Denotational Semantics for Modal Systems S3–S5 Extended by Axioms for Propositional Quantifiers and Identity

Abstract

There are logics where necessity is defined by means of a given identity connective: ${\square\varphi := \varphi\equiv\top}$ (${\top}$ is a tautology). On the other hand, in many standard modal logics the concept of propositional identity (PI) ${\varphi\equiv\psi}$ can be defined by strict equivalence (SE) ${\square(\varphi\leftrightarrow\psi)}$ . 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 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.