On the Contingency of Logic in Possible World Semantics
Iris van der Giessen, Joost J. Joosten, Paul Mayaux, Vicent Navarro Arroyo
TL;DR
The paper addresses how necessity should be interpreted when different possible worlds obey different logics, focusing on a two-logic case with ${\sf CPC}$ and ${\sf IPC}$. It introduces mixed models ${\mathcal{MM}}(\{\mathcal{L}^i\}_{i\in I})$ and their concrete subclass ${\mathcal{CMM}}$, showing that the formulas valid in ${\mathcal MM}({\sf CPC},{\sf IPC})$ coincide with those provable in the intuitionistic modal logic ${\sf iK} + {\sf bem}$, where ${\sf bem}$ enforces Box Excluded Middle. The authors prove soundness and completeness results by relating ${\mathcal{CMM}}$ to birelational models ${\mathcal{BM}}$ for ${\sf iK} + {\sf bem}$ and by constructing translations between concrete mixed models and birelational semantics. They also discuss finite-model considerations and outline future work to broaden the logic-mixing framework to additional logics, and to explore theoretical and philosophical implications of logical contingency in possible-world semantics.
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 $\mathsf{iK} + \mathsf{bem}$.