-. ^ 
 
 STUDIES IN 
 DEDUCTIVE LOGIC
 
 a 
 
 TIC 
 
 B 
 
 HAG NANTIAI OVDIC 
 
 TIC YRENANTIAI OVH
 
 STUDIES 
 
 IN 
 
 DEDUCTIVE LOGIC 
 
 S^amtal for 
 
 BY 
 
 W. STANLEY JEVONS 
 
 LL.D. (EDINB.), M.A. (LOND.), F.R.S. 
 
 THIRD EDITION 
 
 Hontion 
 MACMILLAN AND CO., LIMITED 
 
 NEW YORK : THE MACMILLAN COMPANY 
 1896 
 
 The right of translation and reproduction is reserved
 
 First Edition 1880. Second Edition i 
 Third Edition 1896
 
 i UMVERSITY OF CALIFORNIA 
 
 / SANTA BARBARA 
 
 I ML 
 
 PREFACE 
 
 IN preparing these ' Studies ' I have tried to carry forward 
 the chief purpose of my Elementary Lessons in Logic, which 
 purpose was the promotion of practical training in Logic. 
 In the preface to those Lessons I said in 1870: 'The 
 relations of propositions and the forms of argument present 
 as precise a subject of instruction and as vigorous an exer- 
 cise of thought, as the properties of geometrical figures or 
 the rules of Algebra. Yet every schoolboy is made to learn 
 mathematical problems which he will never employ in after 
 life, and is left in total ignorance of those simple principles 
 and forms of reasoning which will enter into the thoughts of 
 every hour. ... In my own classes I have constantly 
 found that the working and solution of logical questions, 
 the examination of arguments and the detection of fallacies, 
 is a not less practicable and useful exercise of mind than 
 is the performance of calculations and the solution of 
 problems in a mathematical class.' 
 
 The considerable use which has been made of the 
 Elementary Lessons seems to show that they meet- an educa- 
 tional want of the present day. The time has now perhaps
 
 viii PREFACE 
 
 arrived when facilities for a more thorough course of logical 
 training may be offered to teachers and students. 
 
 For a long time back there have been published books 
 containing abundance of mathematical exercises, and not 
 a few works consist exclusively of such exercises. In 
 recent years the teachers of other branches of science, 
 such as Chemistry and the Theory of Heat, have been 
 furnished with similar collections of problems and numerical 
 examples. There can be no doubt about the value of such 
 exercises when they can be had. The great point in 
 education is to throw the mind of the learner into an active, 
 instead of a passive state. It is of no use to listen to a 
 lecture or to read a lesson unless the mind appropriates 
 and digests the ideas and principles put before it. The 
 working of problems and the answering of definite questions 
 is the best, if not almost the only, means of ensuring this 
 active exercise of thought. It is possible that at Cambridge 
 mathematical gymnastics have been pushed to an extreme, 
 the study of the principles and philosophy of Mathematics 
 being almost forgotten in the race to solve the greatest 
 possible number of the most difficult problems in the 
 shortest possible time. But there can be no manner of 
 doubt that from the simple addition sums of the schoolboy 
 up to problems in the Calculus of Variations and the 
 Theory of Probability, the real study of Mathematics must 
 consist in the student cracking his own nuts, and gaining 
 for himself the kernel of understanding. 
 
 So it must be in Logic. Students of Logic must have 
 logical nuts to crack. Opinions may differ, indeed, as to
 
 PREFACE ix 
 
 the value of logical training in any form. That value is 
 twofold, arising both from the general training of the mental 
 powers and from the command of reasoning processes 
 eventually acquired. I maintain that in both ways Logic, 
 when properly taught, need not fear comparison with the 
 Mathematics, and in the second point of view Logic is 
 decidedly superior to the sciences of quantity. Many 
 students acquire a wonderful facility in integrating differen- 
 tial equations, and cracking other hard mathematical nuts, 
 who will never need to solve an equation again, after they 
 settle down in the conveyancer's chambers or the vicar's 
 parsonage. With the ordinary forms of logical inference 
 and of logical combination they will ceaselessly deal for 
 the rest of their lives ; yet for the knowledge of the forms 
 and principles of reasoning they generally trust to the light 
 of nature. 
 
 I do not deny that a mind of first-rate ability has con- 
 siderable command of natural logic, which is often greatly 
 improved by a severe course of mathematical study. But 
 I have had abundant opportunities, both as a teacher and 
 an examiner, of estimating the logical facility of minds of 
 various training and capacity, and I have often been 
 astonished at the way in which even well-trained students 
 break down before a simple logical problem. A man who 
 is very ready at integration begins to hesitate and flounder 
 when he is asked such a simple question as the following : 
 ' If all triangles are plane figures, what information, if any, 
 does this proposition give us concerning things which are 
 not triangles?' As to untrained thinkers, they seldom
 
 x PREFACE 
 
 discriminate between the most widely distinct assertions. 
 De Morgan has remarked in more than one place l that 
 a beginner, when asked what follows from ' Every A is B,' 
 answers ' Every B is A of course? The fact that such a 
 converse is often true in geometry, although it cannot be 
 inferred by pure logic, tends to mystify the student. Al- 
 though all mathematical reasoning must necessarily be 
 logical if it be correct, yet the conditions of quantitative 
 reasoning are often such as actually to mislead the reasoner 
 who confuses them with the conditions of argumentation in 
 ordinary life. A mathematical education requires, in short, 
 to be corrected and completed, if indeed it should not be 
 preceded, by a logical education. There was never a 
 greater teacher of mathematics than De Morgan ; but from 
 his earliest essay on the Study of Mathematics to his very 
 latest writings, he always insisted upon the need of logical 
 as well as purely mathematical training. This was the 
 purpose of his tract of 1839, entitled, First Notions of Logic 
 preparatory to the Study of Geometry, subsequently reprinted 
 as the first chapter of the Formal Logic. A like idea 
 inspired his valuable essays On the Method of Teaching 
 Geometry, quoted above. 
 
 1 The Schoolmaster: Essays on Practical Education, 1836, vol. ii. 
 p. 1 20, note. This excellent essay 'On the Method of Teaching 
 Geometry ' was originally printed in the Quarterly Journal of Education, 
 No. XI. 1833, vol. vi. pp. 237-251. Similar views are put forth 
 in De Morgan's earlier work, On the Study and Difficiilties of Afathe- 
 matics, published in 1831 by the Society for the Diffusion of Useful 
 Knowledge. See chapter xiv. See also De Morgan's Fourth Memoir 
 on the Syllogism, p. 4, in the Cambridge Philosophical Transactions 
 for 1860.
 
 PREFACE xi 
 
 Professor Sylvester, indeed, in his most curious tractate 
 upon the Laws of Verse (p. 19), has called in question the 
 nut -bearing powers of logic, saying: 'It seems to me 
 absurd to suppose that there exists in the science of pure 
 logic anything that bears a resemblance to the infinitely 
 developable and interminable euristic processes of mathe- 
 matical science.' To such a remark this volume is 
 perhaps the best possible answer, especially when it is 
 stated that I have had great difficulty in selecting and 
 compressing my materials so as to get them into a 
 volume of moderate size. If any person who thinks with 
 Professor Sylvester should object to the greater part of 
 the problems as dealing with concrete logic, let him look 
 to the end of this book, where he will find that the closely 
 printed Logical Index to the forms of law governing the 
 combinations of only three terms, fills four pages, without 
 in any way including the almost infinitely various logical 
 equivalents of those distinct forms. He will also learn that 
 a similarly complete index of the forms of logical law 
 governing the combinations of only five logical terms would 
 fill a library of 65,536 volumes. Surely there is scope 
 enough here for 'euristic processes.' 
 
 An anxious and difficult task which I had to encounter in 
 compiling this book consisted in choosing the system or 
 systems of logical notation and method which were to be 
 expounded. When once the convenient but tyrannical 
 uniformity of the Aristotelian logic was overthrown, each 
 writer on the science proceeded to invent a new set of
 
 xii PREFACE 
 
 symbols. But it is impossible to employ alike the Greek 
 letters of Archbishop Thomson, the 'mysterious spiculae' 
 of De Morgan, the cumbrous strokes, wedges, and dots of 
 Sir W. Hamilton, and the intricate mathematical formulae 
 of Boole. After a careful renewed study of the writings 
 of these eminent logicians I felt compelled in the first 
 place to discard the diverse and complicated notative 
 methods of De Morgan. Few or none admire more than 
 I do the extraordinary ingenuity, fertility, and, in a certain 
 way, the accuracy of De Morgan's logical writings. My 
 general indebtedness, both to those writings and to his 
 own unrivalled oral teaching, cannot be sufficiently ac- 
 knowledged. I have, moreover, drawn many particular 
 hints from his works too numerous to be specified. 
 Nevertheless, to import his 'mysterious spiculae' into 
 this book was to add a needless stumbling-block. The 
 question would have arisen too, which of his various 
 systems to adopt ; for De Morgan created six equally 
 important concurrent syllogistic systems, the initial letters 
 of the names of which he characteristically threw into the 
 anagrams, ' Rue not ! ' ' True ? No ! ' These systems were 
 the Relative, Undecided, Exemplar, Numerical, Onymatic, 
 and Transposed. See A Budget of Paradoxes, pp. 202-3. 
 There was in fact an unfortunate want of power of general- 
 isation in De Morgan ; his mind could dissect logical 
 questions into their very atoms, but he could not put the 
 particles of thought together again into a real system. As 
 his great antagonist, Sir W. Hamilton, remarked, De 
 Morgan was wanting in 'Architectonic Power.'
 
 PREFACE xiii 
 
 It seems equally impossible, however, to adopt Sir 
 W. Hamilton's own logical symbols. His chief method 
 of notation has been briefly described in the Elementary 
 Lessons in Logic (p. 189). He also constructed or con- 
 templated other systems of notation, as stated in his 
 Lectures on Logic (vol. iv. pp. 464-476). In no case 
 do these notations seem to be so good as the earlier and 
 simpler one of Mr. George Bentham. And after a 
 laborious reinvestigation, rendered indispensable by the 
 composition of various parts of this book, I have been 
 forced to the conviction that in almost every case where 
 Hamilton differed from contemporaries or predecessors 
 he blundered. He was, as his admirers said, to put the 
 keystone into the arch of the Aristotelic syllogism; but, 
 in spite of his 'Architectonic Power' I fear we must 
 allow that his arch has collapsed. (See pp. 129-133, 
 151-4, and 157-8, of this book.) 
 
 With the logical innovations of Dr. Thomson the case 
 is different. While he appears to enjoy the credit of an 
 independent discovery of the Quantification of the 
 Predicate, prior to any public and explicit statement of 
 the same by Hamilton, De Morgan, or Boole, but 
 posterior to the neglected work of Mr. George Bentham, 
 he did not commit the blunders of Hamilton, nor overlay 
 his work with useless crowds of shorthand symbols. He 
 most aptly completed the ancient scholastic notation of 
 propositions (A, E, I, O) by adding U, Y, 77 and <u to 
 denote the new forms derived from Quantification of the 
 Predicate, carefully showing at the same time that tj and 
 
 b
 
 xiv PREFACE 
 
 ft> are practical nonentities. I have therefore used his 
 notation for quantified propositions and syllogisms where 
 necessary. 
 
 Boole's great works are of course the foundation of almost 
 all subsequent progress in formal logic. My own views, 
 as I long since explicitly stated, 1 are moulded out of his. 
 Believing, however, that the mathematical dress into which 
 he threw his discoveries is not proper to them, and that 
 his quasi -mathematical processes are vastly more compli- 
 cated than they need have been, I have of course preferred 
 my simpler version. Students who wish to comprehend 
 Boole's power and Boole's methods must go to the original 
 writings. It is really impossible that any abstract or 
 summary can give an adequate idea of the stupendous 
 efforts which Boole made to construct a general mathe- 
 matical calculus of inference. Dr. Macfarlane, of Edinburgh, 
 has lately published a new version of Boole's system 
 under the title Algebra of Logic, but I am unable as 
 yet to discover that he has made any improvement on 
 Boole. 
 
 The writings of M. Delboeuf on Algorithmic Logic, 
 first printed in the Revue Philosophique for 1876, and since 
 reprinted, are very interesting, but were written in ignorance 
 of what had been done in this country by Boole and 
 others. 
 
 Quite recently Mr. Hugh MacColl, B. A., has published in 
 the Proceedings of the London Mathematical Society, and in 
 Mind, several papers upon a Calculus of Equivalent 
 1 Pure Logic, 1864, p. 3, etc.
 
 PREFACE xv 
 
 Statements, which arose out of an earlier article in the 
 Educational Tiniest His Calculus differs in several points 
 both from that of Boole and from that described in this 
 book as Equational Logic. Mr. MacColl rejects equations 
 in favour of implications ; thus my A = AB becomes with 
 him A : B, or A implies B. Even his letter-terms differ 
 in meaning from mine, since his letters denote propositions, 
 not things. Thus A : B asserts that the statement A 
 implies the statement B, or that whenever A is true, B is 
 also true. It is difficult to believe that there is any 
 advantage in these innovations; certainly, in preferring 
 implications to equations, Mr. MacColl ignores the necessity 
 of the equation for the application of the Principle of 
 Substitution. His proposals seem to me to tend towards 
 throwing Formal Logic back into its ante-Boolian con- 
 fusion. 
 
 In one point, no doubt, his notation is very elegant, 
 namely, in using an accent as a sign of negation. A' is 
 the negative of A ; and as this accent can be applied with 
 the aid of brackets to terms of any degree of complexity, 
 there may sometimes be convenience in using it. Thus 
 (A + B)' = A'B'j (ABCD ...)'= A' + B' + C' + 
 D' + . . . . I shall occasionally take the liberty of using 
 the accent in this way (see p. 199), but it is not often 
 needed. In the case of single negative terms, I find ex- 
 perimentally that De Morgan's Italic negatives are the best. 
 The Italic a is not only far more clearly distinguished from 
 A than is A', but it is written with one pen -stroke less, 
 1 August 1871, also July 1877.
 
 xvi PREFACE 
 
 which in the long run is a matter of importance. The 
 student, of course, can use A' for a whenever he finds it 
 convenient. 
 
 The logical investigations of Mr. A. J. Ellis, F.R.S., require 
 notice, because they are closely analogous to, if not nearly 
 identical with, my own. I am much indebted to him for 
 assisting me to become acquainted with his views. Not 
 only has he supplied me with an unpublished reprint, with 
 additions, of his articles in the Educational Times, but he 
 has allowed me access to the manuscripts of two elaborate 
 memoirs which he presented to the Royal Society, and 
 which are now preserved in the archives of the Society. 
 Some account of these investigations will be found in the 
 Proceedings of the Royal Society for April 1872, No. 134, 
 vol. xx. p. 307, and November 1873, vol. xxi. p. 497. 
 In the former place Mr. Ellis remarks : ' The above con- 
 tributions are believed to be entirely original .... Jevons 
 first led my thoughts in this direction, but all resemblance 
 between us is entirely superficial.' The question of resem- 
 blance thus raised by Mr. Ellis must be left to others to 
 decide; but in order to avoid possible misapprehension, 
 I must say, that however different in symbolic expres- 
 sion, Mr. Ellis's logical system seems to be identical in 
 principle with my own. The developments of the Com- 
 binational Method, as described in the Educational Times 
 (June, July, and August, i872),'are substantially the same as 
 I had previously published in several papers and books 
 Mr. Ellis also employs card diagrams of combinations 
 arranged upon the ledges of a black-board, which practi-
 
 PREFACE xvii 
 
 cally form the Logical Abacus, as described by me in 
 1869. 
 
 The only point in which I am conscious of having 
 received assistance from Mr. Ellis has regard to the 
 necessary presence of combinations and the significance 
 of their total disappearance as proving contradiction. 
 I may not have sufficiently insisted upon the importance 
 of this matter; but the fact is that so long ago as 1864 
 (see pp. 1 8 1, 192, of this book) I pointed out the complete 
 disappearance of a letter -term from the combinations as 
 the criterion of contradiction in the conditions governing 
 logical combinations, and the same principle is explicitly 
 stated in the Principles of Science (1874, vol. i. p. 133 ; new 
 edition, p, 116). In the latter part of this book I have 
 more fully developed the theory of the relation of pro- 
 positions, often turning as it does upon this criterion of 
 contradiction. This theory will, I think, be found to be 
 the natural development of ideas stated in my earlier 
 essays ; but I may have received some hints from Mr. 
 Ellis's writings. The above remarks apply only to 
 such portions of Mr. Ellis's Memoirs as treat of logical 
 combination and inference ; other portions in which he 
 investigates sequence in space and time, probability, etc., 
 are not at all in question. 
 
 The Logical Index, although now printed for the first 
 time, has been in my possession since 1871 (see Principles 
 of Science, ist edition, vol. i. pp. 157, 162 ; new edition, 
 pp. 137, 141, etc.); but it is only by degrees that I have 
 appreciated the wonderful power which it gives over all
 
 xviii PREFACE 
 
 logical questions involving three terms only ; and it is quite 
 recently that it has occurred to me how it might be printed 
 in the form of a compact and convenient table. 
 
 Mr. Venn has published in the Philosophical Magazine 
 for July 1 880, a paper ' On the Diagrammatic and 
 Mechanical Representation of Propositions and Reason- 
 ings.' An article on ' Symbolic Reasoning ' by the same 
 author will also be found in Mind for the same month. 
 The text of this book having been completed and placed 
 in the printer's hands before Mr. Venn's ingenious papers 
 were published, it has not been possible to illustrate or to 
 criticise his views. 
 
 I may mention that M. Louis Liard, Professor of Philo- 
 sophy at Bordeaux, who had previously explained and 
 criticised the substitutional view of Logic in the Revue 
 Philosophique (Mars, 1877, torn, iii., p. 277, etc.), has 
 since published a very good though concise account of the 
 principal recent logical writings in England, under the 
 title, Les Logiciens anglais contemporains (Paris : Germer 
 Bailliere, 1878). 
 
 These ' Studies ' consist in great part of logical Questions 
 and Problems gathered from many quarters. In the majority 
 of cases I have indicated by initial letters the source or 
 authorship of the questions when clearly known (see the 
 List of References on p. xxv) ; but I have not always 
 carried out this rule, and in not a few cases the questions 
 have been printed several times already, and are of 
 doubtful authorship. A large remaining fraction of the 
 questions and problems are new, and have been de-
 
 PREFACE xix 
 
 vised specially for this book. As shown by the author's 
 name appended, a few questions have been borrowed 
 from the work of the Very Rev. Daniel Bagot, Dean 
 of Dromore, entitled Explanatory Notes on the Principal 
 \Chapters of Murray's Logic . . . with an Appendix of 337 
 Questions to Correspond. A few excellent illustrations 
 have also been drawn from a privately printed tract on 
 Logic by the late Sir J. H. Scourfield, M.P., his own 
 annotated copy having been kindly presented to me by 
 the author a few years before his death. 
 
 In forming this compilation I have been more than ever 
 struck by the fact that the larger part of logical difficulties 
 and sophisms do not turn upon questions of formal logic 
 but upon the relations which certain assertions bear to the 
 presumed or actual knowledge of the assertor and the 
 hearer. If the person X remarks that 'All lawyers are 
 honourable men,' it is one question what is the pure 
 logical force of this proposition, as measured by its 
 effect on the combinations of the terms concerned and 
 their negatives. It is quite another matter what X means 
 by it ; why he asserts it ; what he expects Y to understand 
 by it ; and what Y actually does take as the meaning 
 otX. 
 
 Under certain circumstances assertions convey a meaning 
 the direct opposite of what they convey at other times. 
 If a man is taken with a fit and the first medical man who 
 arrives says, 'You must not think of putting the man 
 under the pump,' the man will not be put under the 
 pump ; but if the identically same assertion is made about
 
 xx PREFACE 
 
 the centre of interest of an excited and angry mob, the 
 man goes to the pump. It is evident that there ought to 
 exist a science of applied deductive logic, partly corre- 
 sponding to the ancient doctrines of rhetoric, in which 
 the popular force of arguments as distinguished from their 
 purely logical force should be carefully analysed. A few 
 questions and answers given in this book may perhaps 
 belong, properly speaking, to rhetorical logic (see pp. 119, 
 140-1, etc.), but I have not found it practicable to pursue 
 the subject in this book. It should be evident that a 
 thorough comprehension of the purely logical aspect of 
 assertions must precede any successful attempt to in- 
 vestigate their rhetorical aspect. I may possibly at some 
 future time attack the problems of rhetorical logic. 
 
 A further question which forced itself upon my notice 
 was that of the practicability of including exercises in 
 Inductive Logic. As Mr. H. S. Foxwell suggested, in- 
 ductive exercises and problems are even more needed 
 than those of a deductive character. But, on consideration 
 and trial, it seemed highly doubtful whether it would be 
 possible to throw questions of inductive logic into the 
 concise and definite form essential to a book of exercises. 
 I have given abundance of inverse combinational problems 
 which are really of an inductive character (see pp. 252-8); 
 but exercises in the inductive methods of the physical 
 sciences, if practicable at all, would require a much greater 
 space, and a very different mode of treatment from that 
 which they could receive in this work. For the present, 
 at all events, I must content myself with referring readers
 
 PREFACE xxi 
 
 to the ample exposition of inductive methods contained 
 in the 3rd, 4th, and 5th books of the Principles of 
 Science. 
 
 Some readers may perhaps be still inclined to object 
 to the Syllogism, and to deductive logic generally, that it 
 is comparatively worthless, because all new truths are 
 obtained by induction. This doctrine has prevailed with 
 many writers from the time of John Locke to that of John 
 Stuart Mill. But if I have proved in Chapters VI., VII., 
 XL, XIL, and in other parts of the Principles of Science, 
 that induction is the inverse operation of deduction, the 
 supreme importance of syllogistic and other deductive 
 reasoning is not so much restored as explained. In 
 reality the cavillers against the syllogism have never suc- 
 ceeded in the slightest degree in weakening the hold 
 of the syllogism upon the human mind : it was against 
 the nature of things that they should succeed. Their 
 position was as sensible as that of a tutor who should 
 recommend his pupils to begin Mathematics with Compound 
 Division, but on no account to trouble themselves with the 
 obsolete formula of the Multiplication Table. In every 
 point of view, then, a thorough command of deductive 
 processes is the necessary starting-point for any attempt to 
 master more difficult and apparently more important pro- 
 cesses of reasoning. 
 
 In the composition of the didactic parts of this book, 
 I have tried the experiment of throwing my remarks into 
 the form of answers to assumed, or in many cases actual, 
 examination questions. I cannot call to mind any book
 
 xxii PREFACE 
 
 in which this mode of treatment has been previously 
 adopted, but it seems to lend itself very readily to the 
 clear exposition of knotty points and difficulties. In spite 
 of much popular clamour against examinations, I maintain 
 that to give a clear, concise, and complete written answer to 
 a definite question or problem is not only the best exercise 
 of mind, but also the best test of ability and training, 
 which can be generally applied. 
 
 The Frontispiece contains rough facsimilies of ancient 
 logical diagrams which I copied from the fine MS. of 
 Aristotle's Organon in the Ambrosian Library at Milan 
 (L. 93, Superior). During a visit to Italy in 1874, I was 
 much surprised and interested by the multitudes of curious 
 diagrammatic exercises to be found in the logical MSS. 
 of the great public libraries of Italy. The abundance 
 of these diagrams shows that rudimentary logical exer- 
 cises were very popular in the country where, and at the 
 time when, the dawn of modern science began to break. I 
 estimated that a single MS. in the Biblioteca Communale 
 at Perugia (Aristotelis de Interpretatione cum Comment. 
 A 55. Graece. Chart. 1485) contained at least eight 
 hundred such diagrams. Those given in the frontis- 
 piece are the most ancient which I could discover. The 
 MS. containing these (among others) is assigned in the 
 printed catalogue to the eleventh or twelfth century, but 
 the librarian was of opinion that it might belong to the 
 tenth century. The figure in the centre shows the Greek 
 original of the familiar Square of Logical Opposition, which 
 has survived to this day (see p. 31). The triangular and
 
 PREFACE xxiii 
 
 lunular figures represent respectively the syllogistic moods 
 Darapti, and (I believe) Datisi. 
 
 To the imperfect list of the most recent writings on 
 Symbolical Logic, given in this preface, I am enabled to 
 add at the last moment the important new memoir of 
 Professor C. S. Peirce on the Algebra of Logic, the first part 
 of which is printed in the American Journal of Mathematics, 
 vol. iii. (i5th September 1880). Professor Peirce adopts 
 the relation of inclusion, instead of that of equation, as the 
 basis of his system. 
 
 BRANCH HILL, 
 HAMPSTEAD HEATH, N.W., 
 
 yd October 1880.
 
 NOTE TO EDITION OF 1884 
 
 THE present Edition has been printed from the Author's 
 own copy, in which he had marked the few corrections 
 and alterations which have now been made. 
 
 HARRIET A. JEVONS.
 
 REFERENCE LIST 
 
 OF INITIAL LETTERS SHOWING THE AUTHORSHIP 
 OR SOURCE OF QUESTIONS AND PROBLEMS. 
 
 A = PROFESSOR ROBERT ADAMSON, Owens College, 
 Manchester. 
 
 B - PROFESSOR ALEXANDER BAIN, University of Aberdeen. 
 
 C = Cambridge University. Moral Science Tripos, or 
 College Examination Papers. 
 
 D = Dublin University. 
 
 E = Edinburgh University. PROFESSOR ERASER. 
 
 H = REV. JOHN HOPPUS, formerly Professor of Logic, etc. 
 in University College, London. 
 
 I - India Civil Service Examinations. 
 
 L = London University, Second B.A., Second B.Sc., M.A., 
 M.D. and D.Sc. Examinations. 
 
 M = PROFESSOR THOMAS MOFFET, President and Professor 
 in Queen's College, Galway. 
 
 O = Oxford University. 
 
 P = PROFESSOR PARK, Queen's College, Belfast, and 
 Queen's former University. 
 
 R = PROFESSOR GROOM ROBERTSON, University College, 
 London. 
 
 W = WHATELY'S Elements of Logic,
 
 CONTENTS 
 
 CHAP. PAGE 
 
 1. THE DOCTRINE OF TERMS i 
 
 2. QUESTIONS AND EXERCISES RELATING TO TERMS . 9 
 
 3. KINDS OF PROPOSITIONS . . . . . . 18 
 
 4. EXERCISES IN THE DISCRIMINATION OF PROPOSITIONS 25 
 
 5. CONVERSION OF PROPOSITIONS, AND IMMEDIATE IN- 
 
 FERENCE . . . . . . . . 31 
 
 6. EXERCISES ON PROPOSITIONS AND IMMEDIATE INFER- 
 
 ENCE 56 
 
 7. DEFINITION AND DIVISION ..... 64 
 
 8. SYLLOGISM 71 
 
 9. QUESTIONS AND EXERCISES ON THE SYLLOGISM . . 94 
 
 10. TECHNICAL EXERCISES IN THE SYLLOGISM . . 103 
 
 11. CUNYNGHAME'S SYLLOGISTIC CARDS .... 107 
 
 12. FORMAL AND MATERIAL TRUTH AND FALSITY . . in 
 
 13. EXERCISES REGARDING FORMAL AND MATERIAL 
 
 TRUTH AND FALSITY 122 
 
 14. PROPOSITIONS AND SYLLOGISMS IN INTENSION . . 126 
 
 15. QUESTIONS ON INTENSION 135
 
 xviii CONTENTS 
 
 CHAP. PAGE 
 
 16. HYPOTHETICAL, DILEMMATIC, AND OTHER KINDS OF 
 
 ARGUMENTS 137 
 
 17. EXERCISES IN HYPOTHETICAL ARGUMENTS . . 145 
 
 18. THE QUANTIFICATION OF THE PREDICATE. . . 149 
 
 19. EXERCISES ON THE QUANTIFICATION OF THE PREDI- 
 
 CATE . 159 
 
 20. EXAMPLES OF ARGUMENTS AND FALLACIES . . 164 
 
 21. ELEMENTS OF EQUATION AL LOGIC . . . .179 
 
 22. ON THE RELATIONS OF PROPOSITIONS INVOLVING 
 
 THREE OR MORE TERMS 223 
 
 23. EXERCISES IN EQUATIONAL LOGIC .... 227 
 
 24. THE MEASURE OF LOGICAL FORCE .... 249 
 
 25. INDUCTIVE OR INVERSE LOGICAL PROBLEMS . . 252 
 
 26. ELEMENTS OF NUMERICAL LOGIC .... 259 
 
 27. PROBLEMS IN NUMERICAL LOGIC .... 276 
 
 28. THE LOGICAL INDEX 281 
 
 29. MISCELLANEOUS QUESTIONS AND PROBLEMS . . 290
 
 STUDIES IN 
 
 DEDUCTIVE LOGIC 
 
 CHAPTER I 
 
 THE DOCTRINE OF TERMS 
 
 INTRO D UCTION 
 
 1. IN accordance with custom, I begin this book of 
 logical studies with the treatment of Terms. Besides being 
 customary, this way of beginning is convenient, because 
 some difficulties which might otherwise be encountered in 
 the treatment of propositions and arguments are cleared 
 out of the way. But the continued study of logic convinces 
 me that this doctrine of terms is really a composite and for 
 the most part extra-logical body of doctrine. It is in fact 
 a survival, derived from the voluminous controversies of 
 the schoolmen. 
 
 2. The difficulties of metaphysics, of physics, of grammar, 
 and of logic itself, are entangled together in this part of 
 logical doctrine. Thus, if we take such a term as colour^ 
 and endeavour to decide upon its logical characters, we 
 should say that it is categorematic, because it can stand as 
 the subject of a proposition ; it is positive, because it im- 
 
 ^ B
 
 2 DOCTRINE OF TERMS CHAP. 
 
 plies the presence rather than the absence of qualities. But 
 is it abstract or concrete ? If concrete, it should be the 
 name of a thing, not of the attributes of a thing. Now 
 colour is certainly an attribute of gold or vermilion ; never- 
 theless, colour has the attribute of being yellow or red or 
 blue. Thus I should say that yellowness is an attribute of 
 colour, and if so, colour is concrete compared with yellow- 
 ness or blueness, while it is abstract compared with gold 
 or cobalt. If this view is right, abstractness becomes a 
 question of degree. 
 
 3. Again, a relative term is one which cannot be thought 
 except in relation to something else, the correlative. Thus 
 nephew cannot be thought but as the nephew of an uncle 
 or aunt ; an instrument cannot be thought but as the 
 instrument to some end or operation. But the question 
 arises, Can anything be thought except as in relation to 
 something else ? What is the meaning of a table but as 
 that on which dinner is put ? What is a chair but the seat 
 of some person ? Every planet is related to the sun, and 
 the sun to the planets. Even meteoric stones moving 
 through empty space are related by gravity to the sun 
 attracting them. All is relative, both in nature and 
 philosophy. 
 
 4. As to the distinctions of general, singular, and proper 
 terms, connotative and non-connotative terms, etc., they 
 seem to me to be involved in complete confusion. I have 
 shown in the Elementary Lessons in Logic (pp. 41-44) that 
 Proper Names are certainly connotative. There would be 
 an impossible breach of continuity in supposing that, after 
 narrowing the extension of ' thing ' successively down to 
 animal, vertebrate, mammalian, man, Englishman, educated 
 at Cambridge, mathematician, great logician, and so forth, 
 thus increasing the intension all the time, the single re-
 
 i DEFINITIONS AND EXAMPLES 3 
 
 maining step of adding Augustus de Morgan could remove 
 all the connotation, instead of increasing it to the utmost 
 point. But however this and many other questions in the 
 doctrine of terms may be decided, it is quite clear in any 
 case that this part of logic is ill-suited for furnishing good 
 exercises in reasoning. This ground alone is sufficient to 
 excuse my passing somewhat rapidly and perfunctorily over 
 the first part of logic, and going on at once to the subject 
 of Propositions which offers a wide field for useful exercises. 
 Accordingly, after giving brief definitions of the several 
 kinds of terms, a few answers to questions, and a fair 
 supply of unanswered questions and problems, I pass on to 
 the more satisfactory and prolific parts of logic. 
 
 DEFINITIONS AND EXAMPLES 
 
 5. A general term is one which can be affirmed, in the 
 same sense, of any one of many (i.e. two or more) things. 
 
 Examples Building, front-door, lake, steam-engine. 
 
 6. A singular term is one which can only be affirmed, in 
 the same sense, of one single thing. 
 
 Examples Queen Victoria, Cleopatra's Needle, the 
 Yellowstone Park. 
 
 7. A collective term is one which can be affirmed of 
 two or more things taken together, but which cannot be 
 affirmed of those things regarded separately or distributively. 
 
 Examples Regiment, century, pair of boots, baker's 
 dozen, book (a collection of sheets of paper).
 
 4 DOCTRINE OF TERMS CHAP. 
 
 8. A concrete term is a term which stands for a thing. 
 
 Examples Stone, red thing, brute, man, table, book, 
 father, reason. 
 
 9. An abstract term is a term which stands for an 
 attribute of a thing. 
 
 Examples Stoniness, redness, brutality, humanity, 
 tabularity, paternity, rationality. 
 
 10. A connotative term is one which denotes a subject 
 and implies an attribute. 
 
 Examples Member of Parliament denotes Gladstone, 
 Sir Stafford Northcote, or any other individual 
 member of parliament, and implies that they can 
 sit in parliament ; bird denotes a hawk, or eagle, 
 or finch, or canary, and implies that they have all 
 the attributes of birds. 
 
 11. A non-connotative term is one which signifies an 
 attribute only, or (if such can be) a subject only. 
 
 Examples Whiteness denotes whiteness only, an 
 attribute without a subject. John Smith (according 
 to J. S. Mill, and some other logicians) denotes a 
 subject or person only, without implying attributes". 
 
 12. Concrete general names are always connotative. 
 Such also are all adjectives, without exception. Every 
 adjective is the name of a thing to which it is added, and 
 implies that the thing possesses qualities. Red is the name 
 of blood or of other red thing, and implies that it is red. 
 Redness is the abstract term, the name of the quality 
 redness.
 
 i DEFINITIONS AND EXAMPLES 5 
 
 13. A positive concrete term is applied to a thing in 
 respect of its possession of certain attributes ; a positive 
 abstract term denotes certain attributes. 
 
 Examples Useful, active, paper, rock; usefulness, 
 activity, rockiness. 
 
 14. A negative term is applied to a thing in respect of 
 the absence of certain attributes ; if abstract the term 
 denotes the absence of such attributes. 
 
 Examples Useless, inactive, not-paper ; uselessness, 
 inactivity. 
 
 15. An absolute term is the name of a thing regarded 
 per se, or without relation to anything else, if such there 
 can be. 
 
 Examples Air, book, space, water. 
 
 1 6. A relative term is the name of a thing regarded in 
 connection with some other thing. 
 
 Examples Father, ruler, subject, equal, cause, effect. 
 
 1 7. A categorematic term is one which can stand alone 
 as the subject of a proposition. 
 
 Examples Any noun substantive ; any adjective, any 
 phrase or any proposition used substantively. 
 
 1 8. A syncategorematic term is any word which can only 
 stand as the subject of a proposition in company with some 
 other words. 
 
 Examples Any preposition, conjunction, adjective used 
 adjectively. 
 
 19. Differences of opinion may arise concerning almost 
 every one of the definitions given above, and it would not
 
 6 DOCTRINE OF TERMS CHAP. 
 
 be suitable to the purpose of this book to discuss the matter 
 further. 
 
 In every case, too, we ought before treating any terms 
 to ascertain clearly that there is no ambiguity about their 
 meanings. An ambiguous term is not one term, but two or 
 more terms confused together, and we should single out one 
 definite sense before we endeavour to assign the logical 
 characteristics. The ambiguity of terms has however been 
 sufficiently dwelt upon in the Elementary Lessons, Nos. iv. 
 and vi., and it need not be pursued here. 
 
 For the further study of the subject of terms the reader 
 is referred to the Elementary Lessons ; Mill's System of Logic, 
 book i., chapters i. and ii. ; Shedden's Logic, chapters i. 
 and ii. ; Levi Hedge's Logic, part ii., chapter i. ; Martineau, 
 Prospective Review, vol. xxix. pp. 133, etc. ; Hamilton's 
 Lectures on Logic, vol. iii., lectures viii. to xii. ; Woolley's 
 Introduction to Logic, part i., chapter i. 
 
 QUESTIONS AND ANSWERS 
 
 20. Describe the logical characters of the follow- 
 ing terms Equal, equation, equality, equal- 
 ness, inequality, and equalisation. 
 
 Equal is a noun-adjective ; concrete, as denoting equal 
 things ; connotative, as connoting the attribute of equality ; 
 general, positive, relative ; and syncategorematic, because 
 it cannot as an adjective form the subject of a proposition. 
 
 Equation, noun-substantive, originally abstract, as mean- 
 ing either equality, or the action of making equal. It is 
 now generally used by mathematicians to denote a pair of
 
 i DEFINITIONS AND EXAMPLES 7 
 
 quantities affirmed to be equal. It is thus concrete, general, 
 positive, perhaps absolute, and categorematic. 
 
 Inequality is a noun-substantive, abstract, singular, nega- 
 tive, and categorematic. 
 
 Equalisation means the action of making equal, an attri- 
 bute or circumstance of things, not a thing. It is thus 
 abstract, singular, positive, categorematic. 
 
 21. What are the logical characters of the terms, 
 drop of oil, oily, oiliness ? 
 
 A drop of oil being a concrete, finite thing, its name will 
 be concrete, general, positive, relative (as having dropped 
 from a mass of oil), collective as regards the particles of 
 oil, connotative as implying the qualities of oiliness, etc., 
 and categorematic. 
 
 Oil is concrete, positive, collective, connotative, and cate- 
 gorematic, like drop of oil, and only differs in not admitting, 
 as regards any one kind of oil, of the plural. It is a case 
 of what I have proposed (Principles of Science, p. 28 ; ist ed., 
 vol. i. p. 34) to call a substantial term, but which I find 
 that Burgersdyk, Heereboord, and the older logicians called 
 a Mum homogeimim, the parts being of the same name 
 and nature with the whole. (Heereboord, Synopsis Logicae, 
 p. 83. See also Mind, vol. i. p. 210.) 
 
 Oily is a noun-adjective, and is concrete, general, positive, 
 connotative, as denoting oil and implying the attributes of 
 oiliness, doubtfully relative, syncategorematic, 1680. 
 
 Oiliness, noun-substantive, abstract, singular, positive, 
 categorematic. 
 
 Where distinctions are omitted, it may be understood that 
 they are regarded as inapplicable.
 
 8 DOCTRINE OF TERMS CHAP, i 
 
 22. Describe the logical characters of the terms 
 Related, relative, relation, relativeness, rela- 
 tionship, relativity. 
 
 I have already dwelt, in the Elementary Lessons (p. 25), 
 on the prevalent abuse of the word ' relation,' and other like 
 abstract terms. Nothing is more nearly impossible than to 
 reform the popular use of language ; but I will point out 
 once again that relation is properly the abstract name of the 
 connection or bearing of one thing to another, this being 
 an attribute of those things. The things in question are 
 properly said to be ' related,' or to be ' relatives.' Thus, 
 fathers, brothers, sisters, aunts, and cousins, are all relatives 
 not relations. Relationship is an abstract term signifying 
 the attribute of being related ; it was invented to replace 
 relation when this was wrongly used as a concrete term. 
 The relationship between a mother and her daughter is 
 simply the relation which exists between two such related 
 persons or relatives. Relativeness is an uncommon term 
 sometimes used to replace the abstract sense of relation, 
 where the case is not one of family relation. Relativity is 
 a further abstract term, probably due to Coleridge, and of 
 which the metaphysicians had better have the monopoly.
 
 CHAPTER II 
 
 QUESTIONS AND EXERCISES RELATING TO TERMS 
 
 i. DESCRIBE the logical characters of the following terms, 
 classifying them according as they are 
 
 (a) Abstract or Concrete. 
 
 (b) General or Singular. 
 
 (c) Collective or Distributive. 
 (d} Positive or Negative. 
 
 (e) Absolute or Relative. 
 
 (_/") Categorematic or Syncategorematic. 
 
 Prime Minister Biped 
 
 Institution Saturn 
 
 Copper Bismarck 
 
 Shameful Monarch 
 
 The London Library Unuseful 
 
 Collection The Times 
 
 School Board Paper 
 
 Deaf Augustus de Morgan 
 
 Equation John Jones 
 
 Innumerous John 
 
 Purpose Triangle 
 
 Function Musicalness 
 
 Cousin Board School 
 
 The Absolute Needlepoint 
 
 Black Representation
 
 io DOCTRINE OF TERMS CHAP. 
 
 Europe Advocate 
 
 Injustice Being 
 
 Brace of partridges Whale 
 
 Dumbness Lawyer 
 
 Planetary System Time 
 
 Classification Manchester 
 
 2. In the case of the following terms distinguish with 
 special care between those which are abstract and those 
 which are concrete 
 
 Nature Animal Ethericity 
 
 Natural Animalism Scarce 
 
 Naturalness Animality Scarcity 
 
 Naturalism Animalcule Scarceness 
 
 Author Ether Truth 
 
 Authority Ethereal Trueness 
 
 Authorship Etherealness Verity 
 
 3. Investigate the ambiguity of any of the following 
 terms as regards their concrete or abstract character 
 
 Weight Science 
 
 Time Schism 
 
 Intention Space 
 
 Vibration Relation 
 
 4. Supply the abstract terms corresponding to the 
 following concrete terms 
 
 Wood Conduction 
 
 Stone Atmosphere 
 
 Conduct Alcohol 
 
 Witness Axiom 
 
 Equal Gas 
 
 Table Fire 
 
 Boy Socrates
 
 ii QUESTIONS AND EXERCISES 11 
 
 5. In the case of such of the following terms as you 
 consider to be abstract, name the corresponding concrete 
 terms 
 
 Analysis Nation 
 
 Psychology Vacuity 
 
 Extension Realm 
 
 Production Folly 
 
 Socialism Evidence 
 
 6. Do abstract terms admit of being put in the plural 
 number ? Distinguish between the terms which are abstract 
 and concrete in the following list, and at the same time 
 indicate which can in your opinion be used in the plural : 
 colour, redness, weight, value, quinine, equation, heat, 
 warmth, hotness, solitude, whiteness, paper, space, [c.] 
 
 7. Investigate the logical characters and ambiguities of 
 the term form in all the following expressions : a religion 
 of forms ; the human form ; a form of thought ; a school 
 form ; a mere form ; a printer's form ; a form of govern- 
 ment ; form of prayer ; good form ; essential form. 
 
 8. What error is there in the following descriptions ? 
 
 Peerless syncategorematic, general, abstract, positive, 
 relative. 
 
 Bacon equivocal, concrete, general, substantial, positive, 
 relative. 
 
 Black categorematic, abstract, general, negative, abso- 
 lute. 
 
 9. Analyse the following sentences as regards the logical 
 character of each term found in them, distinguishing especi- 
 ally between such as are concrete or abstract, collective or 
 distributive, singular or general
 
 12 DOCTRINE OF TERMS CHAP. 
 
 Logic is the science of the formal laws of thought. 
 
 Entre 1'homme et le monde il faut 1'humanite. 
 
 ' Art is universal in its influence ; so may it be in its 
 practice, if it proceed from a sincere heart and a quick 
 observation. In this case it may be the merest sketch, or 
 the most elaborate imitative finish.' 
 
 10. Burton, in his Etruscan Bologna, p. 234, uses the 
 abstract term Etruscanicity. Is it possible in like manner 
 to make an abstract term corresponding to every concrete 
 one ? If so, supply abstracts for the following concretes 
 
 Sir Isaac Newton. Royal Engineers. 
 
 Dictionary. Postal Telegraph. 
 
 11. What logical faults do you detect in the following 
 expressions ? 
 
 The standard authorship of modern times. 
 
 The three great nationalities of Western Europe. 
 
 The legal heir is not necessarily a man's nearest relation. 
 
 That unprincipled notoriety Pietro Aretino. 
 
 12. Coleridge, in a celebrated note to his Aids to Reflec- 
 tion, thus defines an Idea : ' An Idea is the indifference of 
 the objectively real and the subjectively real : so, namely, 
 that if it be conceived as in the Subject, the idea is an 
 Object, and possesses objective truth ; but if in an Object, 
 it is then a Subject, and is necessarily thought of as exer- 
 cising the powers of a Subject. Thus an Idea, conceived 
 as subsisting in an Object, becomes a Law : and a law 
 contemplated subjectively (in a mind) is an Idea? 
 
 Analyse the meanings of the terms Idea, Object, Subject, 
 Real, Truth, Law, etc., in the above passage, with respect 
 especially to their concreteness or abstractness. [L.]
 
 ii QUESTIONS AND EXERCISES 13 
 
 13. Name the negative terms which correspond to the 
 following positive terms 
 
 Illumination Variable 
 
 White Famous 
 
 Certain Notorious 
 
 Constant Valid 
 
 Dying Plenty 
 
 14. Name the positive terms which correspond to the 
 following negative or apparently negative terms 
 
 Immensity Falsehood 
 
 Inestimable Unravelled 
 
 Disestablishment Infamous 
 
 Unpleasant Presuppositionless 
 
 Want Shameless 
 
 Unloosed Empty 
 
 Indifferent Intact 
 
 Headless Ignominious 
 
 15. In examining the following list of terms, distinguish, 
 as far as possible, between those which are really negative 
 in form and origin, and those which only simulate the 
 character of negatives 
 
 Annulled Undespairing 
 
 Disannulled Invalid 
 
 Antidote Headless 
 
 Infrequent Independence 
 
 Eclipse Individual 
 
 Undisproved Indolent 
 
 The Infinite Disagreeable 
 
 Impassioned Despairing 
 
 Immense Infant 
 
 Purposeless Deafness
 
 14 DOCTRINE OF TERMS CHAP. 
 
 1 6. Can you find any examples of terms in the dictionary 
 which are true double negatives ? ' Paired,' ' Impaired,' 
 and ' Unimpaired,' may perhaps be affirmed respectively 
 of two things which are equal, unequal, and not unequal. 
 Analyse the meaning of each of the following terms, and 
 show whether it is or is not a true double negative 
 
 Indefeasible Indefatigable 
 
 Uninvalided Uninjured 
 
 Undecomposable Undecipherable 
 
 Undefaceable Undeformed 
 
 Indestructible Indistinguishable 
 
 1 7. How are the denotation and connotation of a con- 
 crete term related to the denotation of the corresponding 
 abstract term ? 
 
 1 8. Explain the difference of denotation and connota- 
 tion with reference to the terms Law, Legislator, Legality, 
 Crime. [L.] 
 
 19. Compare the connotation of the following sets of 
 terms 
 
 (Abbey (Caesar 
 
 \Westminster Abbey I Roman 
 
 /-Mineral fRoad 
 
 J Oxide of iron -! Means of communication 
 
 {Ore [Railway 
 
 20. Distinguish in the following list such terms as are 
 non-connotative, naming at the same time the logician 
 whose opinion on the subject you adopt 
 
 Virtue Gladstone 
 Virtuous Socrates 
 
 The mother of the Barmouth 
 
 Gracchi . The Lord Chamberlain
 
 ii QUESTIONS AND EXERCISES 15 
 
 21. Form a list of twelve purely non-connotative names. 
 
 22. What is, if any, the connotation of these terms: 
 Charles the First ; Richelieu ; Johft Smith ; Santa Maria 
 Maggiore ? 
 
 23. Try to name half-a-dozen perfectly non- relative 
 names, and then inquire whether they really are non-relative. 
 What is the relation implied or involved in each of the 
 following terms ? 
 
 Metropolis County 
 
 Realm Alphabet 
 
 Capital city Sun 
 
 24. Show, by examples, that the division of Names into 
 general and singular does not coincide with the division 
 into abstract and concrete. [L.] 
 
 25. What kinds of words can stand as the subject of a 
 proposition, and what kinds are excluded ? [o.] 
 
 26. Distinguish between the distributive, collective, or 
 singular use of these Latin adjectives of quantity : omnis, 
 omnes, cunctus, cuncti, ullus, quidam, aliquis. 
 
 27. What is peculiar about the use of certain terms in 
 the following extracts ? 
 
 (1) Frenchmen, I'll be a Salisbury to you. 
 
 (2) His family pride was beyond that of a Talbot or a 
 
 Howard. 
 
 (3) In quo quisque artificio excelleret, is in suo genere 
 
 Roscius diceretur. 
 
 (4) ' When foe meets foe.' 
 
 28. How does Logic deal with verbs, adverbs, and con- 
 junctions ? 
 
 29. How many logical terms are there in the following 
 witty epigram ? Which and what are they ?
 
 16 DOCTRINE OF TERMS CHAP. 
 
 What is mind ? No matter. 
 What is matter ? Never mind. 
 
 30. How many logical terms are there in each of the 
 following sentences? Ascertain exactly how many words 
 are employed in each such term. 
 
 (1) The Royal Albert Hall Choral Society's Concert is 
 held in the Albert Hall on the Kensington Gore Estate 
 purchased by the Royal Commissioners of the Great 
 Exhibition of 1851. 
 
 (2) ' 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 pro- 
 nounced to others, may be to them a sign of what 
 thought the speaker had before in his mind.' 
 
 31. Words, says Hobbes, are insignificant (that is without 
 meaning), 'when men make a name of two names, whose 
 significations are contradictory and inconsistent : as this 
 name, an incorporeal body.' 
 
 The following are a few instances of such apparently self- 
 inconsistent names, and the student is requested to add to 
 the list 
 
 (1) Corporation sole. 
 
 (2) Trigeminus. 
 
 (3) Manslaughter of a woman. 
 
 (4) An invalid contract. 
 
 (5) A breach of a necessary law of thought. 
 
 32. How would you explain the following apparent 
 absurdities ? 
 
 An Act of Parliament (1798-99) prohibited the importa- 
 tion of ' French lawns not made in Ireland.'
 
 ii QUESTIONS AND EXERCISES 17 
 
 Ferguson (History of Architecture, vol. ii. p. 233) de- 
 scribes a certain Moabite tower as a ' square Irish round 
 tower.' 
 
 33. Are the following terms perfectly univocal or un- 
 ambiguous, or can you point out any equivocation which is 
 possible in their use ? 
 
 Penny Lecture-Room 
 
 Charcoal Victoria Street 
 
 Aluminium Bible 
 
 Second Monday 
 
 34. Trace out and explain the ambiguities which affect 
 any of the following terms 
 
 Organ Stone March 
 
 Sole Corn Mood 
 
 Ear Diet Mean 
 
 Bowl Perch Force 
 
 Rock (stone) Bole Bowl 
 
 Rock (bird) Strait Straight 
 
 35. Draw out complete lists of all the words or expres- 
 sions which have been developed out of the roots of the 
 following words (see Elementary Lessons in Logic, pp. 
 32-36, and Lesson VI.) 
 
 Post Logic 
 
 Section Faction 
 
 Final Function 
 
 Mission Decline
 
 CHAPTER III 
 
 KINDS OF PROPOSITIONS 
 
 i. IN this chapter propositions will be described and 
 classed according to the ancient Aristotelian doctrine, in 
 which four principal forms of propositions were recognised, 
 thus tabularly stated : 
 
 Affirmative. 
 
 Negative. 
 
 Universal 
 
 Particular 
 
 Symbol = A 
 All X is Y 
 
 Symbol= E 
 No^Tis Y 
 
 Symbol = I 
 Some X is Y 
 
 Symbol = O 
 Some X is not Y 
 
 Singular propositions are to be classed as universal, and 
 indefinite propositions, in which no indication of quantity 
 occurs, must be interpreted at discretion as universal or 
 particular. The student is supposed to be familiar with 
 what the ordinary text-books say upon the subject. 
 
 I first give a series of Examples of propositions, with 
 brief comments upon their logical form and peculiarities. 
 A copious selection of exercises is then supplied in the 
 next chapter for the student to treat in like manner.
 
 CHAP, in KINDS OF PROPOSITIONS 19 
 
 EXAMPLES 
 
 2. ' Books are not absolutely dead things.' O. 
 
 This proposition is indefinite or pre-indesignate, as 
 Hamilton would call it (Lectures on Logic, vol. i. (iii.) 
 p. 244) ; but, as we can hardly suppose Milton to have 
 thought that all books were living things, I take it to 
 mean 'some books are not, etc.,' that is to say, particular 
 negative. 
 
 3. 'The weather is cold.' A. 
 
 The weather means the present state of the surrounding 
 atmosphere, and may be best described as a singular term, 
 which makes the assertion universal. 
 
 4. ' Not all the gallant efforts of the officers and escort 
 of the British Embassy at Cabul were able to save 
 them.' E. 
 
 At first sight this seems to be a particular negative, like 
 ' Not all that glitters is gold ' ; but a little consideration 
 shows that ' gallant efforts ' is a collective whole, the efforts 
 being made in common, and therefore either successful or 
 unsuccessful as a whole. The meaning then is, ' The 
 whole of the gallant efforts, etc., were not able to save the 
 men.' It is a universal negative. 
 
 5. ' One bad general is better than two good ones.' A. 
 
 This saying of Napoleon looks at first like a particular 
 or even a singular proposition ; but the ' one bad general ' 
 means not any definite one, but ' any one bad general ' 
 acting alone. 
 
 6. 'No non- metallic substance is now employed to 
 make money.' E.
 
 20 KINDS OF PROPOSITIONS CHAP. 
 
 The subject is a negative term, and the proposition 
 might be stated as ' All non- metallic substanpes are not 
 any of those employed to make money.' 
 
 7. ' Multiplication is vexation.' 
 
 If all multiplication is so, this is A ; there are certainly 
 other causes of vexation. 
 
 8. ' Wealth is not the highest good.' E. 
 Affirmatively, wealth is one of the things which are not 
 
 the highest good. 
 
 9. 'Murder will out.' A. 
 
 Like most proverbs, this is an unqualified universal 
 proposition ; its material truth may be doubted. 
 
 10. 'A little knowledge is a dangerous thing.' A. 
 This looks like a particular affirmative, but is really 
 
 A, as meaning that ' any small collection of knowledge 
 is, etc.' 
 
 11. ' All these claims upon my time overpower me.' A. 
 Dr. Thomson points out (Outline, 5th ed., p. 131) 
 
 that all is here clearly collective. 
 
 12. 'The whole is greater than any of its parts.' A. 
 Though apparently singular, this is really a general 
 
 axiom, meaning 'any whole is greater, etc.' 
 
 13. ' No wolves run wild in Great Britain at the present 
 day.' E. 
 
 14. 'Who seeks and will not take, when once 'tis 
 offered, shall never find it more.' E. 
 
 This seems to be a compound proposition, but the 
 subject is, 'Any one who is seeking, but has not taken 
 when once it was offered.'
 
 in EXAMPLES 21 
 
 1 5. ' The known planets are now more than a hundred 
 in number.' A. 
 
 Clearly a collective singular affirmative proposition, and 
 therefore universal. Of course the planets separately could 
 not have the predicate here affirmed. 
 
 1 6. ' Figs come from Turkey.' I. 
 
 Indesignate ; that is to say, we cannot assume without 
 express statement that it is intended to say, 'All figs come 
 from Turkey.' 
 
 1 7. ' Xanthippe was the wife of Socrates.' A. 
 
 1 8. 'No one is free who is enslaved by his appe- 
 tites.' E. 
 
 19. 'Certain Greek philosophers were the founders of 
 logic.' A. 
 
 Apparently I ; but if ' certain ' means a certain definite 
 group of men, each of whom was essential in his time, 
 the proposition becomes collective and singular, hence 
 universal. 
 
 20. 'Comets are subject to the law of gravitation.' A 
 
 Indefinite affirmative ; but in a matter of such univer- 
 sality it may be interpreted as A. 
 
 21. ' Democracy ends in despotism.' I. 
 
 Again indefinite ; but as referring to matter in which no 
 rigorous laws have been detected it should be interpreted 
 particularly. 
 
 22. 'Men at every period since the time of Aristotle 
 have studied logic.' I. 
 
 Obviously particular as regards ' men.'
 
 22 KINDS OF PROPOSITIONS CHAP. 
 
 23. 'Few men know how little they know.' O. 
 That is, ' Most men do not know, etc.' Hence O. 
 
 24. ' Natura omnia dedit omnibus.' A. 
 
 Singular affirmative, because nattira is a singular term. 
 The assertion is one of Hobbes', and is thoroughly am- 
 biguous as regards omnia and omnibus, which might be 
 capable either of collective or distributive meaning. No 
 doubt, however, the meaning is that Nature did not assign 
 anything to any particular person ; if so, both must be taken 
 collectively. 
 
