H. P. Grice, "Notes on J. N. Keynes"

LOGIC; INCLUDING, A GENERALISATION OF LOGICAL PROCESSES IN THEIR APPLICATION TO COMPLEX INFERENCES, by  J. N. KEYNES, M.A., FELLOW OF PEMBROKE COLLEGE, CAMBRIDGE. London: MACMILLAN AND CO. CamMige: PRINTED BY C. J. CLAY, M.A. & SON, AT THE UNIVERSITY PRESS. In addition to problems worked out in detail and unsolved problems, by means of which the philosopher may test his command over logical processes, the following pages contain a somewhat detailed exposition of certain portions of what may be called the “logic,” but the Grecians preferred to refer to as “Athenian dialectic” that seduced the Romans so much. This was necessary in the case of disputed or doubtful points in order that the working out of the problems might be made consistent and intelligible; there were also some points concerning which I was dissatisfied with the method of treatment adopted in the ordinary text-books. At the same time, this volume must be regarded, not as superseding the study of dialectic, but as supplementing it. While certain topics are dealt with in considerable detail others have been omitted ; e.g-., the doctrines of Definition and Division and the Predicables are not touched upon, no definition of the Science itself is given, and no systematic discussion of first  B principles has been introduced. For a general outline of my views on the position of “Athenian dialectic” I may refer the reader to my essay in Mind. For several reasons I should have been glad to rewrite and in some respects to modify this paper; but anything like an adequate treatment of the subject would have enlarged the book considerably beyond the limits that I had assigned to it. I have not endeavoured to distinguish definitely between book-work and problem ; and the unanswered exercises are not separated and placed apart at the end of the chapters, but are introduced at the points at which the student who is systematically working through the book will find himself in a position to solve them. Exercises of a similar character have not been to any considerable extent multiplied, but I believe that no kind of problem relating to the operations of Formal Logic has been overlooked. By reference to sections 261, 262, 281 — 285, the reader will find that the ordinary syllogism admits of problems of some complexity. In the expository portions of Parts I. il. and III., dealing respectively with Terms, Propositions, and Syllogisms, I have in the main followed the traditional lines, though with a few modifications ; e, g.^ in the systematization of immediate inferences, and in some points of detail in connection with the syllogism to which I need not make further reference here. For purposes of illustration Euler's diagrams are employed to a greater extent than is usual in the Oxonian manuals. In Part IV., which contains a generalisation of logical processes in their application to complex inferences, a somewhat new departure is taken. So far as I am aware this part constitutes the first systematic attempt that has been made to deal with formal reasonings of the most complicated character without the aid of mathematical symbols and without abandoning the ordinary non-equational or predicative form of proposition. In this attempt I have met with greater success than I had anticipated; and I believe that the methods which I have formulated will be found to be as easy of application and as certain in obtaining results as the mathematical, symbolical, or diagrammatic methods of Boole, Jevons, Venn and others. The reader may judge of this for himself by comparing with Boole's own solutions the problems discussed in sections 368, 369, 383 — 386; or by solving by different methods other of the problems, e.g,, the very complex one contained in section 408. The book concludes with a general method of solution of what PJevons called the Inverse Problem, and which he himself seemed to regard as soluble only by a series of guesses; Of the Questions and Problems more than half are my own composition. Of the remainder, about a hundred have been taken from various examination papers, and about sixty are from the published writings of Boole, De Morgan, Jevons, Solly, Venn and Whately. In the latter case the name of the author IS appended, generally with a reference to the work from which the example is taken. In the case of problems selected from examination papers, a letter is added indicating their source, as follows: — C.= University of Cambridge ; L.= University of London ; N. = J. S. Nicholson, Professor of Political Economy in the University of Edinburgh ; O. = University of Oxford ; R.=G. Croom Robertson, Professor of Mental Philosophy and Logic in University College, London; V. = J. Venn, Fellow and Lecturer of Gonville and Caius College, Cambridge; W.= J. Ward, Fellow and Assistant Tutor of Trinity College, Cambridge. The logicians to whom I have been chiefly indebted are De Morgan, Jevons and Venn. De Morgan's various logical writings are rendered somewhat formidable and uninviting by reason of the multiplication of symbols and formulae which he is never tired of introducing, and this is probably the reason why they are little read at the present time ; they nevertheless constitute a mine of wealth for all who are interested in the developments of Formal Logic. With Jevons I have continually found myself in disagreement on points of detail, and it is possible that I may give the impression of having taken up a special position of antagonism with regard to him. This is far from being really the case. I believe that since Mill no one else has given such an impetus to the study of Logic, and I hold that in more than one direction he has led the way in new developments of the science that are of great importance. To Mr Venn I am peculiarly indebted, not merely by reason of his published writings, especially his Logic, but also for most valuable suggestions and criticisms given to me while this book was in progress. I am glad to have this opportunity of expressing to him my thanks for the ungrudging help he has afforded me. I am also under great obligation to Miss Martin of Newnham College and to Mr Caldecott of St John's College for criticisms which I have found very helpful. 6, Harvey Road, Cambridge, Brief definitions of word^ name, term, symbol, concept. A word is an articulate sound, or the written equivalent of an articulate sound, which either by itself or in conjunction with other words, constitutes a name, or forms a Sentence. A name is "a word taken at pleasure to serve for a mark which may raise in our mind a thought like to some thought we had before, and which being pronounced to others, may be to them a sign of what thought the speaker had or had not before in his mind." Hobbes, A term is a name regarded as the subject or the predicate of a proposition. A symbolf in its widest signification^ is a sign of any kind ; narrowing our point of view, it is any written sign ; and narrowing it still more, it is a written sign which is employed without the realization at each step of its full signification. Thus, when symbols are used in algebraical reasoning, it is according to certain fixed rules, without reference to or thought of their ulterior meaning. Names may themselves be employed as symbols in this sense. Of course, in the widest sense, all names are symbols. A concept is defined by Sir William Hamilton as " the cognition or idea of the general character or characters, point or points, in which a plurality of objects coincide." In other words, a concept is the mental equivalent of a general name. A categorematic word is one which can by itself be used as a term, i. ^., which can stand alone as the subject or the predicate of a proposition. A syncategorematic word is one which cannot by itself be used as a term, but only in combination with one or more other words. Any noun substantive in the nominative case, or any other part of speech employed as equivalent to a noun substantive, may be used categorematically. Adjectives are sometimes said to be used categorematically by a grammatical ellipsis. In the examples, “The rich are happy," "Blue is an agreeable colour," either a substantive is understood as being qualified by the adjective, or the adjective is used as a substantive, that is, as a mark of something, not merely as a mark qualifying something. Any part of speech, or the inflected cases of nouns substantive, may be used categorematically by a suppositio maierialisy that is, by speaking of the mere word itself as a thing ; for example, " John's is a possessive case," " Rich is an adjective," **With is an English word." Using the word term in the sense in which it was defined in the preceding section, it is clear that we ought not to speak of syncategorematic terms. A general name is a name which is capable of being truly affirmed, in the same sense, of each of an indefinite number of things, real or imaginary. A singular name is a name which is capable of being truly affirmed, in the same sense, of only one thing, real or imaginary. K proper name is a singular name given merely to distinguish an individual person or thing from others, its application after it has been once given being independent of any special attributes that the individual may possess . Thus, “Prime Minister” of England is a general name, since at different times it may be applied to different individuals. We may, for example, talk about ^' the prime ministers of England of the present century." The name is however made singular by the prefix "/^," meaning "the present prime minister," or " the prime minister at the time to which we are referring." Similarly any general name may be made singular ; for example, man, the first man ; star, the pole star. The name God is singular to a monotheist as the name of the Deity, general to a polytheist, or as the name of anything worshipped by anybody. Universe is A proper name might perhaps be defined as "a non-connotative singular name." But this definition presupposes a distinction which is best given subsequently, and it would give rise to a controversy, that also had better be postponed. Compare section 14. 4 terms; x^art I. general in' so' far is we distinguish different kinds of universes, ^.^., the material universe, the terrestrial universe. Sic; it is singular if we mean ^Ae universe, Sjxice is general if we mean a particular portion of space, singular if we mean space in the aggregate. Water is general. PBain takes a different view here; he says, "Names of Material — earth, stone, salt, mercury, water, flame, — are singular. They each denote the entire collection of one species of iraterial" {Logic, Deduction, pp. 48, 49). But when we predicate anything of these terms it is generally of any portion (or of some particular portion) of the material in question, and not of the entire collection of it considered as one aggregate ; thus, if we say, <* Water is composed of oxygen and hydrogen,^' we mean any and every particle of water, and the name has all the distinctive characters of th^ general name. Similarly with regard to the other terms mentioned in the above quotation. It is also to be o1> served that we distinguish different kinds of stone, salt, &c, A name is to be regarded as general if it may be potentially affirmed of more than one, although it accidentally happens that as a matter of fact it can be actually affirmed of only one, e.g., King of England and Spain, We must also note the case in which we are dealing with a name that actually is not applicable to any individual at all; e»g>, “President of the British Republic.” A really singular name is distinguished from these by not being even potentially applicable to more than one individual ; ^.^., the last of the Mohicans, the eldest son of King Edward the First^, ' It seems desirable to make the distinction implied in this para« graph; still I am not sure that it might not in some cases be very difficult to apply it satisfactorily. Nearly all these divisions of names tend to give rise in the last resort to metaphysical difficulties; but, in my opinion, these should as far as possible be avoided in a logical treatise. “Victoria” is the name of more than one individual, and can therefore be truly affirmed of more than one individiiaU Is it therefore general? Mill answers this question in the negative, and rightly, on the ground that the name is not here affirmed of the different individuals in the same sense.: Bain brings out this distinction very clearly in his definition of a general name: '^A general nam& is applicable to a number of things in virtue of their being similar, or having something in common." Victoria is then not general but singular; and it belongs to the sub-class of proper names. Are all collective names singular? A collective name is one which is the name . of a group of things considered as one whole ; e,g.^ regiment, nation, army. A collective name may be singular or general It is the name of a group or collection of things, and so far as it is capable of being truly affirmed in the same sense of only one such group, it is singular ; e.g,^ the 29th regi-ment of foot, “the English nation,” the Bodleian library. But if it is capable of being truly affirmed in the same sense of each of several such groups it is to be regarded as general; e.g,j regiment, nation, library. Bain writes as if a name could be general and singular at the same time,— " Collective names as nation, army, multitude, assembly, universe, are singular; they are plurality combined into unity. But, inasmuch as there are many nations, armieSj assemblies, the names are also general. There being but one 'universe', that term is collective and singular". I should rather say that ^s the above stand, with the possible excepttion of universe, they are not singular at all. . Mill and others imply that there is a distinction between collective and general names* The real distinction however is between the colledtve and distributive use of names. A collective name such as nation, or any name in the plural number, is the name of a collection or group of things. These we may regard as one whole, and something may be predicated of them that is true of them only as a whole; in this case the name is used collectively. On the other hand, the group may be regarded as a series of units, and something may be predicated of these which is true of them only taken individually; in this case the name is used distributively. Also, when anything is predicated of a series of such groups the name is used distributively. The above distinction may be illustrated by the propositions, — All the angles of a triangle are equal to two right angles. All the angles of a triangle are less than two right angles. The subject term is the same in both these cases, but in the first case the predication is true only of the angles all taken together, while in the second it is true only of each of them taken separately; in the first case therefore the term is used collectively, in the second distributively. The peculiarity, then, of a collective name is that it can be used collectively in the singular number, while other names can be used collectively only in the plural number ; compare, for example, the names * clergyman' and 'the Clergy.' Collective names in the plural number may themselves be used distributively, and it is therefore not correct to say that all collective names are singular. It may indeed be held that, while this is true, still when a name is used collectively, it is equivalent to a singular name. For example. The whole army was annihilated, The mob filled the square. But I am doubtful whether even this is true in such a case as the following, — In all cases all the angles of a triangle are equal to two right angles. Please select the terms that are used collectively in the following propositions; also classify the terms contained in these propositions according as they are collective, singular, and general respectively, and find in what way these classes overlap one another: “The Conservatives are in the majority in the House of Lords, All the tribes combined. The nations of the earth rejoiced. Crowds filled all the churches. One generation passeth away and another generation Cometh. Your boxes weigh 140 lbs. The volunteers mustered in considerable numbers. Time flies. True poets are rare. Those who succeeded were few in number. The mob was soon dispersed. Our armies swore terribly in Flanders, The multitude is always in the wrong. Mill defines abstract and concrete names as follows : — "A concrete name is a name which stands for a thing; an abstract name is a name which stands for an attribute of a thing" {JiogiCy i. p. 29)^ In many cases, this distinction is of easy application; for example, triangle is the name of something that possesses the attribute of being bounded by 1 The references are to the ninth edition of Mill's Logic, three straight lines, and it is a concrete name; triangularity is the name of this distinctive attribute of triangles, and it is an abstract name. But there aire other cases to which the application of the distinction is difficult; and an attempt at more precise definition is liable to involve us in metaphysical discussions such as the logician should if possible avoid The first question that arises is what precisely is meant by the word things when it is said that a concrete name is the name of a thing. By a thing, we may mean anything that exists ; but we cannot mean that here, since "attributes" exist, and the distinction between concrete and abstract name^ would vanish. Again, by a thing we may mean a substance ; but substances are contrasted with feelings as well as with attributes^ and this threefold division would make names of feelings neither abstract nor concrete^ which can hardly be intended. With regard to the proper place of names of states of consciousness it would be generally agreed to call them concrete. Thus, while sensibility, the faculty of experiencing sensation, is an abstract name, the name of a sensation itself should be regarded as concrete, being the name of something which possesses attributes, for example, of being pleasurable or painful, of being a sensation of touch or one of hearing. But here again a difficulty arises, since, as pointed out by Mill, in many cases " feelings have no other name than that of the attribute which is grounded on them." For example, by colour we may mean sensations of blue, red, green, &c., or we may mean the attribute which all coloured objects possess in common. In the former case, colour is a concrete name, in the latter an abstract name. Sounds again, is concrete, in so far as it is the name of a sensation, e,g,, "the same sound is in my ears which in those days I heard"; but in the following cases, it should rather be regarded as abstract, — " a tale full of sound and fury," " a name harsh in sound." The matter is still further complicated if Mill's view is taken, and attributes are analysed into sensations, "the distinction which we verbally make between the properties of things and the sensations we receive from them, originating in the convenience of discourse rather than in the nature of what is signified by the terms." For logical purposes however we certainly need not pursue the analysis so far as this. But still another difficulty arises from the fact that we sometimes speak of attributes themselves as having attributes; and so far as this is permissible, we must agree with Professor Jevons that ** abstractness becomes a question of degree." It may be said that civilization is abstract regarded as an attribute of a given state of society, but that it becomes concrete regarded as itself possessing the attribute of progressiveness or the attribute of stationariness. Besides all the above, we have to notice that terms originally abstract are very liable to come to be used as concrete, and this may create further confusion. Thus, Professor Jevons remarks,^-" -^^/chap.il] terms. 15 arily a subject;" (and it was said to connote or signify secondarily the subject). Thus •* white" was regarded as con- notative, whilst the original substances or attributes, as "man" or " whiteness " were called absolute; the former signifying primarily a subject, the latter not signifying a subject at alL Only adjectives and participles therefore (words called by Professor Fowler " attributives ") are connotative in this sense. Mill {LogiCy I. p. 42, note) says that the schoolmen used it in his own sense, though some of their expressions are vague. He quotes James Mill as using it more nearly in the sense ascribed by Mansel to the schoolmen. (iii) Professor Fowler uses the term connotative in a sense different from that of Mill. "A term may be said to denote or designate individuals or groups of individuals, o connote or mean attributes or groups of attributes." In this sense, general names are both connotative and deno- tative; abstract names are connotative but not denotative', (whereas, according to Mill, they are generally speaking denotative but not connotative). This use of the term avoids some difficulties, and I am inclined to regard it as preferable to Mill's. Indeed Mill himself seems to suggest it in one place. He says that James Mill "describes abstract names as being properly concrete names with their connotation dropped : whereas, in his own view, it is the deiioXsXion which would be said to be dropped, what was previously connoted becoming the whole signification " {Logic^ I. p. 42 note). As far as we can I think we should speak merely of the " denotation " and " connotation " of names, rather than of ** denotative " and " connotative names ". ^ Fowler, Deductive Logic^ p. 19. i6 .TERMS. [parti.18.Is' every property possessed by a class- con'- noted by the class-name ? Unfortunately we do not find complete agreement among logicians with regard to the answer that should be given to this question ; and I am inclined to think that in discussing points connected with ** connotation " writers sometimes misunderstand each other, because they do not apprehend that there is fundamental disagreement between them upon this point. I will first give Mill's answer to the question, an answer with which I should myself concur. By the connotation of a class-name he does not mean a/i the properties that may be possessed in common by the class, but only those on account of the possession of which any individual is placed in the class, or called by the name. In other words, we include in the connotation of a class- name only those attributes upon which the classification is founded, and in the absence of any of which we should not regard the name as applicable. For example, although all equilateral triangles are equiangular we should not include equiangularity in the connotation of equilateral triangle; although all kangaroos may happen to be Australian kan- garoos, this is not part of what we mean to imply when we use the name, — an animal subsequently found in the interior of New Guinea, but otherwise possessing all the propertiesofkangarooswould not have the name kangaroo denied to it; although all ruminant animals are cloven- hoofed, we cannot regard cloven -hoofed as part of the meaning of ruminant, and we may say with Mill that were n animal to be discovered which chews the cud, but has ts feet undivided, it would certainly still be called ru- minant. The above meaning of connotation is that to which in CHAP. II.] TERMS. 17 my opinion we should strictly adhere. It is of course open to any one to say that he will include in the connotation of a class name all the properties possessed in common by all members of the class ; but this is simply to use the term in adifferent sense. It is used in this sense by a writer in a recent number of Mind, ** On the connotative side a name means, to usy all those qualities common to the class named with which we are acquainted; — all those properties that are said to be 'involved in our idea' of the thing named. These are the properties that we ascribe to an object when we call it by the name. But, just as the word * man,' for example, denotes every creature, or class of creatures having the attributes of humanity, whether we know him or not, so does the word properly connote the whole of the pro- perties common to the class, whether we know them or not. Many of the facts, known to physiologistsandanatomistsaboutthe constitution of man's brain, for example, are not involved in most men's idea of the brain : the possession of a brain precisely so constituted does not, therefore, form any part of their meaning of the word *man.' Yet surely this is properly connoted by the word" (E. C. Benecke, in Mind, 1881, p. 532). Professor Jevons also uses the term in the same sense. " A term taken in intent (connotation) has for its meaning the whole infinite series of qualities and circum- stances which a thing possesses. Of these qualities or cir- cumstances some may be known and form the description or definition of the meaning; the infinite remainder are unknown" {Pure Logic, p. 4). Professor Bain appears to use the term in an intermediate sense, including in the connotation of a class-name not all the attributes common to the class but all the independent attributes, that is, all that cannot be derived or inferred from others. It ought to be made very clear in any discussion con- K. L. 2 i8 TERMS. [parti. ccrning the connotation of names in which of these several senses we are using the term ** connotation" itself. It may be said that to use the term in Mill's sense, and to make connotation depend on what is intended to be implied by the mere use of the name, is to make it vary with every different speaker. By the same nametwopeople may mean to imply different things, that is, the attributes they would include in the connotation of the name would be different; and not unfrequently some of us may be unable to say precisely what is the meaning that we our- selves attach to the words we use. This is a fact which it is most important to recognise. But for the purposes of formal logic we may assume that every name has a fixed and definite connotation. The object of the definition of names already in use is just to give this ; and in the case of an ideal language properly employed every name would have the same fixed and precise meaning for everybody. 14, Are proper names connotative or non- connotative } On the question here raised Mill speaks decisively, — **The only names of objects which connote nothing are proper names ; and these have, strictly speaking, no signifi- cation" (Logic^ I. p. 36); and most logicians are in agree- ment with him. An opposite view is however taken by Jevons, and some others (r.^., F. H. Bradley, T, Shedden), In one or two places I am inclined to think that Jevons tends somewhat to obscure the point at issue. Thus with reference to Mill he says, — "Logicians have erroneously asserted, as it seems to me, that singular terms are devoid of meaning in intension, the fact being that they exceed all ther terms in that kindofmeaning" (PrindpUs of Science^ I. pp* 3a, 33i with a reference to Mill in the foot-note). CHAP. II.] TERMS. 19 But Mil] distinctly says that some singular names are connotative, e,g,^ the sun, the first emperor of Rome (Logic, I. pp. 34, 5). Again, Jevons says, — " There would be an impossible, breach of continuity in supposing that after narrowing the extension of * thing* successively down to nimal, vertebrate, mammalian, mati, Englishman, educated at Cambridge, mathematician, great logician, and so forth, thus increasing the intension all the time, the single remain- ing step of adding Augustus de Morgan, Professor in Uni- versity College, London, could remove all the connotation, nstead of increasing it to the utmost point" (Studies in Deductive Logic, pp. 2, 3). But every one would allow that we may narrow down the extension of a term till it becomes individualised without destroying its intension or connota- tion ; "the present Professor of Pure Mathematics in Uni- versity College, London" is a singular term, — ^we cannot diminish the extension any further, — ^but it is certainly connotative. We must then clearly understand that the only contro- versy is with regard to what are strictly /r^^r names. Even yet there is a possible source of ambiguity that should be cleared up. If by the connotation of a name we meanailthettributes possessed by the individuals denoted by the name, or even all the independent attributes, Professor Jevons's view may be correct. This does appear to be what Jevons himself means, but it is distinctly not what Mill means, — he means only those attributes which are implied by the name itself. Jevons puts his case as ollows : — "Any proper name, such as John Smith, is almost ithout meaning until we know the John Smith in question. It is true that the name alone connotes the fact that he is a Teuton, and is a male; but, so soon as we know the exact individual it denotes, the name surely implies, also, 2 — 2 20 TERMS. [part I. the peculiar features, form, and character, of that individuaL In fact, as it is only by the peculiar qualities, features, or circumstances of a thing, that we can ever recognise it, no name could have any fixed meaning unless we attached to it, mentally at least, such a definition of the kind of thing denoted by it, that we should know whether any given thing was denoted by it or not. If the name John Smith does not suggest to my mind the qualities of John Smith, how shall I know him when I meet him ? for he certainly does not bear his name written upon his brow '' (Elementary Lessons in Logic, p. 43). A wrong criterion of connotation in Mill's sense is here taken. The connotation of a name isnotthequality or qualities by which I or any one else may happen to recognise the class which it denotes. For example, I may recognise an Englishman abroad by the cut of his clothes^ or a Frenchman by his pronunciation, or a proctor by his bands, or a barrister by his wig ; but I do not mean any of these things by these names, nor do they (in Mill's sense) form any part of the connotation of the names. Compare two such names as ** John Duke Coleridge" and "the Lord Chief Justice of England." They denote the same individual, and I should recognise John Duke Coleridge, and the Lord Chief Justice of England by the same attributes ; but the names are not equivalent, — the one is given as a mere mark of a certain individual to distinguish him from others, and it has no further signification; the other is given on account of the performance of certain functions, which ceasing the name would cease to apply. Surely there is a distinction here, and one which it is important that we should not overlook. Nor is it true that such a name as "John Smith*' connotes "Teuton, male, &c." John Smith might be a race-horse, or a negro, or the pseudonym of a woman, as in CHAP. II.] TERMS, 21 the case of George Eliot. In none of these cases could a name be said to be misapplied as it would if a horse were called a man, or a negro a Teuton,orawoman a male. But it may fairly be said that in a certain sense many proper names do suggest something, that at any rate they were chosen in the first instance for a special reason. For example, Strongi'th'arm, Smith, Jungfrau. Such names however even if in a certain sense connotative when first imposed soon cease to be connotative in the way in which other names are connotative. Their application is in no way dependent on the continuance of the attribute with reference to which they were originally given. As Mill puts it, ^^the name once given is independent of the reason,^* Thus, a man may in his youth have been strong, but we should not continue to call him strong when he is in his dotage ; whilst the name Strongi'th'arm once given would not be taken from him. The name "Smith" may in the first instance have been given because a man plied a certain handicraft, but he would still be called by the same name if he changed his trade, and his descendants continue to be called Smiths whatever their occupations may be. Nor can it be said that the name necessarily implies ancestors of the same name. Proper names of course become connotative when they are used to designate a certain type of person ; for example, a Diogenes, a Thomas, a Don Quixote, a Paul Pry, a Benedick, a Socrates. But, when so used, such names have reallyceasedtobeproper names at all; and they have come to possess all the characters of general names. 15. Discuss the question whether the following terms are respectively connotative or non-connota- 22 TERMS. [part I. tive : — ^Westminster Abbey, the Mikado of Japan,Barmouth. [L.] 16. Enquire whether the following names are respectively connotative or non-connotative: — Caesar, Czar, Lord Beaconsfield, the highest mountain in Europe, Mont Blanc, the Weisshorn, Greenland, the Claimant, the pole star, Homer, a Daniel come to judgment. 17. Can any abstract names possess both deno- tation and connotation.^ In Fowler's use of the term all abstract names are con- notative, that is, they at once suggest or imply attributes ; while none are denotative, that is, they do not denote individuals or groups of individuals. Professor Fowler himself admits that it sounds paradoxical to say that Abstract names are not denotative, but he is of opinion that the employment of the expressions in his sense would simplify the statement and explanation of many logical difficulties. I am inclined to think that the present is a case in point. Mill holds that while most abstract names are non-con- notative, still "even abstract names, though the names only of attributes, may in some instances be justly considered as connotativeforattributesthemselvesmayhaveattributes ascribed to them j and a word which denotes attributes may connote an attribute of those attributes" {Logic, i. p. 33). I have some difficulty in interpreting this passage. Suppose that we have a connotative abstract name denoting the attri- bute A and connoting the attribute B; now a connotative name is always defined by means of its connotation, and we shall therefore define our term by saying that it connotes BCHAP. 11.] TERMS, 23 without any reference whatever to A. What then will dis- tinguish it from the concrete term denoting whatever pos- sesses B ? The solution of the difficulty seems to be that when we talk of one attribute having another ascribed to it, the term denoting it becomes concrete rather than abstract Comparing Mill's definitions of an abstract name and of a connotative name, I fail to understand how the same name can be both \ 18. Explain and discuss the statement: — "In a series of common terms arranged in regular sub- ordination to one another, the denotation and con- notation vary inversely." 19. Explain the following statements : — (a) If a term be abstract, its denotation is the same as the connotation of the corresponding con- crete ? (b) Of the denotation and connotation of a term, one may, both cannot, be arbitrary. {c) Names withindeterminateonnotationare npt to be confounded with names which have more than one connotation. 20. Verbal and Real Propositions. A Verbal Proposition is one in which the connotation  Mr Killick in his Handbook of Milts Logic makes Mill include in the class of connotative names such abstract names as are the names of groups of attributes {e,g.^ humanity). I do not think that MiU himself intended this, nor do I think that the view is a correct one (?.^., accord- ing to Mill's own usage of terms). If an abstract name has both deno- tation and connotation because it is the name of a group of attributes, on what principle shall we distinguish between the attributes that it denotes and those that it connotes ? 24 TERMS. [part I. of the predicate is a part or the whole of the connotation of the subject Bain describes the verbal proposition as "the notion under the guise of the proposition"; and it is certainly convenient to discuss verbal propositions in con- nection with the connotation of names or the intension of concepts. The most important class of verbal propositions are definitions, the essential function of which is to analyse the connotation of names \ The least important class are absolutely tautologous or identical propositions, ^.^., all A is A, a man is a man. Real Propositions, on the other hand, "predicate of a thing some fact not involved in the signification ofthename by which the proposition speaks of it ; some attribute not connoted by that name." The same distinction is also expressed by the pairs of terms, analytic* and synthetic, explicative* and ampliative, essential" and accidental. ^ Besides propositions giving such an analysis more or less com- plete, the following classes of propositions are frequently included under the head of verbal propositions : where the subject and predicate are both proper names, e.g.y Tully is Cicero; where they are dictionary synonyms, e.g., wealth is riches, a story is a tale, charity is love. All such propositions however can hardly be brought under the head of verbal propositions as defined in the text. At any rate if we have decided that a proper name is not connotative, it is clear that in no proposition having a proper name for its subject can the predicate be any part of the connotation of the subject. To include these classes we must define a verbal proposition as a proposition which is wholly concerned with the meaning or application of names, a real proposition as one which is concerned with things or qualities. Even with these definitions, however, while it is a verbal proposition to say that Tully is Cicero (/.^., that these names have the same appli- cation), it is a real proposition to say that Tully is an individual who is also denoted by thenameCicero.^Itshouldbe carefully observed that while the term verbal is some- CHAP. II.] TERMS. 25 21. Which of the following propositions should you regard as Real, and why ? Homer wrote the Iliad, nstinct is untaught ability, Instinct is hereditary experience. [c] "Homer wrote the Iliad" is regarded by Bain as a verbal predication, " We know nothing about Homer except the authorship of the Iliad. We have not a meaning to attach to the subject of the proposition, * Homer', apart from the predicate, * wrote the Iliad.' The affirmation is nothing more than that the author of the Iliad was called Homer" {Logic, Deduction, p. 67). Taking the definition of verbal proposition given in the text, and holding that no proper name is connotative, this view must clearly be ejected. If however by a verbal proposition we mean one that relates in any way to the application of names, (/>., taking the definition given in the note), there may be some- thing to say for it. But is it true that we attach nothing more to "Homer" than ** wrote the Iliad"? Do we not, for example, attach to "Homer" the authorship of other poems, and also an individuality * ? If it is the fact that the Iliad was the work of various authors, as has been times stretched so as to include such a proposition as "Tully is Cicero," this is never the case with the termsanalytic,explicative,essential. These terms are strictly limited to propositions which give no informa- tion whatever (even with regard to the application of names) to any one who is fully acquainted with the connotation or intension of the subject ^ I do not of course mean that this is the connotation of ** Homer," for I hold that no proper names are connotative. I mean that Homer denotes for me a certain individual who was a Greek, who lived prior to a certain date, and who was the author of certain poems other than the Iliad. 26 TERMS. [part i. asserted, would not the proposition become false ? Still, we should perhaps admit that we have here a limiting case. Some light may be thrown on the point thus raised by an answer once sent in by an examinee ; " The accepted opinion is that the Iliad was not written by Homer, but by another man of the same name." "Instinct is untaught ability" and "Instinct is here- ditary experience" may be regarded as verbal and real espectively. 22. Is it a verbal proposition to say that it is hotter in summer than in winter ? Examine the following statements: A free in- stitution is a contradiction in terms ; so is a perfect creature. [v.] 23. If all ;r is j/, and some x is ^, and / is the name of those z's which are ;r; is it a verbal pro- position to say that all/ is^ } [v.] 24. Give one example of each of the following, — (i) a collective generalname,(ii)a singular abstract name, (iii) a connotative abstract name, (iv) a con- otative singular name ; or, if you deny the possi- bility of any of these combinations, state clearly your reasons. CHAPTER III. POSITIVE AND NEGATIVE NAMES. RELATIVE NAMES. 25. Positive and Negative Terms. The essential distinction between positive and negative names as ordinarily understood may be expressed as follows : — a positive name implies the preserue of certain definite attributes; a negative name implies the absence of one or other of certain definite attributes. "Every name," as remarked by De Morgan, "applies to everything positively or negatively " ; for example, every- thing either is or is not a horse. Every name then divides all things in the universe into two classes. Of one of these it is itself the name; and a corresponding name can be framed to denote the other. This pair of names, which between them denote the whole universe, are respectively positive and negative. But which is which ? Which is the negative name, since each {positively denotes a certain class of objects? The distinction lies in the manner in which the class is determined. We may say that in a certain sense a strictly negative name has not an independent connotation of its own ; its denotation is determined by the connotation of thecorrespondingpositivename.Itdenotesanindefinite and unknown class outside a definite and limited class. In other words, by means of its connotation 28 TERMS. [part I. we first mark off the class denoted by the positive name, and then the negative name denotes what is left. The fact that its denotation is thus determined is the distinctive characteristic of the negative name. We have here supposed that between them the positive name and the corresponding negative name exhaust the whole universe. But something different from this is often meant by a negatiye name. Thus De Morgan considers hat parallel and alien are negative names.' "In the formation of language, a great many names are, as to their original signification, of a purely negative character : thus, parallels are only lines which do not meet, aliens are men who are not Britons (i,e.^ in our country)" {Formal Logic^ p. 37). But these names clearly have not the thorough- going negative character that I have just been ascribing to negative names. The difference will be found to consist in this, that in the sense in which alien is a negative name, the positive and negative names (Briton and alien) do not between them exhaust the entire universe, but only a limited universe, namely, in the given case, that constituted by the inhabitants of Great Britain, We may perhaps distinguish between names absolutely negative^wherethereferenceistothe entire universe; and names relatively negative, where the reference is only to some limited universe. Now it will be seen that in the use of such a term as not-white there is a possible ambiguity; we must decide whether in any given instance the name is to be regarded as absolutely or only as relatively negative. Mill chooses the former alternative ; " not-white," he says, '* denotes all things whatever except white things." De Morgan and Bain however consider that in such a case the reference is not to the whole universe but to some particular universe only. Thus, in contrasting white and not-white we are CHAP. III.] TERMS. 29 referring solely to the universe of colour; not-white does not include everything in nature except white things, but only things that are black, red, green, yellow, &c, that is, all toloured things except such as are white \ Whately and Jevons agree with Mill ; and from a logical point of view I think they are right. Or rather I would say that two such terms as S and not-.S' must between them exhaust the universe of discourse, whatever that may be ; and we must not be precluded from making this, if we care to do so, the entire universe of existence. That is, not-5 may be called upon to assume the absolutely negative character*. For if we are unable to denote by not-5 all things whatsoever except iS, it is difficult toseein what way we shall be able to denote these when we have occasion to refer to them. On the other hand, we must also be empowered to indicate a limitation to a particular universe where that is intended. By not-*S then referred to without qualification expressed or implied by the context I would understand the absolute negative of S\ but I should be quite prepared to find a limitation to some more restricted universe in any particular instance. It should be noted that in the case of a limited uni- verse it is sometimes difficult to say which of the pair of contrasted names is really to be regarded as the negative name. For example, De Morgan says that parallel is a negative name, since parallel lines are simply lines that do not meet But we might also define them as lines such that ^ Thus, on Bain's view it would be incorrect to say that an im- material entity such as honesty was not-white. • On this view, "not- white'* might be used to denote not merely coloured things that are not white, but also things that are not coloured at all. It would for example be correct to say that honesty was not- white. 30 TERMS. [part i. if another line be drawn cutting them both, the alternate angles are equal to one another; and then the name appears as a positive name. Similarly in the universe of property, as pointed out by De Morgan, persoruU and realarerespectivelythe negatives of each other ; but if we are to call one positive and the other negative, it is not quite clear which should be which. For a suggestion of Mr Monck's as to the definition of negative terms, see section 29. 26. Privative Names, To the distinction between positive and negative names, MDl adds a class of names called privative. " A privative name is equivalent in its signification to a positive and a negative name taken together; being the name of some- thing which has once had a particular attribute, or for some other reason might have been expected to have it, but which has it not Such is the word dlindy which is not equivalent to nof seeing, or to not capable of seeing, for it would not, except by a poetical or rhetorical figure, be applied to stocks and stones" {Logic, i. p. 44). Perhaps also idle, which Mill gives as a negative, should rather be regarded as a privative term. It does not mean merely "not-working," but "not-working where there is the capacity to work." We should hardly speak of a stone as being **idle.'' The distinction here indicated does not appear to be of logical importance. 27. How far is it true that, as ordinarily under- stood, negative terms have a definite connotation, while in Logic they have not ? So far as it is true, how would you explain the fact } [w.] CHAP.III.]TERMS.2128.Contradictoryandcontraryterms. A positive term and its corresponding negative term are called contradictories, A pair of contradictory terms are so related that between them they exhaust the entire universe o which reference is made, whilst in that universe there is no individual of which both can be at the same time affirmed. The nature of this relation is expressed in the two laws of Contradiction and Excluded Middle. Nothing is at the same time both X and not- A"; Everything is X or not-X For the application of the above to complex terms, see Part iv. The contrary of a term is usually defined as the term denoting that which is furthest removed from it in some particular universe ; e.g., black and white, wise and foolish. Two contraries may in some cases happen to make up between them the whole of the universe in question, e.g., Briton and alien ; but this is not necessary, e.g., black and white. It follows that although two contraries cannot both be true of the same thing at the same time, they may both be false. The above may be called the material contrary. In the case of complex terms, we may also assign a formal con- trary, as is shewn in Part iv. 29. Illustrate Mill's statement that " names which are positive in form are often negative in reality, and others are really positive though their form isnegative."Thefact that a really positive term is sometimes negative in form results from the circumstance that the negative pre-^ fix is sometimes given to the contrary of a term. But we have seen that a term and its contrary may both be positive. 32 TERMS. [part i. For example, pleasant and unpleasant; "the word un- pleasant, notwithstanding its negative form, does not con- note the mere absence of pleasantness, but a less degree of what is signified by the word painful^ which, it is hardly necessary to say, is positive." On the other hand, some names positive in form may be regarded as relatively nega- tive, PART II. PROPOSITIONS. CHAPTER I. KINDS OF PROPOSITIONS. THE QUANTITY AND QUALITY OF PROPOSITIONS. 36. Categorical, Hypothetical and Disjunctive Propositions. For logical purposes, a Proposition may be defined as " a sentence indicative or assertory,*'' (as distinguished, for example, from sentences imperative or exclamatory); in other words, a proposition is a sentence making an affirma- tion or denial, as — All 5is /*, No vicious man is happy. A proposition is Categorical if the affirmation or denial is absolute, as in the above examples. It is Hypothetical if made under a condition, as — If A\& B^ Cis D; Where ignoranceisbliss,'tisfolly to be wise. It is Disjunctive if made with an alternative, as — Either P is ^, or X is Y\ He is either a knave or a fool*. ^ It should be observed that in a disjunctive proposition there may be two distinct subjects asinthefirstoftheabove examples, or only one as in the second. Disjunctive propositions in which there is only one distinct subject are the more amenable to logical treatment. 3— a 36 PROPOSITIONS. [part ii. [The above threefold division is adopted by Mansel. It is perhaps more usual to commence with a twofold division, the second member of which is again subdivided, the term Hypothetical being employed sometimes in a wider and sometimes in a narrower sense. To prevent confusion, it may be helpful to give the following table of the usage of one or two modern logicians with regard to this division. Whately, Mill and Bain : — 1. Categorical. 2. Hypothetical, lypotnencai, r Conditional, or Compound, -I) ! . i- _. 1 1(2) Disjunctive, or Coniolex. V ' •' or Complex Hamilton and Thomson : — 1. Categorical. X-. jv 1 (M Hypothetical 2. Conditional, i) : t^. . ,. ((2) Disjunctive. Fowler (following Boethius) : — 1. Categorical. 2. Conditional ((i) Conjunctive. or Hypothetical. ((2) Disjunctive. Mansel, as I have already remarked, gives at once a threefolddivision,1.Categorical2.HypotheticalrConditional.3.Disjunctive.Hestateshisreasons for his own choice of terms as follows : — " Nothing can be more clumsy than the employ- ment of the word cofiditional in a specific sense, while its Greek equivalent, hypothetical^ is used generically. In Boe- thius, both terms are properly used as synonymous, and generic ; the two species being called conjunctivi^ conjuncti^ CHAP. I.] PROPOSITIONS. 37 or connexiy and disjuncHvi or disjunctu With reference to modern usage, however, it will be better to contract the Greek word than to extend the Latin one. Hypothetical in the following notes, will be used as synonymous with con- dttionar (Hansel's edition oi Aldrich^ p. 103).] 36. A logical analysis of the Categorical Pro- position. In logical analysis, the categorical proposition always consists of three parts, namely, two terms which are united by means of a copula. The subject is that term about which affirmation or denial is made ; it represents some notion already partially deter- mined in our mind, and which it is our aim further to determine. The predicate is that term which is affirmed or denied of the subject; it enables us further to determine the subject, />., to enlarge our knowledge with regard to it. The copula is the link of connection betweenthesubjectandthepredicate,andconsistsofthewordsisoris not ccording as we affirm or deny the latter of the former. In attempting to apply the above analysis to such a proposition as "All that love virtue love angling," we find that, as it stands, the copula is not separately expressed. It may however be written, — subj. cop. pred. All lovers of virtue | are | lovers of angling; and in this form the three diflferent elements of the logical proposition are made distinct. This analysis should always be performed in the case of any proposition that may at first present itself in an abnormal form. A difficulty that may sometimes arise in discriminating the subject 38 PROPOSITIONS. [PART ii. and the predicate is dealt with subsequently,-^ompare section 50. The older logicians distinguished propositions secundi adjacentiSf and propositions tertii adjacentis. In the former, the copula and the predicate are not separated ; ^.^., The man runs. All that love virtue love angling. In the latter, the copula and the predicate are made distinct ; e.g,^ The man is running, All lovers of virtue are lovers of angling. A categorical proposition, therefore, when expressed in exact logical form, is tertii adjacentis. 37. Exponible, capulative^ exclusive^ exceptive pro- posiions. Propositions that are resolvable into more propositions than one have been called exponible^ in consequence of theirsusceptibilityofanalysis.Copulativepropositionsareformedby a direct combination of simple propositions, e,g.