THE DEVELOPMENT OF 
 SYMBOLIC LOGIC 
 
THE DEVELOPMENT 
 
 OF 
 
 SYMBOLIC LOGIC 
 
 A CRITICAL-HISTORICAL STUDY OF 
 THE LOGICAL CALCULUS 
 
 BY 
 
 A. T. SHEARMAN, M.A. 
 
 LONDON 
 WILLIAMS AND NORGATE 
 
 14 HENRIETTA ST., COVENT GARDEN 
 1906 
 
-ec-i^l 
 
PREFACE 
 
 The form that the present work has taken 
 is due to some correspondence that I had 
 with Mr. W. E. Johnson in the year 1903. 
 He pointed out to me the error of think- 
 ing of the various symbolic systems as 
 being radically distinct, and as competing 
 with one another for general acceptance. 
 Rather, he held, it is correct to adopt the 
 view that there is available at the present 
 time what may be called the Logical Cal- 
 culus, and that towards the creation of this 
 Calculus most symbolists have contributed. 
 
 This idea has been worked out in the 
 following pages. I have traced the growth 
 of the subject from the time when Boole 
 originated his generalisations to the time 
 when Mr. Russell, pursuing for the most 
 
 V 
 
 158461 
 
VI Preface 
 
 part the lines laid down by Peano, showed 
 how to deal with a vastly wider range ot 
 problems than Boole ever considered. My 
 attention has been occupied, that is to 
 say, upon the questions to whom we are 
 most indebted for those rules of procedure 
 that may be said now to constitute the Cal- 
 culus, what important differences of opinion 
 have arisen as the subject has been gradually 
 thought out, and which of the conflicting 
 views we find it correct to adopt. The 
 investigation has thus been quite as much 
 critical as historical, for, in demonstrating 
 who have contributed to the creation of the 
 Logical Calculus, it has been necessary con- 
 stantly to point out, in the first place, why 
 certain views have to be rejected as being 
 incorrect, and secondly, wherein one of two 
 suggestions, both of which are excellent, 
 shows an advance upon the other. 
 
 Portions of some of the chapters have 
 appeared in a paper which has been pub- 
 lished in the Proceedings of the Aristotelian 
 
Preface vii 
 
 Society (N.S. vol. v.), and which was entitled 
 " Some Controverted Points in Symbolic 
 Logic." Judging from the kind communi- 
 cations which I have received from several 
 logicians with reference to this paper, I 
 think that the insertion of these sections 
 is likely to prove useful. My opinion upon 
 the questions I then discussed has not, as a 
 result of further thought, undergone any 
 change, but after reading Mr. Russell's 
 work. The Principles of Mathematics^ I have 
 been careful to point out that the view 
 which I expressed as to the relation of 
 Mathematics to Logic is to be regarded as 
 preferable only to the doctrines that were in 
 vogue prior to the time of Peano's analysis 
 of mathematical notions. 
 
 It is unnecessary here to refer to the 
 various writers whom I have considered, 
 since they will be mentioned in the appro- 
 priate places throughout the book. But I 
 should like to say how much I owe in the 
 way of equipment for my task to Dr. Venn 
 
viii Preface 
 
 and Mr. Johnson. It was the former who 
 a good many years ago created in me a taste 
 for the study of SymboHc Logic, and he has 
 ever been ready both to give me informa- 
 tion and to discuss points in which I was 
 interested. Mr. Johnson*s articles in Mind^ 
 by throwing Hght on some of the more 
 difficult questions of Formal Logic, and 
 by exhibiting in a very clear manner 
 the unity running throughout the Logical 
 Calculus, have been to me of the greatest 
 service. 
 
 University College, London, 
 February, 1906. 
 
CONTENTS 
 
 PAGE 
 
 INTRODUCTION i 
 
 CHAPTER I 
 
 SYMBOLS AS REPRESENTING TERMS AND 
 AS REPRESENTING PROPOSITIONS 
 
 Introductory ........ g 
 
 I. On the Primary and on the Exclusive use of 
 
 Literal Symbols . . . . .10 
 
 II. Real and Superficial differences between 
 
 Systems ....... 22 
 
 III. Symbolic Logic and Modals .... 24 
 
 CHAPTER II 
 
 SYMBOLS OF OPERATION 
 
 31 
 
 I. Symbols of Operation are not Essential in 
 Logic ....... 
 
 II. Question whether it is Expedient for the 
 
 Logician to adopt Mathematical Symbols 34 
 
 III. Symbols to denote Logical Addition, Subtrac- 
 
 tion, and Division . . . . .38 
 
 IV. Other Symbols of Operation . . . -Si 
 V. Symbolization of Particular Propositions . 56 
 
 VI. Symbolization of Hypotheticals and of Dis- 
 junctives ....... 59 
 
 « b 
 
c Contents 
 
 CHAPTER III 
 
 THE PROCESS OF SOLUTION 
 
 PAGE 
 
 I. The Solution of the Direct Problem . . 64 
 
 ( I ) Analytical Method 64 
 
 (2) Diagrammatic Method .... 80 
 
 II. The Solution of the Inverse Problem . . 84 
 
 CHAPTER IV 
 
 CONCERNING A CALCULUS BASED ON 
 INTENSION 
 
 Introductory . . . . . . . .91 
 
 I. The Doctrines of Castillon .... 94 
 
 (i) Castillon's Fundamental Notions . . 95 
 
 (2) Illusory Particulars . . . . .111 
 
 (3) Inconvertibility of Real Particulars and of 
 
 Universal Negatives . . . .122 
 
 (4) Castillon's Treatment of Hypothetical and 
 
 of Problematical Judgments . . .124 
 
 (5) Derivation of the Notion of Quantity . .128 
 
 (6) Comparison of Castillon's Symbolism with 
 
 Mrs. Bryant's 131 
 
 II. The General Question of an Intensive 
 
 Logic i33 
 
 (i) A Logic based on Connotation . . .134 
 
 (2) A Logic based on Comprehension . . 138 
 
Contents xi 
 
 CHAPTER V 
 
 THE DOCTRINES OF JEVONS AND OF 
 MR. MacCOLL 
 
 Introductory ....... 
 
 I. Criticism of Jevons .... 
 
 II. Criticism of Mr. MacColl 
 
 ( 1 ) His Employment of Literal Symbols . 
 
 (2) His Treatment of Modal Propositions 
 
 (3) His Doctrine of a Universe of Unrealities 
 
 PAGE 
 161 
 
 CHAPTER VI 
 
 LATER LOGICAL DOCTRINES 
 
 Introductory . . . . . . . .172 
 
 I. Treatment of Multiply-quantified Propositions 173 
 II. The Impossibility of a General Treatment of 
 
 CopuLi^E . . . . . . .183 
 
 III. The New Treatment of Mathematical Con- 
 ceptions . . . . . . .196 
 
 CHAPTER VII 
 THE UTILITY OF SYMBOLIC LOGIC . . 221 
 
 INDEX 233 
 
V 
 
 THE DEVELOPMENT OF 
 SYMBOLIC LOGIC 
 
 INTRODUCTION 
 
 My object in writing this book is to show- 
 that during the last fifty years there has 
 been a definite advance made in SymboKc 
 Logic. By any person commencing the 
 study of this discipHne it is not unnatu- 
 rally soon concluded that there exist several 
 " systems," marked off from one another 
 by fundamental differences. Such systems 
 he is inclined to describe according to the 
 character of the view that the founder 
 entertained as to the import of the pro- 
 position. Thus there is the compartmental 
 view, the predication view, the mutual ex- 
 clusion view, and so on. But subsequent 
 
2 Symbolic Logic 
 
 study enables the reader to perceive that, 
 in adhering to such a conception, he is 
 hiding the points of Hkeness and magni- 
 fying the points of difFerence between the 
 proposed methods of treating the subject, 
 and he is thus led to look rather at the 
 net result of the different efforts. That 
 is to say, instead of continuing to speak 
 of several isolated systems, he proceeds to 
 study the calculus that is nov^ available, 
 and to the construction of which most 
 symbolists are seen to have contributed. 
 
 The calculus that we have said now 
 exists enables us to solve with comparative 
 ease problems relating to qualitative objects.^ 
 We are able, after the performance of certain 
 processes that are in accordance with a few 
 logical laws, to arrive at conclusions respect- 
 ing any class or group of classes that is 
 involved in the premises. These con- 
 
 * If we adopt the Peanesque standpoint we may say that 
 the calculus can deal with either qualitative or quantitative 
 objects. But the statement in the text really expresses the 
 truth of the matter, for before mathematical assertions can be 
 dealt with by Peano they have to be expressed as implications. 
 
Introduction 3 
 
 elusions may show the relation of such 
 class or group to all the other classes in- 
 volved, or we may employ the process of 
 elimination and show the relation to cer- 
 tain only of the other classes. On the 
 other hand, starting with certain con- 
 clusions we are able to discover premises 
 whose presence accounts for them. 
 
 Now, before the time of Boole none of 
 the problems with which the calculus deals 
 could have been solved, unless they were 
 of the very simplest description. He was 
 the first to reach any generalisations of 
 logical doctrines.* There certainly had 
 been some gropings towards the desired 
 end before his time. Venn refers to some 
 of the earlier attempts, and in particular 
 to the work of Lambert. Venn shows that 
 
 * Venn, Symbolic Logic, ed. i. p. xxviii., says that Boole 
 was the originator of a// the higher generalisations. The para- 
 graph is omitted in the second edition. Of course Venn would 
 still hold that Boole was the originator of many of the higher 
 generalisations, but would allow that other logicians have, since 
 the publication of the Lanvs of Thought, generalised in certain 
 directions. The point of Venn's statement is that before 
 Boole's time there were no generalisations at all. 
 
4 Symbolic Logic 
 
 this eighteenth century logician had in the 
 following respects made considerable ad- 
 vances towards the Boolian position. He 
 had recognised that addition, subtraction, 
 multiplication, and division have an ana- 
 logue in Logic, had perceived the inverse 
 nature of the second and fourth of these 
 operations, had enunciated the principal 
 logical laws, had developed simple logical 
 expressions, had seen that his own method 
 would deal with complicated problems, and 
 in one place had observed that the process 
 corresponding to division is an indeter- 
 minate one.* 
 
 Venn also points out that Lambert's 
 coadjutors were Ploucquet and Holland, 
 and that all three took their impulse from 
 Leibnitz and Wolf 
 
 It is certainly desirable in an historical 
 account of the subject to notice these earlier 
 writers, but the fact must not be forgotten 
 that none of them did actually succeed in 
 solving any of the more complicated prob- 
 
 * Venn, Symbolic Logic, Introduction, p. xxxi. 
 
Introduction 5 
 
 lems. I do not think that such writers 
 must be considered as the founders of 
 SvmboHc Logic. Had thev framed anv 
 generaHsations the case would have been 
 different. Or had Boole been acquainted 
 with their ideas, and founded his generalis- 
 ations upon these, it would obviously have 
 been essential to include such writers 
 among those who have assisted in erect- 
 ing the svmbolic structure, but he was 
 not familiar with their work. 
 
 Still, though Boole is the man to whom 
 we are indebted for having first constructed 
 a logical calculus, the one that he produced 
 was of a decidedlv complicated character. 
 Moreover, several processes that torm a 
 highlv important part of Symbolic Logic 
 he did not bring within the scope ot his 
 investigations. And occasionallv he made 
 mistakes. Several minds have contributed 
 in unfolding the svmbolic methods at pre- 
 sent available. We shall have to notice 
 the writers who have made improvements 
 on Boole's procedure, or have corrected 
 
6 Symbolic Logic 
 
 him, or who have grappled with prob- 
 lems that he did not touch. 
 
 My order of exposition is as follows. In 
 the first three chapters I suppose that we 
 are confronted with a complicated set of 
 premises and are required to draw a certain 
 conclusion from them, or that we are given 
 a certain conclusion and are required to 
 assign premises from which it may have 
 been drawn. At each stage in the solution 
 I shall endeavour to show which logician 
 it is who has proposed the best method of 
 procedure to be adopted at that point. In 
 this way I hope to make clear that there 
 has been real development from the time 
 of Boole, and that the principal contri- 
 butors to the development are Venn, 
 Schroder, Keynes, Johnson, Mitchell, C. 
 Ladd-Franklin, and Peirce. 
 
 Then in chapter iv. I prove that those 
 who have been engaged in elaborating 
 the calculus were justified in proceeding 
 by way of an extensive rather than by 
 way of an intensive interpretation of the 
 
Introduction 7 
 
 proposition. Chapter v. is occupied with 
 an analysis of the work of two distin- 
 guished logicians, namely, Jevons and Mr. 
 MacCoU, both of whom have, indeed, 
 proceeded by way of extension,"^ but who 
 have in my opinion fallen into serious 
 errors. Up to this point the investigation 
 is concerned with what may be called the 
 ordinary Symbolic Logic, and with its 
 ordinary employment. In the next chapter 
 I refer (i) to the logicians who have shown 
 how the principles that are utilised for the 
 manipulation of propositions with single 
 quantifications may also be utilised in the 
 case of double and multiple quantifications ; 
 (2) to the view that, though such multiple 
 quantifications may be successfully manipu- 
 lated, it is not possible to treat copulas in a 
 general manner, and so arrive at such a 
 Logic as the expression " Logic of Rela- 
 tives " naturally suggests to the mind ; and 
 
 * Jevons, as Venn points out, professed to interpret the 
 proposition in an intensive manner, but dealt with the subject 
 rather from the extensive standpoint. (See Venn, Symbolic 
 Logic, note on p. 453.) 
 
8 Symbolic Logic 
 
 (3) to Frege, Peano, and Russell, who have 
 shown that, when certain distinctions are 
 made which the older symbolists passed by 
 as unimportant, and when a suitable inter- 
 pretation is given to the conceptions of 
 quantitative mathematics, both the com- 
 prehensiveness and the utility of Symbolic 
 Logic are greatly increased. Finally, having 
 thus traced the development of the subject, 
 I devote a chapter to the consideration of 
 the uses of Symbolic Logic, both in its less 
 and in its more extended application. 
 
CHAPTER I 
 
 SYMBOLS AS REPRESENTING TERMS AND 
 AS REPRESENTING PROPOSITIONS 
 
 The first step in the solution of a com- 
 plicated problem is to take the premises, 
 which are given as a rule in words — as a 
 rule, for the cawing of a rook and the 
 flying of a union-jack may be premises, 
 since they are statements, though of course 
 not propositions,^ — and to put them into 
 some other form, in order that they may 
 be easily manipulated. The customary 
 way is to take letters either to represent 
 the classes or to represent the statements 
 involved. The question then arises as to 
 which of these two the letters should be 
 taken to represent. f Having answered 
 
 * See MacCoIl in Mind, N.S. No. 43. 
 j* The argument in the next few pages has reference to 
 problems of the earUer Symbolic Logic. I hold that in the 
 
 9 
 
lo Symbolic Logic 
 
 this question, we shall be able to see which 
 logicians have been proceeding on the right 
 lines, so far as this point is concerned. 
 
 I. There are here three considerations 
 that must be kept quite distinct if the 
 subject is to be profitably discussed. In 
 the first place, it is possible to affirm that 
 symbols may under one set of conditions 
 represent terms, and under another set of 
 conditions represent propositions, and then 
 it has to be decided which of the two 
 available uses it is expedient primarily to 
 adopt. Secondly, it may be held that it is 
 a matter of indifference whether symbols 
 stand for terms or for propositions. And, 
 
 case of such problems it makes no difference, so long as the 
 appropriate rules are observed, whether we let our symbols 
 stand for terms or for propositions. But, when we come to 
 deal with problems that are not included within the scope of 
 the Boolian treatment, I admit that it is better to let symbols 
 stand primarily for propositions. For, as will be noticed later, 
 the propositions of mathematics which the newer Symbohc 
 Logic has been shown to be able to manipulate consist of 
 implications between propositional functions, and the " molecule " 
 referred to in the latter is a proposition, not a class. 
 
Symbols for Terms and Propositions i i 
 
 in the third place, the opinion may be 
 maintained that only one of the two should 
 be symbolized — on this view it is generally 
 to designate propositions that symbols are 
 exclusively utilised. 
 
 (i) As regards the question of expedi- 
 ency, it has been affirmed that we should 
 commence with the symbolization of pro- 
 positions, for then, firstly, our procedure 
 throughout will be analytical ; and, sec- 
 ondly, we shall avoid the " confusion " that 
 is introduced through the identification of 
 the " physical " combination of propositions 
 into a system with the " chemical " com- 
 bination of subject and predication into a 
 proposition."* 
 
 The former of these reasons is un- 
 doubtedly a strong one, but I am inclined 
 to think that the common method of be- 
 ginning with the consideration of classes, 
 and the operations that may be performed 
 upon them, is the better one to employ. 
 For one thing, the latter procedure is of a 
 
 * Mind, vol. i. N.S. p. 6. 
 
12 Symbolic Logic 
 
 simpler character than the other. But a 
 stronger reason than this is that, during the 
 process of considering the manner in which 
 the analysis of propositions modifies the 
 form of the synthesis, it is necessary to 
 point out that the letters representing pre- 
 dications obey the simple laws of proposi- 
 tional synthesis ; ^ it is, therefore, desirable 
 to be able to refer to an earlier discussion of 
 terms and the operations that may be per- 
 formed upon them. 
 
 With respect to the confusion that it is 
 alleged is likely to arise from our allowing 
 letters originally to represent terms, it is, I 
 think, apt to be exaggerated ; indeed, a 
 careful analysis of what really happens 
 during the employment of literal symbols 
 in the two spheres will show that there is 
 no good reason for confusion in any degree. 
 The fact that contradictories are not the 
 same in both regions has been declared to 
 be a likely source of error. Now it is 
 certainly true that the contradictories in the 
 
 * Mind, vol. i. N.S. p. 352. 
 
Symbols for Terms and Propositions 13 
 
 two cases are different, but this should not 
 involve any uncertainty in the application 
 of the old formulas to the new use. All 
 that is necessary is that we make allowance 
 for the change in the character of the con- 
 tradictory, /.f., we must not admit that 
 propositions are sometimes true and some- 
 times false. 
 
 Again, it has been said that those who 
 utilise the old rules for the new subject- 
 matter will be led actually to confuse a 
 class with a proposition, inasmuch as on 
 the class view the contradictory of x is the 
 class x^ but on the propositional theory the 
 contradictory of the proposition x is the 
 affirmation " x is true." ^ But this criticism 
 loses its force if the distinction is drawn 
 between the truth of a proposition and the 
 statement that the proposition is true. 
 When the old formulas are applied to the 
 new case, the correct procedure is to make 
 the letter symbol represent the truth of a 
 proposition, while such an expression as 
 
 * Mind, vol. i. N.S. p. 17. 
 
14 Symbolic Logic 
 
 a:=1 is used to denote that such a pro- 
 position is true. Hence the contradictory 
 of the truth of x does not leave us with a 
 proposition, but simply with the truth of x. 
 There is thus a perfect analogy between this 
 case and the case where the letters represent 
 classes. And, just as the class x may be 
 declared to exhaust the universe, so it is 
 possible to state that the truth of the pro- 
 position X is the only possibility. In other 
 words, in both cases we may say that ^=1. 
 When writers, who start by making 
 letters stand for classes, come to make such 
 letters stand for the truth of propositions, 
 there is no serious alteration involved, 
 except the one already noticed, in the 
 logical rules that have been established : 
 there is merely another method of interpre- 
 tation put upon the literal symbols. Such 
 logicians argue that the logical machinery 
 may be put to uses other than those for 
 which it was originally intended. For 
 instance, the symbol 1 from meaning the 
 totality of compartments comes to denote 
 
Symbols for Terms and Propositions i 5 
 
 the only possibility, and receives the 
 meaning of no possibility. 
 
 Where the symbolic framework, as elabo- 
 rated from the point of view of the class, 
 does not apply to the new case, the fact is 
 due, as Venn shows, to the circumstance 
 that we have no longer any place in the 
 contradictory for the word "some." In 
 dealing with classes, when it is said that 
 x-\-x=l^ it is meant that both x and x 
 contribute to the total, but on the propo- 
 sition interpretation, the admission of x 
 excludes absolutely the admission of x. 
 Hence, if xy is declared false, we can 
 only say that one of the three xy, xy, xy 
 is true, while, if xy is declared true, then 
 xy, xy, xy must all be false. That is to 
 say, of the formally possible propositional 
 alternants only one can be true. 
 
 (2) But there are some writers who main- 
 tain that it makes no difference whether 
 symbols stand for terms or for propositions. 
 These logicians have to attempt to show 
 that the characters of the contradictories 
 
1 6 Symbolic Logic 
 
 do not vary in the fundamental way that 
 I have just mentioned. Mrs. Ladd- 
 FrankHn, for instance, endeavours to deal 
 with this question by asserting that a pro- 
 position may be true at one time while it 
 is false at another ; * but, as Mr. Johnson 
 remarks, propositions that relate to different 
 times are different propositions. Mrs. Ladd- 
 Franklin asks, " Why exclude from an 
 Algebra which is intended to cover all 
 possible instances of (non-relative) reasoning 
 such propositions as ' sometimes when it 
 rains I am pleased and sometimes when it 
 rains I am indifferent ? ' " But I am not 
 aware that any symbolist wishes to exclude 
 such propositions. Supposing we regard 
 this statement as consisting of two proposi- 
 tions — in contradistinction for the moment 
 to the way in which Mr. Johnson argues, 
 namely, that the particle " and " implies that 
 we have really only one — then the symbolist 
 will, of course, say, " Let x equal the 
 proposition ' sometimes when it rains I 
 
 * Mindy vol. i. N.S. p. 129. 
 
Symbols for Terms and Propositions 17 
 
 am pleased/ and y equal the proposition 
 'sometimes when it rains I am indifferent.'" 
 Here, if these two propositions are true, 
 we shall have x=\ and y = \ respectively ; 
 while, if X is not true, i.e.^ if a; = 0, the 
 verbal rendering will be, " It is not true 
 that sometimes when it rains I am pleased," 
 and similarly with the rendering of j/ = 0. 
 Mrs. Ladd-Franklin argues as though x 
 were made by the symbolist to stand only 
 for such a proposition as " I am always 
 pleased," but, of course, the symbol may 
 stand for any proposition (or rather, truth 
 of any proposition) whatever. 
 
 But though the symbolist can deal with 
 such propositions he will not in conse- 
 quence proceed along the lines that Mrs. 
 Ladd-Franklin thinks Schroder should have 
 followed. She argues that it is not justi- 
 fiable to regard x — <y-\-z as requiring 
 fundamentally different treatment accord- 
 ing as AT, y^ and x stand for terms or for 
 propositions. Schroder had maintained 
 that, when the letters represent pro- 
 
 B 
 
1 8 Symbolic Logic 
 
 positions, it is not possible, as it is on 
 the view that we are deaHng with classes, 
 for X to be divided up between y and z. 
 To this his critic says that in material 
 consequences, such as " if it rains, either 
 I stay in or else I take an umbrella," the 
 proposition is satisfied if there are some 
 instances in which I stay in and some in 
 which I take my umbrella. She fails to 
 observe that the introduction of the word 
 " instances " does away with the special 
 character of the sequence, and reduces the 
 problem to one of class implication. So 
 long as propositions as such are retained, 
 Schroder is undoubtedly correct in saying 
 that X cannot be divided up between y 
 and z. 
 
 Again, Mrs. Ladd- Franklin points out 
 that there is a close resemblance between 
 what she terms logical sequence and the 
 case where the left-hand member of the 
 subsumption stands for a singular subject. 
 But this is not any reason for regarding 
 the question whether we allow literal 
 
Symbols for Terms and Propositions 19 
 
 symbols to stand for terms or for pro- 
 positions as one of indifference. Such a 
 statement as " she is either a queen or a 
 fairy " is one of those Umiting cases, for 
 whose investigation in general we are so 
 much indebted to Dr. Venn. It is quite 
 correct to say that " there seems, in fact, 
 to be a close relationship between the 
 logical sequence between propositions, and 
 the sequence between terms when the 
 subject is singular," but Schroder's general 
 argument is not thereby invalidated : the 
 original formula must be modified to suit 
 the case of the proposition with singular 
 subject and disjunctive predicate, just as 
 there must be modification to meet the 
 case where terms stand for propositions. 
 Mrs. Ladd-Franklin's answer to Schroder, 
 when he asks what can possibly be meant 
 by [a — <b)=d'{-b on the supposition that 
 the letters stand for terms instead of for 
 propositions, appears to me to be quite 
 sound. She says that the verbal render- 
 ing will naturally be as follows : " All 
 
20 Symbolic Logic 
 
 ^ is ^" is-the-same-thing-as "everything 
 is either non-a or else L'' But all that 
 is hereby demonstrated is that the letters 
 in a certain equation may have two 
 different readings : there is here no argu- 
 ment to prove that it is a matter of in- 
 difference, so far as rules of application 
 are concerned, whether our letters in a 
 problem stand for terms or for proposi- 
 tions. 
 
 In stating the facts of the case we have, 
 therefore, to avoid two extremes. On the 
 one hand it is incorrect to say that all 
 the rules apply equally well for both 
 classes and propositions, and on the other 
 hand we need not go so far as to state 
 that the rules are different in the two 
 regions. The former statement is erron- 
 eous ; the latter suggests more disparity 
 between the two procedures than actually 
 exists. 
 
 (3) It has been mentioned also that 
 sometimes it is maintained that symbols 
 should be employed exclusively to repre- 
 
Symbols for Terms and Propositions 21 
 
 sent propositions. Mr. MacColl takes this 
 view of the case. But I can see no valid 
 reason w^hy symbols may not designate 
 now classes and now propositions. The 
 only thing to be remembered is that the 
 rules of procedure are not quite the same 
 in the two cases. If we are to be re- 
 stricted to one only of the two uses, then 
 I think that Venn is justified in saying 
 that symbols should stand for classes rather 
 than for propositions. As regards the 
 question of economy of space in the 
 solution of problems, the evidence seems 
 to show that the class interpretation is 
 to be preferred. Certainly this is the case 
 so far as the representation of the syllogism 
 — after all an important form of reason- 
 ing — is concerned. I do not lay much 
 stress upon the argument based on space- 
 economy. At any rate, we ought not to 
 judge systems by the amount of working 
 that has been offered when the exponents 
 were dealing with certain well-known 
 problems, because as a rule the symbolist 
 
22 Symbolic Logic 
 
 could, if he had so chosen, have made his 
 solution much more compact than he did. 
 Still, seeing that symbolism may at times 
 assist rather than act as a substitute for 
 thought, we need not despise brevity, if 
 thought is thereby rendered the greater 
 assistance. 
 
 I think, then, that it is correct to hold 
 that our letters may represent either classes 
 or propositions, but that v^e must be care- 
 ful to notice that the rules to be adopted 
 in v\Aorking our problems are not the same 
 in the two cases. This view has been 
 adopted by Boole, Venn, and Schroder, 
 who, so far as the point in question is 
 concerned, are thus seen to have been 
 proceeding in the right direction. 
 
 II. A variety of letter symbols has 
 been suggested by these three writers for 
 representing the class or the proposition, 
 and the contradictory of each, namely, 
 XX (Venn), x x^ (Schroder), Xx (Jevons). 
 The selection of different sets of symbols 
 
Symbols for Terms and Propositions 23 
 
 for the class and its contradictory is one 
 of those matters that depend on the 
 logician's individual preferences, and is 
 not of vital importance in marking off 
 one exposition from another. A very- 
 clear distinction must of course be draw^n 
 between real and superficial differences 
 existing among the so-called systems. It 
 is to be regretted that the same symbols 
 are not used by different w^riters in those 
 cases where no principle is involved, for 
 there is no doubt that beginners in the 
 subject are often unnecessarily puzzled 
 on account of the variety that is to 
 be found. The reason that such variety 
 exists is, I think it must be said, 
 that symbolists have frequently not been 
 sufficiently acquainted with the writings 
 of their predecessors. Boole, as we 
 have noticed, was unfamiliar with any- 
 thing that had been done in Symbolic 
 Logic ; and Mr. MacColl in his earlier 
 work furnishes according to Venn " another 
 instance of results independently worked 
 
24 Symbolic Logic 
 
 out, without knowledge of what had 
 been already effected in the same way/' 
 Mr. MacColl's later papers have, how- 
 ever, been prepared with a clear appre- 
 hension of what has been engaging the 
 attention of other writers.^ This want 
 of historical information on the part ot 
 symbolists, besides causing students un- 
 necessary difficulties, has resulted in a good 
 deal of wasted effort, for each logician 
 instead of starting where his predecessor 
 left off, has had to work out all the 
 elementary notions of the subject for 
 himself. 
 
 III. In close connexion with the sub- 
 ject as to whether Symbolic Logic deals 
 primarily or exclusively with propositions 
 is the question as to the kind of propositions 
 to which in any case it must confine 
 itself The symbolist can deal with asser- 
 torics only. It has, however, sometimes 
 
 * See particularly his article in Mind^ N.S. No. 47, 
 p. 355, and his discussions with Mr. Russell and myself with 
 respect to logical existence, Mind, N.S. Nos. 55 and 56. 
 
Symbols for Terms and Propositions 25 
 
 been held that certain other propositions 
 fall within the scope of his treatment. For 
 instance, he is said to be able to manipulate 
 propositions that are "probably true." I 
 think he has nothing; to do with such 
 material, for the simple reason that it does 
 not exist. Mrs. Bryant, in her suggestive 
 paper on " The Relation of Mathematics 
 to General Formal Logic," * still holds 
 to the view that it is a legitimate subject 
 of inquiry when we ask concerning a 
 proposition " how often is it true relative 
 to the total number of cases its occurrence 
 in every one of which would constitute 
 its unconditional truth .? " Two considera- 
 tions show that this question is not an 
 intelligible one. In the first place, it 
 is a mistake to speak of a proposition as 
 being often true, for on each supposed 
 occasion of its truth there would be a 
 new proposition. In the second place, 
 though unconditional truth may well be 
 established from certain true propositions, 
 
 * Proc. ylr'tst. Soc, vol. ii. N.S. p. 121, 
 
26 Symbolic Logic 
 
 this establishment is due simply to the 
 fact that such propositions are true, and 
 not to the fact that they are always true. 
 Mrs. Bryant escapes the mistake of speak- 
 ing of degrees of truth, but she falls into 
 an equally serious error in holding that 
 a proposition may more or less frequently 
 be true. She is quite correct in saying 
 that " a proposition is the assertion of a 
 joint event,'' but when this assertion is 
 once made it is either true or it is false : 
 it cannot be probably true. It may be 
 more or less probable that the events ought 
 to be joined in the way asserted by the pro- 
 position, but such probability is a matter 
 to be taken into consideration before the 
 assertion is made. The error in question 
 arises apparently through the confusion of 
 proposition with event. The probability 
 of an event is certainly measured " by the 
 ratio of the number of cases in which it 
 occurs to the whole number of cases con- 
 sidered," but the probability of the truth 
 of a proposition has no meaning. We 
 
Symbols for Terms and Propositions 27 
 
 may not, as she would allow us, write 
 " proposition " for " event " and " is true " 
 for " occurs." 
 
 Nor can the symbolist manipulate pro- 
 positions respecting probabilities, unless he 
 recognises that he is dealing with an affir- 
 mation of the relation in which a thinker 
 stands to a certain statement. That is to 
 say, the symbolist will still be engaged upon 
 assertoric propositions. Mr. Johnson has 
 made this quite clear. As he expresses it, 
 these assertions about the probability that a 
 predicate is to be attached to a subject relate 
 to a different plane from the one with which 
 pure Logic is concerned. They refer to the 
 obligation under which the thinker finds 
 himself to accept a statement of an assertoric 
 kind, but the propositions that engage the 
 attention of the logician are these assertorics 
 themselves. 
 
 And in the same way that the symbolist 
 cannot without the use of new terms deal 
 with propositions asserting probabilities, so, 
 unless the same procedure is adopted, he 
 
28 Symbolic Logic 
 
 must consider as outside his province many 
 of the kinds of propositions that are 
 mentioned in the very ingenious system 
 that has been elaborated by Mr. MacColl. 
 This logician holds that the symbolist, 
 besides classifying propositions into true 
 and false, may make other classifications 
 according to the necessities of the problem. 
 Thus, in addition to the probable, impro- 
 bable, and even propositions already men- 
 tioned, there are those that are certain, im- 
 possible, and variable, those that are known 
 to be true, known to be false, and neither 
 known to be true nor known to be false, 
 and so on. The objection to this procedure 
 is based on the fact that the considerations 
 according to which such classifications are 
 reached all refer to the relation in which 
 the thinker stands to the proposition, and 
 not to the proposition itself. All such facts 
 as Mr. MacColl has in view can be dealt 
 with in Symbolic Logic, but it is in 
 their case necessary to introduce new terms. 
 Thus, take the case of a proposition A, 
 
Symbols for Terms and Propositions 29 
 
 which we will suppose to be false. We 
 have then symbolically A = 0. Now, sup- 
 pose we introduce the conception involved 
 in the words "it is known," the proposition 
 that we shall have to deal with will be 
 " that i-t is known that A is false is true." 
 It will still be a case of truth or falsehood, 
 but the propositions that we have in mind 
 are not the same. Mr. MacColl is, there- 
 fore, incorrect in stating that his A : B is 
 stronger than A — <B : it is not a matter 
 of strength, it is a matter of an entirely 
 different proposition. 
 
 It will be seen from these considerations 
 why it is that the same writer's recent 
 explanations of his views are unsatisfac- 
 tory.* He maintains, for instance, that 
 his formula [K: x) + {E\ x) : (AB : x) is 
 true, but that he could not use the formula 
 {K: x) + {B: x) = (AB \ x). He grants that 
 the latter is true when A: x means (A + x)^ 
 but not when we have the meaning that 
 he assigns to A : at, viz., (A 4- >^)\ t.e,^ " it 
 
 * Mind, N.S. No. 47, p. 355. 
 
JO Symbolic Logic 
 
 is certain that A implies x.'' In unfolding 
 his view, Mr. MacColl takes an illustration, 
 in which the chances that A is ^ are 3 to 5, 
 that B is ;c are 3 to 5, and that AB is x 
 are 1, and his demonstration that under these 
 circumstances the former of the above for- 
 mulas alone holds good is doubtless sound. 
 But he is not justified in constructing for- 
 mulae upon this plane. At any rate, those 
 that he here constructs form no part of 
 pure Logic, for in this the force of the 
 proposition consists in the definite erasion 
 or salvation of certain compartments. If 
 Mr. MacColl wishes to deal with the data 
 he mentions he should introduce new terms. 
 Pure Logic can take account of the uncer- 
 tainties that such data occasion, but the 
 propositions dealt with will then denote not 
 the relation of the respective letters to x, 
 but the relation of the thinker to each 
 implication. 
 
