{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_6560/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_6560","id":"gutenberg_6560","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/6560","summary":null,"tags":["Category: Philosophy & Ethics","Logic","Mathematics","Philosophy"],"text":"Produced by Arno Peters, David Moynihan, Charles Franks\nand the Online Distributed Proofreading Team.\n\n\n\n\n\n[Transcriber's Note: In this plain-text rendering,\n .'. means therefore\n [alpha], [beta], ..., [Alpha], [Beta], ... for Greek symbols]\n\n\n\n\n\nDEDUCTIVE LOGIC\n\nBY\n\nST. GEORGE STOCK, M.A.\n\nPEMBROKE COLLEGE, OXFORD\n\n\n\n\nPREFACE.\n\nOne critic, who was kind enough to look at this book in manuscript,\nrecommended me to abandon the design of Publishing it, on the ground\nthat my logic was too like all other logics; another suggested to me\nto cut out a considerable amount of new matter. The latter advice I\nhave followed; the former has encouraged me to hope that I shall not\nbe considered guilty of wanton innovation.  The few novelties which I\nhave ventured to retain will, I trust, be regarded as legitimate\nextensions of received lines of teaching.\n\nMy object has been to produce a work which should be as thoroughly\nrepresentative of the present state of the logic of the Oxford Schools\nas any of the text-books of the past. The qualities which I have aimed\nat before all others have been clearness and consistency. For the task\nwhich I have taken upon myself I may claim one qualification--that of\nexperience; since more than seventeen years have now elapsed since I\ntook my first pupil in logic for the Honour School of Moderations, and\nduring that time I have been pretty continuously engaged in studying\nand teaching the subject.\n\nIn acknowledging my obligations to previous writers I must begin with\nArchbishop Whately, whose writings first gave me an interest in the\nsubject. The works of Mill and Hamilton have of course been freely\ndrawn upon. I have not followed either of those two great writers\nexclusively, but have endeavoured to assimilate what seemed best in\nboth. To Professor Fowler I am under a special debt. I had not the\nprivilege of personal teaching from him in logic,--as I had in some\nother subjects; but his book fell into my hands at an early period in\nmy mental training, and was so thoroughly studied as to have become a\npermanent part of the furniture of my mind. Much the same may be said\nof my relation to the late Professor Jevons's Elementary Lessons in\nLogic.  Two other books, which I feel bound to mention with special\nemphasis, are Hansel's edition of Aldrich and McCosh's Laws of\nDiscursive Thought. If there be added to the foregoing Watts's Logic,\nThomson's Outlines of the Laws of Thought, Bain's Deductive Logic,\nJevons's Studies in Deductive Logic and Principles of Science,\nBradley's Principles of Logic, Abbott's Elements of Logic, Walker's\nedition of Murray, Ray's Text-book of Deductive Logic, and\nWeatherley's Rudiments of Logic, I think the list will be exhausted of\nmodern works from which I am conscious of having borrowed. But, not to\nforget the sun, while thanking the manufacturers of lamps and candles,\nI should add that I have studied the works of Aristotle according to\nthe measure of my time and ability.\n\nThis work has had the great advantage of having been revised, while\nstill in manuscript, by Mr. Alfred Robinson, Fellow of New College, to\nwhom I cannot sufficiently express my obligation. I have availed\nmyself to the full of the series of criticisms which he was kind\nenough to send me. As some additions have been made since then, he\ncannot be held in anyway responsible for the faults which less kindly\ncritics may detect.\n\nFor the examples at the end I am mainly indebted to others, and to a\nlarge extent to my ingenious friend, the Rev. W. J. Priest of Merton\nCollege.\n\nMy thanks are due also to my friend and former pupil, Mr. Gilbert\nGrindle, Scholar of Corpus, who has been at the pains to compose an\nindex, and to revise the proofs as they passed through the press.\n\nAnd last, but not least, I must set on record my gratitude to\nCommander R. A. Stock, R.N., one of Her Majesty's Knights of Windsor,\nwithout whose brotherly aid this work might never have been written,\nand would certainly not have assumed exactly its present shape.\n\nOXFORD,\n\n_October_ 22, 1888.\n\n\n\n\nCONTENTS.\n\nPREFACE.\n\nINTRODUCTION, \u00a7\u00a7 1-56.\n\nPART I. Of Terms, \u00a7\u00a7 57-171.\n\n  CHAP. I. Of the Term as distinguished from other words, \u00a7\u00a7 57-76.\n\n  II. Of the Division of Things, \u00a7\u00a7 77-85.\n\n  III. Of the Divisions of Terms, \u00a7\u00a7 86-165.\n\n  IV. Of the Law of Inverse Variation of Extension and Intension, \u00a7\u00a7\n  166-171.\n\nPART II. Of Propositions, \u00a7\u00a7 172-185.\n\n  CHAP. I. Of the Proposition as distinguished from other Sentences,\n  \u00a7\u00a7 172-185.\n\n  II. Of the Copula, \u00a7\u00a7 186-201.\n\n  III. Of the Divisions of Propositions, \u00a7\u00a7 202-273.\n\n  IV. Of the Distribution of Terms, \u00a7\u00a7 274-294.\n\n  V. Of the Quantification of the Predicate, \u00a7\u00a7 295-312.\n\n  VI. Of the Heads of Predicables, \u00a7\u00a7 313-346.\n\n  VII. Of Definition, \u00a7\u00a7 347-384.\n\n  VIII. Of Division, \u00a7\u00a7 385-425.\n\nPART III. Of Inferences, \u00a7\u00a7 426-884.\n\n  CHAP. I. Of Inferences in general, \u00a7\u00a7 426-441.\n\n  II. Of Deductive Inferences, \u00a7\u00a7 442-448.\n\n  III. Of Opposition, \u00a7\u00a7 449-478.\n\n  IV. Of Conversion, \u00a7\u00a7 479-495.\n\n  V. Of Permutation, \u00a7\u00a7 496-502.\n\n  VI. Of Compound Forms of Immediate Inference, \u00a7\u00a7 503-532.\n\n  VII. Of Other Forms of Immediate Inference, \u00a7\u00a7 533-539.\n\n  VIII. Of Mediate Inferences or Syllogisms, \u00a7\u00a7 540-557.\n\n  IX. Of Mood and Figure, \u00a7\u00a7 558-568.\n\n  X. Of the Canon of Reasoning, \u00a7\u00a7 569-581.\n\n  XI. Of the General Rules of Syllogism, \u00a7\u00a7 582-598.\n\n  XII. Of the Determination of the Legitimate Moods of Syllogism, \u00a7\u00a7\n  599-605.\n\n  XIII. Of the Special Rules of the Four Figures, \u00a7\u00a7 606-620.\n\n  XIV. Of the Determination of the Moods that are valid in the Four\n  Figures, \u00a7\u00a7 621-632.\n\n  XV. Of the Special Canons of the Four Figures, \u00a7\u00a7 633-647.\n\n  XVI. Of the Special Uses of the Four Figures, \u00a7\u00a7 648-655.\n\n  XVII. Of the Syllogism with Three Figures, \u00a7\u00a7 656-666.\n\n  XVIII. Of Reduction, \u00a7\u00a7 667-700.\n\n  XIX. Of Immediate Inference as applied to Complex Propositions, \u00a7\u00a7\n  701-730.\n\n  XX. Of Complex Syllogisms, \u00a7\u00a7 731-743.\n\n  XXI. Of the Reduction of the Partly Conjunctive Syllogism, \u00a7\u00a7\n  744-752.\n\n  XXII. Of the Partly Conjunctive Syllogism regarded as all Immediate\n  Inference, \u00a7\u00a7 753-759.\n\n  XXIII. Of the Disjunctive Syllogism, \u00a7\u00a7 760-765.\n\n  XXIV. Of the Reduction of the Disjunctive Syllogism, \u00a7\u00a7 766-769.\n\n  XXV. Of the Disjunctive Syllogism regarded as an Immediate\n  Inference, \u00a7\u00a7 770-777.\n\n  XXVI. Of the Mixed Form of Complex Syllogism, \u00a7\u00a7 778-795.\n\n  XXVII. Of the Reduction of the Dilemma, \u00a7\u00a7 796-797.\n\n  XXVIII. Of the Dilemma regarded as an Immediate Inference, \u00a7\u00a7\n  798,799.\n\n  XXIX. Of Trains of Reasoning, \u00a7\u00a7 800-826.\n\n  XXX. Of Fallacies, \u00a7\u00a7 827-884.\n\nEXERCISES.\n\nINDEX.\n\n\n\n\nINTRODUCTION.\n\n\u00a7 1. LOGIC is divided into two branches, namely--\n\n  (1) Inductive,\n\n  (2) Deductive.\n\n\u00a7 2. The problem of inductive logic is to determine the actual truth\nor falsity of propositions: the problem of deductive logic is to\ndetermine their relative truth or falsity, that is to say, given such\nand such propositions as true, what others will follow from them.\n\n\u00a7 3. Hence in the natural order of treatment inductive logic precedes\ndeductive, since it is induction which supplies us with the general\ntruths, from which we reason down in our deductive inferences.\n\n\u00a7 4. It is not, however, with logic as a whole that we are here\nconcerned, but only with deductive logic, which may be defined as The\nScience of the Formal Laws of Thought.\n\n\u00a7 5. In order fully to understand this definition we must know exactly\nwhat is meant by 'thought,' by a 'law of thought,' by the term\n'formal,' and by 'science.'\n\n\u00a7 6. Thought, as here used, is confined to the faculty of\ncomparison. All thought involves comparison, that is to say, a\nrecognition of likeness or unlikeness.\n\n\u00a7 7. The laws of thought are the conditions of correct thinking. The\nterm 'law,' however, is so ambiguous that it will be well to determine\nmore precisely in what sense it is here used.\n\n\u00a7 8. We talk of the 'laws of the land' and of the 'laws of nature,'\nand it is evident that we mean very different things by these\nexpressions. By a law in the political sense is meant a command\nimposed by a superior upon an inferior and sanctioned by a penalty for\ndisobedience.  But by the 'laws of nature' are meant merely certain\nuniformities among natural phenomena; for instance, the 'law of\ngravitation' means that every particle of matter does invariably\nattract every other particle of matter in the universe.\n\n\u00a7 9. The word 'law' is transferred by a metaphor from one of these\nsenses to the other. The effect of such a command as that described\nabove is to produce a certain amount of uniformity in the conduct of\nmen, and so, where we observe uniformity in nature, we assume that it\nis the result of such a command, whereas the only thing really known\nto us is the fact of uniformity itself.\n\n\u00a7 10. Now in which of these two senses are we using the term 'laws of\nthought'? The laws of the land, it is plain, are often violated,\nwhereas the laws of nature never can be so [Footnote: There is a sense\nin which people frequently speak of the laws of nature being violated,\nas when one says that intemperance or celibacy is a violation of the\nlaws of nature, but here by 'nature' is meant an ideal perfection in\nthe conditions of existence.]. Can the laws of thought be violated in\nlike manner with the laws of the land? Or are they inviolable like the\nlaws of nature?\n\n\u00a7 11. In appearance they can be, and manifestly often are violated-for\nhow else could error be possible? But in reality they can not. No man\never accepts a contradiction when it presents itself to the mind as\nsuch: but when reasoning is at all complicated what does really\ninvolve a contradiction is not seen to do so; and this sort of error\nis further assisted by the infinite perplexities of language.\n\n\u00a7 12. The laws of thought then in their ultimate expression are\ncertain uniformities which invariably hold among mental phenomena, and\nso far they resemble the laws of nature: but in their complex\napplications they may be violated owing to error, as the laws of the\nland may be violated by crime.\n\n\u00a7 13. We have now to determine the meaning of the expression 'formal\nlaws of thought.'\n\n\u00a7 14. The distinction between form and matter is one which pervades\nall nature. We are familiar with it in the case of concrete things. A\ncup, for instance, with precisely the same form, may be composed of\nvery different matter-gold, silver, pewter, horn or what not?\n\n\u00a7 15. Similarly in every act of thought we may distinguish two\nthings--\n\n  (1) the object thought about,\n\n  (2) the way in which the mind thinks of it.\n\nThe first is called the Matter; the second the Form of Thought.\n\n\u00a7 16. Now Formal, which is another name for Deductive Logic, is\nconcerned only with the way in which the mind thinks, and has nothing\nto do with the particular objects thought about.\n\n\u00a7 17. Since the form may be the same, whilst the matter is different,\nwe may say that formal logic is concerned with the essential and\nnecessary elements of thought as opposed to such as are accidental and\ncontingent. By 'contingent' is meant what holds true in some cases,\nbut not in others. For instance, in the particular case of equilateral\ntriangles it is true to say, not only that 'all equilateral triangles\nare equiangular,' but also that 'all equiangular triangles are\nequilateral.' But the evidence for these two propositions is\nindependent. The one is not a formal consequence of the other. If it\nwere, we should be able to apply the same inference to all matter, and\nassert generally that if all A is B, all B is A, which it is notorious\nthat we cannot do.\n\n\u00a7 18. It remains now for the full elucidation of our definition to\ndetermine what is meant by 'science.'\n\n\u00a7 19. The question has often been discussed whether logic is a science\nor an art. The answer to it must depend upon the meaning we assign to\nthese terms.\n\n\u00a7 20. Broadly speaking, there is the same difference between Science\nand Art as there is between knowing and doing.\n\n  Science is systematized knowledge;\n  Art is systematized action.\n  Science is acquired by study;\n  Art is acquired by practice.\n\n\u00a7 21. Now logic is manifestly a branch of knowledge, and does not\nnecessarily confer any practical skill. It is only the right use of\nits rules in thinking which can make men think better. It is\ntherefore, in the broad sense of the terms, wholly a science and not\nat all an art.\n\n\u00a7 22. But this word 'art,' like most others, is ambiguous, and is\noften used, not for skill displayed in practice, but for the knowledge\nnecessary thereto. This meaning is better conveyed by the term\n'practical science.'\n\n\u00a7 23. Science is either speculative or practical. In the first case we\nstudy merely that we may know; in the latter that we may do.\n\n  Anatomy is a speculative science;\n  Surgery is a practical science.\n\nIn the first case we study the human frame in order that we may\nunderstand its structure; in the second that we may assist its\nneeds. Whether logic is a speculative or a practical science depends\nentirely upon the way in which it is treated. If we study the laws of\nthought merely that we may know what they are, we are making it a\nspeculative science; if we study the same laws with a view to deducing\nrules for the guidance of thought, we are making it a practical\nscience.\n\n\u00a7 24. Hence logic may be declared to be both the science and the art\nof thinking. It is the art of thinking in the same sense in which\ngrammar is the art of speaking.  Grammar is not in itself the right\nuse of words, but a knowledge of it enables men to use words\ncorrectly. In the same way a knowledge of logic enables men to think\ncorrectly, or at least to avoid incorrect thoughts. As an art logic\nmay be called the navigation of the sea of thought.\n\n\u00a7 25. The laws of thought are all reducible to the three following\naxioms, which are known as The Three Fundamental Laws of Thought.\n\n  (1) The Law of Identity--\n\n    Whatever is, is;\n\n  or, in a more precise form,\n\n    Every A is A.\n\n  (2) The Law of Contradiction--\n\n    Nothing can both be and not be;\n    Nothing can be A and not A.\n\n  (3) The Law of Excluded Middle--\n\n    Everything must either be or not be;\n    Everything is either A or not A.\n\n\u00a7 26. Each of these principles is independent and self-evident.\n\n\u00a7 27. If it were possible for the law of identity to be violated, no\nviolation of the law of contradiction would necessarily ensue: for a\nthing might then be something else, without being itself at the same\ntime, which latter is what the law of contradiction militates\nagainst. Neither would the law of excluded middle be infringed. For,\non the supposition, a thing would be something else, whereas all that\nthe law of excluded middle demands is that it should either be itself\nor not. A would in this case adopt the alternative of being not A.\n\n\u00a7 28. Again, the violation of the law of contradiction does not\ninvolve any violation of the law of identity: for a thing might in\nthat case be still itself, so that the law of identity would be\nobserved, even though, owing to the law of contradiction not holding,\nit were not itself at the same time. Neither would the law of excluded\nmiddle be infringed. For a thing would, on the supposition, be both\nitself and not itself, which is the very reverse of being neither.\n\n\u00a7 29. Lastly, the law of excluded middle might be violated without a\nviolation of the law of contradiction: for we should then have a thing\nwhich was neither A nor not A, but not a thing which was both at the\nsame time.  Neither would the law of identity be infringed. For we\nshould in this case have a thing which neither was nor was not, so\nthat the conditions of the law of identity could not exist to be\nbroken. That law postulates that whatever is, is: here we have a thing\nwhich never was to begin with.\n\n\u00a7 30. These principles are of so simple a character that the\ndiscussion of them is apt to be regarded as puerile.  Especially is\nthis the case with regard to the law of identity. This principle in\nfact is one of those things which are more honoured in the breach than\nin the observance. Suppose for a moment that this law did not\nhold--then what would become of all our reasoning?  Where would be the\nuse of establishing conclusions about things, if they were liable to\nevade us by a Protean change of identity?\n\n\u00a7 31. The remaining two laws supplement each other in the following\nway. The law of contradiction enables us to affirm of two exhaustive\nand mutually exclusive alternatives, that it is impossible for both to\nbe true; the law of excluded middle entitles us to add, that it is\nequally impossible for both to be false. Or, to put the same thing in\na different form, the law of contradiction lays down that one of two\nsuch alternatives must be false; the law of excluded middle adds that\none must be true.\n\n\u00a732. There are three processes of thought\n\n  (1) Conception.\n\n  (2) Judgement.\n\n  (3) Inference or Reasoning.\n\n\u00a7 33. Conception, which is otherwise known as Simple Apprehension, is\nthe act of forming in the mind the idea of anything, e.g. when we form\nin the mind the idea of a cup, we are performing the process of\nconception.\n\n\u00a7 34. Judgement, in the sense in which it is here used [Footnote:\nSometimes the term 'judgement' is extended to the comparison of\nnameless sense-impressions, which underlies the formation of\nconcepts. But this amounts to identifying judgement with thought in\ngeneral.]  may be resolved into putting two ideas together in the\nmind, and pronouncing as to their agreement or disagreement, e.g. we\nhave in our minds the idea of a cup and the idea of a thing made of\nporcelain, and we combine them in the judgement--'This cup is made of\nporcelain.'\n\n\u00a7 35. Inference, or Reasoning, is the passage of the mind from one or\nmore judgements to another, e.g. from the two judgements 'Whatever is\nmade of porcelain is brittle,' and 'This cup is made of porcelain,' we\nelicit a third judgement, 'This cup is brittle.'\n\n\u00a7 36. Corresponding to these three processes there are three products\nof thought, viz.\n\n  (1) The Concept.\n\n  (2) The Judgement.\n\n  (3) The Inference.\n\n\u00a7 37. Since our language has a tendency to confuse the distinction\nbetween processes and products, [Footnote: E.g. We have to speak quite\nindiscriminately of Sensation, Imagination, Reflexion, Sight, Thought,\nDivision, Definition, and so on, whether we mean in any case a process\nor a product.] it is the more necessary to keep them distinct in\nthought. Strictly we ought to speak of conceiving, judging and\ninferring on the one hand, and, on the other, of the concept, the\njudgement and the inference.\n\nThe direct object of logic is the study of the products rather than of\nthe processes of thought. But, at the same time, in studying the\nproducts we are studying the processes in the only way in which it is\npossible to do so.  For the human mind cannot be both actor and\nspectator at once; we must wait until a thought is formed in our minds\nbefore we can examine it. Thought must be already dead in order to be\ndissected: there is no vivisection of consciousness. Thus we can never\nknow more of the processes of thought than what is revealed to us in\ntheir products.\n\n\u00a7 38. When the three products of thought are expressed in language,\nthey are called respectively\n\n  (1) The Term.\n\n  (2) The Proposition.\n\n  (3) The Inference.\n\n\u00a7 39. Such is the ambiguity of language that we have already used the\nterm 'inference' in three different senses--first, for the act or\nprocess of inferring; secondly, for the result of that act as it\nexists in the mind; and, thirdly, for the same thing as expressed in\nlanguage.  Later on we shall have to notice a further ambiguity in its\nuse.\n\n\u00a7 40. It has been declared that thought in general is the faculty of\ncomparison, and we have now seen that there are three products of\nthought. It follows that each of these products of thought must be the\nresult of a comparison of some kind or other.\n\n  The concept is the result of comparing attributes.\n  The judgement is the result of comparing concepts.\n  The inference is the result of comparing judgements.\n\n\u00a7 41. In what follows we shall, for convenience, adopt the phraseology\nwhich regards the products of thought as clothed in language in\npreference to that which regards the same products as they exist in\nthe mind of the individual. For although the object of logic is to\nexamine thought pure and simple, it is obviously impossible to discuss\nit except as clothed in language. Accordingly the three statements\nabove made may be expressed as follows--\n\n  The term is the result of comparing attributes.\n  The proposition is the result of comparing terms.\n  The inference is the result of comparing propositions.\n\n\u00a7 42. There is an advantage attending the change of language in the\nfact that the word 'concept' is not an adequate expression for the\nfirst of the three products of thought, whereas the word 'term' is. By\na concept is meant a general notion, or the idea of a class, which\ncorresponds only to a common term. Now not only are common terms the\nresults of comparison, but singular terms, or the names of\nindividuals, are so too.\n\n\u00a7 43. The earliest result of thought is the recognition of an\nindividual object as such, that is to say as distinguished and marked\noff from the mass of its surroundings. No doubt the first impression\nproduced Upon the nascent intelligence of an infant is that of a\nconfused whole. It requires much exercise of thought to distinguish\nthis whole into its parts. The completeness of the recognition of an\nindividual object is announced by attaching a name to it. Hence even\nan individual name, or singular term, implies thought or\ncomparison. Before the _child_ can attach a meaning to the word\n'_mother_,' which to it is a singular term, it must have\ndistinguished between the set of impressions produced in it by one\nobject from those which are produced in it by others.  Thus, when\nVergil says\n\n  Incipe, parve puer, risu cognoscere matrem,\n\nhe is exhorting the beatific infant to the exercise of the faculty of\ncomparison.\n\n\u00a7 44. That a common term implies comparison does not need to be\ninsisted upon. It is because things resemble each other in certain of\ntheir attributes that we call them by a common name, and this\nresemblance could not be ascertained except by comparison, at some\ntime and by some one. Thus a common term, or concept, is the\ncompressed result of an indefinite number of comparisons, which lie\nwrapped up in it like so many fossils, witnessing to prior ages of\nthought.\n\n\u00a7 45. In the next product of thought, namely, the proposition, we have\nthe result of a single act of comparison between two terms; and this\nis why the proposition is called the unit of thought, as being the\nsimplest and most direct result of comparison.\n\n\u00a7 46. In the third product of thought, namely, the inference, we have\na comparison of propositions either directly or by means of a\nthird. This will be explained later on. For the present we return to\nthe first product of thought.\n\n\u00a7 47. The nature of singular terms has not given rise to much dispute;\nbut the nature of common terms has been the great battle-ground of\nlogicians. What corresponds to a singular term is easy to determine,\nfor the thing of which it is a name is there to point to: but the\nmeaning of a common term, like 'man' or 'horse,' is not so obvious as\npeople are apt to think on first hearing of the question.\n\n\u00a7 48. A common term or class-name was known to medi\u00e6val logicians\nunder the title of a Universal; and it was on the question 'What is a\nUniversal 7' that they split into the three schools of Realists,\nNominalists, and Conceptualists. Here are the answers of the three\nschools to this question in their most exaggerated form--\n\n\u00a7 49. Universals, said the Realists, are substances having an\nindependent existence in nature.\n\n\u00a7 50. Universals, said the Nominalists, are a mere matter of words,\nthe members of what we call a class having nothing in common but the\nname.\n\n\u00a7 51. Universals, said the Conceptualists, exist in the mind alone,\nThey are the conceptions under which the mind regards external\nobjects.\n\n\u00a7 52. The origin of pure Realism is due to Plato and his doctrine of\n'ideas'; for Idealism, in this sense, is not opposed to Realism, but\nidentical with it. Plato seems to have imagined that, as there was a\nreally existing thing corresponding to a singular term, such as\nSocrates, so there must be a really existing thing corresponding to\nthe common term 'man.' But when once the existence of these general\nobjects is admitted, they swamp all other existences. For individual\nmen are fleeting and transitory--subject to growth, decay and\ndeath--whereas the idea of man is imperishable and eternal. It is only\nby partaking in the nature of these ideas that individual objects\nexist at all.\n\n\u00a7 53. Pure Nominalism was the swing of the pendulum of thought to the\nvery opposite extreme; while Conceptualism was an attempt to hit the\nhappy mean between the two.\n\n\u00a7 54. Roughly it may be said that the Realists sought for the answer\nto the question 'What is a Universal?' in the matter of thought, the\nConceptualists in the form, and the Nominalists in the expression.\n\n\u00a7 55. A full answer to the question 'What is a Universal?' will bring\nin something of the three views above given, while avoiding the\nexaggeration of each. A Universal is a number of things that are\ncalled by the same name; but they would not be called by the same name\nunless they fell under the same conception in the mind; nor would they\nfall under the same conception in the mind unless there actually\nexisted similar attributes in the several members of a class, causing\nus to regard them under the same conception and to give them the same\nname. Universals therefore do exist in nature, and not merely in the\nmind of man: but their existence is dependent upon individual objects,\ninstead of individual objects depending for their existence upon\nthem. Aristotle saw this very clearly, and marked the distinction\nbetween the objects corresponding to the singular and to the common\nterm by calling the former Primary and the latter Secondary\nExistences. Rosinante and Excalibur are primary, but 'horse' and\n'sword' secondary existences.\n\n\u00a7 56. We have seen that the three products of thought are each one\nstage in advance of the other, the inference being built upon the\nproposition, as the proposition is built upon the term. Logic\ntherefore naturally divides itself into three parts.\n\n  The First Part of Logic deals with the Term;\n  The Second Part deals with the Proposition;\n  The Third Part deals with the Inference.\n\n\n\n\n\nPART I.--OF TERMS.\n\n\n\n\nCHAPTER 1.\n\n_Of the Term as distinguished from other words._\n\n\n\u00a7 57. The word 'term' means a boundary.\n\n\u00a7 58. The subject and predicate are the two terms, or boundaries, of a\nproposition. In a proposition we start from a subject and end in a\npredicate (\u00a7\u00a7 182-4), there being nothing intermediate between the two\nexcept the act of pronouncing as to their agreement or disagreement,\nwhich is registered externally under the sign of the copula.  Thus the\nsubject is the 'terminus a quo,' and the predicate is the 'terminus ad\nquem.'\n\n\u00a7 59. Hence it appears that the term by its very name indicates that\nit is arrived at by an analysis of the proposition. It is the\njudgement or proposition that is the true unit of thought and\nspeech. The proposition as a whole is prior in conception to the terms\nwhich are its parts: but the parts must come before the whole in the\nsynthetic order of treatment.\n\n\u00a7 60. A term is the same thing as a name or noun.\n\n\u00a7 61. A name is a word, or collection of words, which serves as a mark\nto recall or transmit the idea of a thing, either in itself or through\nsome of its attributes.\n\n\u00a7 62. Nouns, or names, are either Substantive or Adjective.\n\nA Noun Substantive is the name of a thing in itself, that is to say,\nwithout reference to any special attribute.\n\n\u00a7 63. A Noun Adjective is a name which we are entitled to add to a\nthing, when we know it to possess a given attribute.\n\n\u00a7 64. The Verb, as such, is not recognised by logic, but is resolved\ninto predicate and copula, that is to say, into a noun which is\naffirmed or denied of another, plus the sign of that affirmation or\ndenial. 'The kettle boils' is logically equivalent to 'The kettle is\nboiling,' though it is by no means necessary to express the\nproposition in the latter shape. Here we see that 'boils' is\nequivalent to the noun 'boiling' together with the copula 'is,' which\ndeclares its agreement with the noun 'kettle.' 'Boiling' here is a\nnoun adjective, which we are entitled to add to 'kettle,' in virtue of\ncertain knowledge which we have about the latter. Being a verbal noun,\nit is called in grammar a participle, rather than a mere\nadjective. The word 'attributive' in logic embraces both the adjective\nand participle of grammar.\n\n\u00a7 65. In grammar every noun is a separate word: but to logic, which is\nconcerned with the thought rather than with the expression, it is\nindifferent whether a noun, or term, consists of one word or many. The\nlatter are known as 'many-worded names.' In the following passage,\ntaken at random from Butler's Analogy--'These several observations,\nconcerning the active principle of virtue and obedience to God's\ncommands, are applicable to passive submission or resignation to his\nwill'--we find the subject consisting of fourteen words, and the\npredicate of nine. It is the exception rather than the rule to find a\npredicate which consists of a single word.  Many-worded names in\nEnglish often consist of clauses introduced by the conjunction 'that,'\nas 'That letters should be written in strict conformity with nature is\ntrue': often also of a grammatical subject with one or more dependent\nclauses attached to it, as\n\n  'He who fights and runs away,\n   Will live to fight another day.'\n\n\u00a7 66. Every term then is not a word, since a term may consist of a\ncollection of words. Neither is every word a term. 'Over,' for\ninstance, and 'swiftly,' and, generally, what are called particles in\ngrammar, do not by themselves constitute terms, though they may be\nemployed along with other words to make up a term.\n\n\u00a7 67. The notions with which thought deals involve many subtle\nrelations and require many nice modifications.  Language has\ninstruments, more or less perfect, whereby such relations and\nmodifications may be expressed. But these subsidiary aids to\nexpression do not form a notion which can either have something\nasserted of it or be asserted itself of something else.\n\n\u00a7 68. Hence words are divided into three classes--\n\n  (1) Categorematic;\n\n  (2) Syncategorematic;\n\n  (3) Acategorematic.\n\n\u00a7 69. A Categorematic word is one which can be used by itself as a\nterm.\n\n\u00a7 70. A Syncategorematic word is one which can help to form a term.\n\n\u00a7 71. An Acategorematic word is one which can neither form, nor help\nto form, a term [Footnote: Comparatively few of the parts of speech\nare categorematic.  Nouns, whether substantive or adjective, including\nof course pronouns and participles, are so, but only in their\nnominative cases, except when an oblique case is so used as to be\nequivalent to an attributive.  Verbs also are categorematic, but only\nin three of their moods, the Indicative, the Infinitive, and the\nPotential. The Imperative and Optative moods clearly do not convey\nassertions at all, while the Subjunctive can only figure as a\nsubordinate member of some assertion. We may notice, too, that the\nrelative pronoun, unlike the rest, is necessarily syncategorematic,\nfor the same reason as the subjunctive mood. Of the remaining parts of\nspeech the article, adverb, preposition, and conjunction can never be\nanything but syncategorematic, while the interjection is\nacategorematic, like the vocative case of nouns and the imperative and\noptative moods of verbs, which do not enter at all into the form of\nsentence known as the proposition.].\n\n\u00a7 72. Categorematic literally means 'predicable.'  'Horse,' 'swift,'\n'galloping' are categorematic. Thus we can say, 'The horse is swift,'\nor 'The horse is galloping.'  Each of these words forms a term by\nitself, but 'over' and 'swiftly' can only help to form a term, as in\nthe proposition, 'The horse is galloping swiftly over the plain.'\n\n\u00a7 73. A term then may be said to be a categorematic word or collection\nof words, that is to say, one which can be used by itself as a\npredicate.\n\n\u00a7 74. To entitle a word or collection of words to be called a term, it\nis not necessary that it should be capable of standing by itself as a\nsubject. Many terms which can be used as predicates are incapable of\nbeing used as subjects: but every term which can be used as a subject\n(with the doubtful exception of proper names) can be used also as a\npredicate. The attributives 'swift' and 'galloping' are terms, quite\nas much as the subject 'horse,' but they cannot themselves be used as\nsubjects.\n\n\u00a7 75. When an attributive appears to be used as a subject, it is owing\nto a grammatical ellipse. Thus in Latin we say 'Boni sapientes sunt,'\nand in English 'The good are wise,' because it is sufficiently\ndeclared by the inflexional form in the one case, and by the usage of\nthe language in the other, that men are signified. It is an accident\nof language how far adjectives can be used as subjects. They cease to\nbe logical attributives the moment they are so used.\n\n\u00a7 76. There is a sense in which every word may become categorematic,\nnamely, when it is used simply as a word, to the neglect of its proper\nmeaning. Thus we can say--'\"Swiftly\" is an adverb.' 'Swiftly' in this\nsense is really no more than the proper name for a particular\nword. This sense is technically known as the 'suppositio materialis'\nof a word.\n\n\n\n\nCHAPTER II.\n\n_Of the Division of Things._\n\n\n\u00a7 77. Before entering on the divisions of terms it is necessary to\nadvert for a moment to a division of the things whereof they are\nnames.\n\n\u00a7 78. By a 'thing' is meant simply an object of thought--whatever one\ncan think about.\n\n\u00a7 79. Things are either Substances or Attributes.  Attributes may be\nsub-divided into Qualities and Relations.\n\n                    Thing\n       _______________|_______________\n      |                               |\n  Substance                       Attribute\n                         _____________|____________\n                        |                          |\n                     Quality                    Relation\n\n\u00a7 80. A Substance is a thing which can be conceived to exist by\nitself. All bodies are material substances. The soul, as a thinking\nsubject, is an immaterial substance.\n\n\u00a7 81. An Attribute is a thing which depends for its existence upon a\nsubstance, e.g. greenness, hardness, weight, which cannot be\nconceived to exist apart from green, hard, and heavy substances.\n\n\u00a7 82. A Quality is an attribute which does not require more than one\nsubstance for its existence. The attributes just mentioned are\nqualities. There might be greenness, hardness, and weight, if there\nwere only one green, hard and heavy substance in the universe.\n\n\u00a7 83. A Relation is an attribute which requires two or more substances\nfor its existence, e.g. nearness, fatherhood, introduction.\n\n\u00a7 84. When we say that a substance can be conceived to exist by\nitself, what is meant is that it can be conceived to exist\nindependently of other substances. We do not mean that substances can\nbe conceived to exist independently of attributes, nor yet out of\nrelation to a mind perceiving them. Substances, so far as we can know\nthem, are only collections of attributes. When therefore we say that\nsubstances can be conceived to exist by themselves, whereas attributes\nare dependent for their existence upon substances, the real meaning of\nthe assertion reduces itself to this, that it is only certain\ncollections of attributes which can be conceived to exist\nindependently; whereas single attributes depend for their existence\nupon others. The colour, smoothness or solidity of a table cannot be\nconceived apart from the extension, whereas the whole cluster of\nattributes which constitutes the table can be conceived to exist\naltogether independently of other 'such clusters.  We can imagine a\ntable to exist, if the whole material universe were annihilated, and\nbut one mind left to perceive it. Apart from mind, however, we cannot\nimagine it: since what we call the attributes of a material substance\nare no more than the various modes in which we find our minds\naffected.\n\n\u00a7 85. The above division of things belongs rather to the domain of\nmetaphysics than of logic: but it is the indispensable basis of the\ndivision of terms, to which we now proceed.\n\n\n\n\nCHAPTER III.\n\n_Of the Division of Terms._\n\n\n\u00a7 86. The following scheme presents to the eye the chief divisions of\nterms.\n\n  Term\n    Division of terms according to their place in thought.\n      Subject-Term\n      Attributive\n\n    according to the kind of thing signified.\n      Abstract\n      Concrete\n\n    according to Quantity in Extension.\n      Singular\n      Common\n\n    according to Quality.\n      Positive\n      Privative\n      Negative\n\n    according to number of meanings.\n      Univocal\n      Equivocal\n\n    according to number of things involved in the name.\n      Absolute\n      Relative\n\n    according to number of quantities.\n      Connotative\n      Non-connotative\n\n\n\n_Subject-term and Attributive._\n\n\n\u00a7 87. By a Subject-term is meant any term which is capable of standing\nby itself as a subject, e.g. 'ribbon,' 'horse.'\n\n\u00a7 88. Attributives can only be used as predicates, not as subjects,\ne.g. 'cherry-coloured,' 'galloping.' These can only be used in\nconjunction with other words (syncategorematically) to make up a\nsubject. Thus we can say 'A cherry-coloured ribbon is becoming,' or 'A\ngalloping horse is dangerous.'\n\n\u00a7 89. Attributives are contrivances of language whereby we indicate\nthat a subject has a certain attribute. Thus, when we say 'This paper\nis white,' we indicate that the subject 'paper' possesses the\nattribute whiteness. Logic, however, also recognises as attributives\nterms which signify the non-possession of attributes. 'Not-white' is\nan attributive equally with 'white.'\n\n\u00a7 90. An Attributive then may be defined as a term which signifies the\npossession, or non-possession, of an attribute by a subject.\n\n\u00a7 91. It must be carefully noticed that attributives are not names of\nattributes, but names of the things which possess the attributes, in\nvirtue of our knowledge that they possess them. Thus 'white' is the\nname of all the things which possess the attribute whiteness, and\n'virtuous' is a name; not of the abstract quality, virtue, itself, but\nof the men and actions which possess it. It is clear that a term can\nonly properly be said to be a name of those things whereof it can be\npredicated. Now, we cannot intelligibly predicate an attributive of\nthe abstract quality, or qualities, the possession of which it\nimplies. We cannot, for instance, predicate the term 'learned' of the\nabstract quality of learning: but we may predicate it of the\nindividuals, Varro and Vergil. Attributives, then, are to be regarded\nas names, not of the attributes which they imply, but of the things in\nwhich those attributes are found.\n\n\u00a7 92. Attributives, however, are names of things in a less direct way\nthan that in which subject-terms may be the names of the same\nthings. Attributives are names of things only in predication, whereas\nsubject-terms are names of things in or out of predication. The terms\n'horse' and 'Bucephalus' are names of certain things, in this case\nanimals, whether we make any statement about them or not: but the\nterms 'swift' and 'fiery' only become names of the same things in\nvirtue of being predicable of them. When we say 'Horses are swift' or\n'Bucephalus was fiery,' the terms 'swift' and 'fiery' become names\nrespectively of the same things as 'horse' and 'Bucephalus.' This\nfunction of attributives as names in a secondary sense is exactly\nexpressed by the grammatical term 'noun adjective.' An attributive is\nnot directly the name of anything. It is a name added on in virtue of\nthe possession by a given thing of a certain attribute, or, in some\ncases, the non-possession.\n\n\u00a7 93. Although attributives cannot be used as subjects, there is\nnothing to prevent a subject-term from being used as a predicate, and\nso assuming for the time being the functions of an attributive. When\nwe say 'Socrates was a man,' we convey to the mind the idea of the\nsame attributes which are implied by the attributive 'human.'  But\nthose terms only are called attributives which can never be used\nexcept as predicates.\n\n\u00a7 94. This division into Subject-terms and Attributives may be\nregarded as a division of terms according to their place in\nthought. Attributives, as we have seen, are essentially predicates,\nand can only be thought of in relation to the subject, whereas the\nsubject is thought of for its own sake.\n\n\n\n_Abstract and Concrete Terms_.\n\n\n\u00a7 95. An Abstract Term is the name of an attribute, e.g. whiteness\n[Footnote: Since things cannot be spoken of except by their names,\nthere is a constantly recurring source of confusion between the thing\nitself and the name of it. Take for instance 'whiteness.' The\nattribute whiteness is a thing, the word 'whiteness' is a term.],\nmultiplication, act, purpose, explosion.\n\n\u00a7 96. A Concrete Term is the name of a substance, e.g. a man, this\nchair, the soul, God.\n\n\u00a7 97. Abstract terms are so called as being arrived at by a process of\nAbstraction. What is meant by Abstraction will be clear from a single\ninstance. The mind, in contemplating a number of substances, may draw\noff, or abstract, its attention from all their other characteristics,\nand fix it only on some point, or points, which they have in\ncommon. Thus, in contemplating a number of three-cornered objects, we\nmay draw away our attention from all their other qualities, and fix it\nexclusively upon their three-corneredness, thus constituting the\nabstract notion of 'triangle.' Abstraction may be performed equally\nwell in the case of a single object: but the mind would not originally\nhave known on what points to fix its attention except by a comparison\nof individuals.\n\n\u00a7 98. Abstraction too may be performed upon attributes as well as\nsubstances. Thus, having by abstraction already arrived at the notion\nof triangle, square, and so on, we may fix our attention upon what\nthese have in common, and so rise to the higher abstraction of\n'figure.' As thought becomes more complex, we may have abstraction on\nabstraction and attributes of attributes. But, however many steps may\nintervene, attributes may always be traced back to substances at\nlast. For attributes of attributes can mean at bottom nothing but the\nco-existence of attributes in, or in connection with, the same\nsubstances.\n\n\u00a7 99. We have said that abstract terms are so called, as being arrived\nat by abstraction: but it must not be inferred from this statement\nthat all terms which are arrived at by abstraction are abstract. If\nthis were so, all names would be abstract except proper names of\nindividual substances.  All common terms, including attributives, are\narrived at by abstraction, but they are not therefore abstract terms.\nThose terms only are called abstract, which cannot be applied to\nsubstances at all. The terms 'man' and 'human' are names of the same\nsubstance of which Socrates is a name. Humanity is a name only of\ncertain attributes of that substance, namely those which are shared by\nothers. All names of concrete things then are concrete, whether they\ndenote them individually or according to classes, and whether directly\nand in themselves, or indirectly, as possessing some given attribute.\n\n\u00a7 100. By a 'concrete thing' is meant an individual Substance\nconceived of with all its attributes about it.  The term is not\nconfined to material substances. A spirit conceived of under personal\nattributes is as concrete as plum-pudding.\n\n\u00a7 101. Since things are divided exhaustively into substances and\nattributes, it follows that any term which is not the name of a thing\ncapable of being conceived to exist by itself, must be an abstract\nterm. Individual substances can alone be conceived to exist by\nthemselves: all their qualities, actions, passions, and\ninter-relations, all their states, and all events with regard to them,\npresuppose the existence of these individual substances. All names\ntherefore of such things as those just enumerated are abstract\nterms. The term 'action,' for instance, is an abstract term. For how\ncould there be action without an agent? The term 'act' also is equally\nabstract for the same reason. The difference between 'action' and\n'act' is not the difference between abstract and concrete, but the\ndifference between the name of a process and the name of the\ncorresponding product. Unless acts can be conceived to exist without\nagents they are as abstract as the action from which they result.\n\n\u00a7 102. Since every term must be either abstract or concrete, it may be\nasked--Are attributives abstract or concrete? The answer of course\ndepends upon whether they are names of substances or names of\nattributes. But attributives, it must be remembered, are never\ndirectly names of anything, in the way that subject-terms are; they\nare only names of things in virtue of being predicated of\nthem. Whether an attributive is abstract or concrete, depends on the\nnature of the subject of which it is asserted or denied. When we say\n'This man is noble,' the term 'noble' is concrete, as being the name\nof a substance: but when we say 'This act is noble,' the term 'noble'\nis abstract, as being the name of an attribute.\n\n\u00a7 103. The division of terms into Abstract and Concrete is based upon\nthe kind of thing signified. It involves no reference to actual\nexistence. There are imaginary as well as real substances. Logically a\ncentaur is as much a substance as a horse.\n\n\n\n_Terms._\n\n\n\u00a7 104. A Singular Term is a name which can be applied, in the same\nsense, to one thing only, e.g. 'John,' 'Paris,' 'the capital of\nFrance,' 'this pen.'\n\n\u00a7 105. A Common Term is a name which can be applied, in the same\nsense, to a class of things, e.g. 'man,' 'metropolis,' 'pen.'\n\nIn order that a term may be applied in the same sense to a number of\nthings, it is evident that it must indicate attributes which are\ncommon to all of them. The term 'John' is applicable to a number of\nthings, but not in the same sense, as it does not indicate attributes.\n\n\u00a7 106. Common terms are formed, as we have seen already (\u00a7 99), by\nabstraction, i. e. by withdrawing the attention from the attributes in\nwhich individuals differ, and concentrating it upon those which they\nhave in common.\n\n\u00a7 107. A class need not necessarily consist of more than two\nthings. If the sun and moon were the only heavenly bodies in the\nuniverse, the word 'heavenly body' would still be a common term, as\nindicating the attributes which are possessed alike by each.\n\n\u00a7 108. This being so, it follows that the division of terms into\nsingular and common is as exhaustive as the preceding ones, since a\nsingular term is the name of one thing and a common term of more than\none. It is indifferent whether the thing in question be a substance or\nan attribute; nor does it matter how complex it may be, so long as it\nis regarded by the mind as one.\n\n\u00a7 109. Since every term must thus be either singular or common, the\nmembers of the preceding divisions must find their place under one or\nboth heads of this one.  Subject-terms may plainly fall under either\nhead of singular or common: but attributives are essentially common\nterms.  Such names as 'green,' 'gentle,' 'incongruous' are applicable,\nstrictly in the same sense, to all the things which possess the\nattributes which they imply.\n\n\u00a7 110. Are abstract terms then, it may be asked, singular or common?\nTo this question we reply--That depends upon how they are used. The\nterm 'virtue,' for instance, in one sense, namely, as signifying moral\nexcellence in general, without distinction of kind, is strictly a\nsingular term, as being the name of one attribute: but as applied to\ndifferent varieties of moral excellence--justice, generosity,\ngentleness and so on--it is a common term, as being a name which is\napplicable, in the same sense, to a class of attributes. Similarly the\nterm 'colour,' in a certain sense, signifies one unvarying attribute\npossessed by bodies, namely, the power of affecting the eye, and in\nthis sense it is a singular term: but as applied to the various ways\nin which the eye may be affected, it is evidently a common term, being\nequally applicable to red, blue, green, and every other colour. As\nsoon as we begin to abstract from attributes, the higher notion\nbecomes a common term in reference to the lower. By a 'higher notion'\nis meant one which is formed by a further process of abstraction. The\nterms 'red,' 'blue,' 'green,' etc., are arrived at by abstraction from\nphysical objects; 'colour' is arrived at by abstraction from them, and\ncontains nothing, but what is common to all. It therefore applies in\nthe same sense to each, and is a common term in relation to them.\n\n\u00a7 111. A practical test as to whether an abstract term, in any given\ncase, is being used as a singular or common term, is to try whether\nthe indefinite article or the sign of the plural can be attached to\nit. The term 'number,' as the name of a single attribute of things,\nadmits of neither of these adjuncts: but to talk of 'a number' or 'the\nnumbers, two, three, four,' etc., at once marks it as a common\nterm. Similarly the term 'unity' denotes a single attribute, admitting\nof no shades of distinction: but when a writer begins to speak of 'the\nunities' he is evidently using the word for a class of things of some\nkind or other, namely, certain dramatical proprieties of composition.\n\n\n\nProper _Names_ and _Designations_.\n\n\n\u00a7 112. Singular terms may be subdivided into Proper Names and\nDesignations.\n\n\u00a7 113. A Proper Name is a permanent singular term applicable to a\nthing in itself; a Designation is a singular term devised for the\noccasion, or applicable to a thing only in so far as it possesses some\nattribute.\n\n\u00a7 114. 'Homer' is a proper name; 'this man,' 'the author of the Iliad'\nare designations.\n\n\u00a7 115. The number of things, it is clear, is infinite.  For, granting\nthat the physical universe consists of a definite number of\natoms--neither one more nor one less--still we are far from having\nexhausted the possible number of things. All the manifold material\nobjects, which are made up by the various combinations of these atoms,\nconstitute separate objects of thought, or things, and the mind has\nfurther an indefinite power of conjoining and dividing these objects,\nso as to furnish itself with materials of thought, and also of fixing\nits attention by abstraction upon attributes, so as to regard them as\nthings, apart from the substances to which they belong.\n\n\u00a7 116. This being so, it is only a very small number of things, which\nare constantly obtruding themselves upon the mind, that have singular\nterms permanently set apart to denote them. Human beings, some\ndomestic animals, and divisions of time and place, have proper names\nassigned to them in most languages, e.g. 'John,' 'Mary,' 'Grip,'\n'January,' 'Easter,' 'Belgium,' 'Brussels,' 'the Thames,' 'Ben-Nevis.'\nBesides these, all abstract terms, when used without reference to\nlower notions, are of the nature of proper names, being permanently\nset apart to denote certain special attributes, e.g. 'benevolence,'\n'veracity,' 'imagination,' 'indigestibility, 'retrenchment.'\n\n\u00a7 117. But the needs of language often require a singular term to\ndenote some thing which has not had a proper name assigned to it. This\nis effected by taking a common term, and so limiting it as to make it\napplicable, under the given circumstances, to one thing only.  Such a\nlimitation may be effected in English by prefixing a demonstrative or\nthe definite article, or by appending a description, e.g. 'this pen,'\n'the sofa,' 'the last rose of summer.' When a proper name is unknown,\nor for some reason, unavailable, recourse may be had to a designation,\ne.g. 'the honourable member who spoke last but one.'\n\n\n\n_Collective Terms_.\n\n\n\u00a7 118. The division of terms into singular and common being, like\nthose which have preceded it, fundamental and exhaustive, there is\nevidently no room in it for a third class of Collective Terms. Nor is\nthere any distinct class of terms to which that name can be given. The\nsame term may be used collectively or distributively in different\nrelations. Thus the term 'library,' when used of the books which\ncompose a library, is collective; when used of various collections of\nbooks, as the Bodleian, Queen's library, and so on, it is\ndistributive, which, in this case, is the same thing as being a common\nterm.\n\n\u00a7 119, The distinction between the collective and distributive use of\na term is of importance, because the confusion of the two is a\nfavourite source of fallacy.  When it is said 'The plays of Shakspeare\ncannot be read in a day,' the proposition meets with a very different\nmeasure of acceptance according as its subject is understood\ncollectively or distributively. The word 'all' is perfectly ambiguous\nin this respect. It may mean all together or each separately--two\nsenses which are distinguished in Latin by 'totus' or 'cunctus,' for\nthe collective, and 'omnis' for the distributive use.\n\n\u00a7 120. What is usually meant however when people speak of a collective\nterm is a particular kind of singular term.\n\n\u00a7 121. From this point of view singular terms may be subdivided into\nIndividual and Collective, by an Individual Term being meant the name\nof one object, by a Collective Term the name of several considered as\none. 'This key' is an individual term; 'my bunch of keys' is a\ncollective term.