The Philosophy of Logic of Hugh Maccoll John Spencer the Philosophy of Logic of Hugh Maccoll

·.

THE PHILOSOPHY OF LOGIC OF HUGH MACCOLL JOHN SPENCER THE PHILOSOPHY OF LOGIC OF HUGH MACCOLL

Department of Philosophy Ph.D. Candidate

The Introduction to this study sketches the history of propositional modal logic from Aristotle to the nineteenth century. MacColl's life and influences are also described. The first chapter traces out the development of the concept of implication in MacColl. Bince implication is central to MacColl's logic, this chapter also serves as a commentary on the development of his logic as a whole. A quasi-axiomatic system is presented which represents MacColl's completed system. In developing his system, MacCol1 divided statements into true, false, certain, impossible and variable. The second chapter examines in detail these categories of statements. Chapter Three examines MacColl's theory of logical existence. MacCol1 divided the universe of discourse into the sub-universes of realities and unrealities. By doing so he created a two-sorted theory of quantification. He adrnitted into the uni verse of discourse possible though non-existent objects. The conclusion com pares sorne of C.I. Lewis's central views in logic with those of MacColl. It is argued that MacCol1 anticipated a great deal of Lewis and that it is not implausible to suggest that MacCol1 directly influenced Lewis. It is suggested that MacCol1 should be regarded as one of the founders of modern modal logic. THE PHILOSOPHY OF LOGIC OF HUGH MACCOLL

JOHN R. SPENCER

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements for the degree of Doctor of Philosophy.

Department of Philosophy

McGill University

..... ,', . ., •. _ ._" 1 - -- .:' : ••\ ...... J 1 . -,

cv J obn R. Spencer 1973 PREFACE

This thesis is submitted in partial require ment for the Ph.D. Degree at McGi11 University. It is the first full-Iength study of the logic of Hugh MacColl. After a detailed exarnination of MacColl's logic, it is argued that he is one of the founders of modern modal logic.

This thesis was prepared under the direction of Professor John Trentrnan to whom l would like to express my appreciation for his advice and encouragement. Professor Harry Bracken also read drafts of the thesis and made many helpful suggestions. TABLE OF CONTENTS

Chapter Page

Introduction i xvi

1- Implication 1 2. Division of Statements 58 3. Existential Import 94 4. MacColl's Influence 118 Bibliography 132

" INTRODUCTION

Hugh MacColl developed a symbolic modal logic in the last quarter of the nineteenth century. Although his work was highly original it does fit squarely into a tradition dating from antiquity, a tradition of which

MacColl was almost totally ignorant. In the following

few pages this tradition will be briefly sketched, touching only on those points which relate directly to MacColl.

Modal logic begins, as does formaI logic itself, as a general and systematic science, with Aristotle. In chapters twelve and thirteen of De Interpretationel Aristotle considers the relations between 'possible to bel and

'possible not to bel, 'necessary' and 'impossible' and

'admissible' and 'not admissible'. He first attempts to arrive at the proper negation for 'possible to bel. He considers 'possible not to bel as the negation of 'possible to bel but this is rejected since it is clearly possible for the same thing to be and possibly not to be. If these were contradictory expressions they could not apply to the same thing; hence, 'possibly not to bel cannot be the contradiction of 'possibly to bel. Similarly with 'admissible to bel, 'necessary' and 'impossible' whose negations are 'not admissible to bel, 'not necessary' and 'not impossible'. The

1 Aristotle's Cate ories and De Inter retatione Translated by J. L. Ackrl.ll. Oxford: Clarendon Pres,s, 1963). negation of 'possible not to bel is 'not possible not to bel. After noting this Aristotle then writes:

This is why 'possible to bel and 'possible not to bel may be thought actually to follow from one another. For it is possible for the same thing to be and not to be; such statements are not contradictories of one another. But 'possible to bel and 'not possible to bel never hold together, because they are opposites.2

This is important since it is clear that simply from the fact that 'possible to bel and 'possible not to bel are not contradictories it does not follow that they follow from one another.

Aristotle then raises the question of whether 'possible to bel follows from ' necessary to bel. If it does not then 'not possible to bel would, reasons Aristotle. But, he notes, it cannot be the case that if something is necessary it is not possible. Therefore, 'possible to bel would appear to follow from 'necessary to bel. But Aristotle has already stated that 'possible not to bel follows from 'possible to be'; hence, 'possible not to bel would follow from 'necessary to bel which is absurd and which would also conflict with Aristotle's claim that "what is of necessity is in actuality".3

2 Ibid., p. 61

3 Ibid., p. 68 -iii-

Aristotle does not successfully resolve this difficulty in De Interpretatione; he does not see clearly that he is dealing with two senses of possible, one of which expresses the concept of contingency, the other of not being impossible. He is, however, aware that there is a problem here.

In the Prior Analytics Aristotle clears up this confusion somewhat by noting that there are two senses of 'possible'; a proposition is possible if it is not necessary and, being assumed, results in nothing impossible, and a proposition is possible if it simply follows from one which is necessary.4 He then notes that the expressions lit is not possible to belong', lit is impossible to belong' and 'i-c is necessary not to belong' are either idential or follow from one another. Likewise with the expressions lit is possible to belong', lit is not impossible to belong' and lit is not necessary not to belong'. He then adds "That which is possible then will be not necessary and that which is not necessary will be possible".5 This indicates that the sense of 'possible' Aristotle is working with is that often expressed by 'contingent' rather than that expressed by simply 'not impossible'.

4 Aristotle, Analytica Priora. The Works of Aristotle Translated into English ed. W. D. Ross. Vol. 1. (London: Oxford University Press, 1928) 32a, 18-21 5 Ibid., 32a, 21. - iv-

Although the treatment of modal syllogisms found in the Prior Analytics is mainly beside the point for the purposes of this study (since it is primarily concerned with the development, in MacColl, of a propositional modal logic), several theses of interest to it are found in De Interpretatione. For instance, from 'not possible not to bel follows 'necessary to bel; and from 'not possible to bel, 'impossible to be,.6 These theses are at the center of the first systematic working out of the relations among modal concepts.

The Megarian and Stoic logicians also attempted to work out certain modal ideas and gave definitions of implication using modal concepts. According to Boethius Diodorus gave the following account of modal concepts:

Diodorus defines the possible as that which either is or will be • • • the impossible as that which, being false, will not be true • . • the necessary as that which, being true, will not be false • and the non-necessary as ~hat which either is already or will be false.

The above definitions carry a strong temporal flavour which is also found in MacColl's earliest comments on modal concepts. As is the case with MacColl's early work there is a strong tendency in Diodorus to collapse the distinctions between necessarily true and (merely) true and impossible and

6 Ibid., 22a 7 William Kneale and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 1968) p. 117. - v -

(merely) false. This temporal interpretation is also carried over to the conclusion of Diodorus's Master Argument which is "nothing is possible which neither is nor will be true". 8

Boethius gives the following account of Philo's views which differ greatly from those of Diodorus.

Philo says that the possible is that which by the intrinsic nature of the assertion admits of truth • • • , as when, for example l say that l shall read the Bucolics of Theocritus again today. If no external circumstances prevent·· this, then considered in itself ••• the thing can be affirmed truly. In the same way this same Philo defines the necessary as that which, being true, can never, considered in itself, admit of falsity. The non-necessary he explains as that which, considered in itself, can admit of falsity, and the impossible as that which by itsg intrinsic nature • • • can never admit of truth.

The views expressed here are very close to what is found in MacColl's later writing. Here the modal concepts are defined without the aid of temporal notions and, as Kneale points out, it would appear that possibility, defined as self-consistency, is the basic modal idea while the others are defined in terms of it.

For the purposes of this study one of the most interesting debates in ancient logic was that over the proper

8 Ibid., p. 119

9 Ibid., p. 122 -vi- analysis of conditional propositions. IO Philo defined an implication as a conditional which is true when and only when it does not have a true antecedent and a false consequent. This concept of implication, which is the same as what is now usually referred to as "material implication", appeared paradoxical to several ancient authors just as it did to MacColl in the nineteenth century. It was opposed by Diodorus, who claimed that a conditional proposition is true

"if it neither is or ever was possible for the antecedent to be true and the consequent false".ll Thus Diodorus introduces modal terms into the definition of implication and, since his modal concepts are defined temporaly, these temporal notions enter into his definition. This is, as will be seen, a position close to one that MacColl adopts at one period in his career. A third view was put forward by Chrysippus, whereby a conditional holds whenever the denial of its consequent is incompatible with its antecedent. 12 If 'incompatible' is taken to mean that both the antecedent and the negation of the consequent cannot (logically) be true, then this account of implication cornes very close to what MacColl put forward as his final version of implication.

10 These remarks are based on Benson Mates, Stoic Logic (Berkeley and Los Angeles:U. of California Press, 1961) pp. 42-51 Il Ibid., p. 44 12 Ibid., p. 48 -vii-

This is also close to what is usually, or often, referred to as "strict implication". An exarnp1e quoted by ~1ates is

"if it is day, then it is day".13 This clearly is true for "aIl possible worlds". That is, it does not depend for its truth on any temporal or spatial restrictions. This example, as Mates notes, supports the c1aim that this is the ancient version of what is now called strict implication.

The problem of what is the proper ana1ysis of a conditiona1 also occupied the attention of severa1 Medieval

10gicians. Again we find competing views which are simi1ar to those ear1ier put forward by Philo and Diodorus and again we find that temporal notions are worked into the definition of implication. For instance Buridan defines 'imp1ies' in the fol10wing way:

One proposition is antecedent to another, if it is so related to that other that, both propositions being stated, it is impossible that whatever the first signifies to be so, is so, and whatever the second signifies to be so, is not so. 14

This definition, which is very similar to the definition also given by the pseudo-Scotus, would appear to be that of strict implication. The difficulty with that interpretation, however, is that it is not clear that the Medievals meant by

'impossible' what is meant by that term in the definition of

13 Ibid., p. 48

14 Ernest A. Mody, Truth and Consequence in Mediaeval Logic (Amsterdam: North-Holland, 1953) p. 67 -viii- strict implication.

A distinction was made, however, between those conditionals which are valid "simply" and those which are valid only "as of now", which appears to correspond with the distinction between the Diodorean and the Philonian concepts of implication. Ockham puts this point as follows:

A consequence is valid as of now, when its antecedent can be true and its consequent false at sorne time, though not at this time • • • But a consequence is simply valid when for no time could its antecedent be true and its consequent not true. Thus this consequence is simply valid, 'no animal is running, therefore no man is running', because the antecedent never could be true without the consequent being true. 15

A point of considerable interest here is that, not only did the Medieval logicians give definitions of implication which were re-discovered in the nineteenth century, but they were also aware of the paradoxes which result from the different definitions. Buridan gives as rules for consequences which are "simply" valid: (1) An impossible proposition implies any proposition, and (2) a necessary proposition is implied by any proposition;and for consequences which are valid "as of now": (1) A false proposition implies any proposition and (2) a true proposition is implied by any proposition.16 These rules, of course, correspond to the paradoxes of strict and material implication respectively.

15 Ibid., pp. 74-75 16 Ibid., p. 74 - ix-

The period from the close of the Middle Ages to the nineteenth century saw a general decline in the amount and quality of research in logic. Although work in logic was done during this time, little of it matched the scope and origina1ity of the work of the earlier and 1ater periods. The notable exception during this time was Leibniz. Since, however, most of Leibniz's work in logic remained unpublished unti1 considerably after his death and since he did not work directly on modal logic, his work is of little relevance to a consideration of MacCo1l's antecedents. It should be noted in passing however that the Leibnizian definition of a necessary statement as one which is true in aIl possible wor1ds has played a role in more recent developments in the study of semantics for modal logic.

The nineteenth century witnessed a tremendous revival of interest informaI 1ogic. Foremost among the pioneers during the earlier part of the century was George Boole whose The Mathematical Analysis of Logic (1847) and An Investigation of The Laws of Thought (1854) provided the foundations for much of the work which followed. Boole had sought to construct a calculus of 10gic on an analogy wi th al']ebra which would be free from epistemological and psychological entanglements. His work was influenced by such writers as De Morgan and Sir William Rowan Hamilton17

17 cf. Knea1e and Knea1e, The Development of Logic, p. 404. - ~ but Boole, along with most of his contemporaries, was not very familiar with, and hence not directly influenced by, most of ancient and medieval logic. Boole was more directly influenced by mathematics, and especially probability theory, than he was by earlier efforts in logic.

Indeed the only notable logician during the nineteenth century who was not substantially ignorant of earlier work was C. S. Peirce who had done research into Stoic and Medieval

logic.

The Algebra of Logic, as it had come to be called, developed from Boole to Schroder, who published his main work in the early part of the twentieth century, with important contributions having been made by W. S. Jevons, Peirce and

MacColl in his early writings. Although at first they went unnoticed by logicians the writings of Gottlob Frege in the

last quarter of the nineteenth century came to exert a tremendous influence on logic. Frege had developed a rigorous

deductive system which was based on the logic of propositions

and from which it was hoped that aIl of mathematics could be deduced, or, more precisely, from which a model of mathematics could be deduced. The value of his work was

recognized by Bertrand Russell in Britain who incorporated much of it into his own work which culminated in the writing, with A.N. Whitehead, of Principia Mathematica (1910-1913). - xi -

The important point to be made here concerning MacColl's immediate predecessors and contemporaries is that they aIl were developing extensionalist systems of logic, which were based, first, on the logic of classes where classes are defined extensionally and, later, on the truth values of propositions. It is also important to remember that, with the notable exception of Peirce, they were not very familiar with earlier work in logic, especially the intensional definitions of implication which were discussed by ancient and medieval logicians. AIso, they generally either saw logic as a branch of mathematics or as a basis from which mathematics could be deduced. In either case mathematics and logic were seen as being intimately connected.

Consistent with this approach logicians of this period regarded terms such as 'necessary' and 'possible' as not being logical terms at aIl but rather as epistemological or, in sorne cases,psychological terms, which have no place in formaI logic. For instance in his Begriffsschrift (1879) Frege.wrote the following"

If l term a proposition 'necessary', then l am giving a hint as to my grounds for judgment. But this does not affect the conceptual content of the judgment: and therefore the apodeictic form of a judgment has not for our purposes any significance.

If a proposition is presented as possibl~, then either the speaker is refraining from judgment, and indicating at the same time that he is not acquainted with any laws which the negation of -xii -

the proposition wou1d fo110Wi or else he is saying that the negation of the proposition is in general fa1se. 18

A similar view is expressed by the British 10gician W.E. Johnson, who in 1892 wrote the following:

Moda1ity refers to the grounds on which thè thinker forms his judgment. It, therefore, expresses a relation between the thinker on the one hand and a certain proposition on the other hand. The rea1 terms, then, of the modal proposition are the thinker and his relation to sorne judgment which is propounded to him. Thus the proposition, "s must be plI, asserts (say) that, "Any rational being is bound by his rationa1ity to judge that S is P". 19

It was in such a c1imate of opinion that MacColl wrotei at first he worked within the dominant framework of extensionalist 10gic on1y 1ater moving toward an intensional, or modal, system. When one considers the great developments in logic that occurred in the nineteenth century, it is not surprising that MacCol1's later work, which appeared to many of his contemporaries as being seriously out of step with the new developments, was not weIl received.

Little is known of MacColl's life: he was born in Great Britain in 1836 and at the age of twenty-nine (1865) he moved to Boulogne-sur-Mer in France where he lived until

18 Translations from the Philosophical Writings of Gottlob Frege, by Peter Geach and Max Black (Oxford: Blackwell, 1966) pp. 4-5. 19 W. E. Johnson, "The Logical Calculus", Mind N. S. Vol. l, (1892) pp. 18-19. -xiii-

took his death in 1909 at the age of seventy-three. He student, his degree at the University of London as a private classes, i.e., he did not study at any co11ege, or attend any Arts passing the Matricu1ation exam in 1873, Intermediate 20 He earned in 1874, and the B.A. Pass examination in 1876. If his living by tutoring mathematics in Boulogne-sur-Mer. since he had a private income at a11 it was not substantia1 or he cou1d not afford to subscribe to phi1osophica1 journa1s to purchase many books on logic.

Short1y after he took his degree MacCo11 began Times pub1ishing articles on logic, first in the Educationa1 of the and 1ater his more important papers in the Proceedings first London Mathematica1 Society and in Mind. MacCo11 was 1ed to logic by attempting to solve a prob1em on probabi1ity After which appeared in the Educationa1 Times for 1871. found severa1 frustrating starts on this prob1em MacCo11 He himse1f on logica1 rather than mathematica1 ground. describes this point in his career as fo11owsi- The on1y book on logic that l possessed ,[in 1877] was Prof. Bain's work; and to this l turned. The resernb1ance which my method bore to Boo1e's, as therein described, of course struck me at once; but Boo1e's treatment of the sy110gism was more 1ike1y to put me on the wrong track than to he1p me. As my most e1ementary symbo1s den6tèd statements, not necessari1y connected with quantity

was 20 The information concerning MacCo11's academic record obtained by 1etter from A. H. Wesencraft, reference 1ibrarian, University of London. - xiv-

at aIl, l could not see how the syllogism, with its ever recurring aIl, sorne, none, could be brought within the reach-or-my method. 21 He also notes in his second paper to the Mathematical 22 Society (1878) that he had not yet read Boole. The development of his ideas on logic is discussed in detail in the following chapt~r but it should be noted here that his first important contribution to logic (taking the logic of propositions as basic) was made cornpletely independently of the work of any other logician. Indeed because of his physical isolation and lack of money for books almost aIl of MacColl's work was done with a minimum of contact with other logicians. This may account, in part at least, for his developing a modal system,which was out of step with other developments taking place in logic. However, this can only remain speculation.

MacColl published several papers on logic between

1877 and 1882. He then abandoned aIl research in logic for sixteen years, returning to it in 1896. Articles appeared by him from that point until 1910, one year after his death. In the years between 1882 and 1896 MacColl concentrated on teaching mathematics and also wrote two novelsi Mr. Stranger's Sealed Packet (1888) the story of a trip to the planet Mars in the style of Jules Verne, which was translated into

21 Hugh MacColl, "On the Growth and Use of a Symbolical Language", Memoirs of the Manchester Literary and Philosophical Society Third Series Vol. 7 (l882) p. 243 22 Hugh MacColl, "The Calculus of Equivalent Statements", Proceedings of The London Mathernatical Society Vol. 9 (1877-78) p. 178. - xv - several languages, and Ednor Whitlock (1891).23 MacColl published one other non-logical work, Man's Origin, Destiny and DutY which appeared in 1909.

Shortly after his return to logic MacColl began corresponding with Bertrand Russell. Most of the logical points discussed were subsequently published, but there are sorne revealing comments made in these letters, which date from June 22, 1901 unti1 December 18, 1909, shortly before MacColl's death. It is clear that MacCo11 regretted not having contact with other logicians. In ear1y letters MacColl inquires into the possibility of obtaining a University Post in Great Britain and asks Russell to "kindly think of me" should an opening arise at the University of London. Apparently these inquiries met with no success. Later, in a letter dated January 26, 1905, MacColl remarks that the differences between him and Ruseell appeared to be "more verbal than real" and that if only they could discuss their differences in a meeting they would probably resolve them. There is no record of this meeting having taken place. The letters from MacColl to Russell are now avai1able in the Bertrand Russell Archives in McMaster University. Unfortunately Russell's replies, along with aIl of MacColl's papers, appear to be irretrievably lost. The house in which MacColl lived at the time of his death was comp1etely destroyed in the

23 The Times (London) Wed. 29 December 1909, Obituary columh. - xvi - war of 1939-45, and none of his papers have been discovered e1sewhere. ONE

IMPLICATION

Summary. This chapter begins with a general account of MacColl's approach to logic in his early period, i.e., the years between 1877-1880. It is seen that he was then workinq in the tradition of two-valued extensionalist logic. Workinq within this frarnework he made an important innovation basinq logic on the logic of propositions rather than classes. Problems with his use of such terms as 'statement' and 'proposition', as weIl as the problem of whether he is committed to sorne form of psychologism are noted in passinq. These problems are dealt with in detail in subsequent chapters.

MacColl's definition of the implication relation between propositions is central to his logic. The evolution of his definition from a truth-functional to an intensional definition is traced out in detail.

Where modal terms are used in connection with his definition of implication the clarification of their exact meanings is left for a subsequent chapter. In this chapter they are discussed only enough to make clear what he was attempting to do with his definition of implication.

MacColl's criticisms of· truth-functional systems of logic and the criticisms directed at MacColl by logicians who adhered to truth-functional systems are noted in detail.

- 1 - - 2 -

Finally a quasi-axiomatic system of logic is presented which is claimed to be a faithful reworking of MacColl's logic as it appeared in various journal articles and one book.

MacCol1 gives a very general description of the state of logic and of his role as a logician in the first of his series of papers in Mind. Here he describes logic as "the general science of reasoning in its most abstract sense • • • considered with reference to those general rules and principles of thinking which hold good l whatever be the matter of thought." He laments that logicians are often divided into two warring factions; those who, following Boole, see logic as a branch of mathematics, and traditionalists who, as MacCol1 sees it, feel that

Aristotelian logic is sufficient. He refers to Boole's

Mathematical Analysis of Logic and Laws of Thought as causing ono small trepidation among logicians, who saw their hitherto inviolate territory now for the first time invaded by a foreign power, and with weapons which they had but too much reason to dread".2

This conflict is seen by MacColl as being over which subject logic belongs to, mathematics or philosophy.

MacCo11 characterizes as logicians those philosophers who

1 Hugh MacColl, "Symbolical Reasoning", Mind V. 5 (1880) p. 49. 2 Ibid., p. 46 - 3 - were working within the framework of classical or traditional logic. MacColl, for reasons which will emerge shortly, believed both sides to be in error but asks only to contribute his "humble share as a peacemaker between the two sciences".

The general fault he finds with logicians (the philosophers) is that they, "the successors of the illustrious

Aristotle, [have] not added a single acre to the very restricted possessions bequeathed to them by their great predecessor" 3 while mathematicians suffer by unnecessarily restricting logic to one domain, mathematics, thus neglecting important considerations which arise by looking at "reasoning in general." As this remark about logic not developing at aIl after Aristotle indicates, MacColl, like so many of his contemporaries, was qui te ignorant of the history of logic.

