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 sm of S such that ♢∼sm(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.