\n\n\u00a7 122. A collective term is quite as much the name of one thing as an\nindividual term is, though the thing in question happens to be a\ngroup. A group is one thing, if we choose to think of it as one. For\nthe mind, as we have already seen, has an unlimited power of forming\nits own things, or objects of thought. Thus a particular peak in a\nmountain chain is as much one thing as the chain itself, though,\nphysically speaking, it is inseparable from it, just as the chain\nitself is inseparable from the earth's surface.  In the same way a\nnecklace is as much one thing as the individual beads which compose\nit.\n\n\u00a7 123. We have just seen that a collective term is the name of a group\nregarded as one thing: but every term which is the name of such a\ngroup is not necessarily a collective term. 'London,' for instance, is\nthe name of a group of objects considered as one thing. But 'London'\nis not a collective term, whereas 'flock,' 'regiment,' and 'senate'\nare. Wherein then lies the difference? It lies in this--that flock,\nregiment and senate are groups composed of objects which are, to a\ncertain extent, similar, whereas London is a group made up of the most\ndissimilar objects--streets and squares and squalid slums, fine\ncarriages and dirty faces, and so on. In the case of a true collective\nterm all the members of the group will come under some one common\nname. Thus all the members of the group, flock of sheep, come under\nthe common name 'sheep,' all the members of the group 'regiment' under\nthe common name, 'soldier,' and so on.\n\n\u00a7 124. The subdivision of singular terms into individual and\ncollective need not be confined to the names of concrete things. An\nabstract term like 'scarlet,' which is the name of one definite\nattribute, may be reckoned 'individual,' while a term like 'human\nnature,' which is the name of a whole group of attributes, would more\nfitly be regarded as collective.\n\n\u00a7 126. The main division of terms, which we have been discussing, into\nsingular and collective, is based upon their Quantity in\nExtension. This phrase will be explained presently.\n\n\u00a7 126. We come now to a threefold division of terms into Positive,\nPrivative and Negative. It is based upon an implied two-fold division\ninto positive and non-positive, the latter member being subdivided\ninto Privative and Negative.\n\n                   Term\n      _______________|_______________\n     |                               |\n  Positive                     Non-Positive\n                        _____________|____________\n                       |                          |\n                   Privative                   Negative\n\nIf this division be extended, as it sometimes is, to terms in general,\na positive term must be taken to mean only the definite, or\ncomparatively definite, member of an exhaustive division in accordance\nwith the law of excluded middle (\u00a7 25). Thus 'Socrates' and 'man' are\npositive, as opposed to 'not-Socrates' and 'not-man.'\n\n\u00a7 127. The chief value of the division, however, and especially of the\ndistinction drawn between privative and negative terms, is in relation\nto attributives.\n\nFrom this point of view we may define the three classes of terms as\nfollows:\n\nA Positive Term signifies the presence of an attribute, e.g.: 'wise,'\n'full.'\n\nA Negative Term signifies merely the absence of an attribute,\ne.g. 'not-wise,' 'not-full.'\n\nA Privative Term signifies the absence of an attribute in a subject\ncapable of possessing it, e.g. 'unwise,' 'empty'. [Footnote: A\nprivative term is usually defined to mean one which signifies the\nabsence of an attribute where it was once possessed, or might have\nbeen expected to be present, e.g. 'blind.' The utility of the slight\nextension of meaning here assigned to the expression will, it is\nhoped, prove its justification.]\n\n\u00a7 128. Thus a privative term stands midway in meaning between the\nother two, being partly positive and partly negative--negative in so\nfar as it indicates the absence of a certain attribute, positive in so\nfar as it implies that the thing which is declared to lack that\nattribute is of such a nature as to be capable of possessing it. A\npurely negative term conveys to the mind no positive information at\nall about the nature of the thing of which it is predicated, but\nleaves us to seek for it among the universe of things which fail to\nexhibit a given attribute.\n\nA privative term, on the other hand, restricts us within a definite\nsphere. The term 'empty' restricts us within the sphere of things\nwhich are capable of fulness, that is, if the term be taken in its\nliteral sense, things which possess extension in three dimensions.\n\n\u00a7 129. A positive and a negative term, which have the same matter,\nmust exhaust the universe between them, e.g. 'white' and 'not-white,'\nsince, according to the law of excluded middle, everything must be\neither one or the other. To say, however, that a thing is 'not-white'\nis merely to say that the term 'white' is inapplicable to it.\n'Not-white' may be predicated of things which do not possess extension\nas well as of those which do. Such a pair of terms as 'white' and\n'not-white,' in their relation to one another, are called\nContradictories.\n\n\u00a7 130. Contrary terms must be distinguished from\ncontradictory. Contrary terms are those which are most opposed under\nthe same head. Thus 'white' and 'black' are contrary terms, being the\nmost opposed under the same head of colour. 'Virtuous' and 'vicious'\nagain are contraries, being the most opposed under the same head of\nmoral quality.\n\n\u00a7 131. A positive and a privative term in the same matter will always\nbe contraries, e.g. 'wise' and 'unwise,' 'safe' and 'unsafe': but\ncontraries do not always assume the shape of positive and privative\nterms, but may both be positive in form, e.g. 'wise' and 'foolish,'\n'safe' and 'dangerous.'\n\n\u00a7 132. Words which are positive in form are often privative in\nmeaning, and vice vers\u00e2. This is the case, for instance, with the word\n'safe,' which connotes nothing more than the absence of danger. We\ntalk of a thing involving 'positive danger' and of its being\n'positively unsafe' to do so and so. 'Unhappy,' on the other hand,\nsignifies the presence of actual misery. Similarly in Latin 'inutilis'\nsignifies not merely that there is no benefit to be derived from a\nthing, but that it is _positively injurious_. All such questions,\nhowever, are for the grammarian or lexicographer, and not for the\nlogician. For the latter it is sufficient to know that corresponding\nto every term which signifies the presence of some attribute there may\nbe imagined another which indicates the absence of the same attribute,\nwhere it might be possessed, and a third which indicates its absence,\nwhether it might be possessed or not.\n\n\u00a7 133. Negative terms proper are formed by the prefix 'not-' or\n'non-,' and are mere figments of logic. We do not in practice require\nto speak of the whole universe of objects minus those which possess a\ngiven attribute or collection of attributes. We have often occasion to\nspeak of things which might be wise and are not, but seldom, if ever,\nof all things other than wise.\n\n\u00a7 134. Every privative attributive has, or may have, a corresponding\nabstract term, and the same is the case with negatives: for the\nabsence of an attribute, is itself an attribute. Corresponding to\n'empty,' there is 'emptiness'; corresponding to 'not-full' there may\nbe imagined the term 'not-fulness.'\n\n\u00a7 135. The contrary of a given term always involves the contradictory,\nbut it involves positive elements as well.  Thus 'black' is\n'not-white,' but it is something more besides. Terms which, without\nbeing directly contrary, involve a latent contradiction, are called\nRepugnant, e.g.  'red' and 'blue.' All terms whatever which signify\nattributes that exclude one another may be called Incompatible.\n\n\u00a7 136. The preceding division is based on what is known as the Quality\nof terms, a positive term being said to differ in quality from a\nnon-positive one.\n\n\n\n_Univocal and Equivocal Terms_.\n\n\n\u00a7 137. A term is said to be Univocal, when it has one and the same\nmeaning wherever it occurs. A term which has more than one meaning is\ncalled Equivocal. 'Jam-pot,' 'hydrogen' are examples of univocal\nterms; 'pipe' and 'suit' of equivocal.\n\n\u00a7 138. This division does not properly come within the scope of logic,\nsince it is a question of language, not of thought. From the\nlogician's point of view an equivocal term is two or more different\nterms, for the definition in each sense would be different.\n\n\u00a7 139. Sometimes a third member is added to the same division under\nthe head of Analogous Terms. The word 'sweet,' for instance, is\napplied by analogy to things so different in their own nature as a\nlump of sugar, a young lady, a tune, a poem, and so on. Again, because\nthe head is the highest part of man, the highest part of a stream is\ncalled by analogy 'the head.' It is plainly inappropriate to make a\nseparate class of analogous terms.  Rather, terms become equivocal by\nbeing extended by analogy from one thing to another.\n\n\n\n_Absolute and Relative Terms_.\n\n\n\u00a7 140. An Absolute term is a name given to a thing without reference\nto anything else.\n\n\u00a7 141. A Relative term is a name given to a thing with direct\nreference to some other thing.\n\n\u00a7 142. 'Hodge' and 'man' are absolute terms. 'Husband' 'father,'\n'shepherd' are relative terms. 'Husband' conveys a direct reference to\n'wife,' 'father' to 'Child,' 'shepherd' to 'sheep.' Given one term of\na relation, the other is called the correlative, e.g. 'subject' is\nthe correlative of 'ruler,' and conversely 'ruler' of 'subject.' The\ntwo terms are also spoken of as a pair of correlatives.\n\n\u00a7 143. The distinction between relative and absolute applies to\nattributives as well as subject-terms. 'Greater,' 'near, 'like,' are\ninstances of attributives which everyone would recognise as relative.\n\n\u00a7 144. A relation, it will be remembered, is a kind of attribute,\ndiffering from a quality in that it necessarily involves more\nsubstances than one. Every relation is at bottom a fact, or series of\nfacts, in which two or more substances play a part. A relative term\nconnotes this fact or facts from the point of view of one of the\nsubstances, its correlative from that of the other. Thus 'ruler' and\n'subject' imply the same set of facts, looked at from opposite points\nof view. The series of facts itself, regarded from either side, is\ndenoted by the corresponding abstract terms, 'rule 'and 'subjection.'\n\n\u00a7 145. It is a nice question whether the abstract names of relations\nshould themselves be considered relative terms.  Difficulties will\nperhaps be avoided by confining the expression 'relative _term_'\nto names of concrete things.  'Absolute,' it must be remembered, is a\nmere negative of 'relative,' and covers everything to which the\ndefinition of the latter does not strictly apply. Now it can hardly be\nsaid that 'rule' is a name given to a certain abstract thing with\ndirect reference to some other thing, namely, subjection. Rather\n'rule' and 'subjection' are two names for identically the same series\nof facts, according to the side from which we look at them. 'Ruler'\nand 'subject,' on the other hand, are names of two distinct\nsubstances, but each involving a reference to the other.\n\n\u00a7 146. This division then may be said to be based on the number of\nthings involved in the name.\n\n\n\n_Connotative and Non-Connotative Terms._\n\n\n\u00a7 147. Before explaining this division, it is necessary to treat of\nwhat is called the Quantity of Terms.\n\n\n\n_Quantity of Terms._\n\n\n\u00a7 148. A term is possessed of quantity in two ways--\n\n  (1) In Extension;\n\n  (2) In Intension.\n\n\u00a7 149. The Extension of a term is the number of things to which it\napplies.\n\n\u00a7 150. The Intension of a term is the number of attributes which it\nimplies.\n\n\u00a7 151. It will simplify matters to bear in mind that the intension of\na term is the same thing as its meaning. To take an example, the term\n'man' applies to certain things, namely, all the members of the human\nrace that have been, are, or ever will be: this is its quantity in\nextension. But the term 'man' has also a certain meaning, and implies\ncertain attributes--rationality, animality, and a definite bodily\nshape: the sum of these attributes constitutes its quantity in\nintension.\n\n\u00a7 152. The distinction between the two kinds of quantity possessed by\na term is also conveyed by a variety of expressions which are here\nappended.\n\nExtension = breadth = compass = application = denotation.\n\nIntension = depth = comprehension = implication = connotation.\n\nOf these various expressions, 'application' and 'implication' have the\nadvantage of most clearly conveying their own meaning. 'Extension' and\n'intension,' however, are more usual; and neither 'implication' nor\n'connotation' is quite exact as a synonym for 'intension.' (\u00a7 164.)\n\n\u00a7 153. We now return to the division of terms into connotative and\nnon-connotative.\n\n\u00a7 154. A term is said to connote attributes, when it implies certain\nattributes at the same time that it applies to certain things distinct\ntherefrom. [Footnote: Originally 'connotative' was used in the same\nsense in which we have used 'attributive,' for a word which directly\nsignifies the presence of an attribute and indirectly applies to a\nsubject. In this, its original sense, it was the subject which was\nsaid to be connoted, and not the attribute.]\n\n\u00a7 155. A term which possesses both extension and intension, distinct\nfrom one another, is connotative.\n\n\u00a7 156. A term which possesses no intension (if that be possible) or in\nwhich extension and intension coincide is non-connotative.\n\n\u00a7 157. The subject-term, 'man,' and its corresponding attributive,\n'human,' have both extension and intension, distinct from one\nanother. They are therefore connotative. But the abstract term,\n'humanity,' denotes the very collection of attributes, which was\nbefore connoted by the concrete terms, 'man' and 'human.' In this\ncase, therefore, extension and intension coincide, and the term is\nnon-connotative.\n\n\u00a7 158. The above remark must be understood to be limited to abstract\nterms in their singular sense. When employed as common terms, abstract\nterms possess both extension and intension distinct from one\nanother. Thus the term 'colour' applies to red, blue, and yellow, and\nat the same time implies (i.e. connotes), the power of affecting the\neye.\n\n\u00a7 159. Since all terms are names of things, whether substances or\nattributes, it is clear that all terms must possess extension, though\nthe extension of singular terms is the narrowest possible, as being\nconfined to one thing.\n\n\u00a7 160. Are there then any terms which possess no intension? To ask\nthis, is to ask--Are there any terms which have absolutely no meaning?\nIt is often said that proper names are devoid of meaning, and the\nremark is, in a certain sense, true. When we call a being by the name\n'man,' we do so because that being possesses human attributes, but\nwhen we call the same being by the name, 'John,' we do not mean to\nindicate the presence of any Johannine attributes. We simply wish to\ndistinguish that being, in thought and language, from other beings of\nthe same kind. Roughly speaking, therefore, proper names are devoid of\nmeaning or intension. But no name can be entirely devoid of\nmeaning. For, even setting aside the fact, which is not universally\ntrue, that proper names indicate the sex of the owner, the mere act of\ngiving a name to a thing implies at least that the thing exists,\nwhether in fact or thought; it implies what we may call 'thinghood':\nso that every term must carry with it some small amount of intension.\n\n\u00a7 161. From another point of view, however, proper names possess more\nintension than any other terms. For when we know a person, his name\ncalls up to our minds all the individual attributes with which we are\nfamiliar, and these must be far more numerous than the attributes\nwhich are conveyed by any common term which can be applied to\nhim. Thus the name 'John' means more to a person who knows him than\n'attorney,' 'conservative,' 'scamp,' of 'vestry-man,' or any other\nterm which may happen to apply to him. This, however, is the acquired\nintension of a term, and must be distinguished from the original\nintension. The name 'John' was never meant to indicate the attributes\nwhich its owner has, as a matter of fact, developed. He would be John\nall the same, if he were none of these.\n\n\u00a7 162. Hitherto we have been speaking only of christening-names, but\nit is evident that family names have a certain amount of connotation\nfrom the first. For when we dub John with the additional appellation\nof Smith, we do not give this second name as a mere individual mark,\nbut intend thereby to indicate a relationship to other persons.  The\namount of connotation that can be conveyed by proper names is very\nnoticeable in the Latin language.  Let us take for an example the full\nname of a distinguished Roman--Publius Cornelius Scipio \u00c6milianus\nAfricanus minor. Here it is only the pr\u00e6nomen, Publius, that can be\nsaid to be a mere individual mark, and even this distinctly indicates\nthe sex of the owner. The nomen proper, Cornelius, declares the wearer\nof it to belong to the illustrious gens Cornelia. The cognomen,\nScipio, further specifies him as a member of a distinguished family in\nthat gens. The agnomen adoptivum indicates his transference by\nadoption from one gens to another.  The second agnomen recalls the\nfact of his victory over the Carthaginians, while the addition of the\nword 'minor' distinguishes him from the former wearer of the same\ntitle. The name, instead of being devoid of meaning, is a chapter of\nhistory in itself. Homeric epithets, such as 'The Cloud-compeller,'\n'The Earth-shaker' are instances of intensive proper names. Many of\nour own family names are obviously connotative in their origin,\nimplying either some personal peculiarity, e.g. Armstrong, Cruikshank,\nCourteney; or the employment, trade or calling of the original bearer\nof the name, Smith, Carpenter, Baker, Clark, Leach, Archer, and so on;\nor else his abode, domain or nationality, as De Caen, De Montmorency,\nFrench, Langley; or simply the fact of descent from some presumably\nmore noteworthy parent, as Jackson, Thomson, Fitzgerald, O'Connor,\nMacdonald, Apjohn, Price, Davids, etc. The question, however, whether\na term is connotative or not, has to be decided, not by its origin,\nbut by its use. We have seen that there are some proper names which,\nin a rough sense, may be said to possess no intension.\n\n\u00a7 163. The other kind of singular terms, namely, designations (\u00a7 113)\nare obviously connotative. We cannot employ even the simplest of them\nwithout conveying more or less information about the qualities of the\nthing which they are used to denote. When, for instance, we say 'this\ntable,' 'this book,' we indicate the proximity to the speaker of the\nobject in question. Other designations have a higher degree of\nintension, as when we say 'the present prime minister of England,'\n'the honourable member who brought forward this motion to-night.'\nSuch terms have a good deal of significance in themselves, apart from\nany knowledge we may happen to possess of the individuals they denote.\n\n\u00a7 164. We have seen that, speaking quite strictly, there are no terms\nwhich are non-connotative: but, for practical purposes, we may apply\nthe expression to proper names, on the ground that they possess no\nintension, and to singular abstract terms on the ground that their\nextension and intension coincide. In the latter case it is indifferent\nwhether we call the quantity extension or intension. Only we cannot\ncall it 'connotation,' because that implies two quantities distinct\nfrom one another. A term must already denote a subject before it can\nbe said to connote its attributes.\n\n\u00a7 165. The division of terms into connotative and non-connotative is\nbased on their possession of one quantity or two.\n\n\n\n\nCHAPTER IV.\n\n_Of the Law of Inverse Variation of Extension and Intension._\n\n\n\u00a7 166. In a series of terms which fall under one another, as the\nextension decreases, the intension increases, and vice vers\u00e2. Take for\ninstance the following series--\n\n  Thing\n    |\n  Substance\n    |\n  Matter\n    |\n  Organism\n    |\n  Animal\n    |\n  Vertebrate\n    |\n  Mammal\n    |\n  Ruminant\n    |\n  Sheep\n    |\n  This sheep.\n\nHere the term at the top possesses the widest possible extension,\nsince it applies to everything. But at the same time it possesses the\nleast possible amount of intension, implying nothing more than mere\nexistence, whether in fact or thought. On the other hand, the term at\nthe bottom possesses the greatest amount of intension, since it\nimplies all the attributes of, an individual superadded to those of\nthe class to which it belongs: but its extension is the narrowest\npossible, being limited to one thing.\n\n\u00a7 167. At each step in the descent from the term at the top, which is\ncalled the 'Summum genus,' to the individual, we decrease the\nextension by increasing the intension.  Thus by adding on to the bare\nnotion of a thing the idea of independent existence, we descend to the\nterm 'substance,' This process is known as Determination, or\nSpecialisation.\n\n\u00a7 168. Again, by withdrawing our attention from the individual\ncharacteristics of a particular sheep, and fixing it upon those which\nare common to it with other animals of the same kind, we arrive at the\ncommon term, 'sheep.'  Here we have increased the extension by\ndecreasing the intension. This process is known as Generalisation.\n\n\u00a7 169. Generalisation implies abstraction, but we may have abstraction\nwithout generalisation.\n\n\u00a7 170. The following example is useful, as illustrating to the eye how\na decrease of extension is accompanied by an increase of intension. At\neach step of the descent here we visibly tack on a fresh\nattribute. [Footnote: This example is borrowed from Professor Jevons.]\n\n  Ship\n    |\n  Steam-ship\n    |\n  Screw steam-ship\n    |\n  Iron screw steam-ship\n    |\n  British iron screw steam-ship.\n\nCould we see the classes denoted by the names the pyramid would be\nexactly inverted.\n\n\u00a7 171. The law of inverse variation of extension and intension must of\ncourse be confined to the inter-relations of a series of terms of\nwhich each can be predicated of the other until we arrive at the\nbottom of the scale. It is not meant to apply to the extension and\nintension of the same term. The increase of population does not add to\nthe meaning of 'baby.'\n\n\n\n\n\n\nPART II.--OF PROPOSITIONS.\n\n\n\n\nCHAPTER I.\n\n_Of the Proposition as distinguished from Other Sentences_.\n\n\n\u00a7 172. As in considering the term, we found occasion to distinguish it\nfrom words generally, so now, in considering the proposition, it will\nbe well to begin by distinguishing it from other sentences.\n\n\u00a7 173. Every proposition is a sentence, but every sentence is not a\nproposition.\n\n\u00a7 174. The field of logic is far from being conterminous with that of\nlanguage. Language is the mirror of man's whole nature, whereas logic\ndeals with language only so far as it gives clothing to the products\nof thought in the narrow sense which we have assigned to that term.\nLanguage has materials of every sort lying strewn about, among which\nthe logician has to seek for his proper implements.\n\n\u00a7 175. Sentences may be employed for a variety of purposes--\n\n  (1) To ask a question;\n\n  (2) To give an order;\n\n  (3) To express a feeling;\n\n  (4) To make a statement.\n\nThese various uses give rise respectively to\n\n  (1) The Interrogative Sentence;\n\n  (2) The Imperative Sentence;\n\n  (3) The Exclamatory Sentence;\n\n  (4) The Enunciative Sentence; Indicative Potential.\n\nIt is with the last of these only that logic is concerned.\n\n\u00a7 176. The proposition, therefore, corresponds to the Indicative and\nPotential, or Conditional, sentences of grammar. For it must be borne\nin mind that logic recognises no difference between a statement of\nfact and a supposition. 'It may rain to-morrow' is as much a\nproposition as 'It is raining now.'\n\n\u00a7 177. Leaving the grammatical aspect of the proposition, we must now\nconsider it from the purely logical point of view.\n\n\u00a7 178. A proposition is a judgement expressed in words; and a\njudgement is a direct comparison between two concepts.\n\n\u00a7 179. The same thing may be expressed more briefly by saying that a\nproposition is a direct comparison between two terms.\n\n\u00a7 180. We say 'direct comparison,' because the syllogism also may be\ndescribed as a comparison between two terms: but in the syllogism the\ntwo terms are compared indirectly, or by means of a third term.\n\n\u00a7 181. A proposition may be analysed into two terms and a Copula,\nwhich is nothing more than the sign of agreement or disagreement\nbetween them.\n\n\u00a7 182. The two terms are called the Subject and the Predicate (\u00a7 58).\n\n\u00a7 183. The Subject is that of which something is stated.\n\n\u00a7 184. The Predicate is that which is stated of the subject.\n\n\u00a7 185. Hence the subject is thought of for its own sake, and the\npredicate for the sake of the subject.\n\n\n\n\nCHAPTER II.\n\nOf _the Copula_.\n\n\n\u00a7 186. There are two kinds of copula, one for affirmative and one for\nnegative statements.\n\n\u00a7 187. Materially the copula is expressed by some part of the verb 'to\nbe,' with or without the negative, or else is wrapped up in some\ninflexional form of a verb.\n\n\u00a7 188. The material form of the copula is an accident of language, and\na matter of indifference to logic. 'The kettle boils' is as logical a\nform of expression as 'The kettle is boiling.' For it must be\nremembered that the word 'is' here is a mere sign of agreement between\nthe two terms, and conveys no notion of actual existence.  We may use\nit indeed with equal propriety to express non-existence, as when we\nsay 'An idol is nothing.'\n\n\u00a7 189. When the verb 'to be' expresses existence in fact it is known\nin grammar as 'the substantive verb.' In this use it is predicate as\nwell as copula, as when we say 'God is,' which may be analysed, if we\nplease, into 'God is existent.'\n\n\u00a7 190. We have laid down above that there are two kinds of copula,\naffirmative and negative: but some logicians have maintained that the\ncopula is always affirmative.\n\n\u00a7 191. What then, it may be asked, on this view, is the meaning of\nnegative propositions! To which the answer is, that a negative\nproposition asserts an agreement between the subject and a negative\nterm. When, for instance, we say 'The whale is not a fish,' this would\nbe interpreted to mean 'The whale is a not-fish.'\n\n\u00a7 192. Undoubtedly any negative proposition may be exhibited in an\naffirmative form, since, by the law of excluded middle, given a pair\nof contradictory terms, wherever the one can be asserted, the other\ncan be denied, and vice vers\u00e2. We shall find later on that this\nprinciple gives rise to one of the forms of immediate inference. The\nonly question then can be, which is the more natural and legitimate\nform of expression. It seems simpler to suppose that we assert the\nagreement of 'whale' with 'not-fish' by implication only, and that\nwhat we directly do is to predicate a disagreement between 'whale' and\nthe positive attributes connoted by 'fish.' For since 'not-fish' must\napply to every conceivable object of thought except those which fall\nunder the positive term 'fish,' to say that a whale is a 'not-fish,'\nis to say that we have still to search for 'whale' throughout the\nwhole universe of being, minus a limited portion; which is only a more\nclumsy way of saying that it is not to be found in that portion.\n\n\u00a7 193. Again, the term 'not-fish' must be understood either in its\nintension or in its extension. If it be understood in its intension,\nwhat it connotes is simply the absence of the positive qualities which\nconstitute a fish, a meaning which is equally conveyed by the negative\nform of proposition. We gain nothing in simplicity by thus confounding\nassertion with denial. If, on the other hand, it is to be taken in\nextension, this involves the awkwardness of supposing that the\npredicative power of a term resides in its extensive capacity.\n\n\u00a7 194. We therefore recognise predication as being of two\nkinds--affirmation and negation--corresponding to which there are two\nforms of copula.\n\n\u00a7 195. On the other hand, other logicians have maintained that there\nare many kinds of copula, since the copula must vary according to the\nvarious degrees of probability with which we can assert or deny a\npredicate of a subject. This view is technically known as the doctrine\nof\n\n\n\n_The Modality of the Copula._\n\n\n\u00a7 196. It may plausibly be maintained that the division of\npropositions into affirmative and negative is not an exhaustive one,\nsince the result of an act of judgement is not always to lead the mind\nto a clear assertion or a clear denial, but to leave it in more or\nless doubt as to whether the predicate applies to the subject or\nnot. Instead of saying simply A is B, or A is not B, we may be led to\none of the following forms of proposition--\n\n  A is possibly B.\n  A is probably B.\n  A is certainly B.\n\nThe adverbial expression which thus appears to qualify the copula is\nknown as 'the mode.'\n\n\u00a7 197. When we say 'The accused may be guilty' we have a proposition\nof very different force from 'The accused is guilty,' and yet the\nterms appear to be the same. Wherein then does the difference lie? 'In\nthe copula' would seem to be the obvious reply. We seem therefore\ndriven to admit that there are as many different kinds of copula as\nthere are different degrees of assurance with which a statement may be\nmade.\n\n\u00a7 198. But there is another way in which modal propositions may be\nregarded. Instead of the mode being attached to the copula, it may be\nconsidered as itself constituting the predicate, so that the above\npropositions would be analysed thus--\n\n  That A is B, is possible.\n  That A is B, is probable.\n  That A is B, is certain.\n\n\u00a7 199. The subject here is itself a proposition of which we predicate\nvarious degrees of probability. In this way the division of\npropositions into affirmative and negative is rendered exhaustive. For\nwherever before we had a doubtful assertion, we have now an assertion\nof doubtfulness.\n\n\u00a7 200. If degrees of probability can thus be eliminated from the\ncopula, much more so can expressions of time, which may always be\nregarded as forming part of the predicate. 'The sun will rise\nto-morrow' may be analysed into 'The sun is going to rise to-morrow.'\nIn either case the tense belongs equally to the predicate. It is often\nan awkward task so to analyse propositions relative to past or future\ntime as to bring out the copula under the form 'is' or 'is not': but\nfortunately there is no necessity for so doing, since, as has been\nsaid before (\u00a7 188), the material form of the copula is a matter of\nindifference to logic. Indeed in affirmative propositions the mere\njuxtaposition of the subject and predicate is often sufficient to\nindicate their agreement, e.g. 'Most haste, worst speed,' chalepha\ntha kala. It is because all propositions are not affirmative that we\nrequire a copula at all. Moreover the awkwardness of expression just\nalluded to is a mere accident of language. In Latin we may say with\nequal propriety 'Sol orietur cras' or 'Sol est oriturus cras'; while\npast time may also be expressed in the analytic form in the case of\ndeponent verbs, as 'Caesar est in Galliam profectus'--'Caesar is gone\ninto Gaul.'\n\n\u00a7 201. The copula then may always be regarded as pure, that is, as\nindicating mere agreement or disagreement between the two terms of the\nproposition.\n\n\n\n\nCHAPTER III.\n\n_Of the Divisions of Propositions_.\n\n\n\u00a7 202. The most obvious and the most important division of\npropositions is into true and false, but with this we are not\nconcerned. Formal logic can recognise no difference between true and\nfalse propositions. The one is represented by the same symbols as the\nother.\n\n\u00a7 203. We may notice, however, in passing, that truth and falsehood\nare attributes of propositions and of propositions only. For something\nmust be predicated, i.e.  asserted or denied, before we can have\neither truth or falsehood. Neither concepts or terms, on the one hand,\nnor reasonings, on the other, can properly be said to be true or\nfalse. In the mere notion of a Centaur or of a black swan there is\nneither truth nor falsehood; it is not until we make some statement\nabout these things, such as that 'black swans are found in Australia,'\nor 'I met a Centaur in the High Street yesterday,' that the question\nof truth or falsehood comes in. In such expressions as a 'true friend'\nor 'a false patriot' there is a tacit reference to propositions. We\nmean persons of whom the terms 'friend' and 'patriot' are truly or\nfalsely predicated.  Neither can we with any propriety talk of true or\nfalse reasoning. Reasoning is either valid or invalid: it is only the\npremisses of our reasonings, which are propositions, that can be true\nor false. We may have a perfectly valid process of reasoning which\nstarts from a false assumption and lands us in a false conclusion.\n\n\u00a7 204. All truth and falsehood then are contained in propositions; and\npropositions are divided according to the Quality of the Matter into\ntrue and false. But the consideration of the matter is outside the\nsphere of formal or deductive Logic. It is the problem of inductive\nlogic to establish, if possible, a criterion of evidence whereby the\ntruth or falsehood of propositions may be judged (\u00a7 2).\n\n\u00a7 205. Another usual division of propositions is into Pure and Modal,\nthe latter being those in which the copula is modified by some degree\nof probability. This division is excluded by the view which has just\nbeen taken of the copula, as being always simply affirmative or simply\nnegative.\n\n\u00a7 206. We are left then with the following divisions of\npropositions--\n\n  Proposition\n    according to Form\n      Simple\n\n      Complex\n        Conjunctive\n\tDisjunctive\n\n      Universal\n        Singular\n\tGeneral\n\n    according to Matter\n      Verbal\n      Real\n\n    according to Quantity\n      Universal\n        Singular\n\tGeneral\n\n      Particular\n        Indefinite\n\t(strictly) Particular\n\n    according to Quality\n      Affirmative\n      Negative\n\n\n\n_Simple and Complex Propositions_.\n\n\n\u00a7 207. A Simple Proposition is one in which a predicate is directly\naffirmed or denied of a subject, e.g. 'Rain is falling.'\n\n\u00a7 208. A simple proposition is otherwise known as Categorical.\n\n\u00a7 209. A Complex Proposition is one in which a statement is made\nsubject to some condition, e.g. 'If the wind drops, rain will fall.'\n\n\u00a7 210. Hence the complex proposition is also known as Conditional.\n\n\u00a7 211. Every complex proposition consists of two parts--\n\n  (1) Antecedent;\n\n  (2) Consequent.\n\n\u00a7 212. The Antecedent is the condition on which another statement is\nmade to depend. It precedes the other in the order of thought, but may\neither precede or follow it in the order of language. Thus we may say\nindifferently--'If the wind drops, we shall have rain' or 'We shall\nhave rain, if the wind drops.'\n\n\u00a7 213. The Consequent is the statement which is made subject to some\ncondition.\n\n\u00a7 214. The complex proposition assumes two forms,\n\n  (1) If A is B, C is D.\n\nThis is known as the Conjunctive or Hypothetical proposition.\n\n  (2) Either A is B or C is D.\n\nThis is known as the Disjunctive proposition.\n\n\u00a7 215. The disjunctive proposition may also appear in\nthe form\n\n  A is either B or C,\n\nwhich is equivalent to saying\n\n  Either A is B or A is C;\n\nor again in the form\n\n  Either A or B is C,\n\nwhich is equivalent to saying\n\n  Either A is C or B is C.\n\n\u00a7 216. As the double nomenclature may cause some confusion, a scheme\nis appended.\n\n                 Proposition\n         ____________|_____________\n        |                          |\n      Simple                    Complex\n  (Categorical)               (Conditional)\n                        ___________|__________\n                       |                      |\n                  Conjunctive            Disjunctive.\n                 (Hypothetical)\n\n\u00a7 217. The first set of names is preferable. 'Categorical' properly\nmeans 'predicable' and 'hypothetical' is a mere synonym for\n'conditional.'\n\n\u00a7 218. Let us examine now what is the real nature of the statement\nwhich is made in the complex form of proposition. When, for instance,\nwe say 'If the sky falls, we shall catch larks,' what is it that we\nreally mean to assert?  Not that the sky will fall, and not that we\nshall catch larks, but a certain connection between the two, namely,\nthat the truth of the antecedent involves the truth of the\nconsequent. This is why this form of proposition is called\n'conjunctive,' because in it the truth of the consequent is conjoined\nto the truth of the antecedent.\n\n\u00a7 219. Again, when we say 'Jones is either a knave or a fool,' what is\nreally meant to be asserted is--'If you do not find Jones to be a\nknave, you may be sure that he is a fool.' Here it is the falsity of\nthe antecedent which involves the truth of the consequent; and the\nproposition is known as 'disjunctive,' because the truth of the\nconsequent is disjoined from the truth of the antecedent.\n\n\u00a7 220. Complex propositions then turn out to be propositions about\npropositions, that is, of which the subject and predicate are\nthemselves propositions. But the nature of a proposition never varies\nin thought. Ultimately every proposition must assume the form 'A is,\nor is not, B.'  'If the sky falls, we shall catch larks' may be\ncompressed into 'Sky-falling is lark-catching.'\n\n\u00a7 221. Hence this division turns upon the form of expression, and may\nbe said to be founded on the simplicity or complexity of the terms\nemployed in a proposition.\n\n\u00a7 222. In the complex proposition there appears to be more than one\nsubject or predicate or both, but in reality there is only a single\nstatement; and this statement refers, as we have Seen, to a certain\nconnection between two propositions.\n\n\u00a7 223. If there were logically, and not merely grammatically, more\nthan one subject or predicate, there would be more than one\nproposition. Thus when we say 'The Jews and Carthaginians were Semitic\npeoples and spoke a Semitic language,' we have four propositions\ncompressed into a single sentence for the sake of brevity.\n\n\u00a7 224. On the other hand when we say 'Either the Carthaginians were of\nSemitic origin or argument from language is of no value in ethnology,'\nwe have two propositions only in appearance.\n\n\u00a7 225. The complex proposition then must be distinguished from those\ncontrivances of language for abbreviating expression in which several\ndistinct statements are combined into a single sentence.\n\n\n\n_Verbal and Real Propositions_.\n\n\n\u00a7 226. A Verbal Proposition is one which states nothing more about the\nsubject than is contained in its definition, e.g. 'Man is an animal';\n'Men are rational beings.'\n\n\u00a7 227. A Real Proposition states some fact not contained in the\ndefinition of the subject, e.g. 'Some animals have four feet.'\n\n\u00a7 228. It will be seen that the distinction between verbal and real\npropositions assumes a knowledge of the precise meaning of terms, that\nis to say, a knowledge of definitions.\n\n\u00a7 229. To a person who does not know the meaning of terms a verbal\nproposition will convey as much information as a real one. To say 'The\nsun is in mid-heaven at noon,' though a merely verbal proposition,\nwill convey information to a person who is being taught to attach a\nmeaning to the word 'noon.' We use so many terms without knowing their\nmeaning, that a merely verbal proposition appears a revelation to many\nminds. Thus there are people who are surprised to hear that the lion\nis a cat, though in its definition 'lion' is referred to the class\n'cat.' The reason of this is that we know material objects far better\nin their extension than in their intension, that is to say, we know\nwhat things a name applies to without knowing the attributes which\nthose things possess in common.\n\n\u00a7 230. There is nothing in the mere look of a proposition to inform us\nwhether it is verbal or real; the difference is wholly relative to,\nand constituted by, the definition of the subject. When we have\naccepted as the definition of a triangle that it is 'a figure\ncontained by three sides,' the statement of the further fact that it\nhas three angles becomes a real proposition. Again the proposition\n'Man is progressive' is a real proposition.  For though his\nprogressiveness is a consequence of his rationality, still there is no\nactual reference to progressiveness contained in the usually accepted\ndefinition, 'Man is a rational animal.'\n\n\u00a7 231. If we were to admit, under the term 'verbal proposition,' all\nstatements which, though not actually contained in the definition of\nthe subject, are implied by it, the whole body of necessary truth\nwould have to be pronounced merely verbal, and the most penetrating\nconclusions of mathematicians set down as only another way of stating\nthe simplest axioms from which they started.  For the propositions of\nwhich necessary truth is composed are so linked together that, given\none, the rest can always follow. But necessary truth, which is arrived\nat 'a priori,' that is, by the mind's own working, is quite as real as\ncontingent truth, which is arrived at 'a posteriori,' or by the\nteachings of experience, in other words, through our own senses or\nthose of others.\n\n\u00a7 232. The process by which real truth, which is other than deductive,\nis arrived at 'a priori' is known as Intuition.  E.g. The mind sees\nthat what has three sides cannot but have three angles.\n\n\u00a7 233. Only such propositions then must be considered verbal as state\nfacts expressly mentioned in the definition.\n\n\u00a7 234. Strictly speaking, the division of propositions into verbal and\nreal is extraneous to our subject: since it is not the province of\nlogic to acquaint us with the content of definitions.\n\n\u00a7 235, The same distinction as between verbal and real proposition, is\nconveyed by the expressions 'Analytical' and 'Synthetical,' or\n'Explicative' and 'Ampliative' judgements.\n\n\u00a7 236. A verbal proposition is called analytical, as breaking up the\nsubject into its component notions.\n\n\u00a7 237. A real proposition is called synthetical, as attaching some new\nnotion to the subject.\n\n\u00a7 238. Among the scholastic logicians verbal propositions were known\nas 'Essential,' because what was stated in the definition was\nconsidered to be of the essence of the subject, while real\npropositions were known as 'Accidental.'\n\n\n\n_Universal AND PARTICULAR Propositions_.\n\n\n\u00a7 239. A Universal proposition is one in which it is evident from the\nform that the predicate applies to the subject in its whole extent.\n\n\u00a7 240. When the predicate does not apply to the subject in its whole\nextent, or when it is not clear that it does so, the proposition is\ncalled Particular.\n\n\u00a7 241. To say that a predicate applies to a subject in its whole\nextent, is to say that it is asserted or denied of all the things of\nwhich the subject is a name.\n\n\u00a7 242. 'All men are mortal' is a universal proposition.\n\n\u00a7 243. 'Some men are black' is a particular proposition. So also is\n'Men are fallible;' for here it is not clear from the form whether\n'all' or only 'some' is meant.\n\n\u00a7 244. The latter kind of proposition is known as Indefinite, and must\nbe distinguished from the particular proposition strictly so called,\nin which the predicate applies to part only of the subject.\n\n\u00a7 245. The division into universal and particular is founded on the\nQuantity of propositions.\n\n\u00a7 246. The quantity of a proposition is determined by the quantity in\nextension of its subject.\n\n\u00a7 247. Very often the matter of an indefinite proposition is such as\nclearly to indicate to us its quantity.  When, for instance, we say\n'Metals are elements,' we are understood to be referring to all\nmetals; and the same thing holds true of scientific statements in\ngeneral.  Formal logic, however, cannot take account of the matter of\npropositions; and is therefore obliged to set down all indefinite\npropositions as particular, since it is not evident from the form that\nthey are universal.\n\n\u00a7 248. Particular propositions, therefore, are sub-divided into such\nas are Indefinite and such as are Particular, in the strict sense of\nthe term.\n\n\u00a7 249. We must now examine the sub-division of universal propositions\ninto Singular and General.\n\n\u00a7 250. A Singular proposition is one which has a singular term for its\nsubject, e.g. 'Virtue is beautiful.'\n\n\u00a7 251. A General proposition is one which has for its subject a common\nterm taken in its whole extent.\n\n\u00a7 252. Now when we say 'John is a man' or 'This table is oblong,' the\nproposition is quite as universal, in the sense of the predicate\napplying to the whole of the subject, as when we say 'All men are\nmortal.' For since a singular term applies only to one thing, we\ncannot avoid using it in its whole extent, if we use it at all.\n\n\u00a7 253. The most usual signs of generality in a proposition are the\nwords 'all,' 'every,' 'each,' in affirmative, and the words 'no,'\n'none,' 'not one,' &c. in negative propositions.\n\n\u00a7 254. The terminology of the division of propositions according to\nquantity is unsatisfactory. Not only has the indefinite proposition to\nbe set down as particular, even when the sense manifestly declares it\nto be universal; but the proposition which is expressed in a\nparticular form has also to be construed as indefinite, _so_ that\nan unnatural meaning is imparted to the word 'some,' as used in\nlogic. If in common conversation we were to say 'Some cows chew the\ncud,' the person whom we were addressing would doubtless imagine us to\nsuppose that there were some cows which did not possess this\nattribute. But in logic the word 'some' is not held to express more\nthan 'some at least, if not all.' Hence we find not only that an\nindefinite proposition may, as a matter of fact, be strictly\nparticular, but that a proposition which appears to be strictly\nparticular may be indefinite.  So a proposition expressed in precisely\nthe same form 'Some A is B' may be either strictly particular, if some\nbe taken to exclude all, or indefinite, if the word 'some' does not\nexclude the possibility of the statement being true of all. It is\nevident that the term 'particular' has become distorted from its\noriginal meaning. It would naturally lead us to infer that a statement\nis limited to part of the subject, whereas, by its being opposed to\nuniversal, in the sense in which that term has been defined, it can\nonly mean that we have nothing to show us whether part or the whole is\nspoken of.\n\n\u00a7 255. This awkwardness of expression is due to the indefinite\nproposition having been displaced from its proper position. Formerly\npropositions were divided under three heads--\n\n  (1) Universal,\n\n  (2) Particular,\n\n  (3) Indefinite.\n\nBut logicians anxious for simplification asked, whether a predicate in\nany given case must not either apply to the whole of the subject or\nnot? And whether, therefore, the third head of indefinite propositions\nwere not as superfluous as the so-called 'common gender' of nouns in\ngrammar?\n\n\u00a7 256. It is quite true that, as a matter of fact, any given predicate\nmust either apply to the whole of the subject or not, so that in the\nnature of things there is no middle course between universal and\nparticular. But the important point is that we may not know whether\nthe predicate applies to the whole of the subject or not.  The primary\ndivision then should be into propositions whose quantity is known and\npropositions whose quantity is unknown. Those propositions whose\nquantity is known may be sub-divided into 'definitely universal' and\n'definitely particular,' while all those whose quantity is unknown are\nclassed together under the term 'indefinite.'  Hence the proper\ndivision is as follows--\n\n                  Proposition\n             __________|____________\n            |                       |\n         Definite              Indefinite\n       _____|_______\n      |             |\n  Universal    Particular.\n\n\u00a7 257. Another very obvious defeat of terminology is that the word\n'universal' is naturally opposed to 'singular,' whereas it is here so\nused as to include it; while, on the other hand, there is no obvious\ndifference between universal and general, though in the division the\nlatter is distinguished from the former as species from genus.\n\n\n\n_Affirmative and Negative Propositions._\n\n\n\u00a7 258. This division rests upon the Quality of propositions.\n\n\u00a7 259. It is the quality of the form to be affirmative or negative:\nthe quality of the matter, as we saw before (\u00a7 204), is to be true or\nfalse. But since formal logic takes no account of the matter of\nthought, when we speak of 'quality' we are understood to mean the\nquality of the form.\n\n\u00a7 260. By combining the division of propositions\naccording to quantity with the division according to quality,\nwe obtain four kinds of proposition, namely--\n\n  (1) Universal Affirmative (A).\n\n  (2) Universal Negative (E).\n\n  (3) Particular Affirmative (I).\n\n  (4) Particular Negative (O).\n\n\u00a7 261. This is an exhaustive classification of propositions, and any\nproposition, no matter what its form may be, must fall under one or\nother of these four heads.  For every proposition must be either\nuniversal or particular, in the sense that the subject must either be\nknown to be used in its whole extent or not; and any proposition,\nwhether universal or particular, must be either affirmative or\nnegative, for by denying modality to the copula we have excluded\neverything intermediate between downright assertion and denial. This\nclassification therefore may be regarded as a Procrustes' bed, into\nwhich every proposition is bound to fit at its proper peril.\n\n\u00a7 262. These four kinds of propositions are represented respectively\nby the symbols A, E, I, O.\n\n\u00a7 263. The vowels A and I, which denote the two affirmatives, occur in\nthe Latin words 'affirmo' and 'aio;' E and O, which denote the two\nnegatives, occur in the Latin word 'nego.'\n\n\n\n_Extensive and Intensive Propositions._\n\n\n\u00a7 264. It is important to notice the difference between Extensive and\nIntensive propositions; but this is not a division of propositions,\nbut a distinction as to our way of regarding them. Propositions may be\nread either in extension or intension. Thus when we say 'All cows are\nruminants,' we may mean that the class, cow, is contained in the\nlarger class, ruminant. This is reading the proposition in\nextension. Or we may mean that the attribute of chewing the cud is\ncontained in, or accompanies, the attributes which make up our idea of\n'cow.' This is reading the proposition in intension. What, as a matter\nof fact, we do mean, is a mixture of the two, namely, that the class,\ncow, has the attribute of chewing the cud. For in the ordinary and\nnatural form of proposition the subject is used in extension, and the\npredicate in intension, that is to say, when we use a subject, we are\nthinking of certain objects, whereas when we use a predicate, we\nindicate the possession of certain attributes. The predicate, however,\nneed not always be used in intension, e.g. in the proposition 'His\nname is John' the predicate is not intended to convey the idea of any\nattributes at all. What is meant to be asserted is that the name of\nthe person in question is that particular name, John, and not\nZacharias or Abinadab or any other name that might be given him.\n\n\u00a7 265. Let it be noticed that when a proposition is read in extension,\nthe predicate contains the subject, whereas, when it is read in\nintension, the subject contains the predicate.\n\n\n\n_Exclusive Propositions._\n\n\n\u00a7 266. An Exclusive Proposition is so called because in it all but a\ngiven subject is excluded from participation in a given predicate,\ne.g. 'The good alone are happy,' 'None but the brave deserve the\nfair,' 'No one except yourself would have done this.'\n\n\u00a7 267. By the above forms of expression the predicate is declared to\napply to a given subject and to that subject only. Hence an exclusive\nproposition is really equivalent to two propositions, one affirmative\nand one negative.  The first of the above propositions, for instance,\nmeans that some of the good are happy, and that no one else is so. It\ndoes not necessarily mean that all the good are happy, but asserts\nthat among the good will be found all the happy. It is therefore\nequivalent to saying that all the happy are good, only that it puts\nprominently forward in addition what is otherwise a latent consequence\nof that assertion, namely, that some at least of the good are happy.\n\n\u00a7 268. Logically expressed the exclusive proposition when universal\nassumes the form of an E proposition, with a negative term for its\nsubject\n\n  No not-A is B.\n\n\u00a7 269. Under the head of exclusive comes the strictly particular\nproposition, 'Some A is B,' which implies at the same time that 'Some\nA is not B.' Here 'some' is understood to mean 'some only,' which is\nthe meaning that it usually bears in common language. When, for\ninstance, we say 'Some of the gates into the park are closed at\nnightfall,' we are understood to mean 'Some are left open.'\n\n\n\n_Exceptive Propositions._\n\n\n\u00a7 270. An Exceptive Proposition is so called as affirming the\npredicate of the whole of the subject, with the exception of a certain\npart, e.g. 'All the jury, except two, condemned the prisoner.'\n\n\u00a7 271. This form of proposition again involves two distinct\nstatements, one negative and one affirmative, being equivalent to 'Two\nof the jury did not condemn the prisoner; and all the rest did.'\n\n\u00a7 272. The exceptive proposition is merely an affirmative way of\nstating the exclusive--\n\n  No not-A is B = All not-A is not-B.\n\n  No one but the sage is sane = All except the sage are mad.\n\n\n\n_Tautologous or Identical Propositions_\n\n\n\u00a7 273. A Tautologous or Identical proposition affirms the subject of\nitself, e.g. 'A man's a man,' 'What I have written, I have written,'\n'Whatever is, is.' The second of these instances amounts formally to\nsaying 'The thing that I have written is the thing that I have\nwritten,' though of course the implication is that the writing will\nnot be altered.\n\n\n\n\nCHAPTER IV.\n\n_Of the Distribution of Terms._\n\n\n\u00a7 274. The treatment of this subject falls under the second part of\nlogic, since distribution is not an attribute of terms in themselves,\nbut one which they acquire in predication.\n\n\u00a7 275. A term is said to be distributed when it is known to be used in\nits whole extent, that is, with reference to all the things of which\nit is a name. When it is not so used, or is not known to be so used,\nit is called undistributed.\n\n\u00a7 276. When we say 'All men are mortal,' the subject is distributed,\nsince it is apparent from the form of the expression that it is used\nin its whole extent. But when we say 'Men are miserable' or 'Some men\nare black,' the subject is undistributed.