He makes no direct reference to any non-contemporary works in logic and where he refers to earlier work at aIl he is content to repeat the prejudices of many of his contemporaries.

MacColl, like Boole, wishes to use the methods of mathematics which he sees as having far greater power than the traditional methods of logic. He does not want to construct a system which derives its plausability by considering one interpretation e.g., mathematics, but one which is interpretable in any domain. That is, he wants his

3 Ibid., p. 49 - 4 -

system to be more general than Boole's. For MacCol1 logic is more general than mathematics and includes the latter subjecti hence he deplores Boole's "efforts to squeeze aIl reasoning into the old cast iron formulae constructed specially for numbers and quantities". 4

A brief remark must be made here concerning MacColl's preliminary description of logic as the "general science of reasoning" given in the first paragraph of this chapter. MacColl, on the question of whether logical laws somehow describe psychological processes, is not consistent. In sorne places in his writings it is quite evident that he does not think so while in other places his position is much less clear. For instance in discussing Boole's Laws of Thought MacColl praises him for possessing a "rernarkable analytical insight into the workings of the human mind ••• [and] the subtle laws of its intellectual operations. ,,5

Sorne forrn of psychologism is clearly suggested herei however, in more detailed discussions of the rneaning of "necessity", "possibility" and "variable" as these words are used in MacColl's logic he goes to sorne length to argue that they do not refer (in aIl cases, at least) to the psychological grounds for the assertion containing these words. At this point it is, perhaps, best to leave this as an open question until it can be discussed in sorne detail.

4 Hugh MacColl, "Syrnbolic Reasoning", Mind N.S.V. 6 (1897), p. 505. 5 Ibid., p. 505 - 5 -

Among MacColl's contemporaries writers such as Boole and Jevons, as weIl as MacColl himself, are often referred to as mathematical logicians. However, MacColl did no·t see logic as a method for explicating the ide a of mathematical proof nor did he attempt in any detail to demonstrate that mathematics could be deduced from logic, though he did believe, in a rough and ready manner, that logic included mathematics. For this reason MacColl, and those of his contemporaries for whom it is appropriate, will be referred to as symbolic logicians rather than mathematical logicians throughout this study.

Logicians before the nineteenth century used symbols, as indeed any logician must in order to obtain general results. However, when the expression "symbolic logic" was used in the nineteenth century it was used to characterize the method of Boole and his followers and to contrast that method with that of traditional logic. Indeed logicians such as Boole, Jevons and MacColl were sometimes referred to as "symbolists". This expression was not intended to be simply synonymous with "logicians". For this reason earlier logicians, specifically Ancient and Medieval logicians, will not be considered as symbolic logicians in what follows.

MacColl develops his system in a series. of eight - 6 -

papers entitled "Symbolical Reasoning" which appeared in Mind between 1880 and 1906 and in a series of seven papers, the first of which was entitled "The Calculus of Equivalent Statements and Integration Limits" and the remaining six "On the Calculus of Equivalent Staternents" which appeared in the Proceedings of the London Mathematical Society between 1877 and 1897. His book Symbolic Logic and Its Applications published in 1906, is, in most respects, the cumulation of his earlier researches.

MacColl in his initial publications adopts many of Boole's ideas and much of his symbolisme He d~vides the symbols of symbolic logic into two groups; variables and operators or, to use MacColl's ter.minology, temporary and permanent. The permanent symbols 'x' (conjunction), '+' (disjunction), , . , . (implication), '=' (equivalence) and (negation) are used to construct compound statements out of statement variables (temporary symbols) such as A, B, C etc.

Right from the beginning of his work, however, there are important differences between MacColl and his immediate predecessors. MacColl was the first symbolic logician to found his system on relations among statements rather than classes. He saw this as an important innovation as early as 1877, although it was not recognized as such until sorne time later, and in aIl his writings he is at pains to emphasize this point. In his second paper to the - 7 - mathematical society (1877) MacColl notes that the statement

'AlI X is y' may be written 'x:y' where x denotes the statement that a certain representative individual belongs to the class X and y the statement that it belongs to the class y. 6 Again in his third paper to the Mathematical Society MacColl, while discussing the differences between his logic and that of Boole and Jevons, emphasized that "with me every single letter as weIl as every combination of letters, always denotes a statement". 7 Later he writes:

The simplest, the most general, and the most easily applicable kind of logic is the logic of statements or propositions. To this, and to this alone, can we correctly give the name of pure logic ••• it has the immense advantage of being independent of the accidentaI conventions of language. 8

MacColl, unfortunately, uses words such as 'statement' and 'proposition' to mean different things in different places in his writings. In the one place where he is explicitly trying to explicate these concepts MacColl says that a statement is any sound or symbol used to give information, such as a white flag used to surrender, the warning calI of a rook or a scarecrow placed by a farmer. A proposition, on the other hand, is a linguistic entity.9

6 Hugh MacColl, "On the Calculus of Equivalent Statements", Proceedings of the London Mathematical Society V.9 (1877-78), p. 181. The rel3.tion of this idea to modern quantificational logic will be discussed below. 7 MacColl, "Equivalent Statements", Mathematical Society v. 10 (1878-79), p. 27 8 MacColl, "Symbolic Reasoning", Mind V. Il (1902), p. 352. 9 Ibid. - 8 -

That is, propositions are sentences. There is no special word in MacColl's vocabulary for what it is that a sentence expresses or, to put it another way, MacCol1 does not, in the passage referred to above, admit propositions in the usual sense of the word. However, in practice MacCol1 uses the words 'statement' and 'proposition' interchangeably and means by 'statement' what is usually meant by 'proposition'. That is, MacCol1 would allow that", in sorne way, 'It is raining' and 'Es regent' each express the proposition that it is raining. There are, however, several difficulties in MacColl's position on this issue which will be discussed in the context of Russell's criticisms of MacColl.

The above preliminary account of some of MacColl's attitudes and positions in logic should be sufficient to indicate that for MacCol1 the business of logic is to investigate the logical relationship in which statements stand to one another independently of the special demands of any particular subject matter such as grammar, metaphysics or mathematics. The fundamental relationship, one which MacCol1 sees running through aIl reasoning, is implication. Thus, for MacColl, the basic task of a logician is to develop a satisfactory definition of implication. MacColl begins with a definition of implication which is very close to that of his immediate forerunners in symbolic logic, especially Boole, and ends with an account very similar to sorne of his followers, especially C. I. Lewis. The rest - 9 -

of this chapter will be devoted to tracing this development

and noting some of the features of the system that MacColl

develops on the basis of his definition of implication.

MacColl's interest in symbolic logic was aroused

by a question on probability which appeared in the

Educational Times for 1871. He then thought that his views

on "symbolic language" could be fused with some basic notions

of probability theory to create a symbolic logical calculus which would have important applications in mathematics. 10

The effects of this early interest in probability theory

can be seen in many places in MacColl's writings, beginning with his first account of implication.

MacColl's first account of implication appears

in 1877 when he gives the following rule for use in the

analysis of integration limits:

Let A be any statement whatever, and B be any statement which is implied in A (and which must therefore be true when A is true and false when A is false); or elRe let B be any state ment which is admitted to be true independently of A; then (in either case) we have the equation A = AB~ll

B is implied in A means, simply, that A implies B and the

equation A = AB holds whenever B is true for the reason that

10 MacColl, "Equivalent Statements", Mathematical Society v. 10 (1878-79), p. 27. Il MacColl, "Equivalent Statements",Mathematical Society V. 9 (1877-78), p. 10 - 10 -

a conjunction with a true conjunct resolves to the other

conjunct (s), in this case~, hence A =~. The first case, however, is not as clear cut since MacColl claims that if A is false then B is false as weIl as if A is true then B is true. This rule is not discussed by MacColl nor is it used in such a way as would indicate that MacColl was aware of any difficulties. He does, several years later, develop a system of causal implication which works very much in this manner. The important point here is to note that MacColl thinks implication is an equation which holds on the basis of the truth-values of its constituent statements.

The following year he offers a slightly different definition of implication and introduces his symbol for the implication operator, a colon. Here he

claims that A:B means that the statement ~ implies the s t a t emen t B or th a t wh enever A lS· t rue B'lS a l so t rue. 12 In a note to this he says nIt is evident that the implication A:B and the equation A = AB are equivalent statements". How- ever, here there is no mention of ~~s being false if A is false, marking a small, but important, step from his earlier definition of implication.

Of the several formulas that MacColl lists as valid at this time (1878) one is of particular interest

12 Ibid., p. 177. - Il - herei namely one given as a rule "if AB = 0 then A:B' and B:A,,,13 This formula, which is valid in a system like material implication, makes use of the formula (A' + B) (A:B) which MacColl expressly rejects at a later stage.

That is, from AB = 0 it can be deduced that A:B'and B:A' by using (A' + B) : (A:B) as a rule, since, for MacColl,

AB = 0 is equivalent to A' + B~ (or B'+ A'). It is clear that this much would commit one to the paradoxes of material implication which is what MacColl later expressly tries to avoid.

MacColl's primary reason for introducing his symbol for the implication operator is convenience. He discusses at length in several places the power of a symbolic language over a natural language, describing the relation between the two at one point as having "pretty mu ch the same relation that machine labour has to manual labour". A symbolic language enables "any ordinary mind to obtain by simple mechanical processes results which would be beyond the reach of the strongest intellect if left entirely to its own resources".14 Thus he is at a loss to understand Jevons' criticism that he rejects equation in favour of implications. MacColl's response was that the decision to use A:B or its equivalent the equation A = AB in any particular problem must be made on the "broad grounds of

13 Ibid., p. 178 14 MacColl, "Symbolical Reasoning", Mind V. 5 (1880), p. 45. - 12- practical convenience". As a reductio ad absurdum MacColl notes that a cornplex implication (e.g., one in which the implication, or equation, operator occurs more than once) such as (~.: B) (B: C) (A:C) must be expressed by the cumbersome (A = AB) (B = BC) = (A = AB) (B = BC) (A = AC) in Jevons' system. 15 At this point MacColl sees no very deep t~eoretical gulf between his logic and that of Jevons. But he does believe that his notation is much more efficient and easier to use, hence more powerful as a tool for practical application. He also believes that that is of sorne importance.

The first indication that MacColl will differ radically from his irnmediate predecessors in symbolic logic cornes in his second paper to the mathematical society where, in an attempt to meet an objection of the referee for the paper he introduces the non-implication operator '+'. That is A. B asserts that A does not imply Bi it is equivalent to "the less convenient symbol" (A :!!.)r. In a note to the definition MacColl says that A.f~ asserts "that the truth of B is not a necessary consequence of the truth of A: in other words, it asserts that the statement A is consistent with B', but it makes no assertion as to whether A is consistent with B or not". 16 That is, if a statement

15 Hugh MacColl, "Implicational and Equational Logic", London, Edinbur h and Dublin Philoso hical Ma azine and Journal of Sc~ence v. Il 1881), p. 40. 16 MacColl, "Equivalent Statements", Mathematical Society v. 9 (1877-78), p. 180. - 13 -

A does not imp1y a statement B, then aIl that can be claimed is that A and not -B are consistent and not that A implies not -B. Or, to put it another way, MacColl is rejecting the thesis, which would be valid in a system like materia1 implication, - (p ~ q) ~ (p :. -q) • By introducing

"necessary" and "consistent" into his definition of non~ implication he has here for the first time used modal terms to describe features of his system.

In his first paper in Mind he argues along similar lines that the equivalence (A:B) = (A' + ~) does not hold although the implication (A:B) : (A' + B) does. To illustrate this point he gives the following interpretation. Let A denote the statement "He will persist in his extravagance" and B "He will be ruined". Then A :B may be read as "If he persists in his extravagance he will be ruined" while the disjunction A' + B will be "either he will discontinue his extravagance, or he will be ruined". Now to show that they

, , are not equivalent, s~ys MacColl, consider their denials. The denial of the implication i.e., (A:B) is, on this interpretation, to be read as "He may persist in his extravagance without necessarily being ruined" while the denial of the disjunction A' + B i.e., AB' is to be read as "He will persist in his extravagance, and he will not be ruined". Thus, since the denials are not equivalent the original statements are . 1 t 17 no t equ~va en •

17 MacColl, "Symbolical Reasoning", Mind V. 5 (1880) p. 54. - 14 -

This particular argument, or observation, played an important role in the evolution of MacColl's system and was called into question early by other symbolic logicians. MacColl, in replying to criticisms, was forced to set himself off more sharply from what by then could be considered the mainstream of the development of symbolic logic and to state more clearly what it was that was unique in his system. MacColl's reply went as follows:

Now, l admit at once that, in the ordinary language of life, disjunctive statements are often made which convey, and are intended to convey, a conditional meaning, and further, that the example which l gave in illustration, namely "He will either discontinue his extravagance, or he will be ruined" is one of them.

• • • [However] the real question in dispute is this, does the conditional statement "If a is true b is true", as l define and symbolize it, convey a meaning in any way different from the disjunctive statement "Either a is false or b is true", as l define and it? 18 symbolize -'

This retreat behind the claim "that is how l define and symbolize these concepts" clearly will not do because, for one thing, it makes the original claim completely trivial since one would then be at liberty to define disjunction, say, in any way one saw fit. Also, it would undercut the purpose of MacColl's original point that his

18 Hugh MacColl, "On the Growth and Use of a Symbolical Language", Memoirs of the Manchester Literary and Philosophical Society Third Series V.7 (1882), pp. 236-37. - 15 - interpretation of '+' more adeq~ately represented the facts of disjunction (and implication). If this were not so there would be no point to the (original) example. If this example will not do MacColl's position demands that he show the deficiencies in the example and produce examples that do support his position. He cannot altogether rule out examples from ordinary language since it is from considerations of ordinary language that his account of implication and disjunction derives its initial plausability. On the other hand he is not, of course, committed to saying that his explication of these concepts fits aIl, or even most, of their uses in ordinary language, or, rather, the words used to express these concepts in ordinary language.

MacColl does atternpt to provide examples to support his case but, unfortunately, they are ill chosen. He first divides disjunctive staternents of ordinary language into what he calls conditional disjunctives and unconditional disjunctives. As an example of the former he gives Edward l's remark to Earl Bigod, "By God, sir Earl, you shall either go or hang" which, MacColl notes, could be fairly accurately translated into "If you won't go, l will have you

n hanged • Even here MacColl is uneasy, though, as he notes that the Earl's response, "By God, sir King, l shall neither

n go nor hang , is really stronger than is required since "I may refuse to go wi thout your having me hanged" would be sufficient to deny the king's conditional. - 16 -

As an examp1e of an unconditional disjunctive, that is, a disjunctive that cannot be adequately rendered in conditiona1 form, he gives "We sha11 either go to Brighton or Hastings this summer". However, he soon gives up on this examp1e noting that "in common language" this cou1d be expressed as "If we don't go to Brighton this swmner, we sha11 go to Hastings" or in the same words with 'Brighton' and 'Hastings' reversed. He is then content to write off the prob1em as one be10nging to "ordinary

untechnica1 language". "Even if l have not succeeded in satisfactori1y proving that a:b and a' + b are not synonymous it is safest", writes MacCo11, "to adopt my view in actual practice". 19

This conclusion cou1d hard1y be expected to convince MacCo11's sterner critics. He is, c1ear1y, quite confused as to what conclusions to draw from his examples and, in desperation, fina11y says, in effect, "We1l, my way is best despite this muddle". However, in fairness to

MacCol1, it is not, l think, reading too much into his argument to see what he is after. Indeed his whole position becomes much c1earer in his discussion of the paradoxes of materia1 implication which will be discussed below. What he sees here is that a disjunction can consist of two or more independent statements - that is, independant in the

19 Ibid., pp. 237-238. - 17 -

wide sense of belonging to completely different subject matter - where it would be extremely counter intuitive to talk of one implying the other. For instance if A stands for the statement "2 + 2 = 4" and B "The Thames flows through London" then, sinee the disjunction A + B is true, its corresponding implication A':B would also be true. But what, MacColl would ask, has two and two equalling four to do with the Thames flowing through London? This question will be discussed later but it is important to note now that MacColl saw this as a problem as early as 1878 although he could not state explicitly just what the (alleged) deficiency in the systemes) of logic he was attacking was, nor exactly how his own developing system avoided these problems.

MacColl had not yet begun to appreciate the significance of introducing the modal concepts of necessity and possibility (expressed at this point by the modal term 'may') into the object language of his system. For instance, MacColl can see that his implication operator cannot be defined in terms of any of the other operators he has; indeed he cannot bring negation inside the brackets in a formula such as (A:B) , and is forced to invent a new symbol '7' to express non-implication. It is sorne time before he realizes fully that his concept of implication is not combinatorial and that he will have to introduce operators for necessity and other modal concepts. Indeed, he has no clear idea how his system is different from Boole's although he claims it is fundamentally different. He summarizes their differences - 18 -

as being due to his (MacColl's) use of letters to denote statements not classes, his use of the symbol ':' to denote that the statement following it is true provided that the statement preceding it is true and his use of the accent 20 to express denial. Now, while the first point is important, the latter two are surely not. For that matter, even the first should be considered an improvement on Boole's work and not a fundamental difference.

That he has not yet sharply made the break from Boolean logic (or, perhaps more accurately, two-valued extensional logic) can be seen clearly in the following remarks of MacColl's, made in 1880.

It is easy to see that the implications A:l and O:A give us no information whatever as to the truth or falsehood of A, [where 1 signifies truth and 0 falsehood) but that the equations A = 1 and A~ are the exact equivalents of the implications l:A and A:O respectively, and that from the former we can infer that A is true, and from the latter that A is false.

Consistency of notation inthis algebra of logic requires that the implications A:l and O:A should each be equivalent to 1 whether the staternent A be true or false. 21

The first of the above two paragraphs is a working out of MacColl's clairn, ab ove , that ':' denotes that

20 MacColl, "Equivalent Staternents", Mathematical Society V. 10 (1878-79), p. 27. 21 MacColl, "Symbolical Reasoning", Mind V.5 (1880), pp. 54-55. - 19 - the statement following it is true provided the statement preceding it is, while, as he says, consistency of notation demands that the conclusions in the second paragraph be drawn. What MacColl does not see is that this leaves no room for talk of necessity and possibility as he uses these terms in discussing particular examples of implications. It should also be pointed out here that these two paradoxes contain what later came to be called the paradoxes of material implication; namely that a false statement implies any statement (O:A) and that a true statement is implied by any statement (A:l).

He does not see that this is inconsistent with his denial of the implication (A' + B) : (A:B). Accepting that ~:l, O:A and 0:0 are aIl equivalent to 1, then A:B fails to hold only when A is true and B is false. Similarly A' + B fails to hold only when A is true and B is false, hence, on MacColl's own grounds A' + B cannot be true while A:B is false. Therefore (A' + B) (A:B) holds. It is only when MacColl moves away from a truth-functional account that he escapes this dilemma.

At this point MacColl abandoned his research in logic for sixteen years, returning to it in 1896. One could briefly review the beginning period of MacColl's work in logic by nbting that he made at least one important innovation basing logic on the logic of statements rather than classes; - 20 - he began to develop a theory of quantification (to be discussed more fully below)i he became uneasy about sorne aspects of extensional logic and made the first starts at developing a modal logic, and he made sorne general suggestions that mathematics was contained within logic. Before leaving this period it should be noted that MacColl introduced the

Greek letter Epsilon (E) which denoted the class of staternents "whose truth is taken for granted throughout the Wh 0 1 e 0 f an 1nves. t·19a t· 10n ,,22• This symbol later is used as the symbol for necessity but at this time it is seen as yet another symbolic innovation for the purposes of convenience.

The influence of his early interest in probability theory is still quite evident with MacColl's return to logic although, by this time, he had developed much clearer ideas about the significance of his earlier remarks about implication and, also, about what he took to be paradoxes in two-valued extensional logic.

In his third paper in Mind (1900) MacColl mentions, for the first time, the traditional modalities of

Aristotelean and Medieval logic. He does so by quoting, approvingly, de Morgan's remark that probability was "the unknown God whom the schoolman ignorantly worshipped" when they discussed modality. This indicates once again the

22 MacColl, ~Equivalent Statements", Mathematical Society v. Il (1879-80), p. 114. - 21 - historical ignorance of MacColl and most of his contempor aries in logic. What knowledge MacCol1 had of earlier work came exclusively from text books of logic written in his time which could not be relied upon to present accurate historical information. Many of the questions MacCol1 dealt with were dealt with, often with more sophistication, by Medieval logicians. MacColl's historical allusions are repeated here, as elsewhere in this study, to indicate the degree to which his system was the result of his own work rather than a development of earlier ideas.

On the basis of this analogy with probability MacCol1 introduces the notion of data to explicate the modalities. This is what may be called his intermediate periode During this period aIl the modal opera tors are dyadic rather than monadic. That is, a given statement is necessary, for instance, with respect to sorne data. 'Necessity', 'possibility', 'contingency' and 'impossibility' are aIl relative concepts. What can be necessary with respect to sorne set of data may be impossible or contingent with respect to sorne other set of data. In his later period MacCol1 makes a sharp distinction between dyadic and monadic modal operators. However, discussion of this must wait until MacColl's position has been stated more fully.

MacCol1 uses the following diagram to help bring - 22 - out what he means by the various modalities.

3 Fig. 1 Fig. 2 Fig.

ten Data: Fig. 1, Fig. 2, Fig. 3~ there being always a points in circle E. Let A be the statement that

point P, chosen at random out of E will be in the

circle A.

In Fig. 1 A is a certainty: that is ~E(A represents A) the statement that p will be in E, not the circle

In Fig. 2 A is an impossibility~ i.e., An In Fig. 3 A is a variable~ i.e., AS 23

From consideration of examples of the above type

MacColl arrives at the following definitions.

AE symbolized that A is necessarily trueithe our data. supposition of its falsehood is inconsistent with

That is what l have called relative necessity above.

p. 77. 23 MacColl, "Symbolic Reasoning", Mind N.S.V. 9 (1900), - 23 -

AAEl symbolized that A is true in a particular case but uncertain as a general law. It might, without contradicting the data, turn out false. That is A, as it happens, is true but it could possibly be false.

Al Anl symbolizes that A is false but not impossible. An symbolizes that A is necessarily false; the supposition of its truth is inconsistent with the data.

