A General Theory of Propositional Modal Bundled Modalities

TL;DR

This paper develops a comprehensive theoretical framework for bundled propositional modal operators, including their expressivity, axiomatization, and bisimulation concepts, covering many existing modalities.

Contribution

It introduces a uniform approach to define bisimulations, axiomatizes convex bundled modalities, and applies these to various case studies in modal logic.

Findings

Established a general theory for bundled modalities

Defined bisimulations with Hennessy-Milner property

Axiomatized several specific bundled modalities

Abstract

In studies of bundled modalities, we encode a complex conceptual notion into the semantics of a single modal operator and study its logic. Although there is already a substantial body of work on various concrete bundled operators, we still lack a general understanding of them. In this paper, we provide a general theory of the expressivity and axiomatization of bundled modalities. We offer a uniform way to define bisimulations for arbitrary bundled modalities and justify our definition by the corresponding Hennessy-Milner property. We also define a special class of bundled modalities called convex bundles. This class covers most bundled modalities studied in the literature, and their axiomatizations can be done with the help of convex neighborhood semantics and corresponding representation results. As case studies, we axiomatize the "someone knows" bundle $\bigvee_{a \in A} \Box_a \phi$ over $S5$-models, the "disagreement in group" bundle $\bigvee_{a, b \in A} \Box_a \phi \wedge \Box_b \neg \phi$ over $KD45$-models, and the "belief without knowledge" bundle $B \phi \wedge \neg K \phi$ over $S4.2$-models.