CHAPTER II 
 
 4 
 
 SYMBOLS OF OPERATION 
 
 I. Next as regards the method of 
 connecting the term-symbols. For a long 
 time it was thought to be absolutely 
 necessary to use symbols of operation, but 
 Dr. Keynes has shown that the most 
 complicated problems may be solved 
 with the greatest ease without such use. 
 The words " and " and " or " are amply 
 sufficient in his hands for the connexion 
 of the term-symbols, while to connect 
 the subject-group with the predicate- 
 group he does not depart from the cus- 
 tomary " is." Still, as Mr. Johnson points 
 out, Keynes has hardly developed a logical 
 calculus, for this is characterized by the 
 mechanical application of a few logical 
 
 rules. 
 
 31 
 
32 Symbolic Logic 
 
 But I may say that there is a difference 
 of opinion among logicians as to the best 
 manner in which to describe the advanced 
 work that has been done by Dr. Keynes. 
 On the one hand it is said that he has 
 hardly developed a calculus, and on the 
 other hand the question is asked whether 
 his methods can fairly claim to belong to 
 the Common Logic* Venn thinks that 
 these methods would never have been 
 reached without a training in the earUer 
 symboHc systems, for "the spirit of the 
 methods is throughout of the mathematical 
 type." And Venn, in the second edition 
 of his Symbolic Logic, which appeared after 
 the publication of Keynes' work, repeats 
 the statement made in the first edition to 
 the effect that the want of symmetry in the 
 predication view of the proposition forbids 
 its extension and generalisation. f Thus, if 
 Keynes' work is not a calculus and does not 
 belong to the Common Logic, it is a little 
 
 * See Venn, in Mind, vol. ix. p. 304. 
 •j" Symbolic Logic, 2nd ed. p. 29. 
 
Symbols of Operation 33 
 
 difficult to know how to classify it. My 
 own view is that it is what he claims it to 
 be, a generalisation of (common) logical 
 processes. There are no symbols that 
 are suggestive of Mathematics except the 
 bracket, and none suggestive of earlier 
 symbolic work except x for not-X The 
 distinction between subject and predicate 
 is observed, and the use of the copula is 
 retained. There is generahsation of the 
 various forms of immediate inference com- 
 monly recognised, as well as of mediate 
 arguments involving three or more terms. 
 Whether the processes can be readily 
 described as a calculus is perhaps doubtful. 
 Certainly Keynes does not reach his con- 
 clusions from the mechanical application of 
 a very few fundamental laws, but the rules 
 that he does employ are after all not very 
 numerous, and with a little practice can be 
 applied with almost mechanical facility. I 
 agree with Venn that it is difficult to suppose 
 that such methods would have been reached 
 
 without study of existing symbolic systems, 
 
 c 
 
 >^ OF THE 
 
34 Symbolic Logic 
 
 and there is a distinct resemblance between 
 certain parts of Keynes' treatment of the 
 subject and that given in Schroder's Opera- 
 tions kreis^ to which work frequent reference 
 is made in the notes of the Formal Logic. 
 Still, whatever may have been the history 
 of the growth of the subject in the writer's 
 mind, now that the methods are thus pre- 
 sented I think that they should be regarded 
 as a generalisation of the common logical 
 processes. 
 
 II. Most writers on the subject of 
 Symbolic Logic have undoubtedly intro- 
 duced symbols of operation, and the four 
 following, as is well known, have frequently 
 been used : + , - , x , -^ , to denote re- 
 spectively aggregation, subduction, restric- 
 tion, and the discovery of a class which on 
 restriction by a denominator yields the cor- 
 responding numerator. Of course, other 
 symbols might have been used to designate 
 precisely these operations, and it may be well 
 to ask whether, seeing that these symbols 
 
Symbols of Operation 35 
 
 are employed in another region of thought, 
 it is well to have them employed in both 
 regions. If they had first been used by 
 the class logician, would the thinker who 
 deals with numbers have done wisely in 
 adopting them in his science ? There is 
 no reason, of course, in the nature of 
 things why they should not have been 
 employed in Logic first of all, but they 
 were in use long before the logician began 
 to look around him for some symbols suit- 
 able for the operations he had to perform. 
 Did Boole, therefore, act wisely in making 
 use of these symbols in his solutions ? In 
 some respects he did wisely, and in some 
 he did not. He did wisely because there 
 is some analogy between certain processes 
 of Mathematics and those of Logic ; for 
 instance, the commutative and associative 
 laws are applicable in both regions. And, 
 even in cases where most of all it may be 
 said that the adoption of mathematical 
 symbols is likely to mislead, there is little 
 risk of error if we regard the symbols as 
 
36 Symbolic Logic 
 
 *' representing the operation^ and merely de- 
 noting the result."^ Thus, -, which in 
 
 Mathematics denotes zero, might, regarded 
 solely as a result, be taken in Logic to 
 stand for '' nothing " ; but, when we re- 
 member that the symbol also points to an 
 operation, no confusion need arise. It 
 becomes obvious, that is to say, that we 
 here have the process of finding a class 
 which upon restriction by a gives 0, 
 which class is immediately seen to be a, 
 Boole did wisely also — though perhaps 
 somewhat unconsciously — in that by em- 
 ploying these symbols he directed, as 
 Mr. Johnson has remarked, far more 
 attention to the study of Symbolic Logic 
 than the subject would otherwise have 
 received. 
 
 On the other hand, it may be doubted 
 whether the analogy between the two sets 
 of processes is sufficient to justify the appli- 
 cation of the same symbols ; the formula 
 
 * Mrs. Bryant, Proc. Ar'ist. Soc.j vol. ii. N.S. p. io8. 
 
Symbols of Operation 37 
 
 XX = x, for instance, may always be used in 
 the logical region while being almost en- 
 tirely inapplicable in Mathematics. More- 
 over, had Boole not adopted these symbols 
 there would have been avoided the many 
 disputes concerning the propriety of using 
 them. Without doubt, out of all the 
 controversy on the subject some truth has 
 emerged, but it is probable that, had the 
 relations of classes or of propositions re- 
 ceived the attention that the disputants gave 
 to a comparison of the mathematical and 
 logical processes. Symbolic Logic would 
 have made more rapid strides than it has 
 done. The wonderful mathematical struc- 
 ture was erected without reference to what 
 the logician was doing, or whether he was 
 doing anything, and it may be that the 
 logical structure would have been more im- 
 posing if the builder had concentrated his 
 thought upon his own work, instead of cast- 
 ing side glances to see what was occupying 
 the attention of the mathematician. 
 
38 Symbolic Logic 
 
 III. Much discussion has arisen con- 
 cerning three of these four symbols of 
 operation, and it is stimulating to thought 
 to weigh the arguments that have been 
 advanced in connexion v^ith them. 
 
 (i) First, with regard to the sign +. Boole 
 always used this sign on the understanding 
 that the terms so joined are exclusives. It 
 was his special merit, so it has been affirmed, 
 to improve on the common vagueness. 
 That is to say, if " or " on the popular view 
 means anything from absolute exclusion to 
 identity, then the logician is called upon to 
 improve on the ordinary view when he 
 states his premises in symbolic language. 
 It has also been maintained that there is a 
 very great advantage in adopting the ex- 
 clusive notation, inasmuch as there is then 
 rendered possible the introduction of inverse 
 operations. That is, before ah can be sub- 
 tracted from an aggregate of terms, it must 
 be known that the aggregate contains ah — if 
 the matter were left open there could be no 
 
Symbols of Operation 39 
 
 subtraction. Similarly with division. If a 
 class is to be found which on restriction by 
 a denominator is to yield the numerator, 
 then there must be no indefiniteness as to 
 what this numerator is. 
 
 On the other hand, it is maintained that 
 in the process of expressing the premises 
 in symbolic form much economy of space 
 and time is effected if the non-exclusive 
 method is adopted. Further, on this plan 
 it is possible to arrive at the contradictory 
 by a very simple process. The demonstra- 
 tion of this occupies a prominent position 
 in Schroder's work."^ He proves in the 
 Operationskreis that the contradictory of 
 (ab)^ is [a^ + ^J, and that of [a -^ b)^ is 
 a^^ — in the Vorlesungen the proposition 
 appears as No. 36. Of course, Jevons had 
 previously argued that the individual does 
 often think in the non-exclusive fashion, 
 but this is no reason why such notation 
 should be adopted in the logical calculus. 
 
 ■* See Adamson's excellent critical notice in Mind, vol. x. 
 p. 252, where, however, Schroder is erroneously said to have 
 originated the theorem. 
 
40 Symbolic Logic 
 
 It was for Schroder to point out that by 
 the adoption of the method in the calculus 
 problems could be solved more easily than 
 on the Boolian plan ; and not only would 
 the process be easier, but, what Schroder 
 thinks to be still more important, each step 
 would be intuitively obvious, and justifiable 
 on purely logical grounds. As a result of 
 the long debate, the non-exclusive notation 
 has undoubtedly found favour, and Venn in 
 his second edition adopts it, having come, 
 as he says, to recognise its " brevity and 
 symmetry," but still holding to the view 
 that the question is one of method rather 
 than one of principle. Having thus 
 changed his opinion, Venn has, of course, 
 either to reject all inverse processes, or 
 else to revert to the exclusive notation 
 when dealing with them. 
 
 The confusion which has been stirred up 
 by many of those who have discussed this 
 question is greater, perhaps, than is to be 
 found in any other part of Logic. It is 
 very common to find no distinction made 
 
Symbols of Operation 41 
 
 between (i) what actually takes place in 
 disjunctive thinking, (2) what is the treat- 
 ment of the disjunctive judgment in the 
 text-books that discuss the elementary rules 
 of formal Logic, and (3) what way of deal- 
 ing with the disjunctive is the most service- 
 able for a generaHsed Logic. These three 
 points of view were made clear by Dr. 
 Venn long ago, but they are quite neglected 
 even now in some discussions. For in- 
 stance, Mr. Ross set out recently * " to try 
 to determine the import of the disjunctive 
 judgment, and to find out the exact place 
 which it occupies in the connected whole 
 of logical thought." He then proceeds to 
 criticize Mr. Bradley and Mr. Bosanquet 
 (who are, let it be observed, talking about 
 the manner in which we are thinking when 
 we are thinking disjunctively) by appealing 
 to considerations based on common logical 
 usages. But obviously the practices of the 
 logician can never define the actual form 
 of the judgment. Somewhat later, when 
 
 * Mind, N.S. No. 48, p. 489. 
 
42 Symbolic Logic 
 
 Mr. Ross advances " other considerations 
 which go to show how inexpedient it is to 
 treat the disjunctive judgment as necessarily 
 exclusive," it becomes particularly notice- 
 able that he fails to distinguish between 
 two entirely different questions, one of fact 
 and one of convenience. He actually pro- 
 poses to show how inexpedient it is that 
 alternatives are (in Bradley's view) ex- 
 clusive of each other ! 
 
 To put the matter in the simplest 
 possible form, when Boole meets with some 
 premises involving alternatives, he asks 
 whether he is to regard the alternatives as 
 exclusives or not. Then, if the answer is 
 in the negative, Boole will write down 
 xy -\- xy -\- xy^ where x and^ were the origi- 
 nal non-exclusive alternatives. If Schroder 
 meets with the same premises, he will, of 
 course, also want to know if the alternatives 
 are exclusives, and, when informed that they 
 are not, he will write down x -^ y. Then 
 each symbolist may go to work with his 
 special rules, and each may obtain the 
 
Symbols of Operation 43 
 
 correct solution. Thus it is the person 
 supplying the problem who places the sym- 
 bolist in a position to commence the solu- 
 tion. I should not have put the matter in 
 such an elementary form as this were not 
 the many confusions that still exist a suffi- 
 cient justification. The word " should '* 
 has misled Mr. Ross. It may mean, 
 " How ought I to describe the actual 
 facts in the mind of the individual who 
 is thinking a disjunctive judgment ? " Or 
 it may mean, " How ought I to put down 
 in words or other symbols the facts that 
 constitute the disjunctive thought ? " 
 
 It is relevant here to notice also Mr. 
 Bradley's treatment of the subject of alter- 
 natives. He wishes to show that alter- 
 natives are exclusives, and his procedure 
 is to refer to the state of things when they 
 are not exclusives.^ Evidently, therefore, 
 alternatives can as a matter of fact be 
 either. To put the same thing in other 
 
 * The Principles of Logic ^ p. 124. 
 
44 Symbolic Logic 
 
 words, Mr. Bradley says that when alter- 
 natives are not exclusives we are thinking 
 slovenly. But slovenly thinking is still 
 thinking, though we may readily grant 
 that it is not " always safe." Mr. Bradley 
 seems to have been led to this argument 
 through a confusion of the kind we have 
 just mentioned. He sees difficulties in 
 the way of reasoning if we state the pre- 
 mises symbolically in the non-exclusive 
 manner, and so he argues that those pre- 
 mises must have been given in the exclusive 
 manner. But obviously they may have 
 been given in either form, though we 
 must know which before we can put them 
 down in symbols. When information 
 upon the subject is forthcoming, we can 
 adopt either the exclusive or the non- 
 exclusive method of representation. It has 
 been pointed out that Mr. Ross attempts 
 to show the inexpediency of the fact that 
 alternatives are (in Mr. Bradley's view) 
 exclusives. We now see that Mr. Bradley 
 was led to regard alternatives as exclusives 
 
Symbols of Operation 45 
 
 by reflecting how inexpedient it would 
 be if they are not. 
 
 (2) Concerning the employment of the 
 sign ( — ) some difference of opinion has 
 also arisen. In the first place, it has been 
 pointed out that the sign is not absolutely 
 necessary, since subduction may always be 
 expressed symbolically as restriction. But, 
 though this is true, the reply has reasonably 
 been made that it is frequently more con- 
 venient to employ the minus sign, and that 
 no logical considerations render such em- 
 ployment illegitimate. But it is to be 
 noted that only as denoting subduction is 
 the use of the sign appropriate. If the 
 attempt is made to designate negative terms 
 by prefixing (-) to the positive, only error 
 can result. For, as Venn points out, the 
 tendency then becomes almost irresistible 
 to transfer a term with changed sign to 
 the other side of the equation, and this 
 will mean that a statement is made con- 
 cerning a class about which the premises 
 give no information. 
 
46 Symbolic Logic 
 
 So far all is clear concerning the use of 
 the minus. But sometimes it is employed 
 where the calculus is based on the inten- 
 sive rendering of propositions, and the use 
 in this way deserves some consideration. 
 Castillon has carried out more consistently 
 than any other writer the development of 
 Symbolic Logic on intensive lines, and I 
 shall restrict my remarks here to his treat- 
 ment of the sign in question. What he 
 means by (-) becomes evident when we 
 observe his symbolic representation of the 
 universal negative and of its converse. 
 This proposition appears as S = - A + M, 
 by which he means that the attributes 
 embraced under S are not co-existent with 
 those embraced under A, but are co-existent 
 with those embraced under M.^ Then he 
 affirms that such proposition may be con- 
 verted thus : A = - S + M. Clearly, then, 
 what Castillon means — and he says as much 
 — by the (-) is the mental act of keeping 
 apart, of analysis. But, as he has thus far 
 
 * Sur un nouvel algorithm logique, pp. 9, 10. 
 
Symbols of Operation 47 
 
 been criticized,^ he is supposed in the 
 original proposition to assign to S two 
 aggregates, consisting respectively of nega- 
 tive and positive attributes. But this is 
 what he distinctly avoids doing. When 
 such infinite judgment, as he calls it, is 
 to be designated, he employs the form 
 S = (-A) + M. Moreover, if he had 
 meant what Venn thinks he did, the 
 converse of the universal negative would, 
 of course, have been (- A) = S - M. Is, 
 then, Castillon justified in converting in the 
 way he does ? Obviously not. For to pro- 
 ceed from S=-A + M toA=-S + M 
 is to conclude that A is co-existent with 
 M, a statement which is at variance with 
 the original proposition. So that on in- 
 tensive lines, as these are laid down by 
 Castillon, it is not in general allowable, 
 any more than it is in extensive Logic, 
 to transfer letters with changed sign to 
 the other side of the ( = ).t 
 
 * Venn, Symbolic Logic, 2nd ed. p. 466. 
 I Castillon' s doctrine on this and other questions is examined 
 in detail in chapter iv. 
 
48 Symbolic Logic 
 
 (3) The last sign that need claim our 
 attention is the one corresponding to the 
 ( -f ) of the mathematics of quantity. Has 
 this inverse process any rightful place in 
 Symbolic Logic, or is it a survival of 
 merely historical interest ? I hold that 
 for two reasons the process ought without 
 hesitation to be retained. In the first 
 place, the mental exercise involved in 
 arriving at the comprehension of what 
 is implied in the performance of such 
 inverse operation is, as Venn maintains, 
 of the greatest utility. And, in the second 
 place, the operation is capable of yielding 
 absolutely reliable results. It may be 
 stated in reply to this that, in the perform- 
 ance of the so-called logical division, we 
 utilise symbols that are from the logical 
 standpoint quite meaningless, and that such 
 a procedure is not warrantable ; that, in 
 other words, we should follow on the lines 
 which Schroder has laid down, who makes 
 all intermediate processes intelligible. But 
 in answer to this it is to be noted that a 
 
Symbols of Operation 49 
 
 calculus is a mechanical contrivance for 
 arriving at results that cannot be intuitively 
 reached. Having given our premises we 
 state them in symbolic language, then 
 manipulate this in accordance with a few 
 simple logical laws, and so reach our con- 
 clusion. Whether or not the intermediate 
 results are intelligible is of no importance 
 whatever. Thus, even if the intermediate 
 processes in Logic were unintelligible, as 
 is often affirmed, the inverse operations 
 quite reasonably find their place in the 
 calculus. 
 
 But, as a matter of fact, the stages be- 
 tween the statement of the premises and 
 the arrival at the conclusion are not mean- 
 ingless. Certainly Boole never attempted 
 to assign them a meaning, but Venn has 
 carefully examined all the various forms 
 that arise as a result of " division," and he 
 has shown that they have a perfectly in- 
 telligible logical signification. The words 
 of explanation that are given by Mrs. 
 Bryant as to how imaginary results arise 
 
 D 
 
50 Symbolic Logic 
 
 are not therefore required in the strictly 
 logical realm. She says, " Whenever a 
 subject is reduced to symbolic expression, 
 imaginary results may be expected to 
 appear, and this happens because the 
 operations of thought which the combin- 
 ing symbols represent extend in application 
 beyond the possibilities of the subject- 
 matter." ^ No doubt that sentence throws 
 light on a difficult question. But as Boole's 
 forms have all been assigned a strictly 
 logical explanation by Venn, it cannot be 
 asserted that in Logic there are unintel- 
 ligible expressions that call for considera- 
 tion. There appeared to be such when 
 Boole published his results, but that was 
 only because he did not perform the task 
 of explicitly stating the logical significance 
 of the forms in question. 
 
 To reject inverse processes, as does Mrs. 
 Ladd-Franklin, for instance, is deliberately 
 to throw away useful instruments for solv- 
 ing problems. At the same time, she is 
 
 * Proc. Arht. Soc.^ vol. ii. N.S. p. 131. 
 
Symbols of Operation 51 
 
 unquestionably correct in showing how im- 
 portant is that interpretation of alternatives 
 which will allow of our reaching the con- 
 tradictory with ease. The most satisfac- 
 tory conclusion of the whole matter is that 
 which Venn has formed, namely, to adopt 
 as a rule the non-exclusive rendering, so as 
 to profit by the simple rule for contradic- 
 tion ; but to change to the exclusive 
 notation at times, in order that the advan- 
 tages to be derived from the employment 
 of inverse operations may not be lost. 
 
 IV. We may proceed now to consider 
 certain other symbols of operation that have 
 been suggested as serviceable in the expres- 
 sions of our premises. When the proposi- 
 tions to be dealt with are of the universal 
 affirmative description there is an indefinite 
 element involved, and there have been 
 various proposals for dealing with this. 
 
 Venn considers that the symbol ^ is in 
 every way suitable for the purpose of ex- 
 pressing that the subject is contained some- 
 
52 Symbolic Logic 
 
 where in the predicate. The signification 
 of the symbol may extend from the value 
 to the value 1 ; that is to say, there 
 may be no things existing which are re- 
 presented by the subject-term of the pre- 
 mise, or those things may constitute a 
 portion of the universe of discourse, or they 
 may be co-extensive with what is denoted 
 by the predicate. Boole introduced this 
 symbol into Logic, and with entire appro- 
 priateness, since in Mathematics ^ denotes 
 
 complete indefiniteness. But Boole is not 
 consistent in his use of the symbol, for, 
 though when he is dealing with universals 
 it means with him anything from to 1, 
 he uses it in his representation of particular 
 propositions, and in their case the sig- 
 nification is excluded. The same in- 
 consistency applies to Boole's use of v, 
 
 which is merely his alternative for q. 
 
 Leibnitz, to designate the indefiniteness 
 in question, employs the equation x=:xy,^ 
 
 * See Venn, Symbolic Logic, p. 1 76. 
 
Symbols of Operation 53 
 
 and Jevons prefers this. But, as Venn 
 shows, this is precisely the same as x = ^y. 
 
 For the latter may be expressed as xy = 0, or 
 Ar(l--)/) =0, or x = xy. Again, starting with 
 the form x = xy^ and expressing x in terms 
 
 of y only, Le,, eliminating x trom xy, we 
 
 
 get x = ^y. 
 
 Other writers, in order to denote that 
 there is not necessarily identity between 
 the subject and the predicate, but possibly 
 only subsumption, employ a different form 
 of expression. For instance, C. S. Peirce 
 makes use of — <, which sign suggests 
 by its form the relation involved. It 
 may be said in favour of such symbols 
 that they are nearer to the predicative 
 interpretation of the proposition. Of 
 course, Keynes, who works with the term 
 " is," does not depart at all from the 
 predication point of view, i.e., his method 
 is less artificial than any others. 
 
 As regards these various ways of ex- 
 pressing the universal affirmative, it may 
 
54 Symbolic Logic 
 
 be said that, with the exception of Jevons' 
 form, there is not much difference between 
 them in the matter of suggestiveness and 
 usefulness. It is quite clear, however, that 
 ^ must not be employed, if it is also to 
 
 be employed in representing particular 
 propositions. Jevons' form had better as 
 a rule be discarded, for it introduces the 
 thing defined into the definition. 
 
 It is to be noticed that, if for the uni- 
 versal affirmative we accept the symbolism 
 of Venn and Boole, the import of this 
 proposition is to be regarded as of a 
 negative character. The import of this 
 universal negative is to be similarly in- 
 terpreted. The positive ^ = qJ^ and the 
 negative ^ = 7.y are equivalent, that is to 
 
 say, to xy = and xy = respectively. The 
 significance of this negative rendering 
 was realised by Boole, but was first made 
 conspicuous in the writings of Venn. 
 
 To denote equivalence the symbol ( = ) is, 
 as we have seen, often used. It is to be 
 
Symbols of Operation 55 
 
 observed that this represents equivalence of 
 results, not of operations. I think Mrs. 
 Bryant has brought out this fact more 
 clearly than any one else. She makes it 
 very evident that the signs +, — , x, -^, 
 represent the operation and denote the 
 result.* 
 
 Once more, in the statement of the 
 premises it is frequently necessary to make 
 use of the bracket. Every system has need 
 at times for the bracket. f Even Keynes 
 here makes use of a symbol of operation, 
 as, for instance, in his statement of the 
 Law of Distribution.! Boole v^as the first 
 to employ brackets in Logic on an extensive 
 scale, but the use of the bracket in logical 
 work was not uncommon among earlier 
 writers. For instance, Lambert employs 
 it, especially in his Logische und Philoso- 
 phische Abhandlungen^ and Castillon realises 
 its force in his essays before the Berlin 
 Academy. 
 
 * See p. 36. 
 
 f Mind, N.S. No. I, pp. 4, 5. 
 
 I Formal Logic, 3r(l ed. p. 385. 
 
56 Symbolic Logic 
 
 V. We have discussed the best methods 
 of symbolizing premises where these are 
 universals. But it may be that we have to 
 deal with particulars. How are these to be 
 stated ? And to whom are we indebted 
 for suggesting the best method of stating 
 them ? It has more than once been 
 attempted to express them in much the 
 same way as universals. Thus Boole says 
 that the particular may be represented by 
 the equation v x =: vy. Here there appears 
 to be an identity resembling such an ex- 
 pression as X = y. Similarly Jevons says 
 we may adopt the form AB = AC. But 
 clearly Boole cannot introduce such expres- 
 sions to a treatment resembling that which 
 he employs when dealing with universals, 
 for V in the case of particulars is made by 
 him to exclude the value 0. And Jevons* 
 rendering supposes that the members found 
 in both B and C have more attributes in 
 common than the one of being such 
 
Symbols of Operation 57 
 
 common members, and this supposition is 
 not justifiable.^' 
 
 The best way of dealing with particulars 
 is that expounded by Venn in both editions 
 of his work, namely, to take these pro- 
 positions as denoting the existence of in- 
 dividuals that are found in each of two 
 classes. In his first edition he took the 
 symbol v to designate this existence, so that 
 the particular appeared as xy = v. In the 
 second edition he employs sometimes a form 
 of symbolic procedure closely resembling 
 that proposed by Mrs. Ladd-FrankHn in 
 the Johns Hopkins Studies. Thus to repre- 
 sent "some X is jk" he writes xy > 0, i.e., 
 xy is not nothing. Mrs. Ladd-Franklin's 
 own procedure is the following. She pre- 
 ferred to discard the symbol 0, and to retain 
 only the symbol for denoting everything, 
 viz., 00. Then as this occurred in every 
 proposition it might be neglected. Thus 
 to say "all x is y'' she wrote xy v, and to 
 denote "some x is y" she wrote xyy.j" 
 
 * Venn, Symbolic Logic, p. 392. 
 
 I Johns Hopkins Studies in Logic, p. 25. 
 
58 Symbolic Logic 
 
 Here 00 is understood on the right of the 
 copula, which is a complete or an incom- 
 plete wedge according as the proposition 
 is universal or particular. Mrs. Ladd- 
 Franklin's originality with regard to the 
 treatment of particular propositions consists 
 not in the fact that she represented these as 
 denoting " something/' for such an inter- 
 pretation had already been suggested by 
 Venn, but in her attempt to grapple with 
 the problem of dealing with particulars in 
 a general manner. Her aim was to give 
 formula that would deal with them with 
 as much ease as the formulae that Boole 
 proposed for universals deal with these. 
 But though she realised the full significance 
 of the difficulties, and attempted to sur- 
 mount them, she cannot be said to have 
 succeeded in the attempt. As will presently 
 be shown, the formula that she proposed for 
 elimination in the case of particulars in- 
 volves error. Venn, it may then be said, 
 was the first to recognise the existential 
 character of particulars, while Mrs. Ladd- 
 
Symbols of Operation 59 
 
 Franklin devised a very convenient notation 
 for expressing, and in some respects for 
 dealing with such propositions, but she did 
 not succeed in presenting a general treat- 
 ment of the subject. 
 
 VI. But, supposing, in the next place, 
 that the premises are not stated in categori- 
 cal form, and appear as hypotheticals or dis- 
 junctives, the question arises if they can be 
 expressed in symbolic language. Venn was 
 undoubtedly the first symbolist who went 
 at all fully into this question. He main- 
 tained that hypotheticals and disjunctives 
 may without difficulty be expressed sym- 
 bolically, for they are to be regarded as 
 denoting a certain amount of compart- 
 mental destruction. The symbolist is not 
 interested in the causes that led to the 
 formation of these types of propositions. 
 He does not, that is to say, emphasize the 
 relations in which the individual stands to 
 the environment, and which lead to the 
 utterance of the hypothetical or the dis- 
 
6o Symbolic Logic 
 
 junctive. As in the case of categoricals 
 the only information that we here get is 
 as to what compartments are destroyed, 
 what are saved, and what cannot be said 
 either to be destroyed or to be saved. 
 Venn goes carefully into the consideration 
 of the three kinds of hypothetical, those 
 involving two, three, and four elements 
 respectively, and in each case he determines 
 what effect the proposition has upon the 
 compartmental scheme. In short, accord- 
 ing to Venn, it makes no difference whether 
 the premises are stated in categorical, dis- 
 junctive, or hypothetical form. 
 
 But Mr. Johnson in his discussion of this 
 subject has shown that Venn's investigation 
 is not exhaustive, and that to stay at the 
 point reached by Venn would involve us 
 in error. In the first place, Mr. Johnson 
 distinguishes between conditionals and pure 
 hypothetical. '^ For one thing, in the case 
 of the former, but not in the case of the 
 latter, an equivalent categorical may be 
 * Mind, N.S. No. 1, p. 17. 
 
Symbols of Operation 6i 
 
 obtained whose subject and predicate corre- 
 spond respectively to the hypothesis and 
 consequent.* That is to say, when we 
 have conditionals, the hypothesis and the 
 consequent refer to certain cases of a 
 phenomenon in time or in space, while 
 in a pure hypothetical we have presented 
 an implication between two propositions of 
 independent import. But, further, the pure 
 hypothetical receives minute examination at 
 Mr. Johnson's hands, and he demonstrates 
 that there is no equivalence of any kind 
 between this proposition and a categorical, 
 except in one particular case, namely, where 
 there is a universal antecedent and a par- 
 ticular consequent. In the other three 
 cases it can only be said that a categorical 
 implies, or that it is implied by, a pure 
 hypothetical.f From these considerations 
 it is clear, therefore, that Venn and Boole 
 cannot in their class Logic deal generally 
 with the pure hypothetical, for their pro- 
 
 * Keynes, Formal Logic, 3rd ed. p. 213. 
 •f Mind, N.S. No. 2, p. 242. 
 
62 Symbolic Logic 
 
 cedure is to express the premises of an 
 argument as though these in every case 
 denoted class relationship. But if we 
 follow the course adopted by certain other 
 logicians — MacColl, Johnson, and Russell, 
 for instance — and allow symbols to repre- 
 sent propositions rather than classes, the 
 pure hypothetical can receive adequate 
 treatment. It is to be noted that Keynes 
 asserts ^ that, though pure hypothetical 
 cannot be expressed as categoricals, in 
 which the subject and predicate corre- 
 spond respectively to the hypothesis and 
 consequent, it is always possible to turn 
 the former propositions into categoricals of 
 some sort or other. But if Mr. Johnson's 
 analysis is correct, as it certainly is, this 
 statement of Keynes cannot be accepted : 
 
 * Forma/ Logic, 2nd ed., note on p. 165. Keynes omits 
 this note in the third edition, where he does not refer to 
 the question whether the hypothetical may always be expressed 
 as a categorical of some kind or other. He restricts himself to 
 the remark (p. 213) that reduction to a categorical "is not 
 possible at all (with terms corresponding to the original antece- 
 dent and consequent) in the case of hypotheticals.'* 
 
Symbols of Operation 63 
 
 the change may sometimes, but not in 
 every case, be effected. 
 
 Where the premises contain a disjunc- 
 tion it is important to observe whether 
 this occurs in the subject or in the predi- 
 cate. Venn, Keynes, and Schroder all call 
 attention to this point. Only is there a 
 true disjunctive proposition when the 
 disjunction is found in the predicate. 
 " A or B is C or D," therefore, means 
 " A and B (each of them) is either C 
 or D." 
 
CHAPTER III 
 
 THE PROCESS OF SOLUTION - 
 
 I. (i) Analytical Method, — Having 
 then stated our premises in the way that 
 has been described, the question next arises 
 as to the best method of proceeding with 
 our symbolic expressions. Our object is 
 to determine what is said about one of 
 the letters, or about a combination of 
 them, in terms of the remaining letters, 
 or of a portion of them. The proposi- 
 tions must obviously in some way be 
 combined, so that it may be seen what 
 is the totality of the force they possess. 
 Boole would proceed to bring over all the 
 terms to the left-hand side, and equate to 
 zero. Then, if in each case the left-hand 
 member is an expression that is equal to its 
 square, all such members may be added 
 
 together, and the result be put equal to 0. 
 
 64 
 
The Process of Solution 65 
 
 But, should it happen in any case that a 
 left-hand member is not equal to its square, 
 the process of squaring must be performed 
 before the terms are added to those of the 
 other equations. We thus obtain a single 
 class expression that is equal to zero. The 
 squaring just referred to is necessary in 
 order to obtain co-efficients that are positive, 
 for only in this case will the same con- 
 stituents appear in the resulting equation 
 that occurred in the separate equations.^ 
 
 The characteristic method of Venn, on 
 the other hand, is the following. Suppos- 
 ing the premise appears in the form x = ^y, 
 he restricts the left-hand member by the 
 contradictory of ^, and then equates to zero. 
 In the case of x=y it is necessary both to 
 do this and to equate to zero the result of 
 restricting y by the contradictory of x. The 
 addition of these two equations is a full 
 statement of the information given, i.e., 
 each of the left-hand members being equal 
 to 0, the sum of them will also equal 0. 
 
 * Laivs of Thought^ chap. viii. props. 2 and 3. 
 
 E 
 
66 Symbolic Logic 
 
 This applies, of course, to universals only. 
 With regard to particulars it has been 
 shown above that Boole cannot utilise his 
 rules in their behalf. Venn, however, by- 
 interpreting them as denoting existence, is 
 able to express them in relation to 0. He 
 can show not only what compartments the 
 universals destroy, but what compartments 
 the particulars save. Venn thus shows an 
 advance upon Boole both in dealing with 
 universals in a simpler manner, and in 
 bringing out the full force of particular 
 propositions. 
 
 The process of manipulating the sym- 
 bolized premises having thus begun, it will 
 be well to mention here two formulae that 
 are sometimes useful in effecting a simpli- 
 fication. One of them is due to Boole 
 and the other to Peirce. Boole's is given 
 among his methods of abbreviation, and is 
 (A + x) (B + x) = AB -{- X. The one for which 
 we are indebted to Peirce is as follows : 
 {A+x){B+x)=Ax-{-Bx,^ 
 
 * See Venn, Symbolic Logic, pp. 69, 70. 
 