There are many problems which arise here but it will be convenient to postpone discussion of them until the following chapter. For the moment it will be more use fuI to provide a general "working knowledge" of MacColl's system. A few points must be made, however, before moving on.

First, the two complex statements AAEl and

A'Anl ; that is, A is true and it is false that A is certain (necessary) and A is false and it is false that A is necessarily false (impossible), can be expressed in MacColl's terminology by the much simpler AS , which reads A is variable. This concept of a variable, a statement which is either true or false but neither certain (necessary)24 nor impossible,

24 Sorne ambiguity here is deliberate since MacCol1 uses both of these words to express roughly the same idea. The word • certain' however suggests sorne psychological or (cont'd) - 24 -

24 (cont'd) epistemological grounding. At this stage in his career MacColl is not always careful in distinguishing logical from epistemological, and, in sorne cases, psychological issues. Indeed it is an open question, to sorne extent at least, as to whether there is a logical issue to be separated from the psychological and epistemological in this case hence both words are used to convey MacColl's own ambiguity on this issue. This will be dealt with in the following chapter more fully. - 25 - plays an important role in MacColl's logic and it ought expressed not to be confused with the concept of possibility it by the diamond operator in Lewis type systems. Rather what should be read as a contingency operator since clearly a can be neither A says is ~hat Amay be true or false but certain (necessary}nor impossible. Now, in most modal that E systems if a statement, E' is necessary it follows this is possible; hence, a statement's being possible, in To show sense, does not preclude it from being necessary. the difference in another way, if A is a variable statement transforma then, for MacColl, 50 must A' be variable. This tion does not, of course, hold for possible statements logic. as that word is used in the Lewis systems of modal but it For example "2 + 2 = 4" is possible in this sense, 2 4'11 does not follow that IIIt is not the case that '2 + = attached is possible. Given the traditional central meaning

ll refer to to the tenu "qontingent it does not seem amiss to MacColl's variable operator as a contingency operator, with other, though he does not do 50, and thus avoid confusion or more popular, uses of 'variable'. However when quoting con- paraphrasing MacColl and where the context precludes fusion MacColl's word 'variable' will be used.

MacColl was also aware, at this time, that be 'possibility' as the term has been used above, could of a special expressed in his notation without the introduction - 26 - symbole In other words he was aware that by taking impossibility and negation as primitive he could derive possibility. This is shown in an interesting paragraph in his third paper in the Mind series (1900). Here he writes

The symbol An may be read "It is impossible that A is true". The symbol An t may be read "It is false that it is impossible that A is true"; which may be abbreviated into "It is possible that A is true", or more conveniently still, into "A is possible".

Here possibility is defined in terms of negation and impossibility. Further in the paragraph MacColl makes claims which will be of more interest when comparing MacColl's system to later systems, but it is not without passing interest here.

The symbol AntEEmay be read "It is certain that it is certain that it is false that it is impossible that A is true", which may be abbreviated into "It is certain that A is certainly possible". 25

The above two quotations indicate how one is to read MacColl's symbolism and the last one shows that modalities cannot be reduced in MacColl's system; that is AEE (It is necessary (certain) that A is necessary) cannot be reduced to AE (It is necessary (certain) that A). This suggests that if MacColl's system is to be found equivalent

25 MacColl, "Symbolic Reasoning", Mind N.S.V. 9 (1900), p. 75. - 2"l -

or strongly similar, to any of the later Lewis systems it will be to one of the weaker systems SI - S3.

Although MacColl claims that probability theory provided the inspiration for his logic, and there is no doubt that he did derive sorne sort of inspiration from this source, the effects of this early interest seem, on the whole, to be more harmful than helpful. Indeed, it is not until his interest in this subject lessened that he was able to avoid making many mistakes that impeded the development of his system.

A case in point is the particular use which MacColl makes of "data". For instance in Fig. 3 above the statement A is contingent; that is it may be true or false depending on which point out of the collection is chosen. Suppose the points are named Pl' P2 ••• PlO and that these names are assigned to the points in a random fashion. Suppose also that P4 names a point which is in the Circle E. Now let B be the statement tlp 4 is in the circle E". Presumably MacColl would want to maintain, or, at least, he ought to maintain, that B is also contingent. However, not only is B contingently true, but since "the supposition of its falsehood is inconsistent with our data", it is necessarily (certainly) true, which is absurdo Another example will help make this clear. MacColl gives as an example of a contingent statement "Mrs.Brown is not at home". "If", says MacColl, "at the moment the servant tells - 28 -

me that :'Mrs. Brown is not at home' l happen to see Mrs. Brown walking away in the distance, then l have fresh data and forro E the judgment A In this Càse l say that 'A is certain' because its denial would contradict the data, the evidence of my eyes".26 In general, whenever we have grounds for asserting that a statement is true, then for MacColl, at this time, we have sorne "data" which not only support the assertion but are such that if we accept the "data", then we cannot not accept the statement for "the supposition of its falsehood is inconsistent with our data".

What appears to be happening here is that MacColl is blurring the distinction between (merely) true and necessarily true statements, an error he accuses other logicians of committing and one he claims his system avoids. MacColl, however, is simply confused, and by reading 'variable' as 'contingent' we can avoid this confusion. On independent grounds it appears, as noted above, that this is a correct reading and, as we shall see, in his more formaI working out of his system MacColl defines 'contingent' ('variable') in a way that would not allow a contingent statement's becoming necessary with the addition of sorne new "data".

MacColl is not totally unaware of the possibility of confusion here. Indeed he makes a distinction between

26 Hugh MacColl, Symbolic Logic and its Applications (London: Longmans, Green, 1906), p. 19. - 29 - formally certain and materially certain statements, each of which is grounded on different kinds of data. This dis tinction goes sorne way towards clearing up the confusion, but if MacCol1 took more notice of his own symbolism, the confusion would never arise as it would be obvious that a contingent (variable) statement could never become a necessary statement. This distinction will be discussed in more detail later as will the problem of whether 'certain', especially as it is used in the 'Mrs. Brown is not at home' example, is a psychological or epistemological notion. For the moment, however, it is sufficient to note that these problems are in MacCol1 and that, to sorne extent at least, it appears that the particular way MacCol1 uses the concept of 'data' helps create sorne of these problems. That is, MacCol1 here treats 'data' as any evidence which rela~es to the establishment of the truth or falsehood of a statement, whether the statement is empirical or logical. In the Mrs. Brown case if, in fact, one sees Mrs. Brown walking away in the distance, then, on the assumption that the lady in question cannot be in two places at one time, one knows that she is not at home. Hence, MacCol1 calls the statement "Mrs. Brown is not at home" a certainty. This example, then, is clearly epistemological in nature. It is still the case, however, that to calI this a variable statement (contingent) conflicts with the explication of the modal concepts given a few pages back. It must, on the basis of the evidence, be concluded that MacCo11 was simply confused - 30 -

on this issue, although when writing lists of valid formulas in his symbolism he avoids this confusion.

The period from shortly after 1900 until 1906 marks the completion of MacColl's work in logic. During this time he works out sorne of the implications of his earlier work, gives more detailed criticisms of other systems and presents his own logic in Symbolic Logic and its Applications as a fini shed work. The degree to which he saw himself as opposed to other logicians is brought out weIl by the following remark made in 1903 in Mind •

••• it is hardly an exaggeration to say that no single formula in my system has exactly the same meaning as the formula which is supposed to be its equivalent in other systems. When both are valid, l usually find that mine is the more general and implies the other. 27

MacColl discusses the differences between his system and others by examining how the implication operator functions in the different systems. He takes Peirce's symbol ,-<, as a general representative of the extensiona1ist implication operator and compares it with his own ':'.

Whereas 1 • 1 in 1880 he saw his symbol . as a "brief and convenient" abreviation for the equation (.ê.~ = (ab) by 1903 he declares unequivoca1ly that "their [other symbo1istsJ symbo1 '-<', never does express the same idea as my symbo1

27 MacCol1, "Symbolic Reasoning", Mind N.S.V. 12 (1903) p. 355. - 31 -

':'.11 The symbol '-<', clai~s MacColl, is used ambiguously to mean class inclusion on the one hand, i.e., A-

This point, which could easily be met by the

i' lIother symbolists , is not dwelt upon by MacColl, but it does force him to state Inore clearly exactly what he means by his implication operator ':'. Here he says that A: B is equivalent to IIIt is certain that either A' + B", 29 or, in symbols,

A:B = df. (~' + B)E • Alternatively he defines A:B as being equivalent to (AB,)n; that is, it is impossible that A is true and B is false. On the basis of this definition the expression 'strict implication' will be used to refer to

MacColl's definition of implication. For the same reason the

(alleged) paradoxes which result from this definition will be referred to as paradoxes of strict implication.

MacColl claimed on the basis of his definition of implication that his system was more general, and included the other systems of his day. He did not, however, prove this but simply compared the different definitions of implication, which comparison, he thought, suggested that the

28 Ibid, p. 357 29 Ibid. - 32 - system resu1ting from his definition included the system which resulted from the extensionalist definition of implication. Indeed rigorous deductive systems which would be required to offer a formaI proof of this were unknown to MacCol1, as they were to most of his contemporaries in logic. MacColl's "proof" was as follows:

A-

A B = (A--(B) E

He then notes that these "definitions and comparisons" show that my symbol A:B is formally stronger than and implies their syrnbo1 A-

MacCol1 generally attempts to provide rigorous proofs to show that certain statements are valid or that one statement implies another. Before considering such proofs MacColl's method of proof must first be explained.

" , C" , The syrnbo ls 'T', l' ~, 'n', and '8' (true, false, certain, impossible and contingent) are claimed by MacCo11 to be both predicates which, as the manner of their use indicates, predicate certain properties of statements and operators. Unfortunately, however, in MacColl's proofs he

30 Ibid. - 33 - treats these operators as statements so that, for example To nT asserts that the impossible proposition n is true. carry out this example, the claim that the impossible proposition n, is true is impossible hence the symbol T for impossibility n can be substituted for u. Or, false) to take another example, AT + Al (A is true or A is the is, for MacColl, a certaintYi hence, in a deduction or symbol for certainty ~ can be substituted for A~ + ~l, for any other certainty. This method has serious a deficiencies, not the least of them being that it is mistake to speak of the impossible proposition n when n be is the modal operator for impossibility. This would like, in Lewis' notation, talking of the necessary statement 'c'.

It might be thought that this is merely a Indeed notational point and that n is simply short for An. this is primarily a notational point; it is important, however, since writing n as short for An leads to serious or difficulties when it cornes to proving formulas valid, invalide

To rewrite the first example as it should be n is in MacColl's notation one would have {A )T , i.e~,it writes true that ~ is impossible or, as MacColl normally these expressions, A is impossible. However, the problem, for MacColl, is that we cannot then say that ~ is impossible say is impossible since, for one thing, one might want to - 34 -

nit is necessary that A is impossible" or in MacColl's notation (An)E .• Indeed, this concept of the degree of a formula is one which MacCol1 sees as capable of expression in his system but not in others. There is no mechanism for the reduction of iterated modalities in MacColl's system, as we have seeni hence, a method of proof which used it, to say nothing of treating modal operators as statements, would be out of place. Unfortunately, many of MacColl's '-'proofs" rely on such a method and alrnost aIl of them treat modal operators as statements or, more accurately, as the referent, or extension, of statements. What MacCol1 presents in effect is an (incomplete) decision procedure, not unlike the "fell-swoop" and "fuII-sweepn rnethods Quine uses in Methods of Logic,. where 'E' and 'n' function exactly as 'true' and 'false' in an extensional system. This rnethod can however be modified to present, if not rigorous proofs, at least clear statements of which formulas MacCol1 considered valid and sorne of the reasoning which went into his decisions, especially in complex cases, to regard these formulas as in fact valide The justifica tion for reconstructing MacColl's logic so that, for instance, modal operators are not treated as staternents or the extension of staternents is that in the context in which he uses these proofs it is clear that what he means by saying, for instance, "the impossible proposition n", is really any impossible proposition A. The results are in rnost cases not - 35 - different from what they would be had he not stated it as he did, thus in reconstructing his proofs an effort will be made, where practical, to rewrite them to avoid this error.

~ few proofs in MacColl's original form will also be given so that it will be clear exactly what is being modified.

MacColl's claim that his system includes others is not, of course, the end of the matter. He also feels that his system is better and that this can be .shown by consideration of the paradoxes which occur in a two- valued, truth-functional system of logic, specifically those that have come to be known as the paradoxes of material implication. He begins his attack on these (alleged) paradoxes by discussing a review of Russell's Princip les of Mathematics. 3l Here he notes that Russell is quite right in stating that on the assumption of the equivalence between p~q and -pvq, of any two statements one will imply the other. "But surely", writes MacColl, "the paradoxical conclusion at which he [Russell} arrives should give. logicians pause".32 For instance, consider the two statements "He is a Doctor" (D) and "He is red-haired" CR). Now, is it really the case, asks MacColl, that either R implies D or that D implies R1

The quick answer is, of course, that (~D) v (D~R) is a valid formula for Russell but, since MacColl knows this, it must

31 Hugh MacColl, '" If' and 'Imply'" Mind N.S.V. 17 (1908) pp. 151-152; 453-455. 32 Ibid., p. 151. - 36 - be concluded that MacColl is appealing to sorne intuitive or non-stipulative meaning of 'implies' which, obviously, he feels is captured by the meaning of his symbol:. In other words MacColl is saying "Surely, one's having red hair is independent of one's being a doctor (logically independent); hence, how can one talk of the one statement (R) implying the other (D) or the reverse?

MacColl sets out to show s~~olically that Russell is committed to the claim that either R implies D or D implies R (a claim that Russell would not deny) and, more importantly for our purposes, that he (MacColl) is not committed to such a claim. MacColl's symbolic manipulations here are wrong as will be shown, but they will be stated as he stated them and then reconstructed since an important fact about MacColl's logic will emerge from consideration of this example. MacColl states Russell's reasoning as follows:

(D:R) + (R:D) = (D' + R) + (R' + D) = (D' + D) + (R' + R) =

E + E = E That is, (D:R) + (R:D) where ':' is taken with the sense of

Russell's '~', becomes (~' + R) + (R' + D) by substitution of defined equivalents which then becomes (D' + D) + (R' + R) by distribution. Statements of the form A + A' are regarded as certainties by MacColl; hence, the whole expression resolves into a disjunction of two certainties, which further resolves into a certainty. If the symbol E is read as 'true' - 37 - then there is nothing wrong with the reasoning. It simply points out what no one has denied, that, given Russell's definition of 'implies ' , (D:R) + (R:D) is valide The situation is somewhat more complex when we come to MacColl's statement of his own reasoning in this example; which is,

(D:R) + (R:D) = (D' + R)E + (R' + D)E = (DR,)n + (RD1)n = n +r, = n The first two steps of the reasoning here are simply sub stitutions of defined equivalences. The last two steps, though, are wrong even by MacColl's standards. That is, if any expression governed by the modal operator li resolves to n (which is the only assumption to make here in explaining MacColl's reasoning), then every formula which MacCol1 claimed as valid would also be impossible, which is impossible.

That is, since the modal operator for necessity, ~, can be changed into n with suitable use of negation, every formula governed by ~ could be changed into one governed by n. It is interesting to note that Russell did not seem to notice this mistake in his subsequent reply. What is of more importance, however, is that this does not appear to be simply a careless mistake, though mistake it certainly is, on the part of MacColl. What this indicates is that MacColl's method of "proof", which consists in assuming the formula to be proved and then repeatedly substituting definitional equivalents in an attempt to reduce the formula to either a certainty, impossibility or variable, is seriously flawed. Clearly what he wants to say is that in his system it is - 38 - not the case that for every two statements A and B that either one implies the other. That is, ~acColl wants to reject the formula (AS :BS) + (B S : AS). MacColl also rejected the formula AS = BS as will be seen below. The use of the above example and further examples to be given below, suggests that this is one of the fundamental differences he sees between his system and the others. It was precisely for this reason (that is to show that sorne statements are independent of one another) that MacColl employed contingent statements in his system. He felt this would give logic greater generality and bring out certain logical features that other systems concealed, as the following remarks of MacCollls indicate: ••• it will, I think, be found that in mathematical reasoning the simple rlisjunctive AI + B is always understood to really mean (AI + B)E, which asserts that lit is certain (true in aIl cases) that either A is false or B is true. 1 This being understood, I admit that in mathematics ••• the dis junctive may be considered equivalent to the conditional or implication ••• But in the wider field of general logic the 33 assumption of equivalence is unsafe.

According to MacColl, in mathematics one is, with few exceptions, dealing only with necessary or impossible statements making it unnecessary when constructing a logical system to deal with mathematics to include contingent statements.

33 Ibid., p. 453. - 39 -

Two further examples will illustrate this difference between MacColl and Russell. Suppose A, Band C to be random points on the circumference of a circle. Consider the two sentences 'AB is greater than AC' and 'The angle A is acute'. "Is it not the fact" asks MacColl, "that neither statement implies the other? How then can it be true that 'Either the first implies the second or the second the first. ,?,,34 That is, since it is possible that AB is greater than AC and that the angle A is not acute, how can it be said that 'AB is greater than AC' implies 'A is acute'. Or, sinee it is possible that the angle A is acute and AB is not greater than AC, how can it be that 'A is acute' implies 'AB is greater than AC'?

The second example is somewhat more complex although the point is essentially the same. Here MacColl is explicitly challenging the validity of the following formula;

«A:~) + (B" :C» = (AB:C). He adroits that it is valid as a one way implication; that

is, «A:C) + (~:C» : (AB:C) is valid, but it is not valid

as an equivalence; that is, he rejects (AB:C) : «~:C) + (B:C». He offers the following example to show why he

34 Ibid., p. 152. - 40 - rejects this formula.

Out of the total of ten points marked in the ellipse C and the two circles A and B in the accompanying figure, take a point p at random and let A, Band C as sert respectively, as statements, that Ip is in AI, Ip is in BI and Ip is in CI. The implication AB~C asserts that the point p cannot be in both the circ les A and B without being in the ellipse C, which, in this case, is true. The implication A:C asserts that P cannot be in A without being in C, which is false; and B:C asserts that P cannot be in B without being in C, which is also false. Thus, the alternative (A:C) + (B:C) is false while AB:C is true, hence they canno t b e equ1va. 1 en t • 35

MacColl gives other examples of the same type in support of his aversion to allowing that an implication relation holds between two contingent statements where the denial of the consequent is consistent with the affirmation of the antecedent. However, it will be more worthwhile to

35 MacColl, "Symbolic Reasoning", Mind N.S.V. 12 (1903), p. 362. - 41 - look at criticisms of his position, principally by Russell, not so much in order to see what Russell thought but to get clear on MacColl's position.

Russell responds in detail to two of the examples given above. Of the first, the red-haired doctor example, Russell accuses MacColl of not distinguishing between propositions and propositional functions. He notes that 'He is a doctor' and ~He is red-haired' are not properly propositions until 'he' is specified and cannot therefore be said to imply one another. But if we put 'Mr. Smith' in the place of 'he', then, writes Russell,

••• it is easy to see that one must imply the other, using the word 'imply' in the sense in which l use it ••• l say that 'p' implies 'q' if either 'pl is false or 'q' is true. This is not to be regarded as a proposition but as a definition. Now consider 'Mr. Smith is a doctor' and 'Mr. Smi1;h is red-haired'. Four cases are possible, namely: (1) both are true, (2) both are false, (3) the first is true and the second false, (4) the first is false and the second true. In cases (1) and (2) each implies the other according to the above definition. In case (3) the second implies the first; and in case (4) the first consequences of the above definition of 'implies' together with the fact that a disjunction is true when either or both of its disjuncts is true. Hence, in aIl four cases, at least one of our two propositions implies the other. 36

This response to MacColl's criticisms, which is quoted directly at length to avoid any possible misunderstandings,

36 Bertrand Russell, "' If' and' Imply': A reply to Mr. MacColl", Mind N.S.V. 17 ('1908), p. 301 42 is somewhat curious when it is recalled that it is exactly this definition of implication that MacCo11 argued "should

n give logicians pause • That.is, MacCo11 is qui te aware that given Russell's definition of implication, or, at .. least, of ';:a', the formula ' (p~) v (

greater than AC and that the angle A is not acute or that the angle A is acute and AB is not greater than AC. The mere fact that, in a given exarnple, both staternents are true does not justify the claim that an implication relation holds. The implication A:B expresses "a general law and asserts that it has no exception. Its denial (A:B)' asserts that the law is not in aIl cases valid; it asserts (AB,)nl, that an exception AB' is possible".37 It is this difference as to the definition of implication which is, of course, the issue between MacColl and Russell. Unfortunately Russell does not directly discuss MacColl's definition, nor does he seem to realize that, at least as far as MacColl is concerned, this is the issue between thern. Nevertheless, it should be added that MacColl is not altogether clear on this either. In a letter to Russell dated January 26, 1905 MacColl suggests that the differences between them are more verbal than rea1 and that, if they cou1d ta1k together, they would probably resolve their differences. Later, in 1907, MacColl wrote in Mind, that

The differences between Mx. Russe1l's views and mine are mainly due, l think, to the fact that we reason from different data, because we do not always attach the sarne meanings to sorne words which are vital to the questions at issue. 38

37 MacCo1l, "Symbolic Reasoning", Mind N.S.V. 12 (1903), p.363 38 Hugh MacColl, "Syrnbolic Logic (A Reply}", Mind N.S.V. 16 (1907) p. 470 - 44 - It is true that MacColl and Russell often attach different meanings to the same word but it would be far from the truth to claim that their differences were merely verbal. Indeed MacCol1 had claimed more than twenty years before the above quotations were written, as we have seen, that his system was fundamentally different from other systems of symbolic logic. It can only be speculated as to why MacCol1 here minimizes the differences between them since, in his development as a logician, he saw himself as moving further away from other symbolic logicians. The reason, perhaps, is that MacColl held Russell in very high regard and Russell was, so far as can be determined, the only logician with whom MacCol1 had regular contact (through letters) outside the pages of Mind and, to a lesser degree, publications in other journals.

