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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 groundthat my logic was too like all other logics; another suggested to meto cut out a considerable amount of new matter. The latter advice Ihave followed; the former has encouraged me to hope that I shall notbe considered guilty of wanton innovation. The few novelties which Ihave ventured to retain will, I trust, be regarded as legitimateextensions of received lines of teaching. 

My object has been to produce a work which should be as thoroughlyrepresentative of the present state of the logic of the Oxford Schoolsas any of the text-books of the past. The qualities which I have aimedat before all others have been clearness and consistency. For the taskwhich I have taken upon myself I may claim one qualification--that ofexperience; since more than seventeen years have now elapsed since Itook my first pupil in logic for the Honour School of Moderations, andduring that time I have been pretty continuously engaged in studyingand teaching the subject. 

In acknowledging my obligations to previous writers I must begin withArchbishop Whately, whose writings first gave me an interest in thesubject. The works of Mill and Hamilton have of course been freelydrawn upon. I have not followed either of those two great writersexclusively, but have endeavoured to assimilate what seemed best inboth. To Professor Fowler I am under a special debt. I had not theprivilege of personal teaching from him in logic, --as I had in someother subjects; but his book fell into my hands at an early period inmy mental training, and was so thoroughly studied as to have become apermanent part of the furniture of my mind. Much the same may be saidof my relation to the late Professor Jevons's Elementary Lessons inLogic. Two other books, which I feel bound to mention with specialemphasis, are Hansel's edition of Aldrich and McCosh's Laws ofDiscursive 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'sedition of Murray, Ray's Text-book of Deductive Logic, andWeatherley's Rudiments of Logic, I think the list will be exhausted ofmodern works from which I am conscious of having borrowed. But, not toforget the sun, while thanking the manufacturers of lamps and candles, I should add that I have studied the works of Aristotle according tothe measure of my time and ability. 

This work has had the great advantage of having been revised, whilestill in manuscript, by Mr. Alfred Robinson, Fellow of New College, towhom I cannot sufficiently express my obligation. I have availedmyself to the full of the series of criticisms which he was kindenough to send me. As some additions have been made since then, hecannot be held in anyway responsible for the faults which less kindlycritics may detect. 

For the examples at the end I am mainly indebted to others, and to alarge extent to my ingenious friend, the Rev. W. J. Priest of MertonCollege. 

My thanks are due also to my friend and former pupil, Mr. GilbertGrindle, Scholar of Corpus, who has been at the pains to compose anindex, and to revise the proofs as they passed through the press. 

And last, but not least, I must set on record my gratitude toCommander 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 truthor falsity of propositions: the problem of deductive logic is todetermine their relative truth or falsity, that is to say, given suchand such propositions as true, what others will follow from them. 

§ 3. Hence in the natural order of treatment inductive logic precedesdeductive, since it is induction which supplies us with the generaltruths, from which we reason down in our deductive inferences. 

§ 4. It is not, however, with logic as a whole that we are hereconcerned, but only with deductive logic, which may be defined as TheScience of the Formal Laws of Thought. 

§ 5. In order fully to understand this definition we must know exactlywhat 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 ofcomparison. All thought involves comparison, that is to say, arecognition of likeness or unlikeness. 

§ 7. The laws of thought are the conditions of correct thinking. Theterm 'law, ' however, is so ambiguous that it will be well to determinemore 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 theseexpressions. By a law in the political sense is meant a commandimposed by a superior upon an inferior and sanctioned by a penalty fordisobedience. But by the 'laws of nature' are meant merely certainuniformities among natural phenomena; for instance, the 'law ofgravitation' means that every particle of matter does invariablyattract every other particle of matter in the universe. 

§ 9. The word 'law' is transferred by a metaphor from one of thesesenses to the other. The effect of such a command as that describedabove is to produce a certain amount of uniformity in the conduct ofmen, and so, where we observe uniformity in nature, we assume that itis the result of such a command, whereas the only thing really knownto us is the fact of uniformity itself. 

§ 10. Now in which of these two senses are we using the term 'laws ofthought'? The laws of the land, it is plain, are often violated, whereas the laws of nature never can be so [Footnote: There is a sensein which people frequently speak of the laws of nature being violated, as when one says that intemperance or celibacy is a violation of thelaws of nature, but here by 'nature' is meant an ideal perfection inthe conditions of existence. ]. Can the laws of thought be violated inlike manner with the laws of the land? Or are they inviolable like thelaws of nature?

§ 11. In appearance they can be, and manifestly often are violated-forhow else could error be possible? But in reality they can not. No manever accepts a contradiction when it presents itself to the mind assuch: but when reasoning is at all complicated what does reallyinvolve a contradiction is not seen to do so; and this sort of erroris further assisted by the infinite perplexities of language. 

§ 12. The laws of thought then in their ultimate expression arecertain uniformities which invariably hold among mental phenomena, andso far they resemble the laws of nature: but in their complexapplications they may be violated owing to error, as the laws of theland may be violated by crime. 

§ 13. We have now to determine the meaning of the expression 'formallaws of thought. '

§ 14. The distinction between form and matter is one which pervadesall nature. We are familiar with it in the case of concrete things. Acup, for instance, with precisely the same form, may be composed ofvery different matter-gold, silver, pewter, horn or what not?

§ 15. Similarly in every act of thought we may distinguish twothings--

 (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, isconcerned only with the way in which the mind thinks, and has nothingto 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 andnecessary elements of thought as opposed to such as are accidental andcontingent. By 'contingent' is meant what holds true in some cases, but not in others. For instance, in the particular case of equilateraltriangles it is true to say, not only that 'all equilateral trianglesare equiangular, ' but also that 'all equiangular triangles areequilateral. ' But the evidence for these two propositions isindependent. The one is not a formal consequence of the other. If itwere, we should be able to apply the same inference to all matter, andassert generally that if all A is B, all B is A, which it is notoriousthat we cannot do. 

§ 18. It remains now for the full elucidation of our definition todetermine what is meant by 'science. '

§ 19. The question has often been discussed whether logic is a scienceor an art. The answer to it must depend upon the meaning we assign tothese terms. 

§ 20. Broadly speaking, there is the same difference between Scienceand 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 notnecessarily confer any practical skill. It is only the right use ofits rules in thinking which can make men think better. It istherefore, in the broad sense of the terms, wholly a science and notat all an art. 

§ 22. But this word 'art, ' like most others, is ambiguous, and isoften used, not for skill displayed in practice, but for the knowledgenecessary thereto. This meaning is better conveyed by the term'practical science. '

§ 23. Science is either speculative or practical. In the first case westudy 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 mayunderstand its structure; in the second that we may assist itsneeds. Whether logic is a speculative or a practical science dependsentirely upon the way in which it is treated. If we study the laws ofthought merely that we may know what they are, we are making it aspeculative science; if we study the same laws with a view to deducingrules for the guidance of thought, we are making it a practicalscience. 

§ 24. Hence logic may be declared to be both the science and the artof thinking. It is the art of thinking in the same sense in whichgrammar is the art of speaking. Grammar is not in itself the rightuse of words, but a knowledge of it enables men to use wordscorrectly. In the same way a knowledge of logic enables men to thinkcorrectly, or at least to avoid incorrect thoughts. As an art logicmay be called the navigation of the sea of thought. 

§ 25. The laws of thought are all reducible to the three followingaxioms, 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, noviolation of the law of contradiction would necessarily ensue: for athing might then be something else, without being itself at the sametime, which latter is what the law of contradiction militatesagainst. Neither would the law of excluded middle be infringed. For, on the supposition, a thing would be something else, whereas all thatthe law of excluded middle demands is that it should either be itselfor not. A would in this case adopt the alternative of being not A. 

§ 28. Again, the violation of the law of contradiction does notinvolve any violation of the law of identity: for a thing might inthat case be still itself, so that the law of identity would beobserved, even though, owing to the law of contradiction not holding, it were not itself at the same time. Neither would the law of excludedmiddle be infringed. For a thing would, on the supposition, be bothitself and not itself, which is the very reverse of being neither. 

§ 29. Lastly, the law of excluded middle might be violated without aviolation of the law of contradiction: for we should then have a thingwhich was neither A nor not A, but not a thing which was both at thesame time. Neither would the law of identity be infringed. For weshould in this case have a thing which neither was nor was not, sothat the conditions of the law of identity could not exist to bebroken. That law postulates that whatever is, is: here we have a thingwhich never was to begin with. 

§ 30. These principles are of so simple a character that thediscussion of them is apt to be regarded as puerile. Especially isthis the case with regard to the law of identity. This principle infact is one of those things which are more honoured in the breach thanin the observance. Suppose for a moment that this law did nothold--then what would become of all our reasoning? Where would be theuse of establishing conclusions about things, if they were liable toevade us by a Protean change of identity?

§ 31. The remaining two laws supplement each other in the followingway. The law of contradiction enables us to affirm of two exhaustiveand mutually exclusive alternatives, that it is impossible for both tobe true; the law of excluded middle entitles us to add, that it isequally impossible for both to be false. Or, to put the same thing ina different form, the law of contradiction lays down that one of twosuch alternatives must be false; the law of excluded middle adds thatone 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, isthe act of forming in the mind the idea of anything, e. G. When we formin the mind the idea of a cup, we are performing the process ofconception. 

§ 34. Judgement, in the sense in which it is here used [Footnote:Sometimes the term 'judgement' is extended to the comparison ofnameless sense-impressions, which underlies the formation ofconcepts. But this amounts to identifying judgement with thought ingeneral. ] may be resolved into putting two ideas together in themind, and pronouncing as to their agreement or disagreement, e. G. Wehave in our minds the idea of a cup and the idea of a thing made ofporcelain, and we combine them in the judgement--'This cup is made ofporcelain. '

§ 35. Inference, or Reasoning, is the passage of the mind from one ormore judgements to another, e. G. From the two judgements 'Whatever ismade of porcelain is brittle, ' and 'This cup is made of porcelain, ' weelicit a third judgement, 'This cup is brittle. '

§ 36. Corresponding to these three processes there are three productsof thought, viz. 

 (1) The Concept. 

 (2) The Judgement. 

 (3) The Inference. 

§ 37. Since our language has a tendency to confuse the distinctionbetween processes and products, [Footnote: E. G. We have to speak quiteindiscriminately of Sensation, Imagination, Reflexion, Sight, Thought, Division, Definition, and so on, whether we mean in any case a processor a product. ] it is the more necessary to keep them distinct inthought. Strictly we ought to speak of conceiving, judging andinferring on the one hand, and, on the other, of the concept, thejudgement and the inference. 

The direct object of logic is the study of the products rather than ofthe processes of thought. But, at the same time, in studying theproducts we are studying the processes in the only way in which it ispossible to do so. For the human mind cannot be both actor andspectator at once; we must wait until a thought is formed in our mindsbefore we can examine it. Thought must be already dead in order to bedissected: there is no vivisection of consciousness. Thus we can neverknow more of the processes of thought than what is revealed to us intheir 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 theterm 'inference' in three different senses--first, for the act orprocess of inferring; secondly, for the result of that act as itexists in the mind; and, thirdly, for the same thing as expressed inlanguage. Later on we shall have to notice a further ambiguity in itsuse. 

§ 40. It has been declared that thought in general is the faculty ofcomparison, and we have now seen that there are three products ofthought. It follows that each of these products of thought must be theresult 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 phraseologywhich regards the products of thought as clothed in language inpreference to that which regards the same products as they exist inthe mind of the individual. For although the object of logic is toexamine thought pure and simple, it is obviously impossible to discussit except as clothed in language. Accordingly the three statementsabove 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 thefact that the word 'concept' is not an adequate expression for thefirst of the three products of thought, whereas the word 'term' is. Bya concept is meant a general notion, or the idea of a class, whichcorresponds only to a common term. Now not only are common terms theresults of comparison, but singular terms, or the names ofindividuals, are so too. 

§ 43. The earliest result of thought is the recognition of anindividual object as such, that is to say as distinguished and markedoff from the mass of its surroundings. No doubt the first impressionproduced Upon the nascent intelligence of an infant is that of aconfused whole. It requires much exercise of thought to distinguishthis whole into its parts. The completeness of the recognition of anindividual object is announced by attaching a name to it. Hence evenan individual name, or singular term, implies thought orcomparison. Before the _child_ can attach a meaning to the word'_mother_, ' which to it is a singular term, it must havedistinguished between the set of impressions produced in it by oneobject from those which are produced in it by others. Thus, whenVergil says

 Incipe, parve puer, risu cognoscere matrem, he is exhorting the beatific infant to the exercise of the faculty ofcomparison. 

§ 44. That a common term implies comparison does not need to beinsisted upon. It is because things resemble each other in certain oftheir attributes that we call them by a common name, and thisresemblance could not be ascertained except by comparison, at sometime and by some one. Thus a common term, or concept, is thecompressed result of an indefinite number of comparisons, which liewrapped up in it like so many fossils, witnessing to prior ages ofthought. 

§ 45. In the next product of thought, namely, the proposition, we havethe result of a single act of comparison between two terms; and thisis why the proposition is called the unit of thought, as being thesimplest and most direct result of comparison. 

§ 46. In the third product of thought, namely, the inference, we havea comparison of propositions either directly or by means of athird. This will be explained later on. For the present we return tothe 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 oflogicians. 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 themeaning of a common term, like 'man' or 'horse, ' is not so obvious aspeople are apt to think on first hearing of the question. 

§ 48. A common term or class-name was known to mediæval logiciansunder the title of a Universal; and it was on the question 'What is aUniversal 7' that they split into the three schools of Realists, Nominalists, and Conceptualists. Here are the answers of the threeschools to this question in their most exaggerated form--

§ 49. Universals, said the Realists, are substances having anindependent 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 thename. 

§ 51. Universals, said the Conceptualists, exist in the mind alone, They are the conceptions under which the mind regards externalobjects. 

§ 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, butidentical with it. Plato seems to have imagined that, as there was areally existing thing corresponding to a singular term, such asSocrates, so there must be a really existing thing corresponding tothe common term 'man. ' But when once the existence of these generalobjects is admitted, they swamp all other existences. For individualmen are fleeting and transitory--subject to growth, decay anddeath--whereas the idea of man is imperishable and eternal. It is onlyby partaking in the nature of these ideas that individual objectsexist at all. 

§ 53. Pure Nominalism was the swing of the pendulum of thought to thevery opposite extreme; while Conceptualism was an attempt to hit thehappy mean between the two. 

§ 54. Roughly it may be said that the Realists sought for the answerto the question 'What is a Universal?' in the matter of thought, theConceptualists in the form, and the Nominalists in the expression. 

§ 55. A full answer to the question 'What is a Universal?' will bringin something of the three views above given, while avoiding theexaggeration of each. A Universal is a number of things that arecalled by the same name; but they would not be called by the same nameunless they fell under the same conception in the mind; nor would theyfall under the same conception in the mind unless there actuallyexisted similar attributes in the several members of a class, causingus to regard them under the same conception and to give them the samename. Universals therefore do exist in nature, and not merely in themind of man: but their existence is dependent upon individual objects, instead of individual objects depending for their existence uponthem. Aristotle saw this very clearly, and marked the distinctionbetween the objects corresponding to the singular and to the commonterm by calling the former Primary and the latter SecondaryExistences. Rosinante and Excalibur are primary, but 'horse' and'sword' secondary existences. 

§ 56. We have seen that the three products of thought are each onestage in advance of the other, the inference being built upon theproposition, as the proposition is built upon the term. Logictherefore 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 aproposition. In a proposition we start from a subject and end in apredicate (§§ 182-4), there being nothing intermediate between the twoexcept the act of pronouncing as to their agreement or disagreement, which is registered externally under the sign of the copula. Thus thesubject is the 'terminus a quo, ' and the predicate is the 'terminus adquem. '

§ 59. Hence it appears that the term by its very name indicates thatit is arrived at by an analysis of the proposition. It is thejudgement or proposition that is the true unit of thought andspeech. The proposition as a whole is prior in conception to the termswhich are its parts: but the parts must come before the whole in thesynthetic 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 markto recall or transmit the idea of a thing, either in itself or throughsome 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 athing, when we know it to possess a given attribute. 

§ 64. The Verb, as such, is not recognised by logic, but is resolvedinto predicate and copula, that is to say, into a noun which isaffirmed or denied of another, plus the sign of that affirmation ordenial. 'The kettle boils' is logically equivalent to 'The kettle isboiling, ' though it is by no means necessary to express theproposition in the latter shape. Here we see that 'boils' isequivalent to the noun 'boiling' together with the copula 'is, ' whichdeclares its agreement with the noun 'kettle. ' 'Boiling' here is anoun adjective, which we are entitled to add to 'kettle, ' in virtue ofcertain knowledge which we have about the latter. Being a verbal noun, it is called in grammar a participle, rather than a mereadjective. The word 'attributive' in logic embraces both the adjectiveand participle of grammar. 

§ 65. In grammar every noun is a separate word: but to logic, which isconcerned with the thought rather than with the expression, it isindifferent whether a noun, or term, consists of one word or many. Thelatter 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'scommands, are applicable to passive submission or resignation to hiswill'--we find the subject consisting of fourteen words, and thepredicate of nine. It is the exception rather than the rule to find apredicate which consists of a single word. Many-worded names inEnglish often consist of clauses introduced by the conjunction 'that, 'as 'That letters should be written in strict conformity with nature istrue': often also of a grammatical subject with one or more dependentclauses 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 acollection of words. Neither is every word a term. 'Over, ' forinstance, and 'swiftly, ' and, generally, what are called particles ingrammar, do not by themselves constitute terms, though they may beemployed along with other words to make up a term. 

§ 67. The notions with which thought deals involve many subtlerelations and require many nice modifications. Language hasinstruments, more or less perfect, whereby such relations andmodifications may be expressed. But these subsidiary aids toexpression do not form a notion which can either have somethingasserted 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 aterm. 

§ 70. A Syncategorematic word is one which can help to form a term. 

§ 71. An Acategorematic word is one which can neither form, nor helpto form, a term [Footnote: Comparatively few of the parts of speechare categorematic. Nouns, whether substantive or adjective, includingof course pronouns and participles, are so, but only in theirnominative cases, except when an oblique case is so used as to beequivalent to an attributive. Verbs also are categorematic, but onlyin three of their moods, the Indicative, the Infinitive, and thePotential. The Imperative and Optative moods clearly do not conveyassertions at all, while the Subjunctive can only figure as asubordinate member of some assertion. We may notice, too, that therelative pronoun, unlike the rest, is necessarily syncategorematic, for the same reason as the subjunctive mood. Of the remaining parts ofspeech the article, adverb, preposition, and conjunction can never beanything but syncategorematic, while the interjection isacategorematic, like the vocative case of nouns and the imperative andoptative moods of verbs, which do not enter at all into the form ofsentence 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 byitself, but 'over' and 'swiftly' can only help to form a term, as inthe proposition, 'The horse is galloping swiftly over the plain. '

§ 73. A term then may be said to be a categorematic word or collectionof words, that is to say, one which can be used by itself as apredicate. 

§ 74. To entitle a word or collection of words to be called a term, itis not necessary that it should be capable of standing by itself as asubject. Many terms which can be used as predicates are incapable ofbeing 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 apredicate. The attributives 'swift' and 'galloping' are terms, quiteas much as the subject 'horse, ' but they cannot themselves be used assubjects. 

§ 75. When an attributive appears to be used as a subject, it is owingto a grammatical ellipse. Thus in Latin we say 'Boni sapientes sunt, 'and in English 'The good are wise, ' because it is sufficientlydeclared by the inflexional form in the one case, and by the usage ofthe language in the other, that men are signified. It is an accidentof language how far adjectives can be used as subjects. They cease tobe 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 propermeaning. Thus we can say--'"Swiftly" is an adverb. ' 'Swiftly' in thissense is really no more than the proper name for a particularword. 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 toadvert for a moment to a division of the things whereof they arenames. 

§ 78. By a 'thing' is meant simply an object of thought--whatever onecan think about. 

§ 79. Things are either Substances or Attributes. Attributes may besub-divided into Qualities and Relations. 

 Thing _______________|_______________ | | Substance Attribute _____________|____________ | | Quality Relation

§ 80. A Substance is a thing which can be conceived to exist byitself. All bodies are material substances. The soul, as a thinkingsubject, is an immaterial substance. 

§ 81. An Attribute is a thing which depends for its existence upon asubstance, e. G. Greenness, hardness, weight, which cannot beconceived to exist apart from green, hard, and heavy substances. 

§ 82. A Quality is an attribute which does not require more than onesubstance for its existence. The attributes just mentioned arequalities. There might be greenness, hardness, and weight, if therewere only one green, hard and heavy substance in the universe. 

§ 83. A Relation is an attribute which requires two or more substancesfor its existence, e. G. Nearness, fatherhood, introduction. 

§ 84. When we say that a substance can be conceived to exist byitself, what is meant is that it can be conceived to existindependently of other substances. We do not mean that substances canbe conceived to exist independently of attributes, nor yet out ofrelation to a mind perceiving them. Substances, so far as we can knowthem, are only collections of attributes. When therefore we say thatsubstances can be conceived to exist by themselves, whereas attributesare dependent for their existence upon substances, the real meaning ofthe assertion reduces itself to this, that it is only certaincollections of attributes which can be conceived to existindependently; whereas single attributes depend for their existenceupon others. The colour, smoothness or solidity of a table cannot beconceived apart from the extension, whereas the whole cluster ofattributes which constitutes the table can be conceived to existaltogether independently of other 'such clusters. We can imagine atable to exist, if the whole material universe were annihilated, andbut one mind left to perceive it. Apart from mind, however, we cannotimagine it: since what we call the attributes of a material substanceare no more than the various modes in which we find our mindsaffected. 

§ 85. The above division of things belongs rather to the domain ofmetaphysics than of logic: but it is the indispensable basis of thedivision 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 ofterms. 

 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 standingby 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 inconjunction with other words (syncategorematically) to make up asubject. Thus we can say 'A cherry-coloured ribbon is becoming, ' or 'Agalloping horse is dangerous. '

§ 89. Attributives are contrivances of language whereby we indicatethat a subject has a certain attribute. Thus, when we say 'This paperis white, ' we indicate that the subject 'paper' possesses theattribute whiteness. Logic, however, also recognises as attributivesterms which signify the non-possession of attributes. 'Not-white' isan attributive equally with 'white. '

§ 90. An Attributive then may be defined as a term which signifies thepossession, or non-possession, of an attribute by a subject. 

§ 91. It must be carefully noticed that attributives are not names ofattributes, but names of the things which possess the attributes, invirtue of our knowledge that they possess them. Thus 'white' is thename of all the things which possess the attribute whiteness, and'virtuous' is a name; not of the abstract quality, virtue, itself, butof the men and actions which possess it. It is clear that a term canonly properly be said to be a name of those things whereof it can bepredicated. Now, we cannot intelligibly predicate an attributive ofthe abstract quality, or qualities, the possession of which itimplies. We cannot, for instance, predicate the term 'learned' of theabstract quality of learning: but we may predicate it of theindividuals, Varro and Vergil. Attributives, then, are to be regardedas names, not of the attributes which they imply, but of the things inwhich those attributes are found. 

§ 92. Attributives, however, are names of things in a less direct waythan that in which subject-terms may be the names of the samethings. Attributives are names of things only in predication, whereassubject-terms are names of things in or out of predication. The terms'horse' and 'Bucephalus' are names of certain things, in this caseanimals, whether we make any statement about them or not: but theterms 'swift' and 'fiery' only become names of the same things invirtue of being predicable of them. When we say 'Horses are swift' or'Bucephalus was fiery, ' the terms 'swift' and 'fiery' become namesrespectively of the same things as 'horse' and 'Bucephalus. ' Thisfunction of attributives as names in a secondary sense is exactlyexpressed by the grammatical term 'noun adjective. ' An attributive isnot directly the name of anything. It is a name added on in virtue ofthe possession by a given thing of a certain attribute, or, in somecases, the non-possession. 

§ 93. Although attributives cannot be used as subjects, there isnothing to prevent a subject-term from being used as a predicate, andso assuming for the time being the functions of an attributive. Whenwe say 'Socrates was a man, ' we convey to the mind the idea of thesame attributes which are implied by the attributive 'human. ' Butthose terms only are called attributives which can never be usedexcept as predicates. 

§ 94. This division into Subject-terms and Attributives may beregarded as a division of terms according to their place inthought. Attributives, as we have seen, are essentially predicates, and can only be thought of in relation to the subject, whereas thesubject 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 thingitself and the name of it. Take for instance 'whiteness. ' Theattribute 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, thischair, the soul, God. 

§ 97. Abstract terms are so called as being arrived at by a process ofAbstraction. What is meant by Abstraction will be clear from a singleinstance. The mind, in contemplating a number of substances, may drawoff, or abstract, its attention from all their other characteristics, and fix it only on some point, or points, which they have incommon. Thus, in contemplating a number of three-cornered objects, wemay draw away our attention from all their other qualities, and fix itexclusively upon their three-corneredness, thus constituting theabstract notion of 'triangle. ' Abstraction may be performed equallywell in the case of a single object: but the mind would not originallyhave known on what points to fix its attention except by a comparisonof individuals. 

§ 98. Abstraction too may be performed upon attributes as well assubstances. Thus, having by abstraction already arrived at the notionof triangle, square, and so on, we may fix our attention upon whatthese have in common, and so rise to the higher abstraction of'figure. ' As thought becomes more complex, we may have abstraction onabstraction and attributes of attributes. But, however many steps mayintervene, attributes may always be traced back to substances atlast. For attributes of attributes can mean at bottom nothing but theco-existence of attributes in, or in connection with, the samesubstances. 

§ 99. We have said that abstract terms are so called, as being arrivedat by abstraction: but it must not be inferred from this statementthat all terms which are arrived at by abstraction are abstract. Ifthis were so, all names would be abstract except proper names ofindividual substances. All common terms, including attributives, arearrived at by abstraction, but they are not therefore abstract terms. Those terms only are called abstract, which cannot be applied tosubstances at all. The terms 'man' and 'human' are names of the samesubstance of which Socrates is a name. Humanity is a name only ofcertain attributes of that substance, namely those which are shared byothers. All names of concrete things then are concrete, whether theydenote them individually or according to classes, and whether directlyand in themselves, or indirectly, as possessing some given attribute. 

