Lewis, in the presentation of his calculus of strict implication, contends that this calculus accords with the usual meaning of “implies” such that “p strictly implies q” is synonymous with “q is deducible from p” (pp. 126–127, 235–262). It is the purpose of this paper to present, in 1, certain considerations in the light of which this statement does not hold, and in 2, a new implicative relation upon which a calculus of propositions can be based such that it will accord with the relation of deducibility and from which it is possible to derive the calculi of strict and material implications.

1. Strict implication regarded as synonymous with deducibility. In the presentation of his calculus of strict implication Lewis observes that despite inclusion of both intensional and extensional propositions in this calculus, the meaning of its elements, p, q, r, etc., “remain fixed, whether it is their extensional or their intensional relations which are in question” (p. 120). But if the meaning or content of the elements of this calculus is irrelevant to the logical truth which it contains, then, as Weiss has already remarked, the propositions of this calculus “are really extensionally treated, involving multiple values,” e.g., possibility, on the analogy of truth and falsity. The importance of this fact is that, in consequence, the relation of strict implication is subject to certain paradoxes, such as (19.74) “a proposition which is impossible strictly implies any proposition” or (19.75) “a proposition which is necessarily true is strictly implied by any proposition” (p. 174) or (19.84) “any two necessarily true propositions are strictly equivalent” (p. 176).