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State-based Modal Logics for Free Choice

Maria Aloni, Aleksi Anttila, Fan Yang

TL;DR

The paper develops bilateral state-based modal logic $BSML$ with team semantics to formalize Free Choice phenomena, introducing the nonemptiness atom $NE$ and two extensions: $BSMLI$ with global disjunction $\intd$ and $BSMLO$ with the emptiness operator $OC$. It establishes state-bisimulation-based expressive completeness results: $BSMLI$ captures all state properties invariant under bounded bisimulation, and $BSMLO$ captures all union-closed invariant state properties, while $BSML$ is union-closed and invariant but not complete for the latter class. It then provides sound and complete natural deduction systems for all three logics, using normal-form and realization-based techniques to bridge between the logics and their expressive targets. These results yield a robust, formally rigorous account of Free Choice in a modal-team semantics setting, with connections to inquisitive semantics and potential linguistic applications. The work also suggests avenues for further development, including first-order extensions and applications of the emptiness operator to pragmatic enrichment in semantics.

Abstract

We study the mathematical properties of bilateral state-based modal logic (BSML), a modal logic employing state-based semantics (also known as team semantics), which has been used to account for free choice inferences and related linguistic phenomena. This logic extends classical modal logic with a nonemptiness atom which is true in a state if and only if the state is nonempty. We introduce two extensions of BSML and show that the extensions are expressively complete, and develop natural deduction axiomatizations for the three logics.

State-based Modal Logics for Free Choice

TL;DR

The paper develops bilateral state-based modal logic with team semantics to formalize Free Choice phenomena, introducing the nonemptiness atom and two extensions: with global disjunction and with the emptiness operator . It establishes state-bisimulation-based expressive completeness results: captures all state properties invariant under bounded bisimulation, and captures all union-closed invariant state properties, while is union-closed and invariant but not complete for the latter class. It then provides sound and complete natural deduction systems for all three logics, using normal-form and realization-based techniques to bridge between the logics and their expressive targets. These results yield a robust, formally rigorous account of Free Choice in a modal-team semantics setting, with connections to inquisitive semantics and potential linguistic applications. The work also suggests avenues for further development, including first-order extensions and applications of the emptiness operator to pragmatic enrichment in semantics.

Abstract

We study the mathematical properties of bilateral state-based modal logic (BSML), a modal logic employing state-based semantics (also known as team semantics), which has been used to account for free choice inferences and related linguistic phenomena. This logic extends classical modal logic with a nonemptiness atom which is true in a state if and only if the state is nonempty. We introduce two extensions of BSML and show that the extensions are expressively complete, and develop natural deduction axiomatizations for the three logics.
Paper Structure (10 sections, 52 theorems, 90 equations, 3 figures, 1 table)

This paper contains 10 sections, 52 theorems, 90 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bilateral State-based Modal Logic
  3. Expressive Power and Normal Forms
  4. Bisimulation
  5. Expressive completeness
  6. Axiomatizations
  7. $\BSMLIBF$
  8. $\BSMLOBF$
  9. $\BSMLBF$
  10. Conclusion

Key Result

proposition 1

For any formulas $\theta,\phi$, and $\psi$, if $\phi \equivclosed \psi$, then $\theta[\phi/p] \equivclosed \theta[\psi/p]$.

Figures (3)

  • Figure 1: Models for the sentence Every square is black.
  • Figure 2: Zero-models and $\FC$ in $\BSML$. Worlds are labelled according to what is true in them: $w_p$ stands for a world where only $p$ is true, $w_{pq}$ for one in which only $p$ and $q$ are, etc. The accessibility relation is depicted using arrows.
  • Figure 3: Examples of the semantics

Theorems & Definitions (121)

  • definition 1: Syntax
  • definition 2
  • definition 3: Semantics
  • proposition 1: Replacement w.r.t. strong equivalence
  • proof
  • proof
  • definition 4
  • proof
  • definition 5: Bisimilarity
  • definition 6: Hintikka formulas
  • ...and 111 more