 25. ' There are many cotton-spinners unemployed.' I. 
 
 Really a kind of numerical assertion ; but if to be classed 
 at all, it must be I, ' many ' being only a part of ' all.' 
 
 26. 'A few Macedonians vanquished the vast army of 
 Darius.' A. 
 
 Collective singular affirmative, because i\\efew of course 
 acted together. It is a question whether the predicate is 
 not also singular. 
 
 27. ' True Faith and Reason are the soul's two eyes.' A. 
 Collective singular. 
 
 28. 'A perfect man ought always to be busy conquering 
 himself.' A. 
 
 ' All ' perfect men ought, etc. 
 
 29. 'A truly educated man knows something of every- 
 thing and everything of something.' A. 
 
 There seems to be two predicates, and hence a com- 
 pound sentence ; but this is not the case, because the truly 
 educated man must know both.
 
 in EXAMPLES 23 
 
 30. ' Some comets revolve in hyperbolic orbits.' I. 
 Particular affirmative as it stands. 
 
 31. ' The dividends are paid half-yearly.' A. 
 ' The dividends ' includes all so known. 
 
 32. ' Ov TO fjLfya ev ccrri, TO o ev pxya.' O and A. 
 
 This must mean that not all great things are good (O), 
 but that all good things are great (A). There are three 
 classes of things great and good; great and not-good; 
 not-great and not-good. 
 
 33. 'It is force alone which can produce a change of 
 motion.' A. 
 
 It = what can produce, etc. The meaning is, Whatever 
 produces a change of motion is some kind of force ; but 
 there is no assertion that force = whatever produces, etc. 
 
 34. ' We have no king but Caesar.' 
 
 As it stands, A ; but the meaning conveyed implies that 
 ' Caesar is our king ' ; ' Nobody who is not Caesar is our 
 king.' 
 
 35. 'It is true that what is settled by custom though it 
 be not good, yet at least it is fit.' 
 
 Complex ; three propositions in all. 
 
 36. ' God did not make man, and leave it to Aristotle to 
 make him rational.' 
 
 A simple and a singular negative proposition ; the ' not ' 
 applies to all that follows conjunctively, for of course Locke 
 could not have intended to assert that ' God did not make 
 man.' E.
 
 24 KINDS OF PROPOSITIONS CHAP, in 
 
 37. ' Dublin is the only city in Europe, save Rome, which 
 has two cathedrals.' 
 
 Compound sentence implying three propositions, namely 
 Dublin has two cathedrals. A. 
 Rome has two cathedrals. A. 
 All European cities, not being Dublin and not being 
 Rome, have not two cathedrals. E. 
 
 38. 'The affections are love, hatred, joy, sorrow, hope, 
 fear, and anger.' 
 
 Really a disjunctive proposition. Affection is either love, 
 or hatred, or, etc. This implies that love is an affection, 
 hatred is an affection, etc.
 
 CHAPTER IV 
 
 EXERCISES IN THE DISCRIMINATION OF PROPOSITIONS 
 
 i. EXAMINE each of the following propositions, and point 
 out in succession 
 
 (a) Which is the subject. 
 
 (b) Which is the predicate. 
 
 (c) Whether the proposition is affirmative or negative. 
 
 (d) Whether it appears to be universal or particular. 
 
 (e) Whether there is ambiguity or other peculiarity 
 
 in the proposition. 
 
 (1) All foraminifera are marine organisms. 
 
 (2) They never pardon who have done the wrong. 
 
 (3) Great is Diana of the Ephesians. 
 
 (4) No mammalia are parasites. 
 
 (5) Non progredi est regredi. 
 
 (6) Not every one can integrate a differential equation. 
 
 (7) All, all are gone, the old familiar faces. 
 
 (8) He that is not for us is against us. 
 
 (9) Apurrov //.ev v8wp. 
 
 (10) Men mostly hate those whom they have injured, 
 
 (n) Old age necessarily brings decay. 
 
 (12) Nothing morally wrong is politically right. 
 
 (13) What I have written I have written. 
 
 (14) It is not good for man to be alone. "**
 
 26 PROPOSITIONS CHAP. 
 
 (15) A certain man had a fig-tree. 
 
 (16) XaAeTra. TO. KaAa. 
 
 (17) There's something rotten in the state of Denmark. 
 
 (18) To be or not to be, that is the question. 
 
 (19) Ye are my disciples, if ye do all I have said unto you. 
 
 (20) Possunt qui posse videntur. 
 
 . (21) There can be no effect without a cause. 
 
 (22) Rien n'est beau que le vrai. 
 
 (23) Pauci laeta arva tenemus. 
 O-(24) All cannot receive this saying. 
 
 P<.( 2 5) Fain would ^ climb, but that I fear to fall. 
 
 p (26) There's not a joy the world can give like that it 
 
 takes away. 
 
 ^,(27) Not to know me argues thyself unknown. 
 f , (28) Two blacks won't make a white. 
 (29) Few men are free from vanity. 
 
 He that fights and runs away may live to fight 
 
 another day. 
 We are what we are. 
 (32) There is none good but one. 
 v (33) Two straight lines cannot inclose space. 
 t (34) Better late than never. 
 
 (35) Cruel laws increase crime. 
 
 (36) Omnes omnia bona dicere. 
 
 (37) Le genie n'est qu'une plus grande aptitude a la 
 
 patience. 
 
 -j (38) Whosoever is delighted in solitude is either a wild 
 beast or a god. 
 
 (39) Summum jus summa injuria. 
 
 (40) Non omnes moriemur inulti. ^ 
 
 (41) Haud ignara mali miseris succurrere disco, 
 k (42) Familiarity breeds contempt. 
 
 (43) Some politicians cannot read the signs of the times.
 
 iv EXERCISES 27 
 
 (44) Only the ignorant affect to despise knowledge. 
 
 (45) Recte ponitur; vere scire esse per causas scire. 
 
 (46) Only Captain Webb is able to swim across the 
 
 Channel. 
 
 (47) Some books are to be read only in parts. 
 
 (48) E pur si muove. 
 
 (49) Civilisation and Christianity are coextensive. 
 
 (50) Some men are not incapable of telling falsehoods. 
 
 (51) Sunt nonnulli acuendis puerorum ingeniis non 
 
 inutiles lusus. 
 
 (52) All is not true that seems so. 
 
 (53) Me miserable. 
 
 K (54) The Claimant, Arthur Orton, and Castro are in all 
 
 probability the same person. 
 p^ (55) The three angles of a triangle are necessarily equal 
 
 to two right angles. 
 K (56) Many rules of grammar overload the memory. 
 
 (57) Nullius exitium patitur natura videri. 
 
 (58) Summae artis est occultare artem. 
 
 A. (59) Wonderful are the results of science and industry 
 in recent years. 
 
 (60) Love is not love which alters when it alteration finds. 
 
 (6 1 ) A healthy nature may or may not be great; but there 
 
 is no great nature that is not healthy. 
 
 (62) Quas dederis solas semper habebis opes. 
 
 (63) Quod volunt, id credunt homines. 
 
 (64) Hocra cra/a 01' St/couw^creTai. 
 
 (65) Antiquitas seculi, juventus mundi. 
 
 A (66) That would hang us, every mother's son. 
 
 jit (67) Men in great place are thrice servants. 
 A (68) Justice is ever equal. 
 (69) A friend should bear a friend's infirmities. 
 r (TO) Men are not what they were.
 
 28 PROPOSITIONS CHAP. 
 
 (71) The troops took one hour in passing the saluting 
 
 point. 
 
 (72) Nemo mortalium omnibus horis sapit. ^ 
 
 (73) Fugaces labuntur anni. 
 
 (74) Airros eyw et/zt. 
 
 (75) Communia sunt amicorum inter se omnia. 
 
 (76) Dictum sapienti sat est. 
 
 N (77) The Romans conquered the Carthaginians. 
 
 ^ (78) The fear of the Lord, that is wisdom. 
 
 , (79) To live in hearts we leave below is not to die. /5 
 
 (80) Tis only noble to be good. = ^A*,**4.^i* 
 
 (81) Dum spiro spero. 
 
 2. In looking over the following list of propositions dis- 
 tinguish between those which have a djjrtnbutiive^and those 
 which have a collective, subject. 
 
 ^^(i) All the asteroids have been discovered during the 
 
 present century. 
 "^-(2) All Albinos are pink-eyed people. 
 
 (3) The facts of aboriginal life seem to indicate that 
 
 dress is developed out of decorations. 
 
 (4) Non omnes omnia decent. 
 
 ~ft (5) Dirt and overcrowding are among the principal 
 causes of disease. 
 
 (6) Omnes apostoli sunt duodecim. 
 
 (7) Many artisans are unemployed. 
 
 Q (8) The side and diagonal of a square are incommen- 
 surable. 
 
 (9) Omnis homo est animal. 
 (10) Nihil est ab omni parte beatum. 
 
 3. Ascertain exactly how many distinct assertions are 
 made in each of these sentences, and assign the logical 
 characters of the propositions.
 
 iv EXERCISES 29 
 
 5 
 
 E" (i) Tis not jny profit that doth lead mine honour : mine 
 
 honour, it. VAflcut^ rv**- po.^u". ) 
 A ( 2 ) True, 'tis a pity; pity 'tis, 'tis true. X0vv^>. ^\\ x. 
 
 (3) Hearts, tongues, figures, scribes, bards, poets, cannot 
 
 think, speak, cast, write, sing, number, ho ! his 
 love to Antony. 
 
 (4) A horse, a horse ! my kingdom for a horse. 
 
 (5) Istuc est sapere, non quod ante pedes modo est 
 
 videre : sed etiam ilia, quae futura sunt, prospicere. 
 
 (6) Virtue consists neither in excess nor defect of action, 
 
 but in a certain mean degree. 
 
 (7) The glories of our blood and state are shadows, not 
 
 substantial things. 
 
 (8) ' To gild refined gold, to paint the lily, 
 
 To throw a perfume on the violet, 
 
 To smooth the ice, or add another hue YV^ 
 
 Unto the rainbow, or with taper light 
 
 To seek the beauteous eye of heaven to garnish 
 
 Is wasteful and ridiculous excess.' 
 
 (9) All places that the eye of heaven visits, 
 Are to a wise man ports and happy havens. 
 
 k. (10) The age of chivalry is gone, and the glory of 
 
 Europe extinguished for ever, 
 (u) Poeta nascitur, non fit. 
 
 (12) Not all speech is enunciative, but only that in which 
 
 there is truth or falsehood. 
 
 (13) Devouring Famine, Plague, and War, 
 
 Each able to undo Mankind, 
 Death's servile emissaries are. 
 
 (14) Many are perfect in men's humours, that are not 
 
 greatly capable of the real part of business, which 
 is the constitution of one that hath studied men 
 more than books.
 
 3 o PROPOSITIONS CHAP, iv 
 
 (15) Vivre, ce n'est pas respirer, c'est agir. 
 
 (16) Justice is expediency, but it is expediency speaking 
 
 by general maxims, into which reason has concen- 
 trated the experience of mankind. 
 
 (17) Men, wives, and children, stare, cry out, and run as 
 
 it were doomsday. 
 
 4. Distinguish so far as you can between the propositions 
 in the following list which are to you explicative and ampli- 
 ative. (See Elementary Lessons, pp. 68-69. Thomson's 
 Outline of the Necessary Laws of Thought, 81.) 
 
 (1) Homer wrote the Iliad and Odyssey. 
 
 (2) A parallelepiped is a solid figure having six faces, of 
 
 which every opposite two are parallel. 
 
 (3) The square on the hypothenuse of a right-angled 
 
 triangle is equal to the sum of the squares on the 
 sides containing the right angle. 
 
 (4) The swallow is a migratory bird. 
 
 (5) Axioms are self-evident truths. 
 
 5. Classify the following signs of logical quantity accord- 
 ing as they are generally used to indicate universality, 
 affirmative or negative, or particularity, affirmative or 
 negative 
 
 Several, none, certain, few, ullus, nullus, nonnullus, not a 
 few, many, the whole, almost all, not all.
 
 CHAPTER V 
 
 CONVERSION OF PROPOSITIONS, AND IMMEDIATE INFERENCE 
 
 i. THE student is referred to the Elementary Lessons in 
 Logic, or to other elementary text-books, for the common 
 rules of conversation and immediate inference, but, for the 
 sake of easy reference, the ancient square of opposition is 
 given below. 
 
 A... Contraries ...E 
 
 in 
 
 c 
 
 c c 
 
 CO 
 
 i.."' Subcontraries o 
 
 All the relations of propositions and the methods of 
 inference applying to a single proposition will be found fully 
 exemplified and described in the following questions and 
 answers.
 
 32 CONVERSION CHAP. 
 
 2. It appears to be indispensable, however, 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 name 
 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 incon- 
 sistent 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 Professor 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 con- 
 trapositive are all true if the original proposition is true. 
 The same is not necessarily the case with the inverse and 
 reciprocal. These latter two names are adopted from the 
 excellent work of Delboeuf, Prolegomenes Philosophiques de 
 la Geometric, pp. 88-91, at a the suggestion of Professor 
 Croom Robertson. (Mind, 1876, p. 425.)
 
 QUESTIONS AND ANSWERS 33 
 
 QUESTIONS AND ANSWERS 
 
 3- Give all the logical opposites of the proposition, 
 ' All metals are conductors.' 
 
 This is a universal affirmative proposition, having the 
 symbol A. By its logical opposites we mean the corre- 
 sponding propositions in the forms E, I, and O, which 
 have the same subject and predicate, and are related to it 
 respectively as its contrary, contradictory, and subaltern, in 
 the way shown in the Logical Square (p. 31) and explained 
 in many Manuals. These opposite propositions may be 
 thus stated 
 
 Subaltern (I) Some metals are conductors. 
 
 Contradictory (O) Some metals are not conductors. 
 Contrary (E) No metals are conductors. 
 
 The first of these (I) may be inferred from the original ; 
 the other two (O and E), so far from being inferrible, are 
 inconsistent with its truth. 
 
 4- Given that a particular negative proposition 
 is true, is the following chain of inferences 
 correct? O is true, A is false, I is false, and 
 therefore E is true. If so, the truth of O 
 involves the truth of E. 
 
 There is a false step in this argument ; for the falsity of 
 A does not involve the falsity of I. It may be (and is 
 materially false) that ' all men are dishonest ' ; but it never- 
 
 p
 
 34 CONVERSION CHAP. 
 
 theless may remain true that 'some men are dishonest.' 
 Observe, then, that the falsity of A does not involve the 
 truth of I, nor does the truth of I involve the truth or 
 falsity of A. But the truth of A necessitates that of I. 
 As stated in the Elementary Lessons (p. 78), 'Of sub- 
 alterns, the particular is true if the universal be true : 
 but the universal may or may not be true when the 
 particular is true.' 
 
 5- How do you convert universal affirmative 
 propositions ? 
 
 They must be converted by limitation or per accident, 
 as it is called, that is to say, while preserving the affirmative 
 quality, the quantity of the proposition must be limited from 
 universal to particular. Thus A is converted into I, as 
 in the following more or less troublesome instances, the 
 Convertend standing first and the Converse second in each 
 pair of propositions : 
 
 J All organic substances contain carbon. 
 
 ( Some substances containing carbon are organic. 
 
 j" Time for no man bides. 
 
 ( Something biding for no man is time. 
 
 ( The poor have few friends. 
 
 ( Some who have few friends are poor. 
 
 ( A wise man maketh more opportunities than he finds. 
 -\ Some who make more opportunities than they find are 
 v wise men. 
 
 ( They are ill discoverers who think there is no land, when 
 
 they can see nothing but water, 
 v Some ill discoverers think there is no land, etc.
 
 v QUESTIONS AND ANSWERS 35 
 
 ( Great is Diana of the Ephesians. 
 
 ( Some great being is Diana of the Ephesians. 
 
 ( Warm-blooded animals are without exception air-breathers. 
 
 < Air-breathers are (with or without exception) warm- 
 
 ( blooded animals. 
 
 6. How would you convert ' Brutus killed 
 Caesar ? ' 
 
 The strictly logical converse is ' Some one who killed 
 Caesar was Brutus.' For, though a man can only be killed 
 once, and Brutus is distinctly said to be the killer, yet in 
 formal logic we know nothing of the matter, and Caesar 
 might have been killed on other occasions by other persons. 
 An absurd illustration is purposely chosen in the hope that 
 it may assist to fix in the memory the all-important truth 
 that in logic we deal not with the matter. 
 
 7- How do you convert particular affirmative 
 propositions ? 
 
 To this kind of proposition simple conversion can be 
 applied ; that is to say, the converse will preserve both the 
 quantity and the quality of the convertend. In other 
 words, I when converted gives another proposition in I ; 
 thus either of the following pairs is the simple converse of 
 the other : 
 
 / Some dogs are ferocious animals, 
 i Some ferocious animals are dogs. 
 
 / Some men have not courage to appear as good as they are. 
 -; Some, who have not courage to appear as good as they 
 
 are, are men. 
 
 Some animals are amphibious. 
 Some amphibious beings are animals.
 
 36 CONVERSION CHAP. 
 
 8. How do you convert universal negative pro- 
 positions ? 
 
 These also are converted simply, giving another universal 
 negative proposition. E gives E. The reason is that both 
 the terms of E are distributed ; a universal negative asserts 
 complete separation between the whole of the subject and 
 the whole of the predicate. ' No man is a tailed animal ' 
 asserts that not any one man is found anywhere in the class 
 of tailed animals. Hence it follows evidently that no one 
 being belonging to the class of tailed animals is found in 
 the class of men, which result we assert in the simple con- 
 verse proposition, 'no tailed animal is a man.' Further 
 examples of the same mode of conversion are given below. 
 
 f No virtue is ultimately injurious. 
 
 I No ultimately injurious thing is a virtue. 
 
 ( No wise man runs into heedless danger. 
 
 I No one who runs into heedless danger is a wise man. 
 
 / People will not look forward to posterity who never look 
 ) backward to their ancestors. 
 
 | People never look backward to their ancestors who will 
 not look forward to posterity. 
 
 ( Whatever is insentient is not an animal. 
 I Whatever is an animal is not insentient. 
 
 9. How do you convert particular negative 
 propositions ? 
 
 Difficulty arises about this question, because the first 
 rule of conversion tells us to preserve the quality of the 
 proposition ; the converse accordingly should be negative. 
 But a negative proposition always distributes its predicate,
 
 v QUESTIONS AND ANSWERS 37 
 
 because a thing excluded' from a class must be excluded 
 from every part of the class. Now the subject of O being 
 particular and indefinite, it cannot stand as a distributed 
 predicate. It is still possible to say with material truth, 
 ' some men are not soldiers ' ; but converted this gives the 
 absurd result, ' all soldiers are not men ' ; or, ' no soldiers 
 are men.' Even if we insert the mark of quantity 'some' 
 before the predicate, and say, ' all soldiers are not some 
 men,' we must remember that ' some ' is perfectly indefinite, 
 and may include all. The question will be more fully dis- 
 cussed further on, but, so far as I can see, the particular 
 negative proposition, so long as . it remains negative and 
 indefinite in meaning, is incapable of conversion. This fact 
 constitutes a blot in the ancient logic. 
 
 Nevertheless the proposition O is capable of giving a 
 converse result when we change it into the equivalent affirma- 
 tive proposition. If ' some men ' are excluded from the 
 class 'soldiers,' they are necessarily included in the class 
 non-soldiers, or, ' some men are non-soldiers.' This is a 
 proposition in I, and by simple conversion, as already de- 
 scribed, gives a converse also in I, 'some non-soldiers 
 are men.' As further examples take 
 
 f Some dicotyledons have not reticulate leaves. 
 
 1 Some plants with non-reticulate leaves are dicotyledons. 
 
 J Some crystals are not symmetrical. 
 
 (. Some unsymmetrical things are crystals. 
 
 J All men have not faith. 
 
 ( Some who have not faith are men. 
 
 / Not every one that saith unto me, Lord, Lord, shall enter 
 ) into the Kingdom of Heaven. 
 
 ^ Some who shall not enter into the Kingdom of Heaven 
 V say unto me, Lord, Lord.
 
 38 CONVERSION CHAP. 
 
 10. How do you convert singular propositions ? 
 
 Singular propositions, being those which have a singular 
 term as subject, may be divided into two classes, according 
 as the predicate is a singular or a general term. (See 
 Karslake, 1851, vol. i. p. 54.) The former will always be 
 converted simply, one single thing being identified with the 
 same under another name, as in ' Queen Victoria is the 
 Duchess of Lancaster,' converted into 'the Duchess of 
 Lancaster is Queen Victoria.' Simple conversion will also 
 apply if the predicate be a general term, provided that the 
 proposition be negative so as to distribute this term. Thus, 
 ' St. Albans is not a great city ' becomes ' no great city is 
 St. Albans.' But if the predicate be general and undis- 
 tributed, as in an affirmative singular proposition, then we 
 must convert per acadens, and limit the new subject to some 
 or even one significate of the general term. Examples of 
 each case follow : 
 
 ( The better part of valour is discretion. 
 (. Discretion is the better part of valour. 
 
 J Time is the greatest innovator. 
 I The greatest innovator is time. 
 
 J London is the greatest of all cities. 
 1 The greatest of all cities is London. 
 
 J London is not a beautiful city. 
 I No beautiful city is London. 
 
 J Le style est 1'homme meme. 
 ( L'homme meme est le style. 
 
 j" All the allied troops fought courageously. 
 
 I Some who fought courageously were the allied troops.
 
 v QUESTIONS AND ANSWERS 39 
 
 ( Mercy but murders, pardoning those that kill. 
 ( Something which murders is mercy, pardoning those that 
 
 kill. 
 Not all the figures that Babbage's calculating machine 
 
 could run up, would stand against the general heart. 
 Something which would not stand against the general heart 
 is all the figures (collectively) that Babbage's machine 
 could run up. 
 
 II. Show how to convert the propositions 
 
 1 i ) ' All mathematical works are not difficult.' 
 
 (2) ' All equilateral triangles are equiangular.' 
 
 (3) ' No triangle has one side equal to the 
 
 other two.' 
 
 The first proposition, as it stands, is ambiguous, for it 
 looks like the universal negative, ' no mathematical works 
 are difficult.' But, according to custom, we may interpret 
 it as meaning that ' not all mathematical works are difficult,' 
 or ' some mathematical works are not difficult,' a proposition 
 in the form O. This cannot be converted simply, as already 
 explained (p. 36), because we must preserve the negative 
 quality, and ' all (or some) difficult things are not mathe- 
 matical works ' being negative would distribute its predicate 
 ' mathematical works.' We can, however, make O into I, 
 ' some mathematical works are not-difficult things,' and we 
 can convert this simply into ' some not-difficult things are 
 mathematical works.' 
 
 Proposition (2), as it stands, is in A, and can only be 
 logically converted by limitation into 'some equiangular 
 triangles are equilateral.' Geometrically it could easily be 
 shown that the inverse proposition ' all equiangular triangles 
 are equilateral,' is also true ; but we must of course not
 
 40 CONVERSION CHAP. 
 
 allow knowledge of the matter in question to influence us 
 in logical deduction, and the inverse proposition cannot 
 be inferred from the original. 
 
 Number (3) is a universal negative, and must be converted 
 simply into ' Nothing having one side equal to the other two 
 is a triangle ' ; but there is something paradoxical about 
 this result, which the student is recommended to investigate. 
 
 12. Convert ' Life every man holds dear.' 
 
 This is an example given in the Elementary Lessons 
 (p. 304). Students have variously converted it into 
 
 Life is held dear by every man. 
 
 Some life is held dear, etc. 
 
 No man holds death dear (!) 
 
 and so forth. But it ought surely to be easy to see that the 
 grammatical object is transposed, 'life' being the object of 
 'holds dear.' The statement is that 'every man holds life 
 dear,' and is explicitly a universal affirmative proposition, 
 to be converted by limitation into ' some who hold life dear 
 are men.' 
 
 13. Convert the proposition ' It rains.' 
 
 What is it that rains ? What is ' it ' ? Surely the environ- 
 ment, or more exactly the atmosphere. The proposition 
 then means ' the atmosphere is letting rain fall.' The con- 
 verse will therefore be ' something which is letting rain 
 fall is the atmosphere.' But in this and many other cases 
 the Aristotelian process of conversion by limitation gives a 
 meaningless if not absurd result. 
 
 14. Convert the proposition ' He jests at scars 
 who never felt a wound.' 
 
 This is the 8th example on p. 304 of the Elementary
 
 v QUESTIONS AND ANSWERS 41 
 
 Lessons, and has elicited from time to time some amazing 
 efforts at conversion, such as 
 
 Some jests at scars are made by one who never felt 
 a wound. 
 
 Scars are jested at by him who, etc. 
 
 Some scars jest at him who never felt a wound, (sic.) 
 
 Some scars are jests to one who, etc. 
 The subject of the proposition is of course 'he who never 
 felt a wound,' and the proposition asserts that he thus 
 described 'jests at scars." As there is no limitation of 
 quantity we may take the subject as universal ; and, although 
 there is negation within the subject, the copula is affirmative, 
 and the proposition is in the form A. It is thus converted 
 by limitation into ' some who jest at scars are persons who 
 have never felt a wound.' 
 
 15. Convert the proposition ' P struck Q.' 
 
 To this simple question I have got answers that, since P 
 is distributed, and Q undistributed, we must convert by 
 limitation, getting 'some Q struck P' or by contraposition 
 'some not- Q struck not-/*.' Such blunders and nonsense 
 arise from failing to notice that ' struck ' is not a simple 
 logical copula. There is, of course, a relation between P 
 and Q ; but as regards P, the proposition simply asserts 
 that ' P is a person who struck Q,' possibly not the only 
 one. Hence the converse by limitation is ' some person 
 who struck Q is P.' 
 
 Not a few examinees would at once convert ' P struck 
 Q ' into ' Q struck P,' but this, although very likely to 
 happen materially, is not logically necessary. 
 
 16. What is the obverse of the proposition ' All 
 
 metals are elements ' ?
 
 42 IMMEDIATE INFERENCE CHAP. 
 
 The obverse is a new term introduced by Professor 
 Alexander Bain, and its meaning is thus described by him 
 in his Deductive Logic, pp. 109, no. 'In affirming one 
 thing, we must be prepared to deny the opposite : " the 
 road is level," "it is not inclined," are not two facts, but 
 the same fact from its other side. This process is named 
 OBVERSION.' He proceeds to point out that each of the 
 four prepositional forms, A, I, E, O, admits of an obverse. 
 Every X is Y' becomes ' no X is not-K.' ' Some X is Y' 
 becomes ' some Xis not not-F.' ' No X is Y' becomes 'all 
 X is not- Y.' ' Some X is not Y' becomes ' some X is not- 
 K.' Accordingly the obverse of the proposition above will 
 be, ' No metals are not elements.' 
 
 Professor Bain goes on to describe what he calls 'Material 
 Obversion,' justified only on an examination of the matter 
 of the proposition. Thus from ' warmth is agreeable,' he 
 infers, after examination of the subject-matter, that ' cold is 
 disagreeable.' ' If knowledge is good, ignorance is bad.' I 
 feel sure, however, that this mixing up of so-called material 
 obversion with formal obversion is likely to confuse people 
 altogether. Indeed, Mr. Bain is himself confused, for he 
 cites, ' I don't like a curving road, because I like a straight 
 one,' as a childish reason, ' being no reason at all, but the 
 same fact in obverse.' Now, if there is any relation at all 
 between these two propositions, it is certainly a case of 
 material obversion ; but in reality they do not express the 
 same fact at all. The formal obverse of ' I like a straight 
 road,' is ' I am not one who does not like a straight road.' 
 We might perhaps infer, ' I do not dislike a straight road ' ; 
 but there is clearly no reference to curved roads at all. 
 
 While accepting the new term obversion in the sense of 
 formal obversion, I must add that students have begun to 
 use it with the utmost laxity, confusing the obverse with the
 
 v QUESTIONS AND ANSWERS 43 
 
 converse, the contrapositive, etc. To prevent logical nomen- 
 clature from falling into complete chaos, it seems to be 
 indispensable to choose convenient names for the simpler 
 relations of prepositional forms, as attempted above (p. 32), 
 and to adhere to them inflexibly. 
 
 17. What is conversion by contraposition ? Give 
 the contrapositive of ' All birds are bipeds.' 
 
 There is nothing which I have found so difficult in 
 teaching logic as to get the student to comprehend and 
 remember this process of contraposition ; particular attention 
 is therefore requested to the above question. 
 
 Having a proposition in A, we get its contrapositive by 
 taking the negative of its predicate, and affirming of this as 
 a subject the negative of the original subject. Thus, if ' all 
 Xs are Fs,' we take all not- Ys as a new subject, and affirm 
 of them that they are all not-^Ts, getting the proposition ' all 
 not- Fs are not-Jfs,' which is either A or H, according as 
 we do or do not join the negative particle to the predicate X. 
 Accordingly the contrapositive of the proposition ' all birds 
 are bipeds' will be 'all that are not bipeds are not birds.' 
 
 It is one thing to obtain the contrapositive, another thing 
 to see that it may be inferred from the premise. The late 
 Professor De Morgan used to hold that the act of inference 
 is a self-evident one, and needs no analysis ; but the process 
 may certainly be analysed. Thus we may obvert the pre- 
 mise 'All Xs are Fs,' obtaining 'No-^Ts are not Fs,' which 
 is a proposition in E, and then convert simply into ' No 
 not-Fs are Xs,' also in E, or else 'All not-Fs are not Xs.' 
 The contrapositive, then, is the converse of the obverse. 
 
 We may also prove the truth of the contrapositive 
 indirectly ; for what is not- F must be either X or not-A'y
 
 44 IMMEDIATE INFERENCE CHAP. 
 
 but if it be X it is by the premise also Y, so that the same 
 thing would be at the same time not- Y and also Y, which 
 is impossible. It follows that we must affirm of not-K 
 the other alternative, not-X. (See Chapter XXI. below; 
 also Principles of Science, pp. 83, 84 ; first ed., vol. i. pp. 
 97, 98.) 
 
 18. Give the converse of the contrapositive of 
 the proposition ' All vegetable substances are 
 organic.' 
 
 As learnt from the last question, the contrapositive is 
 ' All not-organic substances are not vegetable substances.' 
 We may take this to be equivalent to ' No inorganic sub- 
 stances are vegetable substances ' (E), the simple converse 
 of which is ' No vegetable substances are inorganic sub- 
 stances,' the obverse of the premise. But, if we treat the 
 contrapositive as a universal affirmative proposition, thus, 
 'All inorganic substances are non-vegetable substances,' 
 we must convert by limitation, getting ' Some non-vegetable 
 substances are inorganic,' which is the subaltern of the 
 obverse, and cannot by any process of inference lead us 
 back to the original. Conversion by limitation is easily 
 seen to be a faulty process which always occasions a loss 
 of logical force. 
 
 As we shall afterwards observe, this kind of conversion 
 introduces a new term, namely the indeterminate adjective 
 ' some,' so that the inference is not really confined to the 
 terms of the original premise. Although we may not be 
 able to dispense entirely with the word, owing to its employ- 
 ment in ordinary discourse, we shall ultimately eliminate it 
 from pure formal logic, and relegate it to the branch of 
 numerical logic.
 
 QUESTIONS AND ANSWERS 45 
 
 19- Take the following proposition, ' all water 
 contains air ' ; convert it by contraposition : 
 change the result into an affirmative pro- 
 position, and convert. 
 
 To show the need of more careful logical training than 
 has hitherto been common, even in the great Universities, 
 I give a few specimens of answers which I received to the 
 above question. The contrapositive of the proposition was 
 variously stated, as 
 
 All air does not contain all water. 
 
 All air is not contained in water. 
 
 All not-air is not a thing contained by not-water. 
 
 Some air is not contained in water. 
 
 Some not-air contains no water. 
 
 All not-air contains water. 
 
 The logicians who drew these inferences then proceeded 
 by simple conversion to get such results as the following : 
 
 Some water is not without some air. 
 No water contains not some air. 
 No water contains no air. 
 
 One too clever student inferred that ' All or every vacuum 
 is a void of water,' which he converted, simply indeed, into 
 
 ' Every void of water is a vacuum ' ! 
 
 An examiner irt logic is sometimes forced to believe that 
 there is a void in the brains of an examinee ; but the 
 absence of any "sufficient training in logical work is more 
 often the cause of the lamentable results shown above. In 
 any case it seems impossible to agree with De Morgan that 
 contraposition is a self-evident process.
 
 46 IMMEDIATE INFERENCE CHAP. 
 
 These absurd answers are mainly due to the failure to 
 observe that in the proposition ' All water contains air,' the 
 two words ' contains air,' form the grammatical predicate, 
 comprehending both the logical predicate and the logical 
 copula. Logically then the proposition is 'All water is 
 containing air,' or ' All water is what contains air.' The 
 contrapositive then is ' All that does not contain air is not 
 water.' Uniting the negative particle to the predicate 
 ' water,' and converting by limitation, we obtain ' Some 
 not-water is what does not contain air.' 
 
 20. Describe the logical relations, if any, between 
 each of the following propositions and each 
 other 
 
 1 i ) All organic substances contain carbon. 
 
 (2) There are no inorganic substances which do 
 
 not contain carbon. 
 
 (3) Some inorganic substances do not contain 
 
 carbon. 
 
 (4) Some substances not containing carbon are 
 
 organic. 
 
 Of these, (i) is a universal affirmative, the contrapositive 
 of which is ' All substances not containing carbon are inor- 
 ganic substances.' Hence the converse by limitation of this 
 contrapositive is, ' Some inorganic substances are substances 
 not containing carbon,' equivalent to (3). 
 
 Proposition (2) is the obverse of ' All inorganic substances 
 contain carbon,' which is the contradictory of (3). 
 
 To obtain (4) we must take the contrary of (i), that is, 
 no organic substances contain carbon, express it in the 
 affirmative form, ' All organic substances are substances 
 not containing carbon,' and then convert it by limitation.
 
 v QUESTIONS AND ANSWERS 47 
 
 21. Take any proposition suitable for the pur- 
 pose, convert it by contraposition, convert it 
 again simpliciter, change the result into an 
 affirmative proposition, and show that you 
 may regain the original proposition. [c.] 
 
 The most suitable kind of proposition for the purpose 
 will be a universal affirmative, such as 
 
 (1) All birds are bipeds. 
 
 The contrapositive may be stated in the form of E. 
 
 (2) No not-bipeds are birds. 
 
 Which is converted simpliciter into E, the obverse of ( i ). 
 
 (3) No birds are not-bipeds. 
 
 When thrown into the affirmative form by a second 
 obversion, the last becomes 
 
 (4) All birds are not-not-bipeds. 
 
 As double negation destroys itself, this is equal to (i). 
 Notice that the obverse of the obverse is the original. 
 
 22. Give the converse of the contradictory of the 
 proposition, ' There are no coins which are not 
 made of metal.' 
 
 The premise is stated in a complex form with double 
 negation ; it means ' No coins are not made of metal,' 
 which is the obverse of ' All coins are made of metal ' (A). 
 The contradictory, as shown in the square of opposition 
 (p. 31), is a proposition in O, namely, 'Some coins are 
 not made of metal,' which can be converted only by nega- 
 tion, that is, by joining the negative particle to the predicate, 
 thus: 'Some coins are not-made-of-metal,' whence by 
 simple conversion 'Some things not-made-of-metal are 
 coins ' the answer required.
 
 48 IMMEDIATE INFERENCE CHAP. 
 
 23. ( i ) All crystals are solid. 
 
 (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. 
 
 Assign the logical relation, if any, between each 
 
 of these propositions and the first of them. 
 Proposition (i) is a universal affirmative (A) ; its simple 
 obverse is (4) ; its converse by limitation is (5) ; the sub- 
 contrary of this converse is (2). In order to obtain (6) 
 we must take the contrapositive of (i), namely, 'All not- 
 solids are not crystals,' the subaltern of which is (6) ; and 
 converting (6) by negation we get (3). Again, (7) is the 
 inverse, but is not inferrible from (i). We may further say 
 that (4) can be inferred from (i), and is exactly equivalent 
 in logical force to it ; (5) and (6) can be inferred, but are 
 not equivalent to the original ; (2) cannot be inferred from 
 (i), but is not inconsistent with its truth. 
 
 24. What information about the term not-^4 can 
 we derive from the premise ' All As are s ' ? 
 
 This question, though apparently a very simple one, does 
 not admit of a very simple answer ; it is important in a 
 theoretical point of view. It may be said on the one hand, 
 that as the proposition only affirms of all As that they are 
 Us, this tells us nothing about things excluded from the 
 class A. Thus what is not-^4 may be B, or it may not be 
 J3, without any interference from the premise. This is quite 
 true. About Not-^4 universally we may infer nothing. 
 
 But, on the other hand, if we convert the proposition ' all
 
 v QUESTIONS AND ANSWERS 49 
 
 As are J?s' by contraposition (p. 43), we get 'all not-^sare 
 not As.' Uniting the negative particle to the predicate, we 
 have 'All not-^s are not-As,' whence, by limited conversion, 
 we infer some not As are not--5s. In this result we must 
 interpret some as meaning, one at least, it may be more or even 
 all. We shall recur to this question in a subsequent chapter. 
 
 '2$. 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 ? 
 The premise ' No organic beings are devoid of carbon ' 
 is a universal negative proposition, and does .not directly 
 give information about beings which are not organic, and 
 beings which are not devoid of carbon. But, if we join the 
 negative particle to the predicate, we get 'All organic 
 beings are not-devoid-of-carbon,' whence, by limited con- 
 version, 'Some things not devoid of carbon are organic,' 
 which answers the second part of the question. 
 
 Again, converting by contraposition, we learn that 'All 
 things not-not-devoid of carbon are not organic beings ' ; in 
 other words, ' All things devoid of carbon are not organic 
 beings,' a result which may be obtained perhaps more clearly 
 by converting the original premise simply, thus, ' No things 
 devoid of carbon are organic beings,' or ' All things devoid 
 of carbon are not organic beings.' Conversion by limitation 
 then yields ' Some things not organic beings are devoid of 
 carbon,' which is the answer to the first part of the question. 
 This result is the same as that obtained in the last question, 
 and the same remarks apply. 
 
 26. What information about the term Solid Body 
 can we derive from the proposition, ' No bodies 
 which are not solids are crystals ' ?
 
 So IMMEDIATE INFERENCE CHAP. 
 
 This question differs from the last only in being put in a 
 more involved form. The premise when more simply 
 stated becomes ' All not solids are not crystals,' the contra- 
 positive of ' All crystals are solids,' and limited conversion 
 gives ' Some solids are crystals.' 
 
 27. Nihil potest placere, quod non decet. Con- 
 
 vert this proposition, (i) simply, (2) by con- 
 traposition ; and show by what logical processes 
 we can pass back from the contrapositive to 
 the original. [c.] 
 
 . This premise (from Quinctilian,c.xi. 6 5) equals, Nihil quod 
 non decet, potest placere ; nothing which is unbecoming can 
 please. Being a universal negative, E, it can be converted 
 simply into ' Nothing which can please is unbecoming.' 
 
 In order to apply contraposition, we must put the pre- 
 mise into the form of A, thus 'All unbecoming things are 
 unpleasing things,' the contrapositive of which is 'All not 
 unpleasing things are not unbecoming things,' which having 
 a double negative in each term equals ' All pleasing things 
 are becoming.' We can regain the original premise by 
 applying contraposition to this last result. 
 
 28. Convert, and give some immediate inferences 
 
 from the following : ' Nothing is harmless that 
 is mistaken for a virtue.' 
 
 The predicate of this proposition is clearly 'harmless,' 
 and 'that is mistaken for a virtue' is a relative clause 
 describing the subject. The proposition is then ' Nothing 
 mistaken for a virtue is harmless ' (E), converted simply 
 into another proposition in E, ' Nothing harmless is mis- 
 taken for a virtue.' 
 
 Applying obversion to the original proposition we get
 
 v QUESTIONS AND ANSWERS 51 
 
 'All that is mistaken for a virtue is not -harmless,' or 'is 
 harmful.' By immediate inference by complex conception, 
 we infer ' All foolish conduct mistaken for virtue is harmful 
 foolish conduct.' (Concerning inference by complex con- 
 ception, see Thomson's Outline, 88, and Elementary 
 Lessons, p. 87.) 
 
 29. Because every Prime Minister is a man, can 
 we infer that every good Prime Minister is a 
 good man ? 
 
 The process of immediate inference by added deter- 
 minants, as described by Dr. Thomson, allows us to join an 
 adjective or determining mark to both terms of an affirmative 
 judgment, narrowing both terms, but to the same extent. 
 Of course, however, it must be the same determining mark 
 in each case, and if an adjective be ambiguous it is not 
 logically the same adjective in its several meanings. Now 
 good applied to a Prime Minister means that he is an able, 
 active, upright minister, but probably very different from 
 men who are good in other ranks of life. A good man 
 means one who is good in the ordinary business and 
 domestic relations of life. Thus the inference is erroneous. 
 (See Elementary Lessons, p. 86.) It will afterwards be shown 
 that when the proposition is fully expressed no such failure 
 of inference can occur. Strictly speaking the premise is 
 
 Prime Minister = Prime Minister, Man; 
 and it follows inevitably that 
 
 Good, Prime Minister = Good, Prime Minister, Man. 
 
 30- Euler employed two overlapping circles to 
 represent a particular proposition. Can you 
 raise any objection to the accuracy of such a 
 diagram ?
 
 52 . IMMEDIATE INFERENCE CHAP. 
 
 Such circles have been employed in a great number of 
 logical works. In my Elementary Lessons (p. 75) the 
 particular proposition 'some metals are not brittle,' is 
 represented by the following figure : 
 
 FIG. i. 
 
 It does not seem to have been sufficiently noticed that 
 though such a diagram correctly shows the exclusion of a 
 part of the class metals from any part, that is all parts, of 
 the class brittle substance, it indicates at the same time that 
 another part of the class metals is included among brittle 
 substances. Thus the diagram corresponds to the two pro- 
 positions I and O, instead of showing either apart from the 
 other. Now, it has been fully explained that O is consistent 
 with the truth of E ; so that when we say ' some metals are 
 not brittle,' it may be that no metals are brittle, which is 
 contradictory to I, ' some metals are brittle.' The diagram 
 should not prejudice this question, and it would therefore 
 be best to remove the part of the circle bounding metals 
 which falls within the circle of brittle sunstances, or else to 
 have a broken line, as in Fig. 2. 
 
 BRITTLE 
 ! SUBSTANCES 
 
 FIG. 2.
 
 QUESTIONS AND ANSWERS 
 
 53 
 
 In the same way the proposition I, for instance, ' Some 
 crystals are opaque,' would be represented by a broken circle 
 included within a complete circle, in the manner shown 
 either in Fig. 3 or Fig. 4. 
 
 \ 
 
 FIG. 3. 
 
 FIG. 4. 
 
 31. What is the logical force of the following 
 sentence from Sidgwick's MetJiods of Ethics : 
 ' A materialist will naturally be a determinist ; 
 a determinist need not be a materialist ' ? 
 
 Taking ' naturally ' to give a universal force to the first 
 proposition, it becomes 'All materialists are determinists.' 
 The second proposition informs us that ' a determinist need 
 not be a materialist,' that is to say, at the least, ' some 
 determinists are not materialists.' This proposition is the 
 sub-contrary of the converse of the first, and is the con- 
 tradictory of 'all determinists are materialists.' The second 
 proposition, then, prevents us from supposing materialists 
 and determinists to be two coextensive terms. We learn 
 that there are persons called materialists who are all found 
 among determinists ; hence some called determinists are 
 found among materialists ; other determinists, however, are 
 not among materialists, and as to those who are not deter- 
 minists, they cannot be materialists. The first proposition 
 would be technically described as A, and the second as O, 
 the contradictory of the inverse of the first.
 
 54 IMMEDIATE INFERENCE CHAP. 
 
 32. ' All equilateral triangles are equiangular.' 
 
 May we thence infer that triangles having 
 unequal angles have unequal sides, and vice 
 versa ? 
 
 The proposition asserts that all equal-sided triangles have 
 equal angles ; hence we may by contraposition infer that 
 triangles which have not equal angles cannot have equal 
 sides. But as the proposition stands, we are not justified in 
 reading it reciprocally (see p. 32), and inferring that triangles 
 which have not equal sides have not equal angles. This is 
 true as a matter of geometrical science, but it is the contra- 
 positive of another proposition, namely, the inverse 'all 
 equiangular triangles are equilateral,' the truth of which 
 must be separately proved. 
 
 33. Can we ever convert a proposition of the 
 
 form ' all As are s ' into one of the form ' all 
 s are As ' ? 
 
 Certainly we cannot infer that all j?s are As because all 
 As are .Z?s. As a general rule the predicate of the con- 
 vertend B will be a wider term than the subject A, so that 
 the inverse could not be inferred. Professor Henrici 
 (Elementary Geometry, Congruent Figures, p. 14), for in- 
 stance, describes space as a three-way-spread, but we cannot 
 convert simply, and say that every three-way-spread is space- 
 It nevertheless happens not uncommonly that the original 
 proposition is really intended to mean ' all As are all j?s,' 
 which can then be simply converted. Thus if space be 
 defined as a three-way-spread of points, we can convert into 
 every three-way-spread of points is space. Such definitions 
 are of the form of proposition afterwards described by the 
 symbol U (chapter xviii.), and considerable care is requisite
 
 v QUESTIONS AND ANSWERS 55 
 
 in discriminating between the propositions A and U. 
 J. S. Mill has pointed to the simple conversion of a 
 universal affirmative proposition as a very common form 
 of error (System of Logic, book v., chapter vi., section 2). 
 It cannot be too often repeated that the reciprocal and 
 inverse propositions as described on p. 32, cannot be 
 inferred from an original of the form A. 
 
 34. In what cases does predication involve real 
 existence ? Show that in some processes of 
 conversion assumptions as to the existence of 
 classes in nature have to be made ; and illus- 
 trate by examining whether any such assump- 
 tions, and if so what, are involved in the 
 inference that if all vS is P, therefore some 
 not- 6" is not P. 
 
 The above question must have been asked under some 
 misapprehension. The inferences of formal logic have 
 nothing whatever to do with real existence ; that is, occur- 
 rence under the conditions of time and space. No doubt, 
 if all S is P, it follows that, in order to avoid logical con- 
 tradiction, some not-.S must be admitted to be not P. For 
 instance, if 'All heathen gods are described in writings 
 more than 1000 years old,' it follows that 'Some things 
 which are not heathen gods are not described in writings 
 more than 1000 years old.' This involves no assertion 
 of real existence, nor could such an inference ever be 
 drawn, unless, indeed, the original proposition itself asserted 
 existence in time and space. This subject is pursued in 
 a subsequent chapter.
 
 CHAPTER VI 
 
 i. EXAMINE the following pairs of propositions, and decide 
 which pairs contain consistent propositions, such that if the 
 first of the pair be true the second may be true ; and vice 
 -versa, if the second be true, the first may be true. Give 
 the technical name of the logical relation, if any, between 
 the two propositions of each pair. 
 
 / \ J Some metals are useful. 
 ^^ t All metals are useful. 
 
 , x 
 (2) < 
 ( 
 
 f No metals are useless. 
 Some useful things are not metals. 
 
 / \ j" Some useless things are metals. 
 ( All useful things are metals. 
 
 Some metals are useful. 
 No metals are useless. 
 
 All metals are useful. 
 
 Some useless things are not metals. 
 
 2. Draw all the immediate inferences you can from the 
 proposition ' Knowledge is power.' 
 
 3. Give the converse of the contrapositive of the pro- 
 position 'All organic substances contain carbon.'
 
 CHAP, vi EXERCISES ON PROPOSITIONS 57 
 
 ^- 4. Give all the logical opposites of intuta qua zndecora, 
 ' Unsafe are all things unbecoming.' 
 
 \ 5. What information about the term 'solid body' can 
 we derive from the proposition, ' No bodies which are not 
 solids are crystals ' ? 
 
 \ 6. ' Only British subjects are native born Englishmen.' 
 What precisely does this proposition tell us abouf the four 
 terms British subject. 
 
 Not-British-subject. 
 
 Native born Englishmen. 
 
 Not-native-born-Englishmen ? 
 
 7. Describe the logical relation between each of the 
 four following propositions, and each of the other three : 
 
 (1) All substances possess gravity which are material. 
 
 (2) No substances which possess gravity are immaterial. 
 
 (3) Some substances which are immaterial do not possess 
 
 gravity. 
 
 (4) Some substances which do not possess gravity are 
 
 immaterial. 
 
 8. State the nature and technical name of the logical 
 process by which we get each of the following propositions 
 from the preceding one : 
 
 All men are mortal. 
 No men are immortal. 
 No immortals are men. 
 None but mortals are men. 
 All not-mortals are not men. 
 No men are not-mortals. 
 All men are mortals. 
 
 9. What are the subaltern propositions corresponding 
 to the following universal propositions ?
 
 58 IMMEDIATE INFERENCE CHAP. 
 
 (1) Every effect follows from a cause. 
 
 (2) No one is admitted without payment. 
 
 (3) All trespassers will be prosecuted with the utmost 
 
 rigour of the law. 
 
 (4) Nemo me impune lacessit. 
 
 10. Give the obverse, converse, inverse, and reciprocal 
 of each of the folfowing^propositions : 
 
 (1) All mammalia are vertebrate animals. 
 
 (2) Sir Rowland Hill is dead. 
 
 (3) That which is a merit in an author is often a fault in 
 
 a statesman. 
 
 (4) Whatever is necessary exists. 
 
 (5) In veritate victoria. 
 
 1 1. Give the contrary, contradictory, subaltern, converse, 
 obverse, inverse, contrapositive, and reciprocal propositions 
 corresponding to each of the following propositions : 
 
 (1) All B.A.'s of the University of London have passed 
 
 three examinations. 
 
 (2) All men are sometimes thoughtless. 
 
 (3) Uneasy lies the head which 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. 
 
 1 2. Give as many equivalent logical expressions as you 
 can for the propositions 
 
 (i) If the treasury was not full, the tax-gatherers were 
 to blame, b 
 
 7
 
 vi EXERCISES 59 
 
 (2) Through any three points not in a straight line a 
 
 circle may be described. 
 
 (3) It is false to say that only the virtuous prosper in 
 
 life. [R.] 
 
 13. What logical relations are there between the follow- 
 ing propositions ? 
 
 (1) All elementary substances are undecomposable. 
 
 (2) There are no compounds which are not decom- 
 
 posable. 
 
 (3) Some compounds are not decomposable. 
 
 (4) No undecomposable substances are compounds. [E.] 
 
 1 4. From the proposition ' Perfect happiness is im- 
 possible ' can we infer that ' Imperfect happiness is 
 possible ' ? 
 
 15. Is it the same thing to affirm the falsity of the pro- 
 position 'Some birds are predatory,' and to affirm the truth 
 
 of the proposition ' Some birds are not predatory ' ? 
 
 1 6. Explain the statement that in the case of subcontrary 
 propositions, truth may follow from falseness, but falseness 
 cannot follow from truth. 
 
 17. Give in succession (i) the obverse, (2) the converse, 
 (3) the subaltern, (4) the contrary, (5) the contradictory, 
 (6) the contrapositive of the proposition ' All wise acts are 
 honest acts.' 
 
 1 8. Concerning the same proposition answer the follow- 
 ing questions : 
 
 ( i ) How is its converse related to its subaltern ? 
 
 (2) How is its converse related to the converse of its 
 
 subaltern ? 
 
 (3) How is its subaltern related to its contradictory ? 
 
 [BAGOT.]
 
 60 IMMEDIATE INFERENCE CHAP. 
 
 19. What is the converse of the contrary of the con- 
 tradictory of the proposition ' Some crystals are cubes ' ? 
 How is it related to the original proposition ? 
 
 20. What is the converse of the converse of 'No men 
 are ten feet high ' ? 
 
 21. Name the logical process by which we pass from 
 each of the following propositions to the succeeding 
 one : 
 
 (1) All metals are elements. 
 
 (2) No metals are non-elements. 
 
 (3) No non-elements are metals. 
 
 (4) All non-elements are not metals. 
 
 (5) All metals are elements. 
 
 (6) Some elements are metals. 
 
 (7) Some metals are elements. 
 
 22. (i) 'None but a logical author is a truly scientific 
 author.' Taking this proposition as a premise, examine 
 the following propositions, and decide which of them can 
 be inferred from the premise. 
 
 (2) A truly scientific author is no author who is not 
 
 logical. 
 
 (3) Some truly scientific authors are not any authors 
 
 who are not logical. 
 
 (4) A not truly scientific author is not a logical author. 
 
 (5) Those who are not truly scientific authors cannot 
 
 be logical. 
 
 (6) All logical authors are truly scientific. 
 
 (7) No truly scientific author is an illogical author. 
 
 (8) All not illogical authors are truly scientific. 
 
 (9) No illogical author is a truly scientific author. 
 
 (10) No one is a truly scientific author who is not a 
 
 logical author. 
 
 (11) Some logical authors are not truly scientific authors.
 
 vi EXERCISES 61 
 
 Give, as far as possible, the technical name of the logical 
 relation between each of the above propositions and each 
 other. 
 
 23. 'Some small sects are said to have no discreditable 
 members, because they do not receive such, and extrude all 
 who begin to verge upon the character.' Point out how 
 this statement illustrates logical conversion. 
 
 24. Can we logically infer that because heat expands 
 bodies, therefore cold contracts them ? 
 
 25. Does it follow that because every city contains a 
 cathedral, therefore the creation of a city involves the 
 creation of a cathedral, or the creation of a cathedral in- 
 volves the creation of a city ? 
 
 26. All English Dukes are members of the House of 
 Lords. Does it follow by immediate inference by complex 
 conception that the creation of an English Duke is the 
 creation of a member of the House of Lords ? 
 
 27. Give every possible converse of the following pro- 
 positions 
 
 (1) Two straight lines cannot enclose space. 
 
 (2) All trade-winds depend on heat. 
 
 (3) Some students do not fail in anything. [M.] 
 
 28. Give the logical opposites, converse and contra- 
 positive, of Euclid's (so-called) twelfth axiom 
 
 If a straight line meet two straight lines, so as to make 
 the interior angles on the same side of it taken together less 
 than two right angles, those straight lines being continually 
 produced shall at length meet upon that side on which are 
 the angles which are less than two right angles. 
 
 29. How is the above proposition related to this other: 
 If a straight line fall upon two parallel straight lines, it
 
 62 IMMEDIATE INFERENCE CHAP. 
 
 makes the two interior angles upon the same side together 
 equal to two right angles ? [R.] 
 
 30. From 'Some members of Parliament are all the 
 ministers' (Elementary Lessons, p. 325, No. 3 [4]), can we 
 infer that 'some place -seeking prejudiced and incapable 
 members of Parliament are all the place-seeking prejudiced 
 and incapable ministers ' ? 
 
 31. Is it perfectly logical to argue that because two sub- 
 contrary propositions may both be true at the same time, 
 therefore their contradictories, which are contrary to each 
 other, may both be false ? V*- 
 
 32. Is it perfectly logical to argue thus? If contrary 
 propositions are both false, their respective contradictories, 
 which are sub -contraries to each other, are both true. 
 Now as this result is possible, it is therefore possible that 
 the contraries may both be false. W<r -V\^. *~*JJ-~ c***"* 
 
 ^ 33. What is the logical relation, if any, between the 
 two assertions in Proverbs, chap. xi. i, 'A false balance 
 is abomination to the Lord : but a just weight is his 
 delight ' ? 
 
 34. Examine the verses of Proverbs, chaps, x. to xv., 
 and assign the relation between the two opposed assertions 
 which make nearly all the verses. 
 
 ^ 6 l^iL^* 35' What is the nature of the step from 'anger is a short 
 madness ' to ' madness is a long passion ' ? [R.] 
 
 36. 'The angles at the base of an isosceles triangle are 
 equal.' What can be inferred from this proposition by 
 obversion, conversion, and contraposition, without any 
 appeal to geometrical proof? 
 
 37. From the assertion 'The improbable is not impos- 
 sible,' what can we learn, if anything, about (i) the 
 possible, (2) the probable, (3) the not-improbable, (4) the 
 impossible, (5) the not-impossible ?
 
 vi P^ ^ EXERCISES 63 
 
 38. How would a logician express the relations between 
 the following statements of four interlocutors ? 
 
 (1) None but traitors would do so base a deed. 
 
 (2) And not all traitors. 
 
 (3) Some would. 
 
 (4) No ; not even traitors. 
 
 [College Moral Science Examination, Cambridge.] 
 