Coming now to the second major criticism advanced against these examples by Russell, we can see that Russell is on much firmer ground. That is, his criticism does not simply miss the point that MacCol1 is trying to make although here again Russell gives no evidence that he clearly sees what MacCol1 is trying to get across. Here Russell accuses MacCol1 of overlooking the distinction between formaI and material implication in examples like the one given above concerning points in two circles and an ellipse. Russell advances this criticism against an earlier example given by MacColl, but, since the point is the same, this - 45 -

criticism would also apply to the exarnple here.

To indicate what he means by "formaI implication" Russell says the following in Principles of Mathematics:

[formaI implications are] such propositions as "x is a man implies x is mortal for aIl values of x" - propositions whose general type is: "fZJ(x) implies W(x) for aIl values of x" where fZJx and 1{ixJ. for aIl values of x, are propositions. ~9

Later in the sarne work he cornes to a discussion of the formula MacColl is questioning~ narnely ~:C : «A:C) + (B:C» to put it in MacColl's notation. Russell writes here that:

••• the above formula can only be truly interpreted in the propositional calculus: in the class calculus it is false. This may be rendered obvious by the following considerations: Let ~x, ~x, Xx be three propositional functions. Then "fZJx, lJIx implies XX" implies, for aIl values of x, that either ~x implies XX or wX implies Xx. But it does not imply that either fZJx implies Xx for aIl values of x, or wx implies XX for aIl values of x. 40 4l Later in Mind Russell makes the sarne point. Clearly, then Russell is saying that MacColl confuses the following two formulas, one of which is valid for Russell and one of which is not:

39 Bertrand Russell, The Principles of Mathematics Second Edition (London: George Allen and Unwin, 1937), p. Il 40 Ibid., p. 22 41 Russell, '''If' and 'Imply''', Mind N.S.V. 12 (1903), p. 301 - 46 -

(1) (x) {(9Jx. 1JjX • ::Jxx) .;). (9Jx;. Xx) v (l/JX:J Xx)}

(2) (x) (9Jx. J/ix • ;:J Xx) .;:). (x) (9Jx;:) Xx) v (x) (lj}X;:) .Xx)

Two questions arise here: first, does MacColl confuse these two formulas such that the one he is objecting to is (2), which Russell would also reject and, second, even if this is the case, does this affect MacColl's main point in this example?

The answer to the first question is yes. An objection might be raised here that since ':' and ,~, do not mean the same thing MacColl's formula cannot be translated into either (1) or (2). This point becomes relevant when considering the second of these two questions but here it is not since the mistake MacColl makes here would be a mistake no matter which meaning one attached to ':' and, also, the point of the example is to give grounds for rejecting a formula such as (1) in the "other" systems of logic, hence MacColl must deal within those systems to show their limitations. Suppose we look back on MacColl's example of the points in the two circles and ellipse and adopt the following interpretation: Uni verse of discourse consists of the two circles, ellipse and ten points, A, B, and C are one place predicates signifying that (1) is a point in A, (1) is a point in B, and (1) is a point in C. It can be seen that MacColl's formula - 47 -

(AB: C). : (A: C) + (~: C) translates into (3) (x) (Ax. Bx. ;» Cx)

.;:1. (x) (Ax ~ Cx) v (x) (Bx ~ Cx). The evidence for this is the ~ay MacColl discusses the consequent (A:C) + (B:C) in the example. There he says that A:C asserts that P cannot be in A without being in C and, independently, P cannot be in B without being in C. To translate back into Russellian notation MacColl is saying that each of the formulas (x)

(Ax =>Cx) and (x) (Bx ;:lCx) is false hence the disjunct

( (A:C) + (B:C) ) is false. But, clearly when the whole disjunct is translated it then becomes (x) (Ax .:IBx) v (x) (Bx ::;)CX) which makes MacColl' s formula equivalent to (2) above which, as mentioned, Russell also rejects. Russell, then, is correct on this point and shows an important deficiency in MacColl's logic in demonstrating this.

Returning now to the second question, above, concerning whether this objection of Russell's, although correct, affects MacColl's main point, one can see that the answer is no. (This is not, of course, to suggest that MacColl is finally correct; only that Russell's.criticisms do not show that he is wrong.) This can be shown as follows: if we give the names Pl' P2 , P3 ••• PlO to the ten points in the figure and suppose that Pl names a point which is in the circle A but not in the ellipse C or circle B and then instantiate formula (1) above we will have

«API BPI) ~ CPl) :> (API :> CPl) v (BPl;;;J CPl) - 48 - which would be valid for Russell. The important point for MacColl, however, is that this ho Ids for Russell only because a false antecedent implies a true (or false) consequent which is precisely what MacColl objects to. MacColl's objections are generally weIl worked out in propositional logic, but when it cornes to quantified logic he is much less precise, leaving openings for Russell to make good criticisms of him without touching main issues between them.

MacColl has a further, and somewhat curious, criticism of formulas such as (1) above. The full implications of the remarks he made concerning this formula will be discussed later; for the moment they will merely be noted. If, in this example, we look at what AB:C says for MacColl it is evident that on the basis of the data AB:f is a certainty.

That is, it cannot not be true. On the other hand (~:C) + (B:C) is contingent (variable) since, by itself and on the basis of the data, it could be false. Now since, for MacColl, a certain statement cannot imply a contingent statement AB:C cannot then imply (A:C) + (B:C).42

Restricting the validity of a formula to one example (data) would certainly limit the use of a logical system (that is, calling AB:C a certainty on the basis of this example) but, fortunately, MacColl does not persist in

42 ll MacColl, IISymbolic Reasoning , Mind N.S.V. 12 (1903) p. 363. - 49 - this, and, generally, a formula is certain if it holds independently of any special data.

The above eX'amples bring out what MacColl finds objectionable about the two-valued truth functional definition of implication. MacColl realized that there were analogous problems with his definition of implication, what have been referred to as the paradoxes of strict implication. MacColl's position on these paradoxes is somewhat ambiguous but before examining his position it must first be shown that he is, in fact, committed to them.

MacColl states the two paradoxical formulas as (1) !l:~ and (2) ~:.f.. which are to be read as "if an impossibility be true any statement x is true, or that an impossibility implies any statement 1andl the statement x (whether true or false) implies any eertainty". 43.

He offers the following proofs; "n:x means (n~,)n, and asserts that nx' is an impossibility, whieh is true, sinee the statement nx' contains the impossible factor n. We prove x:e: as follows: x:e: = (xe:,)n = (xn)n = 1'}n =e:". 44

These statements might weIl be rewritten in the manner indieated previously, so that we would have

43 MaeColl, Symbolic Lagic, p. 13 44 Ibid. - 50. -

. (B:A) where A and B are any two statements. These formulas occur in several places in MacColl's writings and there is no doubt that, given his definition of implication, A:B = (AB,)n , he is committed to them. He appears to take two positions regarding them, however. On the one hand, he seems to suggest that this is the most adequate rendering into symbo1ism of the concept of deducibility, on the other, that this yields the most workable logic for solving prob1ems in mathematics. On balance it would appear that the former is the more important consideration for MacCo11. Sorne evidence for the former view can be found in MacCo11·s discussion of the equivalence (or, more accurate1y, the non-equivalence) of A:B and A' + B encountered above (pp. 15-16). There he suggested that in many uses the disjunction is not equivalent to the implication and it was suggested that this was due to his uneasiness over the c1aim than an implication relationship held between two contingent statements which have nothing, or little, to do with one another. Later (1901) in a letter to Russell he wrote: l do not think that l depart widely from ordinary conventions when l define the (brief) conditional "If A is true B is true" as synonymous with the 10nger and less convenient "The assertion that A is true and (at the same time) B is false is an impossibility".45

45 Letter from Hugh MacColl to Bertrand Russell, June 22, 1901. Bertrand Russe11 Archives, McMaster University. - SI -

This indicates that, according to MacColl, his definition, from which the paradoxes follow, captures sorne "ordinary convention" for the meaning of 'implies' which is not the ordinary or common convention followed by other logicians. In 1903, however, MacColl wrote the following:

Whether my interpretation of this troublesome little conjunction 'if' is the most natural and the most in accordance with ordinary usage, l do not undertake to say; it certainly is the most convenient for the purposes of symbolic logic, and this alone is reason sufficient for its adoption. 46

Later still (1906) he appears to weaken on this last view expressed as he tries to justify calling an implication valid solely on the basis of the modal values of its constituent statements. Here he is discussing two necessary statements A and Beach of which implies the other simply because they are both necessary.

Yet here it is not easy to discover any bond of connection between the two statements A and B; We know the truth [modal valuesj of each statement independently of aIl consideration of the other. We might, it is true, give the appearance of logical deduction somewhat as follows •••

He then gives a short argument which purports to show that '13 + 5 = 18' implies, and is implied by '4 + 6 = 10' and continues as follows:

46 MacColl, "Symbolic Reasoning", Mind N.S.V. 12 (1903), p. 363. - 52 -

Everyone must feel the unreality (from a psychological point of view) of the above argument; yet much of our so-called 'rigorous' mathematical demonstrations are on lines not very dissimilar. 47

This is as close as he cames actually to paradoxes defending his concept of implication as far as the directly are concerned. It would appear from the quotation above, however, that he felt that the paradoxes simply The expressed facts about implication, or deducibility. because feeling that there is something wrong with a system to be it contains these paradoxes is a feeling which ought nor did overcome. Unfortunately MacColl did not consider, anyone put to him, exarnples such as "'2 + 2 = S' implies on a 'Hugh MacColl is a logician'''; exarnples which bring than much stronger psychological feeling of "unreality" n'l3 + 5 = 18' implies '4 + 5 = 10'''. If MacColl replied quotation ta this kind of exarnple in the manner of the 1903 finds given above, it could be objected that since what he is the objectionable in Russell's definition of implication paradoxical air of asserting that an implication relation one holds between two statements having nothing to do with hardly another, a similar objection against his system can be ignored. That is, MacColl cannot, nor can any other logician who objects to a two-valued, extensional of propositional calculus on the basis of the paradoxes

47 MacColl, Symbolic LogiciPP.8l-82. - 53 - material implication, retreat behind the claim that "This is how l define implication for my own symbolic purposes".

As we have seen MacCol1 once did this, but, generally, he tried to avoid the issue of the paradoxes in his own system.

MacCol1 did not set up a formaI system of logic in which primitive terms, axioms, postulates, transformation rules etc., were clearly stated as such. Indeed, it was not until several years after MacColl's death that such a system was constructed for modal logic. How ever, it is also true that he went much beyond appealing to intuition to convince his readers of the validity of the formulas he held were valide Unfortunately, as we have seen, his method of proof was not wholly reliable yet certain important characteristics of a formaI system can be abstracted from MacColl's writings.

It is important to note how much MacCol1 traced out the consequences of his primitive ideas (his definition of implication and, though unstated as such, his formation rules). He sees that his definition of implication leads to the paradoxes of strict implication, and other theses of interest stated belowi he sees the need in his system to reject certain formulas (such as (AB:C) : (A:C) +

(B:f)i he consciously follows certain transformation rules such as substitution of any well-formed formula for a statement variable and substitution of equivalencesi and, as the foregoing indicates, he was especially concerned with - 54 - the question of consistency, or, as MacCol1 put it, he consistency of notation, which suggests strongly that saw his work in logic as constructing a consistent system them in which certain formulas could be proved by showing to be consequences of earlier proved formulas or consequences of his basic definitions.

For these reasons it is justified in concluding like this chapter to exhibit MacColl's system in something the form of an axiomatic system. In attempting to be con faithful to MacCol1 no effort will be made toward the for struction of an elegant, economical system. Since, and example, MacCol1 does not take conjunction as primitive or define disjunction in terms of conjunction and negation, the reverse, both will be stated as primitives (though MacColl, of course, was aware that they were interdefinable).

Earlier a distinction was made between relative words) and absolute (or material and formaI to use MacColl's certainty, impossibility and contingency. This distinction will be examined in sorne detail in the following chapter the but, for the moment, it is enough to note that aIl of following formulas are claimed by MacCol1 to be valid, not absolutely, or formally. That is, their validity is relative to specifie data. - 55 -

Primitive Syrnbo1s

(1) Statement variables: A, B, C •••

(2) Monadic operators: ' (negation) E(necessity) n (Impossibi1ity) e (Contingency) eliminated (3) Dyaadic operators: x (conjunction; usually side in favour of simply placing statements conjoined

by side) , + (disjunction), : (implication), = (equivalence) (4) Brackets: (,)

Formation Ru1es

(1) A statement variable alone is a wff , E n e (2 ) If a is a wff so are a , a , a , a (usually (aS», (3) If a and Sare wff's, then so are (a x S)

(a + S), (a:S), (a = S)

Definitions (1) (a + S) .-. df • (a'S')' S)E (2 ) (a S) .-. di. (aS' ) n • = • df. (a' + (3) (aS) .-. df • (a' + B')' (4) (a' ) E .-. df • an E (5) (a,)n .-. df. a (6 ) (a = S) .-. df • (a:S) (f3;a)

Transformation Rules

(1) Substitution of (strict) equivalences.

(2) Substitution of any wff for a statement variable.

(3) Detachment. - 56 -

AlI of these primitive syrnbols, formation rules, definitions and transformation rules can be found stated or implied in MacColl's writings. However, MacColl did not give any axioms, or what one could fairly interpret as axioms, hence, none will be given here. Also MacColl did not prove theorems in an acceptable manner so what list follows below, instead ofaxioms and theorems, is a of for.mulas MacColl took as valid and important.

48 Some Valid Formulas (1) (A:B) (B:C) : (A:C) This law of the transitivity of implication, called by MacColl "the law of implication", is one formula which could, in fairness, be read as an axiom in his system, since it is used constantly te prove other for.mulas. (2) A(B + C) = AB + AC (3) (A:B) = (B':A') This formula like (1) above is often used in the solution of problems.

(4) (A + A') E

(5) (AA') n

(6) (A E + An + Aa ) E E (7) (A : A) (8) (An · A' ) (9) (AS · A' a)

in 48 Ibid., pp. 7-9. Other lists of valid formulas appear MacColl's writings but the formulas listed here appear with the greatest frequency. - 57 -

(10) (AE: x A:B) : BE:

(11) (Bn x A:B) An

(12) BE:: (A: B)

(13) An: (A:B)

(14) (An Bn ) «A:B) (B:A»

(15) ( (A: B) (B: A) )

(16) (A:B) (A:C) (A:BC)

* (17) (A:B) + (B:A)

* (18) (AB:C) : «A:C) + (B:C» *(19) (A e x A: B) : Be

The first sixteen formulas are ones which are characteristic of MacColl's lagic and turn up often in his writings. The last three, (17 - 19), are important formulas that MacCall rejects as invalide These formulas will he discussed in the last chapter of this study when comparing MacColl' s logic ta later sys·:':'ems. WO

DIVISION OF STATEMENTS

summary. This chapter begins by noting sorne of the problems with MacColl's use of the term 'statement'. MacColl's concepts of certainty, impossibility and contingency are then discussed. The distinction MacCol1 makes between material and formaI modalities is el:plained. Criticisms directed toward MacCol1 from two of his contemporaries, Russell and A. T. Shearrman are noted in detail as are MacColl's responses. The question of whether MacColl's conception of the modalities was temporal is then considered. Finally, MacColl's suggestion for developing an epistemic logic founded on his basic system of logic is noted.

As we have seen MacColl's primary interest was in constructing a logical system based on the logic of state ments. He divided statements into necessary (certain), impossible, and contingent (variable)' as weIl as simply true and false. By so doing he created for himself several of the problems which were briefly touched on earlier. These problems, together with the principal discussions MacCol1 had with his contemporaries on these issues will now be gone into in sorne detail.

- 58 - - 59 -

There is sorne confusion in MacColl's writings as to what is meant by his key words like 'statement', 'certain', which is brought about, in part, by MacColl's using these words to convey slightly different ideas in different places in his writings. This confusion stems from three separate sources: MacColl's logical writings span almost thirty years, and there are major changes from one period to another; MacCol1 is simply careless at times; MacColl relies heavily on context to make clear what he means. This last is the most important for our purposes, and it is one that MacCol1 himself was weIl aware of. Indeed he saw it as a virtue and extended this variableness of meaning to symbols.

• • • [a] princip le underlying my system of notation is the principle that we may vary the meaning of any symbol or arrangement of symbols, provided, firstly, we accompany the change of signification by a new explanatory definition; ••• this variation should not be resorted to wantonly and without cause; ••• The great danger to be guarded against is, of course, • • • ambiguity. The symbol (or combination of symbols) chosen should be such that the context and the general nature of the research must render its meaning unmistakable. l

unfortunately the different meanings one word may take in MacColl's writings are not always made perfectly clear by the contexte In fact inconsistencies arise in several places, which will be brought out in this chapter and l MacColl, "Symbolic Reasoning", Mind V. Il (1902), pp. 362-363. - 60 - where it is possible resolved. One inconsistency occurs with the word 'statement' which MacCol1 defines as "any sound, sign or symbol (or any arrangement of sounds, signs or symbols) emplQyed to give information". For instance, the Union Jack flying from a ship is, for MacColl, a statement which may mean 'This is a British Ship'. Likewise, he wri tes, "The nod [of the head] may mean '1 see him'; the shake of the head '1 do not see him'; the warning 'caw' of the rook, 'a man is coming with a gun' or 'danger approache s, and so on".2 It is also the case, for MacColl, that every statement is either true or false; but if we were to consult a book on heraldry we would not find as part of the description of the Union Jack that it was either true or false, nor would a complete physical description of a head's nodding tell us whether it was true or false; such information would simply be out of place here. In MacColl's definition of 'statement', above, a statement is any ~ound, sign or, symbol, not what a sound, sign or symbol is used to express. His adding "used to give information" to his definition does not alter this since it is still the sound, sign or symbol which is the statement, not the information it is used to give. Nowhere in MacColl's writings is there a clear discussion of statements, propositions and sentences. It is left to the reader to discover what he means by 'statement', the word most frequently used to express what

2 MacColl, Symbolic Logic, p.2. - 61 - is often expressed by using the three words 'statement', 'proposition' and 'sentence'. That is, MacColl, most of the time, makes no distinction between 'statement', 'proposition' and 'sentence'; yet it would be wrong to say, as Russell does, which we shall see shortly, that MacColl makes no distinction between a statement and what is expressed by a statement. It is, primarily, MacColl's definition of 'statement' and his explication of that concept, that crea tes a problem. In his use of statements or statement variables in his logical writings, it is clear that a statement is what is expressed by a symbol rather than the symbol itself, since statements are true, false, certain, impossible and contingent. This last point will be shown more clearly when each of the kinds of statements is discussed. It should be Iloted that when discussing statements below the word is used in the sense of what a sign, sound or symbol expresses rather than as referring to the sign, sound or symbol itself. This is how MacColl generally used the term 'statement' even though his explicit definition of it conflicts with this reading. This policy is adopted for two reasons: first to avoid the danger of misrepresenting MacColl's position technical words are used, generally, in the way he used them except where considerable confusion is likely to occur as was the case with 'variable' in the preceding chapter and, secondly, to avoid confusion when comparing a discussion of a point of MacColl's with direct quotations from his works which state the point under - 62 -

discussion.

As was noted earlier, MacCol1 was the first symbolic logician to found logic on the logic of statements. He felt that by doing so he was only following the natural course of the evolution of language, as the following remarks indicate:

In thus taking statements as the ultimate constituents of symbolic reasoning l believe l am following closely the graduaI evolution of human language from its primitive pre historie forms to its complex developments in the languages, dead or living, of which we have knowledge now. There ~in be little doubt that the language or languages of primeval man, like those of the brutes around him, consisted of simple elementary statements • • • [but were] more or less conventional and therefore capable of indefinite development. 3

Whether or not MacColl's speculations in linguistic anthropology are worthwhile is beside the point here. What is important, however, is that speculations such as these led MacColl to concentrate on statements, rather than classes, and to develop a logic in which different kinds of statements were employed. That is, he believed that not only were statements true or false, which, of course, was a commonplace idea, but that statements had the logical property of being necessary or impossible or contingent, which was far from commonplace and, indeed, as MaCCo11 uses these terms, is still far from commonplace. By doing so he forced sorne of his contemporaries in symbolic logic to come to grips with novel (in that context)

3 Ibid., p. 3. - 63 -

arguments, thus helping them to clarify their own positions, and also, in a fairly direct fashion, helping to bring about contemporary modal logic.

While speculations about the origin of language led MacColl to concentrate on statements, it was his early interest in probability theory which led to his classifying statements in the way he did. As we saw in the preceding chapter, he first introduced his syrnbol for 'certain', 'E', 'to denote aIl staternents whose probability is 1. Sirnilarily, impossible statements were those whose probability was 0 and contingent (variable) statements those whose probability was somewhere between 0 and 1. 4

Further,MacColl claimed, certain statements have the property of lasting and necessary truth; impossible statements that of lasting and necessary falsehood; con tingent statements that of being neither certain nor impossible. 5 A contingent statement has the property of being sometimes true and sometimes false; that is, the contingent statement A, for instance, may be true at one time and false at another. In addition to the above MacColl has a class of meaningless statements; that is, sentences which are grammatically well-forrned but are neither true nor false, e.g., "A small whale can swallow a large conclusion".6

4 MacColl, "Equivalent statements", Mathernatical Society v. 28 (1896-97), p. 555. 5 Ibid., p.' 157. 6 MacColl, Symbolic Logic, p. 10. -64-

One of the difficulties MacCol1 saw with his logic was that if statements were either certain, impossible or contingent with respect to the data under,consideration, this would severely restrict the range of application of his system. In effect for most cases he abandoned this practice of having formulas valid with respect to data. For instance, aIl of the formulas listed as valid at the end of the previous chapter are valid unconditionally. That is, they do not hold only with respect to particular data. Nevertheless, MacCol1 never completely abandoned his early view that "the data" should be taken into account, at least in seme cases, when deciding whether a formula was valide To get round this problem he introduced a distinction between materially valid and formally valid formulas with the former holding with respect to particular data and the latter holding unconditionally. The following remarks of MacColl's will make this clear.

