A syntactical characterization of S5

Extract

This note should be considered as an appendix to a paper by McKinsey. Familiarity with this paper is assumed; its terminology and notation will be used without explanation. McKinsey's main result (theorem 1) is that every class of sentences fulfilling P₁–P₅, with a set of substitutions satisfying A₁–A₄, is a system of modal logic in the sense that the “associated” set, T₂, contains all theorems of S4. It is also proved (theorem 2) that for T₂ to contain all theorems of S5, it is sufficient that the elements of S form a group. The basic idea of the construction is the formalization, by means of D₅, of our intuitive notion of possibility.

In view of the great generality of S, it may be of some interest to give a condition that is both necessary and sufficient for T₂ to contain S5 after some restriction suggested by our intuitive notions about modalities has first been imposed upon S. The following postulate is related to the idea that if a sentence is possible, then its negation is not necessary. Informally speaking, it requires that S be sufficiently “comprehensive” for this idea to become formalizable in terms of S.

P. If ♢α is in T₁, then there exists an element s_m of S such that ♢∼s_m(α) is not in T₁.

I shall now prove that if P is satisfied, the following condition is both necessary and sufficient for T₂ to contain all theorems of S5.

C. If α is in T₁, then ♢s(α) is in T₁ for all elements s of S.