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Inquisitive Neighborhood Logic

Ivano Ciardelli

Abstract

We explore an inquisitive modal logic designed to reason about neighborhood models. This logic is based on an inquisitive strict conditional operator, which quantifies over neighborhoods, and which can be applied to both statements and questions. In terms of this operator we also define two unary modalities that function respectively as a universal and existential quantifier over neighborhoods. We prove that the expressive power of this logic matches the natural notion of bisimilarity in neighborhood models. We show that certain fragments of the language are invariant under certain modifications of the set of neighborhoods, and use this to show that our conditional modality is not definable from the induced unary modalities, and that questions embedded on the right of this conditional are indispensable. We provide a sound and complete axiomatization of our logic, both in general and in restriction to some salient frame classes, and discuss the relations between our logic and other modal logics interpreted over neighborhood models.

Inquisitive Neighborhood Logic

Abstract

We explore an inquisitive modal logic designed to reason about neighborhood models. This logic is based on an inquisitive strict conditional operator, which quantifies over neighborhoods, and which can be applied to both statements and questions. In terms of this operator we also define two unary modalities that function respectively as a universal and existential quantifier over neighborhoods. We prove that the expressive power of this logic matches the natural notion of bisimilarity in neighborhood models. We show that certain fragments of the language are invariant under certain modifications of the set of neighborhoods, and use this to show that our conditional modality is not definable from the induced unary modalities, and that questions embedded on the right of this conditional are indispensable. We provide a sound and complete axiomatization of our logic, both in general and in restriction to some salient frame classes, and discuss the relations between our logic and other modal logics interpreted over neighborhood models.

Paper Structure

This paper contains 20 sections, 32 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Propositional inquisitive logic
  4. Inquisitive modal logics
  5. Inquisitive neighborhood logic
  6. Syntax.
  7. Models.
  8. Semantics.
  9. $\textsf{InqML}_\diamondplus$.
  10. $\textsf{InqML}_\boxplus$.
  11. General case.
  12. Monotonicity properties of fragments
  13. Bisimilarity and expressive power
  14. Bisimilarity.
  15. Expressive power of $\textsf{InqML}_{\Rrightarrow}$.
  16. ...and 5 more sections

Key Result

Proposition 2.8

Every $\alpha\in\mathcal{L}_\mathcal{M}^!$ is truth-conditional. Moreover, every truth-conditional formula $\varphi\in\mathcal{L}_\mathcal{M}$ is equivalent to some declarative $\varphi^!\in\mathcal{L}_\mathcal{M}^!$.

Figures (2)

  • Figure 1: Sets of neighborhoods associated with three worlds.
  • Figure 2: Some conditions on neighborhood frames and corresponding canonical axioms. In the definition of the conditions, variables are implicitly understood to be universally quantified and $wR_\Sigma v$ is defined as $v\in\bigcup\Sigma(w)$. In the axioms, $\varphi$ and $\psi$ stand for arbitrary formulas, while $\alpha$ stands for an arbitrary declarative.

Theorems & Definitions (72)

  • Definition 2.1: Support for InqB
  • Definition 2.2: Truth at a world
  • Definition 2.3: Truth-set
  • Definition 2.4: Truth-conditional formulas
  • Definition 2.6: Declarative operators
  • Definition 2.7: Declaratives
  • Proposition 2.8
  • proof
  • Definition 2.9: Resolutions
  • Proposition 2.10: Normal form
  • ...and 62 more