A proposition is called a formaI certainty when it follows necessarily from our definitions, or our understood linguistic conventions, without further data; and it is called a formaI impossibility, when it is inconsistent with our definitions or Iinguistic conventions. It is called a material certaint~when it follows necessarily from some spec~al data not necessariIy contained in our definitions. Similarly, it is called a material impossibility when it contradicts sorne special datum or data not contained in our definitions. 7

7 Ibid., p. 97 - 65 -

Examples of formally certain statements would include the formulas listed as valid at the end of the previous chapter, statements such as '2 + 2 = 4' etc., and statements such as 'The part is less than the whole'. Formally impossible statements are negations of formally certain statements. On the other hand an example of a materially certain statement would be lia small whale can swallow a large herring" and a materially impossible statement would be 'a small herring can swallow a large whale'. A materially contingent statement would be a statement which is either true or false but neither materially certain nor impossible.

MacColl at one point introduced special notation to distinguish formaI impossibilities and certainties from mate rial impossibilities and certainties. If ~ is a statement, ~a states that ~ is a formaI certainty; ~a that ~ is a formaI impossibility; ~E that ~ is a material certainty; and ~n that ~ is a material impossibility. In MacColl's words ~E says that ~ is always true and ~a that ~ is always true since it would be an inconsistency of language, or a contradiction to our definitions to say that it is false. Similarly, ~n a~serts that ~ is always false and ~a that ~ is always false since it would be a contradiction to our definitions to say that it is true. It is clear from this that ~a implies ~E and ~a implies ~n • That the examples MacCol1 uses to illustrate formaI certainty and impossibility - 66 -

«AB:B+C) and (A+A' :BB') respectively) are drawn from logic is not without significance since he normally uses the sumbols 'E' and 'n' to mean certain and impossible in his logic. Thus where, as for instance in the previous chapter,

a formula is governed by the modal operator E e.g. (A+A,)E, it can be read as "it is formally certain that A or not A".8

This indicates how MacColl applies his principle of "the variableness of meaning of symbols" mentioned above. Whereas 'E' applied to both materially and formally certain statements in most of MacColl's writing (and that is how the symbol was used in the preceding chapter), although, in practice, it almost always applièd to formaI certainties, here it is used to indicate that the formula is a material certainty. This inconsistency in the use of symbols, despite MacColl's being aware of it, leads to several difficulties. For instance in the example of the points in the circles and ellipse cited in the previous chapter, MacColl wanted to deny the validity of (AB:C) : «A:C) + (B:C»i that is, he wanted to deny that it is a formaI certainty. One of the arguments he used was that AB:C on the basis of the data was a certainty while (A:C) + (B:C) was not hence AB:C could not be said to imply(A:C) + (B:C) as certainties can only imply other certainties. However AB:C is a material certainty and, using MacColl's reasoning, the most

8 MacColl, "Equivalent Statements", Mathematical Society, v. 29 (1897-98), p. 98. - 67 - it could then be used to deny is that AB:C : ({A:C) + (B:C» is a material certainty, which, of course, is not the issue. Therefore, this particular argument is beside the point. If MacCol1 had consistently employed his symbols such that one symbol always referred to material certainties and the other to formaI certainties this problem would not arise.

In general, though, MacCol1 used 1 E 1 to indicate that the formula was formally valid, and since it was used that way in the previous chapter, it will continue to be used that way except where it is explicitly mentioned that the formula being considered is a material certainty. For the purposes of this chapter it is more important to note that MacCol1 did distinguish between kinds of certainties and impossibilities than what symbols he used in making this distinction.

The situation with regard to contingent state ments is somewhat more complexe MacCol1 gives as an example of a formally contingent statement A:BC. 9 Clearly this formula is neither certain nor impossible independently of any special data, although given special data the formula may be true or false. If a statement is a material contingency, that is, one which is neither implied nor con tradicted by any specifie data, then it is a formaI contingency. However, if it is a formaI contingency, it does not follow that it is a material eontingency.

9 Ibid., p. 98 - 68 -

This raises the important problem of whether a statement is a contingency only in relation to our ignorance. For instance, in one of the examples in the previous chapter, the formula AB:C occurs. Now, clearly, this formula ought to be regarded as formally contingent by MacColl, but in that particular case it was regarded as materially certain since "the supposition of its denial is inconsistent with our data", i.e., the data in the particular case. It has already been suggested that the problem was with the idea of material certainty (and impossibility) since, for MacColl, once one had grounds for assertinq the truth (or falsehood) of any contingent statement one then had qrounds for asserting that the statement was materially certain (or impossible). This can also be seen after further consideration of one of MacColl's standard examples of a contingent (variable) statement, 'Mrs. Brown is not at home'. MacCol1 says of this example that ••• when we have no data but the mere arrangement of words "Mrs. Brown is not at home", we are justified in calling this proposition, that is to say, this intelligible arrangement of words, a variable, and in asserting AS. If at the moment the servant tells me that "Mrs. Brown is not at home" l happen to see Mrs. Brown walking away in the distance, then l have fresh data and forro the judgment AE • • • In this case l say that "A is certain" because its denial A' ("Mrs. Brown is at home") would contradict my data, the evidence of my eyes. But if, instead of seeing Mrs. Brown walking away in the distance, l see her face peeping cautiously behind a certain through a corner of a window, l obtain fresh data of an opposite kind, and forro the judgment An ••• In this case l say that liA is impossible" because the statement represented by A, namely, "Mrs. Brown is not at home", this - fi9 -

time contradicts my data, which, as before, l obtain through the medium of my two eyes. 10

Here again a statement is considered contingent (variable) only until sufficient evidence is available to determine whether it is true or false; it then becomes either materially certain or materially impossible. Also, in this example, the staternent (proposition, as MacColl uses that word in the above example) 'Mrs. Brown is not at home' would appear to be merely "this intelligible arrangement of words" not what that arrangement of words is used to express. Therefore the same statement may be true at one time and false at another. Indeed MacColl does not shrink from such a claim and Russell is quick to challenge him on this issue.

The situation is made even more complicated by a third example of a contingent statement that MacColl gives, namely 'x = 4,.11 When this statement becomes true or becomes false when appropriate substitutions are made for 'x', it is clear that it is either formally certain, e.g., '4 = 4' or formally impossible, e.g., '5 = 4'. Thus, in this case we have a contingent statement 'x = 4' becoming either formally certain or impossible.

There are several problerns with MacColl's position, as has been indicated, which come out clearly in the criticisms of MacColl given by Russell and MacColl's responses to these

10 MacColl, Symbolic Logic, p. 19 Il MacColl,"Symbolic Reasoning",Mind N.S.V. 6 (1897), p. 496 - 70- - criticisms. One of Russell's main claims, that MacCol1 fails to observe the distinction between a proposition and a propositional function, has already been encountered although here Russell applies this criticism directly to MacColl's notion of contingency. First, then, Russell: As an instance [of a contingent statement (variable)] he [MacColl] gives "Mrs. Brown is not at home". Here it is plain that what is variable primarily is the meaning of the form of words. What is expressed by the form of the words at any given instant is not itself variable; but at another instant something else, itself equally invariable, is expressed by the same form of words. Similarly in other cases. The statement "He is a barrister" expresses a truth in sorne contexts and a falsehood in others. Thus the variability involved is primarily in the meaning of the form of words. Ordinary language emp1oys, for the sake of convenience, many words whose meaning varies with the context or with the time when they are emp10yed ••• It is such forms of words that constitute Mr. MacCol1's"variables". But is this not importing into logic the defects of common speech't ••• when we are told "Mrs. Brown is not at home", we know the time at which this is said, and therefore we know what is meant. But in order to express explicitly the whole of what is meant, it is necessary to add the date, and then the statement is no longer "variable" but always true or always false. and further on the same page Russell adds that MacColl's contingent statements are rea11y propositional functions which are true for sorne values of the variables and false for others: We may say that "x is a barrister" or "Mrs. Brown is not at home at time x" is true for sorne values of x and false for others. Either of these is a propositional function: but neither is a proposition.12

12 Bertrand Russell, "Review of Symbolic Logic and its Applications", Mina N.S.V. 15 (1906), p. 257. - 71 -

A similar point was made earlier by Russell in 1903 in The Principles of Mathematics. Here, it is clear that what he objects to is MacColl's allowing that the same proposition can sometimes be true and at other times false. Mr. MacColl, in an important series of papers has contended for the view that implication and propositions are more fundamental than inclusion and classes; and in this opinion l agree with him. But he does not appear to me to realize adequately the distinction between genuine propositions and such as contain a real variable: thus he is lead to speak of propositions as sometimes true and sometimes false, which of course is impossible with a genuine proposition ••• A proposition, we may say, is anything that is true or that is false. An expression such as "x is a man" is therefore not a proposition, for it is neither true nor fa1se. 13 MacColl's replies to Russell are not a1ways to the point. Indeed their published discussions of each other's work often show a lack of appreciation of what the other is trying to do. MacColl appears to recognize this and, in a letter to Russell, laments the fact that they cannot get together to reso1ve their differences. Unfortunately this meeting never took place.

On the question of propositional functions and propositions MacColl objects to Russe11's remark that 'x is a man' is a propositiona1 function rather than a proposition. In a letter to Russell, dated May 3, 1905, MacColl says that "this is as if you said 'Nothing ought to be called a number un1ess it is odd or even' yet 2/3 is

13 Bertrand Russell, Principles of Mathematics 2nd Edition (London:George Allen and Unwin, 1937), pp. 12-13. - 72 - included in the class of numbers". He goes on to note that aIl grammarians calI 'x is a man' a proposition, and, so far as he can see, that is aIl there is to it. "Your convention suits YOlir system but it does not suit mine". This clearly will not do for Russell as he feels that the distinction'is of great philosophical importance and not simply a matter of convenience in a syrnbolic system. Also, it omits discussion of Russell's central point that, whatever propositions are, they are the bearers of truth and falsity; since 'x is a man' is neither true nor false, it is not a proposition.14 That MacColl seemed to miss entirely Russell's point, in a manner not unlike the way Russell missed what MacColl was trying to do with implication, is borne out by the following remarks MacColl makes concerning variables. • •• certain staternents, such as "Mrs. Brown is not at home", are sometimes used to convey, and do convey, a true information; while at other times they are used (in the very sarne forrn of words) to convey and do convey, a false informa tion. This class of statements l calI variables. Let A denote a statement of this class, and (to fix our ideas) suppose it to have been made on five different occasions: truly on the first, third, and fifth occasions; falsely on the second and fourth. Let the s~rmbols Al, A2, As, A4, As denote these five separate staiements, each expressed in the sarne form of words. The general syrnbol A (not any of the earticular syrnbols Al' A2, As, A4, As) denotes this forrn of words; and therefore, l hold that, though none of the statements Al' A2, can be spoken of separately as sometimes true and sometimes false, yet their general representative A, which may denote any of them, whether true or false, can be thus spoken of. lS

14 Ibid., p. 14 15 MacColl, "Equivalent Statements",Mathematical Society v. 29 (1897-98), p. 99. - 73 -

In these remarks there is considerable confusion. On the one hand we have the letter A denoting

n ~ particular statement, e.g., "MrS. Brown is not at home which on sorne occasions is uttered truly·. and on others falsely. On the other hand we have five different uses, or occurrences, of the same sentence type A and on each occurrence of its use a different statement is made; sorne

of the statements false r some true~ The problem is which of these views, if not both, should be ascribed to MacColl and, then, how does this view stand up against Russell's criticisms. The following remark, from Symbolic Logic and its Applications, though itself perfectly clear, tends to 'cloud the issue further. Here MacColl is discussing con tingent statements (variables)and attempting to meet objections to them: To say that the proposition A is a different proposition when it is false from what it is when it is true, is like saying that Mrs. Brown is a differ~person when she is in from when she is out. 16

Since we can take it that MacColl is firm in his belief that Mrs. Brown is the same person in or out, it follows that, for him, the same statement ('proposition' and 'statement' are, as has been noted, used interchangeably by MacColl) can be at one time true and at another time false. Again, in the passage just above that last quoted, MacColl is quite explicit about contingent statements when considering the statement 'Mrs. Brown is not at home'. It is clear that it

16 MacColl, Symbolic Logic, p. 19 - 74 -

is neither a formaI certainty nor a formaI impossibility so that "when we have no data but the mere arrangement of words, 'Mrs. Brown is not at home', we are justified in calling this proposition, that is to say, this intelligible arrangement of words, a variable,,17 Here the statement is "the mere arrangement of words" , and the distinction MacColl introduced earlier, between the general forro of an expression and the statement made when the expression is used, disappears. It is this feature which Russell objects to 'l1hen he accuses MacColl of overlooking the distinction "between a verbal or symbolic expression and what it means",18 and in this, surely Russell is quite correct. Russell is also correct in noting that MacColl does not always distinguish a proposition from a propositional function. The problem remains, however, of what we are to make of MacColl's contingent statements.

There is sorne indication in the above remarks of MacColl's that by interpreting his system as a tense-logic sorne of the difficulties brought out by Russell could be avoided. 19 For instance, MacColl says that the same statement 'Mrs. Brown is not at home' can be true at one time and false

17 Ibid. 18 Russell, "Review of Symbolic Logic", Mind N.S.V. 15 (1906) p. 256 19 The view that MacColl's logic can be interpreted .te~porally has been suggested by Storrs McCall. See his nHugh MacColl" in Encyclopedia of Philosophy, ed. Paul Edwards, V.4 (New York: MacMillan, 1967), pp. 545-547. - 75 - at another, thereby rejecting Russell's claim that the statement should be read as 'Mrs. Brown is not at home at time x', where once 'x' is specified the statement is then true or false eternally. He also says that to calI the statement

(proposition) 'Mrs. Brown is not at home' a different statement (proposition) when it is true from when it is false is like calling Mrs. Brown a different person when she is out from when she is in. This suggests that the truth value of a statement changes with changes in the fact being referred to although the statement itself does not change, hence the time at which a statement is made is crucial in determining its truth-value. As we have seen, MacColl aiso says that certain statements are always true and impossible statements always false, which is also suggestive of a temporal approach to logic. However, there is clear evidence that this is not what MacColl intended. This is shown clearly in the following passage:

The possibility of converting relations of the past or future into relations of the present is one of the many advantages of pure logic or the logic of statements. Let the symbol A denote the statement "The event a did happen- or let·it denote the statement "Theevent a will happen". In either case we write AT, Al, AE, etc.; that is, A is true, A is false, A is certain, etc. If A= "The event a-did happen-, Then AE asserts that "it is certain that a did happen"; and if A = "The ëVent a will happenr then AT asserts "It is true that a will happen, whether A refers to~he past, present, or future, AT (which replaces A in symbolic reasoning) always refers to the present; and the same may be said of AE , An , AS , and ofAx generally, whatever class of statements x may represent. Thus AT and At are not exactly synonymous with A and A'. 20

20 MacColl, "Symbolic Reasoning", Mind N.S.V. Il (1902), p. 361 - 76 -

The last remark, that there is a difference between saying that A is true and simply A and between A is false and simply not A, suggests another approach to MacColl's contingent statementsi namely, that when we con sider a statement that may b.e either true or false given certain data we are only considering the content of the statement, we are not considering the statement as asserted. On the other hand when we say that A is true, or that A is false we are asserting A or A'. An extension of this reading would also make sense of MacColl's calling contingent statements materially certain :once one had sufficient evidence to calI the statement true, such as in the Mrs. Brown case. For instance we might read MacColl's example in the following way: If we are merely considering the thought of Mrs. Brown's being at home, then that is neither true nor false, and the statement (proposition) 'Mrs. Brown is not at home' may sometimes be asserted and sometimes its negation asserted. In either case we would then calI the asserted staternent either materially certain or impossible. Once it is either materially certain or materially impossible it cannot be otherwisei that is, a material certainty cannot become contingent (variable) or materially impossible at another time.

It appears that MacColl is here hinting at a' distinction of this sort. However, the difficulty with this view is that, for one thing, adding 'it is true' to a state ment is not always sufficient to make an assertion, and, for - 77 - another, MacCol1 includes the symbol for contingency, , 8 ' in the list of those symbols which characterize a statement, such as ' e " , n • etc. That is, to say that A is contingent, A8, is to say more than A; it is to characterize e A in a similar manner as 'e', '~' etc. If A is to be read as synonomous with ~ A, then A8 must also be written with an assertion signe Since the class of statements being considered is the class of contingent statements, denoted by '6', this way out will not do since, then, a contingent statement will not simply be a statement content but will be an asserted statement.

In MacColl's system of logic, i.e., the system outlined at the end of the preceding chapter, formally certain, formally impossible and formally contingent state ments occur. As far as this system of logic is concerned the introduction of materially certain, materially impossible and materially contingent statements is gratuitous, although statements of this type may, for MacColl, play a role in applied logic; that is, in the application of logic to sorne specific problem, such as the example of the two circ les and ellipse encountered above, but they never are unconditionally valid formulas nor negations of unconditionally valid formulas. For this reason sorne of the difficulties with material modalities can be put aside; not that they are unimportant, but because they do not affect the construction of a (more or lessj formaI system of logic which was one of - 78 -

However, it must be remembered MacColl's main purposes. appears in MacColl and that there that these difficulties are MacColl's terms. There still to be no way out of them on to make of MacColl's formaI remains the problem of what certainties and impossibilities. and Russell's criticisms of certainties from his remarks on contingent impossibilities flow directly MacColl's weakest position statements. Indeed, by attacking his remarks on the other two first, that on contingencies, a plausability out of proportion classes of statements derive advanced by Russell. On to the arguments actually he has the following to say: certainties and impossibilities

barrister"] may be Such a form e.g., ["x is a when it is true for aIl values called a certainty we give to x, i.e., when, whatever value of x", is true; it is the resulting proposition values when it is "false for aIl impossible it is neither certain nor of Xii, and variable when shall say that true and impossible. Thus we while applicable to propositions false are alone applicable variable and impossible are certain, and to propositional to ambiguous forms of words functions. 21 adds Further in his review Russell formula, the expression of aIl such for require a symbol MacColl's certainties] we [as author except Frege (so not to be found in any aIl express "~X is true for far as l know) , to calls x". This is what Mr. MacColl values of what the notation AE; but it exhibits explicitly to a namely the fact thatE applies 22 AE conceals, to a proposition. propositional function, not

Logic",Mind N.S.V. 15 (1906), 21 Russell, "Review of Symbolic p. 256 22 Ibid., p. 259 -79-

A quick reading of MacCol1 would provide evidence that ~E of x". This means, for MacColl, "~X is true for aIl values ••• is apparent in such remarks of MacCollls aS"(AI+B)E either asserts that lit is certain (true in aIl cases) that

l Il with Ais false or B is true ." Here Il certain is synonomous for ntrue in aIl cases" or, as Russell would put it "true cornes aIl values of the variables". Even clearer evidence out where MacCo11 is discussing the traditional syllogisme

Let Al' A2, A3 etc., be the individuals forming a class Ai and let BI, B2, B3 , etc., be the individuals forming a Class B. Out of Series Al, A2 etc., let an individual Abe taken at randoIn. The Symbol Afj, on this hypothesis, asserts that A is one of the individuals in the series BI, B2, etc. Hence AfjE, which is an abbreviation for (Afj)E, asserts that Afj is a certainty (E) Thus AfjE may be considered as synonomous with the traditional nAll A is B", or "Every A is aB". Similarly, Afjn, which asserts that Afj is impossible (n), is equivalent to the liNo A is B" of the traditional 109ici While Afjnt denies this, and asserts that "Sorne A is B". 23

This passage and others very like it appear to class support Russellis claim that MacCol1 is dealing with he inclusion, which fact his notation camouflages, when and ascribes the properties of certainty, impossibility contingency to expressions. However here, again, MacCol1 of meaning" is making use of his "principal of the variableness Clearly of symbols and, again, this results in confusion. the MacCbl1 is here using lEI and Inl in a manner not unlike way in which quantifiers are used, and Russellis criticisms

23 MacCbII, nSymbolic Reasoning", Mind N.S.V. Il (1902), p. 353. - 80 -

here are correct, although, it should be noted, if Russell had treated MacColl's formulation sympathetically he would have seen a very strong similarity between this and "the symbol not to be found in any author except Frege". How ever, for the moment this is beside the point. The important point is that MacColl does not always use 'E' and 'n'as he does here. For instance he would calI '2 + 2 = 4' and A + A' certain and '2 + 2 = S'and AA' impossible and not be using 'E' and 'n'in the same way as above. MacCol1 applies 'E', 'n'and 'e' to both propositional functions, in Russell's sense, and propositions. Russell concentrates his attack on the application of these symbols to propositional functions thereby sidestepping the question of whether propositions (statements, in MacColl's terminology) can be characterized as certain, contingent and impossible. Russell simply says that propositions are true or false (only), he does not consider here whether, for example, 'The Thames flows through London' and '2 + 2 = 4' though both true propositions have different logical properties, properties which are brought out by MacColl's use of 'e' and 'E'. MacColl, for his part, invites confusion by not consistently using the symbols 'E', 'n'and 'e' to mean the same thing. He is content to rely upon the context to make his meaning clear; unfortunately this reliance is not always safe, nor does MacColl indicate, in the passages quoted, that he is even aware of a distinction between propositions and propositional functions, to use Russell's terminology. - 81 -

That MacColl is aware of such a distinction is brought out clearly in the following remarks of his. Taking the class of certain, impossible, and contingent statements as primary it is clear, writes MacColl, "that we might develop other analogous three-dimensional systems on similar, yet not identical principles". One of the examples given here by MacCo11 is the following:

••• the symbol u might denote aIl functional statements which, like œS + œ' + S', are true for aIl values of their constituents; v aIl functional statements which, like œB(œ' +S'), are false for aIl values of their constituents' and wall functional statements which, like œ:S or œS+ X, may be true or false according to the values we give to their elementary constituents. 24

It is worth noting that here MacColl uses Greek letters in the examples of certain, impossible and variable "functional propositions". The values of the constituents of œS + œ' + S', for instance, would be statements, A, B, C etc. which statements may be, for MacColl, true, false, certain, impossible or variable. That is, the propositional function, in Russell's sense, may be certain (true for aIl values of its variables) and the statements substituted for the variables may also be certain, not just true or false as Russell would have it. Therefore, certainty, for instance, applies to both propositional functions and propositions for MacColl rather than just to propositional functions as Russell claims.

