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

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'.