§ 100. By a 'concrete thing' is meant an individual Substanceconceived of with all its attributes about it. The term is notconfined to material substances. A spirit conceived of under personalattributes is as concrete as plum-pudding. 

§ 101. Since things are divided exhaustively into substances andattributes, it follows that any term which is not the name of a thingcapable of being conceived to exist by itself, must be an abstractterm. Individual substances can alone be conceived to exist bythemselves: all their qualities, actions, passions, andinter-relations, all their states, and all events with regard to them, presuppose the existence of these individual substances. All namestherefore of such things as those just enumerated are abstractterms. The term 'action, ' for instance, is an abstract term. For howcould there be action without an agent? The term 'act' also is equallyabstract for the same reason. The difference between 'action' and'act' is not the difference between abstract and concrete, but thedifference between the name of a process and the name of thecorresponding product. Unless acts can be conceived to exist withoutagents they are as abstract as the action from which they result. 

§ 102. Since every term must be either abstract or concrete, it may beasked--Are attributives abstract or concrete? The answer of coursedepends upon whether they are names of substances or names ofattributes. But attributives, it must be remembered, are neverdirectly names of anything, in the way that subject-terms are; theyare only names of things in virtue of being predicated ofthem. Whether an attributive is abstract or concrete, depends on thenature 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 nameof 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 uponthe kind of thing signified. It involves no reference to actualexistence. There are imaginary as well as real substances. Logically acentaur is as much a substance as a horse. 

_Terms. _

§ 104. A Singular Term is a name which can be applied, in the samesense, to one thing only, e. G. 'John, ' 'Paris, ' 'the capital ofFrance, ' 'this pen. '

§ 105. A Common Term is a name which can be applied, in the samesense, 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 ofthings, it is evident that it must indicate attributes which arecommon to all of them. The term 'John' is applicable to a number ofthings, but not in the same sense, as it does not indicate attributes. 

§ 106. Common terms are formed, as we have seen already (§ 99), byabstraction, i. E. By withdrawing the attention from the attributes inwhich individuals differ, and concentrating it upon those which theyhave in common. 

§ 107. A class need not necessarily consist of more than twothings. If the sun and moon were the only heavenly bodies in theuniverse, the word 'heavenly body' would still be a common term, asindicating the attributes which are possessed alike by each. 

§ 108. This being so, it follows that the division of terms intosingular and common is as exhaustive as the preceding ones, since asingular term is the name of one thing and a common term of more thanone. It is indifferent whether the thing in question be a substance oran attribute; nor does it matter how complex it may be, so long as itis regarded by the mind as one. 

§ 109. Since every term must thus be either singular or common, themembers of the preceding divisions must find their place under one orboth heads of this one. Subject-terms may plainly fall under eitherhead of singular or common: but attributives are essentially commonterms. Such names as 'green, ' 'gentle, ' 'incongruous' are applicable, strictly in the same sense, to all the things which possess theattributes 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. Theterm 'virtue, ' for instance, in one sense, namely, as signifying moralexcellence in general, without distinction of kind, is strictly asingular term, as being the name of one attribute: but as applied todifferent varieties of moral excellence--justice, generosity, gentleness and so on--it is a common term, as being a name which isapplicable, in the same sense, to a class of attributes. Similarly theterm 'colour, ' in a certain sense, signifies one unvarying attributepossessed by bodies, namely, the power of affecting the eye, and inthis sense it is a singular term: but as applied to the various waysin which the eye may be affected, it is evidently a common term, beingequally applicable to red, blue, green, and every other colour. Assoon as we begin to abstract from attributes, the higher notionbecomes a common term in reference to the lower. By a 'higher notion'is meant one which is formed by a further process of abstraction. Theterms 'red, ' 'blue, ' 'green, ' etc. , are arrived at by abstraction fromphysical objects; 'colour' is arrived at by abstraction from them, andcontains nothing, but what is common to all. It therefore applies inthe 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 givencase, is being used as a singular or common term, is to try whetherthe indefinite article or the sign of the plural can be attached toit. 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 'thenumbers, two, three, four, ' etc. , at once marks it as a commonterm. Similarly the term 'unity' denotes a single attribute, admittingof no shades of distinction: but when a writer begins to speak of 'theunities' he is evidently using the word for a class of things of somekind or other, namely, certain dramatical proprieties of composition. 

Proper _Names_ and _Designations_. 

§ 112. Singular terms may be subdivided into Proper Names andDesignations. 

§ 113. A Proper Name is a permanent singular term applicable to athing in itself; a Designation is a singular term devised for theoccasion, or applicable to a thing only in so far as it possesses someattribute. 

§ 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, grantingthat the physical universe consists of a definite number ofatoms--neither one more nor one less--still we are far from havingexhausted the possible number of things. All the manifold materialobjects, which are made up by the various combinations of these atoms, constitute separate objects of thought, or things, and the mind hasfurther an indefinite power of conjoining and dividing these objects, so as to furnish itself with materials of thought, and also of fixingits attention by abstraction upon attributes, so as to regard them asthings, apart from the substances to which they belong. 

§ 116. This being so, it is only a very small number of things, whichare constantly obtruding themselves upon the mind, that have singularterms permanently set apart to denote them. Human beings, somedomestic animals, and divisions of time and place, have proper namesassigned 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 tolower notions, are of the nature of proper names, being permanentlyset apart to denote certain special attributes, e. G. 'benevolence, ''veracity, ' 'imagination, ' 'indigestibility, 'retrenchment. '

§ 117. But the needs of language often require a singular term todenote some thing which has not had a proper name assigned to it. Thisis effected by taking a common term, and so limiting it as to make itapplicable, under the given circumstances, to one thing only. Such alimitation may be effected in English by prefixing a demonstrative orthe 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, likethose which have preceded it, fundamental and exhaustive, there isevidently no room in it for a third class of Collective Terms. Nor isthere any distinct class of terms to which that name can be given. Thesame term may be used collectively or distributively in differentrelations. Thus the term 'library, ' when used of the books whichcompose a library, is collective; when used of various collections ofbooks, as the Bodleian, Queen's library, and so on, it isdistributive, which, in this case, is the same thing as being a commonterm. 

§ 119, The distinction between the collective and distributive use ofa term is of importance, because the confusion of the two is afavourite source of fallacy. When it is said 'The plays of Shakspearecannot be read in a day, ' the proposition meets with a very differentmeasure of acceptance according as its subject is understoodcollectively or distributively. The word 'all' is perfectly ambiguousin this respect. It may mean all together or each separately--twosenses which are distinguished in Latin by 'totus' or 'cunctus, ' forthe collective, and 'omnis' for the distributive use. 

§ 120. What is usually meant however when people speak of a collectiveterm is a particular kind of singular term. 

§ 121. From this point of view singular terms may be subdivided intoIndividual and Collective, by an Individual Term being meant the nameof one object, by a Collective Term the name of several considered asone. 'This key' is an individual term; 'my bunch of keys' is acollective term. 

§ 122. A collective term is quite as much the name of one thing as anindividual term is, though the thing in question happens to be agroup. A group is one thing, if we choose to think of it as one. Forthe mind, as we have already seen, has an unlimited power of formingits own things, or objects of thought. Thus a particular peak in amountain chain is as much one thing as the chain itself, though, physically speaking, it is inseparable from it, just as the chainitself is inseparable from the earth's surface. In the same way anecklace is as much one thing as the individual beads which composeit. 

§ 123. We have just seen that a collective term is the name of a groupregarded as one thing: but every term which is the name of such agroup is not necessarily a collective term. 'London, ' for instance, isthe 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 acertain extent, similar, whereas London is a group made up of the mostdissimilar objects--streets and squares and squalid slums, finecarriages and dirty faces, and so on. In the case of a true collectiveterm all the members of the group will come under some one commonname. Thus all the members of the group, flock of sheep, come underthe common name 'sheep, ' all the members of the group 'regiment' underthe common name, 'soldier, ' and so on. 

§ 124. The subdivision of singular terms into individual andcollective need not be confined to the names of concrete things. Anabstract term like 'scarlet, ' which is the name of one definiteattribute, may be reckoned 'individual, ' while a term like 'humannature, ' which is the name of a whole group of attributes, would morefitly be regarded as collective. 

§ 126. The main division of terms, which we have been discussing, intosingular and collective, is based upon their Quantity inExtension. 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 divisioninto positive and non-positive, the latter member being subdividedinto 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, orcomparatively definite, member of an exhaustive division in accordancewith the law of excluded middle (§ 25). Thus 'Socrates' and 'man' arepositive, as opposed to 'not-Socrates' and 'not-man. '

§ 127. The chief value of the division, however, and especially of thedistinction drawn between privative and negative terms, is in relationto attributives. 

From this point of view we may define the three classes of terms asfollows:

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 subjectcapable of possessing it, e. G. 'unwise, ' 'empty'. [Footnote: Aprivative term is usually defined to mean one which signifies theabsence of an attribute where it was once possessed, or might havebeen expected to be present, e. G. 'blind. ' The utility of the slightextension of meaning here assigned to the expression will, it ishoped, prove its justification. ]

§ 128. Thus a privative term stands midway in meaning between theother two, being partly positive and partly negative--negative in sofar as it indicates the absence of a certain attribute, positive in sofar as it implies that the thing which is declared to lack thatattribute is of such a nature as to be capable of possessing it. Apurely negative term conveys to the mind no positive information atall about the nature of the thing of which it is predicated, butleaves us to seek for it among the universe of things which fail toexhibit a given attribute. 

A privative term, on the other hand, restricts us within a definitesphere. The term 'empty' restricts us within the sphere of thingswhich are capable of fulness, that is, if the term be taken in itsliteral 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 beeither 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 extensionas well as of those which do. Such a pair of terms as 'white' and'not-white, ' in their relation to one another, are calledContradictories. 

§ 130. Contrary terms must be distinguished fromcontradictory. Contrary terms are those which are most opposed underthe same head. Thus 'white' and 'black' are contrary terms, being themost opposed under the same head of colour. 'Virtuous' and 'vicious'again are contraries, being the most opposed under the same head ofmoral quality. 

§ 131. A positive and a privative term in the same matter will alwaysbe contraries, e. G. 'wise' and 'unwise, ' 'safe' and 'unsafe': butcontraries do not always assume the shape of positive and privativeterms, 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 inmeaning, and vice versâ. This is the case, for instance, with the word'safe, ' which connotes nothing more than the absence of danger. Wetalk 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 athing, but that it is _positively injurious_. All such questions, however, are for the grammarian or lexicographer, and not for thelogician. For the latter it is sufficient to know that correspondingto every term which signifies the presence of some attribute there maybe 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 requireto speak of the whole universe of objects minus those which possess agiven attribute or collection of attributes. We have often occasion tospeak 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 correspondingabstract term, and the same is the case with negatives: for theabsence of an attribute, is itself an attribute. Corresponding to'empty, ' there is 'emptiness'; corresponding to 'not-full' there maybe 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, withoutbeing directly contrary, involve a latent contradiction, are calledRepugnant, e. G. 'red' and 'blue. ' All terms whatever which signifyattributes that exclude one another may be called Incompatible. 

§ 136. The preceding division is based on what is known as the Qualityof terms, a positive term being said to differ in quality from anon-positive one. 

_Univocal and Equivocal Terms_. 

§ 137. A term is said to be Univocal, when it has one and the samemeaning wherever it occurs. A term which has more than one meaning iscalled Equivocal. 'Jam-pot, ' 'hydrogen' are examples of univocalterms; '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 thelogician's point of view an equivocal term is two or more differentterms, for the definition in each sense would be different. 

§ 139. Sometimes a third member is added to the same division underthe head of Analogous Terms. The word 'sweet, ' for instance, isapplied by analogy to things so different in their own nature as alump of sugar, a young lady, a tune, a poem, and so on. Again, becausethe head is the highest part of man, the highest part of a stream iscalled by analogy 'the head. ' It is plainly inappropriate to make aseparate class of analogous terms. Rather, terms become equivocal bybeing extended by analogy from one thing to another. 

_Absolute and Relative Terms_. 

§ 140. An Absolute term is a name given to a thing without referenceto anything else. 

§ 141. A Relative term is a name given to a thing with directreference 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 ofa relation, the other is called the correlative, e. G. 'subject' isthe correlative of 'ruler, ' and conversely 'ruler' of 'subject. ' Thetwo terms are also spoken of as a pair of correlatives. 

§ 143. The distinction between relative and absolute applies toattributives as well as subject-terms. 'Greater, ' 'near, 'like, ' areinstances 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 moresubstances than one. Every relation is at bottom a fact, or series offacts, in which two or more substances play a part. A relative termconnotes this fact or facts from the point of view of one of thesubstances, its correlative from that of the other. Thus 'ruler' and'subject' imply the same set of facts, looked at from opposite pointsof view. The series of facts itself, regarded from either side, isdenoted by the corresponding abstract terms, 'rule 'and 'subjection. '

§ 145. It is a nice question whether the abstract names of relationsshould themselves be considered relative terms. Difficulties willperhaps be avoided by confining the expression 'relative _term_'to names of concrete things. 'Absolute, ' it must be remembered, is amere negative of 'relative, ' and covers everything to which thedefinition of the latter does not strictly apply. Now it can hardly besaid that 'rule' is a name given to a certain abstract thing withdirect reference to some other thing, namely, subjection. Rather'rule' and 'subjection' are two names for identically the same seriesof facts, according to the side from which we look at them. 'Ruler'and 'subject, ' on the other hand, are names of two distinctsubstances, but each involving a reference to the other. 

§ 146. This division then may be said to be based on the number ofthings involved in the name. 

_Connotative and Non-Connotative Terms. _

§ 147. Before explaining this division, it is necessary to treat ofwhat 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 itapplies. 

§ 150. The Intension of a term is the number of attributes which itimplies. 

§ 151. It will simplify matters to bear in mind that the intension ofa 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 humanrace that have been, are, or ever will be: this is its quantity inextension. But the term 'man' has also a certain meaning, and impliescertain attributes--rationality, animality, and a definite bodilyshape: the sum of these attributes constitutes its quantity inintension. 

§ 152. The distinction between the two kinds of quantity possessed bya term is also conveyed by a variety of expressions which are hereappended. 

Extension = breadth = compass = application = denotation. 

Intension = depth = comprehension = implication = connotation. 

Of these various expressions, 'application' and 'implication' have theadvantage 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 andnon-connotative. 

§ 154. A term is said to connote attributes, when it implies certainattributes at the same time that it applies to certain things distincttherefrom. [Footnote: Originally 'connotative' was used in the samesense in which we have used 'attributive, ' for a word which directlysignifies the presence of an attribute and indirectly applies to asubject. In this, its original sense, it was the subject which wassaid to be connoted, and not the attribute. ]

§ 155. A term which possesses both extension and intension, distinctfrom one another, is connotative. 

§ 156. A term which possesses no intension (if that be possible) or inwhich extension and intension coincide is non-connotative. 

§ 157. The subject-term, 'man, ' and its corresponding attributive, 'human, ' have both extension and intension, distinct from oneanother. They are therefore connotative. But the abstract term, 'humanity, ' denotes the very collection of attributes, which wasbefore connoted by the concrete terms, 'man' and 'human. ' In thiscase, therefore, extension and intension coincide, and the term isnon-connotative. 

§ 158. The above remark must be understood to be limited to abstractterms in their singular sense. When employed as common terms, abstractterms possess both extension and intension distinct from oneanother. Thus the term 'colour' applies to red, blue, and yellow, andat the same time implies (i. E. Connotes), the power of affecting theeye. 

§ 159. Since all terms are names of things, whether substances orattributes, it is clear that all terms must possess extension, thoughthe extension of singular terms is the narrowest possible, as beingconfined to one thing. 

§ 160. Are there then any terms which possess no intension? To askthis, is to ask--Are there any terms which have absolutely no meaning?It is often said that proper names are devoid of meaning, and theremark is, in a certain sense, true. When we call a being by the name'man, ' we do so because that being possesses human attributes, butwhen we call the same being by the name, 'John, ' we do not mean toindicate the presence of any Johannine attributes. We simply wish todistinguish that being, in thought and language, from other beings ofthe same kind. Roughly speaking, therefore, proper names are devoid ofmeaning or intension. But no name can be entirely devoid ofmeaning. For, even setting aside the fact, which is not universallytrue, that proper names indicate the sex of the owner, the mere act ofgiving 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 moreintension than any other terms. For when we know a person, his namecalls up to our minds all the individual attributes with which we arefamiliar, and these must be far more numerous than the attributeswhich are conveyed by any common term which can be applied tohim. Thus the name 'John' means more to a person who knows him than'attorney, ' 'conservative, ' 'scamp, ' of 'vestry-man, ' or any otherterm which may happen to apply to him. This, however, is the acquiredintension of a term, and must be distinguished from the originalintension. The name 'John' was never meant to indicate the attributeswhich its owner has, as a matter of fact, developed. He would be Johnall the same, if he were none of these. 

§ 162. Hitherto we have been speaking only of christening-names, butit is evident that family names have a certain amount of connotationfrom the first. For when we dub John with the additional appellationof Smith, we do not give this second name as a mere individual mark, but intend thereby to indicate a relationship to other persons. Theamount of connotation that can be conveyed by proper names is verynoticeable in the Latin language. Let us take for an example the fullname of a distinguished Roman--Publius Cornelius Scipio ÆmilianusAfricanus minor. Here it is only the prænomen, Publius, that can besaid to be a mere individual mark, and even this distinctly indicatesthe sex of the owner. The nomen proper, Cornelius, declares the wearerof it to belong to the illustrious gens Cornelia. The cognomen, Scipio, further specifies him as a member of a distinguished family inthat gens. The agnomen adoptivum indicates his transference byadoption from one gens to another. The second agnomen recalls thefact of his victory over the Carthaginians, while the addition of theword 'minor' distinguishes him from the former wearer of the sametitle. The name, instead of being devoid of meaning, is a chapter ofhistory in itself. Homeric epithets, such as 'The Cloud-compeller, ''The Earth-shaker' are instances of intensive proper names. Many ofour 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 bearerof 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 presumablymore noteworthy parent, as Jackson, Thomson, Fitzgerald, O'Connor, Macdonald, Apjohn, Price, Davids, etc. The question, however, whethera 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 themwithout conveying more or less information about the qualities of thething which they are used to denote. When, for instance, we say 'thistable, ' 'this book, ' we indicate the proximity to the speaker of theobject in question. Other designations have a higher degree ofintension, 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 fromany knowledge we may happen to possess of the individuals they denote. 

§ 164. We have seen that, speaking quite strictly, there are no termswhich are non-connotative: but, for practical purposes, we may applythe expression to proper names, on the ground that they possess nointension, and to singular abstract terms on the ground that theirextension and intension coincide. In the latter case it is indifferentwhether we call the quantity extension or intension. Only we cannotcall it 'connotation, ' because that implies two quantities distinctfrom one another. A term must already denote a subject before it canbe said to connote its attributes. 

§ 165. The division of terms into connotative and non-connotative isbased 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 theextension decreases, the intension increases, and vice versâ. Take forinstance 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 theleast possible amount of intension, implying nothing more than mereexistence, whether in fact or thought. On the other hand, the term atthe bottom possesses the greatest amount of intension, since itimplies all the attributes of, an individual superadded to those ofthe class to which it belongs: but its extension is the narrowestpossible, being limited to one thing. 

§ 167. At each step in the descent from the term at the top, which iscalled the 'Summum genus, ' to the individual, we decrease theextension by increasing the intension. Thus by adding on to the barenotion of a thing the idea of independent existence, we descend to theterm 'substance, ' This process is known as Determination, orSpecialisation. 

§ 168. Again, by withdrawing our attention from the individualcharacteristics of a particular sheep, and fixing it upon those whichare common to it with other animals of the same kind, we arrive at thecommon term, 'sheep. ' Here we have increased the extension bydecreasing the intension. This process is known as Generalisation. 

§ 169. Generalisation implies abstraction, but we may have abstractionwithout generalisation. 

§ 170. The following example is useful, as illustrating to the eye howa decrease of extension is accompanied by an increase of intension. Ateach step of the descent here we visibly tack on a freshattribute. [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 beexactly inverted. 

§ 171. The law of inverse variation of extension and intension must ofcourse be confined to the inter-relations of a series of terms ofwhich each can be predicated of the other until we arrive at thebottom of the scale. It is not meant to apply to the extension andintension of the same term. The increase of population does not add tothe 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 itfrom words generally, so now, in considering the proposition, it willbe well to begin by distinguishing it from other sentences. 

§ 173. Every proposition is a sentence, but every sentence is not aproposition. 

§ 174. The field of logic is far from being conterminous with that oflanguage. Language is the mirror of man's whole nature, whereas logicdeals with language only so far as it gives clothing to the productsof thought in the narrow sense which we have assigned to that term. Language has materials of every sort lying strewn about, among whichthe 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 andPotential, or Conditional, sentences of grammar. For it must be bornein mind that logic recognises no difference between a statement offact and a supposition. 'It may rain to-morrow' is as much aproposition as 'It is raining now. '

§ 177. Leaving the grammatical aspect of the proposition, we must nowconsider it from the purely logical point of view. 

§ 178. A proposition is a judgement expressed in words; and ajudgement is a direct comparison between two concepts. 

§ 179. The same thing may be expressed more briefly by saying that aproposition is a direct comparison between two terms. 

§ 180. We say 'direct comparison, ' because the syllogism also may bedescribed as a comparison between two terms: but in the syllogism thetwo 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 disagreementbetween 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 thepredicate for the sake of the subject. 

CHAPTER II. 

Of _the Copula_. 

§ 186. There are two kinds of copula, one for affirmative and one fornegative statements. 

§ 187. Materially the copula is expressed by some part of the verb 'tobe, ' with or without the negative, or else is wrapped up in someinflexional form of a verb. 

§ 188. The material form of the copula is an accident of language, anda matter of indifference to logic. 'The kettle boils' is as logical aform of expression as 'The kettle is boiling. ' For it must beremembered that the word 'is' here is a mere sign of agreement betweenthe two terms, and conveys no notion of actual existence. We may useit indeed with equal propriety to express non-existence, as when wesay 'An idol is nothing. '

§ 189. When the verb 'to be' expresses existence in fact it is knownin grammar as 'the substantive verb. ' In this use it is predicate aswell as copula, as when we say 'God is, ' which may be analysed, if weplease, 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 thecopula is always affirmative. 

§ 191. What then, it may be asked, on this view, is the meaning ofnegative propositions! To which the answer is, that a negativeproposition asserts an agreement between the subject and a negativeterm. When, for instance, we say 'The whale is not a fish, ' this wouldbe interpreted to mean 'The whale is a not-fish. '

§ 192. Undoubtedly any negative proposition may be exhibited in anaffirmative form, since, by the law of excluded middle, given a pairof contradictory terms, wherever the one can be asserted, the othercan be denied, and vice versâ. We shall find later on that thisprinciple gives rise to one of the forms of immediate inference. Theonly question then can be, which is the more natural and legitimateform of expression. It seems simpler to suppose that we assert theagreement of 'whale' with 'not-fish' by implication only, and thatwhat we directly do is to predicate a disagreement between 'whale' andthe positive attributes connoted by 'fish. ' For since 'not-fish' mustapply to every conceivable object of thought except those which fallunder 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 thewhole universe of being, minus a limited portion; which is only a moreclumsy way of saying that it is not to be found in that portion. 

§ 193. Again, the term 'not-fish' must be understood either in itsintension or in its extension. If it be understood in its intension, what it connotes is simply the absence of the positive qualities whichconstitute a fish, a meaning which is equally conveyed by the negativeform of proposition. We gain nothing in simplicity by thus confoundingassertion with denial. If, on the other hand, it is to be taken inextension, this involves the awkwardness of supposing that thepredicative power of a term resides in its extensive capacity. 

§ 194. We therefore recognise predication as being of twokinds--affirmation and negation--corresponding to which there are twoforms of copula. 

§ 195. On the other hand, other logicians have maintained that thereare many kinds of copula, since the copula must vary according to thevarious degrees of probability with which we can assert or deny apredicate of a subject. This view is technically known as the doctrineof

_The Modality of the Copula. _

§ 196. It may plausibly be maintained that the division ofpropositions into affirmative and negative is not an exhaustive one, since the result of an act of judgement is not always to lead the mindto a clear assertion or a clear denial, but to leave it in more orless doubt as to whether the predicate applies to the subject ornot. Instead of saying simply A is B, or A is not B, we may be led toone 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 isknown as 'the mode. '

§ 197. When we say 'The accused may be guilty' we have a propositionof very different force from 'The accused is guilty, ' and yet theterms appear to be the same. Wherein then does the difference lie? 'Inthe copula' would seem to be the obvious reply. We seem thereforedriven to admit that there are as many different kinds of copula asthere are different degrees of assurance with which a statement may bemade. 

§ 198. But there is another way in which modal propositions may beregarded. Instead of the mode being attached to the copula, it may beconsidered as itself constituting the predicate, so that the abovepropositions 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 predicatevarious degrees of probability. In this way the division ofpropositions into affirmative and negative is rendered exhaustive. Forwherever before we had a doubtful assertion, we have now an assertionof doubtfulness. 

§ 200. If degrees of probability can thus be eliminated from thecopula, much more so can expressions of time, which may always beregarded as forming part of the predicate. 'The sun will riseto-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 oftenan awkward task so to analyse propositions relative to past or futuretime as to bring out the copula under the form 'is' or 'is not': butfortunately there is no necessity for so doing, since, as has beensaid before (§ 188), the material form of the copula is a matter ofindifference to logic. Indeed in affirmative propositions the merejuxtaposition of the subject and predicate is often sufficient toindicate their agreement, e. G. 'Most haste, worst speed, ' chalephatha kala. It is because all propositions are not affirmative that werequire a copula at all. Moreover the awkwardness of expression justalluded to is a mere accident of language. In Latin we may say withequal propriety 'Sol orietur cras' or 'Sol est oriturus cras'; whilepast time may also be expressed in the analytic form in the case ofdeponent verbs, as 'Caesar est in Galliam profectus'--'Caesar is goneinto Gaul. '

§ 201. The copula then may always be regarded as pure, that is, asindicating mere agreement or disagreement between the two terms of theproposition. 