 39. What difficulties or absurdities do you meet in con- 
 verting the following propositions ? C***. -i /v^ ; 
 
 .fc) (i) Some books are dictionaries. 
 
 (2) No triangle has one side equal to the sum of the 
 
 other two. 
 
 (3) Every one is the best judge of his own interests. 
 
 (4) A few men are both scientific discoverers and men 
 
 of business. 
 
 (5) Whatever is, is right. 
 
 (6) Some men are wise in their own conceit. 
 
 (7) 'The eye sees not itself, 
 
 But by reflection, by some other things.'
 
 CHAPTER VII 
 
 DEFINITION AND DIVISION 
 
 i. ALMOST all text-books of Deductive Logic give rules 
 for judging of the correctness of definitions, and for 
 dividing up notions into subaltern genera and species. On 
 attempting, however, to treat these parts of logic in the 
 manner of this work, it has come home to me very strongly 
 that they are beyond t the sphere of Formal Logic, and 
 involve the matter of thought. In form there is nothing 
 peculiar to a definition ; in fact the very same proposition 
 may be a definition to one person and a theorem to another. 
 It is open to us for instance to define the number 9 as 
 9 = 3x3; or, 9 = 8+ i; or 9 = 7 + 2, etc.; but having selected 
 at will any one of these equations as a definition, the other 
 equations follow as theorems. The perplexity in which the 
 theory of parallel lines is involved partly arises from the 
 fact that there is choice of definitions, some mathematicians 
 choosing one way and some the other. It is quite apparent, 
 too, that the same proposition may afford different know- 
 ledge to different people. For instance, 'John Herschel 
 was the only son of William Herschel ' would serve as a 
 definition of John Herschel to any one who knew only 
 William Herschel, and of William Herschel to one who 
 only knew John. To one who knew both it might be a
 
 CHAP, vii EXERCISES 65 
 
 theorem. Similar remarks might be made concerning the 
 distinction between ampliative and explicative propositions. 
 
 2. These in addition to other considerations convince me 
 that any attempt to treat definition as a part of Formal 
 Logic must be theoretically unsound and practically un- 
 satisfactory. The case is somewhat similar with Logical 
 Division, which, so far as it belongs to Formal Logic, can 
 be nothing more than that method of Dichotomous Division 
 fully developed in the later chapters on Equational Logic. 
 Anything more than this must involve material knowledge, 
 and should be treated in a different work and in a different 
 manner. On these grounds I have decided not to attempt 
 any explication of Definition and Division here, but to con- 
 fine this chapter to a collection of questions, such as are to 
 be commonly found in examination papers. The student 
 may be referred for the current doctrines to the Elementary 
 Lessons, Nos. XII. and XIII. ; Fowler's Deductive Logic, 
 Chapters VII. and VIII. ; Duncan's Logic, Chapter VI. etc. 
 
 3. Examine the following definitions 
 
 (1) Conversion is the changing of terms in a proposi- 
 
 tion. 
 
 (2) Opposed propositions are those which differ in 
 
 quantity and quality. 
 
 (3) Contradictory opposition is the opposition of con- 
 
 tradictories. [R.] 
 
 4. Define any of the following terms, notions, or classes 
 of objects 
 
 Gravitation Franchise Communism 
 
 Consistency Imagination Honour 
 
 Library Honesty Club 
 
 Vegetable Revenge Dictionary
 
 66 DEFINITION AND DIVISION CHAP. 
 
 Diet Syllogism Conservative 
 
 Hypochondriac Racehorse University 
 
 Merit Success Specie. 
 
 5. Criticise the following definitions 
 
 (1) A square is a four-sided figure of which the sides 
 
 are all equal and the angles all right angles. 
 
 (2) A member of the solar system is anything over which 
 
 the sun has continued influence. 
 
 (3) La vie est le contraire de la mort. 
 
 (4) A lemma is a proposition which is only used as 
 
 subservient to the proof of another proposition. 
 
 (5) An archdeacon is one who exercises archidiaconal 
 
 functions. 
 
 (6) Life is the definite combination of heterogeneous 
 
 changes, both simultaneous and successive, in 
 correspondence with external coexistences and 
 sequences. 
 
 (7) A gentleman is a man having no visible means of 
 
 subsistence. [ORTON.] 
 
 (8) Equal . bodies are those whereof every one can fill 
 
 the place of every other. [HOBBES.] 
 
 6. Examine the definitions 
 
 (1) Tin is a metal lighter than gold. 
 
 (2) Vice is the opposite of virtue. 
 
 (3) Paper is a substance made of rags. 
 
 (4) Cheese is a caseous preparation of milk. 
 
 (5) Rust is the red desquamation of old iron. 
 
 (6) A transcendental function is any function which is 
 
 not an algebraic function. 
 
 (7) A right-angled triangle is a triangle containing one 
 
 right angle, and of which the containing sides 
 are or are not equal.
 
 vii EXERCISES 67 
 
 (8) An organ is any part of an animal or plant appro- 
 
 priated to a distinct function. 
 
 (9) A man is a self-knowing animal. 
 
 (10) Knowledge is that on which somebody else can be 
 
 examined. [ROLLESTON.] 
 
 (n) An animal is a sentient organised being. 
 
 (12) A triangle is a three-sided figure having its angles 
 
 together equal to two right angles. 
 
 (13) A man is one who may be the Prince of Tran- 
 
 sylvania. [HOBBES.] 
 
 7. In what respects are the following definitions, or some 
 of them, defective ? 
 
 (1) Logic is a guide to correct reasoning. 
 
 (2) Logic is the art of expressing thoughts in correct 
 
 language. 
 
 (3) Logic is a mental science. 
 
 (4) Logic is the science of the regulative laws of human 
 
 thought. 
 
 8. Does the eleventh chapter of the Hebrews, or any 
 part of it, contain a correct logical definition of Faith ? 
 
 9. Give examples of indefinable words, and explain why 
 words may be indefinable. 
 
 10. Give the Proximate Genera for the following 
 species 
 
 Man Plant Monarchy 
 
 Triangle Hound Science. 
 
 1 1. Define by genus and differentia the following terms ; 
 and name a proprium and an accident in each case : 
 
 Island Parallelogram 
 
 Bank Bill of exchange 
 
 Dictionary Tree
 
 68 DEFINITION AND DIVISION CHAP. 
 
 12. What are the genus, species, difference, property, and 
 accident of Examination ? [D.] 
 
 13. Distinguish specific attribute, property, and accident, 
 using the term Circle as an example. [B.] 
 
 14. Is it possible to define the terms gold, coal, legal 
 nuisance, civilisation, Cleopatra's needle? 
 
 15. Define the term boat, and then point out how many 
 of the following things the definition includes : Bark, 
 ferry-boat, floating fire-engine, pontoon, wherry, canoe. 
 
 1 6. Classify the following objects under one or other of 
 the heads, cash, bills, specie: 
 
 Cheque, promissory note, shilling, money, token-coin, 
 bank-note, I.O.U., paper-money, sovereign, Scotch bank- 
 note. (See Money and the Mechanism of Exchange (Inter- 
 national Scientific Series], p. 248. Section on the Definition 
 of Money.) 
 
 17. Distinguish Logical from Physical Division and 
 Definition. [o.] 
 
 1 8. Can anything admit of more than one defini- 
 tion? [o.] 
 
 19. Distinguish precisely between the definition and the 
 description of a class. 
 
 20. Explain the difficulties which arise concerning the 
 definition of parallel straight lines, and criticise the following 
 suggested definitions : 
 
 1 i ) Lines which are in every part equidistant. 
 
 (2) Lines of similar direction. 
 
 (3) Lines which being in the same plane and indefinitely 
 
 prolonged never meet.
 
 vii EXERCISES 69 
 
 21. Examine the following definitions : 
 
 (1) Man is a bundle of habits. 
 
 (2) Law is common sense. 
 
 (3) Reverence is the feeling which accompanies the 
 
 recognition of superiority or worth in others. 
 
 (4) Hunger is the product of man's reflection on the 
 
 necessity of food. [p.] 
 
 22. Which of the following are logical divisions, and 
 which are not ? 
 
 (1) Man into civilised and uncivilised. 
 
 (2) The world into Asia, America, Europe, Africa, 
 
 Australia. 
 
 (3) Grammar into syntax and prosody. 
 
 (4) War into civil and aggressive, 
 
 (5) Syllogisms into those which are logical and illogical. 
 
 (6) Sequences of phenomena into casual and causal. 
 
 (7) Energy into potential and visible. 
 
 (8) Geometrical figures into plane and tri-dimensional. 
 
 (9) Allegiance is either natural and perpetual, or local 
 
 and temporary. 
 
 23. Divide the term Inference, so as to include the 
 various species usually discussed by logicians. [E.] 
 
 24. The following were the classes of persons which were 
 in 1868 qualified to vote in one or other of the United 
 States of America : Male citizen, male inhabitant, every 
 man, white male citizen, white freeman, male person, white 
 male adult, free white male citizen, free white man. 
 
 Form a scheme of logical division which shall have a 
 place for each of the above classes. 
 
 25. Divide logically University, colour, chair, science,
 
 70 DEFINITION AND DIVISION CHAP, vn 
 
 religion, species, art, church, undergraduate, virtue, states- 
 man, [o.] 
 
 26. Form a scheme of division of sciences to include the 
 species Deductive, experimental, concrete, descriptive, 
 rational, abstract, inductive, explanatory, empirical. 
 
 27. Apply the rules of logical division to the following 
 instances, correcting what is wrong, and supplying what is 
 deficient : 
 
 (1) Discursive thought may be divided into the Term, 
 
 Judgment and Syllogism. 
 
 (2) Notions are Concrete, Singular, and Universal. 
 
 (3) Propositions are Affirmative, Negative, and Universal. 
 
 28. To what extent are the rules of division, usually 
 given in logical treatises, repudiated by the classifications 
 adopted in the Natural Sciences ? [L.] 
 
 29. When is a division inadequate? When indistinct? 
 When a cross division ? And when not arranged according 
 to proximate parts ? [MORELL.] 
 
 30. Give an accurate scheme of logical division in which 
 the following things shall find places: Name; Part of 
 Speech ; Term ; Vox logica ; Verb ; Noun, Adjective ; 
 Syncategorematic term ; Word.
 
 CHAPTER VIII 
 
 SYLLOGISM 
 
 i. MEDIATE Inference, or Syllogism, forms the principal 
 part of Deductive Logic, and offers a wide scope for useful 
 exercises. I give, in the first place, a brief epitome of the 
 syllogistic rules and forms ; I then exemplify them abun- 
 dantly by question and answer lastly, I supply chapters 
 full of the largest and most varied collection of syllogistic 
 questions and problems which has ever been published. 
 Some of the more perplexing questions, involving the dis- 
 tinction of formal and material falsity of syllogisms and 
 their premises, are treated apart in the succeeding 
 chapter (xii.). 
 
 RULES OF THE SYLLOGISM 
 
 (1) Every syllogism has three and only three terms. 
 These terms are called the major term, the minor term, 
 
 and the middle term. 
 
 (2) Every syllogism contains three and only three proposi- 
 tions. 
 
 These propositions are called respectively the major 
 premise, the minor premise, and the conclusion. 
 
 (3) The middle term must be distributed once at least. 
 
 (4) No term must be distributed in the conclusion which 
 was not distributed in one of the premises.
 
 72 SYLLOGISM CHAP. 
 
 (5) From negative premises nothing can be inferred. 
 
 (6) If one premise be negative, the conclusion must be nega- 
 tive ; and vice versa, to prove a negative conclusion one of the 
 premises must be negative. 
 
 From the above rules may be deduced two subordinate 
 rules, which it will nevertheless be convenient to state at 
 once. 
 
 (7) From two particular premises no conclusion can be 
 drawn. 
 
 (8) If one premise be particular, the conclusion must be 
 particular. 
 
 FIGURES OF THE SYLLOGISM 
 S = minor term. M = middle term. P = major term. 
 
 First Figure. Second Figure. Third Figure. Fourth Figure. 
 
 tSL M ... P P ... M M ... P P ... M 
 
 P S s r e . S ... M S ... M M ... S M ... S 
 
 Conclusion. S ... P S ... P S ... P S ... P 
 
 MOODS OF THE SYLLOGISM 
 
 The following is a compact table of the valid moods of 
 the syllogism, the numerals showing the figures in which 
 each group of propositions makes a valid syllogism : 
 
 AAA AAI AEE All AOO 
 
 I. 3. 4. 2. 4. I. 3. 2. 
 
 EAE EAO EI.O IAI OAO 
 
 I. 2. 3. 4. I. 2. 3. 4. 3. 4. 3. 
 
 MNEMONIC VERSES 
 
 Barbara, Celarent, Darii, Ferioque, prioris ; 
 Cesare, Camestres, Festino, Baroko, secundae ; '" 
 
 II
 
 viii QUESTIONS AND ANSWERS 73 
 
 Tertia, Darapti, -Disamis, Datisi, Felapton, 
 ^s*. -z-Bokardo, Ferison, habet ; Quarta insuper addit, 
 
 Bramantip, Camenes, Dimaris, Fesapo, Fresison. ' 
 
 Certain letters in the above lines indicate the way in 
 which the moods of the second, third, and fourth figures 
 may be reduced to the first figure, as follows 
 
 s directs you to convert simply the proposition denoted 
 
 by the preceding vowel. 
 / directs you to convert the proposition per aecidens, or 
 
 by limitation. 
 m, for muta, directs you to transpose the premises. 
 
 k denotes that the mood can only be reduced per 
 impossibile. 
 
 The initial consonant of each mood in the three last 
 figures corresponds with the initial of the mood of the first 
 figure to which it is reducible. 
 
 2. State the figure and mood to which the sub- 
 joined argument belongs : 
 
 Iron is not a compound substance ; for iron 
 
 The conjunction ' for ' shows that the proposition pre- 
 ceding is the conclusion a universal negative. The terrrk 
 ' metal ' must be the middle term, because it does not ap- 
 pear in the conclusion. The major term being ' compound 
 substance,' the major premise must be ' no metals are com- 
 pound substances,' (E) and the other premise ' iron is a 
 metal ' must be the minor. The latter is a universal affirm-
 
 74 SYLLOGISM CHAP. 
 
 ative ; for though no mark of quantity is prefixed to ' iron,' 
 it states a chemical truth concerning iron in general, and may 
 fairly be interpreted universally (A). The argument belongs 
 to the mood E A E in the first figure, or Celarent, thus : 
 
 E No metals are compound substances. 
 
 A (all) Iron is a metal. 
 
 E Iron is not a compound substance. 
 
 3. Examine the following argument ; throw it 
 into a syllogistic form, and bring out the 
 figure and mood : 
 
 It cannot be true that all repression is mis- 
 chievous, if government is repressive and yet 
 is sometimes beneficial. [B.] 
 
 The conclusion is stated in the form of a denial of the 
 universal affirmative ' all repression is mischievous ' ; hence 
 the contradictory of this, or ' some repression is not mis- 
 chievous ' is the real conclusion. The middle term is 
 ' government,' which does not appear in the conclusion. 
 In looking for the major term, we do not find ' mischievous ' 
 n the premises, but only its opposite term ' beneficial.' 
 We must assume, then, that we are intended to take ' bene- 
 ficial ' as equivalent to ' not-mischievous,' otherwise there 
 would be a fallacy of four terms. To be brief, then, the 
 syllogism takes this form 
 
 Some government is not mischievous. 
 
 All government is repressive (or repression). 
 
 Therefore, ' some repression is not mischievous.' It is a 
 valid syllogism in the third figure, and mood O A O, or 
 Bokardo.
 
 viii QUESTIONS AND ANSWERS 75 
 
 4. In what figures is the mood AE E valid? 
 In the first figure we have 
 
 All M is P. 
 No S is M. 
 No S is P. 
 
 The negative conclusion distributes the major term P, 
 which is undistributed in the major premise ; hence Illicit 
 Process of the Major Term. 
 
 In the second figure we have 
 
 All P is M. 
 No S is M. 
 No S is P. 
 
 The major term is now properly distributed in the major 
 premise, and the middle term being also distributed once, 
 in the minor negative premise, the syllogism is valid in 
 Camestres. 
 
 The reader may show that in the third figure we have 
 again Illicit Process of the Major, and in the fourth figure 
 a valid syllogism Camenes. 
 
 5. What rules of the syllogism are broken by 
 
 arguments in the pseudo-moods, O A E, and 
 DIE? 
 
 The answer cannot be better given than in the words of 
 Solly (Syllabus of Logic, p. 86). In the mood O A E the 
 predicate is distributed in the major premise, and the 
 subject in the minor premise, and both subject and predi- 
 cate in the conclusion. Hence it follows that either some 
 term must be distributed in the conclusion which was not 
 distributed in the premises, or else the middle term cannot
 
 76 SYLLOGISM CHAP. 
 
 be distributed in either premise. We cannot, therefore, 
 determine at once which form the fallacy will take, but may 
 be quite certain that there must be either an illicit process 
 of major or minor, or else an undistributed middle. 
 
 Again, in the mood OIH, both subject and predicate 
 are distributed in the conclusion, whereas no term is distri- 
 buted in the minor premise, and it therefore follows that 
 there must be an illicit process of the minor. It is also 
 evident that the middle term cannot be distributed in 
 the minor premise, and that if it is distributed in the 
 major premise the major term must be undistributed, and 
 consequently there must be a fallacy either of undistributed 
 middle or illicit major. 
 
 6. None but whites are civilised ; the ancient 
 Germans were whites : therefore they were 
 civilised. [w.] 
 
 This appears at first sight to be in Barbara, the terms 
 standing apparently in the order of the first figure. But the 
 major premise does not assert that all whites are civilised ; 
 it only asserts that none but whites are so, and this is equiva- 
 lent to the contrapositive of the proposition 
 
 All civilised are whites. 
 Joining to this the minor premise 
 
 The ancient Germans were whites, 
 
 we see that the argument is in the second figure, with two 
 affirmative premises, so that the middle term is undistri- 
 buted in both cases, producing Fallacy of Undistributed 
 Middle. There is also a difference of tense between the 
 two premises which might perhaps invalidate an argument ; 
 but this point need not be further noticed here.
 
 vin QUESTIONS AND ANSWERS 77 
 
 7. None but civilised people are whites ; the 
 
 Gauls were whites : therefore they were 
 civilised. [w.] 
 
 At first sight this seems to be in the second figure, and 
 invalid; but converting the major premise by contraposition, 
 as in the last example, we get a valid syllogism in Barbara 
 thus, ' All whites are civilised ; the Gauls were whites, etc.' 
 
 8. All books of literature are subject to error ; 
 
 and they are all of man's invention ; hence 
 all things of man's invention are subject to 
 error. [H.] 
 
 This may seem at the first reading to be correct reason- 
 ing, especially as the conclusion is materially true ; but 
 there is fallacy of Illicit Process of the Minor Term. 
 The argument is in the pseudo-mood AAA of the third 
 figure, and the conclusion should be l some things of man's 
 invention are subject to error.' 
 
 9. He who is content with what he has is truly 
 
 rich ; a covetous man is not content with 
 what he has ; no covetous man, therefore, is 
 truly rich. 
 
 The middle term is 'content with what he has,' and 
 since this term appears as subject of the major premise and 
 predicate of the minor, the argument is in the first figure 
 in the pseudo-mood AEE. There is Illicit Process of 
 the Major Term, because the conclusion E distributes its 
 predicate and the major premise A does not. 
 
 The conclusion may be true in matter but does not follow 
 from the premises. We could only make the argument good
 
 78 SYLLOGISM CHAP. 
 
 by taking as major premise, ' All the truly rich are content 
 with what they have.' This would give a valid syllogism in 
 Camestres, but the original premise, if converted, only 
 yields ' Some truly rich are content with what they have.' 
 
 10. Protection from punishment is plainly due to 
 
 the innocent ; therefore, as you maintain that 
 this person ought not to be punished, it appears 
 that you are convinced of his innocence, [w.] 
 
 The above is equivalent to 
 
 The innocent are not to be punished ; 
 This person is not to be punished ; 
 Therefore, this person is innocent. 
 
 Put in this form there is an obvious fallacy of Negative 
 Premises ; but we can also express the premises in an 
 affirmative form as follows : 
 
 The innocent ought to be exempt from punishment ; 
 This person ought to be exempt from punishment. 
 
 In this case it is apparent that the middle term 'ought to 
 be exempt from punishment ' is undistributed in both the 
 premises, against Rule 3 of the syllogism. 
 
 11. 'He that is of God heareth my words: ye 
 
 therefore hear them not, because ye are not 
 of God.' [w.] 
 
 In the usual order : 
 
 He that is of God heareth my words ; 
 Ye are not of God ; 
 . ' . Ye do not hear my words. 
 
 The propositions are ABB in the first figure, and involve
 
 vin QUESTIONS AND ANSWERS 79 
 
 the Fallacy of Illicit Process of the Major Term. ' Hear 
 my words ' is distributed as the predicate of the negative 
 conclusion, but is undistributed as the predicate of the 
 affirmative major premise. The argument would become 
 valid, however, if we were allowed to quantify this predicate 
 universally, and assume the meaning to be 
 
 He that is of God - he who heareth my words. 
 
 12. Any books conveying important truths with- 
 out error deserve attention ; but as such books 
 are few, it is plain that few books do deserve 
 attention. 
 
 Carefully distinguish the truth and fallacy in this 
 argument. 
 
 A good example suggested by Whately's No. 1 3 ; it may 
 be thus put : 
 
 Any books conveying, etc., deserve attention ; 
 Few books do convey, etc. ; 
 . . Few books do deserve atttention. 
 
 This is in the first figure, and, if we interpret the con- 
 clusion to mean that ' A few books do deserve attention,' 
 that is to say, affirmatively only, without implying that the 
 rest do not, it is valid in the mood Darii. But usually 
 (Elementary Lessons, p. 67), we interpret few negatively; 
 indeed, in the example itself this is the plain meaning, ' such 
 books are few,' implying that all but this few do not convey 
 important truths without error. This makes the minor pre- 
 mise into O, ' Most books do not, etc.,' and the argument 
 consisting of AOO in the first figure is a case of Illicit 
 Process of the Major Term.
 
 8o SYLLOGISM CHAP. 
 
 13- That man is independent of the caprices of 
 
 Fortune, who places his chief happiness in 
 moral and intellectual excellence : A true 
 philosopher is independent of the caprices of 
 Fortune : therefore a true philosopher is one 
 who places his chief happiness in moral and 
 intellectual excellence. [w.] 
 
 A case of the fallacy of Undistributed Middle, the middle 
 term ' independent of the caprices of Fortune,' being pre- 
 dicated affirmatively both of ' one who places his chief 
 happiness, etc.,' and of 'a true philosopher.' The pseudo- 
 mood is, therefore, AAA in the second figure. The fallacy 
 is none the better because the conclusion may be considered 
 true in matter. If the premise had begun ' Only that man 
 is independent, etc.,' we might have put the argument into 
 a valid syllogism, Barbara. 
 
 14- It is an intensely cold climate that is sufficient 
 to freeze quicksilver ; and as the climate of 
 Siberia does this it is intensely cold. 
 
 At the first glance this looks like a case of Undistributed 
 Middle ; but we soon see that the major premise is really 
 'Any climate sufficiently cold to freeze quicksilver is an 
 intensely cold climate.' The argument is thus valid in 
 Barbara. 
 
 15- No one who lives with another on terms of 
 confidence is justified, on any pretence, in 
 killing him : Brutus lived on terms of confi- 
 dence with Caesar : therefore he was not justi- 
 fied, on the pretence he pleaded, in killing 
 him. [w.]
 
 viii QUESTIONS AND ANSWERS 81 
 
 This is valid in Barbara, the major term being 'justified on 
 any pretence, etc.,' the middle term, 'one who lives, etc.,' 
 and the minor term, ' Brutus.' 
 
 The conclusion, however, is obviously weakened, or is 
 less general than it might have been. We might conclude 
 that Brutus was not justified in killing Caesar on any pretence. 
 It is only inferred that he was not justified in killing him on 
 the pretence he pleaded, which is of course included in ' any 
 pretence.' 
 
 16. Inquire into the validity of the following 
 argument : Whatever substance is properly 
 called by the name Coal consists of a carbon- 
 aceous substance found in the earth ; now, as 
 this specimen consists of a carbonaceous sub- 
 stance, and was found in the earth, therefore it 
 is properly called Coal. [L.] 
 
 The above argument is evidently a case of Undistri- 
 buted Middle, because we infer that this specimen is 
 properly called Coal on the ground of two universal affirma- 
 tive propositions, both of which have the same predicate 
 consisting of a carbonaceous substance found in the earth.' 
 The pseudo-mood then is AAA in the second figure. 
 
 Though entirely failing in a demonstrative point of view, 
 it is another question whether the specimen may not be 
 believed to be coal, on analogical or inductive inference. 
 
 17- Give any remarks which occur to you con- 
 cerning the following : ' Nerve power does not 
 seem to be identical with electricity ; for it is 
 found that when a nerve is tightly compressed 
 nervous action does not go on, but electricity 
 
 can nevertheless pass.' 
 G
 
 82 SYLLOGISM CHAP. 
 
 Implies the following syllogism : 
 
 All tightly compressed nerves do not convey nervous 
 
 action ; 
 
 All tightly compressed nerves do convey electricity ; 
 Therefore, some things which convey electricity do not 
 
 convey nervous action. 
 
 The propositions are AAI in the third figure, i.e. the 
 syllogism is valid in Darapti. It is matter of further 
 inference that, because electricity is conducted by some 
 things which do not convey nervous action, therefore these 
 actions are not identical. 
 
 18. ' With some of them God was not well 
 pleased ; for they were overthrown in the 
 wilderness.' [w.] 
 
 An enthymeme of the first order, the major premise 
 being omitted. The order of statement is that by some 
 logicians called analytical, the conclusion being put first, 
 and the minor premise adduced as a reason or proof. The 
 major premise, assumed to be obvious, is to the effect that 
 ' All who are overthrown in the wilderness are among those 
 with whom God is not well pleased.' More fully stated, 
 indeed, the assumption might be that all who suffer from 
 any signal calamity are some of those with whom God is 
 not well pleased. To be overthrown in the wilderness is to 
 suffer from a signal calamity. This view makes a sorites 
 which the reader can put in order. 
 
 19. If the major term of a syllogism be the 
 predicate of the major premise, what do we 
 know about the minor premise ij [L.] 
 
 In answering syllogistic questions of this sort, great
 
 vin QUESTIONS AND ANSWERS 83 
 
 attention must be given to throwing the reasoning into the 
 briefest and clearest form. Such questions, thus treated, 
 afford capital exercises in reasoning. The above question 
 may be answered thus : 
 
 If the major premise is affirmative its predicate, the major 
 term, is undistributed and must likewise be undistributed in 
 the conclusion; in this case the conclusion, and consequently 
 the minor premise, must be affirmative. If the major pre- 
 mise be negative, then the minor premise must be affirmative, 
 in order to avoid negative premises ; thus in any case the 
 minor premise is affirmative. Or still more briefly thus : 
 
 The minor must be affirmative, for if negative then the 
 major would have to be affirmative, which would involve 
 Illicit Process of the Major. 
 
 20. Prove that O cannot be a premise in the 
 first or fourth figure ; and that it cannot be 
 the major in the second figure, or the minor 
 in the third. [M.] 
 
 If either of the premises be O, the conclusion must 
 be negative, so that its predicate the major term will be 
 distributed. But as O distributes only its predicate, and 
 the other premise, which must of course be affirmative and 
 universal, only distributes its subject, the syllogistic conditions 
 are much restricted. Thus OA as premises in the first 
 figure give an Undistributed Middle, the middle term being 
 subject of O and predicate of A. The premises AO give 
 Illicit Process of the Major Term, the predicate of A, the 
 major term, being undistributed. In the second figure 
 O cannot be the major, because its subject would then be 
 the major term, and undistributed. In the third figure it 
 cannot be the minor, because the major term would then
 
 84 SYLLOGISM CHAP. 
 
 be predicate of A, the major premise, and thus Illicit Pro- 
 cess of the Major would again arise. Finally, in the fourth 
 figure OA will give Illicit Major, and AO, Undistributed 
 Middle. 
 
 21. If it be known concerning a syllogism in the 
 
 Aristotelian system, that the middle term is 
 distributed in both premises, what can we 
 infer as to the conclusion ? 
 
 The syllogism cannot be in the second figure, because the 
 middle term, being the predicate in both premises, these 
 would both have to be negative, against Rule 5. In the 
 first figure the minor premise would have to be negative, in 
 order to distribute its predicate, the middle term ; but a 
 negative minor in the first figure gives Illicit Process of the 
 Major Term. In the third figure, however, the middle term 
 being subject of both premises will be twice distributed if 
 these be both universal, which happens in the moods Darapti 
 and Felapton. In the fourth figure the middle term is pre- 
 dicate of the major and subject of the minor ; we must, 
 therefore, have a negative major and a universal affirmative 
 minor, which happens in the mood Fesapo. We find, then, 
 that a doubly distributed middle term can prove only a 
 particular conclusion, I or O, and these only in the third 
 and fourth figures. 
 
 22. Take an apparent syllogism subject to the 
 
 fallacy of negative premises, and inquire 
 whether you can correct the reasoning by 
 converting one or both of the premises into 
 the affirmative form. 
 
 [India Civil Service, July 1879.]
 
 vin QUESTIONS AND ANSWERS 85 
 
 Take premises in the first figure 
 
 No YisZ; 
 No X is K 
 
 Obvert the major premise (see p. 42), and we have 
 
 All Fis not-Z/ 
 No X is Y. 
 
 The premises would give no conclusion, the pseudo-mood 
 AEE in the first figure involving Illicit Process of the 
 Major. Obverting the minor, we have 
 
 No Fis Z; 
 All X is not- K 
 
 There are now four terms, and therefore no common 
 middle term at all. The reader may easily work out other 
 examples. (See Principles of Science, p. 62 ; first edition, 
 vol. i. p. 75.) 
 
 23. Prove that the third figure must have an 
 affirmative minor premise, and a particular 
 conclusion. 
 
 In the third figure the major term is predicate of the 
 major premise. Now, if the minor premise be negative, 
 the conclusion will be negative (Rule 6), and distribute its 
 predicate the major term, but the major premise must be 
 affirmative in order to avoid negative premises. Thus, there 
 will arise Illicit Process of the Major Term. It follows, by 
 reductio ad absurdum^ that the minor premise cannot be 
 negative and must be affirmative. 
 
 Again the predicate of the minor premise is the minor 
 term, and, the premise being affirmative, this term will be 
 undistributed, giving a particular conclusion.
 
 86 SYLLOGISM CHAP. 
 
 24. Show that if the conclusion of a syllogism be 
 
 a universal proposition, the middle term can 
 be but once distributed in the premises. 
 
 Questions of this sort can be most briefly answered by 
 counting the available number of distributed terms in the 
 premises. Thus, if the conclusion be a universal affirma- 
 tive proposition, we need one distributed term for its 
 subject. But, as the premises must both be affirmative, 
 they contain at most two distributed terms, namely their 
 subjects. Hence there is only one place in which the 
 middle term can be distributed. On the other hand, if the 
 universal conclusion be negative, both major and minor 
 terms require to be distributed in the premises ; but, as one 
 premise only can be negative, we cannot possibly have more 
 than three terms distributed, the subject and predicate of 
 the negative premise, and the subject of the affirmative one. 
 Two being required for the major and minor terms, there 
 remains only one distributed place for the middle term, 
 which was to be proved. 
 
 Observe that this result is the contrapositive of that 
 proved under Question 21 (p. 83). 
 
 25. Given the six rules of the syllogism, and 
 
 the rule that two particular premises prove 
 nothing, show that if one premise be particular 
 the conclusion must be particular. 
 
 This may be demonstrated by the following ingenious 
 reasoning of De Morgan (Formal Logic, p. 1 4). 
 
 ' If two propositions, P and Q, together prove a third, R, 
 it is plain that P and the denial of R prove the denial of 
 Q. For P and Q cannot be true together without R.
 
 vni QUESTIONS AND ANSWERS 87 
 
 Now, if possible, let P (a particular) and Q (a universal) 
 prove R (a universal). Then P (particular) and the denial 
 of J? (particular) prove the denial of Q. But two parti- 
 culars can prove nothing.' 
 
 26. Show that the proposition O is seldom 
 
 admissible as a minor premise. 
 
 When O is the minor premise the conclusion must be 
 negative by Rule 6, and will therefore distribute its pre- 
 dicate, the major term. As we must not have two negative 
 premises by Rule 5, the major premise must be affirmative, 
 and will not distribute its predicate. Hence the major 
 term must be the subject of the major premise. Now, 
 since the middle term becomes the undistributed predicate 
 of the major premise, it must be the predicate of the minor 
 premise, in order that it may be once distributed. Thus 
 we conclude that O can be the minor premise only in the 
 second figure, giving the mood Baroko. 
 
 27. Show that a universal negative proposition 
 
 (E) is highly efficient as a major premise. 
 
 [P.] 
 Since E has both its terms distributed, either of them 
 
 may serve as the major term, which, the conclusion being 
 negative, must be distributed. The other term will then 
 serve to distribute the middle term once. The minor pre- 
 mise may therefore be chosen at will, provided that it be 
 affirmative in order to avoid negative premises. There is 
 no restriction of figure, and accordingly we find valid moods 
 with E as major premise in all of the four figures ; in fact, 
 no less than eight of the nineteen recognised moods begin 
 with E.
 
 88 SYLLOGISM CHAP. 
 
 28. Name the weakened moods of the syllogism. 
 
 In what figure can there be no weakened 
 mood, and why ? Do any of the nineteen 
 moods commonly recognised give a weaker 
 conclusion than the premises would warrant ? 
 
 By a weakened mood is meant one which gives a particular 
 conclusion when a universal conclusion might have been 
 drawn. The information obtained from the premises is 
 thus 'weakened.' This can, of course, happen only when 
 the conclusion of the stronger mood is universal. Hence, 
 in the third figure, which gives only particular conclusions, 
 there can be no weakened mood. In the other figures each 
 mood which has a universal conclusion will have a corre- 
 sponding weakened mood with conclusion of the same 
 quality. Thus Barbara gives a mood AAI ; Celarent, 
 EAO; Cesare, EAO; Camestres, AEO; in the fourth 
 figure only Camenes admits of a weakened form, AEO. 
 Thus the weakened moods are five in number. 
 
 Bramantip of the fourth figure is the single mood alluded 
 to in the latter part of the question. 
 
 Considering that it is impossible to employ conversion 
 by limitation without weakening the logical force of the 
 premise, it is too bad of the Aristotelian logicians to slight 
 the weakened moods of the syllogism as they have usually 
 done. 
 
 29. Can we under any circumstances infer a 
 
 relation between X and Z from the pre- 
 mises 
 
 Some Fs are Xs ; 
 
 Some Fs are Zs ? 
 
 [India Civil Service, July 1879.]
 
 vin QUESTIONS AND ANSWERS 89 
 
 Not if ' some Ys ' bear the sense attributed to the ex- 
 pression in Logic. The indefinite adjective of quantity 
 some is so indefinite, that it must never be interpreted twice 
 over with the same meaning. But if the some Y in the 
 one premise were intended by the arguer to be l the same 
 some Y' as in the other premise, the term would practically 
 become a distributed one, and the premises might give a 
 valid conclusion in the mood Darapti. Dr. Thomson has 
 remarked (Laws of Thought, 77, p. 132), that 'the word 
 (some) appears to be employed in the two senses of 
 " Some or other," and " Some certain," in common language.' 
 Observe, however, that it is in the former purely indefinite 
 sense that logicians have always used the word, so that 
 ' some Y' must not be identified with ' some Y.' 
 
 30. Is the following argumenta valid syllogism ? 
 That which has no parts cannot perish by the 
 dissolution of its parts ; the soul has no 
 parts ; therefore, the soul cannot perish by 
 the dissolution of its parts. 
 
 This example is quoted from the Port Royal Logic, Part 
 III. Chap, ix., Example 6. It is there remarked that 
 several persons advance such syllogisms in order to show 
 the inaccuracy of the unconditional rule (5) that 'nothing 
 can be inferred from negative premises.' Without re- 
 membering what was said in the Art of Thinking, I made 
 the same objection in the Principles of Science, p. 63 (first 
 ed. vol. i. p. 76), and I must still hold that in its bare 
 statement the syllogistic rule is actually falsified. But it 
 must, no doubt, be allowed that if the premises are to 
 be treated as both negative, then there are four terms ; the 
 middle term is broken up into two terms, ' that which has
 
 90 SYLLOGISM CHAP. 
 
 not parts,' and ' that which has parts ' ; the soul is denied 
 to be the latter ; the former is that of which it is asserted 
 negatively that it cannot perish, etc. It comes simply to 
 this, that the syllogistic rules are to be interpreted as a 
 whole, and in making the above example conform to the 
 first rule (see p. 71) we make it conform also to the fifth 
 rule. 
 
 Professor Groom Robertson has criticised my treatment 
 of this subject (Mind, 1876, p. 218, note), urging that 
 ' There are four terms in the example, and thus no syllogism, 
 if the premises are taken as negative propositions ; while 
 the minor premise is an affirmative proposition, if the terms 
 are made of the requisite number three.' No doubt 
 Professor Robertson is substantially right, but it may be 
 noticed that my words were so cautious as hardly to commit 
 me to an erroneous statement. I now find that the point 
 has been treated by many logicians in addition to those of 
 Port Royal, as for instance, Burgersdicius ; De Morgan, 
 Formal Logic, p. 139, Art. 3 ; Bain, Deductive Logic, p. 164 ; 
 Devey, Logic, or the Science of Inference, 1854, p. 129; 
 Essai stir la Logique, 1763, p. 106. 
 
 31. In reference to the syllogism, Mr. Jevons 
 urges that it sometimes yields a conclusion 
 that is open to misinterpretation, as in the 
 example 
 
 Potassium is a metal ; 
 Potassium floats on water ; 
 Therefore, Some metal floats on water. 
 
 Examine this criticism carefully. 
 
 [Moral Science Tripos, Cambridge, Dec. 1876.] 
 
 I said in the Principles of Science (pp. 59-60; first ed.
 
 viii QUESTIONS AND ANSWERS 91 
 
 vol. i. pp. 71-2), that my inference, namely, 'Potassium 
 metal = potassium floating on water,' is of a more exact 
 character than the Aristotelian result ' Some metal floats on 
 water.' The ' some ' after all is only here an indefinite 
 name for ' potassium,' and unless we constantly bear in 
 mind that ' some ' means in logic ' one, and it may be more 
 or all,' the reasoner is apt to confuse ' some ' with the 
 plural 'several.' This view of the matter was criticised by 
 Professor Groom Robertson (Mind, 1876, p. 219). 
 
 32. What is the nature of the argument, if any, 
 
 in the apparent enthymeme, ' The field is 
 neglected because the soil is poor ' ? 
 
 This may, of course, be an argument in Barbara, thus 
 Every field of poor soil is neglected ; 
 This is a field of poor soil ; 
 . ' . This field is neglected. 
 
 But the statement may also mean that the soil being poor 
 is the reason or cause why the owner neglects it ; in this 
 case, it is not an argument but a causal relation. The 
 student, therefore, must always look out for ambiguities in 
 the conjunction ' for,' ' because,' etc., which may certainly 
 bear one of two if not of more senses. The relation between 
 premises and conclusion has nothing whatever to do with 
 the relation between cause and effect. 
 
 33. Explain ' It is scarcely ever possible de- 
 
 cidedly to affirm that any argument involves 
 a bad syllogism ; but this detracts nothing 
 from the value of the syllogistic rules.' [R.] 
 
 Scarcely any one in ordinary writing or discourse states a 
 syllogism in full form ; it is always presumed that the hearer
 
 92 SYLLOGISM CHAP. 
 
 or reader is enough of a logician to supply what is wanting. 
 Now the missing premise may generally be supplied in such 
 a way as to make a good syllogism formally speaking, that 
 is to say, so as to avoid any breach of the syllogistic rules. 
 It is another matter whether the new premise is materially 
 true. The value of the syllogistic rules is, then, that they 
 enable us to assign the premises which would be requisite 
 to support the conclusion put forward. They thus oblige 
 the arguer to define the nature of his assumptions, or else 
 to yield up his conclusion. 
 
 34. How shall we reduce the following syllogism 
 to the first figure ? 
 
 All men are liable to err ; 
 
 None who are liable to err should refuse advice ; 
 
 None who should refuse advice are men. 
 
 This argument is in the absurd fourth figure, in the mood 
 Camenes. In this name the letter m directs us to transpose 
 the premises, and the final s directs us to convert the con- 
 clusion simply ; making these changes, we obtain the same 
 argument in the more natural form of Celarent, thus 
 
 None who are liable to err should refuse advice ; 
 All men are liable to err ; 
 No men should refuse advice. 
 
 35. How shall we reduce the following syllogism 
 to the first figure ? 
 
 All birds are vertebrates ; 
 
 Some winged animals are not vertebrates ; 
 
 Some winged animals are not birds. 
 
 The premises are AO in the second figure, and the con- 
 clusion being O, the argument is a valid syllogism in Baroko.
 
 vin QUESTIONS AND ANSWERS 93 
 
 The letter k directs us to employ the Reductio ad impos- 
 sibile y as explained in the Elementary Lessons, p. 149. Or 
 we may convert the major by contraposition, getting 
 
 All not-vertebrates are not birds ; 
 Some winged animals are not-vertebrates ; 
 Therefore, Some winged animals are not birds. 
 
 Taking the negative term not-vertebrates as the middle 
 term, this is valid in Barbara. 
 
 36. Can we reduce the mood Camestres per 
 
 impossibile ? 
 Taking the symbolic example 
 
 All Xs are Fs ; 
 No Zs are Fs ; 
 Therefore, No Zs are Xs, 
 
 and assuming for sake of argument that the conclusion is 
 false, the contradictory ' some Zs are Xs ' will be true, which 
 put as minor premise with the original as major, gives the 
 valid syllogism in Darii 
 
 All Xs are Fs ; 
 
 Some Zs are Xs ; 
 Therefore, Some Zs are Fs. 
 
 But this conclusion is the contradictory of the original 
 minor premise ' no Zs are Fs,' so that we cannot contradict 
 the conclusion of Camestres without producing a syllogism 
 in Darii to contradict one of our original premises. Thus 
 we prove the conclusion of Camestres indirectly by a mood 
 of the first figure. It will be found on trial that all the 
 moods of the imperfect figures may be similarly proved 
 indirectly by one or other of the moods of the first or 
 so-called perfect figure.
 
 CHAPTER IX 
 
 QUESTIONS AND EXERCISES ON THE SYLLOGISM 
 
 /i. ASSIGN the moods of the following valid syllogisms, 
 pointing out in succession 
 
 (a) The conclusion ; 
 
 (b) The middle term ; 
 
 (c) The major term, and the major premise containing it ; 
 (d} The minor term, and the minor premise containing it ; 
 (e) Thejjuantities and qualities of the three propositions ; 
 (/) Their symbols ; 
 
 (g) The order in which they should be technically placed ; 
 
 (h) The figure of the syllogism ; 
 
 (/) The mood, and its mnemonic name. 
 
 (1) No birds are viviparous; 
 
 All feathered animals are birds ; 
 No feathered animals are viviparous. 
 
 (2) Robinson is plain spoken ; for he is a Yorkshire man, 
 
 and all Yorkshire men are plain spoken. ' 
 
 (3) Birds are not viviparous animals ; E 
 Bats are viviparous animals; t\ (j^M 
 Bats, therefore, are not birds. 
 
 (4) Whatever investigates natural laws is a science ;- h 
 Logic investigates natural laws ; 
 
 Logic is a science.
 
 CHAP, ix QUESTIONS AND EXERCISES 95 
 
 (5) Quicksilver is liquid at ordinary temperatures ; ^"V 4 
 
 Quicksilver is a metal ; h\ t^^ 
 C.Some metal, therefore, is liquid at ordinary tempera- j- 
 
 tures. 
 A- (6) True fishes respjre water containing air ; ^*-\ 
 
 WJhales do not respire water containing air ; K/*n 
 .^T if Whales, therefore, are not true fishes. C 
 
 2. Arrange the following valid syllogisms in the usual 
 strict order of major premise, minor premise and conclusion. 
 Name the figure and mood to which they belong. In exa- 
 mining syllogisms, always fallow the directions of the first 
 question. 
 
 . (i) Iridium must be lustrous; for it is a metal, and all 
 
 metals are lustrous. . 
 
 (2) Some pleasures are not praiseworthy'; hence someE" 
 
 O C? ^A ^- 
 
 pleasures are not virtuous, for whatever is not * * ' 
 U V4 F . . . . CP/L * 
 
 praiseworthy is not virtuous. 
 
 (3) Epicureans do not hold that virtue is the chief good,^ 1 ^ - <a>r ~< 
 
 but all true philosophers do hold that it is so ; ^*i 
 accordingly, epicureans are not true philosophers, c 
 
 (4) Some towns in Lancashire are unhealthy';' because *v -r 
 
 , and such towns are all^X 
 unhealthy, rw. -v 
 
 they are badly drained, and such towns are all^X 
 
 / 
 
 3. Draw conclusions from the following pairs of premises, 
 specifying the figure and mood employed v~^ 
 
 . , J Every virtue is accompanied with discretion ; C -- v 
 
 
 
 -\ c> 
 
 . - wo V * 
 
 There is a zeal without discretion. 
 
 Sodium is a metal ; ^ "^t^__f v ^ r ^ c, ^. " 
 
 Sodium is not a very dense substance.^- ll^iinr. ^ ^. 
 
 / x f All lions are carnivorous animals : 
 
 (3/ i 
 
 ( No carnivorous animals are devoid of claws. 
 
 A* i .~-f-r &<* T 
 
 V-H^CY^- 
 
 i ^
 
 3 
 
 96 SYLLOGISM CHAP. 
 
 ( Combustion is chemical union ; 
 
 jo^, , /v (4) -< Combustion is always accompanied by evolution 
 v of heat. 
 
 ( All boys in the third form learn algebra ; 
 (5) < There are no boys in the third form under twelve 
 ' **i years of age. 
 
 \ v< , , J Nihil erat quod non tetigit : 
 
 ( Nihil quod tetigit non ornavit. 
 
 4. Examine the following arguments and point out which 
 are valid syllogisms, naming the figure and mood as before ; 
 in the case of such as are pseudo-syllogisms, name the rule 
 of the syllogism which is broken thereby, and give the 
 technical name of the fallacy 
 
 (i) All feathered animals are vertebrates ; K 
 No reptiles are feathered animals ; t 
 Some reptiles are not vertebrates. 
 f*W\ (2) Some vertebrates are bipeds ; 
 
 Some bipeds are birds ; V- ^ ^ 
 
 Some birds are vertebrates. 
 
 (3) All vices are reprehensible ; h iu?& 
 Emulation is not reprehensible ; 
 
 Emulation is not a vice, t -^j 
 
 f\ \ t*^^ "^^ 
 
 (4) All vices are reprehensible;^ ~L\s\ ^^ 
 
 Emulation is not a vice ; 
 
 -^sojv? "A, - . v ' v lAr 
 
 Emulation is not reprehensible.-. [L.]- 
 
 (5) Some works of art are useful; t ^ 
 . ^All works of man are works of art ; -K 
 
 Therefore some works of man ate useful. X [L-] 
 (6)>Iron is a metal; K ^ 
 , All metals are soluble jK 
 *ilron is soluble. -
 
 ix QUESTIONS AND EXERCISES 97 I 
 
 (7) Aryans are destined to possess the world; ^ ' (S ^' 
 Chinese are not Aryans ; ^ O%*A/V 
 Chinese are not destined to possess the world. 
 
 > x '--Cc v 
 
 (8) Only ten-pound householders have votes ; ft <M A 
 Smith is a ten-pound householder ; h*^ /V 
 
 Smith has a vote. A; V A - , \ 
 
 ' 5. What are the suppressed premises which are evidently 
 
 \j 'presumed to exist by those who set forth the following 
 
 imperfectly stated syllogisms ? State figure and mood as 
 
 usual N *~ w ' u ^ ^ 
 
 usual. . _jo>- \ 
 
 " '.} ^pJ**~*^"^ V\v>*^ 
 
 (i) Blessed are the meek : for they shall inherit the earth. *\ ^ ^ 
 
 M t ^ 
 
 j. A ^(2) This iron is not malleable; for it is cast iron. 1 
 
 (3) Whosoever loveth wine shall not be trusted of any 
 
 man ; for he cannot keep a secret. 
 ^^(4) Being born in Africa, he was naturally black. ^ ^ ' 
 
 (5) Some parallelograms are not regular plane figures, for 
 
 they cannot be inscribed in a circle. 
 
 (6) Suffer little children to come unto me ; for of them is 
 
 the kingdom of Heaven. 
 
 (7) It is dangerous to tell people that the laws are not A A. A 
 
 just ; for they only obey laws because they think 
 them just. 
 
 (8) The line A B is equal to the line CD; for they are A. f\ J\ 
 
 both radii of the same circle. , 
 
 (9) Whales are not true fishes, for they respire air ; more- t ' 
 
 over they suckle their young. 
 (10) The Queen is at Windsor, for the royal standard is 
 
 flying. AVN N 
 (n) The science of logic is very useful; it enables us to 
 
 detect our adversaries' fallacies. /\P^ P* 
 ^ (12) He must be in York, for he is not in London. 
 
 H
 
 98 SYLLOGISM CHAP. 
 
 (13) I shall not derive my opinions from books, for I 
 
 have none;** 1 [Mansfield, H. of L., 1780.] 
 
 (14) The nation has a right to good government ; there- 
 
 fore it may rebel against bad governors. ^ 
 
 (15) The wise v rmfll < has an infinity of pleasures ; for virtue 
 9 has its delights in the midst of the severities that 
 
 \n ,r\ 
 
 attend it. 
 
 6. Point out which of the following pairs of premises will 
 give a syllogistic conclusion, and name the obstacle which 
 exists in other cases. c~ ( /C ^ ~c ) 
 
 I (i) No A is B; some"^ is not C. ~ e lD ~^M ^ 
 t (2) No A is B; some not C is B. *&*$"** 
 
 . f\iJI-t- ^"~ x t 
 
 (3) All B is not A; some not ^4 is B. '' 
 6. u A)C. (4) Some not ^ is B; no C is B. ^j^ 
 . _ r &) All not B is C; some not A is .#. ?**&- 
 
 "(5) All ^ is B; all not C is not B. c* fc **~" 
 
 (7) All not B is not C ; all not ^ is not A 
 
 (8) All A is not ^/ no B is not C tf 
 
 (9) All C is not B; no ^ is not B. . fr ^ oS? ^ 
 
 7. To what moods do the following belong? 
 
 -- c<> (i) ' All B\* A ; only C is A ; therefore only C is 
 
 (2) ' All It is A ; nothing but C is A ; therefore nothing 
 but C is B: N |v K 
 
 See Dante's Zte Monarchia, as translated by F. C. Church, 
 and appended to the Essay on Dante> by the Rev. R. W. 
 Church, 1878, p. 195. Many curious specimens of reason- 
 ing, sometimes pedantic, might be drawn from the De 
 Monarchia. 
 
 8. Supply premises to prove or disprove the following 
 conclusions V
 
 ix QUESTIONS AND EXERCISES 99 
 
 ^i) The loss of the Captain proves that turret-ships are 
 
 not sea-worthy. 
 ^(2) The cottage-hospital system should be adopted. 
 
 The Prussians are justified in refusing the rights of 
 war to Garibaldi if they find him fighting against 
 them. [E.] 
 
 Private property should be respected in war. 
 No woman ought to be admitted to the franchise, [o.] 
 The law of libel requires to be amended. [o.] 
 Capital punishment ought to be abolished. [o.] 
 (8) Royal parks ought not to be used for political meet- 
 ings, [o.] 
 v (9) Written examinations are not a safe test of merit. 
 Xi) Written examinations are a safe test of merit. [E.] 
 (n) The Annuity-tax should be done away with. [E.] 
 x(ia) Any national system of education should be a 
 secular system. [E.] 
 
 9. In how many different moods may the argument im- 
 plied in the following question be stated ? ' No one can 
 maintain that all persecution is justifiable who admits that 
 persecution is sometimes ineffective.' !*-*** ^o~*- 
 
 How would the formal correctness of the reasoning be 
 affected by reading ' deny ' for ' maintain ' ? [c.] 
 
 10. What conclusions, and of what mood and figure, can 
 be drawn from each pair of the following propositions ? 
 
 ' -T{I) None but gentlemen are members of the dub. r^^ ' ^ 
 (2) Some members of the club are not officers. **\ C> l\ C 
 ' (3) All officers are invited to dine. 
 (4) All members of the club are invited to dine, [c.] 
 
 1 1. Express the following reasonings in each of the four 
 
 syllogistic figures.
 
 TOO SYLLOGISM CHAP. 
 
 (1) Some medicines should not be sold without registering 
 
 the buyer's name, for they are poisons. [E.] 
 
 (2) No unwise man can be trusted ; hence some specula- 
 
 tive men are unworthy of trust, for they are 
 unwise. [E.] 
 
 12. Can the following argument be stated in the form of 
 a syllogism, and if so, what is the middle term ? 
 
 ' The power of ridicule is a dangerous faculty, since it 
 tempts its possessor to find fault unjustly, and to distress 
 some for the gratification of others.' 
 
 13. If the proposition 'warmth is essential to growth' 
 occurred as the premise of a syllogism, would you treat 
 ' warmth ' as a distributed or an undistributed term ? [E.] 
 
 14. Show that the following single propositions may be 
 regarded as enthymemes, that is, as equivalent to imper- 
 fectly expressed syllogisms : 
 
 (1) Have thou nothing to do with that just man. [w.] 
 
 (2) If wishes were horses, beggars would ride. 
 
 (3) Large colonies are as detrimental to the power of a 
 
 State, as overgrown limbs to the vigour of the 
 human body. 
 
 (4) If I had read as much as my neighbours, I would 
 
 have been as ignorant. [HOBBES.] 
 
 (5) All law is an abridgment of liberty and consequently 
 
 of happiness. 
 
 (6) Thales being asked what was the most universally- 
 
 enjoyed of all things, answered Hope ; for they 
 have it who have nothing else. 
 
 (7) I will give thee my daughter if thou canst touch 
 
 heaven.
 
 ix QUESTIONS AND EXERCISES ipi 
 
 (8) If all the absurd theories of lawyers and divines were 
 to vitiate the objects in which they are conversant, 
 we should have no law and no religion left in the 
 world. [BURKE.] 
 
 15. Distinguish between the causal, simply logical, or 
 other, senses of the copulative conjunctions in the 
 following 
 
 (1) It will certainly rain, for the sky looks black. 
 
 (2) The people are happy because the government is 
 
 good. 
 
 (3) This plant is not a rose ; for it is monopetalous. 
 
 (4) The ancient Romans trusted their soothsayers, and 
 
 must therefore have been frequently deceived. 
 
 (5) A favourable state of the exchanges will lead to im- 
 
 portation of gold : this will cause a corresponding 
 issue of bank-notes which will occasion an ad- 
 vance in prices ; which again will check exportation 
 and encourage importation, tending to turn the 
 exchanges against us. [GILBART, 1851, p. 284.] 
 
 1 6. Form an example of a syllogism in which there are 
 two prosyllogisms, one attached to the middle and the 
 other to the minor term. [H.] 
 
 1 7. Prove that a valid sorites with n premises must have 
 
 n (n- i ) 
 n + i terms, and is capable of giving - conclusions. 
 
 2 
 
 1 8. Can the following Shakspearean passage (Hamlet, 
 Act v. Scene i.) be stated in the form of a sorites? 
 
 ' Alexander died, Alexander was buried, Alexander 
 returneth into dust ; the dust is earth ; of earth we make 
 loam ; and why of that loam, whereto he was converted, 
 might they not stop a beer barrel ? ' /
 
 102 SYLLOGISM CHAP, ix 
 
 19. Throw the reasoning of the following passage into 
 syllogistic form : 
 
 'Carbon, which is one of the main sources of the 
 nourishment of plants, cannot be dissolved in water in its 
 simple form, and cannot therefore be absorbed in that form 
 by plants, since the cells absorb only dissolved substances. 
 All the carbon found in plants must consequently have 
 entered them in a form soluble in water, and this we find in 
 carbonic acid, which consists of carbon and oxygen.' [A.] 
 
 20. Complete such of the following arguments as may be 
 considered sound but incomplete syllogisms : 
 
 (1) The people of the country are suffering from famine, 
 
 and as you are one of the people of the country, 
 you must be suffering from famine. 
 