The Process of Solution 67 
 
 It may be well also to call attention to 
 the advantages which Schroder's methods of 
 solution have over those of Boole. In the 
 Operationskreis^ and with more elaboration in 
 the Vorlesungen^ Schroder is careful to take 
 no step that is not justifiable on purely logical 
 grounds. All his intermediate processes are 
 intelligible. Jevons was going too far in 
 stating that Boole's methods were " funda- 
 mentally false," * for Venn has pointed out 
 that they are justifiable, but their justifi- 
 cation requires a good deal more than a 
 simple act of intuition. Boole, for instance, 
 applied his rule of development to fractional 
 forms, and it is quite true that the results of 
 that application may be shown to have a 
 definite meaning, a meaning, too, which the 
 symbols employed may be said to suggest. 
 But the suggestion is by no means im- 
 mediate. It needed Venn's careful analysis 
 to bring to light the logical meaning under- 
 lying these symbols. The forms of Boole 
 did not suggest to Jevons their logical in- 
 
 * The Principles of Science, p. 113. 
 
68 Symbolic Logic 
 
 terpretation, but they suggest this at the 
 present day, now that Venn has gone ex- 
 haustively into the subject. 
 
 In the case of Schroder, on the other 
 hand, each step is intuitively obvious, and 
 this obviousness has its advantages. For 
 one thing, the beginner is saved the labour 
 — though this salvation is by no means 
 altogether an advantage — that would have 
 to be spent in tracing out the admissibility 
 of the various steps in the solution. The 
 one method is as logically intelligible as the 
 other, but there is more for the under- 
 standing to do in the one case than in the 
 other, in order to grasp the significance of 
 the process.* Another point in which 
 Schroder excels is that his solutions are 
 
 * I think that Adamson hardly states the case correctly 
 when he says that " the superiority in logical intelligibility of 
 Schroder's solution must be admitted" {Mindy No. x. p. 254), 
 His remark, moreover, seems to suggest that all the inter- 
 mediate processes in a solution ought to be intelligible ; but this 
 is not 80, because a calculus is a means of reaching correct 
 conclusions by means of the mechanical application of a few 
 logical rules, and it is quite possible that in the application of 
 such rules unintelligible elements may temporarily appear. 
 
The Process of Solution 69 
 
 effected with more compactness. Judging 
 the two symbolists by the number of Hnes 
 or pages that they occupy in the solution of 
 the same problem, there is certainly little 
 difference to be noted in this respect,^ and 
 it was probably from such an examination 
 that Adamson was led to state that it is 
 doubtful whether Schroder's methods are so 
 much more compact than Boole's. But on 
 the whole, I think, Schroder in this matter 
 shows an advance upon Boole, while Keynes' 
 methods, which in many respects resemble 
 Schroder's, are neatness itself. 
 
 When the equations have been put into 
 the form of a single sum of terms equated 
 to zero, it is possible to eliminate one or 
 more letters which are not wanted in the 
 result. The first writer to show how 
 elimination in these complicated problems 
 may be effected was Boole. He gave the 
 well-known formula /(l)/(0)=0. That 
 is to say, he developed the expression 
 
 * Laws of Thought, pp. 146- 1 49 ; Der Operations krets des 
 Logikkalkulsy pp. 25-28. 
 
70 Symbolic Logic 
 
 with respect to the term to be eliminated, 
 and then multipHed together the co- 
 efficient of the term and that of its con- 
 tradictory. He also showed that the 
 formula may be extended to the case of 
 the elimination of two or more terms. 
 
 This is undoubtedly one method of 
 eliminating, and one that may frequently 
 be employed. But it is to be noted that 
 it is not the only method, and that it 
 applies only to universals. Venn and 
 Schroder have described other methods 
 for dealing with this problem. Schroder 
 points out that, the equation having been 
 expressed as a sum of terms equated to 
 zero, we may obtain the elimination of 
 a letter if we take all the terms that do 
 not contain the letter, and to them add 
 those which when taken together do not in- 
 volve it. In the expression A;^ + B^ + C = 0, 
 that is to say, the elimination of x will 
 be given by AB + C = 0. And Venn shows 
 how elimination may be effected without 
 bringing over all the terms to the left-hand 
 
The Process of Solution 71 
 
 side. When, for instance, such an equa- 
 tion as njD=xyz + xyx + xyz is given, and 
 we want to eHminate j, what we do is 
 
 to substitute the indefinite symbol ^ for 
 
 y and y in the equation. 
 
 It is important to observe that in deal- 
 ing with the problem of elimination the 
 amount of work required in carrying out 
 the method of Boole is greater than that 
 involved if we follow Schroder. In the 
 case of the earlier logician it is necessary 
 to develop the expression on the left hand 
 with regard to the letters to be eliminated, 
 but in Schroder's method those terms that 
 are free from the letters to be eliminated 
 may be set down at once as part of 
 the required result. As a matter of 
 fact, however, Boole, though he does 
 not in his statement of the rule mention 
 this simplification, avails himself of the 
 shorter method when he comes to work 
 examples.^ 
 
 * See the problem at the bottom of p. 144 in the Laws 
 of Thought^ and his mode of eliminating a. 
 
7^ Symbolic Logic 
 
 Coming now to the case of particulars, 
 the process of elimination was first cor- 
 rectly described by Venn. He shows that 
 the elimination of x from the inequation 
 A;c'4-B^ + C>0 can only yield A + B + C>0, 
 and that in more general symbolic language 
 this result may appear asy(l) y^(0)>0. 
 
 And lastly, suppose that one of the 
 premises is universal, and one is particular. 
 Venn also shows how elimination may be 
 effected in this case. From DAr + E^ + F = 
 and AAr+B^ + C>0, since j? is E and x is D, 
 we may substitute in the inequation and 
 get AD + BE + C>0. This, together with 
 DE + F = from the universal, gives the 
 elimination of x. And Venn here too 
 states in general form the method of dealing 
 with the problem of elimination. His 
 formula appears as follows : — 
 
 I 
 
 /(l)/(0) = 
 
 <^(l)./(l) + <^(0)./(0)>0.* 
 Another formula, one which may be 
 
 * Evidently there is a misprint in Venn's statement of his 
 formula: instead of/(l) he ha8/(«;). (Symbolic Logic ^^. ^SS.) 
 
The Process of Solution 73 
 
 shown to be equivalent to this, has been 
 suggested by Mr. Johnson in an examina- 
 tion question in the Moral Science Tripos,^ 
 This formula runs as follows : If every- 
 thing is f(x) and something is </>(Ar), the 
 elimination of x is given by — 
 
 f Everything is /( 1 ) or /(O), 
 
 ( Something is ^(l)/(l) or </)(0)/(0). 
 
 This problem of dealing with elimination, 
 when there is a combination of universals 
 and particulars, had been attacked by Mrs. 
 Ladd-Franklin, but in her statement of the 
 rule she made a mistake. She says " if the 
 premises include an alternation of particular 
 propositions, the conclusion consists of the 
 partial inclusion of the total co-efficient of 
 X in the particular propositions by the 
 negative of that of x in the universal pro- 
 positions, added to the included combina- 
 tions which are free from x as given." f 
 Such a rule would lead us when dealing 
 
 * Cambridge University Examination Papers, vol. xvi. 
 
 p. 536. 
 
 •j- Johns Hopkins Studies in Logic j p. 45. 
 
74 Symbolic Logic 
 
 with the equation Aat+B^ + Cv and the 
 inequation E^r + Fi^ + Gv to the conclusion 
 EA + C G V . But here we have evidently 
 committed error, for we have concluded 
 that C is something, whereas in the uni- 
 versal it was declared to be nothing.^ 
 
 In tracing to their sources and criticiz- 
 ing the methods that have been suggested 
 for the performance of the process of elimi- 
 nation, it has been necessary to refer to 
 the question of development or expansion. 
 This question has been adequately discussed 
 by all three of the logicians who have done 
 most for the earlier Symbolic Logic, /.^., by 
 Boole, Venn, and Schroder. The last of the 
 three was, I believe, the first to give a geo- 
 metrical justification of the process,t but 
 the analytical treatment of the subject was 
 well treated by Boole at the outset. It was 
 in the Laws of Thought that the now 
 well-known formula f{x) = f(\)x + f{Qi)x 
 first appeared. A companion formula to 
 
 * See the critical notice in Mind for October, 1883. 
 •j" Der Operatlonskreis des Logikkalkuls, p. I4. 
 
The Process of Solution 75 
 
 this has been proposed by Peirce, viz.^ 
 f{x) = {f{l)-\-x}{f{())+x}. This formula 
 may be used when we prefer to have 
 the development as a product of sums 
 rather than as a sum of products.^ 
 
 In order to bring out one or two other 
 points where later logicians show an im- 
 provement on Boole, it may be well to 
 describe somewhat fully one of his solutions. 
 We may take the problem that is worked 
 out on pages 138 and 139 of the Laws of 
 Thought, In the first place, he put the 
 premises in the form ^^ = 0, ^2 = ^' ^^^ ^^ 
 on, i.e.^ in each case he brought over with 
 changed signs all terms that were originally 
 on the right-hand side of the symbol of 
 equality. These equations were then added 
 together, unless V^ or Fg needed squaring, 
 in which case he squared before performing 
 the process of addition. His next step 
 was to make any necessary eliminations, 
 
 * See Venn, Symb. Log. p. 388 ; Peirce in the Amtr. Journ. of 
 Math. vol. iii. p. 38. In Peirce's formula /(qc) is used in- 
 stead of /(I), but the meaning is the same as that of /(I). 
 
76 Symbolic Logic 
 
 and to effect these he proceeded to develop 
 the resulting equation with reference to 
 the letter or letters about which informa- 
 tion was not required : the desired elimi- 
 nation was obtained either by multiplying 
 together the resulting co-efficients, or by 
 constructing the equation Ey + E^l-y) =0, 
 where E is derived by changing into 1 in 
 the original equation the letter about 
 which information is asked, and then 
 eliminating the letters not required, and 
 where E^ comes from changing into the 
 same letter, and making the same elimi- 
 nation. We thus arrive at an equation 
 from which, by means of the application 
 of the principle of expansion to fractional 
 forms, the desired information may be 
 obtained. 
 
 In this process of solution there are 
 several drawbacks, which the more modern 
 methods are able to avoid. We have 
 already noticed certain modes of elimi- 
 nating which are sometimes more con- 
 venient than that proposed by Boole. The 
 
The Process of Solution "j^ 
 
 necessity of squaring may be escaped by 
 adopting Venn's plan of restricting the left- 
 hand member by the contradictory of the 
 right - hand member, and equating the 
 result to zero. Sometimes, again, a resort 
 to diagrams is easier than the plan laid 
 down by Boole, who, strange to say, never 
 appeals in his work to a diagram. And, 
 once more, when in the course of our 
 solution we arrive at a fractional expression 
 with a complicated numerator and denomi- 
 nator, we may, so long as real class terms are 
 
 involved, employ the formula ^- = xy -\--^xy^ 
 
 and thus avoid the liability to error from 
 the 1 and substitution. 
 
 The fact that there is abundance of 
 opportunity of falling into mistakes on the 
 Boolian plan of making this substitution 
 needs perhaps but little emphasis. Those 
 who have worked through several examples 
 by his method will fully realise the liability. 
 Boole himself has not always escaped fall- 
 ing into mistake. His solution, for in- 
 
78 Symbolic Logic 
 
 stance, for ^(1 — /) on p. 145 of the Laws 
 of Thought is certainly wrong, as may easily 
 be shown by working the problem in two 
 or three of the available ways. The answer 
 
 should be irn+~{ir + i). But, though 
 
 Boole's method is long, it has one marked 
 advantage over those proposed by later 
 logicians. That is to say, when the pro- 
 cess that he describes has at last been 
 completed, the who/e solution that can be 
 asked for with regard to the relation of 
 certain letters to certain others has been 
 obtained. For instance, on p. 397 of his 
 Symbolic Logic^ Venn reaches a conclusion 
 
 xz = - ac -)r - dc. Now this ?ives assur- 
 00 ^ 
 
 ance that xz is found somewhere within 
 ac or dc^ but does not tell us whether ac 
 or dc is more extensive than xz. To find 
 out whether this is so, two operations, 
 similar to that by which information 
 concerning xz was obtained, would have 
 to be performed. But, when Boole had 
 once worked out the conclusion concerning 
 
The Process of Solution 79 
 
 xz, there would be no doubt in what way 
 the constituents of the other member ought 
 to be expressed in terms of xz, for the co- 
 efficient 1 or ^ would be decisive upon the 
 point. In other words, when ^ appears as 
 
 a co-efficient in Venn's solution, the ques- 
 tion arises whether the symbol may pos- 
 sibly be equal to 1, and a new piece of 
 work is necessary in order to determine 
 if this is so ; but, when the symbol is 
 found at the conclusion of Boole's process, 
 no such doubt arises, for if a term is there 
 to have the co-efficient 1 this fact will be 
 brought out in the original solution. 
 
 When the method referred to above of 
 obtaining all the denials is employed, it 
 will as a rule be necessary to combine a 
 set of terms with the contradictory of 
 another set of terms. Hence it is desir- 
 able to have a simple means of finding 
 the contradictory of a complicated ex- 
 pression. Here De Morgan was the first 
 to make the way plain. His rule, ex- 
 
8o Symbolic Logic 
 
 tended by Venn, is as follows : For every 
 elementary term substitute the contradic- 
 tory, interchanging at the same time the 
 symbols of addition and multiplication. 
 This rule is mentioned by Schroder on 
 p. 1 8 of the Operations kreis^ but he does 
 not refer to De Morgan. It is also given, 
 without employment of expressions that 
 imply symbols of operation, by Keynes, 
 who alludes to this early work of Schroder. 
 To this rule Venn adds another. He 
 gathers up the expression to be contra- 
 dicted into the form A;c + B;?, and then con- 
 tradicts A and B, thus arriving at A^c + BJc. 
 This procedure is often a simpler one either 
 than applying De Morgan's rule, or than fully 
 expanding the original expression, and then 
 finding the contradictory on the principle 
 that the contradictory of at is 1— at. 
 
 (2) Diagrammatic Method, — We have 
 now somewhat fully discussed the simpli- 
 fications that have been effected in the 
 analytical methods of solving complicated 
 problems. We have next to remark 
 
The Process of Solution 8i 
 
 that pari passu with this simplification 
 there has gone a marked advance in the 
 employment of diagrams, both for the pur- 
 pose of representation and for the purpose 
 of solution. Boole, as we noticed above, 
 never appealed to diagrams. The Eulerian 
 method is entirely unsuitable in the case 
 of complex propositions — is, indeed, not 
 accurate for simple ones. Dr. Keynes has 
 shown the unsuitability of the Eulerian 
 diagrams, and himself has employed a 
 diagrammatic method that is quite accu- 
 rate. But Keynes' procedure, though accu- 
 rate, is much too cumbrous to be of any 
 use except in the case of propositions of 
 the very simplest description. 
 
 Venn set himself as one of his tasks to 
 invent a suitable diagrammatic represen- 
 tation of propositions, and within certain 
 limits he was very successful. He sug- 
 gested the intersection of circles, ellipses, or 
 other closed figures to denote compartments, 
 some of which are destroyed or saved by 
 the premises. This plan works well up 
 
82 Symbolic Logic 
 
 to five terms, and it is possible to use it 
 for six terms. Theoretically, though not 
 practically, it may also be used for any 
 number of terms. It was left for Dr. 
 Marquand to suggest an elegant method 
 of dealing diagrammatically with those 
 cases that involve a very large number of 
 terms. In the case of seven terms, for 
 instance, he forms sixteen rows of squares, 
 eight deep. The rows are alternately (say) 
 d and d. These are marked off in pairs, 
 which are c and f, such pairs into other 
 pairs h and ^, and these, again, into the 
 pairs a and a. Then those squares that 
 constitute the depth of the figure may be 
 marked off similarly with reference to the 
 letters J j/, xx^ and %%, A diagram on this 
 plan for even ten terms would occupy 
 only a page. 
 
 As we remarked a few pages back, dia- 
 grams may sometimes be made of service 
 in the solution of direct problems. Such 
 is the case when the problem is to show 
 that a certain compartment has been en- 
 
The Process of Solution 83 
 
 tirely erased. A diagram constructed on 
 Venn's plan decides at once whether the 
 premises have effected such an erasion. 
 And, where the problem is one of eli- 
 mination, Dr. Marquand's diagrammatic 
 method is serviceable in reaching the 
 desired conclusion. To eliminate _y, for 
 instance, in the problem with seven terms, 
 all that is necessary is to take those groups 
 which involve any of the four pairs of y 
 and j?, add such groups together, and 
 equate the sum to zero.* It may be 
 remarked, however, that, if we adopt Dr. 
 Marquand's diagrammatic scheme, the pro- 
 cess of effecting the erasions is rather a slow 
 one. This arises from the fact that the 
 squares to be erased by each section of the 
 premises do not lie together. 
 
 But, though diagrams are to this extent 
 of service, direct problems are as a rule 
 much more expeditiously worked by literal 
 methods. A diagram takes some time to 
 construct, and often when completed does 
 
 * Venn, Symbolic Lo^ic, p. 374. 
 
84 Symbolic Logic 
 
 not yield at all easily the information we 
 require. In fact, if the symbolist employs 
 diagrams, he is guilty of a somewhat re- 
 trograde movement. For in the case of a 
 figure we have to a certain extent to think 
 out the answer, whereas a genuine calculus 
 supplies us with a mechanical contrivance 
 by which results may be reached with no 
 other thought than that which is required 
 by a person who manipulates a machine.'^ 
 On the whole it must be said that, with 
 the exception of the two cases above men- 
 tioned, diagrams in relation to the solution 
 of direct problems are mainly of interest 
 as giving a pictorial representation of 
 conclusions that have been reached by 
 the analytical method. 
 
 II. So far we have considered the 
 method of finding the description of a 
 letter or letters, mentioned in the premises, 
 in terms of all or of certain only of the 
 remaining letters. But there is also the 
 
 * M'lnd, N.S. vol. i. p. 3. 
 
The Process of Solution 85 
 
 inverse problem, and, as in the case of the 
 direct problem, there have been here, 
 during the period that Symbolic Logic 
 has been studied, many improvements 
 suggested in the methods for effecting a 
 solution. 
 
 Boole does not attempt to deal v^ith the 
 Inverse Problem. Jevons realised w^hat is 
 the nature of the problem, but he did not 
 suggest any general method of obtaining 
 a solution. He drev^ up a table w^hich 
 applies to three terms, and he shov^ed what 
 types of proposition will result in the given 
 destructions.^ But he does not demonstrate 
 how to arrive at a proposition that will 
 occasion the destructions, and he does not 
 attempt to deal with more than three terms. 
 
 It was one of the important contributions 
 of Keynes to the development of Symbolic 
 Logic to suggest three methods by which 
 a solution to this problem can always be 
 obtained. As Venn points out, it cannot be 
 said that there is one and only one solution 
 
 * The Principles of Science^ p. I40. 
 
86 Symbolic Logic 
 
 of these problems ; in other words, each 
 of several solutions will occasion the given 
 destructions. Keynes discovers what terms 
 are saved, and then he throws the proposi- 
 tion into the form : " Everything is X or Y 
 or Z or W," where these letters represent 
 complex terms. Hence " What is not either 
 Z or W is X or Y." Then, if the subject 
 is thrown into the affirmative form, a series 
 of simple propositions may be obtained that 
 are together equivalent to the original pro- 
 position. Another method is to employ 
 the process of obversion and reach the 
 negative form : " Nothing is . . ." This 
 proposition then splits up immediately into 
 a set of simple propositions. A third 
 method is to put the proposition into the 
 form " Everything is . . ." Then select 
 (say) the A and the a members, and form 
 two propositions, " A is . . .,'' " ^ is . . ." 
 Next, in both cases take the B and b mem- 
 bers, so obtaining four propositions. And 
 so proceed until the propositions have no 
 disjunctives in the predicate. 
 
The Process of Solution 87 
 
 A fourth method of dealing with the 
 inverse problem is suggested by Schroder's 
 third corollary to his 36th Proposition,* 
 namely, the method of double negation. 
 This amounts to taking the saved terms 
 and equating them to 1, then contradicting 
 both sides and getting an expression, com- 
 posed of factors, which is equal to 0. This 
 expression may then be multiplied out. 
 Then, if both sides are contradicted again, 
 there will be another expression composed 
 of factors equal to 1. Each factor will 
 consequently be equal to 1, and we arrive 
 at a series of simple propositions. This 
 rule is not given in so many words by 
 Schroder, but it follows immediately from 
 the above-mentioned corollary. 
 
 A considerable advance in simplicity in 
 connexion with this part of the subject 
 was made when Mr. Johnson showed how 
 Keynes' third method may be employed in 
 a semi-diagrammatic manner. Mr. Johnson 
 first of all shows clearly that, by a process 
 
 * Forlesungen, pp. 356-357. 
 
88 Symbolic Logic 
 
 of multiplying out, the Inverse Problem 
 may be solved by a direct process, and that 
 w^e are not, as Jevons maintained we are, 
 here limited to a series of guesses. But the 
 process of multiplying out is long, and the 
 same solution may be obtained if we 
 " represent and by horizontal juxtaposition^ and 
 or by vertical juxtaposition T * That is to 
 say, commencing with the given expression, 
 Mr. Johnson finds what terms are equated 
 to unity. He then takes two lines at right 
 angles to one another, and lets opposite 
 squares represent (say) a and a. Each of 
 these regions may then be divided into 
 similar portions to represent other letters 
 and their contradictories, and so we may 
 proceed until all the letters find their 
 suitable place. Then, if the columns are 
 read downwards, and the letters joined by 
 -f, and if all such sums are multiplied 
 together, an expression will be reached 
 equal to the original one. Hence each one 
 of the factors of this product is equal to 1, 
 
 * Mind, N.S. No. 3, p. 350. 
 
The Process of Solution 89 
 
 and thus simple propositions are at once 
 obtained. Mr. Johnson's originahty here 
 consists in his perceiving that the formula 
 {ac-^dd) = {a + d){d-\-c) could be expressed 
 in this useful way, and in showing in 
 a particularly convincing manner wherein 
 Jevons' error consists in identifying In- 
 duction with the Inverse Problem. The 
 fact that this problem can be solved by a 
 direct process was also made very evident 
 in Mr. Johnson's discussion, but of course 
 Keynes had already practically settled this 
 matter when he proposed the three methods 
 of solution given above. 
 
 It will be noticed that at the end of Mr. 
 Johnson's solution there is an assumption 
 that if the product is equal to 1, then each of 
 the factors is also equal to 1. That this 
 assumption is perfectly justifiable is made 
 clear by reference to one of the truths that 
 was established in the Johns Hopkins Studies 
 by Dr. Mitchell.* He showed that if we 
 have "All x is ^^ " and "All y is a:," i.e., 
 
 * See Venn's detailed critical notice in Mind, October, 1883. 
 
go Symbolic Logic 
 
 xy = and xy = 0, the compound expression 
 of these propositions may be given just 
 as well by {xy + xy + xy) {xy -\- xy + xy) = 1 as 
 by xy + xy = 0. He was the first logician, 
 that is to say, to bring out the fact that the 
 process of multipUcation in the former case 
 is equivalent to the process of addition in 
 the latter. Obviously, therefore, to get 
 back to the separate statement of the 
 premises as equations, with for one of 
 the members, it is necessary that each of 
 the factors {xy + xy + xy) and {xy -\- xy -\- xy) 
 should be equal to 1. 
 
CHAPTER IV 
 
 CONCERNING A CALCULUS BASED ON 
 INTENSION* 
 
 In the critical history, which we have 
 now concluded, of the earlier Symbolic 
 Logic, we have had occasion to note 
 one point on which logicians have been 
 separated into well-marked divisions, viz., 
 the point as to whether symbols should 
 primarily or exclusively be employed to 
 denote classes or to denote propositions. 
 Boole, and Venn after him, used symbols 
 primarily to denote classes or individuals, 
 and secondarily to stand for propositions. 
 
 * By the expression " a Calculus based on Intension " I 
 mean a Logic based on attributes rather than on classes. 
 The term " intension " here is used in a general sense, not 
 in Keynes' special sense of attributes "mentally associated" 
 with a name. On the other hand, " connotation " and " com- 
 prehension " throughout this chapter are taken in the sense that 
 Keynes has assigned to them. 
 
 91 
 
92 Symbolic Logic 
 
 Schroder and the contributors to the Johns 
 Hopkins Studies have advanced the subject 
 on the same Hues. Mr. Johnson in his 
 presentation of the Calculus has used sym- 
 bols in the first place to denote unanalyzed 
 propositions, and, subsequently, v^hen he 
 treats of the synthesis of singly-quantified 
 and of multiply-quantified propositions that 
 follows their analysis, he points out that 
 predications obey the same laws that were 
 originally enumerated as necessary and suf- 
 ficient for the synthesis of unanalyzed 
 propositions. That is to say, symbols 
 here stand primarily for propositions and 
 secondarily for terms. Mr. MacColl also 
 holds that propositions and implication are 
 more fundamental than classes and inclusion.^ 
 We have now to take note of a subject 
 on which all these logicians are in agree- 
 ment : they all hold that Symbolic Logic 
 should adopt an extensive interpretation 
 of the proposition. I think they are un- 
 
 * Mind, No. 17, pp. 49, 51 ; B. Russell, The Principles of 
 Mathematics y p. I2. 
 
On a Calculus Based on Intension 93 
 
 doubtedly correct in holding this view, 
 but the matter, which is one of great 
 importance, calls for careful argument. 
 In the present chapter I shall endeavour 
 to prove that labour is not wisely spent 
 in attempting to elaborate a calculus on 
 intensive lines. My method of proof 
 will be to take the work of the most 
 consistent logician who has proceeded 
 from the intensive standpoint, /.f., Castillon, 
 and to show in detail the errors into which 
 he fell in constructing the various portions 
 of his symbolic structure. It will become 
 evident, I think, as the investigation ad- 
 vances, that a calculus cannot be elaborated 
 on the lines he laid down. Then, having 
 demonstrated where his errors lie, I shall 
 refer to the treatment of the subject given 
 by Venn, with whose conclusions I am 
 in agreement, viz.^ that, when we confine 
 our attention to the attributes that are 
 commonly denoted by the name, a calculus 
 based on intension is impossible, but that, 
 when we take all the attributes common to 
 
94 Symbolic Logic 
 
 the members of the class,* it is possible to 
 reach a calculus, but that this would be un- 
 necessarily long and artificial. That is to 
 say, the logicians mentioned above will 
 be shown to have advanced in the right 
 direction, inasmuch as failure or unneces- 
 sary difficulty is to be met with in the 
 attempt to work out problems by refer- 
 ence to nothing but attributes. 
 
 I. Of the various logicians, whose 
 work is based on an intensive interpretation 
 of the proposition, Lambert is perhaps 
 the most distinguished, but Castillon is the 
 only thinker who, setting out from this 
 point of view, has attempted to be rigor- 
 ously consistent. The memoir, Sur un 
 nouvel algorithme logique^ which Castillon 
 presented to the Berlin Academy in 1803, 
 is in many respects a remarkable pro- 
 duction. In connexion with the attempt 
 that the writer made to utilise symbols 
 to represent the intension of terms, he 
 
 ■^ What Keynes means by ** comprehension/' 
 
On a Calculus Based on Intension 95 
 
 treated of logical problems which have 
 not until quite recently received the atten- 
 tion they deserve, and some of his specula- 
 tions appear even now^ almost unknow^n to 
 the general student of Logic. 
 
 (i) Castillons Fundamental Notions, — And 
 first of all it w^ill be useful to notice w^hat 
 is Castillon's conception of the nature of 
 Symbolic Logic, ^ for he is here to a very 
 large extent unquestionably upon the right 
 lines. Symbolic Logic means a generalised 
 Logic, /.f., it is a general treatment of 
 that aspect of thought w^hich receives 
 a limited treatment in ordinary Formal 
 Logic. The consideration of deduction 
 from this higher point of view^ renders 
 clear many processes that are necessarily 
 described inadequately in the ordinary 
 text-books of Formal Logic ; and, w^hat 
 is still more important, actual errors that 
 have arisen in attempts to treat exclu- 
 sive portions of deductive reasoning are 
 
 * He does not, of course, use this term, which is not, I be- 
 lieve, to be found before the publication of Venn's book in 1881, 
 
g6 Symbolic Logic 
 
 exposed. But the idea of Symbolic Logic 
 also implies the notion of a calculus. That 
 is to say, there is implied such a manipula- 
 tion of symbols that conclusions are reached 
 that would be quite unattainable by intel- 
 lectual intuition. Symbols of some sort are 
 essential in fully conscious reasoning. But 
 as employed in a calculus symbols take on 
 an additional function. They no longer 
 merely help the mind to reach a judgment : 
 they obtain a judgment for a mind that 
 has in the meantime been engaged upon 
 an operation quite different from that of 
 reflecting upon their meaning. As Mr. 
 Johnson has clearly shown, there is in a 
 calculus the intelligent manipulation of 
 symbols by means of which results are un- 
 intelligently reached. This view implies, of 
 course, that a calculus is very different from 
 being merely a symbolical illustration of 
 logical doctrines. Castillon had a clear idea 
 of all these truths. Where he went wrong 
 on this subject — and he has had successors 
 who have fallen into the same mistake — 
 
On a Calculus Based on Intension 97 
 
 was in the following particular. He was 
 of opinion that symbols can in some 
 way discover the forms of judgment, can, 
 for instance, indicate that the infinite judg- 
 ment is not of the essence of Logic, 
 or that naturally all judgment is apo- 
 deictic. But symbols can never discover 
 the forms of judgment, but can only pro- 
 ceed according to their own laws, after 
 the import of the judgment has been un- 
 folded. 
 
 We will proceed to consider the precise 
 description Castillon gives of the nature of 
 the proposition, and to observe the primary 
 use he makes of symbols. Each object that 
 we meet with in the course of our ex- 
 perience forms a mental image, which, 
 though made up of sensations, is at first not 
 recognised as so composed. Subsequently, 
 however, by means of imagination, the 
 individual sensations may be detached from 
 the whole, and a symbol may be employed 
 both for the original complex and for the 
 sensations so separated. Other sensations may 
 
 G 
 
gS Symbolic Logic 
 
 be similarly abstracted, and we thus arrive 
 at such an equation as 0-A-B = C + D. 
 Here O is a congeries of attributes, and the 
 right-hand member represents the attributes 
 that remain when A and B have been 
 abstracted from O. We are in this way, it 
 will be seen, dealing with nothing but 
 attributes, and the symbol that is chosen to 
 denote a combination of attributes in a 
 whole is correctly said to be ( + ), while the 
 symbol (-) is employed to denote that 
 attributes are abstracted from the whole 
 with which we start. The signs ( + ) and 
 (-), that is to say, indicate respectively the 
 combination and the abstraction of attri- 
 butes. It must here be allowed, then, both 
 that there is sound psychological doctrine, 
 in that growth of perception is seen to 
 consist in progress from the obscure to the 
 definite, and that there is appropriateness in 
 the signs that are used to represent the 
 elements and progress of thinking, when 
 this has once commenced. 
 
 But it is necessary to inquire whether 
 
On a Calculus Based on Intension 99 
 
 it is justifiable to use in this way the 
 symbol for equivalence. Venn argues that 
 on the intensive interpretation of proposi- 
 tions such use is inappropriate, since, if a 
 group of attributes is equivalent to another 
 group, there are no means of distinguishing 
 the one from the other.^ This criticism 
 appears to me to be true, but only w^ithin 
 narrow limits, namely, when the two 
 members consist of one group each, and all 
 the attributes of each group are explicitly 
 recognised. But then in the case of most 
 equations there is nothing of this kind. 
 For instance, S = P + M, the form for the 
 universal affirmative in Castillon's sym- 
 bolism, denotes that S consists of a definite 
 attribute, or group of attributes P, and 
 another group M. Here both S and M 
 stand for groups, whose elements are not, 
 as a rule, present before the mind. It is, 
 therefore, surely permissible to say that the 
 group denoted by S is identical with one 
 that is specifically mentioned, together with 
 
 * Symbolic Logic, p. 456. 
 
loo Symbolic Logic 
 
 another group denoted by M. And not 
 only is there the fact that the attributes de- 
 noted by S and by M are not enumerated, 
 but it is obvious that S is regarded as 
 one group, while the right-hand member 
 is composed of two groups : the sign ( = ) 
 is, therefore, quite as appropriate as it is 
 in class Logic if we say that x = xy-\rxy^ 
 where, of course, one class is declared to 
 be identical with its two composing classes. 
 Venn, in urging his conclusion, appears to 
 have confined himself to the consideration 
 of two simple conceptions that are declared 
 to be equivalent. He is correct in saying 
 that many logicians have thought that 
 they were equating attributes, when really, 
 they were merely equating classes whose 
 extension happens to coincide. Of course, 
 any such reference to classes involves 
 inconsistency of treatment. But, when 
 it is said that the attributes designated 
 by S are equivalent to the attributes 
 designated by P, together with those 
 designated by M, there is no more re- 
 
On a Calculus Based on Intension loi 
 
 flection required concerning classes than 
 there is concerning attributes in extension 
 Logic. 
 
 In the next place, we may notice the 
 description that Castillon gives of the laws 
 that lie at the back of formal reasoning. 
 Some logicians have endeavoured to deduce 
 the Law of Contradiction from the Law of 
 Identity, or i:ice versa, but he savs that, if 
 judgments denote respectively the synthesis 
 and the separation of attributes, then 
 every judgment implies both principles. I 
 think that those who deduce one principle 
 from the other can only do so bv some 
 arbitrary assumption that one is " more 
 natural" than the other. "^ But in Loo-ic 
 we are not justified in thus acting according 
 to our preferences. We have to state the 
 actual facts of the case, and in this particular 
 those facts are that the Laws of Tautology 
 and Reciprocity f are equally fundamental : 
 
 * Proc. of the Ar'ist. Soc. N.S. vol. ii. p. I 15. 
 I This term includes the Law of Contradiction and the Law 
 of Excluded Middle. 
 
I02 Symbolic Logic 
 
 the former denotes that it is immaterial to 
 thought whether a thinker duplicates a 
 conception, and the latter that a predication 
 consists, and consists exclusively, of what is 
 left when the conflicting ideal that is in- 
 volved in every judgment has been re- 
 jected.* 
 
 Having maintained to the above extent 
 the independent character of the Laws of 
 Identity and Contradiction, Castillon some- 
 what spoils matters by suggesting that, after 
 all, one of the two principles may be more 
 fundamental than the other. His proposal 
 is to regard the principle of Contradiction 
 as expressing the fact of the composition of 
 a concept, while the principle of Identity 
 denotes the reminiscence of a concept ; for 
 in positing a concept one thinks the presence 
 and not the absence of attributes (Contradic- 
 tion), but in recalling a concept the in- 
 telligence has to retrace the attributes, for 
 otherwise the earlier concept would not be 
 reinstated (Identity). This statement of 
 
 * Mind, N.S. No. i. pp. IO-I2. 
 