^ P is both Q and R (/>., P is Q, P is ^), A is neither JB nor C (/./., A is not Bf A is not C)\ they form one class of exponibles. Exclusive propositions contain some such word as " only," thereby limiting the predicate to the sub- ject ; e.g,^ Only S is F» This may be resolved into S is F^ and P is S. Propositions of this kind also are therefore exponibles. Exceptive propositions limit the subject by such a word as "unless" or "except"; e.g.^ A is X^ unless it happens to be B. These too may perhaps be regarded as exponible propositions. 38. The Quantity and Quality of Propositions. The Quality of a proposition is determined by the copula, being affirmative or negative according as the copula is of the form '*is" or ''is not." Propositions are also divided into universal and partir CHAP. I.] PROPOSITIONS, 39 culavy according as the affirmation or denial is made of the whole or only of a part of the subject. This division of Propositions is said to be according to their Quantity, Combining the two principles of division, we get four fundamental forms of propositions : — (i) theuniversalaffirmative^AllSisPyusuallydenotedbythesymbolA;(2)theparticularffirmative^SomeSisPyusuallyde^notedbythesymbolI;(3)theuniversalnegativeyNoSisPyusuallydenotedbythesybolE ; (4) the particular negativCy Some S is not Py usually denoted by the symbol O. These symbols A, I and E, O are taken from the Latin words affirmo and nego, the affirmative symbols being the first two vowels of the former, and the negative s)mibols the two vowels of the latter. Besides these symbols, it will also be found convenient sometimes to use the following, — SaP=K\\S\sP', SiP = Some S is P; SeP=NoS isP; SoP= Some S is not P. The above are useful when we wish that the symbol which is used to denote the proposition as a whole should also indicate what symbols have been chosen for the subject and the predicate respectively. Thus, MaP=Al\AfisP; PoQ= Some P is not Q. The universal negative should be written in the form No S is Py not All S is not P^ for the latter would usually 40 PROPOSITIONS. [part ii. be understood to be merely particular. Thus, All that glitters is not gold is really an O proposition, and is equi- valent to — Some things that glitter | are not | gold. 39. Indefinite Propositions. According to Quantity, Propositions have sometimesbeendividedinto(i)Universal,(2)Particular,(3)Singular,(4)Indefinite.Singularpropositionsarediscussedinthefollowing section. By an Indefinite Proposition is meant one " in which the Quantity is not explicidy declared by one of the designatory terms «//, every ^ some, many, &c" We may perhaps say with Hamilton that indesignate or preindesignate would be a better term to employ. There can be no doubt that, as Mansel remarks, " The true indefinite proposition is in fact the particular; the statement *some A'\& B^ being applicable to an uncertain number of instances, from the whole class down to any portion of it. For this reason particular pro- positions were called indefinite by Theophrastus " {Aidrich, p. 49). Some indesignate propositions are no doubt intended to be understood as universals, e.g.. Comets are subject to the law of gravitation ; but in such cases before we deal with the proposition logically it is better that the word all should be explicitly prefixed to it. If we are really in doubt with regard to the quantity of the proposition it must logically be regarded as particular. Other designations of quantity besides all and some, e.g, most, are discussed in section 41. The term indefinite has also been applied to propositions in another sense. According to Quality, instead of the two- folddivisiongivenintheprecedingexample,athreefolddivisionis sometimes adopted, namely into affirmative, CHAP. I.] PROPOSITIONS. 41 negative, and infinite or indefinite. For further explanation, see section 44. 40. Singular Propositions. By a Singular or Individual Proposition is meant a pro- position of which the subject is a singular term, one there- fore in which the affirmation or denial is made but of a single specified individual; e.g,y Brutus is an honourable man; Much Ado about Nothing is a play of Shakespeare's ; My boat is on the shore. Singular propositions may usually be regarded as forming a sub-class of Universal propositions, since in every singular proposition the affirmation or denial is of the whole of the subject. Such propositions have however certain pecu- liarities of their own, as we shall note subsequently; e.g,^ they have not like other universal propositions a contrary distinct from their contradictory. Hamilton distinguishes between Universal and Singular Propositions, the predication being in the former case of a Whole Undividedy and in the latter case of a Unit Indivisible, This separation is sometimes useful ; but I think it better not to make it absolute. A singular proposition may without risk of confusion be denoted by one of the symbols A or E ; and in syllogistic inferences, asingularmayalwaysberegardedasequivalent to a universal proposition. The use of independent symbols for affirmative and negative singular propositions would introduce considerable additional com- plexity into the treatment of the Syllogism ; and for this reason alone it seems desirable as a rule to include par- ticulars under universals. We may however divide universal propositions into General and Singular^ and we shall then have terms whereby to call attention to the distinction wherever it may be necessary or useful to do so. 42 PROPOSITIONS. [PART II. There is a certain class of propositions with regard to which there is some difference of opinion as to whether they should be regarded as singular or particular ; for example, such as the following : A certain man had two sons ; A great statesman was present. Mansel {Aldrich, p. 49) decides that they should be dealt with as particulars, and I think rightly, on the ground that if we have two such propositions, " a certain man '' or " a great statesman " being the subject of each, we cannot be sure that the same individual is referred to in both cases. Sometimes however the context may enable us to decide the case differently. There are propositions of another kind with a singular term for subject about which a few words may be said; namely, suchpropositionsasBrowningissometimesobscure;Thatboyissometimesfirstinhisclass.Thesepropositionsmaybetreatedas universal with a somewhat complex predicate, (and it should be noted that in bringing propositions into logical form we are frequently compelled to use very complex predicates) ; thus : — Browning ] is | a poet who is sometimes obscure. That oy | is | a boy who is sometimes first in his class. By a certain transformation however these propositions may also be dealt with as particulars, and such transforma- tion may sometimes be convenient; thus, Some of Browning's writings are obscure, Some of the boy's places in his class are the first places. But when the proposition is thus modi- fied, the subject is no longer a singular term. 41. The logical signification of the words some^ mostyfeiv, all, any. Some may mean merely " some at least," />., not none, or it may carry the further implication, " some at most,'' /., it has the same indefinite character which we logically ascribe to "some"; since the antecedent condition is satisfied if a single A is B. The proposition might indeed be written — If one or more A is B, C is JD. 42. Examine the logical signification of the itali- cised words in the following propositions : — Some are born great- Few Sire chosen^ All is not lost. All men are created equal. All that a man hath will he give for his life. If some A is By some C is D, If any A is By any C is D, IfallAisByallCisD. 48 PROPOSITIONS. [part li. The student must be warned against . treating such a proposition as "If any A is By some C is Z>*' as par- ticular'. Regarded separately the antecedent and the con- sequent in this example are both particular ; but the con- nection between them is affirmed universally, the proposition asserting that " in all cases in which any A is By some C It should be observed that in a considerable number of cases, the hypothetical is of the nature of a singular pro- position, the event referred to in the antecedent being in the nature of things one which can happen but once ; e,g,j If I perish in the attempt, I shall not die unavenged.TotheDisjunctivePropositionwe are unable to apply distinctions of Quality. The proposition. Neither F \s Q nor X i& Y states no alternative, and is therefore not dis- junctive at all. Distinctions of Quantity are however still applicable. Thus, Universal, — Either F\s ^ or -Y is K Particular, — In some cases either jP is ^ or -ST is Y. It is again to be observed that frequently the dis- junctive proposition is of the nature of a singular proposi- tion, the reference being but to a single occasion on which it is asserted that one of the alternatives will hold good. 46. Determine the Quantity and Quality of the following propositions, stating precisely what you regard as the subject and predicate, or in the case ^ I cannot agree with Hamilton (Logic^ i. p. 248), in regarding the following as a particular hypothetical — If some Dodo is, then some animal is. The proposition is a little hard to interpret, but it seems to mean that if there is such a thing as a Dodo, then there is such a thing as an animal; and we must consider that a imiversal connection is here affirmed. CHAP. I.] PROPOSITIONS. 49 of hypothetical propositions, the antecedent and consequent of each : — (i) All men think all men mortal but them- selves. (2) Not to know me argues thyself unknown. (3) To bear is to conquer ourfate.(4)Berkeley,agreatphilosopher,deniedtheexistenceofMatter.(5)Agreatphilosopher has denied the existence of Matter. (6) The virtuous alone are happy. (7) None but Irish were in the artillery. (8) Not every tale we hear is to be believed, (9) Great is Diana of the Ephesians ! (10) All sentences are not propositions. (11) Where there's a will there's a way, (12) Some men are always in the wrong. (13) Facts are stubborn things. (14) He that increaseth knowledge increaseth sorrow. (15) None think the great unhappy, but the great. (16) He can't be wrong, whose life is in the right. (17) Nothing is expedient which is unjust. (18) Mercy but murders, pardoning those that kill. (19) If virtue is involuntary, so is vice. (20) Who spareth the rod, hateth his child. 47. Analyse the following propositions, i,e,, ex- press them in one or more of the strict categorical forms admitted in Logic : — K. L. A 5Q PROPOSITIONS. [part ii. (i) No one can be rich and happy unless he is also temperate and prudent, and not always then. (ii) No child ever fails to be troublesome if ill taught and spoilt (iii) It would be equally false to assert that the rich alone are happy, or that they alone are not. [v.](i) contains two statements which may be reduced to the following forms, — All who are rich and happy | are | temperate and pudent A. Some who are temperateandprudent| are not I rich and happy. O. (ii) may be written, All ill-taught and spoilt children are troublesome. A. iii) Here two statements are given false^ namely, the rich alone are happy ; the rich alone are not happy. We may reduce these false statements to the following, — all who are happy are rich ; all who are not happy are rich. And this gives us these true statements, — Some who are happy are not rich O. Some who are not happy are not rich. O. The original proposition is expressed therefore by means of these two particular negative propositions48. The Distribution of Terms in a Proposition. A term is said to be distributed when reference is made to all the individuals denoted by it ; it is said to be undis- tributed when they are only referred to partially^ />., in- formation is given with regard to a portion of the class denoted by the term, but we are left in ignorance with regard to the remainder of the class. It follows immediately CHAP, I.J PROPOSITIONS. 51' from this definition that the subject is distributed in a universal, and undistributed in a particular, proposition. It can further be shewn that the predicate is distributed in a negative, and undistributed in an affirmative proposition. Thus, if I say. All Sis P, I imply that at any rate some P is 6", but I make no implication with regard to the whole of P, I leave it an open question as to whether there is or is not any P outside the class 5. Similarly if I say. Some S is P, But if I say. No S is Py in excluding the whole of S from /*, I am also excluding the whole of P from 6", and therefore P as well as S is distributed. Again, if I say, Some S is not Py although I make an assertion with regard to a part only of Sy I exclude this part from the whole of Py and therefore the whole of P from it. In this case, then, the predicate is distributed, although the subject is not. Summing up our results we find that A distributes its subject only, I distributes neither its subject nor its predicate,E distributes both its subject and its predicate, O distributes its predicate only, ^y 49. How does the Quality of a Proposition aflfect its Quantity 1 Is the relation a necessary one ? [L.] By the Quantity of a Proposition must here be meant the Quantity of its Predicate, and we have shewn in the preceding section that this is determined by its Quality. The predicate is distributed in negative, undistributed in_ a ffirmativ e^ proposition s. The latter part of the above question refers to Hamilton's doctrine of the Quantification of the Predicate. According to this doctrine, the predicate of an affirmative proposition is sometimes expressly distributed, while the predicate of a 4—2 52 PROPOSITIONS. [PART II. negative proposition is sometimes given undistributed. For example, the following forms are introduced : — Some S is all P, No S is some -P. This doctrine is discussed and illustrated in Part in. chapter 9. 50. In doubtful cases how should you decide which is the subject and which the predicate of a proposition ? [v.] The nature of the distinction between the subject and the predicate of a proposition may be expressed by saying that the su bject is that of whichsomething is affirmed o r denied, the predicate is that which is affirmed or denied ofthe subject; or perhaps still better, the subject is that which we think of as the determined or qualified notion, the predicate that which we think of as the determining or qualifying notion. Now, can we say that the subject always precedes the copula, and that the predicate always follows it? In other words, can we consider the order of the terms to suffice as a criterion? If the proposition is reduced to an equation, as in the doctrine of the quantification of the predicate, I do not see what other criterion we can take; or we might rather say that in this case the distinction between subject and predicate itself fails to hold good. The two are placed on an equality, and we have nothing left by which to distinguish them except the order in which they are stated. This view is indicated by Professor Baynes in his Essay on the New Analytic of Logical Forms. In such a proposition, for example, as "Great is Diana of the CHAP. I.] PROPOSITIONS. 53 Ephesians/' he would call " great " the subject, reading the proposition, however, " (Some) great is (all) Diana of the Ephesians." But leaving this view on one side, we cannot say that the order of terms is always a sufficient criterion. In the proposition just quoted, " Diana of the Ephesians " would generallybeacceptedasthesubject. What further criterion then can be given ? In the case of E and I propositions, (propositions, as will be shewn, which can be simply con- verted), we must appeal to the context or to the question to which the proposition is an answer. If one term clearly conveys information regarding the other term, it is the predicate. It is also more usual that the subject should be read in extension and the predicate in intension. If none of these considerations are decisive, then I should admit that the order of the terms must suffice. In the case of A and O propositions, (propositions, as will be shewn, which cannot be simply converted), a further criterion may be added. From the rules relating to the distribution of terms in a proposition it follows that in affirmative pro- positions the distributed term, (if either term is distributed), is the subject ; whilst in negative propositions, if only one term is distributed, it is the predicate. I am not sure that the inversion of terms ever occurs in the case of an O pro- position ; but in A propositions it is not infrequent. Ap- plying the above to such a proposition as *"* Workers of miracles were the apostles," it is clear that the latter term is distributed while the former is not. The latter term is therefore the subject. A corollary from the rule is that in an affirmative proposition if one andonlyoneterm is singular that is the subject, since a singular is equivalent to a distributed term. This decides such a case as " Great is Diana of the Ephesians." 54 PROPOSITIONS. [part ll. 61. What do you consider to be respectively the subject and the predicate of the following sentences, and why ?(i) Few men attain celebrity. (2) Blessed are the peacemakers.(3) It is mostly the boastful who fail.(4) Clematis is Traveller's Joy. [v.] 62. What do you consider to be the essential distinction between the Subject and Predicate of a proposition ? Apply your answer to the following : — (i) From thence thy warrant is thy sword. (2) That is exactly what I wanted. [v.]CHAPTER II. THE OPPOSITION OF PROPOSITIONS. 53. The Opposition of Categorical Propositions. Two propositions are said to be opposed to each other when they have the same subject and predicate respectively, but differ in quantity or quality or both I Taking the propositions SaP^^ SiPy SePySoP,inpairswe find that there are four possible kinds of relation between them. (i) The pair of propositions may be such that they cannot both be true, and they cannot both be false. This is called contradictory opposition, and subsists between SaP and SoP^ and between SeP and SiP, ^ This definition is given by Aldrich (p. 53 in Mansel's edition). Ueberweg however defines Opposition in such a way as to include only contradiction and contrariety (translation by Lindsay, p. 328); and Mansel remarks that '* Subalterns are improperly classed as opposed propositions" (Aldrich^ p. 59). Professor Fowler follows Aldrich's definition (Deductive Logic^ p. 74), and I think wisely. We want some term to signify this general relation between propositions ; and though it might be possible to find a more convenient term, I do not think hat any confusion is likely to result from the use of the term opposition if the student is careful to notice that it is here used in a technical PROPOSITIONS. [part II. (2) They may be such that they cannot both be true, but they may both be false. This is called contrary oppo- sition. SaP and SeP. (3) They may be such that they cannot both be false, but they may both be true. Subcontrary opposition. StP and SoP, (4) From a given universal proposition, the truth of the particular having the same quality follows, but not vice versa.Thisis subaltern opposition^ the universal being called the subaltemant, and the particular the subaltemate or the subaltern, SaP and SiP. SeP and SoP. All these relations are indicated clearly in the ancient square of opposition. A Contraries I Subcontraries O Propositions mu^t of course be brought to such a form that they have the same subject and the same predicate before we can apply the terms of opposition to them ; for example, All *S is jP and Some P is not S are not contra- dictories. CHAP. II.] PROPOSITIONS* 57 54. On the common view of the opposition of propositions what are the inferences to be drawn (i) from the truth, (2) from the falsity, of each of the four categorical propositions ? [L.] 66. Explain the nature of the opposition between each pair of the following propositions : None but Liberals voted against the motion. Amongst those who voted against the motion were some Liberals. It is untrue that those who voted against the motion were all Liberals. 66. Give the contradictory and the contrary of the following propositions : — (i) A stitch in time saves nine. (2) None but the brave de^^^he fair. (3) He can't be wrong whd^^H^is in the right. (4) The virtuous alone are happy. (i) A stitch in time saves nine. This is to be regarded as a universal affirmative proposition, and we therefore have Contradictory y Some stitches in time do not save nine. I. ~ Contrary^ No stitch in time saves nine. E. (2) None but the brave deserve the fair, = None who are not brave deserve the fair. E. Contradictoryy Some who are not brave deserve the fair. I. Contrary y All who are not brave deserve the fair. A. 8 PROPOSITIONS. [part il (3) He can't be wrong whose life is in the right. E. Contradictory^ Some may be wrong whose lives are in the right. I. Contrary^ All are wrong whose lives are in the right. A. (4) The virtuous alone are happy, = No one who is not virtuous is happy. E. Contradictory^ Some who are not virtuous are happy. I. Contrary^ All who are not virtuous are happy. A. 67. Give the contrary, contradictory, and sub- altern of the following propositions : — (i) All B.A.'s of the University of London have assed three examinations.(2) All men are sometimes thoughtless. (3) Uneasy lies the head that wears a crown. (4) The whole is greater than any of its parts. (5) None but solid bodies are crystals./ (6) He who has been bitten by a serpent is afraid of a rope. (7) He who tries to say that which has never been said before him will probably say that which will never be repeated after him. [Jevons, Studies in Deductive Logic, p. 58.] 68. Explain the technical terms " contrary " and "contradictory," applying them to the following pro- positions : — (i) Few 5 are P.(2)Atany rate, he was not the only one who cheated. (3) Two-thirds of the army are abroad. [v.] CHAP. II.] PROPOSITIONS. 59 It is the same thing to deny the truth of a proposition and to affirm the truth of its contradictory ; and vice versa. The criterion of contradictory opposition is that of the two propositions^ one must he true and the other must be false ; they cannot be true together, but on the other hand no mean is possible between them. The relation between two ontradictories is mutual ; it does not matter which is given true or false, we know that the other is false or true ac- cordingly. Every proposition has its contradictory, which may however be more or less complex in form. The contrary of a given proposition goes beyond mere denial, and sets up a further assertion as far as possible removed from the original assertion. It declares not merely the falsity of the original proposition taken as a whole, but the falsity of every part of it. It follows that if we cannot go beyond the simple denial of the truth of a proposition, then it has no contrary distinct from its contradictory. For example, in order simply to deny the truth of " some S is jP,'' it is necessary to affirm that " no *S is jP,'^ and it is impossible to go further than this in opposition to the given proposition. "Some *S is P" has therefore no contrary as distinguishedfromitscontradictory. We may now apply the terms in question to the given propositions : — ( I ) " Few S are P" = " Most S are not P',' and we might hastily be inclined to say that the contradictory is " Most S are jP." Both these propositions would however be false in the case in which exactly one half S was P. The true contradictory therefore is " At least one half *S' is /I" The ontrary is "All S is P'' Similarly the contradictory of " Most S are P'' is " At least one half S is not P"; and its contrary " No ^S is -P." These examples shew that if we once travel outside the 6a PROPOSITIONS. [part li. limits set by the old logic, and recognise the signs of quantity most and few as well as all and some, we soon become involved in numerical statements. Propositions of the above kind are therefore usually relegated to what has been called numerical logic, a topic discussed at length by De Morgan and to some extent by Jevons. (2) "At any rate, he was not the only one who cheated." A question of interpretation is naturally raised here ; does the statement assert that he cheated, or is this left an open question? We may I think choose the latter alternative. What the speaker intends to lay stress upon is that some others cheated at any rate, whatever may have been the case with him. The contradictory then becomes " No others heated";andwe have no distinct contrary. (3) "Two-thirds of the army are abroad." This may mean "At least two -thirds of the army are abroad," or ** Exactly two-thirds of the army are abroad." On the first interpretation, the contradictory is "Less than two-thirds of the army are abroad"; and the contrary " None of the army are abroad." On the second interpretation, the contradictory is " Not exactly two- thirds of the army are abroad," /. ^., " Either more or less than two -thirds of the army are abroad." With regard to the contrary we are in a certain difficulty ; for we may as it were proceed in two directions, and take our choice between " All the army are abroad " and " None of the army are abroad." I hardly see on what principle we are to choose between these. Fortunately, contrary opposition, unlike contradictory opposition, is of very little logical importance. 59. The Opposition of Singular Propositions. Take the proposition, Socrates is wise. The contra- CHAP. II.] PROPOSITIONS. 6i dictory is — Socrates is not wise ; and so long as we keep to the same terms, we cannot go beyond this simple denial. We have therefore no contrary distinct from the contra- dictory. This opposition of singulars has been called secondary contradiction (ManseFs Aidrich, p. 56). There are indeed two methods of treatment according to which we might findadistinctcontrarynd contradictory in the case of singular propositions, but I think that the above treatment according to which they are not distinguished is preferable to either. (i) We might introduce the material contrary of the pre- dicate instead of its mere contradictory, (compare section 28). Thus we should have — Original proposition, Socrates is wise ; Contradictory, Socrates is not wise ; Contrary, Socrates has not a grain of sense. This might be called the material contrary of the given proposition ^ A fresh term is introduced that could not be formally obtained out of the given proposition. It still remains true that the singular proposition has no formal contrary distinct from its contradictory. (2) Some principle of separation into parts might be introduced according to which the subject would be no longer a whole indivisible ; for example, Socrates might be regarded as having different characteristics at different times or under different conditions. The original proposition would then be read Socrates is always wise, and the contra- dictory would be Socrates is sometimes not wise, while the trary would be Socrates is never wise. Treated in this manner, however, the proposition hardly remains a really singular proposition. ^ The same distinction might be applied to general propositions. 62 PROPOSITIONS. [PART n. 60. Can the ordinary doctrine of the opposition of propositions be applied to hypothetical and dis- junctive propositions ? It has been already shewn that the ordinary distinctions of quantity and quality may be applied to Hypothetical Propositions, and it follows that the ordinary doctrine of opposition will also apply to them. We have UAisB, CisD. A. In some cases in which A is B, C is D. I. If^ is^, CisnotZ>. E. In some cases in which A is B, C is not JD. O. Then, as in the case of Categoricals, — A and I, E and O are subalterns. A and E are contraries. A and O, E and I are contradictories. I and O are subcontraries. There is more danger of contradictories being confused with contraries in the case of Hypotheticals than there is in the case of Categoricals. 7f A is By C is not D is very liable to be given as the contradictory oi If A is Bj C is jD. But it clearly is not its contradictory, so far as they are general propositions^ since both may be false. For ex- ample, the two statements, — If the Times says one thing, the Pall Mall says another; If the Times says one thing, the Pall Mall says the same, /.^., does not say another, — are both false : the two papers are sometimes in agreement and sometimes not If however the Hypothetical proposition is of the nature of a Singular, that is, if the thing referred to in the ante- cedent can happen but once; then as in the case of Singular Categorical propositions, the Contradictory and the Contrary are not to be distinguished. Taking the proposition — If I CHAP. II.] PROPOSITIONS. 63 perish in the attempt, I shall not die unavenged; its con- tradictory may fairly be stated — If I perish in theattempt, I shall die unavenged. We cannotapply distinctions of quality to Disjunctives, and therefore the ordinary doctrineof opposition cannot be applied to them. We may however, find the contradictory and the contrary of a disjunctive proposition, such as A is either B or C Its Contradictory is — In some cases A is neither B nor C; its Contrary — A is neither B nor C We observe then that the contradictory and contrary of a disjunctive are not themselves disjunctive. What has been said with regard to Singular Hypotheticals also appUes mutatismutandis to what may be called Singular Dis- junctives. A point to which our attention is called by the above is that the relation of reciprocity that holds between contra- dictories does not always hold between contraries. If the proposition ^ is the contradictory of the proposition a, then a is also the contradictory of p ; but if 8 is the contrary of a, it does not necessarily follow that a is the contrary of 8. Thus, we have seen that the contrary of ^^A is either B or C" is ''A is neither B nor C" The contrary of the latter however is " A is both B and C," which is not the original proposition over again \ 61. How would you apply the terms contradictory and contrary to the case of complex propositions : e.g., He was certainly stupid ; and, if not mad, either miserably trained, or misled by bad companions } [v.] The criterion of contradictories given in section 58, may be applied to the case of complex propositions. For example, take the complex proposition X is both A and B^ ^ Cf. also the Examples given in section §8. 64 PROPOSITIONS. [part ii. (where J^ is a singular term). Regarded as a whole, this statement is evidently false if X fails to be either one or the other of A and B. It is also clear that it must either be both of them or it must fail to be at least one of them. We have then this pair of contradictories, — \X is both A and B ; \X is either not A or not B, Thus, what we may perhaps call a conjunctive is contra- dicted by a disjunctive, and vice versa. Next take the rather more complex proposition — X is A, and either B or C\ Its contradictory, following the above rule, is X is either not A or neither B nor C Next taketheproposition X is Yy and if it is not Z, // is either Qor R^. It may be reduced to X is y ; and either Z, Qot R, and we at once get the contradictory J^is either not Yox neither Z, Q nor R, It will be noticed that the last example chosen is equi- valent to the one given in the question, the terms of the latter being translated into symbols. The required contra- dictory is therefore — Either he was not stupid, or he was neither mad, miser- ably trained nor misled by bad companions. The application of the term contrary to complex pro- positions is of less interest. We may however consider that we have the contrary of such a proposition when we deny every part of the statement. Thus the contrary of " X is ^ I still assume that the subject of the proposition is a singular term. CHAP, n.] PROPOSITIONS. 65 both A sndB" is ''X is neither A nor jB''; of ''Xis A and either B or C" "^is neither A^ B nor C"; and of the given proposition, "He was neither stupid nor mad nor miserably trained nor misled by bad companions."62. What is the precise meaning of the assertion that a proposition — say "All grasses are edible" — is false ? [Jevons, Studies in Deductive Logic^ p. 116.] Professor Jevons discusses at some length the point here raised, but I find myself quite unable to agree with what he says in connection with it. He commences by givingananswer, which may be called the orthodox one, and which I should certainly hold to be the correct one. When I assert that a proposition is false, I mean that its contradictory is true. The given proposition is of the form A^ and its contradictory is the corresponding O proposition, — Some grasses are not edible. When, there- fore, I say that it is false that all grasses are edible, I mean that some grasses are not edible. Professor Jevons however continues, " But it does not seem to have occurred to logi- cians in general to inquire how far similar relations could be detected in the case of disjunctive and other more com- plicated kinds of propositions. Take, for instance, ^ the assertion that *all endogens are all parallel-leaved plants.' If this be false, what is true ? Apparently that one or more endogens are not parallel-leaved plants, or else that one or more parallel-leaved plants are not endogens. But it may also happen that no endogen is a parallel-leaved plant at all. There are three alternatives, and the simple falsity of the original does not shew which of the possible contra- dictories is true." In this statement, there appear to me to be two errors. In the first place, in saying that one or more endogens areK. L. 5 66 PROPOSITIONS. [PART ll. not parallel-leaved plants, we do not mean to exclude the possibility that no endogen is a parallel-leavedatall.S3rmbolically, Some S is not P does not exclude No S is P. The three alternatives are therefore at any rate reduced to the two first given. But in the second place, I think Professor Jevons is in error in regarding each of these alternatives by itself as a contradictory of the original proposition. The true logical contradictory is the affirma- tion of the truth of one or other of these alternatives. If the original complex proposition is false we certainly know that the new complex proposition limiting us to such alternatives istrue.Thepointat issue may be further illustrated by taking the proposition in question in a symbolic form. All S is all jP is a complex proposition, resolvable into the form, All S is Fy and all P is S. In my view, it has but one contradictory, namely, Either some S is not P, or some P is not S.^ If either of these alternatives holds good, the original statement must in its entirety be false ; and on the other hand, if the latter is false, one at least of these alternatives must be true. Professor Jevons speaks as if Some S is not P were by itself a contradictory of All S is all P. But it is merely inconsistent with it. They may both be false. No doubt in ordinary speech contradictory frequently implies no more than "inconsistent with," and if Professor Jevons means that we should also use the term contradictory in this sense in Logic, the question becomes a verbal one. But he means more than this; he seems to mean that in some cases we can find no proposition that must be true when a given proposition is false. And here I hold that he is rong. ^ The contradictory of " AU .S* is all i"* may also be expressed ** S and P are not coextensive." CHAP. II.] PROPOSITIONS, ^-j If the original proposition is complex, its contradictory will in general be complex too, and possibly still more complex ; but that might naturally be expected. Compare the two preceding sections, where several cases are worked out in detail. The above will I think indicate how misleading is Professor Jevons's further statement, — "It will be shewn in a subsequent chapter that a proposition of moderate complexity has an almost unlimited number of contradic- tory propositions, which are more or less in conflict with the original. The truth of any one or more of these con- tradictories establishes the falsity of the original, but the falsity of the original does not establish the truth of any one or more of its contradictories." No doubt a complex proposition may yield an indefinite number of other propo- sitions the truth of any one of which is inconsistent with its own. But it has only one logical contradictory^ which con- tradictory as suggested above is likely to be a still more complex proposition affirming a number of alternatives one or other of which must hold if the original proposition is "With the point here raised Professor Jevons mixes up another, with regard to which his view is almost more mis- leading. He says, "But the question arises whether there is not confusion of ideas in the usual treatment of this ancient doctrine of opposition, and whether a contradictory of a proposition is not any proposition which involves the falsity of the original, but is not the sole condition of it I apprehend that any assertion is false which is made with- out sufficient grounds. It is false to assert that the hidden side of the moon is covered with mountains, not because we can prove the contradictory, but because we know that the assertor must have made the assertion without evidence. PROPOSITIONS. [PART II. person ignorant of mathematics were to assert that nvolutes are transcendeotal curves,' he would be making se assertion, because, whether they are so or not, he lot know it." Surely in Logic we cannot regard the 1 or falsity of a proposition as depending upon the dedge of the person who affirms it, so that the same osition would now be true, now false. The question lat is truth?" may be an enormously difficult one to 'er absolutely, and I need not say that I shall not npt to deal with it here ; but unless we are allowed to eed from the falsity of "All S is J"' to the truth of [ne ■$ is not P," I do not think we can go far in Logic. 13. Analyse all that Is implied in the assertion le falsity of each of the following propositions : — ■ [l) Roger Bacon was a giant. {2) Descartes died efore Newton was born. (3) Bare assertion is not necessarily the naked (4) All kinds of grasses except one or two species are not poisonous. [Jevons, Studies, p. 124.] 64 Assign precisely the meaning of the assertion that it is false to say that some English soldiers did not behave discreditably in South Africa. [l.] 66. Examine in the case of each of the follow- ing propositions the precise meaning of the assertion that the proposition is false : — (i) Some electricity is generated by friction. CHAP. II.] PROPOSITIONS. 69 (ii) Oxygen and nitrogen are constituents of the air we breathe. (iii) If a straight line falling upon two other straight lines make the alternate angles equal to eachother, these two straight lines shall be parallel. (iv) Actions are either good, bad, or indifferent. /^ tii?/^ ^'^- ^^/ */;^.f>/*nw^ h ^^pf^yy/)r*^ji i'?f iiss^y y/j^j:i^ ^/Vm' W///;/'^-44 f^^,'A« '/< V^^ i/Ht^^^/n nay \^ i^n^A i^ f//;y/i)f^f *^«i/J M M»*/ }/*: <.^^^yj^v.j^/^ fr

tt^% UM4^,iijiUmf p* f^4) 4^^f^*t4ff9 Owr«r*k« ihu^. PROPOSITIONS.Thus^ given a proposition having S for its subjectand P for its predicate, we seek to obtain by immediate inference anew proposition having P for its subject and S for its predicate; and applying this rule to the four fundamental forms of proposition, we get thefollowing table : — Original Proposition* Converse. mS'i&F. A. ^omtPisS. I. Some S is P. I. Some -Pis5. I. No S is P. E. No P is S. E. Some S is not P. 0. (None.) 67. Simple Conversion, and Conversion per ac- cidens. It will be observed that in the case of I and E, the converse is of exactly the same form as the original pro- position (or convertend) ; we do not lose any part of the information given us by the convertend, and we can pass back to it by re-conversion of the converse. The convertend and its converse are equivalent propositions. The con- veifeion in both these cases is said to be simple. In the case of A, it is different; although we start with a universal proposition, we obtain by conversion a particular one only, and by no means of operating upon the converse can we regain the original proposition. The convertend and its converse are not equivalent propositions. This is called conversion /^r accidens^, or conversion by limitation, ^ The conversion of A is said by Mansel to be caUed conversion 72 PROPOSITIONS. [PART II. 68. Particular negfative propositions do not admit of ordinary conversion. It is clear that if the converse is to be a legitimate formal inference from the originalproposition (or convert- end), it must distribute no term that was not distributed in the convertend From this it follows immediately that Some S is not P does not admit of ordinary conversion ; for S which is undistributed in the convertend would be- come the predicate of a negative proposition in the converse, and would therefore be distributed. (I may remind the reader that in what I have called ordinary conversion, with which alone we are now dealing, we do not admit the ontradictory of either the original subject or the original predicate as one of the terms of our converse.) I cannot understand why Professor Jevons should say that the fact that the particular negative proposition is in- capable of ordinary conversion " constitutes a blot in the ancient logic" (Studies in Dedtcctive Logic^ p. 37). We shall find subsequently that just as much can be inferred from the particular negative as from the particular affirmative, (since the latter unlike the former does not admit of con- traposition). Less can be inferred from either of them than can be inferred from the corresponding universal proposition, and this is obviously because the latter gives all the informa- per accidens " because it is not a conversion of the universal per se, but by reason of its containing the particular. For the proposition * Some B is A* is primarily the converse of* Some ^ is ^,' secondarily of * All A is B^" (Mansel's Aldrich, p. 61). Professor Baynes seems to deny that this is the correct explanation of the use of the term {New Analytic of Logical Forms, p. 29) ; but however this may be, I do not think that we can really regard the converse of ^ as obtained through its subaltern. We proceed directly from **A11 A is B" to ** Some Bis A" without the intervention of ** Some A is -ff." CHAP. HI.] PROPOSITIONS. Jz tion given by the particular proposition and more beside. No logic, symbolic or other, can actually obtain more from the given information than the ancient logic does. 69. Give the converse of the following pro- positions : — (i) A stitch in time saves nine. (2) None but the brave deserve the fair. (3) He can*t be wrong whose life is in the right. (4) The virtuous alone are happy. No difficulty can be found in converting or performing other immediate inferences upon any given proposition if it is once brought into logical form, its quantity and quality being determined, and its subject, copula and predicate being definitely distinguished from one another. If this rule is neglected, the most absurd results may be elicited. For example, amongst several curious converses of the first of the above propositions I have had seriously given, — Nine stitches save a stitch in time. Here it is of course entirely overlooked that "save'' cannot be a logical copula. The proposition may be written, All stitches in time I are | things that save nine stitches. This being an A proposition is only convertible per aaidens^ thus, Some things that save nine stitches are stitches in time. The following is wrong, — ^The means of saving nine stitches is a stitch in time ; since there may be other ways of saving " None but the brave deserve the fair." For the converse of this I have had, — The fair deserve none but the brave; and, again, No one ugly deserves the brave. Logically the proposition may be written, No one who is not brave is deserving of the fair. This, being an E proposition, may 74 PROPOSITIONS. [PARTII. be converted simply, giving, No one deserving of the fair is not brave. ** He can't be wrong whose life is in the right" Written in strict logical form, this proposition becomes, — No one whose life is in the right is able to be in the wrong ; and therefore its converse is, — No one who is able to be in the wrong is one whose life is in the right This proposition may now be written in the more natural but not strictly logical form. His life cannot be in the right who can him- self be wrong. " The virtuous alone are happy." In logical form this may be written either, No one who is not virtuous is happy, or All who are happy arevirtuous.Taking it in the first form, the converse is — No one who is happy is not virtuous ; and from this we may again get the second form by changing its quality* — All who are happy are virtuous. The converse of this is, — Some who are virtuous are happy. 70. State in logical form and convert the follow- ing propositions : — (i) There's not a joy the world can give like that it takes away. (2) He jests at scars who never felt a wound. (3) Axioms are self-evident. (4) Natives alone can stand the climate of Africa. (5) Not one of the Greeks at Thermopylae escaped. (6) All that glitters is not gold. [c] 71. Give all the logical opposites of the pro- position : — Some rich men are virtuous ; and also the * Cf. section 73. CHAP. III.] PROPOSITIONS. 75 converse of the contrary of its contradictory. How is the latter directly related to the given proposition } 72. Point out any possible ambiguities in the following propositions, and shew the importance of clearing up such ambiguities for logical purposes : — (i) Some of the candidates have been successful. (ii) Either some gross deception was practised or the doctrine of spiritualism is true. (iii) All are not happy that seem so. (iv) All the fish weighed five pounds. Give the contradictory and (where possible) the converse of each of these propositions. CHAPTER IV. THE OBVERSION AND CONTRAPOSITION OF PROPOSITIONS. 73. The Obversion of Propositions, Obversion is the process of changing the quality of a pro- position without altering its meaning. This change of quality may always be made if at the same time 7ve substitute for the predicate its contradictory. Applying this rule, we have the following table : — Original Proposition, Obverse, All 6" is ^. A. No S is not-^. E. Some S is P, I. Some S is not not--P. 0. No S is P, E. All S is not-^. A. Some S is not P, 0. Some S is not-/*. I. The term Obversion is used by Professor Bain, and it is a convenient one. The process is also called Permutation (Fowler), Aequipollence (Ueberweg), Infinitaiion (Bo wen). Immediate Inference by Privative Conception (Jevons), Contra- version (De Morgan), Contraposition (Spalding). CHAP. IV,] PROPOSITIONS. 'J^ Obversion depends on the supposition that two negatives make an affirmative. De Morgan (Formal Logic, pp. 3, 4) points out that in ordinary speech this is not always strictly true. For example, **not unable" is scarcely used as strictly equivalent to "able," but is understoodto imply a some- what lower degree of ability. " John is able to translate Virgil" is taken to mean that he can translate it well; "Thomas is not unable to translate Virgil" is taken to mean that he can translate it —indifferently. This distinc- tion, however, depends agood deal on the accentuation of the sentence; and it is not one of which Logic can take account. Logically, "-<4 -="" -p="" 0.="" 32="" 38="" 5="" 6-="" 61="" 6="" 74.="" 76.="" 77.="" 78.="" 78="" 79.="" 79="" 8.="" 80.="" 82.="" 82="" 83.="" 83="" 8l="" :-="" :="" a.="" a="" above="" abp.="" according="" accurate="" acquire="" admit="" agreeable.="" agreeable="" aldrich="" all="" allanimals="" alone="" also.="" also="" altered="" always="" an-="" an="" and="" angles="" animals="" annth="" another="" answered="" antecedent.="" antecedent="" any="" anything="" appeal="" appear="" appears="" application="" applying="" apuleius="" are="" aristotle="" arrive="" arrived="" as="" associations="" at.="" at="" attached="" attempt="" b="" bad.="" bain="" base="" be="" becomes="" been="" before="" between="" boethius="" book="" both="" brave="" briefly="" but="" by="" c="" caiegorico="" call="" called="" can="" capella="" carbon.="" carbon="" careful="" case="" cases="" cate="" categorical="" chap.="" cisdy="" ckd="" clusions="" cold="" com-="" con-="" consequent="" consequently="" contain="" containing="" contra-="" contradictory="" contraposition.="" contraposition="" contrapositionthe="" contrapositive.="" contrapositive="" contrapositives="" conver-="" converse="" conversion="" convert="" correspond="" de="" deduce="" defined="" definition="" denial="" denoting="" depending="" describe="" deserve="" dicate="" difference="" difficulties.="" difficulty="" diflference="" dis-="" discover="" discussed="" distinguished="" do="" doctrines="" does="" done="" doubt="" e.="" e="" each="" easy="" educated="" either="" elementary="" employ="" employed="" equal.="" equally="" equivalent.="" establish="" euclid="" every="" evil="" examination="" example="" examples="" expanded.="" explain="" explained="" expression="" f="" failed="" fair.="" feed.="" feed="" ferences="" first="" follow="" followed="" following="" follows="" for="" force="" form="" formal="" forms="" found="" from="" gations="" generalise="" geometrical="" gets="" give="" given="" gives="" good.="" good="" granted="" happy.="" has="" have="" having="" he="" helpful="" here="" higher="" his="" hispanus.="" hiswritings:="" how="" however="" hypothetical="" i.="" i.some="" i="" identity="" if="" igno-="" ignorant="" ii.="" ii="" iii="" immediate="" immediately="" in-="" in="" inclined="" independent="" infer="" inference="" inferences="" inferred="" inferrible="" inorganic="" into="" investi-="" investigation="" is="" isnotp="Some" isosceles="" it="" its="" iv.="" iv="" jevons="" justified="" k.="" kind="" knowledge="" l.="" left="" legitimate="" lessons="" life="" likely="" ll="" logic.="" logic="" logical="" logically="" logicians="" losing="" maintain="" make="" makes="" making="" mansel="" mate-="" material="" matter="" may="" meaning="" means="" meant="" merely="" might="" more="" morgan:="" most="" must="" n="" name.="" nature="" need="" negation.="" negation="" negative="" negatives="" never="" new="" news.="" news="" nine.="" no="" noi-a="" noi-b="" noi-p="" none="" not--="" not--p="" not--y="" not-5.="" not-5="Some" not-5by="" not-="" not-b="" not-i="" not-p="All" not-pby="" not-s.="" not="" note="" notice="" o="" observed="" obtain="" obtainable.="" obtained.="" obtained="" obtaining="" obverse="" obversion.="" obversion="" obvert="" obverted="" of="" ofthe="" ogic="" ognise.="" old="" ome="" on="" one.="" one="" only="" onversion="" open="" or="" orators="" order="" ordinary="" organic="" original="" originally="" other="" others="" otherwise.="" our="" out="" p-="" p.="" p="" pall="" part="" particular="" peace="" petnis="" pis="" plants="" plicated="" point="" pointed="" position.="" position="" positions="" practice="" pre-="" precbe="" preceding="" predi-="" predicate.="" predicate="" presently="" previous="" principle="" pro-="" problem="" proceeding="" process="" productive="" professor="" proof="" proposi-="" proposition.="" proposition:="" proposition="" propositions.="" propositions.in="" propositions:="" propositions="" purposes.="" quality="" question="" quite="" quoted="" r="" rance="" re-="" reducing="" reference="" regard="" regarded="" related="" relations="" remain="" remembered="" require="" respectively="" rest="" result.="" results="" rial="" right.="" riginal="" rule="" s="Some" same="" satisfactorily="" saves="" says="" section="" seek="" separable.="" shall="" shew="" shewn="" should="" simpler="" simplest="" simply="" sion="" solved="" some="" sometimes="" special="" state="" statesmen.="" stitch="" strictly="" subject.="" subject.the="" subject="" substances="" succeed="" such="" sum="" syllogisms.="" syllogistno="" symbols="" table="" taken="" taking="" tbe="" temperature="" text-books="" that="" the="" thefollowing="" their="" them.="" then="" theproposition="" there="" therefore="" these="" they="" think="" this="" those="" though="" thus="" time="" tion="" tions="" tjun="" to="" top.="" transform="" transformation="" traposition="" trapositive="" treated="" triangle="" uaisb="" unchanged.="" unchanged="" uneducated="" up="" us="" usage="" use="" uses="" utility="" v.="" verbal="" vert="" very="" virtuous="" volcanoes="" wallis="" war="" warmth="" way="" we="" well="" what="" whately="" whatever="" when="" whether="" which="" who="" whose="" will="" with="" without="" would="" wrong="" y4="" yield="" yllabus="" z="">, A is not B, E. In some cases in which AisB, C is £>. I. In some cases in which Cis£>, A is B, L None. If A is B, C is not Z?. E. If C is n, A is not . E. In some c^ses in which C is not Z>, A is B, I. In some cases in which ^is^, CisnotZ>. 0. None. In some cases in which C is not A ^ is ^. I. 6—2 84 PROPOSITIONS. [part ii. It must be remembered that we regard the quality of a hypothetical proposition as determined by the quality of the consequent. The obverse of a hypothetical proposition is usually awkward to express. We may however find it if required ; e,g.y the obverse of " If A is B, Cis Z> " is "If A is B, C is not not-Z>." 84. Give the converse and the contrapositive of " If a straight line falling upon two other straight lines make the alternate angles equal to one another, these two straight lines shall be parallel.'