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 — ^)\ 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 ., " 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 Printed by Ballantvne, Hanson Edinburgh &^ London Co. A Catalogue of Williams & Norgate's Publications Divisions of the Catalogue I. THEOLOGY II. PHILOSOPHY, PSYCHOLOGY III. ORIENTAL LANGUAGES, LITERATURE, AND HISTORY IV. PHILOLOGY, MODERN LANGUAGES V. SCIENCE, MEDICINE, CHEMISTRY, ETC. VI. BIOGRAPHY, ARCHAEOLOGY, LITERATURE, MISCEL LANEOUS FULL INDEX OVER PAGE London Williams & Norgate 14 Henrietta Street, Covent Garden, W.C. PAGE 3 2i9 34 39 46 56 INDEX, Abyssinia, Shihab al Din, 37. Alcyonium, Liver/>ool Marine Biol. C. Me»is., 50. Algae, Cooke, 47. 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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 «^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. 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