On the syntactical construction of systems of modal logic

Abstract

When C. I. Lewis developed his theory of strict implication, he left open the question which of his various systems should be regarded as being closest to our intuitions—though he was inclined to favor the system S2. There are to be found in the literature numerous discussions of this question; most of these have condemned S2 as being too strong, and have proposed ways of weakening it.In the present paper I shall attempt to throw some light on this question by setting up a syntactical definition of “possibility.” I shall show that every system of modal logic constructed on the basis of this definition is at least as strong as the Lewis system S4.As the intuitive basis for the syntactical definition of possibility, I take the position that to say a sentence is possible means that there exists a true sentence of the same form. Thus, for example, it would be said that the sentence, “Lions are indigenous to Alaska,” is possible, because of the fact that the sentence, “Lions are indigenous to Africa,” has the same form and is true.