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

TL;DR

This paper develops a denotational semantics for extended modal systems S3--S5 with propositional quantifiers and identity, exploring non-Fregean logic and hyperintensional semantics, and analyzing the role of the Collapse Axiom.

Contribution

It introduces a non-Fregean extension of modal logic with propositional quantifiers, distinguishing necessity from propositional identity and connecting to hyperintensional semantics.

Findings

Models are Boolean prealgebras and their expansions.

PI refines strict equivalence when models are Boolean algebras.

Theories are conservative extensions of S3--S5.

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 $\mathit{SCI}$. 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 $\mathit{SCI}$-models. We show that $\mathit{SCI}$-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 \textit{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.