CHAPTER III. 

_Of the Divisions of Propositions_. 

§ 202. The most obvious and the most important division ofpropositions is into true and false, but with this we are notconcerned. Formal logic can recognise no difference between true andfalse propositions. The one is represented by the same symbols as theother. 

§ 203. We may notice, however, in passing, that truth and falsehoodare attributes of propositions and of propositions only. For somethingmust be predicated, i. E. Asserted or denied, before we can haveeither truth or falsehood. Neither concepts or terms, on the one hand, nor reasonings, on the other, can properly be said to be true orfalse. In the mere notion of a Centaur or of a black swan there isneither truth nor falsehood; it is not until we make some statementabout these things, such as that 'black swans are found in Australia, 'or 'I met a Centaur in the High Street yesterday, ' that the questionof truth or falsehood comes in. In such expressions as a 'true friend'or 'a false patriot' there is a tacit reference to propositions. Wemean persons of whom the terms 'friend' and 'patriot' are truly orfalsely predicated. Neither can we with any propriety talk of true orfalse reasoning. Reasoning is either valid or invalid: it is only thepremisses of our reasonings, which are propositions, that can be trueor false. We may have a perfectly valid process of reasoning whichstarts from a false assumption and lands us in a false conclusion. 

§ 204. All truth and falsehood then are contained in propositions; andpropositions are divided according to the Quality of the Matter intotrue and false. But the consideration of the matter is outside thesphere of formal or deductive Logic. It is the problem of inductivelogic to establish, if possible, a criterion of evidence whereby thetruth 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 degreeof probability. This division is excluded by the view which has justbeen taken of the copula, as being always simply affirmative or simplynegative. 

§ 206. We are left then with the following divisions ofpropositions--

 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 directlyaffirmed 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 madesubject 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 ismade to depend. It precedes the other in the order of thought, but mayeither precede or follow it in the order of language. Thus we may sayindifferently--'If the wind drops, we shall have rain' or 'We shallhave rain, if the wind drops. '

§ 213. The Consequent is the statement which is made subject to somecondition. 

§ 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 inthe 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 schemeis appended. 

 Proposition ____________|_____________ | | Simple Complex (Categorical) (Conditional) ___________|__________ | | Conjunctive Disjunctive. (Hypothetical)

§ 217. The first set of names is preferable. 'Categorical' properlymeans 'predicable' and 'hypothetical' is a mere synonym for'conditional. '

§ 218. Let us examine now what is the real nature of the statementwhich 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 wereally mean to assert? Not that the sky will fall, and not that weshall catch larks, but a certain connection between the two, namely, that the truth of the antecedent involves the truth of theconsequent. This is why this form of proposition is called'conjunctive, ' because in it the truth of the consequent is conjoinedto the truth of the antecedent. 

§ 219. Again, when we say 'Jones is either a knave or a fool, ' what isreally meant to be asserted is--'If you do not find Jones to be aknave, you may be sure that he is a fool. ' Here it is the falsity ofthe antecedent which involves the truth of the consequent; and theproposition is known as 'disjunctive, ' because the truth of theconsequent is disjoined from the truth of the antecedent. 

§ 220. Complex propositions then turn out to be propositions aboutpropositions, that is, of which the subject and predicate arethemselves propositions. But the nature of a proposition never variesin thought. Ultimately every proposition must assume the form 'A is, or is not, B. ' 'If the sky falls, we shall catch larks' may becompressed into 'Sky-falling is lark-catching. '

§ 221. Hence this division turns upon the form of expression, and maybe said to be founded on the simplicity or complexity of the termsemployed in a proposition. 

§ 222. In the complex proposition there appears to be more than onesubject or predicate or both, but in reality there is only a singlestatement; and this statement refers, as we have Seen, to a certainconnection between two propositions. 

§ 223. If there were logically, and not merely grammatically, morethan one subject or predicate, there would be more than oneproposition. Thus when we say 'The Jews and Carthaginians were Semiticpeoples and spoke a Semitic language, ' we have four propositionscompressed into a single sentence for the sake of brevity. 

§ 224. On the other hand when we say 'Either the Carthaginians were ofSemitic 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 thosecontrivances of language for abbreviating expression in which severaldistinct statements are combined into a single sentence. 

_Verbal and Real Propositions_. 

§ 226. A Verbal Proposition is one which states nothing more about thesubject 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 thedefinition of the subject, e. G. 'Some animals have four feet. '

§ 228. It will be seen that the distinction between verbal and realpropositions assumes a knowledge of the precise meaning of terms, thatis to say, a knowledge of definitions. 

§ 229. To a person who does not know the meaning of terms a verbalproposition will convey as much information as a real one. To say 'Thesun is in mid-heaven at noon, ' though a merely verbal proposition, will convey information to a person who is being taught to attach ameaning to the word 'noon. ' We use so many terms without knowing theirmeaning, that a merely verbal proposition appears a revelation to manyminds. Thus there are people who are surprised to hear that the lionis a cat, though in its definition 'lion' is referred to the class'cat. ' The reason of this is that we know material objects far betterin their extension than in their intension, that is to say, we knowwhat things a name applies to without knowing the attributes whichthose things possess in common. 

§ 230. There is nothing in the mere look of a proposition to inform uswhether it is verbal or real; the difference is wholly relative to, and constituted by, the definition of the subject. When we haveaccepted as the definition of a triangle that it is 'a figurecontained by three sides, ' the statement of the further fact that ithas three angles becomes a real proposition. Again the proposition'Man is progressive' is a real proposition. For though hisprogressiveness is a consequence of his rationality, still there is noactual reference to progressiveness contained in the usually accepteddefinition, 'Man is a rational animal. '

§ 231. If we were to admit, under the term 'verbal proposition, ' allstatements which, though not actually contained in the definition ofthe subject, are implied by it, the whole body of necessary truthwould have to be pronounced merely verbal, and the most penetratingconclusions of mathematicians set down as only another way of statingthe simplest axioms from which they started. For the propositions ofwhich necessary truth is composed are so linked together that, givenone, the rest can always follow. But necessary truth, which is arrivedat 'a priori, ' that is, by the mind's own working, is quite as real ascontingent truth, which is arrived at 'a posteriori, ' or by theteachings of experience, in other words, through our own senses orthose 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 seesthat what has three sides cannot but have three angles. 

§ 233. Only such propositions then must be considered verbal as statefacts expressly mentioned in the definition. 

§ 234. Strictly speaking, the division of propositions into verbal andreal is extraneous to our subject: since it is not the province oflogic to acquaint us with the content of definitions. 

§ 235, The same distinction as between verbal and real proposition, isconveyed by the expressions 'Analytical' and 'Synthetical, ' or'Explicative' and 'Ampliative' judgements. 

§ 236. A verbal proposition is called analytical, as breaking up thesubject into its component notions. 

§ 237. A real proposition is called synthetical, as attaching some newnotion to the subject. 

§ 238. Among the scholastic logicians verbal propositions were knownas 'Essential, ' because what was stated in the definition wasconsidered to be of the essence of the subject, while realpropositions were known as 'Accidental. '

_Universal AND PARTICULAR Propositions_. 

§ 239. A Universal proposition is one in which it is evident from theform that the predicate applies to the subject in its whole extent. 

§ 240. When the predicate does not apply to the subject in its wholeextent, or when it is not clear that it does so, the proposition iscalled Particular. 

§ 241. To say that a predicate applies to a subject in its wholeextent, is to say that it is asserted or denied of all the things ofwhich 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 mustbe 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 theQuantity of propositions. 

§ 246. The quantity of a proposition is determined by the quantity inextension of its subject. 

§ 247. Very often the matter of an indefinite proposition is such asclearly to indicate to us its quantity. When, for instance, we say'Metals are elements, ' we are understood to be referring to allmetals; and the same thing holds true of scientific statements ingeneral. Formal logic, however, cannot take account of the matter ofpropositions; and is therefore obliged to set down all indefinitepropositions as particular, since it is not evident from the form thatthey are universal. 

§ 248. Particular propositions, therefore, are sub-divided into suchas are Indefinite and such as are Particular, in the strict sense ofthe term. 

§ 249. We must now examine the sub-division of universal propositionsinto Singular and General. 

§ 250. A Singular proposition is one which has a singular term for itssubject, e. G. 'Virtue is beautiful. '

§ 251. A General proposition is one which has for its subject a commonterm taken in its whole extent. 

§ 252. Now when we say 'John is a man' or 'This table is oblong, ' theproposition is quite as universal, in the sense of the predicateapplying to the whole of the subject, as when we say 'All men aremortal. ' For since a singular term applies only to one thing, wecannot avoid using it in its whole extent, if we use it at all. 

§ 253. The most usual signs of generality in a proposition are thewords '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 toquantity is unsatisfactory. Not only has the indefinite proposition tobe set down as particular, even when the sense manifestly declares itto be universal; but the proposition which is expressed in aparticular form has also to be construed as indefinite, _so_ thatan unnatural meaning is imparted to the word 'some, ' as used inlogic. If in common conversation we were to say 'Some cows chew thecud, ' the person whom we were addressing would doubtless imagine us tosuppose that there were some cows which did not possess thisattribute. But in logic the word 'some' is not held to express morethan 'some at least, if not all. ' Hence we find not only that anindefinite proposition may, as a matter of fact, be strictlyparticular, but that a proposition which appears to be strictlyparticular may be indefinite. So a proposition expressed in preciselythe same form 'Some A is B' may be either strictly particular, if somebe taken to exclude all, or indefinite, if the word 'some' does notexclude the possibility of the statement being true of all. It isevident that the term 'particular' has become distorted from itsoriginal meaning. It would naturally lead us to infer that a statementis limited to part of the subject, whereas, by its being opposed touniversal, in the sense in which that term has been defined, it canonly mean that we have nothing to show us whether part or the whole isspoken of. 

§ 255. This awkwardness of expression is due to the indefiniteproposition having been displaced from its proper position. Formerlypropositions were divided under three heads--

 (1) Universal, (2) Particular, (3) Indefinite. 

But logicians anxious for simplification asked, whether a predicate inany given case must not either apply to the whole of the subject ornot? And whether, therefore, the third head of indefinite propositionswere not as superfluous as the so-called 'common gender' of nouns ingrammar?

§ 256. It is quite true that, as a matter of fact, any given predicatemust either apply to the whole of the subject or not, so that in thenature of things there is no middle course between universal andparticular. But the important point is that we may not know whetherthe predicate applies to the whole of the subject or not. The primarydivision then should be into propositions whose quantity is known andpropositions whose quantity is unknown. Those propositions whosequantity is known may be sub-divided into 'definitely universal' and'definitely particular, ' while all those whose quantity is unknown areclassed together under the term 'indefinite. ' Hence the properdivision 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 soused as to include it; while, on the other hand, there is no obviousdifference between universal and general, though in the division thelatter 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 orfalse. But since formal logic takes no account of the matter ofthought, when we speak of 'quality' we are understood to mean thequality of the form. 

§ 260. By combining the division of propositionsaccording 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 anyproposition, no matter what its form may be, must fall under one orother of these four heads. For every proposition must be eitheruniversal or particular, in the sense that the subject must either beknown to be used in its whole extent or not; and any proposition, whether universal or particular, must be either affirmative ornegative, for by denying modality to the copula we have excludedeverything intermediate between downright assertion and denial. Thisclassification therefore may be regarded as a Procrustes' bed, intowhich every proposition is bound to fit at its proper peril. 

§ 262. These four kinds of propositions are represented respectivelyby the symbols A, E, I, O. 

§ 263. The vowels A and I, which denote the two affirmatives, occur inthe Latin words 'affirmo' and 'aio;' E and O, which denote the twonegatives, occur in the Latin word 'nego. '

_Extensive and Intensive Propositions. _

§ 264. It is important to notice the difference between Extensive andIntensive propositions; but this is not a division of propositions, but a distinction as to our way of regarding them. Propositions may beread either in extension or intension. Thus when we say 'All cows areruminants, ' we may mean that the class, cow, is contained in thelarger class, ruminant. This is reading the proposition inextension. Or we may mean that the attribute of chewing the cud iscontained in, or accompanies, the attributes which make up our idea of'cow. ' This is reading the proposition in intension. What, as a matterof 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 andnatural form of proposition the subject is used in extension, and thepredicate in intension, that is to say, when we use a subject, we arethinking of certain objects, whereas when we use a predicate, weindicate the possession of certain attributes. The predicate, however, need not always be used in intension, e. G. In the proposition 'Hisname is John' the predicate is not intended to convey the idea of anyattributes at all. What is meant to be asserted is that the name ofthe person in question is that particular name, John, and notZacharias 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 inintension, the subject contains the predicate. 

_Exclusive Propositions. _

§ 266. An Exclusive Proposition is so called because in it all but agiven subject is excluded from participation in a given predicate, e. G. 'The good alone are happy, ' 'None but the brave deserve thefair, ' 'No one except yourself would have done this. '

§ 267. By the above forms of expression the predicate is declared toapply to a given subject and to that subject only. Hence an exclusiveproposition is really equivalent to two propositions, one affirmativeand 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. Itdoes not necessarily mean that all the good are happy, but assertsthat among the good will be found all the happy. It is thereforeequivalent to saying that all the happy are good, only that it putsprominently forward in addition what is otherwise a latent consequenceof that assertion, namely, that some at least of the good are happy. 

§ 268. Logically expressed the exclusive proposition when universalassumes the form of an E proposition, with a negative term for itssubject

 No not-A is B. 

§ 269. Under the head of exclusive comes the strictly particularproposition, 'Some A is B, ' which implies at the same time that 'SomeA is not B. ' Here 'some' is understood to mean 'some only, ' which isthe meaning that it usually bears in common language. When, forinstance, we say 'Some of the gates into the park are closed atnightfall, ' we are understood to mean 'Some are left open. '

_Exceptive Propositions. _

§ 270. An Exceptive Proposition is so called as affirming thepredicate of the whole of the subject, with the exception of a certainpart, e. G. 'All the jury, except two, condemned the prisoner. '

§ 271. This form of proposition again involves two distinctstatements, one negative and one affirmative, being equivalent to 'Twoof the jury did not condemn the prisoner; and all the rest did. '

§ 272. The exceptive proposition is merely an affirmative way ofstating 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 ofitself, e. G. 'A man's a man, ' 'What I have written, I have written, ''Whatever is, is. ' The second of these instances amounts formally tosaying 'The thing that I have written is the thing that I havewritten, ' though of course the implication is that the writing willnot be altered. 

CHAPTER IV. 

_Of the Distribution of Terms. _

§ 274. The treatment of this subject falls under the second part oflogic, 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 inits whole extent, that is, with reference to all the things of whichit 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 usedin its whole extent. But when we say 'Men are miserable' or 'Some menare 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 quantityis undefined, and to one whose quantity is definitely limited to apart of its possible extent. 

§ 278. This awkwardness arises from not inquiring first whether thequantity of a term is determined or undetermined, and afterwardsproceeding to inquire, whether it is determined as a whole or part ofits 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 thesestatements and sometimes to contradict the other. 

§ 279. With regard to the quantity of the subject of a proposition nodifficulty can arise. The use of the words 'all' or 'some, ' or of avariety of equivalent expressions, mark the subject as beingdistributed or undistributed respectively, while, if there be nothingto mark the quantity, the subject is for that reason reckonedundistributed. 

§ 280. With regard to the predicate more difficulty may arise. 

§ 281. It has been laid down already that, in the ordinary form ofproposition, the subject is used in extension and the predicate inintension. Let us illustrate the meaning of this by an example. Ifsomeone were to say 'Cows are ruminants, ' you would have a right toask him whether he meant 'all cows' or only 'some. ' You would not byso doing be asking for fresh information, but merely for a moredistinct explanation of the statement already made. The subject beingused in extension naturally assumes the form of the whole or part of aclass. But, if you were to ask the same person 'Do you mean that cowsare all the ruminants that there are, or only some of them?' he wouldhave a right to complain of the question, and might fairly reply, 'Idid not mean either one or the other; I was not thinking of ruminantsas a class. I wished merely to assert an attribute of cows; in fact, Imeant no more than that cows chew the cud. '

§ 282. Since therefore a predicate is not used in extension at all, itcannot possibly be known whether it is used in its whole extent ornot. 

§ 283. It would appear then that every predicate is necessarilyundistributed; and this consequence does follow in the case ofaffirmative propositions. 

§ 284. In a negative proposition, however, the predicate, though stillused in intension, must be regarded as distributed. This arises fromthe nature of a negative proposition. For we must remember that in anyproposition, although the predicate be not meant in extension, italways admits of being so read. Now we cannot exclude one class fromanother without at the same time wholly excluding that other from theformer. To take an example, when we say 'No horses are ruminants, ' themeaning we really wish to convey is that no member of the class, horse, has a particular attribute, namely, that of chewing thecud. 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 givenattribute, we inevitably exclude at the same time any class of thingswhich possess that attribute from the former class. 

§ 285. The difference between the use of a predicate in an affirmativeand in a negative proposition may be illustrated to the eye asfollows. To say 'All A is B' may mean either that A is included in Bor that A and B are exactly co-extensive. 

[Illustration]

§ 286. As we cannot be sure which of these two relations of A to B ismeant, the predicate B has to be reckoned undistributed, since a termis held to be distributed only when we know that it is used in itswhole extent. 

§ 287. To say 'No A is B, ' however, is to say that A falls whollyoutside of B, which involves the consequence that B falls whollyoutside of A. 

[Illustration]

§ 288. Let us now apply the same mode of illustration to theparticular forms of proposition. 

§ 289. If I be taken in the strictly particular sense, there are, fromthe point of view of extension, two things which may be meant when wesay '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, thepredicate is again reckoned undistributed. 

§ 291. If on the other hand 1 be taken in an indefinite sense, so asto admit the possibility of the universal being true, then the twodiagrams which have already been used for A must be extended to 1, inaddition to its own, together with the remarks which we made inconnection with them (§§ 285-6). 

§ 292. Again, when we say 'Some A is not B, ' we mean that some, if notthe whole of A, is excluded from the possession of the attribute B. Ineither case the things which possess the attribute B are whollyexcluded either from a particular part or from the whole of A. Thepredicate 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 thesubject turns upon the quantity of the proposition, whether universalor particular; the question of the distribution or non-distribution ofthe predicate turns upon the quality of the proposition, whetheraffirmative 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 ornon-distribution of the subject only that needs to be taken intoaccount in estimating the quantity of a proposition. 

§ 296. But some logicians have maintained that the predicate, thoughseldom quantified in expression, must always be quantified inthought--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 someB. '

§ 297. If this were so, it is plain that the number of possiblepropositions would be exactly doubled, and that, instead of fourforms, we should now have to recognise eight, which may be expressedas 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 propositionswhich represents the original A, in accordance with the rule that 'Noaffirmative propositions distribute their predicate' (§ 293). 

§ 299. The third represents the original E, in accordance with therule that 'All negative propositions distribute their predicate. '

§ 300. The sixth represents the original I, in accordance with therule that 'No affirmative propositions distribute their predicate. '

§ 301. The seventh represents the original O, in accordance with therule that 'All negative propositions distribute their predicate. '

§ 302. Four new symbols are required, if the quantity of the predicateas well as that of the subject be taken into account in theclassification of propositions. These have been supplied, somewhatfancifully, as follows--

§ 303. The first, 'All A is all B, ' which distributes both subject andpredicate, has been called [upsilon], to mark its extremeuniversality. 

§ 304. The fourth, 'No A is some B, ' is contained in E, and hastherefore been denoted by the symbol [eta], to show its connectionwith E. 

§ 305. The fifth, 'Some A is all B, ' is the exact converse of thesecond, 'All A is some B, ' and has therefore been denoted by thesymbol [Upsilon], which resembles an inverted A. 

§ 306. The eighth is contained in O, as part in whole, and hastherefore had assigned to it the symbol [omega], § 307. The attempt to take the predicate in extension, instead of, asit should naturally be taken, in intension, leads to some curiousresults. Let us take, for instance, the u proposition. Either the signof quantity 'all' must be understood as forming part of the predicateor not. If it is not, then the u proposition 'All A is all B' seemsto contain within itself, not one proposition, but two, namely, 'All Ais B' and 'All B is A. ' But if on the other hand 'all' is understoodto form part of the predicate, then u is not really a general but asingular proposition. When we say, 'All men are rational animals, ' wehave a true general proposition, because the predicate applies to thesubject 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 predicateno longer applies to the subject distributively, but onlycollectively. For it is obvious that 'all rational animals' cannot beaffirmed of every individual in the class, man. What the propositionmeans is that the class, man, is co-extensive with the class, rationalanimal. The same meaning may be expressed intensively by saying thatthe one class has the attribute of co-extension with the other. 

§ 308. Under the head o u come all propositions in which both subjectand predicate are singular terms, e. G. 'Homer was the author of theIliad, ' 'Virtue is the way to happiness. '

§ 309. The proposition [eta] conveys very little information to themind. 'No A is some B' is compatible with the A proposition in thesame matter. 'No men are some animals' may be true, while at the sametime it is true that 'All men are animals. ' No men, for instance, arethe 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 stilltrue that 'Some men are not some rational animals': for no given humanbeing 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 areall rational animals': but, for all that, 'Some rational animals arenot some others. ' This last form of proposition therefore is almostwholly devoid of meaning. 

§ 312. The chief advantage claimed for the quantification of thepredicate is that it reduces every affirmative proposition to an exactequation between its subject and predicate. As a consequence everyproposition would admit of simple conversion, that is to say, ofhaving the subject and predicate transposed without any further changein the proposition. The forms also of Reduction (a term which will beexplained later on) would be simplified; and generally theintroduction of the quantified predicate into logic might be attendedwith certain mechanical advantages. The object of the logician, however, is not to invent an ingenious system, but to arrive at a trueanalysis of thought. Now, if it be admitted that in the ordinary formof proposition the subject is used in extension and the predicate inintension, the ground for the doctrine is at once cut away. For, ifthe predicate be not used in its extensive capacity at all, we plainlycannot be called upon to determine whether it is used in its wholeextent 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 variousthings which can be stated of a subject, viewed in their relation toit. 

§ 316. The treatment of this topic, therefore, as it involves therelation of a predicate to a subject, manifestly falls under thesecond part of logic, which deals with the proposition. It issometimes treated under the first part of logic, as though the headsof predicables were a classification of universal notions, i. E. Commonterms, in relation to one another, without reference to their place inthe proposition. 

§ 317. The heads of predicables are commonly reckonedas 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 arenow used, and afterwards examine the principle on which theclassification is founded and the sense in which they were originallyintended. 

 (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 aclass. 

An Inseparable Accident is one which belongs to all the members of aclass. 

Blackness is a separable accident of man, an inseparable accident ofcoals. 

§ 320. The attributes which belong to anything may be distinguishedbroadly under the two heads of essential and non-essential, oraccidental. By the essential attributes of anything are meant thosewhich are contained in, or which flow from, the definition. Now it maybe questioned whether there can, in the nature of things, be such athing as an inseparable accident. For if an attribute were found tobelong invariably to all the members of a class, we should suspectthat there was some causal connection between it and the attributeswhich constitute the definition, that is, we should suspect theattribute in question to be essential and not accidental. Neverthelessthe term 'inseparable accident' may be retained as a cloak for ourignorance, whenever it is found that an attribute does, as a matter offact, belong to all the members of a class, without there being anyapparent reason why it should do so. It has been observed that animalswhich have horns chew the cud. As no one can adduce any reason whyanimals that have horns should chew the cud any more than animalswhich have not, we may call the fact of chewing the cud an inseparableaccident of horned animals. 

§ 321. The distinction between separable and inseparable accidents issometimes extended from classes to individuals. 

An inseparable accident of an individual is one which belongs to himat all times. A separable accident of an individual is one whichbelongs to him at one time and not at another. 

§ 322. It is an inseparable accident of an individual that he was bornat a certain place and on a certain date. It is a separable accidentof an individual that he resides at a certain place and is of acertain age. 

§ 323. There are some remarks which it may be well to make about theabove five terms before we pass on to investigate the principle uponwhich the division is based. 

§ 324. In the first place, it must of course be borne in mind thatgenus and species are relative terms. No class in itself can be eithera genus or a species; it only becomes so in reference to some otherclass, as standing to it in the relation of containing or contained. 

§ 325. Again, the distinction between genus and difference on the onehand and property on the other is wholly relative to an assumeddefinition. When we say 'Man is an animal, ' 'Man is rational, ' 'Man isprogressive, ' there is nothing in the nature of these statementsthemselves to tell us that the predicate is genus, difference, orproperty respectively. It is only by a tacit reference to the accepteddefinition of man that this becomes evident to us, Similarly, wecannot know beforehand that the fact of a triangle having three sidesis its difference, and the fact of its having three angles aproperty. It is only when we assume the definition of a triangle as athree-sided figure that the fact of its having three angles sinks intoa property. Had we chosen to define it, in accordance with itsetymological meaning, as a figure with three angles, itsthree-sidedness would then have been a mere property, instead of beingthe difference; for these two attributes are so connected togetherthat, whichever is postulated, the other will necessarily follow. 

§ 326. Lastly, it must be noticed that we have not really defined theterm 'accident, ' not having stated what it is, but only what it isnot. It has in fact been reserved as a residual head to cover anyattribute which is neither a difference nor a property. 

§ 327. If the five heads of predicables above given were offered to usas an exhaustive classification of the possible relations in which thepredicate can stand to the subject in a proposition, the first thingthat would strike us is that they do not cover the case in which thepredicate is a singular term. In such a proposition as 'This man isJohn, ' we have neither a predication of genus or species nor ofattribute: but merely the identification of one term with another, asapplying to the same object. Such criticism as this, however, would beentirely erroneous, since no singular term was regarded as apredicate. A predicable was another name for a universal, the commonterm being called a predicable in one relation and a universal inanother-a predicable, extensively, in so far as it was applicable toseveral different things, a universal, intensively, in so far as theattributes indicated were implied in several other notions, as theattributes indicated by 'animal' are implied in 'horse, ' 'sheep, ''goat, ' &c. 

