On a Recent Account of Entailment

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On a Recent Account of Entailment Jonathan Bennett Mind, New Series, Vol. 68, No. 271. (Jul., 1959), pp. 393-395. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28195907%292%3A68%3A271%3C393%3AOARAOE%3E2.0.CO%3B2-8 Mind is currently published by Oxford University Press.

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ON A RECENT ACCOUNT OF ENTAILMENT

MR. P. T. BEACH (following G. H. von Wright's Logical Studies, London, 1957, p. 181) is concerned to develop a concept of entailment in terms of which there is something wrong with the following argument-schema : p. p entails p p. -p entails p p entails p v q p, p v q entails q :. p. p entails q I shall call this the first Lewis argument-schema. Geach says that p entails q if and only if there is an a priori way of getting to know that p > q which is not a way of getting to know either that -p or that q. He expects this to disturb the above argument-schema by falsifying its first premiss, the point being that a truth-table verification of (p. -p) > p will embody a truth-table (p. 7 pj. verification of But there is an a przorz way of getting to know (for any particular proposition P) tha* (P. P) > P which is not also a way of get(P. P) ; namely, performing a truth-ta%le ting to know that P for q. verscation of (p. q) 3 p, and substituting P for p and (P. P) occurring This would show that (P. P) > P without as a line in the proof or occurring in a truth-table with a row of 1's or T's in the crucial column-i.e. without our ' getting to know that ' (P. ,-P) within the meaning of the act. A parallel argument holds against Geach's objection to what I shall call the second Lewis argument-schema : p entails (p. q) v (p. q) (P. 4) V (P. 9) entails p. (q v q) p. (q v q) entails q v q .: p entails p v q la this, the third premiss appears open to objection from Geach, in that a truth-table verification of (p. (q v q)) > (q v q) will embody a truth-table verification of q v q, so that this conditional does not correspond to an entailment in Geach's sense of " entails ". But again there is an a priori way of showing (for any particular propositions P and Q) that (P. (Q v Q)) > (Q v Q) which is not a way of showing that Q v Q ; namely, showing by truth-tables that (p. r) 3 r, and substituting P for p and Q v ,-Q for r. The upshot is that in the Geach-von Wright sense of entailment, any given instance of either of the Lewis argument-schernas holds

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1 " Entailment ", Proceedings qf the Aristotelian Sociefy, suppl. vol. xxxii, 1958.

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J. BENNETT

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good. The reasons for putting this (and its justification) in terms of any rather than a l l 4 . e . for talking of argument-schemas and their instances, rather than of arguments each line of which is an implicitly universally quantified proposition have arisen in the following way. During the discussion of the symposium of which Geach's paper formed a part, another of the symposiasts, Dr. C. Lewy, pointed out that von Wright's admission of " p. q entails p " as a thesis commits him to an acceptance of the first Lewis argument-schema, just as it stands. Geach replied that von Wright had erred in admitting that p. q entails p-that he had probably assumed this to be a thesis because we can prove that (p. q) a p with a proof in whch neither (p. q) nor p is proved on the way ; whereas this doesn't really follow, because " p. q entails p " is, so to speak, a bundle of instances of itself, while the non-proving of (p. q) in the proof of (p. q) 3 p is the non-proving of a bundle, not a bundle of non-provings, and similarly with the non-proving of p. (Explicit use of quantifiers makes the point clear.) The present note steers around this controversy by avoiding any use of (implicitly) quantified entdilment-theses, dealing only in (implicitly) quantified theses of truth-functional logic, and introducing mention of entailment only in metalogical remarks about the results achieved (and the manner of their achievement) by the familiar, truth-functional procedures employed. It also makes it clear that von Wright is vulnerable to Lewy's attack not because of a chance remark but because of the central features of his concept of entailment, and thus that Geach is no better off than von Wright in this respect. Both the Lewis argument-schemas presuppose the transitivity of entailment, and Geach denies his concept this property ; but this does not affect the line of argument advanced here. The reason for this is, roughly speaking, that Geach characterises in a certain way the area of non-transitivity of his concept of entailment, and this area does not overlap either of the Lewis argument-5chemas.l To go into this in full detail would be tedious, especially when the matter can be handled more directly by actually showing in full how we can get to know apriori that P a (Q v Q) is true (for arbitrary P and Q) by a way in which (Q v Q) does not occur as a line in the proof and is not subjected to truth-table analysis : (by truth-tables) 1. p 3 ((P. 4) V (P. "9)) (by truth-tables) 2. (p 3 Up. q) v (P. 19)) 3 (P 3 (q v r ) ) 1 PIP, q/Q = 3 3. P a ((P. Q) v (P. Q))

2 PIP, q/Q, r/- Q = 4

4. ( P 3 ((P. Q) v (P. -- Q))) 3 (P 3 (Q v Q))

4=3>5

5. P a ( Q v N Q )

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This point was made by Lewy in the discussion mentioned above.

ON A RECENT ACCOUNT OF ENTAILMENT

395

This proof holds for any selection of P and Q ; and a similarly general demonstration can be given of any instance of the other Lewis paradox in a way which also satisfies the Geach-von Wright definition of entailment. I do not give proof here, as (apart from the sort of proof mentioned below) the shortest one I can find runs to fifteen lines, and probably could not be shortened much except a t the cost of intolerable complexity of proof-lines and substitution-steps. Anyway, as Dr. Lewy has pointed out to me, either paradox can be derived from the other, in a way satisfying the Geach-von Wright concept of entailment, by contraposition.

JONATHAN BENNETT University of Cambridge