Algebraic semantics for a modal logic close to S1

TL;DR

This paper develops an algebraic semantics for a modal logic close to S1, extending it with a substitution principle and analyzing its relation to other Lewis systems, providing a new perspective on S1's semantics.

Contribution

It introduces a novel algebraic semantics for S1 with a substitution principle, connecting it to S3--S5 and clarifying its position among Lewis modal systems.

Findings

S1+SP is strictly between S1 and S3

S1+SP differs from S2

S1+SP captures propositional identity as strict equivalence

Abstract

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 $\varphi\equiv\psi$ 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.