* [l.] The application of the doctrines of Conversion and Contraposition to Hypothetical Propositions may be illus- trated by means of the above proposition. We must note carefully that it is a universal affirmative, and is therefore only convertible per accidens. This is a point particularly liable to be overlooked where a universal converse can be legitimately inferred (as in the case of the above proposition), though not as an immediate inference. We are in no danger of saying, All men are animals, therefore, all animals are men; but we may be in danger of saying, All equilateral triangles are equiangular, therefore, all equiangular triangles are equilateral. From the point of view however of Formal Logic the latter inference is as erroneous as the former. So far as thgiven proposition is concerned, we have — Converse^ In some cases in which two straight lines are parallel, a straight line falling upon them sKall make the alternate angles equal to one another. Contrapositive^ If two straight lines are not parallel, then a straight line falling upon them shall make the alternate angles not equal to one another. CHAP. IV.] PROPOSITIONS. 85 86. Give the contradictory, the contrary, the converse, and the contrapositive of the following propositions : (i) Things equal to the same thing are equal to oneanother. (2) No one is a hero to his valet. (3) If there is no rain the harvest is never good. (4) None think the great unhappy but the great. (5) Fain would I climb but that I fear to fall. 86. Name the form of each of the following propositions ; and, where possible, give the converse and the contrapositive of each : — (i) Some death is better than some life. {ii) The candidates in each class are not arrangedin order of merit. (iii) Honesty is the best policy.(iv) Not all that tempts your wandering eyes And heedless hearts is lawful prize. (v) If an import duty is a means ofrevenue, it does not afford protection. (vi) Great is Diana of the Ephesians. (vii) All these claims upon my time overpower me. CHAPTER V. THE INVERSION OF PROPOSITIONS. 87. In what cases can we obtain by immediate inference from a given proposition a new proposition having the contradictory of the original subject for its subject, and the original predicate for its predi- A new form of immediate inference is here indicated, by which given a proposition having ^S" for its subject and P for its predicate, we seek to obtain a new proposition having not-^S" for its subject and P for its predicate. If such a proposition can be obtained at all, it will be by a certain combination of the elementary processes of ordinary conversion and obversion. We will take each of the fundamental forms of proposition and see what can be obtained (i) by first converting it, and then performing alternately the operations of obversion and conversion; (2) by first obverting it, and then performing alternately the operations of conversion and obversion. We shall find that in each case we can go on till we reach a particular negative proposition whose turn it is to be converted. (i) The results of performing the processes of con- version and obversion alternately, commencing with the former, are as follows : — CHAP, v.] PROPOSITIONS. 87 ' (i) AWSisP, therefore (by conversion), Some P is ^S", therefore (by obversibn), Some P is not not-S. Here comes the turn for conversion;but we have an O proposition, and can therefore proceed no further. (ii) Some S is F, therefore (by conversion), Some P is 5, therefore (by ob version), Some Pis not not-.S; and we can get no further, (iii) No S is P, therefore (by conversion), No P is 5, herefore (by obversion), All P is not-5, therefore (by conversion). Some not-S is /*, therefore (by obversion), Some not-»S is not not-P. In this case the proposition in italics is the immediate inference that was sought. (iv) Some ^S" is not P, In this case we are not able even to commence our series of operations. (2) The results of performing the processes of Con- version and obversion alternately, commencing with the latter^ are as follows : — (i) All 5 is P, therefore (by obversion), No S is not-P, therefore (by conversion), No not-/* is S^ therefore (by obversion), All not-P is not-5, therefore (by conversion). Some not-^ is not-P^ therefore (by obversion). Some notS is not P, Here again we have obtained the desired form. (ii) Some 6* is P, therefore (by obversion), Some S is not not-jP. SS PROPOSITIONS. [part ii. (iii) TSioSisP, therefore (by obversion), All S is not-P, therefore (by conversion), Some not-jP is S, therefore (by obversion), Some not-P is not not-5. (iv) Some iS is not i', therefore (by obversion), Some S is not-jP, therefore (by conversion), Some not-P is S, therefore(byobversion). Some not-/* is not not-5. We can now answer the question with which we com- menced this enquiry. The required proposition can be obtained only if the given proposition is universal; we then have, according as it is affirmative or negative, — All S is Pp therefore. Some not-iS is not P; No S is P, therefore. Some not-iS is P\ It must be observed that in the case of the former of these we commenced with obversion in order to get the new form, in the latter we commenced with conversion. This form of immediate inference has been more or less casually recognised by various logicians; but I do not remember that it has ever received any distinctive name. Sometimes it has been vaguely classed under contraposition, (compare Jevons, Elementary Lessons in Logic, pp. 185, 6), but it is really as far removed from the process to which that designation has been given as the latter is from ordinary conversion. I venture to suggest the terms Inversion and Inverse*. Thus, Inversion is a process of immediate inference 1 For assumptions respecting "existence" involved in these in- ferences, see chapter 8. * Professor Jevons (carrying out a suggestion of Professor Robert- son's) has introduced the term Inverse in a different sense. I do not however think that for logical purposes we want any new term in the sense in which he uses it ; and I havebeenunable to think of any other equally suitable term for my own purpose, for which a new term really CHAP, v.] PROPOSITIONS. 89 in which from a given proposition we infer another proposition having the contradictory of the original subject for its subject, is needed, if the scheme of immediate inferences by means of conver- sion and obversion is to be made scientifically complete. The term contraverse has occurred to me, but I do not like it so well ; and this again has been appropriated by De Morgan in another sense. Professor Jevons'snomenclature is explained in the following passage from his Studies in Deductive Logic, p. 32 : — *' It appears to be indis- pensable to endeavour to introduce some fixed nomenclature for the relations of propositions involving two terms. Professor Alexander Bain has already made an innovation by using the term obverse, and Professor Hirst, Professor Henrici and other reformers of the teaching of geometry have begun to use the terms converse and obverse in meanings inconsistent with those attached to them in logical science {Mind, 1876, p. 147). It seems needful, therefore, to state in the most explicit way the nomenclature here proposed to be adopted with the concurrence of ProfessoV Robertson. Taking as the original proposition * All A are B,* the following are what we may call the related propositions — Inferrible, Converse, Some B are A, Obverse, No A are not B, Contrapositive, No not B are A, or, all not B are not A, Non-Inferrible. Inverse, All B are A. Reciprocal. All not A are not B, It must be observed that the converse, obverse, and contrapositive are all true if the original proposition is true. The same is not neces- sarily the case with the inverse and reciprocal. These latter two names are adopted from the excellent work of Delboeuf, ProUgomhus Fhilo- sophiques de la GSomitrie, pp. 88 — 91, at the suggestion of Professor Croom Robertson {Mind, 1876, p. 425)." In this scheme what I propose to call the Inverse is not recognised at all. On the other hand, I hardly see why the non-inferrible forms need such a distinct logical recognition as is implied by giving them distinct names ; while except in books on Logic I anticip>ate that the term converse is likely still to be used in its non-logical sense, {i,e,, "All B are ^ " is likely still to be spoken of as the converse of ** All A are and the original predicate for its predicate. In other words, given a proposition having S for subject and P for predicate, we obtain by inversion a new proposition having not-S for subject and P for predicate. We may now sum up the results that have been obtained with regard to immediate inferences. Given two terms iS and Py and admitting their contradictories not-^S" and not-/*, we have eight possible forms of proposition as shewn in the following scheme: — B "). It may be noted that in Jevons's use of terms, the inverse would be the same as the converse in the case of E and I propositions. I imagine also that in consistency there should be yet another term to express the relation of "No not-^ is not-^" or "All not-^ is A^' to "No A is -5"; it is, in the sense in which Jevons uses these terms, neither the Converse, Obverse, Contrapositive, Inverse nor Reciprocal. CHAP, v.] PROPOSITIONS. 91 These propositions may be designated respectively: — (i) The original proposition, (ii) The obverse, (iii) The converse, (iv) The obverted converse, (v) The contrapositive, (vi) The obverted contrapositive, (vii) The inverse, (viii) The obverted inverse. It has been shewn, in sections 66^ 73, 76, and in the above, that if the original proposition is universal, we can infer from it propositions of all the remaining seven forms ; but if it is particular, we can infer only three others. Working out the different cases in detail we have : — A. (i) Original proposition, A/l S is P, (ii) Obverse, M? S is not-P, (iii) Converse, Some P is S, (iv) Obverted converse. Some P is not noi-S. (v) Contrapositive, No not-P is S, (vi) Obverted Contrapositive, All not-P is n^f-S. (vii) Inverse, Some not-S is not P, (viii) Obverted Inverse, Some not-S is not-P, I. (i) Original proposition, Some S is P. (ii) Obverse, Some S is not not-P. (iii) Converse, Some P is S, (iv) Obverted Converse, Some P is not not-S (v) Contrapositive, none can be inferred, (vi) Obverted Contrapositive, none, (vii) Inverse, none, (viii) Obverted Inverse, none. E. (i) Original proposition, JVo S is P, (ii) Obverse, All S is not-P. PROPOSITIONS. (in) Converse, No P is S, (iv) Obverted Converse, All P is not-S. (v) Contrapositive, Some not-P is S. (vi) Obverted Contrapositive, Some not-P is not not-S. (vii) Inverse, Some not-S is P. (viii) Obverted Inverse, Some not-S is not not-P, O. (i) Original proposition, Some S is not P, (ii) Obverse, Some S is not-P. iii) Converse, none can be inferred, (iv) Obverted Converse, none, (v) Contapositive, Some not-P is S. (vi) Obverted Contrapositive, Some not-P is not not-S. (vii) Inverse, none, (viii) Obverted Inverse, none. All the above is summed up in the following Table (using the symbols described in section 38, and denoting not-S by S\ not-P by P'):— Original PropositionConverse Obverted Converse Contrapositive Obverted Contrapositive . . . Obverted Inverse CHAP, v.] PROPOSITIONS. 93 It is worth noticingthatwe can infer the same number of propositions from E as from A (7), from O as from I (3), and the same number of universal propositions from E as from A (3);also in two cases we can get no more from A than from I, and no more from E than from O. 88. Give the inverse of the following proposi- t(i) A stitch in time saves nine. (2) None but the brave deser\'e the fair. (3) He can't be wrong whose life is in the right. (4) The virtuous alone are happy. 89. Assuming that no organic beings are devoid of Carbon, what can we thence infer respectively about beings which are not organic, and things which are not devoid of carbon ? [L.] 90. Make as many Immediate Inferences as you can from the following propositions : — (i) Civilization and Christianity are coextensive. (2) Uneasy lies the head that wears a crown, (3) Your money or your life ! [l.] 91. Write out all the propositions that must be true, and all that must be false, if we grant that (a) A straight line is the shortest distance be- tween two points ; (j8) All the angles of a triangle are equal to two right angles ; (7) Not all the great are happy. [c.] 94 PROPOSITIONS. [part ii. 92. De Morgan says (Fourth Memoir on the Syllogism^ p. 5) of the Laws of Thought : " Every transgression of these laws is an invalid inference ; every valid inference is not a transgression of these laws. But I cannot admit that everything which is not a transgression of these laws is a valid inference.*' Investigate the logical relations between these three assertions. [Jevons, Studies^ p. 301.] 93. Assign the logical relation, if any, between each pair of the following propositions : — (i) All crystals are solids. (2) Some solids are not crystals. (3) Some not crystals are not solids. (4) No crystals are not solids. (5) Some solids are crystals. (6) Some not solids are not crystals. (7) All solids are crystals. [l.] 94. . " All that love virtue love angling." Arrange the following propositions in the four following groups : — (a) Those which can be inferred from the above proposition ; (/8) Those from which it can be inferred ; (7) Those which do not contradict it, but which cannot be inferred from it ; (S) Those which contradict it. CHAP, v.] PROPOSITIONS. 95 (i) None that love not virtue love angling. (li) All that love angling love virtue, (iii) All that love not angling love virtue, (iv) None that love not angling love virtue. (v) Some that love not virtue love angling, (vi) Some that love not virtue love not angling, (vii) Some that love not angling love virtue, (viii) Some that love not angling love not virtue. ^ CHAPTER VI. THE DIAGRAMMATIC REPRESENTATION OF PROPOSITIONS. 96. Methods of illustrating the ordinary processes of Formal Logic by means of Diagrams. Representing the individuals included in any class, or denoted by any name, by a circle, it will be obvious that the five following diagrams represent all possible relations between any two classes : — The force of the different prepositional forms is to ex- clude one or more of these possibilities*. ^ The method of interpreting a proposition by what it excludes or negatives is discussed in more detail in chapter VIII. CHAP. VI.] PROPOSITIONS, All S is P limits us to ( «/» j orSome Sis P to one of the four NoSis P to Some Sis Mi P to one of the three To represent All ^S is i* by a single diagram, thus 98 PROPOSITIONS. [part il. or Some 5 is i' by a single diagram, thus is most misleading; since in each case the proposition really leaves us with other alternatives. This method of employ- ing the diagrams is however adopted by most logicians who have used them, including Sir "William Hamilton {Logic, i. p. 255), and Professor Jevons {Elementary Lessons in Logic, pp. 72 — 75); and the attempt at such simplification has brought their use into undeserved disrepute. Thus, Mr Venn remarks, "The common practice, adopted in so many manuals, of appealing to these diagrams, — Eulerian diagrams as they are often called, — seems to me very questionable. The old four propositions A, E, I, O, do not exactly corre- spond to the five diagrams, and consequently none of the moods in the syllogism can in strict propriety be represented by these diagrams V {Symbolic Logic, p. 15, compare also pp. 424, 425). This is undoubtedly sound as against the use of Euler's circles by Hamilton and Jevons; but I do not admit its force as against their use in the manner described above \ Many of the operations of Formal Logic can be satisfactorily illustrated by their aid; though it is true that they become somewhat cumbrous in relation to the Syllogism. Thus, they may be employed, — (i) To illustrate the distri- bution of the predicate in a proposition. In the case of each of the four fundamental propositions we may shade the part of the predicate concerning which knowledge is given us. We then have, — ^ They are used correctly by Ueberweg. Cf. Lindsay's translation of Ueberweg's System of Logic ^ pp. a 16 — 218. CHAP. VI.] PROPOSITIONS. The result is that with A and I there are cases in which only part of jP is shaded; whereas with E and O, the whole of F is in every case shaded; and it is made clear that negative propositions distribute, while affirmative proposi- tions do not distribute theirpredicates.(2) To illustrate the Opposition of Propositions, Com- paring two contradictory propositions, e,g,y A and O, we see hat they have no case in common, but that between them they exhaust all possible cases. Hence the truth, that two contradictory propositions cannot be true together but that one of them must be true, is brought home to us under a new aspect. Again, comparing two subaltern propositions, e.g., A and I, we notice that the former gives us all the information given by the latter and something more, since it still further limits the possibilities. To make this point the more clear the following table is appended:-— CHAP. VL] PROPOSITIONS. (3) To illustrate the Conversion of Propositions. Thus, it is made quite clear how it is that A admits only of Con- version per accidens. All S is P limits us to one or other of the following The problem of Conversion is — ^What do we know of P in either case? In the first, we have All P is 5, but in the second Some P is *S; /.^., taking the cases indifferently, we have Some P\% S and nothing more. Again, it is made clear how it is that O is inconvertible. Some S is not P limits us to one or other of the following, — What then do we know concerning P} The three cases give us respectively (i) No T' is 5, (ii) Some P is S, and Some P is not 5, (iii) KWPisS. (i) and (iii) are contraries, and (ii) is contradictory to both of them. Hence nothing can be affirmed of P that is true in all three cases indifferently. (4) To illustrate the more complicated forms of imme- diate inference. Taking, for example, the proposition All S is Py we may ask, What does this enable us to assert about 102 PROPOSITIONS. [PARTii: not-/* and not-^S* respectively? We have one or other of these casesWith regard to not-/*, these yield respectively,   We may also prove the truth of the contrapositive (of the proposition All X is Y) indirectly ; for what is not-Y must be either X or not--^; but if it be X it is by the premiss also y, so that the same thing would be at the same time not-F and also F", which is impossible. It follows that we must' affirm of not-F the other alternative, not-X" All the above are interesting, as illustrating the nature of immediate inferences ; but regarded as proofs they labour under the disadvantage of deducing the less complex by means of the more complex. I hardly know what is to be said in favour of the follow- (4) Wolf obtains the subaltern of a universal proposition by a syllogism in Darii^ ^ This is itself an inference by Opposition. 113 PROPOSITIONS. [PARxn. Given AilSisF, we have also Some S is S, by the law of Identity, therefore, Some S is P. (Compare, Mansel, Prolegomena Logica^ p. 217.) (5) "Still more absurd b the elaborate system which Krug, after a hint from Wolf, has constructed, in which all immediate inferences appear as hypothetical syllogisms; a major premiss being supplied in the form, * If all A is By some A is B.^ The author appears to have forgotten, that either this premiss is an additional empirical truth, in which case the immediate reasoning is not a logical process at all ; or it is a formal inference, presupposing the very reasoning to which it is prefixed, and thus begging the whole question" (Mansel, Prolegomena Logical p. 217). 101. How far can the legitimacy of the various processes of Immediate Inference be immediately deduced from the laws of Identity, Contradiction and Excluded Middle? Law of Identity, — A is A. Law of Contradiction, — A is not not--^. Law of Excluded Middle, — A is either B or not--^. We may consider the application of these laws to (i) inferences based on the square of opposition ; (2) obversion ; (3) conversion. (i) The inferences based on the square of opposition may be considered to depend exclusively on the above laws of thought For example, from the truth of All 5 is P we may infer the truth of Some S \% P \rj the law of Identity, and the falsity of some S is not P by the law of Contradiction ; from the falsity of All S is P vre may infer CHAP, viij PROPOSITiaNS. 113 the tnith of some S is not P by the law of Excluded Middle. (2) Obversion also may be based entirely on the laws of Contradiction and Excluded Middle. From All S is P we get No S is not-P by the law of Contradiction; and from No 5 is i' we get All S is not-P by the law of Ex- cluded Middle. (3) The case of Conversion is different ; and I do not see how this process can be based exclusively on these three laws of thought. Mansel holds that it can, but so far as I am able to discover he makes no attempt to establish his position in detail. How, for example, would the application of the three laws of thought prove our iuability to convert an O proposition ? De Morgan appears to me to be perfectly justified in saying,— " When any Writer attempts to shew /low the perception of convertibility ^AisB gives BisA* follows from the principles of identity, difference and excluded middle, I shall be able to judge of the process : as it is, I find that others do not go beyond the simple assertion, and that I myself can detect the petitio prindpii in every one of my own attempts" (Syllabus^ p. 47). The following attempt may be taken as a specimen :— " All A is B^ therefore, some B is A\ for if no B were A^ then A would be both B and not B^ which is impossible.'* It is clear however that conversion is already assumedin this reasoning. If Conversion cannot be based exclusively on the three Laws ofThought, it follows that Contraposition and Inver- sion cannot bebased exclusively on these Laws. 102. Proof of the various rules of Conversion. The question as to what proof should be given of the various rules of conversion has been partially discussed in the two preceding sections. In them we discussed attempts W4' PROPOSITIONS. (PARTMi.> to prove conversions (i) by meanslof syllogisms, (2) by means of thethree laws of identity, contradiction and ex- cluded middle. ■ Bain writes as follows,-— " When we examine carefully* the various processes in Logic, we find them to be material to the very core. Take Conversion, How do we know that, if No -AT is PJ No y is -Y? By examining cases in^ (Jetail, and finding the equivalence t6 be true. Obvious as the inference seems on the mere fonnal ground, we do not; content ourselves with the formal aspect. If we did, we sjiould be as likely to say, All JY is Y gives All Y is X\ Wfe are prevented from this leap merely by the examination ' of cases" {Logic, Deduction, p. 251). The implication here made that the proof of rules of conversion is a kind of inductive proof seems to me unwarranted. The justification of conversion that I should myself give is that in the case of each of the four fundamental ' forms of proposition, its conversion (or in the case of an O ^ proposition, the impossibility of converting it) is self-evident; and that we cannot go beyond this simple statement. Thus, taking an E proposition, I should say that it is self-evident . that if one class is entirely excluded from another class, this second class is entirely excluded from the first. In the case of an A proportion it is clear on reflection that the statement All S y& P may include^ one or other of the two relations of classes, — either S andPcoincident, or P con- - taining S and moire besides, — ^but that these are the only two possible relations to which it can be applied. It is self-evident that in each of these cases some P \% S\ and hence the inference by conversion from an A proposition is shewn to be justified \ In the case of an O proposition, ^ Compare section 95, where these inferences are illustrated by the aid of the Eulerian diagrams. CHAP, VII.] PROPOSITIONS. IIS if we consider all the relationships of classes in which it holds good, we find that nothing is true of P in terms of S in all of them. Hence O is inconvertible*, I may add that I do not see that in the above reasoning we should be assisted by any explicit reference to the three laws of thought; nor that the application of the three laws of thought alone would-be sufficient to give us our results. 103. Without assuming Conversion, how would you logically justify the process of Contraposition ? [c] ^ Again, compare section 95. CHAPTER VIIL PREDICATION AND "EXISTENCE*." 104. Are assuniptions with regard to *' existence " involved in any of the processes of immediate infer-» As pointed out by Mr Venn {Symbolic Logic^ pp. 127, 128), a discussion about "existence" need not in this con- nection involve us in any kind of metaphysical enquiry. " As to the nature of this existence, or what may really be meant by it, we have hardly any need to trouble ourselves, for almost any possible sense in which the logician can understand it will involve precisely the same difficulties and call for the same solution of them. We may leave it to any one to define the existence as he pleases, but when he has done this it will always be reasonable to enquire whether there is anything existing corresponding to the X or Y which constitute our subject and predicate. There can in fact be no fixed tests for this existence, for it will vary widely according to the nature of the subject-matter with which we are concerned in our reasonings^ For in- stance, we may happen to be speaking of ordinary pheno- menal existence, and at the time present; by the distinction ^ It may perhaps be advisable for students, on a first readings to omit this chaptert CAP. VIII.] PROPOSITIONS. ir; in question is then meant nothing more and nothing deeper than what is meant by saying that there are such things as antelopes and elephants in existence, but not such things as unicorns or mastodons. If again we are referring to the sum-total of all that is conceivable, whether real or imagi- nary, then we should mean what is meant by saying that everything must be regarded as existent which does not involve a contradiction in terms, and nothing which does. Or if we were concerned with Wonderland and its occu- pants we need not go deeper down than they do who tell us that March hares exist there. In other words, the inter- pretation of the distinction will vary very widely in different cases, and consequently the tests by which it would have in the last resort to be verified ; but it must always exist as a real distinction, and there is a sufficient identity of sense and application pervading its various significations to enable us to talk of it in common terms." Now, several views may be taken as to what implication with regard to existence, if any, is involved in any given proposition. (i) It may be held that every proposition implies the existence of its Subject, since there is no use in giving infor- mation with regard to a non-existent subject. (2) It may be held that although such existence is gene- rally implied, still it is not so necessarily ; and that at any rate in Formal Logic we ought to leave entirely on one side the question of the existence or the non-existence of the subjects of our propositions. (3) The view is taken by Mr Venn that for purposes of Symbolic Logic, universal propositions should not be regarded as implying the existence of their subjects, but \ki2X particular propositions should be regarded as doing so. This view might be extended to ordinary Formal Logic ^la PROPOSITIONS, [part II.Withoutat ohce deciding' which of these' views is to be pr:efeiTed, we may briefly investigate the consequences which follow from them respectively so far as immediate inferences, are concerned. Fir sty we may take the supposition that every proposition implies the existence of its subject. Thiis, All S\% P implies the existence of 5, and it follows that it also implies the existence of P. No Sv&P implies the existence of 5, and since by the law of excluded middle every S is either Pot not-/', it follows that it also implies the existence of noXrP. But now if from All 6" is /* we are to be allowed to obtain the ordinary immediate inferences, — if, for example, we may infer AH not-/' is not-*S, — the existence of not-/* and nbt-iSare also involved. Similarly, the conversion of No *S is P requires that we posit the existence of P and not-5. On this supposition, then, we find that propositions are not amenable to the ordinary logical operations^ except on the assumption of the existence of classes corresponding not merely to the terms directly involved but also to their contradictories, De Morgan practically adopts this alternative. "By the universe (of a proposition) is meant the collection of all objects which are contemplated as objects about which assertion or denial may take place. Let every name which belongs to the whole universe be excluded as needless: this must be particularly remembered. Let every object which has not the name X {of which there are always some) be conceived as therefore marked with the name x meaning not-^y" (Syllabus^ pp. 12, 13). Compare also Jevons, Pure Logic, pp. 64, 65; Studies in Deductive Logic, p. i8i. Secondly, we may take the supposition that no proposition logically implies the existence of its subject. On this view, the proposition All -S is /* may be read. All S, if there is any S, or, when there is any S, is P] and its full implication CHAP* VIII.] PROPOSITIONS. 119 with regard to existence may be expressed by saying that it denies the existence of. any thing that is at the same time S and not F. In Mr Venn's words, "/^ burden of implication of existence is shifted from the affirmative to the negative form'; that is, it is not the existence of the subject or the predicate (in affirmation) which is implied, but the non-existence of any subject which does not possess the predicate" {Symbolic Logic, p. 141). Similarly,onthis view, No S is F implies the existence neither of S nor of F, but merely denies the existence of anything that is both S and F. Some S is F (or is not F) may be read Some 5, if there is any 5, is F (or is not F). Here we do not even negative or deny the existence of any class absolutely; the sum total of what we affirm with regard to existence is that if any S exists, then some F (or not-F) also exists. Now having got rid of the implication of the existence of the subject in the case of all propositions, we might naturally suppose that in no case in which we make an immediate inference need we trouble ourselves with any question of " existence " at all. On further enquiry, however, we shall find that so far as particulars are obtained, assump- tions with regard to existence are still involved in somQ processes of immediate inference. All S is F2X any rate implies that if there is any S there is also some F, whilst on our present view it does not re- quire that if there is any F there is also some S, But the converse of the given proposition, — Some F is 5, — does imply this. ** If the predicate exists then also the subject exists " must therefore be regarded as an assumption which is involved in the conversion of an A proposition; similarly, in the conversion of an I, and in the contraposition of an £ or of an O proposition. It follows also that in passing from All ^ is /* to Some not-5 is not-/*, we have to assume 12b Propositions. [part h. that if there is any not-S there is also some not-P. It does not appear th^t there is any similar assumption in the conversion of an K proposition ; nor do I think that there is any in the obversion either of A or E, or in the contra- position of A. It might indeed at first sight seem that in passing from No iS is Z' to All S is not-P, we have to assume that if there is any S there is also some hot-P. " But, even on our present supposition, this is necessarily implied in the proposition No S is P itself. If there is any S it is by the law of excluded middle either P or not P; therefore, given that No S is P, it follows immediately that if there is any S there is some not-P. It can also be shewn that since No S is P denies the existence of anything that is both S and Py it implies by itself that if there is any P there is some not-^S; and that since the proposition All S is P denies the existence of anything that is both S and not- P,it implies by itself that if there is any not-P there is some The given supposition then provides for the obversion and contraposition of A, and for the obversion and con- version of E, without any further implication with regard to existence than is contained in the propositions them- selves. But the conversion or inversion of A involves further assumptions, as shewn above ; and the same is true of the contraposition or inversion of E, the conversion of I and the contraposition of O. Now it will be observed that in the first set of cases we obtain by our immediate inference a universal propo- sition ; in the second set a particular one. We may there- fore generalise our results as follows, — On the supposi- tion that no proposition logically implies the existence of its subject we do not require to make any assumption with regard to existence in any process of immediate inference provided that CHAP. VIII.] PROPOSITIONS. 121 // yidds a universal conclusion ; but it is generally other- wise in cases that yield only a particular conclusion. In other words, whenever we are left with a universal con- clusion we need not be afraid that any assumption with regard to existence has been introduced unawares; but whenever We are left with a particular conclusion such an assumption may have been made, and if we find that it is so, this should be explicitly stated TTiirdly, taking Mr Venn's view, which is the same as the preceding so far as universal propositions are concerned, but which regards particular propositions as implpng the existence of their subjects, the result just obtained hardly requires to be modified It must however be observed that on this supposition we cannot even pass from All 5 is F to Some S is Py except under the condition that the existence of S is granted. 106. Shew that in some processes of conversion assumptions as to the existence of classes in nature have to be made; and illustrate by examining whether any such assumptions are involved in the inference that if All 5 is P, therefore Some not-5 is not-P. Concerning this question, Professor Jevons remarks that it "must have been asked under some misapprehension. The inferences of Formal Logic have nothing whatever to do with real existence; that is, occurrence under the con- ditions of time and space *' {Studies in Deductive Logic, p. 55). The question is doubtless somewhat unguarded with regard to the nature of the existence implied, but I think that in any case the discussion in the preceding section shews that it does not admit of being so summarily dis- missed*. Even granting that the formal logician may say ^ What follows is to some extent a repetition of what has been I2a PROPOSITIONS* [PAKT ii> that given the proposition AH .S is F, it is "no concern of his whether or not there are any individuals actually belong- ing to the classes S and -P, nevertheless he must admit that the proposition at least involves that if therie are any ^S* there must be some P, while it does not involve that if there are any F there must be some S. But now convert the propo- sition. We obtain Some F is S, and this does involve that if there are any F there roust be some S, I do not there- fore see how in converting the given proposition this as- sumption can be avoided Thus, from **A11 dragons are serpents", we may infer by conversion **Some serpents are dragons,'' and this proposition implies that if there are; serpents there are also dragons. Similarly, in passing from All 5 is /^ to Some noi-S is not-/*, it must at least be assumed that if S does not constitute the entire universe of discourse, neither does F do so. If we make immediate inferences from hypothetical propositions, the necessity of a similar assumption seems still more obvious. For example, from the true statement that if Governor Musgrave's econo- mic doctrines are correct, Mr Mill makes mistakes in his Political Economy, we can hardly without qualification infer that in some cases in which Mr Mill makes mistakes in his Political Economy, Governor Musgrave's doctrines are cor- rect, since Mr Mill might be sometimes wrong, and never- theless Governor Musgrave always so. In another place {Studies in Deductive Logic, p. 141) Jevons remarks, " I do not see how there is in deductive logic any question about existence"; and with reference to the opposite view taken by De Morgan, he says, " This is One of the few points in which it is possible to suspect him given in the preceding section. The view that I am here especially combating however is that Formal Logic cannot possibly have any concern with questions relating to *' existence." CHAP. VIII.] PROPOSITIONS. 123 of iinsoundnisss." I can however attach no meanibg to Jevons's own " Criterion of Consistency '* {Studies in DeiiU- HveLogiCy p. 181) unless it has some reference to "existence," " It is assumed as a necessary law that every term mu5t have its negative. This was calledthe Law of Infinity in* my first logical essay {Pure Logic ^ P- 65; see also p. 45); but as pointed out by Mr A. J. Ellis, it is assumed by De Mor- gan, in his Syllabus^ Article 16. Thence arises what I pro- pose to call the Criterion of Consistency^ stated as follows : — Any two or more propositions are contrculictory 7vhen, and ofily when, after ail possible substitutions are made, they ouasion the total disappearance of any term, positive or negch live, from the LogiccU Alphabet J' What can this mean but that although we may deny the existence of the combination AB, we cannot without contradiction deny the existence of A itself, or not--^, or B, or not-.5? Indeed, in reference to Jevons's equational logic generally, what can negativing a combination mean but denying its existence ? For example, I take the following quite at random, — " There remain four combinations, ABC, aBC, abC, and abc* But these do not stand on the same logical footing, because if we were to remove ABC, there would be no such thing as A left; and if we were to remove abc there would be no such thing as c left. Now it is the Criterion or condition of logical con- sistency that every separate term and its negative shall remain. Hence there must exist some things which are described by ABC, and other things described by abc^^ {Studies in Deductive Logic, p. 216). With regard to Jevons's criterion of consistency itself, I am hardly prepared to admit it. If I am not allowed to negative X, why should I be allowed to negative AB"^ There is nothing to prevent X from being itself a complex term. In certain combinations indeed it may be convenient124 PROPOSITIONS. [PARTIL to substitute X for AB, or vice versa. It would appear then that what is contradictory when we use a certain set of symbols may not be contradictory when we use another set of symbols. I should say that Jevons's criterion is some- times a convenient assumption to make, but nothing more than this; and it is I think an assumption that should always be explicitly referred to when made. 106. Is a categorical proposition to be regarded as logically implying the existence of its subject } Our answer to this question must depend to some extent on popular usage, and to some extent on logical conveni- ence. So f?ir as universal propositions are concerned, I should be inclined on both grounds to answer it in the In the first place, I do not think that in ordinary speech we always imply the existence of the subjects of our pro- positions. No doubt we usually regard them as existing; but as Mr Venn shews there are undoubtedly exceptions to this rule. ** For instance, assertions about the future do not carry any such positive presumption with them, though the logician would commonly throw them into precisely the same * All X is y type of categorical assertion. * Those who pass this examination are lucky men ' would certainly be tacitly supplemented by the clause *if any such there be.' So too, in most circumstances of our ordinary life, wherever weare clearly talking of an ideal * Perfectly conscientious men think but little of law and rule,* has a sense without implying that there are any such men to be found^ " {Symbolic Logic^ pp. 130, 131). Again, a mathe- * Theaboveseems to me an answer to such a statement as the following: — "In an ordinary proposition the subject is necessarily admitted to exist, either in the real or in s6me imaginary world assumed HAP. VIII.] PROPOSITIONS. J25 matician might assert that a rectilinear figure having a mil- lion equal sides and inscribable in a circle has a million equal angles, without intending to imply the actua) exist- ence of such a figure ; or if I know that A is X, B is K,C is Z, I may affirm that ABC is XVZ without wishing to commit myself to the view that the combination ABC does ever really occur \ Taking complex subjects, and limiting our conception ofexistence as we not unfrequently do to some particular universe, cases of this kind might be multiplied indefinitely. But if it is granted that in ordinary thought the existence of the subject of the proposition sometimes is and some- times is not implied, it follows that since the logician cannotdiscriminate between these cases, he had best content him* self with leaving the question open, that is, he should regard such existence as not necessarily or logicallyimplied. And, further, to adopt this alternative is logically more convenient, since SiO far as the obtaining universal propo- sitions by immediate inference is concerned, we do not on this supposition require any further assumptions with regard to existence in order that such immediate inference may be legitimate. On the other hand, if we take the other alter- for the nonce When we say No stone is alive, or All men are mortal^ we presuppose the existence of stones or of men. Nobody would trouble himself about the possible properties of purely prob- lematic men or stones" {Mind, 1876, pp. 290, 291). But the conclu- sions, " Those who pass this examination are lucky men," " Perfectly conscientious men think but little of law and rule '* may certainly be worth obtaining, although in the' universe to which reference is made, (and in both the cases in question this would be the actual material universe), the subjects of these propositions might be non-existent. ^ Is it not sometimes the case that in order to disprove the existence of some combination, say AB, we establish a self-contradictory pro-r position of the form AB is both C and not-C ? 126 PROPOSITIONS. [PARTir. native and regard categorical propositions as always im- plying the existende of their subjects, we have shewn in section 104 that we require to assume the existence not merely of the actual terms involved in any given proposi- tion, but also of their contradictories. ' •' The importance of the question here raised is mor^ i)articularly manifest when we are dealing' with very complex propositions?, as is shewn by Mr Venn. '• We say then that logically All S is F implies oiily the non-existence of anything that is both S and not-/*; No S is P implies only the non-existence of anything that is both The case oi particular propositions still remains; and here again I am inclined to agree with the view taken by Mr Venn in his Symbolic Logic, namely that such proposi- tions should be regarded as implying the existence of their subjects. The chief grounds for adopting this view is that "•'an assertion confined to 'some' of a class generally resti upon observation or testimony rather than on reasoning or imagination, and therefore almost necessarily postulates existent ata, though the nature of this bbservation and consequent existence is, as already remarked, a perfectly open question" {Symbolic Logic, p. 131). I doubt whether in ordinary speech we ever predicate anything of a non- existent subject unless we do so universally. The principal objection to this view is perhaps the paradox which follows from it, namely that we are not without qualification justified in inferring from All S is F that Some S is F, (since the latter proposition implies the existence of S, while the former does not). It may even be said that this view practically banishes the particular proposition from Logic altogether. Possibly if it were so, it would be no very serious matter. But I do not think that it is so. We have CftAl>. VUI.] l>ROPOSITIONS, 6xily to be careful in using such propositions to note the assumption -involved in their use. The principal value of particulars is in their relation of contradiction to universals of different quality. But their use in this respect is entirely consistent with the above. We have taken the view that the import of All S is P is to deny that there is any S that is not-/*; we are now taking the view that the import of Some S is not F is to iiffirm that there is some S that is not-P. This clearly brings out the contradictory character of the two propositions. Similarlywith I and E. One interesting point ta notice here is that if there is no implication of the existence of the subject in universal pro^. positions we are not actually precluded from asserting to- gether two contraries. We may say All S is Pajid No S is P; but this virtually is to deny the existence of S. ^// S /V P excludes JVb S tsV e;ccludes But these are all possible cases. In other respects, this investigation if pursued might somewhat modify accepted logical doctrines ; but I feel convinced that we should be ultimately left with a consistent whole. The truth is, as Mr Venn has remarked, that most English logicians have made no critical examination at all 12^ PROPOSITIONS. [part II. of the question here raised It may be desirable to return to it briefly in connection with the syllogism. Compare sections 273 — 277. [The above view, which is taken by Mr Venn in respect to Symbolic Logic, and which I have attempted to apply to ordinary Formal Logic, is practically identical with that somewhat recently put forward in a more paradoxical form by Professor Brentano. Compare Mind^ 1876, pp. 289 — 292. *^ Where we say Some man is sick^ Brentano gives as a sulv stitute, There is a sick man. Instead of No stone is aiive^ he puts There is not a live stone. Some man is not learned becomes There is an unlearned man^ Finally, All men are mortal is to be expressed in his system There is not an im- mortal manP'\ 107. Discuss the relation between the propositions All 5 is Z' and All not-5 IS Z'. This is an interesting case to notice in connection with the discussion raised in the preceding sections. All S is P= No S is not-i'= No not-i' is S. All not-^S" is i'= No not-5 is not--P= No not-P is not-5 = All not-i' is S. ' The given propositions come out therefore as contraries. (i) On the view that we ought not to enter into any discussion concerning "existence" in connection with im- mediate inference, we must I suppose rest content with this statement of the case. It seems however sufficiently curious to demand further investigation and explanation. (2) On the view that propositions imply the existence of their subjects, we have shewn in section 104, that we are not justified in passing from All not-^* is P to All not-/* is S unless we assume the existence of nolt-P. But it will CHAP, viii.] PROPOSITIONS. 129 be observed that in the case before us, the given propo- sitions make such an assumption unjustifiable. Since All S is P and All not-S is P, and ever3rthing is either S or not-^ by the law of excluded middle, it follows that nothing is In reducing the given propositions therefore to such a form that they appear as contraries, (and therefore as in- consistent with each other), we assume theverything that taken together they really deny. (3) On the view that at any rate universal propositions do not imply the existence of their subjects, we have shewn in the preceding section, that the propositions No not-/* is S, All not-/* is S, are either inconsistent or else they express the fact that /* constitutes the entire universe of discourse. But thisfact is the verything that is given us by the propo- sitions in their original form. On either of the views (2) or (3), then, the result obtained is satisfactorily accounted for and explained. CHAPTER IX. HYPOTHETICAL AND DISJUNCTIVE PROPOSITIONS. 108. The nature of the logical distinction between Categorical and Hypothetical Propositions. Are the propositions "All B is C" and "If any- thing is B, it is C" equivalent.? or can either be inferred from the other ? Mr Venn holds that the real differentia of Hypothetical Propositions is *'to express human doubt" {Mind, 1879, p. 42). I should myself prefer to express the import of Hypothetical Propositions by saying that they affirm a connection between certain events, whenever they happen or if they ever happen, whilst leaving the question en- tirely open whether or not they do ever happen. The doubt which they imply is rather incidental, than the fundamental or differentiating characteristic belonging to them. Materially indeed I think that they do sometimes imply the actual occurrence of their antecedents. When- ever the connection between the antecedent and the con- sequent in a hypothetical proposition can be inferred from the nature of the antecedent independently of specific experience, (and this may be the more usual case), then the actual happening of the antecedent is not in any sense in- CHAP. IX.] PROPOSITIONS. 131 voived; but if our knowledge of the connection does depend on specific experience, (as it sometimes may), and could not have been otherwise obtained, then such actual happening would appear to be materially involved. For example, the statement, "If we descend into the earth, the temperature increases at a nearly uniform rate of i® Fahr. for every 50 feet of descent down to almost a mile," requires that actual descents into the earth should have been made, for otherwise the truth of the statement could not have been known. It may, however, be replied that the doubt applies to the actual occurrence of the antecedent in a given instance. When I say " If the glass falls, it will rain," I imply doubt as to whether it actually will fall on the occasion to which I am referring. (Compare Venn, Symbolic Logic, pp. 331 — 333.) But may not this be the case also with categorical propo- sitions ? For example, if I am in doubt whether a given plant is an orchid, I may apply the proposition "All orchids have opposite leaves*' in order to resolve my doubt. We have such a case as this whenever categorical propositions are used in the process of diagnosis, and it can hardly be said that wc never doemploycategorical propositions in this mStill, it is clear that the hypothetical proposition does not necessarily imply the actual occurrence of its antecedent; and therefore, if the view is taken that the categorical pro- position does necessarily imply the actual existence of its subject, (compare sections 104, 106), we have a marked distinctionbetween the two kinds of propositions. "If anything is By it is C" cannot be resolved into "All B is C", since thelatter implies the existence of B while the former does not. Another vie^r with regard to categorical propositions, 132 PROPOSITIONS. [PART 11. and the one for which I have expressed a preference, is that they do not necessarily imply, (and therefore do not logically imply), tlie existence of their subjects. On this view, I do not see that we have any logical distinction between hypo- thetical and categorical propositions, except a distinction of form ; that is, they may be resolved into one another. We may say indifferently " All B '\% C or " If anything is B it is C"; " If AisB, C is Z>" or " All cases of A being B are cases of C being £>" Kant denies that we can reduce the hypothetical judg- ment to the categorical form on the following ground : " In categorical judgments nothing is problematical, but every- thing assertative; in hypothetical it is merely the connection between the antecedent and the consequent that is assertative. Hence here we may combine two false judgments." This view has I think been virtually discussed in what I have already said. If the categorical judgment is regarded as affirming not merely a connection between the subject and the predicate but also the existence of the subject, then I admit the force of the above argument, and allow that the hypothetical judgment cannot be reduced to the categorical form. But if the categorical judgment is not regarded as affirming the existence of the subject^ it (like the hypothetical judgment) asserts no more than a connection ; it is no more assertative than the hypothetical judgment, and just as problematic. The non-existence of the subject of the cate- gorical corresponds exactly to the falsity of the antecedent of the hypothetical ; and if in the latter we may combine two false judgments, in the former we may combine two non-existent entities. I may say, If A is By C is Z>, although A is B IS di false judgment ; but similarly I may say any case of A being ^ is a case of C being Dy although the case of A being B is z. non-existent, case. I cannot CHAP. IX.] PROPOSITIONS. 