\n\n\u00a7 277. There is the same ambiguity attaching to the term\n'undistributed' which we found to underlie the use of the term\n'particular.' 'Undistributed' is applied both to a term whose quantity\nis undefined, and to one whose quantity is definitely limited to a\npart of its possible extent.\n\n\u00a7 278. This awkwardness arises from not inquiring first whether the\nquantity of a term is determined or undetermined, and afterwards\nproceeding to inquire, whether it is determined as a whole or part of\nits possible extent.  As it is, to say that a term is distributed,\ninvolves two distinct statements--\n\n  (1) That its quantity is known;\n\n  (2) That its quantity is the greatest possible.\n\nThe term 'undistributed' serves sometimes to contradict one of these\nstatements and sometimes to contradict the other.\n\n\u00a7 279. With regard to the quantity of the subject of a proposition no\ndifficulty can arise. The use of the words 'all' or 'some,' or of a\nvariety of equivalent expressions, mark the subject as being\ndistributed or undistributed respectively, while, if there be nothing\nto mark the quantity, the subject is for that reason reckoned\nundistributed.\n\n\u00a7 280. With regard to the predicate more difficulty may arise.\n\n\u00a7 281. It has been laid down already that, in the ordinary form of\nproposition, the subject is used in extension and the predicate in\nintension. Let us illustrate the meaning of this by an example. If\nsomeone were to say 'Cows are ruminants,' you would have a right to\nask him whether he meant 'all cows' or only 'some.' You would not by\nso doing be asking for fresh information, but merely for a more\ndistinct explanation of the statement already made. The subject being\nused in extension naturally assumes the form of the whole or part of a\nclass. But, if you were to ask the same person 'Do you mean that cows\nare all the ruminants that there are, or only some of them?' he would\nhave a right to complain of the question, and might fairly reply, 'I\ndid not mean either one or the other; I was not thinking of ruminants\nas a class. I wished merely to assert an attribute of cows; in fact, I\nmeant no more than that cows chew the cud.'\n\n\u00a7 282. Since therefore a predicate is not used in extension at all, it\ncannot possibly be known whether it is used in its whole extent or\nnot.\n\n\u00a7 283. It would appear then that every predicate is necessarily\nundistributed; and this consequence does follow in the case of\naffirmative propositions.\n\n\u00a7 284. In a negative proposition, however, the predicate, though still\nused in intension, must be regarded as distributed. This arises from\nthe nature of a negative proposition. For we must remember that in any\nproposition, although the predicate be not meant in extension, it\nalways admits of being so read. Now we cannot exclude one class from\nanother without at the same time wholly excluding that other from the\nformer. To take an example, when we say 'No horses are ruminants,' the\nmeaning we really wish to convey is that no member of the class,\nhorse, has a particular attribute, namely, that of chewing the\ncud. But the proposition admits of being read in another form, namely,\n'That no member of the class, horse, is a member of the class,\nruminant.' For by excluding a class from the possession of a given\nattribute, we inevitably exclude at the same time any class of things\nwhich possess that attribute from the former class.\n\n\u00a7 285. The difference between the use of a predicate in an affirmative\nand in a negative proposition may be illustrated to the eye as\nfollows. To say 'All A is B' may mean either that A is included in B\nor that A and B are exactly co-extensive.\n\n[Illustration]\n\n\u00a7 286. As we cannot be sure which of these two relations of A to B is\nmeant, the predicate B has to be reckoned undistributed, since a term\nis held to be distributed only when we know that it is used in its\nwhole extent.\n\n\u00a7 287. To say 'No A is B,' however, is to say that A falls wholly\noutside of B, which involves the consequence that B falls wholly\noutside of A.\n\n[Illustration]\n\n\u00a7 288. Let us now apply the same mode of illustration to the\nparticular forms of proposition.\n\n\u00a7 289. If I be taken in the strictly particular sense, there are, from\nthe point of view of extension, two things which may be meant when we\nsay 'Some A is B'--\n\n  (1) That A and B are two classes which overlap one another, that is\n  to say, have some members in common, e.g. 'Some cats are black.'\n\n  [Illustration]\n\n  (2) That B is wholly contained in A, which is an inverted way of\n  saying that all B is A, e.g. 'Some animals are men.'\n\n  [Illustration]\n\n\u00a7 290. Since we cannot be sure which of these two is meant, the\npredicate is again reckoned undistributed.\n\n\u00a7 291. If on the other hand 1 be taken in an indefinite sense, so as\nto admit the possibility of the universal being true, then the two\ndiagrams which have already been used for A must be extended to 1, in\naddition to its own, together with the remarks which we made in\nconnection with them (\u00a7\u00a7 285-6).\n\n\u00a7 292. Again, when we say 'Some A is not B,' we mean that some, if not\nthe whole of A, is excluded from the possession of the attribute B. In\neither case the things which possess the attribute B are wholly\nexcluded either from a particular part or from the whole of A.  The\npredicate therefore is distributed.\n\n[Illustration]\n\nFrom the above considerations we elicit the following--\n\n\u00a7 293. Four Rules for the Distribution of Terms.\n\n (1) All universal propositions distribute their subject.\n\n (2) No particular propositions distribute their subject,\n\n (3) All negative propositions distribute their predicate.\n\n (4) No affirmative propositions distribute their predicate.\n\n\n\u00a7 294. The question of the distribution or non-distribution of the\nsubject turns upon the quantity of the proposition, whether universal\nor particular; the question of the distribution or non-distribution of\nthe predicate turns upon the quality of the proposition, whether\naffirmative or negative.\n\n\n\n\nCHAPTER V.\n\n_Of the Quantification of the Predicate._\n\n\n\u00a7 295. The rules that have been given for the distribution of terms,\ntogether with the fourfold division of propositions into A, E, 1, 0,\nare based on the assumption that it is the distribution or\nnon-distribution of the subject only that needs to be taken into\naccount in estimating the quantity of a proposition.\n\n\u00a7 296. But some logicians have maintained that the predicate, though\nseldom quantified in expression, must always be quantified in\nthought--in other words, that when we say, for instance, 'All A is B,'\nwe must mean either that 'All A is all B' or only that 'All A is some\nB.'\n\n\u00a7 297. If this were so, it is plain that the number of possible\npropositions would be exactly doubled, and that, instead of four\nforms, we should now have to recognise eight, which may be expressed\nas follows--\n\n  1. All A is all B. ([upsilon]).\n\n  2. All A is some B. ([Lambda]).\n\n  3. No A is any B. ([Epsilon]).\n\n  4. No A is some B. ([eta]).\n\n  5. Some A is all B. ([Upsilon]).\n\n  6. Some A is some B. ([Iota]).\n\n  7. Some A is not any B. ([Omega]).\n\n  8. Some A is not some B. ([omega]).\n\n\u00a7 298. It is evident that it is the second of the above propositions\nwhich represents the original A, in accordance with the rule that 'No\naffirmative propositions distribute their predicate' (\u00a7 293).\n\n\u00a7 299. The third represents the original E, in accordance with the\nrule that 'All negative propositions distribute their predicate.'\n\n\u00a7 300. The sixth represents the original I, in accordance with the\nrule that 'No affirmative propositions distribute their predicate.'\n\n\u00a7 301. The seventh represents the original O, in accordance with the\nrule that 'All negative propositions distribute their predicate.'\n\n\u00a7 302. Four new symbols are required, if the quantity of the predicate\nas well as that of the subject be taken into account in the\nclassification of propositions. These have been supplied, somewhat\nfancifully, as follows--\n\n\u00a7 303. The first, 'All A is all B,' which distributes both subject and\npredicate, has been called [upsilon], to mark its extreme\nuniversality.\n\n\u00a7 304. The fourth, 'No A is some B,' is contained in E, and has\ntherefore been denoted by the symbol [eta], to show its connection\nwith E.\n\n\u00a7 305. The fifth, 'Some A is all B,' is the exact converse of the\nsecond, 'All A is some B,' and has therefore been denoted by the\nsymbol [Upsilon], which resembles an inverted A.\n\n\u00a7 306. The eighth is contained in O, as part in whole, and has\ntherefore had assigned to it the symbol [omega],\n\n\u00a7 307. The attempt to take the predicate in extension, instead of, as\nit should naturally be taken, in intension, leads to some curious\nresults. Let us take, for instance, the u proposition. Either the sign\nof quantity 'all' must be understood as forming part of the predicate\nor not.  If it is not, then the u proposition 'All A is all B' seems\nto contain within itself, not one proposition, but two, namely, 'All A\nis B' and 'All B is A.' But if on the other hand 'all' is understood\nto form part of the predicate, then u is not really a general but a\nsingular proposition. When we say, 'All men are rational animals,' we\nhave a true general proposition, because the predicate applies to the\nsubject distributively, and not collectively.  What we mean is that\n'rational animal' may be affirmed of every individual in the class,\nman. But when we say 'All men are all rational animals,' the predicate\nno longer applies to the subject distributively, but only\ncollectively.  For it is obvious that 'all rational animals' cannot be\naffirmed of every individual in the class, man. What the proposition\nmeans is that the class, man, is co-extensive with the class, rational\nanimal. The same meaning may be expressed intensively by saying that\nthe one class has the attribute of co-extension with the other.\n\n\u00a7 308. Under the head o u come all propositions in which both subject\nand predicate are singular terms, e.g. 'Homer was the author of the\nIliad,' 'Virtue is the way to happiness.'\n\n\u00a7 309. The proposition [eta] conveys very little information to the\nmind. 'No A is some B' is compatible with the A proposition in the\nsame matter. 'No men are some animals' may be true, while at the same\ntime it is true that 'All men are animals.' No men, for instance, are\nthe particular animals known as kangaroos.\n\n\u00a7 310. The [omega] proposition conveys still less information than the\n[eta]. For [omega] is compatible, not only with A, but with\n[upsilon]. Even though 'All men are all rational animals,' it is still\ntrue that 'Some men are not some rational animals': for no given human\nbeing is the same rational animal as any other.\n\n\u00a7 311. Nay, even when the [upsilon] is an identical proposition,\n[omega] will still hold in the same matter. 'All rational animals are\nall rational animals': but, for all that, 'Some rational animals are\nnot some others.' This last form of proposition therefore is almost\nwholly devoid of meaning.\n\n\u00a7 312. The chief advantage claimed for the quantification of the\npredicate is that it reduces every affirmative proposition to an exact\nequation between its subject and predicate. As a consequence every\nproposition would admit of simple conversion, that is to say, of\nhaving the subject and predicate transposed without any further change\nin the proposition. The forms also of Reduction (a term which will be\nexplained later on) would be simplified; and generally the\nintroduction of the quantified predicate into logic might be attended\nwith certain mechanical advantages. The object of the logician,\nhowever, is not to invent an ingenious system, but to arrive at a true\nanalysis of thought. Now, if it be admitted that in the ordinary form\nof proposition the subject is used in extension and the predicate in\nintension, the ground for the doctrine is at once cut away. For, if\nthe predicate be not used in its extensive capacity at all, we plainly\ncannot be called upon to determine whether it is used in its whole\nextent or not.\n\n\n\n\nCHAPTER VI.\n\n_Of the Heads of Predicables_.\n\n\n\u00a7 313. A predicate is something which is stated of a subject.\n\n\u00a7 314. A predicable is something which can be stated of a subject.\n\n\u00a7 315. The Heads of Predicables are a classification of the various\nthings which can be stated of a subject, viewed in their relation to\nit.\n\n\u00a7 316. The treatment of this topic, therefore, as it involves the\nrelation of a predicate to a subject, manifestly falls under the\nsecond part of logic, which deals with the proposition. It is\nsometimes treated under the first part of logic, as though the heads\nof predicables were a classification of universal notions, i.e. common\nterms, in relation to one another, without reference to their place in\nthe proposition.\n\n\u00a7 317. The heads of predicables are commonly reckoned\nas five, namely,\n\n  (1) Genus.\n\n  (2) Species.\n\n  (3) Difference.\n\n  (4) Property.\n\n  (5) Accident.\n\n\u00a7 318. We will first define these terms in the sense in which they are\nnow used, and afterwards examine the principle on which the\nclassification is founded and the sense in which they were originally\nintended.\n\n  (1) A Genus is a larger class containing under it smaller\n  classes. Animal is a genus in relation to man and brute.\n\n  (2) A Species is a smaller class contained under a larger one. Man\n  is a species in relation to animal.\n\n  (3) Difference is the attribute, or attributes, which distinguish\n  one species from others contained under the same genus. Rationality\n  is the attribute which distinguishes the species, man, from the\n  species, brute.\n\n  N.B. The genus and the difference together make up the Definition of\n  a class-name, or common term.\n\n  (4) A Property is an attribute which is not contained in the\n  definition of a term, but which flows from it.\n\n  A Generic Property is one which flows from the genus.\n\n  A Specific Property is one which flows from the difference.\n\n  It is a generic property of man that he is mortal, which is a\n  consequence of his animality.  It is a specific property of man that\n  he is progressive, which is a consequence of his rationality.\n\n  (5) An Accident is an attribute, which is neither contained in the\n  definition, nor flows from it.\n\n\u00a7 319. Accidents are either Separable or Inseparable.\n\nA Separable Accident is one which belongs only to some members of a\nclass.\n\nAn Inseparable Accident is one which belongs to all the members of a\nclass.\n\nBlackness is a separable accident of man, an inseparable accident of\ncoals.\n\n\u00a7 320. The attributes which belong to anything may be distinguished\nbroadly under the two heads of essential and non-essential, or\naccidental. By the essential attributes of anything are meant those\nwhich are contained in, or which flow from, the definition. Now it may\nbe questioned whether there can, in the nature of things, be such a\nthing as an inseparable accident. For if an attribute were found to\nbelong invariably to all the members of a class, we should suspect\nthat there was some causal connection between it and the attributes\nwhich constitute the definition, that is, we should suspect the\nattribute in question to be essential and not accidental. Nevertheless\nthe term 'inseparable accident' may be retained as a cloak for our\nignorance, whenever it is found that an attribute does, as a matter of\nfact, belong to all the members of a class, without there being any\napparent reason why it should do so. It has been observed that animals\nwhich have horns chew the cud. As no one can adduce any reason why\nanimals that have horns should chew the cud any more than animals\nwhich have not, we may call the fact of chewing the cud an inseparable\naccident of horned animals.\n\n\u00a7 321. The distinction between separable and inseparable accidents is\nsometimes extended from classes to individuals.\n\nAn inseparable accident of an individual is one which belongs to him\nat all times. A separable accident of an individual is one which\nbelongs to him at one time and not at another.\n\n\u00a7 322. It is an inseparable accident of an individual that he was born\nat a certain place and on a certain date.  It is a separable accident\nof an individual that he resides at a certain place and is of a\ncertain age.\n\n\u00a7 323. There are some remarks which it may be well to make about the\nabove five terms before we pass on to investigate the principle upon\nwhich the division is based.\n\n\u00a7 324. In the first place, it must of course be borne in mind that\ngenus and species are relative terms. No class in itself can be either\na genus or a species; it only becomes so in reference to some other\nclass, as standing to it in the relation of containing or contained.\n\n\u00a7 325. Again, the distinction between genus and difference on the one\nhand and property on the other is wholly relative to an assumed\ndefinition. When we say 'Man is an animal,' 'Man is rational,' 'Man is\nprogressive,' there is nothing in the nature of these statements\nthemselves to tell us that the predicate is genus, difference, or\nproperty respectively. It is only by a tacit reference to the accepted\ndefinition of man that this becomes evident to us, Similarly, we\ncannot know beforehand that the fact of a triangle having three sides\nis its difference, and the fact of its having three angles a\nproperty. It is only when we assume the definition of a triangle as a\nthree-sided figure that the fact of its having three angles sinks into\na property. Had we chosen to define it, in accordance with its\netymological meaning, as a figure with three angles, its\nthree-sidedness would then have been a mere property, instead of being\nthe difference; for these two attributes are so connected together\nthat, whichever is postulated, the other will necessarily follow.\n\n\u00a7 326. Lastly, it must be noticed that we have not really defined the\nterm 'accident,' not having stated what it is, but only what it is\nnot. It has in fact been reserved as a residual head to cover any\nattribute which is neither a difference nor a property.\n\n\u00a7 327. If the five heads of predicables above given were offered to us\nas an exhaustive classification of the possible relations in which the\npredicate can stand to the subject in a proposition, the first thing\nthat would strike us is that they do not cover the case in which the\npredicate is a singular term. In such a proposition as 'This man is\nJohn,' we have neither a predication of genus or species nor of\nattribute: but merely the identification of one term with another, as\napplying to the same object. Such criticism as this, however, would be\nentirely erroneous, since no singular term was regarded as a\npredicate. A predicable was another name for a universal, the common\nterm being called a predicable in one relation and a universal in\nanother-a predicable, extensively, in so far as it was applicable to\nseveral different things, a universal, intensively, in so far as the\nattributes indicated were implied in several other notions, as the\nattributes indicated by 'animal' are implied in 'horse,' 'sheep,'\n'goat,' &c.\n\n\u00a7 328. It would be less irrelevant to point out how the classification\nbreaks down in relation to the singular term as subject. When, for\ninstance, we say 'Socrates is an animal,' 'Socrates is a man,' there\nis nothing in the proposition to show us whether the predicate is a\ngenus or a species: for we have not here the relation of class to\nclass, which gives us genus or species according to their relative\nextension, but the relation of a class to an individual.\n\n\u00a7 329. Again, when we say\n\n  (1) Some animals are men,\n\n  (2) Some men are black,\n\nwhat is there to tell us that the predicate is to be regarded in the\none case as a species and in the other as an accident of the subject?\nNothing plainly but the assumption of a definition already known.\n\n\u00a7 330. But if this assumption be granted, the classification seems to\nadmit of a more or less complete defense by logic.\n\nFor, given any subject, we can predicate of it either a class or an\nattribute.\n\nWhen the predicate is a class, the term predicated is called a Genus,\nif the subject itself be a class, or a Species, if it be an\nindividual.\n\nWhen, on the other hand, the predicate is an attribute, the attribute\npredicated may be either the very attribute which distinguishes the\nsubject from other members of the same class, in which case it is\ncalled the Difference, or it may be some attribute connected with the\ndefinition, i.e.  Property, or not connected with it, i.e. Accident.\n\n\u00a7 331. These results may be exhibited in the following scheme--\n\n                         Predicate\n             ________________|_________________\n            |                                  |\n          Class                            Attribute\n      ______|_______                 __________|________\n     |              |               |                   |\n  (Subject a     (Subject a       (The              (Not the\n  common          singular    distinguishing     distinguishing\n  term)            term)        Attribute)         attribute)\n  Genus           Species      Difference\n                                    |___________________\n                                    |                   |\n                                (Connected        (Not connected\n                                 with the             with the\n                                 definition)         definition)\n                                Property            Accident\n\n\u00a7 332. The distinction which underlies this division between\npredicating a class and predicating an attribute (in quid or in quale)\nis a perfectly intelligible one, corresponding as it does to the\ngrammatical distinction between the predicate being a noun substantive\nor a noun adjective. Nevertheless it is a somewhat arbitrary one,\nsince, even when the predicate is a class-name, what we are concerned\nto convey to the mind, is the fact that the subject possesses the\nattributes which are connoted by that class-name. We have not here the\ndifference between extensive and intensive predication, since, as we\nhave already seen (\u00a7 264), that is not a difference between one\nproposition and another, but a distinction in our mode of interpreting\nany and every proposition. Whatever proposition we like to take may be\nread either in extension or in intension, according as we fix our\nminds on the fact of inclusion in a class or the fact of the\npossession of attributes.\n\n\u00a7 333. It will be seen that the term 'species,' as it appears in the\nscheme, has a wholly different meaning from the current acceptation in\nwhich it was defined above. Species, in its now accepted meaning,\nsignifies the relation of a smaller class to a larger one: as it was\noriginally intended in the heads of predicables it signifies a class\nin reference to individuals.\n\n\u00a7 334. Another point which requires to be noticed with regard to this\nfive-fold list of heads of predicables, if its object be to classify\nthe relations of a predicate to a subject, is that it takes no account\nof those forms of predication in which class and attribute are\ncombined.  Under which of the five heads would the predicates in the\nfollowing propositions fall?\n\n  (1) Man is a rational animal.\n\n  (2) Man is a featherless biped.\n\nIn the one case we have a combination of genus and difference; in the\nother we have a genus combined with an accident.\n\n\u00a7 335. The list of heads of predicables which we have been discussing\nis not derived from Aristotle, but from the 'Introduction' of\nPorphyry, a Greek commentator who lived more than six centuries later.\n\n\n\n_Aristotle's Heads of Predicables_.\n\n\n\u00a7 336. Aristotle himself, by adopting a different basis of division,\nhas allowed room in his classification for the mixed forms of\npredication above alluded to. His list contains only four heads,\nnamely,\n\n  Genus ([Greek: g\u00e9nos])\n\n  Definition ([Greek: \u00f2rism\u00f3s])\n\n  Proprium ([Greek: \u00eedion])\n\n  Accident ([Greek: sumbebek\u00f3s])\n\n\u00a7 337. Genus here is not distinguished from difference.  Whether we\nsay 'Man is an animal' or 'Man is rational,' we are equally understood\nto be predicating a genus.\n\n\u00a7 338. There is no account taken of species, which, when predicated,\nresolves itself either into genus or accident. When predicated of an\nindividual, it is regarded as a genus, e.g. 'Socrates is a man'; when\npredicated of a class, it is regarded as an accident, e.g.  'Some\nanimals are men.'\n\n\u00a7 339. Aristotle's classification may easily be seen to be\nexhaustive. For every predicate must either be coextensive with its\nsubject or not, i.e. predicable of the same things. And if the two\nterms coincide in extension, the predicate must either coincide also\nin intension with the subject or not.\n\nA predicate which coincides both in extension and intension with its\nsubject is exactly what is meant by a definition. One which coincides\nin extension without coinciding in intension, that is, which applies\nto the same things without expressing the whole meaning, of the\nsubject, is what is known as a Proprium or Peculiar Property.\n\nIf, on the other hand, the two terms are not co-extensive, the\npredicate must either partially coincide in intension with the subject\nor not. [Footnote: The case could not arise of a predicate which was\nentirely coincided in intension with a subject with which it was not\nco-extensive. For, if the extension of the predicate were greater than\nthat of the subject, its intension would be less, and if less,\ngreater, in accordance with the law of inverse variation of the two\nquantities (\u00a7 166).] This is equivalent to saying that it must either\nstate part of the definition of the subject or not. Now the definition\nis made up of genus and difference, either of which may form the\npredicate: but as the two are indistinguishable in relation to a\nsingle subject, they are lumped together for the present purpose under\nthe one head, genus. When the predicate, not being co-extensive, is\nnot even partially co-intensive with its subject, it is called an\nAccident.\n\n\u00a7 340. Proprium, it will be seen, differs from property.  A proprium\nis an attribute which is possessed by all the members of a class, and\nby them alone, e.g. 'Men are the only religious animals.'\n\n\u00a7 341. Under the head of definition must be included all propositions\nin which the predicate is a mere synonym of the subject, e.g. 'Naso is\nOvid,' 'A Hebrew is a Jew,' 'The skipper is the captain.' In such\npropositions the predicate coincides in extension with the subject,\nand may be considered to coincide in intension where the intension of\nboth subject and predicate is at zero, as in the case of two proper\nnames.\n\n\u00a7 342. Designations and descriptions will fall under the head of\n'proprium' or peculiar property, e.g. 'Lord Salisbury is the present\nprime minister of England,' 'Man is a mammal with hands and without\nhair.' For here, while the terms are coincident in extension, they are\nfar from being so in intension.\n\n\u00a7 343. The term 'genus' must be understood to include not only genus\nin the accepted sense, but difference and generic property as well.\n\n\u00a7 344. These results may be exhibited in the following\nscheme--\n\n                                 Predicate\n                  ___________________|______________\n                  |                                |\n            Coextensive with                      not\n              the subject                      coextensive\n          ________|_________                  _____|________\n          |                |                  |            |\n    Co-intensive          not             partially    not at all\n  with the subject   cointensive        cointensive  [Greek: sumbubek\u00f3s]\n  [Greek: \u00f2rism\u00f3s]   [Greek: \u00eedion]   [Greek: g\u00e9nos]       Accident\n    ______|_____     ______|_____________       |________________\n    |          |     |         |        |       |       |       |\n  Defini-  Synonym Designa- Descrip- Peculiar  Genus Differ- Generic\n   tion              tion     tion   Property         ence  Property\n\n\u00a7 345. Thus Aristotle's four heads of predicables may be split up, if\nwe please, into nine--\n\n  1. Definition \\\n                 > [Greek: \u00f2rism\u00f3s].\n  2. Synonym    /\n\n  3. Designation      \\\n                       |\n  4. Description        > [Greek: \u00eedion].\n                       |\n  5. Peculiar Property/\n\n  6. Genus\t     \\\n                      |\n  7. Difference\t       > [Greek: g\u00e9nos].\n                      |\n  8. Generic Property/\n\n  9. Accident--[Greek: sumbebek\u00f3s].\n\n\u00a7 346. We now pass on to the two subjects of Definition and Division,\nthe discussion of which will complete our treatment of the second part\nof logic. Definition and division correspond respectively to the two\nkinds of quantity possessed by terms.\n\nDefinition is unfolding the quantity of a term in intension.\n\nDivision is unfolding the quantity of a term in extension.\n\n\n\n\nCHAPTER VII.\n\n_Of Definition._\n\n\n\u00a7 347. To define a term is to unfold its intension, i.e. to explain\nits meaning.\n\n\u00a7 348. From this it follows that any term which possesses no intension\ncannot be defined.\n\n\u00a7 349. Hence proper names do not admit of definition, except just in\nso far as they do possess some slight degree of intension: Thus we can\ndefine the term 'John' only so far as to say that 'John' is the name\nof a male person.  This is said with regard to the original intension\nof proper names; their acquired intension will be considered later.\n\n\u00a7 350. Again, since definition is unfolding the intension of a term,\nit follows that those terms will not admit of being defined whose\nintension is already so simple that it cannot be unfolded further. Of\nthis nature are names of simple attributes, such as greenness,\nsweetness, pleasure, existence. We know what these things are, but we\ncannot define them. To a man who has never enjoyed sight, no language\ncan convey an idea of the greenness of the grass or the blueness of\nthe sky; and if a person were unaware of the meaning of the term\n'sweetness,' no form of words could convey to him an idea of it. We\nmight put a lump of sugar into his mouth, but that would not be a\nlogical definition.\n\n\u00a7 351. Thus we see that, for a thing to admit of definition, the idea\nof it must be complex. Simple ideas baffle definition, but at the same\ntime do not require it.  In defining we lay out the simpler ideas\nwhich are combined in our notion of something, and so explain that\ncomplex notion. We have defined 'triangle,' when we analyse it into\n'figure' and 'contained by three lines.'  Similarly we have defined\n'substance' when we analyse it into 'thing' and 'which can be\nconceived to exist by itself.'\n\n\u00a7 352. But when we get to 'thing' we have reached a limit. The Summum\nGenus, or highest class under which all things fall, cannot be defined\nany more than a simple attribute; and for the very good reason that it\nconnotes nothing but pure being, which is the simplest of all\nattributes. To say that a thing is an 'object of thought' is not\nreally to define it, but to explain its etymology, and to reclaim a\nphilosophical term from its abuse by popular language, in which it is\nlimited to the concrete and the lifeless. Again, to define it\nnegatively and to say that a thing is 'that which is not nothing' does\nnot carry us any further than we were before. The law of contradiction\nwarrants us in saying as much as that.\n\n\u00a7 353. Definition is confined to subject-terms, and does not properly\nextend to attributives. For definition is of things through names, and\nan attributive out of predication is not the name of anything. The\nattributive is defined, so far as it can be, through the corresponding\nabstract term.\n\n\u00a7 354. Common terms, other than attributives, ought always to admit of\ndefinition. For things are distributed by the mind into classes owing\nto their possessing certain attributes in common, and the definition\nof the class-name can be effected by detailing these attributes, or at\nleast a sufficient number of them.\n\n\u00a7 355. It is different with singular terms. Singular terms, when\nabstract, admit of definition, in so far as they are not names of\nattributes so simple as to evade analysis.  When singular terms are\nconcrete, we have to distinguish between the two cases of proper names\nand designations.  Designations are connotative singular terms. They\nare formed by limiting a common term to the 'case in hand.'  Whatever\ndefinition therefore fits the common term will fit also the\ndesignation which is formed from it, if we add the attributes implied\nby the limitations. Thus whatever definition fits the common term\n'prime minister' will fit also the singular term 'the present prime\nminister of England' by the addition to it of the attributes of place\nand time which are indicated by the expression. Such terms as this\nhave a definite amount of intension, which can therefore be seized\nupon and expounded by a definition.\n\n\u00a7 356. But proper names, having no original intension of their own,\ncannot be defined at all; whereas, if we look upon them from the point\nof view of their acquired intension, they defy definition by reason of\nthe very complexity of their meaning. We cannot say exactly what\n'John' and 'Mary' mean, because those names, to us who know the\nparticular persons denoted by them, suggest all the most trifling\naccidents of the individual as well as the essential attributes of the\ngenus.\n\n\u00a7 357. Definition serves the practical purpose of enabling us mentally\nto distinguish, or, as the name implies, 'mark off' the thing defined\nfrom all other things whatsoever.  This may seem at first an endless\ntask, but there is a short cut by which the goal may be reached. For,\nif we distinguish the thing in hand from the things which it is most\nlike, we shall, 'a fortiori,' have distinguished it from things to\nwhich it bears a less resemblance.\n\n\u00a7 358. Hence the first thing to do in seeking for a definition is to\nfix upon the class into which the thing to be defined most naturally\nfalls, and then to distinguish the thing in question from the other\nmembers of that class.  If we were asked to define a triangle, we\nwould not begin by distinguishing it from a hawser, but from a square\nand other figures with which it is more possible to confound it. The\nclass into which a thing falls is called its Genus, and the attribute\nor attributes which distinguish it from other members of that class\nare called its Difference.\n\n\u00a7 359. If definition thus consists in referring a thing to a class, we\nsee a further reason why the summum genus of all things cannot be\ndefined.\n\n\u00a7 360. We have said that definition is useful in enabling us to\ndistinguish things from one another in our minds: but this must not be\nregarded as the direct object of the process. For this object may be\naccomplished without giving a definition at all, by means of what is\ncalled a Description. By a description is meant an enumeration of\naccidents with or without the mention of some class-name. It is as\napplicable to proper names as to common terms. When we say 'John Smith\nlives next door on the right-hand side and passes by to his office\nevery morning at nine o'clock,' we have, in all probability,\neffectually distinguished John Smith from other people: but living\nnext, &c., cannot be part of the intension of John Smith, since John\nSmith may change his residence or abandon his occupation without\nceasing to be called by his name.  Indirectly then definition serves\nthe purpose of distinguishing things in the mind, but its direct\nobject is to unfold the intension of terms, and so impart precision to\nour thoughts by setting plainly before us the meaning of the words we\nare using.\n\n\u00a7 361. But when we say that definition is unfolding the intension of\nterms, it must not be imagined that we are bound in defining to unfold\ncompletely the intension of terms. This would be a tedious, and often\nan endless, task. A term may mean, or convey to the mind, a good many\nmore attributes than those which are stated in its definition. There\nis no limit indeed to the meaning which a term may legitimately\nconvey, except the common attributes of the things denoted by it. Who\nshall say, for instance, that a triangle means a figure with three\nsides, and does not mean a figure with three angles, or the surface of\nthe perpendicular bisection of a cone? Or again, that man means a\nrational, and does not mean a speaking, a religious, or an aesthetic\nanimal, or a biped with two eyes, a nose, and a mouth? The only\nattributes of which it can safely be asserted that they can form no\npart of the intension of a term are those which are not common to all\nthe things to which the name applies. Thus a particular complexion,\ncolour, height, creed, nationality cannot form any part of the\nintension of the term 'man.'  But among the attributes common to a\nclass we cannot distinguish between essential and unessential, except\nby the aid of definition itself. Formal logic cannot recognise any\norder of priority between the attributes common to all the members of\na class, such as to necessitate our recognising some as genera and\ndifferentiae and relegating others to the place of properties or\ninseparable accidents.\n\n\u00a7 362. The art of giving a good definition is to seize upon the\nsalient characteristics of the thing defined and those wherefrom the\nlargest number of other attributes can be deduced as consequences. To\ndo this well requires a special knowledge of the thing in question,\nand is not the province of the formal logician.\n\n\u00a7 363. We have seen already, in treating of the Heads of Predicables\n(\u00a7 325), that the difference between genus and difference on the one\nhand and property on the other is wholly relative to some assumed\ndefinition. Now definitions are always to a certain extent arbitrary,\nand will vary with the point of view from which we consider the thing\nrequired to be defined. Thus 'man' is usually contrasted with 'brute,'\nand from this point of view it is held a sufficient definition of him\nto say that he is 'a rational animal,' But a theologian might be more\nanxious to contrast man with supposed incorporeal intelligences, and\nfrom this point of view man would be defined as an 'embodied spirit.'\n\n\u00a7 364. In the two definitions just given it will be noticed that we\nhave really employed exactly the same attributes, only their place as\ngenus and difference has been reversed. It is man's rational, or\nspiritual, nature which distinguishes him from the brutes: but this is\njust what he is supposed to have in common with incorporeal\nintelligences, from whom he is differentiated by his animal nature.\n\n[Illustration]\n\nThis illustration is sufficient to show us that, while there is no\nabsolute definition of anything, in the sense of a fixed genus and\ndifference, there may at the same time be certain attributes which\npermanently distinguish the members of a given class from those of all\nother classes.\n\n\u00a7 365. The above remarks will have made it clear that the intension of\na term is often much too wide to be conveyed by any definition; and\nthat what a definition generally does is to select certain attributes\nfrom the whole intension, which are regarded as being more typical of\nthe thing than the remainder. No definition can be expected to exhaust\nthe whole intension of a term, and there will always be room for\nvarying definitions of the same thing, according to the different\npoints of view from which it is approached.\n\n\u00a7 366. Names of attributes lend themselves to definition far more\neasily than names of substances. The reason of this is that names of\nattributes are primarily intensive in force, whereas substances are\nknown to us in extension before they become known to us in\nintension. There is no difficulty in defining a term like 'triangle'\nor 'monarchy,' because these terms were expressly invented to cover\ncertain attributes; but the case is different with such terms as\n'dog,' 'tree,' 'plant,' 'metal,' and other names of concrete\nthings. We none of us have any difficulty in recognising a dog or\ntree, when we see them, or in distinguishing them from other animals\nor plants respectively. We are therefore led to imagine that we know\nthe meaning of these terms. It is not until we are called upon for a\ndefinition that we discover how superficial our knowledge really is of\nthe common attributes possessed by the things which these names\ndenote.\n\n\u00a7 367. It might be imagined that a common name would never be given to\nthings except in virtue of our knowledge of their common\nattributes. But as a matter of fact, the common name was first given\nfrom a confused notion of resemblance, and we had afterwards to detect\nthe common attributes, when sometimes the name had been so extended\nfrom one thing to another like it, that there were hardly any definite\nattributes possessed in common by the earlier and later members of the\nclass.\n\n\u00a7 368. This is especially the case where the meaning of terms has been\nextended by analogy, e.g. head, foot, arm, post, pole, pipe, &c.\n\n\u00a7 369. But in the progress of thought we come to form terms in which\nthe intensive capacity is everything. Of this kind notably are\nmathematical conceptions. Terms of this kind, as we said before, lend\nthemselves readily to definition.\n\n\u00a7 370. We may lay down then roughly that words are easy or difficult\nof definition according as their intensive or extensive capacity\npredominates.\n\n\u00a7 371. There is a marked distinction to be observed between the\nclasses made by the mind of man and the classes made by nature, which\nare known as 'real kinds.'  In the former there is generally little or\nnothing in common except the particular attribute which is selected as\nthe ground of classification, as in the case of red and white things,\nwhich are alike only in their redness or whiteness; or else their\nattributes are all necessarily connected, as in the case of circle,\nsquare and triangle. But the members of nature's classes agree in\ninnumerable attributes which have no discoverable connection with one\nanother, and which must therefore, provisionally at least, be regarded\nas standing in the relation of inseparable accidents to any particular\nattributes which we may select for the purposes of definition. There\nis no assignable reason why a rational animal should have hair on its\nhead or a nose on its face, and yet man, as a matter of fact, has\nboth; and generally the particular bodily configuration of man can\nonly be regarded as an inseparable accident of his nature as a\nrational animal.\n\n\u00a7 372. 'Real kinds' belong to the class of words mentioned above in\nwhich the extension predominates over the intension. We know well\nenough the things denoted by them, while most of us have only a dim\nidea of the points of resemblance between these things.  Nature's\nclasses moreover shade off into one another by such imperceptible\ndegrees that it is often impossible to fix the boundary line between\none class and another. A still greater source of perplexity in dealing\nwith real kinds is that it is sometimes almost impossible to fix upon\nany attribute which is common to every individual member of the class\nwithout exception. All that we can do in such cases is to lay down a\ntype of the class in its perfect form, and judge of individual\ninstances by the degree of their approximation to it. Again, real\nkinds being known to us primarily in extension, the intension which we\nattach to the names is hable to be affected by the advance of\nknowledge.  In dealing therefore with such terms we must be content\nwith provisional definitions, which adequately express our knowledge\nof the things denoted by them, at the time, though a further study of\ntheir attributes may induce us subsequently to alter the\ndefinition. Thus the old definition of animal as a sentient organism\nhas been rendered inadequate by the discovery that so many of the\nphenomena of sensation can be exhibited by plants,\n\n\u00a7 373. But terms in which intension is the predominant idea are more\ncapable of being defined once for all.  Aristotle's definitions of\n'wealth' and 'monarchy' are as applicable now as in his own day, and\nno subsequent discoveries of the properties of figures will render\nEuclid's definitions unavailable.\n\n\u00a7 374. We may distinguish therefore between two kinds of definition,\nnamely,\n\n  (1) Final.\n\n  (2) Provisional.\n\n\u00a7 375. A distinction is also observed between Real and Nominal\nDefinitions. Both of these explain the meaning of a term: but a real\ndefinition further assumes the actual existence of the thing\ndefined. Thus the explanation of the term 'Centaur' would be a\nnominal, that of 'horse' a real definition.\n\nIt is useless to assert, as is often done, that a nominal definition\nexplains the meaning of a term and a real definition the nature of a\nthing; for, as we have seen already, the meaning of a term is whatever\nwe know of the nature of a thing.\n\n\u00a7 376. It now remains to lay down certain rules for correct\ndefinition.\n\n\u00a7 377. The first rule that is commonly given is that a definition\nshould state the essential attributes of the thing defined. But this\namounts merely to saying that a definition should be a definition;\nsince it is only by the aid of definition that we can distinguish\nbetween essential and non-essential among the common attributes\nexhibited by a class of things. The rule however may be retained as a\nmaterial test of the soundness of a definition, in the sense that he\nwho seeks to define anything should fix upon its most important\nattributes. To define man as a mammiferous animal having two hands, or\nas a featherless biped, we feel to be absurd and incongruous, since\nthere is no reference to the most salient characteristic of man,\nnamely, his rationality. Nevertheless we cannot quarrel with these\ndefinitions on formal, but only on material grounds. Again, if anyone\nchose to define logic as the art of thinking, all we could say is that\nwe differ from him in opinion, as we think logic is more properly to\nbe regarded as the science of the laws of thought. But here also it is\non material grounds that we dissent from the definition.\n\n\u00a7 378. Confining ourselves therefore to the sphere with which we are\nproperly concerned, we lay down the following\n\n\n\n_Rules for Definition._\n\n\n  (1) A definition must be co-extensive with the term defined.\n\n  (2) A definition must not state attributes which imply one another.\n\n  (3) A definition must not contain the name defined, either directly\n  or by implication.\n\n  (4) A definition must be clearer than the term defined.\n\n  (5) A definition must not be negative, if it can be affirmative.\n\nBriefly, a definition must be adequate (1), terse (2), clear (4); and\nmust not be tautologous (3), or, if it can be avoided, negative (5).\n\n\u00a7 379. It is worth while to notice a slight ambiguity in the term\n'definition' itself. Sometimes it is applied to the whole proposition\nwhich expounds the meaning of the term; at other times it is confined\nto the predicate of this proposition. Thus in stating the first four\nrules we have used the term in the latter sense, and in stating the\nfifth in the former.\n\n\u00a7 380. We will now illustrate the force of the above rules by giving\nexamples of their violation.\n\n  Rule 1. Violations. A triangle is a figure with three equal sides.\n\n  A square is a four-sided figure having all its sides equal.\n\nIn the first instance the definition is less extensive than the term\ndefined, since it applies only to equilateral triangles. This fault\nmay be amended by decreasing the intension, which we do by eliminating\nthe reference to the equality of the sides.\n\nIn the second instance the definition is more extensive than the term\ndefined. We must accordingly increase the intension by adding a new\nattribute 'and all its angles right angles.'\n\n  Rule 2. Violation. A triangle is a figure with three sides and three\n  angles.\n\nOne of the chief merits of a definition is to be terse, and this\ndefinition is redundant, since what has three sides cannot but have\nthree angles.\n\n  Rule 3. Violations. A citizen is a person both of whose parents were\n  citizens.\n\n  Man is a human being.\n\n  Rule 4. Violations. A net is a reticulated fabric, decussated at\n  regular intervals.\n\n  Life is the definite combination of heterogeneous changes, both\n  simultaneous and successive, in correspondence with external\n  co-existences and sequences.\n\n  Rule 5. Violations. A mineral is that which is neither animal nor\n  vegetable.\n\n  Virtue is the absence of vice.\n\n\u00a7 381. The object of definition being to explain what a thing is, this\nobject is evidently defeated, if we confine ourselves to saying what\nit is not. But sometimes this is impossible to be avoided. For there\nare many terms which, though positive in form, are privative in force.\nThese terms serve as a kind of residual heads under which to throw\neverything within a given sphere, which does not exhibit certain\npositive attributes. Of this unavoidably negative nature was the\ndefinition which we give of 'accident,' which amounted merely to\nsaying that it was any attribute which was neither a difference nor a\nproperty.\n\n\u00a7 382. The violation of Rule 3, which guards against defining a thing\nby itself, is technically known as 'circulus in definiendo,' or\ndefining in a circle. This rule is often apparently violated, without\nbeing really so. Thus Euclid defines an acute-angled triangle as one\nwhich has three acute angles. This seems a glaring violation of the\nrule, but is perfectly correct in its context; for it has already been\nexplained what is meant by the terms 'triangle' and 'acute angle,' and\nall that is now required is to distinguish the acute-angled triangle\nfrom its cognate species.  He might have said that an acute-angled\ntriangle is one which has neither a right angle nor an obtuse angle:\nbut rightly preferred to throw the same statement into a positive\nform.\n\n\u00a7 383. The violation of Rule 4 is known as 'ignotum per ignotius' or\n'per aeque ignotum.' This rule also may seemingly be violated when it\nis not really so. For a definition may be correct enough from a\nspecial point of view, which, apart from that particular context,\nwould appear ridiculous. From the point of view of conic sections, it\nis correct enough to define a triangle as that section of a cone which\nis formed by a plane passing through the vertex perpendicularly to the\nbase, but this could not be expected to make things clearer to a\nperson who was inquiring for the first time into the meaning of the\nword triangle. But a real violation of the fourth rule may arise, not\nonly from obscurity, but from the employment of ambiguous language or\nmetaphor. To say that 'temperance is a harmony of the soul' or that\n'bread is the staff of life,' throws no real light upon the nature of\nthe definiend.\n\n\u00a7 384. The material correctness of a definition is, as we have already\nseen, a matter extraneous to formal logic. An acquaintance with the\nattributes which terms imply involves material knowledge quite as much\nas an acquaintance with the things they apply to; knowledge of the\nintension and of the extension of terms is alike acquired by\nexperience. No names are such that their meaning is rendered evident\nby the very constitution of our mental faculties; yet nothing short of\nthis would suffice to bring the material content of definition within\nthe province of formal logic.\n\n\n\n\nCHAPTER VIII.\n\n_Of Division._\n\n\n\u00a7 385. To divide a term is to unfold its extension, that is, to set\nforth the things of which it is a name.\n\n\u00a7 386. But as in definition we need not completely unfold the\nintension of a term, so in division we must not completely unfold its\nextension.\n\n\u00a7 387. Completely to unfold the extension of a term would involve\nstating all the individual objects to which the name applies, a thing\nwhich would be impossible in the case of most common terms. When it is\ndone, it is called Enumeration. To reckon up all the months of the\nyear from January to December would be an enumeration, and not a\ndivision, of the term 'month.'\n\n\u00a7 388. Logical division always stops short at classes.  It may be\ndefined as the statement of the various classes of things that can be\ncalled by a common name. Technically we may say that it consists in\nbreaking up a genus into its component species.\n\n\u00a7 389. Since division thus starts with a class and ends with classes,\nit is clear that it is only common terms which admit of division, and\nalso that the members of the division must themselves be common terms.\n\n\u00a7 390. An 'individual' is so called as not admitting of logical\ndivision. We may divide the term 'cow' into classes, as Jersey,\nDevonshire, &c., to which the name 'cow' will still be applicable, but\nthe parts of an individual cow are no longer called by the name of the\nwhole, but are known as beefsteaks, briskets, &c.\n\n\u00a7 391. In dividing a term the first requisite is to fix upon some\npoint wherein certain members of the class differ from others. The\npoint thus selected is called the Fundamentum Divisionis or Basis of\nthe Division.\n\n\u00a7 392. The basis of the division will of course differ according to\nthe purpose in hand, and the same term will admit of being divided on\na number of different principles.  Thus we may divide the term 'man,'\non the basis of colour, into white, black, brown, red, and yellow; or,\non the basis of locality, into Europeans, Asiatics, Africans,\nAmericans, Australians, New Zealanders, and Polynesians; or again, on\na very different principle, into men of nervous, sanguine, bilious,\nlymphatic and mixed temperaments.\n\n\u00a7 393. The term required to be divided is known as the Totum Divisum\nor Divided Whole. It might also be called the Dividend.\n\n\u00a7 394. The classes into which the dividend is split up are called the\nMembra Dividentia, or Dividing Members.\n\n\u00a7 395. Only two rules need be given for division--\n\n  (1) The division must be conducted on a single basis.\n\n  (2) The dividing members must be coextensive with the divided whole.\n\n\u00a7 396. More briefly, we may put the same thing thus--There must be no\ncross-division (1) and the division must be exhaustive (2).\n\n\u00a7 397. The rule, which is commonly given, that each dividing member\nmust be a common term, is already provided for under our definition of\nthe process.\n\n\u00a7 398. The rule that the dividend must be predicable of each of the\ndividing members is contained in our second rule; since, if there were\nany term of which the dividend were not predicable, it would be\nimpossible for the dividing members to be exactly coextensive with it.\nIt would not do, for instance, to introduce mules and donkeys into a\ndivision of the term horse.\n\n\u00a7 399. Another rule, which is sometimes given, namely, that the\nconstituent species must exclude one another, is a consequence of our\nfirst; for, if the division be conducted on a single principle, the\nconstituent species must exclude one another. The converse, however,\ndoes not hold true. We may have a division consisting of mutually\nexclusive members, which yet involves a mixture of different bases,\ne.g. if we were to divide triangle into scalene, isosceles and\nequiangular. This happens because two distinct attributes may be found\nin invariable conjunction.