A General Theory of Propositional Modal 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 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.