795. If a dilemma is correct in form, the conclusion of course +rigorously follows: but a material fallacy often underlies this form +of argument in the tacit assumption that the alternatives offered in +the minor constitute an exhaustive division. Thus the dilemma 'If pain +is severe, it will be brief; and if it last long it will be slight,' +&c., leaves out of sight the unfortunate fact that pain may both be +severe and of long continuance. Again the following dilemma-- + + If students are idle, examinations are unavailing; and, if + they are industrious, examinations are superfluous, + Students are either idle or industrious, + .'. Examinations are either unavailing or superfluous, + +is valid enough, so far as the form is concerned. But the person who +used it would doubtless mean to imply that students could be +exhaustively divided into the idle and the industrious. No deductive +conclusion can go further than its premisses; so that all that the +above conclusion can in strictness be taken to mean is that +examinations are unavailing, when students are idle, and superfluous, +when they are industrious--which is simply a reassertion as a matter +of fact of what was previously given as a pure hypothesis. + + + + +CHAPTER XXVII. + +_Of the Reduction of the Dilemma._ + + +§ 796. As the dilemma is only a peculiar variety of the partly +conjunctive syllogism, we should naturally expect to find it reducible +in the same way to the form of a simple syllogism. And such is in fact +the case. The constructive dilemma conforms to the first figure and +the destructive to the second. + + 1) _Simple Constructive Dilemma_. + + Barbara. + If A is B or if E is F, C is D. All cases of either A being B or E + being F are cases of C being D. + Either A is B or E is F. All actual cases are cases of either + A being B OP E being F. + .'. C is D. .'. All actual cases are cases of C + being D. + + (2) _Simple Destructive_. + + Camstres. + If A is B, C is D and E is F. All cases of A being B are cases of + C being D and E being F. + Either C is not D or E is not F. No actual cases are cases of C being + D and E being F. + .'. A is not B. .'. No actual cases are cases of A + being B. + + (3) _Complex Constructive_. + Barbara. + If A is B, C is D; and if E is F, All cases of either A being B or + G is H. being F are cases of either C being + D or G being H. + Either A is B or E is F. All actual cases are cases of either A + being B or E being F. + .'. Either C is D or G is H. .'. All actual cases are cases of either C + being D or G being H. + + (4) _Complex Destructive_. + + If A is B, C is D; and if E is F, All cases of A being B and E being F + G is H. are cases of C being D and G + being H. + Either C is not D Or G is No actual cases are cases of C being + not H D and G being H. + Either A is not B or E is No actual cases are cases of A being + not F. B and E being F. + +§ 797. There is nothing to prevent our having Darii, instead of +Barbara, in the constructive form, and Baroko, instead of Camestres, +in the destructive. As in the case of the partly conjunctive syllogism +the remaining moods of the first and second figure are obtained by +taking a negative proposition as the consequent of the major premiss, +e.g.-- + + _Simple Constructive_. Celarent or Ferio. + If A is B or if E is F, C is not D No cases of either A being B or E + being F are cases of C being D. + Either A is B or E is F. All (or some) actual cases are cases of + either A being B or E being F + .'. C is not D. .'. All (or some) actual cases are not + cases of C being D. + + + + +CHAPTER XXVIII. + +_Of the Dilemma regarded as an Immediate Inference._ + + +§ 798. Like the partly conjunctive syllogism, the dilemma can be +expressed under the forms of immediate inference. As before, the +conclusion in the constructive type resolves itself into the +subalternate of the major itself, and in the destructive type into the +subalternate of its contrapositive. The simple constructive dilemma, +for instance, may be read as follows-- + + If either A is B or E is F, C is D, + .'. Either A being B or E being F, C is D, + +which is equivalent to + + Every case of either A being B or E being F is a case of C being D. + .'. Some case of either A being B or E being F is a case of C being D. + +The descent here from 'every' to 'some' takes the place of the +transition from hypothesis to fact. + +§ 799. Again the complex destructive may be read thus-- + + If A is B, C is D; and if E is F, G is H, + .'. It not being true that C is D and G is H, it is not + true that A is B and E is F, + +which may be resolved into two steps of immediate inference, namely, +conversion by contraposition followed by subalternation-- + + All cases of A being B and E being F are cases of C being D and G + being H. + .'. Whatever is not a case of C being D and G being H is not a case + of A being B and E being F. + .'. Some case which is not one of C being D and G being H is not a + case of A being B and E being F. + + + + +CHAPTER XXIX. + +_Of Trains of Reasoning._ + + +§ 800. The formal logician is only concerned to examine whether the +conclusion duly follows from the premisses: he need not concern +himself with the truth or falsity of his data. But the premisses of +one syllogism may themselves be conclusions deduced from other +syllogisms, the premisses of which may in their turn have been +established by yet earlier syllogisms. When syllogisms are thus linked +together we have what is called a Train of Reasoning. + +§ 801. It is plain that all truths cannot be established by +reasoning. For the attempt to do so would involve us in an infinite +regress, wherein the number of syllogisms required would increase at +each step in a geometrical ratio. To establish the premisses of a +given syllogism we should require two preceding syllogisms; to +establish their premisses, four; at the next step backwards, eight; at +the next, sixteen; and so on ad infinitum. Thus the very possibility +of reasoning implies truths that are known to us prior to all +reasoning; and, however long a train of reasoning may be, we must +ultimately come to truths which are either self-evident or are taken +for granted. + +§ 802. Any syllogism which establishes one of the premisses of another +is called in reference to that other a Pro-syllogism, while a +syllogism which has for one of its premisses the conclusion of another +syllogism is called in reference to that other an Epi-syllogism. + + + +_The Epicheirema_. + + +§ 803. The name Epicheirema is given to a syllogism with one or both +of its premisses supported by a reason. Thus the following is a +double epicheirema-- + + All B is A, for it is E. + All C is B, for it is F. + .'. All C is A. + + All virtue is praiseworthy, for it promotes the general welfare. + Generosity is a virtue, for it prompts men to postpone self to others. + .'. Generosity is praiseworthy. + +§ 804. An epicheirema is said to be of the first or second order +according as the major or minor premiss is thus supported. The double +epicheirema is a combination of the two orders. + +§ 805. An epicheirema, it will be seen, consists of one syllogism +fully expressed together with one, or, it may be, two enthymemes (§ +557). In the above instance, if the reasoning which supports the +premisses were set forth at full length, we should have, in place of +the enthymemes, the two following pro-syllogisms-- + + (i) All E is A. + All B is E. + .'. All B is A. + + Whatever promotes the general welfare is praiseworthy. + Every virtue promotes the general welfare. + .'. Every virtue is praiseworthy. + + (2) All F is B. + All C is F. + .'. All C is B. + + Whatever prompts men to postpone self to others is a virtue. + Generosity prompts men to postpone self to others. + .'. Generosity is a virtue. + +§ 806. The enthymemes in the instance above given are both of the +first order, having the major premiss suppressed. But there is +nothing to prevent one or both of them from being of the second +order-- + + All B is A, because all F is. + All C is B, because all F is. + .'. All C is A. + + All Mahometans are fanatics, because all Monotheists are. + These men are Mahometans, because all Persians are. + .'. These men are fanatics. + +Here it is the minor premiss in each syllogism that is suppressed, +namely, + + (1) All Mahometans are Monotheists. + + (2) These men are Persians. + + + +_The Sorites_. + + +§ 807. The Sorites is the neatest and most compendious form that can +be assumed by a train of reasoning. + +§ 808. It is sometimes more appropriately called the chain-argument, +and map be defined as-- + + A train of reasoning, in which one premiss of each epi-syllogism is + supported by a pro-syllogism, the other being taken for granted. + +This is its inner essence. + +§ 809. In its outward form it may be described as--A series of +propositions, each of which has one term in common with that which +preceded it, while in the conclusion one of the terms in the last +proposition becomes either subject or predicate to one of the terms in +the first. + +§ 810. A sorites may be either-- + + (1) Progressive, + + or (2) Regressive. + +_Progressive Sorites_. + + All A is B. + All B is C. + All C is D. + All D is E. + .'. All A is E. + +_Regressive Sorites_. + + All D is E. + All C is D. + All B is C. + All A is B. + .'. All A is E. + +§ 811. The usual form is the progressive; so that the sorites is +commonly described as a series of propositions in which the predicate +of each becomes the subject of the next, while in the conclusion the +last predicate is affirmed or denied of the first subject. The +regressive form, however, exactly reverses these attributes; and would +require to be described as a series of propositions, in which the +subject of each becomes the predicate of the next, while in the +conclusion the first predicate is affirmed or denied of the last +subject. + +§ 812. The regressive sorites, it will be observed, consists of the +same propositions as the progressive one, only written in reverse +order. Why then, it may be asked, do we give a special name to it, +though we do not consider a syllogism different, if the minor premiss +happens to precede the major? It is because the sorites is not a mere +series of propositions, but a compressed train of reasoning; and the +two trains of reasoning may be resolved into their component +syllogisms in such a manner as to exhibit a real difference between +them. + +§ 813. The Progressive Sorites is a train of reasoning in which the +minor premiss of each epi-syllogism is supported by a pro-syllogism, +while the major is taken for granted. + +§ 814. The Regressive Sorites is a train of reasoning in which the +major premiss of each epi-syllogism is supported by a pro-syllogism, +while the minor is taken for granted. + + _Progressive Sorites_. + (i) All B is C. + All A is B. + .'. All A is C. + + (2) All C is D. + All A is C. + .'. All A is D. + + (3) All D is E. + All A is D. + .'. All A is E. + + _Regressive Sorites_. + (1) All D is E. + All C is D. + .'. All C is E. + + (2) All C is E. + All B is C. + .'. All B is E. + + (3) All B is E. + All A is B. + .'. All A is E. + +§ 815. Here is a concrete example of the two kinds of sorites, +resolved each into its component syllogisms-- + +_Progressive Sorites_. + + All Bideford men are Devonshire men. + All Devonshire men are Englishmen. + All Englishmen are Teutons. + All Teutons are Aryans. + .'. All Bideford men are Aryans. + + (1) All Devonshire men are Englishmen. + All Bideford men are Devonshire men. + .'. All Bideford men are Englishmen. + + (2) All Englishmen are Teutons. + All Bideford men are Englishmen. + .'. All Bideford men are Teutons. + + (3) All Teutons are Aryans. + All Bideford men are Teutons. + .'. All Bideford men are Aryans. + +_Regressive Sorites._ + + All Teutons are Aryans. + All Englishmen are Teutons. + All Devonshiremen are Englishmen. + All Bideford men are Devonshiremen. + .'. All Bideford men are Aryans. + + (1) All Teutons are Aryans. + All Englishmen are Teutons. + .'