133 see that in the latter of these statements I have committed myself to anything whatever that is not contained in the former. Hamilton also {Logic, i. p. 239) holds that a hypothetical judgment cannot be converted into a categorical. "The thought, A is through B, is wholly different from the thought, A is in B, The judgment, — If God is righteous, then will the wicked be punished, and the judgment, — A righteous God punishes the wicked, are very different, although the matter of thought is the same. In the former judgment, the punishment of the wicked is viewed as a consequent of the righteousness of God ; whereas the latter considers it as an attribute of a righteous God. But as the conse- quent is regarded as something dependent from, — the at- tribute, on the contrary, as somethipg inhering in, it is from two wholly different points of view that the two judgments are formed.'* Now it must certainly be admitted that in any given instance there are reasons why we choose the h)T)othetical mode of expression rather than the cate- gorical, or vice versa ; but the only question that concerns us from a logical point of view is whether precisely the same meaning cannot be expressed in either form. Hamilton would appear to deny not merely that a hypothetical judgment can be converted into a categorical, but also that a categorical can be converted into a hypothetical. But, (leaving on one side the question of the existence of the subject in a categorical proposition, which has already been discussed), can any one who allows that "all orchids have opposite leaves'' deny that "if this plant is an orchid it has opposite leaves"? Can any one who allows that "if there are shg.rpers in the company we ought not to gamble," deny that "all cases in which there are sharpers in the company are cases in which we ought not to gamble"? 134 PROPOSITIONS. [part II. If this is admitted, the logical question is to my mind dis- posed of ^ No doubt hypothetical propositions will frequently look awkward when expressed in the categorical form, but in some cases logical error is more likely to be avoided if we reduce them to this form before manipulating them; and I cannot see how we lose anything, or, (on the view now taken with regard to the existential import of categorical propositions), imply anything that we should not imply, in so dealing with them. I have given examples shewing that the doctrines of opposition and immediate inference may be applied to hypothetical. We shall find that the same is true of the doctrine of syllogism, though it may be useful to frame special rules when we are dealing with propositions expressed in this form. 109. The interpretation of Disjunctive Proposi- tions. There is a difference of opinion among logicians as to ^ Hansel's view upon this question {Aldrich^ pp. 103, 104) is not easy to understand. He admits however that *'^\{ A is B, C is Z? " implies that ** Every case of A being -5 is a case of C being Z?." He even goes so far as to resolve ** If all A is B^ all A is C* into ** All B is C," which is clearly erroneous. His whole treatment of hypo- theticals is puzzling. For example, he says, **The judgment, *If A is By C is />,' asserts the existence of a consequence necessitated by laws other than those of thought, and consequently out of the province of Logic" {Aldrichf p. 236; Prolegomena Logica, p. 230). But «imilarly a categorical proposition may assert a connection not neces- sitated by laws of thought ; and I do not see that we have here any reason for subjecting hypothetical propositions to a peculiar treatment. I am inclined to think that what makes Hansel's discussion of hypo- thetical propositions so difficult is that he attempts to apply to them the strict conceptualist view of Logic, which it is impossible to apply consistently throughout without divesting Logic of all content what- soever.CHAP. IX.] PROPOSITIONS. 135 whether the alternatives in a disjunctive proposition should be regarded as mutually exclusive. For example, in the proposition A is either B or C, there is not general agree- ment as to whether it is logically implied that A cannot be both B and C \ There are at least two questions involved which should be distinguished. (i) In ordinary speech do weintend that the alter- natives in a disjunctive proposition should be necessarily understood as excluding one another ? A very few instances will I think enable us to answer this question in the negative. "Take,for instance, the proposition — 'A peer is either a duke, or a marquis, or an earl, or a viscount, or a baron '...Yet many peers do possess two or more titles, and the Prince of Wales is Duke of Cornwall, Earl of Chester, Baron Renfrew, &c....In the sentence — 'Repent- ance is not a single act, but a habit or virtue,' it cannot be implied that a virtue is not a habit... Milton has the ex- pression in one of his Sonnets — * Unstained by gold or fee,* where it is obvious that if the fee is not always gold, the gold is a fee or bribe. Tennyson has the expression * wreath or anadem,' Most readers would be quite un- certain whether a wreath may be an anadem, or an anadem a wreath, or whether they are quite distinct or quite the same'* (Jevons, Furg Logic, pp. 76, 77).(2) But this does not absolutely settle the question. It may be said: — Granted that in common speech the alternatives of a disjunction may or may not be mutually exclusive, still in Logic we should be more precise, and * Whately, Mansel, Mill, and Jevons would answer this question in the negative ; Kant, Hamilton, Thomson, Boole, Bain, and Fowler in the affirmative. 136 PROPOSITIONS. [part ii. thie statement ^^A is either B or C^ (where it may be both) should be written " A is either B or C or both." This is a question of interpretation or method, and I, do not apprehend that any burning principle is involved in the answer that we may give. For my own part I do not find any reason for diverging from the usage of everyday language. On the other hand, I think that if Logic is to be of practical utility, the less logical forms diverge from those of ordinary speech the better. And further, it conduces to clearness if we make a logical proposition express as little as possible. " A is either B or C, it can- not be both" is best given as two distinct propositions ^ ^ A view strongly opposed to that adopted in the text is taken in a recently published work on the Principles of Logic by Mr Bradley of Merton College, Oxford. His argument is as follows : — *• The com- monest way of regarding disjunction is to take it as a combination ofh)rpotheses. This view in itself is somewhat superficial, and it is possible even to state it incorrectly. * Either A\& B ox C\% D"^ means, we are told, that if A is not B then C is Z?, and if C is not D then A is B, But a moment's reflection shews us that here two cases are omitted. Supposing, in the one case, that A is Bi and supposing, in the other, that C is Z>, are we able in these cases to say nothing at all? Our * either — or' can certainly assure us that, if A is B^ C — D must be false, and that, if C is Z>, then A — B is false. We have not exhausted the disjunctive statement, until we have provided for four possibilities, B and not --5, C and not-C" (Principles of Logic ^ p. I2i). The question raised is really one of interpretation, as I have indicated above; but this is what Mr Bradley will not admit. In my view, it is open to a logician to choose either of the two ways of interpreting a disjunctive proposition, provided that he makes it (juite clear which he has selected ; but I can see no good in dogmatising as in the following passage, — **Our slovenly habits of expression and thought are no real evidence s^ainst the exclusive character of dis- junction. M is ^ or ^' does strictly exclude *^A is both b and r.' When a speaker asserts that a given person is a fool or a rogue, hemay not mean to deny that he is both. But, having qo interest in CHAP. IX,] PROPOSITIONS. 137 Professor Fowler indicates this view in his statement that "it is the object of Logic not to state our thoughts in a condensed form but to analyse them into their simplest elements" (Deductive LogiCy p. 32); though he does not apply it to the case before us. Mansel arguingin favour of the view that I have taken remarks, — "But let us grant for a moment the opposite view, and allow that the proposition, * All C is either A or B^ implies, as a condition of its truth, * No C can be both.' Thus viewed, it is in reality a complex proposition, contain- ing two distinct assertions, each of which may be the ground of two distinct processes of reasoning, governed by two opposite laws. Surely it is essential to all clear thinking, that the two should be separated from each other, and not confounded under one form by assuming the Law of Ex- cluded Middle to be, what it is not, a complex of those of Identity and Contradiction" {Prolegomena Logica, p. 238). Of course if the alternatives are logical contradictories they are logically exclusive, but otherwise in thetreatment of disjunctive propositions in the following pages I donot regard them as being so. If in any case they happen to be materially incompatible, this must be separately stated. 110. From the statement that blood-vessels are either veins or arteries, does it follow logically that a blood-vessel, if it be a vein, is not an artery ? Give your reasons. [L.] shewing that he is both, being perfectly satisfied provided he is one, either b or c, the speaker has not the possibility be in his mind. Ig- noring it as irrelevant, he argues as if it did not exist. And thushemaypractically be right in what he says, though formally his statement is downright false : for he has excluded the alternative bc"*^ (p. 124). 138 PROPOSITIONS. [part ii. 111. Put, if you can, the whole meaning of a dis- junctive proposition (such as, Either A is 5 or C is D) in the form of a single and simple Hypothetical, and prove your expression to be sufficient. [R.] Adopting the view that in a disjunctive proposition the alternatives are not to be regarded as necessarily excluding one another, such a disjunctive proposition as the above is primarily reducible to two hypotheticals, namely, If ^ is not B, C is Z>, and If C is not Z>, A is B. But each of these is the contrapositive of the other, and may therefore be in- ferred from it. Hence the full meaning of the disjunctive is expressed by means oi either oi \h&&Q hypotheticals ^ Professor Croom Robertson called attention to this point in Mind, 1877, p. 266, — "The other form of propo- stion ranged by logicians with the Hypothetical, namely the Disjunctive, may be shewn to be as simple as the pure Hypothetical being in fact a special case of it. The com- mon view is that it involves at least two hypothetical propo- sitions, or, as some say, even four. Thus * Either A is B or C is Z> ' is resolved by some into the four hypotheticals — * Mr Bradley {Principles of Logic, p. 121), lays it down that ** disjunctive judgments cannot really be reduced to hypotheticals " at all ; but I hardly care to disagree with him since he admits all that I should contend for. He distinctly resolves **^ is b or ^" into hypotheticals (p. 130); but, he adds, although the meaning of dis- junctives can thus **be given hypothetically ; we must not go on to argue from this that they trr^ hypothetical" (p. 121). They "declare a fact without any supposition " (p. 122). But so does the hypothetical itself, namely, the connection between the antecedent and the conse- quent. Further, "A combination of hypotheticals surely does not lie in the hypotheticals themselves" (p. 122). Undoubtedly, by means of a combination of hypotheticals, we may make a most categorical state- ment; e.g., l(AisS,CisD; and ii A \& not S, Cis D. CHAP. IX.] PROPOSITIONS. 139 KAisBy CisnotZ)(i), If A is not B, CisD (2), If Cis D, A is not B (3), If C is not I?, AisB (4), — but the first and third of these are rejected by others, and with reason, because they are in fact implied only when the alternatives are logical opposites. The remaining propo- sitions (2) and (4) are, however, the logical contrapositives of one another; and this amounts to saying that either of them dy itself is a full and adequate expression of the original disjunctive." PART III. SYLLOGISMS. CHAPTER I. THE RULES OF THE SYLLOGISM. 112. The Terms oftheSyllogism.Areasoningconsistingof three categorical propositions (of which one is the conclusion), and containing three and only three terms, is called a Categorical Syllogism. Every categorical syllogism then contains three and only three terms, of which two appear in the conclusion and also in one or other of the premisses, and one in the premisses only. That which appears as the predicate of the conclusion, and in one of the premisses, is called the major term; that which appears as the subject of the conclusion, and in one of the premisses, is called the minor term; and that which appears in both the premisses, but not in the conclusion, (being that term by their relations to which the mutual relation of the two other terms is determined), is called the middle term. HAP. I.] SYLLOGISMS. 141 Thus, in the syllogism, — All M is P, All S is M, therefore. All S\s F-y P is the major term, S is the minor term, and M is the middle tenn. [These respective designations of the terms of a syllogism resulted from such a syllogism as, — All M is P, All S is M, therefore, All S is /*, being taken as the type of syllogism. With the exception of the somewhat rare case in which the terms of a propo- sition are coextensive, such a syllogism as the above may be represented by the followingdiagram.Here clearly the major term is the largest in extent, and the minor the smallest,whilethemiddle occupies an intermediate position. But we have no guarantee that the same relation between the terms of a syllogism will hold, when one of the pre- misses is a negative or a particular proposition; d^., the following syllogism, — No^isP, A115isJf, therefore, No S is P^ 42 SYLLOGISMS. [part iil gives as one case where the major term may be the smallest in extent, and the middle the largest Again, the following syllogism, — No M'\%P, Some S is J/, therefore, Some S is not P^ gives as one case where the major term may be the smallest in extent and the minor the largest. With regard to the middle term, however, we may note that although it is not always a middle term in extent, it is always a middle term in the sense that by its means the two other terms are connected, and their mutual relation deter- mined.] 113. The Propositions of the Syllogism. Every categorical syllogism consists of three propositions. Of these one is the conclusion. The premisses are called the major premiss and the minor premiss according as they contain the major term or the minor term respectively. CHAP. I.] SYLLOGISMS. 143 Thus, All M is P, (major premiss), All S is J/, (minor premiss), therefore, All S is F, (conclusion). It is usual, (as in the above syllogism), to state the major premiss first and the conclusion last. 114. The Rules of the Syllogism; and the Deduc- tion of the Corollaries, The rules of the Syllogism as usually stated are as follows ; — (i) Every syllogism contains three and only three terms. (2) Every syllogism consists of three and only three pro- positions. It may be observed that these are not so much rules, as a general description of the nature of the syllogism. A reasoning which does not fulfil these conditions may be formally valid, but we should not call it a syllogism*. The four following rules are really rules in the sense that if, when we have got the reasoning into the form of a syl- logism, they are not fulfilled, then the reasoning is invalid. (3) No one of the three terms of the syllogism must be used ambiguously ; and the middle term must be distributed once at least in the preinisses. This rule is frequently given in the form : " The middle term must be distributed once at least, and must not be ambiguous," (e.g.^ in Jevons, Elementary Lessons, p. 127). * For example, B is greater than C, • A is greater than B, therefore, A is greater than C. Here there are four terms, since the predicate of the second premiss is "greater than ^," and this is not the same as the subject of the first premiss **^.'* . 144 SYLLOGISMS. [part hi. But it is obviousthat we must guard against ambiguous major andambiguous minor as well as against ambiguous middle. If the middle term is distributed in neither of the pre- misses, the syllogism is said to be subject to the fallacy of undistributed middle, (4) No term must he distributed in the conclusion which was not distributed in one of the premisses. The breach of this rule is called illicit process of the major y orillicit process of the minor, as the case may be; or, more briefly, illicit major or illicit minor. (5) From tutonegativepremisses nothing can be inferred, (6) If one premiss is negative, the conclusion must be nega- tive; and to prove a negative conclusion, one of the premisses must be negative. From these rules, three corollaries may be deduced: — (i) From two particular premisses nothing can be in- ferred. Two particular premisses must be either (a) bothnegative, or ()8) both affirmative, or (y) one negative and one affirmative. But in case (a), no conclusion follows by rule 5. In case ()8), since no term can be distributed in two particular affirmative propositions, the middle term cannot be distributed, and therefore no conclusion follows byrule 3. In case (y), if we can have a conclusion it must be nega- tive (rule 6). Themajor term therefore will be distributed in the conclusion ; and hence we must have two terms dis- tributed in the premisses, namely, the middle and the major (rules 3, 4). But a particular negative proposition and a CHAP. I.] SYLLOGISMS. X45 particular affirmative proposition between them distribute only one term. Therefore, no conclusion can be obtained. [De Morgan {Formal Logic, p. 14) proves this corollary as follows : — " Since both premisses are particular in form, the middle term can only enter one of them universally by being the predicate of a negative proposition ; consequently the other premiss must be affirmative, and, being particular, neither of its terms is universal. Consequently both the terms as to which the conclusion is to be drawn enter partially, and theconclusioncan only be a particular affir- mative proposition. But if one of the premisses be negative, the conclusion must be negative* This contradiction shews that the supposition of particular premisses producing a legitimate result is inadmissible."] (ii) If one premiss is particular, so must be the conclusion^. We must have either (a) two negative premisses, but this case is rejected by rule 5 ; or {P) two affirmative premisses; or (y) one affirmative and one negative. In case {P) the premisses, being both affirmative and one of them particular, can distribute but one term between them, This must be the middle term by rule 3. The minor term is therefore undistributed in the premisses, and the conclusion must be particular by rule 4. In case (y) the premisses will between them distribute two and only two terms. These must be the middle by * This and the sixth rule are sometimes combined into the one rule, Concltisio sequitur partem deteriorenty — i.e,^ the conclusion follows the worse or weaker premiss both in quality and in quantity ; a negative being considered weaker than an affirmative, and a particular than a universal* K. L. 10 146 SYLLOGISMS; [part hi. rule 3, and the major by rule 4, (since we have a negative premiss, necessitating a negative conclusion by rule 6, and therefore the distribution of the major term in the conclusion). Again, therefore, the minor cannot be dis- tributed in the premisses, and the conclusion must bepar- ticular by rule 4. [De Morgan {Formal Logic, '^. 14) gives the following very ingenious proof of this corollary: — " If two propositions F and Q, together prove a third, F, it is plain that F and the denial of F, prove the denial of Q. For F and Q can- not be true together without F, Now if possible, let F (a particular) and Q (a universal) prove F (a universal). Then F (particular) and the denial of F (particular) prove the denial of Q, But two particulars can prove nothing."] (iii) From a particular major and a negative minor nothing can be inferred. Since the minor premiss is given negative, the major premiss must by rule 5 beaffirmative. But it is also particular, and it therefore follows that the major term cannot be distri- buted in it. Hence, by rule 4, it must be undistributed in the conclusion,/.^., the conclusion must be affirmative. But also by rule 6, since we have a negative premiss, it must be negative. This contradiction establishes the corollary that under the supposed circumstances no conclusion is possible. 115. Shew by aid of the. syllogistic rules that thepremisses of a syllogism must contain one more distributed term than the conclusion ; also, that there is always thesame number of distributed terms in the predicates of the premisses taken together as in the predicate of the conclusion. Hence deduce CHAP. I.] SYLLOGISMS. 147 the three corollaries. [Cf. Monck, Introduction to Logic, pp. 40, 4l] 116. " When one of the premisses is Particular, the conclusion must be Particular. The transgression of this rule is a symptom of illicit process of the minor." Spalding, Logic, p. 209. Is it the case that we cannot infer a universal conclusion from a parti- cular premiss without committing the fallacy of illicit minor } 117. Illustrate De Morgan's statement that any case which falls under the rule that" from premisses both negative no conclusion can be inferred" may be reduced to a breach of one of the preceding rules. De Morgan {Formal Logic, p. 13) takes two universal negative premisses E, E, In whatever figure they are, they can be reduced by conversion to, — NoPis Jf, No S is M. Then by obversion they become, (without losing any of their force), — All P is not- Jt/; All S is XiOi-M\ and we have undistributed middle. Hence rule 5 is ex- hibited as a corollary from rule 3. An objection may perhaps be taken to the above on the ground that the premisses might also be reduced to,— All M is not-i', Allil/isnot-^; where the middle term is distributed in both premisses. Here however it is to be noted that we have no longer a middle 10 — 2 148 SYLLOGISMS, [part hi. term connecting S andP at all. We shall teturn subsequently to this method of dealing with two negative premisses. The case in which one of the premisses is particular is dealt with by De Morgan {Formal Logtc^ p. 14) as follows : — "Again, No yis X, Some Fs are not Zs, may be converted into Every X is (a thing which is not F), Some (things which are not Zs) are Fs, in which there is no middle term." This is not quite satisfactory, since we may often exhibit a valid syllogism in such a form that there appear to be four terms ; ^.^., I might say, " All M is F^ All S is M, may be converted into All M isFy No S is not-J^, in which there is no middle term." The case in question may however be disposed of by saying that if we can infer nothing from two universal negative premisses, a fortiori we cannot from two negative premisses, one of which is particular, 118. The rule that "if one premiss is negative, the conclusion must be negative," may be established as a corollary from the rule that " from two negative premisses nothing can be inferred.'* The following has been suggested to me by DeMorgan's deduction of corollary ii., (cf. section 114): — If two pro- positions F and Q together prove a third R^ it is plain that F and the denial of R prove the denial of Q, For F and Q cannot be true togetherwithoutR. Now if possible let F (a negative) and Q (an affirmative) prove R (an affirmative). Then F (a negative) and the denial of R (a negative) prove the denial of Q, But two negatives prove nothing. ■vsscsver^^sr-^^nM^^BVS CHAP. I.] SYLLOGISMS. 149 119. Simplification of the Rules of the Syllogism. It would now seem as if the six rules of the syllogism might be simplified Rules i and 2 may be treated as a description of the syllogism rather than as rules for its validity. The part of rule 3 relating to ambiguity may be regarded as contained in the proviso that there shall be only three terms, {t,e,, ifone ofthe terms is ambiguous, we have not really a syllogism according to our definition of syllogism). Rule 5 has been exhibited in section 117 as a corollary from rule 3; and the first part of rule 6 has been shewn in section ii8 to be a corollary from rule 5. We are left then with only three independent rules, — (a) The middle term must be distributed once at least in the premisses ; {P) No term must be distributed in the conclusion un- less it has been distributed in the premisses ; (y) A negative conclusion cannot be inferred from two afiirmative premisses* 120. In reference to the syllogism, it has been urged that the old rule that negative premisses yield no conclusion does not hold true universally, as in the example. Whatever is not metallic is not capable of powerful magnetic influence, carbon is not metallic, therefore, carbon is not capable of powerful magnetic influence. Examine this criticism. [c] Professor Jevons gives this case in his Principles of Science (ist edition, vol. i., p. 76; 2nd edition, p. 6$\ and he states that "the syllogistic rule is actually falsified in its bare and general statement." Professor Croom Robertson has however conclusively shewn (in Mind^ 1876, p. i 19, note) that this apparent ex- I50 SYLLOGISMS. [part hi. ception is no real exception*. *' There zxtfour terms in the example, and thus no syllogism, if the premisses are taken as negative propositions ; while the minor premiss is an affir- mative proposition, if the terms are made of the requisite number three.*' Mr Bradley {Principles of LogiCy p. 254) returns to the position taken by Professor Jevons. In reference to the example given in the above question, he says, " This argu- ment no doubt has qimternio terminorum and is vicious technically, but the fact remains that from two denials you somehow have proved a further denial. * A is not B^ what is not B is not (7, therefore A is not C7'; the premisses are surely negative to start with, and it appears pedantic either to urge on one side that ^A is not--^ ' is simply positive, or on the other that B and not-.^ afford no junction. If from negative premisses I can get my conclusion, it seems idle to object that I have first transformed one premiss ; for that objection does not shew that the premisses are not negative, and it does not shew that I have failed to get my con- clusion." This is somewhat beside the mark ; and if the points on both sides are clearly stated there appears no room for further controversy. On the one hand, it is implicitly admitted both by Professor Jevons (Studies in Deductive Logic, p. 89), and by Mr Bradley, that two negative premisses invalidate a syllogism , i.e., understanding by a syllogism a mediate reasoning containing three and only three terms. Oh the other hand, everyone would allow that from two propositions which may both be regarded as ^ Mr Venn, also, (in the Academy^ Oct. 3, 1874), — "The reply clearly is, that if * not metallic' is to be regarded as the predicate of the minor, then the minor i^ affirmative; if 'metallic' is predicate, then there are four terms." CHAP. I.] SYLLOGISMS. 151 negative, a conclusion may sometimes be obtained; for example, the propositions which constitute the premisses of a syllogism in Barbara^ may be written in a negative form, thus, No M is not-P, No S is not-Jtf, and no doubt the con- clusion — All S is P — still follows. We must not, however, attach undue importance to the distinction between positive and negative propositions. By means of the process of Ob- version, the logician may at will regard any given propo- sition as either positive or negative. [A similar case to that given in the question is dealt with in the Port Royal Logic (Professor Ba)mes's translation, p. 211) as follows ; — ** There are many reasonings, of which all the pro- positions appear negative, and which are, nevertheless, very good, because there is in them one which is negative only in appearance, and in reality affirmative, as we have already shewn, and as we may still further see by this example : That which has no parts cannot perish by the dissoltdion of its parts; The soul has no parts; Therefore^ the soul cannot perish by the dissolution of its parts. There are several who advance such syllogisms to shew that we haveno right to maintain unconditionally this axiom of logic, Nothing can be inferred from pure negatives; but they have not observed that, in sense, the minor of this and such other syllogisms is affirmative, since the middle, which is the subject of the major, is in it the attribute. Now the subject of the major isnot that which has parts, but All S is M, therefore, All S\% P, Cf. section 158. 152 SYLLOGISMS* [part hi. that which has not parts, and thus the sense of the minor is, TIu soul is a thing without partSy which is a proposition affirmative of a negative attribute."] 121. By what means can we obtain a conclusion from the two negative premisses, — No M is P, No J/ is 5.? By obverting the premisses, we have—, All J/ is not-/'. All Mh not-5, therefore, Some not-^S is not-/'\ 122. Take an apparent syllogism subject to the fallacy of negative premisses, and enquire whether you can correct the reasoning by converting one or both of the premisses into the affirmative form. [Je- vons, Studies in Deductive Logic, p. 84.] Both in the Studies and in the Principles of Science (Vol. I., p. 75), Professor Jevons appears to answer this question in the negative. It is certainly not put in an unexceptionable form, but apparently reference is made to the case given in the preeding section. No A is By No A is Cy may be transformed into, — All A IS not-^, All A is not-C; * But this does not invalidate the syllogistic rule that from two nega- tive premisses nothing can be inferred, since so long as both the pre- misses remain negative we have more than three terms and therefore not a syllogism at all. CHAP. I.] SYLLOGISMS. IS3 yielding a conclusion, — Some not- C is not--fi^. [In Jevons's system, this would become, — A »Ac; yielding a conclusion, — Alf = Ac. (Cf. Principles of Science^ voL i., p. 71 j 2nd ed., p. 59).] 123. Given (i) All P is M, (ii) All 5 is J/, (iii) M does not constitute the entire universe of discourse. What conclusion can we infer ? Exhibit the reasoning in the form of an Aristote- lian syllogism. Is the third premiss necessary in order that the conclusion may be obtained ? Make any comments that occur to you in connection with this point From (i) we can obtain by immediate inference. All tiO\.-M\s noXrPy and from (ii) All not- J/ is not-»S ; and these premisses yield the conclusion, — Some not-.^ is not-/*. The reasoning is here exhibited in the form of an Aristote- lian syllogism. Or, we might reason as follows : — Since S and P are both entirely included in M, there must be outside M some not-»S and some not-/* that are coincident ; and this is the same conclusion as before.Now in the latter form of the reasoning it would seem that we have assumed that there is some not-M^ /.^,, that M 154 SYLLOGISMS. [part iii. does not constitute the entire universe of discourse. But the necessity of this assumption was not apparent in our first method of treatment, according to which by a simple process of immediate inference we obtained a perfectly valid syllogism \ The truth appears to be thathere at any rate we have an illustration ofDeMorgan'sview (Formal Logic, p. 112) that in all syllogisms the existence of the middle term is a datum. From the premisses All M is P, All M is 5, we cannot obtain the conclusion Some S\%P without implicitly assuming the existence of M. Take as an example, — All witches ride through the air on broomsticks; All witches are old women; therefore, Some old women ride through the air onbroomsticks. This point is further discussed in sections 273—277. We may note that the reasoning, — All P is M, hWSviM, therefore, Some not-5 is not-i', does not invalidate the syllogistic rule that the middle term must be distributed once at least in the premisses, since as it stands it contains more than three terms and is therefore not a syllogism. 124. Examine the following assertion: ''In no way can a syllogism with two singular premisses be viewed as a genuine syllogistic or deductive inference.^' This assertion is made by Professor Bain, and he illus- trates it {Logic J Deduction, p. 159) by reference to the fol- lowing syllogism : ^ Cofflpare, however, lection 104. CHAP. I.] SYLLOGISMS. 155 Socrates fought at Delium, Socrates was the master of Plato, therefore, The master of Plato fought at Delium. But "the proposition * Socrates was the master of Plato and fought at Delium *, compounded out of the two pre- misses is nothing more than a grammatical abbreviation " ; and the step hence to the conclusion is a mere omission of something that had previously been said. " Now, we never consider that we have made a real inference, a step in ad- vance, when we repeat less than we are entitled to say, or drop from a complex statement some portion not desired at the moment. Such an operation keeps strictly within the domain of Equivalence or Immediate Inference. In no way, therefore, can a syllogism with two singular premisses be viewed as a genuine syllogistic or deductive inference." The above leads up to some very interesting considera- tions, but it proves too much. In the following syllogisms the premises may be similarly compounded together, — all men are mortal, ) ,, . 1 j ^' t ,, . , V all men are mortal and rational: ''all men are rational,) therefore, some rational beings are mortal. all men are mortal,)' ,, • 1 j- 1 • ^1 , ,, . \ all men including kings are mortal : all kings are men, j 00 therefore, all kings are mortal*. 1 With the above, compare the following syllogism, baving two singular premisses : — The Lord Chancellor receives a higher salary than the Prime Minister, Lord Selborne is the Lord Chancellor, therefore, Lord Selborne receives a higher salary than the Prime Minister. The premisses here would similarly, I suppose, be compounded by Professor Bain into "The Lord Chancellor, Lord Selborne, receives a higher salary than the Prime Minister." 156 syllogisms; [part in. Do not Bain's criticisms apply to these syllogisms as much as to the syllogism with two singular premisses ? The method of treatment adopted is indeed particularly ap- plicable to syllogisms in which the middle term is subject in both premisses *; but in any case it is true that the con- clusion of a syllogism contains a part of^ and only a part of, the information contained in the two premisses taken to- gether. Also, we may always combine the two premisses in a single statement; and thus we may always get Bain's result. In other words, in the conclusion of every syllogism "we repeat less than we are entitled to say,'* or, if we care to put it so, **drop from a complex statement some portion not desired at the moment." It may be worth while here to refer to the charge of incompleteness which Professor Jevons (Principles of Science^ i. p» 71) has brought against the ordinary syllogistic conclusion. *' Potassium floats on water, Potassium is a metal," yield, according to him, the conclusion, "Potassium metal is potassium floating on water;" But "Aristotlp would have inferred that some metals float on water. Hence Aristotle's conclusion simply leaves out some of the informa- tion afforded in the premisses; it even leaves us open to interpret the some metals in a wider sense than we are warranted in doing." In reply to this it may be remarked : first, that the Aris- totelian conclusion does not profess to sum up the whole of the information contained in the premisses of the syl- logism; secondly, that some in Logic means merely **not none", "one at least". The conclusion of the above syllo- gism might perhaps better be written "some metal floats on water," or "some metal or metals, &c." Compare Mr Venn", ^ i,e,^ to syllogisms in Figure 5. Cf. section 143. ' '^ Surely, as the old expression * discursive thought' implie3, we CHAP. I.] SYLLOGISMS. 157 in the Academy^ Oct. 3, 1874; also, Professor Croom Robert- son in Mind^ ^876, p. 219. 126. How far does the conclusion of an Aristo- telian syllogism fall short of giving all the informa- tion contained in the premisses? [Jevons, Studies^ P- 215.]126. The connection between the Dictum de omni et nulla and the ordinary rules of syllogism. The Dictum de omni et nulla was given by Aristotle as the axiom on which all syllogistic inference is based. It; applies directly, however, to those syllogisms only in which the major terai is predicate in the major premiss, and the minor term subject in the minor premiss, (/>., to what are called syllogisms in Figure i). The rules of syllogism, on the other hand, apply independently of the position of the terms in the premisses. Nevertheless, it is interesting to trace the connection between them. We shall find all the rules implicitly contained in the Dictum^ but some of them in a less general form, in consequence of the distinction pointed out above. The Dictum may be stated as follows: — "Whatever is predicated, whether affirmatively or negatively, of a term distributed may be predicated in like manner of everything contained under it" designedly pass on from premisses to conclusion, and then drop the premisses from sight. If we want to keep them in sight we can perfectly well retain them as premisses ; if not, if all that we want is the final fact, it is no use to burden our minds or paper with premisses as well as con- clusion. All reasoning is derived from data which under conceivable circumstances might be useful again, but which we are satisfied to recover when we want them." 158 SYLLOGISMS. [part hi. (i) The Dictum provides for three and only three terms; namely, (i) a certain term which must be distributed, (ii) something predicated of this term, (iii) something contained under it. These terms are respectively the middle, major, and minor. We may consider the rule relating to the ambiguity of terms also contained here, since if any term is ambiguous we have practically more than three terms. (2) The Dictum provides for three and only three pro- positions; namely, (i) a proposition predicating something of a term distributed, (ii) a proposition declaring something to be contained under this term, (iii) a proposition making the original predication of the contained term. These pro- positions constitute respectively the major premiss, the minor premiss, and the conclusion of the syllogism. (3) The Dictum prescribes not merely that the middle term shall be distributed once at least in the premisses, but more explicitly that it shall be distributed in the major premiss, — "Whatever is predicated of a term distributed^ [This is really another form of what we shall find to be a special rule of Figure i, namely that the major premiss must be universal. Cf. section 144.] (4) The proposition declaring that something is con- tained under the term distributed must necessarily be an affirmative proposition. The Dictum provides therefore that the premisses shall not be both negative. [It really provides that the minor premiss shall be affirmative, which again is one of the special rules of Figure i.] (5) The words "in like manner" clearly provide against a breach of rule 6, namely that if one premiss is negative, the conclusion must be negative, and vice versa, (6) Illicit process of the major is provided against indi- rectly. We can commit this fallacy only if we have a nega- CHAP. I.] SYLLOGISMS, 159 tive conclusion, but the words "in like manner" declare that if we have a negative conclusion, we must have a nega- tive major premiss, and since in any syllogism to which the Dictum directly applies, the major term is predicate of this premiss, it likewise will be distributed. Illicit process of the minor is simply provided against inasmuch as we are warranted to make our predication in theconclusion only of what has been shewn in the minor premiss to be contained under the middle term. 127. Can the Syllogism be based exclusively on the laws of Identity, Contradiction and Excluded Middle } Mansel answers this question in the affirmative and main- tains {Prolegomena Logica^ p. 222) that "the Principle of Identity is immediately applicable to affirmative moods in any figure, and the Principle of Contradiction to negatives." In order to shew this, he commences by quantifying the predicate (cf. section 217), and taking as an example the syllogism,— All M is some P^ ll S is some M, therefore, All S is some P^ he reads it thus,— " the minor term aU S is identical with a part of My and consequently with a part of that which is given as identical with all J/*, namely some i'." He then takes the syllogism, — All M is some P, Some S is some My therefore. Some S is some Py and, treating it similarly, finds that "the principle immedi- ately applicable to both is the axiom, that what is given as identical with the whole or a part of any concept, must be i6o SYLLOGISMS, [1>ART ill. identical with the whole or a part of that which is identical with the same concept" Passing by the inaccuracy of speaking of the concepts as being identical*, I cannot see that the above axiom is the same as the Principle of Identity, "Every A is AJ^ The syllogism is something riiore than mere subaltern inference ; it involves a passage of thought through a middle term; and it is just this that the Law of Identity as expressed in the formula " Every A is A" ap- pears to me unable to provide for. This. law may tell us that if all Mis P, then some ^is J^; but does it tell us that if all Mis P, therefore S is jP, because it is M'i The Dictum de omni et nullo clearly enunciates the principle involved in syllogistic reasoning; the Law of Identity, if it does so at all, does so less satis- factorily. Or rather I would say that if the Law of Identity is to cover this principle, then it is inadequately expressed in the formula Every A is A\ Similar remarks apply to the attempt to bring syllogisms with negative conclusions under the Principle of Contradiction, "No A is hot-A" ^ It is really the extension of the one concept that is identical with the whole or a part of the extension of the other; and although the comprehension of a concept is practically the concept itself, it is clear that the same is not true of its extension. It has always seemed to me rather curious that the doctrine of the Quantification of the Predicate should have been introduced by writers like Hamilton and Mansel, who lay so much stress on concepts, * I should say the same in reference to ManseVs remark (Prole^ gomena Logica, p. 103), that the Axiom "things that are equal to the same are equal to one another" is only another statement of the Principle of Identity. CHAPTER 11. IMPLE EXERCISES ON THE SYLLOGISM. 128. Explain what is meant by a Syllogism; and put the following argument into syllogistic form : — ** We have no right to treat heat as a substance, for it may be transformed into something which is not heat, and is certainly not a substance at all, namely, mechanical work." [N.] 129. Put the following argument into syllogistic t ^ ^\^ \ form: — How can any one maintain that pain is always an evil, who admits that remorse involves pain, and yet may sometimes be a real good } [v.] 130. It has been pointed out by Ohm that reasoning to the following effect occurs in some works on mathematics: — "A magnitude required for the solution of a problem must satisfy a particular equation, and as the magnitude x satisfies this equa- tion, it is therefore the magnitude required." Examine the logical validity of this argument, [c] 131. If P is a mark of the presence of Q, and R of that of S, and if P and R are never found together, K. L. II i62 SYLLOGISMS. [part hi. . am I right in inferring that Q and S sometimes exist separately ? [v.] The premisses may be stated, — All P is Q, All jR is S, No-Pisi?j and in order to establish the desired conclusion we must be able to infer at least one of the following, — Some Q is not S, Some S is not Q. But neither of these propositions can be inferred, since they distribute respectively S and Q, whilst neither of these terms is distributed in the given premisses. The question is therefore to be answered in the negative. 132. If it is false that the attribute B is ever found coexisting with A, and not less false that the attribute C is sometimes found absent from A, can you assert anything about B in terms of C? [c] 133. Enumerate the cases in which no valid con- clusion can be drawn from two premisses. 134. Shew that (i) If both premisses of a syllogism are affirma- tive, and one but only one of them universal, they will between them distribute only one term ; (ii) If both premisses are affirmative and both universal, they will between them distribute two terms; (iii) If one but only one premiss is negative, CHAP. II.] SYLLOGISMS. 163 and one but only one premiss universal, they will between them distribute two terms; (iv) If one but only one premiss is negative, and both premisses are universal, they will between them distribute three terms, 136. Ascertain how many distributed terms there may be in the premisses of a syllogism more than in the conclusion. [L.] 136. Prove that, when the minor term is predicate in its premiss, the conclusion cannot be A. [l.] 137. If the major term of a syllogism be the predicate of the major premiss, what do we know about the minor premiss ? [l.] 138. How much can you tell about a valid syllogism if you know, — (i) that only the middle term is distributed; (2) that only the middle and minor terms are distributed ; (3) that all three terms are distributed ? [w.] 139. If it be known concerning a syllogism in the Aristotelian system that the middle term is dis- tributed in both premisses, what can we infer as to the conclusion ? [c] If both premisses are affirmative, they can between them distribute only two terms; but by h)rpothesis the middle term is distributed twice in the premisses, the minor term cannot therefore be distributed, and it follows that the conclusion must be particular. j64 syllogisms. [paRt lii. If one of the premisses is negative, we may have three terms distributed in the premisses ; these must, however, be the middle term twice (by hypothesis), and the major term (since the conclusion must now be negative and the major term will therefore be distributed in it); hence the minor term cannot be distributed in the premisses, and it again follows that the conclusion must be particular. But either both premisses willbeaffirmative, or one affirmative and the other negative ; in any case, therefore, we can infer that the conclusion will be particular. [This proof seems preferable to that given by JevonSj Studies in Deductive Logic^ p. 83.] 140. Shew that if the conclusion of a syllogism be a universal proposition, the middle term can be but once distributed in the premisses. [L.] As pointed oiit by Professor Jevons {Studies in Deductive LogiCy p. 85), this proposition is the contrapositive of the result obtained in the preceding section. 141. Shew directly in how many ways it is pos- ible to prove the conclusions SaPy SeP\ point out those that conform immediately to the Dictum de omni et nullo ; and exhibit the equivalence between these and the remainder. [w.] (i) To i[>TOve At/ S is P. Both premisses must be affirmative, and both must be universal. S being distributed in the conclusion, must be distri- buted in the minor premiss, which must therefore be All S is MM not being distributed in the minor must be distri- buted in the major which must therefore be All M is P, CHAP. II.] SYLLOGISMS. i6s SaF can therefore be proved in only one way, namely, All M is F, All SisM, therefore, All S is F-, and this syllogism conforms immediately to the Dictum, (2) To prove No S is F. Both premisses must be universal, and one must be negative while the other is affirmative, />., one premiss must be E and the other A, First, let the major be E, />., either No M is F ox No F is M. In each case the minor must be affirmative and must dis- tribute S ; therefore, it will be All S is M, Secondly^ let the minor be E, /.isJf, All 6* is P, yielding the conclusion All Si^ M. {4) Some S is not M (2), and All iS is Jl[f (4) are therefore true together; but this is self-evidently absurd, since they are contradictories. Hence it has been shewn that the consequence of sup- posing Some S is not F false is a self-contradiction ; and we may therefore infer that it is true. It will be observed that the only explicit syllogism that has been made use of in the above is in Figure i * ; and the ^ Solly (Syllabus of Logic y p. 104) maintains that a full analysis of the reasoning will shew that three distinct syllogisms are really in- volved, — " Let A and B represent the premisses, and C the conclusion of any syllogism. In order to prove C by the indirect method, we commence with assuming that C is not true. The three syllogisms may be then stated as follows : CHAP, iv.] SYLLOGISMS. i8i process is therefore regarded as a reduction of the reasoning to Figure i. This method of reduction is called Reductio ad impossibiky or Reductio per impossibtle^ , ox De duetto ad impossibiky oxDe- ductio ad absurdum. It is the only way of reducing AOO (Figure 2), or OAO (Figure 3), to Figure i, unless we make use of negative terms (as in obversion and contraposition) ; and it was adopted by the old writers in consequence of their objection to negative terms. 158. The mnemonic lines Barbara^ Celarent, &c. The mnemonic verses, (which are spoken of by De Mor- gan as "the magic words by which the different moods have been denoted for many centuries, words which I take to be more full of meaning than any that ever were made "), are usually given as follows, — Barbara^ Celarent^ Darii^ J^erioquQ prioris: CesarCy Camestres^ Festino^ Baroco^ secundae : Tertia, Darapti^ Disamis, Datisij Felaptofty Bocardo, Ferison^ habet : Quarta insuper addit Bramantipj CameneSy DimariSy Fesapo^ Fresison, Each valid mood in every figure, unless it be a subaltern First syllogism : * ^ is ; C is not ; therefore B is not*. Second syllogism : * If ^ is, and C is not, it follows that B is not ; but B is; therefore it is false that Ais and G is not*" Third syllogism : * Either both propositions A is and C is not are false, or else one of them is false; but that A is is not false; therefore that C is not is false, («. ^., C is).' ** I do not see any flaw in this analysis; at any rate it must be admitted that the reasoning involved in Indirect Reduction is highly complex, and since the two moods to which.it is generally applied can also be reduced directly (compare section 159), some modern logicians are inclined to banish it entirely from their treatment of the syllogism. 1 Cf. Hansel'sAldrich,pp. 88^, 89. i82 SYLLOGISMS. [part hi. tnood, is here represented by a separate word ; and in the case of a mood in any of the so-called imperfect figures, (;>., Figures 2, 3, 4), the mnemonic gives full information for its reduction to Figure i, the so-called perfect figure. The only meaningless letters are b (not initial), d, /, n, r, t \ the signification of the remainder is as follows : — The vaivels give the quality and quantity of the propo- sitions of which the syllogism 19 composed ; and therefore really give the syllogism itself. Thus, Camenes being in Figure 4, represents the syllogism, — NoJ/is^S, therefore. No S Is P. The initial letters in the case of Figures 2, 3, 4 shew to which of the moods of Figure i the given mood is to be reduced, namely to that which has the same initial letter. [The letters B, C, 2>, F were chosen for the moods of Figure i as being the first four consonants in the alphabet.] Thus, Camestres is reduced to Celarenty — All /'is J/, v^ 'NoMisS, No hM, ^ All P is M, therefore, No S is P. therefore, No P is S, therefore. No *S is P. s (in the middle of a word) indicates that in the process of reduction the preceding proposition is to be simply con- verted. Thus, in reducing Camestres to Celarenty as shewn above, the minor premiss is simply converted. s (at the end of a word*) shews that the conclusion of the new syllogism has to be simply converted in order to ^ This slight difference in the signification ois and / when they are fined letters is frequently overlooked* CHAP. IV.] YLLOGISMS. 183 obtain the given conclusion. This again is illustrated in the reduction of Camestres, The final s does not affect the con- clusion of Camestres itself, but the conclusion of Celarent to which it is reduced. / (in the middle of a word) signifies that the preceding proposition is to be converted per accidens. Thus, in the reduction of Darapti to Darii^ — KWMv&P, KWM'^P, All M\& Sy Some S is M, therefore, Some S is F» therefore, Some S is P. p (at the end of a word*) implies that the conclusion obtained by reduction is to be converted per accidens. Thus, in Bramantip^ the p obviously cannot affect the / conclusion of the mood itself; it really affects the A conclusion of the syllogism in Barbara which is given by reduction. Thus, — h\\P\%M, .. AllJlfisS, AUil/is^, ^ AW Pis M, therefore, Some SisP, therefore. All P is S, therefore. Some S is P. m indicates that in reduction the premisses have to be transposed, {Metathesis prcemissarum) ; as just shewn in the case of Bramantip, c signifies that the mood is to be reduced indirectly y (/.^., by reductio per impossibile in the manner indicated in the preceding section) ; and the position of the letter indicates that in this process of indirect reduction the first step is to omit the premiss preceding it, /.