§ 328. It would be less irrelevant to point out how the classificationbreaks down in relation to the singular term as subject. When, forinstance, we say 'Socrates is an animal, ' 'Socrates is a man, ' thereis nothing in the proposition to show us whether the predicate is agenus or a species: for we have not here the relation of class toclass, which gives us genus or species according to their relativeextension, 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 theone 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 toadmit of a more or less complete defense by logic. 

For, given any subject, we can predicate of it either a class or anattribute. 

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 anindividual. 

When, on the other hand, the predicate is an attribute, the attributepredicated may be either the very attribute which distinguishes thesubject from other members of the same class, in which case it iscalled the Difference, or it may be some attribute connected with thedefinition, 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 betweenpredicating a class and predicating an attribute (in quid or in quale)is a perfectly intelligible one, corresponding as it does to thegrammatical distinction between the predicate being a noun substantiveor a noun adjective. Nevertheless it is a somewhat arbitrary one, since, even when the predicate is a class-name, what we are concernedto convey to the mind, is the fact that the subject possesses theattributes which are connoted by that class-name. We have not here thedifference between extensive and intensive predication, since, as wehave already seen (§ 264), that is not a difference between oneproposition and another, but a distinction in our mode of interpretingany and every proposition. Whatever proposition we like to take may beread either in extension or in intension, according as we fix ourminds on the fact of inclusion in a class or the fact of thepossession of attributes. 

§ 333. It will be seen that the term 'species, ' as it appears in thescheme, has a wholly different meaning from the current acceptation inwhich it was defined above. Species, in its now accepted meaning, signifies the relation of a smaller class to a larger one: as it wasoriginally intended in the heads of predicables it signifies a classin reference to individuals. 

§ 334. Another point which requires to be noticed with regard to thisfive-fold list of heads of predicables, if its object be to classifythe relations of a predicate to a subject, is that it takes no accountof those forms of predication in which class and attribute arecombined. Under which of the five heads would the predicates in thefollowing 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 theother we have a genus combined with an accident. 

§ 335. The list of heads of predicables which we have been discussingis not derived from Aristotle, but from the 'Introduction' ofPorphyry, 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 ofpredication above alluded to. His list contains only four heads, namely, Genus ([Greek: génos])

 Definition ([Greek: òrismós])

 Proprium ([Greek: îdion])

 Accident ([Greek: sumbebekós])

§ 337. Genus here is not distinguished from difference. Whether wesay 'Man is an animal' or 'Man is rational, ' we are equally understoodto 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 anindividual, it is regarded as a genus, e. G. 'Socrates is a man'; whenpredicated of a class, it is regarded as an accident, e. G. 'Someanimals are men. '

§ 339. Aristotle's classification may easily be seen to beexhaustive. For every predicate must either be coextensive with itssubject or not, i. E. Predicable of the same things. And if the twoterms coincide in extension, the predicate must either coincide alsoin intension with the subject or not. 

A predicate which coincides both in extension and intension with itssubject is exactly what is meant by a definition. One which coincidesin extension without coinciding in intension, that is, which appliesto the same things without expressing the whole meaning, of thesubject, is what is known as a Proprium or Peculiar Property. 

If, on the other hand, the two terms are not co-extensive, thepredicate must either partially coincide in intension with the subjector not. [Footnote: The case could not arise of a predicate which wasentirely coincided in intension with a subject with which it was notco-extensive. For, if the extension of the predicate were greater thanthat of the subject, its intension would be less, and if less, greater, in accordance with the law of inverse variation of the twoquantities (§ 166). ] This is equivalent to saying that it must eitherstate part of the definition of the subject or not. Now the definitionis made up of genus and difference, either of which may form thepredicate: but as the two are indistinguishable in relation to asingle subject, they are lumped together for the present purpose underthe one head, genus. When the predicate, not being co-extensive, isnot even partially co-intensive with its subject, it is called anAccident. 

§ 340. Proprium, it will be seen, differs from property. A propriumis an attribute which is possessed by all the members of a class, andby them alone, e. G. 'Men are the only religious animals. '

§ 341. Under the head of definition must be included all propositionsin which the predicate is a mere synonym of the subject, e. G. 'Naso isOvid, ' 'A Hebrew is a Jew, ' 'The skipper is the captain. ' In suchpropositions the predicate coincides in extension with the subject, and may be considered to coincide in intension where the intension ofboth subject and predicate is at zero, as in the case of two propernames. 

§ 342. Designations and descriptions will fall under the head of'proprium' or peculiar property, e. G. 'Lord Salisbury is the presentprime minister of England, ' 'Man is a mammal with hands and withouthair. ' For here, while the terms are coincident in extension, they arefar from being so in intension. 

§ 343. The term 'genus' must be understood to include not only genusin the accepted sense, but difference and generic property as well. 

§ 344. These results may be exhibited in the followingscheme--

 Predicate ___________________|______________ | | Coextensive with not the subject coextensive ________|_________ _____|________ | | | | Co-intensive not partially not at all with the subject cointensive cointensive [Greek: sumbubekós] [Greek: òrismós] [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, ifwe please, into nine--

 1. Definition \ > [Greek: òrismós]. 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 partof logic. Definition and division correspond respectively to the twokinds 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 explainits meaning. 

§ 348. From this it follows that any term which possesses no intensioncannot be defined. 

§ 349. Hence proper names do not admit of definition, except just inso far as they do possess some slight degree of intension: Thus we candefine the term 'John' only so far as to say that 'John' is the nameof a male person. This is said with regard to the original intensionof 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 whoseintension is already so simple that it cannot be unfolded further. Ofthis nature are names of simple attributes, such as greenness, sweetness, pleasure, existence. We know what these things are, but wecannot define them. To a man who has never enjoyed sight, no languagecan convey an idea of the greenness of the grass or the blueness ofthe 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. Wemight put a lump of sugar into his mouth, but that would not be alogical definition. 

§ 351. Thus we see that, for a thing to admit of definition, the ideaof it must be complex. Simple ideas baffle definition, but at the sametime do not require it. In defining we lay out the simpler ideaswhich are combined in our notion of something, and so explain thatcomplex 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 beconceived to exist by itself. '

§ 352. But when we get to 'thing' we have reached a limit. The SummumGenus, or highest class under which all things fall, cannot be definedany more than a simple attribute; and for the very good reason that itconnotes nothing but pure being, which is the simplest of allattributes. To say that a thing is an 'object of thought' is notreally to define it, but to explain its etymology, and to reclaim aphilosophical term from its abuse by popular language, in which it islimited to the concrete and the lifeless. Again, to define itnegatively and to say that a thing is 'that which is not nothing' doesnot carry us any further than we were before. The law of contradictionwarrants us in saying as much as that. 

§ 353. Definition is confined to subject-terms, and does not properlyextend to attributives. For definition is of things through names, andan attributive out of predication is not the name of anything. Theattributive is defined, so far as it can be, through the correspondingabstract term. 

§ 354. Common terms, other than attributives, ought always to admit ofdefinition. For things are distributed by the mind into classes owingto their possessing certain attributes in common, and the definitionof the class-name can be effected by detailing these attributes, or atleast a sufficient number of them. 

§ 355. It is different with singular terms. Singular terms, whenabstract, admit of definition, in so far as they are not names ofattributes so simple as to evade analysis. When singular terms areconcrete, we have to distinguish between the two cases of proper namesand designations. Designations are connotative singular terms. Theyare formed by limiting a common term to the 'case in hand. ' Whateverdefinition therefore fits the common term will fit also thedesignation which is formed from it, if we add the attributes impliedby the limitations. Thus whatever definition fits the common term'prime minister' will fit also the singular term 'the present primeminister of England' by the addition to it of the attributes of placeand time which are indicated by the expression. Such terms as thishave a definite amount of intension, which can therefore be seizedupon 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 pointof view of their acquired intension, they defy definition by reason ofthe very complexity of their meaning. We cannot say exactly what'John' and 'Mary' mean, because those names, to us who know theparticular persons denoted by them, suggest all the most triflingaccidents of the individual as well as the essential attributes of thegenus. 

§ 357. Definition serves the practical purpose of enabling us mentallyto distinguish, or, as the name implies, 'mark off' the thing definedfrom all other things whatsoever. This may seem at first an endlesstask, 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 mostlike, we shall, 'a fortiori, ' have distinguished it from things towhich it bears a less resemblance. 

§ 358. Hence the first thing to do in seeking for a definition is tofix upon the class into which the thing to be defined most naturallyfalls, and then to distinguish the thing in question from the othermembers of that class. If we were asked to define a triangle, wewould not begin by distinguishing it from a hawser, but from a squareand other figures with which it is more possible to confound it. Theclass into which a thing falls is called its Genus, and the attributeor attributes which distinguish it from other members of that classare called its Difference. 

§ 359. If definition thus consists in referring a thing to a class, wesee a further reason why the summum genus of all things cannot bedefined. 

§ 360. We have said that definition is useful in enabling us todistinguish things from one another in our minds: but this must not beregarded as the direct object of the process. For this object may beaccomplished without giving a definition at all, by means of what iscalled a Description. By a description is meant an enumeration ofaccidents with or without the mention of some class-name. It is asapplicable to proper names as to common terms. When we say 'John Smithlives next door on the right-hand side and passes by to his officeevery morning at nine o'clock, ' we have, in all probability, effectually distinguished John Smith from other people: but livingnext, &c. , cannot be part of the intension of John Smith, since JohnSmith may change his residence or abandon his occupation withoutceasing to be called by his name. Indirectly then definition servesthe purpose of distinguishing things in the mind, but its directobject is to unfold the intension of terms, and so impart precision toour thoughts by setting plainly before us the meaning of the words weare using. 

§ 361. But when we say that definition is unfolding the intension ofterms, it must not be imagined that we are bound in defining to unfoldcompletely the intension of terms. This would be a tedious, and oftenan endless, task. A term may mean, or convey to the mind, a good manymore attributes than those which are stated in its definition. Thereis no limit indeed to the meaning which a term may legitimatelyconvey, except the common attributes of the things denoted by it. Whoshall say, for instance, that a triangle means a figure with threesides, and does not mean a figure with three angles, or the surface ofthe perpendicular bisection of a cone? Or again, that man means arational, and does not mean a speaking, a religious, or an aestheticanimal, or a biped with two eyes, a nose, and a mouth? The onlyattributes of which it can safely be asserted that they can form nopart of the intension of a term are those which are not common to allthe things to which the name applies. Thus a particular complexion, colour, height, creed, nationality cannot form any part of theintension of the term 'man. ' But among the attributes common to aclass we cannot distinguish between essential and unessential, exceptby the aid of definition itself. Formal logic cannot recognise anyorder of priority between the attributes common to all the members ofa class, such as to necessitate our recognising some as genera anddifferentiae and relegating others to the place of properties orinseparable accidents. 

§ 362. The art of giving a good definition is to seize upon thesalient characteristics of the thing defined and those wherefrom thelargest number of other attributes can be deduced as consequences. Todo 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 onehand and property on the other is wholly relative to some assumeddefinition. Now definitions are always to a certain extent arbitrary, and will vary with the point of view from which we consider the thingrequired to be defined. Thus 'man' is usually contrasted with 'brute, 'and from this point of view it is held a sufficient definition of himto say that he is 'a rational animal, ' But a theologian might be moreanxious to contrast man with supposed incorporeal intelligences, andfrom this point of view man would be defined as an 'embodied spirit. '

§ 364. In the two definitions just given it will be noticed that wehave really employed exactly the same attributes, only their place asgenus and difference has been reversed. It is man's rational, orspiritual, nature which distinguishes him from the brutes: but this isjust what he is supposed to have in common with incorporealintelligences, from whom he is differentiated by his animal nature. 

[Illustration]

This illustration is sufficient to show us that, while there is noabsolute definition of anything, in the sense of a fixed genus anddifference, there may at the same time be certain attributes whichpermanently distinguish the members of a given class from those of allother classes. 

§ 365. The above remarks will have made it clear that the intension ofa term is often much too wide to be conveyed by any definition; andthat what a definition generally does is to select certain attributesfrom the whole intension, which are regarded as being more typical ofthe thing than the remainder. No definition can be expected to exhaustthe whole intension of a term, and there will always be room forvarying definitions of the same thing, according to the differentpoints of view from which it is approached. 

§ 366. Names of attributes lend themselves to definition far moreeasily than names of substances. The reason of this is that names ofattributes are primarily intensive in force, whereas substances areknown to us in extension before they become known to us inintension. There is no difficulty in defining a term like 'triangle'or 'monarchy, ' because these terms were expressly invented to covercertain attributes; but the case is different with such terms as'dog, ' 'tree, ' 'plant, ' 'metal, ' and other names of concretethings. We none of us have any difficulty in recognising a dog ortree, when we see them, or in distinguishing them from other animalsor plants respectively. We are therefore led to imagine that we knowthe meaning of these terms. It is not until we are called upon for adefinition that we discover how superficial our knowledge really is ofthe common attributes possessed by the things which these namesdenote. 

§ 367. It might be imagined that a common name would never be given tothings except in virtue of our knowledge of their commonattributes. But as a matter of fact, the common name was first givenfrom a confused notion of resemblance, and we had afterwards to detectthe common attributes, when sometimes the name had been so extendedfrom one thing to another like it, that there were hardly any definiteattributes possessed in common by the earlier and later members of theclass. 

§ 368. This is especially the case where the meaning of terms has beenextended by analogy, e. G. Head, foot, arm, post, pole, pipe, &c. 

§ 369. But in the progress of thought we come to form terms in whichthe intensive capacity is everything. Of this kind notably aremathematical conceptions. Terms of this kind, as we said before, lendthemselves readily to definition. 

§ 370. We may lay down then roughly that words are easy or difficultof definition according as their intensive or extensive capacitypredominates. 

§ 371. There is a marked distinction to be observed between theclasses made by the mind of man and the classes made by nature, whichare known as 'real kinds. ' In the former there is generally little ornothing in common except the particular attribute which is selected asthe ground of classification, as in the case of red and white things, which are alike only in their redness or whiteness; or else theirattributes are all necessarily connected, as in the case of circle, square and triangle. But the members of nature's classes agree ininnumerable attributes which have no discoverable connection with oneanother, and which must therefore, provisionally at least, be regardedas standing in the relation of inseparable accidents to any particularattributes which we may select for the purposes of definition. Thereis no assignable reason why a rational animal should have hair on itshead or a nose on its face, and yet man, as a matter of fact, hasboth; and generally the particular bodily configuration of man canonly be regarded as an inseparable accident of his nature as arational animal. 

§ 372. 'Real kinds' belong to the class of words mentioned above inwhich the extension predominates over the intension. We know wellenough the things denoted by them, while most of us have only a dimidea of the points of resemblance between these things. Nature'sclasses moreover shade off into one another by such imperceptibledegrees that it is often impossible to fix the boundary line betweenone class and another. A still greater source of perplexity in dealingwith real kinds is that it is sometimes almost impossible to fix uponany attribute which is common to every individual member of the classwithout exception. All that we can do in such cases is to lay down atype of the class in its perfect form, and judge of individualinstances by the degree of their approximation to it. Again, realkinds being known to us primarily in extension, the intension which weattach to the names is hable to be affected by the advance ofknowledge. In dealing therefore with such terms we must be contentwith provisional definitions, which adequately express our knowledgeof the things denoted by them, at the time, though a further study oftheir attributes may induce us subsequently to alter thedefinition. Thus the old definition of animal as a sentient organismhas been rendered inadequate by the discovery that so many of thephenomena of sensation can be exhibited by plants, § 373. But terms in which intension is the predominant idea are morecapable of being defined once for all. Aristotle's definitions of'wealth' and 'monarchy' are as applicable now as in his own day, andno subsequent discoveries of the properties of figures will renderEuclid'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 NominalDefinitions. Both of these explain the meaning of a term: but a realdefinition further assumes the actual existence of the thingdefined. Thus the explanation of the term 'Centaur' would be anominal, that of 'horse' a real definition. 

It is useless to assert, as is often done, that a nominal definitionexplains the meaning of a term and a real definition the nature of athing; for, as we have seen already, the meaning of a term is whateverwe know of the nature of a thing. 

§ 376. It now remains to lay down certain rules for correctdefinition. 

§ 377. The first rule that is commonly given is that a definitionshould state the essential attributes of the thing defined. But thisamounts merely to saying that a definition should be a definition;since it is only by the aid of definition that we can distinguishbetween essential and non-essential among the common attributesexhibited by a class of things. The rule however may be retained as amaterial test of the soundness of a definition, in the sense that hewho seeks to define anything should fix upon its most importantattributes. To define man as a mammiferous animal having two hands, oras a featherless biped, we feel to be absurd and incongruous, sincethere is no reference to the most salient characteristic of man, namely, his rationality. Nevertheless we cannot quarrel with thesedefinitions on formal, but only on material grounds. Again, if anyonechose to define logic as the art of thinking, all we could say is thatwe differ from him in opinion, as we think logic is more properly tobe regarded as the science of the laws of thought. But here also it ison material grounds that we dissent from the definition. 

§ 378. Confining ourselves therefore to the sphere with which we areproperly 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); andmust 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 propositionwhich expounds the meaning of the term; at other times it is confinedto the predicate of this proposition. Thus in stating the first fourrules we have used the term in the latter sense, and in stating thefifth in the former. 

§ 380. We will now illustrate the force of the above rules by givingexamples 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 termdefined, since it applies only to equilateral triangles. This faultmay be amended by decreasing the intension, which we do by eliminatingthe reference to the equality of the sides. 

In the second instance the definition is more extensive than the termdefined. We must accordingly increase the intension by adding a newattribute '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 thisdefinition is redundant, since what has three sides cannot but havethree 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, thisobject is evidently defeated, if we confine ourselves to saying whatit is not. But sometimes this is impossible to be avoided. For thereare many terms which, though positive in form, are privative in force. These terms serve as a kind of residual heads under which to throweverything within a given sphere, which does not exhibit certainpositive attributes. Of this unavoidably negative nature was thedefinition which we give of 'accident, ' which amounted merely tosaying that it was any attribute which was neither a difference nor aproperty. 

§ 382. The violation of Rule 3, which guards against defining a thingby itself, is technically known as 'circulus in definiendo, ' ordefining in a circle. This rule is often apparently violated, withoutbeing really so. Thus Euclid defines an acute-angled triangle as onewhich has three acute angles. This seems a glaring violation of therule, but is perfectly correct in its context; for it has already beenexplained what is meant by the terms 'triangle' and 'acute angle, ' andall that is now required is to distinguish the acute-angled trianglefrom its cognate species. He might have said that an acute-angledtriangle is one which has neither a right angle nor an obtuse angle:but rightly preferred to throw the same statement into a positiveform. 

§ 383. The violation of Rule 4 is known as 'ignotum per ignotius' or'per aeque ignotum. ' This rule also may seemingly be violated when itis not really so. For a definition may be correct enough from aspecial point of view, which, apart from that particular context, would appear ridiculous. From the point of view of conic sections, itis correct enough to define a triangle as that section of a cone whichis formed by a plane passing through the vertex perpendicularly to thebase, but this could not be expected to make things clearer to aperson who was inquiring for the first time into the meaning of theword triangle. But a real violation of the fourth rule may arise, notonly from obscurity, but from the employment of ambiguous language ormetaphor. 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 ofthe definiend. 

§ 384. The material correctness of a definition is, as we have alreadyseen, a matter extraneous to formal logic. An acquaintance with theattributes which terms imply involves material knowledge quite as muchas an acquaintance with the things they apply to; knowledge of theintension and of the extension of terms is alike acquired byexperience. No names are such that their meaning is rendered evidentby the very constitution of our mental faculties; yet nothing short ofthis would suffice to bring the material content of definition withinthe province of formal logic. 

CHAPTER VIII. 

_Of Division. _

§ 385. To divide a term is to unfold its extension, that is, to setforth the things of which it is a name. 

§ 386. But as in definition we need not completely unfold theintension of a term, so in division we must not completely unfold itsextension. 

§ 387. Completely to unfold the extension of a term would involvestating all the individual objects to which the name applies, a thingwhich would be impossible in the case of most common terms. When it isdone, it is called Enumeration. To reckon up all the months of theyear from January to December would be an enumeration, and not adivision, of the term 'month. '

§ 388. Logical division always stops short at classes. It may bedefined as the statement of the various classes of things that can becalled by a common name. Technically we may say that it consists inbreaking 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, andalso that the members of the division must themselves be common terms. 

§ 390. An 'individual' is so called as not admitting of logicaldivision. We may divide the term 'cow' into classes, as Jersey, Devonshire, &c. , to which the name 'cow' will still be applicable, butthe parts of an individual cow are no longer called by the name of thewhole, but are known as beefsteaks, briskets, &c. 

§ 391. In dividing a term the first requisite is to fix upon somepoint wherein certain members of the class differ from others. Thepoint thus selected is called the Fundamentum Divisionis or Basis ofthe Division. 

§ 392. The basis of the division will of course differ according tothe purpose in hand, and the same term will admit of being divided ona 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, ona 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 Divisumor Divided Whole. It might also be called the Dividend. 

§ 394. The classes into which the dividend is split up are called theMembra 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 nocross-division (1) and the division must be exhaustive (2). 

§ 397. The rule, which is commonly given, that each dividing membermust be a common term, is already provided for under our definition ofthe process. 

§ 398. The rule that the dividend must be predicable of each of thedividing members is contained in our second rule; since, if there wereany term of which the dividend were not predicable, it would beimpossible for the dividing members to be exactly coextensive with it. It would not do, for instance, to introduce mules and donkeys into adivision of the term horse. 

§ 399. Another rule, which is sometimes given, namely, that theconstituent species must exclude one another, is a consequence of ourfirst; for, if the division be conducted on a single principle, theconstituent species must exclude one another. The converse, however, does not hold true. We may have a division consisting of mutuallyexclusive members, which yet involves a mixture of different bases, e. G. If we were to divide triangle into scalene, isosceles andequiangular. This happens because two distinct attributes may be foundin invariable conjunction. 

§ 400. There is no better test, however, of the soundness of adivision than to try whether the species overlap, that is to say, whether there are any individuals that would fall into two or more ofthe classes. When this is found to be the case, we may be sure that wehave mixed two or more different fundamenta divisionis. If man, forinstance, were to be divided into European, American, Aryan, andSemitic, the species would overlap; for both Europe and Americacontain inhabitants of Aryan and Semitic origin. We have here membersof a division based on locality mixed up with members of anotherdivision, which is based on race as indicated by language. 

§ 401. The classes which are arrived at by an act of division maythemselves be divided into smaller classes. This further process iscalled Subdivision. 

§ 402. Let it be noticed that Rule 1 applies only to a single act ofdivision. The moment that we begin to subdivide we not only may, butmust, adopt a new basis of division; since the old one has, 'exhypothesi, ' been exhausted. Thus, having divided men according to thecolour of their skins, if we wish to subdivide any of the classes, wemust look out for some fresh attribute wherein some men of the samecomplexion differ from others, e. G. We might divide black men intowoolly-haired blacks, such as the Negroes, and straight-haired blacks, like the natives of Australia. 

§ 403. We will now take an instance of division andsubdivision, with a view to illustrating some of thetechnical terms which are used in connection with theprocess. Keeping closely to our proper subject, we willselect 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 thoughtbased on their comparative complexity. The first two of these, namely, the term and the proposition, are then subdivided on the basis oftheir respective quantities. In the case of inference the basis of thedivision is again the degree of complexity. The subdivision of theproposition is carried a step further than that of the others. Havingexhausted 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 theforegoing, the highest class taken is known as the Summum Genus. Thusthe summum genus is the same thing as the divided whole, viewed in adifferent relation. The term which is called the divided whole withreference to a single act of division, is called the summum genuswhenever subdivision has taken place. 

§ 405. The classes at which the division stops, that is, any which arenot subdivided, are known as the Infimae Species. 

§ 406. All classes intermediate between the summum genus and theinfimae species are called Subaltern Genera or Subaltern Species, according to the way they are looked at, being genera in relation tothe classes below them and species in relation to the classes abovethem. 

§ 407. Any classes which fall immediately under the same genus arecalled Cognate Species, e. G. Singular and common terms are cognatespecies of term. 

§ 408. The classes under which any lower class successively falls arecalled Cognate Genera. The relation of cognate species to one anotheris like that of children of the same parents, whereas cognate generaresemble a line of ancestry. 

§ 409. The Specific Difference of anything is the attribute orattributes which distinguish it from its cognate species. Thus thespecific difference of a universal proposition is that the predicateis known to apply to the whole of the subject. A specific differenceis said to constitute the species. 

§ 410. The specific difference of a higher class becomes a GenericDifference with respect to the class below it. A generic differencethen may be said to be the distinguishing attribute of the whole classto which a given species belongs. The generic difference is common tospecies that are cognate to one another, whereas the specificdifference is peculiar to each. It is the generic difference of an Aproposition that it is universal, the specific difference that it isaffirmative. 

§ 411. The same distinction is observed between the specific andgeneric properties of a thing. A Specific Property is an attributewhich flows from the difference of a thing itself; a Generic Propertyis an attribute which flows from the difference of the genus to whichthe thing belongs. It is a specific property of an E proposition thatits predicate is distributed, a generic property that its contrarycannot be true along with it (§ 465); for this last characteristicflows from the nature of the universal proposition generally. 

§ 412. It now remains to say a few words as to the place in logic ofthe process of division. Since the attributes in which members of thesame class differ from one another cannot possibly be indicated bytheir common name, they must be sought for by the aid of experience;or, to put the same thing in other words, since all the infimaespecies are alike contained under the summum genus, their distinctiveattributes can be no more than separable accidents when viewed inrelation to the summum genus. Hence division, being always founded onthe possession or non-possession of accidental attributes, seems tolie wholly outside the sphere of formal logic. This however is notquite the case, for, in virtue of the Law of Excluded Middle, there isalways open to us, independently of experience, a hypotheticaldivision by dichotomy. By dichotomy is meant a division into twoclasses by a pair of contradictory terms, e. G. A division of theclass, 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 allexperience, that men are either white or not. Hence division bydichotomy comes strictly within the province of formal logic. Only itmust be noticed that both sides of the division must be hypothetical. For experience alone can tell us, on the one hand, that there are anymen that are white, and on the other, that there are any but whitemen. 

