Some remarks on semantics and expressiveness of the Sentential Calculus with Identity

TL;DR

This paper explores the semantics and expressiveness of Suszko's Sentential Calculus with Identity, connecting it to Boolean prealgebras, hyperintensional semantics, and modal logic dualities, extending its theoretical framework.

Contribution

It reformulates SCI semantics via Boolean prealgebras, establishes dualities with Lewis-style modal systems, and generalizes Suszko's original approach to S4 and S5 theories.

Findings

Reformulation of SCI semantics using Boolean prealgebras

Establishment of dualities between SCI theories and modal systems

Extension of Suszko's approach to S4 and S5 theories

Abstract

Suszko's Sentential Calculus with Identity SCI results from classical propositional calculus CPC by adding a new connective $\equiv$ and axioms for identity $\varphi\equiv\psi$ (which we interpret here as `propositional identity'). We reformulate the original semantics of SCI in terms of Boolean prealgebras establishing a connection to `hyperintensional semantics'. Furthermore, we define a general framework of dualities between certain SCI-theories and Lewis-style modal systems in the vicinity of S3. Suszko's original approach to two SCI-theories corresponding to S4 and S5 can be formulated as a special case. All these dualities rely particularly on the fact that Lewis' `strict equivalence' is axiomatized by the SCI-principles of `propositional identity'.