^j, the other premiss is to be combined with the contradictory of the conclusion, (Con- versio syllogismi, or ductio per CoHtradictoriani ptopositionem sive per impossibile), c is by some Writers replaced by ^, thus Baroko and Bokardo instead of Baroco and Bdcairdo, ^ See note on the preceding pag0i i84 SYLLOGISMS. [part hi. The following lines are sometimes added to the Verses given above, in order to meet the case of the subaltern moods: — Quinque Subalterni, totidem Generalibus orti, Nomen habent nullum, nee, si bene coUigis, usum\ 169. The direct reduction of Baroco and Bocardo. Mnemonics representing the direct reduction of these moods. ^ The mnemonics have been written in variouft forms.Those given above are from Aldrich, and they are the onesthat are in general use in England. Wallis in his Institutip Logica (1687) gives for Figure 4, Balani^ Cadere^ Digami, Fegano^ Fedibo, P. vanMusschenbroek in his Instiiutiones Logica (1748) gives Barbaric Calenies, Dibatis^ Fes- pamos Fresisom, This variety of forms fpr the moods of Figure 4 was no doubt due to the fact that the recognitipn of this figure at all was quite exceptional until comparatively recently. Compare section 173. According to Ueberweg, the mnemonics run,— Barbara^ Celarent primce, Darii Ferioo^t, CesarCt Camestres^ Fesiino, Baroco secv)ndse. Tertia grande sonans recitat Daraptiy felapton^ Disatnis^ Datisi^ Bocardo^ Ferison, Quaftte Sunt Bamcdip, Calemes, Dimatis^ Fesapo^ Fresison* Mr Carveth Read (Mind^ 1883, p. 440) suggests an ingenious mbdification of the verses, so as to make each mnemonic immediately suggest the figure to which the mood it represents belongs, at the same time abolishing all the unmeaning letters. He takes / as the sign of the first figure, n of the second, r of the third, and / of the fourth. The lines then run Ballcda^ Celallel^ Dalits Fe/ioque prioris. Cisane, Catnesnts^ FesinoHi Banoco secundse. Tertia Darapri, Dnsamis, Darisi^ Feraprc, BocarOf Ferisor habet. Quarta insuper addit BamaiiPt Canutes^ Dimaiis^ FesaptOy FtsistoL Mr Read also suggests mnemonics to indicate the direct reduction of Baroco and Bocardo. Compare the following section. CHAP. IV.] SYLLOGISMS, 185 Baroco : — AU P is M, Some S is not M^ therefore, Some .S" is not F^ may be reduced to Ferio by contrapositing the major pre miss, and obverting the minor premiss, thus, — No not- J/ is Fy Some S is not-JS/", therefore, Some S is not F, Professor Croom Robertson has suggested Faksoko to represent this method of reduction, k denoting obversion, so that ks denotes obversion followed by conversion, (/.^., con- traposition). Whately's word Fakoro (Elements of Logic, p. 97) does not indicate the obversion of the minor premiss (r being with him an unmeaning letter). Bocardo : — Some M is not F, All M\%S, therefore. Some S is not Fy may be reduced to Darii by contrapositing the major premiss and transposing the premisses, thus, All M\%Sy Some TiOXrF is My ■— - - ..I . ■ ■ - — ■ ■ ^■. -.11 - therefore, Some not-/* is S, We have first to convert and then to obvert this conclu- sion, however, in order to get the original conclusion. This process may be indicated by Doksamosky (which again is obviously preferable to Dokamo suggested by Whately, 186 SYLLOGISMS, [part hi. since this word would make it appear as if we immediately obtained the original conclusion in DartT), 160. Shew how to reduce Bramantip by the indirect method. Just as Bocardo and Baroco which are usually reduced indirectly may be reduced directly, so other moods which are usually reduced directly may be reduced indirectly. Bramantip : — All P is M, All M\% S, therefore, Some SS& P\ for, if not, then No 5 is /*; and combining this with the given minor premiss we have a syllogism in Celarenty — No S is P, therefore. No Jfis -P, which yields by conversion No P is M. But this is the contrary of the original major premiss All P is M^ and it is impossible that they should be true together. Hence we infer the truth of the original conclusion. 161. Assuming that any syllogistic reasoning can be expressed in the first Figure, prove that, (omitting the subaltern moods), it can be expressed, directly or indirectly, in any given mood of that Figure. ^ Mr Carveth Read (Mind, i88i, p. 441) uses the letters k and s as above ; but his mnemonics are required also to indicate the figure to which the moods belong (compare the preceding note) ; and he there- fore arrives at Faksnoko and Doksamrosk, Spalding {Logic ^ p. 135) suggests Facoco and Docamoc\ but the processes here indicated by the letter c are not in all cases the same, and thesemnemonics are therefore unsatisfactory. p WJ ■■ I W ■ ■ ■ ■ I^U^B^^^W^^^^PI CHAP, iv.l SYLLOGISMS. 1S7 We may extend the doctrine of reduction, and shew not merely that any syllogism may be reduced to Figure i, but also that it may be reduced to any given mood of that figure, provided it is not a subaltern mood. This position will obviously be established if we can shew that Barbara^ Celarent, Darii and Ferio are mutually reducible to one another. Barbara may be reduced to Celarent by obverting the major premiss and also the new conclusion which is thereby obtained. Thus, All M is /> All S is M, therefore, All S is /*, becomes No M is not-/*, All 5 is J/, therefore, No S is not--P, therefore, All S \% P, Conversely, Celarent is reducible to Barbara; and in a similar manner by obversion of major premiss and con- . elusion Darii and Ferio are reducible to each other. It will now suffice if we can shew that Barbara and Darii are mutually reducible to each other. Obviously the only method possible here is the indirect method. Take Barbara^ MaPy SaM, SaP; for, if not, then we have SoP; and MaP^ SaM^ SoP must be true together. From SoP by first obverting and then converting, (and denoting not-P by P\ we get P'iS, and combining this with SaM we have a syllogism in Darii^ — i88 SYLLOGISMS. [part liu SaM, FiM. P'iM by conversion and obversion becpmes MoP\ and therefore MaP and MoP are true together ; but this is im- possible, since they are contradictories. Therefore, SoP cannot be true, /.^., the truth of SaP is established. Similarly, Darii may be indirectly reduced to Bar2>ara^, MaP, (i) SiM, (ii) SiP, (iii) The contradictory of (iii) is SeP^ from which we obtain PaS\ Combining with (i), we have — PaS MaP,MaS' in Barbara, But from this conclusion we may obtain SeM^ which is the contradictory of (ii)*. 162. Some logicians have .asserted that all the moods of the syllogismare reducible to the form of Barbara. Inquire into the truth of thisassertion, [l.] 163. Making use of any legitimate methods of immediate inference that may be serviceable, shew ^ It has also been maintained, that this reduction is unnecessary, and that, to all intents and purposes, Darii is Barbara, since the *' some S" in the minor is, and is known to be, the same some as in the conclusion. * It would now seem that the Dictum de omni et nulla might if we pleased be reduced to a Dictum de omni ; but it would be vain to pre- tend that any real simpU6cation would be introduced thereby.CHAP. IV.] SYLLOGISMS. 189 how Barbara, Baroco and Be car do may be reduced ostensively to Figure 4. 164. Reduce Ferio to Figure 2, Festino to Figure 3, Felapton to Figure 4. 165. Prove that any mood may be reduced to any other mood provided that the latter contains neither a strengthened premiss nor a weakened con- clusion. 166. Examine the following statement of De Morgan's : — ** There are but six distinct syllogisms. All others are made from them by strengthening one of the premisses, or converting one or both of the premisses, where such conversion is allowable; or else by first making the conversion, and then strengthening one of the premisses." 167. How can you apply the Dictum de omni et nullo to the following syllogism : — Some M is not P, All M is Sy therefore. Some 5 is not P ? 168. How would you apply the Dictum de omni et nullo to the following reasonings } (i) The life of St Paul proves the falsity of the conclusion that only the rich are happy. (2) His weakness might have been foretold from his proneness to favourites, for all weak princes have that failing. [v.] 169. Dicta for the second and third Figures of syllogism corresponding to the Dictum of the first. Thomson i^Laws of Thought, p. 173), and Bowen {Logic^ p. 196), give for Figure 12, a dictum de diverso^ — "If one 190 SYLLOGISMS. [part hi. term is contained in, and another excluded from, a third term, they are mutually excluded"; and for Figure 3, a Dictum de exemplo^ — "Two terms which contain a common part, partly agree, or if one one contains a part which the other does not, they partly differ." The former of these is at least expressed loosely since it would appear to warrant a universal conclusion in Festino and Baroco* Mansel {Aldrtchj p. 86) puts this Dictum in a more satis-, factory form: — "If a certain attribute can be predicated, affirmatively or negatively, of every member of a class, any subject of which it cannot be so predicated, does not belong to the class," This proposition may claim to be axiomatic, and it can be applied directly to any syllogism in Figure 2. The Dictum de eocemplo again as stated above is open to exception. The proposition, " If one term contains a part which another does not they partly differ," applied to No M is Py All M is 5, would appear to justify Some P is not S just as much as Some S is not P, Mansel's amendment here is to give two principles for Figure 3, the Dictum de exemplOj — "If a certain attribute can be affirmed of any portion of the members of a class, it is not incompatible with the distinctive attributes of that class " ; and the Dictum de exceptOy — " If a certain attribute can be denied of any portion of the members of a class, it is not inseparable from the distinctive attributes of that class." But is it essential that in the minor premiss we should be predicating the distinctive attributes of the class as is here implied ? This appears to be a fatal objection to Mansel's dicta for Figure 3. More- over, granted that P is not incompatible with .S", are we there- fore justified in saying Some S is P} I would suggest the following axioms, — " If two terms are both affirmatively predicated of a common third, and one at least of them universally so, they may be par- CHAP, iv;] SYLLOGISMS. 191 tially predicated of each other " ; " If one term is denied while another is affirmed of a common third term, either the denial or the affirmation being universal, the former may be partially denied of the latter." These will I think be found to apply respectively to the affirmative and negative moods of Figure 3, and they may be regarded as axiomatic ; but they are certainly somewhat laboured. 170. Is Redtictio7i an essential part of the doctrine of the syllogism ? According to the original theory of Reduction, the object of the process was to be sure that the conclusion was a valid inference from the premisses. Given a syllogism in Figure i, we are able to test its validity by reference to the Dictum de omni et nullo; but we have no such means of dealing directly with syllogisms in any other figure. Thus, Whately says, — "As it is on the Dictum de omni et nullo that all Reasoning ultimately depends, so, all argun^ents may be in one way or other brought into some one of the four Moods in the First Figure: and a Syllogism is, in that case, said to be reduced'' {Elements of Logic, p. 93). Professor Fowler puts the same position in a more guarded manner, — "As we have adopted no canon for the 2nd, 3rd, and 4th figures, we have as yet no positive proof that the six moods remaining in each of those figures are valid; we merely know that they do not offend against any of the syllogistic rules. But if we can reduce them, /.., a line representing an undistributed term is partly dotted Thus, in the case of All S is P, — the diagram indicates that all S is contained under P, but that we are uncertain as to whether there is or is not any P which is not S,CHAP, v.] SYLLOGISMS, 209 In the case of Some S is not P^ — the diagram indicates that there is S which is not P^ but that we are in ignorance as to the existence of any S that is P. 181. The application of Lambert's diagrammatic scheme to syllogistic reasonings. As applied to syllogisms, the method indicated in the preceding section is much less cumbrous than the Eulerian diagrams*. We may take the following examples : — Barbara P M Baroco ^ Mr Venn (SymbolicLogic^ p. 432) remarks, — "As a whole Lam- bert's scheme seems to me distinctly inferior to the scheme of Euler, and has in consequence been very little employed by other logicians." MrVenn's criticism is chiefly directed against Lambert's representation of the particular affirmative proposition, namely, — The modification, however, which I have here introduced, and which is suggested by Mr Venn himself, meets the objections raised on this ground. K. L. 142IODatisSYLLOGISMS.[PART III.182. . Represent the moods Darii^ Cesare, Daraptiyand Fesapo in Lambert'sscheme.183. Take the premisses of an ordinary syllogism in Barbara, e.g,, all X is F, all Y is Z \ determine precisely and exhaustively what those proposiaffirm, what they deny, and what they leave in doubt, concerning the relations of the terms X, F, Z, [l.] This question can be very well answered by the aid of any of the three diagrammatic schemes which we have just been discussing. Compare also Jevons, Stipes in Deduc- tive Logic^ p. 216. ^J ^ ^1 J iMINlV^^VWIBHipMipiKSr CHAPTER VI. IRREGULAR AND COMPOUND SYLLOGISMS. 184. The Enthymeme. By the Enthymeme^ Aristotle meant what has been called the "rhetorical syllogism" as opposed to the apodeictic, demonstrative, theoretical syllogism. The following is from Hansel's notes to Aldrich (pp. 209 — 211): "The Enthy- meme is defined by Aristotle, trvAAoytcr/tos l^ ciK(^a>v ^ injfieifiiv. The cikos and crrjfi€iov themselves are Propositions ; the former stating a general probability^ the latter a facty which is known to be an indication, more or less certain, of the truth of some further statement, whether of a single fact or of a general behef. The former is a proposition nearly, though not quite, universal ; as *Most men who envy hate': the latter is a singular proposition, which however is not regarded as a sign, except relatively to some other propo- sition,which it is supposed may be inferred from it. The ctKos, when employed in an Enthymeme, will form the major premiss of a Syllogism such as the following : Most men who envy hate, This man envies, therefore. This man (probably) hates. 14 — 2 212 SYLLOGISMS. [part hi. The reasoning is logically faulty; for, the major premiss not being absolutely universal, the middle term is not dis- tributed The (njfUiov will form one premiss of a Syllogism which may be in any of the three figures, as in the following ex- amples : F^re I. All ambitious men are liberal, Pittacus is ambitious, Therefore, Pittacus is liberal. Figure 2. All ambitious men are liberal, Pittacus is liberal, Therefore, Pittacus is ambitious. Figure 3, Pittacus is liberal, Pittacus is ambitious, Therefore, All ambitious men are liberal. The syllogism in the first figure is alone logically valid. In the second, there is an undistributed middle term : in the third, an illicit process of the minor." On this subject the student may be referred to the remainder of the note from which the above extract is taken, and to Hamilton, Discussions, pp. 152 — 156. An enthymeme is now usually defined as a syllogism incompletely stated, one of the premisses or the conclusion being understood but not expi;essed. As has been firequently pointed out, the arguments of everyday life are for the most part enthymematic The same may be said of fallacious arguments, which are seldom completely stated, or their want of cogency would be more quickly recognised. An enthymeme is said to be of the first order when the major premiss is suppressed ; of the second order when the minor premiss is suppressed ; and of the third order when the conclusion is suppressed. _V^mm.vm^ f,mi,fi^ I— jr" -. .i^-Tl* -^ ^.;^Vip^lWrV '^'^ ^< a.li. JP "^l^^amMi^^nSQP^yf^^ CHAP. VI.] SYLLOGISMS. 213Thus, '^Balbus is avaricious,andtherefore, he is un- happy,'' is an enthymeme of the first order; "All avaricious persons are unhappy, and therefore, Balbus is unhappy '^ is an enthymeme of the second order; "All avaricious persons are unhappy, and Balbus is avaricious'' is an enthymeme of the third order. 186. The Polysyllogism ; and the Epicheirema. A chain of syllogisms, that is, a series of syllogisms so linked together that the conclusion of one becomes a pre- miss of another, is called 2i polysyllogtsm. In a polysyllogism, any individual syllogism the conclusion of which becomes the premiss of a succeeding one is called a prosy Uogism ; . any individual syllogism one of the premisses of which is the conclusion of a preceding syllogism is called an epi- syllogism. Thus, —All C'v&DA All BisCA Prosyllogism, herefore. All B h Dyl but. All A isbX episyllogism. therefore. All -4 is Z>. ) The same syllogism may of course be both an episyllo- gism and a prosyllogism, as would be the case with the above episyllogism if the chain were continued further. An epicheirema is a polysyllogism with one or more prosyllogisms briefly indicated only. That is, one or more of the syllogisms of which the polysyllogism is composed is enthymematic. Whately {Logicy p. 117) calls it accord- ingly an enthymematic sentence. The following is an example, £ is D, because it is C, ^ is ^, therefore, A is D, 214 YLLOGISMS. [PART IIL 186. The Sorites. A Sorites is a polysyllogism in which all the conclusions are omitted except the final one; for example, ^is^, ^is (7, CisZ>, £> is E, therefore, ^ is ^. 187. The ordinary Sorites, and the Goclenian Sorites. In the ordinary Sorites, the premiss which contains the subject of the conclusion is stated first; in the Goclenian Sorites it is stated last. Thus, — Ordinary Sorites^ — A is By Bis C, DisE, therefore, A is E, Goclenian Sorites^ — D is Ej CisI?, Bis C, A is B, therefore, A is E, I If, in the case of the ordinary sorites, the argument were drawn out in full, the suppressed conclusions would appear as minor premisses in successive syllogisms. Thus, the ordinary sorites given above may be analysed into the three i following syllogisms, — (i) B is C, ^is^, therefore, A is C; CHAP. VI.] SYLLOGISMS. 215 (2) C is A ^is (7, therefore, A is D; (3) ^ is ^, ^isZ>,therefore, -^ is j£. Here the suppressed conclusion of (i) is seen to be the minor premiss of (2), that of (2) the minor premiss of (3); and so it would go on if the number of propositions con- stituting the Sorites were increased. In the Goclenian Sorites, the premisses are the same, but their order is reversed, and the result of this is that the suppressed conclusions become mq/or premisses in successive syllogisms. Thus the Sorites, — I? is JS, CisD, BisC, A is By therefore, A is JS,may be analysed into the following three syllogisms, — (i) D is £, CisI?,therefore, C is ^ ; (2) C is E, Bis C, therefore, B is E; (3) ^ is ^> ^ is ^, therefore, A is JS, Here the conclusion of (i) becomes the major premiss of (2)3 and so on. 2i6 SYLLOGISMS. [part hi. The ordinary Sorites^ is that which is most usually discussed; but it may be noted that the order of premisses in the Godenian form is that which really corresponds to the customary order of premisses in a simple syllogism. 188. The special rules of the ordinary Sorites. The special rules of the ordinary sorites are,— (i) Only one premiss can be negative; and if one is negative, it must be the last. (2) Only one premiss can be particular; and if one is particular, it must be the first Any ordinary sorites may be represented in skeleton form, the quantity and quality of the premisses bdng left imdetermined, as follows : — ^ What I have called the ordinary Sorites is frequently spoken of as the Aristotelian Sorites ; for example, by ArchbishopThomson {Laws of Thought^ p. 201), and Spalding (Logic, p. 302). Hamilton howeverremarks,— **The name Sorites does not occur in any logical treatise of Aristotle ; nor, as for as I have been able to discover, is there, except in one vague and cursoryallusion, any referenceto what the name is now employed to express"{Lectures on Logic, I. p. 375)* The term Sorites (from c, A is F, A is ^, there- fore, Some Bf and C, and 2^, and ^, are F. (He does not himself give these examples; but that this is what he means may be deduced from his not very lucid statement, — " In Second and Third Figures, there being no subordination of terms, the only Sorites competent is that by repetition of the same middle. In First Figure, there is a new middle term for every new progress of the Sorites ; in Second and Third, only one middle term for any number of extremes. In First Figure, a syllogism only between every second term of the Sorites, the intermediate term constituting the middle term. In the others, every two propositions of the common middle term form a syllogism.'') But it is clear that in the accepted sense of the term these are not sorites at all. In neither of them have we any chain argument, but our conclusion is a mere summation of the conclusions of a number of syllogisms having a common premiss. Hamilton's own definition of sorites, involved as it is, might have saved him from this error. He gives for his definition, — " When, on the common principleofallreasoning,^thatthepart of a part is a part of the whole, — ^we do not stop at the second gradation, or at the part of the high- est part, and conclude that part of the whole, but proceed to some indefinitely remoter part, as Z>, Fj Fy G, If, &c., which, on the general principle, we connect in the conclusion CHAP. VI.] SYLLOGISMS. 219 with its remotest whole, — this complex reasoning is called a ChawrSyliogisM or Sorites^' (Lectures on LogiCy voL i. p. 366). In the above criticism I have followed J. S. Mil^. His own treatment of the question, however, seems open to refutation by the simple method of constructing examples. He considers that the first or last syllogism of a sorites may be in Figure 2 or 3, {e.g,, in Figure 2 we might have ^ is ^, ^ is Cy C v&Dy D is E, No F is E, therefore A is not F), but that it is impossible that all the steps should be in either of these figures. " Every one who understands the laws of the second and third figures (or even the general laws of the syllogism) can see that no more than one step in either of them is admissible in a sorites, and that it must either be the first or the last." But take the following (the suppressed conclusions being nseted in square brackets): — All A is B, NoCisB, [therefore, No A is C\ All D is C, [therefore. No A is D\ AllEisD, [therefore, No A is E\AllFisE,therefore.NoAisF^,^In connection with it, Mill very justly remarks, — " If Sir W. Hamilton had found in any other writer such a misuse of logical language as he is here guilty of, he would have roundly accused him of total ignorance of logical writers" (Examination of Hamilton^ p. 515). 2 This Sorites is analogous to the so-called Aristotelian Sorites, the subject of the conclusion appearing in the premiss stated first. It is to be observed that the rules given in section r88 will not apply except 220 SYLLOGISMS. [part ill. All the syllogisms involved here are in Figure 2, and the sorites itself may I think fairly be said to be in Figure 2. As in the ordinary sorites, the conclusion of each syllogism is the minor of the next The following again may be called a sorites in Figure 3 : — All Bis A, AllBisC, [therefore. Some C is A\ AUCisDy [therefore, Some D is A\ All D is E, therefore. Some E is A^ therefore, Some A is E\ Here the conclusion of each syllogism is the major of the next*. 191. Take any Enthymeme (in the modem sense) and supply premisses so as to expand it into (a) a syllogism, {6) an epicheirema, (c) a sorites; and name the mood, order or variety of each product. [c] 192. Is there any case in which a conclusion can be obtained from two premisses, although the middle term is distributed in neither of them?Theordinarysyllogistic rule relating to the distribution of the middle term does not contemplate the recognition of when the Sorites is in Figure i. For Sorites in Figures 1 and 3, how- ever, other rules might be framed corresponding to the special rules of Figures 2 and 3 in the case of the simple Syllogism. ^ The preceding note applies to this Sorites also. ^ I should admit that such Sorites as the above are not likely to be found in common use. CHAP, vl] syllogisms. 221 any signs of quantity other than all and some. The admis- sion of the sign most yields the valid reasoning, — Most M is F, Most M is S, therefore, Some S is F. We understand most in the sense of more than half, and it clearly follows from the above premisses that there must be some M which is both ^ and F, We cannot however say that in either premiss the term M is distributed. To meet this case, then, the rule with regard to the distribution of the middle term must be amended, if other signs of quantity besides all and some are recognised. Sir W. Hamilton {Logic, vol. 2, p. 362) gives, — "The quantifications of the middle term, whether as subject or predicate, taken together, must exceed the quantity of that term taken in its whole extent"; in other words, we require the idtrortotal distribution of the middle term, in the two premisses taken together. Hamilton then continues some- what too dogmatically, — " The rule of the logicians, that the middle term should be once at least distributed, is untrue. For it is sufficient if, in both the premisses together, its quantification be more than its quantity as a whole, (ultra- total). Therefore, a major part, (a more or most\ in one premiss, and a half in the other, are sufficient to make it effective." De Morgan (Formal Logic, p. 127) writes as follows, — " It is said that in every syllogism the middle term must be universal in one of the premisses, in order that we may be sure that the affirmation or denial in the other premiss may be made of some or all of the things about which affirmation or denial has been made in the first This law, as we shall see, is only a particular case of the truth: it 222 SYLLOGISMS. [part hi. is enough that the two premisses together affirm or deny of more than all the instances of the middle term. If there be a hundred boxes, into which a hundred and one articles of two different kinds are to be put, not more than one of each kind into any one box, some one box, if not more, will have two articles, one of each kind, put into it The common doctrine has it, that an article of one particular kind must be put into every box, and then some one or more of another kind into one or more of the boxes, before it may be affirmed that one or more of different kinds are found together." De Morgan himself works the question out in detail in his treatment of the numerically definite syllogism^ {Formal Logic ^ pp. 141 — 170). 193. The Argument a fortiori and other de- ductive inferences that are not reducible to the ordinary syllogistic form. We may take as an example of the argument a fortiori: B is greater than C, A is greater than B^ therefore, A is greater than C, As this stands, it is clearly not in the ordinary syllogistic form since it contains four terms ; some logicians however profess to reduce it to the ordinary syllogistic form as follows : B is greater than (7, therefore, a greater than B is greater than C, but, ^ is a greater than B^ therefore, A is greater than C. With De Morgan, we may treat this as a mere evasion, or as a petiiio prindpiu The principle of the argument a fortiori is really assumed in passing from " B is greater than C^ to "a greater than B is greater than Cr CHAP. VL] SYLLOGISMS. 223 The following attempted resolution^ must I think be disposed of similarly : Whatever is greater than a greater than C is greater than C, A is greater than a greater than C, therefore, A is greater than C. At any rate, it is clear that this cannot be the whole of the reasoning, since B nolonger appears in the premisses at all. Mansel (Aldrich^ pp. 199, 200) treats the argument a fortiori 2i& a material consequence^ and by this he means, " one in which the conclusionfollows from the premisses solely by the force of the terms," /.^., "from some understood pro- position or propositions, connecting the terms, by the addition of which the mind is enabled to reduce the conse- quence to logical form." He would reduce the argument a fortiori in one of the ways already referred to. This however begs the question that the syllogistic is the only logiccU form. As a matter of fact the cogency of the argu- ment a fortiori is just as intuitively evident as that of a syllogism in Barbara itself. Why should no relation be regarded as formal unless it can be expressed by the word is^ Touching on this case, De Morgan remarks that the formal logician has a right to confine himself to any part of his subject that he pleases; "but he, has no right except the right of fallacy to call that part the whole " {Syllabus^ p. 42). "-^ equals B\ B equals C\ therefore,-^ equals C" is another case to which the same remarks apply. "This is not an instance of common syllogism: the premisses are ^A is an equal of^ ; B is zxs. equal of C So ^ Cf. Hansel's Aldrich^ p. aoo. 224 SYLLOGISMS. [part hi. far as common syllogism is concerned, that *an equal o{ B* is as good for the argument as '^' is a material accident of the meaning of 'equal.' The logicians accordingly, to re- duce this to a common syllogism, state the effect of com- position of relation in a major premiss, and declare that the case before them is an example of that composition in a minor premiss. As in, A is an equal of an equal (of C); Every eqtuil of an equal is an equal; therefore, A is an equal of C. This I treat as a mere evasion. Amongvarious suffi- cient answers this one is enough: men do not think as above. When A=B, B=Cy is made to give A=C, the word equals is a copula in thought, and notanotionattached to a predicate. There are processes which are not those of common syllogism in the logician'smajor premiss above : but waiving this, logic is an analysis of the form of thought, possible and actual, and the logician has no right to declare that other than the actual isactual."(De Morgan, Syllabus^ PP- 31, 2-) There are an indefinite number of other arguments which for similar reasons cannot be reduced to syllogistic form. For example, — X is a contemporary of K, and F of Zj therefore -X" is a contemporary of Z. A is the brother of B^ Bis the brother of C\ therefore, A is the brother of C, We must then reject the claims that have been put for- ward onbehalf of the syllogism to be the exclusive form of all deductive reasoning. As an example of such claims being made, Whately may be quoted. Syllogism, he says, is " the form towhich aU correct reasoning may be ultimately reduced" {Logic^ p. 12). Again, he remarks, *'An argument thus stated regularly and at full length, is called a Syllogism ; which therefore is evidently not a peculiar kind of argument^ but only a peculiar CHAP. VI.] SYLLOGISMS. 225 form of expression, in which every argument maybe stated" {Logic, p. 26) *. Spalding seems to have the same thing in v!ew when he says,— **An inference, whose antecedent is constituted by more propositions than one, isaMediate Inference. The simplest case, that in which the antecedent propositions are two, is the Syllogism, The syllogism is the norm of all inferences whose antecedent is more complex ; and all such inferences may, by those who think it worth while, be resolved into a series of syllogisms" {Logic, p. 158). J. S. Mill endorses these claims. He remarks, — "All valid ratiocination; all reasoning by which from general propositions previously admitted, other propositions equally or less general are inferred \ may be exhibited insome of the above forms," ue,, the syllogistic moods,{Logic, i. p. 191). What is required to fill the logical gap which is created by the admission that the syllogism is not the norm of all valid formal inference has been called the Logic of Rela- tives. The function ofthe Logic of Relatives is to "take account of relations generally, instead of those merely which are indicated by the ordinary logical copula is'\ (Venn, Symbolic Logic, p. 400). The line which this new Logic is likely to take, if it is ever fully worked out, is indicated by the following passage from De Morgan {Syllabus, pp. 30, 31):—"A convertible copula is one in which the copular rela- tion exists between two names both ways : thus 'is fastened to,' *is joined by a road with,' *is equal to,' *is in habit of conversation with,* &c. are convertible copulae. If ^X\s equal to F' then ^Y is equal to X,* &c. A transitive coj^uia, is one in which the copular relation joins X with Z whenever it ^ Cf. alsoWhately, Logic, pp. 24, 5, and p. 34. K. L. 15 n6 SYLLOGISMS.[part ilL joins ^ with Fand Fwith Z. Thus 'is fastened to' is usually understood as a transitive copula : ' J^T is fastened to y and 'F is fastened to Z' give * X is fastened to Z' All the copulse used in thissyUabusaretransitive. The intran- sitive copula cannot be treated without more extensive consideration of the combination of relations than I have now opportunity to give : a second part of this syllabus or an augmented edition, may contain something on this sub- ject" The Student may further be referred to Venn, Symbolic Logic^ pp. 399 — 404. CHAPTER VIL HYPOTHETICAL SYLLOGISMS. 194. The HypotheticalSyllogism and the Hypo- thetico-Categorical Syllogpism. The form of reasoning in which a hypothetical conclusion is inferred from two hypothetical premisses is apparently over- looked by some logicians ; at any rate, it frequently receives no distinct recognition, the term " h)rpothetical syllogism " being limited to the case in which one premiss only is hypothetical. I should however prefer the following definitions : — A Hypothetical Syllogism is a mediate reasoning consist- ing of three propositions in which both the premisses and the conclusion are hypothetical in form j e.g.y—IfCis D.EisF, If A is B, CisD,therefore, If A is B, E is F, A Hypothetico- Categorical Syllogism is a mediate reason- ing consisting of three propositions in which one of the premisses is h3rpothetical in form, while the other premiss and the conclusion are categorical ; e,g,, — If A is By C is Z>, A is By therefore, C is D, IS— 2 228 SYLLOGISMS. [part hi. Thisnomenclature is adopted by Spalding and Ueber- weg, but, as I have already hinted, it is not the most usual Some logicians,{e.g.^ Fowler), call either of the above forms ofreasoning hypothetical syllogisms without distinction. Others, {e, g.^ Jevons), define the hypothetical syllogism so as to include the latter form alone, the former apparently not being regarded by them as a distinct form of reasoning at all This view may be to some extent justified by the very close analogy that exists between the syllogism with two hypothetical premisses and the categorical syllogism; but the difference in form is worth at least a brief discussion. The student should however bear in mind that by the ^^hypothetical syllogism'' in most English works on Logic is meant what has been defined above as the hypothetico- categorical syllogism. 196. Distinction of Figure and Mood in the case of Hypothetical Syllogisms. In the Hypothetical Syllogism, (as defined in the pre- ceding section), the antecedent of theconclusion isequivar lent to the minor term of the categorical syllogism, the consequent of the conclusion to the major term, and the element which does not appear in the conclusion at all to the middle term. Distinctions of mood and figure may be recognbed in precbely the same way as in the case of the categorical syllogism. For example, — Barbara, — If C is D, E is F, If A is B, CUD, therefore, If A is B, E is R Festinoy— If E is F, C is not D. In some cases in which A is B, C is D, herefore. In some cases in which A is B, E is not F. CHAP. VII.] SYLLOGISMS. 229 DarapH,—If C is Dy E is F, IfCisD.AisB, therefore, In soe cases in which A is By E is F. CatneneSy — If E is Fy C is Dy If C is Dy A is not By therefore, If A is By E is not F. In working with hypotheticals it must always be remem- bered that the quality of the proposition is determined by the quality of the consequent. 196. The Reduction of Hypothetical Syllo- gisms. Hypothetical Syllogisms in Figures 2, 3, 4 may be re- duced to Figure i just as in the case of Categorical Syllo- gisms. Thus, the syllogism in Camenes given in the preceding example is reduced as follows to Camestres, — If C is Dy A is not By IfEisFyCisDy therefore, If E is Fy A is not By therefore. If A is By E is not F. According to rule, the premisses have here been trans- ' posed, and the conclusion of the new syllogism is converted in order to obtain the original conclusion.197. Construct Hypothetical Syllogisms in Cesare, Bocardo, FesapOy and reduce them to Figure i. 198. Name the mood and figure of the following : (i) If C is Dy E is not F, In some cases in which A is By C is Dy therefore, In some cases in which A is By E is not F. 330 SYLLOGISMS. [part hi. (2) IfEisF.CisD, IfCisD.A isB, therefore, In some cases in which A is By E is F. Shew that one of these forms may beindirectly- reduced to the other, but not vice versa. Why is this? 199, Name the mood and figure of the following, and shew that either one may be reduced to the other form: — (i) IfEisnotF.CisD, If A is By C is not D, therefore. If A is B, E is F. (2) IfCisDyEisnotF, If AisnotByCis D, therefore, If A is not B^E is not F, 200. The Moods of the Hypothetico-categorical Syllogism. It is usual to distinguish two moods of the hypothetico- categorical syllogism : (i) The modus ponens, (also called the constructive hypo- thetical syllogism), in which the categorical premiss affirms the antecedent of the hypothetical premiss, thereby justifying as a conclusion the affirmation of its consequent. For ex- ample, — If A isB.A is C, AisB, therefore, A is C, (2) The modus tollens^ (also called the destructive hypo- thetical syllogism), in which the categorical premiss denies CHAP. VIL] SYLLOGISMS. 231 theconsequentofthehypotheticalpremiss, thereby justify- ing as a conclusion the denial of its antecedent For ex- ample, — If A isB.Ais C, A is not C^ therefore, A is not B» These may be considered to correspond respectively to Figures i and 2 of the cate^zical syllogism. Thus, the example of modus panens given above may be written, — AU cases if A being B are cases of A being C, This case of A is a case of A being B^ therefore, This case of A is a case of A being C; and we then have a syllogism in Barbara. The following corresponds to Cdarent^ — If A is By A is not C, AisB, therefore, A is not C. The example of modus toUens given above corresponds to Camestres, The following corresponds to Cesare^-^ If A is B, A is not C, AisCy therefore, A is not B, 201. Reduction of the modus tollens to the modus ponens. Any case of modus tollens may be reduced to modus ponens and vice versa. Thus, If A is B, A is C, A is not C^ therefore, A is not By 2S2 SYLLOGISMS. [part iil becomes by contraposition of the h3^othetical premiss, If A is not Cy A is not B^ A is not C, therefore, A is not B; and this is modus ponens. It may be worth noticing here that a categorical syl- logism in Camestres may similarly be reduced to Celarent without transposing the premisses: — All P is M, No Sis M, therefore, No S is P, No notMis P^ All S is not-M, therefore, No Sis P, 202. Shew how the modus ponens may be reduced to the modus tollens, 203. Mention two fallacious modes of arguing from a h)T)othetical major premiss. To what falla- cies in categorical syllogisms do they respectively correspond ? [c] There are two principal fallacies to which we are liable in arguing from a hypothetical major premiss: — (i) It is a fallacy if we regard the affirmation of the consequent as justifying the affirmation of the antecedent For example, If A is B, Ais Q AisC, therefore, A is B\ ^ This would of course be no longer a fallacy if A is B were given as the sole condition of A is C* CHAP. VII.] SYLLOGISMS. 233 (2) It is a fallacy if we regard the denial of the antece^ dent as justifying the denial of the consequent. For ex- ample, If A is By A is C, A is not By therefore, A is not C*. It will easily be seen that these correspond respectively to undistributed middle and illicit major in the case of cate- gorical sylogisms. 204. The claims of the Hypothetico-categorical Syllogism to be regarded as Mediate Inference. Taking the syllogism, — If A is By CisDy but A is By therefore, C is Dy the conclusion is at any rate apparently obtained by a com- bination of two premisses, and the burden of proof certainly seems to lie with those who deny the claims of such an inference as this to be called mediate inference. Professor Bain's arguments, {LogiCy Deductiony p. 117), upon this point are not easy to formulate; but they resolve themselves into one or other or both of the following: — (i) He seems to argue that the so-called hypothetical syllogism is not really mediate inference, because it is "a pure instance of the Law of Consistency"; in other words, because "the conclusion is implied in what has already been stated." But is not this the case in all formal mediate inference? Professor Bain cannot consistently maintain that the categorical syllogism is more than a pure instance ^ See note on the preceding page> 234 SYLLOGISMS. [part iii. of the Law of Consistency; or that the conclusion in such a syllogism is not implied in what h&s already been stated. (2) But he may mean that the conclusion is implied in the h3rpothetical premiss alone. Indeed he goes on to say, " * If the weather continues fine, we shall go into the country ' is transformable into the equivalent form *The weather con- tinues fine, and so we shall go into the country.' Any person affirming the one, does not, in affirming the other, declare a new fact, but the same fact." If this is intended to be understood literally, it is to me a very extraordinary statement Take the following : — If a Russian army lands in Britain, the volunteers will be called out ; . If the sun moves round the earth, modem astronomy is utterly wrong. Are these respectively equivalent to, — the Russians have landed in Britain and so the volunteers are being called out; the sun moves round the earth, and so modem astronomy is utterly wrong ? Besides, if the proposition If A is By C is D implies that A is B, what becomes of the possible reasoning, " But C is not Z>, therefore, A is not B^*} Further arguments in favour of Bain's view' are as follows : — (i) ** Thereis no middle term in the so-called hypo- thetical syllogism." The answer is that there is something in the premisses which does not appear in the conclusion, and thiit this corresponds to the middle term of the cate- gorical syllogism. If we reduce the hypothetical syllogism to the categorical form, this is mote distinctly recognisable. (2) " In the so-called hypothetical syllogism, the minor and the conclusion indifferently change places." This state- ment is erroneous. Taking the syllogism stated at the com- mencement of this section and transposing the so-called minor and the conclusion, we have a fallacy. Compare section 203. CHAP. VII.] SYLLOGISMS. 235 (3) "The major in a so-called hypothetical syllogism consists of two propositions, the categorical major of two terms." This merely telU us that a hypothetical syllogism is not the same in form as a categorical syllogism, but seems to have no bearing on the question whether the so- called hypothetical syllogism is acaseofmediateor of immediate inference. Turning now to the other side of the question, I do not see what satisfactory answers can be given to the following arguments in favour pf regarding the hypothetico-categorical syllogism as a case of mediate inference. In any such syllogism, the two premisses are quite distinct, neither can be inferred from the other,but both are necessary in order that the conclusion m^y be obtained. Again, if we compare with it the inferences which are on all sides admitted to be immediate inferences from the hypothetical proposition, the difference between the two cases is apparent. From Jf A is B^ C is D \ can infer immediately If C is not Dy A is not B ; but I require also to know that C is not D in order to be able to infer that A is not B. It has also been shewn that a reasoning which naturally falls into the form of the hypothetico-categorical syllogism may nevertheless be exhibited in the form ofthe ordinary categorical syllogism, which is admitted to be a case of mediate reasoning. Moreover there arq distinct forms,— the modm ponens and the modus / ; but C is not D ; therefore, A is not B ", — the argument in favour of regarding it as mediate inference is still more forcible; but of course the modus ponens and the modus tollens stand and fell together*. Professor Croom Robertson (Mind^ 1877, p. 264) has suggested an explanation as to the manner in which this controversy may have arisen. He distinguishes the hypo- thetical *'if" from the inferential ^^M^^ the latter being equi- valent to sinu^ seeing thaiy because. No doubt by the aid of a certain accentuation the word " if" may be made to carry with it this force. Professor Robertson quotes a passage from Clarissa Harlowe in which the remark " If you have the value for my cousin that you say you have, you must needs think her worthy to be your wife,'* is explained by the speaker to mean, " Since you have, &c." Using the word in this sense, the conclusion "C is Z>" certainly follows immediately from the bare statement, " If A isB, C isD "; or rather this statement itself affirms the conclusion. We cannot however regard the word "if" as logically carrying with it this inferential implication. When it is so used we ^ In section no I shew further that the H3^othetical Syllogism and the Disjunctive Syllogism also stand and fall together. CHAP, vii.] SYLLOGISMS. 337 have really a condensed mode of expression including two statements in one ; I should indeed turn the argument the other way by saying that in the single statement thus in- terpreted we have a hypothetical syllogism expressed elliptically^ 206. If A is true, B is true ; if iff is true, C is true ; if C is true, D is true. What is theeffect upon the other assertions of supposing successively (i) that D is false ; (2) that C is false ; (3) that B is false ; (4) that A is false ? [Jevons, Studies, p. 146.] 206. Examine the following : If none but B are A, it cannot be possible that any X are V; but all X are Y; therefore Some A are not B. If the reasoning is correct, reduce it to proper syllogistic mood and figure. [v.] 207. Let X, V, Z, P, Q, R, be six propositions : given, {a) Of JST, V, Z, one and only one istrue ; ip) Of P, Qy R, one and only one is true ; {c) If X is true, P is true ; {d) If F is true, Q is true ; {e) If Z is true, R is true ; prove syllogistically, (/) If JST is false, P is false ; (^) If Y is false, Q is false ; (Ji) If Z is false, R is false. ^ Cf. Mansel's Aldrich^ p. 103. CHAPTER VIII. DISJUNCTIVE SYLLOGISMS. 208. The Disjunctive Syllogism.A Disjunctive Syllogism may be defined as a formal reasoning consisting of two premisses and a conclusion, of which one premiss is disjunctive while the other premiss and the conclusion are categorical \ Forexample, A is either B or Cy A is not By therefore, A is C, The categorical premiss in this example denies one of the alternatives stated in the disjunctive premiss, and we ^ Archbishop Thomson's definition of the disjunctive syllogism — " An argument in which there is a disjunctive judgment " (Laws of Thought, p. 197) — ^must I think be regarded as too wide. It would include such a syllogism as the following, — B is either C or Z?, ^is^, therefore, A is either C or Z>. The argument here in no way turns upon the disjunction, and the reasoning may be regarded as an ordinary categorical syllogism in Barbara, the major term being complex. A more general treatment of reasonings involving disjunctive judg- ments is given in Part IV. CHAP. VIII.] SYLLOGISMS. 239 are hence enabled to affirm the other alternative as our con-clusion. This is called themodus tcllendo ponens. Some logicians also recognise as valid a modiisponendo toliens^ in which the categorical premiss affirms one of the alternatives statedin the disjunctive premiss, and the con- clusion denies the other alternative. Thus, A is either B or C, AisB, therefore, A is not C. This proceeds on the assumption that the elements of the disjunction are mutually exclusive, which in my opinion is not necessarily the case^ The recognition or denial of the validity of the modus ponendo tollens depends then upon our interpretation of the disjunctive proposition itself. 209. Comment upon thefollowing definitions of a disjunctive syllogism : — "A disjunctive syllogism is a syllogism of which the major premiss is a disjunctive and the minor a simple proposition, the latter affirming or denying one of the alternatives stated in the former/* "A disjunctive syllogism is a syllogism whose major premiss is a disjunctive proposition." 210. Examine the question whether the force of a Disjunctive Proposition as a premiss in an argument is equivalent to that of a Hypothetical Proposition.[L.]At any rate so far as the disjunctivesyllogism is con- cerned this question must be answered in the affirmative. ^ Cf. section 109. 340 SYLLOGISMS. [part iil A w either B ox C, A is not Bj therefore, A is C; may be resolved into, — If A isnot jE?, ^ is C, A is not Bf therefore, A i» C; or, into, — If A is not Cf A is B, A is not Bf therefore, A is C It may be observed that those who deny the character of mediate reasoning to the hypothetical syllogism must also deny it to the disjunctive syllogism,or else they must refuseto recognise the resolution of the disjunctive proposition into one or more hypothetical propositions. 211* Is it possible to applydistinctions of Figure either to Hypothetical or to Disjunctive Syllogisms ? [c]212. Comment upon Jevons's statement: — ^''It will be noticed that the disjunctive syllogism is governed by totally different rules from the ordinary categorical syllc^ism, since a negative premiss gives an affirmative conclusion in the former, and a negative in the latter/' 218. If all things are either X or Y, and all things are eitherYorZ, what inference can you draw ? (Jevons, Studies, p. 303.] CHAP, viil] syllogisms. 241 214. The Dilemma. The proper place of the Dilemma among Conditional Arguments is made puzzling by the fact that conflicting definitions of the Dilemma are given by different logical writers. It will be useful to comment briefly upon some of thesedefinitions. (i) Mansel (Aldrich^ p. 108) defines theDilemma as "a syllogism having a conditional (h3rpothetical) major premiss with more than one antecedent^ and a disjunc- tive minor." Equivalentdefinitions are given by Whately and Jevons. Three forms of dilemma are recognised by these writers : — i. The Simple Constructive Dilemma. If A is B, Cis Z>j and if ^ is F, Cis D\ But either ^ is ^ or -^ is F\ Therefore, C is D. ii. The Complex Constructive Dilemma.If A is B, Cis D\ and if ^ is F, G is H-, But either -4 is ^ or -^ is F\ Therefore, Either C\% D ot G\% H. iii. The Destructive Dilemma, (alwa)rs Complex). If A is By Cis D\ and ME is F, G is JT- But either C is not Z? or 6^ is not ZT; Therefore, Either A is not B or E'v& not A The Destructive Dilemma is said to be always complex; and the simple form corresponding to the third of the above is certainlyexcluded by the definition given. It would run, — K. L. 16 242 SYLLOGISMS. [part hi. If A is B, Cis Z); and HA is B, Eis F\ But either C is not Z? or -S is not F\ Therefore, A is not B ; and here there is only one antecedentin the major.But the question arises whether such exclusion is not arbitrary, and whether this definition ought not therefore to be rejected. Whately regards the name Dilemma as necessarily implying two antecedents ; but does it not rather imply two alternatives^ each of which is equally distasteful? He goes on toassert that the excluded form is merely a de- structive hypothetical syllogism, similar to the following, — If ^is^, CisZ>; C is not D ; therefore, A is not B, But the two really differ precisely as the simple constructive dilemma, — If Ais B, Cis D\ and if ^ is i?; Cis Z> ; But either -^ is-5 or -S is F\ therefore, C is Z^ ; — differs from the constructive hypothetical syllogism, — If ^is^, C\%P\ AisB; therefore, C is DBesides, it is clear that it is not merely a destructive hypo- thetical, syllogism such as has been already discussed, since the premiss which iscombined with the hypothetical premiss is not categorical but disjunctive \ ^ The argument, — If A Is B, Cis D and jE is F; But either C is not Z? or ^ is not F; Therefore, A is notCHAP. VIII.] SYLLOGISMS. 243 (2) Professor Fowler {Deductive LogiCy p. 116) gives the following : — "There remains the case in which one premiss of the complex syllogism is a conjunctive, (/•^., a hypothetical), and the other a disjunctive proposition, it being of course understood that the disjunctiveproposition deals only with expressions which have already occurred in the conjunctive proposition. This is called a Dilemma?