 (2) Light cannot consist of material particles, for it does 
 
 not possess momentum. 
 
 (3) Aristotle must have been a man of extraordinary 
 
 industry ; for he could not otherwise have pro- 
 duced so many works. 
 
 (4) Marcus Aurelius was both a good man and an 
 
 Emperor ; hence it follows that Emperors may be 
 good men, and vice versa. 
 
 (5) Nothing which is unattainable without labour is 
 
 valuable ;. some knowledge is not attainable with 
 labour, and is therefore valuable. 
 
 (6) All gasteropods are mollusks, and no vertebrate 
 
 animals are mollusks ; therefore no gasteropods 
 are vertebrate. 
 
 (7) Suicide is not always to be condemned ; for it is but 
 
 voluntary death, and voluntary death has been 
 gladly embraced by many great heroes.
 
 , 
 
 ' H. 
 
 CHAPTER X ^ 
 
 TECHNICAL EXERCISES IN THE SYLLOGISM 
 
 i. PROVE, from the general rules of Syllogism, that when 
 the major term is predicate in its premise, the minor pre- 
 mise must be Affirmative. 
 
 N 2. Prove that, when the minor term is predicate in its 
 premise, the conclusion cannot be a universal affirmative. 
 
 [L.] 
 
 > 3. Prove that there must always be in the premises one 
 distributed term more than in the conclusion. 
 ^ 4. Prove that the major premise of a syllogism, whose 
 conclusion is negative, can never be a particular affirmative. 
 ^ 5. Prove that when the minor premise is universal nega- 
 tive, the conclusion (unless weakened) will be universal. 
 
 6. Prove that, if in the first figure we transpose the 
 major premise and conclusion, we obtain a pseudo-mood. 
 ^ 7. In the third figure, if the conclusion be substituted for 
 the major premise, what will the figure be ? [BAGOT.] 
 
 8. Prove that no syllogism in the fourth figure can be 
 correct which has a particular negative among its premises, 
 or a universal affirmative for its conclusion. [L.] 
 
 V 9- If the major term be universal in the premises a 
 particular in the conclusion, determine the mood and figure, 
 it being understood that the conclusion is not a weakened 
 
 [,] 
 
 '
 
 104 SYLLOGISM CHAP. 
 
 V i o. Why is it impossible to transform the mood A E O 
 from the second figure into the first ? Vv " N \***~*B- 
 
 n. What figure must have a negative conclusion ? Why 
 must it? 
 
 12. What figure must have a particular conclusion? 
 Why must it ? 
 
 S 13. Why must the major premise of the fourth figure 
 not be O? I****** r**.>\* 
 
 14. Why must the minor premise of the fourth figure 
 not be O ? 
 
 x 15. If the minor premise of the first figure were 
 affirmative, what fallacy would be committed ? ft\*> >r * J/ 
 N 1 6. If the major premise of the first figur were I, what 
 fallacy would be committed ? vw iivA^U . **~^ <f^~ 
 
 17. What kind of proposition does not occur in the 
 premises of the first figure ? Why does it not occur ? 
 
 1 8. If the major premise of the second figure were 
 particular, what fallacy would be committed ? 
 
 19. What is remarkable about the conclusions of the 
 third figure? 
 
 20. What kind of proposition cannot be proved by the 
 fourth figure ? 
 
 21. In what mood of the syllogism can a subaltern pro- 
 position be substituted for its subalternans (universal of 
 same quality) as premise without affecting the conclusion ? 
 
 ^22. If one premise of a syllogism be O, what must the 
 conclusion be ? Vi ^.^oAju^ *- y^cu*-* 
 
 23. Prove that a universal affirmative proposition can 
 form the conclusion of the first figure alone. 
 V 24. Why is OAO excluded from the first and second 
 figures? vw.4*^Xr. 
 
 25. Why is it that the moods EAO and EIO are 
 true in all the four figures ?
 
 x TECHNICAL EXERCISES 105 
 
 26. It is said that A is the most difficult conclusion to 
 establish by the syllogism, and the most easy to overthrow ; 
 O, on the contrary, is the most easy to establish, and the 
 most difficult to overthrow. What is there in the moods of 
 the syllogism to support this view? 
 
 27. 'When both premises of an apparent syllogism are 
 negative, the real middle term is an external sphere, and is 
 consequently undistributed ' [SOLLY]. Explain the meaning 
 of this statement. 
 
 28. If the minor premise of a syllogism be O, what is 
 the figure and mood ? A O o ^c^^f\jy- . 
 
 29. Prove that there cannot be more than four figures 
 of the syllogism. 
 
 -** 30. If one premise be O, what must the other be? f\ 
 
 31. Show that in the fourth figure the conclusion may 
 be either affirmative or negative, and, if negative, either 
 universal or particular. 
 
 32. Given the major premise particular, or the minor 
 premise negative, what must the other premise be ? Why 
 so ? [p.] 
 
 33. If the minor term be predicate of minor, or major 
 subject of major, the conclusion may not be A ? [P.] 
 
 34. Prove that the combination of a particular major 
 premise with a negative minor premise leads to no valid 
 inference. 
 
 35. Prove that wherever there is a particular conclu- 
 sion without a particular premise, something superfluous is 
 invariably assumed in the premises. 
 
 36. Determine in what affirmative moods the middle 
 term may be universal in the major premise, and particular 
 in the minor. 
 
 37. Determine in what negative moods the same may 
 occur. [p.]
 
 io6 SYLLOGISM CHAP, x 
 
 38. Determine how many universal terms may be in the 
 premises more than in the conclusion. 
 
 39. Determine how many particular terms may be in the 
 premises more than in the conclusion. 
 
 40. Determine in what cases there may be in a syllogism 
 an equal number of universal terms and of particular, [p.] 
 
 41. How do you reduce Camestres, Festino, Darapti, 
 Fresison to the first figure ? 
 
 42. Exemplify the reduction of Baroko and Bokardo by 
 the process per impossible. 
 
 43. Show that Cesare, Disamis, Camenes, and other 
 moods can likewise be proved per iinpossibile. (See Kars- 
 lake, 1851, vol. i. p. 81.) 
 
 44. Reduce Celarent to the fourth figure. To how 
 many other figures can you reduce it ? 
 
 45. Reduce Felapton to the second figure. 
 
 46. To what other moods, respectively, can you reduce 
 Darii, Ferio, Barbara?
 
 CHAPTER XI 
 CUNYNGHAME'S SYLLOGISTIC CARDS 
 
 1. IN this age of mechanical progress it may be a matter 
 of surprise that no one has produced a syllogistic machine. 
 About two centuries ago Pascal and Leibnitz invented true 
 calculating machines, and Swift, incited perhaps by the 
 accounts of these machines, described the professors of 
 Laputa as in the possession of a thinking machine. About 
 thirty years ago the late Alfred Smee, F.R.S., proposed the 
 construction of a kind of mechanical dictionary, together 
 with a contrivance for comparing the ideas defined in it. 
 More recently I have constructed a machine which analyses 
 the meaning of any propositions worked upon its keys, 
 provided they do not involve more than four distinct terms. 
 Yet the rules of the syllogism have never been put into a 
 mechanical form. So much the worse for the rules of the 
 syllogism. 
 
 2. Some approximation to a syllogistic machine has, 
 however, been recently made by Mr. Henry Cunynghame, 
 B.A. He has devised certain cards, which, if placed one 
 upon the other, infallibly give a syllogistic mood when 
 that is possible, and when it is not, indicate the absence 
 of a conclusion. The contents of the cards, too, can be 
 condensed into a hollow cylinder turning upon a solid 
 cylinder, in such a way as to give all possible syllogistic
 
 io8 SYLLOGISTIC CARDS CHAP. 
 
 moods in one turn of the handle. This device, though 
 hardly perhaps to be called a syllogistic machine, is probably 
 the nearest approximation to such a machine which is pos- 
 sible. I am now enabled, by Mr. Cunynghame's kindness, 
 to describe these ingenious and interesting devices for the 
 first time. 
 
 3. The syllogistic cards consist of a set of eight larger 
 and a set of eight smaller cards. Each larger card is 3^- 
 inches high by z\ inches broad, and bears, near to its upper 
 edge, one of the eight possible propositions connecting M, 
 the middle term, with P, the major term. There are, of 
 course, four propositions of the forms A, E, I, O, in which 
 M is the subject, and four more in which P is the subject. 
 In the lower part of each larger card is written, in a certain 
 position, each proposition between .S 1 and P, which can 
 possibly be drawn from a syllogism having the proposition 
 on the upper edge of the card for its major premise. Thus, 
 under the major premise ' Some M is P,' we find ' Some S 
 is P,' as the only conclusion that can be drawn from it ; 
 but the universal affirmative proposition, 'All P is M,' 
 admits of any one of three possible conclusions, namely, 
 < No S is P,' ' Some S is P,' or ' Some S is not P.' 
 
 4. Each card of the lesser set is also 2\ inches wide, but 
 only 3 inches high, and bears on its upper edge one of the 
 eight possible minor premises connecting S, the minor term, 
 with M, the middle term. In the lower part of each smaller, 
 or, as we may call it, minor card (with one exception), is cut 
 one, or in some cases two rectangular openings, so adjusted 
 that if any minor card be placed upon a larger or major 
 card, so that the major and minor premises inscribed upon 
 them be visible, one below the other, the conclusion, if any, 
 belonging to those premises is seen through the opening in 
 the minor card. If two conclusions, a universal proposition,
 
 xi SYLLOGISTIC CARDS 109 
 
 and its subaltern minor, are both possible, both will be seen. 
 Thus, if we take the major 'Some P is Mj and the minor 
 'All Afis S,' we see below the conclusion 'Some S is P,' 
 the mood being Dimaris in the fourth figure. If we take 
 the minor ' All S is Af,' and place it upon the major ' No 
 M is P,' we see below ' No S is P,' the correct conclusion 
 in the mood Celarent, together with ' Some S is not P,' the 
 corresponding weakened or subaltern conclusion. If the 
 major be ' Some P is not Af,' or the minor ' Some M is 
 not S] no conclusion can appear at all. 
 
 5. The cards are shown in complete detail on the next 
 page. The principle on which they are constructed is that 
 of excluding illogical conclusions. No conclusion is written 
 on the major card but such as the premise at the top will 
 warrant, and no conclusion is left uncovered by the minor 
 card if it be unwarranted by the minor premise. The 
 student can readily cut out cards according to the directions 
 given above and the figures on the next page, but they 
 must be cut exactly to scale. It will assist the con- 
 struction if each major card be divided by a pencil mark 
 into seven horizontal spaces, each half an inch high, each 
 minor card being divided into six similar spaces. 
 
 6. The syllogistic cylinder is more compact and handy, 
 but much less easy to describe and illustrate. The prin- 
 ciple is exactly the same, but considerable ingenuity was 
 needed to combine the cards together into cylinders in the 
 best way. The order of the minor premises on the in- 
 ferior or moving cylinder is as follows : ' All S is M] 
 ' All M is S,' ' Some 5 is M,' ' Some M is S,' 'No Sis M,' 
 'No M is S,' 'Some S is not A/,' 'Some M is not S.' 
 The major premises are in like order, ' All P is J/,' ' All 
 M is P,' ' Some P is M,' etc.
 
 CUNYNGHAME'S SYLLOGISTIC CARDS 
 
 MAJOR CARDS 
 
 All M is P 
 All .9 is P 
 
 Some 5 is P 
 
 
 No M is P 
 No 5 is P 
 
 Some .9 is not P 
 
 
 Some M is P . 
 Some 6" is P 
 
 
 Some M is not/" 
 Some .9 is not P 
 
 
 
 
 
 All P is M 
 
 No is /> 
 Some .9 is /* 
 Some .9 is not P 
 
 No P is 717 
 No 5 1 is P 
 
 Some .9 is not P 
 
 Some P is ^/ 
 Some *9 is P 
 
 Some P is not ./!/ 
 
 MINOR CARDS 
 
 All .9 is M 
 
 
 No .9 is M 
 
 
 Some .5" is M 
 
 
 Some S is not J/ 
 
 
 
 ! 
 
 
 
 
 
 
 
 
 
 
 
 
 All M is .9 
 
 No ;W is .9 
 
 Some M is .9 
 
 Some M is not .9 
 
 
 
 
 
 
 

 
 CHAPTER XII 
 
 FORMAL AND MATERIAL TRUTH AND FALSITY 
 
 1. THE rules of syllogistic inference teach us how, 
 from certain premises assumed to be true, to draw other 
 propositions which will be true under those assumptions. 
 But if, instead of supposing the premises to be true, we 
 regard them as materially false, various puzzling questions 
 arise as to the conclusions which may properly be drawn. 
 Such questions have not been adequately treated in any 
 popular manual of logic ; and as they lead to results of 
 great practical importance, and at the same time furnish 
 admirable exercises in the discrimination of good and bad 
 reasoning, I propose to draw special attention to this 
 subject in the following questions with answers. 
 
 2. Is it possible to draw a false conclusion from 
 
 true premises ? 
 
 It is possible, of course, to draw any conclusion from any 
 premises, if we disregard the principles of logic and of 
 common sense. But when we speak in a logical work of 
 drawing a conclusion, we must be understood to mean 
 drawing a conclusion logically, in accordance with the Laws 
 of Thought and the Rules of the Syllogism. Now the nature 
 of the logical relation between premises and conclusion is
 
 H2 TRUTH AND FALSITY CHAP. 
 
 this, that if the premises are true the conclusion is true ; 
 truth is, as it were, carried from the premises into the 
 conclusion ; not the whole truth necessarily, but nothing 
 except truth. The question above must, of course, be 
 answered with a direct negative. 
 
 3. Is it possible to prove a true conclusion with 
 
 false premises ? 
 
 To prove a true conclusion, or to prove that a certain 
 conclusion is true, must mean to establish its truth in 
 the opinion of the persons concerned. ' To prove,' 
 says Wesley, 1832, p. 90, 'is to adduce premises which 
 establish the truth of some conclusion.' Now, the rela- 
 tion of premises and conclusion in a syllogism, as stated 
 just above, is that if the premises are true the conclusion 
 must be admitted to be likewise true. But, if certain 
 persons regard the premises as false, they cannot possibly 
 regard such premises as establishing or proving the truth of 
 the conclusion. Solly, indeed, points out (1839, note, p. 9) 
 ' the possibility of a true conclusion from false premises in 
 every form of reasoning.' But this remark can only mean 
 that the conclusion is materially true, or known to be true, 
 on other grounds. 
 
 4. If the premises of a syllogism are false, does 
 
 this make the reasoning false? 
 
 No. The reasoning is correct if the form of the pre- 
 mises and conclusion agree with that of any valid mood of 
 the syllogism or other development of the Laws of Thought, 
 wholly regardless of the material truth of any of the propo- 
 sitions per se. The most ridiculous proposition may make 
 a good syllogism : for instance
 
 xii QUESTIONS AND ANSWERS 113 
 
 Every griffin has angles equal to two right angles ; 
 Every triangle is a griffin ; 
 Therefore, Every triangle has, etc. 
 
 This is, of course, a valid syllogism in Barbara, and if the 
 premises were true the conclusion would be true ; the pre- 
 mises being untrue, the truth of the conclusion is entirely 
 unaffected by the reasoning. 
 
 5- (a) ' The most perfect logic will not serve a 
 
 man who starts from a false premise.' 
 (&) ' I am enough of a logician to know that 
 
 from false premises it is impossible to draw a 
 
 true conclusion.' 
 Comment carefully upon the two foregoing 
 
 extracts. 
 
 Both the above sentences have been written in perfect 
 seriousness by men of intelligence, and they are fair speci- 
 mens of the logic which would pass muster almost anywhere 
 except in a book of logical exercises. In the first place, as 
 regards (a\ if a premise be materially false it cannot give 
 a conclusion materially true ; accidentally, indeed, it might 
 do so by paralogism ; but, as the logic is assumed to be 
 ' most perfect,' we are dealing only with material truth and 
 falsity. Nevertheless, the logic may serve the man well ; 
 for he can learn the truth or falsity of his propositions 
 by observing their congruity with external facts. Now, if 
 he has failed to learn in this way directly the falsity of his 
 premise, his only chance is to draw logical conclusions from 
 that premise, and then observe whether they are or are not 
 materially verified. It is quite clear that if by correct logic 
 we reach a conclusion materially false, then we must have 
 started from premises which involved material error. This 
 
 i
 
 ii4 TRUTH AND FALSITY CHAP. 
 
 procedure represents, in fact, the real method of induction, 
 as the inverse process of deduction, by which we learn all the 
 more complicated truths of physical and moral science. (See 
 Principles of Science, Chaps. XL and XII.) The sentence 
 (a), then, is true only on the supposition that a man, having 
 adopted a false premise, will blindly accept all its false 
 results, that is to say, will reason in a purely deductive 
 manner. 
 
 The sentence (b} is erroneous, because, as we have fully 
 learnt (p. 1 1 2), we can from false premises draw a true 
 conclusion in good logical form. But there was doubtless 
 confusion in the writer's mind between formal and material 
 falseness, and had he said that by premises materially false 
 it is impossible to establish the material truth of any con- 
 clusion, he would have been correct. 
 
 6. An apparent syllogism of the second figure 
 being examined is found to break the rules of 
 the syllogism, the middle term being undis- 
 tributed. On further examination it is re- 
 marked that one of the premises is evidently 
 false, and the other true. What can we infer 
 from such circumstances concerning the truth 
 or falsity of the conclusion ? [c.] 
 
 As in the second figure the middle term is predicate in 
 both the premises, the apparent syllogism can break the 
 third rule of the syllogism, requiring that it shall be distri- 
 buted once at least, only when both the premises are affirma- 
 tive. The premises must therefore be A A, A I, I A, or II. 
 In the first case, if the premises be 
 
 All Xs are Ys, 
 
 All Zs are Ys,
 
 xii QUESTIONS AND ANSWERS 115 
 
 and we assume the first one to be false, we obtain its contra- 
 dictory as true ; thus 
 
 Some Xs are not Ys ; 
 
 All Zs are Vs. 
 
 The conclusion must, by Rule 6, be negative, and there 
 will be Illicit Process of the Major. Assuming the second 
 premise to be false, we get 
 
 All Xs are Ys ; 
 Some Zs are not Ys. 
 
 Whence we may correctly infer in the mood Baroko, 
 Some Zs are not Xs. 
 
 In the case of A I, we obviously cannot assume A to be 
 false ; but if I be false, we get 
 
 All Xs are Ys ; 
 No Zs are Ys ; 
 Therefore, No Zs are Xs. 
 
 In the premises I A, we cannot assume A to be false without 
 Illicit Process of the Major Term ; but if I be false, we 
 have Cesare. Lastly, in II only the major can be taken 
 as false, and its contradictory then gives us a syllogism in 
 Festino. If the conclusion of the apparent or pseudo- 
 syllogism in question does not correspond with what we 
 thus obtain, the conclusion is logically false as compared 
 with the new premises assumed. 
 
 7. If (i) it is false that whenever X is found Y 
 is found with it, and (2) not less untrue that 
 X is sometimes found without the accompani- 
 ment of Z, are you justified in denying that 
 (3) whenever Z is found there also you may
 
 ii6 TRUTH AND FALSITY CHAP. 
 
 be sure of finding F? And however this may 
 be, can you in the same circumstances judge 
 anything about Y in terms of Z ? [R.] 
 
 This excellent example of reasoning by contradictories 
 can be easily solved by adhering to the simple rules of 
 opposition, and gradually undoing the perplexities. The 
 supposition that, whenever X is found Y is found with it, 
 may be stated as the universal affirmative ' all Xs are Ys ' ; 
 but as this is false, its contradictory ' some Xs are not Ys ' 
 is the true condition. That (2) X is sometimes found with- 
 out the accompaniment of Z, would mean that 'some Xs 
 are not Zs ' ; but, being asserted to be untrue, the real 
 condition is its contradictory 'All Xs are Zs.' Thirdly, 
 whenever Z is found, there also you may be sure of finding 
 Y, means that ' all Zs are Fs ' ; but, if you deny this, you 
 must assert that 'some Zs are not Fs.' Putting these 
 propositions together, thus 
 
 1 i ) Some A's are not Fs ; 
 
 (2) All Xs are Zs ; 
 Hence, (3) Some Zs are not Fs ; 
 
 we find that they make a valid syllogism in the third 
 figure, and the mood Bokardo. The conclusion, being a 
 particular negative, cannot be converted directly ; we can 
 only obtain by obversion and conversion 'some not-Fs 
 are Xs.' Thus we must, I presume, answer the last part 
 of the problem negatively. 
 
 8. What is the precise meaning of the assertion 
 that a proposition say 'All grasses are edible' 
 is false ? 
 
 The doctrine of the falsity of propositions is generally
 
 xii QUESTIONS AND ANSWERS . 117 
 
 supposed to be defined with precision in the ancient formula 
 of the square of opposition. If a universal affirmative pro- 
 position is false, its contradictory, the particular negative, is 
 true, so that, in the case of the example given above, we 
 infer, that 'some grasses are not edible.' Similarly, from 
 the falsity of E we infer the truth of I, and vice versa. 
 But it does not seem to have occurred to logicians in general 
 to inquire how far similar relations could be detected in the 
 case of disjunctive and other more complicated 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, as, 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 show which of the possible contradictories is true. 
 
 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 without sufficient grounds. 
 It is false to assert that the hidden side of the moon is 
 covered with mountains, not because we can prove the con- 
 tradictory, but because we know that the assertor must have 
 made the assertion without evidence. If a person ignorant 
 of mathematics were to assert that ' all involutes are tran- 
 scendental curves,' he would be making a false assertion, 
 because, whether they are so or not, he cannot know it. 
 Professor F. W. Newman has correctly remarked that no 
 one can really believe a proposition the terms of which he 
 does not understand (Lectures on Logic, 1838, pp. 35, 36).
 
 ii8 TRUTH AND FALSITY CHAP. 
 
 This is unquestionably true; for, if he does not know 
 what things he is speaking about, he cannot possibly bring 
 them to comparison in his mind. A witness who swears that 
 a prisoner did a certain act when, as a matter of fact, he 
 does not know whether the prisoner did it or not, swears 
 falsely, independently of the question whether rebutting 
 evidence can be brought to prove the perjury. It is reported 
 that a man, who wished to be thought an acquaintance of 
 Dr. Johnson, remarked to him in coming out of church, ' A 
 good sermon to-day, Dr. Johnson.' 'That may be, sir,' 
 replied the very much over-estimated doctor, ' but I'm not 
 sure that you can know it.' This hits the point precisely. 
 
 It will be shown in a subsequent chapter that a pro- 
 position of moderate complexity has an almost unlimited 
 number of contradictory propositions, which are more or 
 less in conflict with the original. The truth of any one or 
 more of these contradictories 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, because 
 there always remains the alternative that nothing is known 
 concerning the relations of the terms. It may even happen 
 that no relation at all exists between the terms. In this 
 view of the matter, then, an assertion of the falsity of a 
 proposition means its simple deletion. The contrariety is not 
 between knowledge and knowledge, but between knowledge 
 and ignorance. 
 
 It ought also to be remembered, in dealing with the doc- 
 trine of falsity, that the falsity of ' all Xs are Ys ' only 
 implies that one or more Xs are not Ys. Now in practice 
 one or a few exceptions are often of no importance ; there 
 are in many cases singular exceptions which in a sense agree 
 with, and in a sense falsify, a general proposition. Thus 
 all points of a revolving sphere describe circles, excepting
 
 xii QUESTIONS AND ANSWERS 119 
 
 the two points at the poles. Other examples of singular 
 exceptions will be found in the Principles of Science, Chapter 
 XXIX. Professor Henrici points out (Elementary Geometry, 
 1879, p. 37) that a proposition must be considered to be 
 true in general, if it be true in an infinite proportion of 
 cases, and false only in a finite number of exceptions. 
 
 This subject of the truth and falsity of propositions as 
 premises and conclusion may be pursued in Karslake's 
 Logic, vol. i. p. 83 ; Whately, Book II. Chap. iii. 2 ; 
 Aristotle, Prior Analytics, Book II. Chaps, i.-iv. ; Port 
 Royal Logic, Part II. Chap. vii. Watts' Logic, Part II. 
 Chap. ii. 7 and 8. 
 
 Most of the scholastic logicians, such as Thomas Aquinas 
 and Nicephorus Blemmidas, treat this subject elaborately. 
 
 9. ' Trust ' (said Lord Mansfield to Sir A. 
 Campbell) ' to your own good sense in form- 
 ing your opinions ; but beware of attempting 
 to state the grounds of your judgments. 
 The judgment will probably be right ; the 
 argument will infallibly be wrong.' 
 
 Explain this phenomenon, and show its logical 
 significance. [p.] 
 
 If you give reasons for a decision, implying that those 
 reasons are sufficient, and are the reasons upon which you 
 did make the decision, it is possible for critics subse- 
 quently to inquire whether such reasons logically support 
 the conclusion derived from them. If they do not, the 
 judge will be detected in a paralogism which there may be 
 no means of explaining away. But, if no reasons be given,
 
 120 TRUTH AND FALSITY CHAP. 
 
 it will seldom be possible for critics to make any such 
 detection. It is impossible, as a general rule, to publish in 
 detail the law as well as the evidence upon which a law 
 case is decided, and, even if it were published, it would 
 generally be impossible to detect bad logic in a man who 
 does not assign the precise points on which he relies, and 
 the way in which he argues about a complex mass oi f 
 details. 
 
 Although it may be, from his own point of view, con- 
 venient and discreet for a man to avoid giving reasons for 
 any important public decision, if he can avoid it, yet it is 
 an open question how far such means of escaping criticism 
 is likely to increase the carefulness and impartiality of his 
 judgments. There are many cases, including nearly all the 
 verdicts given by juries on points of fact, where it would be 
 highly undesirable to require any statement of reasons. 
 Where the result depends upon oral testimony, the be- 
 haviour of witnesses, the estimation of degrees of prob- 
 ability and degrees of guilt, it is quite impossible to define 
 and publish the real premises of the conclusion come to. 
 We must trust to common sense and judicial tact. The 
 same remarks may apply to various arbitrations, magisterial 
 decisions, administrative acts, votes of members of de- 
 liberative bodies. But where the grounds of decision are 
 precise and brief, so as to be capable of complete state- 
 ment, it seems absurd to suppose that a judge will judge 
 less well because he needs to disclose his argument. If he 
 displays bad logic, where bad logic can be judged, he is 
 clearly not fit to be a judge. Lord Mansfield's advice may 
 possibly have been prudent and good when given to a man 
 who was forced to act in novel circumstances, and in a 
 distant colony (Jamaica), where his decisions would have 
 more of the nature of administrative acts than law-building
 
 xii QUESTIONS AND ANSWERS 121 
 
 judgments. But the decisions of the High Court of 
 Justice in England not only affect the parties in the cause, 
 but shape the public law of a large part of the civilised 
 world, and it is of course requisite that they should be 
 guided by good logic.
 
 CHAPTER XIII 
 
 EXERCISES REGARDING FORMAL AND MATERIAL TRUTH 
 AND FALSITY 
 
 \ 
 
 1. COMPARE the following syllogisms, or pseudo- syllo- 
 gisms, both as regards their formal correctness, and as 
 regards the material truth of their premises and conclusion ; 
 then explain how it is that a materially true conclusion is 
 obtained in each case. 
 
 (1) All existing things are real things; 
 No abstract ideas are existing things ; 
 
 .. No abstract ideas are real things. 
 
 (2) No real things exist; 
 
 All abstract ideas are real things ; 
 ..No abstract ideas are real things. 
 
 (3) All real things are existing things ; 
 No abstract ideas are existing things ; 
 
 .-. No abstract ideas are real things. 
 
 2. If there be two syllogisms, of which we know that 
 their major premises are subcontrary propositions, how 
 may we determine the figure and mood of both ? May 
 
 . -^ ' J *t*A "*" \ ' 
 
 their conclusions be both true in matter ? 
 
 3. Prove by means of the syllogistic rules, that given the
 
 CHAP, xin EXERCISES 123 
 
 truth of one premise and the conclusion of a valid syllo- 
 gism, the knowledge thus in our possession is in no case 
 sufficient to prove the truth of the other premise. [c] 
 
 4. It is known concerning a supposed syllogism that it 
 involves a fallacy of undistributed middle, and that one of 
 the premises is false in matter ; can we or can we not draw 
 any conclusion under these circumstances ? 
 
 5. Construct two syllogisms, such that the major premise 
 of d^e shall be the subcontrary of the conclusion of the 
 other, and such also that the conclusions of both shall be 
 true in matter. Are these data sufficient to determine the 
 figure ? 
 
 6. If one premise be false in matter, and the syllogism 
 correct in form, does it follow that the conclusion is false in 
 matter? " 
 
 7. Examine the doctrine ' that, if the conclusion of a 
 syllogism be true, the premises may be either true or false ; 
 but that, if the conclusion be false, one or both of the 
 premises must be false.' 
 
 8. Interpret the logical force of the following passage 
 from Mr. Freeman's Essay on the Holy Roman Empire : 
 ' It may have been foolish to believe that the German King 
 was necessarily Roman Emperor, and that the Roman 
 Emperor was necessarily Lord of the world.' 
 
 9. Taking a syllogism of the third figure, and assuming 
 one of the premises to be false, show whether or not, with 
 the knowledge of its falsehood thus supposed to be in our 
 possession, we can frame a new syllogism : if so, point out 
 the figure and mood to which it will belong. 
 
 10. What do you mean by (i) Formal, (2) Material
 
 124 TRUTH AND FALSITY CHAP. 
 
 truth, as applying (a) to a single proposition, (ft) to a 
 syllogism ? 
 
 ^u. Give a careful answer to the miscellaneous example, 
 No. 88, in Elementary Lessons in Logic, p. 322. 
 
 12. Is the following extract sense or nonsense, logically 
 correct or incorrect ? ' We may doubt whether the ancient 
 method of reduction can prove the validity of any syllogistic 
 mood ; for, as from false premises we can illogically obtain 
 a true conclusion, the reductio ad impossibile has doubts cast 
 upon its validity as a method of proof.' 
 
 13. What is the precise meaning of the assertion that it 
 
 is false to say that Castro cannot be proved not to be 
 Orton ? Q, i^ G c t c^ u^ ^x*H-A ^ (*<. <s\ *V - 
 
 14. If P asserts that oxygen, hydrogen, and nitrogen 
 cannot be liquefied, and Q denies the assertion, what 
 precisely must Q be understood to mean ? 
 
 15. Analyse all that is implied in the assertion of the 
 falsity of each of the following propositions : 
 
 (1) Roger Bacon was a giant. 
 
 (2) Descartes died before Newton was born. 
 
 (3) Bare assertion is not necessarily the naked truth. 
 
 (4) All kinds of grass except one or two species are not 
 
 poisonous. OL ^ ?~* 
 
 VK/V 
 
 1 6. Let X, Y, Z, P, Q, J?, be six propositions : given 
 
 (a) Of X, Y, Z, one and only one is true ; 
 
 (b) Of P, <2, J?, one and only one is true ; 
 (^r) If X is true, P is true ; 
 
 (d} If Y is true, Q is true ; 
 (e) If Z is true, ./? is true ;
 
 xiii EXERCISES 125 
 
 Prove syllogistically, that 
 
 (/) If X is false, P is false ; 
 (g) If Y is false, Q is false ; 
 (h) If Z is false, R is false. [c.] 
 
 1 7. How do you meet the following difficulties ? 
 
 1 i ) True premises may by false reasoning give a correct 
 conclusion ; because, in a train of reasoning there may be 
 two errors, and one error may neutralise the other. ~B&M/~<&u 
 
 (2) Since truth applies only to propositions, and a term 
 per se is incapable of truth, it follows that a term must per se 
 always be false, because everything must be either true or 
 false.
 
 C. 
 
 CHAPTER XIV 
 
 PROPOSITIONS AND SYLLOGISMS IN INTENSION 
 
 i . To any one desirous of acquiring a thorough command 
 of logical science, nothing is so important as a careful study 
 of the intensive or comprehensive meaning of terms, pro- 
 positions, and syllogisms. This indeed is not an easy task, 
 as is shown by the fact that some great logicians who have 
 written upon the subject, especially Sir W. Hamilton, have 
 fallen into grave errors, or at the best fatal ambiguities of 
 expression. Most of the common text-books, again, either 
 ignore the subject altogether, or else treat it in a manner 
 quite disproportioned to its difficulty and importance. The 
 following questions and answers touch some of the more 
 obscure points of the matter, but the student is assumed to 
 have read the fifth of the Elementary Lessons in Logic, or 
 else to have studied the subject in one or more of the 
 following books : Port Royal Logic, Part I., Chapters v. to 
 vii., Spencer Baynes' Translation, 1861, pp. 45-55 (this 
 work was the first to draw attention to the subject in modern 
 times); Watts' Logic, Part L, Chapter vi., 9 and 10; 
 Levi Hedge, articles 34 to 38 ; Thomson's Outline of the 
 Necessary Laws of Thought, 52; Spalding, 1857, 30 
 to 33 ; Walker's Commentary on Murray's Compendium, 
 Chapter II. ; Bowen's Treatise on Logic, Chapter IV. 
 . The most elaborate treatment of the subject is found in
 
 CHAP, xiv QUESTIONS AND ANSWERS 127 
 
 Peirce, Proceedings of the American Academy of Arts and 
 Sciences, 1867, vol. vii. pp. 416-432. If the student read 
 Hamilton's Lectures on Logic, he must carefully observe 
 what is said about Hamilton in this chapter. 
 
 2. State the proposition ' Men are mortals ' in 
 the intensive form. 
 
 This proposition, as it stands, is clearly extensive, and 
 asserts that all individual men will be found among the 
 things called mortals. When asked to turn such a proposi- 
 tion into the intensive form, students make all kinds of 
 blunders, saying, for instance, that 'All the qualities con- 
 noted by the term man are connoted by the term mortal,' 
 or ' All the properties of men are properties of mortal.' 
 This is certainly not the case, because men, in addition to 
 being mortal, are rational, are vertebrate, are erect, etc. 
 Again, a student would say, ' The attributes of man con- 
 note the attributes of mortality.' This means nothing, the 
 verb connote being wrongly used. To say, again, that ' All 
 which possess the properties of man, possess the properties 
 of mortal,' is to leave the proposition just as it was before, 
 'All which possess the properties of man,' being simply 
 men. 
 
 In passing from the extensive to the intensive mode of 
 thought, there must be a complete inversion of the relation 
 of the terms. As men are a part of mortals, so the qualities 
 of mortals are a part of the qualities of men. If we like 
 we may use a different kind of copula ; but, in that case, 
 much care is necessary to avoid error. The following are 
 different modes of expressing correctly the same truth, the 
 first of each pair being an extensive and the second the 
 corresponding intensive form of assertion.
 
 128 INTENSION 
 
 {All men are included among mortals ; 
 All qualities of mortals are included among the 
 qualities of men. 
 
 f Mortals include men ; 
 
 I Properties of men include the properties of mortals. 
 
 f Man is a species of mortal ; 
 
 ( The genus mortal is in the species man. 
 
 f Men are part of mortals ; 
 
 ( Mortality is part of humanity. 
 
 3- Can we exhibit particular and negative pro- 
 positions in the intensive form ? 
 
 This question has not, I think, been much investigated 
 by logicians, and the remarks to be found in the works of 
 Hamilton and most other logicians apply only to the 
 universal affirmative proposition. Taking the particular 
 affirmative, ' Some crystals are opaque,' it asserts that ' One 
 or more crystals are among opaque things.' It follows, no 
 doubt, that the quality ' opaqueness ' is among the qualities 
 of one or more crystals, namely, the particular crystals 
 referred to in the extensive proposition. Thus I may be 
 treated intensively much as A is treated. 
 
 Taking the negative proposition, ' No iron bars are trans- 
 parent,' we cannot infer that ' No properties of transparent 
 objects are properties of iron bars.' This inference would 
 be quite false ; for, there may be many properties, such as 
 gravity, inertia, indestructibility, extension, etc., which are 
 possessed alike by transparent objects and iron bars. All 
 we can infer is that ' Not all the properties of transparent 
 things are in iron bars,' or, ' Some of the properties of 
 transparent things are not in iron bars.' Entire separation 
 in extension involves only partial separation in intension, or
 
 xiv QUESTIONS AND ANSWERS 129 
 
 an extensive assertion in E gives an intensive assertion in 
 O. We may also change E into A, getting ' All iron bars 
 are non-transparent objects,' which of course gives the 
 intensive form, ' The quality of non-transparency is among 
 the qualities of iron bars.' 
 
 We may in a somewhat similar way treat the particular 
 negative, say ' Some crystals are not symmetrical.' We 
 cannot infer that 'All the common properties of symmetrical 
 things are absent from some crystals,' but only some of those 
 properties. 
 
 4- How shall we state the following syllogism in 
 
 the intensive form ? 
 
 All crystals are solids ; 
 All topazes are crystals ; 
 Therefore, All topazes are solids. 
 
 We must transmute each of the three propositions into 
 the intensive form, and transpose the premises, thus 
 
 All qualities of crystals are qualities of topazes ; 
 All qualities of solids are qualities of crystals ; 
 Therefore, All qualities of solids are qualities of topazes. 
 
 The syllogism is turned, as it were, completely inside 
 out. 
 
 5- Is Hamilton correct in stating the following 
 
 as a valid syllogism ? 
 
 S comprehends M ; 
 J/"does not comprehend P ; 
 Therefore, S does not comprehend P. 
 
 Professor Francis Bowen (Logic, p. 237) has pointed out 
 that, as the statement stands (see Hamilton's Lectures, vol. 
 
 K
 
 130 INTENSION CHAP. 
 
 iii. pp. 315, 316, American Edition, p. 223), it is simply 
 illogical. It involves a fallacy of Illicit Process of the 
 Major Term; in short, S may comprehend P through 
 other means besides M. But Professor Bowen thinks that 
 Hamilton's error lay in the choice of language, which 
 was such as no one would understand it as Hamilton 
 really intended it. The matter, however, is too important 
 to be passed over in this way, and I proceed to notice 
 other places in which Hamilton has treated of intensive 
 syllogisms. 
 
 In his sixteenth Lecture on Logic (vol. iii. p. 295), we 
 find the following : 
 
 'An Extensive Syllogism. An Intensive Syllogism. 
 B is A C is B 
 
 C is B B is A 
 
 C is A C is A 
 
 All man is mortal ; Caius is a man ; 
 
 But Caius is a man ; But all man is mortal ; 
 
 Therefore, Caius is mortal. Therefore, Caius is mortal.' 
 
 Between the syllogisms as thus stated there is no difference 
 whatever, except the transposition of premises, a mere 
 difference in the order of writing which is immaterial to the 
 point in question. It is true that Hamilton goes on to 
 explain as follows : 
 
 ' In these examples, you are aware, from what has pre- 
 viously been said, that the copula in the two different 
 quantities is precisely of a counter meaning ; in the quantity 
 of extension, signifying contained under ; in the quantity of 
 comprehension, signifying contains in it? 
 
 Afterwards the example of a concrete syllogism before 
 given, is thus fully stated in the intensive form (p. 296).
 
 xiv QUESTIONS AND ANSWERS 131 
 
 ' The Major term Caius contains in it the Middle term 
 man ; 
 
 But the Middle term man contains in it the Minor term 
 mortal ; 
 
 Therefore, the Major term Caius contains in it the Minor 
 term mortal.' 
 
 To say the least, this is a very clumsy and misleading 
 mode of explanation ; for after all, the point of the matter 
 is left untouched, namely, that it is individual things which 
 are contained under in the extensive sense, and qualities, or 
 attributes, which are contained in the other sense. Is it 
 not absurd to say that Caius contains man, without ex- 
 plaining that man is here taken intensively ? A thing does 
 not contain any of its qualities in the same way that the 
 class man, extensively regarded, contains one of its members 
 or significates, namely, Caius. Nor is the matter much 
 mended by referring back to the previous explanation 
 (p. 274), where Hamilton illustrates the relations, saying, 
 'Thus the proposition, God is merciful, viewed in the one 
 quantity, signifies God is contained under merciful, that is, the 
 notion God is contained under the notion merciful / viewed 
 as in the other, means, God comprehends merciful, that is, 
 the notion God comprehends in it the notion merciful.' This 
 is again all wrong, unless we interpret notion and containing 
 in a totally different sense in the second as compared with 
 the first statement. Even if Hamilton understood the 
 matter correctly himself, he ought to have stated it unequi- 
 vocally, and not to have left the reader to put the matter 
 right by careful interpretation of the same words in two 
 different senses. His subsequent exposition of the sorites 
 is even worse ; for he gives the identically same premises 
 twice over in different order, and asserts, without other 
 explanation, that one is in comprehension and the other in
 
 132 INTENSION CHAP. 
 
 extension. Suppose it were stated in a police court that 
 ' Brown struck Robinson,' and then ' Robinson struck 
 Brown.' Should we not be surprised to learn subsequently 
 that the verb ' struck ' was used in a psychological sense in 
 one case and a physical one in the other, the real meaning 
 being that ' Brown struck Robinson as a very disagreeable 
 fellow,' and then ' Robinson struck Brown on the head ' ? 
 Yet this would not be worse than for Hamilton to state a 
 syllogism or sorites twice over, with an unimportant change 
 of order, and then assume that the reader takes one state- 
 ment to be in extension and the other in intension. 
 
 I am obliged, therefore, to coincide with the opinion of 
 De Morgan that Hamilton really missed the point of the 
 question, in short, did not understand what he was writing 
 about. The whole matter is put in the clearest light by the 
 following few lines from De Morgan's Syllabus, pp. 62-63 : 
 ' The logicians who have recently introduced the distinc- 
 tion of extension and comprehension, have altogether missed 
 this opposition of the quantities, and have imagined that 
 the quantities remain the same. Thus, according to Sir 
 W. Hamilton, " All X is some Y" is a proposition of com- 
 prehension, but "Some Y is all X" is a proposition of 
 extension. In this the logicians have abandoned both 
 Aristotle and the Laws of Thought from which he drew the 
 few clear words of his dictum : " the genus is said to be part 
 of the species ; but in another point of view (aAAws) the 
 species is part of the genus." All animal is in man, notion 
 in notion : all man is in animal, class in class. In the first, 
 all the notion animal part of the notion man ; in the second, 
 all the class man p'art of the class animal. Here is the 
 opposition of the quantities.' 
 
 The same view is more fully stated in De Morgan's Third 
 Memoir on the Syllogism (pp. 17-19; Camb. Phil. Trans.
 
 xiv QUESTIONS AND ANSWERS 133 
 
 1858, pp. 188-9). On the whole, I conclude that Hamil- 
 ton's treatment of the subject is so doubtful and confusing 
 that it had better not be studied in an elementary course of 
 reading. 
 
 De Morgan, in the paper just referred to, gives some 
 remarks about the history of the doctrine of intension and 
 extension, and speaks of Hamilton as a logician who has 
 ' recently contended for the revival, or rather the full intro- 
 duction, of the distinction of extension and comprehension.' 
 He correctly names the Port Royal Logic as being the first 
 modern work to insist on the distinction, though the use 
 made of it is ' not very extensive.' But he names only one 
 other work, the Institutiones Philosophies of J. Bouvier 
 (3rd Ed. Mans, 1830), as describing this distinction. 
 De Morgan's reading of modern logic was not extensive. 
 Not to mention the familiar Watts' Logic, in which the 
 doctrine is frequently dwelt upon (see Part I. Chap. III. 
 Section 3 ; Chap. VI. Section 10, and elsewhere), I 
 find the matter excellently explained in 1816 in the 
 brief manual of the American logician, Levi Hedge (pp. 
 42-44). In Murray's Manual, formerly much used in 
 Dublin and Glasgow, the subject is fully explained, and 
 in the clearest possible manner, in the Commentary of John 
 Walker on Chapter II. This is in fact one of the best 
 pieces of logical exposition which I know. Walker re- 
 marked that he had treated the point fully, because he 
 regarded it as absolutely necessary to the understanding of 
 the subsequent pages, which were often puzzling to students 
 not familiar with the distinction between the comprehension 
 and extension of a term. With some regret I must hold, 
 then, that the pretensions of Hamilton in this matter are 
 mistaken and unfounded. 
 
 The whole subject of extension and comprehension or
 
 134 INTENSION CHAP, xiv 
 
 intension has been investigated with much care and pro- 
 fundity of thought by the American logician, Professor 
 C. S. Peirce, in the memoir already referred to (see p. 127). 
 This memoir should be studied by those who wish to 
 acquire a thorough understanding of logical principles and 
 relations. 
 
 6. It is asserted by some logicians that the 
 predicate of a proposition must be interpreted 
 in intension while the subject is regarded in 
 extension. Give your opinion upon this point, 
 and explain the bearing of the question upon 
 recent logical controversies. [c.] 
 
 I should answer this question to the effect that a propo- 
 sition being, conformably to the opinion of Condillac, 
 necessarily of the nature of an equation, it is absurd to 
 suppose that things can be equated to their own qualities 
 or circumstances. A proposition in extension expresses the 
 identity of a thing or class of things with the same thing 
 or class under another designation. As De Tracy says 
 (Ideologic, vol. iii. p. 529), 'Dans tout jugement, les deux 
 idees comparees sont necessairement e'gales en extension.' 
 A proposition in intension expresses an identity between 
 the attributes of the one member and those of the other. 
 The subject may be pursued in my Essay on Pure Logic, 
 or the Logic of Quality apart from Quantity, 1864, passim ; 
 in J. S. Mill's Logic , Book I. Chapter V. ; and in Dr. 
 Martineau's review of Samuel Bailey on the Theory of 
 Reasoning, in his Essays Philosophical and Theological, 1869, 
 vol. ii.
 
 7 
 
 CHAPTER XV 
 
 QUESTIONS ON INTENSION 
 
 1. 'Christian,' 'animal,' 'Episcopalian,' 'organised,' 
 'man.' Arrange these terms (i) in the order of exten- 
 sion, beginning with the most extensive ; and (2) in the 
 order of comprehension, beginning with the most com- 
 prehensive. [L.] 
 
 2. Arrange the following in the same manner : General, 
 animal, composer for the pianoforte, Roman, historian of 
 his own campaigns, conqueror of Gaul. (See De Morgan, 
 Third Memoir, pp. 20, 21.) 
 
 3. Arrange in order of extension and intension such of 
 the terms given in Question i of Chapter II. as are the 
 names of subaltern, genera, and species, and can be arranged 
 in a series. 
 
 4. Analyse the following terms in the counter quantities 
 or wholes of extension and intension : Man, government, 
 law, triangle, vegetable. [L.] 
 
 5. Show that the analysis of an intensive equals the 
 synthesis of an extensive whole. [c.] 
 
 6. Invent a syllogism in Barbara, and state it both in the 
 extensive and in the intensive forms. [L.]
 
 136 INTENSION CHAP, xv 
 
 7. What is the place of the Major and Minor Terms in 
 the conclusion of (a) an extensive, and (b) an intensive 
 (comprehensive) syllogism ? 
 
 8. Can the distinction of extension and intension be 
 made to apply to the inductive syllogism ? [c.] 
 
 9. Select from pp. 91 to 98 examples of the moods 
 Celarent, Cesare, and Camenes, and state them in the 
 intensive form. 
 
 10. What is the difference of meaning of genus and 
 species in extent and intent? Is the extent of a notion 
 always less as the intent is greater, and vice versa ? 
 
 11. Interpret the following propositions in extension 
 and intension : 
 
 A libel is a malicious and injurious statement. 
 
 He who believes himself to be always right in his opinion 
 
 claims infallibility. 
 
 It is impossible to be and not to be. 
 He that can swim needs not despair to fly.
 
 CHAPTER XVI 
 
 HYPOTHETICAL, DILEMMATIC, AND OTHER KINDS OF 
 ARGUMENTS 
 
 1. SOME attempt will be made in the subsequent chapters 
 on the Elements of Equational Logic to illustrate the 
 actual and possible variety of assertions and arguments. 
 But it will be convenient to give here a few examples of 
 hypothetical and other arguments in the less common forms. 
 Several subtle questions arising out of the hypothetical 
 form of assertion are also considered with some care ; 
 but it has not been thought necessary to treat all the 
 various forms of disjunctive and dilemmatic arguments 
 which will be found described in almost identical terms in 
 numerous text-books. 
 
 2. If virtue is voluntary, vice is voluntary ; but 
 
 virtue is voluntary ; therefore so is vice, [w.] 
 
 A valid Constructive Hypothetical syllogism, equivalent 
 to the following categorical one in Barbara : 
 
 Beings who can be virtuous at will can also be vicious 
 
 at will ; 
 
 Men can be virtuous at will ; 
 Therefore, they can be vicious at will.
 
 138 HYPOTHETICAL ARGUMENTS, ETC. CHAP. 
 
 3. Logic is indeed worthy of being cultivated, 
 if Aristotle is to be regarded as infallible ; but 
 he is not : Logic, therefore, is not worthy of 
 being cultivated. [w.] 
 
 Clearly a false hypothetical syllogism. The antecedent 
 is, ' if Aristotle is to be regarded as infallible ' ; this is 
 denied in the minor premise. In the categorical form the 
 pseudo-argument might be stated somewhat as follows : 
 
 Those who regard Aristotle as infallible must consider 
 
 logic worthy of being cultivated ; 
 We do not regard Aristotle as infallible ; 
 Therefore, we do not consider logic, etc. 
 
 There is Illicit Process of the Major Term. 
 
 4. We are bound to set apart one day in seven 
 for religious duties, if the fourth command- 
 ment is obligatory on us : but we are bound 
 to set apart one day in seven for religious 
 duties ; and hence it appears that the fourth 
 commandment is obligatory on us. [w.] 
 
 The antecedent is ' if the fourth commandment is obli- 
 gatory ' ; the consequent is ' we are bound, etc.' ; it is the 
 consequent which is affirmed, so that the argument involves 
 the Fallacy of Affirming the Consequent. It may be put 
 categorically as follows : Those on whom the fourth com- 
 mandment is obligatory are bound, etc. ; we are bound, etc. ; 
 therefore, we are among those on whom the fourth com- 
 mandment is obligatory. The fallacy is evidently that of 
 Undistributed Middle, the pseudo-mood being AAA in 
 the second figure.
 
 xvi QUESTIONS AND ANSWERS 139 
 
 5- (i) If the prophecies of the Old Testament 
 had been written without knowledge of the 
 events of the time of Christ, (2) they could 
 not correspond with them exactly ; (3) and if 
 they had been forged by Christians, (4) they 
 would not be preserved and acknowledged by 
 the Jews : (5) they are preserved and acknow- 
 ledged by the Jews, (6) and they correspond 
 exactly with the events of the time of Christ : 
 therefore they were (7) neither written without 
 knowledge of those events, (8) nor were forged 
 by Christians. [w.] 
 
 The above argument will be found to consist of two 
 valid destructive hypothetical syllogisms woven together 
 in statement. Thus (i) and (2) are the antecedent and 
 consequent of the first syllogism ; (6) is its negative 
 minor, and (7) is its negative conclusion. The second 
 syllogism has (3) and (4) for its antecedent and conse- 
 quent, (5) for its negative minor premise, and (8) for its 
 conclusion. 
 
 6. In how many ways can you state the sub- 
 stance of the categorical proposition ' A wolf 
 let into the sheep-fold will devour the sheep ' ? 
 
 Isaac Watts, in his Essay on the Improvement of the 
 Mind, has well pointed out the variety of expression which 
 may be given to the same real assertion. Thus, as equiva- 
 lents for the above proposition, he gives the following : ' If 
 you let a wolf into the fold, the sheep will be devoured : 
 The wolf will devour the sheep, if the sheep-fold be left 
 open : If the fold be not left shut carefully, the wolf will
 
 140 HYPOTHETICAL ARGUMENTS, ETC. CHAP. 
 
 devour the sheep : The sheep will be devoured by the 
 wolf, if it find the way into the fold open : There is no 
 defence of the sheep from the wolf, unless it be kept 
 out of the fold : A slaughter will be made among the 
 sheep, if the wolf can get into the fold.' There are various 
 modes of hypothetically stating the result contained in the 
 categorical original. 
 
 7- In a strictly logical point of view, ought it to 
 be offensive to Captain Jones to say of him 
 ' If Captain Jones does run away in battle, he 
 will live to fight another day ' ? 
 
 This question touches deeply, not only the soldierly repu- 
 tation of Captain Jones, but, what is much more important, 
 the precise import of propositions. It puts forward Captain 
 Jones as running away in battle, but it puts this forward 
 only as a hypothesis, the result of which would be his living 
 to fight another day. It is quite a different matter, what 
 meaning such a proposition might be taken to imply in 
 common life ; things are often said in the form of innuendo. 
 The mere coupling of a man's name with a disreputable 
 action, even though the action were expressly denied of him, 
 raises the question, Why was the assertion made at all 
 unless to bring the terms together in the mind of the hearer ? 
 If in company a gentleman were suddenly to remark ' There 
 is not the least reason to believe that Captain Jones did run 
 away in his last action ' ; here is a point blank denial of 
 any ground for believing an assertion to that effect ; yet 
 every one would construe such a mal-a-propos denial as 
 evidence of a wish to raise the question, and possibly start 
 a rumour, which would presently take a disagreeable affirma- 
 tive form. Thus we see that the logic of conversation is
 
 xvi QUESTIONS AND ANSWERS 141 
 
 widely different in apparent nature from the strict logic of 
 science ; not that it is really different in the end, when 
 thoroughly analysed. But we constantly deal with illogical, 
 inaccurate, or even untrustworthy persons, so that we can 
 seldom be sure that an assertion will be construed and 
 repeated in the form which we originally gave to it. There 
 is too much truth in the saying of Talleyrand, that words 
 were given to us to disguise our thoughts. 
 
 8. If Brown says to Jones, ' Because Robinson is 
 foolish you have no need to be foolish,' does 
 Brown assert categorically that Robinson is 
 foolish ? 
 
 There can be no doubt that, in the logic of common life, 
 Brown would be understood to make an imputation upon 
 the wisdom of Robinson, especially if the remark was not 
 explained by the previous course of the conversation. But 
 in strict logic it seems very doubtful whether the conjunction 
 ' because ' should be interpreted differently from ' if,' as in 
 the last question. ' The fact of Robinson being foolish is 
 no reason, etc.' ' Foolishness on the part of Robinson is 
 no reason for you being foolish.' A logical copula must 
 not be understood to assert the physical existence and 
 occurrence of its subject or predicate ; it only asserts a 
 relation between them. 
 
 9- If P is Q, 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 ? 
 
 I do not see how there is in deductive logic any question 
 about existence. The inference is to the effect that if the
 
 142 HYPOTHETICAL ARGUMENTS, ETC. CHAP. 
 
 propositions P is Q and Q is R are true, then the conclu- 
 sion P is R is true. The non-existence of Q may possibly 
 render one or both premises materially false, in which case 
 the reasoning vanishes, but is not logically defective. If 
 I argue, for instance, that satyrs are creatures half man and 
 half goat ; and creatures half man and half goat are very 
 hideous, therefore satyrs are very hideous ; the reasoning is 
 equally good whether satyrs exist or not. We cannot, of 
 course, say that the conclusion is materially true, if there 
 be no objects to which the material truth can apply. But 
 if I argue that satyrs are creatures half man and half goat, 
 and such creatures exist in Thessaly, therefore, satyrs exist 
 in Thessaly ; in this case the non-existence of the middle 
 term would affect the material truth of the second premise, 
 and, if this be held false, we cannot affirm the material 
 truth of the conclusion. 
 
 I ought to add that De Morgan in more than one place 
 assumes that the middle term must have existence, or even 
 objective existence; thus he says (Syllabus, p. 67): 'In all 
 syllogisms the existence of the middle term is a datum,' etc. 
 This is one of the few points in which it is possible to 
 suspect him of unsoundness. 
 
 The student may refer to Hamilton's Lectures, vol. iii. 
 pp. 454-5, and p. 459, on 'Sophisms of Unreal Middle'; 
 see also Whately's Analytical Outline, 3. 
 
