A General Theory of Propositional Modal Bundled Modalities

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 positive-negative-independent bundles. This class of bundles, together with their duals, cover most bundled modalities studied in the literature, and their axiomatizations can be done with the help of a more abstract notion 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 within 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.