§ 413. What we call a division on a single basis is in reality thecompressed result of a scheme of division and subdivision bydichotomy, in which a fresh principle has been introduced at everystep. 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 meninto white and not-white, and then to have subdivided the men that arenot-white into black and not-black, and so on. From the strictlyformal point of view this division can only be represented asfollows--

 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 seriesshould be negative. We have here to keep the term 'not-red' open, tocover any blue or green men that might turn up. It is only experiencethat enables us to substitute the positive term 'yellow' for'not-red, ' since we know as a matter of fact that there are no men butthose of the five colours given in the original division. 

§ 415. Any correct logical division always admits of being arrived atby the longer process of division and subdivision by dichotomy. Forinstance, the term quadrilateral, or four-sided rectilinear figure, iscorrectly divided into square, oblong, rhombus, rhomboid andtrapezium. 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 mustof course be careful not to include any class which has been alreadysubdivided; but no harm is done by mixing an undivided class, liketrapezium, with the subdivisions of its cognate species. 

§ 417. The two processes of definition and division are intimatelyconnected with one another. Every definition suggests a division bydichotomy, and every division supplies us at once with a completedefinition of all its members. 

§ 418. Definition itself, so far as concerns its content, is, as wehave already seen, extraneous to formal logic: but when once we haveelicited a genus and difference out of the material elements ofthought, we are enabled, without any further appeal to experience, tobase thereon a division by dichotomy. Thus when man has been definedas a rational animal, we have at once suggested to us a division ofanimal into rational and irrational. 

§ 419. Again, the addition of the attributes, rational and irrationalrespectively, to the common genus, animal, ipso facto supplies us withdefinitions of the species, man and brute. Similarly, when wesubdivided rectangle into square and oblong on the basis of theequality or inequality of the adjacent sides, we were by so doingsupplied with a definition both of square and oblong--'A square is arectangle having all its sides equal, ' and 'An oblong is a rectanglewhich has only its opposite sides equal. '

§ 420. The definition of a square just given amounts to the same thingas Euclid's definition, but it complies with a rule which has value asa matter of method, namely, that the definition should state theProximate Genus of the thing defined. 

§ 421. Since definition and division are concerned with the intensionand extension of terms, they are commonly treated of under the firstpart of logic: but as the treatment of the subject implies afamiliarity with the Heads of Predicables, which in their turn implythe proposition, it seems more desirable to deal with them under thesecond. 

§ 422. We have already had occasion to distinguish division fromEnumeration. The latter is the statement of the individual things towhich a name applies. In enumeration, as in division, the wider termis predicable of each of the narrower ones. 

§ 423. Partition is the mapping out of a physical whole into itscomponent parts, as when we say that a tree consists of roots, stem, and branches. In a partition the name of the whole is not predicableof each of the parts. 

§ 424. Distinction is the separation from one another of the variousmeanings of an equivocal term. The term distinguished is predicableindeed of each of the members, but of each in a different sense. Anequivocal term is in fact not one but several terms, as would quicklyappear, if we were to use definitions in place of names. 

§ 425. We have seen that a logical whole is a genus viewed in relationto its underlying species. From this must be distinguished ametaphysical whole, which is a substance viewed in relation to itsattributes, or a class regarded in the same way. Logically, man is apart of the class, animal; metaphysically, animal is contained inman. Thus a logical whole is a whole in extension, while ametaphysical whole is a whole in intension. From the former point ofview species is contained in genus; from the latter genus is containedin 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 seea charred circle on the grass, we infer that somebody has beenlighting a fire there, though we have not seen anyone do it. Thisconclusion is arrived at in consequence of our previous experience ofthe effects of fire. 

§ 427. The term Inference is used both for a process and for a productof thought. 

As a process inference may be defined as the passage of the mind fromone or more propositions to another. 

As a product of thought inference may be loosely declared to be theresult 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. Butthis use of the terms 'antecedent' and 'consequent' must be carefullydistinguished from the use to which they were put previously, todenote the two parts of a complex proposition. 

§ 429. Strictly speaking, the term inference, as applied to a productof thought, includes both the antecedent and consequent: but it isoften used for the consequent to the exclusion of theantecedent. Thus, when we have stated our premisses, we say quitenaturally, 'And the inference I draw is so and so. '

§ 430. Inferences are either Inductive or Deductive. In induction weproceed from the less to the more general; in deduction from the moreto the less general, or, at all events, to a truth of not greatergenerality than the one from which we started. In the former we workup to general principles; in the latter we work down from them, andelicit the particulars which they contain. 

§ 431. Hence induction is a real process from the known to theunknown, whereas deduction is no more than the application ofpreviously existing knowledge; or, to put the same thing moretechnically, in an inductive inference the consequent is not containedin the antecedent, in a deductive inference it is. 

§ 432. When, after observing that gold, silver, lead, and othermetals, are capable of being reduced to a liquid state by theapplication of heat, the mind leaps to the conclusion that the samewill hold true of some other metal, as platinum, or of all metals, wehave then an inductive inference, in which the conclusion, orconsequent, is a new proposition, which was not contained in thosethat went before. We are led to this conclusion, not by reason, but byan instinct which teaches us to expect like results, under likecircumstances. Experience can tell us only of the past: but we allowit 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 whichis done is that which shall be done; and there is no new thing underthe sun. ' Take away this conviction, and the bridge is cut whichconnects the known with the unknown, the past with the future. Thecommonest acts of daily life would fail to be performed, were it notfor this assumption, which is itself no product of the reason. Thusman's intellect, like his faculties generally, rests upon a basis ofinstinct. He walks by faith, not by sight. 

§ 433. It is a mistake to talk of inductive reasoning, as though itwere a distinct species from deductive. The fact is that inductiveinferences are either wholly instinctive, and so unsusceptible oflogical vindication, or else they may be exhibited under the form ofdeductive inferences. We cannot be justified in inferring thatplatinum will be melted by heat, except where we have equal reason forasserting the same thing of copper or any other metal. In fact we arejustified in drawing an individual inference only when we can lay downthe universal proposition, 'Every metal can be melted by heat. ' Butthe moment this universal proposition is stated, the truth of theproposition in the individual instance flows from it by way ofdeductive inference. Take away the universal, and we have no logicalwarrant for arguing from one individual case to another. We do so, aswas said before, only in virtue of that vague instinct which leads usto anticipate like results from like appearances. 

§ 434. Inductive inferences are wholly extraneous to the science offormal logic, which deals only with formal, or necessary, inferences, that is to say with deductive inferences, whether immediate ormediate. These are called formal, because the truth of the consequentis apparent from the mere form of the antecedent, whatever be thenature of the matter, that is, whatever be the terms employed in theproposition or pair of propositions which constitutes theantecedent. In deductive inference we never do more than vary the formof the truth from which we started. When from the proposition 'Brutuswas the founder of the Roman Republic, ' we elicit the consequence 'Thefounder of the Roman Republic was Brutus, ' we certainly have nothingmore in the consequent than was already contained in the antecedent;yet all deductive inferences may be reduced to identities as palpableas this, the only difference being that in more complicated cases theconsequent is contained in the antecedent along with a number of otherthings, whereas in this case the consequent is absolutely all that theantecedent contains. 

§ 435. On the other hand, it is of the very essence of induction thatthere should be a process from the known to the unknown. Widelydifferent as these two operations of the mind are, they are yet bothincluded under the definition which we have given of inference, as thepassage of the mind from one or more propositions to another. It isnecessary to point this out, because some logicians maintain that allinference must be from the known to the unknown, whereas othersconfine it to 'the carrying out into the last proposition of what wasvirtually contained in the antecedent judgements. '

§ 436. Another point of difference that has to be noticed betweeninduction and deduction is that no inductive inference can ever attainmore than a high degree of probability, whereas a deductive inferenceis certain, but its certainty is purely hypothetical. 

§ 437. Without touching now on the metaphysical difficulty as to howwe pass at all from the known to the unknown, let us grant that thereis no fact better attested by experience than this--'That where thecircumstances are precisely alike, like results follow. ' But then wenever can be absolutely sure that the circumstances in any two casesare precisely alike. All the experience of all past ages in favour ofthe daily rising of the sun is not enough to render us theoreticallycertain that the sun will rise tomorrow We shall act indeed with aperfect reliance upon the assumption of the coming day-break; but, forall that, the time may arrive when the conditions of the universeshall have changed, and the sun will rise no more. 

§ 438. On the other hand a deductive inference has all the certaintythat 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 purelyhypothetical. We may feel assured that if the premisses are true, theconclusion is true also. But for the truth of our premisses we have tofall back upon induction or upon intuition. It is not the province ofdeductive logic to discuss the material truth or falsity of thepropositions upon which our reasonings are based. This task is left toinductive logic, the aim of which is to establish, if possible, a testof material truth and falsity. 

§ 439. Thus while deduction is concerned only with the relative truthor falsity of propositions, induction is concerned with their actualtruth or falsity. For this reason deductive logic has been termed thelogic of consistency, not of truth. 

§ 440. It is not quite accurate to say that in deduction we proceedfrom the more to the less general, still less to say, as is oftensaid, that we proceed from the universal to the particular. For it mayhappen that the consequent is of precisely the same amount ofgenerality as the antecedent. This is so, not only in most forms ofimmediate inference, but also in a syllogism which consists ofsingular 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 noessential difference between its laws and those of deduction. 

§ 441. Subjoined is a classification of inferences, which will serveas 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 effectedwithout the intervention of a middle term, which is required inmediate inference. 

§ 444. But the distinction between the two might be conveyed with atleast 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 ofa third. 

§ 445. In that sense of the term inference in which it is confined tothe 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 ofa deductive inference. Wherever we have a conclusion following frommore than two propositions, there will be found to be more than oneinference. 

§ 447. There are three simple forms of immediate inference, namelyOpposition, Conversion and Permutation. 

§ 448. Besides these there are certain compound forms, in whichpermutation is combined with conversion. 

CHAPTER III. 

_Of Opposition. _

§ 449. Opposition is an immediate inference grounded on the relationbetween propositions which have the same terms, but differ in quantityor in quality or in both. 

§ 450. In order that there should be any formal opposition between twopropositions, it is necessary that their terms should be thesame. There can be no opposition between two such propositions asthese--

 (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 forpredicate, and allowed to affix such quantity and quality as weplease, we can of course make up the four kinds of propositionrecognised 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 orfalsity of any one of the four propositions A, E, I, O, what can beascertained with regard to the truth or falsity of the rest, thematter of them being supposed to be the same?

§ 453. The relations to one another of these four propositionsare 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 inquality. 

§ 455. Sub-contrary Opposition is between two particulars which differin quality. 

§ 456. Subaltern Opposition is between two propositions which differonly in quantity. 

§ 457. Contradictory Opposition is between two propositions whichdiffer both in quantity and in quality. 

§ 458. Subaltern Opposition is also known as Subalternation, and ofthe two propositions involved the universal is called the Subalternantand the particular the Subalternate. Both together are calledSubalterns, and similarly in the other forms of opposition the twopropositions involved are known respectively as Contraries, Sub-contraries and Contradictories. 

§ 459. For the sake of convenience some relations are classed underthe head of opposition in which there is, strictly speaking, noopposition at all between the two propositions involved. 

§ 460. Between sub-contraries there is an apparent, but not a realopposition, since what is affirmed of one part of a term may oftenwith truth be denied of another. Thus there is no incompatibilitybetween 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 universalnot only may, but must, be compatible with that of the particular. 

§ 462. Immediate Inference by Relation would be a more appropriatename than Opposition; and Relation might then be subdivided intoCompatible and Incompatible Relation. By 'compatible' is here meantthat there is no conflict between the _truth_ of the twopropositions. Subaltern and sub-contrary opposition would thus fallunder the head of compatible relation; contrary and contradictoryrelation under that of incompatible relation. 

 Relation ______________|_____________ | | Compatible Incompatible ______|_____ _____|_______ | | | | Subaltern Sub-contrary Contrary Contradictory. 

§ 463. It should be noticed that the inference in the case ofopposition is from the truth or falsity of one of the opposedpropositions to the truth or falsity of the other. 

§ 464. We will now lay down the accepted laws of inference with regardto the various kinds of opposition. 

§ 465. Contrary propositions may both be false, but cannot both betrue. Hence if one be true, the other is false, but not vice versâ. 

§ 466. Sub-contrary propositions may both be true, but cannot both befalse. Hence if one be false, the other is true, but not vice versâ. 

§ 467. In the case of subaltern propositions, if the universal betrue, the particular is true; and if the particular be false, theuniversal is false; but from the truth of the particular or thefalsity of the universal no conclusion can be drawn. 

§ 468. Contradictory propositions cannot be either true or falsetogether. Hence if one be true, the other is false, and vice versâ. 

§ 469. By applying these laws of inference we obtain the followingresults--

 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 informationfrom deriving a particular than from denying a universal. Should thisseem surprising, the paradox will immediately disappear, if we reflectthat to deny a universal is merely to assert the contradictoryparticular, whereas to deny a particular is to assert thecontradictory universal. It is no wonder that we should obtain moreinformation from asserting a universal than from asserting aparticular. 

§ 471. We have laid down above the received doctrine with regard toopposition: 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 theirnow accepted logical meaning as indefinite (§ 254), but rather intheir popular sense as 'strict particular' propositions. For if I andO 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 ofthe 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 thatcase the doctrine of contradiction breaks down as well. For I and Omay, on this showing, be false, without their contradictories E and Abeing thereby rendered true. This illustrates the awkwardness, whichwe have previously had occasion to allude to, which ensures fromdividing propositions primarily into universal and particular, insteadof first dividing them into definite and indefinite, and particular (§256). 

§ 472. To be suddenly thrown back upon the strictly particular view ofI and O in the special case of opposition, after having beenaccustomed to regard them as indefinite propositions, is a manifestinconvenience. But the received doctrine of opposition does not evenadhere consistently to this view. For if I and O be taken as strictlyparticular propositions, which exclude the possibility of theuniversal of the same quality being true along with them, we ought notmerely to say that I and O may both be true, but that if one be truethe other must also be true. For I being true, A is false, andtherefore O is true; and we may argue similarly from the truth of O tothe truth of I, through the falsity of E. Or--to put the Same thing ina less abstract form--since the strictly particular proposition means'some, but not all, ' it follows that the truth of one sub-contrarynecessarily carries with it the truth of the other, If we lay downthat some islands only are inhabited, it evidently follows, or ratheris stated simultaneously, that there are some islands also which arenot inhabited. For the strictly particular form of proposition 'Some Aonly is B' is of the nature of an exclusive proposition, and is reallyequivalent to two propositions, one affirmative and one negative. 

§ 473. It is evident from the above considerations that the doctrineof opposition requires to be amended in one or other of twoways. Either we must face the consequences which follow from regardingI and O as indefinite, and lay down that sub-contraries may both befalse, accepting the awkward corollary of the collapse of the doctrineof contradiction; or we must be consistent with ourselves in regardingI and O, for the particular purposes of opposition, as being strictlyparticular, and lay down that it is always possible to argue from thetruth of one sub-contrary to the truth of the other. The latter isundoubtedly the better course, as the admission of I and O asindefinite in this connection confuses the theory of oppositionaltogether. 

§ 474. Of the several forms of opposition contradictory opposition islogically 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'sassertion. If, for instance, some person with whom we were arguingwere to lay down that 'All poets are bad logicians, ' we might betempted in the heat of controversy to maintain against him thecontrary proposition 'No poets are bad logicians. ' This wouldcertainly 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 difficultto refute. 

§ 476. The phrase 'diametrically opposed to one another' seems to beone of the many expressions which have crept into common language fromthe technical usage of logic. The propositions A and O and E and Irespectively are diametrically opposed to one another in the sensethat the straight lines connecting them constitute the diagonals ofthe parallelogram in the scheme of opposition. 

§ 477. It must be noticed that in the case of a singular propositionthere is only one mode of contradiction possible. Since the quantityof such a proposition is at the minimum, the contrary andcontradictory are necessarily merged into one. There is no way ofdenying the proposition 'This house is haunted, ' save by maintainingthe proposition which differs from it only in quality, namely, 'Thishouse is not haunted. '

478. A kind of generality might indeed he imparted even to a singularproposition by expressing it in the form 'A is always B. ' Thus we maysay, 'This man is always idle'--a proposition which admits of beingcontradicted under the form 'This man is sometimes not idle. '

CHAPTER IV. 

_Of Conversion. _

§ 479. Conversion is an immediate inference grounded On thetransposition of the subject and predicate of a proposition. 

§ 480. In this form of inference the antecedent is technically knownas the Convertend, i. E. The proposition to be converted, and theconsequent as the Converse, i. E. The proposition which has beenconverted. 

§ 481. In a loose sense of the term we may be said to have converted aproposition when we have merely transposed the subject and predicate, when, for instance, we turn the proposition 'All A is B' into 'All Bis A' or 'Some A is not B' into 'Some B is not A. ' But thesepropositions plainly do not follow from the former ones, and it isonly with conversion as a form of inference--with Illative Conversionas it is called--that Logic is concerned. 

§ 482. For conversion as a form of inference two rules have been laiddown--

 (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. Aviolation of it involves the fallacy of arguing from part of a term tothe whole. 

§ 484. The second rule is merely a conventional one. We may make avalid inference in defiance of it: but such an inference will be seenpresently 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 thequantity remains the same as before. 

§ 487. We are said to have converted a proposition per accidens, or bylimitation, when the rules for the distribution of terms necessitate areduction 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 'AllA is B' does not mean more than 'All A is some B, ' its proper converseis 'Some B is A. ' For, if we endeavoured to elicit the inference, 'AllB is A, ' we should be distributing the term B in the converse, whichwas not distributed in the convertend. Hence we should be involved inthe fallacy of arguing from the part to the whole. Because 'Alldoctors are men' it by no means follows that 'All men are doctors. '

§ 499. E and I admit of simple conversion, because the quantity of thesubject and predicate is alike in each, both subject and predicatebeing 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 subjectis undistributed and that the proposition is negative. Now, when theproposition is converted, what was the subject becomes the predicate, and, as the proposition must still be negative, the former subjectwould now be distributed, since every negative proposition distributesits predicate. Hence we should necessarily have a term distributed inthe converse which was not distributed in the convertend. From 'Somemen are not doctors, ' it plainly does not follow that 'Some doctorsare not men'; and, generally from 'Some A is not B' it cannot beinferred that 'Some B is not A, ' since the proposition 'Some A is notB' 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 givenmatter a proposition of the form 'All B is A' is true simultaneouslywith 'All A is B. ' Thus it is as true to say that 'All equiangulartriangles are equilateral' as that 'All equilateral triangles areequiangular. ' Nevertheless we are not logically warranted in inferringthe one from the other. Each has to be established on its separateevidence. 

§ 493. On the theory of the quantified predicate the differencebetween simple conversion and conversion by limitation disappears. Forthe quantity of a proposition is then no longer determined solely byreference to the quantity of its subject. 'All A is some B' is of nogreater quantity than 'Some B is all A, ' if both subject and predicatehave an equal claim to be considered. 

§ 494. Some propositions occur in ordinary language in which thequantity of the predicate is determined. This is especially the casewhen the subject is a singular term. Such propositions admit ofconversion 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 theform of the expression as distributed, in the other as undistributed. 

§ 495. Conversion offers a good illustration of the principle on whichwe have before insisted, namely, that in the ordinary form ofproposition the subject is used in extension and the predicate inintension. For when by conversion we change the predicate into thesubject, we are often obliged to attach a noun substantive to thepredicate, in order that it may be taken in extension, instead of, asbefore, 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 animmediate inference grounded on a change of quality in a propositionand a change of the predicate into its contradictory-term. 

§ 497. In less technical language we may say that permutation isexpressing negatively what was expressed affirmatively and vice versâ. 

§ 498. Permutation is equally applicable to all the fourforms 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 excludedmiddle, namely--That one or other of a pair of contradictory termsmust be applicable to a given subject, so that, when one may bepredicated affirmatively, the other may be predicated negatively, andvice versâ (§ 31). 

§ 501. Merely to alter the quality of a proposition would of courseaffect its meaning; but when the predicate is at the same time changedinto its contradictory term, the original meaning of the propositionis retained, whilst the form alone is altered. Hence we may lay downthe following practical rule for permutation--

Change the quality of the proposition and change the predicate intoits contradictory term. 

§ 502. The law of excluded middle holds only with regard tocontradictories. It is not true of a pair of positive and privativeterms, that one or other of them must be applicable to any givensubject. For the subject may happen to fall wholly outside the sphereto which such a pair of terms is limited. But since the fact of a termbeing applied is a sufficient indication of its applicability, andsince within a given sphere positive and privative terms are asmutually destructive as contradictories, we may in all casessubstitute the privative for the negative term in immediate inferenceby permutation, which will bring the inferred proposition more intoconformity with the ordinary usage of language. Thus the concreteinstances 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 immediateinference, 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 andI, and, in this form, admit of simple conversion. We have here twosteps of inference: but the process may be performed at a singlestroke, and is then known as Conversion by Negation. Thus from 'All Ais 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 beseen 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 resolvethemselves 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 tothe exact inferential procedure of permutation followed by simpleconversion. Hence it necessarily applies only to A and 0 propositions, since these when permuted become E and 1, which admit of simpleconversion; 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 withconversion, no matter in what order the two processes may beperformed. If this is not done, inferences quite as legitimate asthose which pass under the title of conversion by negation are leftwithout 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 foundthat the process of simple conversion precedes that of permutation. 

§ 510. In the case of the E proposition a compound inference can bedrawn 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 offact, but means that one of the possible forms of dishonesty among menis that of employing bribery. 

§ 511. If we analyse the preceding, we find that the second step isconversion 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 ofconversion 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 theconverse per accidens of the first--'Some B is A. ' This inference, however, coincides with that from 1 (§ 508), as the similar inferencefrom E (§ 510) coincides with that from 0 (§ 506). 

§ 514. All these inferences agree in the essential feature ofcombining permutation with conversion, and should therefore be classedunder a common name. 

§ 515. Adopting then this slight extension of the term, we defineconversion by negation as--A form of conversion in which the conversediffers in quality from the convertend, and has the contradictory ofone 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 threesteps 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 appearanceby the occurrence of a negative term in the subject or predicate orboth, 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 rightperforming 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 wemay 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 torest 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 onlyin reference to the A proposition, to which the process has hithertobeen confined. Logicians seem to have overlooked the fact thatconversion by contraposition is as applicable to the O as to the Aproposition, though, when expressed in symbols, it presents a moreclumsy 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 thesame 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 bynegation of the original proposition permuted. 

§ 526. Contraposition may also be applied to the E proposition by theuse of conversion per accidens in the place of simple conversion. But, owing to the limitation of quantity thus effected, the result arrivedat is the same as in the case of the O proposition. Thus from 'Nowholesome things are pleasant' we could draw the same inference asbefore. 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 bedefined generally as--A form of conversion in which both subject andpredicate are replaced by their contradictories. 

§ 528. Conversion by contraposition differs in several respects fromconversion 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 ordinaryE proposition except by limitation (§ 526). 

From 'No A is B' we cannot infer 'No not-B is not-A. ' For, if wecould, the contradictory of the latter, namely, 'Some not-B is not-A'would be false. But it is manifest that this is not necessarilyfalse. For when one term is excluded from another, there must benumerous individuals which fall under neither of them, unless itshould so happen that one of the terms is the direct contradictory ofthe 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 inadmitting of full conversion by contraposition, and the form of thatis 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. ' Forthough the proposition holds true as a matter of fact, when A and Bare in part mutually exclusive, yet this is not conveyed by the formof the expression. It may so happen that B is wholly contained underA, while A itself contains everything. In this case it will be truethat 'No not-B is not-A, ' which contradicts the attemptedinference. Thus from the proposition 'Some things are substances' itcannot be inferred that 'Some not-substances are not-things, ' for inthis case the contradictory is true that 'No not-substances arenot-things'; and unless an inference is valid in every case, it is notformally valid at all. 

§ 531. It should be noticed that in the case of the [nu] propositionimmediate inferences are possible by mere contraposition withoutconversion. 

 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 alsonon-equiangular. 

§ 532. The principle upon which this last kind of inference rests isthat when two terms are co-extensive, whatever is excluded from theone 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 abrief allusion to some other forms which have been recognised bylogicians. 

§ 534. Every statement of a relation may furnish us with ail immediateinference in which the same fact is presented from the oppositeside. 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 ofDick'; from 'Bicester is north-east of Oxford' we infer 'Oxford issouth-west of Bicester'; from 'So and so visited the Academy the dayafter he arrived in London' we infer 'So and so arrived in London theday before he visited the Academy'; from 'A is greater than B' weinfer 'B is less than A'; and so on without limit. Such inferences asthese are material, not formal. No law can be laid down for themexcept 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 ofImmediate Inference by Added Determinants, in which from someproposition already made another is inferred, in which the sameattribute 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 mustbe definite qualities, like whiteness, and must in no way involve acomparison. From 'A horse is a quadruped' it may seem at first sightto follow that 'A swift horse is a swift quadruped. ' But we need notgo far to discover how little formal validity there is about such aninference. 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 swiftcompared 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 atall man, ' but only that 'A tall Bushman is a man who is tall for aBushman'; and so on generally. 

§ 537. Very similar to the preceding is the process known as ImmediateInference 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 itdiffers in name rather than in nature, may be explained on theprinciple of Substitution. Starting from the identical proposition, 'The head of a quadruped is the head of a quadruped, ' and being giventhat 'A horse is a quadruped, ' so that whatever is true of 'quadruped'generally we know to be true of 'horse, ' we are entitled to substitutethe narrower for the wider term, and in this manner we arrive at theproposition, The head of a horse is the head of a quadruped. 

