Modal functions in two-valued logic

Abstract

In this paper it will be shown that the fundamental modal functions (“… is necessary,” “… is possible,” etc.) may be represented in the ordinary two-valued logic of propositions, relatively to a subset S of the set K of all propositions of the logic. The only restriction on S is that no member of S can occur in any other member of S. The representation concerns the set SK consisting of all members of K in which occur members of S. Every member of SK may be regarded as a truth-function of the members of S. If S consists of only one member, P, and if f(P) and g(P) are any members of SK, then we define ∣f(P)∣ as (f(P)·f(~P)). The proposition “f(P) is S-necessary” may then be expressed by ∣f(P)· Furthermore we may express “f(P) is S-impossible” by ∣~f(P)∣, abbreviated to f(P)*; we may express “f(P) is S-possible” by abbreviated to f(P); and we may express “f(P) S-strictly implies g(P)” by (f(P)·~g(P))*, abbreviated to f(P)⊰g(P). Thus is obtained a novel analysis of modality, contrasting, on the one hand, with Carnap's syntactic theory of modality, and, on the other hand, with Lewis's view of modal concepts as underivable from other logical concepts. We would maintain that modal concepts are not absolute, but always relative to some S, generally unspecified.In the formal exposition which follows, we shall assume tacitly but not explicitly the important distinction between the symbols we are discussing and symbols denoting or designating the former.