A General Theory of Propositional Modal Bundled Modalities
This work addresses a foundational gap in modal logic for researchers, offering a systematic approach to bundled modalities, though it is incremental as it builds on existing studies.
The paper tackles the lack of a general theory for bundled modalities in logic by providing a uniform framework for defining bisimulations and axiomatizing them, with case studies including axiomatizations for specific bundles like 'someone knows' and 'disagreement in group'.
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 $\bigvee_{a \in A} \Box_a Ï$ over $S5$-models, the "disagreement in group" bundle $\bigvee_{a, b \in A} \Box_a Ï\wedge \Box_b \neg Ï$ over $KD45$-models, and the "belief without knowledge" bundle $B Ï\wedge \neg K Ï$ over $S4.2$-models.