T. J. Smiley; XII.—Entailment and Deducibility, Proceedings of the Aristotelian Society, Volume 59, Issue 1, 1 June 1959, Pages 233–254, https://doi.org/10.1093.

R-Dagger is the theory of relevant implication, Got from the calculus r (see belnap, Jsl, 32, 1-22), By adding machinery for propositional quantification. In r-Dagger define t as for some p, P, F as for all p, P. Then (t, F) is closed in r-Dagger under truth-Functions and relevant implication, Which, When confined to (t, F) acts just like material 'implication.' but r-Dagger admits of many propositions other than t, F. The article also contains polemics against extensionalism and nominalism. (edited).

Following Moore, I use ‘P entails Q’ as a convenient shorthand for ‘Q can be deduced logically from P’, ‘From P, Q follows logically’, ‘There is a logically valid argument with P as sole premise and Q as conclusion’, and the like.1 Apart from a minor point to be raised in Section XVI, distinctions within this cluster do not matter for present purposes. An analysis of the concept of entailment is answerable to careful, educated uses of expressions such as those. (...) An analysis which condemned nearly everything we say about what follows from what simply would not be an analysis of the common concept of entailment. If the concept were inconsistent, some common uses of it would be condemned; but only by standards established by the others. C. I. Lewis maintained this: to say that P entails Q is to say that it is logically impossible that (P & ¬Q).2 If Quine is right, then ‘entails’ and ‘impossible’ are as suspect as all other intensional terms. So perhaps they are; but their uses are not wholly without structure, and there are wrong ways of interrelating them. Lewis’s contention is about the internal geography of the intensional area, not its relations to the surrounding conceptual territory: it is an attempted analysis of one intensional expression in terms of another. I shall argue that Lewis was right, and also - by implication - that his thesis is helpful and clarifying - that is, that it is a genuine analysis. As is well known, Lewis’s analysis implies that each impossible proposition entails every proposition. Accepting the analysis, I accept this result. For one thing, Lewis has an argument for it (I use ‘→’ to abbreviate ‘entails’): (1) P & ¬P.. (shrink)