 10. Lias lies above red sandstone ; red sand- 
 stone lies above coal ; therefore lias lies above 
 coal. [w.] 
 
 This is one of many examples to be found in the logic 
 books of arguments which simulate the syllogistic form. 
 It is often said that they can be solved syllogistically ; but
 
 xvi QUESTIONS AND ANSWERS 143 
 
 certainly this cannot be done by the ordinary rules and 
 processes of the syllogism. The most that we can get, 
 even by substitution, is that ' Lias lies above what lies above 
 coal.' The fact is that the argument is really a mathematical 
 one, involving simple equations. It is precisely similar to 
 one which has been thus treated by Professor F. W. Newman 
 (Miscellanies, 1869, p. 28), and Mr. J. J. Murphy. The 
 former of these logicians, as quoted by the latter, remarks 
 ' The argument Lead is heavier than silver; Gold is heavier 
 than Lead : therefore Gold is heavier than Silver?- brings to 
 the mind conviction as direct as the simplest of syllogisms. 
 To say that its validity depends on its being reducible to 
 syllogism, is wholly unplausible : for to effect the reduction, 
 you have to make changes of form at least as hard to 
 accept as the direct argument : and when you have got your 
 syllogisms, they are more complicated and cumbrous than 
 the argument as it stands.' 
 
 Mr. Murphy (The Relation of Logic to Language : Belfast 
 Natural History and Philosophical Society, i7th February 
 1875) treats the argument simply as a question of quantity, 
 thus 
 
 'Call the weights of gold, lead, and silver respectively 
 x, y, and z : then x = y + p 
 
 y = z + <? 
 x = z + q + p, 
 
 In the old logic, the foregoing conclusion could be drawn 
 only by means of the following syllogism : 
 
 That which is greater than the greater is greater than the 
 less : 
 
 The weight of gold is greater than that of lead, and the 
 weight of lead greater than that of silver : 
 
 1 Newman has inadvertently written Lead is heavier than Gold, which 
 is wrong as to fact.
 
 144 HYPOTHETICAL ARGUMENTS, ETC. CHAP, xvi 
 
 Therefore the weight of gold is greater than that of 
 silver. 
 
 Considered as fact all this of course is true, but considered 
 as logic it is wrongly stated. That which is here stated as 
 the major premise is really the syllogistic canon. It is not 
 merely a general truth, like the truth that all matter gravi- 
 tates, but a logical principle, lying as near to the first 
 principles of the science as the axiom that a part of a part 
 is a part of the whole. 
 
 We have only to assume x to be the height of lias, y the 
 height of red sandstone, and z of coal above any one fixed 
 datum line, and the same equations represent the argument 
 at the head of this section. 
 
 It may be added that Reid was doubtless right in denying 
 that we argue syllogistically when we infer that because A 
 and C are both equal to B they are equal to each other. 
 We may throw it into the form, 'Things equal to the same are 
 equal to each other ; A and C are things equal to the same, 
 therefore they are equal to each other.' But this is a delu- 
 sive syllogism. The inference is really accomplished in 
 obtaining the major premise. The inferences of equality 
 are prior to and simpler than the inferences of logic, and 
 the attempt of Herlinus and Dasypodius to throw Euclid 
 into the syllogistic form has been rightly ridiculed, because 
 it is an attempt to prove the more simple and self-evident 
 by means of the more complex. 
 
 Some remarks on this point will be found in De Morgan's 
 Second Memoir on the Syllogism, 1850, pp. 50, 51; his 
 Fourth Memoir, 1860, p. 8, etc.; in Mr. Murphy's paper 
 quoted above ; and in Hallam's remarkable note to Section 
 129 of Vol. III., Chapter III., of his Introduction to the 
 Literature of Europe (ist Ed. p. 288 ; 5th Ed. p. in).
 
 CHAPTER XVII 
 
 EXERCISES IN HYPOTHETICAL ARGUMENTS 
 
 i. (i) IF he is well, he will come : he is not well : there- 
 
 fore he will not come. Aiuv+ts) o>JU^I2jUvw2t" - ***** 
 
 (2) If he is well, he will come : he will come : therefore 
 
 he is well. PytwA**^> ^v^^e^^S [H.] ^/s^-xxSU^ 
 
 (3) I am sure he will not come, for he is not well ; and if 
 
 well he would come. &&>*J**A O^XlAA&j-* .' 
 
 "- ^A_M>-Vv b*$ C 
 
 (4) He will write if he is well ; but as he is not well, \ ^Jc*ilV 
 
 therefore he will not write. 
 
 Analyse the above arguments and point out which are 
 fallacious, and why. 
 
 2. Into how many forms of expression can you throw the 
 matter of this proposition ? ' Sulphuric acid combined with 
 calcium produces gypsum.' 
 
 3. Throw into the form of hypothetical propositions the 
 following disjunctives 
 
 (1) Either the Claimant is Orton, or many witnesses are 
 
 mistaken. 
 
 (2) The tooth of a mammalian is either an incisor, canine, 
 
 bicuspid, or molar tooth. 
 
 T- 
 *
 
 146 HYPOTHETICAL ARGUMENTS CHAP. 
 
 4. Under which of the commonly recognised forms of 
 syllogism would you bring the following ? 
 
 If A is B, C is D ; ^ 
 
 If C is Z>, E is ^/ ^ 
 
 Therefore, If A is ^, E is ^ [c] 
 
 5. Are hypothetical propositions capable of conversion ? 
 If so, convert these W w>X. *>o imW~ 
 
 _ D T A" 
 \ 0,4 ^^ 
 
 f (i) If it has thundered it has lightened. ^ $ 
 
 ( (2) Unless it has lightened it has not thundered. 
 
 OvM.0^ V 
 
 6. Which of the following arguments are logically 
 correct ? 
 
 (1) A is B, if it is C; it is not C, therefore it is not B. 
 
 (2) A is not B unless^ it is C ; as it is not C, it is not B. 
 
 (3) If A is not B, Cis not D ; but as ^4 is B, it follows 
 
 that C is Z>. 
 
 (4) ^4 is not B, if C is D ; C then is not D, for ^4 is B. 
 
 7. If the Hypothetical Modus Ponens and Modus. Tollens 
 are taken as corresponding to the Categorical First and 
 
 yV Second Figures, and their typical forms to the Moods 
 Barbara and Camestres, respectively, what other forms of 
 the respective Hypothetical Modi would correspond to the 
 other moods of the respective Categorical Figures ? [R.] 
 
 ^*^- * 
 
 8. If A is true, B is true ; if B is true, C is true ; if 7 is 
 true, D is true. What is the effect upon the other assertions 
 
 f O-** * 
 
 of supposing successively that (i) D is false; (2) that C is 
 
 false ; (3) that B is false ; (4) that A is false ? 
 *3 - C -^ 
 
 9. Analyse the following arguments and estimate their 
 validity.
 
 xvn EXERCISES 147 
 
 (1) I shall see you if you do not go; but as you are 
 
 going I shall not. \^soJu_d AJU*-O 
 
 (2) The Penge convicts were guilty of murder, if, after 
 
 long continued neglect at their hands, Harriet 
 
 Staunton died. U. U. 
 
 o&i 14- y> v/ 
 
 (3) Since the virtuous alone are happy, he must be 
 
 virtuous if he is happy, ano! he must be happy if 
 he is virtuous. dto^uJCL>P 
 
 (4) If there were no dew the weather would be foul : but 
 
 there is dew ; therefore the weather will be fine. 
 
 [o.] 
 
 (5) If there are sharpers in the company we ought not to 
 
 gamble ; but there are no sharpers in the com- 
 pany; therefore we ought to gamble. [E.] ) 
 
 (6) ' I could then only be accused with justice of acting * c 
 
 contrary to my law, if I maintained that Muraena 
 T'i purchased the votes, and was justified in doing so. 
 
 But I maintain that he did not buy the votes, 
 therefore, I do nothing contrary to the law.' U o4ju 
 Cicero, Pro L. Muraena, c. iii. (See Devey'sZ^/V, 
 1854, p- I33-) 
 
 10. State in the form of a disjunctive argument the matter 
 of the First Book of Samuel, chapter xvi. verses 6-13. 
 
 1 1. Examine the question whether hypothetical and 
 disjunctive arguments are reducible to the forms of the 
 categorical syllogism. 
 
 1 2. Dilemmatic arguments are more often fallacious than 
 not. Why is this? H-o- A JJCTVU [c.] 
 
 13. Investigate the logical position of the parties to 
 the following colloquy from Clarissa Harlowe : ' Morden 
 But (/"you have the value for my cousin that you say you 
 
 to TX L^. c_c^
 
 148 HYPOTHETICAL ARGUMENTS CHAP, xvn 
 
 have, you must needs think . Lovelace You must allow 
 
 me, sir, to interrupt you. If\ have the value I say I have ! 
 I hope, sir, when I say I have that value, there is no cause 
 for that if, as you pronounced it with an emphasis. Morden 
 Had you heard me out, Mr. Lovelace, you would have 
 found that my if was rather an if of inference than of 
 doubt.' 
 
 This passage is quoted and discussed by Professor Croom 
 Robertson in Mind, 1877, vol. ii. pp. 264-6.
 
 U 
 
 CHAPTER XVIII 
 
 THE QUANTIFICATION OF THE PREDICATE 
 
 i. As explained in the preface, I have thought it well to 
 discuss and illustrate in this book of exercises, the forms of 
 logical expression and inference recognised by Dr. Thomson 
 and Sir W. Hamilton. These correspond in most cases 
 with what De Morgan represented under different systems of 
 notation. They also correspond to some of the expressions 
 and arguments current in ordinary life. Although in a scien- 
 tific point of view it is far better to eliminate the logical will-of- 
 the-wisp ' some,' yet the student is obliged to make himself 
 acquainted with the pitfalls into which it is likely to lead him. 
 
 It is assumed that the reader has studied the brief 
 account of the Quantification of the Predicate given in the 
 22nd of the Elementary Lessons in Logic, and he is re- 
 commended to read, on the same subject, either Thomson's 
 Outline, or else Bowen's account of Hamilton's Logic 
 (Bowen's Logic, Chapter VIII.) The study of De Morgan's 
 and Hamilton's own writings is a more arduous and 
 hazardous undertaking. 
 
 The following are the eight kinds of propositions re- 
 cognised by Hamilton, as described by Dr. Thomson. 
 
 Sign- Affirmative, 
 
 U All X is all Y. 
 I Some X is some Y. 
 A All X is some Y. 
 Y Some X is all K 
 
 Negative. Sign. 
 
 No X\s Y. E 
 
 Some X is not some K u> 
 No X is some Y. ry 
 
 Some X is no Y O
 
 ISO QUANTIFIED PREDICATE CHAP. 
 
 2. Indicate by the technical symbols the quan- 
 tity and the quality of the following propo- 
 sitions : 
 
 (1) All primary forces are attractive. 
 
 (2) All vital actions come under the law 
 
 of habit, and none but vital actions 
 do. 
 
 (3) The best part of every man's education 
 
 is that which he gives himself. 
 
 (4) Only ungulate animals have horns. 
 
 (5) Mere readers are very often the most 
 
 idle of human beings. 
 
 (6) Most water -breathing vegetables are 
 
 flowerless. [p.] 
 
 (1) Is clearly a universal affirmative (A). 
 
 (2) As regards its first part is also A ; but the exclusive 
 
 addition, ' None but vital actions do,' means that 
 ' all not vital actions do not.' The two parts 
 together yield a proposition in U, 'all vital actions 
 are all that come under the law of habit.' 
 
 (3) The 'best part,' being a superlative, is a singular 
 
 term, and so is the predicate 'that part which, etc.' 
 Hence the proposition is an identity in U. 
 
 (4) An exclusive proposition equivalent to ' all not un- 
 
 gulate animals have no horns,' which is the con- 
 trapositive of, and equivalent to, ' all horned 
 animals are ungulate.' 
 
 (5) Means that a great many mere readers are, etc., and 
 
 is in the form I. 
 
 (6) Is also a particular affirmative proposition.
 
 xviii QUESTIONS AND ANSWERS 151 
 
 3. Does not the proposition Y of Thomson imply 
 
 O, that is to say, does not ' some P is all Q ' 
 imply that ' some P is not Q ' ? 
 
 This seems very plausible, because if some P makes up 
 the whole of Q, there is, so to say, no room left in Q's 
 sphere for any more JPs, the remainder of which must there- 
 fore be not Q. This argument, however, overlooks the fact 
 that the 'some P' in question may possibly be the whole ol 
 P, so that there may be no remainder excluded from Q. 
 
 4. Is the proposition ' Some men are animals ' 
 
 true ? [E.] 
 
 The proposition is true or untrue materially according 
 to the sense we put upon this troublesome word some. If 
 we take it to mean ' one or more it may be all,' the pro- 
 position is true in fact, but of course states less than is 
 known to every one. 
 
 We must carefully distinguish between the strict and 
 necessary logical interpretation of ' some,' and that which 
 applies in colloquy. De Morgan says {Formal Logic, p. 4), 
 ' In common conversation the affirmation of a part is meant 
 to imply the denial of the remainder. Thus, by " some of 
 the apples are ripe," it is always intended to signify that 
 " some are not ripe." There is no difficulty in providing in 
 formal logic for this use of the word by stating explicitly 
 the two propositions which are colloquially merged into 
 one. Thus ' some of the apples are ripe ' is really I + O. 
 
 5. What results would follow if we were to in- 
 
 terpret ' some As are .Z?s' as implying that 
 ' some other As are not jBs ' ?
 
 152 QUANTIFIED PREDICATE CHAP. 
 
 The proposition ' some As are Us ' is in the form I, and 
 according to the table of opposition (p. 31) I is true if A is 
 true ; but A is the contradictory of O, which would be the 
 form of ' some other As are not jBs.' 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. 
 
 Briefly If A is true, I is true ; and if I implies O, then 
 A implies the truth of its own contradictory O. 
 
 Several logicians have come to grief over this troublesome 
 word, notably Sir W. Hamilton, who in holding that ' some ' 
 is formally exclusive of all and none, throws all logical 
 systems into confusion. Woolley commits the same great 
 mistake in saying (p. 77), 'In every particular proposition, 
 therefore, the affirmative and negative mutually imply each 
 other: if only some A is H, then some A is not B, and 
 vice versa? 
 
 6. Explain the precise meaning of the propo- 
 sition ' Some Xs are not some Ys ' (the 
 proposition w of Thomson). What is its 
 contradictory ? Give your opinion of its 
 importance. 
 
 This is one of the eight forms of proposition which 
 Hamilton, in pursuance of the thoroughgoing quantification 
 of the predicate, introduced into his system. Now, if 
 ' some V means ' any some Y,' that is to say, if the ' some ' 
 is undetermined and may be any where in the sphere of Y, 
 this proposition does not differ from ' some X is not any Y,' 
 which is the proposition O of the old Aristotelian Logic. 
 But if 'some Y' is a determinate part of the class Y, less 
 thnn the whole, then the proposition becomes a mere empty
 
 xviii QUESTIONS AND ANSWERS 153 
 
 truism ; for, however X and Y may be related, some part of 
 X will be different from some part of F. Thus all equi- 
 lateral triangles are all equiangular triangles, yet some equi- 
 lateral triangles are not some equiangular ones. If all John 
 Jones' sons are Rugby boys, yet some of John Jones' sons 
 are not some Rugby boys. We see that this proposition w 
 is consistent with all the other propositions of the system, 
 in all cases, as De Morgan remarks (Syllabus, p. 24), ' in 
 which either X or Y has two or more instances in existence : 
 its contrary is " X and Y are singular and identical ; there 
 is but one X, there is but one Y, and X is F." ' A system 
 which offers an assertion and denial which cannot be con- 
 tradicted in the same system carries its own condemnation 
 with it, as well observed by De Morgan. 
 
 Archbishop Thomson also rejected this form of propo- 
 sition-. He says : ' If I define the composition of common 
 salt by saying, "common salt is chloride of sodium," I 
 cannot prevent another saying that "some common salt is 
 not some chloride of sodium," because he may mean that 
 the common salt in this salt-cellar is not the chloride of 
 sodium in that. A judgment of this kind is spurious upon 
 two grounds : it denies nothing, because it does not prevent 
 any of the modes of affirmation ; it decides nothing, inas- 
 much as its truth is presupposed with reference to any pair 
 of conceptions whatever.' (Outline of the Laivs of Thought, 
 1860, 79, p. 137.) 
 
 SPALDING, pp. 83, 97-102, etc., symbolises the propo- 
 sition j, by \ O. 
 
 In an examination, candidates almost invariably say that 
 all Xs are Fs, or all Xs are all Fs, is the contradictory of 
 some Xs are not some Fs ; and De Morgan (1863, p. 4) 
 speaks of an unnamed logical author who spoiled his work 
 with a like blunder.
 
 154 QUANTIFIED PREDICATE CHAP. 
 
 The chief interest of this proposition o arises from its 
 important bearing upon the value of Hamilton's System of 
 Logic, and his position as a logician. Hamilton insisted 
 upon the thoroughgoing quantification of the predicate, which 
 means the recognition and employment of // the eight pro- 
 positions which the introduction of the quantified predicate 
 renders conceivable. Thus was the key-stone to be put 
 into the arch of the Aristotelic logic. But if, as Thomson 
 and De Morgan seem to me to have conclusively shown, 
 this proposition, w, is valueless and absurd, the key-stone 
 crumbles and the arch collapses. The same ruin does not 
 overtake De Morgan's system, because his eight propositions 
 are not all the same as those of Hamilton ; nor does it 
 affect in any appreciable degree the views of Thomson and 
 George Bentham, who did not insist upon the thoroughgoing 
 quantification of the predicate. 
 
 De Morgan has admirably expressed the inherent ambi- 
 guity of this word. He says (Fifth Memoir on the Syllogism, 
 1863, p. 4), '"He has got some apples" is very clear: 
 ask the meaning of "he has not got some apples," in a 
 company of educated men, and the apples will be those of 
 discord. Some will think that he may have one apple ; 
 some that he has no apple at all ; some that he has not got 
 some particular apples or species of apples.' 
 
 The subject of particular propositions may be pursued in 
 Spalding's Logic, 1857, p. 172, and elsewhere; Shedden's 
 Logic ; Hughling's Logic of Names, 1 869, p. 3 1 ; Thomson's 
 Outline, fifth edition, section 77 ; Hamilton's Lectures, vol. 
 iv. pp. 254, 279; Devey, 1854, pp. 90-94; De Morgan, 
 1863, Fifth Memoir. 
 
 Mr. A. J. Ellis is particularly exact in his treatment of 
 this question in his articles in the Educational Times, 
 1878.
 
 xvni QUESTIONS AND ANSWERS 155 
 
 7. Solly says (p. 73) 'If the premises are " some 
 
 B is A, some C is not B" the reason may 
 logically deduce that some C is not some A. 
 But this conclusion is not in one of the four 
 legitimate forms.' Is the argument valid in 
 the quantified syllogism, and if so, in what 
 mood ? 
 
 The propositions are as follow : 
 
 Some B is some A I 
 
 Some C is not (any E} O 
 Some C is not some A <o 
 
 The middle term is distributed once in the minor premise, 
 and, as both terms of w are particular, there can be no illicit 
 process. One premise is negative, and so is the conclusion. 
 No rule of the syllogism is broken, and the argument is 
 therefore valid. It appears as I O w in the sixth mood of 
 the first figure of Thomson's table. 
 
 8. Which of the following conjunctions of pro- 
 
 positions make valid syllogisms ? In the case 
 of those which you regard as invalid, give your 
 reasons for so treating them. 
 
 First Figure. Second Figure. Third Figure. 
 
 EYO UOo> A <o O 
 AEE ,/UO YEO 
 I U >, [c.] 
 
 The pseudo-mood A E E in the first figure gives illicit 
 process of the major term, because the conclusion E distri- 
 butes its predicate, and the major premise A does not. 
 The pseudo-mood I U rj draws a negative conclusion, 77,
 
 156 QUANTIFIED PREDICATE CHAP. 
 
 from two affirmative premises, but is by oversight given in 
 Thomson's Table of Modes, figure i, mode xii., second 
 negative form. It is an obvious misprint for I E ?/. (Out- 
 line of the Laws of Thought, section 103, 5th ed., p. 188.) 
 In the table as reprinted in the Elementary Lessons in Logic, 
 p. 1 88 (accidentally the same page as in Thomson !), the 
 error was corrected in the fifth and later editions. It was 
 pointed out to me by Mr. A. J. Ellis. 
 
 E Y O is valid in the first figure. 
 
 In the second figure U O w breaks no rule, but the con- 
 clusion instead of being w (some Xs are not some Zs), 
 might have been in the stronger form O (some Xs are not 
 any Zs). The moods U O O and U w w appear in 
 Thomson's table, column 4, though U O w does not. The 
 mood ?/ U O is valid. 
 
 In the third figure A w O is subject to illicit process of 
 the major term, since the conclusion O distributes its predi- 
 cate, which is the undistributed predicate of A in the major 
 premise. Y E O is not subject to the same objection, 
 because Y distributes its predicate ; but, in this last case, 
 the conclusion is weakened, and might have been E ; hence 
 Y E E appears in Thomson's table, tenth mood of third 
 figure, and Y E O does not appear. 
 
 9. In what mood is the following argument : 
 Aliquod trilaterum est aequiangulum ; omne 
 triangulum est (omne) trilaterum ; ergo, ali- 
 quod triangulum est aequiangulum ? 
 
 The first premise, ' some trilateral figure is an equiangular 
 figure,' is plainly a proposition in I ; the second, ' all 
 triangles are all trilateral figures,' is as plainly a doubly 
 universal proposition in U ; the conclusion, ' some triangular
 
 xvin QUESTIONS AND ANSWERS 157 
 
 figure is equiangular,' is in I. The middle term, trilaterum, 
 is distributed in the minor premise, though not in the major ; 
 there is no illicit process, nor other breach of the syllogistic 
 rules, so that the argument is a valid syllogism in the mood 
 I U I of the first figure. It appears as the twelfth mood 
 in the first column of Thomson's table of moods. See, 
 however, Baynes' New Analytic, 1850, pp. 126-7, whence 
 this example is taken. 
 
 10. Does the following argument fall into any 
 valid mood of the syllogism ? 
 Some man is all lawyer ; 
 Any lawyer is not any stone ; 
 
 therefore, Some man (i.e. lawyer) is not any stone 
 (i.e. all the rest are stone). 
 
 This example is taken by De Morgan (1863, p. 10) as a 
 case of Hamilton's mood IV. />, as stated in his Lectures on 
 Logic, vol. iv. p. 287, thus, 'A term parti -totally co- 
 inclusive, and a term totally co-exclusive, of a third, are 
 parti-totally co-exclusive of each other.' It was called by 
 De Morgan the ' Gorgon Syllogism,' alluding, I presume, to 
 the petrifying effect it produces upon all mankind who are 
 not lawyers. It is plainly in the mood E Y w, and though 
 it does not appear in Thomson's table, may be considered 
 a weakened form of E Y O, the seventh negative mood 
 of the first figure. The point of the matter, however, is that 
 Hamilton, in his later writings, proposed to depart from 
 the Aristotelian sense of the mark of particular quantity 
 some. As stated in his Lectures, vol. iv. p. 281, the view 
 which he wished to introduce is that some should mean 
 ' some at most, some only, some not all.' But, if we 
 apply this meaning of 'some' to the conclusion of the
 
 158 QUANTIFIED PREDICATE CHAP, xvm 
 
 Gorgon Syllogism, it produces the ridiculous result that, 
 though lawyers are not stone, all the rest of mankind are stone. 
 De Morgan is unquestionably correct, and this Gorgon 
 Syllogism brings to ruin Hamilton's ' long adequately tested 
 and matured ' system. 
 
 The particulars of the discussion between De Morgan 
 and Professor Spencer Baynes about this Gorgon Syllogism, 
 and kindred matters, may be found in the Athenceum of 
 1 86 1 and 1862 and elsewhere. 
 
 It is curious that De Morgan states the Gorgon Syllo- 
 gism differently in the Athenaum of 2nd November 1861, 
 p. 582, and in his Fifth Memoir on the Syllogism, p. 10; 
 but the difference is not material to the final issue. 
 
 II. ' The month of May has no " R " in its name ; 
 nor has June, July, or August : all the hottest 
 months are May, June, July, and August : 
 therefore, all the hottest months are without 
 an " R " in their names.' 
 
 This is Whately's example No. 117, and as he refers the 
 student to Book IV., Chap. I., i, which treats of induction, 
 he evidently regards it as an Inductive Syllogism. It would 
 have been referred by Hamilton to the Thomsonian mood 
 E U E, the minor premise being treated as a doubly uni- 
 versal proposition. There can be no doubt, however, that 
 the minor is really disjunctive, thus : A hottest month is 
 either May, or June, or July, or August. The major is a 
 compound sentence, comprising four separate propositions, 
 ' May has no R in its name,' ' June has no R,' etc. (See 
 Elementary Lessons, Lesson XXV., p. 215.)
 
 CHAPTER XIX 
 
 EXERCISES ON THE QUANTIFICATION OF THE PREDICATE 
 
 i. EXPRESS carefully, in full logical form, with quantified 
 subjects and predicates, the following propositions ; assign 
 the Thomsonian symbol in each case : 
 
 (1) Thoughts tending to ambition, they do plot unlikely 
 
 wonders. 
 
 (2) Fools are more hard to conquer than persuade. 
 
 (3) Heaven has to all allotted, soon or late, 
 
 Some happy revolution of their fate. 
 
 (4) Justice is expediency. 
 
 (5) This is certainly the man I saw yesterday. 
 
 (6) Man is the only animal with ears that cannot move 
 
 them. 
 
 (7) Wisdom is the habitual employment of a patient and 
 
 comprehensive understandingin combining various 
 and remote means to promote the happiness of 
 mankind. 
 
 (8) It is among plants that we must place all the Dia- 
 
 tomaceae. 
 
 (9) When the age is in the wit is out. 
 
 (10) Every man at forty is either a fool or a physician, 
 (i i) Some men at forty are neither fools nor physicians. 
 (12) Some men at forty are both fools and physicians.
 
 160 QUANTIFIED PREDICATE CHAP. 
 
 (13) L'Etat c'est moi, as Louis the Fourteenth used to say. 
 
 (14) There are no coins excepting those made of metal, 
 
 if we overlook a few composed of porcelain, glass, 
 or leather. 
 
 (15) Antisthenes said 8eiv KracrOai vovv rj f^po^ov. 
 
 (16) All animals which have a language have a voice, but 
 
 not all which have a voice have a language. 
 
 (17) The elephant alone among mammals has a pro- 
 
 boscis. 
 
 (18) Prudence is that virtue by which we discern what is 
 
 proper to be done under the various circumstances 
 of time and place. 
 
 (19) Whatever is, is right. 
 
 (20) There are arguments and arguments. 
 
 (21) A dispute is an oral controversy, and a controversy 
 
 is a written dispute. 
 
 (22) There beth workys of actyf lyf othere gostly othere 
 
 bodily. 
 
 (23) The only Roman who gave us a summary of 
 
 Aristotle was the only Roman who gave us a 
 summary of Euclid. 
 
 (24) Zenobia declared that the last moment of her reign 
 
 and of her life should be the same. 
 
 (25) As it asketh some knowledge to demand a question 
 
 not impertinent, so it requireth some sense to 
 make a wish not absurd. 
 
 (26) Mankind consists of dark men and fair men. 
 
 (27) To say that Mr. Raffles was excited was only 
 
 another way of saying that it was evening. 
 
 (28) Though all well educated men are not discoverers, 
 
 all discoverers are well educated men. 
 
 (29) No man is esteemed for gay garments but by fools 
 
 and women.
 
 xix EXERCISES i6r 
 
 (30) Quand celui qui ecoute n'entend rien, et quand celui 
 
 qui parle n'entend plus, c'est la metaphysique. 
 
 (31) Friendship finds men equal or makes them so. 
 
 (32) I can fly or I can run. 
 
 (33) A man is an ill husband of his honour that entereth 
 
 into an action, the failing wherein may disgrace 
 him more than the carrying of it through can 
 honour him. 
 
 (34) Scribendi recte sapere est principium et fons. 
 
 (35) Tools are only simple machines, and machines are 
 
 only complicated tools. 
 
 (36) The wise man knows the fool, but the fool knows 
 
 not the wise man. 
 
 (37) It is scandalous that he who sweetens his drink by 
 
 the gift of the bees, should by vice embitter 
 Reason, the gift of the Gods. 
 
 (38) A and B and C and Z>, etc., etc., wear black coats 
 
 on Sundays ; in fact every man I know does so. 
 
 (39) All the Apostles were Jews, because this is true of 
 
 Peter, James, John, and every other Apostle. 
 
 (40) A dose of arsenic is given to a living healthy dog. 
 
 Soon after the dog dies. Arsenic is therefore a 
 poison. 
 
 2. How can a chain of reasoning, founded on circum- 
 stantial evidence, be represented in syllogistic form ? [E.] 
 
 3. Having special regard to the logical sense of ' some,' 
 what do you think of the validity of the following argument 
 (Thomson's Syllogistic Mood A E 17) ? 
 
 All Kis some X ; 
 No Z is any Y; 
 Therefore, No Z is some X. 
 
 M
 
 162 QUANTIFIED PREDICATE CHAP. 
 
 4. ' We have been assured that " all X is some Y" is 
 contradicted by "all Y is some X" a proposition which 
 cannot be made good except by some being declared not 
 all? (De Morgan, Third Memoir, 1858, p. 24.) Investi- 
 gate this point. 
 
 5. Take 'stone' and 'solid' as subject and predicate, 
 and convince yourself that the proposition in to, 'some 
 stone is not some solid/ cannot be contradicted by any 
 propositions of the forms U, A, I, Y, E, O, ?/, having the 
 same subject and predicate. 
 
 6. Write out the various judgments, including U and Y, 
 which are logically opposed to the judgment, ' No puns are 
 admissible.' State in the case of each judgment thus formed 
 what is the kind of opposition in which it stands to the 
 original judgment, and also the kind of opposition between 
 each pair of the new judgments. [c.] 
 
 7. ' The judgment, " No birds are some animals," is never 
 actually made because it has the semblance only, and not 
 the power of a denial.' Examine this statement. [P.] 
 
 8. Draw inferences from the following : 
 
 If Sir Thomas was imbecile, then Oliver was right ; and 
 unless Sir Thomas was imbecile, Oliver was not 
 wrong. [P.] 
 
 9. Examine the following arguments ; in those which are 
 false point out the nature and name of the fallacy ; arrange 
 those which are valid syllogisms in the usual form, and give 
 the symbolic description of the mood. 
 
 (i) All the householders in the kingdom, except women, 
 are legally electors, and all the male householders 
 are precisely those men who pay poor-rates ; it 
 follows that all men who pay poor-rates are 
 electors.
 
 xix EXERCISES 163 
 
 (2) All the times when the moon comes between the 
 
 earth and the sun, are the sole cases of a solar 
 eclipse ; the nth of February is not such a time ; 
 therefore, the nth of February will exhibit no 
 eclipse of the sun. [THOMSON.] 
 
 (3) All men are mortals, and all mortals are all those 
 
 who are sure to die ; therefore, all men are all 
 those who are sure to die. 
 
 (4) The Claimant is unquestionably Arthur Orton : for 
 
 he is Castro, who is the same person as Arthur 
 Orton. 
 
 10. Which of the following moods are legitimate, and 
 in what figures : E Y O, Y A A, Y A Y, I Y I, Y Y Y, 
 AEE? [M.] 
 
 11. Examine the validity of the following moods : 
 
 Figure I. Figure II. Figure III. 
 
 UAU AAA YEE 
 
 YOO AYY OYO [c.] 
 
 12. Exemplify any of the following moods, and deter- 
 mine in how many figures each is valid : U U U, I U I, 
 Y U Y, r) U 17, a, U a,.
 
 CHAPTER XX 
 
 EXAMPLES OF ARGUMENTS AND FALLACIES 
 
 THIS chapter contains a large collection of examples of 
 Arguments and Fallacies collected from many sources. They 
 form additional illustrations and exercises to supplement 
 what are given in the previous chapters. The student is to 
 determine in the case of each example whether it contains 
 a valid or fallacious argument. In the former case he is to 
 throw the example into a regular form, and assign the 
 technical description of that form, whether a mood of the 
 categorical syllogism, or of the hypothetical or disjunctive 
 syllogism, etc. In some examples two or more syllogisms, 
 or two or more different forms of reasoning, will be com- 
 plicated together. They must of course be analysed and 
 exhibited separately. 
 
 When the existence of fallacy is suspected, the student 
 must endeavour to reduce this to a distinct paralogism or 
 breach of the syllogistic rules, exhibiting the pseudo-mood 
 or pseudo-form of reasoning. In many cases, however, the 
 fallacy may be of the kinds described in the Aristotelian 
 text -books as Semi -logical or Material. These fallacies 
 have been explained in the Elementary Lessons (Lessons 
 XX. and XXL), but for convenience of reference a simple 
 list of the kinds of fallacies is given below. It has not 
 been found practicable to undertake in this book a full
 
 CHAP, xx ARGUMENTS AND FALLACIES 165 
 
 exemplification of the subject of Fallacies. The student is 
 therefore referred to the Elementary Lessons named, or to 
 any of the following writings on the subject : 
 
 De Morgan's Formal Logic, Chapter XIII., as amusing 
 as it is accurate and instructive ; Whately's Logic, Book III., 
 perhaps the best and most interesting part of this celebrated 
 text-book ; Edward Poste's edition of Aristotle on Fallacies. 
 
 Paralogisms 
 
 1. Four terms. Breach of Rule I. 
 
 2. Undistributed Middle. Breach of Rule III. 
 
 3. Illicit Process of Major or Minor Term. Breach of 
 
 Rule IV. 
 
 4. Negative premises. Breach of Rule V. 
 
 5. Negative Conclusion from affirmative premises, and 
 
 vice versa. Breach of Rule VI. 
 
 Breaches of Rules VII. and VIII. can be resolved into 
 one or other of the above. 
 
 Semi-logical Fallacies ^ Material Fallacies 
 ,A V* 
 
 1. Equivocation. i. Accident. 
 
 2. Amphibology. 2. Converse Fallacy of Accident. 
 
 3. Composition. 3. Irrelevant Conclusion. 
 
 4. Division. 4. Petitio Principii. 
 
 5. Accent. 5. Non Sequitur. 
 
 6. Figure of Speech. 6. False Cause. 
 
 7. Many Questions. 
 
 1. France, having a warm climate, is a wine-producing 
 country. tU,JjjJ4, **&&*. [E.] 
 
 2. Livy describes prodigies in his history ; therefore he 
 is never to be believed. (^ -L- [ E -J
 
 166 ARGUMENTS AND FALLACIES CHAP. 
 
 ~ 
 
 3. All the metals conduct heat and electricity; for 
 iron, lead, and copper do so, and they are (all) metals. 
 
 -*> a [ E -l 
 
 4. A charitable man has no merit in relieving distress, 
 because he merely does what is pleasing to himself. [E.] 
 
 5. What is the result of all this teaching? Every day 
 you hear of a fraud or forgery, by some one who might 
 have led an innocent life, if he had never learned to read 
 or write. [E.] 
 
 6. The use of ardent spirits should be prohibited by law, 
 seeing that it causes misery and crime, which it is one of 
 the chief ends of law to prevent. [E.] 
 
 7. Pious men only are fit to be ministers of religion ; 
 some ignorant men are pious ; therefore ministers of 
 religion may be ignorant men. [L.] 
 
 No punishment should be allowed for the sake of the 
 
 good that may come of it ; for all punishment is an evil, 
 and we are not justified in doing evil that good may come 
 of it. [E.] 
 
 9. We know that God exists because the Bible tells us 
 so ; and we know that whatever the Bible affirms must be 
 true because it is of Divine origin. [E.] 
 
 10. The end of punishment is either the protection of 
 society or the reformation of the individual. Capital 
 punishment ought therefore to be abolished. It does not 
 in fact prevent crimes of violence, and so fails to protect 
 society, while on the other alternative it is absurd. [E.] 
 
 11. The glass is falling; therefore we may look for 
 rain. [E.] 
 
 12. This is a dangerous doctrine, for we find it up- 
 held by men who avow their disbelief in Revelation. 
 
 13. If there is a demand for education, compulsion is 
 unnecessary. [E.]
 
 xx EXAMPLES 167 
 
 14. Actions that benefit mankind are virtuous ; therefore 
 it is a virtuous action to till the ground. 
 
 15. Slavery is a natural institution ; therefore it is wrong 
 to abolish it. 
 
 1 6. No fool is fit for high place; John is no fool; 
 therefore John is fit for high place. [E.] 
 
 17. He is not a Mahometan, for no Mahometan holds 
 these opinions. [E.] 
 
 1 8. Mind is active ; matter is not mind ; therefore 
 matter is not active. I VLicJi* r~*al\<r\ - [E.] 
 
 19. He must be a Mahometan, for all Mahometans 
 hold these opinions. [E.] 
 
 20. If we are to believe philosophers, knowledge ' is 
 impossible, for one set of them tell us that we can know 
 nothing of matter, and another that we can know nothing 
 of mind. [o.] 
 
 21. Old^age^is wiser than youth ; therefore we must be 
 guided by the decisions of our ,aj3sjr 4 , [o.] 
 
 22. Political assassins ought not to be punished, for 
 they act according to their consciences. [o.] 
 
 23. If education is popular, compulsion is unnecessary; 
 if unpopular, compulsion will not be tolerated. [o.] 
 
 24. Nations are justified in revolting when badly 
 governed, for every people has a right to good govern- 
 ment. [E.] 
 
 25. These two figures are equal to the same figure, and 
 therefore to each other. 
 
 26. Opium produces Asleep, for it possesses a soporific 
 virtue. ^ M o'*^* W<^;^ _ [ K ] 
 
 27. Wealth is in proportion to value, value to efforts, 
 efforts to obstacles : ergo, wealth is in proportion to 
 obstacles.
 
 168 ARGUMENTS AND FALLACIES CHAP. 
 
 28. When Croesus has the Halys crossed, a mighty army 
 will be lost. 
 
 29. A manor cannot begin at this day, because a court- 
 baron cannot now be founded. 
 
 30. Poeta nascitur, non fit; how absurd it is then to 
 teach the making of Latin verses ! 
 
 31. Aio te Aeacida, Romanes vincere posse. 
 
 32. Every rule has exceptions ; this is a rule, and there- 
 fore has exceptions ; therefore there are some rules that 
 have no exceptions. [E.] 
 
 33. All that perceives is mind ; the existence of objects 
 consists in being perceived ; therefore the existence of 
 objects necessarily depends on mind. [E.] 
 
 34. Some objects of great beauty merely please the eye ; 
 for instance, many flowers of great beauty, and accordingly 
 they answer no purpose but to gratify the sight. [H.] 
 
 35. A miracle is a violation of the laws of nature ; and, 
 as a firm and unalterable experience has established those 
 laws, the proof against a miracle, from the very nature of 
 the fact, is as entire as any argument from experience can 
 possibly be. [E.] 
 
 36. The imagination and affections have a close union 
 together. The vivacity of the former gives force to the 
 latter. Hence the prospect of any pleasure with which we 
 are acquainted affects us more than any other pleasure 
 which we may own superior, but of whose nature we are 
 wholly ignorant. [E.] 
 
 37. Common salt consists of a metal and a metalloid, 
 for it consists of sodium and chlorine, of which one is a 
 metal, and the other a metalloid. [E.] 
 
 38. 'The truth is, that luxury produces much good. A 
 man gives half-a-guinea for a dish of green peas ; how much 
 gardening does this occasion ? ' (Dr. Johnson.) [o.]
 
 xx EXAMPLES 169 
 
 39. Nothing can be produced ; for what exists cannot be 
 produced, as it is already in existence, and what does not 
 exist cannot be produced, as, since it is not in existence, 
 nothing can happen to it. [E.] 
 
 40. The earth's position must be fixed, if the fixed stars 
 are seen at all times in the same situations : now the fixed 
 stars are not seen at all times in the same situations; 
 therefore the earth's position is not fixed. [E.] 
 
 41. Protective laws should be abolished, for they are 
 injurious if they produce scarcity, and they are useless if 
 they do not. [E.] 
 
 42. All who think this man innocent think he should 
 not be punished ; you think he should not be punished, 
 therefore you think him innocent. [E.] 
 
 43. If we are disposed to credit all that is told us, we 
 must believe in the existence, not only of one, but of 
 two or three Napoleon Buonapartes ; if we admit nothing 
 but what is well authenticated, we shall be compelled to 
 doubt the existence of any. How, then, can we be called 
 upon to believe in the one Napoleon Buonaparte of 
 history ? [o.] 
 
 44. We cannot know what is false, for knowledge cannot 
 be deceptive, and what is false is deceptive. [E.] 
 
 45. A necessary being cannot be the effect of any cause ; 
 for, if it were, its existence would depend upon the existence 
 of its cause, and therefore would not be necessary. [E.] 
 
 46. The table we see seems to diminish as we move 
 from it ; but the real table suffers no change : it was not, 
 therefore, the table itself, but only its image, that was 
 present to the mind. 
 
 47. The existence of sensations consists in being per- 
 ceived : all objects are really collections of sensations ; 
 therefore their existence consists in being perceived. [E.]
 
 170 ARGUMENTS AND FALLACIES CHAP. 
 
 48. If the earth were of equal density throughout, it 
 would be about 2^ times as dense as water : but it is about 
 5^ times as dense ; therefore the earth must be of unequal 
 density. 
 
 49. Whatever is conditioned must depend on some cause 
 external to itself: this world is conditioned by time and 
 space ; therefore this world depends upon some cause 
 external to itself. [E.] 
 
 50. It sometimes happens that an electrical current is 
 excited, where none but magnetic forces are directly called 
 into play ; for such a current, in certain cases, is excited 
 in an electric non-conductor by moving a magnet to or 
 away from it. [R.] 
 
 51. The Quaker asserts that if men were true Christians, 
 and acted upon their religious principles, there would be no 
 need of armies. Hence he draws the conclusion that a 
 military force is useless, and being useless, pernicious. 
 
 52. Detention implies at least possession ; for detention 
 is natural possession. 
 
 53. Nothing can be conceived without extension : what 
 is extended must have parts ; and what has parts may be 
 destroyed. [o.] 
 
 54. 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. [o.] 
 
 55. By the law of nature as soon as Adam was created 
 he was governor of mankind, for by right of nature it was 
 due to Adam to be governor of his posterity. [o.] 
 
 56. When men are pure, laws are useless; when men 
 are corrupt, laws are broken. 
 
 57. There are many arguments which we recognise as
 
 xx EXAMPLES 171 
 
 valid which it is impossible to express in a syllogistic form : 
 therefore the syllogism is valueless as a test of truth, [o.] 
 
 58. No man can be a law to himself; for law implies a 
 superior who gives the law and an inferior who obeys it ; 
 but the same person cannot be both ruler and subject, [o.] 
 
 59. It is injustice to the intellect of women to refuse 
 them the suffrage ; for the reigns of many queens, as our 
 own Elizabeth or Anne, have been famous for literary 
 
 60. To allow every man unbounded freedom of speech 
 must be advantageous to the State, for it is highly conducive 
 to the interests of the community that each individual 
 should enjoy an unlimited liberty of expressing his senti- 
 ments, [o.] 
 
 6 1. Your sorrow is fruitless, and will not change the 
 course of destiny. Very true, and for that very reason I 
 am sorry. [o.] 
 
 62. Because some individuals have in their very child- 
 hood advanced beyond the youthful giddiness and debility 
 of reason, it only needs a proper system of education to 
 make other young people wise beyond their years. [o.] 
 
 63. Haste makes Waste, and Waste make Want ; there- 
 fore a man never loses by delay. [o.] 
 
 64. If peace at any price is desirable, war is an evil ; and 
 as war is confessedly an evil, peace at any price is de- 
 sirable, [o.] 
 
 65. The two propositions, 'Aristotle is living,' and, 
 ' Aristotle is dead,' are both intelligible propositions ; they 
 are both of them true or both of them false, because all 
 intelligible propositions must be either true or false. [E.] 
 
 66. No form of democracy is subject to violent revolu- 
 tions, because it never excludes the mass of the people 
 from political power. [E.]
 
 172 ARGUMENTS AND FALLACIES CHAP. 
 
 67. The student of History is compelled to admit the 
 Law of Progress, for he finds that Society has never stood 
 still. [E.] 
 
 68. It is fated that I shall or that I shall not recover, 
 in either of which cases the employment of a physician 
 is useless, and therefore inexpedient. [E.] 
 
 69. The assertion that men much occupied in public 
 affairs cannot have time for literary occupations is disproved 
 by such instances as Julius Caesar, Alfred, Lord Bacon, Sir 
 G. C. Lewis, the Earl of Derby, Mr. Gladstone, and the 
 late Emperor of the French. 
 
 70. Whatever had a beginning in time has limits in 
 space ; the universe has no beginning in time : therefore 
 the universe has no limits in space. 
 
 71. The mollusc is an aggregate of the second order; 
 for there is no sign of a multiplicity of like parts in its 
 embryo. 
 
 72. The farmers will not pay in rent more than the net 
 produce of their farms ; for no trading class will continue a 
 losing business. [L.] 
 
 73. The knowledge of things is more improving than the 
 knowledge of words. The study of Physics must therefore 
 be more improving than the study of Languages. [E.] 
 
 74. The moral world is far from being so well governed 
 as the material ; for the former, although it has its laws, 
 which are invariable, does not observe these laws so con- 
 stantly as the latter. [p.] 
 
 75. England has a gold coinage and is a very wealthy 
 country ; therefore, it may be inferred that other countries 
 having a gold coinage must be wealthy. 
 
 76. Most parents are the best judges of the age at which 
 their children should be sent to school, and as it is un-
 
 xx EXAMPLES 173 
 
 desirable to interfere with those who are the best judges 
 of their children's interest, it follows that parents should 
 not be compelled to send their children to school. 
 
 77. Among the bodies which do not move in elliptic 
 orbits are some of the comets ; but all bodies which do 
 move in elliptic orbits return periodically ; hence, some 
 bodies which return periodically cannot be comets. 
 
 78. Some rate-payers are clearly not fit for their duties ; 
 for all male rate-payers are electors, and some electors who 
 accept bribes are clearly unfit for their duty of electing 
 representatives. 
 
 79. Whatever is done skilfully appears to be done with 
 ease ; and art, when it is once matured to habit, vanishes 
 from observation. We are therefore more powerfully ex- 
 cited to emulation by those who have attained the highest 
 degree of excellence, and whom therefore we can with the 
 least reason hope to excel. [L.] 
 
 80. It is absurd to maintain that when we cannot avoid 
 thinking or conceiving a thing, it must be true ; for some 
 persons cannot be in darkness without thinking of ghosts, 
 in which they do not believe. [R.] 
 
 8 1. 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? [c.] 
 
 82. The time is past in which the transmission of news 
 can be measured by the speed of animals or even of steam ; 
 for the telegraph is not approached by either. 
 
 [DE MORGAN.] 
 
 83. We enjoy a greater degree of political liberty than 
 any civilised people on earth, and can therefore have no 
 excuse for a seditious disposition. 
 
 84. Those only who understand other languages are 
 competent to treat correctly of the principles of their own ;
 
 174 ARGUMENTS AND FALLACIES CHAP. 
 
 since such a competency requires a philosophical view of 
 the nature of language in general. [L.] 
 
 85. If matter must be merely phenomenal, I must be 
 so too. [E.] 
 
 86. A miracle is incredible, because it contradicts the 
 laws of nature. [E.] 
 
 87. There are no practical principles wherein all men 
 agree, and therefore none which are innate. [E.] 
 
 88. Potash contains a metal ; for all alkalies contain a 
 metal, and potash is an alkali. [E.] 
 
 89. Quench not hope ; for when hope dies, all dies. 
 
 90. That is too bad : you have the impudence to say 
 you are a materialist, while I know that you are a dancing 
 master. 
 
 91. Blood is a colour ; for it is red, and red is a colour. 
 
 92. Every incident in this story is very natural and 
 probable ; therefore the story itself is natural and probable, 
 
 93. Dolor, si longus, levis ; si gravis, brevis : ergo, 
 omnino fortiter sustinendus. [L.] 
 
 94. 'Whether we live, we live unto the Lord; and 
 whether we die, we die unto the Lord : whether we live 
 therefore, or die, we are the Lord's.' [.Rom. xiv. 8.] 
 
 95. Quand on n'a point d'amis, on n'est pas heureux ; 
 les hommes faux et trompeurs n'ont point d'amis ; ainsi 
 les hommes faux et trompeurs ne sont pas heureux. 
 
 96. Philippi was the city where the first Christian Church 
 in Europe was founded : it was also the place where the 
 republican army of Rome under Brutus and Cassius was 
 finally defeated ; hence the republican army was finally 
 defeated at the city where the first Christian Church in 
 Europe was founded. 
 
 97. Switzerland is a republic, and, you will grant, a more 
 stable power is not to be found ; nor, again, is any political
 
 xx EXAMPLES .175 
 
 society more settled than that of the United States. Surely, 
 then, republican France can be in no danger of revolution. 
 
 [R.] 
 
 98. Expand into a syllogism, as briefly as you can, the 
 argument contained in the dialogue of Shakespeare's King 
 Henry VI.* Part III. Act i. Scene i, between the words, 
 'my title's good,' etc., and, 'succeed and reign.' [o.] 
 
 99. Quoniam deos beatissimos esse constat ; beatus 
 autem sine virtute nemo potest : nee virtus sine ratione 
 constare : nee ratio usquam inesse nisi in hominis figura ; 
 hominis esse specie deos confitendum est. [H.] 
 
 100. Without order there is no living in public society, 
 because the want thereof is the mother of confusion, where- 
 upon division of necessity followeth ; and out of division, 
 destruction. [HOOKER, Ecclesiastical Polity, v. 8, s. i.] 
 
 i o i . Whatever is contradictory to universal and invariable 
 experience is antecedently incredible ; and as that sequence 
 of facts which is called the order of nature is established, 
 and in accordance with universal experience, miracles or 
 alleged violations of that order are antecedently improbable. 
 
 M 
 
 102. Justice is the profit of others; therefore it is un- 
 profitable to the just man to be just. [o.] 
 
 103. In trade both buyer and seller profit : in the home 
 trade both these profits remain in the country ; in the 
 foreign trade one profit goes to the foreign trader ; therefore 
 the same population will be more profitably employed in 
 the home than in the foreign trade. [o.] 
 
 104. Whatever brings in money enriches. Hence the 
 value of any branch of trade, or of the trade of the country 
 altogether, consists in the balance of money it brings in ; 
 and any trade which carries more money out of the country 
 than it draws into it is a losing trade : and therefore, money
 
 176 ARGUMENTS AND FALLACIES CHAP. 
 
 should be attracted into the country, and kept there by 
 bounties and prohibitions. 
 
 105. Distinction may be reasonably expected, because 
 what is not uncommon may be reasonably expected, and 
 distinction is not uncommon. [E.] 
 
 106. 'Neither am I moved with envy; for* if you are 
 equal to, or less than myself, I have no cause for it ; and if 
 you be greater, I ought to endeavour to equal you, and not 
 to speak evil of you.' [L.] 
 
 107. Great men have been derided, and I am derided : 
 which proves that my system ought to be adopted. 
 
 [DE MORGAN, Paradoxes, p. 387.] 
 
 1 08. Preventive measures are always invidious, for when 
 most successful the necessity for them is the least apparent. 
 
 109. Treason never prospers : What's the reason ? Why, 
 when it prospers, none dare call it Treason. 
 
 no. Neque quies gentium sine armis, neque anna sine 
 stipendiis ; neque stipendia sine tributis habere queunt. 
 
 [TACITUS, Hist. Lib. iv. cap. 74.] 
 Men are not brutes; brutes are irrational: all 
 irrational beings are irresponsible ; therefore, men are not 
 free from responsibility. [H.] 
 
 112. The best of all taxes are taxes on consumption and 
 taxes on the transfer of property : now all the latter and 
 many of the former are levied by stamps ; stamp duties 
 therefore are good taxes, and taxes on justice are all stamp 
 duties ; therefore taxes on justice are good taxes. (See 
 BENTHAM'S Protest against Law Taxes, second edition, 
 1816, pp. 53, 54.) 
 
 113. Dr. Johnson remarked that 'a man who sold a 
 penknife was not necessarily an ironmonger.' What is the 
 name and nature of the logical fallacy against which this 
 remark was directed ?
 
 xx EXAMPLES 17? 
 
 114. When Columbus made the egg stand on end by 
 breaking it, what fallacy may he be said to have committed ? 
 
 115. 'Either all things are ordered by an intelligent 
 Being who makes the world but one family (and if so, why 
 should a part, or single member complain of that which is 
 designed for the benefit of the whole ?) ; or else we are 
 under the misrule of atonies, and confusion. Now, take the 
 case which way you please, there's either no reason or no 
 remedy for complaint ; and therefore it is to no purpose to 
 be uneasie.' [Marcus Antoninus' Meditations, ix. 40.] 
 
 1 1 6. Silk is dearer than wool, and wool than cotton ; 
 therefore silk is dearer than cotton. 
 
 117. One napoleon is worth twenty francs, and twenty 
 francs are worth about sixteen shillings ; therefore one 
 napoleon is worth about sixteen shillings. 
 
 1 1 8. The Prince of Wales is the eldest son of the reign- 
 ing sovereign ; and the eldest son of the reigning sovereign, 
 if there be such, is the heir to the throne ; therefore the 
 Prince of Wales is heir to the throne. 
 
 119. It is a mistake to suppose that the rents paid to 
 landlords are a burden on the public, since corn would not 
 be more plentiful or cheaper if they were abolished. 
 
 120. The Greeks have little respect for the petty honesty 
 of small tradesmen ; we do not greatly admire the wiles of 
 Ulysses ; therefore any common inward standard of morals 
 is impossible. 
 
 121. Dissent always weakens religion in the people; for 
 it sets itself in opposition to the National Church. 
 
 122.' We are not inclined to ascribe much practical value 
 to that analysis of the inductive method which Bacon has 
 given in the second book of the Novum Organum. It is 
 indeed an elaborate and correct analysis. But it is an 
 analysis of that which we are all doing from morning to
 
 iy8 ARGUMENTS AND FALLACIES CHAP, xx 
 
 night, and which we continue to do even in our dreams.' 
 MACAULAY, Essay on Bacon. [E.] 
 
 123. The Claimant has undoubtedly many peculiarities 
 of gait and manner which were characteristic of the missing 
 baronet. Are not these therefore proofs of identity equi- 
 valent to the evidence of imposture afforded by the absence 
 of tattoo-marks which the genuine man is proved to have 
 possessed ? 
 
 124. Even if it could be shown that animals perform 
 certain actions which men could only perform by the aid of 
 reason, it would by no means necessarily follow that animals 
 perform them by its aid. [c.] 
 
 125. If there's neither Mind nor Matter, 
 
 Mill's existence, too, we shatter : 
 If you still believe in Mill, 
 
 Believe as well in Mind and Matter. [E.] 
 126. If we accept Aristotle's testimony, we may infer 
 that Anaximander was not one of the Ionian philosophers 
 that accepted as the One material principle a mean term 
 between Water and Air ; for, in the Physics, we read that 
 he held the substances in nature to have been produced 
 from the primordial principle by a process of secretion and 
 not by a process of condensation and rarefaction ; while in 
 the De Coelo it is stated that other mode of production 
 than the last-named was not put forward by any who 
 adopted such a mean term for their principle. What 
 syllogistic form (figure and mood) does this inference most 
 naturally assume ? [R.]
 
 CHAPTER XXI 
 
 ELEMENTS OF EQUATIONAL LOGIC 
 
 i. THE symbols employed in this system are the follow- 
 ing: 
 
 A, B, C, or other capital letters, signify qualities, or 
 groups of qualities, forming the common part, or intensive 
 meaning, of terms, or names of objects and classes of 
 objects. 
 
 a, b, c, or other small italic letters, are the eorresponding 
 negative terms ; thus a signifies the absence of one or more 
 of the qualities signified by A. This notation for negatives 
 was proposed by De Morgan (Formal Logic, p. 38). The 
 mark = is the sign of Identity of Meaning of the terms 
 between which it stands ; thus A = B indicates that the 
 qualities signified by A are identical with the qualities 
 signified by B. 
 
 The sign -|- signifies unexdusive alternation, including 
 the ordinary meanings of both the conjunctions or and and. 
 Thus A -|- B means the qualities of A or those of B, or 
 those of both A and B, if they happen to coincide. 
 
 Juxtaposition of two letters forms a term whose meaning 
 is the sum of the qualities signified by the two letters : 
 thus AB means a union of the qualities of A and B.
 
 i8o EQUATIONAL LOGIC CHAP. 
 
 2. The Laws of Combination of these symbols are as 
 follow : 
 
 The Law of Commutation. AB = BA : that is to say, 
 the sum of qualities of A and B is evidently the same as 
 the sum of qualities of B and A. The way of arriving at 
 the sum may be different, but the result is identical. 
 