. All Englishmen are Aryans. + + (2) All Englishmen are Aryans. + All Devonshiremen are Englishmen. + .'. All Devonshiremen are Aryans. + + (3) All Devonshiremen are Aryans. + All Bideford men are Devonshiremen. + .'. All Bideford men are Aryans. + +§ 816. When expanded, the sorites is found to contain as many +syllogisms as there are propositions intermediate between the first +and the last. This is evident also on inspection by counting the +number of middle terms. + +§ 817. In expanding the progressive form we have to commence with the +second proposition of the sorites as the major premiss of the first +syllogism. In the progressive form the subject of the conclusion is +the same in all the syllogisms; in the regressive form the predicate +is the same. In both the same series of means, or middle terms, is +employed, the difference lying in the extremes that are compared with +one another through them. + +[Illustration] + +§ 818. It is apparent from the figure that in the progressive form we +work from within outwards, in the regressive form from without +inwards. In the former we first employ the term 'Devonshiremen' as a +mean to connect 'Bideford men' with 'Englishmen'; next we employ +'Englishmen' as a mean to connect the same subject 'Bideford men' with +the wider term 'Teutons'; and, lastly, we employ 'Teutons' as a mean +to connect the original subject 'Bideford men' with the ultimate +predicate 'Ayrans.' + +§ 819. Reversely, in the regressive form we first use 'Teutons' as a +mean whereby to bring 'Englishmen' under 'Aryans'; next we use +'Englishmen' as a mean whereby to bring 'Devonshiremen' under the dame +predicate 'Aryans'; and, lastly, we use 'Devonshiremen' as a mean +whereby to bring the ultimate subject 'Bideford men' under the +original predicate 'Aryans.' + +§ 820. A sorites may be either Regular or Irregular. + +§ 821. In the regular form the terms which connect each proposition in +the series with its predecessor, that is to say, the middle terms, +maintain a fixed relative position; so that, if the middle term be +subject in one, it will always be predicate in the other, and vice +versâ. In the irregular form this symmetrical arrangement is violated. + +§ 822. The syllogisms which compose a regular sorites, whether +progressive or regressive, will always be in the first figure. + +In the irregular sorites the syllogisms may fall into different +figures. + +§ 823. For the regular sorites the following rules may +be laid down. + + (1) Only one premiss can be particular, namely, the first, if the + sorites be progressive, the last, if it be regressive. + + (2) Only one premiss can be negative, namely, the last, if the + sorites be progressive, the first, if it be regressive. + +§ 824. _Proof of the Rules for the Regular Sorites_. + + (1) In the progressive sorites the proposition which stands first is + the only one which appears as a minor premiss in the expanded + form. Each of the others is used in its turn as a major. If any + proposition, therefore, but the first were particular, there would + be a particular major, which involves undistributed middle, if the + minor be affirmative, as it must be in the first figure. + + In the regressive sorites, if any proposition except the last were + particular, we should have a particular conclusion in the syllogism + in which it occurred as a premiss, and so a particular major in the + next syllogism, which again is inadmissible, as involving + undistributed middle. + + (2) In the progressive sorites, if any premiss before the last were + negative, we should have a negative conclusion in the syllogism in + which it occurs. This would necessitate a negative minor in the next + syllogism, which is inadmissible in the first figure, as involving + illicit process of the major. + + In the regressive sorites the proposition which stands first is the + only one which appears as a major premiss in the expanded form. + Each of the others is used in its turn as a minor. If any premiss, + therefore, but the first were negative, we should have a negative + minor in the first figure, which involves illicit process of the + major. + +§ 825. The rules above given do not apply to the irregular sorites, +except so far as that only one premiss can be particular and only one +negative, which follows from the general rules of syllogism. But there +is nothing to prevent any one premiss from being particular or any one +premiss from being negative, as the subjoined examples will show. Both +the instances chosen belong to the progressive order of sorites. + + (1) Barbara. + All B is A. + All C is B. + All C is A. + + All B is A. + All C is B. + Some C is D. + All D is E + .'. Some A is E + + [Illustration] + + (2) Disamis. + Some C is D. + All C is A. + Some A is D. + + (3) Darii. + All D is E + Some A is D. + Some A is E. + + (1) Barbara. + All B is C. + All A is B. + All A is C. + + All A is B. + All B is C. + No D is C. + All E is D. + .'. No A is E. + + [Illustration] + + (2) Cesare. + No D is C. + All A is C. + .'. No A is D. + + (3) Camestres. + All E is D. + No A is D. + .'. No A is E. + +§ 826. A chain argument may be composed consisting +of conjunctive instead of simple propositions. This is +subject to the same laws as the simple sorites, to which +it is immediately reducible. + + _Progressive._ _Regressive._ + If A is B, C is D. If E is F, G is H. + If C is D, E is F. If C is D, E is F. + If E is F, G is H. If A is B, C is D. + .'. If A is B, G is H. .'. If A is B, G is H. + + + + +CHAPTER XXX. + +_Of Fallacies_. + + +§ 827. After examining the conditions on which correct thoughts +depend, it is expedient to classify some of the most familiar forms of +error. It is by the treatment of the Fallacies that logic chiefly +vindicates its claim to be considered a practical rather than a +speculative science. To explain and give a name to fallacies is like +setting up so many sign-posts on the various turns which it is +possible to take off the road of truth. + +§ 828. By a fallacy is meant a piece of reasoning which appears to +establish a conclusion without really doing so. The term applies both +to the legitimate deduction of a conclusion from false premisses and +to the illegitimate deduction of a conclusion from any +premisses. There are errors incidental to conception and judgement, +which might well be brought under the name; but the fallacies with +which we shall concern ourselves are confined to errors connected with +inference. + +§ 829. When any inference leads to a false conclusion, the error may +have arisen either in the thought itself or in the signs by which the +thought is conveyed. The main sources of fallacy then are confined to +two-- + + (1) thought, + + (2) language. + +§ 830. This is the basis of Aristotle's division of fallacies, which +has not yet been superseded. Fallacies, according to him, are either +in the language or outside of it. Outside of language there is no +source of error but thought. For things themselves do not deceive us, +but error arises owing to a misinterpretation of things by the +mind. Thought, however, may err either in its form or in its +matter. The former is the case where there is some violation of the +laws of thought; the latter whenever thought disagrees with its +object. Hence we arrive at the important distinction between Formal +and Material fallacies, both of which, however, fall under the same +negative head of fallacies other than those of language. + + | In the language + | (in the signs of thought) + | + Fallacy -| |--In the Form. + |--Outside the language -| + | (in the thought itself) | + | + |--in the Matter. + +§ 831. There are then three heads to which fallacies may be +referred-namely, Formal Fallacies, Fallacies of Language, which are +commonly known as Fallacies of Ambiguity, and, lastly, Material +Fallacies. + +§ 832. Aristotle himself only goes so far as the first step in the +division of fallacies, being content to class them according as they +are in the language or outside of it. After that he proceeds at once +to enumerate the infimć species under each of the two main heads. We +shall presently imitate this procedure for reasons of expediency. For +the whole phraseology of the subject is derived from Aristotle's +treatise on Sophistical Refutations, and we must either keep to his +method or break away from tradition altogether. Sufficient confusion +has already arisen from retaining Aristotle's language while +neglecting his meaning. + +§ 833. Modern writers on logic do not approach fallacies from the same +point of view as Aristotle. Their object is to discover the most +fertile sources of error in solitary reasoning; his was to enumerate +the various tricks of refutation which could be employed by a sophist +in controversy. Aristotle's classification is an appendix to the Art +of Dialectic. + +§ 834. Another cause of confusion in this part of logic is the +identification of Aristotle's two-fold division of fallacies, commonly +known under the titles of In dictione and Extra diotionem, with the +division into Logical and Material, which is based on quite a +different principle. + +§ 835. Aristotle's division perhaps allows an undue importance to +language, in making that the principle of division, and so throwing +formal and material fallacies under a common head. Accordingly another +classification has been adopted, which concentrates attention from the +first upon the process of thought, which ought certainly to be of +primary importance in the eyes of the logician. This classification +is as follows. + +§ 836. Whenever in the course of our reasoning we are involved in +error, either the conclusion follows from the premisses or it does +not. If it does not, the fault must lie in the process of reasoning, +and we have then what is called a Logical Fallacy. If, on the other +hand, the conclusion does follow from the premisses, the fault must +lie in the premisses themselves, and we then have what is called a +Material Fallacy. Sometimes, however, the conclusion will appear to +follow from the premisses until the meaning of the terms is examined, +when it will be found that the appearance is deceptive owing to some +ambiguity in the language. Such fallacies as these are, strictly +speaking, non-logical, since the meaning of words is extraneous to the +science which deals with thought. But they are called +Semi-logical. Thus we arrive by a different road at the same three +heads as before, namely, (1) Formal or Purely Logical Fallacies, (2) +Semi-logical Fallacies or Fallacies of Ambiguity, (3) Material +Fallacies. + +§ 837. For the sake of distinctness we will place the two divisions +side by side, before we proceed to enumerate the infimae species. + + |--In the language + | (Fallacy of Ambiguity) + Fallacy-| + | |--In the Form. + |--Outside the language -| + | + |--In the Matter. + + |--Formal or purely logical. + |--Logical -| + Fallacy-| |--Semi-logical + | (Fallacy of Ambiguity). + |--Material + +838. Of one of these three heads, namely, formal fallacies, it is not +necessary to say much, as they have been amply treated of in the +preceding pages. A formal fallacy arises from the breach of any of the +general rules of syllogism. Consequently it would be a formal fallacy +to present as a syllogism anything which had more or less than two +premisses. Under the latter variety comes what is called 'a woman's +reason,' which asserts upon its own evidence something which requires +to be proved. Schoolboys also have been known to resort to this form +of argument--'You're a fool.' 'Why?' 