^ Under this definition, it is no longer required that thereshall be at least two antecedents in the hypothetical pre- miss ; and hence, four forms are included, namely, the two constructive dilemmas, and a simpleaswellasacomplexdestructive. (3) The following definition is sometimes given: — "The Dilemma (or Trilemma or Polylemma) is a syllogism in which two (or three or more) alternatives are given in one premiss, but in the other it is shewn that in any case the same conclusion follows." This would include the simple constructive dilemma and the simple destructive dilemma, (as already given); but it would not allow that either of the complex dilemmas is must be distinguished from the following, — But C is not Dy and E is not F\ Therefore, A is not B» In the latter of these there is no alternative given at all, and the reasoning is equivalent to two simple hypothetical syllogisms, yielding the same conclusion, namely, — (i) If ^isi?, CisZ); But CisnotZ>; Therefore, A is not B. (1) MA\&B,E\&F'y But E is not F\ Therefore, A is not B, SYLLOGISMS. [part hi. properly so-called, since in each case we are left with the same number of alternatives in the conclusion as are con- tained in the disjunctive premiss. This definition, however, embraces forms that are ex- cluded by both the preceding definitions. For example, If A is, either B ox C is ; But neither B nor C is ; Therefore, A is not*. (4) Hamilton (LogiCj i. p. 350) defines the Dilemma as '^ a syllogism in which thesumption(major)isatoncehypotheticalanddisjunctive,andthesubsumption(minor)sublatesthewholedisjunction, as a consequent, so that the antecedent is sublated in the conclusioa^' This involved definition appears to have chiefly in view the form last given, namely, — If A is, either ^ is or C is ; Neither B is nor C is ; Therefore, A is not ; but it excludes the following, — If A is, C is ; and if B is, C is ; But either ^ is or ^ is ; Therefore, Cis. This however is one of the typical forms of Dilemma according to all the preceding definitions. (5) Thomson {Laws of Thought^ p. 203) gives the follow- ing, — "A dilemma is a syllogism with a conditional (hypo- thetical) premiss, in which either the antecedent or the con- sequent is disjunctive.'' This definition is probably wider than Thomson himself intended It would include such forms as the following :— ^ Cf. Uebcrweg, System of Logic ^ Lindsay's translation, p. 457. i^^*^^" CHAP. VIII.] SYLLOGISMS. 245 If A is -5 or -£ is i^, then C is Z>; But C is not J? ; Therefore, A is not -5, and E is not -F. If ^is^, CisZ^or^isi?-; But A is B; Therefore, C is Z? or jE is /] 215. " Dilemmatic arguments are more often fal- lacious than not." Why is this ? [C] Jevons {Elements of LogiCy p. 168) remarks that " Dilem- matic arguments are more often fallacious than not, because it is seldom possible tofind instances where two alternatives exhaust all the possible cases, unless indeed one of them be the simple negative of the other." In other words, most dilemmatic arguments will be found to contain a false premiss. It is however somewhat misleading to say that a syllogistic argument is fallacious because it contains a false premiss. At any rate, notwithstanding this, the argument itself from the point of view of Formal Logic may be per- fectly cogent. 216. What can be inferred from the premisses, Either A\^BoxC\^ D^ Either C is not Z? or -E is not F ? Exhibit the reasoning in the form of a dilemma. CHAPTER IX. THE QUANTIFICATION OF THE PREDICATE. 217. The eight propositional forms resulting from the explicit Quantification of the Predicate. The fundamental postulate of Logic, according to Sir W. Hamilton, was "that we be allowed to state explicitly in language all that is implicitly contained in thought'^; and since he also maintained that "in thought the predicate is always quantified,'* he made it follow immediately from his postulate, that "in logic, the quantity of the predicate must be expressed, on demand, in language. '' This doctrine of the explicit quantification of the predi- cate led Hamilton to recognise eight distinct propositional forms instead of the customary four : — All S is all F, All S is some Pf SomeS is all F^ Some 5 is some -P, I.No S is any P, E. No S is some P, V*Some S is not any jP, 0. Some 5 is not some P. . The symbols here attached are due to Thomson*, and they are the ones in most common use. i * Thomson however rejects the forms v and «. CHAP. IX.] SYLLOGISMS. 247 The symbols used by Hamilton himself were Afa^ Aft, If a, Ifty Anay Am, Ina, Ini. Here/ indicates an affirmative proposition, n indicates a negative; a means that the cor- responding term is distributed, / that it is undistributed. Spalding's symbols {Logic, p. 83) are A^, A, P, /, E, \E, O, \0. Mr Carveth Read {Theory of Logic, p. 193) suggests A\ A, r, I, E, E^, 0,0^. The equivalence of these various symbols is shewn in the following table : — Thomson. Hamilton. Spalding. Carveth Read. All 5" is all/' All S is some P Some S is all P Some S is some P No S is No S is some P Some S is not any P Some S is not some P io 0^ 218. The meaning to be attached to the word some in the eight prepositional forms recognised by Sir William Hamilton. Professor Baynes, in his authorised exposition of Sir William Hamilton's new doctrine, would at the outset lead 248 SYLLOGISMS. [part iil one to suppose that we have no longer to do with the in- determinate "some" of the Aristotelian Logic, but that this word is now to be used in the more definite sense of ^^some, but not «//." We have seen that the fundamental postulateofLogic on which Hamilton bases his doctrine is " that we be allowed to state explicitly in language, all that is im- plicitly contained in thought " ; and applying this postulate, Mr Baynes {New Analytic of Logical Forms) remarks : — "Predication is nothing more or less than the expression of the relation of quantity in which a notion stands to an individual, or two notions to each other. If this relation were indeterminate — if we were uncertain whether it was of part, or whole, or none — there could be no predication. Since, therefore, the predicate is always quantified in thought, the postulate applies; />., in logic, the quantity of the predicate must be expressed, on demand, in language. For example, if the objects comprised under the subject be some part, but not the whole, of those comprised under the predicate, we write All X is some F, and similarly with other forms." But if it is true that we know definitely the relative extent of subject and predicate, and if "some" is used strictly in the sense of " some but not all," we should have hyitfive propositional forms instead of eight, namely, — All S is all P, All S is some Py Some S is all P, Some S is some P^y No S is any P. We have already shewn (section 95) that the only possible relations between two terms in respect to their extension are given by the five diagrams, — 1Usingsome in the sense here indicated, Sonte S is some P neces- sarily implies Some S is not any P and No S is some P, ^,r^"^f-Vf^ i^ ^CHAP. IX.] SYLLOGISMS. 249 0, 09 These correspond respectively to the above five propo- sitions; and it is clear that on the view indicated by Mr Baynes the eight forms are redundant. This point'^is worked out in detail by Mr Venn (Symbolic Logic, Chap, i.); he shews the utter inadequacy and unscientific character of the Hamiltonian doctrine. I am altogether doubtful whether writers who have adopted the eightfold scheme have themselves recognised the pitfalls that surround the use of the word some. Many passages might be quoted in which they distinctly adopt the meaning — "some, not all." Thus, Thomson {Laws of Thought^ p. 150) makes U and A inconsistent. Bowen {Logic, pp. 169, 170) would pass from I to O by imme- diate inference \ Hamilton himself would agree with Thomson and Bowen on these points ; but he is curiously indecisive on the general question here raised. He remarks {Logic, II. p. 282) that some "is held to be a definite some when the other term is definite," /. e., in A and Y, 17 and O ; but "on the other hand, when both terms are indefinite or particular the some of each is left wholly indefinite,"1 " This sort of Inference," he says, " Hamilton would call Inte- grationy as its effect is, after determining one part, to reconstitute the whole by bringing into view the remaining part." 250 SYLLOGISMS. [part hi. ue,, in I and o)\ This is very confusing, and it would be most difficult to apply the distinction consistently. Hamil- ton himself certainly does not so apply it. For example, on his view it should no longer be the case that two affirmative premisses necessitate an affirmative conclusion; nor that two negative premisses invalidate a syllogism. Thus, the following should be regarded as valid : — All P is some My All M is some 5, therefore. Some S is not any F, No Mi& any F^ Some S is not any M, therefore, Some S is not any F, Such syllogisms as these, however, are not admitted by Hamilton and Thomson. Hamilton's supreme canon of the categorical syllogism {Logic, ii. p. 357) is : — " What worse relation of subject and predicate subsists between either of two terms and a common third term, with which one, at least, is positively related; that relation subsists between the two terms themselves." This clearJy provides ^ Mr Lindsay, however, in expounding Hamilton's doctrine (Ap- pendix to Ueberwe^s System of Logicy p. 580) says more decisively, — ** Since the subject must be equal to the predicate, vagueness in the predesignations must be as far as possible removed. Some is taken as equivalent to some but not all,''^ Spalding {Logic^ p. 184) definitely chooses the other alternative. He remarks that in his own treatise **the received interpretation some at least is steadily adhered to." Mr Carveth Read (Theory of Logic, p. 196) distinguishes two schemes of what he calls Bidesignate Relationships (Quantified Pre- dicates) in one of which the sign Some is understood to mean Some only, and in the other Some at least* In each case, however, he seems to retain eight distinct prepositional forms. CHAP. IX.] SYLLOGISMS. 251 that one premiss at least shall be affirmative, and that an affirmative conclusion should follow from two affirmative premisses. Thomson {Laws of Thought, p. 165) explicitly lays down the same rules. Here then is further evidence of the unscientific nature of the Hamiltonian doctrine. The same subject is pursued further in the three following sections. 219. What results would follow if we were to interpret ' Some A*s are Es ' as implying that ' Some other A's are not ^s * } [Jevons, Studies in Dedtictive Logic, p. 151.] Professor Jevons himself answers this question by say- ing, " The proposition * Some A^s are B*s ' is in the form I, and according to the table of opposition I is true if A is true ; but A is the contradictory of O, which would be the form of 'Some other A^^ are not B^sJ Under such cir- cumstances A could never be true at all, because its truth would involve the truth of its own contradictory, which is absurd." This is turning the criticism the wrong way, and proves too much. It is not true that we necessarily involve our- selves in self-contradiction if we use some in the sense of some only. What should be pointed out is that if we use the word in this sense, the truth of I no longer follows from the truth of A ; but on the other hand these two pro- positions are inconsistent with each other. Taking the five prepositional forms which are obtained by attaching this meaning to some, namely, — All S is all Py All S is some P, Some S is all P, Some S is some P, No S is P, — it should be observed that eachone of these propositions is inconsistent with each of the others, and also that no one is the contradictory of any one of theSYLLOGISMS.[part III. others. If, for example, on this scheme we wish to ex- press the contradictory of U, we can only do so by affirm- ing an alternative between Y, A, I and E. Nothing of all this appears to have been noted by the Hamiltonian writers \ even in the cases in which they ex- plicitly profess to use some in the sense of " some onfy,^* How the above five forms may be expressed by means of the ordinary Aristotelian four forms has been discussed in section 99. 220. If in the eight Hamiltonian forms of pro- position some is used in the ordinary logical sense, what is the precise information given by each of these propositions? Taking the five possible relations between two terms, and numbering them as follows, — (I) (2) (3) we may write against each of the propositional forms the relations which are compatible with it*: — ^ Thomson (Laws of Thought ^ p. 149) gives a scheme of oppo- sition in which I and E appear as contradictories, but A and as contraries. He appears to use some in the sense of some but not all in the case of A and Y only. ' If the Hamiltonian writers had attempted to illustrate their doc- CHAP. IX.]SYLLOGISMS. We have then the following pairs of contradictories, — A,O;Y,Q) I, E. The contradictory of U is obtained by affirming an alternative between 17 and O.We may point out how each of the above would be expressed without the use of quantified predicates : — U = SaFy FaS', A = SaF; Y=FaS; l^SzF; K = SeF; V=FoS; = Sotrine by means of the Eulerian diagrams, they would I thmk either have foimd it to be unworkable, or they would have worked it out to a more distinct and consistent issue.254 SYLLOGISMS. [part hi. What exact information, if any, is given by co is dis- cussed in the following section. 221. The Hamiltonian proposition o), " Some 5 is not some PT The proposition co, "Some S is not some Py^ is not inconsistent with any of the other propositional forms, not even with U, "All S is all P'' For example, "all equi- lateral triangles are all equiangular triangles," yet never- theless "this equilateral triangle is not that equilateral triangle," which is all that w asserts. "Some S is some /*" is indeed always true except when both the subject and the predicate are the name of an individual and the same individual. De Morgan^ {Syllabus, p. 24) points out that its contradictory is, — "aS and P are singular and identi- cal ; there is but one S, there is but one P, and S is /I" It may be said without hesitation that the proposition o) is of absolutely no logical importance. 222. To what extent do the eightforms result- ing from predicating of all or some trains, that they do or do noty stop at all or some stations, coincide in significance with Hamilton's schedule "i In particular, do the objections to " Some A is not some -ff " apply- to the proposition " Some trains do not stop at some stations " ? [v.] 223. Examine Thomson's statement that "1; has the semblance only, and not the power of a denial. True though it is, it does not prevent our making another judgment of the affirmative kind, from the same terms." ^ De Morgan in several passages criticizes with great acuteness the Hamiltonian scheme of propositions. CHAP. IX.] SYLLOGISMS. 255 224. Write out the various judgments, including U and Y, which are logically opposed to the judg- ment : No puns are admissible. State in the case of each judgment thus formed what is the kind of op- position in which it stands to the original judgment, and also the kind of opposition between each pair of the new judgments. [c] 226. Explain precisely how it is that O admits of ordinary conversion if the principle of the Quanti- fication of the Predicate is adopted, although not otherwise. 226. Test the validity of the following syllogisms, and examine whether or not the reasoning contained in those that are valid can be expressed without the use of quantified predicates : — In Figure i, UUUIU77. In Figure 2, 17UO. In Figure 3, YAY, Y,,E. (i) UUU in Figure i is valid: — All Mis all P, All S is all M, therefore, All S is all P. It should be noticed that whenever one of the premisses is U, the conclusion may be obtained by substituting S or P (as the case may be) for Min the other premiss. Without the use of quantified predicates, the above reasoning may be expressed by means of the two syllo- gisms, — AllMisP, AllMisS, Alls is M, AllPisM, therefore, All S is P, therefore, All P is S, 256 SYLLOGISMS. [part iil (2) IU17 in Figure i is invalid, if some is used in its ordinary logical sense. The premisses are Some M is some Fj and All S is all M. We may therefore obtain the legitimate conclusion by substituting S ior M m the major premiss. This yields Sofne S is some P, If, however, some is here used in the sense of som^ only^ No S is some P follows from some S is some Py and the original syllogism is valid, although a n^ative conclusion is obtained from two affirmative premisses. This syllogism is given valid by Thomson {Laws of Thought^ p. 188); but apparently only through a misprint for IE17. Using some in the sense of some onfyy several other syllogisms would be valid that he does not give as such\ (3) ^UO in Figure 2 is valid: — No P is some M^ All Sis all M, therefore, Som^ S is not any P. Without the use of quantified predicates, we can obtain an equivalent argument in Bocardo, thus, — Some M is not Py All Mis Sy therefore, Some S is not P, (4) Y AY in Figure 3 is valid : — Some M is all P, All M is some Sy therefore, Some S is all P. Without quantified predicates the reasoning may be <;xpressed in Barbara, thus, — ^ Cf. section 218. CHAP. IX.] SYLLOGISMS. 257 AllMis S, All Pis My therefore, AH Pis S,, (5) Yi;E in Figure 3 is invalid : — From Some Mis all P, and No Mis some *S, we infer that No S is any P; but this involves illicit process of the minor. 227. Examine the validity of the following nioods : — In Figure i, UAU, YOO, EYO ; In Figure 2, AAA, AYY, UOw; In Figure 3, YEE, OYO, AoO. [c] 228. In what figures, if any, are the following moods valid i Where the conclusion is weakened, point out the fact : — AUI; YAY; UO17; IU9;; UEO. [l.] 229. Is it possible that there should be three propositions such that each in turn is deducible from the other two? [v.] 230. The Figured and the Unfigured Syllogism. The distinction between the figured and the unfigured syllogism is due to Hamilton, and is connected with his doctrine of the Quantification of the Predicate. By a rigid quantification of the predicate the distinction between subject and predicate may be dispensed with ; and such being the case there is no ground left for distinction of figure, (which depends upon the position of the middle term as subject or predicate in the premisses). This K. L. 17 258 SYLLOGISMS. [part hi, gives what Hamilton calls the Unfigured Syllogism, For example, — Any bashfulness and any praiseworthy are not equivalent, All modesty and some praiseworthy are equivalent, therefore. Any bashfulness and any modesty are not equi- valent. All whales and some mammals are equal, All whales and some water animals are equal, therefore. Some mammals and some water animals are equal. There is an approach here towards the Equational Logic. Hamilton gives a distinct canon for the unfigured syllogism as follows : — " In as far as two notions either both agree, or one agreeing the other does not, with a common third notion ; in so far these notions do or do not agree with each other." CHAPTER X. EXAMPLES OF ARGUMENTS AND FALLACIES. 231. Examine technically the following argu- ments : — (i) Those who hold that the Insane should not be punished ought in consistency to admit also that they should not be threatened ; for it is clearly un- just to punish any one without previously threaten- ing him. (2) If he pleads that he did not steal the goods, why, I ask, did he hide them, as no thief ever fails to do .? [v.] 232. Examine technically the following argu- ments : — Knavery and folly always go together ; so, know- ing him to be a fool I distrusted him. If I deny that poverty and virtue are inconsistent, and you deny that they are inseparable, we can at least agree that some poor are virtuous. How can you deny that the infliction of pain is justifiable if punishment is sometimes justifiable and yet always involves pain f [v.] 17 — 2 26o SYLLOGISMS. [part hi. 233. Test the following : — ''If all men were capable of perfection, some would have attained it; but, none having done so, none are capable of it.'* [v.] 234. Examine the following reasoning : — How can you deny that any poor should be re- lieved, when you deny that sickness and poverty are inseparable, and also that any sick should not be relieved ? [v.] 235. In how many different syllogistic moods could you express the reasoning in the following sentence by supplying the proper premisses ? These plants cannot be orchids, for they have opposite leaves. [v.] 236. In how many different moods may the argument implied in the following proposition be stated ? "No one can maintain that all persecution is justifiable who admits that persecution is sometimes ineffective." How would the formal correctness of the reason- ing be affected by reading " deny" for "maintain" f [v.], 237. What conclusions (if any) can be drawn from each pair of the following sentences taken two and two together ? (i) None but gentlemen are members of the club; CHAP. X.] SYLLOGISMS. 261 (2) Some members of the club are not officers ; (3) All members of the club are invited to compete ; (4) All officers are invited to compete. Point out the mood and figure in each case in which you make a valid syllogism; and state yourreasons when you consider that no valid syllogism is possible. [v.] 238. " No wise man is unhappy ; for no dishonest man is wise, and no honest man is unhappy." Examine this inference, and if you think it sound resolve it into a regular syllogism. [w.] 39. Detect the fallacy in the following argu- ment : — " A vacuum is impossible, for if there is nothing between two bodies they must touch." [n.] 240. Write the following arguments in syllogistic form, and reduce them to Figure i : — (a) Falkland was a royalist and a patriot ; there- fore, some royalists were patriots. (fi) All who are punished should be responsible for their actions ; therefore, if some lunatics are not responsible for their actions, they should not be punished. (7) All who have passed the Little-Go have a knowledge of Greek ; hence A. B. cannot have passed the Little-Go, for he has no knowledge of Greek. 1^2 SYLLOGISMS. [part nu 241. Whately says, — " ' Every true patriot is dis- interested, few men are disinterested, therefore few men are true patriots/ might appear at first sight to be in the second figure and faulty ; whereas it is Barbara with the premisses transposed." Do you consider this resolution of the above syllo- gism to be the correct one ? 242. Examine the validity of the following argu- ments : — (a) Old Parr, healthy as the wild animals,attainedtheageof152years; all men might be as healthy as the wild animals; therefore, all men might attain to the age of 152 years. (/9) Most J/ is P, Most 5 is M, therefore, Some S is P. 243. Examine the validity of the following argu- ments : — y (i) Since the end of poetry is pleasure, that cannot be unpoetical with which all are pleased. (ii) It is quite absurd to say " I would rather not exist than be unhappy," for he who says " I will this, rather than that," chooses something. Non-existence, however, is no something, but nothing, and it is impossible to choose rationally when the object to be chosen is nothing. 244. Can the following arguments be reduced to syllogistic form ? CHAP. X.] SYLLOGISMS. 263 (i) The sun is a thing insensible; The Persians worship the sun ; Therefore, the Persians worship a thing insensible. (2) The Divine law commands us to honour kings ; Louis XIV. is a king ; Therefore, the Divine law commands us to honour Louis XIV. [Port Royal Log-ic,] 246. Examine the following arguments; where they are valid, reduce them if you can to syllogistic form ; and where they are invalid, explain the nature of the fallacy : — (i) We ought to believe the Scripture ; Tradition is not Scripture ; Therefore, we ought not to believe tradition. (2) Every good pastor is ready to give his life for his sheep ; Now, there are few pastors in the present daywhoare ready to give their lives for their sheep ; Therefore, there are in the present day few good pastors. ^ (3) Those only who are friends of God are happy; Now, there are rich men who are notfriends of God ; Therefore, there are rich men who are not happy. (4) The duty of a Christian is not to praise those who commit criminal actions ; 264 SYLLOGISMS. [part in. Now, those who engage in a duel commit a criminal action ; Therefore, it is the duty of a Christian not to praise those who engage in duels. (5) The gospel promises salvation to Christians ; Some wicked men are Christians ; Therefore, the gospel promises salvation to wicked men. (6) He who says that you are an animal speaks truly; He who says that you are a goose says that you are an animal ; Therefore, he who says that you are a goose speaks truly. (7) You are not what I am ; I am a man ; Therefore, you are not a man. (8) We can only be happy in this world by aban- doning ourselves to our passions, or by combating them; If we abandon ourselves to them, this is an un- happy state, since it is disgraceful, and we could never be content with it ; If we combat them, this is also an unhappy state, since there is nothing more painful than that inward war which we are continually obliged to carry on with ourselves ; Therefore, we cannot have in this life true happi- ness. CHAP. X.] SYLLOGISMS. 265 (9) Either our soul perishes with the body, and thus, having no feelings, we shall be incapable of any- evil ; or if the soul survives the body, it will be more happy than it was in the body ; Therefore, death is not to be feared. [Port Royal Logic.] 246. Examine the following arguments : — (i) "He that is of God heareth my words: ye therefore hear them not, because ye are not of God." (2) All the fish that the net inclosed were an in- discriminate mixture of various kinds: those that were set aside and saved as valuable, werefishthat the net enclosed : therefore, those that were set aside and saved as valuable, were an indiscriminate mixture of various kinds. (3) Testimony is a kind of evidence which is very likely to be false: the evidence on which most men believe that there are pyramids in Egypt is testimony: therefore, the evidence on which most men believe that there are pyramids in Egypt is very likely to be false. {4) If Paley's system is to be received, one who has no knowledge of a future state has no means of distinguishing virtue and vice : now one who has no means of distinguishing virtue and vice can commit no sin : therefore, if Paley's system is to be received, one who has no knowledge of a future state can commit no sin. (5) If Abraham were justified, it must have been either by faith or by works : now he was not justified 266 SYLLOGISMS. [part hi. by faith (according to James), nor by works (accord- ing to Paul): therefore, Abraham was not justified. (6) For those who are bent on cultivating their minds by diligent study, the incitement of academical honours is unnecessary ; and it is ineffectual, for the idle, and such as are indifferent to mental improve- ment : therefore, the incitement of academical honours is either unnecessary or ineffectual. (7) He who is most hungry eats most ; he who eats least is most hungry : therefore, he who eats least eats most. (8) A monopoly of the sugar-refining business is beneficial to sugar-refiners : and of the corn-trade to corn-growers: and of the silk-manufacture to silk- weavers, &c., &c.; and thus each class of men are benefited by some restrictions. Now all these classes of men make up the whole community : therefore a system of restrictions is beneficial to the community. [Whately.] 247. The following are a few examples in which the reader can try his skill in detecting fallacies, determining the peculiar form of syllogisms, and sup- plying the suppressed premisses of enthymemes. Several of the examples contain more than one syllogism. (i) None but those who are contented with their lot in life can justly be considered happy. But the truly wise man will always make himself contented with his lot in life, and therefore he may justly be considered happy. CHAP. X.] SYLLOGISMS. 267 (2) All intelligible propositions must be either true or false. The two propositions "Caesar is living still" and "Caesar is dead," are both intelligible pro- positions ; therefore they are both true, or both false. (3) Many things are more difficult than to do nothing. Nothing is more difficult to do than to walk on one's head. Therefore, many things are more difficult than to walk on one's head. (4) None but Whigs vote for Mr B. All who vote for Mr B. aretenpoundhouseholders. There- fore none but Whigs are ten-pound householders. (5) If the Mosaic account of the cosmogony is strictly correct, the sun was not created till the fourth day. And if the sun was not created till the fourth day, it could not have been the cause of the alternation of day and night for the first three days. But either the word '* day " is used in Scripture in a different sense to that in which it is commonly ac- cepted now, or else the sun must have been the cause of the alternation of day and night for the first three days. Hence it follows that either the Mosaic account of the cosmogony is not strictly correct, or else the word "day" is used in Scripture in a different sense to that in which it is commonly accepted now. (6) Suffering is a title to an excellent inheritance; for God chastens every son whom He receives. (7) It will certainly rain, for the sky looks very black. [Solly, Syllabus of Logic,] 268 SYLLOGISMS. [part in 248. Examine the following arguments: (i) All the householders in the kingdom, except women, are legally electors, and all the male house- holders are precisely those men who pay poor-rates ; it follows that all men who pay poor-rates are electors. (2) All men are mortals, and all mortals are those who are sure to die; therefore, all men are all those who are sure to die. [Jevons, Studies^ p.162.]249.Statethe following arguments in Logical form, and examine their validity : — (i) Poetry must be either true or false : if the latter, it is misleading ; if the former, it is disguised history, and savours of imposture as trying to pass itself off for more than it is. Some philosophers have therefore wisely excluded poetry from the ideal commonwealth. (2) If we never find skins except as the tegu- ments of animals, we may safely conclude that animals cannot exist without skins.. If colour can- not exist by itself, it follows that neither can any- thing that is coloured exist without colour. So if language without thought is unreal, thought without language must also be so. (3) Had an armistice been beneficial to France and Germany, it would have been agreed upon by those powers; but such has not been the case; it is plain therefore that an armistice would not have been advantageous to either of the belligerents. CHAP. X.] SYLLOGISMS. 269 (4) If we are marked to die, we are enow To do our country loss : and, if to live, The fewer men, the greater share of honour. [o.] 260. Dr Johnson remarked that "a man who sold a penknife was not necessarily an ironmonger." Against what logical fallacy was this remark directed.^ [c] 261. Exhibit the following in syllogistic form; naming the mood and figure ; when possible, reduce them to thefirst figure : (a) The disciples of Wagner overrate him, for he has caused a great reform in dramatic art, and all great reformers are over-esti- mated by their followers, (d) Some undergraduates are guilty of conduct to which no gentleman would stoop; so some undergraduates are not gentlemen. (c) Not all the things we neglect are worthless, for some truths are neglected and none without value. [c] 262. Examine on logical principles the following arguments ; and, if you find any fallacies, name them : {a) The existence of State-officials is unjustifi- able: for since men are by nature equal, it is con- trary to nature that one should govern another. {6) Instinct and reason are opposed : so a good action, if instinctive, is the opposite of that which reason would dictate. [c] 263. Put the following propositions into their simplest Logical form; name the Syllogistic Moods 270 SYLLOGISMS. [part hi. in which they can be proved ; and find premisses that in some Mood will prove them : (i) Not all the unhappy are evildoers. (2) Only the wise are free. [c] 264. Examine the following arguments, pointing out any fallacies that they contain : (a) The more correct the logic, the more certainly will the conclusion be wrong if the premisses are false. Therefore, where the premisses are wholly un- certain the best logician is the least safe guide. {b) The spread of education among the lower orders will make them unfit for their work : for it has always had that effect on those among them who . happen to have acquired it in previous times. if) This pamphlet contains seditious doctrines. The spread of seditious doctrines may be dangerous to the State. Therefore, this pamphlet must be sup- pressed, [c] 255. " To prove that Dissent is wrong you must appeal to the authority of the Church, and this you must base on the Bible ; and you must also deny the supremacy of Conscience. Moreover you, at least, as an Anglican, must ignore the Reformation." How should you draw out fully the argument here implied.? To what extent does it naturally fall into syllogistic form i . [v.] 256. No one can maintain that all republics secure good government who bears in mind that CHAP. X.] SYLLOGISMS. 271 good government is inconsistent with a licentious press. What premisses must be supplied to express the above reasoning in Ferio, Festino and Ferison re- spectively ? [v.] 267.Using any of the forms of Immediate Inference, shew in how many moods the following argument can be expressed : — " Every law is not binding, for some laws are morally bad, and nothing which is so is binding." [l.] 268. State the following reasonings in strict logical form, and estimate their validity : — (a) As thought is existence, what contains no element of thought must be non-existent. (b) Since the laws allow everything that is inno- cent, and avarice is allowed, it is innocent. (c) Timon being miserable is an evil-doer, as happiness springs from well-doing. [l.] 269. Comment carefully upon the following state- ments : — % "The most perfect Logic will not serve a man who starts from a false premiss." "I amenough of a logician to know that from false premisses it is impossible to draw a true conclu- sion." [l.] 260. Might I be satisfied that a particular war was a just one, assuming (what was the fact) that it was popular, and also (what is more doubtful) that all just wars are popular } 272 SYLLOGISMS. [PART ill. Are honours and rewards, public or private, to be pronounced useless, because they cannot influence thestupid, and men of genius rise above them ? Because some persons in the dark cannot help thinking of ghosts, though they do not believe in them, does it follow that it is absurd to maintain that, hen we cannot avoid thinking or conceiving of a thing, it must be true ? [l.] CHAPTER XI. PROBLEMS ON THE SYLLOGISM. 261. Prove by means of the syllogistic rules that, given the truth of one premiss and of the conclusion of a valid syllogism, the knowledge thus in our possession is in no case sufficient to prove the truth of the other premiss. We have to shew that if one premiss and the conclusion of a valid syllogism be taken as a new pair of premisses they do not in any case suffice to establish the other premiss. The premiss given true must be affirmative^ for if it is negative, the original conclusion will be negative, and com- bining these we shall have two negative premisses which can peld no conclusion. The middle term must be distributed in the premiss given true^ for if not it must be distributed in the other premiss, but this being the conclusion of the newsyllogism,itmust also be distributed in the premiss given true or we shall have an illicit process in the new syllogism. Therefore, the premiss given true, being affirmative, and distributing the middle term, cannot distribute the other term which it contains. Neither therefore can this term be distributed in the original conclusion. But this is the K. L. i8 274 SYLLOGISMS. [part in. term which will be the middle term of the new syllogism, and we shall therefore have undistnbuied middle. The given syllogism then being valid, we have shewn it to be impossible that a new syllogism having one of the original premisses and the original conclusion for its pre- misses, with the other original premiss for its conclusion, can be valid also\ 262. Given that in a valid syllogism one premiss is false and the other true, shew that innocase v^rill this suffice to prove the conclusion false". This might be established by taking all possible syllo- gisms, and shewing that the statement holds tnie with regard to each in turn; but this method is clearly to be avoided if possible. It might also be deduced from the proposition established in the preceding example. Let the premisses of a valid syllogism be F and Q and the conclusion R, P and the contradictory of Q will not prove the contradictory of R ; for if so it would follow that P and R would prove Q ; but this has been shewn not to be the case. Another easy solution is obtainable by assuming that ^ Other methods of solution more or less distinct from the above might be given. A somewhat similar problem is discussed by Solly, Syllabus of Logic y pp. 123 — 126, 132 — 136. Hamilton (Logic^ I. p. 450) considers the doctrine "that if the conclusion of a syllogism be true, the premisses may be either true or false, but that if the conclusion be false, one or both of the premisses must be false'' to be extra-logical, if it is not absolutely erroneous. He is clearly wrong, since the doctrine in question admits of a purely formal proof. * This problem might also be stated as follows, — Shew that if for one of the premisses of a valid syllogism we substitute its contradictory, this will not in any case enable us to establish the contradictoryof the original conclusion.CHAP. XI.] SYLLOGISMS. 275 the given syllogism is reduced to Figure i. After such re- duction, it will, in accordance with the special rules of Figure i, have a universal major and an affirmative minor. Then since the contradictory of a universal is particular and of an affirmative negative, if either premiss is given false we have in its place either a particular major or a negative minor. But, (since the syllogism is still in Figure i), in neither of these cases can we draw any conclusion at all, and therefore a fortiori we cannot infer that the original conclusion is false. I add an outline of an independent general solution of the given problem \ Let the following symbols be used: — 7^= premiss given true; F= premiss given false; C — original conclusion; F' = contradictory of F\ C = contradictory of C; a = original syllogism; ^ = syllogism of which the premisses are T and F^ and the conclusion C'\ P = major term; \ J/= middle term;i '^^^'^ ^^^ ^^ *^^ ^^"^^ ^^^ ^^ ^= minor term. ) " ^^^ ^- We have to shew that ^ cannot be a valid syllogism. T cannot be particular, for in this case F would also be particular. T cannot be negative, for in this case F' would also be negative. T then must be universal affirmative. ^ Several steps are omitted, but these the student should carefully fill in for himself. 18—2 276 SYLLOGISMS. [part iir. (i) Let F also be universal affirmative. We may shew that C must also be universal, (/.^., a can- not have a weakened conclusion) ; and it must of course be affirmative. Then in a, 5 and M must be distributed; in )8, P and M must be distributed. But if F distributed M^ M cannot be distributed in ^ ; and if i^ distributed S^ P cannot be distributed in ^. (2) Let F be universal negative. We may again shew that C must be universal. In this case T cannot distribute M\ but neither can F* distribute M, (3) Let F be particular affirmative. C will be universal negative. Therefore, in ^ we must distribute 5, M^ P, But T must distribute M\ it cannot therefore distribute S or P^ one of which must therefore be undistributed in ^. (4) Let F be particular negative. In a, M and P must be distributed; in Pf MandSmust be distributed. But if F distributed M, M cannot be distributed in )8 ; and if -F distributed Py S cannot be distributed in )8. 263. Given a valid syllogism in Figure i, is there any case in which the mere knowledge that we may- start from the contradiction of its premisses will furnish premisses for another valid syllogism } 264. An apparent syllogism of the second figure with a particular premiss is found to break the general rules of the syllogism in this particular only, that the middle term is undistributed. If the particular pre- CHAP* XL] SYLLOGISMS. 277 miss is false and the other true, what do we know about the truth or falsity of the conclusion ? Can an apparent syllogism break all the rules of syllogism at once ? 266. Given the two following statements false: — (i) either all M is all P, or some M is not P\ (ii) some 5 is not M\ — what is all that you can infer, (a) with regard to 5 in terms of P\ (p) with regard to P in terms of 5 ? 266. If (i) it is false that whenever X is found Y is found with it, and (2) not less untrue that X is sometin^es found without the accompaniment of Z^ are you justified in denying that (3) whenever Z is found there also you may be sure of finding Y} And however this may be, can you in the same circum- stances judge anything about Y in terms of Z ? [R.] 267. If whenever X is present,Zisnotabsent,andsometimes when Y is absent, X is present, but if it cannot be said that the absence of X determines anything about either Y or Z, can anything be deter- mined as between Z and Yl [r.] 268. If B is always found to coexist with A, except when X is F, (which it commonly, though not always, is), and if, even in the few cases where X is not F, C is never found absent without B being absent also, can you make any other assertion about a [R.] 278 SYLLOGISMS. [part hi. 269. From P follows Q ; and from R follows 5 ; but Q and 5 cannot both be true ; shew that P and R cannot both be true. (De Morgan.) 270. Given a syllogism, shew in what cases it is possible to reach the same conclusion by substituting for the middle term its contradictory. [w.] [We are supposed here to perform immediate inferences upon our premisses so as to obtain a new middle term which is the contradictory of the original middle term.] 271. What conclusion can be drawn from the following propositions ? The members of the board were all either bond- holders or shareholders, but not both ; and the bond- holders, as it happened, were all on the board, [v.] We have given, — No member of the board is both a bondholder and a shareholder, All bondholders are members of the board; and these premisses yield a conclusion (in Celarent)y Nobondholder is both a bondholder and a shareholder, that is, No bondholder is a shareholder. 272. The following rules were drawn up for a club : — (i) The financial committee shall be chosen from amongst the general committee ; (ii) No one shall be a member both of the general and library committees, unless he be also on the financial committee ; CHAP. XI.] SYLLOGISMS. 279 (iii) No member of the library committee shall be on the financial committee. Is there anything self-contradictory or superfluous in these rules ? [Venn, Symbolic Logic, pp. 261 — 264.] Let -F= member of the financial committee, G = member of the general committee, L = member of the library committee. The above rules then become, — (i) h\\F'\^G\ (ii) If L is G, it is F\ (iii) No L is F. From (ii) and (iii) we obtain (iv) No L is G, The rules may therefore be written, (i) All F is G, (2) No L is G, (3) NoZis/?: But in this form (3) is deducible from (i) and (2). All that is contained therefore in the rules as originally stated may be expressed by (i) and (2); that is, the rules as originally stated were partly superfluous, and they may be reduced to (i) The financial committee shall be chosen from amongst the general committee ; (2) No one shall be a member both of the general and library committees. If (ii) is interpreted as implying that there are individuals whoare on both the general and libraiy committees, then it follows that (ii) and (iii) are inconsistent with each other. 28o SYLLOGISMS. [part III. 273. Are assumptions with regard to "existence" involved in any of the syllogistic processes i We may as in section 104 take three distinct suppositions with regard to the existential implication of propositions, and proceed to answer the above question on the basis of each in turn. The three suppositions are: — (i) All propositions imply the existence both of their subjects, and of their predicates. (2). No propositions imply the existence either of their subjects or of their predicates. (3) Particular propositions imply the existence of their subjects; but universal propositions do not JFirsfy we may take the supposition that every proposition implies the existence both of its subject and of its predicate. In this case, the existence of the major, middle and minor terms is guaranteed by the premisses, and therefore no further assumption with regard to existence is required in order that the conclusion may be legitimately obtained*. Secondly y we may take the supposition that no proposition logically implies the exisierue either of its subject or of its predi- cate. Let the major, middle and minor terms be respectively Py My S, The conclusion will imply that if there is any S there is some P or noti',(accordingas it is affirmative or negative). Will the premisses also necessarily imply this ? It has been shewn in section 141 that a universal affirmative conclusion. All S is Py can only be proved by means of the premisses, — All J/ is P, All iS* is Jf; and it is clear that these premisses themselves necessarily imply that * If however we are to be aUowed to proceed as in section H3, (where from all P is ^» all S is M^ we inferred that some not-^S is not-P)^ we must posit the existence not merely of the terms directly involved, bat also of their contradictories. CHAP. XL] SYLLOGISMS. 281 if there is any S there is some F, No assumption then with regard to existence is involved in syllogistic reasoning if the conclusion is universal affirmative. Again, as shewn in section 141, a universal negative conclusion, No »S is F, can only be proved in the following ways, — (i) No Mis P, (or No P is M), All 5 is M, therefore, No 5 is P, (ii) AUi'isJlf, No_5isJ^ (or No M\% S), therefore, No »S is P. In (i) the minor premiss implies that if S exists then Jf exists, and the major premiss that if Jfexists then not-Z'exists. In (ii) the minor premiss implies that if S exists then not--^ exists, and the major premiss that if not-Jf exists then not-P exists, (as shewn in section 104). It follows then that no assumption is involved if the conclusion is universal negative. Next, let the conclusion be particular. The implication of the conclusion with regard to existence is now contained in the premisses themselves, if the minor premiss is affirma- tive, and if the minor term is the subject oftheminor premiss, and the middle term the subject of the major pr jmiss, {i.e., if the syllogism is in Figure i). The same will be found to hold good on special examination of the moods of Figure 2 which peld particular conclusions. But it is otherwise with regard to the moods of Figures 3 and 4. Take, for example, a syllogism in Darapti^ — ^\M\sP, k\\M\%S, therefore, Some S\s P. 282 SYLLOGISMS. [part hi. The conclusion implies that if S exists P exists; but consistently with the premisses, S may be existent while ^and -P are both non-existent. An implication is therefore contained in the conclusion which is not contained in the premisses themselves. Our results may now be summed up as follows : — On the supposition that no proposition logically implies the existence either of its subject or of its predicate, we do not require to make any assumption with regard to existence in any syllogistic process yielding a universal conclusion in what- ever figure it may bCy nor in any syllogistic process yielding a particular conclusion provided it is in Figure i or Figure 2 ; but it is otherwise if a particular conclusion is obtained in Figure 3 or Figure 4. Thirdly^ taking the supposition that particular proposi- tions imply the existence of their subjects, although universal propositions do not, it will be found that assumptions with regard to existence are involved in syllogistic reasoning in the following and only in the following cases, — (i) In Figures 2 and 4, if the conclusion is particular ; (ii) In Figures i and 3, if the minor premiss is universal and the conclusion particular. The student should for himself fill in the steps necessary to establish this conclusion. 274. " Whatever P and Q may stand for, we may shew a priorithatsomeP is Q, For All PQ is Q by the law of identity, and similarly All PQ is P\ therefore, by a syllogism in Darapti, some P is Qr How would you deal with this paradox ? A solution is afforded by the discussion contained in the preceding section; and this example seems to shew that the enquiry, — how i2x assumptions with regard to exist- CHAP. XL] SYLLOGISMS. 283 ence are involved in syllogistic processes, — ^is not irrelevant or unnecessary. 276. If P is 0, and Q is R, it follows that P is R ; but suppose it to be discovered that no such thing as Q exists, — How is the truth of the conclusion, P is R^ affected by this discovery ? [L.] 276. De Morgan says: — ^"In all syllogfisms the existence of the middle term is a datum'' Inquire into the accuracy of this assertion. What does existence here mean.^ [l.] 277. On the supposition that no proposition logically implies the existence either of its subject or of its predicate, find in what case^ of the Reduction of Syllogisms to Figure i assumptions with regard to existence are involved. 278. Given that the middle term is distributed twice in the premisses of a syllogism, determine directly y (i.e., without any reference to the special rules of the figures, or the possible moods in each figure), in what different moods it might possibly be. The premisses must be either both affirmative,orone affirmative and one negative. In the first case^ both premisses being affirmative can dis- tribute their subjects only. I'he middle term must therefore be the subject in each, and both must be universal This limits us to the one syllogism, — All M\s P, AllJlfis5, therefore, Some S\& P. 84 SYLLOGISMS. [part ill. In the second case^ one premiss being negative, the con- clusion must be negative and will therefore distribute the major term. Hence, the major premiss must distribute the major term, and also (by hypothesis) the middle term. This condition can be fulfilled only by its being one or other of the following, — No M is F^ or No F is M. The major being negative, the minor must be affirmative, and in order to distribute the middle term it must be All Mis S. In this case then we get two syllogisms, namely, — No Mis F,hWMisS, therefore. Some S is not F. NoFisM, All Mis S, therefore, Some S is not F, The given condition limits us therefore to three syllo- gisms, (one affirmative and two negative); and by reference to the mnemonic verses we may now identify these with DarapH and Felapion in Figure 3, and Fesapo in Figure 4. 279. If the major premiss is affirmative, and if the major term is distributed both in premisses and conclusion, while the minor term is undistributed in both, determine directly the mood and figure, [n.] 280. Ifthe major term be distributed in the premisses and undistributed in the conclusion, deter- mine directly the mood and figure. [c] [Professor Jeyons gives this question in the form: "If the major term be universal in the premisses and particular in the conclusion, determine the mood and figure, it being understood that the conclusion is not a weakened one" CHAP. XL] SYLLOGISMS. 285 {Studies in Deductive Logic ^ p. 103) ; but the condition here introduced seems unnecessary, since we are in any case limited to a single syllogism.] 281. Given a valid syllogism with two universal premisses and a particular conclusion, such that if its subaltern is substituted for either of the premisses the same conclusion cannot be inferred, determine the mood and figure of the syllogism. If there is such syllogism, let Sy M, P be its minor^ middle and major terms respectively. Since the conclusion is given particular it must be either Some S is Py or Some S is not P^ Firsty if possible, let it be Some S is P, The only term which we require to distribute in the premisses is M, But since we have two universal premisses, two terms must be distributed in them as subjects \ One of these must be superfluous ; and therefore for one of the premisses we may substitute its subaltern, and still get the same conclusion. The conclusion cannot then be Some S is P, Secondly y if possible, let the conclusion be Some »S is not If the subject of the minor premiss is Sy we may clearly substitute its subaltern without affecting the conclusion. The subject of the minor premiss must therefore be My. which will thus be distributed in this premiss. M cannot also be distributed in the major, or else it is clear that its subaltern might be substituted for the minor and neverthe- 1 We here include the case in which the middle term is itself twice distributed.286 SYLLOGISMS. [part hi. less the same conclusion inferred. The major premiss must therefore be affirmative with M for its predicate. This limits us to the syllogism, — All /'is J/, No J/ is 5, therefore, Some S is not P; and this syllogism, which is AEO in Figure 4, does fulfil the given conditions, for if either premiss is made particular, it becomes invalid The above amounts to a general proof of the proposition laid down in section 147. Every syllogism in which there are two universal premisses with a particular conclusion is a strengthened syllogism^ with the one exception of AEO in Figure 4. [In his studies in Deductive Logic^ p. 105, Jevons gives the following: "Prove that wherever there is a particular conclusion without a particular premiss, something super- fluous is invariably assumed in the premisses." The case of AEO in Figure 4, however, shews that this needs qualifica- tion.] 282. Given two valid syllogisms in the same figure in which the major, middle and minor terms are respectively the same, shew, without reference to the mnemonic verses, that if the minor premisses are subcontraries, the conclusions will be identical. The minor premiss of one of the syllogisms must be O, and the major premiss of this syllogism must therefore be A and the conclusion O. The middle and the major terms having then to be distributed in the premisses, this syllogism is determined, namely, — CHAP. XL] . SYLLOGISMS. 