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

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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 BY 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 con- cepts. 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.
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