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Meaning and Implication Jonathan Bennett Mind, New Series, Vol. 63, No. 252. (Oct., 1954), pp. 451463. Stable URL: http://links.jstor.org/sici?sici=00264423%28195410%292%3A63%3A252%3C451%3AMAI%3E2.0.CO%3B2U Mind is currently published by Oxford University Press.
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11.MEANING
AND IMPLICATION
IT is more than twenty yearb since Everett J. Nelson's article
' Intensional Relations ' first appeared in this journa1,l but it is impossible to tell whether it is still a force to be reckoned with. C. I. Lewis's paradoxes of strict implication, which Professor Nelson was concerned to attack, have certainly found widespread acceptan~e,~ b ut Professor Nelson's rejection of the paradoxes in favour of a concept of implication based on ' connexion of meanings ' seems to have influenced many logicians. Some of the latter who have tried to elaborate the system advanced in ' Intensional Relations ' will be dealt with in the following pages ; but there are others who appear simply to have taken over the theory en bloc (notably Professor Broad in his Exainination of MeTaggart's Philosophy), or who have written of implication in such a way as to suggest that Nelson's theory or something like it must be at the back of their minds (e.g. G. H. von Wright in An Essay in Modal Logic). Thus it seems that the theory's initial plausibility has not been lost on the audience to whom it was addressed, so that a somewhat detailed examination of it, and of some of its philosophical offspring, may still be worth while.
For the sake of completeness, the nature of the paradoxes should be outlined. ' p implies q ' is defined as ' I t is not logically possible that p should be true and q false ', whence it follows that if it is not logically possible that p should be true, then p implies q whatever q may be, and if it is logically impossible that q should be false, then q is implied by p, whatever p may be. Furthermore, since ' p is inconsistent with q ' means ' p implies notq ', an impossible proposition is inconsistent with any proposition, including itself. Everett J. Nelson, ' Intensional Relations ', MIND, 1930, pp. 440453. This acceptance has not been an entirely academic matter. William Kneale ( ' Truths of Logic ', Proceedings of the Aristotelian Society, 19451946, pp. 207234) and K. 'R. Popper (' New Foundations for Logic ', MIND, 1947, pp. 193235) have both used the paradoxes as integral parts of their respective accounts of the nature of logic.
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JONATHAN BENNETT
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Lewis's first attempt to show that there is an ' honesttogoodness ' sense in which these theorems are true l depends, I think, on an ambiguity in the use of the word ' presupposition ', and it need not be discussed here. What is more important is his attempt to deduce the paradoxes from propositions which are undeniably true by modes of inference which are undeniably valid.2 These are the socalled 'independent proofs ' of the paradoxes of strict implication. The first of these proofs, the more important one for present purposes, runs as follows : Take any impossible proposition This implies and also implies Now, (2) implies and (3) (4) implies
p.p P P Pvq 4
(1) (2) (3) (4)
Thus the impossible proposition implies q, whatever q may be. The original statement of this proof employs ' 3 ' for ' implies ', a fact which has been seized upon by some critics as involving a circularity. There is no circularity, however. The real point of the independent proofs is that Lewis is sure that, antecedent to any precise definition of implication, the steps involved in the proofs will be granted by a,ll to be valid.
The article by Nelson mentioned above outlines a system of logic in which the relations of consistency, entailment (which is his version of ' implication as ordinarily understood '), conjunction and disjunction are not determined purely by truthvalues of the propositions concerned, nor purely by modal functions of these values, but by ' t h e meanings of the propositions i n relation to one another '. Lewis, says Nelson, has introduced into his system the intensional notion of possibility, and its derivates, but he uses it only as a function of purely extensional relations. That is, instead of interpreting the intensional disjunction of p and q as the proposition that a disjunctive relation necessarily holds between them, he has interpreted it as the proposition that it is necessary that one of them be true (whether because of an intensional relation between them or because one of them, on its own, must be true). C. I. Ilewis, Survey qf Symbolic Logic (1918), p. 338. C. I. Lewis and C. H. Langford, Symbolic Logic (1932), pp. 250251.
453
MEANING A N D IMPLICATION
Nelson bases his system on the notion of consistency, symbolised by ' o ', which he explains as being the relation which holds between a pair of propositions when there is nothing in their meanings such that the truth of one necessitates the falsity of the other. (The paradoxical theorem, ' () p . 3 . (pop) ' is thus avoided.) Entailment, symbolised by ' I3 ', is defined in terms of consistency, thus : pEq . = .  (PO  q). This sweeps away the paradoxes a t a blow, and n comparison between this and the definition of strict implication,


P3q . = .  0
(P4) sums up the differences between the two systems : Nelson has incorporated Lewis's niodal element in the relation between the two propositions, so that the modalities of the propositions themselves are irrelevant. Nelson symbolises ' p is inconsistent with q ' by ' p/q ', and defines it thus : l?/q . = .  ( ~ 0 4 ) . I n terms of this, intensional logical sum is defined thus p v q . = .  p/  q. Finally, intensional conjunction is explained (not defined) as follows : ' I do not take pq to mean " p is true and q is true ", but simply " p and q ", which is a unit or whole, not simply an aggregate, and expresses the joint force of p and q.' And again, ' pq does not entail r, unless both p and q function together in entailing r '. I n these terms, every proposition is consistent with itself, no proposition entails its contradictory, a necessary proposition is not entailed by any proposition, and so 0n.l For the rejection of the independent proofs of the paradoxes, it would seem to be sufficient to reject either ' p . E pVq ' or ' pqEp ', and the more obvious way seems to be the rejection of the former. Nelson rejects the latter also, however, because the principle of antilogism,
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gives, if we substitute p for r, It is perhaps worthy of