24 MacColl, "Equivalent Statements", Mathematical Society v. 28 (1896-97), p. 555. - 82 -

The statements which MacColl claims are certain are those which, for MacColl, follow from "our definitions and ordinary linguistic conventions" such a

'2 + 2 = 4', 'the part is not greater than the whole' etc. Similarly impossible statements are those which contradict "our definitions and ordinary linguistic conventions". Russell, by concentrating on MacColl's apparent failure to distinguish between propositions and propositional functions and on sorne of the obvious difficulties with contingent statements, does not advance any arguments against MacColl's calling these statements certain or impossible. Therefore, since most of the difficulties Russell sees with MacColl's division of statements can be gotten around, or at least weakened, (the notable exception being his remarks on MacColl's contingent statements) it can be concluded that Russell has not made his case.

MacColl's logic received a different, and more hostile, criticism from another source with the publication in 1906 by A.T. Shearman of The Development of Symbolic Logic. Shearman's object in writing the book was to show that distinct advances had been made in symbolic logic in the fifty years prior ta the publication of his book. He wanted to show what contributions constituted these advances, who had made them and who had impeded the development of logic. The standard used in determining just what should be regarded as an advance was the two-valued extensional logic, which was the dominant loqic at the time Shearman wrote. It is - 8a -

only natural then that MacColl, who opposed this development, should corne off badly in such a work. The bias in this approach is shown in Shearrnan's introductory remarks to his analysis of MacColl's position. Coming now to Mr. MacColl, l wish to point out wherein l think he falls into error. My object in considering his work is to get at the truth on certain debated questions, so that l proceed at once to these. l readily admit that there are several points in which Mr. MacColl • • • agrees with the other writers to whom we have had occasion in previous chapters 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 sorne detail, but with one exception they had been weIl considered by other symbolic 10gicians. 25

What Shearrnan is saying here is that where MacColl agrees with "the other writers" he is correct, but unoriginal, where he disagrees he is wrong. Shearman does not say it in so many words, of course, but when considering Shearrnan's general approach it is hard not to suspect that, for Shearrnan, MacColl is wrong because he differs from "the other writers". This will, perhaps, be brought out more clearly in his detailed criticisms of MacColl.

Shearrnan was a student of W. E. Johnson and was very much influenced by him. It might, therefore, be helpful to glance briefly at what Johnson had to sayon the subject of modalities. In the first of a series of papers Johnson published in Mind in 1892 he writes the following on modality:

25 A.T. Shearrnan, The Development of Symbolic Logic (London: Williams and Norgate, 1906), pp. 149-150. - 84 -

••• perhaps the following will be admitted. Modality refers to the grounds on which the thinker forms his judgment. It, therefore, expresses a relation between the thinker on the one hand and a certain proposition on the other hand. The real terms, then, of the modal proposition are the thinker and his relation to sorne judgment which is propounded to him. Thus, the proposition, "s must be P", asserts (say). that, any rational being is bound by his rationality to judge that "s is pu •••• The modal proposition is, therefore, simply an assertorie on a different plane - concerned with the relation between different sorts of terms. 26

This position, which was held in one form or another by a number of logicians in this period, was modified slightly by Shearman and applied as a criticism of MacColl's work. Shearman saw MacColl's treatment of modalities as one of his "two chief errors", 27 the other being his doctrine of logical existence which will be considered in the next chapter.

Shearman's discussion clearly shows the influence of Johnson. After briefly quoting one source from MacColl on modality he goes on to enunciate what may be called his positive position on this issue: Pure Logic cannot deal separately with these certain propositions: [i.e., MacColl's certainties] it can only deal with the relation in which a thinker stands toward the statement that is certain. and further on the same page

26 W.E. Johnson, "The Logical Calculus", Mind N.S.V. 1 (1892) p. 18-19. -- 27 Shearman, Development of Symbolic Logic, p. 152 - 85 -

As an instance of the way in which statements described by these three terms [i.e., certain, impossible and variable] are to be dealt with take the following: "It is impossible that X is Y". This would appear in such a form as "A thinker who can believe that X is y does not exist". 28

Later in the work Shearman gives an analysis of what the idea of a certain proposition is: For instance, taking a true and taking a certain proposition, these would assume forms such as "AlI 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.- 29

MacCol1 devotes but one brief paragraph at the end of a paper to a reply to the criticisms in Shearman's book, criticisms which MacColl sees as irrelevant "from beginning to end". For instance, says MacColl, Shearman mixes up logic with psychology and defines a statement as impossible when, and only when, nobody can believe it. Now, the belief in witchcraft is not yet dead. It follows therefore from Mr. Shearman's definition that it is still possible for old women to ride through the air on broornsticks, and so long as the belief lasts, the possibility will last also. 30

This reply is not completely satisfactory however, since if anyone is in danger here of mixing logic with psychology, it is MacColl, not Shearman. It is MacColl who says that 'certain' and 'impos~ible' belong to the object language of logic; it is therefore up to MacColl to show that

28 Ibid., p. 153 29 Ibid., p.158 30 MacColl, "Syrnbolic Reasoning", Mind N.S.V.15 (1906), p. 158. - 86 - his idea of 'certain', for instance, is neither psychological nor epistemological in nature, but logical. MacColl's point, however, is that surely a statement's being certain (here 'necessary' would be a better word) or impossible is independent of facts about who, or how many people, believe this or that statement. Just as a statement if it is true remains true whether or not it is generally believed, so a statement if it is necessary is necessary whether or not it is believed. This point will come out more in other examples.

It is interesting that MacColl should close on an example of the impossibility of old ladies flying throuyh the air on broomsticks. Although MacColl, as has been shown, does make a distinction between formaI and material impossibility he is not always careful to observe that distinction. Further, it appears that this is not just a matter of carelessness but that he sometimes finds this distinction suspect. For instance in a letter to Russell, dated June 22, 1901, he observes that With me anything is possible which contradicts no data or definition. Whether it is possible as a matter of actual fact and experience, is a question outside the scope of logic. but he then adds the rider This is perhaps too absolute. Any valid formulae of pure logic should be true independently of truth or falsehood of its constituent statements but in its application to actual realities we should, of course, seek to make our data as accurate as circumstances permit. 31

31 Letter from MacColl to Russell, June 22, 1901. Bertrand Russell Achives, McMaster University. - 87 -

In the above case, owing to the evidence or "actual realities", MacColl is willing to treat the idea that witches fly through the air on broomsticks as impossible. Although his central point here is that a statement is certain or impossible independently of our varying mental attitudes toward it his choice of the witchcraft example only serves to reinforce the claim that impossibility, in this case, has to do with considerations other than logic. MacColl wou Id have been in a better position had he stuck to examples of formaI impossibilities. It is also unfortunate that Shearman ignores, or was unaware of, the many places in his writings where MacColl expressly warns of the dangers of mixing psychology and logic. For instance in Symbolic Logic and its Applications MacColl states explicitly that "formaI logic should net he mixed up with Psychology - its formulae are independent of the varying mental attitude of individuals:: • 32

Shearman never argues directly against MacColl's position; he presents, instead, an alternative analysis of modal propositions, one which may be regarded as the con ventional wisdom of his day. For his part MacColl relies almost wholly on examples in attempting to refute Shearman.

Shearman's next statement on this issue which appeared as a discussion note in Mind in 1907, brings out

32 MacColl, Symbolic Logic, p. 60 - 88 -

clearly the issue between him and MacColl. As Shearman sees it The matter amounts to this: when Mr. MacColl says 'It is certain that X is y' he ho Ids that there is here no reference to any thinker. l hold that there is such a reference. To say that a proposition is certain means that aIl persons are so constituted that the~ must believe that the proposition is true. 3 and further in the same note Shearman discusses one of MacColl's other examples of an impossibility, '2 + 3 = 8'. Now 2 + 3 = a is not an impossibility of the kind that "old women ride through the air on broomsticks" is: the former is a proposition whose truth is inconceivable, while the latter is merely a statement of something which, so far as we know, is physically impossible. Hence, l hold that 2 + 3 = a may be rendered "a person is so constituted that he cannot believe that 2 + 3 = a"or "a person who believes 2 + 3 = a does not exist". l do not hold that the statement of the physical impossibility can be rendered "a person who believes that old women ride through the air on broomsticks does not exist". 34 Here Shearman is correct in noting the distinction between material and formaI impossibility, to use MacColl's terms, but it is doubtful if he would have made much of it here if he had approached MacColl at aIl sympathetically since he must have known that MacColl was weIl aware of this distinction. In fact, aIl Shearman is doing here is repeating his analysis of impossible statements and showing that MacColl's criticisms of this analysis are misplaced since he, Shearman, would not analyze 'It is impossible that old women fly through the air on broomsticks' as "A person who believes

33 A.T. Shearman, "Certainties, Impossibilities and Variables" , Mind N.S.V. 16 (1907), p. 314 34 Ibid., pp. 315-316 - S9 - that old women f1y through the air on broomsticks does not exist'. Obviously, however, MacCo11 wou1d also object on much the same grounds, to analyzing '2 + 3 = S' as -A person who believes '2 + 3 = S' does not exist-. Indeed this is how MacColl respona~ to this argument of Sbeazman's although he chooses to use another examp1e. "To say that a proposition is certain, according to Mr. Shearman, n means that all persons are so constituted that th~ .usr be1ieve that the 2roposition is true-. Now, aIl text-books on trigonometry assert that the ratio which the circumference of a circle has to its diameter lies between 3.141 and 3.142. l hold that this assertion is a certainty, and that it wou1d remain a certainty even if every living human being (Mr. Shearman and l included) believed it to be false. Mr. Shearman, if he sticks to his definition, must ho1d that it is not certain, because a few crazy circle-squarers here and there do not believe it. l once saw a good-sized volume written by a certain Mr. Smith to prove that the exact ratio is 3.125. Should mathematicians, in deference to Mr. Smith's opinion, regard the statement (~ = 3.125) as possible! Mr. Shearman apparently thinks they should: 35 In this rep1y MacCol1 obviously seizes on Shearman's weakest point; namely that an impossible proposition is one that no one actua11y be1ieves. MacColl is quite correct in showing the weakness of tbis position though he does not attempt to work out the implications of Shearman's other formulation; i.e., that an impossible proposition is one which lia person is so constituted that he cannot believe". In fairness to MacCo1l there is no reason for him to try to salvage Shearman' s formulation, however

35 Hugh MacColl, "Symbolic Logic: A Rep1y· , Mind R.S. V. 16 (1907), p. 492. - 90 - poorly put by Shearman. AlI that he is concerned with is that Shearman does not present any good reason for abandoning his own formulation and, in this surely MacColl is correct. That is, Shearman says that an impossible proposition is one no one believes; hence, if someone believes a particular proposition it is not impossible, i.e., it is possible. Now, says MacColl, since sorne people believe the circle can be squared, on Shearman's ter.ms we are now to regard this as possible even though it has been proved to be impossible. For MacColl this is to be simply rejected out of hand; if it is impossible (i.e., logically impossible) to square the circle it matters not who believes otherwise.

This ended the debate between MacColl and Shearman on this issue and, although neither convinced the other, it did help to bring into focus sorne of MacColl's central ideas on the division of statements.

To recapitulate, MacCol1 divides aIl statements into true and false, and, in addition, every statement is either certain, impossible or contingent. MacColl recognizes as weIl that there are propositional functions, or functional propositions to use the expression MacColl prefers, although he is primarily concerned with statements. By 'certain', 'impossible' and 'contingent' MacColl is referring neither, as Russell would have it, to propositional functions (except where he explicitly says sol nor to sorne relation between a 'thinker' and a statement as Shearman claims. By - 91 - the use of the tenns (fonna11y) certain; (formal1y) impossible and (fonna1ly) contingent MacCoI1 believes be is pointing out logical features of statements which a proper system of logic shou1d take into account. None of the criticisms advanced by MacCo11's contemporaries was able to weaken this basic view of MacCo11's, although several of tha~ were correct on sma1ler issues. The above discussion is intended, in part, to show that MacColl was substantially correct in not giving up his position in the face of these criticisms.

In addition to the classes of statements encountered above, MacColl offers suggestions for a logic in

n which the notions of "known to be true", "known to be false , and "neither known to be true or false" function as the primitive operators. This sytem could then be fused with his basic system of modal logic to form what he calls na nine- dimensiona1 sCheme".36 Unfortunate1y, he does not pursue this matter either in his second paper in Mind or in bis sixth paper to the Mathematical Society where he worries that this system "might open psychologica1 or metaphysical questions whose discussion would be out of place in this paper. n37

At any rate it can be seen that MacColl does not lack imagination when it cornes to the possible uses ta

36 MacColl, "Symbolic Reasoning", Mind N.S.V. 6 (1897) p. 509 37 MacColl, "Equivalent Statements'~, Mathematica1 Society v. 28 (1896-97), p. 556 - 92 - which his basic system could be applied. Indeed he feels Boole is partly responsible for impoverishing logic by, seemingly, not leaving the way open for developments of this type. His following remarks in Mind reflect his attitude towards this. l cannot help thinking that the seeming success which attended his [Boole's] efforts to squeeze aIl reasoning into the old castiron formulae constructed specially for numbers and quantities has tempted many other able logicians to waste their energies in the like futile endeavoursi when those energies might have been employed with far greater chances of success in inventing new and independent formulae, more elastic and more suitable for the highly general and widely varying kinds of problems which are destined to enter more and more largely into the ever expanding subject of Symbolic Logic. 38

Again, in the last paper in the "Symbolic Reasoning" series in Mind he laments the fact that most logicians deal only with true and false propositions. This is very mu ch as if one argued that since animaIs are only divisible into two classes, males and females, it is no business of true zoology to consider the respective characteris tics of such creatures as lions, tigers and leopards ••• aIl such attempts to surround symbolic logic by a Chinese wall of exclusion are futile. 39

Although the analogy is somewhat misleading - the point is, after aIl, that perhaps statements can be divided into only true and false while no one has argued that animaIs can be divided only into male and female - it remains that no one

38 MacColl, "Symbolic Reasoning", Mind N.S.V. 6 (1897), p. 505 39 MacColl, "Symbolic Reasoning", Mind N.S.V. 15 (1906),p. 515. - 93 - effectively criticized the central points in MacColl's division of statements and, from where he sat, he could see no reason for excluding from logic what he thought were important considerations. THREE

EXISTENTIAL IMPORT

Surnrnary. MacColl's theory of logical existence is described. This theory was designed to solve the problem associated with traditional logic of whether statements of the forro "AlI A' sare B' s" imply the existence of any A' s. His theory was to be compatible with other positions he took in logic and was to avoid what he took to be paradoxes in extensionalist systems. He divided the universe of discourse into the suh-universes of realities and unrealities, thus creating a two-sorted ~ystem. MacColl's views on quantification are discussed in relation to this two-sorted system. His theory allowed that such things as unicorns, fairies, and even round-squares existed. They belong to the sub-universe of unrealities. His views were subject to sharp criticism from Russell and A.T. Shearman. It is argued that these criticisms were not to the point.

MacColl's opposition to the standard two-valued extensional logic being developed in his day is carried through to the topic of referring or, more specifically, to questions of the existential import of propositions. As we have seen MacCol1 rejected the definition of implication provided by Russell and others largely because that definition contained what were for MacColl unacceptable paradoxes. MacCol1 also sees paradoxes being generated by

- 94 - - 95 - the extensionalist's account of existential import; para doxes which can only be resolved by adapting sorne kind of intensional criteria for the identification of objects. MacOoll sees his position on this matter as being very closely connected with his position on implication; indeed, in each case he rejects an alternative view as being paradoxical and claims that the position he adopts is more fondamental and provides a basis for what he called pure logic. As might be expected this elicits a strong response from Russell who sees MacColl as being committed to the same -contradictions" as he, Russell, sees in Meinong's the ory of objects.

MacColl's theory is stated fully in his sixth paper in the "Symbolic Reasoning" series in Mind and again in bis book Symbolic Logic and its Applications. It is defended and criticized in the Discussion Notes section of Jtind in several issues, including the issue in which Russell's -00 Denoting" first appeared, in 1905 and 1906.

MacColl's aim was to provide a theory which wou1d he compatible with other positions he took, especially vith his definition of implication, and which would solve the problems associated with traditional logic of whether statements of the forro "AlI A is B", "No A is B", "Sorne A is B- and ·Some A is not B" imply the existence of any A's or B·s. MacColl quite explicitly states that the problem - 96 - is whether or not the class A or B exists rather than whether individual Ais or Bis exist. l In the theory he presents to solve this problem, however, it can be seen that he is really concerned with whether these statements imply that Ais or Bis existe In addition he wanted this theory to avoid what he felt were unacceptable paradoxes of extensional logic, such as that resulting from the putative logical equivalence of statements like "AlI round squares are triangles" and "no round squares are triangles".

His approach to this problem is, in spirit, very like his approach to the problem of implication. As we saw earlier, he was not content to allow that a statement A implied (except in the purely stipulative sense of

"materially implied") a statement ~ solely on the grounds of the truth value of one or another, or both, of these statements. Whether two statements happen to be true is, for MacColl, no ground~ for asserting that one implies the other. Similarly with logical existence, whether something named in an argument actually exists in this world is of no concern to MacColl) unicorns and dogs have, for the purposes of logic, the same existence. 2 This is consistent

1 MacColl, "Symbolic Reasoning",Mind N.S.V. 14 (1905), P. 74. 2 An explanation is in order here. In a particular applica tion of logic to sorne uni verse of discourse a distinction between such things as dogs and unicorns can, and probably would, be drawn. One of these, say dogs, would be classed as belonging to the class of "realities", the other to the class of "unrealities". Which would be classed as which however is not a question of logic and it is possible in sorne application for the classification to be reversed. The point for MacColl is that logic must be able to handle both kinds of entities. This will be explained further below. - 97 - with his definition of pure logic as the most general science of reasoning, independent of any special subject matter. If, in a particular application, logic is interpreted to zoology, say, then the distinction between dogs and unicorns may become important. However, in sorne application, or interpretation, it may be important to say such things as "Unicorns have horns" and "Dogs have tails". It is, then, important for MacColl to provide a framework of logical existence under which both of these kinds of objects, existent and non-existent in the zoological sense, can be treated.

For reasons such as these MacColl divides the "Universe of Discourse" or "Symbolic Universe" as he sometimes calls it, into two distinct sub-universes~ the sub- universe of realies, denoted bye, and the sub-universe of 3 unrealities denoted by 0. Together these sub-universes comprise the total universe of discourse and aIl of the objects whose names may enter into an argument belong to one or the other of these sub-universes. The universe of discourse may consist wholly of realities or wholly of unrealities, although, MacColl adds, the second case would

seldom i if ever, be met with in practice, or it may be a mixture of both realities and unrealities. In the first

3 MacCol1 uses the expressions "universe of realities", "universe of unrealities" and "universe of discourse". To avoid confusion the terminology "s ub-universe" is adopted here for the former two. - 98 - two cases the üniverse of discourse is made up of a pure class while in the last of a mixed class. The important point here, however, is that MacColl adroits a sub-universe of unrealities which consists of indefinitely many distinct members such as fairies, unicorns, round-squares, fIat spheres and the like.

The question now is what determines that some thing exists1 MacColl's answer goes as follows:

•.• when any symbol A denotes an individual~ then any intelligible statement the symbol ~(A), containing A, implies that the individual represented by A has a symbolic whether the statement existence~ but individual ~(A) implies that the represented by A has a real depends upon the contexte existence symbol Secondly, when any A denotes a class, then, any intelligible statement symbol ~(A) containing the A implies that the whole symbolic existence; class A has a but whether the statement ~(A) implies that the or Class A is wholly real, wholly unreal, or partly real unreal depends and partly upon the contexte 4 What MacColl means here is that if any symbol occurs as the logical subject of an intelligible expression then that symbol denotes an object~ whether the object is real or unreal is determined not by the logic of the expression but by the context, i.e., by non-logical questions of fact or theory. The word "denotes" is used much more widely by MacColl than by, say, Russell. Unfortunately MacColl never explicitly tries to define or explicate the concept of denoting, nor does he make clear what an "intelligible statement" is. He

4 MacColl, "Symbolic Reasoning", Mind N.S.V. 14 (1905), p. 77 - 99 -

gives as an exarnple of an unintelligible statement "Triangles are virtuous" which would indicate that an intelligible state ment simply is one which is g:cammatically correct and does not contain category mistakes. That is a sentence like "Triangles are virtuous" is not intelligible for MacCol1 since the property of being virtuous or non-virtuous does not apply to triangles. The sentence "round-squares are triangular" however, is considered intelligible, though false, by MacColl. It might weIl be objected that the property of being triangular or non-triangular no more applies to round squares than does that of being virtuous or non-virtuous to triangles. Further, if "round-squares are triangular" is considered intelligible, then it would appear that the only criterion used in determining a statement's intelligibility is one of grammar. This does indeed seem to be the case for MaCColl. •. ·,

A word or symbol denotes an object, for MacColl, if it, the word or symbol, occurs as the subject of a sentence. Thus, since such things as round-squares, flat-spheres and the like exist symbolically (they belong to the sub-universe of unrealities) for MacColl, such statements as "Round-squares are more numerous than flat-spheres" are to be counted among MacColl's intelligible statements. There is nothing in MacColl's writings to suggest sorne alternative interpretation and, unfortunately, MacColl does not face the problem of "round-squares" being simply unintelligible. - 100 -

Another example of MacColl's will help bring out this point. Consider the sentence "The ~ whom you see in the garden is really a bear".5 According to MacColl, the individual in this case denoted by "The man" has a symbolic existence; that is, it belongs to the sub-universe of unrealities. It is an imaginery being, the result of an optical illusion in the case under consideration. Although this example helps to clarify MacColl's position on existence, it is a rather unfortunate example since clearly a sentence like this is different from say, "fairies have magical powers", "Unicorns have one horn", "round-squares are contradictory objects" or "the King of France is bald". The individual denoted by "the man" in the above sentence clearly does exist, or is a member of the sub-universe of realities, as the example indicates. It turns out in the final analysis to be just a queer case of mistaken identity and the sentence could easily be paraphrased as follows "The thing you took to be a man was really a bear". Here "a man" is, by itself, no longer a subject term, and the problem disappears. Clearly this sort of paraphrase is not applicable in the other cases. The problems associated with "The King of France is bald" are not generated by faulty eyesight; nor are "possible objects" gotten rid of so easily.