'Because you are.' When the +conclusion thus merely reasserts one of the premisses, the other must +be either absent or irrelevant. If, on the other hand, there are more +than two premisses, either there is more than one syllogism or the +superfluous premiss is no premiss at all, but a proposition irrelevant +to the conclusion. + +839. The remaining rules of the syllogism are more able to be broken +than the first; so that the following scheme presents the varieties of +formal fallacy which are commonly enumerated-- + + |--Four Terms. + Formal Fallacy-|--Undistributed Middle. + |--Illicit Process. + |--Negative Premisses and Conclusion. + +§ 840. The Fallacy of Four Terms is a violation of the second of the +general rules of syllogism (§ 582). Here is a palpable instance of +it-- + + All men who write books are authors. + All educated men could write books. + .'. All educated men are authors. + +Here the middle term is altered in the minor premiss to the +destruction of the argument. The difference between the actual writing +of books and the power to write them is precisely the difference +between one who is an author and one who is not. + +§ 841. Since a syllogism consists of three terms, each of which is +used twice over, it would be possible to have an apparent syllogism +with as many as six terms in it. The true name for the fallacy +therefore is the Fallacy of More than Three Terms. But it is rare to +find an attempted syllogism which has more than four terms in it, just +as we are seldom tendered a line as an hexameter, which has more than +seven feet. + +§ 842. The Fallacies of Undistributed Middle and Illicit Process have +been treated of under §§ 585, 586. The heading 'Negative Premisses +and Conclusion' covers violations of the three general rules of +syllogism relating to negative premisses (§§ 590-593). Here is an +instance of the particular form of the fallacy which consists in the +attempt to extract an affirmative conclusion out of two negative +premisses-- + + All salmon are fish, for neither salmon nor fish belong to the class + mammalia. + +The accident of a conclusion being true often helps to conceal the +fact that it is illegitimately arrived at. The formal fallacies which +have just been enumerated find no place in Aristotle's division. The +reason is plain. His object was to enumerate the various modes in +which a sophist might snatch an apparent victory, whereas by openly +violating any of the laws of syllogism a disputant would be simply +courting defeat. + +§ 843. We now revert to Aristotle's classification of fallacies, or +rather of Modes of Refutation. We will take the species he enumerates +in their order, and notice how modern usage has departed from the +original meaning of the terms. Let it be borne in mind that, when the +deception was not in the language, Aristotle did not trouble himself +to determine whether it lay in the matter or in the form of thought. + +§ 844. The following scheme presents the Aristotelian classification +to the eye at a glance:-- + + | |--Equivocation. + | |--Amphiboly. + |--In the language -|--Composition. + | |--Division. + | |--Accent. + | |--Figure of Speech. + Modes of -| + Refutation. | |--Accident. + | |--A dicto secundum quid. + | |--Ignoratio Elenchi. + |--Outside the language -|--Consequent. + | |--Petitio Principii. + | |--Non causa pro causa. + | |--Many Questions. + +[Footnote: for "In the language": The Greek is [Greek: para ten lexin], +the exact meaning of which is; 'due to the statement.'] + +§ 845. The Fallacy of Equivocation [Greek: ňmonumía] consists in an +ambiguous use of any of the three terms of a syllogism. If, for +instance, anyone were to argue thus-- + + No human being is made of paper, + All pages are human beings, + .'. No pages are made of paper-- + +the conclusion would appear paradoxical, if the minor term were there +taken in a different sense from that which it bore in its proper +premiss. This therefore would be an instance of the fallacy of +Equivocal Minor. + +§ 846. For a glaring instance of the fallacy of Equivocal Major, we +may take the following-- + + No courageous creature flies, + The eagle is a courageous creature, + .'. The eagle does not fly-- + +the conclusion here becomes unsound only by the major being taken +ambiguously. + +§ 847. It is, however, to the middle term that an ambiguity most +frequently attaches. In this case the fallacy of equivocation assumes +the special name of the Fallacy of Ambiguous Middle. Take as an +instance the following-- + + Faith is a moral virtue. + To believe in the Book of Mormon is faith. + .'. To believe in the Book of Mormon is a moral virtue. + +Here the premisses singly might be granted; but the conclusion would +probably be felt to be unsatisfactory. Nor is the reason far to +seek. It is evident that belief in a book cannot be faith in any sense +in which that quality can rightly be pronounced to be a moral virtue. + +§ 848. The Fallacy of Amphiboly ([Greek: ámphibolía]) is an ambiguity +attaching to the construction of a proposition rather than to the +terms of which it is composed. One of Aristotle's examples is this-- + + [Greek: tň boúlesthai labeîn me toůs polemíous] + +which may be interpreted to mean either 'the fact of my wishing to +take the enemy,' or 'the fact of the enemies' wishing to take me.' The +classical languages are especially liable to this fallacy owing to the +oblique construction in which the accusative becomes subject to the +verb. Thus in Latin we have the oracle given to Pyrrhus (though of +course, if delivered at all, it must have been in Greek)-- + + Aio te, AEacida, Romanos vincere posse. + Pyrrhus the Romans shall, I say, subdue (Whately), + [Footnote: Cicero, De Divinatione, ii. § 116; Quintilian, + Inst. Orat. vii 9, § 6.] + +which Pyrrhus, as the story runs, interpreted to mean that he could +conquer the Romans, whereas the oracle subsequently explained to him +that the real meaning was that the Romans could conquer him. Similar +to this, as Shakspeare makes the Duke of York point out, is the +witch's prophecy in Henry VI (Second Part, Act i, sc. 4), + + The duke yet lives that Henry shall depose. + +An instance of amphiboly may be read on the walls of Windsor +Castle--Hoc fecit Wykeham. The king mas incensed with the bishop for +daring to record that he made the tower, but the latter adroitly +replied that what he really meant to indicate was that the tower was +the making of him. To the same head may be referred the famous +sentence--'I will wear no clothes to distinguish me from my Christian +brethren.' + +§ 849. The Fallacy of Composition [Greek: diaíresis] is likewise a +case of ambiguous construction. It consists, as expounded by +Aristotle, in taking words together which ought to be taken +separately, e.g. + + 'Is it possible for a man who is not writing to write?' + 'Of course it is.' + 'Then it is possible for a man to write without writing.' + +And again-- + + 'Can you carry this, that, and the other?' 'Yes.' + 'Then you can carry this, that, and the other,'-- + +a fallacy against which horses would protest, if they could. + +§ 850. It is doubtless this last example which has led to a convenient +misuse of the term 'fallacy of composition' among modern writers, by +whom it is defined to consist in arguing from the distributive to the +collective use of a term. + +§ 851. The Fallacy of Division ([Greek: diaíresis]), on the other hand, +consists in taking words separately which ought to be taken together, +e.g. + + [Greek: čgó s' ęteka doűlon ônt' čleúteron [Footnote: Evidently the + original of the line in Terence's _Andria_, 37,--feci ex servo + ut esses libertus mihi.], + +where the separation of [Greek: doűlon] from [Greek: ôntra] would lead +to an interpretation exactly contrary to what is intended. + +And again-- + + [Greek: pentékont' ŕndrôn čkatňn lípe dîos Ŕchilleús], + +where the separation of [Greek: ŕndrôn] from [Greek: čkatňn] leads to +a ludicrous error. + +Any reader whose youth may have been nourished on 'The Fairchild +Family' may possibly recollect a sentence which ran somewhat on this +wise--'Henry,' said Mr. Fairchild, 'is this true? Are you a thief and +a liar too?' But I am afraid he will miss the keen delight which can +be extracted at a certain age from turning the tables upon +Mr. Fairchild thus--Henry said, 'Mr. Fairchild, is this true? Are +_you_ a thief and a liar too?' + +§ 852. The fallacy of division has been accommodated by modern writers +to the meaning which they have assigned to the fallacy of +composition. So that by the 'fallacy of division' is now meant arguing +from the collective to the distributive use of a term. Further, it is +laid down that when the middle term is used distributively in the +major premiss and collectively in the minor, we have the fallacy of +composition; whereas, when the middle term is used collectively in the +major premiss and distributively in the minor, we have the fallacy of +division. Thus the first of the two examples appended would be +composition and the second division. + + (1) Two and three are odd and even. + Five is two and three. + .'. Five is odd and even. + + (2) The Germans are an intellectual people. + Hans and Fritz are Germans. + .'. They are intellectual people. + +§ 853. As the possibility of this sort of ambiguity is not confined to +the middle term, it seems desirable to add that when either the major +or minor term is used distributively in the premiss and collectively +in the conclusion, we have the fallacy of composition, and in the +converse case the fallacy of division. Here is an instance of the +latter kind in which the minor term is at fault-- + + Anything over a hundredweight is too heavy to lift. + These sacks (collectively) are over a hundredweight. + .'. These sacks (distributively) are too heavy to lift. + +§ 854. The ambiguity of the word 'all,' which has been before +commented upon (§ 119), is a great assistance in the English language +to the pair of fallacies just spoken of. + +§ 835. The Fallacy of Accent ([Greek: prosodía]) is neither more nor +less than a mistake in Greek accentuation. As an instance Aristotle +gives Iliad xxiii. 328, where the ancient copies of Homer made +nonsense of the words [Greek: tň mčn oú katapútetai ómbro] by writing +[Greek: oű] with the circumflex in place of [Greek: oú] with the acute +accent. [Footnote: This goes to show that the ancient Greeks did not +distinguish in pronunciation between the rough and smooth breathing +any more than their modern representatives.] Aristotle remarks that +the fallacy is one which cannot easily occur in verbal argument, but +rather in writing and poetry. + +§ 856. Modern writers explain the fallacy of accent to be the mistake +of laying the stress upon the wrong part of a sentence. Thus when the +country parson reads out, 'Thou shall not bear false witness +_against_ thy neighbour,' with a strong emphasis upon the word +'against,' his ignorant audience leap [sic] to the conclusion that it +is not amiss to tell lies provided they be in favour of one's +neighbour. + +§ 857. The Fallacy of Figure of Speech [Greek: tň schęma tęs léxeos] +results from any confusion of grammatical forms, as between the +different genders of nouns or the different voices of verbs, or their +use as transitive or intransitive, e.g. [Greek: úgiaínein] has the +same grammatical form as [Greek: témnein] or [Greek: oěkodomeîn], but +the former is intransitive, while the latter are transitive. A sophism +of this kind is put into the mouth of Socrates by Aristophanes in the +Clouds (670-80). The philosopher is there represented as arguing that +[Greek: kápdopos] must be masculine because [Greek: Kleónumos] is. On +the surface this is connected with language, but it is essentially a +fallacy of false analogy. + +§ 858. To this head may be referred what is known as the Fallacy of +Paronymous Terms. This is a species of equivocation which consists in +slipping from the use of one part of speech to that of another, which +is derived from the same source, but has a different meaning. Thus +this fallacy would be committed if, starting from the fact that there +is a certain probability that a hand at whist will consist of thirteen +trumps, one were to proceed to argue that it was probable, or that he +had proved it. + +§ 859. We turn now to the tricks of refutation which lie outside the +language, whether the deception be due to the assumption of a false +premiss or to some unsoundness in the reasoning. + +§ 860. The first on the list is the Fallacy of Accident ([Greek: tň +sumbebekós]). This fallacy consists in confounding an essential with +an accidental difference, which is not allowable, since many things +are the same in essence, while they differ in accidents. Here is the +sort of example that Aristotle gives-- + + 'Is Plato different from Socrates ?' 