On a Calculus Based on Intension 103 
 
 what is involved in the principle of Contra- 
 diction is practically the same as the state- 
 ment given at the end of the last paragraph. 
 But Castillon's account here of the principle 
 of Identity is very unsatisfactory. For the 
 act of recaUing certain attributes, if such 
 recall is occasioned merely by the uncon- 
 scious fusion of certain fresh presentational 
 elements with a portion of the memory 
 train, is certainly not what is meant by 
 the principle of Identity ; while, if the 
 recall is dehberate, then there is contem- 
 poraneously a rejection of certain attri- 
 butes, and the principle of Contradiction 
 is involved. 
 
 The important question now arises 
 whether on this intensive view of Logic 
 it is possible to deal with other than essen- 
 tial attributes, and Castillon has received 
 Venn's approval for consistency in not con- 
 sidering attributes that are accidental. Cas- 
 tillon s universal affirmative is, as we have 
 said, S = A + M, and the universal negative 
 appears as S= - A + M. The former means 
 
I04 Symbolic Logic 
 
 that the object S is the sensation A, plus 
 a group of other sensations M, which 
 remain indeterminate, while the symbols 
 for the universal negative denote that the 
 object S does not comprehend A, that the 
 object S is an indeterminate M without A. 
 Now, at first sight, it does appear that we 
 must confine ourselves here to essential 
 attributes, for in the case of the universal 
 afiimative, at any rate, we just unfold the 
 nature of S, state the group of attributes of 
 which it is composed. But then it seems 
 quite possible to do as Venn hints,^ namely, 
 to regard the essence as consisting of what 
 is usually designated by that term, together 
 with a limited number of other attributes. 
 If Castillon's methods are valid on the 
 narrower interpretation of the word essence 
 they are valid on the wider interpretation. 
 He suggests that, by means of small letters, 
 it is possible to represent accidental attri- 
 butes — thus = A + B + C + D+f+/— and, 
 though he certainly does not recur to 
 
 * Symbolic Logic, p. 463. 
 
On a Calculus Based on Intension 105 
 
 this method of symbolism, I do not think 
 his reason for not doing so was because 
 he felt that there would be anything 
 unsound about such a procedure. In 
 admitting, however, this extended signifi- 
 cation of the term essence, it is to be 
 noted that the number of attributes is 
 limited, for otherwise we should not have 
 an equation, since 00 is not necessarily equal 
 tooo . M, whatever it denotes, is indefinite 
 only, not infinite. 
 
 In close connexion with this discussion 
 is the consideration of the precise interpre- 
 tation to be put upon Castillon's repre- 
 sentation of the universal negative. His 
 procedure here has, I think, been to some 
 extent misunderstood. But it will be 
 better to postpone for a moment the 
 examination of his symbols, and to notice 
 the classification of judgments that he 
 prefers to adopt. The Critical School 
 had admitted infinite terms, so that it could 
 at once throw a universal negative into 
 the form of a universal affirmative. Hence 
 
io6 Symbolic Logic 
 
 that school, Castillon says, tended to regard 
 judgment exclusively as synthesis : the at- 
 tribute comprehended by the subject could 
 be either affirmative or negative. But he 
 urges that it is preferable to divide judg- 
 ments into affirmative and negative, and 
 to reject the class of infinite judgments.^ 
 Castillon's position is rendered the more 
 precise, inasmuch as he states that the 
 classification he adopts, and v^hich he de- 
 scribes as excelling others in both truth 
 and generality, is based upon the definition 
 of judgment given by Wolf: that is to 
 say, judgment is " the act by v^hich the 
 intelligence perceives that two concepts, 
 that are present to it, can be joined or 
 ought to remain separate." Castillon is 
 thus definite enough upon this matter, 
 and it is absolutely essential to remember 
 this, if one is to understand clearly his 
 treatment of the universal negative. 
 
 When, then, it is observed that judg- 
 ment is made to consist of synthesis or of 
 
 * Memoire, p. 1 1. 
 
On a Calculus Based on Intension 107 
 
 analysis, and not of the former only, the 
 explanation that Venn offers of Castillon's 
 universal negative is seen to be misleading.* 
 Venn asserts that the expression S = - A + M 
 indicates that S is composed of two groups 
 of attributes, of v^hich the first is negative 
 and the second is positive. That is to say, 
 he writes the symbols as though the minus 
 sign and the A were bracketed. But this 
 view of the case must be wrong. If such 
 transformation is eff^ected, then we shall be 
 dealing with a universal affirmative, and 
 all judgment will be reducible to synthesis. 
 To get at Castillon^s meaning we must do 
 as he says, namely, take the minus sign in 
 conjunction with the sign of equality. The 
 expression will then be read thus : " S 
 does not contain A but does contain an 
 indeterminate M." I do not mean to argue 
 that Castillon is always consistent in his 
 treatment of this universal negative, for he 
 certainly does sometimes deal with the 
 proposition as though the bracket had been 
 
 * Symbolic Logic , p. 466. 
 
io8 Symbolic Logic 
 
 inserted. Moreover, he admits in so many- 
 words that the infinite affirmative judgment 
 may be reduced to a universal negative, and 
 that the latter may be expressed as the 
 former. But, generally speaking, he does 
 not adopt the method of making a 
 change to the affirmative : he proceeds on 
 the understanding that a proposition which 
 is expressed like the above is to be inter- 
 preted as denoting a separation of A from 
 S. In short, I admit that Castillon is 
 uncertain in his treatment of the universal 
 negative, but think that his intention was 
 always to regard this proposition as a case 
 of decomposition rather than as one of 
 composition. The sign (= — ) with him 
 signifies " is not." 
 
 What Castillon wishes to be understood 
 by the minus sign is well brought out 
 when we look into his treatment of con- 
 version and of syllogism. In some cases 
 of these processes he is able to adhere 
 rigorously to his idea of the import of 
 proposition. For instance, take Celarent^ 
 
On a Calculus Based on Intension 109 
 
 which appears thus : " M = - A + N," 
 "S = M + P," .-. "S = - A + N 4- P." 
 Here S in the conclusion comprehends 
 P, just as was the case in the premise, 
 but does not comprehend A, and yet com- 
 prehends N, since these were the charac- 
 teristics of M in the premise. There is not 
 the slightest need in such a case to change 
 the major premise into a universal affirma- 
 tive infinite judgment. Again, take Cesare : 
 "A = - M + N" (.-. "M ^ - A + N"), 
 " S = M + P," .-. " S = - A + N + P." 
 Just noticing in passing that Castillon gives 
 " M = A + N " as the converse of the 
 major — one of his many accidental mistakes 
 in the use of his symbols — we see that he 
 argues that, since no A is M, but is an 
 indeterminate N, therefore no M is A, 
 but is an indeterminate N. Now, though 
 this attempt at conversion is altogether 
 unjustifiable, as I shall presently show, 
 it helps to bring out the way in which 
 Castillon regarded the meaning of the 
 minus sign. For his conclusion is not 
 
iio Symbolic Logic 
 
 " ( _ M ) = A ■- N," which would be 
 the converse of the universal affirmative, 
 and which would be read "some (- M) 
 is A," but it is " no M is A." On the 
 other hand, for an instance where, in the 
 treatment of the universal negative, he 
 sacrifices principle, take the following : 
 " M = - A + N," " M = S + P,'' there- 
 fore "S + P= -A + N," and hence 
 "S- -A + N- P." Here it is obvious 
 that two things are conceived as iden- 
 tical with M, and are hence concluded 
 to be identical with one another. That is 
 to say, he here interprets the universal 
 negative with the minus sign attached to 
 the A. 
 
 It follows from these considerations that 
 the minus sign in Castillon's calculus is 
 used in two quite different senses. Take 
 his representation of the particular negative, 
 namely, S = — A — M. This denotes that 
 the S attributes are not the A attributes, 
 when the M attributes have been abstracted 
 from the latter. That such must be the 
 
On a Calculus Based on Intension i i i 
 
 correct interpretation is seen from regard- 
 ing the particular positive, A = S - M (the 
 converse of the universal affirmative), w^here 
 the A attributes are equivalent to the S 
 attributes, w^hen the M attributes have 
 been abstracted from those denoted by S. 
 Such, then, being the appropriate inter- 
 pretation of the expression for the parti- 
 cular negative, it is obvious that the minus 
 sign is used with a double signification. 
 In the first place it is coupled with the 
 ( = ) to signify " is not," />., " is to be 
 kept separate from," and in the other case 
 the sign denotes abstraction : the attributes 
 denoted by M are to be eliminated from 
 those denoted by A. 
 
 (2) Illusory Particulars, — We may now 
 pass to an examination of Castillon's peculiar 
 doctrine of the subdivision of particulars. 
 He distinguishes between the real and the 
 illusory particular proposition. The former 
 is the converse of a universal, the latter is 
 true because it is a subaltern. If, that is to 
 say, " S = A + M " represents the universal 
 
1 1 2 Symbolic Logic 
 
 affirmative, it follows that A will be equal 
 to S minus M, and this in Castillon's 
 view will represent what in class Logic 
 is termed the accidental converse. On the 
 other hand, when a particular is said to 
 be true because it is a subaltern, we are 
 stating less than the truth, for we might 
 adhere to the universal. Hence our 
 symbols must bring out this fact, and we 
 shall have such a form as S = A + M. 
 This means that we affirm " some S is A " 
 (S = A - M), whereas it is allowable to 
 say "all S is A " (S = A + M). 
 
 But this conception of the illusory, 
 instead of introducing considerations that 
 involve less than the truth, involves positive 
 error, and must, I think, be definitely 
 rejected. In discussing the matter, the 
 fact that the calculus is being considered 
 from the intensive point of view is to be 
 kept always in mind. Of course, on the 
 class view of the proposition, it is quite 
 allowable, provided that subjects exist, to 
 proceed from A to its subaltern. But 
 
On a Calculus Based on Intension 113 
 
 the same course is not allowable when 
 we are arguing intensively. If the attri- 
 butes denoted by S consist of those denoted 
 by A + M, to state that the attributes 
 denoted by S consist of those denoted by 
 A - M is palpably false. We do not, in 
 taking such a step, arrive at something 
 less than the whole truth : we arrive at 
 a falsity. 
 
 In deciding the fate of this illusory par- 
 ticular the fact has to be noticed that 
 there is no question here of an alternative 
 rendering. The symbol ( + ) may at first 
 sight suggest that the proposition is to be 
 read either as a particular or as a universal. 
 But this is not what Castillon says. His 
 doctrine is that the particular is as a matter 
 of fact taken, where the universal might 
 be. So that an incorrect statement is 
 accepted in place of the correct one. Nor 
 is it a case where the alternative sign is 
 used because the element of doubt enters, 
 because we know that one proposition is 
 to be accepted, but we do not know 
 
 H 
 
114 Symbolic Logic 
 
 which. The illusory neither offers us an 
 alternative nor involves an element of 
 doubt : it compels us to accept a false 
 statement. The attempt, therefore, to 
 get at subalterns in the way adopted by 
 Castillon must be relinquished. 
 
 Hence many of the proofs that Castillon 
 offers for various logical doctrines must 
 be regarded as resting on a false founda- 
 tion. Take, for instance, his method of 
 representing the proof of the converti- 
 bility of particulars affirmative and real. 
 The form for this particular is, as we 
 have seen, A = S-M. Now, he says, 
 since this proposition implies the uni- 
 versal S = A + M — which, however, is not 
 the case, as I shall show presently — and 
 the latter implies the subaltern S = A + M, 
 we reach the desired demonstration. My 
 criticism of this is that he has proceeded 
 from a proposition that is true to one 
 that is false, and that therefore the desired 
 result has not been established. 
 
 After having given this question of 
 
On a Calculus Based on Intension 115 
 
 Castillon's particulars a good deal of con- 
 sideration, I thought that perhaps illu- 
 sories correspond to propositions in which 
 in class Logic the " some " means " some, 
 it may be all," while in real particulars 
 the "some" means "some only." And 
 had this been the correct view of the 
 case there would also have been no possi- 
 bility of making use of illusory parti- 
 culars, for the same system cannot be 
 worked out where there is ambiguity 
 about the meaning of " some " : as Venn 
 would say, the fourfold scheme cannot be 
 made to correspond with the fivefold 
 scheme. But further reflexion upon the 
 matter has led me to dismiss the illusory 
 particular on other grounds, namely, those 
 given above. That " some " in the parti- 
 cular real denotes " some only " there is 
 no doubt whatever. If " all " in it might 
 be substituted for " some," then the con- 
 verse of S = A + M might be A = S + M, 
 which obviously involves a falsity, since, 
 if the components of S are A and M, it 
 
ii6 Symbolic Logic 
 
 is absurd to conclude that A only can be 
 composed of the attributes denoted by S 
 together with those denoted by M. Such 
 being the case, we have not a converse 
 similar to that which is found in ordinary 
 Logic, where, of course, " some " means 
 " some, it may be all." Castillon does not 
 notice this, but thinks that he has per- 
 formed the operation that in common 
 Logic is known as accidental conversion. 
 He could arrive at a converse state- 
 ment concerning " all " — supposing for the 
 moment that the symbol ( = ) would then 
 have a meaning — if M were equal to 
 nothing. But he does not give the 
 slightest hint that he had contemplated 
 this possibility. Always in speaking of 
 M as an indeterminate number of attri- 
 butes, he means that there actually are 
 attributes denoted by this letter. And, as 
 these points are clear enough, it is, I 
 think, equally clear that the illusory par- 
 ticular makes no statement about " some, 
 it may be all " : this proposition is incom- 
 
On a Calculus Based on Intension 117 
 
 patible with the universal, with which 
 Castillon's Logic can ill afford to dis- 
 pense. 
 
 Castillon certainly has a very ingenious 
 argument by which he attempts to show 
 that subalterns should be allowed a place 
 in his calculus. He takes the description 
 of an object to be symbolized thus : 
 = A+B + C + D. Then he says that from 
 this we may conclude that 0-A = B + C + D. 
 Here O-A will denote the species under 
 which the object O is comprised. This 
 is quite reasonable. But he then proceeds 
 to say that since B+C + D are marks of the 
 species, much more will they be the marks 
 of the object O. Hence, if a universal is 
 true, so is the subaltern. The error into 
 which he here falls is obvious. There 
 is no more reason that B + C + D should 
 be among the attributes that are equated 
 to O than that these attributes should be 
 equated to O-A. When it is said that 
 O- A = B + C + D, all that is meant is that 
 if A be taken from the congeries O the 
 
ii8 Symbolic Logic 
 
 remainder will be the right-hand member. 
 If then the attribute A is put back again, 
 the three attributes will be no more 
 characteristic of O than they were of 
 O — A. To argue on h fortiori lines, as 
 does Castillon, is to introduce other than 
 formal considerations. 
 
 It may be noted in passing that on the 
 assumption that in intensive Logic we have 
 only universals and real particulars, and that 
 the latter may be converted in the way 
 Castillon lays down, two important results 
 will follow with respect to arguments in- 
 volving three or more terms. The first 
 result will be that we shall be able to in- 
 clude inferences that have nothing cor- 
 responding to them in the ordinary class 
 Logic, and the second will be that many 
 arguments, which in the latter find a place, 
 can no longer be regarded as valid. As an 
 instance of one of the inferences that would 
 be admitted, take that which is quoted 
 by Venn, and which was exhibited by 
 Castillon in a previous memoir : " some A 
 
On a Calculus Based on Intension 119 
 
 is B, some B is C, therefore some A is C." 
 This, it will be observed, is given in the 
 language of class Logic, but is not, of course, 
 valid in that Logic. According to Castillon, 
 however, the conclusion is quite justifiable, 
 for each of the three propositions may be 
 expressed by means of conversion as a 
 universal, and then we have "B = A + M, 
 C = B + P, .-. C = A + M + P." As an in- 
 stance of a demonstration that Castillon be- 
 lieves may be admitted, but which must be 
 rejected, take the following: "M = A + N, 
 M = S + P (or S = M-P), .-. S = A + N-P : 
 />., making +N-P=+Q, we conclude 
 S = A + Q." The illusory particulars here 
 being inadmissible, this form of argument 
 must be regarded as untenable. 
 
 In drawing inferences involving illusory 
 particulars, Castillon, it may be remarked, 
 is sometimes led by the associations of 
 quantitative mathematics into further error. 
 For instance, he does not hesitate to change 
 the sign + into ±. It is true he does not 
 mean anything by the change : in present- 
 
I20 Symbolic Logic 
 
 ing the sixth mood of the third figure, he 
 says that the subaltern of M = S + P is 
 M = S±P, so that he clearly makes no dis- 
 tinction between this and M = S + P. But 
 he ought not to make such a change, for 
 what his reversed symbols must actually 
 be taken to mean, when strictly interpreted, 
 is that he is using a universal proposi- 
 tion where he is warranted in using a 
 particular, a course that is obviously un- 
 justifiable. 
 
 I have attempted to prove that Castillon's 
 illusory particular must be entirely rejected, 
 for it cannot be retained without involving 
 the logician in self-contradiction. But it is 
 necessary here to observe that Dr. Venn 
 has expressed the opinion that Castillon 
 did actually reject these propositions. Dr. 
 Venn, soon after the publication of the first 
 edition of his Symbolic Logic^ was able to 
 obtain a copy of Castillon's memoir, and 
 sent a short account of the system to Mind,^ 
 The account is substantially the same as 
 
 * Vol. vi. p. 448. 
 
On a Calculus Based on Intension 121 
 
 that given in Venn's second edition, but 
 contains a statement to the effect that in 
 the memoir particulars are divided into two 
 kinds, one of v^hich Castillon " rejects " as 
 " illusoires." But in w^hat sense can such 
 rejection be said to have been made ? 
 lUusories are used in all arguments, just as 
 much as are particulars real. When the 
 demonstration of the validity of the con- 
 version of real particulars is offered, no 
 hesitation is felt in making use, as we have 
 seen, of illusories ; and constantly in the 
 proofs of the syllogistic moods illusory 
 particulars occupy an important place. In 
 several cases we are told that precisely the 
 same results are reached if for an illusory 
 a real is substituted. I think it must be 
 concluded that Castillon admitted both kinds 
 of particulars. The illusories were not re- 
 jected, as Venn affirms they were, and, as I 
 have attempted to show, they ought to have 
 been. They were certainly regarded as 
 never making the best of themselves ; but 
 the fact that they might have developed 
 
122 Symbolic Logic 
 
 into universals and did not do so was not 
 considered a reason why they should not be 
 employed in logical proofs, when it was 
 found convenient to resort to them. 
 
 (3) Inconvertibility of Real Particulars 
 and of Universal Negatives, — I said a few 
 pages back that Castillon is not justified 
 in drawing an inference from A = S - M 
 to S = A + M.^ Of course he is quite 
 justified, as we have seen, in proceed- 
 ing from the latter to the former, but if 
 he commences with the former he may not 
 proceed to the latter. To attempt to draw 
 this second inference is equivalent to the 
 attempt to get an A proposition by the 
 conversion of an I proposition. Had 
 Castillon been arguing in class language he 
 would never, of course, have attempted to 
 draw such a conclusion, but, as when he 
 started with the intensive representation of 
 a universal affirmative he obtained the form 
 for the particular by taking over the letters 
 
 * I am indebted to Mr. Johnson for directing my attention 
 to this important point. 
 
On a Calculus Based on Intension 123 
 
 with changed sign, he thought he might 
 start with the form for the particular 
 affirmative, and then transfer letters with 
 changed signs. We shall see in the next 
 paragraph that he is not justified in per- 
 forming the process corresponding to con- 
 version of the universal negative, so that 
 not in general, but only in a special in- 
 stance, is he justified in changing signs and 
 transferring terms to the other side of the 
 sign of equality. 
 
 Castillon's universal negative need not 
 detain us long. The main thing to notice 
 in addition to what has already been men- 
 tioned is the reason for its inconvertibility. 
 The proposition is symbolized thus : — 
 S = — A + M, which means, as we have 
 explained, that the attributes denoted by 
 S are to be separated from those denoted 
 by A, but are to be regarded as co-existing 
 with those denoted by M. Now, such 
 being the case, it is quite unwarrantable to 
 conclude that the attributes denoted by A 
 may consist of those denoted by M. But 
 
124 Symbolic Logic 
 
 this would be asserted if we were to admit 
 the converse A = - S + M, a procedure that 
 Castillon considers to be valid. 
 
 From the above discussions it will be 
 seen that Castillon is not justified in making 
 use in his calculus of more than the follow- 
 ing : universal affirmatives, real particulars 
 (in which "some" means "some only"), 
 and universal negatives, of which the 
 second and third are inconvertible. 
 
 (4) Castillon s Treatment of Hypothetical 
 and of Problematical Judgments, — It is 
 necessary finally to consider two questions 
 with which the logician arguing on in- 
 tensive lines, as much as he who proceeds 
 from the point of view of the class or of 
 the proposition, is concerned, namely, the 
 questions as to the relation of hypothetical 
 and problematical judgments to categorical, 
 and the possibility of there really being any 
 quantitative element involved in a pro- 
 position. 
 
 To commence with the former subject. 
 Venn remarks that it need hardly be said 
 
On a Calculus Based on Intension 125 
 
 that the distinction between hypothetical 
 and categorical is, on the intensive view, 
 rejected. And this is undoubtedly Cas- 
 tillon's view of the case. But I should like 
 to point out that, though it is attempted 
 in this way to get rid of the purely hypo- 
 thetical element, Castillon evidently feels 
 that there is something wrong in the pro- 
 cedure, and it is interesting to watch the 
 device that he adopts in order to escape 
 from the difficulty. " In the case of the 
 hypothetical judgment it is evidently 
 necessary that the intelligence has the per- 
 ception that, if the attribute A belongs to 
 the subject, the attribute B also belongs to 
 it, and the intelligence can only have this 
 perception in so far as it perceives that the 
 concept of the subject comprehends, or can 
 comprehend^ that of the attribute A, and the 
 latter the concept of the attribute B." I 
 have italicized the words by means of which 
 this conceptualist logician escapes from 
 the difficulty of the situation. His intro- 
 duction of these few words shows that he 
 
126 Symbolic Logic 
 
 recognises that in a hypothetical what we 
 really have is, as Venn would say, a known 
 conjunction of two phenomena, but we are 
 not sure whether there is an instance of the 
 pair before us. 
 
 In proceeding to consider the facts re- 
 lating to problematical judgments, we can- 
 not but be struck by the circumstance that, 
 though Castillon had not arrived at an 
 adequate account of these judgments, his 
 efforts were being made in the right direc- 
 tion. Maimon had been willing to accept 
 the problematical judgment, but Castillon 
 holds that this should not be considered as 
 part of the material that is manipulated by 
 the logician,^ for the so-called judgment 
 has reference to the state of the mind 
 previous to the formation of a judgment, 
 to the preliminary indecision as to whether 
 an attribute does or does not belong to a 
 subject. Castillon was here quite faithful 
 to his principles. For he could, strictly 
 speaking, only make a categorical statement 
 
 * He means, of course, the person who treats of "pure" Logic. 
 
On a Calculus Based on Intension 127 
 
 concerning the subject and its attributes, 
 and if feeling prevented his doing this there 
 would be, of course, no proposition. There 
 was with him no alternative but to form 
 a certain subject-predicate combination and 
 to refuse to form it. Where he failed to 
 come up to modern thought upon the 
 subject was in not perceiving that, besides 
 having an implication, we may have a 
 statement as to the relation in which the 
 thinker stands to that implication. The 
 statement of this relation would supply- 
 material upon which the pure logician could 
 work. Sometimes the facts occasioning the 
 thinker's mental attitude may be such that 
 they admit of being stated in quantitative 
 terms, sometimes such that they can be 
 stated only by means of such vague terms as 
 " probable," " possible." When numerical 
 elements enter, the statement must be handed 
 over to the mathematician, but, when non- 
 quantitative terms are retained, the pro- 
 position can be dealt with in Logic, where it 
 will be necessary to symbolize the relation 
 
128 Symbolic Logic 
 
 in which the thinker stands to the impli- 
 cation with which he is confronted. 
 
 (5) Derrcation of the Notion of ^lantity. — 
 We now come to what is perhaps the 
 most striking of the proposed changes in 
 logical doctrine that are involved in the 
 course of treating the proposition from 
 the intensive standpoint. In all ordinary 
 presentations of Logic the division of pro- 
 positions into universal and particular is 
 regarded as obviously justifiable. And the 
 same division is made in the common nota- 
 tions with which the svmbolist is familiar. 
 It is true that, with this classification, the 
 singular proposition gives rise to some dis- 
 cussion, but the symbolist does not hesitate 
 to regard such proposition as a special case 
 of the universal, i.e., the class may shrink 
 down to an individual. Further, the classi- 
 fication is not invalidated when, in dealing 
 with multiply-quantified propositions, a dis- 
 tinction has to be made between "some or 
 other" and "a certain some." This limi- 
 tation and this subdivision, in fact, only 
 
On a Calculus Based on Intension 129 
 
 bring out more prominently the apparently- 
 indispensable character of the distinction 
 between the universal and the particular. 
 It is, therefore, strange to hear Castillon 
 denying that propositions can be divided 
 up in this way. The equation S = A + M 
 carries with it, he says, no information 
 either of universality or of particularity : 
 what we have is merely a statement that 
 the subject S comprehends the attributes A 
 together with those denoted by M. In 
 order to join to this statement the idea 
 of universality or of particularity some 
 other act is required than this act of syn- 
 thesis. And of course he would argue that 
 the converse A = S — M has similarly no 
 notion of quantity attaching to it. 
 
 It must be confessed that, in arguing in 
 this manner, Castillon is proceeding in 
 accordance with his principles. But, when 
 he comes to justify his opinion that the 
 notion of quantity is derived from a syllo- 
 gistic process, it is at once seen that his 
 reasoning involves the very quantitative 
 
1^0 Symbolic Logic 
 
 element, the origin of the idea of which 
 he wishes to explain. He agrees — with 
 certain reservations — to the syllogism by 
 which Maimon endeavours to make the 
 desired deduction. The notion of univer- 
 sality, for instance, is derived from such 
 an argument as the following : the con- 
 cept " man " comprehends the attribute 
 " animal," the representation of such and 
 such an individual, say Caius, Titius, &c., 
 comprehends the concept "man," hence the 
 representation of any individual whatever 
 comprehends the concept " animal '' ; i.e., 
 all men are animals. Castillon objects to 
 regard this reasoning as containing no 
 notion of quantity, for he considers that 
 at least the second premise contains the 
 notion of unity. He is inclined, moreover, 
 to believe that every judgment carries with 
 it the notion of " one " and " two," since 
 there is supposed the concept of the subject 
 and the concept of the attribute. But 
 with this qualification of the statement of 
 Maimon, who had maintained that the 
 
On a Calculus Based on Intension 131 
 
 above reasoning contains no notion what- 
 ever of quantity, Castillon agrees that it is 
 possible thus to arrive at the conception 
 of universality. But it is obvious, when 
 we consider his argument, that it involves 
 a petitio principii^ for, unless every attribute 
 of B were among those of A, wc could 
 not infer that the C attributes, which are 
 some of the B attributes, are among those 
 of A. Directly a three-term argument of 
 this description is analyzed we see that 
 there is no longer any rigid rejection of 
 the notions of universality and particu- 
 larity. Even when we perform such a 
 simple process as that of conversion it is 
 obvious that the notion of a part is im- 
 plied. Indeed, Castillon says as much : — 
 " S= A + M gives A = S - M, which in- 
 dicates that A does not comprehend S, 
 but a part of S, that which remains when 
 from S one abstracts M." 
 
 (6) Comparison of Castillon s Symbolism 
 with Mrs, Bryant' s. — Before proceeding to 
 more general considerations, I may here call 
 
132 Symbolic Logic 
 
 attention to the points of correspondence 
 between Castillon's symbolism and that 
 briefly sketched by Mrs. Bryant.* For 
 the A proposition, when read intensively, 
 Mrs. Bryant gives ^--a-^+Z^^^-^, and 
 hence a-f (go -/S) = 00 , which may be in- 
 terpreted the act of comprehension pre- 
 dicates a and not-/3 of a class, and thus 
 reaches the same result as if oo were pre- 
 dicated of the class. That is to say, 
 the class ab does not exist. Castillon's 
 form is S = A + M. This is evidently the 
 same as a+(GO ~/S) = go, for the latter may 
 be written thus: a = /3+ao~GO, where, as 
 Mrs. Bryant shows, the oo - 00 is either 
 zero or positive ; but Castillon, as we have 
 pointed out, regards only the positive value. 
 Mrs. Bryant's form for the universal negative 
 is ^-°-/^ = 6'~°°, and this when written 
 a=-^ + ao is also clearly the same as 
 S=-A + M, except that I do not think, 
 for the reasons I have mentioned, that 
 Castillon's M did or could mean an infinite 
 
 * Loc. dt.f p. 130. 
 
On a Calculus Based on Intension 133 
 
 number of attributes : the M is only 
 indefinite. As Castillon, when critically 
 examined, has nothing corresponding to 
 the I and O of the ordinary Logic, it is not 
 possible to institute a comparison between 
 the two symbolisms as regards particular 
 propositions. So far as one can tell from 
 the matter of her paper, and from the fact 
 that she quotes the first edition of Venn's 
 Symbolic Logic, Mrs. Bryant, when writing, 
 was not aware of the work done by 
 Castillon ; but, in any case, her treatment of 
 the subject is original. To the extent, how- 
 ever, that there is identity between the con- 
 clusions of the two logicians, it is of course 
 impossible — supposing my criticisms of 
 Castillon to be valid — to proceed on the 
 lines suggested by Mrs. Bryant. 
 
 II. We have now examined somewhat 
 fully the principles underlying Castillon's 
 procedure, and the investigation shows that 
 a calculus is not workable on the plan he 
 unfolded. He proposed a system where 
 
134 Symbolic Logic 
 
 the notions of universality and particularity 
 could be reached only by means of a petitio 
 principii^ where universal negatives could not 
 be converted, v^here on conversion of a 
 particular affirmative v^e reach a universal 
 affirmative, where " some " is inaccurately 
 employed, and where it is not possible 
 consistently to deal with hypotheticals. 
 Such a system is certainly one that cannot 
 lay claim to general acceptance. 
 
 (i) A Logic based on Connotation, — But 
 it will be well to look for a moment at 
 the question of an intensive Logic without 
 reference to Castillon's work. My view on 
 the general question is in close agreement 
 with that which Venn reached in his 
 chapter on this subject.'^ That is to say, 
 where the attributes are taken that are de- 
 noted by the name, the available stock of 
 propositions is too limited to make it worth 
 while to attempt to elaborate a calculus. 
 The only proposition in the ordinary Logic 
 that would find a counterpart in this inten- 
 
 ♦ Symbolic Lo^'tc, p. 453, 
 
On a Calculus Based on Intension 135 
 
 sive scheme is the universal affirmative. 
 To represent the universal negative it v^ould 
 not do to refer to two different groups of 
 attributes, and to represent the particular 
 propositions it would not do to have two 
 groups of attributes, of which some were 
 found in both groups. 
 
 This is the meaning of intension that 
 Couturat has in mind when he holds that 
 Symbolic Logic can only be built up 
 from the standpoint of extension, and Mr. 
 Russell adopts the same meaning when 
 he asserts that though, if we must choose 
 either pure intension or pure extension as 
 a starting-point, Couturat's view is correct, 
 we may commence by assuming an inter- 
 mediate position, and that this course is 
 necessary if we wish to avoid self-contra- 
 diction, and if we wish to deal with infinite 
 classes.^ That it is this meaning of inten- 
 sion which these logicians tacitly adopt is 
 made clear from considering Couturat's 
 statement that an examination of the 
 
 '^ The Principles of Mathematics^ p. 66, 
 
136 Symbolic Logic 
 
 system of Leibnitz " proves that algorith- 
 mic Logic — t.e.^ exact and rigorous Logic 
 — cannot be founded on the confused and 
 vague ^ consideration of comprehension ; it 
 has only succeeded in being constituted 
 by Boole because he made it rest on the 
 exclusive consideration of extension/' f 
 
 Now intension can be confused and 
 vague only when it is said to embrace 
 the conventionally fixed attributes, | and not 
 when the totality of the attributes is held 
 to constitute the intension. In the case 
 of some names there may be always un- 
 certainty what is the conventionally fixed 
 number of attributes, and in the case of 
 others, even though at one period the 
 intension is well known, there will gene- 
 
 * Italics mine. 
 
 •j" L,a Log'tque de Leibnitz^ p. 387. 
 
 :{: These two adjectives might indeed be applied to that 
 conception of intension which includes the attributes " that are 
 mentally associated with a name, whether or not they are 
 actually impHed by it," i.e.., to the conception of intension as 
 Keynes uses this word ; but that Leibnitz — and so Couturat — 
 was not thinking of such attributes is clear, I think, from the 
 example which is quoted immediately in the text. 
 
On a Calculus Based on Intension 137 
 
 rally soon be uncertainty, owing to the 
 changes that scientific researches effect on 
 popular thought ; but the totality of the 
 attributes, positive and negative, known and 
 unknown, is obviously a fixed quantity. 
 
 Another way of showing that Couturat 
 has in mind the view of intension here 
 ascribed to him is to consider the ex- 
 pressions of Leibnitz, for we shall thus see 
 what precisely it is that Couturat thinks 
 cannot be made the basis of a generalised 
 Logic. Now Leibnitz says if A repre- 
 sents " triangle," and B represents " equi- 
 lateral," then A + B represents the concept 
 " equilateral triangle," * />., the attributes 
 denoted by this term are conceived of as 
 two in number, viz., triangularity and 
 equilateralness. But this is the conven- 
 tional idea of an equilateral triangle : the 
 totality of the attributes embraces many 
 more attributes than these, e,g,, the quahty 
 of being equiangular, and the quality of 
 having each angle equal to sixty degrees. 
 
 * La Logique de Leiiniiz, p. 376. 
 