\n\n\u00a7 400. There is no better test, however, of the soundness of a\ndivision than to try whether the species overlap, that is to say,\nwhether there are any individuals that would fall into two or more of\nthe classes. When this is found to be the case, we may be sure that we\nhave mixed two or more different fundamenta divisionis. If man, for\ninstance, were to be divided into European, American, Aryan, and\nSemitic, the species would overlap; for both Europe and America\ncontain inhabitants of Aryan and Semitic origin. We have here members\nof a division based on locality mixed up with members of another\ndivision, which is based on race as indicated by language.\n\n\u00a7 401. The classes which are arrived at by an act of division may\nthemselves be divided into smaller classes.  This further process is\ncalled Subdivision.\n\n\u00a7 402. Let it be noticed that Rule 1 applies only to a single act of\ndivision. The moment that we begin to subdivide we not only may, but\nmust, adopt a new basis of division; since the old one has, 'ex\nhypothesi,' been exhausted. Thus, having divided men according to the\ncolour of their skins, if we wish to subdivide any of the classes, we\nmust look out for some fresh attribute wherein some men of the same\ncomplexion differ from others, e.g. we might divide black men into\nwoolly-haired blacks, such as the Negroes, and straight-haired blacks,\nlike the natives of Australia.\n\n\u00a7 403. We will now take an instance of division and\nsubdivision, with a view to illustrating some of the\ntechnical terms which are used in connection with the\nprocess. Keeping closely to our proper subject, we will\nselect as an instance a division of the products of thought,\nwhich it is the province of logic to investigate.\n\n                  Product of thought\n            _______________|____________________________\n           |                     |                      |\n         Term               Proposition             Inference\n       ____|___            ______|_____            _____|______\n      |        |          |            |          |            |\n  Singular  Common    Universal   Particular  Immediate     Mediate\n                       ___|___      ___|___\n                      |       |    |       |\n                      A       E    I       O\n\nHere we have first a threefold division of the products of thought\nbased on their comparative complexity. The first two of these, namely,\nthe term and the proposition, are then subdivided on the basis of\ntheir respective quantities. In the case of inference the basis of the\ndivision is again the degree of complexity. The subdivision of the\nproposition is carried a step further than that of the others. Having\nexhausted our old basis of quantity, we take a new attribute, namely,\nquality, on which to found the next step of subdivision.\n\n\u00a7 404. Now in such a scheme of division and subdivision as the\nforegoing, the highest class taken is known as the Summum Genus. Thus\nthe summum genus is the same thing as the divided whole, viewed in a\ndifferent relation. The term which is called the divided whole with\nreference to a single act of division, is called the summum genus\nwhenever subdivision has taken place.\n\n\u00a7 405. The classes at which the division stops, that is, any which are\nnot subdivided, are known as the Infimae Species.\n\n\u00a7 406. All classes intermediate between the summum genus and the\ninfimae species are called Subaltern Genera or Subaltern Species,\naccording to the way they are looked at, being genera in relation to\nthe classes below them and species in relation to the classes above\nthem.\n\n\u00a7 407. Any classes which fall immediately under the same genus are\ncalled Cognate Species, e.g. singular and common terms are cognate\nspecies of term.\n\n\u00a7 408. The classes under which any lower class successively falls are\ncalled Cognate Genera. The relation of cognate species to one another\nis like that of children of the same parents, whereas cognate genera\nresemble a line of ancestry.\n\n\u00a7 409. The Specific Difference of anything is the attribute or\nattributes which distinguish it from its cognate species. Thus the\nspecific difference of a universal proposition is that the predicate\nis known to apply to the whole of the subject. A specific difference\nis said to constitute the species.\n\n\u00a7 410. The specific difference of a higher class becomes a Generic\nDifference with respect to the class below it.  A generic difference\nthen may be said to be the distinguishing attribute of the whole class\nto which a given species belongs. The generic difference is common to\nspecies that are cognate to one another, whereas the specific\ndifference is peculiar to each. It is the generic difference of an A\nproposition that it is universal, the specific difference that it is\naffirmative.\n\n\u00a7 411. The same distinction is observed between the specific and\ngeneric properties of a thing. A Specific Property is an attribute\nwhich flows from the difference of a thing itself; a Generic Property\nis an attribute which flows from the difference of the genus to which\nthe thing belongs. It is a specific property of an E proposition that\nits predicate is distributed, a generic property that its contrary\ncannot be true along with it (\u00a7 465); for this last characteristic\nflows from the nature of the universal proposition generally.\n\n\u00a7 412. It now remains to say a few words as to the place in logic of\nthe process of division. Since the attributes in which members of the\nsame class differ from one another cannot possibly be indicated by\ntheir common name, they must be sought for by the aid of experience;\nor, to put the same thing in other words, since all the infimae\nspecies are alike contained under the summum genus, their distinctive\nattributes can be no more than separable accidents when viewed in\nrelation to the summum genus. Hence division, being always founded on\nthe possession or non-possession of accidental attributes, seems to\nlie wholly outside the sphere of formal logic.  This however is not\nquite the case, for, in virtue of the Law of Excluded Middle, there is\nalways open to us, independently of experience, a hypothetical\ndivision by dichotomy. By dichotomy is meant a division into two\nclasses by a pair of contradictory terms, e.g. a division of the\nclass, man, into white and not-white. Now we cannot know,\nindependently of experience, that any members of the class, man,\npossess whiteness; but we may be quite sure, independently of all\nexperience, that men are either white or not. Hence division by\ndichotomy comes strictly within the province of formal logic. Only it\nmust be noticed that both sides of the division must be hypothetical.\nFor experience alone can tell us, on the one hand, that there are any\nmen that are white, and on the other, that there are any but white\nmen.\n\n\u00a7 413. What we call a division on a single basis is in reality the\ncompressed result of a scheme of division and subdivision by\ndichotomy, in which a fresh principle has been introduced at every\nstep. Thus when we divide men, on the basis of colour, into white,\nblack, brown, red and yellow, we may be held to have first divided men\ninto white and not-white, and then to have subdivided the men that are\nnot-white into black and not-black, and so on. From the strictly\nformal point of view this division can only be represented as\nfollows--\n\n                            Men\n          ___________________|_____\n         |                         |\n  White (if any)          Not-white (if any)\n                  _________________|_____\n                 |                       |\n         Black (if any)           Not-black (if any)\n                       __________________|____\n                      |                       |\n              Brown (if any)           Not-brown (if any)\n                          ____________________|____\n                         |                         |\n                  Red (if any)              Not-red (if any).\n\n\u00a7 414. Formal correctness requires that the last term in such a series\nshould be negative. We have here to keep the term 'not-red' open, to\ncover any blue or green men that might turn up. It is only experience\nthat enables us to substitute the positive term 'yellow' for\n'not-red,' since we know as a matter of fact that there are no men but\nthose of the five colours given in the original division.\n\n\u00a7 415. Any correct logical division always admits of being arrived at\nby the longer process of division and subdivision by dichotomy. For\ninstance, the term quadrilateral, or four-sided rectilinear figure, is\ncorrectly divided into square, oblong, rhombus, rhomboid and\ntrapezium.  The steps of which this division consists are as follows--\n\n                    Quadrilateral\n                __________|_________\n               |                    |\n         Parallelogram          Trapezium\n          _____|_____________________\n         |                           |\n     Rectangle                 Non-rectangle\n      ___|___                   _____|_____\n     |       |                 |           |\n  Square   Oblong           Rhombus     Rhomboid.\n\n\u00a7 416. In reckoning up the infimae species in such a scheme, we must\nof course be careful not to include any class which has been already\nsubdivided; but no harm is done by mixing an undivided class, like\ntrapezium, with the subdivisions of its cognate species.\n\n\u00a7 417. The two processes of definition and division are intimately\nconnected with one another. Every definition suggests a division by\ndichotomy, and every division supplies us at once with a complete\ndefinition of all its members.\n\n\u00a7 418. Definition itself, so far as concerns its content, is, as we\nhave already seen, extraneous to formal logic: but when once we have\nelicited a genus and difference out of the material elements of\nthought, we are enabled, without any further appeal to experience, to\nbase thereon a division by dichotomy. Thus when man has been defined\nas a rational animal, we have at once suggested to us a division of\nanimal into rational and irrational.\n\n\u00a7 419. Again, the addition of the attributes, rational and irrational\nrespectively, to the common genus, animal, ipso facto supplies us with\ndefinitions of the species, man and brute. Similarly, when we\nsubdivided rectangle into square and oblong on the basis of the\nequality or inequality of the adjacent sides, we were by so doing\nsupplied with a definition both of square and oblong--'A square is a\nrectangle having all its sides equal,' and 'An oblong is a rectangle\nwhich has only its opposite sides equal.'\n\n\u00a7 420. The definition of a square just given amounts to the same thing\nas Euclid's definition, but it complies with a rule which has value as\na matter of method, namely, that the definition should state the\nProximate Genus of the thing defined.\n\n\u00a7 421. Since definition and division are concerned with the intension\nand extension of terms, they are commonly treated of under the first\npart of logic: but as the treatment of the subject implies a\nfamiliarity with the Heads of Predicables, which in their turn imply\nthe proposition, it seems more desirable to deal with them under the\nsecond.\n\n\u00a7 422. We have already had occasion to distinguish division from\nEnumeration. The latter is the statement of the individual things to\nwhich a name applies. In enumeration, as in division, the wider term\nis predicable of each of the narrower ones.\n\n\u00a7 423. Partition is the mapping out of a physical whole into its\ncomponent parts, as when we say that a tree consists of roots, stem,\nand branches. In a partition the name of the whole is not predicable\nof each of the parts.\n\n\u00a7 424. Distinction is the separation from one another of the various\nmeanings of an equivocal term. The term distinguished is predicable\nindeed of each of the members, but of each in a different sense. An\nequivocal term is in fact not one but several terms, as would quickly\nappear, if we were to use definitions in place of names.\n\n\u00a7 425. We have seen that a logical whole is a genus viewed in relation\nto its underlying species. From this must be distinguished a\nmetaphysical whole, which is a substance viewed in relation to its\nattributes, or a class regarded in the same way. Logically, man is a\npart of the class, animal; metaphysically, animal is contained in\nman. Thus a logical whole is a whole in extension, while a\nmetaphysical whole is a whole in intension. From the former point of\nview species is contained in genus; from the latter genus is contained\nin species.\n\n\n\n\n\nPART III.--OF INFERENCES.\n\n\n\n\nCHAPTER I.\n\n_Of Inferences in General_.\n\n\n\u00a7 426. To infer is to arrive at some truth, not by direct experience,\nbut as a consequence of some truth or truths already known. If we see\na charred circle on the grass, we infer that somebody has been\nlighting a fire there, though we have not seen anyone do it. This\nconclusion is arrived at in consequence of our previous experience of\nthe effects of fire.\n\n\u00a7 427. The term Inference is used both for a process and for a product\nof thought.\n\nAs a process inference may be defined as the passage of the mind from\none or more propositions to another.\n\nAs a product of thought inference may be loosely declared to be the\nresult of comparing propositions.\n\n\u00a7 428. Every inference consists of two parts--\n\n  (1) the truth or truths already known;\n\n  (2) the truth which we arrive at therefrom.\n\nThe former is called the Antecedent, the latter the Consequent. But\nthis use of the terms 'antecedent' and 'consequent' must be carefully\ndistinguished from the use to which they were put previously, to\ndenote the two parts of a complex proposition.\n\n\u00a7 429. Strictly speaking, the term inference, as applied to a product\nof thought, includes both the antecedent and consequent: but it is\noften used for the consequent to the exclusion of the\nantecedent. Thus, when we have stated our premisses, we say quite\nnaturally, 'And the inference I draw is so and so.'\n\n\u00a7 430. Inferences are either Inductive or Deductive.  In induction we\nproceed from the less to the more general; in deduction from the more\nto the less general, or, at all events, to a truth of not greater\ngenerality than the one from which we started. In the former we work\nup to general principles; in the latter we work down from them, and\nelicit the particulars which they contain.\n\n\u00a7 431. Hence induction is a real process from the known to the\nunknown, whereas deduction is no more than the application of\npreviously existing knowledge; or, to put the same thing more\ntechnically, in an inductive inference the consequent is not contained\nin the antecedent, in a deductive inference it is.\n\n\u00a7 432. When, after observing that gold, silver, lead, and other\nmetals, are capable of being reduced to a liquid state by the\napplication of heat, the mind leaps to the conclusion that the same\nwill hold true of some other metal, as platinum, or of all metals, we\nhave then an inductive inference, in which the conclusion, or\nconsequent, is a new proposition, which was not contained in those\nthat went before. We are led to this conclusion, not by reason, but by\nan instinct which teaches us to expect like results, under like\ncircumstances. Experience can tell us only of the past: but we allow\nit to affect our notions of the future through a blind belief that\n'the thing that hath been, it is that which shall be; and that which\nis done is that which shall be done; and there is no new thing under\nthe sun.' Take away this conviction, and the bridge is cut which\nconnects the known with the unknown, the past with the future. The\ncommonest acts of daily life would fail to be performed, were it not\nfor this assumption, which is itself no product of the reason. Thus\nman's intellect, like his faculties generally, rests upon a basis of\ninstinct. He walks by faith, not by sight.\n\n\u00a7 433. It is a mistake to talk of inductive reasoning, as though it\nwere a distinct species from deductive. The fact is that inductive\ninferences are either wholly instinctive, and so unsusceptible of\nlogical vindication, or else they may be exhibited under the form of\ndeductive inferences. We cannot be justified in inferring that\nplatinum will be melted by heat, except where we have equal reason for\nasserting the same thing of copper or any other metal.  In fact we are\njustified in drawing an individual inference only when we can lay down\nthe universal proposition, 'Every metal can be melted by heat.' But\nthe moment this universal proposition is stated, the truth of the\nproposition in the individual instance flows from it by way of\ndeductive inference. Take away the universal, and we have no logical\nwarrant for arguing from one individual case to another. We do so, as\nwas said before, only in virtue of that vague instinct which leads us\nto anticipate like results from like appearances.\n\n\u00a7 434. Inductive inferences are wholly extraneous to the science of\nformal logic, which deals only with formal, or necessary, inferences,\nthat is to say with deductive inferences, whether immediate or\nmediate. These are called formal, because the truth of the consequent\nis apparent from the mere form of the antecedent, whatever be the\nnature of the matter, that is, whatever be the terms employed in the\nproposition or pair of propositions which constitutes the\nantecedent. In deductive inference we never do more than vary the form\nof the truth from which we started. When from the proposition 'Brutus\nwas the founder of the Roman Republic,' we elicit the consequence 'The\nfounder of the Roman Republic was Brutus,' we certainly have nothing\nmore in the consequent than was already contained in the antecedent;\nyet all deductive inferences may be reduced to identities as palpable\nas this, the only difference being that in more complicated cases the\nconsequent is contained in the antecedent along with a number of other\nthings, whereas in this case the consequent is absolutely all that the\nantecedent contains.\n\n\u00a7 435. On the other hand, it is of the very essence of induction that\nthere should be a process from the known to the unknown. Widely\ndifferent as these two operations of the mind are, they are yet both\nincluded under the definition which we have given of inference, as the\npassage of the mind from one or more propositions to another. It is\nnecessary to point this out, because some logicians maintain that all\ninference must be from the known to the unknown, whereas others\nconfine it to 'the carrying out into the last proposition of what was\nvirtually contained in the antecedent judgements.'\n\n\u00a7 436. Another point of difference that has to be noticed between\ninduction and deduction is that no inductive inference can ever attain\nmore than a high degree of probability, whereas a deductive inference\nis certain, but its certainty is purely hypothetical.\n\n\u00a7 437. Without touching now on the metaphysical difficulty as to how\nwe pass at all from the known to the unknown, let us grant that there\nis no fact better attested by experience than this--'That where the\ncircumstances are precisely alike, like results follow.' But then we\nnever can be absolutely sure that the circumstances in any two cases\nare precisely alike. All the experience of all past ages in favour of\nthe daily rising of the sun is not enough to render us theoretically\ncertain that the sun will rise tomorrow We shall act indeed with a\nperfect reliance upon the assumption of the coming day-break; but, for\nall that, the time may arrive when the conditions of the universe\nshall have changed, and the sun will rise no more.\n\n\u00a7 438. On the other hand a deductive inference has all the certainty\nthat can be imparted to it by the laws of thought, or, in other words,\nby the structure of our mental faculties; but this certainty is purely\nhypothetical. We may feel assured that if the premisses are true, the\nconclusion is true also. But for the truth of our premisses we have to\nfall back upon induction or upon intuition. It is not the province of\ndeductive logic to discuss the material truth or falsity of the\npropositions upon which our reasonings are based. This task is left to\ninductive logic, the aim of which is to establish, if possible, a test\nof material truth and falsity.\n\n\u00a7 439. Thus while deduction is concerned only with the relative truth\nor falsity of propositions, induction is concerned with their actual\ntruth or falsity. For this reason deductive logic has been termed the\nlogic of consistency, not of truth.\n\n\u00a7 440. It is not quite accurate to say that in deduction we proceed\nfrom the more to the less general, still less to say, as is often\nsaid, that we proceed from the universal to the particular. For it may\nhappen that the consequent is of precisely the same amount of\ngenerality as the antecedent. This is so, not only in most forms of\nimmediate inference, but also in a syllogism which consists of\nsingular propositions only, e.g.\n\n  The tallest man in Oxford is under eight feet.\n  So and so is the tallest man in Oxford.\n  .'. So and so is under eight feet.\n\nThis form of inference has been named Traduction; but there is no\nessential difference between its laws and those of deduction.\n\n\u00a7 441. Subjoined is a classification of inferences, which will serve\nas a map of the country we are now about to explore.\n\n                            Inference\n        ________________________|__________\n       |                                   |\n   Inductive                           Deductive\n                          _________________|_______________\n                         |                                 |\n                     Immediate                          Mediate\n              ___________|__________                 ______|______\n             |                      |               |             |\n           Simple               Compound          Simple       Complex\n       ______|________________      |         ______|_____________|_\n      |           |           |     |        |            |         |\n  Opposition Conversion Permutation |   Conjunctive  Disjunctive Dilemma\n                                    |\n                           _________|________\n                          |                  |\n                      Conversion         Conversion\n                          by                 by\n                       Negation           position\n\n\n\n\nCHAPTER II.\n\n_Of Deductive Inferences._\n\n\n$ 442. Deductive inferences are of two kinds--Immediate and Mediate.\n\n\u00a7 443. An immediate inference is so called because it is effected\nwithout the intervention of a middle term, which is required in\nmediate inference.\n\n\u00a7 444. But the distinction between the two might be conveyed with at\nleast equal aptness in this way--\n\nAn immediate inference is the comparison of two propositions directly.\n\nA mediate inference is the comparison of two propositions by means of\na third.\n\n\u00a7 445. In that sense of the term inference in which it is confined to\nthe consequent, it may be said that--\n\nAn immediate inference is one derived from a single proposition.\n\nA mediate inference is one derived from two propositions conjointly.\n\n\u00a7 446. There are never more than two propositions in the antecedent of\na deductive inference. Wherever we have a conclusion following from\nmore than two propositions, there will be found to be more than one\ninference.\n\n\u00a7 447. There are three simple forms of immediate inference, namely\nOpposition, Conversion and Permutation.\n\n\u00a7 448. Besides these there are certain compound forms, in which\npermutation is combined with conversion.\n\n\n\n\nCHAPTER III.\n\n_Of Opposition._\n\n\n\u00a7 449. Opposition is an immediate inference grounded on the relation\nbetween propositions which have the same terms, but differ in quantity\nor in quality or in both.\n\n\u00a7 450. In order that there should be any formal opposition between two\npropositions, it is necessary that their terms should be the\nsame. There can be no opposition between two such propositions as\nthese--\n\n  (1) All angels have wings.\n\n  (2) No cows are carnivorous.\n\n\u00a7 451. If we are given a pair of terms, say A for subject and B for\npredicate, and allowed to affix such quantity and quality as we\nplease, we can of course make up the four kinds of proposition\nrecognised by logic, namely,\n\n  A. All A is B.\n\n  E. No A is B.\n\n  I. Some A is B.\n\n  O. Some A is not B.\n\n\u00a7 452. Now the problem of opposition is this: Given the truth or\nfalsity of any one of the four propositions A, E, I, O, what can be\nascertained with regard to the truth or falsity of the rest, the\nmatter of them being supposed to be the same?\n\n\u00a7 453. The relations to one another of these four propositions\nare usually exhibited in the following scheme--\n\n      A . . . . Contrary . . . . E\n      . .                      . .\n      .   .                  .   .\n      .     .              .     .\n      .       .          .       .\n      .         .      .         .\n      .           .  .           .\n  Subaltern  Contradictory    Subaltern\n      .           .  .           .\n      .         .      .         .\n      .       .          .       .\n      .     .              .     .\n      .   .                  .   .\n      . .                      . .\n      I . . . Sub-contrary . . . O\n\n\u00a7 454. Contrary Opposition is between two universals which differ in\nquality.\n\n\u00a7 455. Sub-contrary Opposition is between two particulars which differ\nin quality.\n\n\u00a7 456. Subaltern Opposition is between two propositions which differ\nonly in quantity.\n\n\u00a7 457. Contradictory Opposition is between two propositions which\ndiffer both in quantity and in quality.\n\n\u00a7 458. Subaltern Opposition is also known as Subalternation, and of\nthe two propositions involved the universal is called the Subalternant\nand the particular the Subalternate. Both together are called\nSubalterns, and similarly in the other forms of opposition the two\npropositions involved are known respectively as Contraries,\nSub-contraries and Contradictories.\n\n\u00a7 459. For the sake of convenience some relations are classed under\nthe head of opposition in which there is, strictly speaking, no\nopposition at all between the two propositions involved.\n\n\u00a7 460. Between sub-contraries there is an apparent, but not a real\nopposition, since what is affirmed of one part of a term may often\nwith truth be denied of another. Thus there is no incompatibility\nbetween the two statements.\n\n  (1) Some islands are inhabited.\n\n  (2) Some islands are not inhabited.\n\n\u00a7 461. In the case of subaltern opposition the truth of the universal\nnot only may, but must, be compatible with that of the particular.\n\n\u00a7 462. Immediate Inference by Relation would be a more appropriate\nname than Opposition; and Relation might then be subdivided into\nCompatible and Incompatible Relation. By 'compatible' is here meant\nthat there is no conflict between the _truth_ of the two\npropositions. Subaltern and sub-contrary opposition would thus fall\nunder the head of compatible relation; contrary and contradictory\nrelation under that of incompatible relation.\n\n                         Relation\n              ______________|_____________\n             |                            |\n        Compatible                   Incompatible\n       ______|_____                  _____|_______\n      |            |                |             |\n  Subaltern  Sub-contrary       Contrary  Contradictory.\n\n\u00a7 463. It should be noticed that the inference in the case of\nopposition is from the truth or falsity of one of the opposed\npropositions to the truth or falsity of the other.\n\n\u00a7 464. We will now lay down the accepted laws of inference with regard\nto the various kinds of opposition.\n\n\u00a7 465. Contrary propositions may both be false, but cannot both be\ntrue. Hence if one be true, the other is false, but not vice vers\u00e2.\n\n\u00a7 466. Sub-contrary propositions may both be true, but cannot both be\nfalse. Hence if one be false, the other is true, but not vice vers\u00e2.\n\n\u00a7 467. In the case of subaltern propositions, if the universal be\ntrue, the particular is true; and if the particular be false, the\nuniversal is false; but from the truth of the particular or the\nfalsity of the universal no conclusion can be drawn.\n\n\u00a7 468. Contradictory propositions cannot be either true or false\ntogether. Hence if one be true, the other is false, and vice vers\u00e2.\n\n\u00a7 469. By applying these laws of inference we obtain the following\nresults--\n\n  If A be true, E is false, O false, I true.\n\n  If A be false, E is unknown, O true, I unknown.\n\n  If E be true, O is true, I false, A false.\n\n  If E be false, O is unknown, I true, A unknown.\n\n  If O be true, I is unknown, A false, E unknown.\n\n  If O be false, I is true, A true, E false.\n\n  If I be true, A is unknown, E false, O unknown.\n\n  If I be false, A is false, E true, O true.\n\n\u00a7 470. It will be seen from the above that we derive more information\nfrom deriving a particular than from denying a universal. Should this\nseem surprising, the paradox will immediately disappear, if we reflect\nthat to deny a universal is merely to assert the contradictory\nparticular, whereas to deny a particular is to assert the\ncontradictory universal. It is no wonder that we should obtain more\ninformation from asserting a universal than from asserting a\nparticular.\n\n\u00a7 471. We have laid down above the received doctrine with regard to\nopposition: but it is necessary to point out a flaw in it.\n\nWhen we say that of two sub-contrary propositions, if one be false,\nthe other is true, we are not taking the propositions I and O in their\nnow accepted logical meaning as indefinite (\u00a7 254), but rather in\ntheir popular sense as 'strict particular' propositions. For if I and\nO were taken as indefinite propositions, meaning 'some, if not all,'\nthe truth of I would not exclude the possibility of the truth of A,\nand, similarly, the truth of O would not exclude the possibility of\nthe truth of E. Now A and E may both be false. Therefore I and O,\nbeing possibly equivalent to them, may both be false also. In that\ncase the doctrine of contradiction breaks down as well. For I and O\nmay, on this showing, be false, without their contradictories E and A\nbeing thereby rendered true.  This illustrates the awkwardness, which\nwe have previously had occasion to allude to, which ensures from\ndividing propositions primarily into universal and particular, instead\nof first dividing them into definite and indefinite, and particular (\u00a7\n256).\n\n\u00a7 472. To be suddenly thrown back upon the strictly particular view of\nI and O in the special case of opposition, after having been\naccustomed to regard them as indefinite propositions, is a manifest\ninconvenience. But the received doctrine of opposition does not even\nadhere consistently to this view. For if I and O be taken as strictly\nparticular propositions, which exclude the possibility of the\nuniversal of the same quality being true along with them, we ought not\nmerely to say that I and O may both be true, but that if one be true\nthe other must also be true. For I being true, A is false, and\ntherefore O is true; and we may argue similarly from the truth of O to\nthe truth of I, through the falsity of E. Or--to put the Same thing in\na less abstract form--since the strictly particular proposition means\n'some, but not all,' it follows that the truth of one sub-contrary\nnecessarily carries with it the truth of the other, If we lay down\nthat some islands only are inhabited, it evidently follows, or rather\nis stated simultaneously, that there are some islands also which are\nnot inhabited. For the strictly particular form of proposition 'Some A\nonly is B' is of the nature of an exclusive proposition, and is really\nequivalent to two propositions, one affirmative and one negative.\n\n\u00a7 473. It is evident from the above considerations that the doctrine\nof opposition requires to be amended in one or other of two\nways. Either we must face the consequences which follow from regarding\nI and O as indefinite, and lay down that sub-contraries may both be\nfalse, accepting the awkward corollary of the collapse of the doctrine\nof contradiction; or we must be consistent with ourselves in regarding\nI and O, for the particular purposes of opposition, as being strictly\nparticular, and lay down that it is always possible to argue from the\ntruth of one sub-contrary to the truth of the other. The latter is\nundoubtedly the better course, as the admission of I and O as\nindefinite in this connection confuses the theory of opposition\naltogether.\n\n\u00a7 474. Of the several forms of opposition contradictory opposition is\nlogically the strongest. For this three reasons may be given--\n\n  (1) Contradictory opposites differ both in quantity and in quality,\n  whereas others differ only in one or the other.\n\n  (2) Contradictory opposites are incompatible both as to truth and\n  falsity, whereas in other cases it is only the truth _or_\n  falsity of the two that is incompatible.\n\n  (3) Contradictory opposition is the safest form to adopt in\n  argument. For the contradictory opposite refutes the adversary's\n  proposition as effectually as the contrary, and is not so hable to a\n  counter-refutation.\n\n\u00a7 475. At first sight indeed contrary opposition appears stronger,\nbecause it gives a more sweeping denial to the adversary's\nassertion. If, for instance, some person with whom we were arguing\nwere to lay down that 'All poets are bad logicians,' we might be\ntempted in the heat of controversy to maintain against him the\ncontrary proposition 'No poets are bad logicians.' This would\ncertainly be a more emphatic contradiction, but, logically considered,\nit would not be as sound a one as the less obtrusive contradictory,\n'Some poets are not bad logicians,' which it would be very difficult\nto refute.\n\n\u00a7 476. The phrase 'diametrically opposed to one another' seems to be\none of the many expressions which have crept into common language from\nthe technical usage of logic. The propositions A and O and E and I\nrespectively are diametrically opposed to one another in the sense\nthat the straight lines connecting them constitute the diagonals of\nthe parallelogram in the scheme of opposition.\n\n\u00a7 477. It must be noticed that in the case of a singular proposition\nthere is only one mode of contradiction possible. Since the quantity\nof such a proposition is at the minimum, the contrary and\ncontradictory are necessarily merged into one. There is no way of\ndenying the proposition 'This house is haunted,' save by maintaining\nthe proposition which differs from it only in quality, namely, 'This\nhouse is not haunted.'\n\n478. A kind of generality might indeed he imparted even to a singular\nproposition by expressing it in the form 'A is always B.' Thus we may\nsay, 'This man is always idle'--a proposition which admits of being\ncontradicted under the form 'This man is sometimes not idle.'\n\n\n\n\nCHAPTER IV.\n\n_Of Conversion._\n\n\n\u00a7 479. Conversion is an immediate inference grounded On the\ntransposition of the subject and predicate of a proposition.\n\n\u00a7 480. In this form of inference the antecedent is technically known\nas the Convertend, i.e. the proposition to be converted, and the\nconsequent as the Converse, i.e. the proposition which has been\nconverted.\n\n\u00a7 481. In a loose sense of the term we may be said to have converted a\nproposition when we have merely transposed the subject and predicate,\nwhen, for instance, we turn the proposition 'All A is B' into 'All B\nis A' or 'Some A is not B' into 'Some B is not A.' But these\npropositions plainly do not follow from the former ones, and it is\nonly with conversion as a form of inference--with Illative Conversion\nas it is called--that Logic is concerned.\n\n\u00a7 482. For conversion as a form of inference two rules have been laid\ndown--\n\n  (1) No term must be distributed in the converse which was not\n  distributed in the convertend.\n\n  (2) The quality of the converse must be the same as that of the\n  convertend.\n\n\u00a7 483. The first of these rules is founded on the nature of things. A\nviolation of it involves the fallacy of arguing from part of a term to\nthe whole.\n\n\u00a7 484. The second rule is merely a conventional one.  We may make a\nvalid inference in defiance of it: but such an inference will be seen\npresently to involve something more than mere conversion.\n\n\u00a7 485. There are two kinds of conversion--\n\n  (1) Simple.\n\n  (2) Per Accidens or by Limitation.\n\n\u00a7 486. We are said to have simply converted a proposition when the\nquantity remains the same as before.\n\n\u00a7 487. We are said to have converted a proposition per accidens, or by\nlimitation, when the rules for the distribution of terms necessitate a\nreduction in the original quantity of the proposition.\n\n\u00a7 488.\n\n  A can only be converted per accidens.\n\n  E and I can be converted simply.\n\n  O cannot be converted at all.\n\n\u00a7 489. The reason why A can only be converted per accidens is that,\nbeing affirmative, its predicate is undistributed (\u00a7 293). Since 'All\nA is B' does not mean more than 'All A is some B,' its proper converse\nis 'Some B is A.' For, if we endeavoured to elicit the inference, 'All\nB is A,' we should be distributing the term B in the converse, which\nwas not distributed in the convertend.  Hence we should be involved in\nthe fallacy of arguing from the part to the whole. Because 'All\ndoctors are men' it by no means follows that 'All men are doctors.'\n\n\u00a7 499. E and I admit of simple conversion, because the quantity of the\nsubject and predicate is alike in each, both subject and predicate\nbeing distributed in E and undistributed in I.\n\n\n     / No A is B.\n  E <\n     \\ .'. No B is A.\n\n     / Some A is B.\n  I <\n     \\ .'. Some B is A.\n\n\u00a7 491. The reason why O cannot be converted at all is that its subject\nis undistributed and that the proposition is negative. Now, when the\nproposition is converted, what was the subject becomes the predicate,\nand, as the proposition must still be negative, the former subject\nwould now be distributed, since every negative proposition distributes\nits predicate. Hence we should necessarily have a term distributed in\nthe converse which was not distributed in the convertend. From 'Some\nmen are not doctors,' it plainly does not follow that 'Some doctors\nare not men'; and, generally from 'Some A is not B' it cannot be\ninferred that 'Some B is not A,' since the proposition 'Some A is not\nB' admits of the interpretation that B is wholly contained in A.\n\n[Illustration]\n\n\u00a7 492. It may often happen as a matter of fact that in some given\nmatter a proposition of the form 'All B is A' is true simultaneously\nwith 'All A is B.' Thus it is as true to say that 'All equiangular\ntriangles are equilateral' as that 'All equilateral triangles are\nequiangular.' Nevertheless we are not logically warranted in inferring\nthe one from the other. Each has to be established on its separate\nevidence.\n\n\u00a7 493. On the theory of the quantified predicate the difference\nbetween simple conversion and conversion by limitation disappears. For\nthe quantity of a proposition is then no longer determined solely by\nreference to the quantity of its subject. 'All A is some B' is of no\ngreater quantity than 'Some B is all A,' if both subject and predicate\nhave an equal claim to be considered.\n\n\u00a7 494. Some propositions occur in ordinary language in which the\nquantity of the predicate is determined. This is especially the case\nwhen the subject is a singular term.  Such propositions admit of\nconversion by a mere transposition of their subject and predicate,\neven though they fall under the form of the A proposition, e.g.\n\n  Virtue is the condition of happiness.\n  .'. The condition of happiness is virtue.\n\nAnd again,\n\n  Virtue is a condition of happiness.\n  .'. A condition of happiness is virtue.\n\nIn the one case the quantity of the predicate is determined by the\nform of the expression as distributed, in the other as undistributed.\n\n\u00a7 495. Conversion offers a good illustration of the principle on which\nwe have before insisted, namely, that in the ordinary form of\nproposition the subject is used in extension and the predicate in\nintension. For when by conversion we change the predicate into the\nsubject, we are often obliged to attach a noun substantive to the\npredicate, in order that it may be taken in extension, instead of, as\nbefore, in intension, e.g.\n\n  Some mothers are unkind.\n  .'. Some unkind persons are mothers.\n\nAgain,\n\n  Virtue is conducive to happiness.\n  .'. One of the things which are conducive to happiness is virtue.\n\n\n\n\nCHAPTER V.\n\n_Of Permutation._\n\n\n\u00a7 496. Permutation [Footnote: Called by some writers Obversion.] is an\nimmediate inference grounded on a change of quality in a proposition\nand a change of the predicate into its contradictory-term.\n\n\u00a7 497. In less technical language we may say that permutation is\nexpressing negatively what was expressed affirmatively and vice vers\u00e2.\n\n\u00a7 498. Permutation is equally applicable to all the four\nforms of proposition.\n\n  (A) All A is B.\n      .'. No A is not-B (E).\n\n  (E) No A is B.\n      .'. All A is not-B (A).\n\n  (I) Some A is B.\n      .'. Some A is not not-B (O).\n\n  (O) Some A is not B.\n      .'. Some A is not-B (I).\n\n\u00a7 499, Or, to take concrete examples--\n\n  (A) All men are fallible.\n      .'. No men are not-fallible (E).\n\n  (E) No men are perfect.\n      .'. All men are not-perfect (A).\n\n  (I) Some poets are logical.\n      .'. Some poets are not not-logical (O).\n\n  (O) Some islands are not inhabited.\n      .'. Some islands are not-inhabited (I).\n\n\u00a7 500. The validity of permutation rests on the principle of excluded\nmiddle, namely--That one or other of a pair of contradictory terms\nmust be applicable to a given subject, so that, when one may be\npredicated affirmatively, the other may be predicated negatively, and\nvice vers\u00e2 (\u00a7 31).\n\n\u00a7 501. Merely to alter the quality of a proposition would of course\naffect its meaning; but when the predicate is at the same time changed\ninto its contradictory term, the original meaning of the proposition\nis retained, whilst the form alone is altered. Hence we may lay down\nthe following practical rule for permutation--\n\nChange the quality of the proposition and change the predicate into\nits contradictory term.\n\n\u00a7 502. The law of excluded middle holds only with regard to\ncontradictories. It is not true of a pair of positive and privative\nterms, that one or other of them must be applicable to any given\nsubject. For the subject may happen to fall wholly outside the sphere\nto which such a pair of terms is limited. But since the fact of a term\nbeing applied is a sufficient indication of its applicability, and\nsince within a given sphere positive and privative terms are as\nmutually destructive as contradictories, we may in all cases\nsubstitute the privative for the negative term in immediate inference\nby permutation, which will bring the inferred proposition more into\nconformity with the ordinary usage of language. Thus the concrete\ninstances given above will appear as follows--\n\n  (A) All men are fallible.\n      .'. No men are infallible (E).\n\n  (E) No men are perfect.\n      .'. All men are imperfect (A).\n\n  (I) Some poets are logical.\n      .'. Some poets are not illogical (O).\n\n  (O) Some islands are not inhabited.\n      .'. Some islands are uninhabited (I).\n\n\n\n\nCHAPTER VI.\n\n_Of Compound Forms of Immediate Inference._\n\n\n\u00a7 503. Having now treated of the three simple forms of immediate\ninference, we go on to speak of the compound forms, and first of\n\n\n\n_Conversion by Negation._\n\n\n\u00a7 504. When A and O have been permuted, they become respectively E and\nI, and, in this form, admit of simple conversion. We have here two\nsteps of inference: but the process may be performed at a single\nstroke, and is then known as Conversion by Negation. Thus from 'All A\nis B' we may infer 'No not-B is A,' and again from 'Some A is not B'\nwe may infer 'Some not-B is A.' The nature of these inferences will be\nseen better in concrete examples.\n\n\u00a7 505.\n\n  (A) All poets are imaginative.\n      .'. No unimaginative persons are poets (E).\n\n  (O) Some parsons are not clerical.\n      .'. Some unclerical persons are parsons (I).\n\n\u00a7 506. The above inferences, when analysed, will be found to resolve\nthemselves into two steps, namely,\n\n  (1) Permutation.\n\n  (2) Simple Conversion.\n\n  (A) All A is B.\n      .'. No A is not-B (by permutation).\n      .'. No not-B is A (by simple conversion).\n\n  (O) Some A is not B.\n      .'. Some A is not-B (by permutation).\n      .'. Some not-B is A (by simple conversion).\n\n\u00a7 507. The term conversion by negation has been arbitrarily limited to\nthe exact inferential procedure of permutation followed by simple\nconversion. Hence it necessarily applies only to A and 0 propositions,\nsince these when permuted become E and 1, which admit of simple\nconversion; whereas E and 1 themselves are permuted into A and 0,\nwhich do not. There seems to be no good reason, however, why the term\n'conversion by negation' should be thus restricted in its meaning;\ninstead of being extended to the combination of permutation with\nconversion, no matter in what order the two processes may be\nperformed. If this is not done, inferences quite as legitimate as\nthose which pass under the title of conversion by negation are left\nwithout a name.\n\n\u00a7 508. From E and 1 inferences may be elicited as follows--\n\n  (E) No A is B.\n      .'. All B is not-A (A).\n\n  (I) Some A is B.\n      .'. Some B is not not-A (O).\n\n  (E) No good actions are unbecoming.\n      .'.  All unbecoming actions are not-good (A).\n\n  (I) Some poetical persons are logicians.\n      .'. Some logicians are not unpoetical (O).\n\nOr, taking a privative term for our subject,\n\n  Some unpractical persons are statesmen.\n  .'. Some statesmen are not practical.\n\n\u00a7 509. When the inferences just given are analysed, it will be found\nthat the process of simple conversion precedes that of permutation.\n\n\u00a7 510. In the case of the E proposition a compound inference can be\ndrawn even in the original order of the processes,\n\n  No A is B.\n  .'. Some not-B is A.\n\n  No one who employs bribery is honest.\n  .'. Some dishonest men employ bribery.\n\nThe inference here, it must be remembered, does not refer to matter of\nfact, but means that one of the possible forms of dishonesty among men\nis that of employing bribery.\n\n\u00a7 511. If we analyse the preceding, we find that the second step is\nconversion by limitation.\n\n  No A is B.\n  .'. All A is not-B (by permutation).\n  .'. Some not-B is A (by conversion per accidens).\n\n\u00a7 512. From A again an inference can be drawn in the reverse order of\nconversion per accidens followed by permutation--\n\n  All A is B.\n  .'. Some B is not not-A.\n\n  All ingenuous persons are agreeable.\n  .'. Some agreeable persons are not disingenuous.\n\n\u00a7 513. The intermediate link between the above two propositions is the\nconverse per accidens of the first--'Some B is A.' This inference,\nhowever, coincides with that from 1 (\u00a7 508), as the similar inference\nfrom E (\u00a7 510) coincides with that from 0 (\u00a7 506).\n\n\u00a7 514. All these inferences agree in the essential feature of\ncombining permutation with conversion, and should therefore be classed\nunder a common name.\n\n\u00a7 515. Adopting then this slight extension of the term, we define\nconversion by negation as--A form of conversion in which the converse\ndiffers in quality from the convertend, and has the contradictory of\none of the original terms.\n\n\u00a7 516. A still more complex form of immediate inference is known as\n\n\n\n_Conversion by Contraposition._\n\n\nThis mode of inference assumes the following form--\n\n  All A is B.\n  .'. All not-B is not-A.\n\n  All human beings are fallible.\n  .'. All infallible beings are not-human.\n\n\u00a7 517. This will be found to resolve itself on analysis into three\nsteps of inference in the following order--\n\n  (1) Permutation.\n\n  (2) Simple Conversion.\n\n  (3) Permutation.\n\n\u00a7 518. Let us verify this statement by performing the three steps.\n\n  All A is B.\n  .'. No A is not-B (by permutation).\n  .'. No not-B is A (by simple conversion).\n  .'. All not-B is not-A (by permutation).\n\n  All Englishmen are Aryans.\n  .'. No Englishmen are non-Aryans.\n  .'. No non-Aryans are Englishmen.\n  .'. All non-Aryans are non-Englishmen.\n\n\u00a7 519. Conversion by contraposition may be complicated in appearance\nby the occurrence of a negative term in the subject or predicate or\nboth, e.g.\n\n  All not-A is B.\n  .'. All not-B is A.\n\nAgain,\n\n  All A is not-B.\n  .'. All B is not-A.\n\nLastly,\n\n  All not-A is not-B.\n  .'. All B is A.\n\n\u00a7 520. The following practical rule will be found of use for the right\nperforming of the process--\n\n  Transpose the subject and predicate, and substitute for each its\n  contradictory term.\n\n\u00a7 521. As concrete illustrations of the above forms of inference we\nmay take the following--\n\n  All the men on this board that are not white are red.\n  .'. All the men On this board that are not red are white.\n\nAgain,\n\n  All compulsory labour is inefficient.\n  .'. All efficient labour is free (=non-compulsory).\n\nLastly,\n\n  All inexpedient acts are unjust.\n  .'. All just acts are expedient.\n\n\u00a7 522. Conversion by contraposition may be said to\nrest on the following principle--\n\n  If one class be wholly contained in another, whatever is external to\n  the containing class is external also to the class contained.\n\n  [Illustration]\n\n\u00a7 523. The same principle may be expressed intensively as follows:--\n\n  If an attribute belongs to the whole of a subject, whatever fails to\n  exhibit that attribute does not come under the subject.\n\n\u00a7 524. This statement contemplates conversion by contraposition only\nin reference to the A proposition, to which the process has hitherto\nbeen confined. Logicians seem to have overlooked the fact that\nconversion by contraposition is as applicable to the O as to the A\nproposition, though, when expressed in symbols, it presents a more\nclumsy appearance.\n\n  Some A is not B.\n  .'. Some not-B is not not-A.\n\n  Some wholesome things are not pleasant.\n  .'. Some unpleasant things are not unwholesome.\n\n\u00a7 525. The above admits of analysis in exactly the same way as the\nsame process when applied to the A proposition.\n\n  Some A is not B.\n  .'. Some A is not-B (by permutation).\n  .'. Some not-B is A (by simple conversion).\n  .'. Some not-B is not not-A (by permutation).\n\nThe result, as in the case of the A proposition, is the converse by\nnegation of the original proposition permuted.\n\n\u00a7 526. Contraposition may also be applied to the E proposition by the\nuse of conversion per accidens in the place of simple conversion. But,\nowing to the limitation of quantity thus effected, the result arrived\nat is the same as in the case of the O proposition. Thus from 'No\nwholesome things are pleasant' we could draw the same inference as\nbefore. Here is the process in symbols, when expanded.\n\n  No A is B.\n  .'. All A is not-B (by permutation).\n  .'. Some not-B is A (by conversion per accidens).\n  .'. Some not-B is not not-A (by permutation).\n\n\n\u00a7 527. In its unanalysed form conversion by contraposition may be\ndefined generally as--A form of conversion in which both subject and\npredicate are replaced by their contradictories.\n\n\u00a7 528. Conversion by contraposition differs in several respects from\nconversion by negation.\n\n  (1) In conversion by negation the converse differs in quality from\n  the convertend: whereas in conversion by contraposition the quality\n  of the two is the same.\n\n  (2) In conversion by negation we employ the contradictory either of\n  the subject or predicate, but in conversion by contraposition we\n  employ the contradictory of both.\n\n  (3) Conversion by negation involves only two steps of immediate\n  inference: conversion by contraposition three.\n\n\u00a7 529. Conversion by contraposition cannot be applied to the ordinary\nE proposition except by limitation (\u00a7 526).\n\nFrom 'No A is B' we cannot infer 'No not-B is not-A.' For, if we\ncould, the contradictory of the latter, namely, 'Some not-B is not-A'\nwould be false. But it is manifest that this is not necessarily\nfalse. For when one term is excluded from another, there must be\nnumerous individuals which fall under neither of them, unless it\nshould so happen that one of the terms is the direct contradictory of\nthe other, which is clearly not conveyed by the form of the expression\n'No A is B. 'No A is not-A' stands alone among E propositions in\nadmitting of full conversion by contraposition, and the form of that\nis the same after it as before.\n\n\u00a7 530. Nor can conversion by contraposition be applied at all to I.\n\n[Illustration]\n\nFrom 'Some A is B' we cannot infer that 'Some not-B is not-A.' For\nthough the proposition holds true as a matter of fact, when A and B\nare in part mutually exclusive, yet this is not conveyed by the form\nof the expression.  It may so happen that B is wholly contained under\nA, while A itself contains everything. In this case it will be true\nthat 'No not-B is not-A,' which contradicts the attempted\ninference. Thus from the proposition 'Some things are substances' it\ncannot be inferred that 'Some not-substances are not-things,' for in\nthis case the contradictory is true that 'No not-substances are\nnot-things'; and unless an inference is valid in every case, it is not\nformally valid at all.\n\n\u00a7 531. It should be noticed that in the case of the [nu] proposition\nimmediate inferences are possible by mere contraposition without\nconversion.\n\n  All A is all B.\n  .'. All not-A is not-B.\n\nFor example, if all the equilateral triangles are all the equiangular,\nwe know at once that all non-equilateral triangles are also\nnon-equiangular.\n\n\u00a7 532. The principle upon which this last kind of inference rests is\nthat when two terms are co-extensive, whatever is excluded from the\none is excluded also from the other.\n\n\n\n\nCHAPTER VII.\n\n_Of other Forms of Immediate Inference._\n\n\n\u00a7 533. Having treated of the main forms of immediate inference,\nwhether simple or compound, we will now close this subject with a\nbrief allusion to some other forms which have been recognised by\nlogicians.\n\n\u00a7 534. Every statement of a relation may furnish us with ail immediate\ninference in which the same fact is presented from the opposite\nside. Thus from 'John hit James' we infer 'James was hit by John';\nfrom 'Dick is the grandson of Tom' we infer 'Tom is the grandfather of\nDick'; from 'Bicester is north-east of Oxford' we infer 'Oxford is\nsouth-west of Bicester'; from 'So and so visited the Academy the day\nafter he arrived in London' we infer 'So and so arrived in London the\nday before he visited the Academy'; from 'A is greater than B' we\ninfer 'B is less than A'; and so on without limit. Such inferences as\nthese are material, not formal. No law can be laid down for them\nexcept the universal postulate, that\n\n  'Whatever is true in one form of words is true in every other form\n  of words which conveys the same meaning.'