The above preliminary account of logical exist ence answers the question, for MacColl, of whether the statement

5 Ibid. - 101 -

"AlI Ais are Bis" commits one the existence of any Ais. The answer, • obviously, is yes. One is committed to the symbolic existence of Ais. Whether or not there really are any Ais in this world is, for MacColl, not a question of logic. As Storrs McCa11 has pointed out,6 what MacCol1 is claiming here is that "AlI Ais are Bis" implies "Some Ais are Bis" but "Some Ais are Bis" does not mean that there actually exists any Ais as members of the sub-universe of realities. Perhaps a concrete example will help make this clear. For MacCol1 "AlI unicorns have horns" implies "Some unicorns have horns" but "Some unicorns have horns" does not imply that there exists as members of the sub-universe of realities any unicorns. Both "AlI unicorns have harns" and "Some unicorns have horns" imply that unicorns exist either as members of the sub-universe of realities or the sub-universe of unrealities. Which of these sub-universes unicorns are members of is not, for MacColl, a question of logic. Logic for him is concerned with inference, not with what things exist in this world.

This concept of existence also allows MacColl to avoid the problem of whether logic presupposes a uni verse of at least one member. For instance in judging a thesis of the form " :1 x <$l5x v-f2}x)" MacColl would not be led to any metaphysical position which claimed, as a necessary truth, that

6 Storrs McCall, "Connexive Implication and the Syllogism" , Mind N.S.V. 76 (1967), pp. 348-350. - 102 - at least one thing existed. It is not that he would shrink from such a claim; merely that he wanted to keep pure logic free from such entanglements.

One of several interesting curiosities of MacCollls logic should perhaps be mentioned here, a curiosity which may be a result of MacCollls isolation from other logicians during most of his working life and of his rather scant knowledge of the history of logic. Despite the fact that one of the explicit reasons for MacCollls formulation of this theory of logical existence is to solve problems raised by the interpretation of the traditional syllogism, he held that, technically, aIl of the traditional forms of the syllogism are invalide This is so, claims MacColl, since they are stated as if each of the premises and the conclusion are asserted. For example, consider the following valid argu ment form: AlI Ais are Bis lûl Bis are Cis Therefore AlI A's are C's The validity of this argument form depends upon internaI analysis of its propositions. With this major point MacCol1 does not disagree. What he claims is that if, for instance, it were false that nAlI Ais are Bis" then the conclusion would not follow given the way in which the argument form is stated. That is according to MacCol1 the conclusion "Therefore AlI Ais are C'sn is asserted in this form and if one of the - 103 - premises is false, the conclusion may weIl be false. MacColl is drawing a distinction which is sometimes expressed by the terminology "valid" and "sound", a sound argument being one that is valid and has true premises, hence, a true conclusion. MacCol1 claims that the fonn in which the syllogism is given indicates that the argument forms are sound, rather than valide He urges that they be written in the form "if AlI A's are B's and AlI B's are Crs, then AlI A's are C's". This is clearly a minor point since the argument forms were always considered valid rather than sound. However, MacColl apparently took this quite seriously and offered the following example to illustrate his point • ••• Suppose a General, whose mind, during his past University days, had been over-imbued with the traditional logic, were in wartime to say, in speaking of an untried and possibly innocent prisoner, 'He is a SPYi therefore he must be shot, , and that this order were carried out to the letter. Could he afterwards exculpate himself by saying that it was aIl an unfortunate mistake, due to the deplorable ignorance of his subordinatesi that if these had, like him, received ••• a logical education, they woulü have known at once that what he really meant was 'if he is a spy, he must be shot'.7 This example which was written toward the end of MacColl's career (1906) further illustrates how iconoclastic he had becomei not only wer~ aIl of his contemporaries wrong in important respects but traditional logic needed to be rewritten. This example was not commented upon by any of MacColl's contemporaries.

7 MacColl, Symbolic Logic, pp. 48-49. - 104 -

Returning now to the question of logical existence we might take a closer look at what MacColl means by "Symbolic Universe" or "Universe of Discourse". In his book, Symbolic Logic and its Applications, which is his last statement of the matter, he writes:

Let S denote our Symbolic Universe or ~Universe of Discourse", consisting of aIl the things Sl, S2etc., real., unreal, existent, or non existent, expressly-mëntioned or tacitlY- understood, in our argument or discourse • • • Now there are two mutually complementary classes which are so often spoken of i.n logic that it is convenient to designate them bl' special symbols; these are the class of individuals which, in the given circumstances, have a real existence, and the class of individuals which, in the given circumstances, have not a real existence. The first class is the class e, made up of the individuals el, e2, etc. To this class belongs every individual of which, in the given circumstances one can truly say "It exists"- that is to say, not merely symbolically but really • • • The second class is the class 0, made up of the individuals 01' 02 etc. To this class belongs every individual of which, in the given circumstances we can truly say "It does not exist". ••• In no case, however, in fixing the limits of the class e, must the context or circumstances be overlooked. 8

What things exist, i.e., in the sense of "logically exists" is made clearer in MacColl's discussion of the Universe of discourse in his Sixth Mind paper. Here he writes: ••• This Symbolic Universe or "Universe of Discourse", may enlarge as the argument proceeds, seizing, appropriating and firmly retaining every new entity (not excepting self-contradictory entities ••• ) which we designate by a symbole 9

8 Ibid., pp. 42-43. 9 MacColl, "Symbolic Reasoning", Mind N.S.V. 14 (1905), p. 76 - 105 -

There are, as can be readily seen, sorne difficulties in the notion of a uni verse of discourse consisting of aIl and only those things "expressly mentioned", which precludes the possibility of stipulating that the universe of discourse consists of, say, "aIl men" or "aIl positive integers" or "everything" etc. One difficulty would be that one would have expressly to mention each individual in the universe of dis course for that thing to be a member of the universe of discourse, an undertaking which would prove impossible to carry out in sorne cases. In addition aIl universally quantified expressions which were valid would be valid only relative to sorne particular universe of discourse which would contain only those objects "expressly mentioned". This would serve to severely limit those expressions. Fortunately MacColl in practice does not strictly follow this course; hence, we can safely ignore this particular difficulty. The above paragraph from MacColl also points out explicitly what was clear from examples used earlier, narnely that round-squares, fIat spheres and the like do count as entities for MacColl.

It is important, however, to note that MacColl's system is "two-sorted,,;lO that is, the quanti fiers range over two sorts of variables, those which denote "real" objects

10 This terminology is used by Nicholas Rescher in connection wi th MacCol1. See his "On the Logic of Existence and Denotation" Philosophical Review V. 68 (1959), pp. 157- 180. -106-

is a and those which denote "unreal" objects. Since there danger of reading later developments in logic into MacColl's work here, it is necessary to explain MacColl's concept of a quantifier in his own terms.

MacColl never developed a really use fuI notation fiers for quantified logic nor did he have symbols for quanti as such. Nevertheless he did come very close to developing such a notation and very clearly he had an idea of quantifica 1878 in tion not too unlike modern views on the subject. In the Proceedings of the London Mathematical Society MacColl wrote: The statement AlI X is y may be denoted by the implication x:y in which x denotes the statement that a certain representative individual belongs to the class X, and y denotes the statement that he [it1 belongs to the class Y ••• The number of individuals belonging to a class may be known or unknown, constant or varying, finite or infinite • • • • the s1:atement no X is y will thus be expressed by the implication x:y' ••• some X is y will be expressed by the non implication x+y' ••• Sorne means one individual at least • ••• some X is not Y will be expressed by the non implication y+x,.ll

Later when MacColl had abandoned his symbol for non-implication he rewrote ('+') and had introduced symbols for modal concepts as the standard four statements of traditional syllogistic

Il MacColl, "Equivalent Statements", Mathematical Society v. 9 (1877-78), p. l8L.

.; . - 107 -

follows:

Let S be any indiv~dual taken at random out of our Symbolic Uni verse, or Uni verse of Discourse, and let x, y, z respectively denote the three propositions SX, sY, sz. Thin _ x~~ y', z' must respectively denote S , S y, S •••• with these conventions we get - (A) Every (or aIl) X isxY = SX:sy = x:y = (xy')~ (0) Sorne Xis not X =_.{S :SY)' = (x :y~' = (xy') n (E) No X is y = S.n.: S-:[ = x' :y' = (xy) (I) Sorne X is y = (Sx:S-y)' (x:y')' (xy)-n In this way we can express every syllogism of the traditional logic in ter.ms of x, y, z, which represent three propositions having the same subject S, but different predicates X, Y, Z.12

It can be seen here that MacColl cornes quite close to having a separate variable for individuals and one for predicates; for instance the first step in MacColl's analysis of "AlI X is Y", which is sX: sY, could be read as "for every S, if S is X, then S is Y". Unfortunately, the next step in his analysis, ~:~, blurs this distinction although in his remarks he notes that lower case x, y, and z, which he sometimes called "unrestricted variables", represent propositions having the same subject but different predicates.

The move from ~:~ to (~,)n and the analogous moves in the other cases above simply follow from his definitions of implication, conjunction, negation, etc.

MacColl's version then, of the standard four

12 MacColl, Symbolic Logie, pp. 44~45. - 108 -

statement 13 forms should be read as follows: (A) It is impossible for something to be both an X and a not-y; (0) It is not impossible, i.e., it is possible, for something to be both an X and a not-Yi (E) It is impossible for something to be both an X and a Yi and (I) It is not impossible, i.e., it is possible, for something to be both an X and a Y.

MacColl's final version of quantification (MacColl, of course, did not use the term) should be seen as an extension of his 1878 views, an extension which is directly related to other developments in his logic from the earlier period to the later. It is not the result of any direct influence from other logicians.

Returning now to the question of what things count as objects in MacColl's system we can conclude that something exists that corresponds to every grammatical noun, in the sense of "symbolically exists". This is especially clear from the above translation into MacColl's symbolism of the standard forms of classical logic. For instance "Sorne X is y" becomes "It is not impossible that sorne X is a Y", or, in other words, i t is possible that sorne X is Y. (AlI symbolic transformations stated or implied

13 These are not claimed to be equivalent to the four standard forms, since with the addition of the modal terms they would be too weak for that purpose. Rather they are how the analogous statement forros in MacColl's system are to be written. -109-

to hold for MacColl follow the rules set forth for his system in the first chapter above). Thus, for something to exist, in this sense, it merely has to be possible to ascribe sorne property ta it. Or, as Rescher has stated in discussing Maccoll,14 something exists if sorne non-trivial property, i.e., a property which does not apply to everything, can be ascribed to it. Hence e.g.,unicorns exist since sorne non- trivial property, that of having one horn, can be ascribed to the "possible object" unicorn. Again, whether something actually exists, that is belongs to the class e, (sub-universe of realities), depends upon the contexte

MacColl never makes perfectly clear what he means by "context", but from the use he makes of that term it appears that he means what is usually meant by "interpretation". That is, if we interpret logic to, say, zoology, then the facts, theories, assumptions etc. of that science determine what things belong to the sub-universe of realities. Pure logic, as MacColl uses that expression, must remain neutral with respect to what things actually existe By containing both a sub-universe of realities and a sub-universe of unrealities logic can be used to demonstrate the logical relations which hold in both kinds of worlds, the actual and the possible.

14 Rescher, "Existence and Denotation" Philosophical Review v. 68 (1959), see especially pp. 174-179. -110-

What MacColl considered a major advantage of his system over other (symbolic) systems of his time is that sentences such as "AlI round squares are triangles" or "AlI unicorns have one horn" will come out false in his system (if interpreted to the class of realities). Thus it avoids what he took to be further paradoxes of extensional systems. For instance, when considering "AlI A is B" MacColl makes the following claim: [when Boolian logiciansl .•• define 0 (or any other symbol) as indicating non-existence, and then assert that the equivalence (O)=OA) is always true, whatever the class Amay be, they appear to me to make an assertion which cannot easily be reconciled with their data or definitions. For suppose the class 0 to consist of the three unrealities 0 1 , O2, 0 3 and the class A to consist of 0 3 , el' e2, e3 (one unreality and three realities), the class OA common to both contains but one 15 individual, the unreality 0 3 •

Here it is apparent that the "mistake" Boo:tian logicians make is only a mistake on MacColl's assumptions. That is, the logicians MacColl is attacking would not accept his division of the universe of discourse into the sub- universe of realities and the sub-universe of unrealities, where the latter is composed of several distinct members. It is only on the assumption that they would accept MacColl's division of the universe of discourse into these two sub- universes that the formulae 'O=OA' is in jeopardy. However, by avoiding one mistake they commit another as far as MacColl

15 MacColl, "Symbolic Reasoning", Mind N.S.V. 14 (1905), p. 78. -111- is concerned, for the concept of the null class determines that any quantified conditional whose antecedent is false for aIl values of its variables is true. Hence, for instance, "AlI round-squares are triangles" is true since there are no round-squares. Similarly "no round-squares are triangles" is true. Bence, the two sentences are logical1y equivalent. This MacColl regards as a paradox analogous with (and determined by) the paradoxes of material implication discussed above.

The main responses from MacColl's contemporaries were again from Russell and A.T. Shearman. The spirit of Russell's criticisms, which were made in 1905 and 1906, can, perhaps, best be seen in the following remarks Russell published in 1919. Logic, l should maintain, must no more admit a unicorn than zoology can; for logic is concerned with the real world just as truly as zoology, though with its more abst: ~ct and general features ••• There is only one world the "real" world ••• A robust sense of reality is very necessary in framing a correct analysis of propositions about unicorns, golden mountains, round-squares, and other such pseudo-objects. 16

These remarks, as weIl as others of the same type written earlier, were directed as much at MacColl as they were at Meinong. Indeed in nOn Denoting" Russell accuses MacColl of following Meinong's the ory of objects "which we

16 Bertrand Russell, Introduction to Mathematical Philosophy (London: George Allen and Unwin, 1919), pp. 169-170. -112- have seen reason to reject because it conflicts with the 1aw of contradiction".17 However, in his detailed discussion of MacColl, as we shall see, Russell sirnply points out that imaginery or "possible-though-non-existent" objects can be dispensed with by using his theory of descriptions. He does not, un fortunately, discuss MacCo1l's objections to his own view, which view has the consequence that staternents such as "AlI triangles are round-squares" and "No triangles are round squares" are logically equivalent.

The principal point that Russell rnakes against MacCo11 is that there are two distinct senses of "exist" which MacColl fails to note. There is (a) the sense of existence which occurs in philosophy and daily 1ife in which existence is predicated on an individual such as in "God exists", "Socrates existed", and so on, and (b) the sense of existence which is used in symbolic logic where to say that A exists means that A is a class which has at least one rnernber. These senses of 'exist' are as distinct, says Russell, "as stocks in a flower-garden and stocks on the stock exchange"and only one, 'exists' in the sense of being a c1ass with at 1east one mernber, belongs to logic. 18

17 Bertrand Russell, "On Denoting" Reprinted in l.M. Copi and J.A. Gould eds. Contemporary Readings in Logical Theory (New York: MacMillan, 1967), p. 104 18 Bertrand Russell, "The Existential lrnport of Propositions" , Mind N.S.V. 14 (1905), p. 398 - 113 -

This point of Russell's is somewhat confused. Here he is saying quite explicitly that something exists, in the sense of 'exists' that is relevant to logic, if, and only if, it is a class having at least one member. Hence, only classes exist, and whatever is not a class does not exist, in this sense. However, for Russell the null class does exist and it, by àefinition, has no members. Also it is obvious that what classes exist will for Russell depend upon which individuals existe That is, if unicorns exist in sense Ca) above, then the class of unicorns will exist in sense Cb). These two senses of 'exist' are quite clearly related; in exactly what way it is not important to deterrnine here. There is some arnbiguity here in Russell's comments since he is concerned with the existence of individuals although he keeps stating the problem is if it were a question of what classes existed, quite apart from what individuals existe This point will be returned to below.

A.T. Shearrnan, in his study of the development of logic, offers a similar criticism:

••• The question then arises how it is possible that two such notions of existence [Ca) and Cb) above] should ever be confounded, and the answer is to be found by considering those classes which have members, 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 sorne classes which have members and these do not appear in such world, e.g., the class of numbers, or the class of mathematical 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 - 114 -

say, that c1ass that has no mernbers. 19

There are at 1east two responses MacColl could give to this particular criticism a1though he does not give either. First it is evident that Russell and Shearman beg the question. Simply to say that 'exist' in logic means to be a c1ass with at least one mernber, with the consequence that there is only one null c1ass to which the classes of round-squares, unicorns and the rest are logically equivalent, will not do since it is just that idea which MacColl is questioning. For MacColl, to exist logically, to exist syrnbo1ically in the sense exp1ained above.

The second response touches on a more substantive issue. According to Russell's and Shearman's argument the question of whether something logica11y exists, iXlat is, is a class with at 1east one member, is not determined by logic but by other (phi1osophical) considerations. For instance, we know that horses exist, hence for Russell and Shearman we know that the c1ass of horses exists, has at least one mernber, since we have encountered horses. We know that the c1ass centaur, is equivalent to the nu11 class, since we know from considerations other than logic that centaurs are mythical creatures. Therefore, for Russell and Shearman the class of horses logica11y exists, whi1e the class of centaurs

19 A.T. Shearman, The Development of Symbo1ic Logic, pp. 164-165. -115- does note However, MacColl's response here could simply be that the class of centaurs has members, namely aIl possible centaursi hence, it also exists in the same sense as the class of horses. MacColl need not dispute Russell's and Shearman's claim that to exist logically is to be a class with at least one member. He merely needs to point out that the question of which classes exist (have members) depends upon what philosophical position one takes regarding the existence of individuals. The dispute between MacCol1 and Russell occurs at this level. The issue between them is clearly not over what classes exist, but over what individuals existe Once again we find Russell attempting to settle the difference by invoking a stipulative definition, and, as in the case of 'implies', this method of argument simply will not do since it is the basic definitions, and the uses they are put to, that MacCol1 is challenging.

MacColl does, of course, draw a distinction between the existence of objects like tables, chairs and horses, which may be said to "appear in the phenomenal world~ to use Shearman's words, and objects like unicorns, centaurs and the like, which do note The first of these types belongs to the sub-universe of realities, the second to the sub universe of unrealities. The main difference between Russell and MacColl is that MacColl is claiming that the totality of objects is wider than the-totality of things which actually -116- existe That he is cornmitted to this position by his definition of implication and his (rough) working out of quantified logic has been shown above. In this context Russell does not attack these views directly nor does he attack their consequences in an attempt to formulate a reductio ad adsurdum argument.

MacColl's own defence of his position against Russell's criticisms is very limited. He does not, first of aIl, attempt any detailed refutation of the major points of Russell's "On Denoting" but relies, almost wholly, on pointing out the paradoxical results of Russell's definition of the null class. As far as Russell's claim that MacCo11 confuses two senses of "exist" goes,MacColl replies

That the word "existence" ••• has various meanings is quite truei but l 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. 20

This, again, illustrates how MacColl and Russe11 managed to talk past each other. Quite rightly Russe11 did not accept this reply as conclusive. MacColl's theory of logical existence is, then , consistent with his other positions in logic. By creating a two-sorted theory of quantification and thereby admitting possible objects into the

20 MacColl, "The Existential Import of Propositions", Mind N.S.V. 14 (1905), p. 401 - 117 - universe of discourse he is able to offer solutions to certain problerns in the philosophy of logic. Whether these solutions are, finally, satisfactory is an open question. It is true, however, that none of MacColl's contemporaries offered good reasons for his abandoning this theory. Although MacColl's reasons for ignoring Russell's criticisrns rnay not always have been sound, good reasons rnay be given on behalf of MacColl for putting thern aside. FOUR

MACCOLL'S INFLUENCE

MacColl's influence during his lifetime was fairly slight; he was often referred to by Schroder but usually in connection with other logicians; when he was singled out, it was to be criticized. As we have seen, Russell was quite familiar with MacColl's works, but on those issues which MacColl took most seriously he and Russell were in strong dis agreement. This was so mainly because MacColl was the only symbolic logician working towards a symbolic modal system; attitudes and aims like MacColl's were not viewed sympathetically during the period. which saw the dramatic development of extensionalist logic. lndeed in Shearman's work, quoted above, MacColl was dismissed as completely unimportant although, fortunately for MacColl, this attitude was not shared by logicians such as Russell and Schroder.

After the publication of Principia Mathematica works by C. l. Lewis began appearing, developing a modal logic which would be rigorously deductive on the model of Principia. Lewis's first publications on this topic appear before the 1914-1918 War in Mind and in the Journal of Philosophy, Psychology and Scientific Method (later the Journal of Philosophy) and contain criticisms of extensionalist

-118- -119- logic, and the first steps at constructing an axiomatic modal logic. Lewis's work on this subject culminated in Symbolic Logic which was written with C.H. Langford, and was published in 1932 with a second edition in 1959. In their preface to this work Lewis and Langford acknowledge their debt to Tarski, Lukasiewicz, Wittgenstein and parry among others, but there is no mention of the writings of MacColl, though, we know from Lewis's Survey of Symbolic Logic, first published in 1918, that Lewis was familiar with MacColl's writings. In fact MacColl is not referred to once in the whole Symbolic Logic, even though the first chapter is a short history of the subject.

The many sources of Lewis's thought which led to his developing several modal calculi will not be gone . . into here. However, fa~rly extensive quotations from his writings will be examined. These quotations express sorne of his attitudes toward logic, his specifie criticisms of material implication, his definition of implication, and sorne of the reasons why he thought his systemes) was the better when compared with the system of material implication. These remarks will be compared with remarks made by MacColl manY.years earlier, most of which have already been encountered in the preceding chapters. By doing this it will be seen that the minimum claim one can make is that MacColl anticipated a great deal of Lewis (it must be kept in -120- mind that MacColl did not have the advantage of Principia llathematica) while a not implausible conjecture would be that he influenced Lewis in a fairly direct fashion. The point of this is to try to show that MacColl should be seen as an important figure in the development of symbolic modal logic. That he is not now generally so regarded will be shown later. AlI quotations from Lewis are taken from the Dover (1959) edition of Symbolic Logic and from those parts of that work written solely by Lewis.