138 Symbolic Logic 
 
 Thus Couturat is thinking of the number 
 of attributes implied by the name, or of 
 the connotation in the sense that this word 
 is used by Mill and Keynes. And this 
 must be the meaning that Mr. Russell 
 adopts, since in stating that Couturat's is 
 the correct position, were there not an 
 intermediate one available, he is evidently 
 thinking of extension and intension in the 
 same sense as is Couturat.* 
 
 (2) A Logic based on Comprehension, — The 
 only way to reach an intensive Logic 
 would be, as Venn says, to take all the 
 attributes that are common to the mem- 
 bers of a class. It would then be possible 
 to draw a diagram that would be similar 
 to the one used in compartmental Logic, 
 
 * I may here remark that, though with Mr. Russell I should 
 hold that it is useful for the symbolist for the purpose of 
 defining infinite classes to retain the conception of intension, I 
 think it is somewhat misleading to say that Symbolic Logic has 
 its lair in a position intermediate between pure intension and 
 pure extension. When the calculus comes to be worked, it is 
 necessary definitely to take up either the one position or the 
 other, and, as the reasoning of this chapter shows, the extensive 
 interpretation is the appropriate one to be adopted, 
 
On a Calculus Based on Intension 139 
 
 but whose compartments would represent 
 combinations of attributes instead of groups 
 of individuals. 
 
 But, in order that such a calculus could 
 be developed, it is to be noted that certain 
 important assumptions would have to be 
 made, and that even with the help of these 
 it would not be possible to deal with all the 
 processes corresponding to those which are 
 found in class Logic. The assumptions in 
 question are that negative attributes may be 
 freely admitted, and that every combination 
 of attributes mentioned in our scheme does 
 not necessarily exist. "^ The process that 
 cannot be symbolized on intensive lines is 
 that of class subtraction. Even for addition 
 no intensive logician has suggested any 
 symbolism, but Venn has pointed out that 
 this operation can be represented by means 
 of a symbol denoting alternation, for in- 
 stance by the symbol *- . In case of sub- 
 traction no symbol could be correctly used, 
 for, though we might place a symbol 
 
 * Venn, Symbolic Logic, pp. 469-473. 
 
140 Symbolic Logic 
 
 between two groups of attributes, we 
 should not be dealing with an operation 
 that affected attributes, but with one that 
 affected the corresponding classes. I pre- 
 sume, however, that class subtraction could 
 be dealt with by means of the following de- 
 vice. Supposing that the premises are given 
 in a form that involves the subtraction of 
 classes, we could turn this expression into 
 one denoting multiplication, inasmuch as 
 x — xy^xy. Then this product could be 
 symbolized by the addition of positive and 
 negative attributes. 
 
 Now, if the rules of class Logic have 
 reference to the four processes of addition, 
 subtraction, multiplication, and division, and 
 if these processes can either directly or 
 indirectly be represented by symbols that 
 stand for attributes, it seems to me to be 
 demonstrated that the rules of class Logic 
 can be adapted to deal with the respective 
 groups of attributes. M. Couturat affirms 
 that a Logic based wholly on intension is 
 impossible, and Mr. Russell agrees with 
 
On a Calculus Based on Intension 141 
 
 him.^ But though, when the narrower 
 conception of the word intension is taken, 
 this impossibiHty may certainly be estab- 
 lished, I do not think, for the reasons 
 that I have given, that there cannot be a 
 calculus on the adoption of the wider 
 interpretation. I am here only discussing 
 the possibility of the case, not the natural- 
 ness or simplicity of such a calculus. In 
 both of these qualities such an intensive 
 Logic as the one described would be far 
 inferior to that which deals with classes in 
 a direct manner. 
 
 To sum up the results of this chapter. 
 We have shown that the most consistent 
 reasoner from the intensive standpoint was 
 led into many and serious errors, and that a 
 calculus cannot be elaborated in the way 
 that he described. Then we observed that, 
 if our attention is confined exclusively to 
 the attributes that are commonly denoted 
 by the name, we can deal only with 
 universal affirmatives, so that nothing of the 
 
 * The Principles of Mathematics^ p. 66. 
 
142 Symbolic Logic 
 
 nature of a calculus can be reached on this 
 view of intension. And, finally, we have 
 seen that, when all the attributes common 
 to the members of the class denoted by 
 the name are taken as our starting-point, 
 it would be possible to reach correct 
 conclusions, but that the process would be 
 long and artificial, when compared with the 
 one in which it is classes or propositions 
 that are symbolized. 
 
CHAPTER V 
 
 THE DOCTRINES OF JEVONS AND OF 
 MR, MacCOLL 
 
 It was explained in the Introduction that 
 the object of this work is to show that 
 during the last fifty years there has been 
 a distinct advance made in Symbolic Logic. 
 In the first three chapters we were occu- 
 pied in tracing the earlier portion of this 
 development. In the fourth chapter we 
 demonstrated that the logicians who have 
 effected the advance were justified in 
 taking an extensive view of the import of 
 the proposition. The present chapter will 
 be occupied with an examination of the 
 work of two logicians, viz.^ Jevons and 
 Mr. MacColl, who have proceeded by 
 way of extension, but who have, I think, 
 
 fallen into several serious errors. 
 
 143 
 
144 Symbolic Logic 
 
 Of these two writers the former un- 
 questionably exercised in England, at any 
 rate, a greater influence than any other 
 logician of his time, while the latter has 
 in all his work shown an ability and 
 inventiveness of a very high order. In 
 spite of these facts, however, I cannot 
 but think that Jevons and Mr. MacColl 
 have not assisted to any great extent in 
 erecting the symbolic structure that is at 
 present available. In the case of Jevons 
 the reason of this seems to be that he 
 was wanting in the power of originating 
 important logical generalisations, and that 
 he failed to appreciate the full signifi- 
 cance of the work done by other logicians. 
 The smallness in the number of Mr. Mac- 
 Coirs contributions to the creation of a useful 
 calculus is apparently due to his conviction 
 that it is impossible for him to co-operate 
 with other symbolists, since their pro- 
 cedure involves, in his opinion, many limi- 
 tations and errors. It becomes necessary, 
 therefore, for us to look at the work of 
 
The Doctrines of Jevons 145 
 
 these two logicians, and to make evident, 
 in the first place, that the reputation of 
 Jevons must not be based upon the fact 
 that he contributed in any important de- 
 gree to the creation of a Symbolic Logic, 
 and, secondly, that Mr. MacCoU's pro- 
 cesses have not the advantage that he 
 claims for them, but that they are based 
 on views that imply errors from which 
 the ordinary symbolic logician is free. 
 
 I. To begin then with Jevons. It 
 will not be necessary in his case to go 
 into very great detail, since most of his 
 deficiencies have been sufficiently examined 
 in various parts of Venn's Symbolic Logic, 
 But I have drawn up as full a statement 
 of the case as I have been able to reach. 
 
 Jevons' doctrine of the superiority of the 
 equation x = xy to represent the universal 
 affirmative is erroneous, for this form is 
 immediately reducible to x= ^y ov x:=v y. 
 It is impossible to adopt his method of 
 
 ■Fw^ft>^i:;v K 
 
 •-THE ^A 
 
 I UMIVERSITY ;/ 
 
1^6 Symbolic Logic 
 
 denoting particular propositions, for, though 
 he avoids the difficulty apparent in the 
 
 Boolian system, where ^ is taken to denote 
 complete indefiniteness, such escape is 
 effected by employing the postulate that 
 no term whatever shall be equivalent to 0. 
 This would exclude the possibility of a 
 calculus, for a collection of consistent pro- 
 positions may eventually be found to have 
 established the entire destruction of a certain 
 term.* I should agree with this criticism 
 of Venn's, but I do not think that Jevons 
 would have done so ; he would probably 
 have replied that if such collections of 
 propositions resulted in such a destruction 
 then the group was not perfectly consistent. 
 Again, we have already seen that Jevons' 
 argument against using the exclusive nota- 
 tion in Logic is not valid, though, since his 
 time, this method of dealing with alterna- 
 tives has been largely adopted : his point 
 was that we do often think in the non- 
 
 * Venn, Symbolic Logics p. 156. 
 
The Doctrines of Jevons 147 
 
 exclusive manner, but this is no reason why 
 we should do so in our symbolic reasoning. 
 He certainly drew up a table by which a 
 type of proposition may be reached for the 
 solution of the inverse problem in the case 
 of three terms, but he did not assist in 
 removing the difficulty involved in solving 
 the inverse problem in general. More- 
 over, his doctrine that Induction is to be 
 identified with this inverse method is quite 
 erroneous, for, as Mr. Johnson has most 
 perspicuously shown, the series of pro- 
 positions that Jevons desires to reach are 
 only determinants of the data — are, that 
 is to say, neither more general nor more 
 conjectural than the data. Jevons' concep- 
 tion of Boole's idea of the scope of Mathe- 
 matics was, previous to the second edition 
 of the Principles of Science^ altogether mis- 
 taken, and hence the attempts in the earlier 
 edition to "divest his (Boole's) system of 
 a mathematical dress" could not result in 
 much that is useful.^ But even in the 
 
 * G. B. Halsted, in Mindy No. 9, p. 134. 
 
148 Symbolic Logic 
 
 second edition the inaccurate notion has 
 only partially disappeared. Boole's is now a 
 quasi-mathematical system : it still requires 
 " the manipulation of mathematical symbols 
 in a very intricate and perplexing manner." 
 Jevons, in holding the view that the process 
 of subtraction is useless because the same 
 operation can be represented as one of 
 restriction, passes over the fact that each 
 may be useful at times. His objection that, 
 because he admits the Law of Unity into 
 his system, it was necessary for Boole to 
 do the same is without force, since Boole 
 was not guilty of any inconsistency in the 
 
 omission. Jevons declared that ^ cannot 
 
 be understood without reference to the 
 mathematics of quantity, an assertion which 
 is refuted from the simplest logical con- 
 siderations : the expression represents " the 
 class of which if we take ' no part ' we 
 obtain 'nothing.'" I do not profess that 
 this list is complete, but it must be 
 confessed that, though Jevons stimulated 
 
The Doctrines of Mr. MacColl 149 
 
 logical thought much more extensively 
 than most men are enabled to do, his 
 actual contributions to the development 
 of Symbolic Logic were few and re- 
 latively unimportant. His great powers 
 were, in short, less successfully occupied 
 in the logical than in the mathematical 
 realm. In pure economic theory and in 
 currency investigations, where in both cases 
 the argument is almost entirely concerning 
 quantities, his work is of the utmost value, 
 and has placed him in the very first rank 
 of thinkers upon such subjects. 
 
 II. Coming now to Mr. MacColl, I 
 wish to point out wherein I think he falls 
 into error. My object in considering his 
 work is to get at the truth on certain 
 debated questions, so that I proceed at 
 once to these. I readily admit that there 
 are several points in which Mr. MacColl — 
 and the same remark applies to Jevons — 
 agrees with the other writers to whom 
 we have had occasion in previous chapters 
 
i^o Symbolic Logic 
 
 to refer. Of course, if Mr. MacColl had 
 been the first to give prominence to these 
 points, in which there is agreement, it 
 would have been necessary for us to dwell 
 upon them here in some detail, but with 
 one exception they had been well con- 
 sidered by other symbolic logicians. 
 
 (i) Mr, Mac Co IPs Employment of Literal 
 Symbols, — The question that Mr. MacColl 
 was the first to bring to the front is that 
 respecting the use of literal symbols to 
 denote propositions rather than to denote 
 classes. In his papers published in the 
 Proceedings of the London Mathematical 
 Society^ and in his contribution to Mind 
 in 1880, he clearly showed that symbols 
 may be employed in this way.^ And, 
 inasmuch as the newer Symbolic Logic 
 regards the process of symbolizing pro- 
 
 * Mr. MacColl was not the first person to utilise symbols 
 in this manner, for sometimes letters are made to stand for 
 propositions by Boole, De Morgan, and others, but he un- 
 doubtedly gave prominence to such employment, and, more- 
 over, as stated immediately in the text, he considered that 
 symbols should always stand for propositions. 
 
The Doctrines of Mr. MacColl 151 
 
 positions as more fundamental than that 
 of symboHzing classes, we are indebted 
 to Mr. MacColl for emphasizing the less 
 usual application of symbols. 
 
 At the same time, even on this point 
 I think that Mr. MacColl goes astray. 
 In the first place, his view is that symbols 
 should be restricted to the prepositional 
 use.* But, so far as the earlier Symbolic 
 Logic is concerned, no such restriction is 
 necessary. I have already argued this 
 point. t Provided we employ the appro- 
 priate rules, it makes no difference whether 
 the problems solved by Venn, for instance, 
 are treated in the one way or in the other. 
 And, in the second place, Mr. MacColl 
 allows his symbols indiscriminately to re- 
 present propositions and prepositional func- 
 tions ; I but, in so far as he has done so, 
 he has not assisted in producing the newer 
 Symbolic Logic, for in this it is a matter 
 
 * Mind, No. 17, p. 49 ; Venn, Symbolic Logic, p. 492. 
 
 •j- pp. 10-22. 
 
 ;j: Russell, The Principles of Mathematics, pp. 12, 22. 
 
152 Symbolic Logic 
 
 of fundamental importance to draw a clear 
 line of distinction between the two uses. 
 Mr. MacCoirs view, therefore, of the 
 prepositional use of symbols is both un- 
 necessarily at variance with the older, and 
 does not fit in with the more recent 
 doctrine. 
 
 (2) Mr, MacCoirs Treatment of Modal 
 Propositions, — Mr. MacColFs two chief 
 errors consist in his treatment of modal 
 and kindred propositions, and in his doc- 
 trine of logical existence. Each of these 
 questions may now be carefully discussed.* 
 
 The subject of modals is constantly turn- 
 ing up in Mr. MacColl's writings, but 
 perhaps he has nowhere more clearly stated 
 his view as to the treatment of such pro- 
 positions than in his second and fifth papers 
 in Mind^ and to these we may give our 
 chief attention. In his second paper he 
 
 * The former was referred to in the first chapter, and my 
 general opinion upon the subject was stated, but Mr. MacCoU's 
 work claims more detailed examination than was possible in 
 that place. 
 
 t N.S. N08. 24 and 47. 
 
The Doctrines of Mr. MacColl 153 
 
 asserts that " sometimes we have data or 
 premises P which are 7iot always certain 
 or admitted to be true." But this con- 
 ception of certainty impHes a relation that 
 Mr. MacColl has not observed. It implies 
 an obligation on the part of a thinker 
 to accept the truth of an assertoric proposi- 
 tion. Pure Logic cannot deal separately 
 with these certain propositions : it can only 
 deal with the relation in which a thinker 
 stands towards the statement that is certain. 
 Similar remarks apply to the treatment of 
 propositions that Mr. MacColl classes as 
 variable or as impossible. We do not in 
 the case of these employ special rules. 
 As an instance of the way in which 
 statements described by these three terms 
 are to be dealt with, take the following : 
 " It is impossible that x is yT This would 
 appear in such a form as " A thinker who 
 can believe that ;c is ^ does not exist.'* 
 That is to say, statements that are cer- 
 tainties, impossibilities, or variables may all 
 appear in the form AB = 0. 
 
154 Symbolic Logic 
 
 It is because he has not perceived the 
 method of deahng with these statements 
 respecting probabiHties that Mr. MacColl 
 frequently falls into the mistake of speaking 
 of propositions as sometimes true and some- 
 times false. It may sometimes be the case 
 that the phenomenon a is followed by the 
 phenomenon b^ but it is not the case that 
 the proposition p is sometimes true and 
 sometimes false. Like Mrs. Ladd-Franklin 
 and Mrs. Bryant, Mr. MacColl confuses 
 events with statements. That this is so 
 is made very clear from his interpretation 
 
 of H. Here each of these letters represents 
 
 a statement, and the expression is " called 
 a causal implication, as it indicates some 
 causal connexion between « and ^S." ^ But 
 a statement cannot be the cause of another 
 statement : the term " cause " has reference 
 to two phenomena, not to two propositions 
 about phenomena. 
 
 The same considerations show why Mr. 
 
 * Mind, N.S. No. 24, p. 498. 
 
The Doctrines of Mr. MacColl 155 
 
 MacColl's use of the term " strength," 
 when appUed to propositions, is decidedly 
 inappropriate. When a implies /3, but ^ 
 does not imply a, a is said to be stronger 
 than ,8. Of course the distinction between 
 what implies and what is implied by is of 
 fundamental importance. But the differ- 
 ence is one of kind, not of strength. The 
 latter term suggests that an entity which 
 is under consideration possesses different 
 amounts of force at different times. But an 
 argument containing the so-called weaker 
 proposition would be an entirely different 
 argument from one containing the stronger. 
 The same statement cannot be said at one 
 time to reach the strength of an impossi- 
 bility, and at another to sink down to the 
 weakness of a certainty.^ 
 
 It will now be apparent what was meant 
 by saying that Mr. MacColl confuses pro- 
 positions with propositional functions. The 
 variable " a implies /3,'' which he calls a 
 proposition, is, as Mr. Russell points out, 
 
 * Minc/y N.S. No. 24, p. 499. 
 
156 Symbolic Logic 
 
 a propositional function : the statement 
 does not affirm truth or falsehood, but 
 when special values are given to the x in a 
 and P we get a proposition. Mr. MacColl 
 maintains that " a implies /5 " is true under 
 certain circumstances. The two views are, 
 therefore, radically distinct. In one it is 
 held that " a implies /3 " may on certain 
 occasions be spoken of as being true, 
 while in the other it is held that this 
 implication is neither true nor false. And, 
 since the newer Symbolic Logic proceeds 
 on the understanding that the distinction 
 between propositions and propositional 
 functions must be constantly observed, it 
 follows that Mr. MacColl cannot be said 
 to have definitely assisted in the advance 
 that has recently been made. 
 
 It cannot be said that Mr. MacColl in 
 his fifth paper has made his position on 
 this question more tenable. He there 
 institutes a comparison between his views 
 and those of other symbolists, in the course 
 of which he says : " I divide propositions 
 
The Doctrines of Mr. MacColl 157 
 
 not only into true and false, but into 
 various other classes according to the neces- 
 sities of the problem treated ; as, for ex- 
 ample, into certain^ impossible^ variable ; or 
 into known to be true^ known to be false ^ 
 neither known to be true nor known to be false ; 
 or into formal certainties^ formal impossi- 
 bilities^ formal variables (/.^., those which 
 are neither) ; or into probable^ improbable^ 
 even (i,e,^ with chance even) ; and so on 
 ad libitum^ But reflexion shows that 
 every proposition which he has in view, 
 when taken in conjunction with the fact 
 that it occurs in the respective class, gives 
 rise to a true or a false statement, since 
 what is in each case stated is the relation 
 in which a thinker stands to an asser- 
 tion, and the statement of this relation 
 is an assertoric proposition. Hence all 
 Mr. MacColl's propositions can be dealt 
 with by the rules of ordinary Symbolic 
 Logic. 
 
 Mr. MacColl thinks that other sym- 
 bolists make no difference between the 
 
158 Symbolic Logic 
 
 true and the certain, and between the false 
 and the impossible. But it is quite clear 
 that the assertorics in each pair are dif- 
 ferent. For instance, taking a true and 
 taking a certain proposition, these would 
 assume forms such as " All the angles of 
 a triangle are equal to two right angles," 
 and " A thinker is so constituted that he 
 must believe that the angles of a triangle 
 are equal to two right angles." Variable 
 propositions are not overlooked by the 
 ordinary symbolist, but he cannot accept 
 the view that they " are possible, but un- 
 certain, propositions whose chance of being 
 true is some proper fraction between 
 and 1." While there is a meaning in 
 speaking of the chance that a phenomenon 
 will occur, there is no meaning in saying 
 that the chance that a proposition is true 
 is greater or less. At least the only mean- 
 ing that such an expression could have 
 would be where an individual was known 
 to be a partial deceiver : we could then 
 of course speak of the chance that some 
 
The Doctrines of Mr. MacColl 159 
 
 assertion of his would be true. But this 
 is not what Mr. MacColl means. 
 
 From these arguments it will be clear 
 that it is not correct to say " that the 
 whole world of new ideas opened up by 
 this exponential or predicative system of 
 notation is a world with which they (ordi- 
 nary symbolists) are utterly unable to deal ; 
 the bare attempt on the part of logicians 
 would lead to a general break-up of all 
 the systems now taught, and a recasting of 
 the whole of logic on different principles." 
 I hope that I have made it evident, in the 
 first place, that it is inexpedient to speak 
 of many antagonistic systems rather than 
 of a calculus that has evolved as a result of 
 efforts in different directions, and, secondly, 
 that this calculus can deal with all the 
 statements that Mr. MacColl has in view. 
 
 Mr. MacColl attributes the non-adop- 
 tion of his doctrines to the perversity of 
 human nature in general and of profes- 
 sional logicians in particular.^ But this 
 
 * Mind, N.S. No. 47, p. 356. 
 
i6o Symbolic Logic 
 
 cannot be the true cause. One has only 
 to study the writings of such men as Venn, 
 Johnson, Keynes, and Russell, to see that 
 every really valuable logical truth is readily 
 welcomed. For instance, Venn long held 
 to the view that it is better to draw up 
 rules on the understanding that symbols 
 joined by (+) are exclusives. He thought 
 it highly important to keep to this ren- 
 dering, because it was essential for the 
 introduction of those inverse processes to 
 which he attached such great value. But 
 in spite of his preference he writes in his 
 second edition : " I shall now adopt the 
 other, or non-exclusive notation : — partly, 
 I must admit, because the voting has gone 
 this way, and in a matter of procedure 
 there are reasons for not standing out 
 against such a verdict."* Then, again, Mr. 
 Johnson readily accepts Keynes' methods 
 of solving the Inverse Problem, and with 
 one of them produces a still more effec- 
 tive way of reaching the solution. This 
 
 * Symbolic Logicy 2nd ed. p. 46. 
 
The Doctrines of Mr. MacColl i6i 
 
 improvement Keynes inserts in a subse- 
 quent edition. Once more, Boole, Venn, 
 and Schroder (in the Operation skreis) ex- 
 press their premises as terms equated to 
 zero. Then Dr. Mitchell shows that it 
 is possible, instead of equating to zero, to 
 equate to unity, and Venn, in his review 
 of the Johns Hopkins Studies^^ and in the 
 second edition of his Symbolic Logic^ adopts 
 this suggestion, while Mr. Johnson states 
 that Dr. Mitchell, by the introduction of 
 certain processes, among which comes this 
 one of taking the affirmative form of ex- 
 pressing premises, has been " enabled both 
 to simplify and to extend the range of 
 logical symbolism in a most suggestive 
 way." t If such writers reject Mr. Mac- 
 ColFs doctrine the cause must be found, I 
 think, not in their prejudices, but in its 
 untenability. 
 
 (3) Mr. Mac Col Ps Doctrine of a Universe of 
 Unrealities, — I will now examine the other 
 important point on which I think Mr. 
 
 * Mind for Oct., 1883. t Mind, N.S. No. 2, p. 241. 
 
 L 
 
1 62 Symbolic Logic 
 
 MacColl falls into error, viz,^ that respecting 
 logical existence. His views on this ques- 
 tion were fully stated in his sixth paper in 
 Mind^^ but he called further attention to the 
 subject in a note in the following number 
 of the review. t In the subsequent number 
 Mr. Russell and I gave our reasons for hold- 
 ing that the doctrine expounded in those 
 places contained fundamental errors. Mr. 
 MacColl has replied to both criticisms, and 
 in the last number of Mind\ I referred to 
 the points in this reply. The subject may, 
 therefore, be said to have been pretty fully 
 discussed. I shall here briefly state Mr. 
 MacColl's opinion, and shall then mention 
 the two arguments, quite distinct ones, by 
 which in my opinion it has been refuted. 
 
 We are told in the sixth paper that 
 " we assume our Symbolic Universe (or 
 ' Universe of Discourse ') to consist of our 
 
 * N.S. No. 53, p. 74. 
 
 + Mr. MacColl has also given a short summary of his views 
 on this subject in his recently published Symbolic Logic and Its 
 ylppllcattons^ pp. 76-78. 
 
 X Jan., 1906, p. 143. 
 
The Doctrines of Mr. MacColl 163 
 
 universe of realities, e^, e^, e^, etc., together 
 with our universe of unreaUties, 0^ 0.^, O3, 
 etc., w/ien both these enter into our argument. 
 But when our argument deals only with 
 realities^ then our Symbolic Universe S^, Sg, 
 S3, etc., and our Universe of realities, 
 e ^ e ^ e ^ etc., will be the same ; there will 
 be no universe of unrealities 0^, Og, O^, etc. 
 Similarly, our Symbolic Universe may con- 
 ceivably, but hardly ever in reality, coincide 
 with our universe of unrealities." This 
 statement very definitely represents Mr. 
 MacCoirs view on the subject : there are 
 two universes, one consisting of realities and 
 the other of unrealities, and the Symbolic 
 Universe may, according to the argument, 
 consist of either or of both. Among 
 realities will come " the man whom you see 
 in the garden " and " my uncle," when we 
 utter such a proposition as " The man whom 
 you see in the garden is my uncle " ; but, if 
 we say, " The man whom you see in the 
 garden is really a bear,'' we shall be speak- 
 ing firstly of an unreality and then of a 
 
164 Symbolic Logic 
 
 reality.* Such objects as " round squares " 
 are unrealities. 
 
 Now here, as Mr. Russell has pointed 
 outjf two quite different things are con- 
 fused, viz,^ the things that exist in a 
 philosophical sense and the things that 
 exist in a logical sense. To say that a 
 thing exists in the former sense means that 
 the thing has phenomenal existence, or 
 other existence of a philosophical character, 
 whereas " to say that A exists in a logical 
 sense means that A is a class which has at 
 least one member." The question then 
 arises how it is possible that two such 
 notions of existence should ever be con- 
 founded, and the answer is to be found by 
 considering those classes which have mem- 
 bers, and whose members do exist in the 
 philosophical sense. For instance, the class 
 horse is one which has members, and these 
 appear in the phenomenal world. But then 
 there are some classes which have members 
 
 * Mind, N.S. No. 53, p. 77. 
 I Mtndy N.S. No. 55, p. 398. 
 
The Doctrines of Mr. MacColl 165 
 
 and these do not appear in such world, e.g.^ 
 the class of numbers, or the class of mathe- 
 matical principles. The difficulty that Mr. 
 MacColl raises with regard to centaurs, 
 round squares, and so on, is solved by 
 noticing that classes of such things are 
 identical with the null-class, that is to say, 
 the class that has no members. 
 
 Having thus explained the nature and 
 origin of the confusion between the two 
 kinds of existence, Mr. Russell is able to 
 show that in the logical sense of the term 
 existence (for with the other sense the 
 logician has nothing whatever to do) the I 
 and O propositions require that there should 
 be at least one value of x for which x is S, 
 that is to say, that S should exist, whereas 
 in the case of A and E such existence is 
 not necessary. 
 
 Thus Mr. Russell's method of demonstrat- 
 ing that Mr. MacColl is involved in error 
 amounts to making the fact indisputable 
 that Mr. MacColl has identified two totally 
 different things, viz,^ philosophical and 
 
1 66 Symbolic Logic 
 
 logical reality. Another way of proving 
 that Mr. MacColl's position is untenable is 
 to show that it involves him either in self- 
 contradiction or in the necessity of making 
 unjustifiable assumptions. This was the 
 line of argument that I pursued in the dis- 
 cussion, and which I will here describe, but 
 by a somewhat different method from the 
 one previously adopted. In the first place, 
 then, it is certainly self-contradictory to 
 speak of two universes of discourse. The 
 Universe of Discourse in Symbolic Logic 
 means all the things that we are talking 
 about, and there cannot be two such groups 
 of "all." Within the Universe of Dis- 
 course there may certainly be two com- 
 partments, one of realities and the other of 
 unrealities, but this is a very different thing 
 from saying that there may be two universes. 
 The question is one of principle, not one 
 of mere words. 
 
 Next, consider the passages in which 
 Mr. MacCoU has made unjustifiable assump- 
 tions. He believes that his fundamental 
 
The Doctrines of Mr. MacColl 167 
 
 division into realities and unrealities sup- 
 plies a method of getting rid of certain 
 paradoxes that ordinary symbolists have 
 to accept. He says that, whereas these 
 thinkers are led to state " every round 
 square (a null class) is a triangle/' he can 
 say "no round square is a triangle." But 
 such a universal negative can be reached 
 only by labelling some of our compart- 
 ments real and some unreal, and to do 
 this two premises are assumed, viz.^ " no 
 round squares are real," and " all triangles 
 are real." It is surely quite apparent 
 that, having arrived at the possible com- 
 partments, which are indicated in the case 
 of two terms, we have no right without 
 further information to go over such com- 
 partments and state that some of them 
 have, and some of them have not, exist- 
 ence in a philosophical sense. 
 
 I think these arguments are quite suf- 
 ficient to show that Mr. MacColFs doc- 
 trine on the subject of existence cannot 
 be accepted. But he accuses ordinary 
 
i68 Symbolic Logic 
 
 symbolists of becoming involved in error 
 in holding their view. It is necessary, 
 therefore, finally to prove that he is mis- 
 taken in making this assertion. In his 
 note in Mind^ Mr. MacColl holds that 
 it cannot be right to say that the formula 
 0A = will apply whatever A may be. 
 For, let A stand for " existent." Then 
 we shall have " every non-existence is 
 existent," and this, he says, is absurd. 
 But there is no absurdity here. For with 
 two terms and " existent " the universe 
 of discourse is necessarily divided into 
 four compartments, namely, not-existent, 
 existent, not-0 existent, not-0 not-existent. 
 Whether the four may be expressed as 
 less than four is not a point that we need 
 here consider. Now, when we say " every 
 non-existence is existent," what happens 
 is that the first of these compartments 
 is erased. This implies no absurdity. 
 Where self-contradiction would come in 
 would be if we were to say that this or 
 
 * N.S. No. 54, p. 295, 
 
The Doctrines of Mr. MacColl 169 
 
 any other compartment was both erased 
 and occupied. So that, as Mr. MacColl 
 has not shown that ordinary symbolists 
 are guilty of self-contradiction in stating 
 that every non-existence is existent, he 
 has not proved that the formula (0A = 0) 
 cannot be accepted. 
 
 I may notice also in passing the argu- 
 ment advanced by Mr. MacColl in his 
 criticism of the ordinary employment of 
 1 and in prepositional Logic* His 
 object is to show that such usage leads 
 to absurdity. To do this he commences 
 by affirming that since 1 and denote 
 true and false propositions respectively, 
 these symbols represent two mutually ex- 
 clusive classes of propositions. Hence the 
 definition — < 1 should assert that every 
 false proposition is a true proposition, 
 which is absurd. My reply to this is 
 that it rests on a misunderstanding. For 
 1 and never do represent true and 
 false propositions, and consequently two 
 * M'lnd^ N.S. No. 47, p. 357. 
 
I/O Symbolic Logic 
 
 mutually exclusive classes of propositions. 
 The symbols denote respectively the only 
 possibility and no possibility : we do not 
 refer to a class at all. The introduction 
 here of the definition — < 1 is, therefore, 
 altogether unjustifiable. 
 
 From these considerations I think that 
 the case against Mr. MacColl on the 
 subject of existence must be said to have 
 been established. On the one hand, he 
 has been show^n to have been wanting in 
 discrimination between two totally different 
 things, and, on the other, his statements 
 have been demonstrated either to involve 
 him in self-contradiction or to rest on 
 unjustifiable assumptions. Also the charge 
 that he brings against ordinary symbolists 
 of unwarrantably generalising is seen to be 
 without foundation. 
 
 But before leaving this subject I think 
 
 it desirable to clear up a point on which 
 
 the reader of Mr. MacCoU's reply to 
 
 Mr. Russell * may still feel uncertain. 
 
 * Mind, N.S. No. 55, p. 401. 
 
The Doctrines of Mr. MacColl 171 
 
 Mr. MacColl says : " that the word existence, 
 like many others, has various meanings 
 is quite true ; but I cannot admit that 
 any of these ' lies wholly outside Symbolic 
 Logic' Symbolic Logic has a right to 
 occupy itself with any question whatever 
 on which it can throw any light." It 
 would thus appear that Mr. Russell's 
 symbolism cannot deal with certain prob- 
 lems with which a calculus may be 
 expected to deal. But we may be quite 
 sure that when Mr. Russell said that 
 some meanings of existence lie "wholly 
 outside Symbolic Logic" he did not mean 
 that the logician cannot manipulate pro- 
 positions that give information respecting 
 the various kinds of existence. What 
 was meant was that Symbolic Logic, in 
 occupying itself " with any question what- 
 ever on which it can throw any light," — 
 questions of existence among others — does 
 not adopt any special meaning of existence 
 that may be found in Philosophy. 
 
CHAPTER VI 
 
 LATER LOGICAL DOCTRINES 
 
 In the present chapter I propose to deal 
 with the following topics : (i) the doctrine 
 of multiple quantification, (2) the impos- 
 sibility of establishing a Logic of Relatives 
 in the sense of a generalised treatment of 
 copulas, and (3) the new Symbolic Logic, 
 the ideal of whose exponents is to demon- 
 strate that there exists a logical calculus 
 which is capable of dealing with any prob- 
 lems whatsoever of a deductive character. 
 When these topics have been unfolded 
 the arguments contained in this book 
 will have been brought to a conclusion, 
 t\e,, we shall have demonstrated that 
 there has been a real advance from the 
 year 1854 to the year 1903, when 77/^ 
 
 Laws of lUiought and Hhe Principles of 
 
 172 
 
Later Logical Doctrines 173 
 
 Mathematics were respectively published. 
 The elucidation of the first and third topic 
 is of obvious importance for our purpose. 
 The discussion of the second is of indirect 
 assistance : v^e shall show that modern 
 logicians have been justified in maintaining 
 that " no Formal Logic really treats of 
 Relatives in general qua Relatives." ^ 
 
 L First of all, then, we will refer to 
 the doctrine of multiple quantification. This 
 question was taken up by Mr. Peirce in the 
 American Journal of Mathematics 'SiXii. in the 
 Johns Hopkins Studies in Logic, by Dr. 
 Mitchell in the latter work, and more 
 recently by Mr. Johnson in Mind, The idea 
 of multiple quantifications, as Mr. Johnson 
 shows, naturally follows from starting our 
 logical investigations with the consideration 
 of singular or molecular propositions. We 
 may, that is to say, synthesize two of these, 
 and get such a simple statement as that 
 " Socrates is mortal and Greek ? " Here the 
 
 * M'md, N.S, No. I, p. 26. 
 
