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Yonemitsu, Naoto
1951.
On systems of strict implication.
Tohoku Mathematical Journal,
Vol. 3,
Issue. 1,
Article contents
Implication and deducibility
Published online by Cambridge University Press: 12 March 2014
Extract
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).
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- Copyright © Association for Symbolic Logic 1936
References
1 All page numbers appearing concurrently with the text refer to Lewis, C. I. and Langford, C. H., Symbolic logic, New York, 1932Google Scholar.
2 Weiss, Paul, On alternative logics, The philosophical review, vol. 42 (1933), p. 522CrossRefGoogle Scholar.
3 Lewis, C. I., Mind and the world order, New York, 1929, p. 435Google Scholar.
4 Principia mathematica, Camb. Univ. Press, 1925, vol. 1, p. 107Google Scholar.
5 Although Lewis sometimes reads “p⊰q” as “p logically implies q” and “p=q” as “p and q are logically equivalent,” we should restrict our readings of these expressions to “p strictly implies q” and “p and q are strictly equivalent” respectively. The fact that Lewis has used the historic equivalence sign “=“ for purposes of convenience in the calculus of strict implication should lend no support to the claim that “p⊰q·⊰p” is synonymous in every respect to the relation “q is deducible from p and p is deducible from q.” Failure to observe this point leads to inevitable confusion and error. In order to distinguish the symbol for strict equivalence “p = q” from the symbol for logical equivalence “p = q” in this paper, the horizontal bars in the latter symbol will be of a bolder typographical face, and in order to distinguish the symbol for strict consistency “p o q” from the symbol for logical consistency “O(pq)” in this paper, the circle in the latter symbol will be of a bolder typographical face and will be placed in front of the expression over which it extends.
6 In a paper entitled Consistency and independence in postulational technique, Philosophy of science, vol. 3 (1936), no. 2Google Scholar, the required differentiation between consistency and possibility was first observed and effected by defining consistency in terms of the possibly possible. However, this procedure is open to the objection that it employs a familiar term in an unfamiliar signification.
7 Demonstration of the existence principles of this calculus require certain other rules of procedure comparable to those specified for the calculus of strict implication. Cf. Symbolic logic, pp. 181–184.
8 The definition of number, as given by Frege and Russell, was equally unfamiliar, but its unfamiliarity was not considered ground for its rejection. Cf., e.g., Russell, Bertrand, Our knowledge of the external world, New York, 1929, pp. 221–223Google Scholar.
9 One interesting point in connection with this fact has to do with the determination of iterated modalities. Lewis has asked the question “if a proposition is necessary, does it follow that it is necessarily necessary? … Here is an opportunity to make a contribution to logic, because nobody has ever answered these questions” (The monist, vol. 42 (1932), p. 502Google Scholar). The answer which the calculus of logical implication suggests is that if “p ∾ q” is regarded as synonymous with “q is deducible from p” then the necessary necessity of a proposition does not follow from its necessity—and this fact does not require the repudiation of all necessarily necessary propositions. Cf. Symbolic logic, pp. 497–499.
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