\n\n\u00a7 535. There is a sort of inference which goes under the title of\nImmediate Inference by Added Determinants, in which from some\nproposition already made another is inferred, in which the same\nattribute is attached both to the subject and the predicate, e.g.,\n\n  A horse is a quadruped.\n  .'. A white horse is a white quadruped.\n\n\u00a7 536. Such inferences are very deceptive. The attributes added must\nbe definite qualities, like whiteness, and must in no way involve a\ncomparison. From 'A horse is a quadruped' it may seem at first sight\nto follow that 'A swift horse is a swift quadruped.' But we need not\ngo far to discover how little formal validity there is about such an\ninference. From 'A horse is a quadruped' it by no means follows that\n'A slow horse is a slow quadruped'; for even a slow horse is swift\ncompared with most quadrupeds.  All that really follows here is that\n'A slow horse is a quadruped which is slow for a horse.' Similarly,\nfrom 'A Bushman is a man' it does not follow that 'A tall Bushman is a\ntall man,' but only that 'A tall Bushman is a man who is tall for a\nBushman'; and so on generally.\n\n\u00a7 537. Very similar to the preceding is the process known as Immediate\nInference by Complex Conception, e.g.\n\n  A horse is a quadruped.\n  .'. The head of a horse is the head of a quadruped.\n\n\u00a7 538. This inference, like that by added determinants, from which it\ndiffers in name rather than in nature, may be explained on the\nprinciple of Substitution. Starting from the identical proposition,\n'The head of a quadruped is the head of a quadruped,' and being given\nthat 'A horse is a quadruped,' so that whatever is true of 'quadruped'\ngenerally we know to be true of 'horse,' we are entitled to substitute\nthe narrower for the wider term, and in this manner we arrive at the\nproposition,\n\n  The head of a horse is the head of a quadruped.\n\n\u00a7 539. Such an inference is valid enough, if the same caution be\nobserved as in the case of added determinants, that is, if no\ndifference be allowed to intervene in the relation of the fresh\nconception to the generic and the specific terms.\n\n\n\n\nCHAPTER VIII.\n\n_Of Mediate Inferences or Syllogisms._\n\n\n\u00a7 540. A Mediate Inference, or Syllogism, consists of two\npropositions, which are called the Premisses, and a third proposition\nknown as the Conclusion, which flows from the two conjointly.\n\n\u00a7 541. In every syllogism two terms are compared with one another by\nmeans of a third, which is called the Middle Term. In the premisses\neach of the two terms is compared separately with the middle term; and\nin the conclusion they are compared with one another.\n\n\u00a7 542. Hence every syllogism consists of three terms, one of which\noccurs twice in the premisses and does not appear at all in the\nconclusion. This term is called the Middle Term. The predicate of the\nconclusion is called the Major Term and its subject the Minor Term.\n\n\u00a7 543. The major and minor terms are called the Extremes, as opposed\nto the Mean or Middle Term.\n\n\u00a7 544. The premiss in which the major term is compared with the middle\nis called the Major Premiss.\n\n\u00a7 545. The other premiss, in which the minor term is compared with the\nmiddle, is called the Minor Premiss.\n\n\u00a7 546. The order in which the premisses occur in a syllogism is\nindifferent, but it is usual, for convenience, to place the major\npremiss first.\n\n\u00a7 547. The following will serve as a typical instance of a syllogism--\n\n                 Middle term     Major term   \\\n  Major Premiss. All mammals are warm-blooded  |  Antecedent\n                                                >    or\n                 Minor term    Middle term     |  Premisses\n  Minor Premiss. All whales are mammals       /\n\n                 Minor term     Major term   \\  Consequent or\n             .'. All whales are warm-blooded  > Conclusion.\n\n\u00a7 548. The reason why the names 'major, 'middle' and 'minor' terms\nwere originally employed is that in an affirmative syllogism such as\nthe above, which was regarded as the perfect type of syllogism, these\nnames express the relative quantity in extension of the three terms.\n\n[Illustration]\n\n\u00a7 549. It must be noticed however that, though the middle term cannot\nbe of larger extent than the major nor of smaller extent than the\nminor, if the latter be distributed, there is nothing to prevent all\nthree, or any two of them, from being coextensive.\n\n\u00a7 550. Further, when the minor term is undistributed, we either have a\ncase of the intersection of two classes, from which it cannot be told\nwhich of them is the larger, or the minor term is actually larger than\nthe middle, when it stands to it in the relation of genus to species,\nas in the following syllogism--\n\n  All Negroes have woolly hair.\n  Some Africans are Negroes.\n  .'. Some Africans have woolly hair.\n\n  [Illustration]\n\n\u00a7 551. Hence the names are not applied with strict accuracy even in\nthe case of the affirmative syllogism; and when the syllogism is\nnegative, they are not applicable at all: since in negative\npropositions we have no means of comparing the relative extension of\nthe terms employed. Had we said in the major premiss of our typical\nsyllogism, 'No mammals are cold-blooded,' and drawn the conclusion 'No\nwhales are cold-blooded,' we could not have compared the relative\nextent of the terms 'mammal' and 'cold-blooded,' since one has been\nsimply excluded from the other.\n\n  [Illustration]\n\n\u00a7 552. So far we have rather described than defined the syllogism. All\nthe products of thought, it will be remembered, are the results of\ncomparison. The syllogism, which is one of them, may be so regarded in\ntwo ways--\n\n  (1) As the comparison of two propositions by means of a third.\n\n  (2) As the comparison of two terms by means of a third or middle\n  term.\n\n\u00a7 553. The two propositions which are compared with one another are\nthe major premiss and the conclusion, which are brought into\nconnection by means of the minor premiss. Thus in the syllogism above\ngiven we compare the conclusion 'All whales are warm-blooded' with the\nmajor premiss 'All mammals are warm-blooded,' and find that the former\nis contained under the latter, as soon as we become acquainted with\nthe intermediate proposition 'All whales are mammals.'\n\n\u00a7 554. The two terms which are compared with one another are of course\nthe major and minor.\n\n\u00a7 555. The syllogism is merely a form into which our deductive\ninferences may be thrown for the sake of exhibiting their\nconclusiveness. It is not the form which they naturally assume in\nspeech or writing. Practically the conclusion is generally stated\nfirst and the premisses introduced by some causative particle as\n'because,' 'since,' 'for,' &c. We start with our conclusion, and then\ngive the reason for it by supplying the premisses.\n\n\u00a7 556. The conclusion, as thus stated first, was called by logicians\nthe Problema or Quaestio, being regarded as a problem or question, to\nwhich a solution or answer was to be found by supplying the premisses.\n\n\u00a7 557. In common discourse and writing the syllogism is usually stated\ndefectively, one of the premisses or, in some cases, the conclusion\nitself being omitted. Thus instead of arguing at full length\n\n  All men are fallible,\n  The Pope is a man,\n  .'. The Pope is fallible,\n\nwe content ourselves with saying 'The Pope is fallible, for he is a\nman,' or 'The Pope is fallible, because all men are so'; or perhaps we\nshould merely say 'All men are fallible, and the Pope is a man,'\nleaving it to the sagacity of our hearers to supply the desired\nconclusion. A syllogism, as thus elliptically stated, is commonly,\nthough incorrectly, called an Enthymeme. When the major premiss is\nomitted, it is called an Enthymeme of the First Order; when the minor\nis omitted, an Enthymeme of the Second Order; and when the conclusion\nis omitted an Enthymeme of the Third Order.\n\n\n\n\nCHAPTER IX.\n\n_Of Mood and Figure._\n\n\n\u00a7 558. Syllogisms may differ in two ways--\n\n  (1) in Mood;\n\n  (2) in Figure.\n\n\u00a7 559. Mood depends upon the kind of propositions employed. Thus a\nsyllogism consisting of three universal affirmatives, AAA, would be\nsaid to differ in mood from one consisting of such propositions as EIO\nor any other combination that might be made. The syllogism previously\ngiven to prove the fallibility of the Pope belongs to the mood\nAAA. Had we drawn only a particular conclusion, 'Some Popes are\nfallible,' it would have fallen into the mood AAI.\n\n\u00a7 560. Figure depends upon the arrangement of the terms in the\npropositions. Thus a difference of figure is internal to a difference\nof mood, that is to say, the same mood can be in any figure.\n\n\u00a7 561. We will now show how many possible varieties there are of mood\nand figure, irrespective of their logical validity.\n\n\u00a7 562. And first as to mood.\n\nSince every syllogism consists of three propositions, and each of\nthese propositions may be either A, E, I, or O, it is clear that there\nwill be as many possible moods as there can be combinations of four\nthings, taken three together, with no restrictions as to\nrepetition. It will be seen that there are just sixty-four of such\ncombinations.  For A may be followed either by itself or by E, I, or\nO.  Let us suppose it to be followed by itself. Then this pair of\npremisses, AA, may have for its conclusion either A, E, I, or O, thus\ngiving four combinations which commence with AA. In like manner there\nwill be four commencing with AE, four with AI, and four with AO,\ngiving a total of sixteen combinations which commence with\nA. Similarly there will be sixteen commencing with E, sixteen with I,\nsixteen with O--in all sixty-four. It is very few, however, of these\npossible combinations that will be found legitimate, when tested by\nthe rules of syllogism.\n\n\u00a7 563. Next as to figure.\n\nThere are four possible varieties of figure in a syllogism, as may be\nseen by considering the positions that can be occupied by the middle\nterm in the premisses. For as there are only two terms in each\npremiss, the position occupied by the middle term necessarily\ndetermines that of the others. It is clear that the middle term must\neither occupy the same position in both premisses or not, that is, it\nmust either be subject in both or predicate in both, or else subject\nin one and predicate in the other. Now, if we are not acquainted with\nthe conclusion of our syllogism, we do not know which is the major and\nwhich the minor term, and have therefore no means of distinguishing\nbetween one premiss and another; consequently we must Stop here, and\nsay that there are only three different arrangements possible. But, if\nthe Conclusion also be assumed as known, then we are able to\ndistinguish one premiss as the major and the other as the minor; and\nso we can go further, and lay down that, if the middle term does not\nhold the same position in both premisses, it must either be subject in\nthe major and predicate in the minor, or else predicate in the major\nand subject in the minor.\n\n\u00a7 564. Hence there result\n\n  _The Four Figures._\n\nWhen the middle term is subject in the major and predicate in the\nminor, we are said to have the First Figure.\n\nWhen the middle term is predicate in both premisses, we are said to\nhave the Second Figure.\n\nWhen the middle term is subject in both premisses, we are said to have\nthe Third Figure.\n\nWhen the middle term is predicate in the major premiss and subject in\nthe minor, we are said to have the Fourth Figure.\n\n\u00a7 565. Let A be the major term; B the middle. C the minor.\n\n  Figure I. Figure II. Figure III. Figure IV.\n  B--A       A--B        B--A        A--B\n  C--B       C--B        B--C        B--C\n  C--A       C--A        C--A        C--A\n\nAll these figures are legitimate, though the fourth is comparatively\nvalueless.\n\n\u00a7 566. It will be well to explain by an instance the meaning of the\nassertion previously made, that a difference of figure is internal to\na difference of mood. We will take the mood EIO, and by varying the\nposition of the terms, construct a syllogism in it in each of the four\nfigures.\n\n  I.\n    E No wicked man is happy.\n    I Some prosperous men are wicked.\n    O .'. Some prosperous men are not happy.\n\n  II.\n    E No happy man is wicked.\n    I Some prosperous men are wicked.\n    O .'. Some prosperous men are not happy.\n\n  III.\n    E No wicked man is happy.\n    I Some wicked men are prosperous.\n    O .'. Some prosperous men are not happy.\n\n  IV.\n    E No happy man is wicked.\n    I Some wicked men are prosperous.\n    O .'. Some prosperous men are not happy.\n\n\u00a7 567. In the mood we have selected, owing to the peculiar nature of\nthe premisses, both of which admit of simple conversion, it happens\nthat the resulting syllogisms are all valid. But in the great majority\nof moods no syllogism would be valid at all, and in many moods a\nsyllogism would be valid in one figure and invalid in another. As yet\nhowever we are only concerned with the conceivable combinations, apart\nfrom the question of their legitimacy.\n\n\u00a7 568. Now since there are four different figures and sixty-four\ndifferent moods, we obtain in all 256 possible ways of arranging three\nterms in three propositions, that is, 256 possible forms of syllogism.\n\n\n\n\nCHAPTER X.\n\n_Of the Canon of Reasoning._\n\n\n& 569. The first figure was regarded by logicians as the only perfect\ntype of syllogism, because the validity of moods in this figure may be\ntested directly by their complying, or failing to comply, with a\ncertain axiom, the truth of which is self-evident. This axiom is known\nas the Dictum de Omni et Nullo. It may be expressed as follows--\n\n  Whatever may be affirmed or denied of a whole class may be affirmed\n  or denied of everything contained in that class.\n\n\u00a7 570. This mode of stating the axiom contemplates predication as\nbeing made in extension, whereas it is more naturally to be regarded\nas being made in intension.\n\n\u00a7 571. The same principle may be expressed intensively as follows--\n\n  Whatever has certain attributes has also the attributes which\n  invariably accompany them .[Footnote: Nota notae est nota rei\n  ipsius. 'Whatever has any mark has that which it is a mark of.'\n  Mill, vol. i, p. 201,]\n\n\u00a7 572. By Aristotle himself the principle was expressed in a neutral\nform thus--\n\n  'Whatever is stated of the predicate will be stated also of the\n  subject [Footnote: [Greek: osa kat\u00e0 to\u00fb kategoroum\u00e9nou l\u00e9getai p\u00e1nta ka\u00ec\n  kat\u00e0 to\u00fb hypokeim\u00e9nou rhaet\u00e9setai]. Cat. 3, \u00a7 I].'\n\nThis way of putting it, however, is too loose.\n\n\u00a7 573. The principle precisely stated is as follows--\n\n  Whatever may be affirmed or denied universally of the predicate of\n  an affirmative proposition, may be affirmed or denied also of the\n  subject.\n\n\u00a7 574. Thus, given an affirmative proposition 'Whales are mammals,' if\nwe can affirm anything universally of the predicate 'mammals,' as, for\ninstance, that 'All mammals are warm-blooded,' we shall be able to\naffirm the same of the subject 'whales'; and, if we can deny anything\nuniversally of the predicate, as that 'No mammals are oviparous,' we\nshall be able to deny the same of the subject.\n\n\u00a7 575. In whatever way the supposed canon of reasoning may be stated,\nit has the defect of applying only to a single figure, namely, the\nfirst. The characteristic of the reasoning in that figure is that some\ngeneral rule is maintained to hold good in a particular case. The\nmajor premiss lays down some general principle, whether affirmative or\nnegative; the minor premiss asserts that a particular case falls under\nthis principle; and the conclusion applies the general principle to\nthe particular case. But though all syllogistic reasoning may be\ntortured into conformity with this type, some of it finds expression\nmore naturally in other ways.\n\n\u00a7 576. Modern logicians therefore prefer to abandon the Dictum de Omni\net Nullo in any shape, and to substitute for it the following three\naxioms, which apply to all figures alike.\n\n\n\n_Three Axioms of Mediale Inference._\n\n\n  (1) If two terms agree with the same third term, they agree with one\n  another.\n\n  (2) If one term agrees, and another disagrees, with the same third\n  term, they disagree with one another.\n\n  (3) If two terms disagree with the same third term, they may or may\n  not agree with one another.\n\n\u00a7 577. The first of these axioms is the principle of all affirmative,\nthe second of all negative, syllogisms; the third points out the\nconditions under which no conclusion can be drawn.  If there is any\nagreement at all between the two terms and the third, as in the cases\ncontemplated in the first and second axioms, then we have a conclusion\nof some kind: if it is otherwise, we have none.\n\n\u00a7 578. It must be understood with regard to these axioms that, when we\nspeak of terms agreeing or disagreeing with the same third term, we\nmean that they agree or disagree with the same part of it.\n\n\u00a7 579. Hence in applying these axioms it is necessary to bear in mind\nthe rules for the distinction of terms.  Thus from\n\n  All B is A,\n  No C is B,\n\nthe only inference which can be drawn is that Some A is not C (which\nalters the figure from the first to the fourth).  For it was only part\nof A which was known to agree with B.  On the theory of the quantified\npredicate we could draw the inference No C is some A.\n\n\u00a7 580. It is of course possible for terms to agree with different\nparts of the same third term, and yet to have no connection with one\nanother. Thus\n\n  All birds fly.\n  All bats fly.\n\nBut we do not infer therefrom that bats are birds or vice vers\u00e2.\n\n\u00a7 581. On the other hand, had we said,--\n\n  All birds lay eggs,\n  No bats lay eggs,\n\nwe might confidently have drawn the conclusion\n\n  No bats are birds\n\nFor the term 'bats,' being excluded from the whole of the term 'lay\neggs,' is thereby necessarily excluded from that part of it which\ncoincides with 'birds.'\n\n[Illustration]\n\n\n\n\nCHAPTER XI.\n\n_Of the Generad Rules of Syllogism._\n\n\n\u00a7 582. We now proceed to lay down certain general rules to which all\nvalid syllogisms must conform.  These are divided into primary and\nderivative.\n\nI. _Primary_.\n\n  (1) A syllogism must consist of three propositions only.\n\n  (2) A syllogism must consist of three terms only.\n\n  (3) The middle term must be distributed at least once in the\n  premisses.\n\n  (4) No term must be distributed in the conclusion which was not\n  distributed in the premisses.\n\n  (5) Two negative premisses prove nothing.\n\n  (6) If one premiss be negative, the conclusion must be negative.\n\n  (7) If the conclusion be negative, one of the premisses must be\n  negative: but if the conclusion be affirmative, both premisses must\n  be affirmative.\n\nII. _Derivative_.\n\n  (8) Two particular premisses prove nothing.\n\n  (9) If one premiss be particular, the conclusion must be particular.\n\n\u00a7 583. The first two of these rules are involved in the definition of\nthe syllogism with which we started.  We said it might be regarded\neither as the comparison of two propositions by means of a third or as\nthe comparison of two terms by means of a third.  To violate either of\nthese rules therefore would be inconsistent with the fundamental\nconception of the syllogism.  The first of our two definitions indeed\n(\u00a7 552) applies directly only to the syllogisms in the first figure;\nbut since all syllogisms may be expressed, as we shall presently see,\nin the first figure, it applies indirectly to all.  When any process\nof mediate inference appears to have more than two premisses, it will\nalways be found that there is more than one syllogism.  If there are\nless than three propositions, as in the fallacy of 'begging the\nquestion,' in which the conclusion simply reiterates one of the\npremisses, there is no syllogism at all.\n\nWith regard to the second rule, it is plain that any attempted\nsyllogism which has more than three terms cannot conform to the\nconditions of any of the axioms of mediate inference.\n\n\u00a7 584.  The next two rules guard against the two fallacies which are\nfatal to most syllogisms whose constitution is unsound.\n\n\u00a7 585.  The violation of Rule 3 is known as the Fallacy of\nUndistributed Middle.  The reason for this rule is not far to seek.\nFor if the middle term is not used in either premiss in its whole\nextent, we may be referring to one part of it in one premiss and to\nquite another part of it in another, so that there will be really no\nmiddle term at all.  From such premisses as these--\n\n  All pigs are omnivorous,\n  All men are omnivorous,\n\nit is plain that nothing follows. Or again, take these premisses--\n\n  Some men are fallible,\n  All Popes are men.\n\nHere it is possible that 'All Popes' may agree with precisely that\npart of the term 'man,' of which it is not known whether it agrees\nwith 'fallible' or not.\n\n\u00a7 586. The violation of Rule 4 is known as the Fallacy of Illicit\nProcess. If the major term is distributed in the conclusion, not\nhaving been distributed in the premiss, we have what is called Illicit\nProcess of the Major; if the same is the case with the minor term, we\nhave Illicit Process of the Minor.\n\n\u00a7 587. The reason for this rule is that if a term be used in its whole\nextent in the conclusion, which was not so used in the premiss in\nwhich it occurred, we would be arguing from the part to the whole. It\nis the same sort of fallacy which we found to underlie the simple\nconversion of an A proposition.\n\n\u00a7 588. Take for instance the following--\n\n  All learned men go mad.\n  John is not a learned man.\n  .'. John will not go mad.\n\nIn the conclusion 'John' is excluded from the whole class of persons\nwho go mad, whereas in the premisses, granting that all learned men go\nmad, it has not been said that they are all the men who do so. We have\nhere an illicit process of the major term.\n\n\u00a7 589. Or again take the following--\n\n  All Radicals are covetous.\n  All Radicals are poor.\n  .'. All poor men are covetous.\n\nThe conclusion here is certainly not warranted by our premisses. For\nin them we spoke only of some poor men, since the predicate of an\naffirmative proposition is undistributed.\n\n\u00a7 590. Rule 5 is simply another way of stating the third axiom of\nmediate inference. To know that two terms disagree with the same third\nterm gives us no ground for any inference as to whether they agree or\ndisagree with one another, e.g.\n\n  Ruminants are not oviparous.\n  Sheep are not oviparous.\n\nFor ought that can be inferred from the premisses, sheep may or may\nnot be ruminants.\n\n\u00a7 591. This rule may sometimes be violated in appearance, though not\nin reality. For instance, the following is perfectly legitimate\nreasoning.\n\n  No remedy for corruption is effectual that does not render it\n  useless.\n  Nothing but the ballot renders corruption useless.\n   .'. Nothing but the ballot is an effectual remedy for corruption.\n\nBut on looking into this we find that there are four terms--\n\n  No not-A is B.\n  No not-C is A.\n  .'. No not-C is B.\n\nThe violation of Rule 5 is here rendered possible by the additional\nviolation of Rule 2. In order to have the middle term the same in both\npremisses we are obliged to make the minor affirmative, thus\n\n  No not-A is B.\n  All not-C is not-A.\n  .'. No not-C is B.\n\n  No remedy that fails to render corruption useless is effectual.\n  All but the ballot fails to render corruption useless.\n  .'. Nothing but the ballot is effectual.\n\n\u00a7 592. Rule 6 declares that, if one premiss be negative, the\nconclusion must be negative. Now in compliance with Rule 5, if one\npremiss be negative, the other must be affirmative. We have therefore\nthe case contemplated in the second axiom, namely, of one term\nagreeing and the other disagreeing with the same third term; and we\nknow that this can only give ground for a judgement of disagreement\nbetween the two terms themselves--in other words, to a negative\nconclusion.\n\n\u00a7 593. Rule 7 declares that, if the conclusion be negative, one of the\npremisses must be negative; but, if the conclusion be affirmative,\nboth premisses must be affirmative. It is plain from the axioms that a\njudgement of disagreement can only be elicited from a judgement of\nagreement combined with a judgement of disagreement, and that a\njudgement of agreement can result only from two prior judgements of\nagreement.\n\n\u00a7 594. The seven rules already treated of are evident by their own\nlight, being of the nature of definitions and axioms: but the two\nremaining rules, which deal with particular premisses, admit of being\nproved from their predecessors.\n\n\u00a7 595. Proof of Rule 8.--_That two particular premisses prove\nnothing_.\n\nWe know by Rule 5 that both premisses cannot be negative. Hence they\nmust be either both affirmative, II, or one affirmative and one\nnegative, IO or OI.\n\nNow II premisses do not distribute any term at all, and therefore the\nmiddle term cannot be distributed, which would violate Rule 3.\n\nAgain in IO or OI premisses there is only one term distributed,\nnamely, the predicate of the O proposition.  But Rule 3 requires that\nthis one term should be the middle term. Therefore the major term must\nbe undistributed in the major premiss. But since one of the premisses\nis negative, the conclusion must be negative, by Rule 6. And every\nnegative proposition distributes its predicate. Therefore the major\nterm must be distributed where it occurs as predicate of the\nconclusion. But it was not distributed in the major premiss. Therefore\nin drawing any conclusion we violate Rule 4 by an illicit process of\nthe major term.\n\n\u00a7 596. Proof of Rule 9.--_That_, _if_ one _premiss be\nparticular_, _the conclusion must be particular_.\n\nTwo negative premisses being excluded by Rule 5, and two particular by\nRule 8, the only pairs of premisses we can have are--\n\n  AI, AO, EI.\n\nOf course the particular premiss may precede the universal, but the\norder of the premisses will not affect the reasoning.\n\nAI premisses between them distribute one term only.  This must be the\nmiddle term by Rule 3. Therefore the conclusion must be particular, as\nits subject cannot be distributed,\n\nAO and EI premisses each distribute two terms, one of which must be\nthe middle term by Rule 3: so that there is only one term left which\nmay be distributed in the conclusion. But the conclusion must be\nnegative by Rule 4. Therefore its predicate must be distributed.\nHence its subject cannot be so. Therefore the conclusion must be\nparticular.\n\n\u00a7 597. Rules 6 and 9 are often lumped together in a single\nexpression--'The conclusion must follow the weaker part,' negative\nbeing considered weaker than affirmative, and particular than\nuniversal.\n\n\u00a7 598. The most important rules of syllogism are summed up in the\nfollowing mnemonic lines, which appear to have been perfected, though\nnot invented, by a medi\u00e6val logician known as Petrus Hispanus, who was\nafterwards raised to the Papal Chair under the title of Pope John XXI,\nand who died in 1277--\n\n  Distribuas medium, nec quartus terminus adsit;\n  Utraque nec praemissa negans, nec particularis;\n  Sectetur partem conclusio deteriorem,\n  Et non distribuat, nisi cum praemissa, negetve.\n\n\n\n\nCHAPTER XII.\n\n_Of the Determination of the Legitimate Moods of Syllogism._\n\n\n\u00a7 599. It will be remembered that there were found to be 64 possible\nmoods, each of which might occur in any of the four figures, giving us\naltogether 256 possible varieties of syllogism. The task now before us\nis to determine how many of these combinations of mood and figure are\nlegitimate.\n\n\u00a7 600. By the application of the preceding rules we are enabled to\nreduce the 64 possible moods to 11 valid ones.  This may be done by a\nlonger or a shorter method. The longer method, which is perhaps easier\nof comprehension, is to write down the 64 possible moods, and then\nstrike out such as violate any of the rules of syllogism.\n\n   AAA   -AEA-  -AIA-  -AOA-\n  -AAE-   AEE   -AIE-  -AOE-\n   AAI   -AEI-   AII   -AOI-\n  -AAO-   AEO   -AIO-   AOO\n\n  -EAA-  -EEA-  -EIA-  -EOA-\n   EAE   -EEE-  -EIE-  -EOE-\n  -EAI-  -EEI-  -EII-  -EOI-\n   EAO   -EEO-   EIO   -EOO-\n\n[Illustration]\n\n\u00a7 601. The batches which are crossed are those in which the premisses\ncan yield no conclusion at all, owing to their violating Rule 6 or 9;\nin the rest the premises are legitimate, but a wrong conclusion is\ndrawn from each of them as are translineated.\n\n\u00a7 602. IEO stands alone, as violating Rule 4. This may require a\nlittle explanation.\n\nSince the conclusion is negative, the major term, which is its\npredicate, must be distributed. But the major premiss, being 1, does\nnot distribute either subject or predicate. Hence IEO must always\ninvolve an illicit process of the major.\n\n\u00a7 603. The II moods which have been left valid, after being tested by\nthe syllogistic rules, are as follows--\n\n  AAA.  AAI.  AEE.  AEO.  AII.  AOO.\n  EAE.  EAO.  EIO.\n  IAI.\n  OAO.\n\n\u00a7 604. We will now arrive at the same result by a shorter and more\nscientific method. This method consists in first determining what\npairs of premisses are valid in accordance with Rules 6 and g, and\nthen examining what conclusions may be legitimately inferred from them\nin accordance with the other rules of syllogism.\n\n\u00a7 605. The major premiss may be either A, E, I or O.  If it is A, the\nminor also may be either A, E, I or O. If it is E, the minor can only\nbe A or I. If it is I, the minor can only be A or E. If it is O, the\nminor can only be A.  Hence there result 9 valid pairs of premisses.\n\n  AA.  AE.  AI.  AO.\n  EA.  EI.\n  IA.  IE.\n  OA.\n\nThree of these pairs, namely AA, AE, EA, yield two conclusions apiece,\none universal and one particular, which do not violate any of the\nrules of syllogism; one of them, IE, yields no conclusion at all; the\nremaining five have their conclusion limited to a single proposition,\non the principle that the conclusion must follow the weaker part.\nHence we arrive at the same result as before, of II legitimate moods--\n\n  AAA.   AAI.   AEE.   AEO.   EAE.   EAO.\n  AII.   AOO.   EIO.   IAI.   OAO.\n\n\n\n\n\nCHAPTER XIII.\n\n_Of the Special Rules of the Four Figures_.\n\n\n\u00a7 606. Our next task must be to determine how far the 11 moods which\nwe arrived at in the last chapter are valid in the four figures. But\nbefore this can be done, we must lay down the\n\n\n\n_Special Rules of the Four Figures_.\n\n\n  FIGURE 1.\n\n  Rule 1, The minor premiss must be affirmative.\n\n  Rule 2. The major premiss must be universal.\n\n\n  FIGURE II.\n\n  Rule 1. One or other premiss must be negative.\n\n  Rule 2. The conclusion must be negative.\n\n  Rule 3. The major premiss must be universal.\n\n\n  FIGURE III.\n\n  Rule 1. The minor premiss must be affirmative.\n\n  Rule 2. The conclusion must be particular.\n\n\n  FIGURE IV.\n\n  Rule 1. When the major premiss is affirmative, the minor must be\n  universal.\n\n  Rule 2. When the minor premiss is particular, the major must be\n  negative.\n\n  Rule 3, When the minor premiss is affirmative, the conclusion must\n  be particular.\n\n  Rule 4. When the conclusion is negative, the major premiss must be\n  universal.\n\n  Rule 5. The conclusion cannot be a universal affirmative.\n\n  Rule 6. Neither of the premisses can be a particular negative.\n\n\n\u00a7 607. The special rules of the first figure are merely a reassertion\nin another form of the Dictum de Omni et Nullo. For if the major\npremiss were particular, we should not have anything affirmed or\ndenied of a whole class; and if the minor premiss were negative, we\nshould not have anything declared to be contained in that class.\nNevertheless these rules, like the rest, admit of being proved from\nthe position of the terms in the figure, combined with the rules for\nthe distribution of terms (\u00a7 293).\n\n\n\n_Proof of the Special Rules of the Four Figures._\n\n\nFIGURE 1.\n\n\u00a7 608. Proof of Rule 1.--_The minor premiss must be affirmative_.\n\n  B--A\n  C--B\n  C--A\n\nIf possible, let the minor premiss be negative. Then the major must be\naffirmative (by Rule 5), [Footnote: This refers to the General Rules\nof Syllogism.] and the conclusion must be negative (by Rule 6). But\nthe major being affirmative, its predicate is undistributed; and the\nconclusion being negative, its predicate is distributed. Now the major\nterm is in this figure predicate both in the major premiss and in the\nconclusion. Hence there results illicit process of the major\nterm. Therefore the minor premiss must be affirmative.\n\n\u00a7 609. Proof of Rule 2.--_The major premiss must be universal._\n\nSince the minor premiss is affirmative, the middle term, which is its\npredicate, is undistributed there. Therefore it must be distributed in\nthe major premiss, where it is subject. Therefore the major premiss\nmust be universal.\n\n\nFIGURE II.\n\n\u00a7 610. Proof of Rule 1,--_One or other premiss must be negative_.\n\n  A--B\n  C--B\n  C--A\n\nThe middle term being predicate in both premisses, one or other must\nbe negative; else there would be undistributed middle.\n\n\u00a7 611. Proof of Rule 2.--_The conclusion must be negative._\n\nSince one of the premisses is negative, it follows that the conclusion\nalso must be so (by Rule 6).\n\n\u00a7 612. Proof of Rule 3.--_The major premiss must be universal._\n\nThe conclusion being negative, the major term will there be\ndistributed. But the major term is subject in the major\npremiss. Therefore the major premiss must be universal (by Rule 4).\n\n\nFIGURE III.\n\n\u00a7 613. Proof of Rule 1.--_The minor premiss must be affirmative._\n\n  B--A\n  B--C\n  C--A\n\nThe proof of this rule is the same as in the first figure, the two\nfigures being alike so far as the major term is concerned.\n\n\u00a7 614. Proof of Rule 2.--_The conclusion must be particular_.\n\nThe minor premiss being affirmative, the minor term, which is its\npredicate, will be undistributed there. Hence it must be undistributed\nin the conclusion (by Rule 4).  Therefore the conclusion must be\nparticular.\n\n\nFIGURE IV.\n\n\u00a7 615. Proof of Rule I.--_When the major premiss is affirmative,\nthe minor must be universal_.\n\nIf the minor were particular, there would be undistributed\nmiddle. [Footnote: Shorter proofs are employed in this figure, as the\nstudent is by this time familiar with the method of procedure.]\n\n\u00a7 616. Proof of Rule 2.--_When the minor premiss is particular, the\nmajor must be negative._\n\n  A--B\n  B--C\n  C--A\n\nThis rule is the converse of the preceding, and depends upon the same\nprinciple.\n\n\u00a7 617. Proof of Rule 3.--_When the minor premiss is affirmative, the\nconclusion must be particular._\n\nIf the conclusion were universal, there would be illicit process of\nthe minor.\n\n\u00a7 618. Proof of Rule 4.--_When the conclusion is negative, the major\npremiss must_ be universal.\n\nIf the major premiss were particular, there would be illicit process\nof the major.\n\n\u00a7 619. Proof of Rule 5.--_The conclusion CANNOT be A UNIVERSAL\naffirmative_.\n\nThe conclusion being affirmative, the premisses must be so too (by\nRule 7). Therefore the minor term is undistributed in the minor\npremiss, where it is predicate. Hence it cannot be distributed in the\nconclusion (by Rule 4).  Therefore the affirmative conclusion must be\nparticular.\n\n\u00a7 620. Proof of Rule 6.--_Neither of the premisses can lie a,\nPARTICULAR NEGATIVE_.\n\nIf the major premiss were a particular negative, the conclusion would\nbe negative. Therefore the major term would be distributed in the\nconclusion. But the major premiss being particular, the major term\ncould not be distributed there. Therefore we should have an illicit\nprocess of the major term.\n\nIf the minor premiss were a particular negative, then, since the major\nmust be affirmative (by Rule 5), we should have undistributed middle.\n\n\n\n\nCHAPTER XIV\n\n_Of the Determination of the Moods that are valid in the Four\nFigures._\n\n\n\u00a7 621. By applying the special rules just given we shall be able to\ndetermine how many of the eleven legitimate moods are valid in the\nfour figures.\n\n$622. These eleven legitimate moods were found to be\n\n  AAA.  AAI.  AEE.  AEO.  AII.  AOO.  EAE.\n  EAO.  EIO.  IAI.  OAO.\n\n\nFIGURE 1.\n\n\u00a7 623. The rule that the major premiss must be universal excludes the\nlast two moods, IAI, OAO. The rule that the minor premiss must be\naffirmative excludes three more, namely, AEE, AEO, AOO.\n\nThus we are left with six moods which are valid in the first figure,\nnamely,\n\n  AAA.  EAE.  AII.  EIO.  AAI.  EAO.\n\n\nFIGURE II.\n\n\u00a7 624. The rule that one premiss must be negative excludes four moods,\nnamely, AAA, AAI, AII, IAI. The rule that the major must be universal\nexcludes OAO. Thus we are left with six moods which are valid in the\nsecond figure, namely,\n\n  EAE.  AEE.  EIO.  AOO.  EAO.  AEO.\n\n\nFIGURE III.\n\n\u00a7 625. The rule that the conclusion must be particular confines us to\neight moods, two of which, namely AEE and AOO, are excluded by the\nrule that the minor premiss must be affirmative.\n\nThus we are left with six moods which are valid in the third figure,\nnamely,\n\n  AAI.  IAI.  AII.  EAO.  OAO.  EIO.\n\n\nFIGURE IV.\n\n\u00a7 626. The first of the eleven moods, AAA, is excluded by the rule\nthat the conclusion cannot be a universal affirmative.\n\nTwo more moods, namely AOO and OAO, are excluded by the rule that\nneither of the premisses can be a particular negative.\n\nAII violates the rule that when the major premiss is affirmative, the\nminor must be universal.\n\nEAE violates the rule that, when the minor premiss is affirmative, the\nconclusion must be particular.  Thus we are left with six moods which\nare valid in the fourth figure, namely,\n\n  AAI.  AEE.  IAI.  EAO.  EIO.  AEO.\n\n\u00a7 627. Thus the 256 possible forms of syllogism have been reduced to\ntwo dozen legitimate combinations of mood and figure, six moods being\nvalid in each of the four figures.\n\n  FIGURE I.   AAA.  EAE.  AII.  EIO.  (AAI.  EAO.)\n\n  FIGURE II.  EAE.  AEE.  EIO.  AGO.  (EAO.  AEO.)\n\n  FIGURE III. AAI.  IAI.  AII.  EAO.   OAO.  EIO.\n\n  FIGURE IV.  AAI.  AEE.  IAI.  EAO.   EIO. (AEO.)\n\n\u00a7 628. The five moods enclosed in brackets, though valid, are\nuseless. For the conclusion drawn is less than is warranted by the\npremisses. These are called Subaltern Moods, because their conclusions\nmight be inferred by subalternation from the universal conclusions\nwhich can justly be drawn from the same premisses. Thus AAI is\nsubaltern to AAA, EAO to EAE, and so on with the rest.\n\n\u00a7 629. The remaining 19 combinations of mood and figure, which are\nloosely called 'moods,' though in strictness they should be called\n'figured moods,' are generally spoken of under the names supplied by\nthe following mnemonics--\n\n  Barbara, Celarent, Darii, Ferioque prioris;\n  Cesare, Camestres, Festino, Baroko secund\u00e6;\n  Tertia Darapti, Disamis, Datisi, Felapton,\n  Bokardo, Ferison habet; Quarta insuper addit\n  Bramantip, Camenes, Dimaris, Fesapo, Fresison:\n  Quinque Subalterni, totidem Generalibus orti,\n  Nomen habent nullum, nee, si bene colligis, usum.\n\n\u00a7 630. The vowels in these lines indicate the letters of the mood. All\nthe special rules of the four figures can be gathered from an\ninspection of them. The following points should be specially noted.\n\nThe first figure proves any kind of conclusion, and is the only one\nwhich can prove A.\n\nThe second figure proves only negatives.\n\nThe third figure proves only particulars.\n\nThe fourth figure proves any conclusion except A.\n\n\u00a7 631. The first figure is called the Perfect, and the rest the\nImperfect figures. The claim of the first to be regarded as the\nperfect figure may be rested on these grounds--\n\n  1. It alone conforms directly to the Dictum de Omni et Nullo.\n\n  2. It suffices to prove every kind of conclusion, and is the only\n  figure in which a universal affirmative proposition can be\n  established.\n\n  3. It is only in a mood of this figure that the major, middle and\n  minor terms are to be found standing in their relative order of\n  extension.\n\n\u00a7 632. The reason why a universal affirmative, which is of course\ninfinitely the most important form of proposition, can only be proved\nin the first figure may be seen as follows.\n\n_Proof that A can only be established in figure I._\n\nAn A conclusion necessitates both premisses being A propositions (by\nRule 7). But the minor term is distributed in the conclusion, as being\nthe subject of an A proposition, and must therefore be distributed in\nthe minor premiss, in order to which it must be the subject. Therefore\nthe middle term must be the predicate and is consequently\nundistributed. In order therefore that the middle term may be\ndistributed, it must be subject in the major premiss, since that also\nis an A proposition. But when the middle term is subject in the major\nand predicate in the minor premiss, we have what is called the first\nfigure.\n\n\n\n\nCHAPTER XV.\n\n_Of the Special Canons of the Four Figures._\n\n\n\u00a7 633. So far we have given only a negative test of legitimacy, having\nshown what moods are not invalidated by running counter to any of the\nspecial rules of the four figures. We will now lay down special canons\nfor the four figures, conformity to which will serve as a positive\ntest of the validity of a given mood in a given figure. The special\ncanon of the first figure--will of course be practically equivalent to\nthe Dictum de Omni et Nullo. All of them will be expressed in terms of\nextension, for the sake of perspicuity.\n\n\n\n_Special Canons of the Four Figures._\n\n\nFIGURE 1.\n\n\u00a7 634. CANON. If one term wholly includes or excludes another, which\nwholly or partly includes a third, the first term wholly or partly\nincludes or excludes the third.\n\nHere four cases arise--\n\n  [Illustration]\n\n  (1) Total inclusion (Barbara).\n\n  All B is A.\n  All C is B.\n  .'. All C is A.\n\n\n  [Illustration]\n\n  (2) Partial inclusion (Darii).\n\n  All B is A.\n  Some C is B.\n  .'. Some C is A.\n\n\n  [Illustration]\n\n  (3) Total exclusion (Celarent).\n\n  No B is A.\n  All C is B.\n  .'. No C is A.\n\n\n  [Illustration]\n\n  (4) Partial exclusion (Ferio).\n\n  No B is A.\n  Some C is B.\n  .'. Some C is not A.\n\n  FIGURE II.\n\n\u00a7 635. CANON. If one term is excluded from another, which wholly or\npartly includes a third, or is included in another from which a third\nis wholly or partly excluded, the first is excluded from the whole or\npart of the third.\n\nHere we have four cases, all of exclusion--\n\n  (1) Total exclusion on the ground of inclusion in an excluded term\n  (Cesare).\n\n    [Illustration]\n\n    No A is B.\n    All C is B.\n    .'. No C is A.\n\n  (2) Partial exclusion on the ground of a similar partial inclusion\n  (Festino).\n\n    [Illustration]\n\n    No A is B.\n    Some C is B.\n    .'. Some C is not A.\n\n  (3) Total exclusion on the ground of exclusion from an including\n  term (Camestres).\n\n    [Illustration]\n\n    All A is B.\n    No C is B.\n    .'. No C is A.\n\n  (4) Partial exclusion on the ground of a similar partial exclusion\n  (Baroko).\n\n    [Illustration]\n\n    All A is B.\n    Some C is not B.\n    .'. Some C is not A.\n\n  FIGURE III.\n\n\u00a7 636. CANON. If two terms include another term in common, or if the\nfirst includes the whole and the second a part of the same term, or\nvice vers\u00e2, the first of these two terms partly includes the second;\nand if the first is excluded from the whole of a term which is wholly\nor in part included in the second, or is excluded from part of a term\nwhich is wholly included in the second, the first is excluded from\npart of the second.\n\nHere it is evident from the statement that six cases arise--\n\n  (1) Total inclusion of the same term in two others\n  (Darapti).\n\n    [Illustration]\n\n    All B is A.\n    All B is C.\n    .'. some C is A.\n\n  (2) Total inclusion in the first and partial inclusion\n  in the second (Datisi).\n\n    [Illustration]\n\n    All B is A.\n    Some B is C.\n    .'. some C is A.\n\n  (3) Partial inclusion in the first and total inclusion in\n  the second (Disamis).\n\n    [Illustration]\n\n    Some B is A.\n    All B is C.\n    .'. some C is A.\n\n  (4) Total exclusion of the first from a term which is\n  wholly included in the second (Felapton).\n\n    [Illustration]\n\n    No B is A.\n    All B is C.\n    .'. some C is not A.\n\n  (5) Total exclusion of the first from a term which is\n  partly included in the second (Ferison).\n\n    [Illustration]\n\n    No B is A.\n    Some B is C.\n    .'. some C is not A.\n\n  (6) Exclusion of the first from part of a term which\n  is wholly included in the second (Bokardo).\n\n    [Illustration]\n\n    Some B is not A.\n    All B is C.\n    .'. Some C is not A.\n\n  FIGURE IV.\n\n\u00a7 637. CANON. If one term is wholly or partly included in another\nwhich is wholly included in or excluded from a third, the third term\nwholly or partly includes the first, or, in the case of total\ninclusion, is wholly excluded from it; and if a term is excluded from\nanother which is wholly or partly included in a third, the third is\npartly excluded from the first.\n\nHere we have five cases--\n\n  (1) Of the inclusion of a whole term (Bramsntip).\n\n    [Illustration]\n    All A is B.\n    All B is C.\n    .'. Some C is (all) A.\n\n  (2) Of the inclusion of part of a term (DIMARIS).\n\n    [Illustration]\n\n    Some A is B.\n    All B is C.\n    .'. Some C is (some) A,\n\n  (3) Of the exclusion of a whole term (Camenes).\n\n    [Illustration]\n\n    All A is B.\n    No B is C.\n    .'. No C is A.\n\n  (4) Partial exclusion on the ground of including\n  the whole of an excluded term (Fesapo).\n\n    [Illustration]\n\n    No A is B.\n    All B is C.\n    .'. Some C is not A.\n\n  (5) Partial exclusion on the ground of including\n  part of an excluded term (Fresison).\n\n    [Illustration]\n\n    No A is B.\n    Some B is C.\n    .'. Some C is not A.\n\n\u00a7 638. It is evident from the diagrams that in the subaltern moods the\nconclusion is not drawn directly from the premisses, but is an\nimmediate inference from the natural conclusion. Take for instance AAI\nin the first figure. The natural conclusion from these premisses is\nthat the minor term C is wholly contained in the major term A. But\ninstead of drawing this conclusion we go on to infer that something\nwhich is contained in C, namely some C, is contained in A.\n\n  [Illustration]\n\n  All B is A.\n  All C is B.\n  .'. all C is A.\n  .'. some C is A.\n\nSimilarly in EAO in figure 1, instead of arguing that the whole of C\nis excluded from A, we draw a conclusion which really involves a\nfurther inference, namely that part of C is excluded from A.\n\n  [Illustration]\n\n  No B is A.\n  All C is B.\n  .'. no C is A.\n  .'. some C is not A.\n\n\u00a7 639. The reason why the canons have been expressed in so cumbrous a\nform is to render the validity of all the moods in each figure at once\napparent from the statement. For purposes of general convenience they\nadmit of a much more compendious mode of expression.\n\n\u00a7 640. The canon of the first figure is known as the Dictum de Omni et\nNullo--\n\n  What is true (distributively) of a whole term is true of all that it\n  includes.\n\n\u00a7 641. The canon of the second figure is known as the Dictum de\nDiverse--\n\n  If one term is contained in, and another excluded from a third term,\n  they are mutually excluded.\n\n\u00a7 642. The canon of the third figure is known as the Dictum de Exemplo\net de Excepto--\n\n  Two terms which contain a common part partly agree, or, if one\n  contains a part which the other does not, they partly differ.\n\n\u00a7 643. The canon of the fourth figure has had no name assigned to it,\nand does not seem to admit of any simple expression. Another mode of\nformulating it is as follows:--\n\n  Whatever is affirmed of a whole term may have partially affirmed of\n  it whatever is included in that term (Bramantip, Dimaris), and\n  partially denied of it whatever is excluded (Fesapo); whatever is\n  affirmed of part of a term may have partially denied of it whatever\n  is wholly excluded from that term (Fresison); and whatever is denied\n  of a whole term may have wholly denied of it whatever is wholly\n  included in that term (Camenes).\n\n\u00a7 644. From the point of view of intension the canons of the first\nthree figures may be expressed as follows.\n\n\u00a7 645. Canon of the first figure. Dictum de Omni et Nullo--\n\n  An attribute of an attribute of anything is an attribute of the\n  thing itself.\n\n\u00a7 646. Canon of the second figure. Dictum de Diverso--\n\n  If a subject has an attribute which a class has not, or vice versa,\n  the subject does not belong to the class.\n\n\u00a7 647. Canon of the third figure.\n\n  1. Dictum de Exemplo--\n\n    If a certain attribute can be affirmed of any portion of the\n    members of a class, it is not incompatible with the distinctive\n    attributes of that class.\n\n  2. Dictum de Excepto--\n\n    If a certain attribute can be denied of any portion of the members\n    of a class, it is not inseparable from the distinctive attributes\n    of that class.\n\n\n\n\nCHAPTER XVI.\n\n_Of the Special Uses of the Four Figures._\n\n\n\u00a7 648. The first figure is useful for proving the properties of a\nthing.\n\n\u00a7 649. The second figure is useful for proving distinctions between\nthings.\n\n\u00a7 650. The third figure is useful for proving instances or exceptions.\n\n\u00a7 651. The fourth figure is useful for proving the species of a genus.\n\nFIGURE 1.\n\n\u00a7 652.\n\n  B is or is not A.\n  C is B.\n  .'. C is or is not A.\n\nWe prove that C has or has not the property A by predicating of it B,\nwhich we know to possess or not to possess that property.\n\n  Luminous objects are material.\n  Comets are luminous.\n  .'. Comets are material.\n\n  No moths are butterflies.\n  The Death's head is a moth.\n  .'. The Death's head is not a butterfly.\n\nFIGURE II.\n\n\u00a7 653.\n\n  A is B. A is not B.\n  C is not B. C is B.\n  .'. C is not A. .'. C is not A.\n\nWe establish the distinction between C and A by showing that A has an\nattribute which C is devoid of, or is devoid of an attribute which C\nhas.\n\n  All fishes are cold-blooded.\n  A whale is not cold-blooded.\n  .'. A whale is not a fish.\n\n  No fishes give milk.\n  A whale gives milk.\n  .'. A whale is not a fish.\n\nFIGURE III.\n\n\u00a7 654.\n\n  B is A. B is not A.\n  B is C. B is C.\n  .'. Some C is A. .'. Some C is not A.\n\nWe produce instances of C being A by showing that C and A meet, at all\nevents partially, in B. Thus if we wish to produce an instance of the\ncompatibility of great learning with original powers of thought, we\nmight say\n\n  Sir William Hamilton was an original thinker.\n  Sir William Hamilton was a man of great learning.\n  .'. Some men of great learning are original thinkers.\n\nOr we might urge an exception to the supposed rule about Scotchmen\nbeing deficient in humour under the same figure, thus--\n\n  Sir Walter Scott was not deficient in humour.\n  Sir Walter Scott was a Scotchman.\n  .'. Some Scotchmen are not deficient in humour.\n\nFIGURE IV.\n\n\u00a7 655.\n\n  All A is B, No A is B.\n  All B is C. All B is C.\n  .'. Some C is A .'.Some C is not A.\n\nWe show here that A is or is not a species of C by showing that A\nfalls, or does not fall, under the class B, which itself falls under\nC. Thus--\n\n  All whales are mammals.\n  All mammals are warm-blooded.\n  .'. Some warm-blooded animals are whales.\n  No whales are fishes.\n  All fishes are cold-blooded.\n  .'. Some cold-blooded animals are not whales.\n\n\n\n\nCHAPTER XVII.\n\n_Of the Syllogism with three figures._\n\n\u00a7 656. It will be remembered that in beginning to treat of figure (\u00a7\n565) we pointed out that there were either four or three ligures\npossible according as the conclusion was assumed to be known or\nnot. For, if the conclusion be not known, we cannot distinguish\nbetween the major and the minor term, nor, consequently, between one\npremiss and another. On this view the first and the fourth figures are\nthe same, being that arrangement of the syllogism in which the middle\nterm occupies a different position in one premiss from what it does in\nthe other.  We will now proceed to constitute the legitimate moods and\nfigures of the syllogism irrespective of the conclusion.\n\n\u00a7 657. When the conclusion is set out of sight, the number of possible\nmoods is the same as the number of combinations that can be made of\nthe four things, A, E, I, O, taken two together, without restriction\nas to repetition. These are the following 16:--\n\n  AA   EA   IA   OA\n  AE  -EE-  IE  -OE-\n  AI   EI  -II- -OI-\n  AO  -EO- -IO- -OO-\n\nof which seven may be neglected as violating the general rules of the\nsyllogism, thus leaving us with nine valid moods--\n\n  AA.  AE.  AI.  AO.  EA.  EI.  IA.  IE.  OA.\n\n\u00a7 658. We will now put these nine moods successively into the three\nfigures. By so doing it will become apparent how far they are valid in\neach.\n\n\u00a7 659. Let it be premised that\n\n  when the extreme in the premiss that stands first is predicate in\n  the conclusion, we are said to have a Direct Mood;\n\n  when the extreme in the premiss that stands second is predicate in\n  the conclusion, we are said to have an Indirect Mood.\n\n\u00a7 660. FIGURE 1.\n\n  _Mood AA._\n  All B is A.\n  All C is B.\n  .'. All C is A, or Some A is C, (Barbara & Bramantip).\n\n  _Mood AE._\n  All B is A.\n  No C is B.\n  .'. Illicit Process, or Some A is not C, (Fesapo).\n\n  _Mood AI._\n  All B is A.\n  Some C is B.\n  .'. Some C is A, or Some A is C. (Darii & Disamis).\n\n  _Mood AO._\n  All B is A.\n  Some C is not B.\n  .'. Illicit Process, (Ferio).\n\n  _Mood EA._\n  No B is A.\n  All C is B.\n  .'. No C is A, or No A is C, (Celarent & Camenes).\n\n  _Mood EI._\n  No B is A.\n  Some C is B.\n  .'. Some C is not A, or Illicit Process.\n\n  _Mood IA._\n  Some B is A.\n  All C is B.\n  .'. Undistributed Middle.\n\n  _Mood IE._\n  Some B is C. Some B is not A.\n  No A is B. All C is B.\n  .'. Illicit Process, or Some C is not A, (Fresison).\n\n  _Mood OA._\n  Some B is not A.\n  All C is B.\n  .'. Undistributed Middle.\n\n\u00a7 661. Thus we are left with six valid moods, which yield four direct\nconclusions and five indirect ones, corresponding to the four moods of\nthe original first figure and the five moods of the original fourth,\nwhich appear now as indirect moods of the first figure.\n\n\u00a7 662. But why, it maybe asked, should not the moods of the first\nfigure equally well be regarded as indirect moods of the fourth? For\nthis reason-that all the moods of the fourth figure can be elicited\nout of premisses in which the terms stand in the order of the first,\nwhereas the converse is not the case. If, while retaining the quantity\nand quality of the above premisses, i. e. the mood, we were in each\ncase to transpose the terms, we should find that we were left with\nfive valid moods instead of six, since AI in the reverse order of the\nterms involves undistributed middle; and, though we should have\nCelarent indirect to Camenes, and Darii to Dimaris, we should never\narrive at the conclusion of Barbara or have anything exactly\nequivalent to Ferio. In place of Barbara, Bramantip would yield as an\nindirect mood only the subaltern AAI in the first figure. Both Fesapo\nand Fresison would result in an illicit process, if we attempted to\nextract the conclusion of Ferio from them as an indirect mood. The\nnearest approach we could make to Ferio would be the mood EAO in the\nfirst figure, which may be elicited indirectly from the premisses of\nCAMENES, being subaltern to CELARENT. For these reasons the moods of\nthe fourth figure are rightly to be regarded as indirect moods of the\nfirst, and not vice vers\u00e2.