'Yes.' 'Is Socrates a man ?' + 'Yes.' 'Then Plato is different from man.' + +To this we answer--No: the difference of accidents between Plato and +Socrates does not go so deep as to affect the underlying essence. To +put the thing more plainly, the fallacy lies in assuming that whatever +is different from a given subject must be different from it in all +respects, so that it is impossible for them to have a common +predicate. Here Socrates and Plato, though different from one another, +are not so different but that they have the common predicate 'man.' +The attempt to prove that they have not involves an illicit process of +the major. + +§ 861. The next fallacy suffers from the want of a convenient name. It +is called by Aristotle [Greek: tň áplos tóde ę pę légestai kaě mč +kupíos] or, more briefly, [Greek: tň áplôs ę mé], or [Greek: tň pę kaí +áplôs], and by the Latin writers 'Fallacia a dicto secundum quid ad +dictum simpliciter.' It consists in taking what is said in a +particular respect as though it held true without any restriction, +e.g., that because the nonexistent ([Greek: tň mč ôn]) is a matter of +opinion, that therefore the non-existent is, or again that because the +existent ([Greek: tň ôn]) is not a man, that therefore the existent is +not. Or again, if an Indian, who as a whole is black, has white teeth, +we should be committing this species of fallacy in declaring him to be +both white and not-white. For he is only white in a certain respect +([Greek: pę]), but not absolutely ([Greek: ŕplôs]). More +difficulty, says Aristotle, may arise when opposite qualities exist in +a thing in about an equal degree. When, for instance, a thing is half +white and half black, are we to say that it is white or black? This +question the philosopher propounds, but does not answer. The force of +it lies in the implied attack on the Law of Contradiction. It would +seem in such a case that a thing may be both white and not-white at +the same time. The fact is--so subtle are the ambiguities of +language--that even such a question as 'Is a thing white or +not-white?' straightforward, as it seems, is not really a fair one. We +are entitled sometimes to take the bull by the horns, and answer with +the adventurous interlocutor in one of Plato's dialogues--'Both and +neither.' It may be both in a certain respect, and yet neither +absolutely. + +§ 862. The same sort of difficulties attach to the Law of Excluded +Middle, and may be met in the same way. It might, for instance, be +urged that it could not be said with truth of the statue seen by +Nebuchadnezzar in his dream either that it was made of gold or that it +was not made of gold: but the apparent plausibility of the objection +would be due merely to the ambiguity of language. It is not true, on +the one hand, that it was made of gold (in the sense of being composed +entirely of that metal); and it is not true, on the other, that it was +not made of gold (in the sense of no gold at all entering into its +composition). But let the ambiguous proposition be split up into its +two meanings, and the stringency of the Law of Excluded Middle will at +once appear-- + + (1) It must either have been composed entirely of gold or not. + + (2) Either gold must have entered into its composition or not. + +§ 863. By some writers this fallacy is treated as the converse of the +last, the fallacy of accident being assimilated to it under the title +of the 'Fallacia a dicto simpliciter ad dictum secundum quid.' In this +sense the two fallacies may be defined thus. + +The Fallacy of Accident consists in assuming that what holds true as a +general rule will hold true under some special circumstances which may +entirely alter the case. The Converse Fallacy of Accident consists in +assuming that what holds true under some special circumstances must +hold true as a general rule. + +The man who, acting on the assumption that alcohol is a poison, +refuses to take it when he is ordered to do so by the doctor, is +guilty of the fallacy of accident; the man who, having had it +prescribed for him when he was ill, continues to take it morning, +noon, and night, commits the converse fallacy. + +§ 864. There ought to be added a third head to cover the fallacy of +arguing from one special case to another. + +§ 865. The next fallacy is Ignoratio Elenchi [Greek: člégchou +âgnoia]. This fallacy arises when by reasoning valid in itself one +establishes a conclusion other than what is required to upset the +adversary's assertion. It is due to an inadequate conception of the +true nature of refutation. Aristotle therefore is at the pains to +define refutation at full length, thus-- + +'A refutation [Greek: ęlegchos] is the denial of one and the same--not +name, but thing, and by means, not of a synonymous term, but of the +same term, as a necessary consequence from the data, without +assumption of the point originally at issue, in the same respect, and +in the same relation, and in the same way, and at the same time.' + +The ELENCHUS then is the exact contradictory of the opponent's +assertion under the terms of the law of contradiction. To establish by +a syllogism, or series of syllogisms, any other proposition, however +slightly different, is to commit this fallacy. Even if the substance +of the contradiction be established, it is not enough unless the +identical words of the opponent are employed in the +contradictory. Thus if his thesis asserts or denies something about +[Greek: lópion], it is not enough for you to prove the contradictory +with regard to [Greek: ěmátion]. There will be need of a further +question and answer to identify the two, though they are admittedly +synonymous. Such was the rigour with which the rules of the game of +dialectic were enforced among the Greeks! + +§ 866. Under the head of Ignoratio Elenchi it has become usual to +speak of various forme of argument which have been labelled by the +Latin writers under such names as 'argumentum ad hominem,' 'ad +populum,' 'ad verecundiam,' 'ad ignorantiam,' 'ad baculum'--all of +them opposed to the 'argumentum ad rem' or 'ad judicium.' + +§ 867. By the 'argumentum ad hominem' was perhaps meant a piece of +reasoning which availed to silence a particular person, without +touching the truth of the question. Thus a quotation from Scripture +is sufficient to stop the mouth of a believer in the inspiration of +the Bible. Hume's Essay on Miracles is a noteworthy instance of the +'argumentum ad hominem' in this sense of the term. He insists strongly +on the evidence for certain miracles which he knew that the prejudices +of his hearers would prevent their ever accepting, and then asks +triumphantly if these miracles, which are declared to have taken place +in an enlightened age in the full glare of publicity, are palpably +imposture, what credence can be attached to accounts of extraordinary +occurrences of remote antiquity, and connected with an obscure corner +of the globe? The 'argumentum ad judicium' would take miracles as a +whole, and endeavour to sift the amount of truth which may lie in the +accounts we have of them in every age. [Footnote: On this subject see +the author's _Attempts at Truth_ (Trubner & Co.), pp. 46-59.] + +§ 868. In ordinary discourse at the present day the term 'argumentum +ad hominem' is used for the form of irrelevancy which consists in +attacking the character of the opponent instead of combating his +arguments, as illustrated in the well-known instructions to a +barrister--'No case: abuse the plaintiff's attorney.' + +§ 869. The 'argumentum ad populum' consists in an appeal to the +passions of one's audience. An appeal to passion, or to give it a less +question-begging name, to feeling, is not necessarily amiss. The heart +of man is the instrument upon which the rhetorician plays, and he has +to answer for the harmony or the discord that comes of his +performance. + +§ 870. The 'argumentum ad verecundiam' is an appeal to the feeling of +reverence or shame. It is an argument much used by the old to the +young and by Conservatives to Radicals. + +§ 871. The 'argumentum ad ignorantiam' consists simply in trading on +the ignorance of the person addressed, so that it covers any kind of +fallacy that is likely to prove effective with the hearer. + +§ 872. The 'argumentum ad baculum' is unquestionably a form of +irrelevancy. To knock a man down when he differs from you in opinion +may prove your strength, but hardly your logic. + +A sub-variety of this form of irrelevancy was exhibited lately at a +socialist lecture in Oxford, at which an undergraduate, unable or +unwilling to meet the arguments of the speaker, uncorked a bottle, +which had the effect of instantaneously dispersing the audience. This +might be set down as the 'argumentum ad nasum.' + +§ 873. We now come to the Fallacy of the Consequent, a term which has +been more hopelessly abused than any. What Aristotle meant by it was +simply the assertion of the consequent in a conjunctive proposition, +which amounts to the same thing as the simple conversion of A (§ 489), +and is a fallacy of distribution. Aristotle's example is this-- + + If it has rained, the ground is wet. + .'. If the ground is wet, it has rained. + +This fallacy, he tells us, is often employed in rhetoric in dealing +with presumptive evidence. Thus a speaker, wanting to prove that a man +is an adulterer, will argue that he is a showy dresser, and has been +seen about at nights. Both these things however may be the case, and +yet the charge not be true. + +§ 874. The Fallacy of Petitio or Assumptio Principii [Greek: tň čn +ŕrchę aěteîstai or lambánein] to which we now come, consists in an +unfair assumption of the point at issue. The word [Greek: aěteîstai], +in Aristotle's name for it points to the Greek method of dialectic by +means of question and answer. This fact is rather disguised by the +mysterious phrase 'begging the question.' The fallacy would be +committed when you asked your opponent to grant, overtly or covertly, +the very proposition originally propounded for discussion. + +§ 875. As the question of the precise nature of this fallacy is of +some importance we will take the words of Aristotle himself +(Top. viii. 13. §§ 2, 3). 