Denotational semantics for modal systems S3--S5 extended by axioms for propositional quantifiers and identity

There are logics where necessity is defined by means of a given identity connective: $\mathrm{\square }\phi :=\phi \equiv \mathrm{\top }$ ($\mathrm{\top }$ is a tautology). On the other hand, in many standard modal logics the concept of propositional identity (PI) $\phi \equiv \psi$ can be defined by strict equivalence (SE) $\mathrm{\square }(\phi \leftrightarrow \psi )$. All these approaches to modality involve a principle that we call the Collapse Axiom (CA): “There is only one necessary proposition.” In this paper, we consider a notion of PI which relies on the identity axioms of Suszko’s non-Fregean logic $SCI$. Then $S3$ proves to be the smallest Lewis modal system where PI can be defined as SE. We extend $S3$ to a non-Fregean logic with propositional quantifiers such that necessity and PI are integrated as non-interdefinable concepts. CA is not valid and PI refines SE. Models are expansions of $SCI$-models. We show that $SCI$-models are Boolean prealgebras, and vice-versa. This associates Non-Fregean Logic with research on Hyperintensional Semantics. PI equals SE iff models are Boolean algebras and CA holds. A representation result establishes a connection to Fine’s approach to propositional quantifiers and shows that our theories are $\mathit{\text{conservative}}$ extensions of $S3$–$S5$, respectively. If we exclude the Barcan formula and a related axiom, then the resulting systems are still complete w.r.t. a simpler denotational semantics.