174 Symbolic Logic 
 
 subjects of the synthesized propositions are 
 the same, and the predicates are different. 
 Or we may have a set of such singular 
 propositions with the same predicate and 
 different subjects. These yield the proposi- 
 tions " Every S is / " and " Some S is />," 
 according as the synthesis of the singular 
 propositions is of a determinative or of 
 an alternative description. Then, as it is 
 possible to have a determinative or an 
 alternative synthesis of two universals or of 
 two particulars, or to have either of these 
 kinds of synthesis when one proposition is 
 universal and one particular, there will be 
 presented for consideration a total of six cases. 
 Three of these, viz,^ the determinative syn- 
 theses, cover the ground of syllogistic reason- 
 ing. Finally, instead of having to synthesize 
 propositions with one aggregate of subjects, 
 it is possible that we may have to deal with 
 two or with more than two aggregates. It 
 is in the last case that the proposition is 
 said to contain multiple quantifications. 
 One of the problems here met with. 
 
Later Logical Doctrines 175 
 
 which Mr. Johnson has shown how to solve, 
 concerns the method of synthesizing these 
 multiply-quantified propositions. Another 
 problem is where we are given such a syn- 
 thesis and have to find the least determinate 
 alternant that implies the given synthesis, 
 or the most determinate determinant that 
 the synthesis implies. He draws attention 
 to the fact that in the solution of the first 
 problem the important point to remember 
 is that " the external quantification must 
 be regarded primarily as quantified subject, 
 and all that is internal to it as the predica- 
 tion for that subject." ^ For instance, the 
 synthesis of " All nis love some it may be 
 different ns " and " All rns serve all ns^' is 
 "All ms (love some it may be different n s and 
 serve all n s)^' and this reduces to " All nis 
 (love and serve some it may be different ns 
 and serve all ns)r Here the "All ms^' 
 which is the external quantification, is the 
 subject, and what is internal to it, />., what 
 is in the bracket, is the predicate. With 
 * Mind, N.S. No. 3, p. 353. 
 
176 Symbolic Logic 
 
 regard to the second problem, in selecting 
 a determinant from a synthesis of multiply- 
 quantified propositions the expression must 
 first be stated as a series of prepositional 
 alternants. Then each of these may be 
 synthesized into a single proposition. In 
 effecting this synthesis we must make the 
 particular quantifications as far as possible 
 external to the universal, the reason being 
 that we want to get as determinate a deter- 
 minant as we can, and it is a principle that 
 internal synthesis has potency over external. 
 Then, finally, we must reject unnecessary 
 symbols, and make our selection from the 
 resulting determinants. From this rule the 
 rule for obtaining the least determinate 
 alternant may be found if we interchange 
 the words determinative and alternative, and 
 the words universal and particular.^ 
 
 It will be seen that here we are dealing 
 
 with a subject upon which Boole and Venn 
 
 give no hints. So that Peirce, Mitchell, 
 
 and Johnson, in unfolding the doctrine, have 
 
 * MW, N.S. No. 3, pp. 347, 356, 357. 
 
Later Logical Doctrines 177 
 
 made distinct contributions to the Logical 
 Calculus. But it is to be noted that, though 
 these writers have all treated this subject, 
 Mr. Johnson shows an advance upon Mr. 
 Peirce and Dr. Mitchell in the following 
 respects. Mr. Peirce did not make it at all 
 plain that this doctrine of multiply-quanti- 
 fied propositions is a natural continuation of 
 the doctrine of singly-quantified propositions. 
 He first of all worked out the theory of 
 relative addition and relative multiplication, 
 and showed how double quantifications may 
 be dealt with by means of these processes. 
 Then, when he came to deal with cases 
 where " relative and non-relative operations 
 occur together," and with those involving 
 plural relations, i.e., relations subsisting 
 between three or more objects, he argued 
 from the point of view of the singular 
 proposition."^ According to his treatment, 
 therefore, it would appear that there is no 
 unity running throughout the Logical Cal- 
 
 * Mind,'^.S. No. 2, pp. 249, 250 ; Johns Hopkins Studies 
 in l^ogic, p. :ioo. 
 
 M 
 
178 Symbolic Logic 
 
 cuius : Boolian principles apply to singly- 
 quantified propositions, and to cases of 
 multiply-quantified propositions, but double 
 quantifications are dealt with on different 
 principles. Mr. Johnson, on the other 
 hand, has shown that the Boolian principles 
 are applicable throughout the whole treat- 
 ment of the three kinds of propositions. 
 Mr. Peirce's relative addition and relative 
 multiplication may be expressed in the form 
 of ordinary Boolian addition and multi- 
 plication. For instance, " x loves some 
 benefactor of y " may be read " for some z 
 that z is loved by x and is a benefactor of 
 jy,'* while " X loves all but the benefactors 
 of y '' may be read " for every z that z is 
 loved by a: or is a benefactor of ^." It was 
 clearly an advance when Mr. Johnson 
 showed that, since nothing is involved in 
 the Calculus but pure synthesis and pure 
 negation, there is no need for treating in 
 any exceptional manner the cases of double 
 quantification. 
 
 As regards Dr. Mitchell, whose treat- 
 
 ^, 
 
Later Logical Doctrines 179 
 
 ment of doubly-quantified propositions is 
 admirable, and of whom it has justly 
 been said that his work contains " the 
 most important simplification of the Boolian 
 Logic that has appeared," * the one fault 
 to be found is that he seems always to 
 consider that time is the secondary dif- 
 ferentiating mark.f He says : " Let \J 
 stand for the universe of class terms, as 
 before, and let V represent the universe 
 of time," I and in his examples V in each 
 case has this signification. But, as Mr. 
 Johnson holds, there is no need whatever 
 to limit the reference in this way. We 
 can, for instance, deal with such proposi- 
 tions as "every x loves every ^," just as 
 much as we can with such propositions 
 as " all the Browns were ill during every 
 part of the year." Mr. Johnson in un- 
 folding his own doctrine avoids this 
 restriction, and treats the subject in a 
 
 * Mind, N.S. No. 2, p. 240. 
 
 t Mind, N.S. No. 2, p. 247 n, 
 
 \ Johns Hopkins Studies in Logic, p. 87. 
 
i8o Symbolic Logic 
 
 perfectly general way. The Calculus as 
 presented by him is thus of a more useful 
 character than that for which we are in- 
 debted to Dr. Mitchell. 
 
 During the discussion that has occupied 
 this section I have not referred to the work 
 of De Morgan. But he was undoubtedly 
 the first to deal with the subject that we 
 have been considering,^ and it will be pos- 
 sible now to see how far he had advanced. f 
 
 De Morgan was firmly of opinion that 
 a generalised Logic ought to consider 
 all the formal laws of relation, and that 
 syllogism is to be considered as one par- 
 ticular form of relation. Having asserted 
 this conviction, he proceeded to deal with 
 
 * See the Trans, of the Camh. Ph'ilosoph. Soc. vol. x. p. 331. 
 
 f That the work of pioneering was not easy may be inferred 
 from his assertion : " I have had to work my way through trans- 
 formations as new to my own mind, as far as the separation of 
 form is concerned, as the common moods of syllogism to the 
 beginner. If there be any person who can see at a glance, and 
 with justifiable confidence, what classes of men, including 
 women, are specified in * the non-ancestors of all non-descend- 
 ants of z,' I should not like to submit to his criticism the con- 
 fusions and blunders through which I arrived at the following 
 results." [Loc. cit. p. 334.) 
 
Later Logical Doctrines i8i 
 
 syllogism from the point of view of the 
 relational proposition. First he drew up 
 a table which shows what are the converse, 
 the contrary, the converse of the contrary, 
 and the contrary of the converse, of the 
 propositions that express such compounded 
 relations as are involved in the conclusion 
 of a syllogism. He deals with this question 
 of conversion and contrariety in quite the 
 modern way. Then he affirms that the 
 " supreme law of syllogism of three terms, 
 the law which governs every possible case, 
 and to which every variety of expression 
 must be brought before inference can be 
 made, is this : any relation of X to Y 
 compounded with any relation of Y to Z 
 gives a relation of X to Z," or that " the 
 universal and all-containing form of syllo- 
 gism is seen in the statement of X . . LMZ 
 is the necessary consequence of X . . LY 
 and Y . . MZ.'' ^ When this idea of 
 relation is brought to bear on syllogism, 
 he shows that Figure is important, " but 
 
 * Loc, cit. p. 347. 
 
1 82 Symbolic Logic 
 
 not as connected with the place of the 
 middle term. Whether we say X . . LY 
 or LY . . X, the figure is the same. 
 Change of figure can be effected only by 
 conversion of relation." Having explained 
 this, he exhibits in tabular form the 
 conclusions that may be reached with two 
 premises expressing relations. Here each 
 figure has four " phases/' determined by the 
 quality only of the premises. And, finally, 
 he shows how the ordinary syllogism with 
 quantified subject is resolvable into the 
 simple relational propositions that he has 
 been discussing. 
 
 Thus it will be seen that De Morgan 
 had comprehended the all-important char- 
 acter of singular propositions and of their 
 synthesis, for the propositions that he 
 combines in his second table are really 
 the molecular propositions upon which Mr. 
 Johnson lays so much emphasis. And De 
 Morgan, having shown how two of these 
 may be synthesized, lays it down, though 
 not in so many words, that syntheses of 
 
Later Logical Doctrines 183 
 
 these propositions, when consisting of dif- 
 ferent subjects and the same predicate, 
 yield the quantified propositions with which 
 ordinary Formal Logic is concerned. But 
 he did not get any further than this. That 
 is to say, he did not show that the same 
 principles will explain how inferences may 
 be obtained from propositions involving 
 double or multiple quantifications. This 
 subject was first treated by Mr. Peirce and 
 Dr. Mitchell, but in different ways, and 
 has been presented in a complete form 
 by Mr. Johnson, who, in the course of 
 demonstrating that inferences of the kind 
 in question rest upon the principles which 
 we have shown De Morgan was able up 
 to a certain point to apply, has made it 
 clear how the writings of the two American 
 logicians may be brought into harmony. 
 
 IL Perhaps there is no term in 
 Logic which the reader is Ukely to find 
 so perplexing as the term " Logic of Rela- 
 tives." He not unreasonably supposes 
 
184 Symbolic Logic 
 
 when he comes to this part of the subject 
 that he is going to consider all those 
 expressions whose subject and predicate are 
 not connected by the copula " is," but by 
 the many other words or phrases that 
 frequently join these fundamental portions 
 of a proposition. Such general treatment of 
 copulce is certainly what the term in 
 question suggests to the mind, and this is 
 the extension that De Morgan at any rate 
 had in view. But in modern logical works 
 this investigation is given up as hopeless, 
 and instead of it we are introduced to the 
 subject of multiple quantifications. Of 
 course, such alteration in the subject-matter 
 need not have involved any confusion, and 
 some writers have made it perfectly clear 
 to their readers that the problem investi- 
 gated is no longer the wider one. But 
 Mr. Peirce calls the new inquiry by the 
 old name " Logic of Relatives," and such 
 a procedure is very misleading. "^ 
 
 * Johns Hopkins Studies in Logic^ p. 192; American Jour, 
 of Math, vol. iii. 
 
Later Logical Doctrines 185 
 
 The important question at once arises 
 whether the larger investigation is bound 
 to be fruitless, and, if so, why such is the 
 case. I think that a general treatment of 
 copuls cannot be undertaken bv the logi- 
 cian, because we need in every case to have 
 a piece of special information given us 
 beyond the propositions that form the 
 premises. Such information is necessary 
 whether the conclusion is reached syllogis- 
 tically or intuitively without the use of 
 syllogism. That such additional proposi- 
 tion is required before copula other than 
 " is " can be brought under the rules of syl- 
 logism is very clear. Take the case men- 
 tioned by Jevons. He says : " If I argue, 
 for instance, that because Daniel Bernoulli 
 was the son of John, and John the brother 
 of James, therefore Daniel w^as the nephew 
 of James, it is not possible to prove 
 this conclusion by any simple logical pro- 
 cess " ; we need also to be informed that 
 the son of a brother is a nephew. Again, 
 to take a case mentioned by Venn : " If the 
 
1 86 Symbolic Logic 
 
 distance of A and of B from C is exactly a 
 mile, that of A from B (the relation de- 
 sired) may be anything not exceeding two 
 miles " ; here the additional proposition 
 would have to contain information concern- 
 ing the angular measurements of the triangle 
 made by joining the points occupied by 
 the three persons, and to declare in general 
 terms what, under such circumstances, is 
 the distance between two persons situated 
 as are A and B. In still more indefinite cir- 
 cumstances of relation we should have to 
 possess a still more complicated piece of 
 information along with the original state- 
 ments. Now, since we must undoubtedly 
 reject the doctrine that was once frequently 
 held on this subject, viz,^ that such an 
 argument as "A equals B, B equals C, 
 therefore A equals C," is, when put in 
 another form, an actual case of syllogistic 
 reasoning, — the opponents of such a view 
 were quite right when they argued that 
 this putting into another form involves a 
 petitio principii: De Morgan, for instance, 
 
Later Logical Doctrines 187 
 
 made this reioinder, and Keynes is in 
 agreement with him — before all possible 
 premises of the kind in question can be 
 dealt with svllogistically there will be 
 needed an infinite number ot such special 
 pieces of information, and this amounts to 
 saying that a general treatment of relatives 
 is impossible. If, on the other hand, the 
 validity of such arguments as we are con- 
 sidering is declared not to be established by 
 means of syllogism, but to be as intuitively 
 evident as the validity of Barbara itsell, the 
 statement means, I take it, that in each case 
 there is involved a separate dictum, corre- 
 sponding to the dictum of the svllogism. 
 Since, however, the number of such cases is 
 unlimited, there will be an infinite number 
 of dicta in our Logic, which again is 
 impossible. 
 
 The wav out of the difficulty appears to 
 be the following. It must be admitted 
 that such propositions as the above are not 
 susceptible of being so manipulated that 
 they shall be put into syllogistic form. 
 
1 88 Symbolic Logic 
 
 Also it is absurd to suppose that we have at 
 our disposal an infinite number of major 
 premises or of dicta. Hence the general 
 treatment of copulas is impossible. But 
 what we can do is to admit an arbitrary 
 number of general propositions other than 
 the dictum de omni^ and the propositions thus 
 admitted allow of our dealing with a limited 
 number of arguments like the above. There 
 is a special group of such statements of 
 great importance, and they occur in the 
 region of quantitative mathematics. I refer 
 to the axioms of Geometry. From the 
 pre-Peanesque point of view these may be 
 regarded either as the assumptions that are 
 necessary in order to allow of the appli- 
 cation of syllogistic reasoning to propositions 
 of that science, or as of the nature of dicta^ 
 i.e., statements that allow of our drawing 
 conclusions by reference to them, and 
 without employment of the dictum de omni. 
 But, inasmuch as some of the assumptions 
 are used only occasionally, it seems de- 
 cidedly better not to speak of them as 
 
Later Logical Doctrines 189 
 
 dicta, but as propositions that are required 
 for the employment of the syllogism to 
 the material that is the subject of the 
 argument. 
 
 It may indeed be said that the syllogistic 
 treatment of relative reasoning is the appro- 
 priate one on another ground, viz,^ because 
 there is between the dictum de omni and the 
 other general propositions a difference of 
 such a kind as to give unique importance to 
 the former. It is sometimes asked, as by 
 De Morgan,* whether the axioms of mathe- 
 matics are not " equally necessary, equally 
 self-evident, equally incapable of demonstra- 
 tion out of more simple elements " with the 
 dictum^ and, if so, whether the two are not 
 equally important ? My view is that, what- 
 ever may be the character of the two kinds 
 of axioms as regards derivation and self- 
 evidence, they are not of equal importance. 
 For in all reasoning concerning quantities 
 the dictum de omni may be employed, while 
 in reasoning concerning qualities, where, of 
 
 "^ Trans, Camb, Philosophy Soc, vol, x. j). 338, 
 
190 Symbolic Logic 
 
 course, the dictum is also needed, the axioms 
 of quantitative mathematics afford no assist- 
 ance. De Morgan in another place * en- 
 deavours to show that questions of equality 
 and of identity are formally on an equal 
 footing, since " the word equals is a copula 
 in thought, and not a notion attached to a 
 predicate ^"^ and that '' logic is an analysis of 
 the form of thought, possible and actual, 
 and the logician has no right to declare 
 that other than the actual is actual." The 
 answer to this appears to be that, though 
 the individual does actually regard the 
 " equals " as a copula, he does so only by a 
 process of abbreviation : the form when 
 fully expressed is one of identity. The 
 logician is not bound to treat as of funda- 
 mental importance each kind of abbreviation 
 that mankind has adopted. It is enough 
 for him to deal with the fully expressed 
 form, and to explain, as we have done 
 above, that in the case of arguments con- 
 cerning quantities what we really have is 
 
 * Syllabus of a Proposed System of Logic, pp. 31, 32. 
 
Later Logical Doctrines 191 
 
 a syllogistic process plus some material 
 assumptions. 
 
 In this discussion we have been consider- 
 ing cases in which only three terms are 
 involved, and the matter has been regarded 
 from the point of view of ordinary Formal 
 Logic. In this narrower region the dictum 
 is unique. But from such statements it is 
 not to be concluded that we shall not when 
 discussing the generaHsation of logical pro- 
 cesses reject the dictum. It will be rejected, 
 however, not because it is not in a unique 
 way of a formal character, but because it 
 applies to only three terms, and we must 
 adopt axioms that are "necessary and 
 sufficient" for dealing with arguments of 
 any degree of complexity. 
 
 At first sight the above statement of the 
 case appears perhaps to agree with the view 
 that Boole adopted. But there is really no 
 such agreement. Boole held that general 
 logic is quantitative mathematics with the 
 quantity element left out, that is to say, 
 class logic and quantitative mathematics 
 
192 Symbolic Logic 
 
 participate in the nature of general logic, 
 and have in addition their own special 
 characteristics. It seems to me, on the 
 other hand, that there are not two species 
 of the genus general logic : there is one 
 logic, and that is class or propositional 
 logic, and all that there is in mathematics 
 is such logic, together with some material 
 assumptions concerning quantitative objects. 
 No argument whatever can be carried on 
 in quantitative mathematics without the 
 explicit or implicit application of class or 
 propositional logic at every step. Certainly 
 Boole appeared to establish two species of 
 reasoning, when he applied the symbols of 
 mathematics to the manipulation of argu- 
 ments involving classes ; but what he was 
 really doing was to show how qualitative 
 reasoning, if we employ in it symbols 
 analogous to those that represent quantita- 
 tive objects and processes, may be extended 
 far beyond the limits of the old syllogistic 
 arguments. To put the matter in a word, 
 even from the standpoint of pre-Peanes(^ue 
 
Later Logical Doctrines 193 
 
 notions it is better to recognise only the so- 
 called specific logic of quality, and to regard 
 quantitative reasoning as merely qualitative 
 reasoning together w^ith certain assumptions 
 concerning the relations of quantities. As 
 Dr. Shadv^orth H. Hodgson says,"^* formal 
 logic " is a system wholly unrestricted in its 
 range," or, as he adds, class Logic is " the 
 Logic of the whole nature of any and every 
 object of thought, of its What, rl ea-nv, of its 
 ^z//^, which includes both its ^ak and its 
 ^antumr That is to say, class Logic has 
 to do with the relation of classes whether 
 qualitatively or quantitatively determined. 
 
 It need hardly be said that though Jevons 
 speaks of the necessity of there being 
 additional information, before the pro- 
 position that I have quoted from him can 
 be manipulated, he does not make any 
 general statement on the subject. And he 
 evidently considers that all such arguments 
 form a class distinct from the miscellaneous 
 selection which he brings forward in 
 
 * Proc, Ar'tst, Soc. N.S. vol. ii. pp. 135, 136. 
 
 N 
 
194 Symbolic Logic 
 
 illustration of his principle of Substitution. 
 My view is rather that his illustrations are 
 special cases of relative reasoning, and 
 that this is not in general possible except 
 on the lines that I have endeavoured to 
 indicate. 
 
 So long as we do not make use of the 
 doctrines which have been unfolded by 
 Peano, the above is, I believe, the best way 
 to regard the reasoning that is involved in 
 quantitative mathematics. Peano's method, 
 as will be explained in the next section, 
 allows of our dispensing with these arbitrary 
 assumptions : all the material that is dis- 
 cussed in Mathematics is regarded as ex- 
 pressible in terms of variables and logical 
 constants, and so as susceptible of being 
 manipulated by the rules of Symbolic Logic. 
 Such a way of approaching the subject is 
 a great improvement. At the same time 
 it is obviously desirable to observe what 
 is the proper way to regard mathematical 
 reasoning if we are confined to the pre- 
 Peanesque point of view. The important 
 
Later Logical Doctrines 195 
 
 matter, however, for our present purpose 
 is to notice that this discussion about 
 Mathematics arose because we found that 
 it is impossible to estabhsh a Logic of 
 Relatives in the sense of a general treat- 
 ment of copula. 
 
 When it is stated, as was the case at the 
 commencement of this section, that the 
 expression " Logic of Relatives " as gener- 
 ally used refers only to the operations per- 
 formed upon propositions involving multiple 
 quantifications, it is not meant to suggest 
 that this investigation is not important. 
 On the contrary, as will have been gathered 
 from the earlier portion of this chapter, 
 I think that we have here a development 
 of the greatest interest. An investigation 
 of the principles, according to which results 
 concerning multiple quantifications may be 
 reached, naturally follows the study of the 
 subject-matter of ordinary Symbolic Logic, 
 in which, of course, we are concerned with 
 singly-quantified propositions. 
 
196 Symbolic Logic 
 
 III. We have now traced the develop- 
 ment of SymboHc Logic up to the furthest 
 point that was reached before the work done 
 by Frege, Peano, and Russell. I propose in 
 this section to make clear in what way 
 these writers have shown that the subject 
 can deal in a direct manner with material 
 with which it was supposed up to their 
 time that it could deal either not at all, or 
 only in an indirect manner. We have seen 
 in the last section that Boole, Venn, and 
 Schroder could not bring within the scope 
 of their procedure the deductions of Mathe- 
 matics, except by making arbitrary assump- 
 tions concerning the way certain quantities 
 are related. That is to say, mathematical 
 material could be dealt with by these 
 logicians only in an indirect manner. The 
 alternative to this procedure — so far as 
 pre-Peanesque doctrines are concerned — is 
 to adopt Kant's view that mathematical 
 demonstrations owe their certainty, just as 
 do syllogistic inferences, to special exer- 
 
Later Logical Doctrines 197 
 
 cises of an intuitional faculty. The validity 
 of the dictum de omni and the validity of 
 the axioms of Mathematics must on this 
 viev^ be held to stand on an equal footing : 
 each must be held to be intuitively obvious. 
 There v^ould thus be tw^o branches of de- 
 ductive reasoning, that concerned with quali- 
 tative and that concerned with quantitative 
 objects. 
 
 Concerning these two views it is to be 
 noted that they are not satisfactory. The 
 first obviously involves us in an uncer- 
 tain number of assumptions. The second 
 7nay be based on a faulty philosophical 
 analysis. For there are many philosophers 
 who hold that the facts of experience do 
 not justify us in asserting the existence of a 
 faculty such as that which Kant described, 
 and, therefore, do not permit us to declare 
 that the validity of the truths assumed in 
 Mathematics is intuitively obvious. Such 
 philosophers explain the certainty felt in 
 adopting these truths as due to the accu- 
 mulated results of special experiences. 
 
198 Symbolic Logic 
 
 Now both of these explanations of the 
 mathematical axioms cannot be true^ though 
 both Kantians and their critics may con- 
 tinue to hold that their respective statements 
 correctly account for the characteristics of 
 mathematical reasoning. 
 
 In short, since the earlier Symbolic Logic 
 could not deal in a satisfactory manner with 
 the deductions of Mathematics, and since it 
 may be that mathematical axioms do not 
 really rest on intuition, the subject of the 
 relation of Logic to Mathematics was, 
 previous to the time of Peano, in a state 
 that demanded careful consideration. This 
 consideration has been given by Peano and 
 his followers. What they have done is to 
 take mathematical ideas and analyze them, 
 so as to arrive at the general notions that 
 are involved in the various parts of the 
 science. Then these notions are all shown 
 to be expressible in terms of variables and 
 logical constants, the latter being notions 
 that are all definable in terms of a speci- 
 fied number of indefinables. The resulting 
 
Later Logical Doctrines 199 
 
 propositions are susceptible of being dealt 
 with by the ordinary rules of Symbolic 
 Logic, and in this way Mathematics is 
 seen to derive its validity from the fact 
 that it depends on principles of a logical 
 character. 
 
 This view of the subject has been fully 
 expounded by Mr. Russell in his work T!he 
 Principles of Mathematics, He thinks that 
 the analysis of mathematical conceptions 
 reveals the fact that there are some eight or 
 nine of these general notions, and he main- 
 tains that the logical constants by which 
 such notions may be expressed require only 
 these indefinables : implication between pro- 
 positions, relation of a member to a class 
 of which it is a member, the notion of 
 such that,, the notion of relation, and the 
 notions involved in formal implication.* 
 
 To put the matter in a sHghtly different 
 form, Mr. Russell is occupied throughout 
 his book in justifying his definition of 
 Pure Mathematics, which runs as follows : 
 
 * The Principles of Mathematics^ p. 1 1. 
 
200 Symbolic Logic 
 
 " Pure Mathematics is the class of all pro- 
 positions of the form "" p implies q^ where 
 p and q are propositions containing one 
 or more variables, the same in the two 
 propositions, and neither p nor q contains 
 any constants except logical constants.'' ^ 
 Here the notion of implication connecting 
 p and q involves some of the indefinables of 
 Symbolic Logic, and, when we come to 
 observe of what p and q are composed, we 
 find that they are composed of variables, 
 the same in each, and of certain logical 
 constants that are expressible in terms of 
 the other indefinables. 
 
 Now on this view it is clear that the 
 relation in which Symbolic Logic stands to 
 Mathematics is quite changed. Previous 
 to the work of Peano the two disciplines 
 were regarded as occupied respectively with 
 qualitative and with quantitative objects. 
 Symbolic Logic could not deal directly 
 with the material upon which Mathe- 
 matics is occupied, and Mathematics could 
 
 ■*^ Loc. c'tt. p. 3. 
 
Later Logical Doctrines 201 
 
 not of course solve problems with respect 
 to qualitative objects. But on the new 
 view Symbolic Logic can solve both kinds 
 of problems, or, more correctly, the two 
 kinds are reduced to one, and with this 
 Symbolic Logic deals. Symbolic Logic 
 may thus be regarded as synonymous with 
 deductive reasoning. 
 
 Perhaps the most important of the notions 
 of the new doctrine is that respecting the 
 relation of an individual to the class of 
 which it is a member. The older sym- 
 bolist considered that there is no need to 
 make a distinction between this relation 
 and that of a class to a wider class ; in 
 other words, he held that the singular pro- 
 position may be treated as a special case of 
 the universal. Venn, for instance, adopts 
 this view, and applies it in working the 
 example on p. 345 of his Symbolic Logic, 
 As it happens, there is no difficulty in that 
 problem in passing over the distinction in 
 question, for the premises do not contain 
 any alternatives in the predicates. But, 
 
202 Symbolic Logic 
 
 when disjunctive predicates do occur, it 
 becomes at once apparent that the con- 
 tradictory of the predicate is not of 
 the same description as the contradic- 
 tory where classes are involved. When 
 we are dealing throughout with classes 
 the contradictory of (x-^-y) is xy, and 
 this may be combined with the subject- 
 term, and the whole be equated to zero. 
 But when the subject-term is singular the 
 contradictory of the predicate (x+y) will 
 not be xy, but either this combination or 
 one of the two impossible combinations of 
 the three xy^ xy, xy ; or, as Vailati puts the 
 matter, in the case oi cea + b^ where a and b 
 are classes, it follows that one of the two, 
 cea or ceb^ is true, whereas in c>a-\-b^ 
 where c h z class, there is no need for 
 either c > a or c > b to be true.^' Still 
 another way of distinguishing the relation 
 where an individual is involved from the 
 relation of classes is to say that the relation 
 in the latter but not in the former case is 
 
 ■*■ Revue de Metaphysique et de Morale^ vol. vii. No. I, p. 97. 
 
Later Logical Doctrines 203 
 
 transitive."*' Peano was the first to give 
 prominence to this unique character of the 
 reference of an individual to a class, but 
 Frege in the Grundgetze der Arithmetik had 
 recognised the existence of the great dif- 
 ference between the two relations. f The 
 importance of making the distinction in 
 question becomes especially observable when 
 the attempt is made to bring mathematical 
 conceptions within the scope of Symbolic 
 Logic. For these conceptions are found 
 to be equivalent to implications between 
 prepositional functions, /.f., between classes, 
 whose elements are propositions stating the 
 relation of an individual to a class of which 
 it is a member. 
 
 It may here be remarked that Mr. 
 Whitehead holds that Mathematics is to be 
 identified with deductive reasoning,:]: and 
 that the discipline can take the place of all 
 intellectual exercises, except philosophy, 
 
 * The Principles of Mathematics^ p. 19. 
 
 t Russell, loc. c'lt. pp. 19, 512. 
 
 % A Treatise on Universal Algebra^ toI. i. p. viii. 
 
204 Symbolic Logic 
 
 inductive reasoning, and imaginative com- 
 positions. In the earlier portion of this 
 statement Mr. Whitehead does not identify 
 SymboHc Logic w^ith deductive reason- 
 ing, but makes the former a species of the 
 latter. It will thus be seen that he does 
 not take up precisely the same position 
 as Mr. Russell, v^ho identifies deductive 
 reasoning with Symbolic Logic, and repre- 
 sents this discipline as somewhat wider in 
 character than Mathematics. The second 
 portion of the above statement needs, I 
 think, some elucidation. I presume Mr. 
 Whitehead means that Symbolic Logic 
 (when the term is used as equivalent to 
 deductive reasoning) cannot at the same 
 time work problems and enunciate the 
 principles upon which it rests, and in this 
 I should agree with him. But this 
 hardly justifies us in stating that philo- 
 sophy is excluded from Symbolic Logic. 
 For the latter can manipulate philosophical 
 arguments quite as much as any others. 
 That is to say, if our premises give informa- 
 
Later Logical Doctrines 205 
 
 tion of a philosophical character we can 
 deal with them, and we shall arrive in our 
 conclusion at information of a similar 
 character. 
 
 As regards inductive reasoning, the ques- 
 tion arises whether this is really anything 
 different from deductive reasoning. There 
 is a good deal to be said for Mr. Russell's 
 view when he remarks : " What is called 
 induction appears to me to be either dis- 
 guised deduction or a mere method of 
 making plausible guesses."* I do not 
 myself consider that this expresses the 
 truth of the matter, for the process of 
 generalisation is, as Venn makes quite 
 clear in his 'Empirical Logic^ as important 
 as the original insight or the subsequent 
 verification. Mr. Russell's view, however, 
 is that in induction we are really concerned 
 with statements similar to those which are 
 found in pure Mathematics. He would 
 thus consider that the statement " If the 
 conditions of the Method of Difference 
 
 * Loc, clt, p. II w. 
 
2o6 Symbolic Logic 
 
 are fulfilled, the cause that will always 
 produce the phenomenon under investi- 
 gation has been discovered " is precisely 
 similar to such a statement as : " If two 
 sides of a triangle are equal each to each, 
 and if the included angles are equal, the 
 bases are equal/' And just as the principles 
 of Peano will allow of our manipulating 
 the latter proposition by means of Sym- 
 bolic Logic, so Russell would argue that 
 by the same means may the former be 
 manipulated. 
 
 The other doctrine, asserted by Mr. 
 Whitehead, namely, that Symbolic Logic 
 cannot be of service in the production of 
 imaginative literature, is undoubtedly true. 
 The statement holds whether we are 
 speaking of the earlier or of the later 
 application of the Calculus. For imagina- 
 tion is most effectively employed when the 
 faculty of pure reason is, comparatively 
 speaking, quiescent, and, therefore, when 
 the Calculus, which is a substitute for 
 the exercise of pure reason, is not being 
 
Later Logical Doctrines 207 
 
 employed. In other words, the associative 
 forces are more important for enabUng the 
 individual to reach conceptions of beauty, 
 while, for reaching the true, such forces 
 must be held in check, and there must be 
 an observance of logical rule. 
 
 In order to make clear the way in which 
 the school of Peano treats mathematical 
 conceptions we may refer to the subject of 
 integers and of their addition and multipli- 
 cation. Mr. Russell demonstrates that it is 
 possible to define numbers in such a way 
 that they are seen to be susceptible of being 
 manipulated by the rules of Logic. A 
 number may be defined as a class of similar 
 classes, />., of classes whose members are 
 correlated one to one ; ^ and, since classes 
 may be logically treated, numbers are 
 brought within the scope of Pure Logic. 
 Thus the various arithmetical operations 
 that are performed upon numbers may be 
 expressed in logical language. For in- 
 stance, addition is definable thus : " If k 
 
 * The Principles of Mathematics ^ pp. 113, 115. 
 
2o8 Symbolic Logic 
 
 be a class of classes no two of which have 
 any common terms (called for short an 
 exclusive class of classes), then the arith- 
 metical sum of the numbers of the various 
 classes of k is the number of terms in the 
 logical sum of k^ ^ Mr. Russell calls 
 special attention to the fact that in bring- 
 ing addition of numbers within the scope 
 of Pure Logic it is essential to speak of 
 the numbers as classes of classes. Unless 
 there is such reference to the latter, we 
 should never be able to take account of 
 a repeated term. If a sum of numbers 
 be spoken of merely as a class of numbers, 
 then, supposing the number 1 is repeated, 
 we cannot take account of this repetition, 
 since in Logic 1 + 1 = 1. On the other 
 hand, when we have in mind a class of 
 classes, if 1 occurs in one class, and again in 
 another class, such number may be taken 
 account of in each case, for all the terms of 
 the one class are different from the terms of 
 the other. 
 
 * Loc, cit. p. 1 1 8. 
 
Later Logical Doctrines 209 
 
 Just as addition of integers may be ex- 
 pressed as a logical sum of terms of two 
 or more classes, so multiplication may be 
 expressed as a sum of terms of a single 
 class. The definition of the latter process 
 as presented by Mr. Russell, who follows 
 Mr. Whitehead, is as follows : " Let k be 
 a class of classes, no two of which have any 
 term in common. Form what is called 
 the multiplicative class of k, ie,, the class 
 each of whose terms is a class formed by 
 choosing one and only one term from each 
 of the classes belonging to k. Then the 
 number of terms in the multiplicative class 
 of k is the product of all the numbers of 
 the various classes composing k.'' This 
 definition has the advantage of introducing 
 no order among the numbers multiplied, 
 and it applies both to finite and to infinite 
 classes."* 
 
 It is thus seen that from the Peanesque 
 point of view numbers may be defined as 
 classes of a certain kind, and addition and 
 
 * The Principles of Mathematics, p. 1 1 9. 
 