'People seem to beg the question in five +ways. First and most glaringly, when one takes for granted the very +thing that has to be proved. This by itself does not readily escape +detection, but in the case of "synonyms," that is, where the name and +the definition have the same meaning, it does so more +easily. [Footnote: Some light is thrown upon this obscure passage by a +comparison with Cat. I. § 3, where 'synonym' is defined. To take the +word here in its later and modern sense affords an easy +interpretation, which is countenanced by Alexander Aphrodisiensis, but +it is flat against the usage of Aristotle, who elsewhere gives the +name 'synonym,' not to two names for the same thing, but to two things +going under the same name. See Trendelenberg on the passage.] + +Secondly, when one assumes universally that which has to be proved in +particular, as, if a man undertaking to prove that there is one +science of contraries, were to assume that there is one science of +opposites generally. For he seems to be taking for granted along with +several other things what he ought to have proved by itself. + +Thirdly, when one assumes the particulars where the universal has to +be proved; for in so doing a man is taking for granted separately what +he was bound to prove along with several other things. Again, when +one assumes the question at issue by splitting it up, for instance, +if, when the point to be proved is that the art of medicine deals with +health and disease, one were to take each by itself for granted. + +Lastly, if one were to take for granted one of a pair of necessary +consequences, as that the side is incommensurable with the diagonal, +when it is required to prove that the diagonal is incommensurable with +the side.' + +§ 876. To sum up briefly, we may beg the question in five ways-- + + (1) By simply asking the opponent to grant the point which requires + to be proved; + + (2) by asking him to grant some more general truth which involves + it; + + (3) by asking him to grant the particular truths which it involves; + + (4) by asking him to grant the component parts of it in detail; + + (5) by asking him to grant a necessary consequence of it. + +§ 877. The first of these five ways, namely, that of begging the +question straight off, lands us in the formal fallacy already spoken +of (§ 838), which violates the first of the general rules of +syllogism, inasmuch as a conclusion is derived from a single premiss, +to wit, itself. + +§ 878. The second, strange to say, gives us a sound syllogism in +Barbara, a fact which countenances the blasphemers of the syllogism in +the charge they bring against it of containing in itself a petitio +principii. Certainly Aristotle's expression might have been more +guarded. But it is clear that his quarrel is with the matter, not with +the form in such an argument. The fallacy consists in assuming a +proposition which the opponent would be entitled to deny. Elsewhere +Aristotle tells us that the fallacy arises when a truth not evident by +its own light is taken to be so. [Footnote: [Greek: Ôtan tň mč dí +aůtoű gnostňn dí aůtoű tis čpicheiraę deiknúnai, tót' aěteîtai tň čx +ŕrchęs.]. Anal. Pr. II. 16. § I ad fin.] + +§ 879. The third gives us an inductio per enumerationem simplicem, a +mode of argument which would of course be unfair as against an +opponent who was denying the universal. + +§ 880. The fourth is a more prolix form of the first. + +§ 881. The fifth rests on Immediate Inference by Relation (§ 534). + +§ 882. Under the head of petitio principii comes the fallacy of +Arguing in a Circle, which is incidental to a train of reasoning. In +its most compressed form it may be represented thus-- + + (1) B is A. + C is B. + .'. C is A. + + (2) C is A. + B is C. + .'. B is A. + +§ 883. The Fallacy of Non causa pro causa ([Greek: tň mč aîtion] or +[Greek: aîtoin]) is another, the name of which has led to a complete +misinterpretation. It consists in importing a contradiction into the +discussion, and then fathering it on the position controverted. Such +arguments, says Aristotle, often impose upon the users of them +themselves. The instance he gives is too recondite to be of general +interest. + +§ 884. Lastly, the Fallacy of Many Questions ([Greek: tň tŕ déo +črotémata ęn poieîn]) is a deceptive form of interrogation, when a +single answer is demanded to what is not really a single question. In +dialectical discussions the respondent was limited to a simple 'yes' +or 'no'; and in this fallacy the question is so framed as that either +answer would seem to imply the acceptance of a proposition which would +be repudiated. The old stock instance will do as well as +another--'Come now, sir, answer "yes" or "no." Have you left off +beating your mother yet?' Either answer leads to an apparent +admission of impiety. + +A late Senior Proctor once enraged a man at a fair with this form of +fallacy. The man was exhibiting a blue horse; and the distinguished +stranger asked him--'With what did you paint your horse?' + + + + +EXERCISES. + + +These exercises should be supplemented by direct questions upon the +text, which it is easy for the student or the teacher to supply for +himself. + + +PART I. + + +CHAPTER I. + +Classify the following words according as they are categorematic, +syncategorematic or acategorematic;-- + + come peradventure why + through inordinately pshaw + therefore circumspect puss + grand inasmuch stop + touch sameness back + cage disconsolate candle. + + +CHAPTER II. + +Classify the following things according as they are substances, +qualities or relations;-- + + God likeness weight + blueness grass imposition + ocean introduction thinness + man air spirit + Socrates raillery heat + mortality plum fire. + + +CHAPTER III. + +1. Give six instances each of-attribute, abstract, singular, +privative, equivocal and relative terms. + +2. Select from the following list of words such as are terms, and +state whether they are (1) abstract or concrete, (2) singular or +common, (3) univocal or equivocal:-- + + van table however + enter decidedly tiresome + very butt Solomon + infection bluff Czar + short although Caesarism + distance elderly Nihilist. + +3. Which of the following words are abstract terms?-- + + quadruped event through + hate desirability thorough + fact expressly thoroughness + faction wish light + inconvenient will garden + inconvenience volition grind. + +4. Refer the following terms to their proper place under each of the +divisions in the scheme:-- + + horse husband London + free lump empty + liberty rational capital + impotent reason Capitol + impetuosity irrationality grave + impulsive double calf. + +5. Give six instances each of proper names and designations. + +6. Give six instances each of connotative and non-connotative terms. + +7. Give the extension and intension of-- + + sermon animal sky + clock square gold + sport fish element + bird student fluid + art river line + gas servant language + + +CHAPTER IV. + +Arrange the following terms in order of extension--carnivorous, thing, +matter, mammal, organism, vertebrate, cat, substance, animal. + + * * * * * + + +PART II. + + +CHAPTER I. + +Give a name to each of the following sentences:-- + + (1) Oh, that I had wings like a dove! + + (2) The more, the merrier. + + (3) Come rest in this bosom, my own stricken deer. + + (4) Is there balm in Gilead? + + (5) Hearts may be trumps. + + +CHAPTER II. + +Analyse the following propositions into subject, copula and +predicate:-- + + (1) He being dead yet speaketh. + + (2) There are foolish politicians. + + (3) Little does he care. + + (4) There is a land of pure delight. + + (5) All's well that ends well. + + (6) Sweet is the breath of morn. + + (7) Now it came to pass that the beggar died. + + (8) Who runs may read. + + (9) Great is Diana of the Ephesians. + + (10) Such things are. + + (11) Not more than others I deserve. + + (12) The day will come when Ilium's towers shall perish. + + +CHAPTER III. + +1. Express in logical form, affixing the proper symbol:-- + + (1) Some swans are not white. + + (2) All things are possible to them that believe. + + (3) No politicians are unprincipled. + + (4) Some stones float on water. + + (5) The snow has melted. + + (6) Eggs are edible. + + (7) All kings are not wise. + + (8) Moths are not butterflies. + + (9) Some men are born great. + + (10) Not all who are called are chosen. + + (11) It is not good for man to be alone. + + (12) Men of talents have been known to fail in life. + + (13) 'Tis none but a madman would throw about fire. + + (14) Every bullet does not kill. + + (15) Amongst Unionists are Whigs. + + (16) Not all truths are to be told. + + (17) Not all your efforts can save him. + + (18) The whale is a mammal. + + (19) Cotton is grown in Cyprus. + + (20) An honest man's the noblest work of God. + + (21) No news is good news. + + (22) No friends are like old friends. + + (23) Only the ignorant affect to despise knowledge. + + (24) All that trust in Him shall not be ashamed. + + (25) All is not gold that glitters. + + (26) The sun shines upon the evil and upon the good. + + (27) Not to go on is to go back. + + (28) The king, minister, and general are a pretty trio. + + (29) Amongst dogs are hounds. + + (30) A fool is not always wrong. + + (31) Alexander was magnanimous. + + (32) Food is necessary to life. + + (33) There are three things to be considered, + + (34) By penitence the Eternal's wrath's appeased. + + (35) Money is the miser's end. + + (36) Few men succeed in life. + + (37) All is lost, save honour. + + (38) It is mean to hit a man when he is down. + + (39) Nothing but coolness could have saved him. + + (40) Books are generally useful. + + (41) He envies others' virtue who has none himself. + + (42) Thankless are all such offices. + + (43) Only doctors understand this subject. + + (44) All her guesses but two were correct. + + (45) All the men were twelve. + + (46) Gossip is seldom charitable. + +2. Give six examples of indefinite propositions, and then quantify +them according to their matter. + +3. Compose three propositions of each of the following kinds:-- + + (1) with common terms for subjects; + + (2) with abstract terms for subjects; + + (3) with singular terms for predicates; + + (4) with collective terms for predicates; + + (5) with attributives in their subjects; + + (6) with abstract terms for predicates. + + +CHAPTER IV. + +1. Point out what terms are distributed or undistributed in the +following propositions:-- + + (1) The Chinese are industrious. + + (2) The angle in a semi-circle is a right angle. + + (3) Not one of the crew survived. + + (4) The weather is sometimes not propitious. + +The same exercise may be performed upon any of the propositions in the +preceding list. + +2. Prove that in a negative proposition the predicate must be +distributed. + + +CHAPTER V. + +Affix its proper symbol to each of the following propositions:-- + + (1) No lover he who is not always fond. + + (2) There are Irishmen and Irishmen. + + (3) Men only disagree, + Of creatures rational. + + (4) Some wise men are poor. + + (5) No Popes are some fallible beings. + + (6) Some step-mothers are not unjust. + + (7) The most original of the Roman poets was Lucretius. + + (8) Some of the immediate inferences are all the forms of + conversion. + + +CHAPTER VI. + +1. Give six examples of terms standing one to another as genus to +species. + +2. To which of the heads of predicables would you refer the following +statements? And why? + + (1) A circle is the largest space that can be contained by one line. + + (2) All the angles of a square are right angles. + + (3) Man alone among animals possesses the faculty of laughter. + + (4) Some fungi are poisonous. + + (5) Most natives of Africa are negroes. + + (6) All democracies are governments. + + (7) Queen Anne is dead. + + +CHAPTER VII. + +1. Define the following terms-- + + Sun inn-keeper tea-pot + hope anger virtue + bread diplomacy milk + carpet man death + sincerity telescope mountain + poverty Senate novel. + +2. Define the following terms as used in Political Economy-- + + Commodity barter value + wealth land price + money labour rent + interest capital wages + credit demand profits. + +3. Criticise the following as definitions-- + + (1) Noon is the time when the shadows of bodies are shortest. + + (2) Grammar is the science of language. + + (3) Grammar is a branch of philology. + + (4) Grammar is the art of speaking and writing a language with + propriety. + + (5) Virtue is acting virtuously. + + (6) Virtue is that line of conduct which tends to produce happiness. + + (7) A dog is an animal of the canine species. + + (8) Logic is the art of reasoning. + + (9) Logic is the science of the investigation of truth by means of + evidence. + + (10) Music is an expensive noise. + + (11) The sun is the centre of the solar system. + + (12) The sun is the brightest of those heavenly bodies that move + round the earth. + + (13) Rust is the red desquamation of old iron. + + (14) Caviare is a kind of food. + + (15) Life is the opposite of death. + + (16) Man is a featherless biped. + + (17) Man is a rational biped. + + (18) A gentleman is a person who has no visible means of + subsistence. + + (19) Fame is a fancied life in others' breath. + + (20) A fault is a quality productive of evil or inconvenience. + + (21) An oligarchy is the supremacy of the rich in a state. + + (22) A citizen is one who is qualified to