The first five chapteLs of Symbolic Logic are given over to an introductory treatment of the subject of symbolic logic. It is of interest here to note that in that section of the book the paradoxes of material implication are mentioned several times and towards the end of the fifth chapter Lewis remarks that nIt is especially desirable that the logic of propositions should be so developed that the usual meaning of 'implies', which is intensional, should be

ll l included •

In chapter VI Lewis sets about developing such a system. Among the definitions is included the following definition of strict implication:

1 C.I. Lewis and C.H. Langford, Symbolic Logic 2nd Edition (New York: Dover publications, 1959), p. 120 2 Ibid., p. 124 - 121 - which is identical with MacColl'8 definition of 'implies',

(A:B) = (AB,)n.

Lewis shows that if p strictly implies q then p also materially implies q.

By virtue of this principle [he adds] whenever a strict implication can be asserted, the corresponding material implication can also be asserted. The converse does not hold: strict implication is a narrower relation than material implications; the assertion of a strict implication is a stronger statement than the assertion of the corresponding material implication. 3

Compare this with MacColl's claim, which has been encountered earlier, that

••• my symbol A:B is formally strong~r than and implies their symbol A~B,just as A is formally stronger than and implies AL. Thus, my symbol A:B never coincides in meaning with their symQol A-

Among the formulas Lewis rejects,i.e., shows or claims that they do not hold for strict implication, are the following:

pq.-.:J • r :,: p.-!J. q-a r, 5 6 and p~q .v. q-3P (this is stated as an Existence postulate in the form

(3p,q) : -(p~q) .-(p~-q».

3 Ibid., p. 137

4 MacColl, "Symbolic Reasoning", Mind N.S.V. 12 (1903), p.357

5 Lewis and Langford, Symbolic Logic, p. 146

6 Ibid., p. 179 - 122 -

As has been noted MacColl found it necessary to reject the following formulas: (AB:C) : «A:C) + (B:C» and (A:B) + (B:A) the first of which is identical to the first of the Lewis formulas above, while the second of MacColl's formulas expresses what is intended by the existence po:~tulate, i.e., that there are independent propositions.

Another important feature of strict implication as Lewis sees it is that while in material implications "a merely true proposition is indistinguishable from one which is logically necessary;and ••• a merely false proposition is indistinguishable from one which is self-contradictory or absurd ,[in] terms of strict implication ••• the logically necessary is distinguished from the merely true, and the logically impossible from the merely false." 7

In 1903 MacColl had written these very similar comments: They [the extensionalist logicians] divide pro positions into two classes, and two only, the true and the false. l divide propositions not only into true and false, but into various other classes according to the necessities of the problem treated; as for example into certain [necessary] impossible, variable [contingent] They make no distinction between the true and the certain [necessary], [or] between the false and the impossible; so that in their system, every uncertain [not necessary] proposition is false, and every possible [not impossible] proposition true. B

7 Ibid., p. 143. 8 MacColl, "Symbolic Reasoning" ~Mind N.S.V. 12 (1903), p. 356 -123-

This formed the basis for MacColl's claim that his system was more general and included the others. While the two valued truth-functional logic can be applied with profit to mathematics where only necessary and impossible statements are encountered, it cannot be generally applied since it is not the case that in every application only these classes of statements are to be found.

It is weIl known that Lewis as weIl as MacColl dwelt a great deal on the paradoxes of mate rial implication. Indeed for both of these logicians what they took to be absurd consequences of the system of material implication played a large p~rt in determining their own systems. These consequences, which both claim their systems avoid, center on the fact that in systems of material implication no two propositions are consistent and independent. That is, of any two propositions at least one will imply the other. Indeed this is not a startling fact; no one who would embrace a system of material implication would be unaware of this consequence, but both MacColl and Lewis pointed it out at every opportunity, which indicates the great importance that they both gave it.

Two of MacColl's examples, one of which was used earlier, will be repeated here.

MacColl notes that, according to Russell, at least one of the two propositions "He is a Doctor" and "He is -124- red-hairedll implies the other: that is, '(R ~ D) v (D ~ R) , is true. For him neither implies the other. He then comments: Thus,Mr. Russell, arguing correctly from the customary convention of logicians, arrives at the strange conclusion that we may conclude from a man's re~ hair that he is a doctor, or from his being a doctor that (whatever appearances may say to the contrary) his hair is red. My argument, founded on what seems to me a more natural convention, and one more in accordance with ordinary linguistic usage, arrives at the (to me) self-evident result, that in neither case does the conclusion follow from the premises - that an Englishman may be red-haired without being a doctor, and that he may also be a doctor without being red-haired. 9 Again responding to a criticism made by Russell MacColl gives a similar example: Suppose, for example, we find it predicted ••• that "this year a great war will take place in Europe", and also that "This year a disastrous earthquakewill take place in Europe" •••• Let W denote the first proposition and Ethe second. It is surely an awkward assumption (or convention) that leads here to the con clusion that "either W implies E or else E implies W". War in Europe does not necessarily imply a disastrous earthguake the same year in Europe: nor does a disastrous earthquake in Europe necessarily imply a great war the same year in Europe. 10 ---

Lewis expresses the same point in the following way: Suppose and p and q be sorne pair of true but independent propositions - neither deducible from the other. For example, let p = "Vinegar tastes sour" and q = "Sorne men have beards".

9 MacColl, "'If" and 'Imply''', Mind N.S.V. 17 (1908), p. 152 10 Ibid., p. 453. -125-

Since any truth-implication, pIq, will hold when p and q are both true, it will hold in this case. For these meanings, pIq will not express a tautology: the statement "It is not the case that ('Vinegar tastes sour' is true and 'Sorne men have beards' is false)" is a true statement; but it is not tautological or necessarily true since it has a conceivable alternative. 1 1

For both MacColl and Lewis, then, the inadequacy of material implication is shown by noting that a relation of material implication holds between two obviously unrelated propositions. Both are also aware of paradoxes in their own systems, that a necessary proposition implies any proposition and an impossible proposition is implied by any proposition. Lewis explicitly declares these to be simple facts about entailment while MacColl is somewhat coy on the point. After noting the paradoxes, MacColl claims that readers who are upset by them probably think that they mean that an impossible statement implies its own necessity, which of course would be absurde It is not at aIl clear who MacColl thinks would ~ interpret the paradoxes in this way. At any rate he simply points out that they are not to be so interpreted but does not discuss exactly how they are to be interpreted. 12

The kinds of claims MacColl makes on behalf of his system have been extensively noted above and here again

Il Lewis and Langford, Symbolic Logic, p. 243 12 MacColl, Symbolic Logic, p. 13. -126- we find a strong similarity between the views of MacColl and Lewis. Lewis claims that in his systems the implication operator captures the meaning of 'deducible from'; that is,

"p~q" is claimed to be synonomous with "q is deducible from pll.13 He notes that while the laws of aIl formaI systems are equally true, it is only those laws of strict implication which have any useful application to inference. Pragmatically the laws of truth-value systems are, for Lewis, "'false' or unacceptable in the sense that they have no useful application to inference". 14 MacColl also stresses the pragmatic value of his system over others. For instance in 1903 he wrote the following: Whether my interpretation of this troublesome little conjunction if is the MOSt natural and the MOSt in accordance with ordinary usage, l do not undertake to say; it certainly is the MOSt convenient for the purposes of symbolic logic, and this alone is reason sufficient for its adoption. 15

It was mentioned above that if MacColl's system were to be found equivalent or strongly analogous to, any of the Lewis systems it would be to one of the weaker l6 systems SI or S2' or perhaps S3. Since MacColl did not develop an axiomatic system, comparisons of this type are very difficult to make; for instance a difference between S2 and

S3 is that in one, S2' the expression" (p; q)..., (DP~ Dq) Il

13 Lewis and Langford, Symbolic Log:ic, p. 247 14 Ibid. , p. 263 15 MacColl, "Symbolic Reasoning",Mind N.S.V. 12 (1903), p. 363 16 This is substantially the view stated by Storrs McCall in his "Hugh MacColl" in The Encyclopedia of Philosophy ed. Paul Edwards (New York: MacMillan, 1967) V.4, pp. 545-547. -127- is treated as a rule while in the other, S3' this expression is an axiome Since MacColl did not make this kind of distinction it is impossible to say definitely that his system is equivalent to S3 rather than S20r the reverse. Any more elaborate reconstruction of MacColl's logic than the one given on pp. 55-56 above which would show conclusively that MacColl's system was, or was not, equivalent to sorne Lewis system would involve arbitrary choices; that is, one would haY,;-e "CO make decisions which are not stated or implied in MacColl's writings. Such decisions would not suit the purposes of this study. What one can say, however, is that in general terms MacColl wanted to get a system like S2 or S3. This weaker claim can be supported by noting that MacColl reje"cts certain formulas, such as .ap~ 0<> p17 18 and OP "QQ P which are found in the stronger Lewis systems. Furthermore, although MacColl does not have a consistency postulate as such, it is an important feature of his system that if two statements A and B are conjoined and either A or B is fmpossible, then the conjunction is an impossibility. Thus if the conjunction is not impossible, then neither Anor B is impossible, i.e., both are possible. This would indicate that MacColl's logic is stronger than SI since that system does not contain the consistency postulate.

It is interesting to note that Lewis believed

17 MacColl, "Question for Logicians", Mind N.S.V. 9 (1900) p. 144. 18 MacColl, "Symbolic Reasoning",Mind N.S.V. 9 (1900), p. 75 -128-

that the weaker systems are the most appropriate systems for logical purposes as opposed to the mathematical interests 19 which may be served by the various systems.

This brief comparison of Lewis and MacColl is not intended to be a commentary on Lewis nor to suggest that Lewis simply reworked MacColl using Principia Mathematica as a model. Such.a claim would be grossly unfair to Lewis. It is intended to show, however, that a number of points which Lewis stressed and considered very important were also stressed sometimes in much the same language, by MacColl many years earlier. For instance both had much the same view about the paradoxes of material implication (and also about those of strict implication); both saw the need for logic to have more general application than in their view systems of material implication have; both had the same definition of (strict) implication; both saw the need for rejecting many of the same, or similar formulas; both stressed the pragmatic value of their systems for much the same reasons. Several other similarities could be mentioned here. The stress has been put on those which are most significant. While the originality and scope of Lewis's work should not be minimized, it is perhaps safe to say that Lewis's own assessment of MacColl, that "the fundarnental ideas and procedures •••

19 Lewis and Langford, Symbolic Logic, p. 502 -129-

[of MacCo1l's system] suggest somewhat the system of Strict Implication" ,20 is somewhat understated. In fairness to Lewis it should be noted that in the original edition of A Survey of Symbo1ic Logic (1918) he notes that "The fundamental ideas of the system [Survey System] are simi1ar to those of fij:lcCol1' s Symbo1ic Logic and i ts Applications. ,,21 The section of the book in which this remark occurred was de1eted fram subsequent editions of the work - which, in its abreviated form, is primari1y historica1 - 1eaving on1y the weaker comment about MacCo11's work. These comments constitute the who1e of Lewis' discussion of MacCo1l.

As has been noted MacCol1 emphasized ear1y in his career that logic shou1d be based on the logic of statements and he was the first modern symbo1ic logician to make this c1aim strong1y. He then proceeded to deve10p a system based on the logic of statements but included the modal notions of necessary, contingency and possibi1ity. With few exceptions recent historians, in the very 1imited space they have given MacCo11, have tended to point out, correct1y, that MacCol1 was the first to base logic on propositions

20 C.I. Lewis, A Survey of Symbo1ic Logic (New York: Dover Publications, 1960), p. 108 21 C.I. Lewis, A (Berkeley: University of p. 292 -130- b ut h ave gosse 1 d over h 1S" wor k on mo d a 1 1 Og1C. " 22 For instance Bochenski, while discussing Megarian definitions of implication, claims that "another definition is also first found arnong the Megarians • • • and this time as their main concept of implication, arnong the Scholastics .. . and is re-introduced by Lewis in 1918.,,23 This is, of course, the definition of strict implication which MacColl had put forward many years before Lewis. In his treatrnent of MacColl Bochenski relies on early tests (1877/78) of MacColl's and thus sees him as part of the development of the two-valued Boolian algebra from Boole to Schroder. MacColl's role as a founder of modern modal logic is overlooked. This general assessment is also true of Jorgensen's A Treatise of FormaI Logic which was first published in 1931. Although MacColl is discussed in several places in this work nowhere is it suggested that he was attempting to move logic away from the current truth-functional systems.

22 Perhaps the most notable exception is Storrs McCall who gives a description of MacColl's work in modal logic in his article on MacColl in The Encyclopedia of Philosophy V. 4, pp. 545-547. Storrs McCall also discusses MacColl in connection with modal logic in "Connexive Implication and the Syllogism" Mind N.S.V. 76 (1967), pp. 346- 356. MacColl's work-rn-modal logic is also discussed by Nicholas Rescher in "On the Logic of Existence and Denotation" Philosophical Review V. 68 (1969), pp. 176-179 23 I.M. Bochenski, A History of FormaI Logic Trans. and ed. by Ivo Thomas (Notre Darne, Indiana: University of Notre Darne Press, 1961), p. 15 -131-

For instance Jorgensen devotes most of a page 24 to MacColl's criticism of the syllogism, which has been dis cussed above, i.e., that since aIl of the statements in the syllogism are, or appear to be, asserted, the syllogism is invalid whenever at least one of the statements is false. But he does not consider those parts of MacColl's work which are most significant for an understanding of MacColl's system as a whole. In Kneale & Kneale's The Development of Logic (1902) MacColl is briefly mentioned once in connection with modal logic, and this oeeurs in the context of a discussion of Lewis's logie. Here Kneale repeats Lewis's assessment of MacColl almost exactly. He writes:

In his [Lewis's1 definition of implication he followed H. MacColl, who had included sorne suggestions for modal logic in Symbolic Logic and its Applications, published in 1906; and, like MaeColl, he admitted that according to this deiinition an impossible proposition must imply every proposition and a necessary proposition be implied by every proposition. 25

As this study has, l think, shown to say simply that MacColl "included sorne suggestions for modal logic" would be something like saying that Boole had ineluded sorne suggestions for an extensional algebra of logic. Both left a great deal to be done; nevertheless, each did much toward providing the groundwork for later developments.

24 Jorgen Jorgensen, A Treatise of FormaI Logic 3 vols. (New York: Russell and Russell, 1962), V. 3, p. 279 25 William and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 1968), p. 549. - 132 -

BIBLIOGRAPHY

Aristotle. Analytica priora. The Works of Aristotle trans-

~ated into English. Edited by W. D. Ross. Vol. 1. London: Oxford University Press, 1928. Categories and De Interpretatione. Translated J. L. Ackrill. Oxford: Clarendon Press, 1963. Bain, Alexander. Logic, 2 vols. London: Longmans, Green, Reader & Dyer, 1870. Bochenski, I.M. Ancient FormaI Logic. Amsterdam: North Holland Publishing Company, 1951. A History of FormaI Logic. Translated and Edited by Ivo. Thomas. Notre Dame, Indiana: University of Notre Dame Press, 1961. Boehner, Philotheus. Medieval Logic: An Outline of its Development from 1250 - c. 1400. Manchester: Manchester University Press, 1966. Boole, George. The Mathematical Analysis of Logic. Cambridge: Cambridge University Press, 1847. An Investigation of The Laws of Thought on which are founded The Mathematical Theories of Logic and Probabilities. New York: Dover Publications, 1958. Bowne, G.D. The Philosophy of Logic 1880-1908. The Hague: Mouton & Co., 1966. Copi, Irving M. and Gould, James A. eds. Contemporary Readings in Logical Theory. New York: Macmillan, 1967. - 133 -

De Morgan, Augustus. On The Syllogism and other Logica1

Writings. Edited with an introduction by Peter

Heath. London: Routledge & Kegan Paul, 1966.

Frege, Gottlob. Translations From the Philosophical Writings

of Gottlob Frege. Trans. by Peter Geach and Max

Black. Oxford: Basil Blackwell, 1966.

Geach, Peter. "Assertion". Philosophica1 Review. Vol. 74

(1965) •

Hughes, G.E. and Cresswell, M.J. An Introduction to Modal

Logic. London: Methuen, 1968.

Jevons, W. Stanley. Elementary Lessons in Logi.~ London:

Macmillan & COG, 1965.

Johnson, W.E. "The Logical Calculus". Mind N.S. Vol. l

(1892) •

Jorgensen, Jorgen. A Treatise of FormaI Logic. 3 vols.

New York: Russell and Russell, 1962.

Kneale, William, and Kneale, Martha. The Development of Logic.

Oxford: Clarendon Press, 1968.

Lewis, Clarence Irving. "The Calculus of Strict Implication".

Mind, N.S. Vol. 23 (1914).

A Survey of Syrnbolic Logic. Berkeley: University of California Press, 1918.

A Survey of Syrnbolic Logic. New York: Dover

Publications, 1960.

and Langford, Cooper Harold. Syrnbolic Logic.

Second Edition. New York: Dover Publications, 1959. - 134 -

Mates, Benson. Stoic Logic. Berkeley and Los Angeles:

University of California Press, 1961.

McCall, Storrs. Aristotle's Modal Syllogisms. Amsterdam:

North-Holland Publishing Co., 1963

"Connexive Implication and the Syllogism". Mind.

N.S. Vol. 76 (1967)

"MacColl". The Encyclopedia of Philosophy.

8 Vols. Ed. Paul Edwards. New York: MacMillan, 1967.

MacCo11, Hugh. "Symbolical or abbreviated Language, with an application to Mathematical Probability". Educationa1

Times. (London) Vol. 28 (1877).

"The Calculus of Equivalent Statements and

Integration Limits". Proceedings of the London

Mathematical Society. Vol. 9 (1877-78) •

"The Calculus of Equivalent Statements (second

paper) ". Proceedings of the London Mathematica1

Society. Vol. 9 (1877-78)

"The Calculus of Equivalent Statements (third

paper) ".Proceedings of the London Mathematical Society.

Vol. 10 (1878-79)

"The Calculus of Equivalent Statements (fourth

paper)" • Proceedings of the London Mathematical

Society. Vol. Il (1879-80).

"Symbolical Reasoning". Mind. Vol. 5 (1880) •

"Implicational and Equational Logic". The London,

Edinburgh and Dublin Philosophical Magazine and

Journal of Science. Series 5. Vol. Il (1881). - 135 -

MacColl, Hugh. "On the Growth and Use of a Symbo1ica1

Language". Memoirs of the Manchester Literary and

Philosophical Society. Third Series. Vol. 7 (1882).

"The Calculus of Equivalent Statements. (Fifth

paper)". Proceedings of the London Mathematical

Society. Vol. 28 (1896-7).

"The Calculus of Equivalent Statements. (Sixth

Paper) " • Proceedings of the London Mathematical

Society. Vol. 28 (1896-7).

"Symbolical Reasoning. (ll)n. Mind. N.S. Vol. 6

(1897)

"The Calculus of Equivalent Statements. (seventh

paper)". Proceedings of the London Mathematical

Society. Vol. 29 (1897-8).

"Exp1anatory Note and Correctionn • Proceedings of

the London Mathematica1 Society. Vol. 30 (1898-9)

"Review of Whitehead's Universa1 Algebra". Mind

N.S. Vol. 8 (1899).

hS:i~bo1ica1 Reasoning (111.) n Mind. N.S. Vol. 9

(1900) •

"Question for Logiciansn• Mind. N.S. Vol. 9 (1900).

"Symbo1ic Reasoning (IV)". Mind. N.S. Vol. Il (1902).

" Symbolic Reasoning (V) ". Mind. N.S. Vol. 12 (1903).

"Symbo1ic Logic". The Athenaeum. No. 4007

(August 13, 1904).

"Symb1ic Reasoning (VI)". Mind. N.S. Vol. 14 (1905). - 136 -

MacColl, Hugh. "Existential Import". Mind. N.S. Vol. 14

(1905) •

"Symbolic Reasoning" (VII)". Mind. N.S. Vol. 14

(1905) • "Note in Reply to Bertrand Russell". Mind.

N.S. Vol. 14 (1905). "The Existential Import of Propositions". Mind.

N.S. Vol. 14 (1905).

Symbolic Logic and its Applications. London: Longmans, Green, and Co., 1906.

"Symbolic Reasoning (VIII)". Mind. N.S. Vol. 15

(1906). "Symbolic Logic (A Rep1y)". Mind. N.S. Vol. 16

(1907) • "If and Imp1y". Mind. N.S. Vol. 17 (1908) "Linguistic Misunderstandings". Mind. N.S. Vol.

19, (1910). Letters to Bertrand Russell 1901-1909. Bertrand Russell Archives, McMaster University. Moo.dy,Ernest. Truth and Consequence in Mediaeva1 Logic.

Amsterdam: North-Ho11and Pub1ishing Co., 1953. Nidditch, P. H. The Deve10pment of Mathematica1 Logic. Monographs in Modern Logic. London: Rout1edge and

Kegan Paul, 1966. Peirce, C.S. Co11ected Papers. 8 Vols. Ed. C. Hartshorne, P. Weiss and A.W. Burks. Cambridge: Harvard

University Press, 1931-58. - 137 -

Quine, Willard Van Or.man. Methods of Logic. Revised edition.

New York: Holt, Rinehart and Winston, 1964.

ll Rescher, Nicholas. "On the Logic of Existence and Denotation •

Philosophical Review. (Vol. 68) (1959).

Russell, Bertrand. "Existential Import". Mind. N.S. Vol.

14 (1905).

"On Denoting". Mind. N.S. Vol. 14 (1905).

n "Review of Symbolic Logic and its Applications •

Mind. N.S. Vol. 15 (1906).

"If and Imply". Mind. N.S. Vol. 17 (1908). Introduction to Mathematical Philosophy. London:

George Allen and Unwin Ltd., 1919.

The Principles of Mathematics. 2nd Edition. London:

George Allen and Unwin, 1937. ______, and Whitehead, A.N. Principia Mathematica.

3 Vols. Cambridge: Cambridge University Press,

1910-1913.

SChr8der, Ernst. Vorlesongen Uber Die Algebra der Logik.

3 Vols. Leipzig: Druck und Verlag Von B.G. Teubner,

1890-1895.

n Shearman, A.T. "Mr. MacColl's Views on Logical Existence •

Mind. N.S. Vol. 15 (1906). The Development of Symbolic Logic. London:

Williams and Norgate, 1906. "Certainties, Impossibilities and Variables". Mind.

N.S. Vol. 16 (1907).