2IO Symbolic Logic 
 
 multiplication of numbers may be expressed 
 as logical addition of terms. In a similar 
 way it may be shown that the other pro- 
 cesses which are performed upon numbers 
 may be expressed as logical processes. 
 And, just as numbers may be dealt with by 
 the pure logician, so, Mr. Russell shows, 
 may be the other general mathematical 
 notions that analysis reveals to be funda- 
 mental. 
 
 I have mentioned three writers who 
 expound such doctrines as we have just 
 been considering. My object has been to 
 show the advanced position that Frege, 
 Peano, and Russell have assumed with 
 regard to the application of Symbolic 
 Logic. It must not, however, be sup- 
 posed that there are not considerable 
 differences between the views of these 
 writers. The points in which there is 
 disagreement are numerous, and call for 
 some consideration. 
 
 As regards the two latter writers there 
 IS a difference, in the first place, as to the 
 
Later Logical Doctrines 211 
 
 notions that are to be taken as indefinable, 
 and, secondly, as to the propositions that 
 are to be taken as primitive. We have 
 already enumerated the indefinables as 
 given by Mr. Russell. Peano's are as 
 follows : " Class, the relation of an indi- 
 vidual to a class of which it is a member, 
 the notion of a term, implication where 
 both propositions contain the same vari- 
 ables, />., formal implication, the simul- 
 taneous affirmation of two propositions, 
 the notion of definition, and the negation 
 of a proposition." * These are for the 
 most part the same as those given by 
 Mr. Russell, but the latter writer rightly 
 prefers to take as ultimate the simulta- 
 neous affirmation of all the propositions of 
 a class, and maintains that formal and 
 material implication should be both men- 
 tioned among the indefinables. Moreover, 
 he thinks there is no need to make nega- 
 tion a primitive idea, since, if we start 
 with propositions rather than with classes, 
 
 * The Principles of Mathematics, P- 27. 
 
212 Symbolic Logic 
 
 it is possible to show that all propositions 
 respecting negation are only other forms 
 of the principle of Reduction. This prin- 
 ciple runs thus : " If /> implies p and q 
 implies q^ then ' "/> implies q'' implies /> ' 
 implies ^," where of course nothing is 
 involved but implication.* 
 
 Coming to primitive propositions, Mr. 
 Russell urges that it is not desirable to 
 have one merely to describe the product 
 of two classes, but that instead of such 
 proposition one should be stated respecting 
 the product of a class of classes ; for, 
 unless the latter procedure be adopted, we 
 cannot describe the product of the classes 
 contained in an infinite class. Similarly, 
 there is no reason to have two axioms to 
 say respectively that ah is contained in a 
 and b^ and that ba is contained in each 
 of these : if we speak of a class of classes, 
 then when we obtain the logical product 
 of the contained classes, there will be no 
 reference to the order of the terms. f 
 
 * On this last point see loc. at. pp. 17, 31. j- Loc. c'tt. p. 30. 
 
Later Logical Doctrines 213 
 
 Finally, in connexion with his discussion 
 of primitive propositions, Peano is led to 
 define the null-class as the class which is 
 contained in every class. Mr. Russell 
 does not disagree with this definition, 
 but he points out that, in order to accept 
 it, Peano must be explicit as to what is 
 meant by the assertion that " x is zn a " 
 implies "a; is a ^ for all values of x, 
 Peano hesitates to say whether this im- 
 plication does or does not involve that x 
 must be an a^ but inclines to adopt the 
 view that it does. But, if such were the 
 case, " ' X is a ' implies ' ^ is a y for 
 all values of x " (which, following Peano, 
 we must take as equivalent to — < i) 
 could not be taken as a definition of 0, 
 for to utter such an implication would 
 be to say that both has and has not 
 members. On the other hand, if we 
 adopt the view that in the implication 
 " ' ;c is an a' implies *" x is a /^ ' for all 
 values of x " we are not confined to the 
 xs that actually are aSy no contradic- 
 
214 Symbolic Logic 
 
 tion is involved in dealing with the null- 
 class.^' 
 
 From these considerations I think it may 
 safely be said that, w^hile Peano has gone 
 further than his predecessors, on the above- 
 mentioned points Mr. Russell occupies the 
 correct position, and so shows an advance 
 upon Peano. 
 
 The other logician who has seen that 
 Symbolic Logic may deal with more pro- 
 blems than were contemplated by Boole, 
 Venn and Schroder, is Frege. This writer's 
 importance has until lately been quite over- 
 looked. Venn, whose historical researches 
 in Logic have usually been so productive 
 of valuable results, passes him over with 
 but scant attention, merely remarking that 
 " Here again we have an instance of an 
 ingenious man working out a scheme — in 
 this case a very cumbrous one — in apparent 
 ignorance that anything better of the kind 
 had ever been attempted before. . . . The 
 obvious defect in this scheme is the inor- 
 
 * Loc* cit, pp. 32, 38. 
 
Later Logical Doctrines 215 
 
 dinate amount of space demanded for its 
 display. Nearly half a page is sometimes 
 expended on an implication which, with 
 any reasonable notation, could be com- 
 pressed into a single line." * This obser- 
 vation upon Frege's cumbrousness is no 
 doubt fully justified, but his logical doc- 
 trines, as Mr. Russell has shown, are 
 deserving of much more attention than 
 Venn has given to them. 
 
 Frege's chief excellence consists, I think, 
 firstly, in recognising the importance of 
 considering propositions and their impli- 
 cations as fundamental,t and, secondly, in 
 perceiving that, without making arbitrary 
 assumptions, it is possible to bring certain 
 mathematical arguments within the scope 
 of Symbolic Logic. In the first chapter I 
 contended that, so far as we are concerned 
 with the problems discussed (say) by Venn, 
 it does not matter whether symbols repre- 
 
 * Symbolic Logic, pp. 493, 494. 
 
 f It should be noted that Mr. MacColl had drawn atten- 
 tion to this importance two years before the Begriffsschrift 
 was published. 
 
2i6 Symbolic Logic 
 
 sent terms or represent propositions, but 
 I quite admit that when we attempt to 
 extend the scope of Symbolic Logic it is 
 preferable to let symbols represent pro- 
 positions rather than classes."* Frege's pro- 
 positional calculus closely resembles that 
 adopted by Mr. Russell. There are, how- 
 ever, certain important differences between 
 the two. For instance, Frege considers 
 that if a occurs in a proposition, the latter 
 may always be resolved into a and a state- 
 ment about a^ whereas Mr. Russell holds 
 that, though such analysis is possible in the 
 case of an expression of the form " Socrates 
 is a man,'' the analysis is not possible when 
 we have to deal with such a statement as 
 " Socrates is a man implies Socrates is a 
 mortal." f 
 
 With respect to Frege's extension of the 
 scope of Symbolic Logic it is to be observed 
 that he made a start in the direction that 
 Peano and Russell have pursued with such 
 
 * See note on p. 9. 
 t Loc. cit. pp. 84, 85. 
 
Later Logical Doctrines 217 
 
 important results. Especially noteworthy 
 is his treatment of cardinal numbers. 
 These he defines very much as they are 
 defined above, i.e.^ he shows that they are 
 really classes. But he makes the strange 
 assertion in the Grundlagen der Arithmetik 
 that numbers cannot be applied to actually 
 existing things, but only to class-concepts.^ 
 I think with Mr. Russell that this view 
 is incorrect, since, \w those cases where 
 Frege says that the same set of objects may 
 have difi^erent numbers, it is not really the 
 same objects that are being considered. 
 
 That there are several points of difference 
 between Frege's doctrine and that set forth 
 by Mr. Russell is very certain, but, as the 
 latter writer remarks, they are few as com- 
 pared with the points of agreement. Of 
 the truth of one statement of fundamental 
 importance both writers are fully convinced, 
 namely, that a " proposition concerning 
 every does not necessarily result from enu- 
 meration of the entries in a catalogue." 
 
 * The Principles of Mathematics y p. 519. 
 
21 8 Symbolic Logic 
 
 They hold as against Mill, and as against 
 Kelly — who in attacking Frege practically 
 takes up Mill's position — that general pro- 
 positions may frequently be established by 
 means of a procedure which does not in- 
 volve previous reference to the individual 
 cases that are covered by the universal 
 term.* 
 
 In concluding this account of the newer 
 doctrines, I think it important to show that 
 their acceptance is in agreement with the 
 view that I endeavoured to unfold in the 
 second section of this chapter. I there 
 maintained that it has been correctly held 
 that a Logic of Relatives is impossible. In 
 the present section we have seen that Mr. 
 Russell takes as one of his indefinables 
 the notion of relation. It may appear, 
 therefore, that there is here involved a 
 contradiction. But such is not the case. 
 What was proved above to be impossible is 
 a logic that can deal with all copute in 
 a general manner, just as ordinary Formal 
 
 * The Principles of Mathematics, p. 522. 
 
Later Logical Doctrines 219 
 
 Logic can by means of the dictum deal with 
 all propositions whose terms are connected 
 by the copula "is." It was shown that, 
 owing to the infinite number of different 
 kinds of copula that exist, there cannot be 
 a dictum by reference to which any two 
 or more propositions containing any of 
 these copute can be synthesized. There 
 cannot, in other words, be a logic that 
 rests upon what may be called a Universal 
 Dictum. But it is quite true that the 
 subject of relations, when by this is meant 
 the doctrine of multiple quantifications, 
 forms part of the material with which the 
 symboHc logician has to deal. The notion 
 of relation will thus for the logician be ulti- 
 mate, and the propositions that contam 
 the quantitative terms as subjects may be 
 synthesized in the way that was de- 
 scribed above in the first section. Thus 
 mathematical notions may quite well in- 
 volve the notion of relation, and so be 
 treated logically, and yet a Logic of Rela- 
 tives in any other sense than the doctrine of 
 
220 Symbolic Logic 
 
 multiple quantifications may be an impossi- 
 bility. 
 
 And, once more, there is no contradiction 
 between Mr. Russell's view that classes, 
 owing to the fact that they sometimes 
 consist of an infinite number of terms, 
 should be defined by reference to intension, 
 and Mr. Johnson's doctrine that, when 
 multiply-quantified propositions come to 
 be synthesized, there is a reference to 
 the individuals that constitute the classes.*' 
 This reference is essential whether the 
 classes contain an infinite or a finite number 
 of members, and whether the classes are 
 formed by enumeration of objects or by 
 reference to attributes. 
 
 * See on this point an important note by Mr. Johnson in 
 Mind, N.S. No. i. p. 28. 
 
CHAPTER VII 
 
 THE UTILITT OF SYMBOLIC LOGIC 
 
 I PROPOSE in this concluding chapter to 
 indicate briefly what is the utiHty of the 
 Symbolic Logic whose development we 
 have traced in the preceding pages. Of the 
 educational advantages arising from the 
 concentration of thought, that the discip- 
 line demands, it is impossible to speak too 
 highly. On all sides the educational utility 
 of mathematical ^ study is recognised, but I 
 venture to state that Symbolic Logic takes 
 no second place in this respect. Probably, 
 also, every one would allow that the general- 
 ised treatment of thought throws much 
 light upon problems that appear in the 
 special or syllogistic treatment. As re- 
 gards the direct utility of the discipline, 
 
 * I am here using this term in the sense in which it was 
 understood previous to the time of Peano. 
 
 221 
 
222 Symbolic Logic 
 
 the question is somewhat complex. It may 
 readily be granted that natural science 
 cannot make any direct use of Symbolic 
 Logic. Mathematics is absolutely necessary 
 for an insight into many of Nature's laws, 
 but natural science is not immediately 
 furthered by the rules of the logical calculus. 
 Jevons seemed to think that the facts point 
 in the other direction, for he held that 
 science is advanced by means of the Substi- 
 tution of Similars. But the truth is that 
 science must supply the premises upon 
 which the symbolic logician may bring to 
 bear his mechanical contrivances. 
 
 It is, I think, quite true to assert that 
 Jevons believed that Symbolic Logic assists 
 in the advancement of science. His posi- 
 tion on the subject is, however, not always 
 perfectly clear. As Mr. E. C. Benecke 
 points out,* while in the former part of 
 the Principles of Science a calculus is elabo- 
 rated, there is no reference to such con- 
 struction in the later parts, where the 
 
 ♦ Proc. of Arist. Soc, N.S. vol. ii. p. 141. 
 
The Utility of Symbolic Logic 223 
 
 methods employed in the various sciences 
 are discussed. Still, Jevons definitely 
 asserted that " the Substitution of Similars 
 is a phrase which seems aptly to ex- 
 press the capacity of mutual replace- 
 ment existing in any two objects which 
 are like or equivalent to a sufficient 
 degree," * and " in every act of inference 
 or scientific method we are engaged about 
 a certain identity, sameness, similarity, like- 
 ness, resemblance, analogy, equivalence or 
 equality apparent between two objects." f 
 Nothing could be clearer than these state- 
 ments. We must not, therefore, I think, 
 regard Jevons as first developing a logi- 
 cal calculus, and then as proceeding to 
 deal with scientific methods. Rather the 
 whole of his Principles of Science has to do 
 with the methods of science (as Croom 
 Robertson says, " the Methods, rather 
 than the Principles, of Science, would, 
 perhaps, be a more appropriate title for 
 
 * The Principles of Science^ p. 17. 
 j* Loc. cit. p. I. 
 
224 Symbolic Logic 
 
 the book as it stands'*), and the latter 
 portion of the volume is engaged not 
 upon an investigation quite distinct from 
 that which occupies the former part, but 
 with the work of ascertaining " when and 
 for what purposes a degree of similarity- 
 less than complete identity is sufficient 
 to warrant substitution." This substitu- 
 tion is all along held to be the funda- 
 mental process. 
 
 To resume the main discussion of this 
 section, we have said that Symbolic Logic 
 does not directly lead us to any new truths 
 in natural science. It is, however, by no 
 means the case that no new truth at all, but 
 only a recognition in another form of the 
 information contained in the premises is 
 reached by means of the calculus. For 
 what is a new truth ? It is an accurate 
 subject-predicate combination that an indi- 
 vidual forms, but which has never till then 
 been formed in the history of the race. 
 Now such a combination may be reached 
 deductively or inductively. It was a new 
 
The Utility of Symbolic Logic 225 
 
 truth when the conclusion of Euc. I. 47 
 was for the first time reached, just as it was 
 a new truth when Adams and Leverrier 
 discovered the planet Neptune. In a 
 second sense a truth may be said to be new 
 when, though well known to science, the 
 full force of the subject-predicate combina- 
 tion is for the first time grasped by the 
 mind of a student. Here again the above- 
 mentioned combinations may take equal 
 rank in their claims to be designated new. 
 And, just as in pure Mathematics the 
 results may constitute new truths in both of 
 the above senses, so in Symbolic Logic we 
 may be said in the same senses to reach a 
 new truth. For instance, the difficult prob- 
 lem that was first solved by Boole * gave a 
 result that was true and altogether new, and 
 this solution, which is well known to all 
 symbolists, is the occasion of the experience 
 of a new truth in the mind of each student 
 of the subject. 
 
 Moreover, though it be correct, as we 
 
 * Boole, Laws of Thought, pp. 146- 148. 
 
226 Symbolic Logic 
 
 have seen, to say that SymboHc Logic 
 cannot directly assist the individual in his 
 scientific pursuits or in his daily affairs, the 
 indirect help of the discipline in such 
 spheres of practice is by no means insig- 
 nificant. Mankind is consciously or semi- 
 consciously much occupied with questions 
 that turn upon the relations of classes, so 
 that the manner of looking at things which 
 the logical study makes habitual cannot 
 fail to be of service in practical concerns. 
 Instead of confining himself to things that 
 are seen, the logician spontaneously is led to 
 regard the things that are not seen. It has 
 become a custom with him to consider the x 
 as of equal value with the x. The truth is 
 not that his logically developed habits are 
 not applicable to the affairs of practical life, 
 but rather that he will so weigh the pros 
 and cons of a question that his active forces 
 will be apt to suffer from a certain paralysis. 
 The man of strong will, who is possessed of 
 a vivid idea of one aspect of a practical 
 problem, is much more likely to achieve a 
 
The Utility of Symbolic Logic 227 
 
 great deal than the man who sees accurately 
 both sides. Hence the dilemma faces us 
 whether it is better to act vigorously, and 
 accomplish much that has to be revised and 
 largely undone, or to produce only a small 
 amount, but such as needs little alteration. 
 
 Now, if the study of Symbolic Logic 
 is thus indirectly of use in natural science 
 and in ordinary affairs, then a fortiori the 
 study is of service to the philosopher. For 
 I take it that we philosophize rather in 
 order to know than in order to act, and 
 therefore in Philosophy there is no danger 
 whatever arising from seeing the other side 
 of a question. I think, moreover, that the 
 principles of Symbolic Logic point in a 
 striking manner to the fact that in Philo- 
 sophy we can reach nothing simpler than 
 a duality, however far we press our inves- 
 tigations. Attempts to reduce the world 
 to unity — to God, to Self, to Nature, 
 for instance — appear to be doomed to fail. 
 In this extreme case our 1 means the 
 totality of the existent, the universe in the 
 
228 Symbolic Logic 
 
 common acceptation of that term. As 
 before, x-\-x = l of necessity, and with this 
 necessity we are obliged to stop. We 
 cannot estabHsh the existence of x only, for 
 at the outset of an attempt at such demon- 
 stration we become involved in self-con- 
 tradiction. For instance, let x stand for 
 "God," then x will stand for " not-God." 
 Now, if we attempt to demonstrate the 
 non-existence of x^ we shall be proceeding 
 in an absurd manner, for we shall be 
 assuming, if not ourselves, at any rate our 
 reasoning, which evidently is a part of the 
 X, An opponent of this argument might 
 perhaps affirm that the human proof may 
 well be regarded as a form of Divine 
 reasoning. God would thus be proving 
 His own exclusive existence. But it is 
 obvious that the circumstances under which 
 such Divine ratiocination would be taking 
 place would be such that a human thinker 
 was recognising the argument as his own 
 construction. Hence the human thinker 
 and his thought would still be distinct from 
 
The Utility of Symbolic Logic 229 
 
 the Divine. And, similarly, in our other 
 efforts to reach unity, the argument is based 
 on the assumption of an ultimate duality. 
 
 It should be observed here that I do 
 not maintain that the existence of this 
 duality can be proved by logical rule, but 
 only that the principles of Symbolic Logic 
 point to the circumstance that we cannot 
 demonstrate the fact of unity. That is to 
 say, if a Calculus of the kind v^e have de- 
 scribed be accepted, such acceptance in- 
 volves the necessity of our being content 
 to stop short v^ith a philosophical duality. 
 In other v^ords, if wt adopt the formula 
 x-\-x = l — as we do throughout the whole of 
 the Calculus — then in the limit, />., where 1 
 means the totality of the existent, whatever 
 object X may represent, there will always 
 appear along with it another object x. 
 
 The remarks that we have made with 
 respect to the utility of the ordinary Sym- 
 bolic Logic apply also to the so-called Logic 
 of Relatives. In this further study we do 
 not arrive at anything more general or con- 
 
230 Symbolic Logic 
 
 jectural than the multiply-quantified pro- 
 positions with which we start. There is 
 here, therefore, no instrument by which 
 the problems of natural science may be 
 solved. But the educational advantage and 
 indirect assistance of the study, and the 
 possibility of reaching new truths, in the 
 sense that we have just mentioned, are the 
 same as in the case of the Symbolic Logic 
 that deals with singly-quantified proposi- 
 tions. 
 
 Up to this point Symbolic Logic cannot 
 be said to be able to reach new scientific 
 truths. But, when we consider the relation 
 in which Mathematics has in recent years 
 been shown to stand to the discipline, and 
 when it is remembered that Mathematics 
 is essential in many scientific investigations, 
 it is observable that Symbolic Logic is no 
 longer confined to the uses just mentioned. 
 New propositions in certain sciences are 
 now seen to be derivable by applying 
 logical rules to propositions that represent 
 mathematical notions. Part of Mr. White- 
 
The Utility of Symbolic Logic 231 
 
 head's object in his work, A 'Treatise on 
 Universal Algebra^ is to demonstrate the 
 scientific value of Symbolic Logic. "^ The 
 book is occupied in exhibiting the Algebras 
 "both as systems of symbolism, and also as 
 engines for the investigation of the possi- 
 bilities of thought and reasoning connected 
 with the abstract general idea of space." 
 
 And just as, on the modern view of the 
 relation of Logic to Mathematics, Sym- 
 bolic Logic may reach new truths in 
 science, so by the same means may truths 
 that are already known to science be 
 reached by the mind of a student. This is 
 well brought out in a suggestive article by 
 Mr. T. P. Nunn.f He shows that when 
 the formula for finding the Centigrade 
 reading from the Fahrenheit reading is re- 
 garded purely as a statement, such formula 
 may, by means of the Logical Calculus, 
 be transformed into another, by which, 
 
 * He does not, however, as I have pointed out (p. 204), 
 take up quite the same position as Mr. Russell, 
 •j" See the periodical School, vol. iv. No. 22, 
 
232 Symbolic Logic 
 
 given the Centigrade reading, the Fahren- 
 heit reading may be obtained. Here there 
 is a direct appHcation of logical rules to 
 the verbal elements of a statement. A 
 student, therefore, by making use of these 
 rules, is enabled in such cases to reach truths 
 which are new to him, and which are ol a 
 scientific character. 
 
INDEX 
 
 Accidental attributes, Cas- 
 tillon's symbolization of, 
 104. 
 
 Adamson, R., on intermediate 
 processes, 68 n. ; on Schro- 
 der's solutions, 68 n. 
 
 Addition, Russell's definition 
 of, 207 ; symbolization of, 
 
 139- 
 Alternants, least determinate, 
 
 175. 
 
 Alternatives, Boole's symboliza- 
 tion of, 42 ; exclusive method 
 of representing, 38 ; F. H. 
 Bradley on, 43 ; non-exclu- 
 sive method of representing, 
 39 ; Schroder's symboliza- 
 tion of, 42 ; Venn's sym- 
 bolization of, 51, 160. 
 
 " And," W. E. Johnson on 
 particle, 16. 
 
 Assertorics, Symbolic Logic 
 confined to, 24. 
 
 Attributes, abstraction of, 98 ; 
 combination of, 98 ; con- 
 ventionally fixed, 136 ; ex- 
 istence of, 139 ; intension 
 constituted by totality of, 
 136; negative, 139. 
 
 Axioms, De Morgan on 
 mathematical, 190. 
 
 Axioms in Geometry, pre- 
 Peanesque treatment of, 
 188. 
 
 Benecke, E. C, criticism of 
 Jevons by, 222. 
 
 Boole, G., brackets used by, 
 
 55 ; calculus constructed by, 
 5 ; formula of simplifica- 
 tion adopted by, 66 ; frac- 
 tional forms used by, 67 ; 
 inconsistency of, 52 ; Jevons' 
 criticism of, 67, 147 ; mathe- 
 matical symbols used by, 
 35 ; method of elimination 
 adopted by, 69, 71 ; on 
 logical expansion, 74 ; on 
 symbolization of particulars, 
 
 56 ; problem worked by, 75 ; 
 process of squaring adopted 
 by, 65 ; progress since time 
 of, 6 ; propositions sym- 
 bolized by, 150 n. ; repre- 
 sentation of premises by, 
 161 ; symbolists anterior to, 
 3, 4 ; symbolization of alter- 
 natives by, 42 ; transposition 
 of terms by, 64 ; Venn's 
 explanation of forms used 
 by, 67 ; view of general logic 
 adopted by, 191. 
 
 Boolian principles, Johnson 
 on application of, 178. 
 
 Brackets, use of, 55. 
 
 Bradley, F. H., treatment of 
 alternatives by, 43. 
 
 Bryant, S., confusion of state- 
 ments and events by, 154; 
 intensive symbolism adopted 
 by, 131 : on origin of 
 imaginary results, 49 ; on 
 propositions frequently true, 
 25, 26. 
 
 233 
 
234 
 
 Index 
 
 Calculus, available logical, 
 2 ; Boole's, 5 ; nature of a, 
 31, 48, 49, 96. 
 
 Castillon, G. F., brackets used 
 by, 55 ; conception of Sym- 
 bolic Logic held by, 95 ; 
 consistency of, 94 ; erroneous 
 proofs offered by, 114; in- 
 consistency of, 102 ; invalid 
 inferences admitted by, 118 ; 
 memoir by, 94 ; on classifi- 
 cation of judgments, 105; on 
 conversion of particulars, 
 122; on hypotheticals, 124; 
 on illusory particulars, iii ; 
 on infinite judgments, 47 ; 
 on logical laws, loi ; on 
 nature of propositions, 97 ; 
 on notion of quantity, 129; 
 on syllogism, 108 ; on sym- 
 bolization of accidental at- 
 tributes, 104 ; on symboliza- 
 tion of essential attributes, 
 103 ; on transposition of 
 terms, 123; on use of minus 
 sign, 46 ; representation of 
 universal negative by, 46 ; 
 S. Bryant's symbolism com- 
 pared with that of, 131; 
 treatment of problematical 
 judgments by, 126; treat- 
 ment of universal negative 
 by, 123; use of word " some " 
 by, 115; Venn's criticism of, 
 99,107,120. 
 
 " Certain," distinction between 
 "true" and, 158. 
 
 " Certainties," common treat- 
 ment of, 158; nature of 
 MacColl's, 153. 
 
 Chance, propositions and, 158. 
 
 Class-concepts, application of 
 number to, 217. 
 
 Class of classes, number as, 
 208. 
 
 Classes, individuals distin- 
 
 guished from, 202, 203 ; 
 symbolization of, 11, 12. 
 
 Compartment, universe dis- 
 tinguished from, 166. 
 
 Compartments, salvation of, 
 66. 
 
 Comprehension, Logic based 
 on, 138; signification of 
 term, 91. 
 
 Conclusions, character of 
 Venn's, 78. 
 
 Conditionals, hypotheticals dis- 
 tinguished from, 60 ; nature 
 of, 60, 61. 
 
 Connotation, Keynes' view of, 
 138; Logic based on, 134; 
 Mill's view of, 138; signi- 
 fication of term, 91. 
 
 Contradictory, De Morgan's 
 rule for finding, 79, 80 ; 
 symbol for, 22 ; Venn's rule 
 for finding, 80, 
 
 Contradictories, interpretation 
 of symbols representing, 
 13; two types of, 202. 
 
 Conversion, Castillon on, 108, 
 122. 
 
 Copulae, general treatment of, 
 184, 218. 
 
 Couturat, L., examination of 
 Leibnitz by, 136 ; on a Logic 
 of intension, 140; on mean- 
 ing of intension, 135. 
 
 Critical School, infinite terms 
 admitted by, 105. 
 
 Deductive reasoning. White- 
 head on scope of, 203, 
 204. 
 
 De Morgan, A., advance made 
 by, 182; notion of Figure 
 adopted by, 181 ; on a 
 generalised Logic, 180; on 
 axioms of Mathematics, 190 ; 
 on contrariety of relational 
 propositions, 181 ; on con- 
 
Index 
 
 235 
 
 version of relational proposi- 
 tions, 181 ; on finding con- 
 tradictories, 79, 80 ; on 
 nature of Logic, 190; on 
 supreme law of syllogism, 
 181 ; propositions symbolized 
 by, 150 n. 
 Destruction, compartmental, 
 
 59- 
 
 Determinants, most deter- 
 minate, 175. 
 
 Diagrams, disadvantages of, 
 83 ; employment of, ']']^ 81 ; 
 solutions effected by, 82. 
 
 Dicta in relative reasoning, 
 187. 
 
 Dictum de oinni^ importance 
 of, 189 ; sphere of, 191. 
 
 Disjunction, interpretation of, 
 
 Disjunctive predicates, con- 
 tradictories of, 202 ; Vailati 
 on, 202. 
 
 Disjunctives, symbolization of, 
 
 .5?-. 
 Division, exclusive notation 
 
 and, 39 ; sign of, 48. 
 Double quantifications, Mit- 
 chell on, 178, 179; Peirce 
 on, 177. 
 
 Elimination, Boole's method 
 of, 69, 71 ; C. Ladd- 
 FrankHn's formula for, 58, 
 73 ; diagrams useful for, 83 ; 
 in case of particular proposi- 
 tions, 72 ; Johnson's formula 
 for, 73 ; Schroder's method 
 of, 70, 71 ; Venn's formula 
 for, 70, 72. 
 
 Equations, transposition of 
 terms in, 45. 
 
 Equivalence, symbol for, 54, 
 99. 
 
 Essential attributes, 103. 
 
 Eulerian diagrams, 81. 
 
 Event, proposition distin- 
 guished from, 26, 27. 
 
 Events, statements confused 
 with, 154. 
 
 Existence, logical, 164 ; Mac- 
 Coll on, 171 ; philosophical, 
 164. 
 
 Existence of attributes, 139. 
 
 Existential import of univer- 
 sal, 54. 
 
 Expansion, analytical treat- 
 ment of, 74 ; Boole's formula 
 for, 74 ; geometrical justifica- 
 tion of, 74 ; Peirce's formula 
 for, 75. 
 
 "False," distinction between 
 "impossible" and, 158. 
 
 Figure, De Morgan's notion 
 of, 181. 
 
 Formal Logic, S. H. Hodgson 
 on nature of, 193. 
 
 Formulae, modification of, 19. 
 
 Fractional forms, Boole's, 67. 
 
 Frege, G., differences between 
 Russell and, 216; on dis- 
 tinction between individuals 
 and classes, 203; on forma- 
 tion of general propositions, 
 217 ; on fundamental char- 
 acter of implication, 215 ; 
 on scope of Symbolic Logic, 
 215 ; prepositional calculus 
 of, 216; treatment of car- 
 dinal numbers by, 217 ; 
 Venn's criticism of, 214. 
 
 General Logic, Boole's view 
 of, 191. 
 
 General propositions, forma- 
 tion of, 217, 
 
 Generalised Logic, 95 ; De 
 Morgan on a, 180. 
 
 Habits, logically developed, 
 226. 
 
236 
 
 Index 
 
 Hodgson, S. H., on nature of 
 Formal Logic, 193. 
 
 Holland, G. J., Lambert's co- 
 adjutor, 4. 
 
 Hypotheticals, Castillon on, 
 124; conditionals distin- 
 guished from, 60 ; nature of, 
 61 ; symbolization of, 59 ; 
 treatment of, 61, 62. 
 
 Illusory particulars, Castillon 
 on. III. 
 
 Imaginary results, origin of, 
 50. 
 
 Implication, Frege on funda- 
 mental character of, 215. 
 
 Implications, relation of 
 thinker to, 30. 
 
 " Impossible," distinction be- 
 tween " false" and, 158. 
 
 Inconsistency, Boole's, 52. 
 
 Indefinables, logical constants 
 imply, 198 ; Peano's list of, 
 211 ; Russell's list of, 199. 
 
 Indefiniteness, symbol for. 
 
 Induction, connexion of 
 Peano's principles with, 
 206 ; Inverse Problem dis- 
 tinguished from, 89 ; Jevons 
 on nature of, 147 ; pure 
 Mathematics and, 205 ; 
 Russell on nature of, 205 ; 
 Whitehead on position of, 
 204. 
 
 Infinite classes, synthesis of 
 propositions containing, 220. 
 
 Infinite judgments, Castillon's 
 treatment of, 47. 
 
 Infinite terms. Critical School 
 admits, 105. 
 
 Integers, logical treatment of, 
 207. 
 
 Intension, assumptions in logic 
 of, 139; changes produced 
 in, 136; classes determined 
 
 by, 220 ; Couturat's interpre- 
 tation of term, 135 ; Leib- 
 nitz's interpretation of term, 
 137; Logic based on, 140; 
 Russell's interpretation of 
 term, 135; signification of 
 term, 91. 
 Intensive Logic, minus sign in. 
 
 Intermediate processes, in- 
 telligibility of, 48, 49. 
 
 Internal synthesis, potency of, 
 176. 
 
 Inverse operations, possibility 
 of, 38 ; reliable results of, 48 ; 
 utility of, 48. 
 
 Inverse Problem, improve- 
 ments in solution of, 85 ; in- 
 duction distinguished from, 
 89; Jevons' treatment of, 
 85, 147; Johnson's solution 
 of, 87, 160 ; Keynes' solution 
 of, 86, 160, 161 ; Schroder's 
 methods of dealing with, 
 87 ; variety of answers to, 
 85. 
 
 Jevons, W. S., criticism of 
 Boole by, 67, 147 ; interpre- 
 tation of propositions by, 
 7 ;i^. ; on Inverse Problem, 
 147 ; on Law of Unity, 148; 
 on nature of induction, 147 ; 
 on relative reasoning, 185; 
 on symbolization of alter- 
 natives, 146 ; on Substitution 
 of Similars, 224 ; on symboliz- 
 ation of particulars, 56; on 
 thinking in non-exclusive 
 manner, 39 ; representation 
 of universal affirmative by, 
 145 ; symbol for indefinite- 
 ness used by, 52 ; symbol- 
 ization of particulars by, 
 146. 
 
 Johns Hopkins Studies^ 89. 
 
Index 
 
 237 
 
 Johnson, W. E., criticism of 
 Peirce by, 178 ; formula for 
 elimination given by, 7^ ; 
 Inverse Problem solved 
 by, 87 ; Mitchell criticized 
 by, 179; on application of 
 Boolian principles, 178 ; on 
 hypothetical and condi- 
 tionals, 60 ; on modal pro- 
 positions, 27 ; on molecular 
 propositions, 173 ; on nature 
 of a calculus, 31, 96 ; on 
 particle "and," 16; on 
 selecting determinants, 176; 
 on synthesis of multiply- 
 quantified propositions, 175. 
 
 Judgment, symbols cannot dis- 
 cover forms of. 97 ; Wolf's 
 definition of, 106. 
 
 Judgments, Castillon's classifi- 
 cation of, 105. 
 
 Kant, critics of, 197 ; view 
 of mathematical reasoning 
 adopted by, 196. 
 