\n\n$663. FIGURE II.\n\n\n  _Mood AA._\n  All A is B.\n  All C is B.\n  .'. Undistributed Middle.\n\n  _Mood AE._\n  All A is B.\n  No C is B.\n  .'. No C is A, or No A is C, (Camestres & Cesare).\n\n  _Mood AI._\n  All A is B.\n  Some C is B.\n  .'. Undistributed Middle.\n\n  _Mood AO._\n  All A is B.\n  Some C is not B.\n  .'. Some C is not A, (Baroko), or Illicit Process.\n\n  _Mood EA._\n  No A is B.\n  All C is B.\n  .'. No C is A, or No A is C, (Cesare & Carnestres).\n\n\n  _Mood EI_\n  No A is B.\n  Some C is B.\n  .'. Some C is not A, (Festino), or Illicit Process.\n\n  _Mood IA._\n  Some A is B.\n  All C is B.\n  .'. Undistributed Middle.\n\n   _Mood IE._\n   Some A is B.\n   No C is B.\n   .'. Illicit Process, or Some A is not C, (Festino).\n\n   _Mood OA._\n   Some A is not B.\n   All C is B.\n   .'. Illicit Process, or Some A is not C, (Baroko).\n\n\u00a7 664. Here again we have six valid moods, which yield four direct\nconclusions corresponding to Cesare, CARNESTRES, FESTINO and\nBAROKO. The same four are repeated in the indirect moods.\n\n\u00a7 665. FIGURE III.\n\n  _Mood AA._\n  All B is A.\n  All B is C.\n  .'. Some C is A, or Some A is C, (Darapti).\n\n  _Mood AE._\n  All B is A.\n  No B is C.\n  .'. Illicit Process, or Some A is not C, (Felapton).\n\n  _Mood AI._\n  All B is A,\n  Some B is C.\n  .'. Some C is A, or Some A is C, (Datisi & Disamis).\n\n  _Mood AO._\n  All B is A.\n  Some B is not C.\n  .'. Illicit Process, Or Some A is not C, (Bokardo).\n\n  _Mood EA._\n  No B is A.\n  All B is C.\n  .'. Some C is not A, (Felapton), or Illicit Process.\n\n  _Mood EI._\n  No B is A.\n  Some B is C.\n  .'. Some C is not A, (Ferison), or Illicit Process.\n\n  _Mood IA._\n  Some B is A.\n  All B is C.\n  .'. Some C is A, Or Some A is C, (Disamis & Datisi).\n\n  _Mood IE._\n  Some B is A.\n  No B is C.\n  .'. Illicit Process, or Some A is not C, (Ferison).\n\n  _Mood QA._\n  Some B is not A.\n  All B is C.\n  .'. Some C is not A, (Bokardo), or Illicit Process.\n\n\u00a7 666. In this figure every mood is valid, either directly or\nindirectly. We have six direct moods, answering to Darapti, Disamis,\nDatisi, Felapton, Bokardo and Ferison, which are simply repeated by\nthe indirect moods, except in the case of Darapti, which yields a\nconclusion not provided for in the mnemonic lines. Darapti, though\ngoing under one name, has as much right to be considered two moods as\nDisamis and Datisi.\n\n\n\n\nCHAPTER XVIII.\n\n_Of Reduction._\n\n\n\u00a7 667. We revert now to the standpoint of the old logicians, who\nregarded the Dictum de Omni et Nullo as the principle of all\nsyllogistic reasoning. From this point of view the essence of mediate\ninference consists in showing that a special case, or class of cases,\ncomes under a general rule. But a great deal of our ordinary reasoning\ndoes not conform to this type. It was therefore judged necessary to\nshow that it might by a little manipulation be brought into conformity\nwith it. This process is called Reduction.\n\n\u00a7 668. Reduction is of two kinds--\n\n  (1) Direct or Ostensive.\n\n  (2) Indirect or Ad Impossibile.\n\n\u00a7 669. The problem of direct, or ostensive, reduction is this--\n\n  Given any mood in one of the imperfect figures (II, III and IV) how\n  to alter the form of the premisses so as to arrive at the same\n  conclusion in the perfect figure, or at one from which it can be\n  immediately inferred. The alteration of the premisses is effected by\n  means of immediate inference and, where necessary, of transposition.\n\n\u00a7 670. The problem of indirect reduction, or reductio (per\ndeductionem) ad impossibile, is this--Given any mood in one of the\nimperfect figures, to show by means of a syllogism in the perfect\nfigure that its conclusion cannot be false.\n\n\u00a7 671. The object of reduction is to extend the sanction of the Dictum\nde Omni et Nullo to the imperfect figures, which do not obviously\nconform to it.\n\n\u00a7 672. The mood required to be reduced is called the Reducend; that to\nwhich it conforms, when reduced, is called the Reduct.\n\n\n\n_Direct or Ostensive Reduction._\n\n\n\u00a7 673. In the ordinary form of direct reduction, the only kind of\nimmediate inference employed is conversion, either simple or by\nlimitation; but the aid of permutation and of conversion by negation\nand by contraposition may also be resorted to.\n\n\u00a7 674. There are two moods, Baroko and Bokardo, which cannot be\nreduced ostensively except by the employment of some of the means last\nmentioned. Accordingly, before the introduction of permutation into\nthe scheme of logic, it was necessary to have recourse to some other\nexpedient, in order to demonstrate the validity of these two\nmoods. Indirect reduction was therefore devised with a special view to\nthe requirements of Baroko and Bokardo: but the method, as will be\nseen, is equally applicable to all the moods of the imperfect figures.\n\n\u00a7 675. The mnemonic lines, 'Barbara, Celarent, etc., provide complete\ndirections for the ostensive reduction of all the moods of the second,\nthird, and fourth figures to the first, with the exception of Baroko\nand Bokardo. The application of them is a mere mechanical trick, which\nwill best be learned by seeing the process performed.\n\n\u00a7 676. Let it be understood that the initial consonant of each name of\na figured mood indicates that the reduct will be that mood which\nbegins with the same letter.  Thus the B of Bramantip indicates that\nBramantip, when reduced, will become Barbara.\n\n\u00a7 677. Where m appears in the name of a reducend, me shall have to\ntake as major that premiss which before was minor, and vice versa-in\nother words, to transpose the premisses, m stands for mutatio or\nmetathesis.\n\n\u00a7 678. s, when it follows one of the premisses of a reducend,\nindicates that the premiss in question must be simply converted; when\nit follows the conclusion, as in Disamis, it indicates that the\nconclusion arrived at in the first figure is not identical in form\nwith the original conclusion, but capable of being inferred from it by\nsimple conversion. Hence s in the middle of a name indicates something\nto be done to the original premiss, while s at the end indicates\nsomething to be done to the new conclusion.\n\n\u00a7 679. P indicates conversion per accidens, and what has just been\nsaid of s applies, mutatis mutandis, to p.\n\n\u00a7 680. k may be taken for the present to indicate that Baroko and\nBokardo cannot be reduced ostensively.\n\n\u00a7 681. FIGURE II.\n\n  Cesare.        \\       /  Celarent.\n  No A is B.      \\  =  /   No B is A.\n  All C is B.     /     \\   All C is B.\n  .'. No C is A. /       \\  .'. No C is A.\n\n  Camestres.     \\       /  Celarent.\n  All A is B.     \\  =  /   No B is C.\n  No C is B.      /     \\   All A is B.\n  .'. No C is A. /       \\  .'. No A is C.\n                            .'. No C is A.\n\n  Festino.                    Ferio.\n  No A is B.          \\     / No B is A.\n  Some C is B.         | = |  Some C is B.\n  .'. Some C is not A./     \\ .'. Some C is not A.\n                      [Baroko]\n\n\u00a7 682. FIGURE III.\n\n  Darapti.         \\     /   Darii.\n  All B is A.       \\ = /    All B is A.\n  All B is C.       /   \\    Some C is B.\n  .'. Some C is A. /     \\   Some C is A.\n\n  Disamis.         \\     /   Darii.\n  Some B is A.      \\ = /    All B is C.\n  All B is C.       /   \\    Some A is B.\n  .'. Some C is A. /     \\   .'. Some A is C.\n                             .'. Some C is A.\n\n  Datisi.          \\     / Darii.\n  All B is A.       \\ = /  All B is A.\n  Some B is C.      /   \\  Some C is B.\n  .'. Some C is A. /     \\ .'. Some C is A.\n\n  Felapton.            \\     / Ferio.\n  No B is A.            \\ = /  No B is A.\n  All B is C.           /   \\  Some C is B.\n  .'. Some C is not-A. /     \\ .'. Some C is not-A.\n\n  [Bokardo].\n\n  Ferison.             \\     / Ferio.\n  No B is A.            \\ = /  No B is A.\n  Some B is C.          /   \\  Some C is B\n  .'. Some C is not A. /     \\ .'. Some C is not A.\n\n\u00a7 683. FIGURE IV.\n\n  Bramantip.      \\     / Barbara.\n  All A is B.      \\ = /  All B is C.\n  All B is C.      /   \\  All A is B.\n  .. Some C is A. /     \\ .. All A is C.\n                          .'. Some C is A.\n\n  Camenes              Celarent\n  All A is B   \\     / No B is C.\n  No B is C.    | = |  All A is B.\n  .. No C is A./     \\ .'. No A is C.\n                       .'. No C is A.\n\n  Dimaris.                Darii.\n  Some A is B.    \\     / All B is C.\n  All B is C.      | = |  Some A is B.\n  .'. Some C is A./     \\ .'. Some A is C.\n                          .'. Some C is A.\n\n  Fesapo.                     Ferio.\n  No A is B.          \\     / No B is A.\n  All B is C.          | = |  Some C is B.\n  .'. Some C is not A./     \\ .'. Some C is not A.\n\n  Fresison.                   Ferio.\n  No A is B.          \\     / No B is A.\n  Some B is C.         | = |  Some C is B.\n  .'. Some C is not A./     \\ .'. Some C is not A.\n\n\u00a7 684. The reason why Baroko and Bokardo cannot be reduced ostensively\nby the aid of mere conversion becomes plain on an inspection of\nthem. In both it is necessary, if we are to obtain the first figure,\nthat the position of the middle term should be changed in one\npremiss. But the premisses of both consist of A and 0 propositions, of\nwhich A admits only of conversion by limitation, the effect of which\nwould be to produce two particular premisses, while 0 does not admit\nof conversion at all,\n\nIt is clear then that the 0 proposition must cease to be 0 before we\ncan get any further. Here permutation comes to our aid; while\nconversion by negation enables us to convert the A proposition,\nwithout loss of quantity, and to elicit the precise conclusion we\nrequire out of the reduct of Boltardo.\n\n  (Baroko) Fanoao.            Ferio.\n  All A is B.         \\     / No not-B is A.\n  Some C is not-B.     | = |  Some C is not-B.\n  .'. Some C is not-A./     \\ .'. Some C is not-A.\n\n  (Bokardo) Donamon.          Darii.\n  Some B is not-A.    \\     / All B is C.\n  All B is C.          | = |  Some not-A is B\n  .'. Some C is not-A./     \\ .'. Some not-A is C.\n                              .'. Some C is not-A.\n\n\u00a7 685. In the new symbols, Fanoao and Donamon, [pi] has been\nadopted as a symbol for permutation; n signifies conversion by\nnegation. In Donamon the first n stands for a process which resolves\nitself into permutation followed by simple conversion, the second for\none which resolves itself into simple conversion followed by\npermutation, according to the extended meaning which we have given to\nthe term 'conversion by negation.' If it be thought desirable to\ndistinguish these two processes, the ugly symbol Do[pi]samos[pi] may\nbe adopted in place of Donamon.\n\n\u00a7 686. The foregoing method, which may be called Reduction by\nNegation, is no less applicable to the other moods of the second\nfigure than to Baroko. The symbols which result from providing for its\napplication would make the second of the mnemonic lines run thus--\n\n  Benare[pi], Cane[pi]e, Denilo[pi], Fano[pi]o secundae.\n\n\u00a7 687. The only other combination of mood and figure in which it will\nbe found available is Camenes, whose name it changes to Canene.\n\n\u00a7 688.\n\n  (Cesare) Benarea.           Barbara.\n  No A is B.          \\     / All B is not-A.\n  All C is B.          | = |  All C is B.\n  .'. No C is A.      /     \\ .'. All C is not-A.\n                              .'. No C is A.\n\n  (Camestres) Cane[pi]e.      Celarent.\n  All A is B.         \\     / No not-B is A.\n  No C is B.           | = |  All C is not-B.\n  .'. No C is A.      /     \\ .'. No C is A.\n\n  (Festino) Denilo[pi].       Darii.\n  No A is B.          \\     / All B is not-A.\n  Some C is B.         | = |  Some C is B.\n  .'. Some C is not A./     \\ .'. Some C is not-A.\n                              .'. Some C is not A.\n\n  (Camenes) Canene.           Celarent.\n  All A is B.         \\     / No not-B is A.\n  No B is C.           | = |  All C is not-B.\n  .'. No C is A.      /     \\ .'. No C is A.\n\n\u00a7 689. The following will serve as a concrete instance of Cane[pi]e\nreduced to the first figure.\n\n  All things of which we have a perfect idea are perceptions.\n  A substance is not a perception.\n  .'. A substance is not a thing of which we have a perfect idea.\n\nWhen brought into Celarent this becomes--\n\n  No not-perception is a thing of which we have a perfect idea.\n  A substance is a not-perception.\n  .'. No substance is a thing of which we have a perfect idea.\n\n\u00a7 690. We may also bring it, if we please, into Barbara, by permuting\nthe major premiss once more, so as to obtain the contrapositive of the\noriginal--\n\n  All not-perceptions are things of which we have an imperfect idea.\n  All substances are not-perceptions.\n  .'. All substances are things of which we have an imperfect idea.\n\n\n\n_Indirect Reduction._\n\n\n\u00a7 691. We will apply this method to Baroko.\n\n  All A is B.           All fishes are oviparous.\n  Some C is not B.      Some marine animals are not oviparous.\n  .'. Some C is not A.  .'. Some marine animals are not fishes.\n\n\u00a7 692. The reasoning in such a syllogism is evidently conclusive: but\nit does not conform, as it stands, to the first figure, nor\n(permutation apart) can its premisses be twisted into conformity with\nit. But though we cannot prove the conclusion true in the first\nfigure, we can employ that figure to prove that it cannot be false, by\nshowing that the supposition of its falsity would involve a\ncontradiction of one of the original premisses, which are true ex\nhypothesi.\n\n\u00a7 693. If possible, let the conclusion 'Some C is not A' be\nfalse. Then its contradictory 'All C is A' must be true. Combining\nthis as minor with the original major, we obtain premisses in the\nfirst figure,\n\n  All A is B,   All fishes are oviparous,\n  All C is A,   All marine animals are fishes,\n\nwhich lead to the conclusion\n\n  All C is B,   All marine animals are oviparous.\n\nBut this conclusion conflicts with the original minor, 'Some C is not\nB,' being its contradictory. But the original minor is ex hypothesi\ntrue. Therefore the new conclusion is false. Therefore it must either\nbe wrongly drawn or else one or both of its premisses must be false.\nBut it is not wrongly drawn; since it is drawn in the first figure, to\nwhich the Dictum de Omni et Nullo applies.  Therefore the fault must\nlie in the premisses. But the major premiss, being the same with that\nof the original syllogism, is ex hypothesi true. Therefore the minor\npremiss, 'All C is A,' is false. But this being false, its\ncontradictory must be true. Now its contradictory is the original\nconclusion, 'Some C is not A,' which is therefore proved to be true,\nsince it cannot be false.\n\n\u00a7 694. It is convenient to represent the two syllogisms in\njuxtaposition thus--\n\n    Baroko.                  Barbara.\n  All A is B.              All A is B.\n  Some C is not B.     \\/  All C is A.\n  .'. Some C is not A. /\\  All C is B.\n\n\u00a7 695. The lines indicate the propositions which conflict with one\nanother. The initial consonant of the names Baroko and Eokardo\nindicates that the indirect reduct will be Barbara. The k indicates\nthat the O proposition, which it follows, is to be dropped out in the\nnew syllogism, and its place supplied by the contradictory of the old\nconclusion.\n\n\u00a7 696. In Bokardo the two syllogisms will stand thus--\n\n    Bokardo.                Barbara.\n  Some B is not A.    \\ / All C is A.\n  All B is C.          X  All B is C.\n  .'. Some C is not A./ \\ .'. All B is A.\n\n\u00a7 697. The method of indirect reduction, though invented with a\nspecial view to Baroko and Bokardo, is applicable to all the moods of\nthe imperfect figures. The following modification of the mnemonic\nlines contains directions for performing the process in every\ncase:--Barbara, Celarent, Darii, Ferioque prioris; Felake, Dareke,\nCeliko, Baroko secundae; Tertia Cakaci, Cikari, Fakini, Bekaco,\nBokardo, Dekilon habet; quarta insuper addit Cakapi, Daseke, Cikasi,\nCepako, Ces\u00efkon.\n\n\u00a7 698. The c which appears in two moods of the third figure, Cakaci\nand Bekaco, signifies that the new conclusion is the contrary, instead\nof, as usual, the contradictory of the discarded premiss.\n\n\u00a7 699. The letters s and p, which appear only in the fourth figure,\nsignify that the new conclusion does not conflict directly with the\ndiscarded premiss, but with its converse, either simple or per\naccidens, as the case may be.\n\n\u00a7 700. l, n and r are meaningless, as in the original lines.\n\n\n\n\nCHAPTER XIX.\n\n_Of Immediate Inference as applied to Complex Propositions._\n\n\n\u00a7 701. So far we have treated of inference, or reasoning, whether\nmediate or immediate, solely as applied to simple propositions. But it\nwill be remembered that we divided propositions into simple and\ncomplex. I t becomes incumbent upon us therefore to consider the laws\nof inference as applied to complex propositions. Inasmuch however as\nevery complex proposition is reducible to a simple one, it is evident\nthat the same laws of inference must apply to both.\n\n\u00a7 702. We must first make good this initial statement as to the\nessential identity underlying the difference of form between simple\nand complex propositions.\n\n\u00a7 703. Complex propositions are either Conjunctive or Disjunctive (\u00a7\n214).\n\n\u00a7 704. Conjunctive propositions may assume any of the four forms, A,\nE, I, O, as follows--\n\n  (A) If A is B, C is always D.\n  (E) If A is B, C is never D.\n  (I) If A is B, C is sometimes D.\n  (O) If A is B, C is sometimes not D.\n\n\u00a7 705. These admit of being read in the form of simple propositions,\nthus--\n\n  (A) If A is B, C is always D = All cases of A being B are cases of C\n  being D. (Every AB is a CD.)\n\n  (E) If A is B, C is never D = No cases of A being B are cases of C\n  being D. (No AB is a CD.)\n\n  (I) If A is B, C is sometimes D = Some cases of A being B are cases\n  of C being D. (Some AB's are CD's.)\n\n  (O) If A is B, C is sometimes not D = Some cases of A being B are\n  not cases of C being D.  (Some AB's are not CD's.)\n\n\u00a7 706. Or, to take concrete examples,\n\n  (A) If kings are ambitious, their subjects always suffer.\n  = All cases of ambitious kings are cases of subjects suffering.\n\n  (E) If the wind is in the south, the river never freezes.\n  = No cases of wind in the south are cases of the river freezing.\n\n  (I) If a man plays recklessly, the luck sometimes goes against him.\n  = Some cases of reckless playing are cases of going against one.\n\n  (O) If a novel has merit, the public sometimes do not buy it.\n  = Some cases of novels with merit are not cases of the public buying.\n\n\u00a7 707. We have seen already that the disjunctive differs from the\nconjunctive proposition in this, that in the conjunctive the truth\nof the antecedent involves the truth of the consequent, whereas in the\ndisjunctive the falsity of the antecedent involves the truth of the\nconsequent.  The disjunctive proposition therefore\n\n  Either A is B or C is D\n\nmay be reduced to a conjunctive\n\n  If A is not B, C is D,\n\nand so to a simple proposition with a negative term for subject.\n\n  All cases of A not being B are cases of C being D.\n  (Every not-AB is a CD.)\n\n\u00a7 708. It is true that the disjunctive proposition, more than any\nother form, except U, seems to convey two statements in one\nbreath. Yet it ought not, any more than the E proposition, to be\nregarded as conveying both with equal directness. The proposition 'No\nA is B' is not considered to assert directly, but only implicitly,\nthat 'No B is A.' In the same way the form 'Either A is B or C is D'\nought to be interpreted as meaning directly no more than this, 'If A\nis not B, C is D.' It asserts indeed by implication also that 'If C is\nnot D, A is B.' But this is an immediate inference, being, as we shall\npresently see, the contrapositive of the original. When we say 'So and\nso is either a knave or a fool,' what we are directly asserting is\nthat, if he be not found to be a knave, he will be found to be a\nfool. By implication we make the further statement that, if he be not\ncleared of folly, he will stand condemned of knavery. This inference\nis so immediate that it seems indistinguishable from the former\nproposition: but since the two members of a complex proposition play\nthe part of subject and predicate, to say that the two statements are\nidentical would amount to asserting that the same proposition can have\ntwo subjects and two predicates. From this point of view it becomes\nclear that there is no difference but one of expression between the\ndisjunctive and the conjunctive proposition.  The disjunctive is\nmerely a peculiar way of stating a conjunctive proposition with a\nnegative antecedent.\n\n\u00a7 709. Conversion of Complex Propositions.\n\n  A  / If A is B, C is always D.\n     \\ .'. If C is D, A is sometimes B.\n\n  E  / If A is B, C is never D.\n     \\ .'. If C is D, A is never B.\n\n  I  / If A is S, C is sometimes D.\n     \\ .'. If C is D, A is sometimes B.\n\n\u00a7 710. Exactly the same rules of conversion apply to conjunctive as to\nsimple propositions.\n\n\u00a7 711. A can only be converted per accidens, as above.\n\nThe original proposition\n\n  'If A is B, C is always D'\n\nis equivalent to the simple proposition\n\n  'All cases of A being B are cases of C being D.'\n\nThis, when converted, becomes\n\n  'Some cases of C being D are cases of A being B,'\n\nwhich, when thrown back into the conjunctive form, becomes\n\n  'If C is D, A is sometimes B.'\n\n\u00a7 712. This expression must not be misunderstood as though it\ncontained any reference to actual existence.  The meaning might be\nbetter conveyed by the form\n\n  'If C is D, A may be B.'\n\nBut it is perhaps as well to retain the other, as it serves to\nemphasize the fact that formal logic is concerned only with the\nconnection of ideas.\n\n\u00a7 713. A concrete instance will render the point under discussion\nclearer. The example we took before of an A proposition in the\nconjunctive form--\n\n  'If kings are ambitious, their subjects always suffer'\n\nmay be converted into\n\n  'If subjects suffer, it may be that their kings are ambitious,'\n\ni.e. among the possible causes of suffering on the part of subjects is\nto be found the ambition of their rulers, even if every actual case\nshould be referred to some other cause. It is in this sense only that\nthe inference is a necessary one. But then this is the only sense\nwhich formal logic is competent to recognise. To judge of conformity\nto fact is no part of its province. From 'Every AB is a CD' it follows\nthat ' Some CD's are AB's' with exactly the same necessity as that\nwith which 'Some B is A' follows from 'All A is B.' In the latter case\nalso neither proposition may at all conform to fact.  From 'All\ncentaurs are animals' it follows necessarily that 'Some animals are\ncentaurs': but as a matter of fact this is not true at all.\n\n\u00a7 714. The E and the I proposition may be converted simply, as above.\n\n\u00a7 715. O cannot be converted at all. From the proposition\n\n  'If a man runs a race, he sometimes does not win it,'\n\nit certainly does not follow that\n\n  'If a man wins a race, he sometimes does not run it.'\n\n\u00a7 716. There is a common but erroneous notion that all conditional\npropositions are to be regarded as affirmative. Thus it has been\nasserted that, even when we say that 'If the night becomes cloudy,\nthere will be no dew,' the proposition is not to be regarded as\nnegative, on the ground that what we affirm is a relation between the\ncloudiness of night and the absence of dew. This is a possible, but\nwholly unnecessary, mode of regarding the proposition. It is precisely\non a par with Hobbes's theory of the copula in a simple proposition\nbeing always affirmative. It is true that it may always be so\nrepresented at the cost of employing a negative term; and the same is\nthe case here.\n\n\u00a7 717. There is no way of converting a disjunctive proposition except\nby reducing it to the conjunctive form.\n\n\u00a7 718. _Permutation of Complex Propositions_.\n\n  (A) If A is B, C is always D.\n      .'. If A is B, C is never not-D. (E)\n\n  (E) If A is B, C is never D.\n      .'. If A is B, C is always not-D. (A)\n\n  (I) If A is B, C is sometimes D.\n      .'. If A is B, C is sometimes not not-D. (O)\n\n  (O) If A is B, C is sometimes not D.\n      .'. If A is B, C is sometimes not-D. (I)\n\n\u00a7 719.\n\n  (A) If a mother loves her children, she is always kind to them.\n      .'. If a mother loves her children, she is never unkind to\n      them. (E)\n\n  (E) If a man tells lies, his friends never trust him.\n      .'. If a man tells lies, his friends always distrust him. (A)\n\n  (I) If strangers are confident, savage dogs are sometimes friendly.\n      .'. If strangers are confident, savage dogs are sometimes not\n      unfriendly. (O)\n\n  (O) If a measure is good, its author is sometimes not popular.\n      .'. If a measure is good, its author is sometimes\n      unpopular. (I)\n\n\u00a7 720. The disjunctive proposition may be permuted as it stands\nwithout being reduced to the conjunctive form.\n\n  Either A is B or C is D.\n  .'. Either A is B or C is not not-D.\n\n  Either a sinner must repent or he will be damned.\n  .'. Either a sinner must repent or he will not be saved.\n\n\u00a7 721. _Conversion by Negation of Complex Propositions._\n\n  (A) If A is B, C is always D.\n      .'. If C is not-D, A is never B. (E)\n\n  (E) If A is B, C is never D.\n      .'. If C is D, A is always not-B. (A)\n\n  (I) If A is B, C is sometimes D.\n      .'. If C is D, A is sometimes not not-B. (O)\n\n  (O) If A is B, C is sometimes not D.\n      .'. If C is not-D, A is sometimes B. (I)\n\n  (E per acc.) If A is B, C is never D.\n      .'. If C is not-D, A is sometimes B. (I)\n\n  (A per ace.) If A is B, C is always D.\n      .'. If C is D, A is sometimes not not-D. (O)\n\n\u00a7 722.\n\n  (A) If a man is a smoker, he always drinks.\n      .'. If a man is a total abstainer, he never smokes. (E)\n\n  (E) If a man merely does his duty, no one ever thanks him.\n      .'. If people thank a man, he has always done more than his\n      duty. (A)\n\n  (I) If a statesman is patriotic, he sometimes adheres to a party.\n      .'. If a statesman adheres to a party, he is sometimes not\n      unpatriotic. (O)\n\n  (O) If a book has merit, it sometimes does not sell.\n      .'. If a book fails to sell, it sometimes has merit. (I)\n\n  (E per acc.) If the wind is high, rain never falls.\n      .'. If rain falls, the wind is sometimes high. (I)\n\n  (A per acc.) If a thing is common, it is always cheap.\n      .'. If a thing is cheap, it is sometimes not uncommon. (O)\n\n\u00a7 723. When applied to disjunctive propositions, the distinctive\nfeatures of conversion by negation are still discernible. In each of\nthe following forms of inference the converse differs in quality from\nthe convertend and has the contradictory of one of the original terms\n(\u00a7 515).\n\n\u00a7 724.\n\n  (A) Either A is B or C is always D.\n      .'. Either C is D or A is never not-B. (E)\n\n  (E) Either A is B or C is never D.\n      .'. Either C is not-D or A is always B. (A)\n\n  (I) Either A is B or C is sometimes D.\n      .'. Either C is not-D or A is sometimes not B. (O)\n\n  (O) Either A is B or C is sometimes not D.\n      .'. Either C is D or A is sometimes not-B. (I)\n\n\u00a7 725.\n\n  (A) Either miracles are possible or every ancient historian is\n      untrustworthy.\n      .'. Either ancient historians are untrustworthy or miracles are\n      not impossible. (E)\n\n  (E) Either the tide must turn or the vessel can not make the port.\n      .'. Either the vessel cannot make the port or the tide must\n      turn. (A)\n\n  (1) Either he aims too high or the cartridges are sometimes bad.\n      .'. Either the cartridges are not bad or he sometimes does not\n      aim too high. (0)\n\n  (O) Either care must be taken or telegrams will sometimes not be\n      correct.\n      .'. Either telegrams are correct or carelessness is sometimes\n      shown. (1)\n\n\u00a7 726. In the above examples the converse of E looks as if it had\nundergone no change but the mere transposition of the\nalternative. This appearance arises from mentally reading the E as an\nA proposition: but, if it were so taken, the result would be its\ncontrapositive, and not its converse by negation.\n\n\u00a7 727. The converse of I is a little difficult to grasp.  It becomes\neasier if we reduce it to the equivalent conjunctive--\n\n  'If the cartridges are bad, he sometimes does not aim too high.'\n\nHere, as elsewhere, 'sometimes' must not be taken to mean more than\n'it may be that.'\n\n\u00a7 728. _Conversion by Contraposition of Complex Propositions._\n\nAs applied to conjunctive propositions conversion by contraposition\nassumes the following forms--\n\n  (A) If A is B, C is always D.\n      .'. If C is not-D, A is always not-B.\n\n  (O) If A is B, C is sometimes not D.\n      .'. If C is not-D, A is sometimes not not-B.\n\n  (A) If a man is honest, he is always truthful.\n      .'. If a man is untruthful, he is always dishonest.\n\n  (O) If a man is hasty, he is sometimes not malevolent.\n      .'. If a man is benevolent, he is sometimes not unhasty.\n\n\u00a7 729. As applied to disjunctive propositions conversion by\ncontraposition consists simply in transposing the two alternatives.\n\n  (A) Either A is B or C is D.\n      .'. Either C is D or A is B.\n\nFor, when reduced to the conjunctive shape, the reasoning would run\nthus--\n\n  If A is not B, C is D.\n  .'. If C is not D, A is B.\n\nwhich is the same in form as\n\n  All not-A is B.\n  .'. All not-B is A.\n\nSimilarly in the case of the O proposition\n\n  (O) Either A is B or C is sometimes not D.\n      .'. Either C is D or A is sometimes not B.\n\n\u00a7 730. On comparing these results with the converse by negation of\neach of the same propositions, A and 0, the reader will see that they\ndiffer from them, as was to be expected, only in being permuted. The\nvalidity of the inference may be tested, both here and in the case of\nconversion by negation, by reducing the disjunctive proposition to the\nconjunctive, and so to the simple form, then performing the process as\nin simple propositions, and finally throwing the converse, when so\nobtained, back into the disjunctive form. We will show in this manner\nthat the above is really the contrapositive of the 0 proposition.\n\n  (O) Either A is B or C is sometimes not D.\n\n  = If A is not B, C is sometimes not D.\n\n  = Some cases of A not being B are not cases of C being D. (Some A is\n  not B.)\n\n  = Some cases of C not being D are not cases of A being B. (Some\n  not-B is not not-A.)\n\n  = If C is not D, A is sometimes not B.\n\n  = Either C is D or A is sometimes not B.\n\n\n\n\nCHAPTER XX.\n\n_Of Complex Syllogisms_.\n\n\n\u00a7 731. A Complex Syllogism is one which is composed, in whole or part,\nof complex propositions.\n\n\u00a7 732. Though there are only two kinds of complex proposition, there\nare three varieties of complex syllogism.  For we may have\n\n  (1) a syllogism in which the only kind of complex proposition\n  employed is the conjunctive;\n\n  (2) a syllogism in which the only kind of complex proposition\n  employed is the disjunctive;\n\n  (3) a syllogism which has one premiss conjunctive and the other\n  disjunctive.\n\nThe chief instance of the third kind is that known as the Dilemma.\n\n                       Syllogism\n        ___________________|_______________\n        |                                 |\n     Simple                            Complex\n  (Categorical)                      (Conditional)\n                     _____________________|_______________\n                     |                    |              |\n                Conjunctive          Disjunctive      Dilemma\n               (Hypothetical)\n\n\n\n_The Conjunctive Syllogism_.\n\n\n\u00a7 733. The Conjunctive Syllogism has one or both premisses conjunctive\npropositions: but if only one is conjunctive, the other must be a\nsimple one.\n\n\u00a7 734. Where both premisses are conjunctive, the conclusion will be of\nthe same character; where only one is conjunctive, the conclusion will\nbe a simple proposition.\n\n\u00a7 735. Of these two kinds of conjunctive syllogisms we will first take\nthat which consists throughout of conjunctive propositions.\n\n\n\n_The Wholly Conjunctive Syllogism_.\n\n\n\u00a7 736. Wholly conjunctive syllogisms do not differ essentially from\nsimple ones, to which they are immediately reducible. They admit of\nbeing constructed in every mood and figure, and the moods of the\nimperfect figures may be brought into the first by following the\nordinary rules of reduction. For instance--\n\n    Cesare.                              Celarent.\n\n  If A is B, C is never D.     \\     / If C is D, A is never B.\n  If E is F, C is always D.     | = |  If E is F, C is always D.\n  .'. If E is F, A is never B. /     \\ .'. If E is F, A is never B.\n\n  If it is day, the stars never shine.\\   /If the stars shine, it is never day.\n  If it is night, the stars always     \\=/ If it is night, the stars always\n  shine.                               / \\ shine.\n  .'. If it is night, it is never day /   \\.'. If it is night, it is never day.\n\n    Disamis.                             Darii.\n  If C is D, A is sometimes B. \\     / If C is D, E is always F.\n  If C is D, E is always F.     | = |  If A is B, C is sometimes D.\n  If E is F, A is sometimes B. /     \\ .'. If A is B, E is sometimes F.\n                                       .'. If E is F, A is sometimes B.\n\n  If she goes, I sometimes go.    \\     / If she goes, he always goes,\n  If she goes, he always goes.     | = |  If I go, she sometimes goes.\n  .'. If he goes, I sometimes go. /     \\ .'. If I go, he sometimes goes.\n                                          .'. If he goes, I sometimes go.\n\n\n\n_The Partly Conjunctive Syllogism._\n\n\n\u00a7 737. It is this kind which is usually meant when the Conjunctive or\nHypothetical Syllogism is spoken of.\n\n\u00a7 738. Of the two premisses, one conjunctive and one simple, the\nconjunctive is considered to be the major, and the simple premiss the\nminor. For the conjunctive premiss lays down a certain relation to\nhold between two propositions as a matter of theory, which is applied\nin the minor to a matter of fact.\n\n\u00a7 739. Taking a conjunctive proposition as a major premiss, there are\nfour simple minors possible. For we may either assert or deny the\nantecedent or the consequent of the conjunctive.\n\n      Constructive Mood.                Destructive Mood.\n  (1) If A is B, C is D.            (2) If A is B, C is D.\n      A is B.                           C is not D.\n      .'. C is D.                       .'. A is not B.\n\n  (3) If A is B, C is D.            (4) If A is B, C is D.\n      A is not B.                       C is D.\n      No conclusion.                    No conclusion.\n\n\u00a7 740. When we take as a minor 'A is not B ' (3), it is clear that we\ncan get no conclusion. For to say that C is D whenever A is B gives us\nno right to deny that C can be D in the absence of that\ncondition. What we have predicated has been merely inclusion of the\ncase AB in the case CD.\n\n[Illustration]\n\n\u00a7 741. Again, when we take as a minor, 'C is D' (4), we can get no\nuniversal conclusion. For though A being B is declared to involve as a\nconsequence C being D, yet it is possible for C to be D under other\ncircumstances, or from other causes. Granting the truth of the\nproposition 'If the sky falls, we shall catch larks,' it by no means\nfollows that there are no other conditions under which this result can\nbe attained.\n\n\u00a7 742. From a consideration of the above four cases we elicit the\nfollowing\n\n\n\n_Canon of the Conjunctive Syllogism._\n\n\nTo affirm the antecedent is to affirm the consequent, and to deny the\nconsequent is to deny the antecedent: but from denying the antecedent\nor affirming the consequent no conclusion follows.\n\n\u00a7 743. There is a case, however, in which we can legitimately deny the\nantecedent and affirm the consequent of a conjunctive proposition,\nnamely, when the relation predicated between the antecedent and the\nconsequent is not that of inclusion but of coincidence--where in fact\nthe conjunctive proposition conforms to the type u.\n\nFor example--\n\n  _Denial of the Antecedent_.\n  If you repent, then only are you forgiven.\n  You do not repent.\n  .'. You are not forgiven.\n\n  _Affirmation of the Consequent_.\n  If you repent, then only are you forgiven.\n  You are forgiven.\n  .'. You repent.\n\n\n\n\nCHAPTER XXI.\n\n_Of the Reduction of the Partly Conjunctive Syllogism._\n\n\n\u00a7 744. Such syllogisms as those just treated of, if syllogisms they\nare to be called, have a major and a middle term visible to the eye,\nbut appear to be destitute of a minor. The missing minor term is\nhowever supposed to be latent in the transition from the conjunctive\nto the simple form of proposition. When we say 'A is B,' we are taken\nto mean, 'As a matter of fact, A is B' or 'The actual state of the\ncase is that A is B.' The insertion therefore of some such expression\nas 'The case in hand,' or 'This case,' is, on this view, all that is\nwanted to complete the form of the syllogism. When reduced in this\nmanner to the simple type of argument, it will be found that the\nconstructive conjunctive conforms to the first figure and the\ndestructive conjunctive to the second.\n\n    _Constructive Mood_.    _Barbara_.\n\n  If A is B, C is D. \\     / All cases of A being B are cases of\n                      \\ = /    C being D.\n  A is B.             /   \\  This is a case of A being B.\n  .'. C is D.        /     \\ .'. This is a case of C being D.\n\n  _Destructive Mood._     Camestres.\n\n  If A is B, C is D. \\     / All cases of A being B are cases of\n                      \\ = /    C being D.\n  C is not D.         /   \\  This is not a case of C being D.\n  .'. A is not B.    /     \\ .'. This is not a case of A being B.\n\n\u00a7 745. It is apparent from the position of the middle term that the\nconstructive conjunctive must fall into the first figure and the\ndestructive conjunctive into the second.  There is no reason, however,\nwhy they should be confined to the two moods, Barbara and\nCarnestres. If the inference is universal, whether as general or\nsingular, the mood is Barbara or Carnestres; if it is particular, the\nmood is Darii or Baroko.\n\n  Barbara.                         Camestres.\n  If A is B, C is always D. \\      If A is B, C is always D.    \\\n  A is always B.             \\     C is never D.                 \\\n  .'. C is always D.          \\    .'. A is never B.              \\\n                               |                                   |\n  If A is B, C is always D.   /    If A is B, C is always D.      /\n  A is in this case B.       /     C is not in this case D.      /\n  .'. C is in this case D.  /      .'. A is not in this case B. /\n\n  Darii.                           Baroko.\n\n  If A is B, C is always D.        If A is B, C is never D.\n  A is sometimes B.                C is sometimes not D.\n  .'. C is sometimes D.            .'. A is sometimes not B.\n\n\u00a7 746. The remaining moods of the first and second figure are obtained\nby taking a negative proposition as the consequent in the major\npremiss.\n\n  Celarent.                      Ferio.\n  If A is B, C is never D.       If A is B, C is never D.\n  A is always B.                 A is sometimes B.\n  .'. C is never D.              .'. C is sometimes not D.\n\n  _Cesare_.                 Festino.\n  If A is B, C is never D.       If A is B, C is never D.\n  C is always D.                 C is sometimes D.\n  .'. A is never B.              .'. A is sometimes not B.\n\n\u00a7 747. As the partly conjunctive syllogism is thus reducible to the\nsimple form, it follows that violations of its laws must correspond\nwith violations of the laws of simple syllogism. By our throwing the\nillicit moods into the simple form it will become apparent what\nfallacies are involved in them.\n\n  _Denial of Anteceded_.\n\n  If A is B, C is D. \\     / All cases of A being B are cases of C\n                      \\ = /     being D.\n  A is not B.         /   \\  This is not a case of A being B.\n  .'. C is not D.    /     \\ .'. This is not a case of C being D.\n\nHere we see that the denial of the antecedent amounts to illicit\nprocess of the major term.\n\n\u00a7 7481 _Affirmation of Consequent_.\n\n  If A is B, C is D. \\     / All Cases of A being B are cases of C\n                      | = |     being D.\n  C is D.            /     \\ This is a case of C being D.\n\nHere we see that the affirmation of the consequent amounts to\nundistributed middle.\n\n\u00a7 749. If we confine ourselves to the special rules of the four\nfigures, we see that denial of the antecedent involves a negative\nminor in the first figure, and affirmation of the consequent two\naffirmative premisses in the second. Or, if the consequent in the\nmajor premiss were itself negative, the affirmation of it would amount\nto the fallacy of two negative premisses. Thus--\n\n  If A is B, C is not D. \\      / No cases of A being B are cases of C\n                          | =  |      being D.\n  C is not D.            /      \\ This is not a case of C being D.\n\n\u00a7 750. The positive side of the canon of the conjunctive\nsyllogism--'To affirm the antecedent is to affirm the consequent,'\ncorresponds with the Dictum de Omni. For whereas something (viz. C\nbeing D) is affirmed in the major of all conceivable cases of A being\nB, the same is affirmed in the conclusion of something which is\nincluded therein, namely, 'this case,' or 'some cases,' or even 'all\nactual cases.'\n\n\u00a7 751. The negative side--'to deny the consequent is to deny the\nantecedent'--corresponds with the Dictum de Diverse (\u00a7 643). For\nwhereas in the major all conceivable cases of A being B are included\nin C being D, in the minor 'this case,' or 'some cases,' or even 'all\nactual cases' of C being D, are excluded from the same notion.\n\n\u00a7 752. The special characteristic of the partly conjunctive syllogism\nlies in the transition from hypothesis to fact.  We might lay down as\nthe appropriate axiom of this form of argument, that 'What is true in\nthe abstract is true--in the concrete,' or 'What is true in theory is\nalso true in fact,' a proposition which is apt to be neglected or\ndenied. But this does not vitally distinguish it from the ordinary\nsyllogism. For though in the latter we think rather of the transition\nfrom a general truth to a particular application of it, yet at bottom\na general truth is nothing but a hypothesis resting upon a slender\nbasis of observed fact. The proposition 'A is B' may be expressed in\nthe form 'If A is, B is.' To say that 'All men are mortal' may be\ninterpreted to mean that 'If we find in any subject the attributes of\nhumanity, the attributes of mortality are sure to accompany them.'\n\n\n\n\nCHAPTER XXII.\n\n_Of the Partly Conjunctive Syllogism regarded as an Immediate\nInference_.\n\n\n\u00a7 753. It is the assertion of fact in the minor premiss, where we have\nthe application of an abstract principle to a concrete instance, which\nalone entitles the partly conjunctive syllogism to be regarded as a\nsyllogism at all.  Apart from this the forms of semi-conjunctive\nreasoning run at once into the moulds of immediate inference.\n\n\u00a7 754. The constructive mood will then be read in this way--\n\n  If A is B, C is D,\n  .'. A being B, C is D.\n\nreducing itself to an instance of immediate inference by subaltern\nopposition--\n\n  Every case of A being B, is a case of C being D.\n  .'. Some particular case of A being B is a case of C being D.\n\n\u00a7 755. Again, the destructive conjunctive will read as follows--\n\n  If A is B, C is D,\n  .'. C not being D, A is not B.\n\nwhich is equivalent to\n\n  All cases of A being B are cases of C being D.\n  .'. Whatever is not a case of C being D is not a case of A being B.\n  .'. Some particular case of C not being D is not a case of A being\n  B.\n\nBut what is this but an immediate inference by contraposition, coming\nunder the formula\n\n  All A is B,\n  .'. All not-B is not-A,\n\nand followed by Subalternation?\n\n\u00a7 756. The fallacy of affirming the consequent becomes by this mode of\ntreatment an instance of the vice of immediate inference known as the\nsimple conversion of an A proposition. 'If A is B, C is D' is not\nconvertible with 'If C is D, A is B' any more than 'All A is B' is\nconvertible with 'All B is A.'\n\n\u00a7 757. We may however argue in this way\n\n  If A is B, C is D,\n  C is D,\n  .'. A may be B,\n\nwhich is equivalent to saying,\n\n  When A is B, C is always D,\n  .'. When C is D, A is sometimes B,\n\nand falls under the legitimate form of conversion of A per accidens--\n\n  All cases of A being B are cases of C being D.\n  .'. Some cases of C being D are cases of A being B.\n\n\u00a7 758. The fallacy of denying the antecedent assumes the following\nform--\n\n  If A is B, C is D,\n  .'. If A is not B, C is not D,\n\nequivalent to--\n\n  All cases of A being B are cases of C being D.\n  .'. Whatever is not a case of A being B is not a case of C being D.\n\nThis is the same as to argue--\n\n  All A is B,\n  .'. All not-A is not-B,\n\nan erroneous form of immediate inference for which there is no special\nname, but which involves the vice of simple conversion of A, since\n'All not-A is not-B' is the contrapositive, not of 'All A is B,' but\nof its simple converse 'All B is A.'\n\n\u00a7 759. The above-mentioned form of immediate inference, however\n(namely, the employment of contraposition without conversion), is\nvalid in the case of the U proposition; and so also is simple\nconversion. Accordingly we are able, as we have seen, in dealing with\na proposition of that form, both to deny the antecedent and to assert\nthe consequent with impunity--\n\n  If A is B, then only C is D,\n  .'. A not being B, C is not D;\n\nand again, C being D, A must be B.\n\n\n\n\nCHAPTER XXIII.\n\n_Of the Disjunctive Syllogism_.\n\n\n\u00a7 760. Roughly speaking, a Disjunctive Syllogism results from the\ncombination of a disjunctive with a simple premiss. As in the\npreceding form, the complex proposition is regarded as the major\npremiss, since it lays down a hypothesis, which is applied to fact in\nthe minor.\n\n\u00a7 761. The Disjunctive Syllogism may be exactly defined as follows--\n\n  A complex syllogism, which has for its major premiss a disjunctive\n  proposition, either the antecedent or consequent of which is in the\n  minor premiss simply affirmed or denied.\n\n\u00a7 762. Thus there are four types of disjunctive syllogism possible.\n\n_Constructive Moods._\n\n  (1) Either A is B or C is D. (2) Either A is B or C is D.\n      A is not B.                  C is not D.\n      .'. C is D.                  .'. A is B.\n\n  Either death is annihilation or we are immortal.\n  Death is not annihilation.\n  .'. We are immortal.\n\n  Either the water is shallow or the boys will be drowned.\n  The boys are not drowned.\n  .'. The water is shallow.\n\n_Destructive Moods_.\n\n  (3) Either A is B or C is D. (4) Either A is B or C is D.\n      A is B.                      C is D.\n      .'. C is not D.              .'. A is not B.\n\n\u00a7 763. Of these four, however, it is only the constructive moods that\nare formally conclusive. The validity of the two destructive moods is\ncontingent upon the kind of alternatives selected. If these are such\nas necessarily to exclude one another, the conclusion will hold, but\nnot otherwise. They are of course mutually exclusive whenever they\nembody the result of a correct logical division, as 'Triangles are\neither equilateral, isosceles or scalene.'  Here, if we affirm one of\nthe members, we are justified in denying the rest. When the major thus\ncontains the dividing members of a genus, it may more fitly be\nsymbolized under the formula, 'A is either B or C.' But as this admits\nof being read in the shape, 'Either A is B or A is C,' we retain the\nwider expression which includes it. Any knowledge, however, which we\nmay have of the fact that the alternatives selected in the major are\nincompatible must come to us from material sources; unless indeed we\nhave confined ourselves to a pair of contradictory terms (A is either\nB or not-B). There can be nothing in the form of the expression to\nindicate the incompatibility of the alternatives, since the same form\nis employed when the alternatives are palpably compatible. When, for\ninstance, we say, 'A successful student must be either talented or\nindustrious,' we do not at all mean to assert the positive\nincompatibility of talent and industry in a successful student, but\nonly the incompatibility of their negatives--in other words, that, if\nboth are absent, no student can be successful. Similarly, when it is\nsaid, 'Either your play is bad or your luck is abominable,' there is\nnothing in the form of the expression to preclude our conceiving that\nboth may be the case.\n\n\u00a7 764. There is no limit to the number of members in the disjunctive\nmajor. But if there are only two alternatives, the conclusion will be\na simple proposition; if there are more than two, the conclusion will\nitself be a disjunctive. Thus--\n\n  Either A is B or C is D or E is F or G is H.\n  E is not F.\n  .'. Either A is B or C is D or G is H.\n\n\u00a7 765. The Canon of the Disjunctive Syllogism may be laid down as\nfollows--\n\n  To deny one member is to affirm the rest, either simply or\n  disjunctively; but from affirming any member nothing follows.\n\n\n\n\nCHAPTER XXIV.\n\n_Of the Reduction of the Disjunctive Syllogism._\n\n\u00a7 766. We have seen that in the disjunctive syllogism the two\nconstructive moods alone are formally valid. The first of these,\nnamely, the denial of the antecedent, will in all cases give a simple\nsyllogism in the first figure; the second of them, namely, the denial\nof the consequent, will in all cases give a simple syllogism in the\nsecond figure.\n\n  _Denial of Antecedent_ = Barbara.\n\n    Either A is B or C is D.\n    A is not B.\n    .'.C is D\n\n  is equal to\n\n    If A is not B, C is D.\n    A is not B.\n    .'. C is D.\n\n  is equal to\n\n    All cases of A not being B are cases of C being D.\n    This is a case of A not being B.\n    .'. This is a case of C being D.\n\n\n  _Denial of Consequent_ = Camestres.\n\n    Either A is E or C is D.\n    C is not D.\n    .'. A is B.\n\n  is equal to\n\n    If A is not B, C is D.\n    C is not D.\n    .'. A is B.\n\n  is equal to\n\n    All cases of A not being B are cases of C being D.\n    This is not a case of C being D.\n    .'. This is not a case of A being B.\n\n\u00a7 767. The other moods of the first and second figures can be obtained\nby varying the quality of the antecedent and consequent in the major\npremiss and reducing the quantity of the minor.\n\n\u00a7 768. The invalid destructive moods correspond with the two invalid\ntypes of the partly conjunctive syllogism, and have the same fallacies\nof simple syllogism underlying them. Affirmation of the antecedent of\na disjunctive is equivalent to the semi-conjunctive fallacy of denying\nthe antecedent, and therefore involves the ordinary syllogistic\nfallacy of illicit process of the major.\n\nAffirmation of the consequent of a disjunctive is equivalent to the\nsame fallacy in the semi-conjunctive form, and therefore involves the\nordinary syllogistic fallacy of undistributed middle.\n\n  _Affirmation of Antecedent_ = _Illicit Major_.\n\n    Either A is B or C is D.\n    A is B.\n    .'. C is not D.\n\n  is equal to\n\n    If A is not B, C is D.\n    A is B.\n    .'. C is not D.\n\n  is equal to\n\n    All cases of A not being B are cases of C being D.\n    This is not a case of A not being B.\n    .'. This is not a case of C not being D.\n\n\n  _Affirmation of Consequent_ = _Undistributed Middle_.\n\n    Either A is B or C is D.\n    C is D.\n\n  is equal to\n\n    If A is not B, C is D.\n    C is D.\n\n  is equal to\n\n    All cases of A not being B are cases of C being D.\n    This is a case of C being D.\n\n\u00a7 769. So far as regards the consequent, the two species of complex\nreasoning hitherto discussed are identical both in appearance and\nreality. The apparent difference of procedure in the case of the\nantecedent, namely, that it is affirmed in the partly conjunctive, but\ndenied in the disjunctive syllogism, is due merely to the fact that in\nthe disjunctive proposition the truth of the consequent is involved in\nthe falsity of the antecedent, so that the antecedent being\nnecessarily negative, to deny it in appearance is in reality to assert\nit.\n\n\n\n\nCHAPTER XXV.\n\n_The Disjunctive Syllogism regarded as an Immediate Inference_.\n\n\n\u00a7 770. If no stress be laid on the transition from disjunctive\nhypothesis to fact, the disjunctive syllogism will run with the same\nfacility as its predecessor into the moulds of immediate inference.\n\n\u00a7 771.\n\n  _Denial of Antecedent_.       Subalternation.\n\n  Either A is B or C is D,       Every case of A not being B\n                                 is a case of C being D.\n  .'. A not being B, C is D.     .'. Some case of A not being B\n                                 is a case of C being D.\n\n\u00a7 772.\n\n  _Denial of Consequent_.      Conversion by Contraposition\n                                    + Subalternation.\n\n  Either A is B or C is D.       All cases of A not being B\n                                 are cases of C being D.\n  .'. C not being D, A is B      .'. All cases of C not being D are\n                                 cases of A being B.\n                                 .'. Some case of C not being D is\n                                 a case of A being B.\n\n\u00a7 773. Similarly the two invalid types of disjunctive syllogism will\nbe found to coincide with fallacies of immediate inference.\n\n\u00a7 774.\n\n  _Affirmation of Antecedent_.   Contraposition without\n                                      Conversion.\n\n  Either A is B or C is D.         All cases of A not being B are\n                                   cases of C being D.\n  .'. A being B, C is not D        .'. All cases of A being B are\n                                   cases of C not being D.\n\n\u00a7 775. The affirmation of the antecedent thus comes under the\nformula--\n\n  All not-A is B,\n  .'. All A is not-B,\n\na form of inference which cannot hold except where A and B are known\nto be incompatible. Who, for instance, would assent to this?--\n\n  All non-boating men play cricket.\n  .'. All boating men are non-cricketers.\n\n\u00a7 776.\n\n  _Affirmation of Consequent_.   Simple Conversion of A.\n\n  Either A is B or C is D.          All cases of A not being B are\n                                    cases of C being D.\n  .'.C being D, A is not B.         .'. All cases of C being D are\n                                    cases of A not being B.\n\n\u00a7 777. We may however argue in this way--\n\n                                     Conversion of A per accidens.