§ 539. Such an inference is valid enough, if the same caution beobserved as in the case of added determinants, that is, if nodifference be allowed to intervene in the relation of the freshconception to the generic and the specific terms. 

CHAPTER VIII. 

_Of Mediate Inferences or Syllogisms. _

§ 540. A Mediate Inference, or Syllogism, consists of twopropositions, which are called the Premisses, and a third propositionknown as the Conclusion, which flows from the two conjointly. 

§ 541. In every syllogism two terms are compared with one another bymeans of a third, which is called the Middle Term. In the premisseseach of the two terms is compared separately with the middle term; andin the conclusion they are compared with one another. 

§ 542. Hence every syllogism consists of three terms, one of whichoccurs twice in the premisses and does not appear at all in theconclusion. This term is called the Middle Term. The predicate of theconclusion is called the Major Term and its subject the Minor Term. 

§ 543. The major and minor terms are called the Extremes, as opposedto the Mean or Middle Term. 

§ 544. The premiss in which the major term is compared with the middleis called the Major Premiss. 

§ 545. The other premiss, in which the minor term is compared with themiddle, is called the Minor Premiss. 

§ 546. The order in which the premisses occur in a syllogism isindifferent, but it is usual, for convenience, to place the majorpremiss 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' termswere originally employed is that in an affirmative syllogism such asthe above, which was regarded as the perfect type of syllogism, thesenames express the relative quantity in extension of the three terms. 

[Illustration]

§ 549. It must be noticed however that, though the middle term cannotbe of larger extent than the major nor of smaller extent than theminor, if the latter be distributed, there is nothing to prevent allthree, or any two of them, from being coextensive. 

§ 550. Further, when the minor term is undistributed, we either have acase of the intersection of two classes, from which it cannot be toldwhich of them is the larger, or the minor term is actually larger thanthe 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 inthe case of the affirmative syllogism; and when the syllogism isnegative, they are not applicable at all: since in negativepropositions we have no means of comparing the relative extension ofthe terms employed. Had we said in the major premiss of our typicalsyllogism, 'No mammals are cold-blooded, ' and drawn the conclusion 'Nowhales are cold-blooded, ' we could not have compared the relativeextent of the terms 'mammal' and 'cold-blooded, ' since one has beensimply excluded from the other. 

 [Illustration]

§ 552. So far we have rather described than defined the syllogism. Allthe products of thought, it will be remembered, are the results ofcomparison. The syllogism, which is one of them, may be so regarded intwo 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 arethe major premiss and the conclusion, which are brought intoconnection by means of the minor premiss. Thus in the syllogism abovegiven we compare the conclusion 'All whales are warm-blooded' with themajor premiss 'All mammals are warm-blooded, ' and find that the formeris contained under the latter, as soon as we become acquainted withthe intermediate proposition 'All whales are mammals. '

§ 554. The two terms which are compared with one another are of coursethe major and minor. 

§ 555. The syllogism is merely a form into which our deductiveinferences may be thrown for the sake of exhibiting theirconclusiveness. It is not the form which they naturally assume inspeech or writing. Practically the conclusion is generally statedfirst and the premisses introduced by some causative particle as'because, ' 'since, ' 'for, ' &c. We start with our conclusion, and thengive the reason for it by supplying the premisses. 

§ 556. The conclusion, as thus stated first, was called by logiciansthe Problema or Quaestio, being regarded as a problem or question, towhich a solution or answer was to be found by supplying the premisses. 

§ 557. In common discourse and writing the syllogism is usually stateddefectively, one of the premisses or, in some cases, the conclusionitself 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 aman, ' or 'The Pope is fallible, because all men are so'; or perhaps weshould merely say 'All men are fallible, and the Pope is a man, 'leaving it to the sagacity of our hearers to supply the desiredconclusion. A syllogism, as thus elliptically stated, is commonly, though incorrectly, called an Enthymeme. When the major premiss isomitted, it is called an Enthymeme of the First Order; when the minoris omitted, an Enthymeme of the Second Order; and when the conclusionis 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 asyllogism consisting of three universal affirmatives, AAA, would besaid to differ in mood from one consisting of such propositions as EIOor any other combination that might be made. The syllogism previouslygiven to prove the fallibility of the Pope belongs to the moodAAA. Had we drawn only a particular conclusion, 'Some Popes arefallible, ' it would have fallen into the mood AAI. 

§ 560. Figure depends upon the arrangement of the terms in thepropositions. Thus a difference of figure is internal to a differenceof 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 moodand figure, irrespective of their logical validity. 

§ 562. And first as to mood. 

Since every syllogism consists of three propositions, and each ofthese propositions may be either A, E, I, or O, it is clear that therewill be as many possible moods as there can be combinations of fourthings, taken three together, with no restrictions as torepetition. It will be seen that there are just sixty-four of suchcombinations. For A may be followed either by itself or by E, I, orO. Let us suppose it to be followed by itself. Then this pair ofpremisses, AA, may have for its conclusion either A, E, I, or O, thusgiving four combinations which commence with AA. In like manner therewill be four commencing with AE, four with AI, and four with AO, giving a total of sixteen combinations which commence withA. Similarly there will be sixteen commencing with E, sixteen with I, sixteen with O--in all sixty-four. It is very few, however, of thesepossible combinations that will be found legitimate, when tested bythe rules of syllogism. 

§ 563. Next as to figure. 

There are four possible varieties of figure in a syllogism, as may beseen by considering the positions that can be occupied by the middleterm in the premisses. For as there are only two terms in eachpremiss, the position occupied by the middle term necessarilydetermines that of the others. It is clear that the middle term musteither occupy the same position in both premisses or not, that is, itmust either be subject in both or predicate in both, or else subjectin one and predicate in the other. Now, if we are not acquainted withthe conclusion of our syllogism, we do not know which is the major andwhich the minor term, and have therefore no means of distinguishingbetween one premiss and another; consequently we must Stop here, andsay that there are only three different arrangements possible. But, ifthe Conclusion also be assumed as known, then we are able todistinguish one premiss as the major and the other as the minor; andso we can go further, and lay down that, if the middle term does nothold the same position in both premisses, it must either be subject inthe major and predicate in the minor, or else predicate in the majorand subject in the minor. 

§ 564. Hence there result

 _The Four Figures. _

When the middle term is subject in the major and predicate in theminor, we are said to have the First Figure. 

When the middle term is predicate in both premisses, we are said tohave the Second Figure. 

When the middle term is subject in both premisses, we are said to havethe Third Figure. 

When the middle term is predicate in the major premiss and subject inthe 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 comparativelyvalueless. 

§ 566. It will be well to explain by an instance the meaning of theassertion previously made, that a difference of figure is internal toa difference of mood. We will take the mood EIO, and by varying theposition of the terms, construct a syllogism in it in each of the fourfigures. 

 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 ofthe premisses, both of which admit of simple conversion, it happensthat the resulting syllogisms are all valid. But in the great majorityof moods no syllogism would be valid at all, and in many moods asyllogism would be valid in one figure and invalid in another. As yethowever we are only concerned with the conceivable combinations, apartfrom the question of their legitimacy. 

§ 568. Now since there are four different figures and sixty-fourdifferent moods, we obtain in all 256 possible ways of arranging threeterms 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 perfecttype of syllogism, because the validity of moods in this figure may betested directly by their complying, or failing to comply, with acertain axiom, the truth of which is self-evident. This axiom is knownas 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 asbeing made in extension, whereas it is more naturally to be regardedas 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 neutralform thus--

 'Whatever is stated of the predicate will be stated also of the subject [Footnote: [Greek: osa katà toû kategorouménou légetai pánta kaì katà toû 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, ' ifwe can affirm anything universally of the predicate 'mammals, ' as, forinstance, that 'All mammals are warm-blooded, ' we shall be able toaffirm the same of the subject 'whales'; and, if we can deny anythinguniversally of the predicate, as that 'No mammals are oviparous, ' weshall 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, thefirst. The characteristic of the reasoning in that figure is that somegeneral rule is maintained to hold good in a particular case. Themajor premiss lays down some general principle, whether affirmative ornegative; the minor premiss asserts that a particular case falls underthis principle; and the conclusion applies the general principle tothe particular case. But though all syllogistic reasoning may betortured into conformity with this type, some of it finds expressionmore naturally in other ways. 

§ 576. Modern logicians therefore prefer to abandon the Dictum de Omniet Nullo in any shape, and to substitute for it the following threeaxioms, 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 theconditions under which no conclusion can be drawn. If there is anyagreement at all between the two terms and the third, as in the casescontemplated in the first and second axioms, then we have a conclusionof some kind: if it is otherwise, we have none. 

§ 578. It must be understood with regard to these axioms that, when wespeak of terms agreeing or disagreeing with the same third term, wemean that they agree or disagree with the same part of it. 

§ 579. Hence in applying these axioms it is necessary to bear in mindthe 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 (whichalters the figure from the first to the fourth). For it was only partof A which was known to agree with B. On the theory of the quantifiedpredicate we could draw the inference No C is some A. 

§ 580. It is of course possible for terms to agree with differentparts of the same third term, and yet to have no connection with oneanother. 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 'layeggs, ' is thereby necessarily excluded from that part of it whichcoincides with 'birds. '

[Illustration]

CHAPTER XI. 

_Of the Generad Rules of Syllogism. _

§ 582. We now proceed to lay down certain general rules to which allvalid syllogisms must conform. These are divided into primary andderivative. 

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 ofthe syllogism with which we started. We said it might be regardedeither as the comparison of two propositions by means of a third or asthe comparison of two terms by means of a third. To violate either ofthese rules therefore would be inconsistent with the fundamentalconception 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 processof mediate inference appears to have more than two premisses, it willalways be found that there is more than one syllogism. If there areless than three propositions, as in the fallacy of 'begging thequestion, ' in which the conclusion simply reiterates one of thepremisses, there is no syllogism at all. 

With regard to the second rule, it is plain that any attemptedsyllogism which has more than three terms cannot conform to theconditions of any of the axioms of mediate inference. 

§ 584. The next two rules guard against the two fallacies which arefatal to most syllogisms whose constitution is unsound. 

§ 585. The violation of Rule 3 is known as the Fallacy ofUndistributed Middle. The reason for this rule is not far to seek. For if the middle term is not used in either premiss in its wholeextent, we may be referring to one part of it in one premiss and toquite another part of it in another, so that there will be really nomiddle 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 thatpart of the term 'man, ' of which it is not known whether it agreeswith 'fallible' or not. 

§ 586. The violation of Rule 4 is known as the Fallacy of IllicitProcess. If the major term is distributed in the conclusion, nothaving been distributed in the premiss, we have what is called IllicitProcess of the Major; if the same is the case with the minor term, wehave Illicit Process of the Minor. 

§ 587. The reason for this rule is that if a term be used in its wholeextent in the conclusion, which was not so used in the premiss inwhich it occurred, we would be arguing from the part to the whole. Itis the same sort of fallacy which we found to underlie the simpleconversion 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 personswho go mad, whereas in the premisses, granting that all learned men gomad, it has not been said that they are all the men who do so. We havehere 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. Forin them we spoke only of some poor men, since the predicate of anaffirmative proposition is undistributed. 

§ 590. Rule 5 is simply another way of stating the third axiom ofmediate inference. To know that two terms disagree with the same thirdterm gives us no ground for any inference as to whether they agree ordisagree 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 maynot be ruminants. 

§ 591. This rule may sometimes be violated in appearance, though notin reality. For instance, the following is perfectly legitimatereasoning. 

 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 additionalviolation of Rule 2. In order to have the middle term the same in bothpremisses 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, theconclusion must be negative. Now in compliance with Rule 5, if onepremiss be negative, the other must be affirmative. We have thereforethe case contemplated in the second axiom, namely, of one termagreeing and the other disagreeing with the same third term; and weknow that this can only give ground for a judgement of disagreementbetween the two terms themselves--in other words, to a negativeconclusion. 

§ 593. Rule 7 declares that, if the conclusion be negative, one of thepremisses must be negative; but, if the conclusion be affirmative, both premisses must be affirmative. It is plain from the axioms that ajudgement of disagreement can only be elicited from a judgement ofagreement combined with a judgement of disagreement, and that ajudgement of agreement can result only from two prior judgements ofagreement. 

§ 594. The seven rules already treated of are evident by their ownlight, being of the nature of definitions and axioms: but the tworemaining rules, which deal with particular premisses, admit of beingproved from their predecessors. 

§ 595. Proof of Rule 8. --_That two particular premisses provenothing_. 

We know by Rule 5 that both premisses cannot be negative. Hence theymust be either both affirmative, II, or one affirmative and onenegative, IO or OI. 

Now II premisses do not distribute any term at all, and therefore themiddle 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 thatthis one term should be the middle term. Therefore the major term mustbe undistributed in the major premiss. But since one of the premissesis negative, the conclusion must be negative, by Rule 6. And everynegative proposition distributes its predicate. Therefore the majorterm must be distributed where it occurs as predicate of theconclusion. But it was not distributed in the major premiss. Thereforein drawing any conclusion we violate Rule 4 by an illicit process ofthe major term. 

§ 596. Proof of Rule 9. --_That_, _if_ one _premiss beparticular_, _the conclusion must be particular_. 

Two negative premisses being excluded by Rule 5, and two particular byRule 8, the only pairs of premisses we can have are--

 AI, AO, EI. 

Of course the particular premiss may precede the universal, but theorder of the premisses will not affect the reasoning. 

AI premisses between them distribute one term only. This must be themiddle term by Rule 3. Therefore the conclusion must be particular, asits subject cannot be distributed, AO and EI premisses each distribute two terms, one of which must bethe middle term by Rule 3: so that there is only one term left whichmay be distributed in the conclusion. But the conclusion must benegative by Rule 4. Therefore its predicate must be distributed. Hence its subject cannot be so. Therefore the conclusion must beparticular. 

§ 597. Rules 6 and 9 are often lumped together in a singleexpression--'The conclusion must follow the weaker part, ' negativebeing considered weaker than affirmative, and particular thanuniversal. 

§ 598. The most important rules of syllogism are summed up in thefollowing mnemonic lines, which appear to have been perfected, thoughnot invented, by a mediæval logician known as Petrus Hispanus, who wasafterwards 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 possiblemoods, each of which might occur in any of the four figures, giving usaltogether 256 possible varieties of syllogism. The task now before usis to determine how many of these combinations of mood and figure arelegitimate. 

§ 600. By the application of the preceding rules we are enabled toreduce the 64 possible moods to 11 valid ones. This may be done by alonger or a shorter method. The longer method, which is perhaps easierof comprehension, is to write down the 64 possible moods, and thenstrike 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 premissescan yield no conclusion at all, owing to their violating Rule 6 or 9;in the rest the premises are legitimate, but a wrong conclusion isdrawn from each of them as are translineated. 

§ 602. IEO stands alone, as violating Rule 4. This may require alittle explanation. 

Since the conclusion is negative, the major term, which is itspredicate, must be distributed. But the major premiss, being 1, doesnot distribute either subject or predicate. Hence IEO must alwaysinvolve an illicit process of the major. 

§ 603. The II moods which have been left valid, after being tested bythe 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 morescientific method. This method consists in first determining whatpairs of premisses are valid in accordance with Rules 6 and g, andthen examining what conclusions may be legitimately inferred from themin accordance with the other rules of syllogism. 

§ 605. The major premiss may be either A, E, I or O. If it is A, theminor also may be either A, E, I or O. If it is E, the minor can onlybe A or I. If it is I, the minor can only be A or E. If it is O, theminor 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 therules of syllogism; one of them, IE, yields no conclusion at all; theremaining 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 whichwe arrived at in the last chapter are valid in the four figures. Butbefore 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 reassertionin another form of the Dictum de Omni et Nullo. For if the majorpremiss were particular, we should not have anything affirmed ordenied of a whole class; and if the minor premiss were negative, weshould not have anything declared to be contained in that class. Nevertheless these rules, like the rest, admit of being proved fromthe position of the terms in the figure, combined with the rules forthe 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 beaffirmative (by Rule 5), [Footnote: This refers to the General Rulesof Syllogism. ] and the conclusion must be negative (by Rule 6). Butthe major being affirmative, its predicate is undistributed; and theconclusion being negative, its predicate is distributed. Now the majorterm is in this figure predicate both in the major premiss and in theconclusion. Hence there results illicit process of the majorterm. 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 itspredicate, is undistributed there. Therefore it must be distributed inthe major premiss, where it is subject. Therefore the major premissmust 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 mustbe 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 conclusionalso 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 bedistributed. But the major term is subject in the majorpremiss. 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 twofigures 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 itspredicate, will be undistributed there. Hence it must be undistributedin the conclusion (by Rule 4). Therefore the conclusion must beparticular. 

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 undistributedmiddle. [Footnote: Shorter proofs are employed in this figure, as thestudent is by this time familiar with the method of procedure. ]

§ 616. Proof of Rule 2. --_When the minor premiss is particular, themajor must be negative. _

 A--B B--C C--A

This rule is the converse of the preceding, and depends upon the sameprinciple. 

§ 617. Proof of Rule 3. --_When the minor premiss is affirmative, theconclusion must be particular. _

If the conclusion were universal, there would be illicit process ofthe minor. 

§ 618. Proof of Rule 4. --_When the conclusion is negative, the majorpremiss must_ be universal. 

If the major premiss were particular, there would be illicit processof the major. 

§ 619. Proof of Rule 5. --_The conclusion CANNOT be A UNIVERSALaffirmative_. 

The conclusion being affirmative, the premisses must be so too (byRule 7). Therefore the minor term is undistributed in the minorpremiss, where it is predicate. Hence it cannot be distributed in theconclusion (by Rule 4). Therefore the affirmative conclusion must beparticular. 

§ 620. Proof of Rule 6. --_Neither of the premisses can lie a, PARTICULAR NEGATIVE_. 

If the major premiss were a particular negative, the conclusion wouldbe negative. Therefore the major term would be distributed in theconclusion. But the major premiss being particular, the major termcould not be distributed there. Therefore we should have an illicitprocess of the major term. 

If the minor premiss were a particular negative, then, since the majormust be affirmative (by Rule 5), we should have undistributed middle. 

CHAPTER XIV

_Of the Determination of the Moods that are valid in the FourFigures. _

§ 621. By applying the special rules just given we shall be able todetermine how many of the eleven legitimate moods are valid in thefour 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 thelast two moods, IAI, OAO. The rule that the minor premiss must beaffirmative 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 universalexcludes OAO. Thus we are left with six moods which are valid in thesecond figure, namely, EAE. AEE. EIO. AOO. EAO. AEO. 

FIGURE III. 

§ 625. The rule that the conclusion must be particular confines us toeight moods, two of which, namely AEE and AOO, are excluded by therule 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 rulethat the conclusion cannot be a universal affirmative. 

Two more moods, namely AOO and OAO, are excluded by the rule thatneither of the premisses can be a particular negative. 

AII violates the rule that when the major premiss is affirmative, theminor must be universal. 

EAE violates the rule that, when the minor premiss is affirmative, theconclusion must be particular. Thus we are left with six moods whichare valid in the fourth figure, namely, AAI. AEE. IAI. EAO. EIO. AEO. 

§ 627. Thus the 256 possible forms of syllogism have been reduced totwo dozen legitimate combinations of mood and figure, six moods beingvalid 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, areuseless. For the conclusion drawn is less than is warranted by thepremisses. These are called Subaltern Moods, because their conclusionsmight be inferred by subalternation from the universal conclusionswhich can justly be drawn from the same premisses. Thus AAI issubaltern to AAA, EAO to EAE, and so on with the rest. 

§ 629. The remaining 19 combinations of mood and figure, which areloosely called 'moods, ' though in strictness they should be called'figured moods, ' are generally spoken of under the names supplied bythe following mnemonics--

 Barbara, Celarent, Darii, Ferioque prioris; Cesare, Camestres, Festino, Baroko secundæ; 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. Allthe special rules of the four figures can be gathered from aninspection of them. The following points should be specially noted. 

The first figure proves any kind of conclusion, and is the only onewhich 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 theImperfect figures. The claim of the first to be regarded as theperfect 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 courseinfinitely the most important form of proposition, can only be provedin 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 (byRule 7). But the minor term is distributed in the conclusion, as beingthe subject of an A proposition, and must therefore be distributed inthe minor premiss, in order to which it must be the subject. Thereforethe middle term must be the predicate and is consequentlyundistributed. In order therefore that the middle term may bedistributed, it must be subject in the major premiss, since that alsois an A proposition. But when the middle term is subject in the majorand predicate in the minor premiss, we have what is called the firstfigure. 

CHAPTER XV. 

_Of the Special Canons of the Four Figures. _

§ 633. So far we have given only a negative test of legitimacy, havingshown what moods are not invalidated by running counter to any of thespecial rules of the four figures. We will now lay down special canonsfor the four figures, conformity to which will serve as a positivetest of the validity of a given mood in a given figure. The specialcanon of the first figure--will of course be practically equivalent tothe Dictum de Omni et Nullo. All of them will be expressed in terms ofextension, for the sake of perspicuity. 

_Special Canons of the Four Figures. _

FIGURE 1. 

§ 634. CANON. If one term wholly includes or excludes another, whichwholly or partly includes a third, the first term wholly or partlyincludes 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 orpartly includes a third, or is included in another from which a thirdis wholly or partly excluded, the first is excluded from the whole orpart 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 thefirst includes the whole and the second a part of the same term, orvice 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 whollyor in part included in the second, or is excluded from part of a termwhich is wholly included in the second, the first is excluded frompart 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 anotherwhich is wholly included in or excluded from a third, the third termwholly or partly includes the first, or, in the case of totalinclusion, is wholly excluded from it; and if a term is excluded fromanother which is wholly or partly included in a third, the third ispartly 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 theconclusion is not drawn directly from the premisses, but is animmediate inference from the natural conclusion. Take for instance AAIin the first figure. The natural conclusion from these premisses isthat the minor term C is wholly contained in the major term A. Butinstead of drawing this conclusion we go on to infer that somethingwhich 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 Cis excluded from A, we draw a conclusion which really involves afurther 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 aform is to render the validity of all the moods in each figure at onceapparent from the statement. For purposes of general convenience theyadmit of a much more compendious mode of expression. 

§ 640. The canon of the first figure is known as the Dictum de Omni etNullo--

 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 deDiverse--

 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 Exemploet 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 offormulating 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 firstthree 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 athing. 

§ 649. The second figure is useful for proving distinctions betweenthings. 

§ 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 anattribute which C is devoid of, or is devoid of an attribute which Chas. 

 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 allevents partially, in B. Thus if we wish to produce an instance of thecompatibility of great learning with original powers of thought, wemight 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 Scotchmenbeing 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 Afalls, or does not fall, under the class B, which itself falls underC. 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 ligurespossible according as the conclusion was assumed to be known ornot. For, if the conclusion be not known, we cannot distinguishbetween the major and the minor term, nor, consequently, between onepremiss and another. On this view the first and the fourth figures arethe same, being that arrangement of the syllogism in which the middleterm occupies a different position in one premiss from what it does inthe other. We will now proceed to constitute the legitimate moods andfigures of the syllogism irrespective of the conclusion. 

§ 657. When the conclusion is set out of sight, the number of possiblemoods is the same as the number of combinations that can be made ofthe four things, A, E, I, O, taken two together, without restrictionas 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 thesyllogism, 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 threefigures. By so doing it will become apparent how far they are valid ineach. 

§ 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 directconclusions and five indirect ones, corresponding to the four moods ofthe 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 firstfigure equally well be regarded as indirect moods of the fourth? Forthis reason-that all the moods of the fourth figure can be elicitedout of premisses in which the terms stand in the order of the first, whereas the converse is not the case. If, while retaining the quantityand quality of the above premisses, i. E. The mood, we were in eachcase to transpose the terms, we should find that we were left withfive valid moods instead of six, since AI in the reverse order of theterms involves undistributed middle; and, though we should haveCelarent indirect to Camenes, and Darii to Dimaris, we should neverarrive at the conclusion of Barbara or have anything exactlyequivalent to Ferio. In place of Barbara, Bramantip would yield as anindirect mood only the subaltern AAI in the first figure. Both Fesapoand Fresison would result in an illicit process, if we attempted toextract the conclusion of Ferio from them as an indirect mood. Thenearest approach we could make to Ferio would be the mood EAO in thefirst figure, which may be elicited indirectly from the premisses ofCAMENES, being subaltern to CELARENT. For these reasons the moods ofthe fourth figure are rightly to be regarded as indirect moods of thefirst, 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 directconclusions corresponding to Cesare, CARNESTRES, FESTINO andBAROKO. 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 orindirectly. We have six direct moods, answering to Darapti, Disamis, Datisi, Felapton, Bokardo and Ferison, which are simply repeated bythe indirect moods, except in the case of Darapti, which yields aconclusion not provided for in the mnemonic lines. Darapti, thoughgoing under one name, has as much right to be considered two moods asDisamis and Datisi. 

CHAPTER XVIII. 

_Of Reduction. _

§ 667. We revert now to the standpoint of the old logicians, whoregarded the Dictum de Omni et Nullo as the principle of allsyllogistic reasoning. From this point of view the essence of mediateinference consists in showing that a special case, or class of cases, comes under a general rule. But a great deal of our ordinary reasoningdoes not conform to this type. It was therefore judged necessary toshow that it might by a little manipulation be brought into conformitywith 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 (perdeductionem) ad impossibile, is this--Given any mood in one of theimperfect figures, to show by means of a syllogism in the perfectfigure that its conclusion cannot be false. 

§ 671. The object of reduction is to extend the sanction of the Dictumde Omni et Nullo to the imperfect figures, which do not obviouslyconform to it. 

§ 672. The mood required to be reduced is called the Reducend; that towhich it conforms, when reduced, is called the Reduct. 

_Direct or Ostensive Reduction. _

§ 673. In the ordinary form of direct reduction, the only kind ofimmediate inference employed is conversion, either simple or bylimitation; but the aid of permutation and of conversion by negationand by contraposition may also be resorted to. 