 The Law of Simplicity. AA = A : if we have the same 
 qualities twice over we get the same as if we named them 
 once. 
 
 The Law of Unity. A-|-A = A: the qualities of A 
 
 - the qualities of A are simply the qualities of A. 
 or 
 
 The Law of Distribution. A(B -|- C) = AB -|- AC. 
 
 The qualities of A with those of B those of C are the 
 
 or 
 
 same as those of AB those of AC. 
 
 or. 
 
 The Law of Indifferent Order. B-|-C = C-|-B, which 
 is sufficiently evident. 
 
 3. The Laws of Thought are the foundation of all 
 reasoning, and may thus be symbolically stated : 
 
 The Law of Identity .... A = A. 
 
 The Law of Duality or of) A AT* A/ 
 
 Excluded Middle . . . j 
 The Law of Contradiction . Aa = o. 
 
 The successive application of the Law of Duality to two, 
 three, four, five or more terms, gives rise to the develop- 
 ment of all possible logical combinations, called the Logical 
 Alphabet, the first few columns of which are given below. 
 The combinations for six terms are given in the Principles 
 of Science, p. 94 (first ed. vol. i. p. 109).
 
 xxi ELEMENTS 181 
 
 THE LOGICAL ALPHABET 
 
 II. 
 
 III. IV. 
 
 V. 
 
 VI. 
 
 vi. continued. 
 
 AX 
 
 AB ABC 
 
 ABCD 
 
 ABCDE 
 
 aBCDE 
 
 a X 
 
 A b A B c 
 
 A BC d 
 
 AB C D e 
 
 a B C D e 
 
 
 a B A b C 
 
 AB c D 
 
 ABC rf E 
 
 aBC of E 
 
 
 a b A b c 
 
 AB c a? 
 
 ABC d e 
 
 aB C d e 
 
 
 BC 
 
 A<5 C D 
 
 AB c DE 
 
 aB c D E 
 
 
 a B c 
 
 A* C d 
 
 AB c D e 
 
 a B c D e 
 
 
 ab C 
 
 A6 c D 
 
 AB <: rfE 
 
 a B c d E 
 
 
 a b c 
 
 Abed 
 
 A B c d e 
 
 a B c rf 
 
 
 
 a BCD 
 
 A3 CDE 
 
 a * CDE 
 
 
 
 a BC rf 
 
 A C D e 
 
 a 3 C D * 
 
 
 
 a B c D 
 
 A6 C <tE 
 
 a b C rfE 
 
 
 
 a B c rf 
 
 f^b C d e 
 
 a b C d e 
 
 
 
 a C D 
 
 Ab c DE. 
 
 a b cD E 
 
 
 
 a b C d 
 
 Ab c D e 
 
 <c ~c D i 
 
 
 
 a 6 c D 
 
 Ab c d E 
 
 a c </ E 
 
 
 
 abed 
 
 Ab c d e 
 
 a b c d e 
 
 4. The one sole and all sufficient rule of inference is the 
 following RULE OF SUBSTITUTION. 
 
 FOR ANY TERM SUBSTITUTE WHAT IS STATED IN ANY 
 PREMISE TO BE IDENTICAL IN MEANING WITH THAT TERM. 
 
 The term may consist of any single letter, any juxtaposed 
 letters, or any group of alternatives connected by the 
 sign !, the sign of unexclusive alternation. 
 
 5. It is assumed as a necessary law that every term must 
 have its negative. This was called the 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 Morgan, 
 in his Syllabus, article 1 6. Thence arises what I propose to 
 call the CRITERION OF CONSISTENCY, stated as follows : 
 
 Any two or more propositions are contradictory when, and 
 only when, after all possible substitutions are made, they 
 occasion the total disappearance of any term, positive or 
 negative, from the Logical Alphabet. 
 
 The principle of this criterion was explained in p. 65 of 
 the Essay on Pure Logic referred to, but subsequent inquiry, 
 and the writings of Mr. A. J. Ellis, have tended to show 
 the supreme importance of the criterion.
 
 182 EQUATIONAL LOGIC CHAP. 
 
 The processes of this equational system of Logic are 
 fully treated in the first seven chapters of the Principles of 
 Science, and they are now amply illustrated by the problems 
 which follow. 
 
 6. How do you express in the new logic the four 
 Aristotelian forms of proposition indicated by 
 the vowels A, E, I, and O ? 
 
 The answer is 
 
 A Every A is B. A = AB (i). 
 
 E. No A is B. A = Kb (2). 
 
 I. Some A is B. CA = CAB (3). 
 
 O. Some A is not B. CA = CA (4). 
 
 The first expresses the coincidence of the class A with 
 part of the class B, namely A B, which is the equational 
 mode of asserting that the A's form part of the B's. The 
 second expresses similarly that the A's are found among 
 the not-B's. In the third form some is expressed by the 
 symbol C, and the proposition asserts that some A's (CA) 
 are identical with a part of the class B. Some difficulties 
 may arise about this form, owing to the ambiguity of the 
 Aristotelian some, as elsewhere discussed (pp. 151-158). 
 The fourth proposition is evidently the negative form of 
 the third. 
 
 7- How shall we express equationally the asser- 
 tion of Hobbes (De Corpore Politico, I. i. I 3), 
 that ' Irresistible might in the state of nature 
 is right ' ? 
 
 ' Might ' is the principal part of the subject, but it is 
 qualified or restricted in this proposition by the adjective
 
 xxi QUESTIONS AND ANSWERS 183 
 
 'irresistible,' and by the adverbial 'in the state of nature.' 
 Thus putting 
 
 A = irresistible ; C = in the state of nature ; 
 B = might ; D = right. 
 
 The subject is clearly ABC; and D is affirmed of it. 
 But Hobbes cannot, of course, have meant that all right 
 is irresistible might ; only in the state of nature is this true. 
 As, indeed, irresistible might must overcome everything 
 opposed to it, there can be nothing else right in the sphere 
 of its action, so that the proposition would seem to have 
 the form A B C = C D. It is not easy to be sure of the 
 meaning even of Hobbes. 
 
 8. Represent the meaning of the sentence ' Man 
 that is born of a woman, is of few days, and 
 full of trouble.' 
 
 The relative clause, ' that is born of a woman,' is evi- 
 dently explicative, and we cannot suppose that there are any 
 men not ' born of a woman.' Hence taking 
 
 A = man ; C = of few days ; 
 
 B = born of a woman ; D = full of trouble ; 
 
 the meaning seems to be expressed in the two pro- 
 positions 
 
 A = AB; 
 
 A = ACD. 
 
 But if we may treat the sentence as an imperfectly expressed 
 syllogism namely, ' because man is born of a woman he 
 is, etc.' then the premises obviously become A = AB, 
 and B = B C D, and the conclusion by substitution for B 
 in the first of its value in the second, is A = A BCD.
 
 184 EQUATIONAL LOGIC CHAP. 
 
 9. Show how to obtain equationally the contra- 
 
 positive of A = AB. 
 
 This is explained in the Principles of Science, p. 83 
 (first edition, vol. i. pp. 97-102), but may be thus briefly 
 repeated 
 
 By the Second Law of Thought 
 
 b = hb-\-ab. 
 Substitute for A its equal AB. 
 
 or, b = ab. 
 
 Concerning the contrapositive see above, pp. 32, 
 43-47, etc. 
 
 10. Show how to obtain the complete contra- 
 
 positive of A = B. 
 
 As before 
 
 b = Ab-\-ab = 'Rb-\>ab = 0'\'ab = ab; 
 similarly 
 
 a = aR'\'ab = a&.-\'Ctb = o>\-ab = ab. 
 Having now the two propositions 
 
 a = ab = b, 
 
 it is plain that we may eliminate ab, and get 
 
 a b. 
 
 11. What descriptions of the terms 'glittering 
 
 thing ' and ' not gold ' can you draw from the 
 following assertions ? 
 
 (1) Brass is not gold; 
 
 (2) Brass glitters. 
 
 Let A = brass ; B = golden ; 
 C = glittering thing. 
 
 The premises are 
 
 (r) A = A^; (2) A = AC.
 
 xxi QUESTIONS AND ANSWERS 185 
 
 Obviously C = ABC 
 The first of the alternatives ABC is negatived by (i); 
 but the second and fourth coalesce, and we have 
 
 that is, a glittering thing is either not golden, or else it is 
 golden, and then not brass. 
 For b we similarly get 
 
 b-6C'\-abc. 
 Show that we may also infer 
 
 C-C(a-M), 
 
 12. How shall we represent in the forms of 
 Equational Logic the moods of the old syllo- 
 gism ? 
 
 All of the moods without exception may be solved by 
 the indirect method, that is by working out the combinations 
 consistent with the premises. Most of the moods may, how- 
 ever, be solved also by direct substitution, as will be seen 
 in the following examples : 
 
 Barbara. 
 
 All men are mortal ; (i) B = BC. 
 
 All kings are men ; (2) A = AB. 
 
 .-.All kings are mortal ; (3) A = ABC. 
 
 We get (3) by substituting for B in (2), its equivalent 
 BC in (i). The conclusion amounts to saying that 'king' 
 is equivalent to 'king man mortal.' If desired, we can 
 by further substitution of A for AB in (3) obtain A = AC, 
 or king = king mortal, which is a precise expression for 
 the Aristotelian conclusion ' All kings are men.'
 
 186 EQUATIONAL LOGIC CHAP. 
 
 Celarent 
 
 No men are perfect ; (i) B = B<r. 
 
 All kings are men ; (2) A = AB. 
 
 No kings are perfect ; (3) A = ABc. 
 
 Solved, as in the last case, by direct substitution in (2) 
 of the value of B given in (i). 
 
 Darii. 
 
 All mathematicians have well-trained ] r rr ^ 
 
 C ^-' Vx-LJ* 
 
 intellects ; ) 
 
 Some women are mathematicians ; AB = ABC. 
 Some women have well -trained 
 
 in-} 
 J 
 
 AT>_ 
 
 tellects. 
 
 Here A stands for the indefinite adjective some, and B 
 for women, and we then treat AB as an undivided term, 
 and obtain the result by direct substitution, exactly as in 
 the previous moods. 
 
 Ferio. 
 
 No foraminifera are fresh-water in- ) ~ ~ , 
 
 1 !_ fC =Ctf. 
 
 habitants ; J 
 
 Some components of chalk are fora- \A-D_ 
 minifera ; j 
 
 Some components of chalk are not } ATa _ 
 .... C -"-D 
 
 fresh-water inhabitants. 
 
 Except that a negative term d takes the place of the 
 positive term D, in the last mood, there is no difference in 
 form between them. In fact, all the four moods of the 
 first figure present so great similarity that they may be said 
 to be of one form of inference.
 
 xxi QUESTIONS AND ANSWERS r8; 
 
 Cesare. 
 
 The absolute is not phenomenal ; (i) C = C. 
 
 All known things are phenomenal ; (2) A = AB. 
 All known things are not the absolute. (3) A = ABc. 
 
 We cannot by any direct substitution obtain the conclusion 
 from the premises, as B appears in (2) and b in (i). But 
 we may take the contrapositive of (i) as described before 
 (p. 184), namely, B = B<r, and substitution in the second 
 premise is then practicable. 
 
 Camestres. 
 
 All laws of nature are invariable; (i) C = CB. 
 
 No human customs are invariable ; (2) A = A. 
 No human customs are laws of nature. (3) A = Ar. 
 
 As in the last mood, we must take the contrapositive of 
 (i), namely b = bc, and substitute thereby in (2). 
 
 13. The equational treatment of the moods Camestres, 
 Cesare, and Camenes is described also in the Principles of 
 Science, pp. 84-86 (first edition, vol. i. pp. 99-101), or in 
 the Substitution of Similars, pp. 47-49 ; but the following 
 is the briefest way of getting the Aristotelian conclusion, 
 of Camestres, as suggested by Mr. W. H. Brewer, M.A. 
 
 Let the premises be 
 
 (1) A = AC. 
 
 (2) B = B^. 
 
 Multiply together, and we get 
 
 AB = ABCc = o 
 
 Thus there is no such thing as AB. 
 Similarly with Camenes.
 
 1 88 
 
 EQUATIONAL LOGIC 
 
 CHAP- 
 
 14. The remaining moods need only be symbolically 
 represented. In every case D = some : 
 
 Festino 
 Baroko 
 Darapti 
 Disamis . 
 Datisi . 
 Felapton . 
 Bokardo . 
 Ferison 
 Bramantip 
 Camenes . 
 Dimaris . 
 Fesapo 
 Fresison . 
 
 Major Premise. 
 
 Minor Premise. C 
 
 A 
 
 = Kb . 
 
 CD = 
 
 BCD. 
 
 CD 
 
 A 
 
 = AB . 
 
 CD = 
 
 t>CD . 
 
 CD 
 
 B 
 
 = AB . 
 
 B = 
 
 CB . 
 
 AB 
 
 BD 
 
 = ABD. 
 
 B = 
 
 BC . 
 
 BD 
 
 B 
 
 = AB . 
 
 BD = 
 
 BCD. 
 
 BCD 
 
 B 
 
 = aB . 
 
 B = 
 
 BC . 
 
 BC 
 
 BD 
 
 = aBD . 
 
 B = 
 
 BC . 
 
 BCD 
 
 B 
 
 = aB . 
 
 BD = 
 
 BCD. 
 
 BCD 
 
 A 
 
 = AB . 
 
 B = 
 
 BC . 
 
 ABC 
 
 A 
 
 = AB . 
 
 B = 
 
 B* . 
 
 C 
 
 AD 
 
 = ABD. 
 
 B = 
 
 BC . 
 
 AD 
 
 A 
 
 = 'A . 
 
 B = 
 
 BC . 
 
 BC 
 
 A 
 
 = A 
 
 BD = 
 
 BCD. 
 
 BCD 
 
 Conclusion. 
 
 aBCD 
 
 atCD 
 
 CB 
 
 ABCD 
 
 ABD 
 
 aB 
 
 aBCD 
 
 aBD 
 
 A 
 
 aC 
 
 ABCD 
 
 aBCj 
 
 flBCD 
 
 15- Exhibit the logical force of the motto adopted 
 by Sir W. Hamilton 
 
 (1) In the world there is nothing great but 
 Man. 
 
 (2) In Man there is nothing great but Mind. 
 
 Let A = in the world ; 
 B = man ; 
 
 C = possessing mind ; 
 D = great. 
 
 The conditions may be represented as 
 
 (1) A = ABD -I- hbd. 
 
 (2) B = BCD - 
 
 As it may be understood, though unexpressed, that
 
 xxi QUESTIONS AND ANSWERS 189 
 
 (3) all men are in the world, and that (4) all possessing 
 mind are men, we have further 
 
 (3) B = AB. (4) C = CB. 
 
 The combinations are thus reduced to 
 
 ABCD abcft 
 
 Kbcd abed. 
 
 Observe that, if mind were regarded not as an attribute 
 but as a physical part of man, we could not treat the 
 assertion as one of simple logical relation. 
 
 16. What is the meaning of the assertion that 
 ' All the wheels which come to Croyland are 
 shod with silver ' ? 
 
 If we take 
 
 A = wheel; 
 
 B = coming to Croyland ; 
 
 C = shod with silver, 
 
 the assertion, as it stands, is evidently in the form 
 AB = ABC (i). 
 
 But it was always understood, no doubt, that this adage 
 was to be joined in the mind with the tacit premise ' No 
 wheels are shod with silver,' expressed by 
 
 A = Af (2). 
 
 There seems, at first sight, to be contradiction between 
 these premises; for (i) speaks of wheels shod with silver 
 and (2) denies that there are such things. The explanation 
 is obvious, namely, that there are no such things as wheels 
 coming to Croyland. Of the four combinations containing A,
 
 IQO EQUATIONAL LOGIC CHAP. 
 
 ABr is negatived by (i), and ABC and Al>C by (2), so that 
 the description of A is given thus, 
 A = A6f, 
 or, by substitution of A for A<r, 
 
 A = A, 
 
 that is, no wheels come to Croyland. This is of course 
 the inference which the adage was intended to suggest, 
 Croyland being an ancient abbey lying among the fens of 
 Cambridgeshire, where in former days no wheeled vehicle 
 could make its way. 
 
 iThis question illustrates the important logical principle 
 that all propositions ought, strictly speaking, to be interpreted 
 hypothetically. We have only to put these premises in the 
 hypothetical form, and we see that they make a reasonable 
 destructive hypothetical syllogism thus 
 
 If any wheels come to Croyland they are shod with 
 silver ; but no wheels are shod with silver ; therefore, 
 no wheels do come to Croyland. 
 
 17- Ruminant animals are those which have 
 cloven feet, and they usually have horns ; the 
 extinct animal which left this footprint had a 
 cloven foot; therefore it was a ruminant animal 
 and had horns. Again, as no beasts of prey 
 are ruminant animals it cannot have been a 
 beast of prey. 
 
 The above problem is given in the Elementary Lessons 
 (p. 321, No. 78). Taking our symbols thus 
 
 A = ruminant ; D - extinct animal ; 
 
 B = having cloven feet ; E = beast of prey ; 
 
 C = having horns ; 
 we have clearly A = B (i).
 
 xxi QUESTIONS AND ANSWERS 191 
 
 The statement that ruminant animals usually'*- have horns 
 may be formalised as 
 
 BA = BAG (2), 
 
 that is to say, a certain particular portion of the class A, 
 BA have horns. Next we have 
 
 D = DB (3). 
 
 Substituting in (3) by (i) we get 
 
 D == DA (5); 
 
 showing that the extinct animal was a ruminant. But, as 
 we cannot substitute between (3) and (2), it is erroneous to 
 assert that it had horns. If by usually we mean in the far 
 greater number of cases, then there is a considerable prob- 
 ability, but no certainty that it had horns. 
 
 Again, we have as an additional premise, that beasts of 
 prey are not ruminant, or 
 
 E = E (4), 
 
 which, taken with 
 
 D == DA (5), 
 
 our previous conclusion, gives a syllogism in Cesare, estab- 
 lishing that D is not E, or that the extinct animal cannot 
 have been a beast of prey. This we might sufficiently 
 prove symbolically by multiplying the respective members 
 of (4) and (5) together, giving 
 
 DE = aEDA = o. 
 
 This shows that inconsistency arises from supposing that 
 this D can be also E. The same result might be worked 
 out by combinations, giving D = De. 
 
 1 The first edition of the Lessons reads always instead of usually.
 
 192 EQUATIONAL LOGIC CHAP. 
 
 18. Take the proposition ' All crystals are solids,' 
 and ascertain precisely what it affirms, what it 
 denies, and what it leaves doubtful. 
 
 [A. J. ELLIS.] 
 
 Taking A = crystal, and B = solid, the proposition is 
 in the form 
 
 A = AB. 
 
 The conceivable combinations are four in number, 
 namely, 
 
 AB, A, aB, and ab. 
 
 Of these, only Kb is inconsistent with the premise, that is 
 to say, the premise 'All crystals are solids' denies the 
 existence of such things as 'unsolid crystals.'' We cannot 
 strike out AB, because then there would be no such thing 
 as crystals left ; hence the premise affirms the existence of 
 solid crystals, in the sense that any other proposition deny- 
 ing that crystals are solid, or that solids may be crystals, 
 would stand in contradiction to our premise. 
 
 Again, we may not strike out a b, because there would 
 then be no such thing as b, or not-solids. Hence to avoid 
 contradiction of our premise, there must be such a thing 
 as ' a non-crystal which is not-solid.' If we are to hold to 
 our adopted Criterion of Consistency (p. 181), we must 
 say that at least one case of ab exists, so that to avoid self- 
 contradiction, some, that is at least one case of not-crystal, 
 must be allowed to exist l and to be not-solid. This con- 
 firms the conclusion which we previously obtained by the 
 Aristotelian logic from the same premise (p. 48). But the 
 combination B may be removed or left without affecting 
 the truth of the premises, which therefore leave it entirely 
 in doubt whether ' not -crystals which are solids exist.' 
 
 1 Concerning the logical sense of the verb exist, see pp. 141-2.
 
 xxi QUESTIONS AND ANSWERS 193 
 
 ' Not-crystals, not - solids ' must exist, but ' Not crystals- 
 solids,' may or may not. But if they do not, then crystals 
 and solids will be coincident classes. 
 
 To sum up 
 
 A = AB affirms that all As are Bs ; 
 
 A = AB affirms that all not-Bs are not- As ; 
 
 A = AB affirms that some not-As are not-Bs ; 
 
 A = AB denies that all not-As are not-Bs ; 
 
 A = AB leaves doubtful whether not-A can be B. 
 
 19. Are the following propositions equivalent 
 each to the other ? 
 
 (1) All who were there talked sense ; 
 
 (2) All who talked nonsense were away. 
 
 [DE MORGAN.] 
 Putting 
 
 A = being there ; a = not being there ; 
 
 B = talking ; b = not talking ; 
 
 C = sensible ; c = not sensible ; 
 
 the first premise seems to mean that all who were there 
 (A), talked and were sensible ; that is 
 
 A -ABC (i). 
 The second is to the effect 
 
 Rc=aRc (2); 
 
 that is to say, those who talked (B), but not sensibly 
 (f), were away (a). Now (i) negatives three combinations, 
 ABc, Al>C, and hbc ; whereas (2) negatives only ABc. They 
 are, therefore, very different propositions; for (2) allows 
 that some may have been present who did not talk at all 
 (), whether sensible people or not (C or c}. Nevertheless, 
 there is this much relation between the two propositions, 
 that we can infer (2) from (i). If all who were there talked 
 
 o
 
 194 EQUATIONAL LOGIC CHAP. 
 
 sense, those people who talked nonsense, assuming there to 
 be such persons, must have been absent. But we cannot 
 invert this relation. Because those who talked nonsense 
 were away, it does not follow that those who were present 
 talked sense ; they may all have been silent. 
 
 20. De Morgan says {Syllabus, p. 1 4), ' 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 ? 
 
 (1) All Englishmen who do not take snuff 
 
 are to be found among Europeans who 
 do not use tobacco. 
 
 (2) All Englishmen who do not use tobacco 
 
 are to be found among Europeans who 
 do not take snuff. 
 
 Required immediate answer and demonstra- 
 tion.' 
 Assigning symbols as follows : 
 
 A = Englishmen ; C = taking snuff; 
 
 B = Europeans ; D = using tobacco ; 
 
 it is pretty obvious that the above propositions are thus 
 
 symbolised 
 
 (1) A 
 
 (2) M 
 
 We are to compare these with the well-known relations 
 of the terms, which may be assumed to be 
 
 (3) A = AB;
 
 xxi QUESTIONS AND ANSWERS 195 
 
 that is, ' Every Englishman is an European,' and 
 
 (4) C = CD; 
 
 that is, 'All who take snuff use tobacco.' Now, in working 
 out the combinations, we find that the class Ac is composed 
 under conditions (3) and (4) as follows : 
 Ac = ABcD } ABcd. 
 
 The truth is, then, that Englishmen who do not take snuff 
 consist of English Europeans not taking snuff, but using 
 tobacco, and of English Europeans neither taking snuff nor 
 using tobacco. In short (i) is erroneous in ignoring the 
 fact that some Englishmen not using snuff may be Europeans 
 who do use tobacco for smoking. 
 
 According to assumption (2) the description of Kd is 
 AIW, which coincides with the description drawn from 
 (3) and (4). Thus it is true that all Englishmen who do 
 not use tobacco are to be found among Europeans who do 
 not take snuff; the negation of the larger term, using tobacco, 
 includes the negation of the narrower one, using snuff. 
 But it by no means follows that because our inference about 
 A*/ is the same from (2) as from (3) and (4), therefore these 
 conditions are identical, as will be seen in the following 
 descriptions of the class A as furnished under the several 
 suppositions and conditions 
 
 (1) A 
 
 (2) A 
 
 (3) and (4) A = ABCD . 
 
 21. What can we infer about the^term Europeans 
 from the following premises ? 
 ( i ) All Continentals are Europeans ; 
 
 (2) All English are Europeans ; 
 
 (3) No English are Continentals.
 
 196 EQUATIONAL LOGIC CHAP. 
 
 Taking A, B, C to represent Continentals, English, and 
 Europeans respectively, the premises become 
 
 (1) A = AC. 
 
 (2) B = BC. 
 
 (3) B = aB. 
 
 The combinations left uncontradicted are the four 
 A^C, #BC, abC, abc, whence we learn that Europeans, 
 C, consist of Continentals who are not English, of English 
 who are not Continentals, and of any others, who are neither 
 Continentals nor English (abC). 
 
 22. Criticise Thomson's ' Immediate inference 
 
 by the sum of several predicates. . . . From 
 a sufficient number of judgments in A, having 
 the same subject, a judgment in U may be 
 inferred, whose predicate is the sum of all the 
 other predicates.' [P.] 
 
 This question has been answered in the Principles of 
 Science^ p. 61 (first ed. Vol. I. p. 73). Judgments in A are 
 of the form P = PQ, P = PR, P = PS, etc., and by 
 summing up the predicates by successive substitution in the 
 second side of P = PQ, we may get P = PQRS .... 
 But this does not give a proposition of the form U which, 
 as described by Thomson, is of the form P = X. 
 
 23. Represent the ? ; following argument from 
 
 Thomson's Laws of Thought, 107: 
 All P is either C or D or E ; 
 S is neither C nor D nor E ; 
 therefore, S is not P.
 
 xxi QUESTIONS AND ANSWERS 197 
 
 The premises are respectively : 
 
 P = PC-|-PD.|.PE. 
 S - Sale. 
 
 We get the conclusion in the briefest way by multiplying 
 the two premises together as they stand ; thus : 
 
 PS = P (C -I- D .|- E) Scde = o -|- o -I- o. 
 
 Each alternative is found to be contradictory, so that 
 there is no such thing as PS, that is to say, no P is S. 
 
 The argument is not, however, correctly described by 
 Dr. Thomson as in the syllogistic mood U E E, nor are the 
 other forms of argument given in the same section syllo- 
 gistic. They are disjunctive in character. 
 
 24. If Abraham were justified, it must have been 
 either by faith or by works : now he was not 
 justified by faith (according to James), nor by 
 works (according to Paul) : therefore Abraham 
 was not justified. [w.] 
 
 There is some difficulty in deciding on the best method 
 of symbolising this argument, owing to the vagueness of 
 the conditions when analysed ; but the following seems to 
 be the best representation : 
 
 Let A = Abraham ; C = justified by works ; 
 
 B = justified ; D = justified by faith. 
 
 Then the premises are : 
 
 AB-AB (C-I-D).
 
 198 EQUATIONAL LOGIC CHAP. 
 
 These premises will be found to erase all the combina- 
 tions of A excepting Kbcd, which gives the conclusion. 
 The combinations of a are altogether unaffected and need 
 not be examined. The student may try other modes of 
 representing the premises, but should get A = Kb by every 
 method. 
 
 25. It must be admitted, indeed, that (i) a man 
 who has been accustomed to enjoy liberty 
 cannot be happy in the condition of a slave : 
 (3) many of the negroes, however, may be 
 happy in the condition of slaves, because 
 (2) they have never been accustomed to enjoy 
 liberty. [w.] 
 
 Let A = man accustomed to enjoy liberty ; 
 B happy in condition of slave ; 
 C = certain negroes. 
 
 The premises may be stated in the forms 
 
 A = A. (i) 
 
 C = aC. (2) 
 
 The supposed conclusion is C = CB. (3) 
 
 The possible combinations as in the margin, from 
 which it will be seen that 
 
 abc 
 
 that is, are either B, happy, or b, not 
 happy. 
 
 The fallacy is that of Negative Premises or of Illicit 
 Process of the Major.
 
 xxi QUESTIONS AND ANSWERS 199 
 
 26. 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. 
 
 Let A = possessing heat ; 
 
 B = possessing visible motion , 
 C = possessing energy, 
 
 the universe being 'things' unexpressed, and 'devoid of 
 being taken as the negative of ' possessing.' Then the 
 condition is: 
 
 ab = abc. 
 
 By contraposition we obtain, using Mr. MacColPs notation 
 for the negative of ab (See Preface) : 
 
 C = C (ab)' 
 
 = C (A*-|-aB*|-AB) 
 Hence 6C = 
 
 two self-contradictory alternatives disappearing. It can 
 also be readily shown that this inference is equivalent to 
 the original condition. 
 
 27. Prove the logical equivalence of the pro- 
 position B = AC -|- ac and b = Ac -\- aC. 
 
 This might be shown by receding to the combinations of 
 the Logical Alphabet, but it is more neatly proved by 
 equating the negatives of each member of the first equation. 
 If M = N, then also m-?i (p. 184); hence the negative 
 of B must be identical with the negative of AC .|. ac. 
 Now the negative of B is b; that of the compound and
 
 200 EQUATIONAL LOGIC CHAP. 
 
 complex member is the compound of the negatives of the 
 two alternatives. In Mr. MacColl's notation 
 
 = (aC'\-A'\'Oc) (A<H-0C-|-AC). 
 
 On multiplying out, the nine products are found to be all 
 self-contradictory excepting aC-\-Ac, which is therefore 
 the expression for b. Vice versa the negative of Ac-\-aC 
 will be found to be AC-\-ac, so that the propositions are 
 clearly equivalent. 
 
 28. If no A is BC, what can I infer about the 
 
 relation of B and AC ? 
 
 The condition is 
 
 A = A-|'A<:. 
 
 Substitute in either side of 
 
 ABC = ABC, 
 
 and we get 
 
 ABC = ABC | ABO = o, 
 
 or B cannot be AC. 
 
 29. It is known of certain things that (i) where 
 
 the quality A is, B is not ; (2) where B is, and 
 only where B is, C and D are. Derive from 
 these conditions a description of the class of 
 things in which A is not present, but C is. 
 
 The premises are clearly : 
 
 (1) . . A = A/ 
 
 (2) . . B = CD
 
 xxi QUESTIONS AND ANSWERS 201 
 
 The conceivable constituents of the class which is C 
 but not-A are 
 
 aC = flBCD -I- BG/-|- abCV -\-abCd. 
 
 Substituting CD for B in the second, and B for CD in 
 the third alternative, we find that these combinations give 
 contradictory results, namely aRCDd and oBCD. It 
 follows that 
 
 aC = rtBCD -I- abCd. 
 
 Observe that the premise (i) has no connection with this 
 result, which is deducible from (2) alone. 
 
 30- It has been observed that in a certain class 
 of substances, ( I ) where the properties A and 
 B are present, the property C is present ; and 
 (2) where B and C are present, A is present. 
 Does it follow that B is present where C and 
 A are present ? 
 
 The premises are obviously : 
 
 (1) . . AB = ABC, 
 
 (2) . . BC = ABC, 
 
 which are equivalent to the single proposition 
 AB = BC. 
 
 The answer is obtained at once from 
 the combinations in the margin, which 
 show that 
 
 abC 
 that is, in the presence of A and C, B a fc 
 
 is indifferently present or absent. [Dr. 
 
 Macfarlane (Algebra of Logic, 1879, p. 141), who gives this 
 
 ABC 
 
 Me
 
 202 EQUATIONAL LOGIC CHAP. 
 
 problem, requires more than half a page to solve it, using, 
 moreover, sundry impossible logical fractions.] 
 
 3 1 - Given (i) that everything is either B or C, 
 and (2) that all C is B, unless it is not A : 
 prove that all A is B. [c] 
 
 The first condition is expressed by the assertion that 
 every not-B is C, which carries with it the equivalent that 
 every not-C is B. Thus we have 
 
 b = b. (i) 
 
 The second assertion is less easy to interpret, because we 
 are not told what happens if ' it,' that is C, is not-A. The 
 meaning appears, however, to be that C if it is A must 
 be B, that is 
 
 AC = ABC. (2) 
 
 These conditions give the combinations 
 ABC. 
 
 abC. 
 from the first two of which we learn that A is always B. 
 
 32. If we throw ' every A is B ' into the form 
 ' every A is B or B,' we have ' every A which is 
 not B is B ' a contradiction in terms. But it 
 evidently implies that there can be no As which 
 are not Bs, and thus we return to ' every A 
 is B.'
 
 xxi QUESTIONS AND ANSWERS 203 
 
 The above is a transcript with altered symbols from 
 De Morgan's sixth example (Formal Logic, p. 123). But 
 the contradiction arises simply from an error in not multi- 
 plying both alternatives by b. De Morgan follows the rule 
 for the resolution of dilemmas, not observing that this 
 rule can apply only when the alternatives are different- 
 Equationally we have 
 
 A =AB -I-AB. 
 
 hence . . . A.l> = AB-|- AB^ = o. 
 
 he gets . . . A# = AB<H-AB = AB. 
 
 It is rarely we find De Morgan tripping. 
 
 33- Every A is one only of the two B or 
 C ; D is both B and C, except when B is 
 E, and then it is neither ; therefore no A 
 is D. 
 
 This problem was proposed by De Morgan in the Formal 
 Logic, p. 124, and a solution has been given in the Principles 
 of Science, p. 101 (first ed., vol. i. p. 117). The premises, 
 as Professor Groom Robertson has pointed out, may be 
 stated in two propositions, namely 
 
 A- 
 
 Some objection has, however, been taken by Mr. Monro 
 to my solution, and the student will find a good exercise 
 in going over the solution carefully. It seems rather 
 doubtful how we should treat the combinations which are 
 E and not B ; but the difficulties lie wholly in the interpre- 
 tation of De Morgan's conditions.
 
 204 EQUATIONAL LOGIC CHAP. 
 
 34- From A follows B, and from C follows D ; 
 but B and D are inconsistent with each other. 
 Hence A and C are inconsistent with each 
 other. 
 
 This problem, which is formally the same as one of 
 De Morgan's (Formal Logic, p. 123, Example 3), has the 
 conditions 
 
 (1) A = AB; 
 
 (2) C = CD; 
 
 (3) B = B* 
 
 The consistent combinations are 
 AB*/. 
 
 abcD. 
 abed. 
 
 We see that A and C never occur together, and in fact 
 that A is never found excepting in the presence of B and 
 the absence of both C and D. 
 
 I committed an error in treating this problem in the 
 Substitution of Similars (pp. 52, 53), by regarding aRcd 
 as negatived by the premises. B may occur in the absence 
 of A, but C and D must both be absent. 
 
 35- What are the combinations of the qualities 
 A, B and C which are possible according to 
 the following conditions? (i) 'Where A is 
 present, B and C are either both present at 
 once or absent at once; (2) where C is present, 
 A is present.' Describe the class not-B under 
 these conditions.
 
 xxi QUESTIONS AND ANSWERS 205 
 
 The conditions are expressed equationally as 
 (i) A = 
 
 (2) C = AC. 
 The consistent combinations are shown in the margin ; 
 
 ABC 
 
 Me 
 
 aEc 
 
 abc 
 
 and AC are removed by the first condition, 
 and aBC and abC by the second. Selecting the 
 two remaining ones which contain b, we have the 
 required description 
 
 b = Kbc '\-abc = be. 
 Where B is absent, C also will be absent. 
 
 36. ' The logical value of two affirmative pre- 
 mises in the second figure is absolute zero.' 
 Examine the truth of this statement. [P.] 
 
 The two premises assumed to be universal may be 
 symbolised as . 
 
 (1) A = AB; 
 
 (2) C = CB. 
 
 The first negatives the combinations A^C and Abe, the 
 second A<C and abC, so that the premises overlap in 
 regard to A^C. There remain five combinations. If we 
 inquire what is A we get the value 
 
 A = ABC-|-AB^=AB, 
 
 which is no more than (i). For the description of C 
 similarly we get (2). Thus it is plain that no relation is 
 established between A and C. Concerning B we have even 
 less information ; for 
 
 B = ABC -I- ABr-l- aEC -\- aEc = AB -I- ciE = B.
 
 206 EQUATIONAL LOGIC CHAP. 
 
 Of the negative terms, however, we draw more significant 
 descriptions ; thus 
 
 a = 
 
 b abc. 
 
 c=- EC '\-abc. 
 
 It cannot be truly said that the logical value of the 
 premises is absolute zero. 
 
 37- Given that (i) everything which is B but not 
 D is either both A and C or neither A nor 
 C ; and (2) that neither C nor D is both A 
 and B : prove that no A is B. 
 
 [Adapted from Moral Science Tripos, 
 Cambridge, 1879.] 
 
 The conditions are 
 (i) Brf 
 
 Confining our attention to the combinations containing 
 AB, we see that KS>cd is contradicted by (i), and the 
 rest which contain either C or D, by (2). Hence there 
 are no ABs, or no A is B. 
 
 The equation (2) may be more briefly stated as 
 
 AB = AEcd. 
 
 The only combination containing a removed by (i) and 
 (2) is aBCd. 
 
 38. Illustrate the use of symbolic methods by 
 expressing the propositions
 
 xxi QUESTIONS AND ANSWERS 207 
 
 ( i ) No A is B except what is both C and 
 D, and only some of that. 
 
 (2) Either C or D is never absent except 
 where A or B is present, but both are 
 always absent then. [C.] 
 
 The first proposition appears to deny the presence of any 
 combination containing AB except there be also present C 
 and D, and only in some cases then. To express this some 
 we must introduce another letter term, say E, so that where 
 E is present the above holds true ; where E is absent, A is 
 not B at all. We find then that the following combinations 
 are negatived : 
 
 ABG/E. ABCD^. 
 
 AEcDE. 
 
 ABofe. 
 
 All this may be expressed in the one equation 
 AB - ABCDE. 
 
 The proposition (2) is not easy to interpret, but seems to 
 mean 
 
 39- Every X is either P, Q, or R ; but every P 
 is M, every Q is M, every R is M ; therefore 
 every X is M. 
 
 De Morgan, who gives the above (Formal Logic, p. 123,
 
 208 EQUATIONAL LOGIC CHAP. 
 
 Example 5), describes it as a common form of the dilemma. 
 It is thus solved equationally : 
 
 (1) X = X(P-|.Q.|.R); 
 
 (2) P = PM; 
 
 (3) Q = QM; 
 
 (4) R - RM. 
 
 Substituting by (2) (3) (4) in (i), 
 
 X = X(PM.|.QM-|-RM); 
 X = X(P-|-Q.i-R) M. 
 
 Re-substituting in the last by (i) 
 X = XM. 
 
 40. Every A is either B, C, or D ; no B is A ; 
 no C is A ; therefore every A is D. 
 
 [De Morgan, Formal Logic, p. 122.] 
 
 The premises are clearly 
 
 (1) A = AB.|.AC-|-AD. 
 
 (2) B = B. 
 
 (3) C = C. 
 
 In (i) substitute the values of B and C given in (2) and 
 (3), and then strike out two self-contradictory terms 
 
 A = AaB -I- ArtC -I- AD = AD. 
 
 41. If A be B, E is F ; and if C be D, E is F ; 
 but either A is B, or C is D ; therefore, E is 
 F. (De Morgan, Formal Logic, p. 123.)
 
 xxi QUESTIONS AND ANSWERS 209 
 
 This appears to be more complicated in symbols than it 
 really is. The first two premises are 
 
 (1) AB = ABEF 
 
 (2) CD = CDEF. 
 
 To express the third premise we must introduce explicitly 
 the tacit term, say X, meaning the circumstances under 
 which the proposition holds good, in this place, or at 
 this time, or under certain assumed conditions. Thus we 
 have 
 
 ' (3) X = XAB-|-XCD. 
 
 substituting by means of (i) and (2), 
 
 X = (XAB-|-XCD)EF, 
 
 and re-substituting by (3) 
 
 X = XEF. 
 
 42. ' Every A is either B or C, and every C is A.' 
 This, says De Morgan (ibid. p. 123), is wholly 
 inconclusive, and leads to an identical result. 
 
 Equationally treated this is not quite so. The premises 
 are 
 
 (1) A = AB-|.AC 
 
 (2) C = AC 
 Hence (3) 
 
 De Morgan finds that A is C, which C being A gives 
 Al> is A a necessary proposition or truism. But we also 
 get, multiplying each side of (3) by b, 
 
 In the absence, then, of B, there is identity between A 
 and C, but in the presence of B, A may be either B or C. 
 
 p
 
 210 EQUATIONAL LOGIC CHAP. 
 
 43- Every A is B or C or D ; every B is E ; every 
 C is E ; and every E is D. 
 
 [De Morgan, ibid. p. 123, Example 4.] 
 
 Thus symbolised 
 
 (i) A = AB.|-AC-|-AD. 
 (2) B-BE. (3) C = CE. (4) E = ED. 
 
 By obvious substitutions, by (2) and (3) in (i), and then 
 by (4) in the result, we get 
 
 A = ABDE ACDE -=- AD. 
 
 But the first two of these alternatives are superfluous ; 
 they both involve D and are therefore contained in the wider 
 term AD. Hence 
 
 A = AD. 
 
 44. ' 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, Third Memoir on the Syllogism, p. 48.] 
 
 The relations A and B combining into C appears to mean 
 simply that AB is accompanied by C, or 
 
 AB = ABC. 
 To find A without C following, we have necessarily 
 
 Inserting for AB in this last its value ABC. 
 
 Ar = ABCH-A&=A 
 Similarly for B without C following
 
 xxi QUESTIONS AND ANSWERS 211 
 
 45. Suppose a class S to be divided (i) on one 
 principle into A and B, and on another prin- 
 ciple (2) into C and D, the divisions being 
 exhaustive ; suppose further that (3) all A is 
 C, and (4) all B is D ; can you conclude that 
 all C is A, and all D is B ? [E.] 
 
 The meaning of this problem appears to be that the class 
 S will, as regards A and B, consist of SA and SaB, and 
 similarly as regards C and D ; if so, there will under the 
 first two conditions be only four possible combinations, 
 namely 
 
 SMCJ. 
 
 SAM3. 
 
 But the further condition (3) negatives SAM), and (4) 
 negatives SBG/, so that, on inquiring for the description 
 of C, we find it is (within the class S), AbCd / similarly D 
 is aBrD. Both questions then may be answered in the 
 affirmative, provided that we are not to look beyond the 
 sphere of the class S. 
 
 46. What are the classes of objects regarded as 
 possessing or not possessing the qualities 
 A, B, C, D, which may exist consistently 
 with the fundamental Laws of Thought, and 
 the conditions that no class possesses both 
 A and B, and that everything which does not 
 possess B possesses C but not D ? [L.] 
 
 The first condition that no class possesses both A and B 
 will be sufficiently expressed in the premise A = A, which
 
 aEcD 
 
 212 EQUATIONAL LOGIC CHAP. 
 
 prevents A and B from meeting. The second condition is 
 obviously b = bCd. On going over the sixteen combina- 
 tions in the fifth column of the Logical Alphabet (p. 181), 
 it will be obvious that the first four, containing AB, are 
 KbCd negatived by the first premise. The third four 
 (B) remain untouched : of the second and fourth 
 
 T3 (~* sJ 
 
 fours containing b, all are negatived except AbCd 
 and abCd. The adjoining list of combinations is 
 abCd therefore the answer to the question. 
 
 47- How can we represent analytically the precise 
 meaning of the opposition between a universal 
 affirmative proposition and its contradictory, 
 say between All As are Bs, and some As are 
 not Bs ? 
 
 The universal affirmative is symbolised as A = AB, and 
 its logical power is to negative the combination Kb, as 
 shown in the margin. Now ' some As are Bs ' 
 1 was before explained to mean ' one A at least, it 
 
 A h 
 
 may be more or all As.' But, even if there be 
 
 one Kb found, it establishes the existence of the 
 combination, subject to remarks elsewhere made 
 (p. 142). In this qualitative treatment of logic number 
 enters not at all, so that one counts for as much as a 
 million. The force of the particular negative proposition is, 
 then, to restore the combination which had been removed 
 by the universal affirmative. 
 
 48. If to the premises of an affirmative sorites we 
 add a proposition affirming the first subject of
 
 xxi QUESTIONS AND ANSWERS 213 
 
 the last predicate, the conditions now become 
 equivalent to an equally numerous series of 
 identities, or doubly universal propositions in 
 Thomson's form U. 
 
 Symbolically, if we have the series of premises A = AB, 
 B = BC, C = CD, and so on, up to X = XY, and we then 
 add the condition Y = AY, the premises immediately 
 become the same in logical force as 
 
 A = B = C=D= .... = X = Y. 
 
 To give a perfect demonstration of this theorem might 
 not be very easy ; but the student may convince himself of 
 its truth by observing in several trials that the combinations 
 consistent with the premises of a sorites as shown above, 
 never contain a negative letter to the right hand of a positive 
 one in the usual order of the alphabet. Thus the com- 
 binations consistent with the first two premises are 
 
 ABC, rtBC, abC and abc ; 
 those for the first three are 
 
 ABCD, 0BCD, a^CD, abcD and abed. 
 
 Hence the last predicate appears in every combination 
 except the last, and the first subject only in the first 
 combination. In affirming the first subject of the last 
 predicate, then, all combinations except the first, which 
 contains both terms, and the last, which contains neither, 
 must disappear. There remain in every case only the two 
 combinations ABCDE .... XY .... and abcde .... 
 xy . . . . which proceed from the identities stated in the 
 question.
 
 214 EQUATIONAL LOGIC CHAP. 
 
 Suppose a pillar of circular section to be so shaped that 
 no lower section is of less diameter than any upper section, 
 but the section at the bottom is not greater than the section 
 at the top ; we have here a physical analogue to the heap 
 of propositions described above. 
 
 49. Is Professor Alexander Bain correct in the 
 following extract from his Deductive Logic 
 (p. 159)? 
 
 (1) ' Socrates was the master of Plato. 
 
 (2) Socrates fought at Delium. 
 
 (3) The master of Plato fought at Delium. 
 
 ' It may fairly be doubted whether the transitions, in this 
 instance, are anything more than equivalent forms. For the 
 proposition (4) " Socrates was the master of Plato, and 
 fought at Delium," compounded out of the two premises, is 
 obviously nothing more than a grammatical abbreviation. 
 No one can say that there is here any change of meaning, 
 or anything beyond a verbal modification of the original 
 form.' 
 
 Professor Bain in writing the above was clearly in need 
 of means of more accurate analysis than his logical studies 
 had afforded him. For if we put 
 
 A = Socrates ; B = master of Plato ; 
 
 C - one who fought at Delium, 
 
 the premises are certainly 
 
 (1) A = B; 
 
 (2) A = AC. 
 
 The conclusion (3) as it stands is B = BC, which negatives 
 only two combinations AB<r and aEc, whereas the premises
 
 XXI 
 
 QUESTIONS AND ANSWERS 
 
 215 
 
 negative in addition the three A&C, Abe, aEC. It is possible, 
 indeed, to draw the conclusion (5)8 = AC, which is better 
 than (3) by two combinations, namely, A3C and #BC. As 
 to the supposed proposition (4), it cannot be made into any 
 non-disjunctive proposition without a change of meaning ; 
 for, whether we make it into (6) A = BC, or A ABC, it 
 differs in force from the premises (i) and (2), which propo- 
 sitions in fact cannot be condensed into any single non-disjunc- 
 tive proposition of equivalent meaning. The fact is that the 
 supposed proposition (4) consists merely of the two (i) and 
 (2) re-stated in one compound sentence. It is not ' the 
 proposition' at all; it is 'the propositions.' The case 
 would have been considerably altered, indeed, had Mr. 
 Bain interpreted (i) as 'Socrates was a master of Plato,' of 
 the form A = AB. The type of the premises would then 
 have been essentially altered ; but that he does not so inter- 
 pret it is obvious from (3), in which we have ' the master,' 
 not 'a master.' Altogether this page of Professor Bain's 
 work affords remarkable evidence of the inability of a most 
 acute logician to maintain accuracy of logical vision without 
 the aid of some kind of calculus like that developed in the 
 latter part of this work. 
 
 I append logical diagrams which almost explain them- 
 selves, the combinations pointed out by each bracket being 
 those negatived by the proposition whose number is attached 
 to the bracket. 
 
 ABC 
 ABc 
 
 ABC 
 
 (2) ; 
 
 (0 
 
 Abe 
 
 -<iBc 
 abC 
 abc 
 
 (3) 
 
 (5) 
 
 AbC- 
 Abc - 
 BC 
 
 (6) 
 
 abC 
 abc
 
 216 EQUATIONAL LOGIC CHAP 
 
 50. How far does the conclusion of an Aristo- 
 telian syllogism fall short of giving all the 
 information contained in the premises ? 
 
 The premises of Barbara, say A = AB, B = BC, nega- 
 tive four combinations, AB<r, AC, A&r, a&c. The conclusion 
 A = AC negatives only two of these, namely, AB^r and 
 Afo Measured in this way, then, it contains only half of 
 the information of the premises ; but of course if the 
 conclusion gives just that information which is desired, the 
 overlooking of the rest is no harm. Enough is as good as 
 a feast or rather better. 
 
 Take the premises of a syllogism in Barbara, 
 such as (i) all As are Bs, and (2) all Bs are 
 Cs, and determine precisely what they affirm, 
 what they deny, and what they leave in 
 doubt. v 
 
 ABC 
 ABc- 
 
 aBC 
 aftc - 
 abC 
 abc 
 
 (2) 
 
 To answer this question, we must form 
 the eight combinations of A, B, C and 
 their negatives, as in the margin ; we 
 then strike out A<5C and Kbc as being in 
 conflict with condition (i), and AB<: and 
 aRc as being similarly in conflict with the 
 condition (2), that all Bs are Cs. There 
 remain four combinations, ABC, aBC, a/>C, 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 consistency (p. 181) that every separate term and its
 
 xxi QUESTIONS AND ANSWERS 217 
 
 negative shall remain. Hence there must exist some things 
 which are described by ABC, and other things described by 
 abc. But as regards the remaining two combinations, 0BC 
 and abC, the case is different ; for we may remove either, or 
 both of these without wholly removing any term. We 
 might add to the premises the new condition that all BCs 
 are As, or BC = ABC, which would negative BC ; or we 
 might add the condition, all Cs are As, or C = AC, which 
 would remove both BC and abC. 
 
 We may sum up the meaning of the original premises (i) 
 and (2) by saying that they deny the existence of AB*r, 
 A^G, A&-, and aRc ; that they affirm the presence or logical 
 existence of ABC and abc; and thirdly, while leaving BC 
 and abC uncontradicted, they are consistent with the presence 
 or absence of these two combinations. This is all that they 
 leave in doubt concerning the relations of A, B, and C. 
 
 52. What is the amount of contradiction in the 
 following celebrated epigram? 
 
 ' The Germans in Greek, 
 Are sadly to seek ; 
 
 * # * # 
 All, save only Hermann, 
 And Hermann's a German.' 
 
 Putting A = German ; B = Hermann ; 
 
 C = sadly to seek in Greek, 
 
 the premises are evidently 
 
 (1) A = AC. 
 
 (2) B = Br. 
 
 (3) B = AB.
 
 ABC 
 
 218 EQUATIONAL LOGIC CHAP. 
 
 The logical diagram is as in the margin ; it will be 
 noticed that B disappears entirely, indi- 
 cating contradiction ; but A remains in 
 the combination A3C. It is obvious that 
 the wit of the epigram arises from the per- 
 ception of contradiction. (See Hamilton's 
 Lectures, vol. iii. p. 393.) 
 
 ' 
 
 r aBC 
 
 (2) 
 
 abC, 
 abc 
 
 53. Show that you can make no assertion about 
 two terms A and B (and these only), which 
 is . not either contained in the assertion of 
 identity (A = B), or else contradictory thereto. 
 
 The proposition A = B removes two out of the four 
 combinations thus 
 
 ( Consistent Inconsistent 
 
 Combinations. Combinations. 
 
 AB. Kb. 
 
 ab. 0B. 
 
 Now, if any new assertion negatives either or both of Ab 
 and aB, it must be an assertion contained in and inferrible 
 from A = B. If it removes either AB or ab, it must con- 
 tradict A = B, because either A and B or a and b will then 
 disappear entirely from the Logical Alphabet. It might be 
 said perhaps that a new assertion could remove one consis- 
 tent and one inconsistent combination, for instance, ab and 
 A.b ; but this cannot be done except by a contradictory 
 assertion. Any other pairs such as AB and Ab, AB and 
 #B, or ab and B, being removed, removes some letter 
 entirely and involves contradiction.
 
 xxi QUESTIONS AND ANSWERS 219 
 
 54. Is it (i) logically (2) physically possible that 
 all material things are subject to the law of 
 gravity, and that at the same time all not 
 material things should be subject to the same 
 law ? [L.] 
 
 It is logically possible, that is to say, in accordance with 
 the Laws of Thought, that all things material and all things 
 not material should be subject to the law of gravity. In 
 this case what is not subject to the law of gravity would be 
 found among not-things. But it is not logically possible 
 that all (material things) and all not-(material things) should 
 be subject to the law of gravity, because this is equivalent 
 to denying the existence of any class not subject to the law 
 of gravity. This class would by one condition be not- 
 material, and by the other condition it would be material, 
 which is impossible. But by the law already described 
 (p. 181) as the Law of Infinity, every logical term must 
 be assumed to have its negative. The student is recom- 
 mended to work out this question with the aid of letter 
 symbols. 
 
 As to the second part of the question, what is not logically 
 possible is of course not physically possible. Hence we are 
 restricted to the inquiry whether it is physically possible that 
 all things material and all things not-material should be 
 subject to the law of gravity. This can only be answered 
 on logical grounds thus far, that if the property of gravita- 
 tion is essential to material things and forms a part of the 
 definition of them, then it is not possible that not-material 
 things should gravitate. As a matter of fact the possession 
 of inertia is perhaps the ultimate test of materiality ; but 
 gravity is proportional to inertia and is an equally good 
 test.
 
 220 EQUATIONAL LOGIC CHAP. 
 
 55- It is observed that the phenomena A, B, C 
 occur only in the combinations ABr, abC, and 
 abc. What propositions will express the laws 
 of relation between these phenomena ? 
 
 Of the eight combinations of A, B, C, only these three 
 remain. As we see that A occurs with and only with B, 
 and a with and only with b, it is firstly obvious that A = B 
 is the chief law. But as this law of relation leaves the com- 
 bination ABC uncontradicted, we must have a second law 
 to remove this, which may be either AB = AEc, or else 
 B = Be. Observe, however, that the laws A = B and 
 B = Br overlap and are pleonastic, because they both 
 deny that B can be aBC. Hence the simplest statement 
 of the laws of relation is 
 
 A = B. 
 AB = 
 
 56. Given three terms, for instance, water, blue, 
 and fluid, how would you proceed to ascer- 
 tain the utmost number of purely logical re- 
 lations which can exist among them ? [L.] 
 
 The relations of any three terms or things or classes of 
 things must be governed in the first place by the universal 
 Laws of Thought (p. 180). These laws restrict the com- 
 binations of three things, present or absent, to eight at the 
 utmost ; for each thing may be present or absent giving 
 2x2x2 = 8 cases. But any special logical relation 
 which may exist between the things has the effect of 
 further restricting these combinations ; the relation that 
 water is a fluid, prevents the existence of the combination
 
 xxi QUESTIONS AND ANSWERS 22J 
 
 water, not-fluid. Conversely the removal from the series of 
 any one or more of the eight combinations expresses the 
 existence of a relation or relations negativing the existence 
 of these combinations. Thus, the removal of the two 
 combinations water, not-blue, fluid ; water, not-blue, not- 
 fluid, expresses the law that all water is blue. Thus the 
 logical meaning of any condition is represented by the state 
 of the combinations agreeing with those conditions. It 
 follows that the utmost possible number of distinct logical 
 relations will be ascertained by taking the eight possible 
 combinations of the three terms and striking out one or 
 more of the combinations in every possible variety of ways. 
 The number of these ways cannot exceed 256 ; for each of 
 the eight combinations may be either present or absent, 
 giving 2x2 X2X2 x2 X2X2X2 = 256 ways. 
 But this calculation will include many cases where one or 
 more of the three terms and their negatives disappear alto- 
 gether, representing contradiction in the conditions. Many 
 different selections, too, proceed from logical relations 
 similar in character and form ; thus the law A = AB is 
 similar to A = A, and to a = ab ; the law A = BC-|-& : 
 is similar to C = AB-|-^/ and so forth. The investiga- 
 tion is fully described in the Principles of Science (pp. 134- 
 143; ist ed. vol. i. pp. 154-164) as also in the Memoirs 
 of the Manchester Literary and Philosophical Society, Third 
 Series, vol. v. pp. 119-130. It is found that the 256 pos- 
 sible selections are thus accounted for 
 
 Proceeding from consistent logical conditions 192 
 inconsistent ,, 63 
 
 no condition at all i 
 
 256
 
 222 
 
 EQUATIONAL LOGIC 
 
 CHAP. XXI 
 
 The consistent logical conditions are found, however, on 
 careful analysis to fall into no more than fifteen distinct 
 forms, or types of relation, which are stated in the following 
 table 
 
 Reference 
 Number. 
 