287 All P is M, Some S is not My therefore, Some S is not P, Since the other syllogism is to be in the same figure, its minor premiss must be Some S\% M the major must there- fore be universal, and in order to distribute the middle term it must be negative. The syllogism then is also determined, namely, — ^oP\%M, Some S is M^ therefore, Some S is not P. The conclusions of the two syllogisms are thus shewn to be identical. 283. Given two valid syllogisms in the same figure in which the major, middle and minor terms are respectively the same, shew, without reference to the mnemonic verses, that if the minor premisses are contradictories, the conclusions will not be contra- dictories. 284. Is it possible that there should be a validsyllogism such that, each of the premisses being con- verted, a new syllogism is obtainable giving a conclu- sion in which the old major and minor terms have changed places } Prove the correctness of your answer by general reasoning, and if it is in the affirmative, determine the syllogism or syllogisms fulfilling the given con- ditions. If such a syllogism is possible, it cannot have two affir- mative premisses, or (since A can only be converted per 288 SYLLOGISMS, [part hi. auidcns) we should have two particulax premisses in the new syllogism. Therefore, the original syllogism must have one negative premiss. This cannot be O, since O is inconvertible. Therefore, one premiss of the original syllogism must be E. Firsty let this be the major premiss. Then the minor premiss mustbe affirmative, and its converse being a par- ticular affirmative will not distribute either of its terms. But this converse will be the major premiss of the new syllogism, which also must have a negative conclusion. We should then have illicit major in the new syllogism, and this suppo- sition will not give us the desired result Secondly^ let the minor premiss of the original syllogism be E. The major premiss in order to distribute the old major term must be A, with the major term as subject. We get then the following,satisfying the given conditions : — AllT'isi?/, No Mis 5, or No S is M, therefore. No 6' is F, or Some S is not F, that is, we really have four syllogisms, such that both pre- misses being converted, thus, — No iS is M, or No M is .S, Some M is F, — we have a new syllogism giving a conclusion in which the old major and minor terms have *' would be read " Anything that is both A and B is both C and Z>." (2) "What is either ^ or ^" is a complex term result- ing fromthe disjunctive combination of the simple terms A and^\ In what follows it must be remembered that I have adopted the view, that logically the alternatives in a dis- junction, (unless they are formal contradictories), are non- exclusive. Thus, if wespeak of anything as being "^ or B ^ we do not exclude the possibility of its being both A and B^ (compare section 109). In other words "-4 or j9" does not exclude " ABT The force of a disjunctive term when it is the subject of a proposition should be carefully noted. ** Anything that is either /* or Q is ^," or "whatever is either P ox Q is R^'^ may sometimes for the sake of brevity be written "/* or Q is Rr The latter expression, however, might also be inter- preted to mean "one of the two P ox Q'y& R, but we do notknow which"; and in consequence of this possible am- biguity, the more definite mode of statement, "Whatever is either P ox Qh R'" is to be preferred. A complex term may of course involve both conjunctive and disjunctive combination : e.g,, " AB or CI?" It is to be noted that the statement that anything is ^'A or B and ^ This kind of complex term is called by Jevons a plural term {Pure Logic ^ p. 25). Sofar as it requires a distinctive name I think I should preferto call it a disjunctive term, ^ The subject of this proposition is to be regarded as a single dis- junctive term. The same meaning might be given by saying "P and Q are -^," but in this case I should consider that we have two distinct subjects, and two propositions elliptically expressed. 19 2 292 COMPLEX INFERENCES. [part iv. at the same time C or 2? " is equivalent to the statement that it is " ^ C or AD or BC or BD:' We speak of 2i proposition as being complex if either its subject or its predicate is a complex term. 287. In a complex term the order of combination is indifferent. This is true whether the combination be conjunctive or disjunctive.Thus, AB and BA are precisely the same terms. It isobviously the same thing if we speak of anything as being both A and By or if we speak of it as being both B and A. Again "^ or ^" and "j9 or A^* have precisely the same signification. It is the same thing to speak of anything as being ^ or ^ as to speak of it as being B or A. 288. The Opposition of Complex Terms. We shall findit convenient to denote the contradictory of any simpleterm by the corresponding small letter. Thus for not- A we write ^, for not-^ we write b, A and a there- fore denote between them the whole universe of discourse (whatever that may be), but they denote nothing in common. In other words, whatever A may designate, it is necessarily true that Everything (in the universe of discourse) \& A ox a\ and that A is not a. It also follows that Aa necessarily represents a non-existent class ; what is both A and noi-A cannot have a place in any universe. However complex a term may be, we can always find its contradictory by applying the criterion laid down in section 28. "A pair of contradictory terms are so related that between them they exhaust the entire universe to which reference is made, whilst there is no individual of which they can both be at the same time affirmed." ^K^F^^mmimrmr^^^^t^^W m mtm'-l^fl^mt «■ ■■ i i ^w^im^Hp^va^ CHAP. I.] COMPLEX INFERENCES. 293 Now whatever is not AB must be either a or d, whilst nothing that is AB can be either of these; and vice versa, (AB, \a or ^, are therefore apairofcontradictories. Similarly, A or By jiby are a pair of contradictories. If, then, two simple terms are conjunctively combined into a complex term, the contradictory of this complex term is given by disjunctively combining the contradictories of the simple terms. And, conversely, if two simple terms are dis-junctively combined into a complex term, the contradictory of this complex term is given by conjunctively combining thecontradictories of the simple terms. In each case, we substitute for the simple terms involved their contradictories, and (as the case may be) change and for or^ or or for and. But howevercomplex a term may be, it must consist of a seriesof conjunctive and disjunctive combinations, and it may be successively resolved into the combination of pairs of relatively simple terms till it is at last shewn to result from the combination of absolutely simpleterms. For example, — ABC or DE or FG results from the disjunctive combination of the pair, — [ABC or DE, \ FG', ABC or DE resxilts from the disjunctive combination of the pair, — iABC, \DE\ 294 COMPLEX INFERENCES. [part iv. FG results from the conjunctive combination of the pair, — KG', and similarly we may resolve ABC, DE. We may hence deduce the following general rule for obtaining the contradictory of any complex term : — For each simple terminvolved^ substitute its contradictory ;everywhere change and for or, and or for and *. This rule is of simple application, and in what follows will be found to be ofveryconsiderableimportance. Its full force will be made more apparent later on. Thus the contradictory of A or BC is a and (^ or r), Le,y ab or ac. The contradictory of ABC or ABD is {a or b or ^)and {a or b or d\ which, as we shall presently shew', is resolvable into a ox b ox cd. In such statements as the above, the use of brackets is necessary to avoid ambiguity. Thus, a ox b ox c and a ox b or d might be read a ox b ox ca ox b ox d-, but we really mean that each term in the second set of alternatives is to be conjunctively combined with each term in the first set of alternatives. ^ In applying this rule, the information given by two such proposi- tions as "AT isP," " F is P," if stated in the form of a single pro- position, must be expressed " What is either X ox F is P," not ** X and Y are P." Compare section 286. ' Cf. section 196. CHAP, I.J COMPLEX INFERENCES. 295 Two terms may be inconsistentwithout being contra- dictories; i.e. J they cannot both be affirmed of anything, but it may be that thereare some things of which neither can beaffirmed. Thus, we can say that, whatever A, B and C may stand for, ^^AB is not ^C," (since if AB were ^C it would involve something being at the same time both B and XioXrB) ; but we cannot say that, whatever A^ B and C may stand for, "Everything is AB or ^C," (since something might be Ahc^ which is neither AB nor bC). If a con- jimctive termcontains a term which is the contradictory of a term contained in another conjunctive term then it follows that these two conjunctive terms are inconsistent. If two conjunctive terms are such that every term inone has corresponding to it in the other its contradictory, these two terms may be regarded as logical contraries, (com- pare the definition of contrary terms given in section 28}. Thus, AhC^ aBc may be spoken of as contraries. 289. The Development of Terms by means of the Law of Excluded Middle. By the Law of Excluded Middle, Everthing is B or b, and therefore, A is AB or Ab, Again, Everything is C or ^; therefore, AB is ABC or ABc, and Ab is AbC or Abc\ therefore, A is ABC or ABc or AbC or Abe, This is called the development of a term with reference to other terms ; thus, A is here developed with reference to B and C Compare Jevons, Pure Logic, p. 37. He calls any two alternatives which are the same, except as regards one term in each which are contradictories, a dual term. Thus, ^^AB or Ab^^ is a dual term as regards B, CHAPTER 11. THE SIMPLIFICATION OF COMPLEX PROPOSITIONS. 290. Types of Complex Propositions. Complex Propositions may be divided, — Firsty (as in the case of simple propositions), according as they are affirmative or negative ; e.g.y All AB is C ox D\ NoABis CoxD. Secondly^ (also as in the case of simplepropositions), according as they are universal or particular ; e.g,, All AB is C ox D ^ Some AB is C or D. We shall deal very little with particular complex propo- sitions, and it will frequently be found convenient to write universalcomplex propositions in the indefinite form. Thus, by AB is C or Z? we understand, — All AB is C or D. Thirdly, according as only the subject or only the predicate or both subject and predicate are complex terms ; e,g,, AB is C, Ai&B ox C, AB is C or D, CHAP. II.]COMPLEX INFERENCES. 297 Fourthly y according as there is or is not (ot) conjunctive combination in the subject, — e,g.y A IS C or D^ AB is C or Z> ; ()8) conjunctive combination in the predicate, — e,g,^ AB is C, ^^is CD\ (y) disjunctive combination in the subject, — e,g,y A is CDy Whatever is either ^ or ^ is CD ; (8) disjunctive combination in the predicate, — e.g., AB is C, AB is C or D, 291. The Resolution of Complex into relatively Simple Propositions. Affirmative. Affirmative complex propositions may be immediately resolved into relatively simple ones, so far as there is conjunctive combination in the predicate, or dis- junctive combination in the subject. Thus, — (i) XisAB is obviously resolvable into the two propositions, — ^is^, A'is^. (2) Whatever is either X or Fis A, is obviously resolvable into the two propositions, — ^is^, Fis^. 298 COMPLEX INFERENCES. [PART iv. Negative, Negative complex propositions may be im- mediately resolved into relatively simple ones, so far as there is disjunctive combination either in the subject or in the predicate. Thus, (i) Nothing that is either X or F is -^ is obviously resolvable into, — No X is A, No Y is A. (2) NoJTis^or^ is obviously resolvable into, — No X is Ay No^is^, The difference between affirmative and negative proposi- tions here must be carefully noticed. So far as there is con- junctive combination in the subject or disjunctive combina- tion in the predicate of an affirmativeproposition, or con- junctive combination either in the subject or in the predicate of a negative proposition, we cannot immediately resolve it into simpler propositions. Even in these cases, however, complex propositions may be resolved into relatively simple ones in a more roundabout way, namely, by the aid of obversion or con- traposition, as will be shewn subsequently. Compare espe- cially chapter v. 292. The Equivalence of Propositions. Two propositions are equivalent if each can be inferred from the other. Similarly, two sets of propositions are equivalent if every member of each set can be inferred from the other set chap.il] complex inferences. 299 When we omit terms from a proposition, or introduce fresh terms, or when in any way we obtain a proposition or set of propositions from another proposition or set of pro- positions, we should carefully distinguish two cases: — Firsts where the force of the original statement is un- affected, so that we can pass back from the new proposi* tion orpropositions to the original proposition or proposi- tions. Secondly^ where the force of the original statement is weakened, so that we cannot pass back from the new pro- position or propositions to the original proposition or pro- positions. In many cases it is of very great importance to know whether in a process of manipulation we have or have not lost any of the information originally given us. 293. The Omission of Terms from a Complex Proposition, the force of the assertion remaining unaf- fected. (i) // is superfluous for any simple term to appear more than once in a conjunctive term. Thus AA merely denotes the class A, ABB merely denotes the class AB, Such terms in their original form are tautologous, and the repetition of the term should therefore be struck out. Compare Boole, Laws of Thought^ p. 31, and Jevons, Pure Logic^ p. 15. (2) In a series of cUtematives it is superfluous for any given alternative to be repeated. To say that anything is "-4 or A'' is to say that it is -4; to say that anything is "-4 or BC or j9C" is to say that it is "-^ or BC^\ The repetition of an alternative should 300 COMPLEX INFERENCES. [part iv. therefore always be struck out Compare Jevons, Pure Logic y p. 26. (3) In a universal negative proposition it is superfluous for the same term to appear in every alternative in the subject and also in every alternative in the predicate, that is, in such a case it may be omitted either from the subject or from the predicate. For example, to say that No AB is -4 C is precisely the same as to say that No AB is C, or that No Bv& AC, For to say that No AB is -^C is the same thing as to deny that anything is ABAC; but, as shewn above, the repetition of the term A is superfluous, and the statement may therefore be reduced to the denial that anything is ABC And this may equally well be expressed by saying No AB is C, or No^ is^C. Compare also Chapter in, On the Conversion * of Complex Propositions. Similarly, No AB is AC or AD may be reduced to No AB is C or D, or to No B is AC ox AD, (4) If in an affirmative proposition a term that appears in every alternative in the subject appears also in any alterna- tive in the predicate, it may be dropped from the latter without affecting the force of the statement, A is AB may, (as shewn in section 291), be resolved into A is A, AisB. But ^ is ^ is a merely identical proposition and gives no information. AisAB may therefore be reduced to the single proposition ^is^. CHAP. II.] COMPLEX INFERENCES. 301 Similarly, AB is AC or BC may be reduced to AB is C or C, and therefore, as shewn above, to AB is a (5) If one of a series of alternatives is merely a stibdivi- sion of another of the alternatives it may be omitted without destroying any of the force of the original assertion. In other words, in adisjunctive term, "any alternative may be re- moved, of which a part forms another alternative," (com- pare Jevons, Pure Logic^ p. 26). Thus, AB is a subdivision of Ay and "-^ or AB^^ may therefore be reduced to "^." This may be shewn as follows : — X IS A ox AB] but, AB is A, therefore, X is Aox Ay therefore, Xis A-y and conversely, \i X is Ay since, by the law of excluded middle, A is AB or Aby it follows that X is Ab or AB] but, Ab is Ay therefore, Xis A ox AB, Similarly, X is AB or ABC or BD may be simplified by the omission of ABCy becoming X is AB or BD. (6) Anyterm which represents a non-existent class may bviously be dropped from a series of alternatives without altering the force of the proposition ; this is the case with 302 COMPLEX INFERENCES, [part iv. a termwhich involves a self-contradiction. " Aa *' means that which is bothA and not--^, but by the law of Contra- diction, no such class is possible. Such a term as Aa may therefore always be dropped. It follows, therefore, that if X\% A or Bb^ X\^ A\ and it is clear that there is here no weakening of the force of the original proposition. (7) Reduction of dual terms, (Compare section 289.) Another simplification is possible where we have two alternatives, one of which contains a term which is the contradictory of a term contained in the other, the remain- ing terms in each being the same; ^.^., ABC or ABc. These may be replaced by a single term containing only the ele- ments which are common to both the original alternatives, without any of the force of theoriginal proposition being lost. Thus, ''ABC or ABc'' may be replaced by "^^." For ABC and ABc are both AB\ and conversely, by the law of excluded middle, AB is ABC or ABc Thus, X is AB orAh ; but, AB is Ay and Ah is A ; therefore, X is A, We have also, Xis A ; but, A is AB or Ah, therefore, X is AB or Ah, (8) If in a series of cUtematives occurring either in the subject or in the predicate of a proposition, the contradictory cf any given alternative appears comhined with other terms in other alternatives it may he omitted from the latter without altering the force of the assertion. Thus, " A or aB '* may be replaced by " A or B,' and vice versa. CHAP. II.] COMPLEX INFERENCES. 30^ For, given X is -^ or aB\ since aB is B^ it follows that X is -4 or B. And, conversely, given X\% A or B; since B is AB or aB, it follows that X is ^ or AB or aB ; but, AB is -4, therefore, .Y is -<4 -4="" -4i5rf="" -4rf="" -="" -propositions="" -st="" -x="" .="" 11.="" 1="" 20="" 288.="" 289="" 293="" 294.="" 295.="" 296.="" 297.="" 298.="" 299.="" 2="" 300.="" 301.="" 302.="" 303.="" 304.="" 304="" 305.="" 305="" 306.="" 307.="" 307="" 308.="" 309.="" 309="" 310.="" 312="" 315="" 393="" 3io="" 3o6="" 3o8="" :="" a.="" a:="" a="" aa="" ab.="" ab="" abb="" abc="" abd="" abdr="" abe.="" abe="" above="" aby="" ac="" acd="" acis="" acj="" acy="" ad="" ade="" adopted="" ady="" ae="" aey="" affected.="" affecting="" affirm-="" affirma-="" affirmative="" affirmed="" affirming="" affirms="" again="" ah="" all="" already="" also="" alter-="" altering="" alternative="" alternatives.="" alternatives="" although="" always="" an="" analysed="" and="" andc="" any="" anything="" appeared="" appears="" application="" applying="" aproof="" are="" as="" asesta-blished="" 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itself="" iv.="" j="" jffrf="" jt="" just="" keep="" kind="" l.="" laid="" law="" laws="" length="" less="" loss="" lost="" made="" manipulation="" may="" mayalso="" mean="" means="" merely="" middle="" might="" more="" must="" n="" natives="" nature="" necessarily="" necessary="" negative="" negatives="" neither="" new="" no="" non-="" non-existent.="" nonexistenceof="" nor="" not-="" not-c="" not-z="" not="" nothing="" notice="" notices="" now="" numbers="" o="" obtain="" obtained="" obvious.="" obvious="" occurring="" of="" omission="" omitted="" on="" one="" or="" orae="" orbc="" ordinary="" original="" otacjd="" other="" othesis="" ox="" p="" pair="" pairs="" part="" particular.="" particular="" pass="" passage="" pf="" place="" plification.="" position="" positions="" pq="" pqr="" pre-="" preceding="" predi-="" predicate.="" predicate="" premisses.="" preted="" previously="" principles="" pro-="" proceedalways="" process="" processes="" promised="" proof="" proposition.="" proposition="" propositions.="" propositions="" prs="" ps="" q.="" q="" qr="" qrs="" questions="" r="" re-="" reasoning="" reduced="" regard="" relation="" replaced="" resolved="" result="" reverse="" rf="" rfor="" rj="" rrf="" rule="" s:i="" s="" same="" sametimeaband="" say="" second.="" second="" section.="" section="" sections.="" sections="" selves="" separate="" series="" set.="" set="" shew="" shewn="" should="" sidered.="" sim-="" similarly="" simple="" simplest="" simplification="" simplify="" simply="" since="" some="" something="" sphere="" state="" statement.="" statement="" steps="" still="" student="" subdivisions="" subject="" substituted="" t.e="" t="" taken="" term.="" term="" termmay="" terms.="" terms="" tf="" than="" that="" the="" theequivalence="" their="" them-="" then="" there="" thereby="" therefore.="" therefore="" they="" thing="" this="" those="" though="" thought="" three="" thrown="" thus="" time="" tion.="" tive="" to="" transferred="" treated="" troublesome="" true="" two="" types="" ue.y="" uni-="" universal="" unless="" us="" use="" usual="" v.="" v="" valent="" value="" various="" versa.="" versa="" versal="" version="" vice="" view="" viii.="" was="" we="" weakened="" were="" what="" whatever="" when="" where="" wherever="" whether="" which="" whichatermthat="" will="" with="" within="" without="" words="" would="" written="" x="" xis="" xy="" y="" yields=""> by d). It is worth while pointing out that from All A is BC'wq may obtain by conversion Some B is AC^ and Some C is AB ; but in complicated inferences we shall hardly ever have occasion to convert affirmative propositions in this way. We shall find however that to counterbalance this, the process ofcontraposition is particularly valuable in its application to complex universal affirmative propositions. 311. Shew clearly that if No De is ABc, then No ABcD \se\ if No ^ is Bdk, then No Bd is ck ; if No AbDFisK, then No AbcDE is FK\ if ABCis EF, then ABCG is BE\ if No AbDE is bCE, then No CDEF is AbH. CHAPTER IV. THE OBVERSION OF COMPLEX PROPOSITIONS. 312. The Obversion of Propositionscontaining more than two terms. The doctrine of Obversion is immediately applicable to Complex Propositions ; and we require no modification of our former definition of Obversion. From any given pro- position we may infer a new one by changing its quality and taking as a new predicate the contradictory of the original predicate. The proposition thus obtained is called the obverse of the original proposition. The only difficulty connected with the obversion of complex propositions consists in finding the contradictory of a complex term. We have, however, in section 288, given a simple rule for finding the contradictory of any complex term : — For each simple term involved^ substitute its contra- dictory ; write oxA/or or, and 01 for and. Applying this rule to ^^AB or ^," we have "(flf or b) and {A or B),'^ i,e,, *'Aa or Ab or aB or Bb*'; but since Aa and Bb involve self-contradiction, they may, as shewn in section 293, be omitted. The obverse, therefore, of "All X is AB or ab » is "No X is Ab or aBJ' CHAP. IV.] COMPLEX INFERENCES. 315 313. Find the obverse of each of the following propositions : — (i) A is BC, (2) A is BC or DE, (3) No A is BcE or BCF, (4) No ^ is ^ or bcDEfox bcdEF. (i) ^^A is BC gives at once " No -4 is ^ or cr (2) 'M is ^C or Z)^" gives "No A is (<^ or c) and at the same time (d or ^)." As already pointed out, it may be necessary to use brackets in this way to avoid ambiguity. Without brackets, however, and avoiding all chance of ambiguity, we may write the above, — "No A is bd or be or cd or ce^ The student should make it very clear to him- self that these two forms are really equivalent. (3) " No A is BcE or BCFP Here by the application of the general rule we have as the contradictory of the predicate, — "(^ or C or i) and at the same time ip or c or/)."What is "both b and h'' is of course "^," and we have no moreinformation about a thing if we are told that it is "both b and ^" than if we are told that it is simply "^"; it has also been already pointed out that such a term as Cc must represent what is non-existent, and therefore when it is given as one among several alternatives it may be neglected ; again as shewn in section 293 such an expression as "^ or be or bf or Cf* may be simplified to "^ or CfP Remembering these three points, we find that " (p or C or e) and {p or c or/)" maybe written "^ or Cfox ce or ef'^ For the obverse of the given proposition, we have, therefore, "-^ is ^ or Cfox ce or ^" (4) " No ^ is ^ or bcDEf or bcdEF:' The obverse is, — "^ is b and {B or C or d or e or F) and {B or C or Doxe or/)"; U., "^ is bC or bDFox be or bdp 3i6 COMPLEX INFERENCES. [part iv. 314. Find the obverse of each of the following propositions : — i) Nothing IS JT, ForZ; [2) JSfis A6 or aC; [3) W is XZ or Yz or YZ or Xy or xZ; [4) A6 is CDEf or Ci/ or rZy orcdE ; ;5) No De is ABC or Adc; [6) No -4 is CV/ or cDor^^rf.315. No citizen is at once a voter, a house- holder and a lodger ; nor is there any citizen who isneither of the three. Every citizen is either a voter but not a house- holder, or a householder and not a lodger, or a lodger without a vote. Are these statements precisely equivalent ? [v.] It may be shewn that each of these statements is the logical obverse of the other. They are therefore precisely equivalent. Let F= voter, z/ = not voter 3 ir= householder, h = not householder; Z = lodger, /= not lodger. The first of the given statements is No Citizen is VlfZ or v/il; therefore (by obversion). Every citizen is either z; or ^ or / and is also either V or If or Z; therefore (combining these possibilities), Every citizen is either Ifv or Zv or Vh or Zh or VI or HI, But (by the law of Excluded Middle), Ifv is either /fZv orIHv; CHAP. IV.] COMPLEX INFERENCES. 317 therefore,Hv is Lv or HL Similarly, Lh is ^ or Zz^; and VI is HI or Vh. Therefore, Every citizen is Vh or HI ox Lv^ which is the second of the given ^atements. Again, starting from this second statement, it follows {by obversion) that No citizen is at the same time v or Hy h ox L, I or V'y therefore, No citizen is vh or vL or HL, and at the same time / or Vy therefore. No citizen is vhl or VHL^ whichbrings us back to the first of the given statements. 316. Shew that the two followingpropositions are equivalent : — No ^ IS ^ or BC or BD or DE, X is aBcd or obDe or abd. CHAPTER V. THE CONTRAPOSITION OF COMPLEX PROPOSITIONS. 317. The application of the term Contraposition to propositions containingmore than two terms. According to ouroriginaldefinition, we contraposit a proposition when we infer from it a new proposition which has the contradictory of the old predicate for its subject and the old subject for its predicate. Thus, ** No not--5 is A" is the contrapositive of "All A isj9"; **A11 not--5 is not-A'* is its obverted contrapositive. Similarly, the contrapositive of "-4 is ^ or C" would be "No dc is A", the obverted contrapositive '*dc is a". The contrapositive of "-4 is BC" would be "No ^ or ^ is AJ* It will be observed, therefore, that the old rule for obtain- ing the contrapositive still applies, namely, — first obvert the given proposition, and then convert it. The contrapositive of a negative proposition is as before particular, and may be practically neglected. Thefollowing simple rule may then be given for ob- taining the obverted contrapositive of a universal affirmativeproposition : — Take as a new subject the contradictory of the old predicate^ and as a newpredicate the contradictory of the CHAP, v.] COMPLEX INFERENCES. 319 old subject^ the proposition still remaining affirmative. For example^ — A is BCjtherefore, whatever is ^ or ^ is a. A is B ox C, therefore,be is a. A isBC ox Ef therefore, whatever is be or ee is a* So far I have been discussing what may be called the full contrapositive of a complex proposition ; and starting with a universal affirmative we can pass back from such a contrapositive to the original proposition. In other words, any universal affirmative proposition and its full contra- positive are equivalent propositions. Inrelation to complex propositions, however, we shall find it convenient to give the term Contraposition an ex- tended meaning. We may say that we have a process of Contraposition when from a given proposition we infer a new one in which the contradictory of a term that appeared in the predicate of the original proposition now appears in the subject^ or the contradictory of a term that appeared in the subject of the original proposition now appears in the predicate. We may distinguish ^z^r operations which will be in- cluded under thisdefinition : (i) The operation of obtaining the full contrapositive of a given proposition, as above described and defined.(2) From ^^A is BC ox E^\ we may infer "whatever is be or ce is fl"; but in a given application it may be sufficient for ustoknow that " be is a '\ and although this is not the full contrapositive of the original proposition, we may regard it as immediately obtained from the original pro- position by a process ofcontraposition. With reference to this case, the following general rule may be given, — If one or other of a series of alternatives is 320 COMPLEX INFERENCES.[partIV. predicated of a subject^ the contradictory of this subject may be predicated of any term that is incompatible withall these alter- natives. Thus, if " A is PqR OTpRS*\ we may infer that "/j is fl"; since /^ is neither PqR nor pRS, and therefore what isps cannot beA, But we have not here the full contra- positive ofthegiven proposition, and we could not pass back from ''ps is a'' to "^ is PqR or pRS'\ but only to ''A is Pot S:* (3) From the proposition "^ is B or C", it follows that if A is not B, it is C;but this is expressed by the proposition *^Ab is C", and the contradictory of a term that originally appeared in the predicate now appears in the subject, —/.d,according to the above definition we have a process of contraposition. This process might also bedescribed as the omission of one or more of a series of alternatives in the predi- cate by a further particularisation of the subject. With reference to this case, the followinggeneral rule may be given, — If any term X is combined with every al- ternative in the subject of a proposition^ every altemaiive in the predicate which contains the contradictory of this term may be omitted^. Thus, from Whatever is -^ or -5 is C or DX orEx^ wemayobviously infer Whatever is AX or BX is C or D. ^ This may sometimes result in the disappearance of all thealterna- tives; and the meaning of such result is that we now have a non- existent subject. Thus, given P is ABCD or Abed or aBCd, if we particularise the subject by making itPbC^ we find that all the alternatives in the predicate disappear. The interpretation is that the class PbC \& non-existent, u e,^ l^o P \%bC\ a conclusion which of course might also have been obtained directly from the given proposi- tion. CHAP, v.] COMPLEX INFERENCES. 331 (4) The last operation to which reference is made above is the reverse of that which we have just discussed. From the proposition ^^AB is C", we may infer "A is d or C". This may be described as a generalisation of the subject by the additionof one or more alternatives in the predicate* But it is also clear that it comes under the extended mean- ing that we have given to the term Contraposition. To meet this case, the following general rule may be given, — Any term that appears in the subject of a proposition may be dropped therefrom^ if its contradictory is at the same time added as an additional cUtemative in the predicate. The following may be taken as typical examples of all the operations that we now include under contraposition: — AB is CD or de\ therefore,^rj/, anything that is either cD or cE or dE is a or by (the full contrapositive, obverted, according to our original definition); secondly f cE is a or b} thirdly, ABD is C; fourthly, A is b or CD or de. Combinations of the third and fourth operations give AcD is b ; Ad is b ot e\ &c The first of the above being called the Full Contra- positive of the given proposition, the remaining inferences may be called Partial Contrapositives, according to our extended definition of contraposition. In each case, some term disappears from the subject or from the predicate of the original proposition, and is replaced by its contradictory in the predicate or the subject accordingly. Only in the full contrapositive, however, is every term thus transposed. K. L. 21 322 COMPLEX INFERENCES. [part iv. I do not think that any confusion need result from the nomenclature now proposed, since the extended use of the term Contraposition can be applied only to complex propo- sitions. There is still only one kind of Contraposition possible in the case of the categorical proposition containing but two terms. The great importance of Contraposition as we are now dealing with it in connection with complex propositions is that by its means, given a universal affirmative proposition of any complexity , we may obtain separate information with regard to any term thai appears in the subject^ or with regard to the contradictory of any term that appears in the predicate^ or with regard to any combination of these terms. Thus, given "-X'F is P or Qr^\ by the process described as the generalisation of the subject, we have X\&y ox P ox Qr, The particularisation of the subject gives XYp is Qr, XYq is P, &c. ; and by the combination of these processes, we have J^is^or Qr-^ &c. Again, the full contrapositive of the original proposition is Whatever is/<^ oxpR is x oxy; from which we have pisx oxy ox Qr, qisxoxyoxPf &c.CHAP, v.]COMPLEX INFERENCES. 323 318. Given "All D that is either 5 or C is ^," shew that " Everything that is not--4 is either not--ff and not-C or else it is not-Z>." [De Morgan.] This example and the five following examples are adapt- ed from De Morgan, Syllabus^ p. 42. They are also given by Jevons, Studies, p. 241, in connection with his Equational Logic. They are all simple exercises in Contraposition. We have, What is either BD or CD is A^ therefore, a is (^ or d) and {c or //), therefore, a is be or d. 319. Given " All A is either BC or BD;' shew that "All that is not-B and not- C is not-^" and "All that is noi'D is not-^." [De Morgan.] 320. If A is either BC or i?, and if whatever is BC or D is Ay shew that whatever is not- A is not-Z> and also either not-ff ornot-C, and whatever is not-Z> and at the same time not-B or not-C is noX-A. [De Morgan.] 321. If whatever is B or CD or CE is A^ what- do we know about not-^ ? [De Morgan.] 322. If whatever is either B or C and at thesametimeeither D or ^ is Ay what do we know about not--4 ? [De Morgan.] 323. If that which is A or BC and is also D or EF is Xy what is all that we know about not-JSf ? [De Morgan.] What is {A or BC) and (Z> or EF) is X\ therefore (by contraposition), ;k; is ^^ or o^ or ^ or ^ 21 — 2 324 COMPLEX INFERENCES. [part iv. [TTifere is apparently a misprint in Jevons's transcription of this example {Studies^ p. 241). He uses different letters, but his implied solution is " x is either ab or c and it is also either de Qxf\ I cannot see how this is obtainable.] 324. To say that whatever is devoid of the pro- perties of A must have those either of B or of D^ or else be devoid of those of C, is the same as to say that what is devoid of the properties of B and Z>, but possesses those of C, must have A. Prove this. [Jevons, Studies^ p. 239.] 325. Shew that "C is -^^ or aB'' is equivalent to the two propositions ^^AB is r" and ^^ab is c^\ [Jevons, Studies, p. 239.] 326. Prove the equivalence of the following as- sertions : — (i) Every gem is either rich or rare. (2) Every gem which is not rich is rare. (3) Every gem which is not rare is rich. (4) Everything which is neither rich nor rare is not a gem. [Jevons, Studies, p. 229.] 327. If that which is devoid of heat and devoid of visible motion is devoid of energy, it follows that what is devoid of visible motion but possesses energy cannot be devoid of heat. [Jevons, Studies, p. 199.] 328. If the relations A and B combine into C, it is clear that A without C following means that there is not B, and that B without C following means that there is not A. [De Morgan.] CHAP, v.] COMPLEX INFERENCES. 325329. Any one who wishes to test himself and his friends upon the question whether analysis of the forms of enunciation would be useful or not, may try himself and them on the following question : — Is either of the following propositions true, and if either, which ? (i) AH Englishmen who do not take snuff are to be found among Europeans who do not use tobacco. ) All Englishmen who do not use tobacco are to be found among Europeans who do not take snuff. Required immediate answer and demonstration. [De Morgan.] 330. Is the student of logic, generally speaking, prepared rapidly to analyse the two following pro- positions, and to say whether or no they must be identical, if the identity of synonyms be granted } (i) The suspicion of a nation is easily excited, as well against its more civilised as against its more warlike neighbours, and such suspicion is with diffi- culty removed. (2) When we see a nation either backward to suspect its neighbour, or apt to be satisfied by ex- planations, we may rely upon it that the neighbour is neither the more civilised nor the more warlike of the two. [De Morgan.] 331. Infer all that you possibly can by way of Contraposition or otherwise, from the assertion, All A that is neither B nor C is X. [r.] 326 COMPLEX INFERENCES. [part iv. The given proposition may be written Abe is X; and taking it as it stands, the converse is Some X is Abe, and the contrapositive (obverted) x \s a ox B ox C. We may also get a number of other propositions with which we may proceed in the same way; e,g.y — Abx is Cy Acx is By &c. Confining ourselves however to such universal proposi- tions as can be obtained, the problem may be solved generally as follows : — Thegiven proposition may be thrown into the form, — Nothing is at the same time Ay by c and x\ and we see that it is symmetrical with regard to the terms Ay by Cy X. We are sure then that anything that is true of A is true mutatis mutandis of by c and Xy that anything that is true of Ab is true mutatis mutandis of any pair of the terms, and similarly for combinations three and three to- gether. We have at once the four symmetrical propositions, — A\^Box CoxX'y (i) b \% a ox C ox X; (2) ^ is ^z or ^ or X; (3) X IS a ox B ox C, (4) ' Then from (i) we have Ab is C or X', (i) and the five corresponding propositions are, — Ac IS B ox X] (ii) Ax is B ox C; (Hi) be is a ox X; (iv) bx is a or C; (v) ex is a or B» (vi) CHAP, v.] COMPLEX INFERENCES. 327 Again from (i), — Abe is Xy (which is the original proposition), (a) and we have, similarly, — Abx is C\ (fi) Acx is B; (y) bcx is a, (8) It should be noted that the following are pairs of contra- positives, — (I) (8), (2) (y), (3) 03), (4) (a), CO (vO, (ii) (v), (iii) (iv). 332. Find the full contrapositive of each of the following propositions : — A is BCDe or bcDe ; AB is CD or cDE or de ; Whatever is AB or 6C is aCd or Acd; Where A is present along with either B ox C,D is present and C absent or D and E are both absent ; Whatever is ABCorabc is DEF or def. 333. Compare the logical force of the following propositions : — . (i) All voters who are not lodgers are house- holders who pay rates ; (2) No one who is not a lodger and who does not pay rates is a voter ; (3) A voter who is a householder is not a lodger; (4) A householder who does not pay rates is not a voter ; (5) All who pay rates or are householders are voters; 328 COMPLEX INFERENCES. [part iv. (6) Anyone who is not a householder or who being a householder does not pay rates is either not a voter or else he is a lodger ; (7) All who have a vote pay rates ; (8) Anyone who has no vote is either not a rate- payer or not a householder. 334. If A is either B or C, shew that whatisnotBis either C or not A. 336. What is the difference between the assertion that A is BC and the pair of assertions that 6 is a, and ^ is ^ ? [Jevons, Studies, p. 239.] 336. KA unless it is B is either CD oxEF, shew that not-C is either not-^ or B or EF. 337. Establish the following, — (i) Where B is absent, either A and C are both present or A and D are both absent; therefore, where C is absent, either B is present or D is absent. (ii) Where A is present and also either B or E^ either C is present and D absent or C is absent and D present; therefore, where C and D are either both present or both absent, either A is absent or B and E are both absent. (lii) Where A is present, either B and C are both present or C is present D being absent or C is present F being absent or H is present ; therefore, where C is absent, A cannot be present H being absent. 338. Among plane figures the circle is the only curve of equalcurvature. Shew that this is the same CHAP, v.] COMPLEX INFERENCES. 329 as to assert that a plane figure must either be a curve of equal curvature, in which case it is also a circle, or else, not a circle and then not a curve of equal curva- ture. [Jevons, Studies^ p. 235.] Let F= plane figure, C= circle, E = curve of equal curvature. "Among plane figures the circle is the only curve of equal curvature," may be expressed by PC is CE^ and Fc is ce, "A plane figure must either be a curve of equalcurvature, in which case it is also a circle, or else, not a circle and then not a curve of equal curvature," becomes F is CE or ce. It is immediately obvious that the two statements are equivalent. 339. " Similar figures consist of all figures whose corresponding angles are equal and whose sides are proportional." Give all the propositions involving not more terms, which can be inferred from the above. Give also one proposition equivalent to it [L.] Let F= similar figures, Q = figures whose corresponding angles are equal, R = figures whose sides are proportional. The given statement may be resolved into the two pro- positions, — All F isQR, All QR is F. From these, by contraposition, we may infer, — / is ^ or r; Q is FR oxpr\ «^iiW 33P COMPLEX INFERENCES. IpaRt iv. R is FQ oxpq\ FQ is R] FR is Q'y pQisr; pR is q\ Qr'isp', qR is/; qr is p. Fq and Fr represent non-existent classes ; and we have no information with regard to pq and pr. This I think affords a complete solution of the first part of the question. We may obtain a statement equivalent to the given statement, by taking the full contrapositives of the two propositions into whichweresolved it and then combining them. Thus, F is QR, ^q or r is/, QR is F, =/ is q or r; and these are combined in the statement that/ consists of all things that are q or r. "Figures that are not similar consist of all figures whose corresponding angles are not equal or whose sides are not proportional." 340. Given AisBC, what, if anything, do you know concerning the classes AB, A6, AC, Ac, a, aB, ad, aC, ac, B, BC, Be, b, bC, be, C, e? A is BC, therefore, by conversion. Some BC is A. (i) By contraposition, we may obtain the two propositions, No ^ is ^, (2) No e is A. (3) CHAP, v.] COMPLEX INFERENCES. 331 Then by once more obverting and converting, Some « is ^ ) (^) Some ais c, ) We cannot combine these into " Some a is dc" since we do not know that the same a is referred to in both cases. The other forms which can be obtained are inreality only weakened forms of one or other of the above. By (3) No c is A, L e,y Nothing is Ac^ (5) therefore {afartwri). No B is Ac, (6) Similarly, by (2), No b is Ay u e,, Nothing is Aby (7) therefore (a/orfiori), No C is Ab. (8) Again, by obversion of the original proposition, No A is Cy therefore (a fortiori) ^ No AB is c* (9) Similarly, No ^ C is ^. (10) Also from (2), No^Cis^, (11) and No ^r is y^. (12) Similarly, from (3), No Be is A, (13) 347. Given that PQr is ^5^ or ^*Z? or aCDE or BCdeF or bCdf or CDEF or rf^, what is all that you know concerning the classes, — A, a, B, b, C, c, D, d, E, e, F,f> 348. The Inversion of Complex Propositions. We might define Inversion in connection with complex propositions as a process by which from a given proposition we infer a new one in which some term in the subject is replaced by its contradictory. I have not, however, thought it worth while to give any detailed discussion of inversion here, because this process always results in par- ticular propositions; these are of small importance at the best, and they involve assumptions concerning the existenceof their subjects, which are so inconvenient whenthese subjects are very complex, that they are best neglected alto We cannot obtain any information with regard to the remaining classes aB^ ab, aCy ac, [As already indicated, I should consider that (i) and (4) involve assumptions with regard to " existence." Without any such assumptions,however, we can obtain all the re- maining inferences. We may regard (4) as obtained by inversion of the original proposition. Cf. section 348.] 341. Assuming that armed steamvessels consist of the armed vessels of the Mediterranean and the steam-vessels not of the Mediterranean, inquire whe- ther we can thence infer the following results : — 332COMPLEX INFERENCES. [part IV. (i) There are no armed vessels except steam- vessels in the Mediterranean. (2) All unarmed steam-vessels are in the Medi- terranean. (3) All steam-vessels not of the Mediterranean are armed. (4) The vessels of the Mediterranean consist of all unarmed steam-vessels, any number of armed steam-vessels, and any number of unarmed vessels without steam. [Jevons, Studies, p. 231, from Boole.] 342. If AB is either Cd or cDe, and also either eF or //, and if the same is true of BII, what do we know of that which is E ? We have given, — What is AB or ^^is {Cd or cBe) and (eFor If); therefore. What is AB or ^ZTis CdeFox cJDeF or Cdffor cDeH; therefore, What is ABE or BHE is CdH\ therefore, E is CdHox b or ah. 343. If A that is B is either P ox Q and also either R or S^ and if the same is true of A that is both C and D, what is all that we know about that which s neither P hor 6" ? 344. Given that whatever is PQ or A P is bCD or abdE or aBCdE or Abed, shew that, — ; (i) abP is CD or dE or q\ (2) DP is bC or aq ; (3) Whatever is B or Cd or cD \s a or p\ (4) B\sC or p or aq ; CHAP, v.] COMPLEX INFERENCES. 333 (5) Cd is a or p;(6) AB is p\ (7) li aeis c or d itisp or q\ (8) If BP is c or D or e it is aq. 346. Given A is BC or BDE or BDF, infer de- scriptions of the following terms Ace, Acf,ABcD. [Jevons, Studies, pp. 237, 238.] In accordance with rules aheady laid down, wehaveimmediately,— Ace is BDF-y Acfis BDE ; ABcD isEor R 346. Given b is CDe or Acd or adeF or acdEFy infer descriptions of -4, ^, CZ?, cD, de. gether, unless a very special treatment is accorded to them. 334 COMPLEX INFERENCES. [part it. 349. Summary of the results obtainable by Con- version, Obversion, and Contraposition. (i) By Conversion of a universal negative we can obtain separate information with regard to any term that appears either in the subject or in the predicate, or with regard to any combination of these terms. For example, No AB is CD ; therefore, No A is BCZ>, No C is ABI?, ^o JBD is AC. (2) By Obversion we can change any proposition from the affirmative to the negative form, or vice versa. For example, AB is CD or EF', therefore, No AB is ce or cf ox de ox df, Noi'is QR'y therefore, Z' is ^ or r. (3) By Contraposition of a universal affirmative we can obtain information with regard to any term that appears in the subject, or with regard to the contradictory of any term that appears in the predicate, or with regard to any combi- nation of these terms. For example, AB is CD or EF] therefore, -^ is ^ or CD or EF^ CIS a ox b ox EF, Be is a or CD, ce is a or b, Ad/ is by &c. CHAPTER VI. THE COMBINATION OFCOMPLEX PROPOSITIONS. 360. The Combination of Universal Affirmative Propositions the subjects of which do not contain Contradictories. X is Pj or P^ or ... or P^^ yis Ci or^, or ... or Q^, may be taken as types of two such propositions. By combining them we have XY\% (P^ or P^OT ... or PJ), and also (QiOT Q^or ...or ^J; t.e.f XYis P^Q^ or P^Q^ or ... or P^Q^ or P^Q^ or P^Q, or ... or P^Q^ or or P^Q, or P^Q, or ... or P^Q^. If the subject of both the original propositions had beeu X, then of course we should have X is P^Q, or P^Q^ or &c In this case, the new proposition is equivalentto the two propositions with which we started, />., we could pass back 336 COMPLEX INFERENCES. [part iv. from it to them. But when the subjects of the original propositions are not the same, the new proposition is not equivalent to them. In combining propositions, the student should never lose an opportunity of simplifying his results ; and such opportunities will be found to be of continual recurrence. The following are examples : (i) ^ is C or jD, B'\scE\ therefore, AB is cDEy since thecombination of C and cE is self-destructive. (2) ^ is ^ or C, A is r orZ> ; therefore, A is Be or BD or CD. (3) X is AB or bee, YhaBCox DE] therefore, XY is ABDE ; for again it will be found that all the other combinations in the predicate contain contradictories. (4) X\% A 01 Be or 2?, y is aB or Be or Cd\ therefore, -STFis Be or aBD ox ACd. The alternatives in full are AaB or ABe or A Cd or aBe or Be or BeCd or aBD or BeD or CdD. But AaBy BeCdy CdD represent non-existentclasses and may therefore be omitted. ABe, aBe, BeD are merely partial repetitions oi Be, and therefore they too may be omitted. Compare section 293. After a very little practice, the student will find it unnecessary to write out the alternatives in full. CHAP. VI.] COMPLEX INFERENCES. 337 (S) XisA or bd or eE^ F is ^C or aBe or d\ therefore, XY\& AC oibdoxAdor cdE. 361. The Combination of Universal Negative Propositionsthe subjects of which do not contain Contradictories. No AT is/', No ris c, may be taken as types of two such propositions. By combining them we have simply, — No -YF is either /'or Q. The following are examples : (i) No A IS cdy No -5 is Cor ^; therefore, No AB is either C or ^ or cd. (2) No A is hCy No A is Cd'y therefore, No A is be or Cd. (3) No X is either aB or aC or bC ox aE or bEy No Y is either Ad or Ae or bd or be or ^^ or r<(? ; therefore, No XY is either aB or « C or ^C or aE or ^-£ or Ad or ^^ or ^// or be or cd or ce ', therefore, No XY\^ either a or b or dor e^. (4) No X IS abd or a Cd, • No Yis be or bJD or ACD; therefore. No XY is abd or ^CaT or be or ^i? or A CD, (5) No Jif is aBC or^z CZ? or aBe or izZV, No Fis AeD or a^Z? or ^^Z> oraDE or ^Z>^ ; therefore. No XY\% aBC or aD or ^77 or ^^^'. ^ Of. section 302. * Cf. section 303. K. L. 22 33» COMPLEX INFERENCES. [partiv.382. The Combination of Propositions the sub- jects of which contain Contradictories. Such propositions cannot be direcdy combined in the manner just discussed. If AB is i>, and ^C is ^, we are not really ipven any inllMiDation by beii^ told that what is both AB said bC'isDE. To avoid this diflicul^ we must by partial contia- pontion remove bath the contradictories into the predicates of their respective propositions. Thusy the propositions **AB is 2>" and ^bC is E^ may be tedaceto the forms **-^ is ^ or D" and ^CisBcfr E"; and we then have by combining them, **AC is bE cr BD ixDE"". Starting with such a pair of propositions as the above, it is requisite to take both the contradictories into the predicates, or we shall still be left with a merely identical proposition* For example, combining ^AB is If* and ** C isBor E"*, we have ""ABC is B or D or E^, wbidk obviously tells us nothing. If, however, the propositions can be reduced to such a form that the subject terms so far as they are not contrail dictories are the same, thepredicates also being the same; then we may obtain a new proposition by just omitting the contradictory terms. Thus if we have propositions of the form AB is CfAbis C, we may infer (since A is AB orAb) that A is C. The same result is also obtainable by means of thende previously given, — AB is C, and Ab is C, therefore, ^ is ^ or C, and ^ is ^ or £7, therefore, -<^ is -^C or ^C or C, therefore, ^ is C CHAPTER VII. INFERENCES FROM COMBINATIONS OF COMPLEX PROPOSITIONS. 