exercise deliberative and + judicial functions. + + (23) Length is that dimension of a solid which would be measured by + the longest line. + + (24) An eccentricity is a peculiar idiosyncrasy. + + (25) Deliberation is that species of investigation which is + concerned with matters of action. + + (26) Memory is that which helps us to forget. + + (27) Politeness is the oil that lubricates the wheels of society. + + (28) An acute-angled triangle is one which has an acute angle. + + (29) A cause is that without which something would not be. + + (30) A cause is the invariable antecedent of a phenomenon. + + (31) Necessity is the mother of invention. + + (32) Peace is the absence of war. + + (33) A net is a collection of holes strung together. + + (34) Prudence is the ballast of the moral vessel. + + (35) A circle is a plane figure contained by one line. + + (36) Superstition is a tendency to look for constancy where + constancy is not to be expected. + + (37) Bread is the staff of life. + + (38) An attributive is a term which cannot stand as a subject. + + (39) Life is bottled sunshine. + + (40) Eloquence is the power of influencing the feelings by speech or + writing. + + (41) A tombstone is a monument erected over a grave in memory of the + dead. + + (42) Whiteness is the property or power of exciting the sensation of + white. + + (43) Figure is the limit of a solid. + + (44) An archdeacon is one who exercises archidiaconal functions. + + (45) Humour is thinking in jest while feeling in earnest. + + +CHAPTER VIII. + +1. Divide the following terms-- + + Soldier end book + church good oration + apple cause school + ship government letter + vehicle science verse. + +2. Divide the following terms as used in Political Economy-- + + Requisites of production, labour, consumption, stock, wealth, + capital. + +3. Criticise the following as divisions-- + + (1) Great Britain into England, Scotland, Wales, and Ireland. + + (2) Pictures into sacred, historical, landscape, and mythological. + + (3) Vertebrate animals into quadrupeds, birds, fishes, and reptiles. + + (4) Plant into stem, root, and branches. + + (5) Ship into frigate, brig, schooner, and merchant-man. + + (6) Books into octavo, quarto, green, and blue. + + (7) Figure into curvilinear and rectilinear. + + (8) Ends into those which are ends only, means and ends, and means + only. + + (9) Church into Gothic, episcopal, high, and low. + + (10) Sciences into physical, moral, metaphysical, and medical. + + (11) Library into public and private. + + (12) Horses into race-horses, hunters, hacks, thoroughbreds, ponies, + and mules. + +4. Define and divide-- + + Meat, money, virtue, triangle; + +and give, as far as possible, a property and accident of each. + + +PART III. + + +CHAPTERS I-III. + +1. What kind of influence have we here? + + The author of the Iliad was unacquainted with writing. + Homer was the author of the Iliad. + .'. Homer was unacquainted with writing. + +2. Give the logical opposites of the following propositions-- + + (1) Knowledge is never useless. + + (2) All Europeans are civilised. + + (3) Some monks are not illiterate. + + (4) Happy is the man that findeth wisdom. + + (5) No material substances are devoid of weight. + + (6) Every mistake is not culpable. + + (7) Some Irishmen are phlegmatic. + +3. Granting the truth of the following propositions, what other +propositions can be inferred by opposition to be true or false? + + + (1) Men of science are often mistaken. + + (2) He can't be wrong, whose life is in the right. + + (3) Sir Walter Scott was the author of Waverley. + + (4) The soul that sinneth it shall die. + + (5) All women are not vain. + +4. Granting the falsity of the following propositions, what other +propositions can be inferred by opposition to be true or false?-- + + (1) Some men are not mortal. + + (2) Air has no weight. + + (3) All actors are improper characters. + + (4) None but dead languages are worth studying. + + (5) Some elements are compound. + + +CHAPTER IV. + +1. Give, as far as possible, the logical converse of each of the +following propositions-- + + (1) Energy commands success. + + (2) Mortals cannot be happy. + + (3) There are mistakes which are criminal. + + (4) All's well that ends well. + + (5) Envious men are disliked. + + (6) A term is a kind of word or collection of words. + + (7) Some Frenchmen are not vivacious. + + (8) All things in heaven and earth were hateful to him. + + (9) The square of three is nine. + + (10) All cannot receive this saying. + + (11) P struck Q. + + (12) Amas. + +2. 'More things may be contained in my philosophy than exist in heaven +or earth: but the converse proposition is by no means true.' Is the +term converse here used in its logical meaning? + + +CHAPTER V. + +Permute the following propositions-- + + (1) All just acts are expedient. + + (2) No display of passion is politic. + + (3) Some clever people are not prudent. + + (4) Some philosophers have been slaves. + +The same exercise may be performed upon any of the propositions in the +preceding lists. + + +CHAPTER VI. + +1. Give the converse by negation of-- + + (1) All women are lovely. + + (2) Some statesmen are not practical. + + (3) All lawyers are honest. + + (4) All doctors are skilful. + + (5) Some men are not rational. + +2. Give the contrapositive of-- + + (1) All solid substances are material. + + (2) All the men who do not row play cricket. + + (3) All impeccable beings are other than human, + + (4) Some prejudiced persons are not dishonest. + +3. Prove indirectly the truth of the contrapositive of 'All A is B.' + +4. Criticise the following as immediate inferences-- + + (1) All wise men are modest. + .'. No immodest men are wise. + + (2) Some German students are not industrious. + .'. Some industrious students are not Germans. + + (3) Absolute difference excludes all likeness. + .'. Any likeness is a proof of sameness. + + (4) None but the brave deserve the fair. + .'. All brave men deserve the fair. + + (5) All discontented men are unhappy. + .'. No contented men are unhappy. + + (6) Books being a source of instruction, our knowledge must come + from our libraries. + + (7) All Jews are Semitic. + .'. Some non-Semitic people are not Jews. + +5. Show by what kind of inference each of the subjoined propositions +follows from + + All discontented men are unhappy. + + (1) All happy men are contented. + + (2) Some discontented men are unhappy. + + (3) Some contented men are happy. + + (4) Some unhappy men are not contented. + + (5) No discontented men are happy. + + (6) Some happy men are contented. + + (7) Some contented men are not unhappy. + + (8) Some unhappy men are discontented. + + (9) No happy men are discontented. + + (10) Some discontented men are not happy. + + (11) Some happy men are not discontented. + + (12) None but unhappy men are discontented. + +From how many of these propositions can the original one be derived? +And why not from all? + + +CHAPTER VII. + +What kind of inference have we here?-- + + (1) None but the ignorant despise knowledge. + .'. No wise man despises knowledge. + + (2) A is superior to B. + .'. B is inferior to A. + + +CHAPTER VIII. + +Fill up the following enthymemes, mentioning to which order they +belong, and state which of them are expressed in problematic form-- + + (1) I am fond of music: for I always like a comic song. + + (2) All men are born to suffering, and therefore you must expect + your share. + + (3) Job must have committed some secret sins: for he fell into + dreadful misfortunes. + + (4) Latin was the language of the Vestals, and therefore no lady + need be ashamed of speaking it. + + (5) None but physicians came to the meeting. There were therefore no + nurses there. + + (6) The human soul extends through the whole body, for it is found + in every member. + + (7) No traitor can be trusted, and you are a traitor. + + (8) Whatever has no parts does not perish by the dissolution of its + parts. Therefore the soul of man is imperishable. + +Is the suppressed premiss in any case disputable on material grounds? + + +CHAPTERS IX-XVIII. + +Refer the following arguments to their proper mood and figure, or show +what rules of syllogism they violate. + + (1) No miser is a true friend, for he does not assist his friend + with his purse. + + (2) Governments are good which promote prosperity. + The government of Burmah does not promote prosperity. + .'. It is not a good government. + + (3) Land is not property. + Land produces barley. + .'. Beer is intoxicating. + + (4) Nothing is property but that which is the product of man's hand. + The horse is not the product of man's hand. + .'. The horse is not property. + + (5) Some Europeans at least are not Aryans, because the Finns are + not. + + (6) Saturn is visible from the earth, and the moon is visible from + the earth. Therefore the moon is visible from Saturn. + + (7) Some men of self-command are poor, and therefore some noble + characters are poor. + + (8) Sparing the rod spoils the child: so John will turn out very + good, for his mother beats him every day. + + (9) Some effects of labour are not painful, since every virtue is an + effect of labour. + + (10) The courageous are confident and the experienced are + confident. Therefore the experienced are courageous. + + (11) No tale-bearer is to be trusted, and therefore no great talker + is to be trusted, for all tale-bearers are great talkers. + + (12) Socrates was wise, and wise men alone are happy: therefore + Socrates was happy. + + +II. + +1. From the major 'No matter thinks' draw, by supplying the minor, the +following conclusions-- + + (1) Some part of man does not think. + + (2) The soul of man is not matter. + + (3) Some part of man is not matter. + + (4) Some substance does not think. + +Name the figured mood into which each syllogism falls. + +2. Construct syllogisms in the following moods and figures, stating +whether they are valid or invalid, and giving your reasons in each +case-- + + AEE in the first figure; EAO in the second; IAI in the third; AII in + the fourth. + +3. Prove that 'Brass is not a metal,' using as your middle term +'compound body.' + +4. Construct syllogisms to prove or disprove-- + + (1) Some taxes are necessary. + + (2) No men are free. + + (3) Laws are salutary. + +5. Prove by a syllogism in Bokardo that 'Some Socialists are not +unselfish,' and reduce your syllogism directly and indirectly. + +6. Prove the following propositions in the second figure, and reduce +the syllogisms you use to the first-- + + (1) All negroes are not averse to education. + + (2) Only murderers should be hanged. + +7. Prove in Baroko and also in Ferio that 'Some Irishmen are not +Celts.' + +8. Construct in words the same syllogism in all the four figures. + +9. Invent instances to show that false premisses may give true +conclusions. + + +III. + +1. What moods are peculiar to the first, second, and third figures +respectively? + +2. What moods are common to all the figures? + +3. Why can there be no subaltern moods in the third figure? + +4. What is the only kind of conclusion that can be drawn in all the +figures? + +5. Show that IEO violates the special rules of all the figures. + +6. In what figures is AEE valid? + +7. Show that AEO is superfluous in any figure. + +8. Prove that O cannot be a premiss in the first figure, nor a minor +premiss anywhere but in the second. + +9. Show that in the first figure the conclusion must have the quality +of the major premiss and the quantity of the minor. + +10. Why do the premisses EA yield a universal conclusion in the first +two figures and only a particular one in the last two? + +11. Show that AAI is the only mood in the fourth figure in which it is +possible for the major term to be distributed in the premiss and +undistributed in the conclusion. + +12. Why are the premisses of Fesapo and Fresison not transposed in +reduction like those of the other moods of the fourth figure? + + +IV. + +1. Why is it sufficient to distribute the middle term once only? + +2. Prove that