 Keynes, J. N., compact 
 methods of, 69 ; employ- 
 ment of diagrams by, 81 ; 
 Formal Logic, 34 ; on conno- 
 tation, 138; on meaning of 
 intension, 136 ;/., predica- 
 tive standpoint adopted by, 
 53 ; solution of Inverse 
 Problem by, 85, 160, 161 ; 
 symbols of operation not 
 used by, 31 ; use of bracket 
 by, 55- 
 
 Ladd-Franklin, C, answer 
 to Schroder by, 19, 20 ; 
 confusion of statements and 
 events by, 154; criticism of 
 Schroder by, 17, 18 ; for- 
 mula for elimination pro- 
 posed by, 58, 73 ; inverse 
 processes rejected by, 50 ; 
 
 on logical sequence, 18, 
 19 ; on use of literal 
 symbols, 16; symbolization 
 of particulars by, 57. 
 
 Lambert, J. H., brackets used 
 by? 55; coadjutors of, 4; 
 intensive system of, 94 ; 
 work done by, 3, 4. 
 
 Law of Unity, Jevons' view of, 
 148. 
 
 Laws of Thought, problem 
 worked in, 75. 
 
 Leibnitz, Couturat's examina- 
 tion of, 136 ; influence of, 
 4 ; on meaning of inten- 
 sion, 136 ;/., 137 ; symbol 
 for indefiniteness used by, 
 
 Limiting case, instance of, 
 19. 
 
 Literal symbols, MacColl's use 
 of, 150 : primary use of, 
 10 ; variety of, 22. 
 
 Logic, connotation as the 
 basis of, 134 ; De Morgan 
 on nature of, 190 ; general- 
 isation of common, 33 ; use 
 of mathematical symbols in, 
 
 35- . . 
 
 Logic of intension, possi- 
 bility of, 140; unnatural- 
 ness and complexity of, 
 141. 
 
 Logic of relatives, notion of 
 relation and, 219 ; Peirce on, 
 184 ; utility of, 229. 
 
 Logical calculus, problems 
 solved by, 2. 
 
 Logical constants, nature of, 
 198. 
 
 Logical existence, 164, 171 ; 
 MacColl's views on, 162. 
 
 Logical laws, Castillon on, 
 
 lOI. 
 
 Logical machinery', double use 
 of, 14. 
 
238 
 
 Index 
 
 mathe- 
 
 Logical processes; 
 
 matical and, 36. 
 Logical sequence, singular 
 
 propositions and, 18, 19. 
 
 MacColl, H., criticism by, 
 168, 169; formula adopted 
 by, 29 ; independent work 
 of, 23 ; on classification of 
 propositions, 28 ; on ex- 
 istence, 171 ; on statements 
 and propositions, 9 n. ; 
 on Universe of Discourse, 
 162 ; on use of literal 
 symbols, 21, 150; symboliza- 
 tion of propositional func- 
 tions by, 151 ; treatment of 
 modals by, 152. 
 
 Maimon, S., on notion of uni- 
 versality, 130; on proble- 
 matical judgments, 126. 
 
 Marquand, H., employment of 
 diagrams by, 82. 
 
 Material consequences, C. 
 Ladd-Franklin on, 18. 
 
 Mathematics, Logic involved 
 in, 192 ; new treatment of 
 propositions in, 9 n. ; Peano's 
 treatment of, 194 ; relation 
 of Symbolic Logic to, 200 ; 
 Russell's analysis of concep- 
 tions of, 199 ; Russell's defi- 
 nition of, 199, 200; validity 
 of, 199. 
 
 Mathematical conceptions, an- 
 alysis of, 198, 199. 
 
 Mathematical reasoning, Kant 
 on, 196. 
 
 Mathematical symbols, Boole's 
 use of, 35. 
 
 Methods of solution compared, 
 76. 
 
 Mill, J. S., on connotation, 
 138. 
 
 Minus sign, 45 ; Castillon's 
 double use of, 1 10. 
 
 Mitchell, O. H., Johnson's 
 criticism of, 179; on change 
 of terms into factors, 89 ; 
 on double quantifications, 
 178, 179 ; representation of 
 premises by, 161. 
 
 Modals, MacColl's treatment 
 of, 152; Pure Logic and, 
 153 ; Johnson's treatment of, 
 
 -7- 
 Molecular propositions, 173. 
 
 Multiple quantifications, 173. 
 
 Multiplication, Russell's defini- 
 tion of, 209. 
 
 Multiply - quantified proposi- 
 tions, synthesis of, 175. 
 
 Negation, Russell on, 211. 
 
 Negative attributes, 139. 
 
 Negative interpretation of pro- 
 positions, 54. 
 
 Negative terms, symbolization 
 of, 45. 
 
 New truths, nature of, 224. 
 
 Null-class, 165 ; Peano's defini- 
 tion of, 213. 
 
 Numbers, Frege's treatment 
 of, 217 ; Russell's definition 
 of cardinal, 207. 
 
 Nunn, T. P., on utility of Sym- 
 bolic Logic, 231. 
 
 Particular propositions, 
 compartments saved by, 
 66 ; elimination in case of, 
 72 ; existential character of, 
 57, 58 ; Jevons' representa- 
 tion of, 146 ; symbolization 
 of, 56 ; Venn's symbolization 
 of, 57. 
 
 Peano, G., connexion of in- 
 duction with principles of, 
 206 ; indefinables enumerated 
 by, 211 ; null-class defined 
 by, 213 : on distinction 
 between individuals and 
 
^UNIVERSITY ) 
 Index 
 
 239 
 
 classes, 203 ; primitive pro- 
 positions enumerated by, 
 212; Russell's criticism of, 
 211, 212; treatment of 
 Mathematics by, 194. 
 
 Peirce, C. S., formula of 
 simplification adopted by, 
 66; Johnson's criticism of, 
 178; on double quantifica- 
 tions, 177 ; on Logic of 
 Relatives, 184 ; on logical 
 expansion, 75 ; on plural 
 relations, 177 ; symbol for 
 subsumption used by, 53. 
 
 Philosopher, Symbolic Logic 
 useful to, 227. 
 
 Philosophy, Whitehead on 
 relation of Symbolic Logic 
 to, 204. 
 
 Philosophical duality, Symbolic 
 Logic and, 227. 
 
 Philosophical existence, 164. 
 
 Ploucquet, G., Lambert's coad- 
 jutor, 4. 
 
 Plural relations, Peirce on, 
 
 " Possibilities," nature of 
 MacColl's, 153. 
 
 Potency, syntheses vary in, 
 176. 
 
 Premises, symbolic representa- 
 tion of, 9 ; Venn's treat- 
 ment of, 65. 
 
 Primitive propositions, Peano's, 
 212. 
 
 Probability of an event, 26. 
 
 Probabilities, propositions re- 
 specting, 27. 
 
 Problematical judgments, Cas- 
 tillon on, 126. 
 
 Proof, Castillon's methods of. 
 
 Propositions, Castillon's de- 
 scription of, 97 ; chance 
 and, 158 ; combination of, 
 64; concerning probabilities, 
 
 27 ; extensive interpreta- 
 tion of, 92 ; existential 
 import of, 54 ; MacColTs 
 classification of, 28 ; indefi- 
 nite element in, 51 ; negative 
 interpretation of, 54 ; non- 
 existence of probably true, 
 25 ; physical combination 
 of, 11; predication view 
 of, 32; strength of, 155; 
 symbolization of, 150 n. 
 
 Propositional calculus, Frege's, 
 216. 
 
 Propositional functions, Rus- 
 sell on, 155 ; s>Tnbolization 
 of, 151. 
 
 Pure Logic, formulae of, ^o ; 
 modals and, 153. 
 
 Pure Mathematics, induction 
 and, 205. 
 
 Quantity, Castillon on notion 
 of, 129. 
 
 Realities, Universe of, 163. 
 
 Relation, mathematics and 
 notion of, 219. 
 
 Relational propositions, con- 
 trary of, 181 ; converse 
 of, 181. 
 
 Relative addition, Peirce on, 
 
 Relative multiplication, Peirce 
 on, 177. 
 
 Relative reasoning, dicta in, 
 187 ; Jevons on, 185 ; syl- 
 logistic treatment of, 186 ; 
 Venn on, 185. 
 
 Restriction, subduction may 
 be replaced by, 45. 
 
 Results, equivalence of, 55. 
 
 Ross, G. R. T., on disjunctive 
 judgments, 41. 
 
 Rules, mechanical application 
 of, y:,. 
 
240 
 
 Index 
 
 Russell, B., definition of addi- 
 tion given by, 207 ; definition 
 of numbers given by, 207 ; 
 differences between Frege 
 and, 216 ; mathematical in- 
 definables enumerated by, 
 199 ; on analysis of mathe- 
 matical conceptions, 199 ; on 
 definition of multiplication, 
 209 ; on distinction between 
 individuals and classes, 202, 
 203 ; on formation of general 
 propositions, 217 ; on logical 
 existence, 171 ; on meaning 
 of intension, 135 ; on nature 
 of induction, 205 ; on posi- 
 tion occupied by Symbolic 
 Logic, 138 n. ; on propositions 
 respecting existence, 171 ; 
 on propositional functions, 
 155 ; Peano criticized by, 
 211, 212 ; Pure Mathematics 
 defined by, 199, 200; view 
 of negation adopted by, 
 211. 
 
 Schroder, E., Adamson's 
 criticism of, 68 n. ; C. Ladd- 
 Franklin's answer to, 19, 
 20 ; compact methods of, 
 69 ; Inverse Problem solved 
 by methods of, 87 ; method 
 of elimination adopted by, 
 70, 71 ; on intelligibility of 
 intermediate processes, 40, 
 48 ; on logical expansion, 
 74; on method of finding 
 contradictory, 39; on secon- 
 dary use of literal sym- 
 bols, 17, 18 ; Operationskreis^ 
 34 ; representation of pre- 
 mises iDy, 161 ; symboliza- 
 tion of alternatives by, 
 42. 
 
 Schroder's methods, advan- 
 tages of, 67. 
 
 Science, Symbolic Logic and, 
 
 230. 
 Self-contradiction, nature of, 
 
 168. 
 Simplification, Boole's formula 
 
 of, 66 ; Peirce's formula of, 
 
 66. 
 Singular propositions, 173; 
 
 logical sequence and, 18, 
 
 19 ; treatment of, 201. 
 Solutions, diagrammatic, 82 ; 
 
 space occupied by, 21, 22. 
 " Some," Castillon's use of 
 
 word, 115. 
 Squaring, Boole's process of, 
 
 65. 
 Statements, events confused 
 
 with, 154; propositions dis- 
 tinguished from, 9. 
 Strength of propositions, 155. 
 Subalterns, 112, 117. 
 Subduction, restriction may 
 
 replace, 45. 
 Substitution, liability to error 
 
 in, -jZ. 
 Substitution of Similars, science 
 
 and, 222. 
 Subsumption, symbols for, 
 
 53- 
 Subtraction, exclusive notation 
 
 and, 38 ; intensive treat- 
 ment of, 140 ; symbolization 
 
 of, 139- 
 
 Syllogism, Castillon's treat- 
 ment of, 108 ; supreme law 
 of, 181. 
 
 Symbols, exclusive use of, 151 ; 
 primary use of literal, 10 ; 
 results denoted by, 36 ; 
 operations represented by, 
 
 36. . ^ 
 
 Symbols of operation, frequent 
 
 use of, 34 ; Keynes dispenses 
 
 with, 31. 
 Symbolic Logic, Castillon's 
 
 notion .of, 95 ; contributors 
 
Index 
 
 241 
 
 to development of, 6 ; 
 educational advantages of, 
 221 ; Frege's conception of 
 scope of, 215; imaginative 
 literature and, 206 ; indirect 
 utility of, 226 ; Jevons on 
 utility of, 222 ; new truths 
 reached by, 224 ; Nunn on 
 utility of, 231 ; philosophical 
 duality and, 227 ; relation 
 of Mathematics to, 200 ; 
 Russell's view of position of, 
 138 n. ; science and, 222, 
 230; use to philosopher of, 
 227 ; Whitehead on utility 
 of, 230, 231. 
 
 Symbolic Universe, MacColl 
 on, 162. 
 
 Symbolists, pre-Boolian, 3, 
 
 4- 
 
 Synthesis of molecular proposi- 
 tions, 173. 
 
 Synthesis of multiply-quantified 
 propositions, 175. 
 
 Systems, logical, i ; real and 
 superficial differences be- 
 tween, 23. 
 
 Terms, Boole's transposition 
 
 of, 64. 
 Things, application of numbers 
 
 to, 217. 
 Transposition of terms, 123. 
 " True," distinction between 
 
 "certain" and, 158. 
 Truth, prepositional, 13. 
 
 Universal affirmative, Jevons' 
 representation of, 145. 
 
 Universal dictum, non-exist- 
 ence of, 219, 
 
 Universal negative, Castillon's 
 representation of, 46, 105, 
 123. 
 
 Universe, compartment dis- 
 tinguished from, 166. 
 
 Universe of Discourse, Mac- 
 Coll on, 162. 
 Unrealities, Universe of, 163. 
 
 Vailati, G., on disjunctive 
 predicates, 202. 
 
 " Variables," common treat- 
 ment of, 158 ; nature of Mac- 
 Coil's, 153. 
 
 Venn, J., Boole's forms ex- 
 plained by, 67 ; Castillon 
 criticized by, 99 ; conclu- 
 sions reached by, 78 ; criti- 
 cism of Frege by, 214 ; 
 employment of diagrams by, 
 81 ; formula for elimination 
 given by, 72 ; interpretation 
 of Castillon by, 47, 120 ; 
 method of elimination des- 
 cribed by, 70 ; on Boole's ori- 
 ginality, 3 71. ; on Castillon's 
 universal negative, 107 ; on 
 elimination from particulars, 
 72; on exclusive use of literal 
 symbols, 21 ; on finding 
 contradictories, 80 ; on 
 hypothetical and disjunc- 
 tives, 59 ; on intelligibility 
 of intermediate processes, 
 49; on interpretation of alter- 
 natives, 40, 160 ; on Keynes' 
 methods, 32 ; on Logic based 
 on comprehension, 138; on 
 negative interpretation of 
 universals, 54 ; on perform- 
 ance of inverse processes, 40 ; 
 on relative reasoning, 185 ; 
 on secondary use of literal 
 symbols, 15 ; on symboliza- 
 tion of negative terms, 45 ; on 
 symbolization of particulars, 
 57 ; on treatment of singular 
 propositions, 201 ; treatment 
 of premises by, 65, 161. 
 
 Whitehead, A. N., on mul- 
 Q 
 
242 
 
 Index 
 
 tiplication, 209 ; on position 
 of inductive reasoning, 204 ; 
 on relation of Mathematics 
 to deductive reasoning, 203 ; 
 on relation of Philosophy to 
 Symbolic Logic, 204 ; on re- 
 lation of Symbolic Logic to 
 
 imaginative literature, 206 ; 
 on scope of deductive reas- 
 oning, 203, 204 ; on utility 
 of Symbolic Logic, 230, 
 231. 
 Wolf, Ch., influence of, 4 ; on 
 judgment, 106. 
 
 THE END 
 
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INDEX, 
 
 Abyssinia, Shihab al Din, 37. 
 Alcyonium, Liver/>ool Marine Biol. C. 
 
 Me»is., 50. 
 Algae, Cooke, 47. 
 
 America, Creation Myths of, Curtin, 57. 
 Americans, The, Miinsterberg, 30. 
 Anarchy and Law, Brewster, 29. 
 Anatomy, Cleland, 47 ; Cunninghavt 
 
 Mettioirs, 48 ; Lockwood, 51. 
 Anthropolog-y, Prehistoric, Avebtiry, 56 ; 
 
 Engelhardt, 57 ; Laing, 59. 
 Evolution of Religion, Farnell, 57. 
 Apocalypse, Bleek, 8 ; Clark, 16. 
 Apostles and Apostolic Times, Dob- 
 
 schiitz, 4 ; Hausrath, 18 ; Weinel, 
 
 5 ; Weizsdcker, 7 ; Zeller, 9. 
 Statutes of, edit. G. Horner, 26. 
 Apostolic Succession, Clark, 16. 
 Arabic, Grammar, Socin, 37. 
 
 Poetry, Faizidlah Bhai, 35 ; Lyall, 35 ; 
 
 Ndldeke, 36. 
 Ascidia, Liverpool Ma?-ine Biol. Mems., 
 
 50. 
 Assyrian, Dictionary, Muss-Arnolt, 36 ; 
 
 No7-ris, 36. 
 Grammar, Delitzsch, 34. 
 Language, Delitzsch, 34. 
 Assyriology, Brown, 56 ; Delitzsch, 10, 
 
 34 ; Evans, 35 ; Sayce, 13 ; Schrader, 
 9- 
 
 Astigmatic Tests, Praj/, 52 ; Snellen, 54. 
 Astronomy, Cunninghavt Mems., V., 
 
 48 ; Memoirs of Roy. Ast'ronom. 
 Soc, 62. 
 
 Augustine, St., Confessions of, Harnack, 
 
 18. 
 Babylonia, i^^^ Assyriology. 
 Belief, Religious, Upton, 15. 
 Beneficence, Negative and Positive, 
 Spencer, Principles of Ethics, II., 31, 
 Bible, 16. 
 
 .S"^^ a/i^(7 Testament. 
 Beliefs about, Savage, 25. 
 Hebrew Texts, 19. 
 History of Text, Weir, 27. 
 Plants, Henslow, 19. 
 Problems, Cheyne, 12. 
 Bibliography, Bibliographical Register,56. 
 Biology, Basiian, 46 ; Liveypool Marine 
 
 Biol. Mems., 50 ; Spencer, 31. 
 Botany, Bentham and Hooker, 46 ; 
 Church, 47 ; Cooke, 47 ; Grevillea, 
 
 49 ; /our. of the Linnean Soc, 49 ; 
 Prior, 52. 
 
 Brain, Cimningham Me7its., VII., 48. 
 Buddha, Buddhism, Davids, 14 ; Hardy, 
 
 35 ; Oldenberg, 36. 
 Calculus, Harnack, 49. 
 
 Canons of Athanasius, TVjt^ d?^ Trans. 
 
 Soc, 38. 
 Cardium, Liverpool Mar-ine Biol. Mems., 
 
 so- 
 Celtic, j^t «^<? Irish. 
 
 Stokes, 43 ; Sullivan, 44. 
 Heathendom, Rhys, 15. 
 
 Ceremonial Institutions, Spencer, Princ. 
 
 of Sociology, II., 31. 
 Chaldee, Grammar, Tnrpie, 38. 
 
 Lexicon, Fuerst, 35. 
 Chemistry, Berzelius, 46 ; Diitmar, 48 ; 
 Faraday, 49 ; Van't Hoff, 49. 
 Hindu, ^aj', 52. 
 Christ, Early Christian Conception of, 
 Pfleiderer, 13, 23. 
 Life of, Keifn, 8. 
 
 No Product of Evolution, Henslow, 19. 
 Study of, Robinson, 24. 
 Teaching of, Harnack, 6, 11. 
 The Universal, Beard, 16. 
 Christianity, Evolution of, Gz7/, 18. 
 
 History of, Baur, 8 ; Dobschiitz, 4 ; 
 Harnack, 6, 11, 18; Hausrath, 8, 
 19 ; Johnson, 20 ; Mackay, 21 ; 
 IVernle, 3. 
 in Talmud, Herford, 19. 
 Liberal, Rdville, 11. 
 Roots of, Martineau, 22. 
 Simplest form of, D7-uvimond, 14. 
 Spread of, Harnack, 5. 
 What is? Harnack, 6, 11. 
 Church, Catholic, Renan, 14. 
 
 Christian, Baur, 8; Cla?-k, 16; Z)^3- 
 
 schiitz, 4 ; Hatch, 14 ; IVernle, 4. 
 Coming, H%inter, 20. 
 Civic, Apathy, Hunter, 20. 
 Codex Palatino-Vaticanus, rt?^*^ Z^c- 
 
 tiires, III., 44. 
 Co6.mra,LiverpoolMarineBiol.Mems.,s^. 
 Coleoptera, Murray, 52. 
 Communion of Christian with God, Herr- 
 mann, 19. 
 Comte, Spencer, 32. 
 Constellations, Primitive, Broion, 56. 
 Cornish, Stokes, 43. 
 Creed, Christian, 16. 
 Crown Theological Library, 10. 
 Cuneiform Inscriptions, Schrader, 9. 
 Daniel and his Prophecies, C //. //. 
 Wright, 28. 
 Critical Commentary on, C //. //. 
 Wright, 28. 
 Danish Dictionary, Rosing, 43. 
 Darwinism, Schunnan, 30. 
 Denmark, Engelhardt, 57. 
 Doctrine and Principle, Beeby, 16. 
 Dogma, History' of, Harnack, 5. 
 
 of Virgin Birth, Lobstein, 10. 
 Domestic Institutions, Spencer, Princ. 
 
 of Sociology, I., 31. 
 Duck Tribes, Morphology of, Cunning- 
 ham Mems., VI., 48. 
 Dutch, Cape, Oordt, 43 ; Werner, 45. 
 Dynamics, Cnnninghavt Mems., IV., 48. 
 
 Chemical, Van't Hoff, 49. 
 Ecclesiastes, Taylor, 26. 
 Ecclesiastical Institutions, Spencer, 
 Princ. of Sociology, III., 31, 32. 
 of Holland, Wicksteed, 27. 
 Echinus, Liverpool Marine Biol. Mems., 
 50. 
 
INDEX— continued. 
 
 42 
 
 Economy, Political, Mackenzie^ 30. 
 
 Education, Herbert, 57 ; Lodge, 
 Spencer, 32. 
 
 Educational Works, see Special Cata- 
 logue. 
 
 Egypt, Religion of, Renouf, 15. 
 
 Egyptian Grammar, E7»ian, 35. 
 
 Enoch, Book of. Gill, 18. 
 
 Epidemiology, Trans. 0/ Epidemiolog. 
 Soc, 55. 
 
 Epizootic Lymphangitis, Treatise on, 
 Pallin, 52. 
 
 Ethics, and Religion, Martineau, 22. 
 Data of, Spencer, Principles of E., I., 
 
 Induction of, Spencer, Principles of E., 
 
 I., 31. 
 
 Kantian, Schurman, 30. 
 
 of Evolution, Schurman, 30. 
 
 of Individual Life, Spencer, Principles 
 
 ofE., I., 31. 
 of Reason, Laurie, 29. 
 Principles of, Spencer, 31. 
 Ethnologfy, Cunningha?)i Menis., X., 48. 
 Evolution, Spencer, 31, 32. 
 
 of the Idea of God, D' Alviella, 14. 
 of Religious Thought, U Alviella, 15. 
 Exodus, Hoerning, 20. 
 Ezekiel, Mosheh ben Shesheth, 22. 
 Faith, Herrmatin, 12 ; Rix, 24 ; Wim- 
 
 nter, 27. 
 Fisheries, British, Johnstone, 49. 
 Flinders Petrie Papyri, Cunningham 
 
 Mems., VI 1 1., IX., 48. 
 Flora of Edinburgh, Sonntag, 54. 
 French, Bo'ielle, 40 ; Delbos, 41 ; Engine, 
 41 ; Hugo, 41, 42 ; Rogct, 43 ; also 
 Special Education Catalogue. 
 Literature, Roget, 43. 
 Novels, Ar-niy Series, 39. 
 Fungi, Cooke, 47 ; Grevillea, 49. 
 Genera Plantarum, Bentham and 
 
 Hooker, 46. 
 Genesis, Hebrew Texts, 19, 35 ; Wright, 
 
 C. H. H., 27. 
 Geography, Ancient, Kiepert, 58. 
 Geometry, Spencer, W. G., 54. 
 German, Literature, Nibelungenlied, 
 42 ; Phillipps, 43. 
 Novels, Arfuy Series, 39. 
 Germany, Marcks, 59. 
 God, Idea of, D' Alviella, 14. 
 Gospel, First, Plain Commentary, 23. 
 
 Fourth, Dru7>imond, 17 ; Tayler, 26. 
 Gospels, Lost and Hostile, Gould, 18. 
 
 Old and New Certainty, Robinson, 24. 
 Greek, Modern, Zompolides, 45. 
 Gymnastics, Medical, Sch?-eber, 54. 
 Health, Herbert, 49. 
 Hebrew, Biblical, Kennedy, 35. 
 Language, Delitzsch, 34. 
 Lexicon, Fuerst, 35. 
 New School of Poets, Albrecht, 36. 
 Scriptures, Sharpe, 25. 
 Story, Peters, 23. 
 Synonyms, Kefinedy, 35. 
 Text of O.T., ^^^i>, 27. 
 Texts, 19, 35- 
 
 Hebrews, History of, Kittel, 6 ; Peters, 
 12 ; Sharpe, 25. 
 Religion of, Kuenen, 9 ; Montefiore, 14. 
 Heterogenesis, Bastian, 46. 
 Hibbert Lectures, 14, 15. 
 Hygiene : How to Live, Caton, 47. 
 Hymns, Jones, 20. 
 Icelandic, Lil/a, 42 ; F^^a Gluvis Saga, 
 
 . 44- 
 
 Dictionary, Zoega, 45. 
 
 Grammar, Bayldon, 40. 
 Individualism, Spencer, Man z/. State, 32. 
 Irish, Atkinson, 40; ^£i^^ of Ballymote, 
 
 40 ; i5o(?/^ <y^ Leinster, 41 ; Hogan, 
 
 41 ; Leabhar Breac, 42 ; Leabhar 
 na H- Uidhri, 42 ; O' Grady, 43 ; 
 Stokes, 43 ; Todd Lectures, 44 ; 
 Yellow Book of Lecan, 45. 
 
 Isaiah, Diettrich, 34 ; Hebrew Texts, 19, 
 
 35- 
 Israel, History of, Kittel, 6 ; Peters, 23 ; 
 Sharpe, 25. 
 Religion of, Kuenen, g. 
 in Egypt, Wright, G. H. H., 28. 
 Jeremiah, Mosheh ben Shesheth, 22. 
 Jesus, Life of, Keiin, 8. 
 The Real, Vickers, 27. 
 Times of, Hausrath, 8. 
 6"^^ a/5tf Christ. 
 Job, Book of, Ewald, 8 ; Hebrew Text, 
 19, 35 ; Wright, G. H. H, 28. 
 Rabbinical Comment, on, Z'^jr^ &* 
 Trans. Soc, 38. 
 Justice, Spencer, Princ. of Ethics, II., 
 
 31, 32. 
 Kant, Schurman, 30. 
 Kindergarten, Goldammer, 57. 
 Knowledge, Evolution of, Perrin, 30. 
 Labour, Harrison, 57 ; Schloss, 59 ; 
 
 Vynne, 60. 
 Leabhar Breac, 42; Atkinson, 40; 
 
 Hogan, 41. 
 Leprosy, Abraham, 46. 
 Life and Matter, Lodge, 21. 
 Lives of the Saints, Hogan, 41. 
 Logarithms, 5a«^, 53 ; Schroen, 54 ; 
 
 ^t'^«, 55- ^ 
 London Library Catalogue, 57. 
 Lumbar Curve, Ctmningham Mems., 
 
 II., 48. 
 Mahabharata, Sdrensen, 37. 
 Malaria, Annett, 46 ; Boyce, 47 ; Duttoft, 
 
 43 ; Mems. of Liverpool School of 
 
 Tropical Medicine, 51 ; ^d?^j, 55 ; 
 
 Stephens, 54. 
 Maori, Dictionary, Williams, 45. 
 
 Manual, Maori, 42. 
 Materialism, Martineau, 22. 
 Mathematics, Hamack, 49 ; Spencer, 54. 
 
 5^£? «/^(? Logarithms. 
 Mediaeval Thought, /"t?^/?, 23. 
 Mesca Ulad, Todd Lectures, I., 44. 
 Metaphysics, Laurie, 29. 
 Mexico, Religions of, Reville, 15. 
 Micah, Book of, Taylor, 26. 
 Microscopy, Journal of the Roy. Micro. 
 
 Soc. , 50 ; Journal of the Quekett 
 
 Micro. Club, 50. 
 
INDEX— ( 
 
 Midrash, Christianity in, Iler/ord, 19. 
 Mineral Systems, Chapman, 47. 
 Monasticistn, Harnack, 18. 
 Mosquitoes, Mems. of Liverpool School 
 
 0/ Trop. Medicine, 51. 
 Mythology, American, Curtin, 57. 
 Greek, Broivn, 56 ; St. Clair, 59. 
 Northern, Stephens, 60. 
 Myxomycetes, Cooke, 47. 
 Natural Selection, Spencer, 32. 
 Nautical Terms, Delbos, 41. 
 Nennius, The Irish, Ilogan, 41. 
 New Guinea, Cunningham Mems., X., 48. 
 New Testament, see Testament, 26. 
 New Testament Times, Hausrath, 8, 19. 
 Nitidulariae, Murray, 52. 
 Norwegian Dictionary, Larsen, 42 ; 
 
 Rosing; 43. 
 Ophthalmic Tests, Pray, 52 ; Snellen, 54. 
 Origins, Christian, Johtison, 20. 
 
 of Religion, Hibbert Lectures, 14, 15. 
 Pali, Dipavamsa, 34 ; Milanda Panho, 
 36 ; Vinaya Pitakam, 38. 
 Handbook, Frankfurter, 35. 
 Miscellany, 36. 
 Pathology, Inflammation Idea in, /vrt«- 
 
 soni, 52. 
 Paul, St., ^««^. 8; Pfeiderer, 9; 
 
 IVeinel, 5. 
 Persian, Avesti Pahlavi, 34. 
 
 Grammar, Plaits, 37. 
 Peru, Religions of, Reville, 15. 
 Philo Judaeus, Dnanmond, 29. 
 Philosophy, 29. 
 
 and Experience, Hodgson, 29. 
 Jewish Alexandrian, Drtim/notid, 29. 
 of Religion, Pfleiderer, 9. 
 Reorganisation of, Hodgson, 29. 
 Religion of, Perrin, 22. 
 Synthetic, Collins, 29 ; Spencer, 31. 
 Phyllotaxis, Church, 47. 
 Plague, Boghurst, 46. 
 Political Institutions, Spencer, Prmc. of 
 
 Sociology, II., 31. 
 Prayers, Common Prayer, 16 ; Jones, 
 20 ; Personal, 22 ; Sadler, 24 ; T'^w 
 Services, 26 ; Vizard, 27. 
 Prehistoric Man, Avebury, 56; Engel- 
 
 hardt, 57 *, Laing, 59. 
 Printing at Brescia, Peddle, 59. 
 Professionallnstitutions, Spencer, Princ. 
 
 of Sociology, III., 31' 
 Profit-sharing, Schloss, 59. 
 Prophets of O.T., Ewald, 8. 
 Protestant Faith, Hermann, 12 ; 
 
 Rtoille, II. 
 Psalms, Hebrew Texts, 19, 35. 
 and Canticles, T^w Services, 26. 
 Commentary, Ewald, 8. 
 Psychology, y»/z«rf, 30; Scripture, 30; 
 IVundt, 33. 
 of Belief, Pikler, 30. 
 Principles of, Spencer, 31. 
 Reconciliation, Hens low, 19. 
 Reformation, Beard, 14. 
 Religion, Child and, 13. 
 
 History of, Kuenen, 9, \/^;RSville, 9, 15. 
 of Philosophy, Perrin, 23. 
 
 continued. 
 
 Religion, Philosophy of, Pfleiderer, 9. 
 
 Struggle for Light, Wimmer, 11. 
 
 ^'^■^ «to Christianity, History of. 
 Religions, National and Universal, 
 Kuenen, 21. 
 
 of Authority, Sabatier, 4. 
 Resurrection, Macan, 22 ; Marchant, 22. 
 Reviews and Periodical Publications, 
 
 61. 
 Rigveda, IVallis, 38. 
 Rome, Renan, 15. 
 Runes, Stephens, 60. 
 Ruth, ^Fr4-A^, C. //. //., 27. 
 Sanitation, in Cape Coast Town, Taylor, 
 
 55- 
 
 in Para, Notes, 52. 
 Sanscrit, Ahhidha^iaratnamala, 34; 
 
 Sorensen, 37. 
 Self-Aid, in War, Caver-hill, 47. 
 Sermons, Beard, 16 ; Broadbent, 16. 
 Services, Common Prayer, 16 \ Jones, 20 ; 
 
 7"^« Services, 26. 
 Silva Gadelica, O'Grady, 43. 
 Social Dynamics, Mackenzie, 30. 
 
 Statics, Spencer, 32. 
 Sociology, Descriptive, Spencer, 32. 
 
 Principles of, Spencer, 31. 
 
 Study of, Spencer, 32. 
 Solomon, Song of, Reville, 23. 
 South Place Ethical Society, Conway, 
 
 17- 
 Spanish Dictionary, Velasquez, 44. 
 Spinal Cord, Bruce, 47. 
 Sternum, Paterson, 52. 
 Storms, Piddington, 52. 
 Sun Heat, Cunninghavi Mems., III., 48. 
 Surgery, System of, &<?« Bergmann, 46. 
 Syriac, Ber-nstein, 34 ; Diettrich, 34 ; 
 
 Noldeke, 36. 
 Taal, Afrikander, Oordt, 42 ; Werner, 45. 
 Talmud, Christianity in, Herford, 19. 
 Tennyson, JF^/^, 60. 
 Testament, New, Commentary, Protes- 
 ta7it Commentary, 9. 
 Textual Criticism, Nestle, 7. 
 Times, Hausrath, 8, 19. 
 6"<r^ rt/^o Gospels. 
 Testament, Old, Cuneiform Inscriptions, 
 Schrader, 9. 
 Literature of, Katttzsch, 21. 
 Test Types, /"r^^, 52 ; Snellen, 54. 
 Theism, r^y^-o', 27. 
 Theological Translation Library, 3. 
 Theology, Analysis of, ^/^^, 18. 
 
 History of, Pfleiderer, 9. 
 Truth, Search for, Horton, 20. 
 Trypanosomiasis, Button, 48. 
 Virgil, Henry, 57. 
 Virgin Birth, Lobstein, 10. 
 Weissmann, Spencer, 32. 
 Woman's Labour, Englishwoman's 
 Review, 57; Harrison, 57 ; I'ynne, 60. 
 Suffrage, Blackburn, 56. 
 Yellow Fever, Durham, 48. 
 Zoology, Fasciculi Malayenses, 49 ; 
 Journal of the Linnean Soc, 49 ; 
 Liverpool Marine Biology Com- 
 tnittee Mems., 50. 
 
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