\n  Either A is B or C is D.           All cases of A not being B\n                                     are cases of C being D.\n  .'. C being D, A is sometimes B.   .'. Some cases of C being D are\n                                     cases of A not being B.\n\n  The men who pass this examination must have either talent or industry.\n  .'. Granting that they are industrious, they may be without talent.\n\n\n\n\nCHAPTER XXVI.\n\n_Of the Mixed Form of Complex Syllogism_.\n\n\n\u00a7 778. Under this head are included all syllogisms in which a\nconjunctive is combined with a disjunctive premiss. The best known\nform is\n\n\n\n_The Dilemma_.\n\n\n\u00a7 779. The Dilemma may be defined as--\n\n  A complex syllogism, having for its major premiss a conjunctive\n  proposition with more than one antecedent, or more than one\n  consequent, or both, which (antecedent or consequent) the minor\n  premiss disjunctively affirms or denies.\n\n\u00a7 780. It will facilitate the comprehension of the dilemma, if the\nfollowing three points are borne in mind--\n\n  (1) that the dilemma conforms to the canon of the partly conjunctive\n  syllogism, and therefore a valid conclusion can be obtained only by\n  affirming the antecedent or denying the consequent;\n\n  (2) that the minor premiss must be disjunctive;\n\n  (3) that if only the antecedent be more than one, the conclusion\n  will be a simple proposition; but if both antecedent and consequent\n  be more than one, the conclusion will itself be disjunctive.\n\n\u00a7 781. The dilemma, it will be seen, differs from the partly\nconjunctive syllogism chiefly in the fact of having a disjunctive\naffirmation of the antecedent or denial of the consequent in the\nminor, instead of a simple one. It is this which constitutes the\nessence of the dilemma, and which determines its possible\nvarieties. For if only the antecedent or only the consequent be more\nthan one, we must, in order to obtain a disjunctive minor, affirm the\nantecedent or deny the consequent respectively; whereas, if there be\nmore than one of both, it is open to us to take either course. This\ngives us four types of dilemma.\n\n\u00a7 782.\n\n  (1). _Simple Constructive._\n\n    If A is B or C is D, E is F.\n    Either A is B or C is D.\n    .'. E is F.\n\n  (2). _Simple Destructive._\n\n    If A is B, C is D and E is F.\n    Either C is not D or E is not F.\n    .'. A is not B.\n\n  (3). _Complex Constructive._\n\n    If A is B, C is D; and if E is F, G is H.\n    Either A is B or E is F.\n    .'. Either C is D or G is H.\n\n  (4). _Complex Destructive_.\n\n    If A is B, C is D; and if E is F, G is H.\n    Either C is not D or G is not H.\n    .'. Either A is not B or E is not F.\n\n\u00a7 783.\n\n  (1). _Simple Constructive_.\n\n    If she sinks or if she swims, there will be an end of her.\n    She must either sink or swim.\n    .'. There will be an end of her.\n\n  (2). _Simple Destructive_.\n\n    If I go to Town, I must pay for my ticket and pay my hotel bill.\n    Either I cannot pay for my ticket or I cannot pay my hotel bill.\n    .'. I cannot go to Town.\n\n  (3). _Complex Constructive_.\n\n    If I stay in this room, I shall be burnt to death, and if I jump\n    out of the window, I shall break my neck.\n    I must either stay in the room or jump out of the window.\n    .'. I must either be burnt to death or break my neck.\n\n  (4). _Complex Destructive_.\n\n    If he were clever, he would see his mistake; and\n    if he were candid, he would acknowledge it.\n    Either he does not see his mistake or he will not acknowledge it.\n    .'. Either he is not clever or he is not candid.\n\n\u00a7 784. It must be noticed that the simple destructive dilemma would\nnot admit of a disjunctive consequent. If we said,\n\n  If A is B, either C is D or E is F,\n  Either C is not D or E is not F,\n\nwe should not be denying the consequent. For 'E is not F' would make\nit true that C is D, and 'C is not D' would make it true that E is F;\nso that in either case we should have one of the alternatives true,\nwhich is just what the disjunctive form 'Either C is D or E is F'\ninsists upon.\n\n\u00a7 785. In the case of the complex constructive dilemma the several\nmembers, instead of being distributively assigned to one another, may\nbe connected together as a whole--thus--\n\n  If either A is B or E is F, either C is D or G is H.\n  Either A is B or E is F.\n  .'. Either C is D or G is H.\n\nIn this shape the likeness of the dilemma to the partly conjunctive\nsyllogism is more immediately recognisable.  The major premiss in this\nshape is vaguer than in the former. For each antecedent has now a\ndisjunctive choice of consequents, instead of being limited to\none. This vagueness, however, does not affect the conclusion. For, so\nlong as the conclusion is established, it does not matter from which\nmembers of the major its own members flow.\n\n\u00a7 786. It must be carefully noticed that we cannot treat the complex\ndestructive dilemma in the same way.\n\n  If either A is B or E is F, either C is D or G is H.\n  Either C is not D or G is not H.\n\nSince the consequents are no longer connected individually with the\nantecedents, a disjunctive denial of them leaves it still possible for\nthe antecedent as a whole to be true. For 'C is not D' makes it true\nthat G is H, and 'G is not H' makes it true that C is D. In either\ncase then one is true, which is all that was demanded by the\nconsequent of the major. Hence the consequent has not really been\ndenied.\n\n\u00a7 787. For the sake of simplicity we have limited the examples to the\ncase of two antecedents or consequents.  But we may have as many of\neither as we please, so as to have a Trilemma, a Tetralemma, and so\non.\n\nTRILEMMA.\n\n  If A is B, C is D; and if E is F, G is H; and if K is L, M is N.\n  Either A is B or E is F or K is L.\n  .'. Either C is D or G is H or K is L.\n\n\u00a7 788. Having seen what the true dilemma is, we shall now examine some\nforms of reasoning which resemble dilemmas without being so.\n\n\u00a7 789. This, for instance, is not a dilemma--\n\n  If A is B or if E is F, C is D.\n  But A is B and E is F.\n  .'. C is D.\n\n  If he observes the sabbath or if he refuses to eat pork, he is a\n  Jew.\n  But he both observes the sabbath and refuses to eat pork.\n  .'. He is a Jew.\n\nWhat we have here is a combination of two partly conjunctive\nsyllogisms with the same conclusion, which would have been established\nby either of them singly.  The proof is redundant.\n\n\u00a7 790. Neither is the following a dilemma--\n\n  If A is B, C is D and E is F.\n  Neither C is D nor E is F.\n  .'. A is not B.\n\n  If this triangle is equilateral, its sides and its angles will be\n  equal.\n  But neither its sides nor its angles are equal.\n  .'. It is not equilateral.\n\nThis is another combination of two conjunctive syllogisms, both\npointing to the same conclusion. The proof is again redundant. In this\ncase we have the consequent denied in both, whereas in the former we\nhad the antecedent affirmed. It is only for convenience that such\narguments as these are thrown into the form of a single\nsyllogism. Their real distinctness may be seen from the fact that we\nhere deny each proposition separately, thus making two independent\nstatements--C is not D and E is not F. But in the true instance of the\nsimple destructive dilemma, what we deny is not the truth of the two\npropositions contained in the consequent, but their compatibility; in\nother words we make a disjunctive denial.\n\n\u00a7 791. Nor yet is the following a dilemma--\n\n  If A is B, either C is D or E is F.\n  Neither C is D nor E is F.\n  .'. A is not B.\n\n  If the barometer falls there will be either wind or rain.\n  There is neither wind nor rain.\n  .'. The barometer has not fallen.\n\nWhat we have here is simply a conjunctive major with the consequent\ndenied in the minor. In the consequent of the major it is asserted\nthat the two propositions, 'C is D' and 'E is F' cannot both be false;\nand in the minor this is denied by the assertion that they are both\nfalse.\n\n\u00a7 792. A dilemma is said to be rebutted or retorted, when another\ndilemma is made out proving an opposite conclusion. If the dilemma be\na sound one, and its premisses true, this is of course impossible, and\nany appearance of contradiction that may present itself on first sight\nmust vanish on inspection. The most usual mode of rebutting a dilemma\nis by transposing and denying the consequents in the major--\n\n  If A is B, C is D; and if E is F, G is H.\n  Either A is B or E is F.\n  .'. Either C is D or G is H.\n\nThe same rebutted--\n\n  If A is B, G is not H; and if E is F, C is not D.\n  Either A is B or E is F.\n  .'. Either G is not H or C is not D.\n    = Either C is not D or G is not H.\n\n\u00a7 793. Under this form comes the dilemma addressed by the Athenian\nmother to her son--'Do not enter public life: for, if you say what is\njust, men will hate you; and, if you say what is unjust, the gods will\nhate you' to which the following retort was made--'I ought to enter\npublic life: for, if 1 say what is just, the gods will love me; and,\nif 1 say what is unjust, men will love me.' But the two conclusions\nhere are quite compatible. A man must, on the given premisses, be both\nhated and loved, whatever course he takes. So far indeed are two\npropositions of the form\n\n  Either C is D or G is H,\n  and Either C is not D or G is not H,\n\nfrom being incompatible, that they express precisely the same thing\nwhen contradictory alternatives have been selected, e.g.--\n\n  Either a triangle is equilateral or non-equilateral.\n  Either a triangle is non-equilateral or equilateral.\n\n\u00a7 794. Equally illusory is the famous instance of rebutting a dilemma\ncontained in the story of Protagoras and Euathlus\n(Aul. Gell. Noct. Alt. v. 10), Euathlus was a pupil of Protagoras in\nrhetoric. He paid half the fee demanded by his preceptor before\nreceiving lessons, and agreed to pay the remainder when he won his\nfirst case.  But as he never proceeded to practise at the bar, it\nbecame evident that he meant to bilk his tutor. Accordingly Protagoras\nhimself instituted a law-suit against him, and in the preliminary\nproceedings before the jurors propounded to him the following\ndilemma--'Most foolish young man, whatever be the issue of this suit,\nyou must pay me what I claim: for, if the verdict be given in your\nfavour, you are bound by our bargain; and if it be given against you,\nyou are bound by the decision of the jurors.'  The pupil, however, was\nequal to the occasion, and rebutted the dilemma as follows. 'Most\nsapient master, whatever be the issue of this suit, I shall not pay\nyou what you claim: for, if the verdict be given in my favour, I am\nabsolved by the decision of the jurors; and, if it be given against\nme, I am absolved by our bargain.' The jurors are said to have been so\npuzzled by the conflicting plausibility of the arguments that they\nadjourned the case till the Greek Kalends. It is evident, however,\nthat a grave injustice was thus done to Protagoras. His dilemma was\nreally invincible. In the counter-dilemma of Euathlus we are meant to\ninfer that Protagoras would actually lose his fee, instead of merely\ngetting it in one way rather than another. In either case he would\nboth get and lose his fee, in the sense of getting it on one plea, and\nnot getting it on another: but in neither case would he actually lose\nit.\n\n\u00a7 795. If a dilemma is correct in form, the conclusion of course\nrigorously follows: but a material fallacy often underlies this form\nof argument in the tacit assumption that the alternatives offered in\nthe minor constitute an exhaustive division. Thus the dilemma 'If pain\nis severe, it will be brief; and if it last long it will be slight,'\n&c., leaves out of sight the unfortunate fact that pain may both be\nsevere and of long continuance. Again the following dilemma--\n\n  If students are idle, examinations are unavailing; and, if\n  they are industrious, examinations are superfluous,\n  Students are either idle or industrious,\n  .'. Examinations are either unavailing or superfluous,\n\nis valid enough, so far as the form is concerned. But the person who\nused it would doubtless mean to imply that students could be\nexhaustively divided into the idle and the industrious. No deductive\nconclusion can go further than its premisses; so that all that the\nabove conclusion can in strictness be taken to mean is that\nexaminations are unavailing, when students are idle, and superfluous,\nwhen they are industrious--which is simply a reassertion as a matter\nof fact of what was previously given as a pure hypothesis.\n\n\n\n\nCHAPTER XXVII.\n\n_Of the Reduction of the Dilemma._\n\n\n\u00a7 796. As the dilemma is only a peculiar variety of the partly\nconjunctive syllogism, we should naturally expect to find it reducible\nin the same way to the form of a simple syllogism. And such is in fact\nthe case. The constructive dilemma conforms to the first figure and\nthe destructive to the second.\n\n  1) _Simple Constructive Dilemma_.\n\n                                      Barbara.\n  If A is B or if E is F, C is D.   All cases of either A being B or E\n                                       being F are cases of C being D.\n  Either A is B or E is F.          All actual cases are cases of either\n                                       A being B OP E being F.\n  .'. C is D.                       .'. All actual cases are cases of C\n                                       being D.\n\n  (2) _Simple Destructive_.\n\n                                       Camstres.\n  If A is B, C is D and E is F.      All cases of A being B are cases of\n                                        C being D and E being F.\n  Either C is not D or E is not F.   No actual cases are cases of C being\n                                        D and E being F.\n  .'. A is not B.                    .'. No actual cases are cases of A\n                                        being B.\n\n  (3) _Complex Constructive_.\n                                          Barbara.\n  If A is B, C is D; and if E is F,     All cases of either A being B or\n     G is H.                               being F are cases of either C being\n                                           D or G being H.\n  Either A is B or E is F.              All actual cases are cases of either A\n                                           being B or E being F.\n  .'. Either C is D or G is H.          .'. All actual cases are cases of either C\n                                           being D or G being H.\n\n  (4) _Complex Destructive_.\n\n  If A is B, C is D; and if E is F,   All cases of A being B and E being F\n  G is H.                                are cases of C being D and G\n                                         being H.\n  Either C is not D Or G is           No actual cases are cases of C being\n     not H                               D and G being H.\n  Either A is not B or E is           No actual cases are cases of A being\n  not F.                                 B and E being F.\n\n\u00a7 797. There is nothing to prevent our having Darii, instead of\nBarbara, in the constructive form, and Baroko, instead of Camestres,\nin the destructive. As in the case of the partly conjunctive syllogism\nthe remaining moods of the first and second figure are obtained by\ntaking a negative proposition as the consequent of the major premiss,\ne.g.--\n\n  _Simple Constructive_.                Celarent or Ferio.\n  If A is B or if E is F, C is not D  No cases of either A being B or E\n                                         being F are cases of C being D.\n  Either A is B or E is F.            All (or some) actual cases are cases of\n                                         either A being B or E being F\n  .'. C is not D.                     .'. All (or some) actual cases are not\n                                         cases of C being D.\n\n\n\n\nCHAPTER XXVIII.\n\n_Of the Dilemma regarded as an Immediate Inference._\n\n\n\u00a7 798. Like the partly conjunctive syllogism, the dilemma can be\nexpressed under the forms of immediate inference.  As before, the\nconclusion in the constructive type resolves itself into the\nsubalternate of the major itself, and in the destructive type into the\nsubalternate of its contrapositive.  The simple constructive dilemma,\nfor instance, may be read as follows--\n\n  If either A is B or E is F, C is D,\n  .'. Either A being B or E being F, C is D,\n\nwhich is equivalent to\n\n  Every case of either A being B or E being F is a case of C being D.\n  .'. Some case of either A being B or E being F is a case of C being D.\n\nThe descent here from 'every' to 'some' takes the place of the\ntransition from hypothesis to fact.\n\n\u00a7 799. Again the complex destructive may be read thus--\n\n  If A is B, C is D; and if E is F, G is H,\n  .'. It not being true that C is D and G is H, it is not\n      true that A is B and E is F,\n\nwhich may be resolved into two steps of immediate inference, namely,\nconversion by contraposition followed by subalternation--\n\n  All cases of A being B and E being F are cases of C being D and G\n  being H.\n  .'. Whatever is not a case of C being D and G being H is not a case\n  of A being B and E being F.\n  .'. Some case which is not one of C being D and G being H is not a\n  case of A being B and E being F.\n\n\n\n\nCHAPTER XXIX.\n\n_Of Trains of Reasoning._\n\n\n\u00a7 800. The formal logician is only concerned to examine whether the\nconclusion duly follows from the premisses: he need not concern\nhimself with the truth or falsity of his data. But the premisses of\none syllogism may themselves be conclusions deduced from other\nsyllogisms, the premisses of which may in their turn have been\nestablished by yet earlier syllogisms. When syllogisms are thus linked\ntogether we have what is called a Train of Reasoning.\n\n\u00a7 801. It is plain that all truths cannot be established by\nreasoning. For the attempt to do so would involve us in an infinite\nregress, wherein the number of syllogisms required would increase at\neach step in a geometrical ratio. To establish the premisses of a\ngiven syllogism we should require two preceding syllogisms; to\nestablish their premisses, four; at the next step backwards, eight; at\nthe next, sixteen; and so on ad infinitum. Thus the very possibility\nof reasoning implies truths that are known to us prior to all\nreasoning; and, however long a train of reasoning may be, we must\nultimately come to truths which are either self-evident or are taken\nfor granted.\n\n\u00a7 802. Any syllogism which establishes one of the premisses of another\nis called in reference to that other a Pro-syllogism, while a\nsyllogism which has for one of its premisses the conclusion of another\nsyllogism is called in reference to that other an Epi-syllogism.\n\n\n\n_The Epicheirema_.\n\n\n\u00a7 803. The name Epicheirema is given to a syllogism with one or both\nof its premisses supported by a reason.  Thus the following is a\ndouble epicheirema--\n\n  All B is A, for it is E.\n  All C is B, for it is F.\n  .'. All C is A.\n\n  All virtue is praiseworthy, for it promotes the general welfare.\n  Generosity is a virtue, for it prompts men to postpone self to others.\n  .'. Generosity is praiseworthy.\n\n\u00a7 804. An epicheirema is said to be of the first or second order\naccording as the major or minor premiss is thus supported. The double\nepicheirema is a combination of the two orders.\n\n\u00a7 805. An epicheirema, it will be seen, consists of one syllogism\nfully expressed together with one, or, it may be, two enthymemes (\u00a7\n557). In the above instance, if the reasoning which supports the\npremisses were set forth at full length, we should have, in place of\nthe enthymemes, the two following pro-syllogisms--\n\n  (i) All E is A.\n      All B is E.\n      .'. All B is A.\n\n  Whatever promotes the general welfare is praiseworthy.\n  Every virtue promotes the general welfare.\n   .'. Every virtue is praiseworthy.\n\n  (2) All F is B.\n      All C is F.\n       .'. All C is B.\n\n  Whatever prompts men to postpone self to others is a virtue.\n  Generosity prompts men to postpone self to others.\n   .'. Generosity is a virtue.\n\n\u00a7 806. The enthymemes in the instance above given are both of the\nfirst order, having the major premiss suppressed.  But there is\nnothing to prevent one or both of them from being of the second\norder--\n\n  All B is A, because all F is.\n  All C is B, because all F is.\n  .'. All C is A.\n\n  All Mahometans are fanatics, because all Monotheists are.\n  These men are Mahometans, because all Persians are.\n  .'. These men are fanatics.\n\nHere it is the minor premiss in each syllogism that is suppressed,\nnamely,\n\n  (1) All Mahometans are Monotheists.\n\n  (2) These men are Persians.\n\n\n\n_The Sorites_.\n\n\n\u00a7 807. The Sorites is the neatest and most compendious form that can\nbe assumed by a train of reasoning.\n\n\u00a7 808. It is sometimes more appropriately called the chain-argument,\nand map be defined as--\n\n  A train of reasoning, in which one premiss of each epi-syllogism is\n  supported by a pro-syllogism, the other being taken for granted.\n\nThis is its inner essence.\n\n\u00a7 809. In its outward form it may be described as--A series of\npropositions, each of which has one term in common with that which\npreceded it, while in the conclusion one of the terms in the last\nproposition becomes either subject or predicate to one of the terms in\nthe first.\n\n\u00a7 810. A sorites may be either--\n\n  (1) Progressive,\n\n  or (2) Regressive.\n\n_Progressive Sorites_.\n\n  All A is B.\n  All B is C.\n  All C is D.\n  All D is E.\n  .'. All A is E.\n\n_Regressive Sorites_.\n\n  All D is E.\n  All C is D.\n  All B is C.\n  All A is B.\n  .'. All A is E.\n\n\u00a7 811. The usual form is the progressive; so that the sorites is\ncommonly described as a series of propositions in which the predicate\nof each becomes the subject of the next, while in the conclusion the\nlast predicate is affirmed or denied of the first subject. The\nregressive form, however, exactly reverses these attributes; and would\nrequire to be described as a series of propositions, in which the\nsubject of each becomes the predicate of the next, while in the\nconclusion the first predicate is affirmed or denied of the last\nsubject.\n\n\u00a7 812. The regressive sorites, it will be observed, consists of the\nsame propositions as the progressive one, only written in reverse\norder. Why then, it may be asked, do we give a special name to it,\nthough we do not consider a syllogism different, if the minor premiss\nhappens to precede the major? It is because the sorites is not a mere\nseries of propositions, but a compressed train of reasoning; and the\ntwo trains of reasoning may be resolved into their component\nsyllogisms in such a manner as to exhibit a real difference between\nthem.\n\n\u00a7 813. The Progressive Sorites is a train of reasoning in which the\nminor premiss of each epi-syllogism is supported by a pro-syllogism,\nwhile the major is taken for granted.\n\n\u00a7 814. The Regressive Sorites is a train of reasoning in which the\nmajor premiss of each epi-syllogism is supported by a pro-syllogism,\nwhile the minor is taken for granted.\n\n  _Progressive Sorites_.\n  (i) All B is C.\n      All A is B.\n      .'. All A is C.\n\n  (2) All C is D.\n      All A is C.\n      .'. All A is D.\n\n  (3) All D is E.\n      All A is D.\n      .'. All A is E.\n\n  _Regressive Sorites_.\n  (1) All D is E.\n      All C is D.\n      .'. All C is E.\n\n  (2) All C is E.\n      All B is C.\n      .'. All B is E.\n\n  (3) All B is E.\n      All A is B.\n      .'. All A is E.\n\n\u00a7 815. Here is a concrete example of the two kinds of sorites,\nresolved each into its component syllogisms--\n\n_Progressive Sorites_.\n\n  All Bideford men are Devonshire men.\n  All Devonshire men are Englishmen.\n  All Englishmen are Teutons.\n  All Teutons are Aryans.\n  .'. All Bideford men are Aryans.\n\n  (1) All Devonshire men are Englishmen.\n      All Bideford men are Devonshire men.\n      .'. All Bideford men are Englishmen.\n\n  (2) All Englishmen are Teutons.\n      All Bideford men are Englishmen.\n      .'. All Bideford men are Teutons.\n\n  (3) All Teutons are Aryans.\n      All Bideford men are Teutons.\n      .'. All Bideford men are Aryans.\n\n_Regressive Sorites._\n\n  All Teutons are Aryans.\n  All Englishmen are Teutons.\n  All Devonshiremen are Englishmen.\n  All Bideford men are Devonshiremen.\n  .'. All Bideford men are Aryans.\n\n  (1) All Teutons are Aryans.\n      All Englishmen are Teutons.\n      .'. All Englishmen are Aryans.\n\n  (2) All Englishmen are Aryans.\n      All Devonshiremen are Englishmen.\n      .'. All Devonshiremen are Aryans.\n\n  (3) All Devonshiremen are Aryans.\n      All Bideford men are Devonshiremen.\n      .'. All Bideford men are Aryans.\n\n\u00a7 816. When expanded, the sorites is found to contain as many\nsyllogisms as there are propositions intermediate between the first\nand the last. This is evident also on inspection by counting the\nnumber of middle terms.\n\n\u00a7 817. In expanding the progressive form we have to commence with the\nsecond proposition of the sorites as the major premiss of the first\nsyllogism. In the progressive form the subject of the conclusion is\nthe same in all the syllogisms; in the regressive form the predicate\nis the same. In both the same series of means, or middle terms, is\nemployed, the difference lying in the extremes that are compared with\none another through them.\n\n[Illustration]\n\n\u00a7 818. It is apparent from the figure that in the progressive form we\nwork from within outwards, in the regressive form from without\ninwards. In the former we first employ the term 'Devonshiremen' as a\nmean to connect 'Bideford men' with 'Englishmen'; next we employ\n'Englishmen' as a mean to connect the same subject 'Bideford men' with\nthe wider term 'Teutons'; and, lastly, we employ 'Teutons' as a mean\nto connect the original subject 'Bideford men' with the ultimate\npredicate 'Ayrans.'\n\n\u00a7 819. Reversely, in the regressive form we first use 'Teutons' as a\nmean whereby to bring 'Englishmen' under 'Aryans'; next we use\n'Englishmen' as a mean whereby to bring 'Devonshiremen' under the dame\npredicate 'Aryans'; and, lastly, we use 'Devonshiremen' as a mean\nwhereby to bring the ultimate subject 'Bideford men' under the\noriginal predicate 'Aryans.'\n\n\u00a7 820. A sorites may be either Regular or Irregular.\n\n\u00a7 821. In the regular form the terms which connect each proposition in\nthe series with its predecessor, that is to say, the middle terms,\nmaintain a fixed relative position; so that, if the middle term be\nsubject in one, it will always be predicate in the other, and vice\nvers\u00e2. In the irregular form this symmetrical arrangement is violated.\n\n\u00a7 822. The syllogisms which compose a regular sorites, whether\nprogressive or regressive, will always be in the first figure.\n\nIn the irregular sorites the syllogisms may fall into different\nfigures.\n\n\u00a7 823. For the regular sorites the following rules may\nbe laid down.\n\n  (1) Only one premiss can be particular, namely, the first, if the\n  sorites be progressive, the last, if it be regressive.\n\n  (2) Only one premiss can be negative, namely, the last, if the\n  sorites be progressive, the first, if it be regressive.\n\n\u00a7 824. _Proof of the Rules for the Regular Sorites_.\n\n  (1) In the progressive sorites the proposition which stands first is\n  the only one which appears as a minor premiss in the expanded\n  form. Each of the others is used in its turn as a major.  If any\n  proposition, therefore, but the first were particular, there would\n  be a particular major, which involves undistributed middle, if the\n  minor be affirmative, as it must be in the first figure.\n\n  In the regressive sorites, if any proposition except the last were\n  particular, we should have a particular conclusion in the syllogism\n  in which it occurred as a premiss, and so a particular major in the\n  next syllogism, which again is inadmissible, as involving\n  undistributed middle.\n\n  (2) In the progressive sorites, if any premiss before the last were\n  negative, we should have a negative conclusion in the syllogism in\n  which it occurs. This would necessitate a negative minor in the next\n  syllogism, which is inadmissible in the first figure, as involving\n  illicit process of the major.\n\n  In the regressive sorites the proposition which stands first is the\n  only one which appears as a major premiss in the expanded form.\n  Each of the others is used in its turn as a minor. If any premiss,\n  therefore, but the first were negative, we should have a negative\n  minor in the first figure, which involves illicit process of the\n  major.\n\n\u00a7 825. The rules above given do not apply to the irregular sorites,\nexcept so far as that only one premiss can be particular and only one\nnegative, which follows from the general rules of syllogism. But there\nis nothing to prevent any one premiss from being particular or any one\npremiss from being negative, as the subjoined examples will show. Both\nthe instances chosen belong to the progressive order of sorites.\n\n  (1) Barbara.\n  All B is A.\n  All C is B.\n  All C is A.\n\n  All B is A.\n  All C is B.\n  Some C is D.\n  All D is E\n   .'. Some A is E\n\n  [Illustration]\n\n  (2) Disamis.\n  Some C is D.\n  All C is A.\n  Some A is D.\n\n  (3) Darii.\n  All D is E\n  Some A is D.\n  Some A is E.\n\n  (1) Barbara.\n  All B is C.\n  All A is B.\n  All A is C.\n\n  All A is B.\n  All B is C.\n  No D is C.\n  All E is D.\n   .'. No A is E.\n\n  [Illustration]\n\n  (2) Cesare.\n  No D is C.\n  All A is C.\n   .'. No A is D.\n\n  (3) Camestres.\n  All E is D.\n  No A is D.\n   .'. No A is E.\n\n\u00a7 826. A chain argument may be composed consisting\nof conjunctive instead of simple propositions. This is\nsubject to the same laws as the simple sorites, to which\nit is immediately reducible.\n\n  _Progressive._                _Regressive._\n  If A is B, C is D.                If E is F, G is H.\n  If C is D, E is F.                If C is D, E is F.\n  If E is F, G is H.                If A is B, C is D.\n   .'. If A is B, G is H.            .'. If A is B, G is H.\n\n\n\n\nCHAPTER XXX.\n\n_Of Fallacies_.\n\n\n\u00a7 827. After examining the conditions on which correct thoughts\ndepend, it is expedient to classify some of the most familiar forms of\nerror. It is by the treatment of the Fallacies that logic chiefly\nvindicates its claim to be considered a practical rather than a\nspeculative science.  To explain and give a name to fallacies is like\nsetting up so many sign-posts on the various turns which it is\npossible to take off the road of truth.\n\n\u00a7 828. By a fallacy is meant a piece of reasoning which appears to\nestablish a conclusion without really doing so.  The term applies both\nto the legitimate deduction of a conclusion from false premisses and\nto the illegitimate deduction of a conclusion from any\npremisses. There are errors incidental to conception and judgement,\nwhich might well be brought under the name; but the fallacies with\nwhich we shall concern ourselves are confined to errors connected with\ninference.\n\n\u00a7 829. When any inference leads to a false conclusion, the error may\nhave arisen either in the thought itself or in the signs by which the\nthought is conveyed. The main sources of fallacy then are confined to\ntwo--\n\n  (1) thought,\n\n  (2) language.\n\n\u00a7 830. This is the basis of Aristotle's division of fallacies, which\nhas not yet been superseded. Fallacies, according to him, are either\nin the language or outside of it. Outside of language there is no\nsource of error but thought. For things themselves do not deceive us,\nbut error arises owing to a misinterpretation of things by the\nmind. Thought, however, may err either in its form or in its\nmatter. The former is the case where there is some violation of the\nlaws of thought; the latter whenever thought disagrees with its\nobject. Hence we arrive at the important distinction between Formal\nand Material fallacies, both of which, however, fall under the same\nnegative head of fallacies other than those of language.\n\n           | In the language\n           | (in the signs of thought)\n           |\n  Fallacy -|                         |--In the Form.\n           |--Outside the language  -|\n           | (in the thought itself) |\n                                     |\n                                     |--in the Matter.\n\n\u00a7 831. There are then three heads to which fallacies may be\nreferred-namely, Formal Fallacies, Fallacies of Language, which are\ncommonly known as Fallacies of Ambiguity, and, lastly, Material\nFallacies.\n\n\u00a7 832. Aristotle himself only goes so far as the first step in the\ndivision of fallacies, being content to class them according as they\nare in the language or outside of it.  After that he proceeds at once\nto enumerate the infim\u00e6 species under each of the two main heads. We\nshall presently imitate this procedure for reasons of expediency.  For\nthe whole phraseology of the subject is derived from Aristotle's\ntreatise on Sophistical Refutations, and we must either keep to his\nmethod or break away from tradition altogether. Sufficient confusion\nhas already arisen from retaining Aristotle's language while\nneglecting his meaning.\n\n\u00a7 833. Modern writers on logic do not approach fallacies from the same\npoint of view as Aristotle. Their object is to discover the most\nfertile sources of error in solitary reasoning; his was to enumerate\nthe various tricks of refutation which could be employed by a sophist\nin controversy. Aristotle's classification is an appendix to the Art\nof Dialectic.\n\n\u00a7 834. Another cause of confusion in this part of logic is the\nidentification of Aristotle's two-fold division of fallacies, commonly\nknown under the titles of In dictione and Extra diotionem, with the\ndivision into Logical and Material, which is based on quite a\ndifferent principle.\n\n\u00a7 835. Aristotle's division perhaps allows an undue importance to\nlanguage, in making that the principle of division, and so throwing\nformal and material fallacies under a common head. Accordingly another\nclassification has been adopted, which concentrates attention from the\nfirst upon the process of thought, which ought certainly to be of\nprimary importance in the eyes of the logician.  This classification\nis as follows.\n\n\u00a7 836. Whenever in the course of our reasoning we are involved in\nerror, either the conclusion follows from the premisses or it does\nnot. If it does not, the fault must lie in the process of reasoning,\nand we have then what is called a Logical Fallacy. If, on the other\nhand, the conclusion does follow from the premisses, the fault must\nlie in the premisses themselves, and we then have what is called a\nMaterial Fallacy. Sometimes, however, the conclusion will appear to\nfollow from the premisses until the meaning of the terms is examined,\nwhen it will be found that the appearance is deceptive owing to some\nambiguity in the language. Such fallacies as these are, strictly\nspeaking, non-logical, since the meaning of words is extraneous to the\nscience which deals with thought. But they are called\nSemi-logical. Thus we arrive by a different road at the same three\nheads as before, namely, (1) Formal or Purely Logical Fallacies, (2)\nSemi-logical Fallacies or Fallacies of Ambiguity, (3) Material\nFallacies.\n\n\u00a7 837. For the sake of distinctness we will place the two divisions\nside by side, before we proceed to enumerate the infimae species.\n\n          |--In the language\n          | (Fallacy of Ambiguity)\n  Fallacy-|\n          |                        |--In the Form.\n          |--Outside the language -|\n                                   |\n                                   |--In the Matter.\n\n                      |--Formal or purely logical.\n          |--Logical -|\n  Fallacy-|           |--Semi-logical\n          |             (Fallacy of Ambiguity).\n          |--Material\n\n838. Of one of these three heads, namely, formal fallacies, it is not\nnecessary to say much, as they have been amply treated of in the\npreceding pages. A formal fallacy arises from the breach of any of the\ngeneral rules of syllogism. Consequently it would be a formal fallacy\nto present as a syllogism anything which had more or less than two\npremisses. Under the latter variety comes what is called 'a woman's\nreason,' which asserts upon its own evidence something which requires\nto be proved.  Schoolboys also have been known to resort to this form\nof argument--'You're a fool.' 'Why?' 'Because you are.' When the\nconclusion thus merely reasserts one of the premisses, the other must\nbe either absent or irrelevant.  If, on the other hand, there are more\nthan two premisses, either there is more than one syllogism or the\nsuperfluous premiss is no premiss at all, but a proposition irrelevant\nto the conclusion.\n\n839. The remaining rules of the syllogism are more able to be broken\nthan the first; so that the following scheme presents the varieties of\nformal fallacy which are commonly enumerated--\n\n                 |--Four Terms.\n  Formal Fallacy-|--Undistributed Middle.\n                 |--Illicit Process.\n                 |--Negative Premisses and Conclusion.\n\n\u00a7 840. The Fallacy of Four Terms is a violation of the second of the\ngeneral rules of syllogism (\u00a7 582). Here is a palpable instance of\nit--\n\n  All men who write books are authors.\n  All educated men could write books.\n  .'. All educated men are authors.\n\nHere the middle term is altered in the minor premiss to the\ndestruction of the argument. The difference between the actual writing\nof books and the power to write them is precisely the difference\nbetween one who is an author and one who is not.\n\n\u00a7 841. Since a syllogism consists of three terms, each of which is\nused twice over, it would be possible to have an apparent syllogism\nwith as many as six terms in it.  The true name for the fallacy\ntherefore is the Fallacy of More than Three Terms. But it is rare to\nfind an attempted syllogism which has more than four terms in it, just\nas we are seldom tendered a line as an hexameter, which has more than\nseven feet.\n\n\u00a7 842. The Fallacies of Undistributed Middle and Illicit Process have\nbeen treated of under \u00a7\u00a7 585, 586.  The heading 'Negative Premisses\nand Conclusion' covers violations of the three general rules of\nsyllogism relating to negative premisses (\u00a7\u00a7 590-593). Here is an\ninstance of the particular form of the fallacy which consists in the\nattempt to extract an affirmative conclusion out of two negative\npremisses--\n\n  All salmon are fish, for neither salmon nor fish belong to the class\n  mammalia.\n\nThe accident of a conclusion being true often helps to conceal the\nfact that it is illegitimately arrived at.  The formal fallacies which\nhave just been enumerated find no place in Aristotle's division. The\nreason is plain.  His object was to enumerate the various modes in\nwhich a sophist might snatch an apparent victory, whereas by openly\nviolating any of the laws of syllogism a disputant would be simply\ncourting defeat.\n\n\u00a7 843. We now revert to Aristotle's classification of fallacies, or\nrather of Modes of Refutation. We will take the species he enumerates\nin their order, and notice how modern usage has departed from the\noriginal meaning of the terms. Let it be borne in mind that, when the\ndeception was not in the language, Aristotle did not trouble himself\nto determine whether it lay in the matter or in the form of thought.\n\n\u00a7 844. The following scheme presents the Aristotelian classification\nto the eye at a glance:--\n\n              |                   |--Equivocation.\n              |                   |--Amphiboly.\n              |--In the language -|--Composition.\n              |                   |--Division.\n              |                   |--Accent.\n              |                   |--Figure of Speech.\n  Modes of   -|\n  Refutation. |                        |--Accident.\n              |                        |--A dicto secundum quid.\n              |                        |--Ignoratio Elenchi.\n              |--Outside the language -|--Consequent.\n              |                        |--Petitio Principii.\n              |                        |--Non causa pro causa.\n              |                        |--Many Questions.\n\n[Footnote: for \"In the language\": The Greek is [Greek: para ten lexin],\nthe exact meaning of which is; 'due to the statement.']\n\n\u00a7 845. The Fallacy of Equivocation [Greek: \u00f2monum\u00eda] consists in an\nambiguous use of any of the three terms of a syllogism. If, for\ninstance, anyone were to argue thus--\n\n  No human being is made of paper,\n  All pages are human beings,\n  .'. No pages are made of paper--\n\nthe conclusion would appear paradoxical, if the minor term were there\ntaken in a different sense from that which it bore in its proper\npremiss. This therefore would be an instance of the fallacy of\nEquivocal Minor.\n\n\u00a7 846. For a glaring instance of the fallacy of Equivocal Major, we\nmay take the following--\n\n  No courageous creature flies,\n  The eagle is a courageous creature,\n  .'. The eagle does not fly--\n\nthe conclusion here becomes unsound only by the major being taken\nambiguously.\n\n\u00a7 847. It is, however, to the middle term that an ambiguity most\nfrequently attaches. In this case the fallacy of equivocation assumes\nthe special name of the Fallacy of Ambiguous Middle. Take as an\ninstance the following--\n\n  Faith is a moral virtue.\n  To believe in the Book of Mormon is faith.\n  .'. To believe in the Book of Mormon is a moral virtue.\n\nHere the premisses singly might be granted; but the conclusion would\nprobably be felt to be unsatisfactory.  Nor is the reason far to\nseek. It is evident that belief in a book cannot be faith in any sense\nin which that quality can rightly be pronounced to be a moral virtue.\n\n\u00a7 848. The Fallacy of Amphiboly ([Greek: \u00e1mphibol\u00eda]) is an ambiguity\nattaching to the construction of a proposition rather than to the\nterms of which it is composed. One of Aristotle's examples is this--\n\n  [Greek: t\u00f2 bo\u00falesthai labe\u00een me to\u00f9s polem\u00edous]\n\nwhich may be interpreted to mean either 'the fact of my wishing to\ntake the enemy,' or 'the fact of the enemies' wishing to take me.' The\nclassical languages are especially liable to this fallacy owing to the\noblique construction in which the accusative becomes subject to the\nverb.  Thus in Latin we have the oracle given to Pyrrhus (though of\ncourse, if delivered at all, it must have been in Greek)--\n\n  Aio te, AEacida, Romanos vincere posse.\n  Pyrrhus the Romans shall, I say, subdue (Whately),\n  [Footnote: Cicero, De Divinatione, ii. \u00a7 116; Quintilian,\n  Inst. Orat. vii 9, \u00a7 6.]\n\nwhich Pyrrhus, as the story runs, interpreted to mean that he could\nconquer the Romans, whereas the oracle subsequently explained to him\nthat the real meaning was that the Romans could conquer him. Similar\nto this, as Shakspeare makes the Duke of York point out, is the\nwitch's prophecy in Henry VI (Second Part, Act i, sc. 4),\n\n  The duke yet lives that Henry shall depose.\n\nAn instance of amphiboly may be read on the walls of Windsor\nCastle--Hoc fecit Wykeham. The king mas incensed with the bishop for\ndaring to record that he made the tower, but the latter adroitly\nreplied that what he really meant to indicate was that the tower was\nthe making of him. To the same head may be referred the famous\nsentence--'I will wear no clothes to distinguish me from my Christian\nbrethren.'\n\n\u00a7 849. The Fallacy of Composition [Greek: dia\u00edresis] is likewise a\ncase of ambiguous construction. It consists, as expounded by\nAristotle, in taking words together which ought to be taken\nseparately, e.g.\n\n  'Is it possible for a man who is not writing to write?'\n  'Of course it is.'\n  'Then it is possible for a man to write without writing.'\n\nAnd again--\n\n  'Can you carry this, that, and the other?' 'Yes.'\n  'Then you can carry this, that, and the other,'--\n\na fallacy against which horses would protest, if they could.\n\n\u00a7 850. It is doubtless this last example which has led to a convenient\nmisuse of the term 'fallacy of composition' among modern writers, by\nwhom it is defined to consist in arguing from the distributive to the\ncollective use of a term.\n\n\u00a7 851. The Fallacy of Division ([Greek: dia\u00edresis]), on the other hand,\nconsists in taking words separately which ought to be taken together,\ne.g.\n\n  [Greek: \u00e8g\u00f3 s' \u00eateka do\u00fblon \u00f4nt' \u00e8le\u00fateron [Footnote: Evidently the\n  original of the line in Terence's _Andria_, 37,--feci ex servo\n  ut esses libertus mihi.],\n\nwhere the separation of [Greek: do\u00fblon] from [Greek: \u00f4ntra] would lead\nto an interpretation exactly contrary to what is intended.\n\nAnd again--\n\n  [Greek: pent\u00e9kont' \u00e0ndr\u00f4n \u00e8kat\u00f2n l\u00edpe d\u00eeos \u00c0chille\u00fas],\n\nwhere the separation of [Greek: \u00e0ndr\u00f4n] from [Greek: \u00e8kat\u00f2n] leads to\na ludicrous error.\n\nAny reader whose youth may have been nourished on 'The Fairchild\nFamily' may possibly recollect a sentence which ran somewhat on this\nwise--'Henry,' said Mr.  Fairchild, 'is this true? Are you a thief and\na liar too?'  But I am afraid he will miss the keen delight which can\nbe extracted at a certain age from turning the tables upon\nMr. Fairchild thus--Henry said, 'Mr. Fairchild, is this true? Are\n_you_ a thief and a liar too?'\n\n\u00a7 852. The fallacy of division has been accommodated by modern writers\nto the meaning which they have assigned to the fallacy of\ncomposition. So that by the 'fallacy of division' is now meant arguing\nfrom the collective to the distributive use of a term. Further, it is\nlaid down that when the middle term is used distributively in the\nmajor premiss and collectively in the minor, we have the fallacy of\ncomposition; whereas, when the middle term is used collectively in the\nmajor premiss and distributively in the minor, we have the fallacy of\ndivision. Thus the first of the two examples appended would be\ncomposition and the second division.\n\n  (1) Two and three are odd and even.\n      Five is two and three.\n      .'. Five is odd and even.\n\n  (2) The Germans are an intellectual people.\n      Hans and Fritz are Germans.\n      .'. They are intellectual people.\n\n\u00a7 853. As the possibility of this sort of ambiguity is not confined to\nthe middle term, it seems desirable to add that when either the major\nor minor term is used distributively in the premiss and collectively\nin the conclusion, we have the fallacy of composition, and in the\nconverse case the fallacy of division. Here is an instance of the\nlatter kind in which the minor term is at fault--\n\n  Anything over a hundredweight is too heavy to lift.\n  These sacks (collectively) are over a hundredweight.\n  .'. These sacks (distributively) are too heavy to lift.\n\n\u00a7 854. The ambiguity of the word 'all,' which has been before\ncommented upon (\u00a7 119), is a great assistance in the English language\nto the pair of fallacies just spoken of.\n\n\u00a7 835. The Fallacy of Accent ([Greek: prosod\u00eda]) is neither more nor\nless than a mistake in Greek accentuation. As an instance Aristotle\ngives Iliad xxiii. 328, where the ancient copies of Homer made\nnonsense of the words [Greek: t\u00f2 m\u00e8n o\u00fa katap\u00fatetai \u00f3mbro] by writing\n[Greek: o\u00fb] with the circumflex in place of [Greek: o\u00fa] with the acute\naccent. [Footnote: This goes to show that the ancient Greeks did not\ndistinguish in pronunciation between the rough and smooth breathing\nany more than their modern representatives.]  Aristotle remarks that\nthe fallacy is one which cannot easily occur in verbal argument, but\nrather in writing and poetry.\n\n\u00a7 856. Modern writers explain the fallacy of accent to be the mistake\nof laying the stress upon the wrong part of a sentence. Thus when the\ncountry parson reads out, 'Thou shall not bear false witness\n_against_ thy neighbour,' with a strong emphasis upon the word\n'against,' his ignorant audience leap [sic] to the conclusion that it\nis not amiss to tell lies provided they be in favour of one's\nneighbour.\n\n\u00a7 857. The Fallacy of Figure of Speech [Greek: t\u00f2 sch\u00eama t\u00eas l\u00e9xeos]\nresults from any confusion of grammatical forms, as between the\ndifferent genders of nouns or the different voices of verbs, or their\nuse as transitive or intransitive, e.g. [Greek: \u00fagia\u00ednein] has the\nsame grammatical form as [Greek: t\u00e9mnein] or [Greek: o\u00eckodome\u00een], but\nthe former is intransitive, while the latter are transitive. A sophism\nof this kind is put into the mouth of Socrates by Aristophanes in the\nClouds (670-80).  The philosopher is there represented as arguing that\n[Greek: k\u00e1pdopos] must be masculine because [Greek: Kle\u00f3numos] is. On\nthe surface this is connected with language, but it is essentially a\nfallacy of false analogy.\n\n\u00a7 858. To this head may be referred what is known as the Fallacy of\nParonymous Terms. This is a species of equivocation which consists in\nslipping from the use of one part of speech to that of another, which\nis derived from the same source, but has a different meaning. Thus\nthis fallacy would be committed if, starting from the fact that there\nis a certain probability that a hand at whist will consist of thirteen\ntrumps, one were to proceed to argue that it was probable, or that he\nhad proved it.\n\n\u00a7 859. We turn now to the tricks of refutation which lie outside the\nlanguage, whether the deception be due to the assumption of a false\npremiss or to some unsoundness in the reasoning.\n\n\u00a7 860. The first on the list is the Fallacy of Accident ([Greek: t\u00f2\nsumbebek\u00f3s]). This fallacy consists in confounding an essential with\nan accidental difference, which is not allowable, since many things\nare the same in essence, while they differ in accidents. Here is the\nsort of example that Aristotle gives--\n\n  'Is Plato different from Socrates ?' 'Yes.' 'Is Socrates a man ?'\n  'Yes.' 'Then Plato is different from man.'\n\nTo this we answer--No: the difference of accidents between Plato and\nSocrates does not go so deep as to affect the underlying essence. To\nput the thing more plainly, the fallacy lies in assuming that whatever\nis different from a given subject must be different from it in all\nrespects, so that it is impossible for them to have a common\npredicate. Here Socrates and Plato, though different from one another,\nare not so different but that they have the common predicate 'man.'\nThe attempt to prove that they have not involves an illicit process of\nthe major.\n\n\u00a7 861. The next fallacy suffers from the want of a convenient name. It\nis called by Aristotle [Greek: t\u00f2 \u00e1plos t\u00f3de \u00ea p\u00ea l\u00e9gestai ka\u00ec m\u00e8\nkup\u00edos] or, more briefly, [Greek: t\u00f2 \u00e1pl\u00f4s \u00ea m\u00e9], or [Greek: t\u00f2 p\u00ea ka\u00ed\n\u00e1pl\u00f4s], and by the Latin writers 'Fallacia a dicto secundum quid ad\ndictum simpliciter.' It consists in taking what is said in a\nparticular respect as though it held true without any restriction,\ne.g., that because the nonexistent ([Greek: t\u00f2 m\u00e8 \u00f4n]) is a matter of\nopinion, that therefore the non-existent is, or again that because the\nexistent ([Greek: t\u00f2 \u00f4n]) is not a man, that therefore the existent is\nnot. Or again, if an Indian, who as a whole is black, has white teeth,\nwe should be committing this species of fallacy in declaring him to be\nboth white and not-white. For he is only white in a certain respect\n([Greek: p\u00ea]), but not absolutely ([Greek: \u00e0pl\u00f4s]). More\ndifficulty, says Aristotle, may arise when opposite qualities exist in\na thing in about an equal degree. When, for instance, a thing is half\nwhite and half black, are we to say that it is white or black? This\nquestion the philosopher propounds, but does not answer. The force of\nit lies in the implied attack on the Law of Contradiction. It would\nseem in such a case that a thing may be both white and not-white at\nthe same time. The fact is--so subtle are the ambiguities of\nlanguage--that even such a question as 'Is a thing white or\nnot-white?' straightforward, as it seems, is not really a fair one. We\nare entitled sometimes to take the bull by the horns, and answer with\nthe adventurous interlocutor in one of Plato's dialogues--'Both and\nneither.' It may be both in a certain respect, and yet neither\nabsolutely.\n\n\u00a7 862. The same sort of difficulties attach to the Law of Excluded\nMiddle, and may be met in the same way.  It might, for instance, be\nurged that it could not be said with truth of the statue seen by\nNebuchadnezzar in his dream either that it was made of gold or that it\nwas not made of gold: but the apparent plausibility of the objection\nwould be due merely to the ambiguity of language.  It is not true, on\nthe one hand, that it was made of gold (in the sense of being composed\nentirely of that metal); and it is not true, on the other, that it was\nnot made of gold (in the sense of no gold at all entering into its\ncomposition). But let the ambiguous proposition be split up into its\ntwo meanings, and the stringency of the Law of Excluded Middle will at\nonce appear--\n\n  (1) It must either have been composed entirely of gold or not.\n\n  (2) Either gold must have entered into its composition or not.\n\n\u00a7 863. By some writers this fallacy is treated as the converse of the\nlast, the fallacy of accident being assimilated to it under the title\nof the 'Fallacia a dicto simpliciter ad dictum secundum quid.' In this\nsense the two fallacies may be defined thus.\n\nThe Fallacy of Accident consists in assuming that what holds true as a\ngeneral rule will hold true under some special circumstances which may\nentirely alter the case.  The Converse Fallacy of Accident consists in\nassuming that what holds true under some special circumstances must\nhold true as a general rule.\n\nThe man who, acting on the assumption that alcohol is a poison,\nrefuses to take it when he is ordered to do so by the doctor, is\nguilty of the fallacy of accident; the man who, having had it\nprescribed for him when he was ill, continues to take it morning,\nnoon, and night, commits the converse fallacy.\n\n\u00a7 864. There ought to be added a third head to cover the fallacy of\narguing from one special case to another.\n\n\u00a7 865. The next fallacy is Ignoratio Elenchi [Greek: \u00e8l\u00e9gchou\n\u00e2gnoia]. This fallacy arises when by reasoning valid in itself one\nestablishes a conclusion other than what is required to upset the\nadversary's assertion. It is due to an inadequate conception of the\ntrue nature of refutation.  Aristotle therefore is at the pains to\ndefine refutation at full length, thus--\n\n'A refutation [Greek: \u00ealegchos] is the denial of one and the same--not\nname, but thing, and by means, not of a synonymous term, but of the\nsame term, as a necessary consequence from the data, without\nassumption of the point originally at issue, in the same respect, and\nin the same relation, and in the same way, and at the same time.'\n\nThe ELENCHUS then is the exact contradictory of the opponent's\nassertion under the terms of the law of contradiction. To establish by\na syllogism, or series of syllogisms, any other proposition, however\nslightly different, is to commit this fallacy. Even if the substance\nof the contradiction be established, it is not enough unless the\nidentical words of the opponent are employed in the\ncontradictory. Thus if his thesis asserts or denies something about\n[Greek: l\u00f3pion], it is not enough for you to prove the contradictory\nwith regard to [Greek: \u00ecm\u00e1tion]. There will be need of a further\nquestion and answer to identify the two, though they are admittedly\nsynonymous. Such was the rigour with which the rules of the game of\ndialectic were enforced among the Greeks!