Aristotelian and Mathematical Logic

ARISTOTELIAN AND 1\iATHEMATICAL.LOGIC "l X ~T is the difference between the traditional ArisV V ~~:elian logic and the newer, so-.called mathematical logic? 1 To such a question Jhe mathematical logicians have a ready answer: the older logic is but a part of the new, and a part so small as to be really only a" fragment," 2 and a fragment so paltry as to be "entirely insignificant." a And what then is the value of Aristotelian logic? Really none at all, unless it be of a certain pedagogical value as a means of introducing the student to mathematical logic,4 or unless it be valued purely for its historical interest as a somewhat extraordinary and very long-enduring cultural monument. As such, it might merit mention, and perhaps even treatment, in a contemporary text-book; 5 but, of course, it should not any longer be taken very seriously. And what do th~ Aristotelian logicians say to all this? Naturally, they don't like it; and yet the unfortunate thing is that they have not said much by way of rebuttal. Why not? Is it because in the nature of the case no rebuttal can be given? 1 This tenn " mathematical logic " is doubtless none too fortunate, simply because, many other names, signifying exactly the same body of material, have gained currency along with the name " mathematical logic." Indeed, this body of material, as Prof. Lewis has remarked (Lewis and Langford, Symbolic Logic, New York, 1982, p. S), "has not yet acquired any single and well-understood name. It is called ' mathematical logic ' as often as ' symbolic logic,' and the designations 'exact logic,' 'formal logic,' and logistic are also used. None of these is completely satisfactory; all of them attempt to convey a certain difference of this subject from the logic which comes down to us from Aristotle and. was given its traditional fonn by the medieval scholastics." • Alfred Tarski, Introduction to Logic. New York, 1946, p. 19. • Ibid. • Eaton in his General Logic gives this justification for his treatment of traditional logic. • Quine, W. V., Short Course in Logic, Cambridge, 1946, p. 87. 50 ARISTOTELIAN AND MATHEMATICAL LOGIC 51 This is what the mathematical logicians all confidently assume, and they seem pretty well to have convinced others of the fact as welL And yet it is still at least a possibility that the reason no rebuttal has been given is because as yet no champions have come forth from the camp of the Aristotelians to give it. But why have champions not been forthcoming? One reason might be that the presentations of Aristotelian logic that have been given in modern text-books have in no wise done justice to the subject. Thus, stressing formal logic-and even this in a very watered-down fashion-to the almost complete exclusion of material logic, they have quite generally failed to make clear what the nature of logic is, what its significance is, what the peculiar subject matter is with which logic should concern itself. As a result, the mathematical logicians have had little trouble brushing the whole discipline aside as if of no moment, and the Aristotelians for their part have found themselves unhappily embarrased for want of really ready ammunition.6 Nevertheless, for all this, the Aristotelian tradition is quite capable of presenting a remark~bly dear and unambiguous account of the scope and nature of logic. Moreover, once this account is honestly considered and is set over against the notso -clear and rather ambiguous account of the general character of logic that is given in mathematical logic, it may become apparent that the one discipline certainly is not subsumable under the other as a mere insignificant part of the whole. On the contrary, the very nature and purpose of logic as conceived by the one discipline may turn out to be so radically different from what it is as conceived by the other, that to call both of 6 These strictures may seem extreme, and the author recognizes that he speaks subject to correction. And yet he ventures to suggest that no really full-fledged treatment of Aristotelian logic has been brought out since John of St...