§ 674. There are two moods, Baroko and Bokardo, which cannot bereduced ostensively except by the employment of some of the means lastmentioned. Accordingly, before the introduction of permutation intothe scheme of logic, it was necessary to have recourse to some otherexpedient, in order to demonstrate the validity of these twomoods. Indirect reduction was therefore devised with a special view tothe requirements of Baroko and Bokardo: but the method, as will beseen, is equally applicable to all the moods of the imperfect figures. 

§ 675. The mnemonic lines, 'Barbara, Celarent, etc. , provide completedirections for the ostensive reduction of all the moods of the second, third, and fourth figures to the first, with the exception of Barokoand Bokardo. The application of them is a mere mechanical trick, whichwill best be learned by seeing the process performed. 

§ 676. Let it be understood that the initial consonant of each name ofa figured mood indicates that the reduct will be that mood whichbegins with the same letter. Thus the B of Bramantip indicates thatBramantip, when reduced, will become Barbara. 

§ 677. Where m appears in the name of a reducend, me shall have totake as major that premiss which before was minor, and vice versa-inother words, to transpose the premisses, m stands for mutatio ormetathesis. 

§ 678. S, when it follows one of the premisses of a reducend, indicates that the premiss in question must be simply converted; whenit follows the conclusion, as in Disamis, it indicates that theconclusion arrived at in the first figure is not identical in formwith the original conclusion, but capable of being inferred from it bysimple conversion. Hence s in the middle of a name indicates somethingto be done to the original premiss, while s at the end indicatessomething to be done to the new conclusion. 

§ 679. P indicates conversion per accidens, and what has just beensaid of s applies, mutatis mutandis, to p. 

§ 680. K may be taken for the present to indicate that Baroko andBokardo 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 ostensivelyby the aid of mere conversion becomes plain on an inspection ofthem. In both it is necessary, if we are to obtain the first figure, that the position of the middle term should be changed in onepremiss. But the premisses of both consist of A and 0 propositions, ofwhich A admits only of conversion by limitation, the effect of whichwould be to produce two particular premisses, while 0 does not admitof conversion at all, It is clear then that the 0 proposition must cease to be 0 before wecan get any further. Here permutation comes to our aid; whileconversion by negation enables us to convert the A proposition, without loss of quantity, and to elicit the precise conclusion werequire 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 beenadopted as a symbol for permutation; n signifies conversion bynegation. In Donamon the first n stands for a process which resolvesitself into permutation followed by simple conversion, the second forone which resolves itself into simple conversion followed bypermutation, according to the extended meaning which we have given tothe term 'conversion by negation. ' If it be thought desirable todistinguish these two processes, the ugly symbol Do[pi]samos[pi] maybe adopted in place of Donamon. 

§ 686. The foregoing method, which may be called Reduction byNegation, is no less applicable to the other moods of the secondfigure than to Baroko. The symbols which result from providing for itsapplication 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 willbe 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]ereduced 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 permutingthe major premiss once more, so as to obtain the contrapositive of theoriginal--

 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: butit does not conform, as it stands, to the first figure, nor(permutation apart) can its premisses be twisted into conformity withit. But though we cannot prove the conclusion true in the firstfigure, we can employ that figure to prove that it cannot be false, byshowing that the supposition of its falsity would involve acontradiction of one of the original premisses, which are true exhypothesi. 

§ 693. If possible, let the conclusion 'Some C is not A' befalse. Then its contradictory 'All C is A' must be true. Combiningthis as minor with the original major, we obtain premisses in thefirst 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 notB, ' being its contradictory. But the original minor is ex hypothesitrue. Therefore the new conclusion is false. Therefore it must eitherbe 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, towhich the Dictum de Omni et Nullo applies. Therefore the fault mustlie in the premisses. But the major premiss, being the same with thatof the original syllogism, is ex hypothesi true. Therefore the minorpremiss, 'All C is A, ' is false. But this being false, itscontradictory must be true. Now its contradictory is the originalconclusion, '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 injuxtaposition 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 oneanother. The initial consonant of the names Baroko and Eokardoindicates that the indirect reduct will be Barbara. The k indicatesthat the O proposition, which it follows, is to be dropped out in thenew syllogism, and its place supplied by the contradictory of the oldconclusion. 

§ 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 aspecial view to Baroko and Bokardo, is applicable to all the moods ofthe imperfect figures. The following modification of the mnemoniclines contains directions for performing the process in everycase:--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, Cesïkon. 

§ 698. The c which appears in two moods of the third figure, Cakaciand Bekaco, signifies that the new conclusion is the contrary, insteadof, 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 thediscarded premiss, but with its converse, either simple or peraccidens, 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, whethermediate or immediate, solely as applied to simple propositions. But itwill be remembered that we divided propositions into simple andcomplex. I t becomes incumbent upon us therefore to consider the lawsof inference as applied to complex propositions. Inasmuch however asevery complex proposition is reducible to a simple one, it is evidentthat the same laws of inference must apply to both. 

§ 702. We must first make good this initial statement as to theessential identity underlying the difference of form between simpleand 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 theconjunctive proposition in this, that in the conjunctive the truthof the antecedent involves the truth of the consequent, whereas in thedisjunctive the falsity of the antecedent involves the truth of theconsequent. 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 anyother form, except U, seems to convey two statements in onebreath. Yet it ought not, any more than the E proposition, to beregarded as conveying both with equal directness. The proposition 'NoA 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 Ais not B, C is D. ' It asserts indeed by implication also that 'If C isnot D, A is B. ' But this is an immediate inference, being, as we shallpresently see, the contrapositive of the original. When we say 'So andso is either a knave or a fool, ' what we are directly asserting isthat, if he be not found to be a knave, he will be found to be afool. By implication we make the further statement that, if he be notcleared of folly, he will stand condemned of knavery. This inferenceis so immediate that it seems indistinguishable from the formerproposition: but since the two members of a complex proposition playthe part of subject and predicate, to say that the two statements areidentical would amount to asserting that the same proposition can havetwo subjects and two predicates. From this point of view it becomesclear that there is no difference but one of expression between thedisjunctive and the conjunctive proposition. The disjunctive ismerely a peculiar way of stating a conjunctive proposition with anegative 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 tosimple 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 itcontained any reference to actual existence. The meaning might bebetter 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 toemphasize the fact that formal logic is concerned only with theconnection of ideas. 

§ 713. A concrete instance will render the point under discussionclearer. The example we took before of an A proposition in theconjunctive 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 isto be found the ambition of their rulers, even if every actual caseshould be referred to some other cause. It is in this sense only thatthe inference is a necessary one. But then this is the only sensewhich formal logic is competent to recognise. To judge of conformityto fact is no part of its province. From 'Every AB is a CD' it followsthat ' Some CD's are AB's' with exactly the same necessity as thatwith which 'Some B is A' follows from 'All A is B. ' In the latter casealso neither proposition may at all conform to fact. From 'Allcentaurs are animals' it follows necessarily that 'Some animals arecentaurs': 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 conditionalpropositions are to be regarded as affirmative. Thus it has beenasserted that, even when we say that 'If the night becomes cloudy, there will be no dew, ' the proposition is not to be regarded asnegative, on the ground that what we affirm is a relation between thecloudiness of night and the absence of dew. This is a possible, butwholly unnecessary, mode of regarding the proposition. It is preciselyon a par with Hobbes's theory of the copula in a simple propositionbeing always affirmative. It is true that it may always be sorepresented at the cost of employing a negative term; and the same isthe case here. 

§ 717. There is no way of converting a disjunctive proposition exceptby 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 standswithout 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 distinctivefeatures of conversion by negation are still discernible. In each ofthe following forms of inference the converse differs in quality fromthe 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 hadundergone no change but the mere transposition of thealternative. This appearance arises from mentally reading the E as anA proposition: but, if it were so taken, the result would be itscontrapositive, and not its converse by negation. 

§ 727. The converse of I is a little difficult to grasp. It becomeseasier 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 contrapositionassumes 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 bycontraposition 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 runthus--

 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 ofeach of the same propositions, A and 0, the reader will see that theydiffer from them, as was to be expected, only in being permuted. Thevalidity of the inference may be tested, both here and in the case ofconversion by negation, by reducing the disjunctive proposition to theconjunctive, and so to the simple form, then performing the process asin simple propositions, and finally throwing the converse, when soobtained, back into the disjunctive form. We will show in this mannerthat 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, thereare 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 conjunctivepropositions: but if only one is conjunctive, the other must be asimple one. 

§ 734. Where both premisses are conjunctive, the conclusion will be ofthe same character; where only one is conjunctive, the conclusion willbe a simple proposition. 

§ 735. Of these two kinds of conjunctive syllogisms we will first takethat which consists throughout of conjunctive propositions. 

_The Wholly Conjunctive Syllogism_. 

§ 736. Wholly conjunctive syllogisms do not differ essentially fromsimple ones, to which they are immediately reducible. They admit ofbeing constructed in every mood and figure, and the moods of theimperfect figures may be brought into the first by following theordinary 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 orHypothetical Syllogism is spoken of. 

§ 738. Of the two premisses, one conjunctive and one simple, theconjunctive is considered to be the major, and the simple premiss theminor. For the conjunctive premiss lays down a certain relation tohold between two propositions as a matter of theory, which is appliedin the minor to a matter of fact. 

§ 739. Taking a conjunctive proposition as a major premiss, there arefour simple minors possible. For we may either assert or deny theantecedent 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 wecan get no conclusion. For to say that C is D whenever A is B gives usno right to deny that C can be D in the absence of thatcondition. What we have predicated has been merely inclusion of thecase AB in the case CD. 

[Illustration]

§ 741. Again, when we take as a minor, 'C is D' (4), we can get nouniversal conclusion. For though A being B is declared to involve as aconsequence C being D, yet it is possible for C to be D under othercircumstances, or from other causes. Granting the truth of theproposition 'If the sky falls, we shall catch larks, ' it by no meansfollows that there are no other conditions under which this result canbe attained. 

§ 742. From a consideration of the above four cases we elicit thefollowing

_Canon of the Conjunctive Syllogism. _

To affirm the antecedent is to affirm the consequent, and to deny theconsequent is to deny the antecedent: but from denying the antecedentor affirming the consequent no conclusion follows. 

§ 743. There is a case, however, in which we can legitimately deny theantecedent and affirm the consequent of a conjunctive proposition, namely, when the relation predicated between the antecedent and theconsequent is not that of inclusion but of coincidence--where in factthe 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 theyare 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 ishowever supposed to be latent in the transition from the conjunctiveto the simple form of proposition. When we say 'A is B, ' we are takento mean, 'As a matter of fact, A is B' or 'The actual state of thecase is that A is B. ' The insertion therefore of some such expressionas 'The case in hand, ' or 'This case, ' is, on this view, all that iswanted to complete the form of the syllogism. When reduced in thismanner to the simple type of argument, it will be found that theconstructive conjunctive conforms to the first figure and thedestructive 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 theconstructive conjunctive must fall into the first figure and thedestructive conjunctive into the second. There is no reason, however, why they should be confined to the two moods, Barbara andCarnestres. If the inference is universal, whether as general orsingular, the mood is Barbara or Carnestres; if it is particular, themood 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 obtainedby taking a negative proposition as the consequent in the majorpremiss. 

 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 thesimple form, it follows that violations of its laws must correspondwith violations of the laws of simple syllogism. By our throwing theillicit moods into the simple form it will become apparent whatfallacies 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 illicitprocess 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 toundistributed middle. 

§ 749. If we confine ourselves to the special rules of the fourfigures, we see that denial of the antecedent involves a negativeminor in the first figure, and affirmation of the consequent twoaffirmative premisses in the second. Or, if the consequent in themajor premiss were itself negative, the affirmation of it would amountto 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 conjunctivesyllogism--'To affirm the antecedent is to affirm the consequent, 'corresponds with the Dictum de Omni. For whereas something (viz. Cbeing D) is affirmed in the major of all conceivable cases of A beingB, the same is affirmed in the conclusion of something which isincluded therein, namely, 'this case, ' or 'some cases, ' or even 'allactual cases. '

§ 751. The negative side--'to deny the consequent is to deny theantecedent'--corresponds with the Dictum de Diverse (§ 643). Forwhereas in the major all conceivable cases of A being B are includedin C being D, in the minor 'this case, ' or 'some cases, ' or even 'allactual cases' of C being D, are excluded from the same notion. 

§ 752. The special characteristic of the partly conjunctive syllogismlies in the transition from hypothesis to fact. We might lay down asthe appropriate axiom of this form of argument, that 'What is true inthe abstract is true--in the concrete, ' or 'What is true in theory isalso true in fact, ' a proposition which is apt to be neglected ordenied. But this does not vitally distinguish it from the ordinarysyllogism. For though in the latter we think rather of the transitionfrom a general truth to a particular application of it, yet at bottoma general truth is nothing but a hypothesis resting upon a slenderbasis of observed fact. The proposition 'A is B' may be expressed inthe form 'If A is, B is. ' To say that 'All men are mortal' may beinterpreted to mean that 'If we find in any subject the attributes ofhumanity, the attributes of mortality are sure to accompany them. '

CHAPTER XXII. 

_Of the Partly Conjunctive Syllogism regarded as an ImmediateInference_. 

§ 753. It is the assertion of fact in the minor premiss, where we havethe application of an abstract principle to a concrete instance, whichalone entitles the partly conjunctive syllogism to be regarded as asyllogism at all. Apart from this the forms of semi-conjunctivereasoning 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 subalternopposition--

 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, comingunder 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 oftreatment an instance of the vice of immediate inference known as thesimple conversion of an A proposition. 'If A is B, C is D' is notconvertible with 'If C is D, A is B' any more than 'All A is B' isconvertible 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 followingform--

 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 specialname, 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, ' butof its simple converse 'All B is A. '

§ 759. The above-mentioned form of immediate inference, however(namely, the employment of contraposition without conversion), isvalid in the case of the U proposition; and so also is simpleconversion. Accordingly we are able, as we have seen, in dealing witha proposition of that form, both to deny the antecedent and to assertthe 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 thecombination of a disjunctive with a simple premiss. As in thepreceding form, the complex proposition is regarded as the majorpremiss, since it lays down a hypothesis, which is applied to fact inthe 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 thatare formally conclusive. The validity of the two destructive moods iscontingent upon the kind of alternatives selected. If these are suchas necessarily to exclude one another, the conclusion will hold, butnot otherwise. They are of course mutually exclusive whenever theyembody the result of a correct logical division, as 'Triangles areeither equilateral, isosceles or scalene. ' Here, if we affirm one ofthe members, we are justified in denying the rest. When the major thuscontains the dividing members of a genus, it may more fitly besymbolized under the formula, 'A is either B or C. ' But as this admitsof being read in the shape, 'Either A is B or A is C, ' we retain thewider expression which includes it. Any knowledge, however, which wemay have of the fact that the alternatives selected in the major areincompatible must come to us from material sources; unless indeed wehave confined ourselves to a pair of contradictory terms (A is eitherB or not-B). There can be nothing in the form of the expression toindicate the incompatibility of the alternatives, since the same formis employed when the alternatives are palpably compatible. When, forinstance, we say, 'A successful student must be either talented orindustrious, ' we do not at all mean to assert the positiveincompatibility of talent and industry in a successful student, butonly the incompatibility of their negatives--in other words, that, ifboth are absent, no student can be successful. Similarly, when it issaid, 'Either your play is bad or your luck is abominable, ' there isnothing in the form of the expression to preclude our conceiving thatboth may be the case. 

§ 764. There is no limit to the number of members in the disjunctivemajor. But if there are only two alternatives, the conclusion will bea simple proposition; if there are more than two, the conclusion willitself 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 asfollows--

 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 twoconstructive moods alone are formally valid. The first of these, namely, the denial of the antecedent, will in all cases give a simplesyllogism in the first figure; the second of them, namely, the denialof the consequent, will in all cases give a simple syllogism in thesecond 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 obtainedby varying the quality of the antecedent and consequent in the majorpremiss and reducing the quantity of the minor. 

§ 768. The invalid destructive moods correspond with the two invalidtypes of the partly conjunctive syllogism, and have the same fallaciesof simple syllogism underlying them. Affirmation of the antecedent ofa disjunctive is equivalent to the semi-conjunctive fallacy of denyingthe antecedent, and therefore involves the ordinary syllogisticfallacy of illicit process of the major. 

Affirmation of the consequent of a disjunctive is equivalent to thesame fallacy in the semi-conjunctive form, and therefore involves theordinary 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 complexreasoning hitherto discussed are identical both in appearance andreality. The apparent difference of procedure in the case of theantecedent, namely, that it is affirmed in the partly conjunctive, butdenied in the disjunctive syllogism, is due merely to the fact that inthe disjunctive proposition the truth of the consequent is involved inthe falsity of the antecedent, so that the antecedent beingnecessarily negative, to deny it in appearance is in reality to assertit. 

CHAPTER XXV. 

_The Disjunctive Syllogism regarded as an Immediate Inference_. 

§ 770. If no stress be laid on the transition from disjunctivehypothesis to fact, the disjunctive syllogism will run with the samefacility 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 willbe 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 theformula--

 All not-A is B, . '. All A is not-B, a form of inference which cannot hold except where A and B are knownto 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 aconjunctive is combined with a disjunctive premiss. The best knownform 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 thefollowing 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 partlyconjunctive syllogism chiefly in the fact of having a disjunctiveaffirmation of the antecedent or denial of the consequent in theminor, instead of a simple one. It is this which constitutes theessence of the dilemma, and which determines its possiblevarieties. For if only the antecedent or only the consequent be morethan one, we must, in order to obtain a disjunctive minor, affirm theantecedent or deny the consequent respectively; whereas, if there bemore than one of both, it is open to us to take either course. Thisgives 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 wouldnot 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 makeit 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 severalmembers, instead of being distributively assigned to one another, maybe 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 conjunctivesyllogism is more immediately recognisable. The major premiss in thisshape is vaguer than in the former. For each antecedent has now adisjunctive choice of consequents, instead of being limited toone. This vagueness, however, does not affect the conclusion. For, solong as the conclusion is established, it does not matter from whichmembers of the major its own members flow. 

§ 786. It must be carefully noticed that we cannot treat the complexdestructive 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 theantecedents, a disjunctive denial of them leaves it still possible forthe antecedent as a whole to be true. For 'C is not D' makes it truethat G is H, and 'G is not H' makes it true that C is D. In eithercase then one is true, which is all that was demanded by theconsequent of the major. Hence the consequent has not really beendenied. 

§ 787. For the sake of simplicity we have limited the examples to thecase of two antecedents or consequents. But we may have as many ofeither as we please, so as to have a Trilemma, a Tetralemma, and soon. 

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 someforms 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 conjunctivesyllogisms with the same conclusion, which would have been establishedby 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, bothpointing to the same conclusion. The proof is again redundant. In thiscase we have the consequent denied in both, whereas in the former wehad the antecedent affirmed. It is only for convenience that sucharguments as these are thrown into the form of a singlesyllogism. Their real distinctness may be seen from the fact that wehere deny each proposition separately, thus making two independentstatements--C is not D and E is not F. But in the true instance of thesimple destructive dilemma, what we deny is not the truth of the twopropositions contained in the consequent, but their compatibility; inother 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 consequentdenied in the minor. In the consequent of the major it is assertedthat 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 bothfalse. 

§ 792. A dilemma is said to be rebutted or retorted, when anotherdilemma is made out proving an opposite conclusion. If the dilemma bea sound one, and its premisses true, this is of course impossible, andany appearance of contradiction that may present itself on first sightmust vanish on inspection. The most usual mode of rebutting a dilemmais 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 Athenianmother to her son--'Do not enter public life: for, if you say what isjust, men will hate you; and, if you say what is unjust, the gods willhate you' to which the following retort was made--'I ought to enterpublic 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 conclusionshere are quite compatible. A man must, on the given premisses, be bothhated and loved, whatever course he takes. So far indeed are twopropositions 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 thingwhen 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 dilemmacontained in the story of Protagoras and Euathlus(Aul. Gell. Noct. Alt. V. 10), Euathlus was a pupil of Protagoras inrhetoric. He paid half the fee demanded by his preceptor beforereceiving lessons, and agreed to pay the remainder when he won hisfirst case. But as he never proceeded to practise at the bar, itbecame evident that he meant to bilk his tutor. Accordingly Protagorashimself instituted a law-suit against him, and in the preliminaryproceedings before the jurors propounded to him the followingdilemma--'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 yourfavour, 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, wasequal to the occasion, and rebutted the dilemma as follows. 'Mostsapient master, whatever be the issue of this suit, I shall not payyou what you claim: for, if the verdict be given in my favour, I amabsolved by the decision of the jurors; and, if it be given againstme, I am absolved by our bargain. ' The jurors are said to have been sopuzzled by the conflicting plausibility of the arguments that theyadjourned the case till the Greek Kalends. It is evident, however, that a grave injustice was thus done to Protagoras. His dilemma wasreally invincible. In the counter-dilemma of Euathlus we are meant toinfer that Protagoras would actually lose his fee, instead of merelygetting it in one way rather than another. In either case he wouldboth get and lose his fee, in the sense of getting it on one plea, andnot getting it on another: but in neither case would he actually loseit. 

§ 795. If a dilemma is correct in form, the conclusion of courserigorously follows: but a material fallacy often underlies this formof argument in the tacit assumption that the alternatives offered inthe minor constitute an exhaustive division. Thus the dilemma 'If painis 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 besevere 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 whoused it would doubtless mean to imply that students could beexhaustively divided into the idle and the industrious. No deductiveconclusion can go further than its premisses; so that all that theabove conclusion can in strictness be taken to mean is thatexaminations are unavailing, when students are idle, and superfluous, when they are industrious--which is simply a reassertion as a matterof 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 partlyconjunctive syllogism, we should naturally expect to find it reduciblein the same way to the form of a simple syllogism. And such is in factthe case. The constructive dilemma conforms to the first figure andthe 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 ofBarbara, in the constructive form, and Baroko, instead of Camestres, in the destructive. As in the case of the partly conjunctive syllogismthe remaining moods of the first and second figure are obtained bytaking 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 beexpressed under the forms of immediate inference. As before, theconclusion in the constructive type resolves itself into thesubalternate of the major itself, and in the destructive type into thesubalternate 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 thetransition 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 theconclusion duly follows from the premisses: he need not concernhimself with the truth or falsity of his data. But the premisses ofone syllogism may themselves be conclusions deduced from othersyllogisms, the premisses of which may in their turn have beenestablished by yet earlier syllogisms. When syllogisms are thus linkedtogether we have what is called a Train of Reasoning. 

§ 801. It is plain that all truths cannot be established byreasoning. For the attempt to do so would involve us in an infiniteregress, wherein the number of syllogisms required would increase ateach step in a geometrical ratio. To establish the premisses of agiven syllogism we should require two preceding syllogisms; toestablish their premisses, four; at the next step backwards, eight; atthe next, sixteen; and so on ad infinitum. Thus the very possibilityof reasoning implies truths that are known to us prior to allreasoning; and, however long a train of reasoning may be, we mustultimately come to truths which are either self-evident or are takenfor granted. 

§ 802. Any syllogism which establishes one of the premisses of anotheris called in reference to that other a Pro-syllogism, while asyllogism which has for one of its premisses the conclusion of anothersyllogism is called in reference to that other an Epi-syllogism. 

_The Epicheirema_. 

§ 803. The name Epicheirema is given to a syllogism with one or bothof its premisses supported by a reason. Thus the following is adouble 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 orderaccording as the major or minor premiss is thus supported. The doubleepicheirema is a combination of the two orders. 

§ 805. An epicheirema, it will be seen, consists of one syllogismfully expressed together with one, or, it may be, two enthymemes (§557). In the above instance, if the reasoning which supports thepremisses were set forth at full length, we should have, in place ofthe 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 thefirst order, having the major premiss suppressed. But there isnothing to prevent one or both of them from being of the secondorder--

 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 canbe 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 ofpropositions, each of which has one term in common with that whichpreceded it, while in the conclusion one of the terms in the lastproposition becomes either subject or predicate to one of the terms inthe 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 iscommonly described as a series of propositions in which the predicateof each becomes the subject of the next, while in the conclusion thelast predicate is affirmed or denied of the first subject. Theregressive form, however, exactly reverses these attributes; and wouldrequire to be described as a series of propositions, in which thesubject of each becomes the predicate of the next, while in theconclusion the first predicate is affirmed or denied of the lastsubject. 

§ 812. The regressive sorites, it will be observed, consists of thesame propositions as the progressive one, only written in reverseorder. 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 premisshappens to precede the major? It is because the sorites is not a mereseries of propositions, but a compressed train of reasoning; and thetwo trains of reasoning may be resolved into their componentsyllogisms in such a manner as to exhibit a real difference betweenthem. 

§ 813. The Progressive Sorites is a train of reasoning in which theminor 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 themajor 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 manysyllogisms as there are propositions intermediate between the firstand the last. This is evident also on inspection by counting thenumber of middle terms. 

§ 817. In expanding the progressive form we have to commence with thesecond proposition of the sorites as the major premiss of the firstsyllogism. In the progressive form the subject of the conclusion isthe same in all the syllogisms; in the regressive form the predicateis the same. In both the same series of means, or middle terms, isemployed, the difference lying in the extremes that are compared withone another through them. 

[Illustration]

§ 818. It is apparent from the figure that in the progressive form wework from within outwards, in the regressive form from withoutinwards. In the former we first employ the term 'Devonshiremen' as amean to connect 'Bideford men' with 'Englishmen'; next we employ'Englishmen' as a mean to connect the same subject 'Bideford men' withthe wider term 'Teutons'; and, lastly, we employ 'Teutons' as a meanto connect the original subject 'Bideford men' with the ultimatepredicate 'Ayrans. '

§ 819. Reversely, in the regressive form we first use 'Teutons' as amean whereby to bring 'Englishmen' under 'Aryans'; next we use'Englishmen' as a mean whereby to bring 'Devonshiremen' under the damepredicate 'Aryans'; and, lastly, we use 'Devonshiremen' as a meanwhereby to bring the ultimate subject 'Bideford men' under theoriginal predicate 'Aryans. '

§ 820. A sorites may be either Regular or Irregular. 

§ 821. In the regular form the terms which connect each proposition inthe series with its predecessor, that is to say, the middle terms, maintain a fixed relative position; so that, if the middle term besubject in one, it will always be predicate in the other, and viceversâ. In the irregular form this symmetrical arrangement is violated. 