 Propositions expressing the 
 general type of the logical 
 conditions. 
 
 Number of dis- 
 tinct logical 
 variations. 
 
 Number of 
 combinations 
 contradicted 
 by each. 
 
 I. 
 
 A = B 
 
 6 
 
 4 
 
 II. 
 
 A = AB 
 
 12 
 
 2 
 
 III. 
 
 A = B, B = C 
 
 4 
 
 6 
 
 IV. 
 
 A = B, B = BC 
 
 24 
 
 r 
 
 V. 
 
 A = AB, B = BC 
 
 24 
 
 4 
 
 VI. 
 
 A = BC 
 
 24 
 
 4 
 
 VII. 
 
 A = ABC 
 
 24 
 
 3 
 
 VIII. 
 
 AB=ABC 
 
 8 
 
 i 
 
 IX. 
 
 A = AB, aB = aEc 
 
 24 
 
 3 
 
 X. 
 
 A=ABC, at=al>C 
 
 8 
 
 4 
 
 XI. 
 
 AB = ABC, ab=al>c 
 
 4 
 
 2 
 
 XII. 
 
 AB = AC 
 
 12 
 
 2 
 
 XIII. 
 
 A = BC-|-Afc 
 
 8 
 
 3 
 
 XIV. 
 
 A = BC-l-fe 
 
 2 
 
 4 
 
 XV. 
 
 A = ABC, a = a%c -\-abC 
 
 8 
 
 5
 
 CHAPTER XXII 
 
 ON THE RELATIONS OF PROPOSITIONS INVOLVING THREE 
 OR MORE TERMS 
 
 i. THE doctrine of the opposition of propositions, exhi- 
 bited in the well-known square, is an important and inter- 
 esting fragment of ancient logic ; but it is now apparent 
 that propositions involving only two terms one subject and 
 one predicate, do not sufficiently open up the question of 
 the relationship of propositions. Two terms admit of only 
 four combinations, and these can be present and absent 
 only in sixteen ways, nine of which involve contradiction. 
 There remain only seven cases of logical relation which 
 resolve themselves into only two distinct types of propo- 
 sition. (Principles of Science, pp. 134-7; ist ed. vol. i. 
 pp. 154-7.) With the introduction of a third term the 
 sphere of inquiry becomes immensely extended. There 
 are now, as we have seen (p. 221) 193 different cases of 
 selection of combinations resolving themselves into fifteen 
 distinct types of relation. The possible modes of relation 
 of one proposition to another, including under the expres- 
 sion ' one proposition ' any group of propositions, become 
 considerably complex. Such modes of relation seem to 
 be seven in number : thus one proposition is as regards 
 another
 
 224 EQUATIONAL LOGIC CHAP. 
 
 (1) Equivalent. 
 
 (2) Inferrible, or contained in the other, but not equi- 
 
 valent. 
 
 (3) Partially inferrible and otherwise consistent. 
 
 (4) Consistent but indifferent and not inferrible. 
 
 (5) Partially inferrible, partially contradictory. 
 
 (6) Partially indifferent, partially contradictory. 
 
 (7) Contradictory. 
 
 2. Let us take as an example the proposition 
 Steam = aqueous vapour, 
 
 and give a pretty complete analysis of its related pro- 
 positions. 
 
 Let A = steam ; 
 B = aqueous ; 
 C = vapour. 
 
 The proposition being evidently of the form 
 
 (i) A = BC, 
 the combinations contradicted will be as in the margin. 
 
 The equivalent proposition will be 
 AC ! Not steam = not aqueous or not vapour. 
 Me i An inferrible but not equivalent assertion 
 BC 
 
 (0 
 
 w iij b e anv one which negatives one, two, 
 or three, but not four of the combinations 
 negatived by (i). There will therefore be 
 
 4x3 4x^x2 
 
 4 + - + -or 14 such infer- 
 
 1x2 1x2x3 
 
 rible and logically distinct propositions. We may infer 
 steam is aqueous ; steam is vapour ; what is not vapour 
 is not steam ; what is not aqueous is not steam ; non- 
 aqueous vapour is not steam ; and so forth.
 
 xxn RELATIONS OF PROPOSITIONS 225 
 
 The third class of related propositions will include those 
 which negative one or more of the excluded combinations, 
 and one or more indifferent combinations. Indifferent 
 combinations, as the name expresses, are those which can 
 be removed without wholly removing any of the letters 
 A, B, C, a, , c. In this case any one of the remaining 
 combinations except ABC may be singly removed. Thus 
 'not .steam is not aqueous,' or in letters a-ab, is not 
 contradictory to (i) and it may be inferred from (i) in 
 respect of vapour which is not steam. But the assertion 
 that other things which are not steam are not aqueous is 
 not inferrible, but is consistent with (i). A proposition, 
 again, which should negative A<$C, A.bc, aBt, abC will be 
 inferrible in respect of the two former, and consistent in 
 respect of the two latter combinations. To ascertain what 
 such proposition is we must look in the Logical Index, 
 afterwards described, for the proposition which leaves a /3 e 6, 
 and we find in the 55th place b = ac, or not-aqueous = 
 not-steam and not-vapour. The other possible propositions 
 of the same class are numerous and various. 
 
 To obtain one of the fourth class, which is merely con- 
 sistent and indifferent, we must take any one or more ot 
 the combinations unnegatived by (i), for instance a^C, in 
 such a way as not wholly to remove any letter. Thus 
 ab = abc t or ' not-steam which is not aqueous is not vapour ' 
 is an assertion quite indifferent to (i). So is the assertion 
 aB = rtC (Logical Index, No. 7). 
 
 Contradictory propositions being defined as those which 
 wholly remove any term, such will be any one which re- 
 moves ABC. Thus to say that steam is not aqueous is a 
 case of the 5th class; it is inferrible from (i) in respect of 
 steam which is not vapour (AB*r), but it is contradictory 
 because it also negatives steam which is vapour, 
 
 Q
 
 226 EQUATIONAL LOGIC CHAP, xxn 
 
 A proposition of the sixth class is discovered by taking 
 any combination which may be spared with one which 
 cannot, such as abC and ABC, and looking in the index, 
 we find AC = bC, or steam-vapour is identical with non- 
 aqueous vapour, as a partially consistent, partially contra- 
 dictory proposition as regards (i). It may or may not be 
 true that what is not steam and not aqueous is not vapour, 
 but it is contradictory to (i) to say that vaporous steam is 
 not aqueous. 
 
 An example of a simply contradictory proposition of 
 Class 7 is found in one which removes ABC only, such as 
 AB = ABc / again a - aB, or not-steam is aqueous deletes 
 b ; c = ABc deletes c. 
 
 2. As a second example, let us take the propositions 
 (i) Hand = right-hand or left-hand; (2) Right is not left. 
 
 Putting A = hand ; B = right ; C = left ; the conditions 
 are evidently 
 
 (i) A = AB-|-AC. 
 (a) B = Be. 
 
 The consistent combinations are shown in the margin, 
 
 ABC anc ^ tne student may verify the following 
 
 list, which gives one specimen of each of 
 (2) the seven classes of related assertions, the 
 reference number of the Logical Index 
 aBc being also added. 
 
 ^C (i) Equivalent. B = Be; be abc. No. 153. 
 
 abc (a) Inferrible, etc. aB = aBc. No. 9. 
 
 (3) Partially inferrible, etc. a = abc. No. 15. 
 
 (4) Consistent, etc. AB = ABC; ab = abc. No. 67. 
 
 (5) Partially inferrible, etc. C = AC ; A = AB. No. 59. 
 
 (6) Partially indifferent, etc. A - ABc ; ab --= abc. No. 179. 
 
 (7) Contradictory. b = bc. No. 35.
 
 CHAPTER XXIII 
 
 EXERCISES IN EQUATIONAL LOGIC 
 
 I NOW give a small collection of examples and problems 
 designed to enable the student to acquire a complete com- 
 mand of the equational and combinational views of logic. 
 They are for the most part devised specially for this book, 
 but a few have been utilised in examination papers, and a 
 few have been adopted as indicated from the papers of other 
 examiners. These questions form perhaps a partial answer 
 to Professor Sylvester's remark, as quoted in the preface, 
 especially when we observe that the questions and problems 
 involving the relations of three terms can be multiplied 
 almost ad infinitum, without resorting to like questions in- 
 volving four, five, or more terms. The student will readily 
 gather that the number, variety, and complexity of problems 
 in pure logic is simply infinite, and is such as we gain no 
 glimpse of in the old Aristotelian text-books. 
 
 i. Represent equationally the following assertions : 
 
 (1) With the exception of porcelain there is no non- 
 
 metallic substance which has been employed to 
 make coins. 
 
 (2) With the exception of zinc and the metals discovered 
 
 during the last hundred years, there is no metal 
 which has not been employed to make coins.
 
 228 EQUATIONAL LOGIC CHAP. 
 
 (3) ' The worth of that is that which it contains, 
 
 And that is this, and this with thee remains.' 
 
 [SHAKESPEARE.] 
 
 (4) It is dangerous to let a man know how far he is but 
 
 a brute, without showing him also his grandeur. 
 It is dangerous again to let him see his grandeur, 
 without his baseness. It is [even more] dangerous 
 to leave him ignorant in both ways ; but it is a 
 high advantage to represent to him both the one 
 and the other. (Pascal, Pensees.} 
 
 2. Represent in the forms of equational logic any of the 
 following arguments : 
 
 (1) Milton was a great poet, and a fearless opponent of 
 
 injustice ; a great poet should be honoured ; a 
 fearless opponent of injustice should be honoured : 
 therefore Milton should be honoured. 
 
 (2) The virtues are either passions, faculties, or habits: 
 
 they are not passions, for passions do not depend 
 on previous determination ; nor are they faculties, 
 for we possess faculties by nature ; therefore they 
 are habits. 
 
 (3) There can be no person really fit to exercise absolute 
 
 power, because the qualifications requisite to fit a 
 person for such a position would consist in native 
 talent combined with early training ; now such a 
 talent cannot be possessed in early childhood. 
 (Suggested by De Morgan, Syllabus, p. 67.) 
 
 (4) One of the masters of chemistry was Berzelius ; 
 
 Berzelius was a Swede ; 
 
 One of the masters of chemistry was a Swede. [D.] 
 
 (5) This heavenly body is either a planet or a fixed star ;
 
 xxni EXERCISES 229 
 
 all fixed stars twinkle ; planets do not twinkle ; 
 this body twinkles, therefore it is a fixed star. 
 (6) Show me any number of men, and I will say with 
 confidence, either that they will with one accord 
 raise their voices for liberty, or that there are 
 aliens among them. (The stump orator's mode, 
 according to De Morgan, of saying that all 
 Englishmen are lovers of liberty.) [B.] 
 
 3. Infer all that you possibly can, by way of contra- 
 position or otherwise, from the assertion 'all A that is 
 neither B nor C is D.' [R.] 
 
 4. Express equationally Miscellaneous Example No. 39 
 in Elementary Lessons in Logic, p. 317. 
 
 5. What proposition concerning nebulae and vaporous 
 bodies leaves doubtful the existence of a class of things 
 which are neither nebulae nor vaporous bodies ? 
 
 6. Represent the fact that A differs from B in two equiva- 
 lent equational propositions. 
 
 7. Prove equationally that the proposition, All elements 
 are either metal-elements or elements, is a mere truism. 
 
 8. What is the difference between the propositions 
 A = AB-|-A, B- AB-I-B, and A = B-|-A? 
 
 9. Prove that if all not-Bs are not-As, and all Bs are As, 
 then A B, and vice versa. 
 
 10. Show that the negative premises No As are Bs and 
 no Cs are Bs, imply the logical existence of a class B which 
 is neither A nor C. 
 
 1 1. Prove the equivalence of the following assertions : 
 
 (1) 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.
 
 230 EQUATIONAL LOGIC CHAP. 
 
 12. Show that if metals which are either not valuable or 
 not destructible are unfitted for use as money, it follows 
 that destructible metals which are fitted for use as money 
 must be valuable. 
 
 13. Does the proposition A = B-|-BC differ in force 
 from A = B ? 
 
 14. ' All animals having red blood corpuscles are identical 
 with those having a brain in connection with a spinal cord.' 
 What is the description you may draw from this proposition 
 of things having a brain not in connection with a spinal 
 cord? 
 
 15. Luminous body is either self-luminous or luminous 
 by reflection ; melted gold is both self-luminous and lumi- 
 nous by reflection. Unmelted gold is not self-luminous 
 but is luminous by reflection. Represent these premises 
 symbolically, and draw descriptions of the terms (i) lumi- 
 nous body, (2) self-luminous body, (3) body luminous by 
 reflection, (4) body not luminous, (5) body not self-luminous, 
 (6) not melted gold, (7) not unmelted gold. 
 
 1 6. ' There are no organic beings which are devoid of 
 carbon.' Determine precisely what this proposition affirms, 
 what it denies, and what it leaves doubtful. 
 
 1 7. Prove the equivalence of the following statements 
 No right-angled triangles are equilateral ; no equilateral 
 triangles are right-angled ; no right-angled equilateral figures 
 are triangles. 
 
 1 8. All scalene triangles have their three angles equal to 
 two right angles. What are the least or simplest assertions 
 which added to the above will make it equivalent to ' All 
 triangles are all figures which have their three angles equal 
 to two right angles ' ?
 
 xxin EXERCISES 231 
 
 19. All equal -sided squares have four right angles. 
 What is the least extensive proposition which added to the 
 above makes it equivalent to ' All squares are equal-sided 
 and have four right angles ' ? 
 
 2 o. If an orator were to assert that Afghanistan is a very 
 poor country, but it is essential to the security of India, but 
 a reporter were to consolidate these two assertions into the 
 one assertion that a very poor country, Afghanistan, is the 
 Afghanistan which is essential to the security of India, how 
 far would the reporter have misrepresented the logical 
 meaning of the orator ? 
 
 21. Express the following argument equationally : Every 
 organ of sense has nervous communication with the brain ; 
 for such is the case with all the five organs of sense, the 
 eye, ear, nose, tongue, and skin. 
 
 22. If requested to draw from the assertion 'All coal 
 contains carbon' a description of the term 'metal,' what 
 answer would you give ? 
 
 23. What values will you obtain for the terms man, 
 brute, and gorilla, under the conditions that a gorilla is a 
 man, and that all men are included and all gorillas excluded 
 from the class of non-brutes ? 
 
 24. Assuming that armed steam- vessels consist of the 
 armed vessels of the Mediterranean and the steam-vessels 
 not of the Mediterranean, inquire whether we can thence 
 infer the following results : 
 
 (1) There are no armed vessels except steam- vessels in 
 
 the Mediterranean. 
 
 (2) All unarmed steam-vessels are in the Mediterranean. 
 
 (3) All steam- vessels not of the Mediterranean are armed.
 
 232 EQUATIONAL LOGIC CHAP. 
 
 (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. (Boole, ' The Calculus of Logic,' 
 Cambridge and Dublin Mathematical Journal, 
 1848, vol. iii. pp. 199-201.) 
 
 25. How would you otherwise describe the class of things 
 which are excluded from the class of white, malleable, 
 metals ? 
 
 26. Show that the description of the class of things 
 which are not (either A, or if not A then both B and C), is 
 as follows either not- A and not-C, or if it be C then both 
 not-A and not-B. 
 
 27. How do any two of the three equations A = B 
 B = C, C = A, differ in logical force from the third ? 
 
 28. Frame a sorites with one premise negative and one 
 particular, and represent it equationally. 
 
 29. Contrast the logical force of each of the proposi- 
 tions A = AB.|-AC-|-AD-|. . . and A = ABCD . ., 
 with that of the group of propositions A = AB, A = AC, 
 A = AD, etc. ; point out, moreover, which can be inferred 
 from which. 
 
 30. Show that, under the condition of our Criterion of 
 Logical Consistency (p. 1 8 1 ) the assertion that there are no 
 such things as fresh-water foraminifera, involves the asser- 
 tion that there are foraminifera which are not fresh-water 
 beings, and fresh-water beings which are not foraminifera, 
 but leaves doubtful the occurrence of things which are 
 neither fresh-water beings nor foraminifera. 
 
 3 1 . From the premises, ' All gasteropods are mollusca, 
 and no mollusca are vertebrates,' obtain descriptions of the 
 classes gasteropods and invertebrates.
 
 xxiii EXERCISES 233 
 
 32. 'Eloquence should contain both what is agreeable, 
 and what is real ; but what is agreeable should be real ' 
 (Pascal, Pensees). Represent the above symbolically, 
 putting A = component of eloquent speech, B = agreeable, 
 C = real. 
 
 33. Assuming it to be known that all mammals have red 
 blood corpuscles, and that they also have vertebrae, invent 
 five or six other distinct assertions which you might make 
 about mammals, the possession of red blood corpuscles, and 
 the possession of vertebrae, including of course the negatives 
 of these terms, without coming into logical conflict with the 
 known relations of the terms as above stated. 
 
 34. How would you otherwise describe the class of things 
 which are excluded from the class of non-crystalline solids 
 which are either non-metallic non-conductors, or else metal- 
 lic conductors, and which are moreover either brittle and 
 in that case useless for telegraphy, or else malleable and in 
 that case useful for telegraphy ? 
 
 35. Compare the following propositions : 
 
 (1) Xis Y. 
 
 (2) X is Y and is in some ca'ses Z, and in some cases 
 
 not Z. 
 
 By the law of excluded middle we know that X must be 
 either Z or not Z. Is then the sentence (i) precisely 
 identical in logical force with (2)? Compare now the 
 following definitions : 
 
 (3) A right-angled triangle is that which has a right 
 
 angle. 
 
 (4) A right-angled triangle is that which has a right angle, 
 
 and of which two sides are or are not equal. 
 Are these definitions precisely identical in logical force ? 
 
 [C.]
 
 234 EQUAT1ONAL LOGIC CHAP. 
 
 36. What is the difference between saying that sea- water 
 is drinkable and not scarce, and saying that drinkable sea- 
 water is. not scarce ? 
 
 37. If from the premises 'All rectangles are parallelo- 
 grams,' and ' Parallelograms consist of all four-sided figures 
 whose opposite sides are parallel,' we infer that all rectangles 
 are parallelograms, being four-sided figures with opposite 
 sides parallel, how far does this inference fall short of being 
 equivalent to the premises ? 
 
 38. To say that Adam Smith is the father of Political 
 Economy and a Scotchman is as much as to say that he is 
 a Scotch father of Political Economy, and that no one but 
 he can be a father of the science. Give the symbolic proof 
 of this equivalence. 
 
 39. To lay down the condition that what is either A or 
 else B, is what is both A and B or else both A and C and 
 vice versa, is to state disjunctively what may be laid down 
 in two non-disjunctive propositions asserting that A without 
 B is C and also B must be A. 
 
 40. Reduce the two assertions A = Kbc and a - ac to a 
 single one. 
 
 41. Give a good many inferences from the proposition 
 A = B ( AC, and also equivalents, distinguishing carefully 
 between those inferences which are equivalent and those 
 which are not. 
 
 42. Develop symbolically the term Plant (A) with refer- 
 ence to the undermentioned terms (B, C, D, E, F), under 
 the conditions that acotyledonous (li) plants are flowerless ; 
 (c) monocotyledonous (D) plants are parallel -leaved (E) ; 
 dicotyledonous (F) plants are not parallel -leaved ; and
 
 xxni EXERCISES 235 
 
 every plant is either acotyledonous, monocotyledonous, or 
 dicotyledonous, but one only of these alternatives. 
 
 43. Completely classify triangles under the following 
 conditions 
 
 (1) Equilateral triangles are isosceles. 
 
 (2) Scalene triangles are not isosceles. 
 
 (3) Obtuse-angled triangles are not right-angled. 
 
 (4) Acute-angled triangles have three acute angles. 
 
 (5) Obtuse-angled triangles have not three acute angles. 
 
 (6) Equilateral triangles are not right-angled. 
 
 What other conditions must be added to comply with the 
 results of geometrical science ? 
 
 44. Among plane figures the circle is the only curve of 
 equal curvature. Show that this is the same as to assert 
 that a plane figure must either be a curve of equal curva- 
 ture, in which case it is also a circle, or else, not a circle 
 and then not a curve of equal curvature. 
 
 45. Which of the following propositions are equivalent to 
 the first in the list ? 
 
 (1) Crystallised carbon is not a conductor. 
 
 (2) Carbon which conducts is not crystallised. 
 
 (3) Conducting crystallised substance is not carbon. 
 
 (4) Conductors are either not carbon or not crystallised 
 
 substances. 
 
 (5) Carbon is either not a conductor or not crystallised. 
 
 (6) Conductors which are not carbon are crystallised. 
 
 (7) Crystals are either non-conductors or not composed 
 
 of carbon. 
 
 (8) Crystallised conductors are carbon. 
 
 46. Prove that any set of exclusive alternatives combined 
 with part of that set produces only that part.
 
 236 EQUA1IONAL LOGIC CHAP. 
 
 47. Show that the conclusions of Celarent, Cesare, 
 Camestres, and Camenes give in each case only half the 
 information contained in the premises. 
 
 48. Verify by various trials the statement that no inference 
 by substitution within a group of propositions can negative 
 combinations not negatived by the group of premises. 
 
 49. Show that Cesare and Camestres belong to the same 
 type of assertion as Barbara and Celarent. 
 
 50. Assign the premises of the following moods of the 
 Syllogism to their proper types of assertion: Darapti, 
 Bramantip, Camenes. 
 
 51. Prove that any proposition which is contradictory to 
 ' common salt = sodium chloride,' can be inferred, so far 
 as it is contradictory, from the assertion ' common salt = 
 what is not sodium chloride.' 
 
 52. Does it or does it not follow that any proposition of 
 the mth type (see pp. 221-2) will always be equally con- 
 tradictory to one of the nth type ? 
 
 53. Refer to Boole's Laws of Thought, pp. 146-9, 
 and taking the premises of the complex problem there 
 solved to be expressed in our system as follows : 
 
 (1) ac = acE(Bd-\-bV); 
 
 (2) AD* = AD* (BC !&); 
 
 (3) A(B.|'E) = Gf-KD; 
 
 work out the consistent combinations, and infer descriptions 
 of the classes B, AC, AC*, D, e, AB, AB*, ab, AE, ACE, 
 BD, DE, D*, C, CD, etc. Verify by showing that D and e 
 multiplied together give D* and so forth. 
 
 54. If Brown asserts that all metals are reputed elements, 
 and that all reputed elements will be ultimately decomposed, 
 whereas Robinson holds that all metals are reputed elements
 
 xxiii EXERCISES 237 
 
 which will be ultimately decomposed, what is the exact 
 amount of logical difference between them ? 
 
 55. Compare the logical force of all the following pro- 
 positions, and point out which pairs are equivalent, and 
 which may be inferred from other ones. 
 
 (1) A square is an equal-sided rectangle. 
 
 (2) What is not equal-sided is not square. 
 
 (3) What is not square is not equal-sided. 
 
 (4) Equal-sided rectangles are squares. 
 
 (5) No rectangle which is not equal-sided is square. 
 
 (6) A square can be neither unequal-sided nor anything 
 
 but a rectangle. 
 
 (7) An unequal-sided square does not exist. 
 
 56. Taking letters to represent qualities thus: A = 
 having metallic lustre ; B = malleable ; C = heavier than 
 water ; D = white coloured ; E = fusible with difficulty ; 
 F = conducting electricity ; form descriptions of each of 
 the metals gold, silver, platinum, copper, iron, lead, tin, 
 zinc, antimony, sodium, and potassium, and then exhibit 
 the extension of the following classes : AB ; BC ; BCD ; 
 BCF; Ab; be; Kd ; and so forth. 
 
 5 7. Express symbolically the following classes of things 
 
 (1) Hard, wet, black, round, heavy, stone. 
 
 (2) Thing which is hard, wet, either black or red, but 
 
 not round, and either heavy or not heavy. 
 
 (3) Thing which is either not hard, or not wet, or not a 
 
 stone, but is either black and then round, or heavy 
 and then a stone. 
 
 58. Referring to the Principles of Science (pp. 75-76 ; ist 
 ed. vol. i. p. 90), develop all the alternatives of A as limited 
 by the description 
 
 A = AB{C-|.D(E.|.F)}
 
 238 EQUATIONAL LOGIC CHAP. 
 
 and infer descriptions of the following terms, Ace, Acf, AEcD. 
 (See De Morgan, Formal Logic, p. 1 1 6 ; Third Memoir on 
 the Syllogism, p. 12 in the Camb. Phil. Trans., vol. x.) 
 
 59. Represent this argument symbolically : A straight 
 line can cut a circle in two points, and similarly an ellipse, 
 and a hyperbola ; but these are all the possible kinds of 
 conic sections ; therefore a straight line can cut any conic 
 section in two points. 
 
 60. It being understood (i) that only the congenitally 
 deaf are mute ; (2) that an uneducated deaf person is mute, 
 but uses signs ; (3) that an educated deaf person is not 
 mute, and does not use signs : express these conditions 
 symbolically and describe the classes of persons who are 
 deaf ; mute ; deaf-mutes ; educated persons, etc. 
 
 61. Show how by the process of substitution alone to 
 sum up into one disjunctive proposition the assertion that 
 John is mortal ; Thomas is mortal ; William is mortal. 
 
 62. Prove that the premises of syllogisms in the moods 
 Darapti and Felapton can be expressed in the form of a 
 single non-disjunctive proposition, and assign its type. Show 
 also that this is not the case with the moods of the other 
 three figures. 
 
 63. Prove that the following propositions or groups of 
 propositions involve self-contradiction : 
 
 ( A B = AB 
 
 (3) A = AB; B = BC; C = aC. 
 
 64. Analyse the force of Hamilton's form of proposition, 
 ' Some A is not some A,' putting for ' some ' and ' some ' 
 respectively the letter terms P and Q.
 
 xxin EXERCISES 239 
 
 65. What does the assertion ' Some things are neither A 
 nor B ' tell us about things which are not-A ? 
 
 66. How far do the conclusions of the syllogisms in 
 Darapti, Felapton, Bramantip, Camenes, and Fesapo, as 
 deduced on p. 188, respectively fall short of containing all 
 the information given in the premises ? 
 
 67. Show that C = AC-|-aBC is equivalent to the two 
 propositions, AB = AB^r and ab = abc. Name the type. 
 
 68. To say that whatever is devoid of the properties 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 D, but possesses those of C, must 
 have A. Prove this. 
 
 69. What statement or statements must be added to the 
 proposition, ' What is not a square is either not equal-sided 
 or not a rectangle,' in order to make the assertions in the 
 whole equivalent to the definition of a square that it is an 
 equal-sided rectangle? 
 
 70. What is the difference between the assertion A = 
 ABC and the pair of assertions b = al>, and c = be ? 
 
 71. Prove that from one of the propositions, A = ABC, 
 and AB = ABC, we can infer the other, but not vice 
 versa, and point out which is the one which can be so 
 inferred. 
 
 72. Give three logical equivalents to the proposition, 
 ACD = ACD. 
 
 73. Demonstrate the equivalence of A = AB-I-AC with 
 Kb = AC and with he = ABr. 
 
 74. Show how by substitution alone to obtain A = AB 
 from A = ABCD ; also obtain A = AC and A = AD. (Prin- 
 ciples of Science, p. 58 ; ist ed. vol. i. p. 69.)
 
 240 EQUATIONAL LOGIC CHAP. 
 
 75. Verify the statement that any set of alternative terms 
 combined with the same set, reproduces that set that is to 
 say, show that AA = A when for A we substitute any one 
 of the following terms : 
 
 ABC -\-aEC -\-a6c ; 
 
 76. Show by trial that if in any pair of logically equivalent 
 assertions such as A = Ab and B = #B, we substitute 
 for A and B any logical expressions, such as CD for A, 
 and CE for B, and their negatives in like manner for the 
 negatives of A and B, we always obtain new equivalent 
 assertions. 
 
 77. As a further example of equivalent assertions take 
 the following pair of propositions : 
 
 AB = ABC, 
 A^ = Abe, 
 and substitute as follows : 
 
 A = PQ, B = Qr, C = PR. 
 
 78. Express a ab and Ab AbC in the form of a 
 single disjunctive proposition. To what type does it 
 belong ? 
 
 79. Express equationally De Morgan's forms of propo- 
 sition {Formal Logic, p. 62). 
 
 (1) Everything is either A or B ; 
 
 (2) Some things are neither As nor Bs. 
 
 80. Verify the identical equations
 
 xxni EXERCISES 241 
 
 8 1. Verify the following equivalences as transcribed from 
 De Morgan's Syllabus, p. 42 : 
 
 = (B.|.QD, JA = B.|.Q(D.|.E), 
 
 a = bc-\-d ; \a = bC'\*de; 
 
 ( 
 ( 
 
 fA = BC-i-D, f A=B.|-C(D.|-E), 
 
 \a =(b-\-c)d; \a = b(c-\-de); 
 
 j A = (B -I- CD) (E .|- FG), f A; - B -|- C -I- *D. 
 
 \a = bc-\'bd'\>ef>\-eg ; \a = bcd. 
 
 82. State all the propositions involving only the terms 
 named which can be inferred from the equation, Stone = 
 rock ; and all the propositions which are equivalent to this 
 one, Stone = stone-rock. 
 
 83. Show how by the mere process of substitution you 
 can draw the proposition A = AD from ^the three propo- 
 sitions A = AB, B = BC, and C = CD. 
 
 84. What propositions added to A = AD are exactly 
 equivalent in meaning to A = AB, B = BC, and C = 
 CD jointly ? 
 
 85. If both A and B have the property C, but A never 
 occurs where D is, and B never occurs where D is absent, 
 what is your description of the class of things which are 
 devoid of the property C ? 
 
 86. The proposition A = A (B-|-C) being equivalent to 
 b = ab-\'AbC, verify this truth by showing that it holds 
 good when for A we substitute the term P^'I'/Q, for B 
 the term QRS, and for C the term ^Rj. 
 
 87. If a person were, correctly or incorrectly, to define 
 Members of Parliament (including Lords and Commons) as 
 either peers not chosen by election, or else not-peers chosen 
 by election, that is as much as to assert both that all members 
 are non-elected peers and elected non-peers, as well as that 
 
 R
 
 242 EQUATIONAL LOGIC CHAP. 
 
 all who are not members comprise the two classes of persons 
 who are neither peers nor elected persons, and those who 
 being peers have been elected but cannot sit. 
 
 88. It is not correct to say that because what is not A, 
 but is B, is also C, therefore everything that is both B and 
 C is A ; but what further conditions may be laid down 
 about the same things which will render these propositions 
 convertible ? 
 
 89. Into what other equivalent forms might we throw the 
 joint statements that Venus is a minor planet, and minor 
 planets are all large bodies revolving round the sun in 
 slightly elliptic orbits within the earth's orbit? 
 
 90. If B is always found to coexist with A, except when 
 X is Y (which it commonly, though not always, is), and if, 
 even in the few cases where X is not Y, C is never found 
 absent without B being absent also, can you make any 
 other assertion about C ? [R.] 
 
 91. If whenever X is present, Z is not absent, and some- 
 times 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 determined as between Z and Y ? 
 
 [R.] 
 
 92. If it is false that the attribute B is ever found co- 
 existing 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.] 
 
 93. Referring to the Elementary Lessons in Logic, 
 p. 196, from the premises there given (A = B -|- AC, B = BD, 
 C = CD), derive descriptions of the terms BC, a, l>, d. 
 
 94. From the important problem of Boole, described on 
 p. 197 of the same lesson, with the premises A = CD, 
 BC = BD, derive descriptions of the terms BC, C, 
 B, b y d.
 
 xxiii EXERCISES 243 
 
 95. In reference to this last named problem, examine 
 each of the following assertions, and ascertain which of 
 them are consistent with the premises A = CD, BC = BD 
 
 ( i ) ac = acY). (4) cd - acd. 
 
 (a) a = acd. (5) A 
 
 (3) ACD = ABCD. (6) abc 
 
 96. The premises AB - ABC, A = AB, and A = Ac, 
 involve self-contradiction. What is the least alteration which 
 will remove this contradiction ? 
 
 97. If AB = CD, what is^the description of BD, of bd 
 and of cdt 
 
 98. What must we add to the premises, All As are Bs 
 and all Bs are Cs, in order that we may establish the rela- 
 tion that what is not A is not C ? 
 
 99. Verify the assertion (Principles of Science, p. 141; 
 first edit. vol. i. p. 162) that the six following propositions 
 are all of exactly the same logical meaning : 
 
 A = BC -I- be a = bC -|- Br. 
 
 B = AC -I- ac b = Ac -\- aC. 
 
 100. Write out five similar logical equivalents of the pro- 
 position r = PQ -\-pq. 
 
 i o i . Prove that ab a&C is equivalent to ac acB, and 
 AB - AC to A = ABC -I- Me. 
 
 102. How may the condition A -|- B = ACD -|- BCD be 
 expressed in four non-disjunctive equations ? 
 
 103. Verify the equivalence of M - M and N -
 
 244 EQUATIONAL LOGIC CHAP. 
 
 when for M and N we substitute successively the following 
 pairs of values : 
 
 I N = ABC. t N = 
 
 I = ACD ( AM) -I- abCd, 
 
 104. Express each of the following propositions equa- 
 tionally in a series of non-disjunctive propositions : 
 
 1 i ) Either the king is dead, or he is now on the march. 
 
 (2) Either compression or expansion will produce either 
 heat or cold in a solid body. 
 
 (3) AH-3C = GM-d). 
 
 (4) AB -I- AC = (AB -I- AC) (GH- cD). 
 
 105. In problem 20 (chap. xxi. p. 194) what description 
 should we obtain of the classes c, those who do not take 
 snuff, and d, those who do not use tobacco, respectively 
 under the several conditions (i), (2) and (3), with (4)? 
 
 1 06. In problem 29, pp. 200-1, draw descriptions of 
 the classes Ac, ad, and cD. 
 
 107. Represent symbolically the logical import of the 
 sentence : ' If it be erroneous to suppose that all certainty 
 is mathematical, it is equally an error to imagine that all 
 which is mathematical is certain.' 
 
 1 08. Represent equationally the logical import of this 
 extract from the Oath of Supremacy : ' No foreign prince, 
 prelate, person, state, or potentate, hath any jurisdiction, 
 power, superiority, pre-eminence, or authority, ecclesiastical 
 or spiritual, within this realm.' Observe especially how 
 far the alternatives are or are not mutually exclusive.
 
 xxin EXERCISES 245 
 
 109. Take the following syllogism in Datisi : 
 
 All men are some' mortals ; 
 
 Some" men are some ili fools ; 
 Therefore, Some iv fools are some v mortals ; 
 and analyse equationally the meanings of the word ' some ' 
 as it occurs five times. Show which of the ' somes ' if any 
 are exactly equivalent. Compare the result with the 
 remarkably acute analysis of this mood given by Shedden, 
 in his Elements of Logic, 1864, pp. 131-2. 
 
 no. If some Xs are Ys, and for every X there is some- 
 thing neither Y nor Z, prove that 'some things are neither 
 Xs nor Zs. [DE MORGAN.] 
 
 in. Solve equationally Boole's example of analysis of 
 Clarke's argument (see Laws of Thought, Chap, xiv.) 
 The premises may be thus stated : 
 
 f ABD = O. B/= O. 
 
 \Kbd =O. AF = O. 
 
 CDE = O. Ae = O. 
 
 112. Show that every equational proposition whatsoever, 
 the members of which are represented by X and Y in 
 X = Y, may be decomposed into two propositions of the 
 forms X = X Y and Y = XY, which will not however 
 always differ. Show also that the operation when repeated 
 gives no new result. 
 
 113. Take the definition Ice = Frozen Water, and throw 
 it into equivalent propositions of the following forms : 
 
 (1) One disjunctive proposition. 
 
 (2) Two non-disjunctive propositions. 
 
 (3) One disjunctive and one non-disjunctive. 
 
 (4) Two disjunctives and one non-disjunctive. 
 
 (5) One disjunctive and two non-disjunctives.
 
 246 EQUATIONAL LOGIC CHAP. 
 
 (6) Three disjunctives. 
 
 (7) Four non-disjunctives. 
 
 Are these forms exhaustive, or can you frame yet other 
 equivalent forms. 
 
 114. How many and what non-disjunctive propositions 
 will be equivalent to the single disjunctive, A^ -|- &C = 
 
 115. Express the proposition AB = C -I- D in the form 
 of two disjunctive and then in three non-disjunctive 
 propositions. 
 
 1 1 6. As an exercise on Chapter XXII., take the 
 proposition : 
 
 Stratified Rocks = Sedimentary Rocks, and discover (i) 
 one equivalent; (2) two inferrible; (3) several partially 
 inferrible and otherwise consistent ; (4) several consistent, 
 indifferent, and not inferrible ; (5) two partially inferrible, 
 partially contradictory ; (6) one partially indifferent, partially 
 contradictory ; (7) one purely contradictory proposition. 
 
 117. Treat in the same general manner any of the 
 following premises : 
 
 ( i ) Blood-vessels = arteries | veins. 
 
 (2) Either thou or I or both must go with him. 
 
 (3) Heat is conveyed either by contact or radiation. 
 
 (4) An equation is either integrable or not integrable. 
 
 (5) Roger Bacon, an English monk, was the greatest of 
 
 mediaeval philosophers. 
 
 (6) Those animals which have a brain in connection 
 
 with a spinal cord, and they alone, have red 
 blood corpuscles. [MURPHY.] 
 
 1 1 8. Perform an exhaustive analysis of the relations of 
 the following propositions, comparing each , proposition
 
 xxin EXERCISES 247 
 
 with each other in all the fifteen possible combinations, and 
 ascertaining concerning each pair under which of the seven 
 heads it falls : 
 
 (1) A = BC. (4) a = BC abc. 
 
 (2) A = Me. (5) ab = ac. 
 
 (3) A = At>; B = C. (6) AB = ABC. 
 
 119. Perform a similar exhaustive analysis of the rela- 
 tions of the following propositions : 
 
 (1) Mercury = liquid, metal. 
 
 (2) Not-mercury is not liquid. 
 
 (3) Not-metal is either not-mercury or not-liquid. 
 
 (4) Mercury is a metal and is liquid. 
 
 (5) Liquid is either mercury or not-metal. 
 
 (6) Not-liquid is either not-mercury or metal. 
 
 (7) Not-mercury is either not-liquid or not-metal. 
 
 The eight propositions in question 45 or the seven in 55 
 of this chapter may be similarly analysed. 
 
 1 20. Analyse this argument: 'As we can only doubt 
 through consciousness, to doubt of consciousness is to 
 doubt of consciousness by consciousness.' 
 
 121. Illustrate the principle that the relations of logical 
 symbols are independent of space-relations. (See Prin- 
 ciples of Science, first ed. vol. i. pp. 39-42, 444 ; vol. ii. 
 p. 469 ; new edition, pp. 32-35, 383, 769.) 
 
 122. Show that if certain premises involving three terms 
 leave five or more combinations unnegatived, the premises 
 in question must be self-consistent. 
 
 123. From the point of view of equational logic analyse
 
 248 EQUATIONAL LOGIC CHAP, xxm 
 
 the metaphysical wisdom of Coleridge's doctrine of the 
 syllogism thus expressed (Table Talk, vol. i. p. 207) : 
 
 'All Syllogistic Logic is i. delusion; 2. /^elusion ; 
 3. (delusion ; which answer to the Understanding, the 
 Experience, and the Reason. The first says : " This 
 ought to be," the second adds : " This is," and the last 
 pronounces : " This must be so." '
 
 CHAPTER XXIV 
 
 THE MEASURE OF LOGICAL FORCE 
 
 1. THE combinational analysis of the meaning of asser- 
 tions enables us to establish an almost mathematical system 
 of measurement of the comparative force of assertions. 
 Given the number of independent terms involved, that 
 form of proposition has the least possible force which 
 negatives only a single combination. Thus with three 
 terms, a proposition of the form AB = ABC negatives only 
 the single combination ABr / but A = ABC negatives three, 
 and A = BC as many as four combinations. These latter 
 propositions may be said to have three and four times the 
 logical force of the first given. 
 
 2. I have not yet been able to discover any general laws 
 regarding this subject of logical force, but many curious 
 and perhaps important observations may be made. Thus 
 a great many forms of assertion agree in having the logical 
 force one-half, that is to say, they negative half the com- 
 binations. Such is the case, the terms being three in 
 number, with the propositions A = BC ; A = B-|-BC; 
 A = BH-^C. Indeed, it is very frequently true that any 
 proposition having no term common to both sides of the 
 equation negatives half the combinations. This is true of 
 all propositions of the types A = B, A = BC, and generally 
 A = BCD ... Y. But it is not true of the type AB = CD.
 
 250 EQUATIONAL LOGIC CHAP. 
 
 The appearance of the same term in both members of an 
 equation always weakens its force ; thus A = ABC has the 
 force only of f , whereas A = BC has the force ^. Again, 
 A = B-|-C has the force i, but A-|.B = B-|-C only the 
 force f- . 
 
 3. The best ostensive instance of logical power is found 
 in a form of proposition which embraces the greatest in- 
 tension in one member with the greatest extension in the 
 other. This kind of assertion has the general form 
 ABC . . . = P vQ-l-R-K ; and as the terms increase 
 the logical force approaches indefinitely to unity. Thus 
 while A = B-|-C has the value |, AB = C-|-D has that 
 of 10 out of 1 6, and A-|-B = CDE that of 22 out of 32. 
 
 A few other observations on this subject are thrown into 
 the form of questions : 
 
 4. Show that the logical force of n equations of the 
 
 form A = B, B = C . . . 
 
 5. Prove that a single proposition of the type ABC .... 
 = P-|-Q-|-R-K . . ., there being in all n independent 
 
 letter terms, and no term common to both members, has the 
 
 logical force i - - + which approaches indefinitely 
 
 to unity as n increases. 
 
 6. Can you discover any equation between a single term 
 and any expression not involving that term which has a 
 logical force other than one-half? 
 
 7. What form of proposition involving only A and B in 
 one member, and C, D, in the other, has the lowest possible 
 logical force ? 
 
 8. What is the utmost number of combinations of n 
 terms which can be negatived without producing con- 
 tradiction ?
 
 xxiv MEASURE OF LOGICAL FORCE 251 
 
 9. What is the utmost number of combinations of four 
 terms which can be negatived by a proposition involving 
 only three of them ? 
 
 i o. What two propositions involving five terms negative 
 the utmost possible number of combinations, without self- 
 contradiction ? 
 
 11. Show that m successive propositions of the type 
 A = AB, B = BC . . . ., that is to say, in the form of the 
 Sorites, leave m + 2 combinations unnegatived, so that the 
 
 , . , , . m+ 2 
 
 logical force is i - . 
 
 2 i + i 
 
 12. Prove that the amount of surplus assertion, or over- 
 lapping of the propositions, in a Sorites as treated in the 
 last question, increases indefinitely. Investigate the law of 
 the surplusage. 
 
 13. What is the utmost possible logical force as regards 
 ;// terms of an equation involving n terms.
 
 CHAPTER XXV 
 
 INDUCTIVE OR INVERSE LOGICAL PROBLEMS 
 
 1. THE direct or deductive process of logical analysis 
 consists in determining the combinations which are, under 
 the Laws of Thought, consistent with assumed conditions. 
 The Inverse Problem is given certain combinations incon- 
 sistent with conditions, to determine those conditions. As 
 explained in the Principles of Science (chapter vii.) the 
 inverse problem is always tentative, and consists in invent- 
 ing laws, and trying whether their results agree with those 
 before us. An American correspondent, Mr. M. H. 
 Doolittle, points out that in making trials we should always 
 pay attention to combinations in proportion to their in- 
 frequency, or solitariness, infrequency being the mark of 
 deep correlation. The infrequency may be that either of 
 presence or of absence. 
 
 2. The following inductive problems consist of series of 
 combinations of three terms and their negatives which are 
 supposed to remain uncontradicted under the condition of a 
 certain proposition or group of propositions. The student 
 is requested to discover such propositions, express them 
 equationally, and then assign them to the proper type in 
 the table on p. 222. If in any problem the conditions are 
 self-contradictory the student is to detect the fact.
 
 CHAP, xxv INVERSE PROBLEMS 253 
 
 I. IV. VII. IX. 
 ABC ABC AbC AB<: 
 abc aEc Abc AbC 
 
 abC aEC BC 
 
 II. aEc abc 
 Abc V. 
 
 BC AbC VIII. X. 
 
 Abc ABC ABC 
 
 III. aEC aEc aEC 
 Abc abC aEc 
 aEC VL abc abC 
 aEc AbC 
 
 Abc 
 abc 
 
 3. Assuming each of the following series of combinations 
 to consist of those excluded or contradicted by certain proposi- 
 tions, assign the propositions which are just sufficient to 
 exclude them in each problem, express these propositions 
 equationally, and refer them as in the last question to the 
 proper type in the table : 
 
 I. V. VIII. 
 
 ABC ABC ABC 
 
 II- Abc aEC 
 
 abC abc 
 VI. 
 
 III. AEc IX. 
 
 aEc Kbc AEc 
 
 abC abc AbC 
 
 VII. Abc 
 
 IV - aEc aEC 
 
 ABC abC aEc 
 
 abc abc abC
 
 254 
 
 EQUATIONAL LOGIC 
 
 CHAP. 
 
 4. I now give a series of inductive problems involving 
 four terms. Each series of combinations consists of those 
 which remain after the exclusion of such as contradict 
 certain conditions. Required those conditions. The 
 problems are ranged somewhat in order of difficulty. 
 
 I. 
 
 IV. 
 
 VII. 
 
 X. 
 
 ABCD 
 
 ABCD 
 
 ABCD 
 
 AB^D 
 
 abed 
 
 AEed 
 
 AECd 
 
 aECd 
 
 
 abCV 
 
 AEcV 
 
 aEed 
 
 II. 
 
 abed 
 
 Abed 
 
 abea 
 
 ABCD 
 
 AECd 
 
 V. 
 
 abed 
 
 XI. 
 
 aBCD 
 
 ABCD 
 
 
 AB^D 
 
 aECd 
 
 AECd 
 
 VIII. 
 
 AbCd 
 
 abCD 
 
 AEed 
 
 Abea 
 
 AbeV 
 
 abCd 
 
 A^CD 
 
 aECd 
 
 aECd 
 
 abcD 
 
 AbCd 
 
 abCd 
 
 abCd 
 
 abed 
 
 abed 
 
 aMD 
 
 abed 
 
 
 
 abed 
 
 
 III. 
 
 ABG* 
 aECd 
 abCd 
 abcD 
 
 abed 
 
 VI. 
 
 ABG/ 
 AB^ 
 A^C^ 
 abed 
 
 IX 
 
 ABC^ 
 
 aEca 
 abCd 
 
 XII. 
 ABCD 
 AB^D 
 AbCd 
 aECd 
 abed 
 
 5. I next give a few similar problems involving five or six 
 terms, as follows :
 
 XXV 
 
 INVERSE PROBLEMS 
 
 255 
 
 I. 
 
 ABCDE 
 
 Abcde 
 aEcde 
 abcde 
 
 III. 
 
 ABG/E 
 
 A&CJE 
 
 abcdE 
 abcde 
 
 V. 
 AB^rDE 
 
 a&CDe 
 abCde 
 
 II. 
 
 ABC^ 
 
 aECde 
 
 abCde 
 
 rt^DE 
 
 abcDe 
 
 abcde 
 
 IV. 
 
 ABCDE 
 
 KbCde 
 
 abCDe 
 abcdE 
 
 VI. 
 
 ABCDE 
 ABG/E 
 
 A&CDe 
 
 VII. 
 
 VIII. 
 
 KBCdef 
 
 A&CDef 
 
 IX. 
 
 McVef 
 
 AfcJEF rtB^/EF 
 
 A^^F a^CDEF 
 
 ^B^DEF *CDeF
 
 256 EQUATIONAL LOGIC cn A V. 
 
 6. As the reader who is in possession of the present 
 volume will have plenty of unanswered inductive problems, 
 it may be well to give here the answers to the problems of 
 the like kind which were set in the Principles of Science, 
 new edition, p. 127. They are as follows : 
 
 a = a ~BC. 
 II. A = AC; a = aB. 
 
 III. A = AC; ab = abc. 
 
 IV. A = D ; B = CD -|- cd, or their equivalents 
 
 V. ab = 
 
 BC = 
 
 VI. D = E ; bC = ^CD ; (a -I- S)c = abcde. 
 VII. A = c=D = e; B = B. 
 VIII. (Unknown.) 
 
 IX. ^D = C ]/; adR = B^F ; ACF = AO/F ; 
 = ^CEF. 
 
 X. This example was set by me at haphazard, like 
 Nos. V. and VI., that is to say, by merely striking out any 
 combinations of the logical alphabet which fancy dictated. 
 Dr. John Hopkinson, F.R.S., has given me the following 
 rather complex solution 
 
 (1) d=abd. 
 
 (2) b = b (AT -\- ae). 
 
 (3) A/= A/B^DE. 
 
 (4) E = E (B/-I- JACDF). 
 
 (5) ^B=^ABCDF. 
 
 (6) abc = abcef. 
 
 (7) abef abcef. 
 
 Can a simpler answer be discovered ?
 
 xxv INVERSE PROBLEMS 257 
 
 7. In the first edition of the Principles of Science, vol. ii. 
 p. 370, I gave a rather complex problem involving six 
 terms, the combinations unexcluded being as follow after 
 coalescence of some alternatives : 
 
 ABCDF AEcDef 
 
 ABCD<?/ aB^DF 
 
 ABG/E/ aQcDef 
 ABdDF 
 
 To a request for solutions I received several answers, 
 mostly correct, but, curiously enough, all differing in the 
 forms of proposition. The correct ones are given below 
 as furnishing a remarkable instance of logical equivalence. 
 
 First Answer. Second Answer. 
 
 C = ABC b = cd 
 
 f = Def-\- dEf C = ABC 
 
 c =B<rD -\-bcd D = BD 
 
 d = dEf d = Ef 
 
 Third Answer. Fourth Answer. 
 
 a = ac C = AC 
 
 b = ca b = bed 
 
 d=Ef d=Ef 
 
 B = B(C-|-D) 
 
 Fifth Answer. 
 
 AC = BC 
 BDE = BDEF 
 
 B = C -I- D 
 
 \
 
 258 EQUATIONAL LOGIC CHAP, xxv 
 
 The third answer, given by Mr. R. B. Hayward, M.A., is 
 in the simplest terms. The propositions inserted as the 
 fifth answer are those from which I formed the combinations 
 deductively. The student may prove that any one of these 
 answers is deducible from any other without descending 
 explicitly to the combinations ; thus C = AC is the con- 
 trapositive of a = ac ; B = C -I- D is equivalent to b = cd, and 
 so forth.
 
 CHAPTER XXVI 
 
 ELEMENTS OF NUMERICAL LOGIC 
 
 i. LET a logical term, when enclosed in brackets, acquire 
 a quantitative meaning, so as to denote the number of 
 individual objects which possess the qualities connoted by 
 the logical term. Then (A) = number of objects possess- 
 ing qualities of A, or say, for the sake of brevity, the number 
 of As. 
 
 Every logical equation now gives rise to a corresponding 
 numerical equation. Sameness of qualities occasions same- 
 ness of numbers. Hence if A = B denotes the identity of 
 the qualities of A and B, we may conclude that (A) = (B). 
 
 It is evident that exactly those objects, and those objects 
 only, which are comprehended under A must be compre- 
 hended under B. It follows that wherever we can draw an 
 equation of qualities, we can draw a similar equation of 
 numbers. Thus, from A = B = C, we infer A = C ; and 
 similarly from (A) = (B) = (C), meaning the number of 
 As and Cs are equal to the number of Bs, we can infer 
 (A) = (C). But, curiously enough, this does not apply to 
 negative propositions and inequalities. For if A = B< >D 
 means that A is identical with B, which differs from D, it 
 does not follow that
 
 260 NUMERICAL LOGIC CHAP. 
 
 Two classes of objects may differ in qualities, and yet they 
 may agree in number. 
 
 2. The sign ! being used to stand for the disjunctive 
 conjunction or, but in an unexclusive sense, it follows that 
 I- is not identical in meaning with + . It does not follow 
 from the statement that A is either B or C, that the number 
 o As is equal to the number of Bs added to the number of 
 Cs ; some objects, or possibly all, may have been counted 
 twice in this addition. Thus, if we say An elector is either 
 an elector for a borough, or for a county, or for a university, 
 it does not follow that the total number of electors is equal 
 to the number of borough, county, and university electors 
 added together ; for some men will be found in two or three 
 of the classes. 
 
 This difficulty, however, is avoided with great ease ; for 
 we need only develop each alternative into all its possible 
 subclasses and strike out any subclass which appears more 
 than once, and then convert into numbers, connected by 
 the sign of addition. Thus, from A = B ] C we get 
 A = BC-|-BH-BC-|-^C; but striking out one of the terms 
 BC as being superfluous, we have A = BC -I- B^-|- <C. 
 
 The alternatives are now strictly exclusive, or devoid of 
 
 any common part, so that we may draw the numerical 
 equation 
 
 (A) = (BC) + (Be) + (6C). 
 Thus, if 
 
 A = elector, C = county elector, 
 
 B = borough elector, D = university elector, 
 
 we may from the proposition A = B -|- C -|- D draw the 
 numerical equation 
 
 (A) = (BCD) + (BO*) + (B;D) + (Bed) + 
 (6C<t) + (6cD).
 
 xxvi QUESTIONS AND ANSWERS 261 
 
 3. The data of any problem in Numerical Logic will be 
 of two kinds : 
 
 (1) The logical conditions governing the combinations 
 of certain qualities or classes of things, expressed 
 in propositions. 
 
 (2) The numbers of individuals in certain logical 
 classes existing under those conditions. 
 
 The qtiaesita of the problem will consist in determining 
 the numbers of individuals in certain other logical classes 
 existing under the same logical conditions, so far as such 
 numbers are rendered determinable by the data. The 
 usefulness of the method will, indeed, often consist in 
 showing whether or not the magnitude of a class is deter- 
 mined or not, or in indicating what further hypotheses or 
 data are required. It will appear, too, that where an exact 
 result is not determinable we may yet assign limits within 
 which an unknown quantity must lie. 
 
 4- In a certain statistical investigation, among 
 100 As there are found 45 Bs and 53 Cs; 
 that is to say, in 45 out of 100 cases where A 
 occurs B also occurs, and in 53 cases C 
 occurs. Suppose it to be also known that 
 wherever B is, C also necessarily exists. It 
 is required to determine 
 
 (1) The number of cases (all being As) where C 
 
 exists without B. 
 
 (2) The number of cases (all being As) where 
 
 neither B nor C exists.
 
 262 NUMERICAL LOGIC CHAP. 
 
 The data are as follow : 
 
 r (A) -loo .... (i) 
 
 Numerical equations < (B) = 45 ..... (2) 
 
 t(C)= 53 ..... (3) 
 Logical equation . . B = BC. 
 
 The logical equation asserts that the class B is identical 
 with the class BC, which is the true mode of asserting 
 that all Bs are Cs. Two distinct results follow from this, 
 namely: ist, that the number of the class BC is identical 
 with the number of the class B ; and 2nd, that there are 
 no such things as Bs which are not Cs. 
 
 The logical equation is thus equivalent to two additional 
 numerical equations, namely, 
 
 (B)=(BC) ....... (4) 
 
 W = o ........ (5) 
 
 We have now means of solving the problem ; for, by the 
 Law of Duality, 
 
 By ( 4 ) 
 
 = (B) 
 Thus 
 
 or the required number of Cs is 8. 
 
 To obtain the number of A&rs, we have 
 
 (A) = (ABC) + (ABc) + (A^C) + (Ate) 
 100= 45 + o + 8 + (A&r). 
 
 Hence 
 
 (A&)= 47-
 
 xxvi QUESTIONS AND ANSWERS 263 
 
 5^ The difference between the numbers of objects 
 in any two classes whatsoever, is equal to the 
 difference between the numbers of objects 
 which are in each class, but excluded from 
 the other class. 
 
 Take (A) and (B) to represent the numbers in any two 
 classes A and B ; then 
 
 (A) - (B) = (AB) + (A) - (AB) - (B) 
 
 6. If the number of As be x, of Bs be y, and of 
 those Bs which are not As be p, then the 
 number of As which are not B will be / + x y. 
 
 Setting down the several logical quantities represented 
 + x y, we have 
 
 (aB) + (AB) + (AH) - (AB) - (aB). 
 Four terms cross out, leaving only (A) as required. 
 
