On the Contingency of Logic in Possible World Semantics

TL;DR

This paper explores how different logical systems can coexist in possible world semantics by introducing mixed models, revealing that the valid formulas align with an extended intuitionistic modal logic.

Contribution

It introduces mixed models combining classical and intuitionistic logic in possible world semantics and characterizes their valid formulas with an extended intuitionistic modal logic.

Findings

Valid formulas correspond to $ extsf{iK} + extsf{bem}$ logic.

Established soundness and completeness for mixed models.

Linked mixed models to birelational models for the logic.

Abstract

This paper investigates the contingency of logic within the framework of possible world semantics. Possible world semantics captures the meaning of necessitation, i.e., a statement is necessarily true if it holds in all possible worlds. Standard Kripkean semantics assumes that all possible worlds are governed by one single logic. We relax this assumption and introduce mixed models, in which different worlds may obey different logical systems. The paper provides a first case study where we mix classical propositional logic ($\mathsf{CPC}$) and intuitionistic propositional logic ($\mathsf{IPC}$) in the possible world semantics. We define the class of mixed models $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$, together with a subclass of concrete mixed models ($\mathcal{CMM}$), and establish their semantic properties. Our main result shows that the set of formulas valid in $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ corresponds exactly to the intuitionistic modal logic $\mathsf{iK}$ extended with the Box Excluded Middle axiom ($\mathsf{iK} + \mathsf{bem}$). To demonstrate this, we prove soundness and completeness results linking $\mathcal{M}\mathcal{M}(\mathsf{CPC}, \mathsf{IPC})$ and $\mathcal{CMM}$, and birelational models for $\mathsf{iK} + \mathsf{bem}$.