from two affirmative premisses you cannot get a negative +conclusion. + +3. Prove that there must be at least one more term distributed in the +premisses than in the conclusion. + +4. Prove that the number of distributed terms in the premisses cannot +exceed those in the conclusion by more than two. + +5. Prove that the number of undistributed terms in the premisses +cannot exceed those in the conclusion by more than one. + +6. Prove that wherever the minor premiss is negative, the major must +be universal. + +7. Prove that wherever the minor term is distributed, the major +premiss must be universal. + +8. If the middle term be twice distributed, what mood and figure are +possible? + +9. If the major term of a syllogism be the predicate of the major +premiss, what do we know about the minor premiss? + +10. When the middle term is distributed in both premisses, what must +be the quantity of the conclusion? + +11. Prove that if the conclusion be universal, the middle term can +only be distributed once in the premisses. + +12. Show how it is sometimes possible to draw three different +conclusions from the same premisses. + + +CHAPTER XIX. + +1. Convert the following propositions-- + + (1) If a man is wise, he is humble. + + (2) Where there is sincerity there is no affectation. + + (3) When night-dogs run, all sorts of deer are chased. + + (4) The nearer the Church, the further from God. + + (5) If there were no void, all would be solid. + + (6) Not to go on is sometimes to go back. + +2. Express in a single proposition-- + + If he was divine, he was not covetous; and if he was covetous, he + was not divine. + +3. Exhibit the exact logical relation to one another of the following +pairs of propositions-- + + (1) If the conclusion be false, the premisses are false. If the + conclusion be true, the premisses are not necessarily true. + + (2) If one premiss be negative, the conclusion must be negative. + + If the conclusion be negative, one of the premisses must be + negative. + + (3) The truth of the universal involves the truth of the particular. + + The falsity of the particular involves the falsity of the universal. + + (4) From the truth of the particular no conclusion follows as to the + universal. + + From the falsity of the universal no conclusion follows as to the + particular. + + (5) If the conclusion in the fourth figure be negative, the major + premiss must be universal. + + If the major premiss in the fourth figure be particular, the + conclusion must be affirmative. + + (6) If both premisses be affirmative, the conclusion must be + affirmative. + + If the conclusion be negative, one of the premisses must be + negative. + +4. 'The Method of Agreement stands on the ground that whatever +circumstance can be eliminated is not connected with the phenomenon by +any law; the Method of Difference stands on the ground that whatever +circumstance cannot be eliminated is connected with the phenomenon by +a law.' Do these two principles imply one another? + + +CHAPTERS XX-XXVIII. + +1. Fill up the following enthymemes, and state the exact nature of the +resulting syllogism-- + + (1) If Livy is a faultless historian, we must believe all that he + tells us; but that it is impossible to do. + + (2) If they stay abroad, the wife will die; while the husband's + lungs will not stand the English climate. It is to be feared + therefore that one must fall a victim. + + (3) He is either very good, very bad, or commonplace. But he is not + very good. + + (4) Either a slave is capable of virtue or he is not. + .'. Either he ought not to be a slave or he is not a man. + + (5) Does not his feebleness of character indicate either a bad + training or a natural imbecility? + + (6) Those who ask shan't have; those who don't ask don't want. + + (7) If a man be mad, he deviates from the common standard of + intellect. + .'. If all men be alike mad, no one is mad. + + (8) 'I cannot dig; to beg I am ashamed.' + +2. 'The infinite divisibility of space implies that of time. If the +latter therefore be impossible, the former must be equally so.' +Formulate this argument as an immediate inference. + +3. Examine the following arguments-- + + (1) If we have a dusty spring, there is always a good wheat + harvest. We shall therefore have a poor harvest this year, for the + spring has not been dusty. + + (2) Virtues are either feelings, capacities, or states; and as they + are neither feelings nor capacities, they must be states. + + (3) Everything must be either just or unjust. + Justice is a thing, and is not unjust. + .'. Justice is just. + + Similarly justice is holy. + But the virtues of knowledge, justice, courage, temperance, and + holiness were declared to be different from one another. + .'. Justice is unholy and holiness unjust. + + +CHAPTER XXIX. + +Formulate the following trains of reasoning, resolve them into their +component parts, and point out any violations of the rules of +syllogism which they may contain-- + + (1) No Church Institutions are useful; for they teach religious + matters, not business matters, which latter are useful, being + profitable. + + (2) Mr. Darwin long ago taught us that the clover crop is dependent + on the number of maiden ladies in the district. For the ladies keep + cats, and the cats destroy the field-mice, which prey on the bees, + which, in their turn, are all-important agents in the fertilisation + of the clover flowers. + + (3) Athletic games are duties; for whatever is necessary to health + is a duty, and exercise is necessary to health, and these games are + exercise. + + (4) The iron-trade leads to the improvement of a new country; for + furnaces require to be fed with fuel, which causes land to be + cleared. + + (5) 'Is stone a body?' 'Yes.' 'Well, is not an animal a body?' + 'Yes,' 'And are you an animal?' 'It seems so.' 'Then you are a + stone, being an animal.' + + (6) If A is B, C is D. + If E is F, G is H. + But if A is B, E is F. + .'. If C is D, G is sometimes H. + + (7) The soul is not matter. + My arm is not myself. + + (8) Honesty deserves reward and a negro is a + fellow-creature. Therefore an honest negro is a fellow-creature + deserving of reward. + + +CHAPTER XXX. + +1. Point out any ambiguities which underlie the following +propositions-- + + (1) Every one who has read the book in French will recommend those + who have not to read it in English. + + (2) I will not do this because he did it. + + (3) These are all my books. + + (4) By an old statute of the date of Edward III it was accorded + 'that Parliament should be holden every year once or more often if + need be.' + + (5) They found Mary and Joseph and the babe lying in a manger. + + (6) The king and his minister are feeble and unscrupulous. + + (7) Heres meus uxori meae triginta pondo vasorum argenteorum dato, + quae volet. + +2. Examine the following arguments, formulating them when sound, and +referring them, when unsound, to the proper head of fallacy-- + + (1) We know that thou art a teacher come from God; for no man can do + these signs that thou doest, except God be with him. S. John iii. 2. + + (2) 'Sir Walter Scott's novels have ceased to be popular.' 'Well, + that's only because nobody reads them.' + + (3) What we produce is property. + The sheriff produces a prisoner. + .'. A prisoner is property. + + (4) As all metals are not necessarily solid, we may expect some + metals to be liquid. + + (5) Moses was the son of Pharaoh's daughter. + .'. Moses was the daughter of Pharaoh's son. + + (6) If Aeschines took part in the public rejoicings over the success + of my policy, he is inconsistent in condemning it now; if he did + not, he was a traitor then. + + (7) It is wrong to stick knives into people. + .'. Surgeons ought to be punished. + + (8) If a thing admits of being taught, there must be both teachers + and learners of it. + .'. If there are neither teachers nor learners of a thing, that + thing does not admit of being taught. + + (9) It is unnecessary to lend books, if they are common, and wrong + to lend them, if they are rare. Therefore books should not be lent + from public libraries. + + (10) Seeing is believing. + .'. What is not seen cannot be believed. + + (11) St. Paul was not of Jewish blood, for he was a Roman citizen. + + (12) To call you an animal is to speak the truth. + To call you an ass is to call you an animal. + .'. To call you an ass is to speak the truth. + + (13) Pain chastens folly. A life of ease must therefore be one of + folly incurable. + + (14) We cannot be happy in this world; for we must either indulge + our passions or combat them. + + (15) It must be clear to the most unlettered mind that, as all + things were originally created by the Deity, including the hair on + our heads and the beards on our faces, there can be no such thing as + property. + + (16) The crime was committed by the criminal. + The criminal was committed by the magistrate. + .'. The crime was committed by the magistrate. + + (17) General councils are as likely to err as the fallible men of + whom they consist. + + (18) Dead dogs are heavier than living ones, because vitality is + buoyant. + + (19) Deliberation is concerned with actions. + Actions are means. + .'. Deliberation is concerned with means. + + (20) 'No beast so fierce but has a touch of pity; But I have none: + therefore I am no beast.' + + (21) Practical pursuits are better than theoretical. + .'. Mathematics are better than logic. + + (22) Death must be a good. For either the soul, ceasing to be, + ceases ta suffer, or, continuing to be, lives in a better state. + + (23) What is right should be enforced by law. + .'. Charity should be so enforced. + + (24) All animals were in the Ark. + .'. No animals perished in the Flood. + + (25) If he robs, he is not honourable. + If he pays all his dues, he does not rob. + .'. If he pays all his dues, he is honourable. + + (26) A dove can fly a mile in a minute. + A swallow can fly faster than a dove. + .'. A swallow can fly more than a mile in a minute. + + (27) 'I must soap myself, because it's Sunday.' + 'Then do you only soap yourself on Sunday.' + + (28) If the charge is false, the author of it is either ignorant or + malicious. But the charge is true. Therefore he is neither. + + (29) All the angles of a triangle are equal to two right angles. + The angle at the vertex is an angle of a triangle. + .'. It is equal to two right angles. + + (30) Si gravis sit dolor, brevis est; si longus, levis. Ergo + fortiter ferendus. + + (31) You are not what I am. + I am a man. + .'. You are not a man. + + (32) The extension of the franchise is necessary, for it is + imperative that the right of voting should be granted to classes who + have hitherto not possessed this privilege. + + (33) If Hannibal is really victorious, he does not need supplies; + while, if he is deluding us, we ought certainly not to encourage him + by sending them. Livy, xxiii. 13. § 5. + + (34) Laws must punish, and punishment hurts. + All laws therefore are hurtful. + + (35) The sun is an insensible thing. + The Persians worship the sun. + .'. The Persians worship an insensible thing. + + (36) Some ores are not metals; for they are not fluids, + and some metals are not fluids. + + (37) All the Grecian soldiers put the Persians to flight. + .'. Every Grecian soldier could rout the Persians. + + (38) The resurrection of Jesus Christ is either an isolated fact or + else admits of parallel. But if it be an isolated fact, it cannot be + rendered probable to one who denies the authority of Christianity; + and, if it admit of parallel, it no longer proves what is + required. Therefore it is either incapable of being substantiated or + else makes nothing for the truth of Christianity. + + (39) The resurrection of Christ in the flesh and his ascension into + heaven were events either intrinsically incredible in their nature + or not. If the former, the prevalent belief in them can only be + accounted for by miracles; if the latter, they ought to be believed + even without miracles. St. Aug. De Civ. Dei, xxii. 8. + + (40) Only contented people are wise. Therefore the tramp contented + in his rags is necessarily a wise man. + + (41) Four-legged things are brutes. + Tables are four-legged things. + .'