\n\n\u00a7 866. Under the head of Ignoratio Elenchi it has become usual to\nspeak of various forme of argument which have been labelled by the\nLatin writers under such names as 'argumentum ad hominem,' 'ad\npopulum,' 'ad verecundiam,' 'ad ignorantiam,' 'ad baculum'--all of\nthem opposed to the 'argumentum ad rem' or 'ad judicium.'\n\n\u00a7 867. By the 'argumentum ad hominem' was perhaps meant a piece of\nreasoning which availed to silence a particular person, without\ntouching the truth of the question.  Thus a quotation from Scripture\nis sufficient to stop the mouth of a believer in the inspiration of\nthe Bible.  Hume's Essay on Miracles is a noteworthy instance of the\n'argumentum ad hominem' in this sense of the term. He insists strongly\non the evidence for certain miracles which he knew that the prejudices\nof his hearers would prevent their ever accepting, and then asks\ntriumphantly if these miracles, which are declared to have taken place\nin an enlightened age in the full glare of publicity, are palpably\nimposture, what credence can be attached to accounts of extraordinary\noccurrences of remote antiquity, and connected with an obscure corner\nof the globe? The 'argumentum ad judicium' would take miracles as a\nwhole, and endeavour to sift the amount of truth which may lie in the\naccounts we have of them in every age. [Footnote: On this subject see\nthe author's _Attempts at Truth_ (Trubner & Co.), pp. 46-59.]\n\n\u00a7 868. In ordinary discourse at the present day the term 'argumentum\nad hominem' is used for the form of irrelevancy which consists in\nattacking the character of the opponent instead of combating his\narguments, as illustrated in the well-known instructions to a\nbarrister--'No case: abuse the plaintiff's attorney.'\n\n\u00a7 869. The 'argumentum ad populum' consists in an appeal to the\npassions of one's audience. An appeal to passion, or to give it a less\nquestion-begging name, to feeling, is not necessarily amiss. The heart\nof man is the instrument upon which the rhetorician plays, and he has\nto answer for the harmony or the discord that comes of his\nperformance.\n\n\u00a7 870. The 'argumentum ad verecundiam' is an appeal to the feeling of\nreverence or shame. It is an argument much used by the old to the\nyoung and by Conservatives to Radicals.\n\n\u00a7 871. The 'argumentum ad ignorantiam' consists simply in trading on\nthe ignorance of the person addressed, so that it covers any kind of\nfallacy that is likely to prove effective with the hearer.\n\n\u00a7 872. The 'argumentum ad baculum' is unquestionably a form of\nirrelevancy. To knock a man down when he differs from you in opinion\nmay prove your strength, but hardly your logic.\n\nA sub-variety of this form of irrelevancy was exhibited lately at a\nsocialist lecture in Oxford, at which an undergraduate, unable or\nunwilling to meet the arguments of the speaker, uncorked a bottle,\nwhich had the effect of instantaneously dispersing the audience. This\nmight be set down as the 'argumentum ad nasum.'\n\n\u00a7 873. We now come to the Fallacy of the Consequent, a term which has\nbeen more hopelessly abused than any.  What Aristotle meant by it was\nsimply the assertion of the consequent in a conjunctive proposition,\nwhich amounts to the same thing as the simple conversion of A (\u00a7 489),\nand is a fallacy of distribution. Aristotle's example is this--\n\n  If it has rained, the ground is wet.\n  .'. If the ground is wet, it has rained.\n\nThis fallacy, he tells us, is often employed in rhetoric in dealing\nwith presumptive evidence. Thus a speaker, wanting to prove that a man\nis an adulterer, will argue that he is a showy dresser, and has been\nseen about at nights. Both these things however may be the case, and\nyet the charge not be true.\n\n\u00a7 874. The Fallacy of Petitio or Assumptio Principii [Greek: t\u00f2 \u00e8n\n\u00e0rch\u00ea a\u00ecte\u00eestai or lamb\u00e1nein] to which we now come, consists in an\nunfair assumption of the point at issue.  The word [Greek: a\u00ecte\u00eestai],\nin Aristotle's name for it points to the Greek method of dialectic by\nmeans of question and answer. This fact is rather disguised by the\nmysterious phrase 'begging the question.' The fallacy would be\ncommitted when you asked your opponent to grant, overtly or covertly,\nthe very proposition originally propounded for discussion.\n\n\u00a7 875. As the question of the precise nature of this fallacy is of\nsome importance we will take the words of Aristotle himself\n(Top. viii. 13. \u00a7\u00a7 2, 3). 'People seem to beg the question in five\nways. First and most glaringly, when one takes for granted the very\nthing that has to be proved. This by itself does not readily escape\ndetection, but in the case of \"synonyms,\" that is, where the name and\nthe definition have the same meaning, it does so more\neasily. [Footnote: Some light is thrown upon this obscure passage by a\ncomparison with Cat. I. \u00a7 3, where 'synonym' is defined. To take the\nword here in its later and modern sense affords an easy\ninterpretation, which is countenanced by Alexander Aphrodisiensis, but\nit is flat against the usage of Aristotle, who elsewhere gives the\nname 'synonym,' not to two names for the same thing, but to two things\ngoing under the same name. See Trendelenberg on the passage.]\n\nSecondly, when one assumes universally that which has to be proved in\nparticular, as, if a man undertaking to prove that there is one\nscience of contraries, were to assume that there is one science of\nopposites generally. For he seems to be taking for granted along with\nseveral other things what he ought to have proved by itself.\n\nThirdly, when one assumes the particulars where the universal has to\nbe proved; for in so doing a man is taking for granted separately what\nhe was bound to prove along with several other things.  Again, when\none assumes the question at issue by splitting it up, for instance,\nif, when the point to be proved is that the art of medicine deals with\nhealth and disease, one were to take each by itself for granted.\n\nLastly, if one were to take for granted one of a pair of necessary\nconsequences, as that the side is incommensurable with the diagonal,\nwhen it is required to prove that the diagonal is incommensurable with\nthe side.'\n\n\u00a7 876. To sum up briefly, we may beg the question in five ways--\n\n  (1) By simply asking the opponent to grant the point which requires\n  to be proved;\n\n  (2) by asking him to grant some more general truth which involves\n  it;\n\n  (3) by asking him to grant the particular truths which it involves;\n\n  (4) by asking him to grant the component parts of it in detail;\n\n  (5) by asking him to grant a necessary consequence of it.\n\n\u00a7 877. The first of these five ways, namely, that of begging the\nquestion straight off, lands us in the formal fallacy already spoken\nof (\u00a7 838), which violates the first of the general rules of\nsyllogism, inasmuch as a conclusion is derived from a single premiss,\nto wit, itself.\n\n\u00a7 878. The second, strange to say, gives us a sound syllogism in\nBarbara, a fact which countenances the blasphemers of the syllogism in\nthe charge they bring against it of containing in itself a petitio\nprincipii. Certainly Aristotle's expression might have been more\nguarded. But it is clear that his quarrel is with the matter, not with\nthe form in such an argument. The fallacy consists in assuming a\nproposition which the opponent would be entitled to deny. Elsewhere\nAristotle tells us that the fallacy arises when a truth not evident by\nits own light is taken to be so. [Footnote: [Greek: \u00d4tan t\u00f2 m\u00e8 d\u00ed\na\u00f9to\u00fb gnost\u00f2n d\u00ed a\u00f9to\u00fb tis \u00e8picheira\u00ea deikn\u00fanai, t\u00f3t' a\u00ecte\u00eetai t\u00f2 \u00e8x\n\u00e0rch\u00eas.]. Anal. Pr. II. 16. \u00a7 I ad fin.]\n\n\u00a7 879. The third gives us an inductio per enumerationem simplicem, a\nmode of argument which would of course be unfair as against an\nopponent who was denying the universal.\n\n\u00a7 880. The fourth is a more prolix form of the first.\n\n\u00a7 881. The fifth rests on Immediate Inference by Relation (\u00a7 534).\n\n\u00a7 882. Under the head of petitio principii comes the fallacy of\nArguing in a Circle, which is incidental to a train of reasoning. In\nits most compressed form it may be represented thus--\n\n  (1) B is A.\n      C is B.\n      .'. C is A.\n\n  (2) C is A.\n      B is C.\n      .'. B is A.\n\n\u00a7 883. The Fallacy of Non causa pro causa ([Greek: t\u00f2 m\u00e8 a\u00eetion] or\n[Greek: a\u00eetoin]) is another, the name of which has led to a complete\nmisinterpretation. It consists in importing a contradiction into the\ndiscussion, and then fathering it on the position controverted. Such\narguments, says Aristotle, often impose upon the users of them\nthemselves.  The instance he gives is too recondite to be of general\ninterest.\n\n\u00a7 884. Lastly, the Fallacy of Many Questions ([Greek: t\u00f2 t\u00e0 d\u00e9o\n\u00e8rot\u00e9mata \u00ean poie\u00een]) is a deceptive form of interrogation, when a\nsingle answer is demanded to what is not really a single question. In\ndialectical discussions the respondent was limited to a simple 'yes'\nor 'no'; and in this fallacy the question is so framed as that either\nanswer would seem to imply the acceptance of a proposition which would\nbe repudiated. The old stock instance will do as well as\nanother--'Come now, sir, answer \"yes\" or \"no.\" Have you left off\nbeating your mother yet?'  Either answer leads to an apparent\nadmission of impiety.\n\nA late Senior Proctor once enraged a man at a fair with this form of\nfallacy. The man was exhibiting a blue horse; and the distinguished\nstranger asked him--'With what did you paint your horse?'\n\n\n\n\nEXERCISES.\n\n\nThese exercises should be supplemented by direct questions upon the\ntext, which it is easy for the student or the teacher to supply for\nhimself.\n\n\nPART I.\n\n\nCHAPTER I.\n\nClassify the following words according as they are categorematic,\nsyncategorematic or acategorematic;--\n\n  come        peradventure    why\n  through     inordinately    pshaw\n  therefore   circumspect     puss\n  grand       inasmuch        stop\n  touch       sameness        back\n  cage        disconsolate    candle.\n\n\nCHAPTER II.\n\nClassify the following things according as they are substances,\nqualities or relations;--\n\n  God         likeness       weight\n  blueness    grass          imposition\n  ocean       introduction   thinness\n  man         air            spirit\n  Socrates    raillery       heat\n  mortality   plum           fire.\n\n\nCHAPTER III.\n\n1. Give six instances each of-attribute, abstract, singular,\nprivative, equivocal and relative terms.\n\n2. Select from the following list of words such as are terms, and\nstate whether they are (1) abstract or concrete, (2) singular or\ncommon, (3) univocal or equivocal:--\n\n  van        table        however\n  enter      decidedly    tiresome\n  very       butt         Solomon\n  infection  bluff        Czar\n  short      although     Caesarism\n  distance   elderly      Nihilist.\n\n3. Which of the following words are abstract terms?--\n\n  quadruped        event           through\n  hate             desirability    thorough\n  fact             expressly       thoroughness\n  faction          wish            light\n  inconvenient     will            garden\n  inconvenience    volition        grind.\n\n4. Refer the following terms to their proper place under each of the\ndivisions in the scheme:--\n\n  horse          husband         London\n  free           lump            empty\n  liberty        rational        capital\n  impotent       reason          Capitol\n  impetuosity    irrationality   grave\n  impulsive      double          calf.\n\n5. Give six instances each of proper names and designations.\n\n6. Give six instances each of connotative and non-connotative terms.\n\n7. Give the extension and intension of--\n\n  sermon          animal      sky\n  clock           square      gold\n  sport           fish        element\n  bird            student     fluid\n  art             river       line\n  gas             servant     language\n\n\nCHAPTER IV.\n\nArrange the following terms in order of extension--carnivorous, thing,\nmatter, mammal, organism, vertebrate, cat, substance, animal.\n\n       *       *       *       *       *\n\n\nPART II.\n\n\nCHAPTER I.\n\nGive a name to each of the following sentences:--\n\n  (1) Oh, that I had wings like a dove!\n\n  (2) The more, the merrier.\n\n  (3) Come rest in this bosom, my own stricken deer.\n\n  (4) Is there balm in Gilead?\n\n  (5) Hearts may be trumps.\n\n\nCHAPTER II.\n\nAnalyse the following propositions into subject, copula and\npredicate:--\n\n  (1) He being dead yet speaketh.\n\n  (2) There are foolish politicians.\n\n  (3) Little does he care.\n\n  (4) There is a land of pure delight.\n\n  (5) All's well that ends well.\n\n  (6) Sweet is the breath of morn.\n\n  (7) Now it came to pass that the beggar died.\n\n  (8) Who runs may read.\n\n  (9) Great is Diana of the Ephesians.\n\n  (10) Such things are.\n\n  (11) Not more than others I deserve.\n\n  (12) The day will come when Ilium's towers shall perish.\n\n\nCHAPTER III.\n\n1. Express in logical form, affixing the proper symbol:--\n\n  (1) Some swans are not white.\n\n  (2) All things are possible to them that believe.\n\n  (3) No politicians are unprincipled.\n\n  (4) Some stones float on water.\n\n  (5) The snow has melted.\n\n  (6) Eggs are edible.\n\n  (7) All kings are not wise.\n\n  (8) Moths are not butterflies.\n\n  (9) Some men are born great.\n\n  (10) Not all who are called are chosen.\n\n  (11) It is not good for man to be alone.\n\n  (12) Men of talents have been known to fail in life.\n\n  (13) 'Tis none but a madman would throw about fire.\n\n  (14) Every bullet does not kill.\n\n  (15) Amongst Unionists are Whigs.\n\n  (16) Not all truths are to be told.\n\n  (17) Not all your efforts can save him.\n\n  (18) The whale is a mammal.\n\n  (19) Cotton is grown in Cyprus.\n\n  (20) An honest man's the noblest work of God.\n\n  (21) No news is good news.\n\n  (22) No friends are like old friends.\n\n  (23) Only the ignorant affect to despise knowledge.\n\n  (24) All that trust in Him shall not be ashamed.\n\n  (25) All is not gold that glitters.\n\n  (26) The sun shines upon the evil and upon the good.\n\n  (27) Not to go on is to go back.\n\n  (28) The king, minister, and general are a pretty trio.\n\n  (29) Amongst dogs are hounds.\n\n  (30) A fool is not always wrong.\n\n  (31) Alexander was magnanimous.\n\n  (32) Food is necessary to life.\n\n  (33) There are three things to be considered,\n\n  (34) By penitence the Eternal's wrath's appeased.\n\n  (35) Money is the miser's end.\n\n  (36) Few men succeed in life.\n\n  (37) All is lost, save honour.\n\n  (38) It is mean to hit a man when he is down.\n\n  (39) Nothing but coolness could have saved him.\n\n  (40) Books are generally useful.\n\n  (41) He envies others' virtue who has none himself.\n\n  (42) Thankless are all such offices.\n\n  (43) Only doctors understand this subject.\n\n  (44) All her guesses but two were correct.\n\n  (45) All the men were twelve.\n\n  (46) Gossip is seldom charitable.\n\n2. Give six examples of indefinite propositions, and then quantify\nthem according to their matter.\n\n3. Compose three propositions of each of the following kinds:--\n\n  (1) with common terms for subjects;\n\n  (2) with abstract terms for subjects;\n\n  (3) with singular terms for predicates;\n\n  (4) with collective terms for predicates;\n\n  (5) with attributives in their subjects;\n\n  (6) with abstract terms for predicates.\n\n\nCHAPTER IV.\n\n1. Point out what terms are distributed or undistributed in the\nfollowing propositions:--\n\n  (1) The Chinese are industrious.\n\n  (2) The angle in a semi-circle is a right angle.\n\n  (3) Not one of the crew survived.\n\n  (4) The weather is sometimes not propitious.\n\nThe same exercise may be performed upon any of the propositions in the\npreceding list.\n\n2. Prove that in a negative proposition the predicate must be\ndistributed.\n\n\nCHAPTER V.\n\nAffix its proper symbol to each of the following propositions:--\n\n  (1) No lover he who is not always fond.\n\n  (2) There are Irishmen and Irishmen.\n\n  (3) Men only disagree,\n      Of creatures rational.\n\n  (4) Some wise men are poor.\n\n  (5) No Popes are some fallible beings.\n\n  (6) Some step-mothers are not unjust.\n\n  (7) The most original of the Roman poets was Lucretius.\n\n  (8) Some of the immediate inferences are all the forms of\n  conversion.\n\n\nCHAPTER VI.\n\n1. Give six examples of terms standing one to another as genus to\nspecies.\n\n2. To which of the heads of predicables would you refer the following\nstatements? And why?\n\n  (1) A circle is the largest space that can be contained by one line.\n\n  (2) All the angles of a square are right angles.\n\n  (3) Man alone among animals possesses the faculty of laughter.\n\n  (4) Some fungi are poisonous.\n\n  (5) Most natives of Africa are negroes.\n\n  (6) All democracies are governments.\n\n  (7) Queen Anne is dead.\n\n\nCHAPTER VII.\n\n1. Define the following terms--\n\n  Sun          inn-keeper   tea-pot\n  hope         anger        virtue\n  bread        diplomacy    milk\n  carpet       man          death\n  sincerity    telescope    mountain\n  poverty      Senate       novel.\n\n2. Define the following terms as used in Political Economy--\n\n  Commodity    barter     value\n  wealth       land       price\n  money        labour     rent\n  interest     capital    wages\n  credit       demand     profits.\n\n3. Criticise the following as definitions--\n\n  (1) Noon is the time when the shadows of bodies are shortest.\n\n  (2) Grammar is the science of language.\n\n  (3) Grammar is a branch of philology.\n\n  (4) Grammar is the art of speaking and writing a language with\n  propriety.\n\n  (5) Virtue is acting virtuously.\n\n  (6) Virtue is that line of conduct which tends to produce happiness.\n\n  (7) A dog is an animal of the canine species.\n\n  (8) Logic is the art of reasoning.\n\n  (9) Logic is the science of the investigation of truth by means of\n  evidence.\n\n  (10) Music is an expensive noise.\n\n  (11) The sun is the centre of the solar system.\n\n  (12) The sun is the brightest of those heavenly bodies that move\n  round the earth.\n\n  (13) Rust is the red desquamation of old iron.\n\n  (14) Caviare is a kind of food.\n\n  (15) Life is the opposite of death.\n\n  (16) Man is a featherless biped.\n\n  (17) Man is a rational biped.\n\n  (18) A gentleman is a person who has no visible means of\n  subsistence.\n\n  (19) Fame is a fancied life in others' breath.\n\n  (20) A fault is a quality productive of evil or inconvenience.\n\n  (21) An oligarchy is the supremacy of the rich in a state.\n\n  (22) A citizen is one who is qualified to exercise deliberative and\n  judicial functions.\n\n  (23) Length is that dimension of a solid which would be measured by\n  the longest line.\n\n  (24) An eccentricity is a peculiar idiosyncrasy.\n\n  (25) Deliberation is that species of investigation which is\n  concerned with matters of action.\n\n  (26) Memory is that which helps us to forget.\n\n  (27) Politeness is the oil that lubricates the wheels of society.\n\n  (28) An acute-angled triangle is one which has an acute angle.\n\n  (29) A cause is that without which something would not be.\n\n  (30) A cause is the invariable antecedent of a phenomenon.\n\n  (31) Necessity is the mother of invention.\n\n  (32) Peace is the absence of war.\n\n  (33) A net is a collection of holes strung together.\n\n  (34) Prudence is the ballast of the moral vessel.\n\n  (35) A circle is a plane figure contained by one line.\n\n  (36) Superstition is a tendency to look for constancy where\n  constancy is not to be expected.\n\n  (37) Bread is the staff of life.\n\n  (38) An attributive is a term which cannot stand as a subject.\n\n  (39) Life is bottled sunshine.\n\n  (40) Eloquence is the power of influencing the feelings by speech or\n  writing.\n\n  (41) A tombstone is a monument erected over a grave in memory of the\n  dead.\n\n  (42) Whiteness is the property or power of exciting the sensation of\n  white.\n\n  (43) Figure is the limit of a solid.\n\n  (44) An archdeacon is one who exercises archidiaconal functions.\n\n  (45) Humour is thinking in jest while feeling in earnest.\n\n\nCHAPTER VIII.\n\n1. Divide the following terms--\n\n  Soldier    end           book\n  church     good          oration\n  apple      cause         school\n  ship       government    letter\n  vehicle    science       verse.\n\n2. Divide the following terms as used in Political Economy--\n\n  Requisites of production, labour, consumption, stock, wealth,\n  capital.\n\n3. Criticise the following as divisions--\n\n  (1) Great Britain into England, Scotland, Wales, and Ireland.\n\n  (2) Pictures into sacred, historical, landscape, and mythological.\n\n  (3) Vertebrate animals into quadrupeds, birds, fishes, and reptiles.\n\n  (4) Plant into stem, root, and branches.\n\n  (5) Ship into frigate, brig, schooner, and merchant-man.\n\n  (6) Books into octavo, quarto, green, and blue.\n\n  (7) Figure into curvilinear and rectilinear.\n\n  (8) Ends into those which are ends only, means and ends, and means\n  only.\n\n  (9) Church into Gothic, episcopal, high, and low.\n\n  (10) Sciences into physical, moral, metaphysical, and medical.\n\n  (11) Library into public and private.\n\n  (12) Horses into race-horses, hunters, hacks, thoroughbreds, ponies,\n  and mules.\n\n4. Define and divide--\n\n  Meat, money, virtue, triangle;\n\nand give, as far as possible, a property and accident of each.\n\n\nPART III.\n\n\nCHAPTERS I-III.\n\n1. What kind of influence have we here?\n\n  The author of the Iliad was unacquainted with writing.\n  Homer was the author of the Iliad.\n  .'. Homer was unacquainted with writing.\n\n2. Give the logical opposites of the following propositions--\n\n  (1) Knowledge is never useless.\n\n  (2) All Europeans are civilised.\n\n  (3) Some monks are not illiterate.\n\n  (4) Happy is the man that findeth wisdom.\n\n  (5) No material substances are devoid of weight.\n\n  (6) Every mistake is not culpable.\n\n  (7) Some Irishmen are phlegmatic.\n\n3. Granting the truth of the following propositions, what other\npropositions can be inferred by opposition to be true or false?\n\n\n  (1) Men of science are often mistaken.\n\n  (2) He can't be wrong, whose life is in the right.\n\n  (3) Sir Walter Scott was the author of Waverley.\n\n  (4) The soul that sinneth it shall die.\n\n  (5) All women are not vain.\n\n4. Granting the falsity of the following propositions, what other\npropositions can be inferred by opposition to be true or false?--\n\n  (1) Some men are not mortal.\n\n  (2) Air has no weight.\n\n  (3) All actors are improper characters.\n\n  (4) None but dead languages are worth studying.\n\n  (5) Some elements are compound.\n\n\nCHAPTER IV.\n\n1. Give, as far as possible, the logical converse of each of the\nfollowing propositions--\n\n  (1) Energy commands success.\n\n  (2) Mortals cannot be happy.\n\n  (3) There are mistakes which are criminal.\n\n  (4) All's well that ends well.\n\n  (5) Envious men are disliked.\n\n  (6) A term is a kind of word or collection of words.\n\n  (7) Some Frenchmen are not vivacious.\n\n  (8) All things in heaven and earth were hateful to him.\n\n  (9) The square of three is nine.\n\n  (10) All cannot receive this saying.\n\n  (11) P struck Q.\n\n  (12) Amas.\n\n2. 'More things may be contained in my philosophy than exist in heaven\nor earth: but the converse proposition is by no means true.' Is the\nterm converse here used in its logical meaning?\n\n\nCHAPTER V.\n\nPermute the following propositions--\n\n  (1) All just acts are expedient.\n\n  (2) No display of passion is politic.\n\n  (3) Some clever people are not prudent.\n\n  (4) Some philosophers have been slaves.\n\nThe same exercise may be performed upon any of the propositions in the\npreceding lists.\n\n\nCHAPTER VI.\n\n1. Give the converse by negation of--\n\n  (1) All women are lovely.\n\n  (2) Some statesmen are not practical.\n\n  (3) All lawyers are honest.\n\n  (4) All doctors are skilful.\n\n  (5) Some men are not rational.\n\n2. Give the contrapositive of--\n\n  (1) All solid substances are material.\n\n  (2) All the men who do not row play cricket.\n\n  (3) All impeccable beings are other than human,\n\n  (4) Some prejudiced persons are not dishonest.\n\n3. Prove indirectly the truth of the contrapositive of 'All A is B.'\n\n4. Criticise the following as immediate inferences--\n\n  (1) All wise men are modest.\n      .'. No immodest men are wise.\n\n  (2) Some German students are not industrious.\n      .'. Some industrious students are not Germans.\n\n  (3) Absolute difference excludes all likeness.\n      .'. Any likeness is a proof of sameness.\n\n  (4) None but the brave deserve the fair.\n      .'. All brave men deserve the fair.\n\n  (5) All discontented men are unhappy.\n      .'. No contented men are unhappy.\n\n  (6) Books being a source of instruction, our knowledge must come\n  from our libraries.\n\n  (7) All Jews are Semitic.\n      .'. Some non-Semitic people are not Jews.\n\n5. Show by what kind of inference each of the subjoined propositions\nfollows from\n\n  All discontented men are unhappy.\n\n  (1) All happy men are contented.\n\n  (2) Some discontented men are unhappy.\n\n  (3) Some contented men are happy.\n\n  (4) Some unhappy men are not contented.\n\n  (5) No discontented men are happy.\n\n  (6) Some happy men are contented.\n\n  (7) Some contented men are not unhappy.\n\n  (8) Some unhappy men are discontented.\n\n  (9) No happy men are discontented.\n\n  (10) Some discontented men are not happy.\n\n  (11) Some happy men are not discontented.\n\n  (12) None but unhappy men are discontented.\n\nFrom how many of these propositions can the original one be derived?\nAnd why not from all?\n\n\nCHAPTER VII.\n\nWhat kind of inference have we here?--\n\n  (1) None but the ignorant despise knowledge.\n      .'. No wise man despises knowledge.\n\n  (2) A is superior to B.\n      .'. B is inferior to A.\n\n\nCHAPTER VIII.\n\nFill up the following enthymemes, mentioning to which order they\nbelong, and state which of them are expressed in problematic form--\n\n  (1) I am fond of music: for I always like a comic song.\n\n  (2) All men are born to suffering, and therefore you must expect\n  your share.\n\n  (3) Job must have committed some secret sins: for he fell into\n  dreadful misfortunes.\n\n  (4) Latin was the language of the Vestals, and therefore no lady\n  need be ashamed of speaking it.\n\n  (5) None but physicians came to the meeting. There were therefore no\n  nurses there.\n\n  (6) The human soul extends through the whole body, for it is found\n  in every member.\n\n  (7) No traitor can be trusted, and you are a traitor.\n\n  (8) Whatever has no parts does not perish by the dissolution of its\n  parts. Therefore the soul of man is imperishable.\n\nIs the suppressed premiss in any case disputable on material grounds?\n\n\nCHAPTERS IX-XVIII.\n\nRefer the following arguments to their proper mood and figure, or show\nwhat rules of syllogism they violate.\n\n  (1) No miser is a true friend, for he does not assist his friend\n  with his purse.\n\n  (2) Governments are good which promote prosperity.\n      The government of Burmah does not promote prosperity.\n      .'. It is not a good government.\n\n  (3) Land is not property.\n      Land produces barley.\n      .'. Beer is intoxicating.\n\n  (4) Nothing is property but that which is the product of man's hand.\n      The horse is not the product of man's hand.\n      .'. The horse is not property.\n\n  (5) Some Europeans at least are not Aryans, because the Finns are\n  not.\n\n  (6) Saturn is visible from the earth, and the moon is visible from\n  the earth. Therefore the moon is visible from Saturn.\n\n  (7) Some men of self-command are poor, and therefore some noble\n  characters are poor.\n\n  (8) Sparing the rod spoils the child: so John will turn out very\n  good, for his mother beats him every day.\n\n  (9) Some effects of labour are not painful, since every virtue is an\n  effect of labour.\n\n (10) The courageous are confident and the experienced are\n confident. Therefore the experienced are courageous.\n\n (11) No tale-bearer is to be trusted, and therefore no great talker\n is to be trusted, for all tale-bearers are great talkers.\n\n (12) Socrates was wise, and wise men alone are happy: therefore\n Socrates was happy.\n\n\nII.\n\n1. From the major 'No matter thinks' draw, by supplying the minor, the\nfollowing conclusions--\n\n  (1) Some part of man does not think.\n\n  (2) The soul of man is not matter.\n\n  (3) Some part of man is not matter.\n\n  (4) Some substance does not think.\n\nName the figured mood into which each syllogism falls.\n\n2. Construct syllogisms in the following moods and figures, stating\nwhether they are valid or invalid, and giving your reasons in each\ncase--\n\n  AEE in the first figure; EAO in the second; IAI in the third; AII in\n  the fourth.\n\n3. Prove that 'Brass is not a metal,' using as your middle term\n'compound body.'\n\n4. Construct syllogisms to prove or disprove--\n\n  (1) Some taxes are necessary.\n\n  (2) No men are free.\n\n  (3) Laws are salutary.\n\n5. Prove by a syllogism in Bokardo that 'Some Socialists are not\nunselfish,' and reduce your syllogism directly and indirectly.\n\n6. Prove the following propositions in the second figure, and reduce\nthe syllogisms you use to the first--\n\n  (1) All negroes are not averse to education.\n\n  (2) Only murderers should be hanged.\n\n7. Prove in Baroko and also in Ferio that 'Some Irishmen are not\nCelts.'\n\n8. Construct in words the same syllogism in all the four figures.\n\n9. Invent instances to show that false premisses may give true\nconclusions.\n\n\nIII.\n\n1. What moods are peculiar to the first, second, and third figures\nrespectively?\n\n2. What moods are common to all the figures?\n\n3. Why can there be no subaltern moods in the third figure?\n\n4. What is the only kind of conclusion that can be drawn in all the\nfigures?\n\n5. Show that IEO violates the special rules of all the figures.\n\n6. In what figures is AEE valid?\n\n7. Show that AEO is superfluous in any figure.\n\n8. Prove that O cannot be a premiss in the first figure, nor a minor\npremiss anywhere but in the second.\n\n9. Show that in the first figure the conclusion must have the quality\nof the major premiss and the quantity of the minor.\n\n10. Why do the premisses EA yield a universal conclusion in the first\ntwo figures and only a particular one in the last two?\n\n11. Show that AAI is the only mood in the fourth figure in which it is\npossible for the major term to be distributed in the premiss and\nundistributed in the conclusion.\n\n12. Why are the premisses of Fesapo and Fresison not transposed in\nreduction like those of the other moods of the fourth figure?\n\n\nIV.\n\n1. Why is it sufficient to distribute the middle term once only?\n\n2. Prove that from two affirmative premisses you cannot get a negative\nconclusion.\n\n3. Prove that there must be at least one more term distributed in the\npremisses than in the conclusion.\n\n4. Prove that the number of distributed terms in the premisses cannot\nexceed those in the conclusion by more than two.\n\n5. Prove that the number of undistributed terms in the premisses\ncannot exceed those in the conclusion by more than one.\n\n6. Prove that wherever the minor premiss is negative, the major must\nbe universal.\n\n7. Prove that wherever the minor term is distributed, the major\npremiss must be universal.\n\n8. If the middle term be twice distributed, what mood and figure are\npossible?\n\n9. If the major term of a syllogism be the predicate of the major\npremiss, what do we know about the minor premiss?\n\n10. When the middle term is distributed in both premisses, what must\nbe the quantity of the conclusion?\n\n11. Prove that if the conclusion be universal, the middle term can\nonly be distributed once in the premisses.\n\n12. Show how it is sometimes possible to draw three different\nconclusions from the same premisses.\n\n\nCHAPTER XIX.\n\n1. Convert the following propositions--\n\n  (1) If a man is wise, he is humble.\n\n  (2) Where there is sincerity there is no affectation.\n\n  (3) When night-dogs run, all sorts of deer are chased.\n\n  (4) The nearer the Church, the further from God.\n\n  (5) If there were no void, all would be solid.\n\n  (6) Not to go on is sometimes to go back.\n\n2. Express in a single proposition--\n\n  If he was divine, he was not covetous; and if he was covetous, he\n  was not divine.\n\n3. Exhibit the exact logical relation to one another of the following\npairs of propositions--\n\n  (1) If the conclusion be false, the premisses are false.  If the\n  conclusion be true, the premisses are not necessarily true.\n\n  (2) If one premiss be negative, the conclusion must be negative.\n\n  If the conclusion be negative, one of the premisses must be\n  negative.\n\n  (3) The truth of the universal involves the truth of the particular.\n\n  The falsity of the particular involves the falsity of the universal.\n\n  (4) From the truth of the particular no conclusion follows as to the\n  universal.\n\n  From the falsity of the universal no conclusion follows as to the\n  particular.\n\n  (5) If the conclusion in the fourth figure be negative, the major\n  premiss must be universal.\n\n  If the major premiss in the fourth figure be particular, the\n  conclusion must be affirmative.\n\n  (6) If both premisses be affirmative, the conclusion must be\n  affirmative.\n\n  If the conclusion be negative, one of the premisses must be\n  negative.\n\n4. 'The Method of Agreement stands on the ground that whatever\ncircumstance can be eliminated is not connected with the phenomenon by\nany law; the Method of Difference stands on the ground that whatever\ncircumstance cannot be eliminated is connected with the phenomenon by\na law.' Do these two principles imply one another?\n\n\nCHAPTERS XX-XXVIII.\n\n1. Fill up the following enthymemes, and state the exact nature of the\nresulting syllogism--\n\n  (1) If Livy is a faultless historian, we must believe all that he\n  tells us; but that it is impossible to do.\n\n  (2) If they stay abroad, the wife will die; while the husband's\n  lungs will not stand the English climate. It is to be feared\n  therefore that one must fall a victim.\n\n  (3) He is either very good, very bad, or commonplace.  But he is not\n  very good.\n\n  (4) Either a slave is capable of virtue or he is not.\n  .'. Either he ought not to be a slave or he is not a man.\n\n  (5) Does not his feebleness of character indicate either a bad\n  training or a natural imbecility?\n\n  (6) Those who ask shan't have; those who don't ask don't want.\n\n  (7) If a man be mad, he deviates from the common standard of\n  intellect.\n  .'. If all men be alike mad, no one is mad.\n\n  (8) 'I cannot dig; to beg I am ashamed.'\n\n2. 'The infinite divisibility of space implies that of time.  If the\nlatter therefore be impossible, the former must be equally so.'\nFormulate this argument as an immediate inference.\n\n3. Examine the following arguments--\n\n  (1) If we have a dusty spring, there is always a good wheat\n  harvest. We shall therefore have a poor harvest this year, for the\n  spring has not been dusty.\n\n  (2) Virtues are either feelings, capacities, or states; and as they\n  are neither feelings nor capacities, they must be states.\n\n  (3) Everything must be either just or unjust.\n      Justice is a thing, and is not unjust.\n      .'. Justice is just.\n\n      Similarly justice is holy.\n      But the virtues of knowledge, justice, courage, temperance, and\n      holiness were declared to be different from one another.\n      .'. Justice is unholy and holiness unjust.\n\n\nCHAPTER XXIX.\n\nFormulate the following trains of reasoning, resolve them into their\ncomponent parts, and point out any violations of the rules of\nsyllogism which they may contain--\n\n  (1) No Church Institutions are useful; for they teach religious\n  matters, not business matters, which latter are useful, being\n  profitable.\n\n  (2) Mr. Darwin long ago taught us that the clover crop is dependent\n  on the number of maiden ladies in the district. For the ladies keep\n  cats, and the cats destroy the field-mice, which prey on the bees,\n  which, in their turn, are all-important agents in the fertilisation\n  of the clover flowers.\n\n  (3) Athletic games are duties; for whatever is necessary to health\n  is a duty, and exercise is necessary to health, and these games are\n  exercise.\n\n  (4) The iron-trade leads to the improvement of a new country; for\n  furnaces require to be fed with fuel, which causes land to be\n  cleared.\n\n  (5) 'Is stone a body?' 'Yes.' 'Well, is not an animal a body?'\n  'Yes,' 'And are you an animal?' 'It seems so.' 'Then you are a\n  stone, being an animal.'\n\n  (6) If A is B, C is D.\n  If E is F, G is H.\n  But if A is B, E is F.\n  .'. If C is D, G is sometimes H.\n\n  (7) The soul is not matter.\n  My arm is not myself.\n\n  (8) Honesty deserves reward and a negro is a\n  fellow-creature. Therefore an honest negro is a fellow-creature\n  deserving of reward.\n\n\nCHAPTER XXX.\n\n1. Point out any ambiguities which underlie the following\npropositions--\n\n  (1) Every one who has read the book in French will recommend those\n  who have not to read it in English.\n\n  (2) I will not do this because he did it.\n\n  (3) These are all my books.\n\n  (4) By an old statute of the date of Edward III it was accorded\n  'that Parliament should be holden every year once or more often if\n  need be.'\n\n  (5) They found Mary and Joseph and the babe lying in a manger.\n\n  (6) The king and his minister are feeble and unscrupulous.\n\n  (7) Heres meus uxori meae triginta pondo vasorum argenteorum dato,\n  quae volet.\n\n2. Examine the following arguments, formulating them when sound, and\nreferring them, when unsound, to the proper head of fallacy--\n\n  (1) We know that thou art a teacher come from God; for no man can do\n  these signs that thou doest, except God be with him. S. John iii. 2.\n\n  (2) 'Sir Walter Scott's novels have ceased to be popular.' 'Well,\n  that's only because nobody reads them.'\n\n  (3) What we produce is property.\n      The sheriff produces a prisoner.\n      .'. A prisoner is property.\n\n  (4) As all metals are not necessarily solid, we may expect some\n  metals to be liquid.\n\n  (5) Moses was the son of Pharaoh's daughter.\n      .'. Moses was the daughter of Pharaoh's son.\n\n  (6) If Aeschines took part in the public rejoicings over the success\n  of my policy, he is inconsistent in condemning it now; if he did\n  not, he was a traitor then.\n\n  (7) It is wrong to stick knives into people.\n      .'. Surgeons ought to be punished.\n\n  (8) If a thing admits of being taught, there must be both teachers\n      and learners of it.\n      .'. If there are neither teachers nor learners of a thing, that\n      thing does not admit of being taught.\n\n  (9) It is unnecessary to lend books, if they are common, and wrong\n  to lend them, if they are rare.  Therefore books should not be lent\n  from public libraries.\n\n  (10) Seeing is believing.\n       .'. What is not seen cannot be believed.\n\n  (11) St. Paul was not of Jewish blood, for he was a Roman citizen.\n\n  (12) To call you an animal is to speak the truth.\n       To call you an ass is to call you an animal.\n       .'. To call you an ass is to speak the truth.\n\n  (13) Pain chastens folly. A life of ease must therefore be one of\n  folly incurable.\n\n  (14) We cannot be happy in this world; for we must either indulge\n  our passions or combat them.\n\n  (15) It must be clear to the most unlettered mind that, as all\n  things were originally created by the Deity, including the hair on\n  our heads and the beards on our faces, there can be no such thing as\n  property.\n\n  (16) The crime was committed by the criminal.\n       The criminal was committed by the magistrate.\n       .'. The crime was committed by the magistrate.\n\n  (17) General councils are as likely to err as the fallible men of\n  whom they consist.\n\n  (18) Dead dogs are heavier than living ones, because vitality is\n  buoyant.\n\n  (19) Deliberation is concerned with actions.\n       Actions are means.\n       .'. Deliberation is concerned with means.\n\n  (20) 'No beast so fierce but has a touch of pity; But I have none:\n  therefore I am no beast.'\n\n  (21) Practical pursuits are better than theoretical.\n       .'. Mathematics are better than logic.\n\n  (22) Death must be a good. For either the soul, ceasing to be,\n  ceases ta suffer, or, continuing to be, lives in a better state.\n\n  (23) What is right should be enforced by law.\n       .'. Charity should be so enforced.\n\n  (24) All animals were in the Ark.\n       .'. No animals perished in the Flood.\n\n  (25) If he robs, he is not honourable.\n       If he pays all his dues, he does not rob.\n       .'. If he pays all his dues, he is honourable.\n\n  (26) A dove can fly a mile in a minute.\n       A swallow can fly faster than a dove.\n       .'. A swallow can fly more than a mile in a minute.\n\n  (27) 'I must soap myself, because it's Sunday.'\n       'Then do you only soap yourself on Sunday.'\n\n  (28) If the charge is false, the author of it is either ignorant or\n  malicious. But the charge is true.  Therefore he is neither.\n\n  (29) All the angles of a triangle are equal to two right angles.\n       The angle at the vertex is an angle of a triangle.\n       .'. It is equal to two right angles.\n\n  (30) Si gravis sit dolor, brevis est; si longus, levis.  Ergo\n  fortiter ferendus.\n\n  (31) You are not what I am.\n       I am a man.\n       .'. You are not a man.\n\n  (32) The extension of the franchise is necessary, for it is\n  imperative that the right of voting should be granted to classes who\n  have hitherto not possessed this privilege.\n\n  (33) If Hannibal is really victorious, he does not need supplies;\n  while, if he is deluding us, we ought certainly not to encourage him\n  by sending them.  Livy, xxiii. 13. \u00a7 5.\n\n  (34) Laws must punish, and punishment hurts.\n  All laws therefore are hurtful.\n\n  (35) The sun is an insensible thing.\n       The Persians worship the sun.\n       .'. The Persians worship an insensible thing.\n\n  (36) Some ores are not metals; for they are not fluids,\n  and some metals are not fluids.\n\n  (37) All the Grecian soldiers put the Persians to flight.\n       .'. Every Grecian soldier could rout the Persians.\n\n  (38) The resurrection of Jesus Christ is either an isolated fact or\n  else admits of parallel. But if it be an isolated fact, it cannot be\n  rendered probable to one who denies the authority of Christianity;\n  and, if it admit of parallel, it no longer proves what is\n  required. Therefore it is either incapable of being substantiated or\n  else makes nothing for the truth of Christianity.\n\n  (39) The resurrection of Christ in the flesh and his ascension into\n  heaven were events either intrinsically incredible in their nature\n  or not. If the former, the prevalent belief in them can only be\n  accounted for by miracles; if the latter, they ought to be believed\n  even without miracles. St.  Aug. De Civ. Dei, xxii. 8.\n\n  (40) Only contented people are wise. Therefore the tramp contented\n  in his rags is necessarily a wise man.\n\n  (41) Four-legged things are brutes.\n       Tables are four-legged things.\n       .'. Tables are brutes.\n\n  (42) The apparent volcanoes in the moon are not volcanoes; for\n  eruptions are produced by gases only, and there are no gases in the\n  moon.\n\n  (43) To read the Scriptures is our duty. Therefore the Captain was\n  wrong in punishing the helmsman for reading the Bible at the time\n  when the ship struck.\n\n  (44) The divine law orders that kings should be honoured.\n       Louis Quatorze is a king.\n       .'. The divine law orders that Louis Quatorze should be honoured.\n\n  (45) Those who desire the same object are unanimous.\n       Caesar and Pompey both desire the same object, namely, supreme\n       power.\n       .'. They are unanimous.\n\n  (46) Either the ministers left at home will be ciphers or they will\n  not be ciphers. If they are ciphers, cabinet government, which is\n  equivalent to constitutional government, will receive a rude blow.\n  If they are not ciphers, the cabinet will be considering matters of\n  the utmost importance in the absence, and the gratuitous absence, of\n  two of its most important members. 'The Standard,' Wed. June 5,\n  1878.\n\n  (47) One patent stove saves half the ordinary amount of\n  fuel. Therefore two would save it all.\n\n  (48) One number must win in the lottery.\n       My ticket is one number.\n       .'. It must win.\n\n  (49) All good shepherds are prepared to lay down their lives for the\n       sheep.\n       Few in this age are so prepared.\n       .'. Few in this age are good shepherds.\n\n  (50) You cannot define the sun; for a definition must be clearer\n  than the thing defined, and nothing can be clearer than the source\n  of all light.\n\n  (51) To give the monopoly of the home market to the produce of\n  domestic industry ... must in almost all cases be either a useless\n  or a hurtful regulation. If the produce of domestic can be brought\n  there as cheap as that of foreign industry, the regulation is\n  evidently useless; if it cannot, it is generally hurtful. Adam\n  Smith, Wealth of Nations, Bk. iv. ch. 2.\n\n  (52) Verberare est actio.\n       Ergo et vapulare.\n\n  (53) The ages of all the members of this family are over 150.\n       The baby is a member of this family.\n       .'. Its age is over 150.\n\n  (54) Romulus must be an historical person; because it is not at all\n  likely that the Romans, whose memory was only burdened with seven\n  kings, should have forgotten the most famous of them, namely, the\n  first.\n\n  (55) All scientific treatises that are clear and true deserve\n       attention.\n       Few scientific treatises are clear and true.\n       .'. Few scientific treatises deserve attention.\n\n  (56) The Conservative Government is an expensive one; for, on their\n  going out of Office, there was a deficit.\n\n  (57) A man is forbidden to marry his brother's wife, or, in other\n  words, a woman is forbidden to marry her husband's brother, that is,\n  a woman is directly forbidden to marry two brothers. Therefore a man\n  may not marry two sisters, so that a man may not marry his wife's\n  sister.\n\n\n\n\nINDEX.\n\n\nThe references refer to the sections.\n\nAbstraction, 97.\n\nAcategorematic words, 71.\n\nAccent, Fallacy of, 855.\n\nAccident, 318.\n\nAccident, Fallacy of, 860.\n\nA dicto secundum quid, Fallacy of, 861.\n\nAmphiboly, Fallacy of, 848.\n\nAntecedent\n  of a complex proposition, 212.\n  of an inference, 428.\n\nA posteriori Truth, 232.\n\nA priori Truth, 231.\n\n'A' Propositions, 260.\n  conversion of, 489.\n\nArguing in a circle, 882.\n\nArgumentum ad hominem, etc., 867.\n\nArt, 20.\n\nAttribute, 81 sqq.\n  Essential and non-essential, 320.\n\nAttributives, 88 sqq.\n\n\nBasis of Division, 391.\n\n\nCategorematic words, 71.\n\nCirculus in definiendo, 382.\n\nCommon Terms, 105.\n  how formed, 99.\n  nature of, 48.\n\nComplex Proposition, 209.\n  conversion of, 709.\n  conversion by contraposition of, 728.\n  conversion by negation of, 721.\n  divided into conjunctive and disjunctive, 214.\n  permutation of, 718.\n\nComplex Syllogism, 731.\n  mixed form of, 778.\n\nComposition, Fallacy of, 849.\n\nConcept, 36, 40 sqq.\n\nConception, 33.\n\nConceptualists, 51.\n\nConclusion, 540.\n  predicate of, 542.\n  subject of, 542.\n\nConjunctive Syllogisms, 733.\n  canon of, 742.\n  reduction of partly, 744.\n  partly conjunctive syllogisms as an immediate inference, 753.\n\nConnotation of Terms, 148.\n\nConsequent of a complex proposition, 213.\n  of an inference, 428.\n\nConsequent, Fallacy of, 873.\n\nContingent, 17.\n\nContradiction, Law of, 25 sqq.\n\nContradictory Propositions, 458.\n  Terms, 129.\n\nContrary Propositions, 458.\n  Terms, 130.\n\nConverse, 480.\n\nConversion, 479.\n  of complex propositions, 709.\n  by contraposition, 516.\n  illative, 481.\n  by negation, 504.\n  per accidens, 487.\n  simple, 486.\n  rules of, 482.\n\nConvertend, 480.\n\nCopula, 58, 64, 186 sqq.\n  modality of, 196.\n\nCorrelatives, 142.\n\n\nDeduction and Induction, difference of, 431 sqq.\n\nDeductive Inference, 442.\n\nDeductive Logic, definition of, 4.\n\nDefinition of Terms, 347 sqq.\n  of Aristotle ([Greek: \u00f2rism\u00f3s]), 336.\n  final, 374.\n  nominal, 375.\n  provisional, 374.\n  real, 375.\n  rules of, 378.\n\nDenotation of Terms, 152.\n\nDescription, 360.\n\nDesignations, 112.\n\nDetermination, 167.\n\nDictum de omni et nullo, 569.\n  de diverso, 641.\n  de exemplo et excepto, 642.\n\nDifference, 318, 358.\n  generic, 410.\n  specific, 409.\n\nDilemma, 732, 779.\n  rebutted, 792.\n  reduction of, 796.\n  regarded as an immediate inference, 798.\n\nDisjunctive Syllogism, 760.\n  canon of, 765.\n  reduction of, 766.\n  regarded as an immediate inference, 770.\n\nDistinction, 424.\n\nDistribution of Terms, 274.\n  four rules for, 293.\n\nDivided whole, 393.\n\nDividing members, 394.\n\nDivision, 385 sqq.\n  by dichotomy, 412.\n  rules of, 395.\n\nDivision, Fallacy of, 851.\n\nDivision of Propositions, 206.\n  of terms, 86.\n  of things, 77.\n\n\nEnthymeme, incorrectly so-called, 557.\n\nEnumeration, 387, 422.\n\nEpicheirema, 803.\n\nEpisyllogism, 802.\n\n'E' Propositions, 260.\n  conversion of, 490.\n\nEquivocation, Fallacy of, 845.\n\nExcluded Middle, Law of, 25 sqq., 502.\n\nExtension of Terms, 149 sqq., 166 sqq.\n\n\nFallacy, 827 sqq.\n  of ambiguity, 831.\n  definition of, 828.\n  formal, 838.\n  logical, 836.\n  material, 831, 836.\n  of undisturbed middle, 585.\n\nFigure of Speech, Fallacy of, 857.\n\nFigures, of a Syllogism, 558.\n  special canons of, 633.\n  special rules of, 606.\n  special uses of, 648.\n\nFormal Logic, 16.\n\nFour Terms, Fallacy of, 840.\n\nFundamentum Divisionis, 391.\n\n\nGeneralisation, 168.\n\nGenus, 318.\n  as used by Aristotle, 336.\n  cognate, 408.\n  proximate, 420.\n  subaltern, 406.\n  summum, 167, 404.\n\n\nHeads of Predicables, 313.\n  as given by Aristotle, 336.\n\n\n'Ideas' of Plato, 52.\n\nIdentity, Law of, 25 sqq.\n\nIgnoratio Elenchi, Fallacy of, 865.\n\nIgnotum per ignotius, 383.\n\nIllicit Process, Fallacy of, 586.\n\nImmediate Inference, 442 sqq.\n  by added determinants, 535.\n  by complex conception, 537.\n  applied to complex propositions, 701.\n\nImmediate Inference, compound forms of, 503.\n  partly conjunctive syllogisms regarded as, 753.\n  by conversion, 479.\n  disjunctive syllogisms regarded as, 770.\n  by opposition, 462.\n  by permutation, 496.\n\nInduction, differing from Deduction, 430 sqq.\n\nInductive Logic, 2, 204.\n\nInferences in general, 426.\n  classification of, 441.\n  deductive, 442.\n  inductive, 430.\n\nIntimae species, 405.\n\nIntension of Terms, 150, 166.\n\nIntuition, 232.\n\nInverse Variation, Law of, 166.\n\n'I' Propositions, 260.\n  conversion of, 490.\n\n\n'Judgement,' various meanings of, 32, 36.\n\n\n'Law,' ambiguities of the word, 7 sqq.\n\n\nMajor Premiss, 544.\n\nMajor Term, 542.\n\nMany Questions, Fallacy of, 884.\n\nMediate Inferences or Syllogisms, 444, 540 sqq.\n  axioms of, 576.\n\nMembra Dividentia, 394.\n\nMiddle Term, 541.\n  position of, in a syllogism, 563.\n\nMinor Premiss, 545.\n\nMinor Term, 542.\n\nModality, Question of, 196.\n\nMode, the, 196.\n\nMoods of a Syllogism, 558.\n  determination of the legitimate, 599.\n  subaltern, 628.\n  valid in the Four Figures, 621.\n  mnemonics of, valid in Four Figures, 629.\n\n\nName, definition of, 61.\n\nNegative Premisses and Conclusion, Fallacy of, 842.\n\nNominalists, 50, 54.\n\nNon causa pro causa, Fallacy of, 883.\n\nNouns, 62.\n\n\nOpposition, 449 sqq.\n  contradictory, 457.\n  contrary, 454.\n  laws of, 464.\n  subaltern, 456.\n  sub-contrary, 455.\n\n'O' Propositions, 260.\n  conversion of, 491.\n\n\nPartition, 423.\n\nPermutation, 496 sqq.\n  of Complex Propositions, 718.\n\nPetitio Principii, Fallacy of, 874.\n\nPredicable, 314.\n\nPredicate of a Proposition, 58, 184.\n  read in extension, 307.\n  quantification of, 295 sqq.\n  quantity of, 281, 494.\n\nPredication, 194.\n  in quid or in quale, 332.\n\nPremisses, 540.\n  major, 544.\n  minor, 545.\n\nPrimary Existences, 55.\n\nProblema, the, 556.\n\nProper Names, 113.\n\nProperty, 318.\n  generic, 411.\n  specific, 411.\n\nProposition, 172 sqq.\n  accidental, 238.\n  affirmative, 258.\n  complex or conditional, 209.\n  conjunctive or hypothetical, 214, 704.\n  conversion of, 479.\n  definition of, 178.\n  disjunctive, 214.\n  divisions of, 206.\n  essential, 238.\n  exceptive, 270.\n  exclusive, 266.\n  extensive, 264.\n  general, 251.\n  indefinite, 244.\n  intensive, 264.\n  modal, 205.\n  negative, 258.\n  particular, 240.\n  pure, 205.\n  quality of, 258.\n  quantity of, 246.\n  real or synthetical, 227.\n  simple or categorical, 207.\n  singular, 250.\n  tautologous or identical, 273.\n  universal, 239.\n  verbal or analytical, 224.\n\nProprium, 336.\n\nPro-syllogism, 802.\n\n\nQuaestio, the, 556.\n\nQuality, a, 82.\n\nQuality of the matter, 204.\n  of propositions, 258.\n\nQuantification of the Predicate, 295 sqq., 493.\n\nQuantity of propositions, 258.\n  of terms, 148.\n\n\nRealists, 49.\n\nReal Kinds, 371.\n\nReasoning or Inference, 35.\n  the canon of, 560.\n  trains of, 800.\n\nReduction of propositions, 667.\n  of the dilemma, 796.\n  of disjunctive syllogisms, 766.\n  indirect, 691.\n  mnemonics for, 697.\n  ostensive or direct, 673.\n  of partly conjunctive syllogisms, 744.\n\nRelation, a, 83, 144.\n\nRelation, immediate inference by, 462.\n  compatible and incompatible, 462.\n\n\nScience, 20.\n\nSecondary Existences, 55.\n\nSimple Apprehension, 33.\n\nSorites, the, 807 sqq.\n\nSpecialisation, 167.\n\nSpecies, 318.\n  cognate, 407.\n  infimae, 405.\n  subaltern, 406.\n\nSubalternant, 458.\n\nSubalternate, 458.\n\nSubalternation, 458.\n\nSubalterns, 458.\n\nSub-contraries, 458.\n\nSub-division, 401.\n\nSubject, 58, 183.\n  how used, 264.\n  quantity of, 279.\n\nSubstance, 80, 84.\n\nSummum Genus, 167, 404.\n\nSuppositio Materialis, 76.\n\nSyllogism, 546 sqq.\n  complex, 731.\n  in common discourse, 557.\n  conjunctive, 733.\n  definition of, 552.\n  disjunctive, 760.\n  general rules of, 582.\n  figures of, 560, 563.\n  with three figures, 656.\n  legitimate moods of, 599 sqq.\n  mnemonics for, 598.\n  moods of, 559, 562.\n\nSyncntegorematic words, 70.\n\nSynonym, 345.\n\n\nTerm, 57 sqq.\n  absolute, 140.\n  abstract, 95.\n  analogous, 139.\n  attributive, 88.\n  collective, 118.\n  common, 105.\n  concrete, 96.\n  connotative, 147.\n  contradictory, 129.\n  contrary, 130.\n  definition of, 347.\n\nTerms, distribution of, 275.\n  distributive and collective use of, 119.\n  division of, 86.\n  equivocal, 137.\n  incompatible, 135.\n  individual, 121.\n  major, middle, and minor, 542.\n  negative, 126.\n  non-connotative, 147.\n  positive, 126.\n  privative, 126.\n  quantity of, 148.\n\nTerms, relative, 141.\n  repugnant, 135.\n  singular, 43, 104.\n  subject, 87.\n  undistributed, 277.\n  univocal, 137.\n\n\nUniversals, nature of, 48, 55.\n\n'U' Propositions, 297.\n\n\nVerb, 64.\n\n\nWords, their relation to terms, 65 sqq,\n\n\n\nTHE END.","title":"Deductive Logic","tradition":"philosophy-ethics","translated_text":null,"word_count":74460}