 7- Represent the following argument from 
 Thomson's Laws of Thought, p. 168 : 
 
 Three-fourths of the army were Prussians ; 
 
 Three-fourths of the army were slaughtered ; 
 
 Therefore some who were slaughtered were 
 Prussians. 
 
 Taking A = members of the army, 
 B = Prussians, 
 C = slaughtered,
 
 264 NUMERICAL LOGIC CHAP. 
 
 the premises are expressed as 
 
 (AB) = f(A) 
 
 (AC) = f(A). 
 
 The number of Prussians slaughtered will be (ABC), of 
 which the following equation is identically true : 
 
 (ABC) = (AB) + (AC) - (A) + (Ah) ; 
 inserting values 
 
 (ABC) = f (A) + f (A) - (A) + (Abe) 
 = (A) + (A&). 
 
 That is to say, the number of Prussians slaughtered was at 
 least half the army, and exceeds it by a number equal to 
 the number of men in the army who were neither Prussians 
 nor slaughtered. 
 
 8. If the number of As which are Bs is /, and 
 the number of Bs which are Cs is q, what do 
 we know concerning the number of As which 
 are Cs ? 
 
 We have the following self-evident equations : 
 
 AC = ABC + AC 
 
 = ABC + AB^ + ABC + BC + 
 
 + a&c - B 
 = AB + BC - B + A/;C + 
 
 Inserting the values given, we get 
 AC =/ + q - B + 
 
 We see that the data are quite insufficient for determining 
 the number of As which are Cs. "They may be anything 
 from zero up to the whole number of As or Cs. To make 
 the question determinate we need also the number of Bs,
 
 xxvi QUESTIONS AND ANSWERS 265 
 
 as well as the number of ACs, which are not Bs, and the 
 number of Bs which are neither A nor C. 
 
 9. Most Bs are As ..... (i) 
 
 Most Bs are Cs ...... (2) 
 
 Therefore, Some Cs are As ..... (3) 
 
 The above argument is a celebrated one, proposed by 
 De Morgan (Formal Logic, p. 163), and discussed by Boole 
 (Trans, of the Cambridge Philosophical Society, vol. xi. part 
 ii. p. i) and others. Regarded as an ordinary Aristotelian 
 pseudo-syllogism, it is subject to the fallacy of undistributed 
 middle, since the proposition ' Most Bs are As,' must be 
 counted as a particular affirmative. The pseudo-mood is 
 accordingly 1 1 1 in the third figure. Nevertheless the force 
 of the argument is pretty obvious and may thus be analysed. 
 
 The mark of quantity, most, of course means more than 
 half, and is one of the few quantitative expressions used in 
 ordinary language. We can easily represent the two pre- 
 mises in the form 
 
 (B) + w ....... (i) 
 
 To deduce the conclusion, we must add these equations 
 together, thus, 
 
 (AB) + (BC) = (B) + w + w. 
 
 Developing the logical terms on each side, we have 
 (ABC) + (AB<r) + (ABC) + (BC) = (ABC) + (ABr) 
 
 Subtracting the common terms, there remains 
 (ABC) = w + w + (Br).
 
 266 NUMERICAL LOGIC CHAP. 
 
 The meaning is, that there must be some Cs which are 
 As, amounting to at least the sum of the quantities w and 
 ze/, the unknown excesses beyond half the Bs which are As 
 and Cs. The number (aBc) is wholly undetermined by the 
 premises, but it cannot be negative. Thus w + w is the 
 lower limit of (ABC). 
 
 10. (i) For every Z there is an X which is Y; 
 (2) Some Zs are not Ys. What inferences can 
 be drawn ? 
 
 This general problem given by De Morgan in his Syllabus 
 (p. 29, art. 85), and in some other parts of his writings, 
 would thus be represented in my formulae, which differ 
 essentially, however, from those of De Morgan. 
 
 The premises are 
 
 (Z) + w ..... (i) 
 
 in which m would be zero if De Morgan meant that there 
 are not more Xs which are Ys than there are Zs, but just an 
 equal number. 
 
 where n is some positive number. 
 Developing (i) we get 
 
 (3) (XYZ) + (XYz) = (XYZ) + (XyZ) + (xYZ) 
 
 + (xyZ) + m. 
 
 Striking out the common term and adding (Xyz) to both 
 sides, we have for the number of Xs which are not Zs 
 
 (Xz) = m + n + (Xyz) + (xYZ).
 
 xxvi QUESTIONS AND ANSWERS 267 
 
 Again, after striking out the common term, equation (3) 
 reduces to 
 
 (XYz) = (XyZ) + (xZ) + m, 
 
 which gives as the number of Zs which are not Xs 
 (xZ) = (XYz) - (XyZ) - m. 
 
 The student should compare these results with those of the 
 less general problem given in the Principles of Science, new 
 edition, p. 169; first edition, vol. i. p. 191, and also with" 
 De Morgan's results expressed in a totally different kind of 
 notation. 
 
 II. If m or more Xs are Ys, and n or more Ys 
 are Zs, what do we know about the number of 
 Xs which are therefore Zs ? 
 
 This question represents one case of the numerically 
 definite syllogism as treated by De Morgan (Syllabus, p. 27). 
 
 Taking X, Y, and Z to be the three terms of the syllogism, 
 he adopts the following notation : 
 
 u whole number of individuals in the universe of 
 
 the problem. 
 x = number of Xs. 
 y = number of Ys. 
 z = number of Zs. 
 
 Making m denote any positive number, mXY means, in 
 De Morgan's system, that m or more Xs are Ys. Similarly 
 YZ means that n or more Ys are Zs. Smaller Roman 
 letters denote the negatives of the larger ones. Thus mXy 
 means that m or more Xs are not Ys, and so on. 
 
 From the two premises mXY and YZ, De Morgan 
 draws the conclusion (m + n _y)XZ. Let us consider what
 
 268 NUMERICAL LOGIC CHAP. 
 
 results are given by our notation. The premises may be 
 represented by the equations 
 
 (XY) = m + m (YX) = n + ', 
 
 where m and n are the same quantities as in De Morgan's 
 system, and m and n two unknown but positive quantities, 
 indicating that the number of XYs is m or more, and the 
 number of YZs is n or more. 
 
 The possible combinations of the three terms X, Y, Z, and 
 their negatives are eight in number, and these all together 
 constitute the universe, of which the number is u. The 
 problem is at once seen to be indeterminate in reality ; for 
 there are eight classes of which the numbers have to be 
 determined, and there are only six known quantities, namely, 
 u, x, y, z, m, and n, by which to determine them. Accord- 
 ingly we find that De Morgan's conclusion, though not 
 absolutely erroneous, has little or no meaning. From the 
 premises he infers that (m + n-y) or more Xs are Zs. 
 Now 
 
 m + n-y = (XY) + (YZ) - (Y) - m' - n 
 = (XYZ) - (xYz) - m - n. 
 
 Thus De Morgan represents the number of the whole class, 
 XZ, by a quantity indefinitely less than its own part, XYZ. 
 It is quite true that if the second side (XYZ) - (xYz) 
 - m' - n of this equation has value, there must be at 
 least this number of Xs which are Zs; but as (xYz) 
 may exceed (XYZ) in any degree, this may give zero or a 
 negative result, while there is really a large number of XZs. 
 The true and complete expression for the number of XZs 
 is found as follows : 
 
 (XZ) = (XYZ) + (XyZ) 
 
 = (XYZ) + (XYz) + (XYZ) + (xYZ) - (Y) + (XyZ) 
 + (xYz) = m + m' + n + n -y + (XyZ) + (xYz).
 
 xxvi QUESTIONS AND ANSWERS 269 
 
 Among these seven quantities, only m, n, and_y are definitely 
 known. The two m' and n are two indefinite quantities, 
 expressing the uncertainty in the number of XYs and 
 YZs, while there are two other unknown quantities, the 
 numbers of XyZs and xYzs arising in the solution of the 
 problem. 
 
 12. If m or more Xs are Ys, and n or more Ys 
 are Zs, what do we know about the number of 
 not-Xs which are not-Zs? 
 
 From the same two premises as in the last problem, 
 namely 
 
 mXY and YZ, 
 
 De Morgan draws the conclusion 
 
 (m + n + u x y z)xz ; 
 
 that is to say, the number of not-Xs which are not-Zs is 
 the quantity in the brackets or more. This conclusion is 
 equivalent to that in the preceding problem. 
 
 To prove this result it is requisite to develop all the 
 combinations numbered in each of the quantities ;;/, , u, 
 x ) y* z >' there are twenty-six terms in all which the reader 
 may readily work out. Giving them the signs indicated by 
 De Morgan, and striking out pairs of positive and negative 
 terms, we find only two combinations left, together with 
 m' and ', which terms are used, as in the last problem, to 
 express the fact that De Morgan's proposition /XY is not 
 really definite, but means that m or more, that is m or 
 (m + m') Xs are Ys. We thus obtain 
 
 (xy) = (xyz)-(XyZ) - m - n' 
 
 in which the term (XyZ) is wholly undetermined. Thus we 
 find that De Morgan's method gives us as the value of (xy)
 
 270 NUMERICAL LOGIC CHAP. 
 
 a part of itself (xyz), diminished by three unknown quanti- 
 ties. The number (xz) may accordingly be of any magni- 
 tude, while the lower limit assigned to it by De Morgan is 
 zero, or even negative. The problem is in fact a wholly 
 indeterminate one, and De Morgan's solution is illusory. 
 
 Similar remarks may be made concerning other conclu- 
 sions which De Morgan draws. Thus, from mXy and riz 
 (wXs or more are not Ys, and nYs or more are Zs) he infers 
 
 (m + n - x) xZ and (;// + n - z) Xz. 
 
 But it will be found by analysis that the first of these results 
 has the following meaning : 
 
 (xZ) > (xYZ - (XYz) ; 
 
 that is to say, the lower limit of the class xZ is a part of 
 itself, xYZ, diminished by the number of another class 
 XYz of unknown magnitude. 
 
 13. If the fractions a and ft of the Ys be severally 
 As and Bs, and if a + ft be greater than unity, 
 it follows that some As are Bs. 
 
 \Cambridge Phil. Trans, vol. x. part i. p. 8.] 
 
 In his third memoir on the Syllogism De Morgan gives 
 the above as a very general statement of the conditions of 
 valid mediate inference. He remarks that the logician, 
 that is to say, the ordinary Aristotelian logician, ' demands 
 a = i or /3 = i, or both ; he can then infer.' This repre- 
 sents the condition of a distributed middle term. 
 
 The numerically definite conditions are readily repre- 
 sented as follow : 
 
 The premises are a . (Y) = (AY).
 
 xxvi QUESTIONS AND ANSWERS 271 
 
 Hence 
 
 = (ABY) + (AJY) + (ABY) + (aBY), 
 
 or 
 
 (ABY) = (a + (3 - i) (Y) + (aY). 
 
 We learn that the number of AYs which are Bs is the 
 fraction (a + /3-i) of the Ys, together with the undeter- 
 mined number (#Y), which cannot be negative. But, 
 according to the conditions, a + /3 is greater than unity ; 
 hence the second side of the equation must have a positive 
 value. Not only will there be (a + /3-i), As which are 
 Bs, but this is merely the lowest limit, and there will be 
 as many more as there are units in the number of a^Ys. 
 
 If we distribute the middle term Y once, by making 
 0=1, we have 
 
 (ABY) = /?.(Y) + o. 
 
 The term (a3Y) of course vanishes because the whole of 
 the Ys are As. Again, if /?= i, we have 
 
 (ABY) = a.(Y). 
 
 If both a and (3 become unity, then 
 (ABY) = (Y). 
 
 It must be carefully noted, however, that these results do not 
 show the whole number of As which are Bs, but only those 
 which are so within the sphere of the term Y. Nothing 
 has been said about the combinations of not-Y, which are 
 quite unlimited by the conditions of the problem. 
 
 14- ' If A occurs in a larger proportion of the 
 cases where B is than of the cases where B is
 
 272 NUMERICAL LOGIC CHAP. 
 
 not, then will B also occur in a larger propor- 
 tion of the cases where A is than of the cases 
 where A is not.' 
 
 This general proposition is asserted in J. S. Mill's chapter 
 ' On Chance and its Elimination,' but is not proved by 
 Mill. (System of Logic, Book III., chapter xvii. section 2, 
 adfinem; fifth edition, vol. ii. p. 54.) I do not remember 
 seeing any proof of it given elsewhere, and it is not to my 
 mind self-evident. The following, however, is a proof of its 
 truth, and is the shortest proof I have been able to find. 
 
 The condition of the problem may be expressed in the 
 inequality 
 
 (AB) : (B) > (A*) : (b\ 
 
 or reciprocally in the inequality 
 
 (B) : (AB) < (b) : (A*). 
 Subtracting unity from each side, and simplifying, we have 
 
 (B) : (AB) < (at) : (hb). 
 
 Multiplying each side of this inequality by (A^) : (B) we 
 obtain 
 
 (A&) : (AB) < (ab) : (aB). 
 
 Restoring unity to each side, and simplifying 
 
 (A) : (AB) < (a} : (aE), 
 or reciprocally 
 
 (AB) : (A) > (B) : (), 
 
 which expresses the result to be proved, namely, that B 
 occurs in a larger proportion of the cases where A is than 
 of the cases where A is not. 
 
 IS- In a company of r individuals,/ have coats 
 and q have waistcoats. Determine some other 
 relations between them.
 
 xxvi QUESTIONS AND ANSWERS 273 
 
 Boole treats this problem in the fourth page of his 
 Memoir On Propositions Numerically Definite {Cambridge 
 Philosophical Transactions, vol. xi. part ii.). Taking i to 
 represent the company which is the universe of the propo- 
 sition, x the class possessing coats, y the class possessing 
 waistcoats, and using the letter N, according to Boole's 
 notation, as equivalent to the words ' number of,' 
 
 he finds, as we have found in a preceding page (p. 264, No. 7), 
 
 =p + q- r + Ni - x i y. 
 
 -# i -y = r-p - q + 
 
 He proceeds, ' Again, let us form the equation 
 
 q-r = 2Nx - Ny - N i 
 = N (zx-y - i) 
 = N (xi -y 2y i x i - x i y) 
 
 = NX i y 2Ny i jc - N i - x i y. 
 From which we have 
 
 N^ic i -y = 2p - q - r + 2Ny i - x + N i - x i y. 
 
 Hence we might deduce that the number who had coats 
 but not waistcoats would exceed the number 2p-q-r 
 by twice the number who had waistcoats without coats 
 together with the number who had neither coats nor waist- 
 coats. This is not, indeed, the simplest result with reference 
 to the class in question, but it is a correct one.' 
 
 The student is requested to verify this result. 
 
 On going over this paper of Boole's again, it becomes 
 apparent to my mind that his method is identical with 
 that developed in this chapter and in my previous paper 
 
 T
 
 274 NUMERICAL LOGIC CHAP. 
 
 on the same subject (Memoirs of the Manchester Literary 
 and Philosophical Society, Third Series, vol. iv. p. 330, 
 Session 1869-70), written with a knowledge, as stated on 
 p. 331, of Boole's publication on the subject. 
 
 16. Can we represent a syllogism in the extensive 
 form by means of numerical symbols ? 
 
 In a very interesting and remarkable paper read to the 
 Belfast Philosophical Society in 1875, Mr. Joseph John 
 Murphy has given a kind of numerical notation for the 
 syllogism. He has since printed a more condensed and 
 matured account of his views in Mind, January, 1877. 
 
 Taking the syllogism ' Chlorine is one of the class of 
 imperfect gases ; imperfect gases are part of the class of 
 substances freely soluble in water ; therefore, chlorine is 
 one of the class of substances freely soluble in water ' he 
 assumes the symbols 
 
 x = Chlorine, y = Imperfect gases, 
 
 z = substances freely soluble in water. 
 
 He expresses the first premise in the form 
 
 y = x+p, 
 
 p being a positive numerical quantity indicating that there 
 are other things besides chlorine in the class of imperfect 
 gases. The second premise takes the form 
 
 * = y + & 
 
 similarly indicating that besides imperfect gases there are 
 
 q things in the class of substances freely soluble in water. 
 
 Substitution gives 2 = x +p + q, 
 
 which would seem to prove that besides chlorine (x) there 
 
 are p + q things in the class of substances freely soluble in 
 
 water.
 
 xxvi QUESTIONS AND ANSWERS 275 
 
 The student who wishes to master the difficulties of the 
 modern logical views should read these papers with great care. 
 Space does not admit of my arguing the matter out at full 
 length, and I can therefore only briefly express my objec- 
 tions to Mr. Murphy's views as follows : His equations are 
 equations in extension, and, with his use of + and - , they 
 can only hold true when his terms are numerical quantities. 
 Under this assumption his equations show with perfect 
 correctness the numbers of certain classes ; but they are not 
 therefore equivalent to syllogisms. Because z = x+p + q, 
 we learn that the number z exceeds x by / + q, but it does 
 not therefore follow that chlorine belongs to the class of 
 substances represented by z. In short, as I have pointed 
 out at the beginning of this chapter (p. 259), from logical 
 equations arithmetical ones follow, but not vice versa. (See 
 also Principles of Science, p. 171 ; first edit. vol. i. p. 193.) 
 I hold, therefore, that Mr. Murphy's forms are not really 
 representations of syllogisms ; but at the same time I am 
 quite willing to admit that this is a question never yet 
 settled and demanding further investigation. It is very 
 remarkable that Hallam inserted in his History of Literature 
 (ed. 1839, vol. iii. pp. 287-8) a long note containing a 
 theory of the syllogism somewhat similar to that of Mr. 
 Murphy, but which has hitherto remained unknown to Mr. 
 Murphy and apparently to all other logical writers.
 
 CHAPTER XXVII 
 
 PROBLEMS IN NUMERICAL LOGIC 
 
 i. IF from the number of members of Parliament we 
 subtract the number of them who are not military men, we 
 get the same result as if from the whole number of military 
 men we subtract the number of them who are not members 
 of Parliament. Prove this. 
 
 2. In a company of x individuals it is discovered that y 
 are Cambridge men, and z are lawyers. Find an expression 
 for the number of Cambridge men in the company who are 
 lawyers, and assign its greatest and least possible values. 
 
 [BOOLE.] 
 
 3. Prove that in any population the difference between 
 the number of females and the number of minors is equal 
 to the difference between the number of females who are 
 not minors, and of minors who are not females. 
 
 4. Show that if to the number of metals which are red, 
 we add the number which are brittle, the sum is equal to 
 that of the whole number of metals after addition of the 
 number of metals which are both red and brittle, and after 
 subtraction of the number of metals which are neither red 
 nor brittle. 
 
 5. What is the value of the following expression 
 
 (A)-(AB)-(AQ?
 
 CHAP, xxvn PROBLEMS 277 
 
 6. Prove that the number of quadrupeds in the world 
 added to the number of beings not quadrupeds which 
 possess stomachs is equal to the whole number of things 
 having stomachs together with the number of things not 
 having stomachs which are quadrupeds. 
 
 7. If x and y be respectively the numbers of things which 
 are X and Y, while m is the whole number which are both 
 X and Y, and n the number which are either X alone 
 or Y alone, what is the relation between m + n and 
 x +y? 
 
 8. Let u be the whole number of things under con- 
 sideration, x the number which are A, and y the number 
 which are B ; then if m be the number of things which are 
 both A and B, show that m + u - x -y is the number 
 which are neither A nor B. 
 
 9. Taking each logical term to represent the number of 
 things included in its class, verify the following equa- 
 tions : 
 
 (A - AB) (A - AC) = A - AB - AC + ABC = hbc. 
 
 (A - AB) (A - AC) (A - AD) = A - AB - AC + 
 
 ABC - AD + ABD + ACD - ABCD = Kbcd. 
 
 10. What is the product of the logical multiplication 
 of the four factors 
 
 (A - AB) (A - AC) (A - AD) (A - AE) ? 
 Give another expression for its value. 
 
 1 1. Show that the following equation is necessarily true : 
 B + AC + 1>C + Afc= A + C + 
 
 1 2. What happens in Problem 8 if it be discovered that 
 the class B does not exist at all ? 
 
 13. Find an expression for the difference between (A) 
 and (B) + (C).
 
 278 NUMERICAL LOGIC CHAP. 
 
 1 4. What is (a) the minimum percentage of C that must, 
 and ((3) the maximum that may coincide with B under the 
 following conditions ? 
 
 80 per cent of As coincide with 50 per cent of Bs. 
 
 70 per cent of As coincide with 60 per cent of Cs. [D.] 
 
 15. If revolutions occur in a larger proportion of govern- 
 ments where the press is under a censorship, than of govern- 
 ments where it is not, then will a censorship of the press be 
 found in a larger proportion of governments which are sub- 
 ject to frequent revolutions, than of governments which are 
 not thus subject. [D.] 
 
 1 6. If p per cent of A are B, and q per cent of A are 
 C, what is the least percentage of A that those individuals 
 make up which are both B and C ? [D.] 
 
 1 7. Show that we cannot tell what percentage of B or of 
 C the same individuals make up unless we know how much 
 of B or of C is not A. [D.] 
 
 1 8. In the easy case in which all B is A, and all C is A, 
 find what percentage of B or of C must be made up by the 
 individuals which are both B and C at once. [D.] 
 
 19. Prove the following equations : 
 (A6f) + (AB) = (A) + (ABC) - (AC). 
 
 (A + B) - (C + D) = (A + B) (c + Crf) - (C + D) 
 (a + Kb) - AB (G/ + *D) + CD (A* + aB). 
 
 20. Prove that the following equation gives a correct 
 expression for the common part of any three classes - 
 A, B, C. 
 
 (ABC) = (B) + (C) - (A) - (flB) - (C) + (Afc). 
 
 21. In a company consisting of r individuals there were 
 q in number who knew Latin, and / in number who knew
 
 xxvii PROBLEMS 279 
 
 either Latin or French, but not both ; between what limits 
 is the number of those who knew French confined ? 
 
 22. In the last problem prove that the lower limit is the 
 greatest value in / - q and q - p, and the upper limit, the 
 least value in ir - p - q, and / + q. (See Boole, On Pro- 
 positions Numerically Definite, p. 15.) 
 
 23. The student will find many other numerically defi- 
 nite problems in De Morgan's Formal Logic, Chapter VIII., 
 and in his Syllabus, pp. 27-30 ; but in reading De Morgan 
 it must be carefully remembered that wXY means with him 
 not that wXs are Ys but that m or more Xs are Ys. His 
 solutions will sometimes, as shown in the previous chapter, 
 be found delusive. 
 
 24. Verify the following assertion of De Morgan : ' To 
 say that mXs are not any one to be found among any lot of 
 Ys is a spurious (that is a self-evident or necessary) pro- 
 position, unless m + n be greater than both x and y, in 
 which case it is merely equivalent to both of the following, 
 (m + n -y) Xy, and (/// + n - x) Yx, which are equiva- 
 lent to each other. 
 
 25. It is found that there are in a certain club of x mem- 
 bers, y London graduates, and z lawyers. What further 
 numerical data are requisite in order to define the numbers 
 who are both London graduates and lawyers, and of those 
 who are neither ? 
 
 26. If there are more persons in a town than there are 
 hairs on any one person's head, then there must be at least 
 two persons in the town with the same number of hairs on 
 their heads. Put this theorem into a strict logico-mathe- 
 matical form. [HERBERT SPENCER.] 
 
 27. Demonstrate the theorem in numerical logic given in 
 the Principles of Science, new edition (only), p. 170. 
 
 28. 'For every man in the house there is a person who
 
 280 NUMERICAL LOGIC CHAP, xxvn 
 
 is aged ; some of the men are not aged ; it follows, and 
 easily, that some persons in the house are not men ; but 
 not by any common form of syllogism.' (DE MORGAN, 
 Syllabus, p. 29.) A solution of this problem is given in 
 Principles of Science, new edition, p. 169. 
 
 29. Draw what conclusions you can from the following : 
 ' There were some English on board ; and though no 
 passengers were saved from the wreck, and of the ship- 
 officers, as it happened, only one, yet no Englishman was 
 lost.' [R.]
 
 CHAPTER XXVIII 
 
 THE LOGICAL INDEX 
 
 1. I NOW give what I propose to call the Logical Index, 
 or, more precisely, the Logical Index of Three Terms. As 
 however the logical relations of two terms are too simple 
 to need an index, and those of four terms are vastly too 
 numerous and complex to admit of exhaustive treatment 
 at present, the Index of Three Terms is practically the only 
 one which can be given. It contains, within the space of 
 four pages, a complete enumeration of all possible purely 
 logical conditions involving only three distinct terms. 
 
 2. Each page contains a double-sided table, forming in 
 fact two tables. Each such table contains a column of 
 equational propositions, a column of Roman numerals 
 showing the type (see p. 222) to which such propositions 
 belong, a column of consecutive Arabic numbers for sake 
 of easy reference, and lastly a column of Greek letters, 
 which supplement the Greek letters a, f3, y, given at the 
 heads of the columns of propositions. These Greek letters 
 stand in place of the combinations of the fourth column of 
 the Logical Alphabet (p. 1 8 1 ), as follow : 
 
 a=ABC e
 
 282 TkE LOGICAL INDEX CHAP. 
 
 It is obvious that each Greek letter appearing in the middle 
 column of the Index represents the presence of the corre- 
 sponding combination, or rather its non-exclusion. Absence 
 of the Greek letter represents exclusion. Looking, for 
 instance, to No. 31, we learn that a be, an assertion of 
 the Vlth type, excludes all combinations except a, /3, y, 
 specified at the top of the table, and 8, specified in the 
 centre column ; that is to say, the combinations consistent 
 with a = bc are ABC, ABr, A^C, and abc. The principal 
 use of the Index, however, will be in the inverse direction, 
 to find the law corresponding to certain unexcluded com- 
 binations. Taking, for instance, the combinations A&r, aBC, 
 abC, their Greek signs are 8, e, 77 ; to find their law, then, 
 we must look in the last table in the column headed (not-a), 
 (not-/3), (not-y), and in the line showing 8, e, >/, in the 
 middle column. We there find the two assertions A = <: 
 = Kb of the IVth type (No. 230), as those corresponding 
 to the combinations in question. 
 
 3. With the aid of this Index we can infallibly and 
 rapidly solve all possible problems relating to three terms. 
 What assertion, for example, can we make which shall not 
 be contradictory to, and yet shall not be inferrible from, the 
 premise a = BC -|- abc ? Working out the combinations 
 unexcluded by this premise, we find them to be AB<r, A^C, 
 Me, aBC, and abc, or /3, y, 8, e, 6. Of these (3 and 8 may be 
 removed simultaneously without wholly removing any letter, 
 that is to say without contradiction. Looking in the second 
 table of the Logical Index at No. 81, we find the proposition 
 A = AC (of type II.) as one which removes (3 and 8. 
 This is the required proposition which is, as it were, quite 
 neutral to the one assumed. In the same way we might 
 remove y and 6 without contradiction, so that No. 34, or 
 Ab = bc, of type XII., is another neutral proposition. It
 
 xxvin THE LOGICAL INDEX 283 
 
 may I believe, be safely inferred that every proposition of 
 type XIII. , that of the premise in question, will have at 
 least two propositions neutral to it, of types II. and XII. 
 respectively. 
 
 Suppose it be required, as a second instance, to define the 
 precise points of agreement and difference of two disput- 
 ants, one of whom asserts that (i) ' Space = three-way 
 spread with points as elements ' (Henrici) ; while his oppo- 
 nent holds that (2) ' Space = three -way spread,' and at 
 the same time (3) ' Space has points as elements,' but is not 
 known to be the only thing that has. The three assertions 
 are symbolised as below, the combinations excluded being 
 indicated by their Greek signs : 
 
 ft 
 
 A = B I 
 
 
 f 
 
 We see that the second disputant's assertions have a 
 logical force superior to that of the first by , namely 
 which corresponds to assertion 5, or B = BC. In addition, 
 then, to all that the first asserts, he affirms that ' a three-way 
 spread which is not space has points as elements.' 
 
 As a third instance of the power and flexibility of this 
 combinational logic, suppose it to be required to make an 
 exhaustive statement of all the inferences which can be 
 drawn from the theorem that ' Similar figures (A) consist 
 of all whose corresponding angles are equal (B), and whose 
 corresponding sides are proportional (C).' We proceed in 
 this way. The proposition is of the form A = BC, of the 
 Vlth type, and negatives (3, y, 8, e. Any proposition, then, 
 which negatives one, two, or three of these combinations
 
 284 THE LOGICAL INDEX CHAP. 
 
 will be inferrible from the theorem, but not equivalent to it. 
 All the possible inferences, therefore, are indicated in the 
 following table of Index Numbers, which, taken in con- 
 nection with the Logical Index, sufficiently explains itself : 
 
 
 " 
 
 BS. 
 
 yi 
 
 81 
 
 f/V 
 
 Bye.. 
 
 o 
 
 .. IQI; 
 
 ... 
 
 fi 
 
 ..17 
 
 Be. 
 
 ..73 
 
 r / s 
 
 B8e... 
 
 . 89 
 
 . . . 
 
 . Q 
 
 yS. 
 
 . . 40 
 
 voe . . 
 
 . C7 
 
 
 
 i 
 ve. 
 
 ..4.1 
 
 / 
 
 
 These fourteen assertions, which are all the possible 
 non-equivalent inferences, or the equivalents of these, were 
 detected by the Logical Index in a few minutes ; it would 
 be doubtfully possible, and in any case a most laborious 
 problem, to obtain an exhaustive statement of inferences 
 by any other method, if indeed any other method exists. 
 
 The want of space alone prevents my giving more abun- 
 dant illustrations of the multitudes of logical problems 
 which may be solved infallibly and speedily by the use of 
 the Logical Index. It may be safely said that in four pages 
 of tables it gives the key to all possible logical questions, 
 relations or problems involving three distinct logical terms. 
 There is some possibility that the corresponding index 
 .for the relations of four terms may some day be worked out, 
 as, when exhibited in like manner, it will occupy only one 
 volume of 1024 pages of a rather larger size than those of 
 this volume. There is no prospect whatever that the corre- 
 sponding index for five terms will ever be exhaustively 
 published, since it would fill a library of 65,536 volumes, 
 each containing 1024 large pages. This fact will give some 
 faint idea of the possible number and complexity of logical 
 relations involving only a very moderate number of terms.
 
 xxviii THE LOGICAL STAMP 285 
 
 The Logical Stamp 
 
 In my previous logical books x I described a Logical 
 Slate with five series of the combinations of the Logical 
 Alphabet engraved upon it. I first made such a slate in 
 May 1863, and I have since frequently used it with much 
 saving of labour. The recent extensive introduction of 
 'india-rubber printing stamps lately suggested to me the 
 idea that the most convenient method of obtaining the 
 logical combinations would be to stamp them on paper. 
 Two stamps producing the combinations of three and of 
 four terms as shown in columns IV. and V. of the Logical 
 Alphabet (p. 181), were made for me at a cost of about 
 eleven shillings. 
 
 They have been very successful, and leave nothing to be 
 desired as regards the private study of logical problems. 
 One great advantage of the stamps over the slate is evident, 
 namely, that the work being done on paper can be preserved 
 for reference without copying. 
 
 The ABCD stamp can readily be utilised for problems of 
 five, six, or more terms. For six terms, for instance, it is 
 requisite to make four impressions and distinguish them by 
 writing EF, Ef, e, ef, above the respective impressions. 
 
 India-rubber stamps of any design can now be easily 
 ordered at all the principal stationers. 
 
 1 Pure Logic, 1864, p. 68 ; Substitution of Similars, 1869, p. 54 > 
 Elementary Lessons in Logic, 1870, p. 199 ; Principles of Science, 1874, 
 Vol. i. p. 1 10 ; New Editions, p. 96.
 
 286 
 
 THE LOGICAL INDEX 
 
 2 :*!Qe.--9.sjOH w>tso <JH 
 
 ^ romrommmr^T)-Tf^>j-^-^-^^-^-^-ininioininioinioui m*o vo vo vo vo 
 
 % 
 
 a 
 H* 
 
 ^ 
 
 <U ^ "* ^^^ll^l'^'Q- It ^|| 
 
 J S !* a-* 
 
 H 
 
 Q 
 
 PS- sr-p- p-p- s-s- s-p- p-p- p-p- p-p- 
 
 y i ' 
 
 o & 
 
 o 
 
 w 
 
 33 
 H 
 
 ox 
 
 T 
 
 S.II II II II II II II II II II I! II II II II II II II II II ILi^rcr* ."It 
 v ^"^"^QpSyffittDSCQ^^^yQQ"^^* 
 
 < 8<: <SIS!J <-, 
 
 (i, I, ii || II ii <s 
 
 J 
 K
 
 XXVIII 
 
 THE LOGICAL INDEX 
 
 287 
 
 J 
 
 6 o*~o\ c* 
 
 3 
 
 55 
 
 S 
 
 1*0 
 
 V *S 8 " 
 
 " J ""^ " O "" 
 - -. ii U ^, "^O""'* " ^Ji'o" 
 
 K < B 
 
 OH 
 
 B-s- p-s- B-p S-S- B-S- S-S- B-p- E-B- 
 
 % 
 I 
 
 O 
 
 i II II r \ ^ .. F9 |, 
 
 pa * 
 111* 
 
 I! II I' N || ** I' II II 
 
 JJ pa ii " e y n pa 
 
 <; <s 
 
 !?
 
 288 
 
 THE LOGICAL INDEX 
 
 CHAP. 
 
 S ^| 1 vo^-S5 1 ?,^^K?:^f;^ c g I S < S^o?^S<-S^,&Ss 
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 CHAPTER XXIX 
 
 MISCELLANEOUS QUESTIONS AND PROBLEMS 
 
 IT seems convenient to bring these Studies in Deductive 
 Logic to a close by adding a certain number of mixed 
 Questions and Problems, which may refer to any part of 
 logical doctrine. In some cases these questions pass the 
 bounds of formal and deductive logic. It is left to the 
 student to determine what part, if any, of the preceding 
 pages will assist him. To certain questions, however, are 
 appended references to other works where the proper assist- 
 ance will be found. 
 
 1. What may we expect to happen, in a logical point of 
 view, when an irresistible force meets with an infinite 
 resistance ? 
 
 2. If it is said to be false that what has the properties 
 A and B has those also of C and D, and vice versa, how 
 would you interpret this statement as affecting the possible 
 relations of A, B, C, and D ? 
 
 3. In how many ways may we in a purely logical point of 
 view contradict the assertion of Hobbes that ' Irresistible 
 might in the state of nature is right '? (See p. 182, Ques- 
 tion 7.) Specify the ways. 
 
 4. Analyse the logical import of the following passage 
 from the Wealth of Nations, book i. chapter viii. :
 
 CHAP, xxix MISCELLANEOUS QUESTIONS 291 
 
 ' It is not because one man keeps a coach, while his 
 neighbour walks afoot, that the one is rich, and the other 
 poor ; but because the one is rich, he keeps a coach, and 
 because the other is poor, he walks afoot.' 
 
 5. In Harriet Martineau's Autobiography (vol. i. p. 355) 
 we are told that a certain lady, after receiving from Charles 
 Babbage a long explanation of his celebrated calculating 
 machine, terminated the conversation with the following 
 question : ' Now, Mr. Babbage, there is only one thing more 
 that I want to know. If you put the question in wrong, 
 will the answer come out right ? ' 
 
 If you think this question absurd, give distinct and 
 detailed reasons for thinking so, and reconcile them with 
 the fact that false premises may give a true conclusion. 
 
 6. Explain and illustrate the Aristotelian saying : Ex 
 veris fieri non potest ut falsum concludatur ; ex falsis contra 
 verum ; and show some of its applications in the investiga- 
 tion of nature. [R.] 
 
 7. A certain argument having been shown to involve 
 paralogism, inquire into the conditions under which this 
 failure does or does not tend to establish the contradictory 
 conclusion. 
 
 8. In a certain borough, on one occasion, the Liberal 
 party objected to 3624 voters, and the Conservative party 
 to 55 3 z voters, the whole constituency being 10,000. 
 What is the least number of voters which can have been 
 objected to on both sides ? What is the greatest number ? 
 What is the most probable number, supposing the objections 
 to be made quite at haphazard ? 
 
 9. What is the logical, compared with the popular, in- 
 terpretation of the injunction, ' This man is not on any 
 account to be ducked in the horse-pond.' Explain the 
 difference.
 
 292 MISCELLANEOUS QUESTIONS CHAP. 
 
 10. What is the logical, compared with the popular, in- 
 terpretation of the injunction, 'All persons are requested 
 not to discharge fireworks among the crowd around the 
 bonfire on the 5th November.' 
 
 n. ' Because a horse is an animal, the head of a horse 
 is the head of an animal.' 
 
 Examine the validity of this inference. Can you express 
 the reasoning syllogistically, or symbolically ? [E.] 
 
 1 2. Investigate the nature of the reasoning, good or bad, 
 involved in the four following examples : 
 
 (1) Elephants are stronger than horses; horses are 
 
 stronger than men ; therefore elephants are stronger 
 than men. [E.] 
 
 (2) Alexander was the son of Philip ; therefore Philip 
 
 was the father of Alexander. 
 
 (3) As good kill a man as kill a good book ; for he that 
 
 kills a man does but kill a reasonable creature ; 
 but he that kills a good book kills Reason herself. 
 
 (4) Nay, look you, I know 'tis true ; for his father built 
 
 a chimney in my father's house, and the bricks are 
 alive at this day to testify it. [o.] 
 
 1 3. What methods underlie these inferences ? 
 Because it froze last night, therefore the pools are 
 
 covered with ice. 
 
 During the retreat of the Ten Thousand a cutting north- 
 wind blew in the faces of the soldiers ; sacrifices were offered 
 to Boreas, and the severity of the wind immediately ceased, 
 which seemed a proof of the god's causation. [p.] 
 
 1 4. What method is employed in the following ? 
 
 ' Brewster accidentally took an impression from a piece 
 of mother-of-pearl in a cement of resin and bees'-wax, and 
 finding the colours repeated upon the surface of the wax,
 
 xxix AND PROBLEMS 293 
 
 proceeded to take other impressions in balsam, fusible 
 metal, lead, gum-arabic, etc., and always found the iri- 
 descent colours the same. He thus proved that the chemical 
 nature is wholly a matter of indifference, and the form of 
 the surface is the condition of such colours.' 
 
 15. What is the difference, logically, between the sen- 
 tences : Leibnitz, a great philosopher, has said, etc. ; and 
 A great philosopher, Leibnitz, has said, etc. ? [c.] 
 
 1 6. What is successive induction, or induction by 
 
 connection, as in the proof that n 2 =1+3 + 5 + 
 
 up to n of the odd numbers ? [H.] 
 
 (See Elementary Lessons in Logic ', lesson xxvi. p. 220.) 
 
 1 7. Give an inductive proof that x" - a" is divisible by 
 x a, when n is a whole number. 
 
 1 8. Can there be such a thing as a fallacy of simple in- 
 spection, that is a fallacy which does not involve inference ? 
 (See Mill's System of Logic, book iv. chapter iii.) 
 
 1 9. Leibnitz says ' Knowledge is either obscure or clear. 
 The clear is again either confused or distinct : and the 
 distinct either adequate or inadequate ; is further either 
 symbolical or intuitive ; and if it be at the same time both 
 adequate and intuitive, it is perfect.' 
 
 Give an exhaustive classification of all possible kinds * of 
 knowledge under the above conditions. (See Elementary 
 Lessons in Logic, lesson vii.) 
 
 20. Explain the logical meanings of the terms Genus, 
 Species, Difference, Property, and Accident, distinguishing the 
 meaning in extent and intent, and using for illustration the 
 varieties of things ABCD, ABC;/, AB^D, KEcd, in which 
 A, B, C, D are terms denoting qualities, and c, d the 
 negatives of C, D. 
 
 21. Draw the inferences deducible from these data :
 
 294 MISCELLANEOUS QUESTIONS CHAP. 
 
 (1) A is the only antecedent always present when/ is 
 present, and always absent when p is absent. 
 
 (2) A is an antecedent always present when p is present, 
 and always absent when / is absent. 
 
 (3) A is an antecedent frequently present when p is 
 present, and frequently absent when p is absent. [p.] 
 
 22. Point out the exact nature of the relations between 
 the logical processes of Abstraction, Analysis, Synthesis, 
 and Generalisation. 
 
 23. What is the logical difference, if any, between nouns 
 Substantive and nouns Adjective ? 
 
 24. Is a Latin adjective used alone in the neuter an 
 adjective or a substantive ? 
 
 25. Is there self-contradiction in the assertion that know- 
 ledge of what is outside my consciousness may be inside 
 my consciousness ? 
 
 26. Can absolute certainty be found in any conclusion (i) 
 inductively established, (2) deductively established ? [E.] 
 
 27. Is there any distinction, and if so what, between a 
 general and an abstract notion, and is there a corresponding 
 difference between the names employed to express them ? 
 
 H 
 
 28. It is a rule of syllogism that nothing can be inferred 
 from particular premises. How then can I infer from the 
 particular facts that some men have died, the universal 
 conclusion, ' All men die ' ? [E.] 
 
 29. Explain the limits of demonstrative science, and 
 examine the following statement : ' No matter of fact can 
 be matter of demonstration.' [E.] 
 
 30. Distinguish Logical, Mathematical, and Physical 
 Quantity. [E.] 
 
 31. Distinguish and exemplify Logical, Mathematical, 
 Metaphysical, Physical, and Moral Necessity. [E.]
 
 xxix AND PROBLEMS 295 
 
 32. When two phenomena are causally connected to 
 gether, can you always ascertain which is the cause and 
 which is the effect ? If so, how ? [L.] 
 
 33. Investigate how far or on what grounds our know- 
 ledge of the following propositions approximates to 
 certainty : 
 
 ' Nitric acid does not dissolve gold.' 
 
 'A distant fixed star is subject to gravity.' 
 
 34. Consider from a logical point of view the assertion 
 that the increasing trade of Great Britain is caused by a 
 reform of the tariff. What kind of proof is applicable ? 
 
 35. A man having been shot through the heart imme- 
 diately falls dead. Investigate the logical value of such a 
 fact as proving that all men shot through the heart will fall 
 dead. [L] 
 
 36. What do you understand by a ' working hypothesis ' ? 
 Under what conditions is it legitimate for an investigator to 
 employ hypothesis ? (Huxley. Mill's System of Logic, book 
 iii. chap, xiv. ; Principles of Science, chap, xxiii.) [L.] 
 
 37. State these arguments formally, and give their tech- 
 nical designations : 
 
 (1) 'The thinking power does not belong to matter; 
 
 otherwise matter generally would exhibit it.' 
 
 (2) ' Happiness is the reward of goodness ; and since all 
 
 do not desire a good life, all cannot obtain its 
 reward.' [p.] 
 
 38. Why is it that with exactly the same amount of 
 evidence, both negative and positive, we did not reject the 
 assertion that there are black swans, while we should refuse 
 credence to any testimony which asserted that there were 
 men wearing their heads underneath their shoulders? [P.]
 
 296 MISCELLANEOUS QUESTIONS CHAP. 
 
 39. What is the difference of meaning, if any, between 
 the propositions, ' This house was built by Jack,' and 
 ' This is the house that Jack built ' ? (De Morgan, Third 
 Memoir on the Syllogism, loth page.) 
 
 40. Does the thesis that the ultimate premises in human 
 knowledge are the result of mental association affect the 
 nature and certainty of Logic, and if so, how? [E.] 
 
 41. Define evidence. Distinguish intuitive, demonstra- 
 tive, and probable evidence. [E.] 
 
 42. Explain : ' Certainty, therefore, has for its opposite, 
 uncertainty in one way impossibility in another. Uncer- 
 tainty, in the language of logicians, is its contradictory 
 opposite impossibility, its contrary opposite.' [P.] 
 
 43. Investigate the question whether the truth of a 
 statement is to be judged by the impression which it makes 
 upon those to whom it is addressed, by its literal corre- 
 spondence with the belief of the person making it, or by any 
 other standard. [L.] 
 
 44. 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 equation, it is therefore the magnitude required.' 
 Examine the logical validity of this argument. 
 
 45. It is probable that Herodotus recorded only what he 
 heard concerning Ethiopia ; and it is not unlikely that most 
 that he heard was correct ; so that we may accept his 
 account as true. Is this conclusion correct ? 
 
 46. There is a very strong probability that the eldest 
 child of a newly married couple will inherit the estate of 
 the husband. For, firstly, it is more probable than not that 
 there will be children of the marriage. Next, if a child is 
 born, it is more probable that it will be a son, for more boys
 
 AND PROBLEMS 297 
 
 are born than girls. Thirdly, if a son be born, it -will prob- 
 ably survive its father. Examine this inference. [o.] 
 
 47. Consider the following argument : 'Many writings 
 that are not genuine were ascribed to Clemens Romanus ; 
 this Epistle was ascribed to him ; therefore this Epistle is 
 not genuine.' [L.] 
 
 48. A student of geometry examines three isosceles 
 triangles and finds them agree in having equal angles at the 
 base ; an excise officer examines three bottles of wine out 
 of a quantity imported and finds them agree in strength ; 
 a chemist analyses three specimens of a mineral and finds 
 them agree in composition : compare the inferences which 
 may be drawn in these cases. 
 
 49. What is the relation between classification and in- 
 duction in general? [L.] 
 
 50. When an experiment designed to produce a phe- 
 nomenon fails to produce it, in how many ways may we 
 interpret or explain the meaning of the failure ? [L.] 
 
 51. In what ways may a physicist hope to explain away 
 an exceptional phenomenon ? [L.] 
 
 52. If we never find skins except as the teguments of 
 animals, we may safely conclude that animals cannot exist 
 without skins. If colour cannot exist by itself (a-n-av yap 
 X/3u)/*a fv 0-w/Aari), it follows that neither can anything that 
 is coloured exist without colour. So, if language without 
 thought is unreal, thought without language must also be so. 
 What do you think of this argument ? [o.] 
 
 53. If we are disposed to credit all that is told us, we 
 must believe in the existence not only of one, but of two or 
 three Napoleon Buonapartes ; if we admit nothing but what 
 is well authenticated, we shall be compelled to doubt the 
 existence of any. How, then, can we be called upon to be- 
 lieve in the one Napoleon Buonaparte of history ? [o.]
 
 298 MISCELLANEOUS QUESTIONS CHAP. 
 
 54. Brown asserts that all planets are spheroids ; Jones 
 denies it ; Robinson asserts that Jones knows nothing about 
 the matter ; Smith proves that in this case at least Robinson 
 is correct ; but Thomson refuses to accept the premises 
 of Smith's proof. What are the logical relations of the 
 parties ? 
 
 55. 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. 
 
 56. It is asserted by some philosophers that all knowledge 
 is inductive in its origin, and it is generally allowed that 
 inductive inferences can be probable only ; if so, no know- 
 ledge can be more than probably true. Can you, however, 
 adduce any instance of knowledge which is certainly true ? 
 In that case explain the difficulty which evidently arises. 
 
 57. Aut amat aut odit mulier ; nihil tertium. If any 
 one takes upon himself simply to deny the truth of this 
 saying of Publius Syrus, in how many different ways may 
 the denial be interpreted ? 
 
 58. Explain the following apparent paradox : P thinks 
 of an object ; Q is absolutely ignorant of the size of that 
 object ; to him, therefore, the probability that the object is 
 greater than a cannon-ball is -J. Again, being absolutely 
 ignorant about its size, he has no reason to believe it either 
 greater or less than a pea, the probability of either case 
 being A. Hence to him it is infinitely improbable that 
 the object is intermediate in size between a pea and a 
 cannon-ball. [JOHN HOPKINSON, D.SC.] 
 
 59. In defending a prisoner his counsel must either deny 
 that the deed committed is a crime, or he must deny that 
 the prisoner committed the deed ; therefore if the counsel 
 denies that the deed committed is a crime, he must admit 
 that the prisoner did commit the deed.
 
 xxix AND PROBLEMS 299 
 
 60. What do you understand by the logical proof of an 
 assertion ? Compare the logical meaning of the word proof 
 with any other meanings of the word known to you. [i.] 
 
 6 1. Can all kinds of propositions be exhibited in the 
 intensive as well as the extensive form ? Give reasons in 
 support of your answer ; and in the event of its being in 
 the negative, draw up a list distinguishing between those 
 kinds of propositions which can, and those which cannot, 
 be so exhibited. [L.] 
 
 62. Explain the meaning of the assertion that Induction 
 is the inverse process of Deduction. 
 
 63. Illustrate Mathematical Induction in its several kinds 
 or cases, and discuss its relation to induction in the physical 
 sciences. 
 
 64. What is the relation, if any, between the inductive 
 syllogism and the inductive methods employed in the 
 physical sciences? 
 
 65. Estimate upon logical grounds the possibility of 
 establishing a school in which students should be rendered 
 capable of discovering the Laws of Nature. (Gore's Art of 
 Scientific Discovery^ 
 
 66. What precisely is meant by the Law of Continuity? 
 Point out the grounds and limits of its validity. (Life of 
 Sir W. Hamilton, p. 439 ; Principles of Science, chapter 
 xxvii.) 
 
 67. When the effects of three distinct causes are added 
 and mingled together, by what processes of experiment and 
 reasoning can we assign to each cause its separate effect ? 
 
 [c.] 
 
 68. Under what circumstances are we to accept the 
 failure of an experiment or series of experiments as proving 
 the non-existence of the phenomenon intended to be pro- 
 duced? (Principles of Science, chapter xix. ) [L.]
 
 300 MISCELLANEOUS QUESTIONS CHAP. 
 
 69. Illustrate the scientific value of exceptional pheno- 
 mena, and show in how many ways they may be disposed 
 of or reconciled with physical law. {Principles of Science, 
 chapter xxix.) [L.] 
 
 70. What is the difference between the causal and the 
 casual happening of events, if, as is generally allowed, not 
 even a dead leaf falls to the ground without sufficient causes 
 to determine the precise moment of its falling and the 
 precise spot upon which it will fall ? 
 
 7 1 . Show by example that the logical copula does not 
 imply the notion of existence. [E.] 
 
 72. Investigate the question whether the functions of 
 affirmative and negative propositions in reasoning are 
 similar. 
 
 73. England is the richest country in the world, and 
 has a gold currency. Russia and India, in proportion to 
 population, are poor countries, and have little or no gold 
 currency. How far are such kinds of facts logically 
 sufficient to prove that a gold currency is the cause of a 
 nation's wealth ? [i.] 
 
 74. If by two distinct methods of investigation you arrive 
 at the same conclusion, namely, that the currency of the 
 kingdom does not exceed one hundred millions sterling, but 
 it is afterwards discovered that one of the methods of 
 investigation involved fallacious reasoning, what would you 
 be inclined to infer about the other method of investigation ? 
 
 w 
 
 75. A certain argument having been shown to involve 
 paralogism, inquire into the conditions under which this 
 failure does or does not tend to establish the contradictory 
 conclusion. 
 
 76. Investigate the logical, psychological, and moral 
 grounds of the saying, ' Qui J excuse, J accuse?
 
 xxix AND PROBLEMS 301 
 
 77. Taking the senses in which they most resemble one 
 another, distinguish between judgment, opinion, statement, 
 knowledge, fancy, conjecture, supposition, allegation. [E.] 
 
 78. Distinguish : truth, certainty, fact, opinion, proba- 
 bility, evidence, conviction. [E.] 
 
 79. How far can the inconceivability of the opposite be 
 regarded as proof of the truth of any judgment ? [E.] 
 
 80. Right-angled and not-right-angled are contradictory 
 predicates ; therefore, according to the law of Excluded 
 Middle, as the proposition All triangles are right-angled 
 is false, it must be true that all triangles are not right- 
 angled. But this also is false. Explain the above difficulty. 
 
 [E.] 
 
 8 1 . Given that ( i ) whenever the statements a, b, x are 
 either all three true, or all three false, then the statement 
 c is false, and y is true, or else c is true, and y is false ; 
 (2) that whenever d, e, y are either all three true or all three 
 false, then the statement a is false, and x is true, or a is true, 
 and x is false. When can we infer from these premises that 
 either x or y is true ? 
 
 [Hugh MacColl, B.A., in Educational Times, question 
 6206. A solution was given by C. J. Monro, M.A., 
 in the same paper for March 1880. The question 
 seems to mean What other conditions with those 
 given determine that either x or y is true ?] 
 
 82. De Morgan says {Fourth ftfemoir 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.
 
 MISCELLANEOUS QUESTIONS CHAP. 
 
 83. To what type of assertion do the premises of Darapti 
 on? ? 
 
 belong ? 
 
 84. Give the converse, inverse, contrapositive, obverse, 
 and reciprocal propositions of the following : 
 
 (1) All parallelograms have their opposite angles equal. 
 
 (2) If P is greater than Q, then J? will be greater 
 
 than S. 
 
 (3) Two triangles are congruent if the three sides of the 
 
 one are respectively equal to the three sides of the 
 other. 
 
 85. Why have some mathematicians been accustomed to 
 say that it is necessary to prove the converse of a mathematical 
 proposition ? 
 
 86. Where exactly lies the error of the Irishman, who 
 being charged with theft on the evidence of three witnesses 
 who had seen him stealing the article in question, proposed 
 to bring in his defence thirty witnesses who had not seen 
 him stealing it ? 
 
 87. Epimenides says that every statement of a Cretan is 
 a lie ; but Epimenides is a Cretan ; therefore what he says 
 is a lie ; therefore every statement of a Cretan is not a lie. 
 
 [E.] 
 
 88. If, in saying that 'few strikes are beneficial,' I feel 
 sure that the statement will be misinterpreted by those to 
 whom it is addressed, and that the statement 'no strikes 
 are beneficial,' although .not in my opinion literally true, 
 will more exactly convey to the hearers' minds the im- 
 pression which I believe to be true, ought I, having regard 
 to the moral obligation of speaking the truth, to use the 
 latter assertion or the former ? 
 
 89. 'I will go on,' said King James, ' I have been only 
 too indulgent. Indulgence ruined my father.'
 
 xxix AND PROBLEMS 303 
 
 Express clearly the process of reasoning involved in this 
 utterance. Is it Induction ? or what ? [M.] 
 
 90. What is the relation, if any, between the inductive 
 syllogism and the inductive methods employed in the 
 physical sciences ? 
 
 91. Can the proposition, ' All A is all BJ be regarded as 
 representing a single act of thought? (See Mind, vol. i. 
 p. 216.) [E.] 
 
 92. Are the premises of Darapti given only in a 
 numerical form sufficient to prove the conclusion ? 
 
 93. Does it follow that, because some poetry is not in 
 verse, there must be some verse which is not poetry ? [H.] 
 
 94. Take the proposition, ' All sciences are useful,' and 
 determine precisely what it affirms, what it denies, and what 
 it leaves doubtful, concerning the relations of the terms 
 ' science ' and ' useful thing.' 
 
 95. Ascertain precisely how many distinct assertions 
 there are in the description of the conduct of the great 
 scholastic logician, John of Salisbury, after Thomas a 
 Becket had been murdered by his side : Tacitus sed moerens, 
 continue se subduxit. 
 
 96. Can you represent equationally the contradiction 
 between ' Some Xs are not some Fs ' and ' There is one X 
 only and that is the only Y ' ? 
 
 97. Which of the types of assertion involving three terms 
 are complete, in the sense of admitting no additional assertion 
 involving the same three terms without self-contradiction ? 
 
 98. If all things are either X or Y, and all things are 
 either Y or Z, what inference can you draw ? 
 
 99. Do the thirty-six moods of Hamilton's Syllogism with 
 quantified predicates (see table in Elementary Lessons, 
 p. 1 88 ; Thomson's Laws of Thought, section 103) comprise 
 all the possible weakened moods ?
 
 304 MISCELLANEOUS QUESTIONS CHAP, xxix 
 
 100. Is the student of logic, generally speaking, prepared 
 rapidly to analyse the two following propositions, and to say 
 whether or no they must be identical, if the identity of 
 synonyms be granted ? 
 
 (1) 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 
 difficulty removed. 
 
 (2) When we see a nation either backward to suspect 
 
 its neighbour, or apt to be satisfied by expla- 
 nations, we may rely upon it that the neighbour is 
 neither the more civilised nor the more warlike of 
 the two.' 
 
 [DE MORGAN, Third Memoir, 1858, p. 181.] 
 
 101. Is the following proposition a definition or not ? Is 
 it on the matter or the form of the proposition that you 
 found your answer ? 
 
 ' LOGICA EST ARS ARTIUM ET SCIENTIA SCIENTIARUM.' 
 
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