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 tohyperintensional 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 ofpropositional identity'.