A General Theory of Propositional Modal Bundled Modalities

2026-03-27

AI Summary

This work addresses the lack of a unified theoretical framework for bundled modalities in propositional modal logic, which has hindered systematic analyses of their expressivity and axiomatization. The paper proposes a general framework that defines bisimulation relations for arbitrary bundled modalities and introduces the class of convex bundled modalities together with their convex neighborhood semantics, thereby unifying expressivity analysis and axiomatization. It innovatively establishes a method for verifying the Hennessy–Milner property and successfully provides complete axiomatizations for three representative bundled modalities—“someone knows,” “group disagreement,” and “belief without knowledge”—corresponding respectively to S5, KD45, and S4.2 models. This constitutes the first systematic theoretical foundation for this area.

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 $\bigvee_{a \in A} \Box_a φ$ over $S5$-models, the "disagreement in group" bundle $\bigvee_{a, b \in A} \Box_a φ\wedge \Box_b \neg φ$ over $KD45$-models, and the "belief without knowledge" bundle $B φ\wedge \neg K φ$ over $S4.2$-models.

Problem

• bundled modalities
• expressivity
• axiomatization
• modal logic
• propositional logic

Innovation

• bundled modalities
• bisimulation
• convex bundles
• axiomatization
• neighborhood semantics