§ 822. The syllogisms which compose a regular sorites, whetherprogressive or regressive, will always be in the first figure. 

In the irregular sorites the syllogisms may fall into differentfigures. 

§ 823. For the regular sorites the following rules maybe 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 onenegative, which follows from the general rules of syllogism. But thereis nothing to prevent any one premiss from being particular or any onepremiss from being negative, as the subjoined examples will show. Boththe 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 consistingof conjunctive instead of simple propositions. This issubject to the same laws as the simple sorites, to whichit 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 thoughtsdepend, it is expedient to classify some of the most familiar forms oferror. It is by the treatment of the Fallacies that logic chieflyvindicates its claim to be considered a practical rather than aspeculative science. To explain and give a name to fallacies is likesetting up so many sign-posts on the various turns which it ispossible to take off the road of truth. 

§ 828. By a fallacy is meant a piece of reasoning which appears toestablish a conclusion without really doing so. The term applies bothto the legitimate deduction of a conclusion from false premisses andto the illegitimate deduction of a conclusion from anypremisses. There are errors incidental to conception and judgement, which might well be brought under the name; but the fallacies withwhich we shall concern ourselves are confined to errors connected withinference. 

§ 829. When any inference leads to a false conclusion, the error mayhave arisen either in the thought itself or in the signs by which thethought is conveyed. The main sources of fallacy then are confined totwo--

 (1) thought, (2) language. 

§ 830. This is the basis of Aristotle's division of fallacies, whichhas not yet been superseded. Fallacies, according to him, are eitherin the language or outside of it. Outside of language there is nosource of error but thought. For things themselves do not deceive us, but error arises owing to a misinterpretation of things by themind. Thought, however, may err either in its form or in itsmatter. The former is the case where there is some violation of thelaws of thought; the latter whenever thought disagrees with itsobject. Hence we arrive at the important distinction between Formaland Material fallacies, both of which, however, fall under the samenegative 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 bereferred-namely, Formal Fallacies, Fallacies of Language, which arecommonly known as Fallacies of Ambiguity, and, lastly, MaterialFallacies. 

§ 832. Aristotle himself only goes so far as the first step in thedivision of fallacies, being content to class them according as theyare in the language or outside of it. After that he proceeds at onceto enumerate the infimæ species under each of the two main heads. Weshall presently imitate this procedure for reasons of expediency. Forthe whole phraseology of the subject is derived from Aristotle'streatise on Sophistical Refutations, and we must either keep to hismethod or break away from tradition altogether. Sufficient confusionhas already arisen from retaining Aristotle's language whileneglecting his meaning. 

§ 833. Modern writers on logic do not approach fallacies from the samepoint of view as Aristotle. Their object is to discover the mostfertile sources of error in solitary reasoning; his was to enumeratethe various tricks of refutation which could be employed by a sophistin controversy. Aristotle's classification is an appendix to the Artof Dialectic. 

§ 834. Another cause of confusion in this part of logic is theidentification of Aristotle's two-fold division of fallacies, commonlyknown under the titles of In dictione and Extra diotionem, with thedivision into Logical and Material, which is based on quite adifferent principle. 

§ 835. Aristotle's division perhaps allows an undue importance tolanguage, in making that the principle of division, and so throwingformal and material fallacies under a common head. Accordingly anotherclassification has been adopted, which concentrates attention from thefirst upon the process of thought, which ought certainly to be ofprimary importance in the eyes of the logician. This classificationis as follows. 

§ 836. Whenever in the course of our reasoning we are involved inerror, either the conclusion follows from the premisses or it doesnot. 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 otherhand, the conclusion does follow from the premisses, the fault mustlie in the premisses themselves, and we then have what is called aMaterial Fallacy. Sometimes, however, the conclusion will appear tofollow from the premisses until the meaning of the terms is examined, when it will be found that the appearance is deceptive owing to someambiguity in the language. Such fallacies as these are, strictlyspeaking, non-logical, since the meaning of words is extraneous to thescience which deals with thought. But they are calledSemi-logical. Thus we arrive by a different road at the same threeheads as before, namely, (1) Formal or Purely Logical Fallacies, (2)Semi-logical Fallacies or Fallacies of Ambiguity, (3) MaterialFallacies. 

§ 837. For the sake of distinctness we will place the two divisionsside 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 notnecessary to say much, as they have been amply treated of in thepreceding pages. A formal fallacy arises from the breach of any of thegeneral rules of syllogism. Consequently it would be a formal fallacyto present as a syllogism anything which had more or less than twopremisses. Under the latter variety comes what is called 'a woman'sreason, ' which asserts upon its own evidence something which requiresto be proved. Schoolboys also have been known to resort to this formof argument--'You're a fool. ' 'Why?' 'Because you are. ' When theconclusion thus merely reasserts one of the premisses, the other mustbe either absent or irrelevant. If, on the other hand, there are morethan two premisses, either there is more than one syllogism or thesuperfluous premiss is no premiss at all, but a proposition irrelevantto the conclusion. 

839. The remaining rules of the syllogism are more able to be brokenthan the first; so that the following scheme presents the varieties offormal 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 thegeneral rules of syllogism (§ 582). Here is a palpable instance ofit--

 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 thedestruction of the argument. The difference between the actual writingof books and the power to write them is precisely the differencebetween one who is an author and one who is not. 

§ 841. Since a syllogism consists of three terms, each of which isused twice over, it would be possible to have an apparent syllogismwith as many as six terms in it. The true name for the fallacytherefore is the Fallacy of More than Three Terms. But it is rare tofind an attempted syllogism which has more than four terms in it, justas we are seldom tendered a line as an hexameter, which has more thanseven feet. 

§ 842. The Fallacies of Undistributed Middle and Illicit Process havebeen treated of under §§ 585, 586. The heading 'Negative Premissesand Conclusion' covers violations of the three general rules ofsyllogism relating to negative premisses (§§ 590-593). Here is aninstance of the particular form of the fallacy which consists in theattempt to extract an affirmative conclusion out of two negativepremisses--

 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 thefact that it is illegitimately arrived at. The formal fallacies whichhave just been enumerated find no place in Aristotle's division. Thereason is plain. His object was to enumerate the various modes inwhich a sophist might snatch an apparent victory, whereas by openlyviolating any of the laws of syllogism a disputant would be simplycourting defeat. 

§ 843. We now revert to Aristotle's classification of fallacies, orrather of Modes of Refutation. We will take the species he enumeratesin their order, and notice how modern usage has departed from theoriginal meaning of the terms. Let it be borne in mind that, when thedeception was not in the language, Aristotle did not trouble himselfto determine whether it lay in the matter or in the form of thought. 

§ 844. The following scheme presents the Aristotelian classificationto 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: òmonumía] consists in anambiguous use of any of the three terms of a syllogism. If, forinstance, 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 theretaken in a different sense from that which it bore in its properpremiss. This therefore would be an instance of the fallacy ofEquivocal Minor. 

§ 846. For a glaring instance of the fallacy of Equivocal Major, wemay 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 takenambiguously. 

§ 847. It is, however, to the middle term that an ambiguity mostfrequently attaches. In this case the fallacy of equivocation assumesthe special name of the Fallacy of Ambiguous Middle. Take as aninstance 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 wouldprobably be felt to be unsatisfactory. Nor is the reason far toseek. It is evident that belief in a book cannot be faith in any sensein which that quality can rightly be pronounced to be a moral virtue. 

§ 848. The Fallacy of Amphiboly ([Greek: ámphibolía]) is an ambiguityattaching to the construction of a proposition rather than to theterms of which it is composed. One of Aristotle's examples is this--

 [Greek: tò boúlesthai labeîn me toùs polemíous]

which may be interpreted to mean either 'the fact of my wishing totake the enemy, ' or 'the fact of the enemies' wishing to take me. ' Theclassical languages are especially liable to this fallacy owing to theoblique construction in which the accusative becomes subject to theverb. Thus in Latin we have the oracle given to Pyrrhus (though ofcourse, 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 couldconquer the Romans, whereas the oracle subsequently explained to himthat the real meaning was that the Romans could conquer him. Similarto this, as Shakspeare makes the Duke of York point out, is thewitch'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 WindsorCastle--Hoc fecit Wykeham. The king mas incensed with the bishop fordaring to record that he made the tower, but the latter adroitlyreplied that what he really meant to indicate was that the tower wasthe making of him. To the same head may be referred the famoussentence--'I will wear no clothes to distinguish me from my Christianbrethren. '

§ 849. The Fallacy of Composition [Greek: diaíresis] is likewise acase of ambiguous construction. It consists, as expounded byAristotle, in taking words together which ought to be takenseparately, 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 convenientmisuse of the term 'fallacy of composition' among modern writers, bywhom it is defined to consist in arguing from the distributive to thecollective 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: ègó s' êteka doûlon ônt' èleúteron [Footnote: Evidently the original of the line in Terence's _Andria_, 37, --feci ex servo ut esses libertus mihi. ], where the separation of [Greek: doûlon] from [Greek: ôntra] would leadto an interpretation exactly contrary to what is intended. 

And again--

 [Greek: pentékont' àndrôn èkatòn lípe dîos Àchilleús], where the separation of [Greek: àndrôn] from [Greek: èkatòn] leads toa ludicrous error. 

Any reader whose youth may have been nourished on 'The FairchildFamily' may possibly recollect a sentence which ran somewhat on thiswise--'Henry, ' said Mr. Fairchild, 'is this true? Are you a thief anda liar too?' But I am afraid he will miss the keen delight which canbe extracted at a certain age from turning the tables uponMr. 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 writersto the meaning which they have assigned to the fallacy ofcomposition. So that by the 'fallacy of division' is now meant arguingfrom the collective to the distributive use of a term. Further, it islaid down that when the middle term is used distributively in themajor premiss and collectively in the minor, we have the fallacy ofcomposition; whereas, when the middle term is used collectively in themajor premiss and distributively in the minor, we have the fallacy ofdivision. Thus the first of the two examples appended would becomposition 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 tothe middle term, it seems desirable to add that when either the majoror minor term is used distributively in the premiss and collectivelyin the conclusion, we have the fallacy of composition, and in theconverse case the fallacy of division. Here is an instance of thelatter 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 beforecommented upon (§ 119), is a great assistance in the English languageto the pair of fallacies just spoken of. 

§ 835. The Fallacy of Accent ([Greek: prosodía]) is neither more norless than a mistake in Greek accentuation. As an instance Aristotlegives Iliad xxiii. 328, where the ancient copies of Homer madenonsense of the words [Greek: tò mèn oú katapútetai ómbro] by writing[Greek: oû] with the circumflex in place of [Greek: oú] with the acuteaccent. [Footnote: This goes to show that the ancient Greeks did notdistinguish in pronunciation between the rough and smooth breathingany more than their modern representatives. ] Aristotle remarks thatthe fallacy is one which cannot easily occur in verbal argument, butrather in writing and poetry. 

§ 856. Modern writers explain the fallacy of accent to be the mistakeof laying the stress upon the wrong part of a sentence. Thus when thecountry 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 itis not amiss to tell lies provided they be in favour of one'sneighbour. 

§ 857. The Fallacy of Figure of Speech [Greek: tò schêma tês léxeos]results from any confusion of grammatical forms, as between thedifferent genders of nouns or the different voices of verbs, or theiruse as transitive or intransitive, e. G. [Greek: úgiaínein] has thesame grammatical form as [Greek: témnein] or [Greek: oìkodomeîn], butthe former is intransitive, while the latter are transitive. A sophismof this kind is put into the mouth of Socrates by Aristophanes in theClouds (670-80). The philosopher is there represented as arguing that[Greek: kápdopos] must be masculine because [Greek: Kleónumos] is. Onthe surface this is connected with language, but it is essentially afallacy of false analogy. 

§ 858. To this head may be referred what is known as the Fallacy ofParonymous Terms. This is a species of equivocation which consists inslipping from the use of one part of speech to that of another, whichis derived from the same source, but has a different meaning. Thusthis fallacy would be committed if, starting from the fact that thereis a certain probability that a hand at whist will consist of thirteentrumps, one were to proceed to argue that it was probable, or that hehad proved it. 

§ 859. We turn now to the tricks of refutation which lie outside thelanguage, whether the deception be due to the assumption of a falsepremiss or to some unsoundness in the reasoning. 

§ 860. The first on the list is the Fallacy of Accident ([Greek: tòsumbebekós]). This fallacy consists in confounding an essential withan accidental difference, which is not allowable, since many thingsare the same in essence, while they differ in accidents. Here is thesort 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 andSocrates does not go so deep as to affect the underlying essence. Toput the thing more plainly, the fallacy lies in assuming that whateveris different from a given subject must be different from it in allrespects, so that it is impossible for them to have a commonpredicate. 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 ofthe major. 

§ 861. The next fallacy suffers from the want of a convenient name. Itis called by Aristotle [Greek: tò áplos tóde ê pê légestai kaì mèkupíos] or, more briefly, [Greek: tò áplôs ê mé], or [Greek: tò pê kaíáplôs], and by the Latin writers 'Fallacia a dicto secundum quid addictum simpliciter. ' It consists in taking what is said in aparticular respect as though it held true without any restriction, e. G. , that because the nonexistent ([Greek: tò mè ôn]) is a matter ofopinion, that therefore the non-existent is, or again that because theexistent ([Greek: tò ôn]) is not a man, that therefore the existent isnot. 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 beboth white and not-white. For he is only white in a certain respect([Greek: pê]), but not absolutely ([Greek: àplôs]). Moredifficulty, says Aristotle, may arise when opposite qualities exist ina thing in about an equal degree. When, for instance, a thing is halfwhite and half black, are we to say that it is white or black? Thisquestion the philosopher propounds, but does not answer. The force ofit lies in the implied attack on the Law of Contradiction. It wouldseem in such a case that a thing may be both white and not-white atthe same time. The fact is--so subtle are the ambiguities oflanguage--that even such a question as 'Is a thing white ornot-white?' straightforward, as it seems, is not really a fair one. Weare entitled sometimes to take the bull by the horns, and answer withthe adventurous interlocutor in one of Plato's dialogues--'Both andneither. ' It may be both in a certain respect, and yet neitherabsolutely. 

§ 862. The same sort of difficulties attach to the Law of ExcludedMiddle, and may be met in the same way. It might, for instance, beurged that it could not be said with truth of the statue seen byNebuchadnezzar in his dream either that it was made of gold or that itwas not made of gold: but the apparent plausibility of the objectionwould be due merely to the ambiguity of language. It is not true, onthe one hand, that it was made of gold (in the sense of being composedentirely of that metal); and it is not true, on the other, that it wasnot made of gold (in the sense of no gold at all entering into itscomposition). But let the ambiguous proposition be split up into itstwo meanings, and the stringency of the Law of Excluded Middle will atonce 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 thelast, the fallacy of accident being assimilated to it under the titleof the 'Fallacia a dicto simpliciter ad dictum secundum quid. ' In thissense the two fallacies may be defined thus. 

The Fallacy of Accident consists in assuming that what holds true as ageneral rule will hold true under some special circumstances which mayentirely alter the case. The Converse Fallacy of Accident consists inassuming that what holds true under some special circumstances musthold 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, isguilty of the fallacy of accident; the man who, having had itprescribed 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 ofarguing from one special case to another. 

§ 865. The next fallacy is Ignoratio Elenchi [Greek: èlégchouâgnoia]. This fallacy arises when by reasoning valid in itself oneestablishes a conclusion other than what is required to upset theadversary's assertion. It is due to an inadequate conception of thetrue nature of refutation. Aristotle therefore is at the pains todefine refutation at full length, thus--

'A refutation [Greek: êlegchos] is the denial of one and the same--notname, but thing, and by means, not of a synonymous term, but of thesame term, as a necessary consequence from the data, withoutassumption of the point originally at issue, in the same respect, andin the same relation, and in the same way, and at the same time. '

The ELENCHUS then is the exact contradictory of the opponent'sassertion under the terms of the law of contradiction. To establish bya syllogism, or series of syllogisms, any other proposition, howeverslightly different, is to commit this fallacy. Even if the substanceof the contradiction be established, it is not enough unless theidentical words of the opponent are employed in thecontradictory. Thus if his thesis asserts or denies something about[Greek: lópion], it is not enough for you to prove the contradictorywith regard to [Greek: ìmátion]. There will be need of a furtherquestion and answer to identify the two, though they are admittedlysynonymous. Such was the rigour with which the rules of the game ofdialectic were enforced among the Greeks!

§ 866. Under the head of Ignoratio Elenchi it has become usual tospeak of various forme of argument which have been labelled by theLatin writers under such names as 'argumentum ad hominem, ' 'adpopulum, ' 'ad verecundiam, ' 'ad ignorantiam, ' 'ad baculum'--all ofthem opposed to the 'argumentum ad rem' or 'ad judicium. '

§ 867. By the 'argumentum ad hominem' was perhaps meant a piece ofreasoning which availed to silence a particular person, withouttouching the truth of the question. Thus a quotation from Scriptureis sufficient to stop the mouth of a believer in the inspiration ofthe Bible. Hume's Essay on Miracles is a noteworthy instance of the'argumentum ad hominem' in this sense of the term. He insists stronglyon the evidence for certain miracles which he knew that the prejudicesof his hearers would prevent their ever accepting, and then askstriumphantly if these miracles, which are declared to have taken placein an enlightened age in the full glare of publicity, are palpablyimposture, what credence can be attached to accounts of extraordinaryoccurrences of remote antiquity, and connected with an obscure cornerof the globe? The 'argumentum ad judicium' would take miracles as awhole, and endeavour to sift the amount of truth which may lie in theaccounts we have of them in every age. [Footnote: On this subject seethe author's _Attempts at Truth_ (Trubner & Co. ), pp. 46-59. ]

§ 868. In ordinary discourse at the present day the term 'argumentumad hominem' is used for the form of irrelevancy which consists inattacking the character of the opponent instead of combating hisarguments, as illustrated in the well-known instructions to abarrister--'No case: abuse the plaintiff's attorney. '

§ 869. The 'argumentum ad populum' consists in an appeal to thepassions of one's audience. An appeal to passion, or to give it a lessquestion-begging name, to feeling, is not necessarily amiss. The heartof man is the instrument upon which the rhetorician plays, and he hasto answer for the harmony or the discord that comes of hisperformance. 

§ 870. The 'argumentum ad verecundiam' is an appeal to the feeling ofreverence or shame. It is an argument much used by the old to theyoung and by Conservatives to Radicals. 

§ 871. The 'argumentum ad ignorantiam' consists simply in trading onthe ignorance of the person addressed, so that it covers any kind offallacy that is likely to prove effective with the hearer. 

§ 872. The 'argumentum ad baculum' is unquestionably a form ofirrelevancy. To knock a man down when he differs from you in opinionmay prove your strength, but hardly your logic. 

A sub-variety of this form of irrelevancy was exhibited lately at asocialist lecture in Oxford, at which an undergraduate, unable orunwilling to meet the arguments of the speaker, uncorked a bottle, which had the effect of instantaneously dispersing the audience. Thismight be set down as the 'argumentum ad nasum. '

§ 873. We now come to the Fallacy of the Consequent, a term which hasbeen more hopelessly abused than any. What Aristotle meant by it wassimply 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 dealingwith presumptive evidence. Thus a speaker, wanting to prove that a manis an adulterer, will argue that he is a showy dresser, and has beenseen about at nights. Both these things however may be the case, andyet the charge not be true. 

§ 874. The Fallacy of Petitio or Assumptio Principii [Greek: tò ènàrchê aìteîstai or lambánein] to which we now come, consists in anunfair assumption of the point at issue. The word [Greek: aìteîstai], in Aristotle's name for it points to the Greek method of dialectic bymeans of question and answer. This fact is rather disguised by themysterious phrase 'begging the question. ' The fallacy would becommitted 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 ofsome importance we will take the words of Aristotle himself(Top. Viii. 13. §§ 2, 3). 'People seem to beg the question in fiveways. First and most glaringly, when one takes for granted the verything that has to be proved. This by itself does not readily escapedetection, but in the case of "synonyms, " that is, where the name andthe definition have the same meaning, it does so moreeasily. [Footnote: Some light is thrown upon this obscure passage by acomparison with Cat. I. § 3, where 'synonym' is defined. To take theword here in its later and modern sense affords an easyinterpretation, which is countenanced by Alexander Aphrodisiensis, butit is flat against the usage of Aristotle, who elsewhere gives thename 'synonym, ' not to two names for the same thing, but to two thingsgoing under the same name. See Trendelenberg on the passage. ]

Secondly, when one assumes universally that which has to be proved inparticular, as, if a man undertaking to prove that there is onescience of contraries, were to assume that there is one science ofopposites generally. For he seems to be taking for granted along withseveral other things what he ought to have proved by itself. 

Thirdly, when one assumes the particulars where the universal has tobe proved; for in so doing a man is taking for granted separately whathe was bound to prove along with several other things. Again, whenone 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 withhealth and disease, one were to take each by itself for granted. 

Lastly, if one were to take for granted one of a pair of necessaryconsequences, as that the side is incommensurable with the diagonal, when it is required to prove that the diagonal is incommensurable withthe 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 thequestion straight off, lands us in the formal fallacy already spokenof (§ 838), which violates the first of the general rules ofsyllogism, inasmuch as a conclusion is derived from a single premiss, to wit, itself. 

§ 878. The second, strange to say, gives us a sound syllogism inBarbara, a fact which countenances the blasphemers of the syllogism inthe charge they bring against it of containing in itself a petitioprincipii. Certainly Aristotle's expression might have been moreguarded. But it is clear that his quarrel is with the matter, not withthe form in such an argument. The fallacy consists in assuming aproposition which the opponent would be entitled to deny. ElsewhereAristotle tells us that the fallacy arises when a truth not evident byits own light is taken to be so. [Footnote: [Greek: Ôtan tò mè díaùtoû gnostòn dí aùtoû tis èpicheiraê deiknúnai, tót' aìteîtai tò èxàrchês. ]. Anal. Pr. II. 16. § I ad fin. ]

§ 879. The third gives us an inductio per enumerationem simplicem, amode of argument which would of course be unfair as against anopponent 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 ofArguing in a Circle, which is incidental to a train of reasoning. Inits 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: tò mè aîtion] or[Greek: aîtoin]) is another, the name of which has led to a completemisinterpretation. It consists in importing a contradiction into thediscussion, and then fathering it on the position controverted. Sucharguments, says Aristotle, often impose upon the users of themthemselves. The instance he gives is too recondite to be of generalinterest. 

§ 884. Lastly, the Fallacy of Many Questions ([Greek: tò tà déoèrotémata ên poieîn]) is a deceptive form of interrogation, when asingle answer is demanded to what is not really a single question. Indialectical discussions the respondent was limited to a simple 'yes'or 'no'; and in this fallacy the question is so framed as that eitheranswer would seem to imply the acceptance of a proposition which wouldbe repudiated. The old stock instance will do as well asanother--'Come now, sir, answer "yes" or "no. " Have you left offbeating your mother yet?' Either answer leads to an apparentadmission of impiety. 

A late Senior Proctor once enraged a man at a fair with this form offallacy. The man was exhibiting a blue horse; and the distinguishedstranger asked him--'With what did you paint your horse?'

EXERCISES. 

These exercises should be supplemented by direct questions upon thetext, which it is easy for the student or the teacher to supply forhimself. 

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, andstate whether they are (1) abstract or concrete, (2) singular orcommon, (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 thedivisions 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 andpredicate:--

 (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 quantifythem 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 thefollowing 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 thepreceding list. 

2. Prove that in a negative proposition the predicate must bedistributed. 

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 tospecies. 

2. To which of the heads of predicables would you refer the followingstatements? 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 otherpropositions 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 otherpropositions 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 thefollowing 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 heavenor earth: but the converse proposition is by no means true. ' Is theterm 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 thepreceding 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 propositionsfollows 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 theybelong, 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 showwhat 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, thefollowing 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, statingwhether they are valid or invalid, and giving your reasons in eachcase--

 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 notunselfish, ' and reduce your syllogism directly and indirectly. 

6. Prove the following propositions in the second figure, and reducethe 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 notCelts. '

8. Construct in words the same syllogism in all the four figures. 

9. Invent instances to show that false premisses may give trueconclusions. 

III. 

1. What moods are peculiar to the first, second, and third figuresrespectively?

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 thefigures?

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 minorpremiss anywhere but in the second. 

9. Show that in the first figure the conclusion must have the qualityof the major premiss and the quantity of the minor. 

10. Why do the premisses EA yield a universal conclusion in the firsttwo 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 ispossible for the major term to be distributed in the premiss andundistributed in the conclusion. 

12. Why are the premisses of Fesapo and Fresison not transposed inreduction 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 negativeconclusion. 

3. Prove that there must be at least one more term distributed in thepremisses than in the conclusion. 

4. Prove that the number of distributed terms in the premisses cannotexceed those in the conclusion by more than two. 

5. Prove that the number of undistributed terms in the premissescannot exceed those in the conclusion by more than one. 

6. Prove that wherever the minor premiss is negative, the major mustbe universal. 

7. Prove that wherever the minor term is distributed, the majorpremiss must be universal. 

8. If the middle term be twice distributed, what mood and figure arepossible?

9. If the major term of a syllogism be the predicate of the majorpremiss, what do we know about the minor premiss?

10. When the middle term is distributed in both premisses, what mustbe the quantity of the conclusion?

11. Prove that if the conclusion be universal, the middle term canonly be distributed once in the premisses. 

12. Show how it is sometimes possible to draw three differentconclusions 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 followingpairs 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 whatevercircumstance can be eliminated is not connected with the phenomenon byany law; the Method of Difference stands on the ground that whatevercircumstance cannot be eliminated is connected with the phenomenon bya law. ' Do these two principles imply one another?

CHAPTERS XX-XXVIII. 

1. Fill up the following enthymemes, and state the exact nature of theresulting 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 thelatter 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 theircomponent parts, and point out any violations of the rules ofsyllogism 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 followingpropositions--

 (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, andreferring 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: òrismós]), 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.