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The modal systems S1–S3 were introduced by C. I. Lewis as logics for strict implication. While there are Kripke semantics for S2 and S3, there is no known natural semantics for S1. We extend S1 by a Substitution Principle (SP) which generalizes a reference rule of S1. In system S1 + SP, the relation of strict equivalence ϕ ≡ ψ satisfies the identity axioms of R. Suszko's non-Fregean logic adapted to the language of modal logic (we call these axioms the axioms of propositional identity). This enables us to develop a framework of algebraic semantics which captures S1 + SP as well as the Lewis systems S3–S5. So from the viewpoint of algebraic semantics, S1 + SP turns out to be an interesting modal logic. We show that S1 + SP is strictly contained between S1 and S3 and differs from S2. It is the weakest modal logic containing S1 such that strict equivalence is axiomatized by propositional identity.

Algebraic semantics for a modal logic close to S1