. Tables are brutes. + + (42) The apparent volcanoes in the moon are not volcanoes; for + eruptions are produced by gases only, and there are no gases in the + moon. + + (43) To read the Scriptures is our duty. Therefore the Captain was + wrong in punishing the helmsman for reading the Bible at the time + when the ship struck. + + (44) The divine law orders that kings should be honoured. + Louis Quatorze is a king. + .'. The divine law orders that Louis Quatorze should be honoured. + + (45) Those who desire the same object are unanimous. + Caesar and Pompey both desire the same object, namely, supreme + power. + .'. They are unanimous. + + (46) Either the ministers left at home will be ciphers or they will + not be ciphers. If they are ciphers, cabinet government, which is + equivalent to constitutional government, will receive a rude blow. + If they are not ciphers, the cabinet will be considering matters of + the utmost importance in the absence, and the gratuitous absence, of + two of its most important members. 'The Standard,' Wed. June 5, + 1878. + + (47) One patent stove saves half the ordinary amount of + fuel. Therefore two would save it all. + + (48) One number must win in the lottery. + My ticket is one number. + .'. It must win. + + (49) All good shepherds are prepared to lay down their lives for the + sheep. + Few in this age are so prepared. + .'. Few in this age are good shepherds. + + (50) You cannot define the sun; for a definition must be clearer + than the thing defined, and nothing can be clearer than the source + of all light. + + (51) To give the monopoly of the home market to the produce of + domestic industry ... must in almost all cases be either a useless + or a hurtful regulation. If the produce of domestic can be brought + there as cheap as that of foreign industry, the regulation is + evidently useless; if it cannot, it is generally hurtful. Adam + Smith, Wealth of Nations, Bk. iv. ch. 2. + + (52) Verberare est actio. + Ergo et vapulare. + + (53) The ages of all the members of this family are over 150. + The baby is a member of this family. + .'. Its age is over 150. + + (54) Romulus must be an historical person; because it is not at all + likely that the Romans, whose memory was only burdened with seven + kings, should have forgotten the most famous of them, namely, the + first. + + (55) All scientific treatises that are clear and true deserve + attention. + Few scientific treatises are clear and true. + .'. Few scientific treatises deserve attention. + + (56) The Conservative Government is an expensive one; for, on their + going out of Office, there was a deficit. + + (57) A man is forbidden to marry his brother's wife, or, in other + words, a woman is forbidden to marry her husband's brother, that is, + a woman is directly forbidden to marry two brothers. Therefore a man + may not marry two sisters, so that a man may not marry his wife's + sister. + + + + +INDEX. + + +The references refer to the sections. + +Abstraction, 97. + +Acategorematic words, 71. + +Accent, Fallacy of, 855. + +Accident, 318. + +Accident, Fallacy of, 860. + +A dicto secundum quid, Fallacy of, 861. + +Amphiboly, Fallacy of, 848. + +Antecedent + of a complex proposition, 212. + of an inference, 428. + +A posteriori Truth, 232. + +A priori Truth, 231. + +'A' Propositions, 260. + conversion of, 489. + +Arguing in a circle, 882. + +Argumentum ad hominem, etc., 867. + +Art, 20. + +Attribute, 81 sqq. + Essential and non-essential, 320. + +Attributives, 88 sqq. + + +Basis of Division, 391. + + +Categorematic words, 71. + +Circulus in definiendo, 382. + +Common Terms, 105. + how formed, 99. + nature of, 48. + +Complex Proposition, 209. + conversion of, 709. + conversion by contraposition of, 728. + conversion by negation of, 721. + divided into conjunctive and disjunctive, 214. + permutation of, 718. + +Complex Syllogism, 731. + mixed form of, 778. + +Composition, Fallacy of, 849. + +Concept, 36, 40 sqq. + +Conception, 33. + +Conceptualists, 51. + +Conclusion, 540. + predicate of, 542. + subject of, 542. + +Conjunctive Syllogisms, 733. + canon of, 742. + reduction of partly, 744. + partly conjunctive syllogisms as an immediate inference, 753. + +Connotation of Terms, 148. + +Consequent of a complex proposition, 213. + of an inference, 428. + +Consequent, Fallacy of, 873. + +Contingent, 17. + +Contradiction, Law of, 25 sqq. + +Contradictory Propositions, 458. + Terms, 129. + +Contrary Propositions, 458. + Terms, 130. + +Converse, 480. + +Conversion, 479. + of complex propositions, 709. + by contraposition, 516. + illative, 481. + by negation, 504. + per accidens, 487. + simple, 486. + rules of, 482. + +Convertend, 480. + +Copula, 58, 64, 186 sqq. + modality of, 196. + +Correlatives, 142. + + +Deduction and Induction, difference of, 431 sqq. + +Deductive Inference, 442. + +Deductive Logic, definition of, 4. + +Definition of Terms, 347 sqq. + of Aristotle ([Greek: ňrismós]), 336. + final, 374. + nominal, 375. + provisional, 374. + real, 375. + rules of, 378. + +Denotation of Terms, 152. + +Description, 360. + +Designations, 112. + +Determination, 167. + +Dictum de omni et nullo, 569. + de diverso, 641. + de exemplo et excepto, 642. + +Difference, 318, 358. + generic, 410. + specific, 409. + +Dilemma, 732, 779. + rebutted, 792. + reduction of, 796. + regarded as an immediate inference, 798. + +Disjunctive Syllogism, 760. + canon of, 765. + reduction of, 766. + regarded as an immediate inference, 770. + +Distinction, 424. + +Distribution of Terms, 274. + four rules for, 293. + +Divided whole, 393. + +Dividing members, 394. + +Division, 385 sqq. + by dichotomy, 412. + rules of, 395. + +Division, Fallacy of, 851. + +Division of Propositions, 206. + of terms, 86. + of things, 77. + + +Enthymeme, incorrectly so-called, 557. + +Enumeration, 387, 422. + +Epicheirema, 803. + +Episyllogism, 802. + +'E' Propositions, 260. + conversion of, 490. + +Equivocation, Fallacy of, 845. + +Excluded Middle, Law of, 25 sqq., 502. + +Extension of Terms, 149 sqq., 166 sqq. + + +Fallacy, 827 sqq. + of ambiguity, 831. + definition of, 828. + formal, 838. + logical, 836. + material, 831, 836. + of undisturbed middle, 585. + +Figure of Speech, Fallacy of, 857. + +Figures, of a Syllogism, 558. + special canons of, 633. + special rules of, 606. + special uses of, 648. + +Formal Logic, 16. + +Four Terms, Fallacy of, 840. + +Fundamentum Divisionis, 391. + + +Generalisation, 168. + +Genus, 318. + as used by Aristotle, 336. + cognate, 408. + proximate, 420. + subaltern, 406. + summum, 167, 404. + + +Heads of Predicables, 313. + as given by Aristotle, 336. + + +'Ideas' of Plato, 52. + +Identity, Law of, 25 sqq. + +Ignoratio Elenchi, Fallacy of, 865. + +Ignotum per ignotius, 383. + +Illicit Process, Fallacy of, 586. + +Immediate Inference, 442 sqq. + by added determinants, 535. + by complex conception, 537. + applied to complex propositions, 701. + +Immediate Inference, compound forms of, 503. + partly conjunctive syllogisms regarded as, 753. + by conversion, 479. + disjunctive syllogisms regarded as, 770. + by opposition, 462. + by permutation, 496. + +Induction, differing from Deduction, 430 sqq. + +Inductive Logic, 2, 204. + +Inferences in general, 426. + classification of, 441. + deductive, 442. + inductive, 430. + +Intimae species, 405. + +Intension of Terms, 150, 166. + +Intuition, 232. + +Inverse Variation, Law of, 166. + +'I' Propositions, 260. + conversion of, 490. + + +'Judgement,' various meanings of, 32, 36. + + +'Law,' ambiguities of the word, 7 sqq. + + +Major Premiss, 544. + +Major Term, 542. + +Many Questions, Fallacy of, 884. + +Mediate Inferences or Syllogisms, 444, 540 sqq. + axioms of, 576. + +Membra Dividentia, 394. + +Middle Term, 541. + position of, in a syllogism, 563. + +Minor Premiss, 545. + +Minor Term, 542. + +Modality, Question of, 196. + +Mode, the, 196. + +Moods of a Syllogism, 558. + determination of the legitimate, 599. + subaltern, 628. + valid in the Four Figures, 621. + mnemonics of, valid in Four Figures, 629. + + +Name, definition of, 61. + +Negative Premisses and Conclusion, Fallacy of, 842. + +Nominalists, 50, 54. + +Non causa pro causa, Fallacy of, 883. + +Nouns, 62. + + +Opposition, 449 sqq. + contradictory, 457. + contrary, 454. + laws of, 464. + subaltern, 456. + sub-contrary, 455. + +'O' Propositions, 260. + conversion of, 491. + + +Partition, 423. + +Permutation, 496 sqq. + of Complex Propositions, 718. + +Petitio Principii, Fallacy of, 874. + +Predicable, 314. + +Predicate of a Proposition, 58, 184. + read in extension, 307. + quantification of, 295 sqq. + quantity of, 281, 494. + +Predication, 194. + in quid or in quale, 332. + +Premisses, 540. + major, 544. + minor, 545. + +Primary Existences, 55. + +Problema, the, 556. + +Proper Names, 113. + +Property, 318. + generic, 411. + specific, 411. + +Proposition, 172 sqq. + accidental, 238. + affirmative, 258. + complex or conditional, 209. + conjunctive or hypothetical, 214, 704. + conversion of, 479. + definition of, 178. + disjunctive, 214. + divisions of, 206. + essential, 238. + exceptive, 270. + exclusive, 266. + extensive, 264. + general, 251. + indefinite, 244. + intensive, 264. + modal, 205. + negative, 258. + particular, 240. + pure, 205. + quality of, 258. + quantity of, 246. + real or synthetical, 227. + simple or categorical, 207. + singular, 250. + tautologous or identical, 273. + universal, 239. + verbal or analytical, 224. + +Proprium, 336. + +Pro-syllogism, 802. + + +Quaestio, the, 556. + +Quality, a, 82. + +Quality of the matter, 204. + of propositions, 258. + +Quantification of the Predicate, 295 sqq., 493. + +Quantity of propositions, 258. + of terms, 148. + + +Realists, 49. + +Real Kinds, 371. + +Reasoning or Inference, 35. + the canon of, 560. + trains of, 800. + +Reduction of propositions, 667. + of the dilemma, 796. + of disjunctive syllogisms, 766. + indirect, 691. + mnemonics for, 697. + ostensive or direct, 673. + of partly conjunctive syllogisms, 744. + +Relation, a, 83, 144. + +Relation, immediate inference by, 462. + compatible and incompatible, 462. + + +Science, 20. + +Secondary Existences, 55. + +Simple Apprehension, 33. + +Sorites, the, 807 sqq. + +Specialisation, 167. + +Species, 318. + cognate, 407. + infimae, 405. + subaltern, 406. + +Subalternant, 458. + +Subalternate, 458. + +Subalternation, 458. + +Subalterns, 458. + +Sub-contraries, 458. + +Sub-division, 401. + +Subject, 58, 183. + how used, 264. + quantity of, 279. + +Substance, 80, 84. + +Summum Genus, 167, 404. + +Suppositio Materialis, 76. + +Syllogism, 546 sqq. + complex, 731. + in common discourse, 557. + conjunctive, 733. + definition of, 552. + disjunctive, 760. + general rules of, 582. + figures of, 560, 563. + with three figures, 656. + legitimate moods of, 599 sqq. + mnemonics for, 598. + moods of, 559, 562. + +Syncntegorematic words, 70. + +Synonym, 345. + + +Term, 57 sqq. + absolute, 140. + abstract, 95. + analogous, 139. + attributive, 88. + collective, 118. + common, 105. + concrete, 96. + connotative, 147. + contradictory, 129. + contrary, 130. + definition of, 347. + +Terms, distribution of, 275. + distributive and collective use of, 119. + division of, 86. + equivocal, 137. + incompatible, 135. + individual, 121. + major, middle, and minor, 542. + negative, 126. + non-connotative, 147. + positive, 126. + privative, 126. + quantity of, 148. + +Terms, relative, 141. + repugnant, 135. + singular, 43, 104. + subject, 87. + undistributed, 277. + univocal, 137. + + +Universals, nature of, 48, 55. + +'U' Propositions, 297. + + +Verb, 64. + + +Words, their relation to terms, 65 sqq, + + + +THE END. + + + + + + +End of the Project Gutenberg EBook of Deductive Logic, by St. George Stock + +*** END OF THE PROJECT GUTENBERG EBOOK DEDUCTIVE LOGIC *** + +This file 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