353. Problem. — Given any proposition, and any term JT, to discriminate between the cases in which the proposition does, and those in which it does not, afford information with regard to this term. We may assume that the original proposition is not an identical proposition. If it is negative, let it by obversion be made affirmative. Then, written in its most general form, it will be Whatever is P^P^...ot QiQ^'-or &c is A^A^ ... or B^B^.,^ or &C. As shewn in section 291, this may be resolved into the independent propositions : — PyP^ ... is A^A^ ... or BJS^ ... or &c. ; Q^Q^ ... is A^A^ ... or B^B^ ... or &c. ; &a &C. &c; in none of which is there any disjunction in the subject We may deal with these propositions separately, and if any one of them affords information with r^ard to Xj then of course the original proposition does so. We have then to consider a proposition of the form P^P^ ... P^ is A^A^ ... or B^B^ ... or &c. 22 — 2 340 COMPLEX INFERENCES. [part IV. From this by contraposition we get, — Everything is A^A^ ... or B,B^ ... or &c. orp^ or/, ... or/,; aiiA\\sace,Xh A^A,...OT B^B, ...or&cor/, or/, ... or/,. We may now strike out all alternatives in the predicate which contain x. If they all contain x, then the information afforded us with regard to X is that it is non-existent. If some alternatives are left, then the proposition will a£ford information concerning X unless, when the predicate has been simplified to the fullest possible extent, one of the alternatives is itself J?' uncombined with any other term, in which case it is clear that we are left with a merely identical proposition. Now one of these alternatives will be .y in any of the following cases, and only in these cases : — Mrst, If one of the alternatives in the predicate of the ori^nal proposition, when reduced to the affirmative form, isX Secondly, If any set of alternatives in the predicate of the original proposition, when reduced to the affirmative form, constitute a development of X; {since "AXaz aX" is equivalent to X; "ABX or A^X or aBX or abX" is also equivalent to X; and SO on). Thirdly, If one of the alternatives in the predicate of the .original proposition, when reduced to the affirmative form, contains X in combination solely with some term or terms appearing also in the subject ; since in this case such alternative is equivalent to X simply. For example, "AB is AX ox D"i3 equivalent to "AB is Xor I?." CHAP. VII.] COMPLEX INFERENCES. 34i By contraposition of this proposition in its original form we have, — Everything is AX ox D or a or d, but, (cf. section 293), *^AXor a^' is equivalent to "Yor a.*' Fourthly^ If one of the terms originally contained in the subject is x\ since in that case we should after contra- position have -Y as one of the alternatives in the predicate. The above may now be summed up in the propo- sition : — Any non-identical proposition will afford information with regard to any term X^ unless (after it has been brought to the affirmative form\ {j) one of the alternatives in the predicate is X^ or (2) any set of alternatives in the predicate constitute a development of -ST, or (3) any alternative in the predicate con- tains X in combination with such terms only as appearlso in every alternative in the subject^ or (4) every alternative in the subject contains x. If, after the proposition has been reduced to the affinna- tive form, the simplifications noticed in section 293 have been effected, then the criterion becomes more simply, — Any non-identical proposition will afford information with regard to any term X, unless y {after it has been brought to the affirmative form^ and its predicate so simplified that it contains no superfluous terms) ^ one of the alternatives in the predicate is Xy or every alternative in the subject contains x. If instead of -X" we have a complex term XYZ, then no part of this term must appear as an alternative in the predi- cate, and there must be at least one alternative in the subject which does not contain the contradictory of any part of this complex term : 1.^., no alternative in the predi- cate must htXy F, or Z, and some alternative in the subject must contain neither x^ y, nor z. The above criterion is of simple application. 42 COMPLEX INFERENCES. [part iv. 364. Say, by inspection, which of the following propositions give information concerning A, aB, b; bCdy respectively : — Ab is bCdox c\ bd is A or bC or abc ; Whatever is aox B v& c oxD\ Whatever is -^4^ or be is bE or cE ox e\ X is AX or ab or Be or Cd. 355* Problem. — ^Given any number of propositions involving any number of terms, tO' determine what is all the information that they jointly afford with regard to any given term or combination of terms that they contain. The great majority of direct problems* involving com- plex propositions may be brought under the above general form. If the student will turn to Boole, Jevons, or Venn, he will find that it is by them treated as the central problem of Symbolic Logic. A general method of solution is as follows: — Let X be the term concerning which information is desired. Find what information each proposition gives separately with regard to JT, thus obtaining a new set of propositions of the form X is P^ or P^ or ... or P^. This is always possible by the aid of the rules given in the preceding chapters*. It should be remembered that in section 353 we have discriminated the cases in which any given proposition really affords information with regard to X. ^ Inverse problems are discussed in chapter xii. ' The importance of these rules, especially of those relating to Con- traposition, is now made more apparent. . . CHAP. VII.] COMPLEX INFERENCES. 343 Those propositions which do not do so may of course be altogether left out of account. Next combine the propositions thus obtained in the manner indicated in section 350. This will give the desired solution. The method might be varied by bringing the proposi- tions to the form, — No -X'is Q^ or Q^ or ... or ^», then combining as in section 351, and finally obverting the result It will depend on the form of the original proposi- tions whether this Variation is desirable ^ If information is desired with regard to several terms, it may be found convenient to bring all the propositions to the form, — Everything is P^ or F^ ... or F^; and to combine them at once, getting in a single proposition a summation of all the information given by the separate propositions taken together. From this we may immediately obtain all that is known concerning X by leaving out every alternative that contains ;r, all that is known concerning V by leaving out every alternative that contains j', and so on. The following may be taken as a simple example of the method. It is adapted from Boole (Laws of Thought^ p* 1x8). ** Given ist, that wherever the properties A and B are combined, either the property C, or the property Z>, is present also; but they are not jointly present: 2nd, that wherever the properties B and C are combined, the pro- perties A and D are eitherboth present with them, or both ^ The second method bears a somewhat close resemblance to Jevons's Indirect Method ; though it is not quite the same. The first method however is quite distinct from Jevons'smethod. 344 COMPLEX INFERENCES. [part iv. absent Shew that where A is present, either j5 or C is absent" The premisses may be written, — AB is Cd or cD\ (i) BC is AD or ad. (2) Then, we may immediately obtain, — from (i), A\%b 01 Cd or cD\ • • • and from (2), ^ is ^ or ^ or D^\ therefore (by combining these), ^ is 3 or €D\ therefore, ^ is ^ or ^; which is the desired result This is a simple example; but many more complicated ones will be found in the following pages. The method here described will I think in nearly every case be found less laborious than that employed by Jevons*, — namely, the writing down all the possible a priori alterna- tives so far as the terms involved are concerned, and then striking out those that are inconsistent with the premisses! Also, while it neither requires that propositions shall be reduced to the form of equations, nor involves the use of mathematical symbols or diagrams, I have not in practice found it less effective than the methods of Boole and Venn*. I shall further endeavour to shew in subsequent sec- lions, how special results may frequently be obtained in a still simpler way by the aid of various formal processes. In some of the examples that follow both the general method and special methods are employed. ' An intermediate step might be introduced here, namely, ABC is D, * Pure Logic, pp. 44, 45 ; Principles ofScience^ chapter vi. * At the same time of course these methods have a peculiar interest and significance of their own. CHAP. VII.]COMPLEX INFERENCES. 345 While the special methods are as a rule to be preferred when they have been discovered, it generally takes some time and ingenuity to discover them. On the other hand, the general methodabove described may be always imme- diately applied without any preliminary study of the case.Also, while special methods are useful to establish given results, we can ordinarily be satisfied that we have a com- plete solution with regard to any term only when we have employed the general method. CHAP. VIII.] COMPLEX INFERENCES. 347 68. Given (i) P is QR, (2) /is qr; shew that (3) Q is RP, (4) R is PQ. 369. Given (i) R is P ox pq, (2) ^ is i? or Pr, (3) qR is P\ shew that/ is ^r. 360. Whenever X is present, F and Z are both present; and whenever X is absent, Y and Z are both absent. What can thence be inferred with regard to the relation between Y and Z ? 361. (i) Wherever there is smoke there is also fire or light ; (2) Wherever there is light and smoke there is also fire ; (3) There is no fire without either smoke or light. Given the truth of the above propositions, what is all that you can infer with regard to (i) circumstances where there is smoke ; (ii) circumstances where there IS not smoke ; (iii)circumstances where there is not light ? [W.] Let A = circumstances where there is smoke, B = circumstances where there is light, C= circumstances where there is fire. The premisses are, — ^ is ^ or C, (i) AB is C, (2) Cis^or^.' (3) (i) and (2) yield Ai% C, By (3) ab is c\ therefore^ ^ is ^ or c. By (i) and (3), ^ is « or C, and also A or c\ therefore, b is AC or ac. 348 COMPLEX INFERENCES. [part iv. We have then, —(i) Where there is smoke, there is fire; (ii) Where there is not smoke,there is either light or there is no fire; (iii) Where there is no light, there is cither both fire and smoke or neither fire nor smoke. 362. Shew the equivalence between the two sets of propositions, — (i) A is BC, Bis AC, C IS AB. (ii) A is BC, a is be. Bis ACi C is AB, give by contraposition a is be, a is be gives by contraposition B is A, C is A; and since A is BC, we have Bis ACf C is AB. 863. Shew the equivalence between the following sets of propositions : — (i) b is aC, c is aB ; (2) A is BC, b is aC\ (3) AisBC, c is aB. 364. Shew the equivalence between the following sets of propositions : — (i) a is BC, bis AC, Cis Ab or aB\ CHAP. VIII.] COMPLEX INFERENCES. 349(2) a Is BC, B is Ac or aCy c is AB ; (3) A is Be or ^C, ^ is .4 C ^ is AB. 366. ^ IS Be or 5C, 6 is AC, e is AB. Shew that all the information given by the combination of these propositions is also given by the propositions, — A is 6 or e, 6 is A,e is AB or a6. 366. Apply the method of solution described in section 355 to ordinary syllogisms in Barbara, Cesare, Camenes. Barbara has for its premisses, — (i) Mis P, . (2) S is M. By (i), Sis m 01 P\ by (2), 5 is J/; therefore, S is MP\ therefore, 5 is -P. Cesar e, (i) No -Pis Jlf, (2) 5 is Jf. By (i), Sism or/; by (2), SisM\ therefore, SisMp\ therefore, S is not P. Camenes, (i) Pis M, (2) No ilf is .S. By(i), Si&Moxp'y by (2), 5 is ni'y therefore, Sismp', therefore, No S is P. I^KSfr350 COMPLEX INFERENCES. [partiv.367. Assign propositions concerning the terms gem, rich, rare^ whose aggregate force shall be such that no further assertion can be made about the same terms without contradicting the propositions assigned. [L.] Jevons regards any two or more propositions as incon- sistent when they involve the total disappearance of any term, positive or negative, {Studies in Deductive Z^V, p. 181); and in dealing with such a question as the above, it is perhaps a convenient criterion to take. It is equivalent in this instance to the assumption that there exist gems and not gems, rich things and not rich things, rare things and not rare things. It is an assumption, however, and as such should always be explicitly stated, when made. Com- pare Part II. Chapter viii. The reason why it is convenient to make it here is that, (as shewn in section 106), on the supposition that All S i^ F does not itself imply the exist- ence of »S, All S is /*and No .S is -P are either inconsistent, or between them deny the existence of S. We must therefore exclude the latter possibility, if we wish to be able to say definitely that two such propositions are in- consistent The given question is now solved by the three proposi- tions : — (i) All gems are rich and rare ; (2) All rich thingsarerare gems ; (3) All rare things are rich gems. We can make no further assertion regarding these terms or their negatives which are not either implied by the above, or else inconsistent with them ; since the only classes which they allow to remain are rich rare gems and not rich not rare not gems. CHAP. VIII.] COMPLEX INFERENCES. 35^ The two propositions, — (i) All gems are rich and rare ; (ii) AH things not gems are neither rich nor rare; also afford a solution. The equivalence between (i), (2), (3), and (i), (ii), has been already shewn in section 362. The student should now find other sets of propositions, not equivalent to the above, which also afford a solution of the given problem. A I CHAPTER IX. PROBLEMS INVOLVING FOUR TERUS. 368. Suppose that an analysis of the properties of a particular class of substances has led to thefollowing general conclusions, viz.: 1st, That wherever the properties A and B are combined, either the property C, or the property ZJ is present also ; but they are not jointly present 2nd, That wherever the properties B and C are combined, the properties A and D are either both present with them, or both absent. 3rd, That wherever the properties A and B are both absent, the properties C and D are both absent also ; and vice versd, where the properties C and D are both absent, A and B are both absent also. Let it then be required from the above to deter- mine what may be concluded in any particular in- stance from the presence of the property A with respect to the presence or absence of the properties B and C, paying no regard to the property D. [Boole, Laws of Thought, pp. 118 — 120 ; compare also Venn, Symbolic Logic^ pp. 276 — 278.] CHAP. IX.] COMPLEX INFERENCES. 353 One solution has already been given in section 355. We might also proceed as follows. The premisses are : AB is Cd or cD, (i) B C IS AD ox ad^ (ii) ab is cd^ =(iii) cd is ak (iv) By (i), No AB is CDy therefore, No A is BCD, (i) By (ii), No BC is Ad, therefore, No A is BCd, (2) Combining (i) and (2), we have, —No A is BC, t.e,, All -4 is ^ or c. This solves the problem as set Boole also shews that All dC is A. This is a contra- positive of (iii). We have not required to make use of (iv) at all. 369. Given the same premisses as in the preceding section, prove that :— (i) Wherever the property C is found, either the property A or the property B will be found with it, but not both of them together, (2) If the property B is absent, either A and C will be jointly present, or C will be absent. (3) If A and C are jointly present, B will be absent. [Boole, Laws of Thought, p. 129.] First, By (i), ABC is d, hy{ii), ABC is D; Le,, there is no such thing as ABC, L e,, Cis aox b^ . L. 23 354 COMPLEX INFERENCESL [paxt rr. Also, hj contraposition of (iii), CisA or B; therefore, C\% At or aB. f i) Secondly^ By (iii), ^ is ^ or r, therefore, ^ is ^ Cor/: (2) Thirdly^ We have shewn that it follows from (i) and ^^> that there is no such thing as ABC^ ihereforcy AC is t* (3) 370. It is known of certain things that (i) where the quality ^ is, iP is not ; (2) where B is, and only where B is, C and D are. Derive from these con- ditions a description of the class of thii^ in whidh A is not present, but C is*. (Jevons, Studies, p« 20Q.] The premisses are, — (i) ^is^; (2) Bis CD', (3) CD is B. (i) aflfords no information with regard to aC CC sec- tion 353. But by (2), ; 3"d hy (3), aC ]sBord; therefore, aC is BD or td 371. Taking the same premisses as .in the pre- vious section, draw descriptions of the classes Ae, ab^ and cD, [Jevons, Studies, p. 244.] We have (i) Aisb\ (2) B is CD% (3) CDisB. By (i), ^^is^; by (2), ^^ is ^ ; (3) aflfords no information with r^pard to ^/; CHAP. IX.] COMPLEX INFERENCES. 355 By {3)9 ^^ is c or if. By (i), cDis aox d; by (2), cl?is d; therefore, cJ? is d. We can obtain no farther information with regard to a^ and cD. The desired results, therefore, are, — ad is COT d'y cD is b, 372. There is a certain class of things from which A picks out the ' X that is Z^ and the Y that is not Zl and B picks out from the remainder * the Z which is Y andthe X that is not K' It is then found thatnothing is left but the class ' Z which is not X! The whole of this class is however left. What can be determined about the class originally ? [Venn, Symbolic Logic, pp. 267, 8.] The chief difficulty in this problem consists in the accu- rate statement of the premisses. Call the original class W. We then have, — ^is XZor Yz or PZor Xy or xZ, (i) :«:Zis m (2) No xZ is WXZox WYz or WYZ or WXy, i.e.,(leaving out such part of this statement as is merely identical), NoorZis WYZ (3) We may nowproceed as follows : — By (3), lioJVisxYZ; (4) By (i), No ^is xyz. (5) 23—2 3S6 COMPLEX INFERENCES. [part iv. Combinii^ this with (4}, we find that the class did not originally contain any not- A" that was either both Y and Z or neither Y nor Z. (a) affords no information regarding the class W, except that everything that is Z but not X is contained within it The student may however notice that from this proposition in conjunction with (3), it may be deduced thatall YZ isX 373. At a certain town where an examination is held, it is known that, (i) Every candidate is either a junior who does not take Latin, or a Senior who takes Composition, (2) Every junior candidate takes either Latin or Composition. (3) AH candidates who take Composition, also take Latin, and are juniors. Shew that if this be so there can be no candidates there. [Venn, Symbolic Logic, pp. 270, i.] Let X = candidate, A = junior, sothat a = senior, B =taking Latin, C = talcing Composition. We then have, — X Is Ab or aC ; (i) XA is 3 or C; (2) XCkAS, (3) {2)znA(z)giwtXAi%B; herefore, No X\s Ab; also by (3), No X is aC. It therefore follows from (i) that there can be no such thing as X CHAP, ixj COMPLEX INFERENCES, 357^ 374 Given (i );ir IS j/^; {2)ZW\sy] {i)y\sZW\ (4) zisW) is) XW is YZ; shew that (i) Nothing is W, (ii) Everything is XYZ. [Venn, Symbolic Logic^ pp. 271, 2.] By (5), XWh YZ; but by (2), No ^is YZ; therefore, Nothing is XWl By (i), X W is yz; but by (3), Nothing is yz ; therefore. Nothing is x W. But WisXWoxxW; therefore, Nothing is W. (i) By{i\xisyz]^ but by (3), Nothing is^^; therefore. Nothing is x, By(3)>7is W, and by (4), z is W; but by (i), Nothing is W; therefore. Nothing isy ox z; therefore. Nothing is x,y or z; />., Everything is -i^yZ. (ii) 376. If thriftlessness and poverty are inseparable, and virtue and misery are incompatible, and if thrift be a virtue, can any relation be proved to exist be- tween misery and poverty ? If moreover all thriftless people are either virtuous or not miserable, what follows ? [v.] Let A = thriftless, ^ = poor, C= virtuous, Z> = miserable. 35« COMPLEX INFERENCES. [part iv. Then the premisses as first given may be written, — AisB\ (i) ^is^', (2) No CisA (3) ais C. (4) Can we now find any relation between B and Z> ? We may proceed by finding aU that we can assert con- cerning B and I? respectively. By (2), -^ is ^ ; by (3), B is COT d*; therefore, B is Ac or Ad; therefore, If ^ is 27, it is Ac ; i,e,f If poverty is accompanied by misery, it is also accom* panied by thriftlessness, and it is not accompanied by virtue. Agsun, by (i), Z>isa or B; by (2), Dis A or ^; therefore, Z> is AB or ad; by (3), 27 is^:; therefore, JD isA Be or a3c; by (4), Z> is A or C; therefore, £> is A Be; i,e,f Misery is always accompanied by poverty; or, misery is never found unaccompanied by poverty. This result might also be obtained by two ordinary syllogisms in Barbara^ as follows : ^ If we adopted the equational rendering of propositions, which however I have intentionally avoided, *M is iff*' and *^B is A " would of course be summed up in *^A=B,** In cases of this kind, the equa- tional rendering is at its best. ' (i) gives no information regarding B\ and, so far as iff is con- cerned, (4) merely repeats part of the information given by (2). CHAP. IX.] COMPLEX INFERENCES. 359 By (3), -C> is r; by (4), cisA', therefore, D is A; by (i), A is jB; therefore, jD is jB. If to the given premisses we now add ($) A is C or d, we find that jD is both A and a, sl result which must be interpreted as affirming thenon-existence of J9; />., There is no such thing as misery. It will be a more complete answer to the latter part of the question to note the full result of combining all the given premisses. By (i), Everything is a or jB ; . by(2),Everything is ^ or ^ ; therefore, Everything is AB ov ad; by (3), Everything "is c or d; therefore. Everything is AB^ or ABd or alfc or add ; by (4),Everything is -^ or C; therefore, Everything is A Be or ABd or adCd; by(5),Everything is « or C or ^; therefore, Everything is ABd or adCd,This gives us : — A is Bd; a is dCd; B is Ad; b is aCd; C is ABd or ahd; c is ABd; There is no such thing as D ; d is AB or dbC, 36o COMPLEX INFERENCES. [part iv. 376. A given class is made up of those who are not male guardians, nor female ratepayers, nor lodgers who are neither guardians nor ratepayers. How can we simplify the description of this class if we know- that all guardians are ratepayers, that every person who is not a lodger is either a guardian or a rate- payer, and that all male ratepayers are guardians ? [v.] Let X^iht given class, A = male, ^ = guardian, C= ratepayer, D - lodger. Then X is made of those who are not AB nor ^ C nor bcD ; that is, X is made up of those who are aBc or AbC or acd or Abd or bed. But we are told that, — (i) B is C; (2) ^ is ^ or C; (3) AC is B. From (i), it follows that there is no aBc; from (2), that there is no bed; from (3), that there is no -^^C; from (2) and (3) taken together that there is no Abd; from (i) and (2) taken together that there is no acd. It therefore follows that the given class is itself non- existent. We might arrive at the same result as follows : — By (i). Everything is ^ or C; by (2), Everything is -5 or C or Z> ; therefore. Everything is bl> or C; CHAP. IX.] COMPLEX INFERENCES. 361 by (3), Everything is ^ or ^ or^; therefore, Everything is abD or bcD ox aC or BC\ but abD is aC or bcD S and BC is AB or aC^\ therefore, Everything is aC or bcD or AB\ which again shews that the given class is non-existent. 377. Given that everything that is Q but not 5 is either both P and R or neither P nor R and that neither R nor 5 is both P and (2, shew that no P isQ. 378. Where C is present, A, B and -O are all present; where D is present, A^ B and C are either all three present or all three absent. Shew that when either A or 5 is present, C and D are either both present or both absent. How much of the given information is superfluous so far as the desired conclusion is concerned ? 379. Every voter is both a ratepayer and oc- cupier, or not a ratepayer at all. If any voter who pays rates is an occupier, then he is on the list. No voter on the list is both a ratepayer and an occupier. Examine the results of combining these three statements. [v.] 380. At a certain examination, all the candidates who were entered for Latin were also entered for either ^ Since, by the law of Excluded Middle, aJbD is ahCD or aihcD, 5 2 Since, by the law of Excluded Middle, BC is ABG or oBQ. 362 COMPLEX INFERENCES.  [part IV.Greek, French, or German, but not for more than one of these languages ; all the candidates who were not entered for German were entered for two at eastofthe other languages ; no candidate who was entered for both Greek and French was entered for German, but all candidates who were entered for neither Greek nor French were entered for Latin. Shew that all the candidates were entered fortwo of the four languages but none for more than two. 381. AB is A ad is cd, c is ABD or abd, D is AB. AH the information given hy these propositions is also given by the propositions, — ABC is Dy abd is Cy c is AD or abd^ D is AB or acorBc\ and vice versd. 382. Shew that the following sets of propositions are equivalent : — (i) ^ is ^ or ^; ^ is aCd\ els aB; D is c. (2) A is BC; b is aC; Cis A Bd or abd, (3) A is B\ Bis A or c\ cis aB\ D is c. (4) b is aC\ A isC\ Cis d; aC is b. (5) cis aB\Dis aB; A is B ; aB is c (6) A IS BC; BCis A ; D isBc; b is CCHAPTER X. PROBLEMS INVOLVING FIVE TERMS. 383. Let the observation of a classofnaturalproductions be supposed to have led to the following general results. 1st. That in whichsoever of these productions theproperties A and C are missing, the property E is found, together with one of the properties B and D, but not with both. 2nd. That wherever the properties A and D arefound while £ is missing, the properties B and C willeitherbothbe found, or both be missing. 3rd. That wherever the property A is found in conjunction with either B or E, or both of them, there either the property C or the property I> will be found, but not both ofthem.And conversely, wherever the property C or D is found singly, there the property A will be found in conjunction with either B or E, or both of them. Shew that it follows that In whatever substances tlie property A is founds there will also be found eitlur t/te property C or t/ie property D^ but not both^ or else the 364 COMPLEX INFERENCES. [part iv. properties By C, and Z>, will all be wanting.And con- versely, where either the property C or the property D is found singly y or the properties By C, and Dy are together missingy there the property A will be found, [Boole, Laws of Thought, pp. 146 — 148. Cp. also Venn, Symbolic Logic, pp. 280, 281.] The premisses are as follows : — I St, AllacisBdE or bDE*, (i) 2nd, All ADe is BC or bc\ (ii) 3rd, All AB is Cd or cD\ (iii) All AEis Cd or cD\ (iv)AW CdisAB or AE; (v) ll cD is AB or AE. (vi) We are required to prove : — All A is Cd or cD or bed; (a) AW Cd is A; (/?) AllcZ>isA; (y) AW bed isA. (8) First, By (iii) and (iv). If -^ is ^ or ^ it is Cd or cD; therefore, A is Cd or cZ> or ^^. (i) By (ii), Ae is either BC or ^r or d; therefore, Abe is ^^ or d; therefore, Abe is bee or ^^^. (2) By (v), Cd is ^ or E; therefore, Cis B or £> or E; therefore (by contraposition), bde is c; therefore, bde is bed; therefore. If Abe is bde it is bed. (3) Again by (vi), eZ> is B or E; therefore (as above), bee is d;therefore, If Abe is bee it is ^^.^. (4) CHAP. X.] COMPLEX INFERENCES. 365 Therefore, by (2), (3), and (4), Abe is bed; therefore from (i), 4is either Cd or eD or bed, (a) Seeondlyy (fi) and (y) follow immediately from (v) and (vi). T/iirdfy, from (i), we have directly, No ae is bd; therefore (by conversion).No ^^^is a; therefore, All bed is A. (8) The first of the desired results might also be obtained as follows : — As before we may shew that A is Cd or eD or be; and we therefore have what is required if we can shew that Abe is ed. By (ii), Abe is e or d; by (v), Abe is ^ or Z>; therefore, Abe is e; by (vi), Abe is C or d; therefore, Abe is ed. We have here employed a modificationofthe general method described in section 355. e might also by this method obtain a complete solution of the problemsofar as A is concerned. (i) gives no information whatever with regard to A \But by (ii), A is BC or be or d or E\ by (iii), -4is ^ or Cd or cD\ therefore, A isCd or be or bd or bE or cDE\ by (iv), -^ is Cd or r2?or ^; therefore, A is C// or cDE or^^Z? or ^^^ or bde; by (v), -4 is ^ or ^ or ^ or Z>; therefore, A is ^Z>J? or bcD or ^r^ or ^G/ or CdE\ ^ Since or a. (i) By (i), ^^ is Bd or ^Z>; therefore, No ac is -5Z>; therefore, No aB is rZ>. (2) By(v), C^is^; therefore (by contraposition), ^z is ^ or Z>;therefore, No a is Cd; therefore, No aB is Cd. (3) By (2) and (3), No aB is cD or Cd; therefore, All aB is cd or CZ?. Combining this result with (i), we have, — B is A Cd or Acl? or acd or «CZ>. (a) Secondly, From (i) we have directly, acd is -5^^ ; therefore, acd is -^. (p) Thirdly, By (ii), No ADe is bC; therefore, No -^C is bDe. (i) By (iii). No AB is CD; therefore. No -^Cis BD. (2) By (iv)No-^^is CZ?; therefore. No ^ C is DE. (3) Therefore, by (i), (2), and (3), \i AC is D it is neither be nor B ox E; but (by the law of excluded middle), All ^ C is either B ox E ox be; therefore, No ACis D ; therefore. All ACis d. (y) 368 COMPLEX INFERENCES, [part iv. Lastly, By (vi), cjD is A ; therefore (by contraposition), ac is d, (8) For complete solutions with regard to B, acd, A C, ac, see the following section. 386. Given the same premisses as in section 383, obtain complete solutions with regard to B^ acd, AC, ac. Complete solutions with regard to B, acd, AC, ac, may- be obtained by the general method described in section 355 as follows : —Brst, By (i), BisdEorAorC; by (ii), ^ is C or a or d ox E] therefore, -^ isC or dE or Ad or AE\ y (iii), B is Cd or cD ox a-, therefore, B is Cd ox aC or adE or AcDE\ (iv) gives no information with regard to B that is not already given by (iii) ; by (v), ^ is -^ or ^ or 27; therefore, B i% AcDE or ACd or aCD or acdE» (vi) gives no further information with regard to B, The above includes the special solution given by Boole, — B is acd or aCD or AcD or ACd. Secondly, By (i) acd is BE. None of the other propositions give any information with regard to acd^. This then is the complete solution so far as acd is concerned. ^ Since the subjects of all these propositions contain either A, C, or D. Cf. section 353CHAP.x.] COMPLEX INFERENCES. 369 Thirdly, By (ii), ACisBordorE; by (iii), AC is d 01 d; therefore, AC is dor dE; by (iv), AC is d or e; therefore, ACis d; by (v), ^C is ^ or ^ or 2?; therefore, AC is Bd or dE. Lastly, By (i), ac is BdE or bDE ; by (vi), acisd) therefore, ac is BdE, 387. Every A is one only of the two B ox C,D IS both B and C except when B is E and then it is neither ; therefore no A is A [De Morgan.] This example, originally given by De Morgan, (using however different letters), and taken by Professor Jevons to illustrate his symbolical method, {Principles of Science, VoL i, p. 117; Studies in DeductiveLogic, p.203), is chosen by Professor Groom Robertson to shew that "the most com- plex problems can, as special logical questions, be more easily and shortly dealt with upon the principles and with the recognised methods of the traditional logic'* than by evons's system,  The mention of E as E has no bearing on the special conclusion A is not L> and may be dropt, while the impli- cation is kept in view ; otherwise, for simplification, let BC stand for *both B and C,* and be for * neither B nor C* The premisses then are, — (i) L> is either BC or be, (2) ^ is neither BC nor bcj which is a well-recognised form of Dilemma with the con- clusion A is not L>. Or, by expressing (2) as A is not K. L. 24 370 COMPLEX INFERENCES. [PAKT IV. either BC or ic, the conclusion may be got in Camestres. If it be objected that we have here by the traditional pro- cesses got only a special conclusion, it is a sufficient reply that any conclusion by itself must be special. What other conclusion from these premisses is the common logic power- less toobtain?" {Mind, 1876, p. 222.) The solution is also obtainable as follows, — By the first premiss, A is Be or bC, and by the second, AhBCoxbcatd; therefore, A is Bed or bCd, therefore, A is d. Professor Robertson's solution is in this case preferable. But I append the above as a further illustration of my own method. Compared with the problem of Boole's just dis- cussed or with the problems that follow, this of De Morgan's is not particularly complex. 388. Suppose it known that, — (1) Where B is present and C absent, either D is present or E is absent; (2) Where A and Dare present and C absent, B is present ; (3) Where B is absent and C present, A is present ; {4) Where C and D are present, A is absent or B is present ; {5) Where E is present and D absent, A and C are not both present nor are B and C both absent ; (6) Where B is present and D absent, C is absent ; (7)Where A is present and E absent, either B or D is present ; , GHAP. X.] COMPLEX INFERENCES. 371 then we can shew that, — i) Where A is present, B is present ; (ii) Where B is absent, C is absent ; (ill) Where C is present, D is present ; (iv) Where D is absent, E is absent. First, By (2), AcD is ^ ; by (4), CD \% ^^ is de; therefore, ^ is -4 or de» By(2),^Z>is Cor^ is be or E) by (7), -^^Z> is -5 or ^; therefore, -4^Z> is j9j5 or ^^. (11) Secondly^ (iii) may be inferred from (ii). By (8), CD is ^Z> ; by (9), Cd is d^^; therefore, C is ^Z> or ad. (12) By (8), E is Ab or e or d; by (9), -£ is -<4 -4="" 38o="" abd="" ae="" aed="" aicb="" aicii="" aj="" by="" c="" cl="" complex="" cor="" d="" dj="" e="" inferences.="" is-fforflor="" is="" j="" l="" n="" or="" part="" therefore=""> ; therefore, £ is iC or AB^D. (13) By (8), e is AB or ^ or rf; by (9), f is ^ or C or d; therefore, e is AB or Ac or if; by (10), ^is.i^Cor J"; therefore, e is ^^/)or^fZ> or aBCd; by (1 r), « is 5 or a or C or d; therefore, e is ABCD or ^ic^ or a^Crf; therefore, e is ^C or Adi.\D. (14) Thirdly, (!) may be inferred from (iii). By (13). -ff-^ is/); by (14), ffi-is/>; therefore, ^ is i? ; therefore, AcisD; by (ts), AC is D; therefore, .^ is Z>. (i) By(i3),) By (ii), j9 is ^; by (iii), Bd is ^ ; therefore, B is ^ is ^^ifi?-; by (iv), dd is ^, But, (by the law of excluded middle), ^ is -5 or 2? or dd; therefore, A is BCdE/ 01 bDeFox bde^ K. L. 25 386 COMPLEX INFERENCES. [part iv. (8) By (2), a is bed; by (ii), D is AbF. But, e isAd or a or D ; therefore, e is Ad orad^dor AbDF\ and by (i}| ^ is^; therefore, ^ isAbd or o^^ or AbDF. (9) By (2), ^ is o^^/i^; therefore, -^is ^^^if or A. By (i) and (3), Fis be; therefore, Fis abcdor Abe. (10) By(3), ^is^; by(i),.5i8^Ca£:; by (2), ^ is d^^^if» But,/is -^^ or ^ or tf ; therefore, /is Abe or ABCdE or a^^^fc; and by (ii),/is^; therefore, /is Abde or ABCdE or a^r^ is absent, A , ^, and Z^ are present, while C and F are absent, and z//<;^ versd; (4) Where 5 is absent, C, £", and .P are present, while D is absent ; (5) WhereD and F are both absent, B and 2s are present while C is absent. [Jevons, Pure Logic, pp. 66, 67.] CHAP. XL] COMPLEX INFERENCES. 389 404. With respect tocertain classes of pheno- mena, it is observed that, — (i) Where B is absent, E is present, but C, Dy and F are absent ; (ii) Where B is present while D is absent, A, C, and E are present, but F is absent ; (iii) \{B and D are both present, E is not present F being absent, nor is C present A being absent. It may hence be deduced that, — (i) If -4 is absent, C\s absent. (2) If 5 is present, either Cor Z? is present.(3)1{ B, Dy and E are all present, F is present. (4) If C is absent, B and D are either both pre- sent orboth absent. (5) If C and D are bothabsent, B is absent. (6) If C is present, A and B are both present. (7) li D is present, B ispresent. (8) If D is absent, E is present but F is absent. (9) If F is absent, D,and £" cannot both be pre- sent or both absent, 406.Given, — (i) aB is c or D; (2) BE is DF or cdF; (3) C isaB or ^^5" or D ; (4) ^Z> is ^or /^ ; (5) bf is ^/ is C; therefore, Bed is K, (iii) By(9), ^Z^i^/^is^ori; By (5),^ is Zr or 7; therefore, AcDFh H ox K\ ^y {Z\ ABF\% h 01 K\ therefore, ABcDFis K] therefore, ABcD is/ or K\ By (4), Acf\% d] therefore, AcD is F; therefore, ABcD is K) But by (iii), ABcd is K\ therefore, ABc is K\ And by (ii), ABC is iT; therefore, AB is -/T. (iv) By (5), hi is ^C or Cd or C/*;therefore, bDF'is Hox 7; By (9), ^ZPT'is Hox i or iT; therefore, AhDF is ^^ or ^. (v) By (7), bDEFisgox h] therefore, AbDEF\s g or K', therefore, AbDEFG is K', therefore, DEFG is a ox B or JC; By {2), AbC is DEFG; therefore, AbC is -/T. (vi) 394 COMPLEX INFERENCES. [part iv. By (lo), AbcDEF'is G or // or AT; By {^\ bDEF \^ g or h; therefore, AbcDEF'is h or K\ By(v), AbDF\%HoxK', therefore, AbcDEFis K, therefore, AbcJDE is /or K; By (4), AcD is F; therefore, AbcDExs K, (vii) By (i), be is JDE or Z^or hi; and by (5), be is If or /; therefore, be isZ>-^ or Df] By (4), u4^ is d or ^; therefore, Abe is Z7-£; therefore, -<4 -="" 125.="" 144="" 252.="" 2="" 2nd="" 396="" 397="" 398="" 409.="" 410.="" :="" a="" ab="" aba="" abc="" abcde="" abce.="" abce="" abe="" abedey="" absolute="" ace="" admit="" after="" agree="" agreeing="" aimed="" alternatives="" although="" always="" an="" and="" answer="" any="" apparently="" appears="" are="" as="" ascertain="" assuming="" at="" b="BC," bd="" be="" becomes="" before="" belong="" binations="" ble="" blem="" but="" butby="" by="" c.="" c="e," called="" can="" casually="" categorical="" categoricals="" cer-="" certain="" chapter="" cipal="" combination="" combinations.="" combinations="" compare="" comparing="" complete="" complex="" conditions="" conjunctive="" consist="" consistent="" consists="" contains="" contemplates="" corn-="" could="" count="" data.="" deductive="" described="" determine="" difficulty="" discover="" disjunctive="" disjunctives="" do="" does="" down="" e="" easy="" ed.="" either="" embodied="" enter="" equal="" equiva-="" equivalent="" every="" everything="" extracts="" extreme="" f="" facility="" fairly="" far="" few="" fewer="" find="" finding="" following="" follows="" for="" force="" foundthat="" from="" fulfilled="" general="" generally="" give="" given="" greater="" guesses="" hardly="" has="" have="" having="" he="" here="" hich="" him="" however="" i.="" i="" if="" in="" indetermi-="" indicated="" indirect="" inductive="" infallibly="" inference="" inferences.="" instance="" into="" inventing="" inverse.="" inverse="" inverseproblem="" involve="" involved="" involves="" involving="" is="" isjr="" isted.="" it.chap.="" it="" its="" iv.="" iv="" jevons.="" jevons="" jt="" just="" law="" laws="" lay="" lent="" let="" logic="" logical="" matter="" may="" mean="" meets="" method="" more="" most="" much="" must="" my="" myself="" n="" namely="" nate="" nature="" necessary="" need="" no="" nor="" not="" now="" number="" obey.="" observing="" obtain="" of="" oftheform="" on="" once="" one="" or="" order="" others="" out="" p.="" p="" part="" plicity="" point="" positions="" possi-="" possible="" precisely="" predicate.="" predicate="" predicates="" prefer="" premisses.="" premisses="" prin-="" principles="" pro-="" probably="" problem.="" problem="" problems.="" proceeded.="" process="" professor="" propo-="" proposition="" propositions="" rather="" reader="" recourse="" regard="" regarded="" relations="" remaining="" remember="" require="" resolution="" respectively.="" results="" right="" rinciples="" rule="" satisfactory="" say="" science="" section="" seek="" series="" set="" sets="" shall="" should="" sim-="" simple="" simpler="" since="" single="" sitions="" small="" smaller="" so-="" so="" solution.="" solution="" solutions.="" solutions="" solve="" some="" speaking="" speaks="" standard="" state="" strictly="" strike="" studies="" sub-jects="" subject="" such="" tain="" taken="" tentative.="" tentative="" terms.="" terms="" test="" than="" that="" the="" thededuced="" their="" then="" thepjo-.="" therefore="" they="" think="" this="" those="" three="" tives="" to="" together="" trial="" trials="" trouble="" true="" trying="" tudies="" two="" ubcf="" us.="" v.="" v="" vi="" view="" vii="" vol.="" we="" what="" when="" whether="" which="" will="" with="" withthreealterna-="" would="" xii.="">., Given a proposition limiting us to a number of complex alternatives to find a set of propositions In- volving as simple relations as possible which shall be equivalent to it. The data maybe written in the form, — Everything is /* or Q or 5 or T or &c., where /*, Q, &c., arc complex terms. By contraposition ' we may bring over one or more of these complex terms from the predicate into the subject, so that we have, — What is neither P nor 5 nor &c. is Q orT'or &c. The selection of certain terms for transposition in this way is arbitrary, (and it is here that the indeterminate- ness of the problembecomes apparent) ; but it will gene- rally be found best to take two or three which have as much in common aspossible. *'What is neither /'nor S nor &c. is Qox Tor &c" will immediately resolve itself into a scries of propositions, which taken together give all the information originally given'. If any of these are themselves very complex we may proceed with them in the same way. We may then suppose ourselves left with a series of fairly simple propo- sitions ; but it will probably be found that some of these merely repeat information given by others, so that they may be omitted. We may find to what extent this is the case, by adopting any one of the three following methods : — First, byleaving out each proposition in turn, and de- termining (by the ordinary rules) what the remainder by combination give concerning its subject. If we find that ^ Cf. lection 317. ^ .Cf. chapter ii. CHAP.xii.] COMPLEX INFERENCES. 399 it addsnothing to the information that they give it may be omitted. Secondly, by bringing each proposition tothe form, — Nothing is X^ or X^ orX«, and then comparing it with the combination of the re- mainder. Thirdly^ by writing down all possible combinations after Jevons's plan, {Pure Logic, p. 46; Principles of Scietue, Chapter vi ; Studies, p. 181), and noting which are excluded by each proposition in turn. If a proposition excludes no combination that is not also excluded by other proposi- tions it may be omitted. We are now left with a series of propositions which are mutually independent. By further comparison however we shall probably find that some of them may be still further simplified. When such simplification has been carried as far as possible we shall have our final solution. This may be verified by recombining the propositions that we have obtained, by which operation we ought to arrive again at the series of alternatives with which we started. To illustrate the abovemethod, four examples follow which are worked out in full detail. I. For our first example we may take one of those chosen by Jevons in the extract quoted in section 409, Given the proposition that " Everything is ABC or Abe oiaBC or abC;' find a set of propositions involving as simple relations as possible which shall be equivalent to it. {400COMPLEX INFERENCES. [part iv. By contraposition, What is neither ABC nor Ahc is aBC or abC\ therefore, What is a or Be or bC is aBC or abC\ />., [a is C, i5^ is not, \bC\%a. *^ Be is not" is reducible to "^ is C"; and this proposition and "a is C" may be combined into "r is ^^." Our solution therefore is, — V is Ab^ ,^Cis fl. By combining these propositions it will be found that we regain the original proposition. II. We may next take the more complex example contained in the extract from Jevons quoted in section 409, The given alternatives are, — ACe, aBCe, aBedEyabCCj abcEu Therefore, What is not aBcdE or abcE is ACe or aBCe or abCe\ therefore, {A is Ce ; C is Ae or aBe or abe ; ^ is ^C or aBC or abC\ BD is ACe or aCe ; these propositions are immediately reducible to, — CHAP, xil] COMPLEX INFERENCES. 401 and they may be further resolved into, — E IS ac] c IS aE\ BD is Ce, This solution again may be verified by re-combination. III. The following problem is from Jevons, Frindples of Science^2nd ed., p. 127, (Problem v.). The given alternatives are, — ABCD, ABCd, ABcd, AbCD, AbcD, aBCD, aBcD, aBcdy abCd, Then, by contraposition, what is neither of the four following, — ABCD, ABCd, aBcDy aBcd, must be one of the remainder. But " What is neither ABCD, ABCd, aBcD nor aBcd^' is equivalent to ** What is neither ABC nor aBc^^ and that is equivalent to " What is b or Ac or aCr Therefore, What is b ox Ac ox aC is A Bed or AbCD or AbcD or aB CD or abCd. K. L. 26 402 COMPLEX INFERENCES. [part iv. From this we have our first resolution of the given information in the three propositions: —(i)disA£>OTaCd; (2) Ac is Bd or l^I? ; (3) aCis BD or bd. Each ofthese may again be broken up intotwo pro- positions ; — h is AD or aCd^ becomes If b is not ADy it x&aCd\ that is, (i) ab is Cd^ (ii) bd is aC, (2) may similarly be broken up into, — (iii) ABc is d^ (iv) Acd is B ; and (3) into, — (v) aBC is Z>, (vi) tf CZ> is ^. But (iv) is inferrible from (ii), and (vi) is inferrible from(i); (iv) and (vi) may therefore be omitted. We have then for our final solution, — (i) ab is Cdy (2) bd is aC, (3) ^^^is^, (4) aBC is jD. This is practically equivalent to the solution given in Jevons, Studies^ p. 256. We may now verify it as follows : — By (i), Everything is A or B ox Cd; By (2), Everything is aC or B or !>-, therefore, Everything is AD or aCd or B ) \CHAP. XII.] COMPLEX INFERENCES. 403 By (3), Everything is a or ^ or C ox d\ therefore, Everything is AbD or A CD or aB or aCd ox BC oxBd; By (4), Everything is -r4 or ^ or ^r or Z> ; therefore, Everything is ABC or ABd or AdD or ACI> or aBc or aBZ> oradCd or BCD or Bed. But, AdD is ^^CZ> or ^^^2?. Expandingallthe terms similarly, we have, — Everything is A BCD, or ABCd, or ABcd, or AbCD, or AbcDy or a^Cj9, or aBcDy or ^^^^, or tf^CV/. These are precisely the alternatives that were originally given us. IV. The following example is also from Jevons, Prin- ciples of Science, 2nd Edition, p. 127, (Problem viii). In his Studies, p. 256, he speaks of the solution as unknown; and I am, therefore, the more interested in shewing that a fairly simple solution, involving no more than Jive categorical propositions, may be obtained by the application of the general rule formulated in this section. The given alternatives are, — ABCDE, ABCDe, ABCde, ABcde, 26 — 2 404 COMPLEX INFERENCES. [part iv. AbCDE, AbcdE, Ahcdey aBCDCy aBCde^ aBcDe^ abCDCy abCdE, abcDe, abcdE. Therefore, What is neither ABCDE nor ABCDe nor ABCde nor Ahcde nor aBCDe nor aBCde is ABcde or AbCDE or AbcdE or aBcDe or (Z^CZ?^ or abCdE or ^^Z? is either bC or ^2^. I CHAP. XII.] COMPLEX INFERENCES. 405 Therefore, our proposition becomes dE, or hC, or Bc^ or cDy or tfjB, or abcj or ace^ "What is is either ABcde, or AbCDEy or AbcdE^ or aBcDe^ - or abCDcy or a^CVf^, or abcDe^ or abcdE''\ and this is resolvable into the following set of pro- positions, — (i) ^^ is ab or ^^; (2) ^C is ADE or <7z -="" abe="" ade="" ae="" is="" j="" or="" z=""> ; (7) iw^ is D* Of these, (2)may be broken up into, — (8) AbCisDE; (9) bCDE\%A\ (10) bCde is non-existent. But (9) may be inferred from (5), and (10) may be in- ferred from (6) and (8); (8) may therefore be substituted for (2). ST M ^^~7 _• r.« m^_ 3 * • -L JT.: J -* ' - ^Si_*^kA* .r 3 -- ^ ^ 3 ^' i -J' - .»'-•_ ::j? -t. .:^-r s r "■HTr'Pi i " "~ ■*■ ■* - JL ' " " J '— -a 'L5 ^'jir.-^n ILSV- 3«. »^^L ^& .^ ^^»m WS. sx-srj jr^ jtjt CHAP. XII.] COMPLEX INFERENCES. 407 By (ii), Everything is a or ^ or r or DE\ therefore, Everything is a or BD or Be or be or cD or a or DE-, By (iii), Everything is a or C or d]therefore, Everything is a or BCD or BCe or Bde or bed or CDE or ^^^; By (iv), Everything is A or ^^ or c, therefore, Ever)rthing is A BCD or A CDE or a^^ or ae or j^Cif or Bde or ^^^ or ^^^; By (v), Ever3rthing is A or BC or Z> or E) therefore, Everything is ABCD ox ABde or Abed or ^CZ?^ or Aede or tf^d?^ or a2>^ or -^CV or bedE. But ABCD is ABCDE or ABCDe, and so with the others. Expanding the terms in this way, we have, — Everything is ABCDE or ABCDe or ABCde or ABede or AbCDE or AbcdE or Abcde or aBCDe 4o8 COMPLEX INFERENCES. [part iV. ^ The student will immediately recognize that this is equivalent to our former solution. Equationally it would be written Ab—c. or aBCde or aBcDe orabCDe or abCdE I or abcDe • | or abcdE, These are again the alternativeswith which we com- menced. 411. Another Method of Solution of the Inverse Problem. Another method of solving the Inverse Problem, sug- gested to me (in a slightly- different form) by Mr Venn, is to writedownthe original complex proposition in the negative | form, /. e,y to obvert it, before resolving it. It has already i been shewn that a negative proposition with a disjunctive | predicate, may be immediately broken up into a set of simpler propositions. In some cases, especially where the number of destroyed combinations as compared with those that are saved is small, this plan is of easier application than that given in the preceding section. To illustrate this method we may take two or three of the examples already discussed. j I. Everything is ABC or Abe or aBC or abC\ \ therefore, by ob version. Nothing is AbC or Bcoi ac\ and this proposition is at once resolvable into, — i 'Ab is Cy I € is Ab \ CHAP.xil.] COMPLEX INFERENCES. 409 II. Everything is ACe or aBCe or aBcdE or ahCe or abcE\ therefore, by obversion, Nothing is AE or CE or BDE or -<4r :="" a="" abcd="" abcjd="" ac="" ae="" amp="" as="" b="" bcd="" bd="" be="" before.="" c="" ce="" ci="" cis="" cox="" d.="" e="" everything="" fe="" fol-="" ibdis="" iii.="" is="" lows="" may="" no="" noi="" or="" ox="" proposition="" reached="" resolved="" same="" solution="" successively="" that="" the="" this="" we=""> or Adc£> or aBCD or dfuff^Z> or ^^rrf or c^G/; therefore, by obversion. Nothing is Ahd or bed or ABcD or tf^r or a^2? or aBCd'y and this proposition may be successively resolved as follows : — No bd is A or c; 1^0 ABc is £>; No ab is cox D; No aBC is d. bd is aC; A Be is d ; ^ab is Cd; [aBC is I>. 4IO COMPLEX INFERENCES. [part IV. This again repeats our original solution. It is curious that in each of the above cases we should by independent methods have attained the same result. 412. It IS observed that the phenomena AyB.C occur only in the combinations A Be, abC, and abc. What propositions will express the laws of relation between these phenomena } [Jevons, Studies, p. 219.] Everything is A Be or ahC or ahe. Noticing that "^C or abc'^ isequivalent to ab, we have by contraposition, What is not ab is ABc ; that is, What \^ A 01 B is ABc \ that is, (A is Bc^ (A IS Bc^ \B is Ac. 413. Find propositions thatleave only the follow- ing combinations, — A BCD, ABcD, AbCd, aBCd, abed. [Jevons, Studies, p. 254.] Jevons gives this as the most difficult of his series of in- verse problems involving four terms. It may be solved as follows: — Everything is ABCD or ABcD or AbCd or aBCd or abed. Noticing that ABCD or ABcD is equivalent to ABDy we have, What is neither AbCd nor aBCd is ABD or cUfcd. Therefore, What is AB or ab or e ox D is ABD or abed-y and this is resolvable into the four propositions, — 'ABi^D, (i) ab is ed, (2) e is ABD or abdy (3) ^D is AB. (4) But by (4) D is AB, and by (2)abisd'y therefore (3) may be reduced to ^ is 2? or ab, i.e., ed is ab. CHAP.xn.] COMPLEX INFERENCES. 4" Our set of propositions may therefore be reduced to, — 'AB is D, ah is cdy cd is ab^ J) is AB \ 414 It is observed that the phenomena A, B, C, Dy Ef F are present or absent only in the combi- nations,— ^^CZ^F, ABCDef, ABCdEf, ABcDF, ABcDef, aBcDF, aBcDefy bcdEf, What propositions will express the laws of relation between these phe- nomena ? [Jevons, Studies, p. 257.] Jevons gives five solutions more or less differing from one another, and all expressed equationaliy. The following is still another solution expressed in the ordinary preposi- tional forms : — BEf\% Cd, b is d, C is AB, d is Ef. 416. Resolve the proposition " Everything is one or other of the following, — ABCDeF, ABcDEf, AbCDEF, AbCDeF, AbcDeF, « ^ Written equationaliy, this solution would appear still simpler; namely, — ' AB^D,ab^c(L 413 COMPLEX INFERENCES. [part iv. aBCDEf, aBcDEf, abCDeF^ abCdeFt abcDeff abcdefl* into a series of simple propositions.[Jevons, Principles of Science ^ 2nd ed., p. 127, (Problem x.).] The following is a solution : — (i) ABE\%cDf\ (2) AcDF'\%be\ (3) aF\%bCe\ (4) bf is ace ; (5) ^is^; (6) ^is abc.  This is rather less complex than the solution by Dr John Hopkinson given in Jevons, Studies^ p. 256, namely : — (i) disab) (2) b is AF or ae; (3) A/ is jBcDF; (4) E is B/orAbCDF; (5) Be is A CDF; (6) abc is ^; (7) abe/is c. It will be a useful exercise for the student to shew that these two sets of propositions are really equivalent 416. How many and what non-disjunctive pro- positions are equivalent to the statement that " What is either A b or bC is Cd or cD, and vice versd "? [Jevons, Studies, p. 246,] CHAP. XII.] COMPLEX INFERENCES. 413 We have given, — 'Ad is CdoxcD, (i) bCv& CdotcD, (2) Cd'\% Ab oibC^ (3) ^cD is Ab or bC. (4) (i) may be resolved into, — {Abe is D, (s) \AbD is c. (6) (2) becomes bC'isd, (7) (3) may be resolved into, — (a Cd is b, (8) iBCisD. (9) (4) may be resolved into, — \ac is dy (10) Be is d. (11) But (6) may be inferred from (7); and (8) from (9). We therefore have for our solution : — Abe is Z>, ^Cis dy BC is D, ac is dy ^Be is d. 417. The following is a further series of inverse problems, which should be solved by the methods indicated in sections 410 and 4x1. In each case we have given a complex proposition which it is desired to resolve into a series of relatively simple propositions. (i) Everything is ABCD or aBCD or aBCd or abCd or abcD or abed. I 414 COMPLEX INFERENCES. [part iv. (2) Everything is A BCD or AbCd or aBcD or abed, (3) Everything is AbCD or AbCd or Abed or aBed or abCD or ^j^Crf or a^r^. (4) Everything is AbeDE or aBCd or aBCE or a-ffrrf or a^^4? or ^^CV or abee or ^^Z?^ or ^ ^rfi? or BcdE or ^C!Z?^. (5) Everything is ABCE or -4-ff^^ or ABeE or -/i5rf(? or Abed or ^^CjE" or ^fo^ or abdE or «^d& or BCde. (6) Everything is A BCDE or ABCdE or -4 i5r2?i? or ABeDe or -^-ff^^ or AbCdE or -4^crf^ or aBCDE or aBCde or abCDE or abeDe. {fj Everything is ^5Z>^ or ^5i?F or ^rZ?^ or -/}^^ or tf^Z>^ or aBDF or ^3C!Z> or ^^Ci/ or abcD or fl&rf or aCDE or « Ci?^ or ^Ci/jE" or aCde or ^rZ>^ or aDEFox aDEf or tf/?^F or aDefox BcDF or ^^^Z?* or (8) Everything is AbdE or Abef or -43F or Aedefox aBDF ox abCFox aCdE or tf^