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A General Theory of Propositional Modal Bundled Modalities

Yifeng Ding, Yuanzhe Yang

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.

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 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 over -models, the "disagreement in group" bundle over -models, and the "belief without knowledge" bundle over -models.

Paper Structure

This paper contains 13 sections, 22 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Language and Semantics
  3. Bisimulations for Bundled Modalities
  4. Convex Bundled Modalities
  5. Convex Neighborhood Semantics
  6. Case Studies
  7. Someone Knows and Disagreement in Group
  8. Belief without Knowledge
  9. Conclusion
  10. First-Order Undefinability of Bundled Bisimulation
  11. Proof for Proposition \ref{['prop.cvbis']}
  12. Proofs for Lemma \ref{['lem.cano1']} and Lemma \ref{['lem.cano2']}
  13. Proof Sketch for Lemma \ref{['lem.unr']}

Key Result

Proposition 3.2

(AC) For any model $\mathcal{M},w$, any $\tau \in \mathcal{T}_\mathbf{A}$ and $\phi \in \mathcal{L}_{\textnormal{\Circle}}$,

Figures (1)

  • Figure 1: A schematic picture of the Kripke frame $(W,R)$ of $\mathcal{M}$ (the valuation is empty), in which $2^\mathbb{N}$ is represented by a segment of the real line.

Theorems & Definitions (61)

  • Definition 2.1: Language
  • Definition 2.2: Kripke Models
  • Definition 2.3: Bundle Terms
  • Definition 2.4: Bundled Semantics
  • Definition 3.1: Generated Neighborhood
  • Proposition 3.2
  • proof
  • Definition 3.3: Domain and Completion
  • Proposition 3.4
  • Definition 3.5: Bundled Bisimulation
  • ...and 51 more