A New Modal Framework for Epistemic Logic

TL;DR

This paper introduces a versatile modal framework for epistemic logic that incorporates existential quantification within modalities, enhancing expressiveness while maintaining desirable logical properties.

Contribution

It proposes a general, expressive modal framework with a new modality $ox^x$ that captures existential quantification, extending standard epistemic logic.

Findings

The framework retains properties like finite-tree-model property.

It provides a van Benthem-like characterization relative to first-order modal logic.

The logic over S5 frames is axiomatized with intuitive axioms.

Abstract

Recent years witnessed a growing interest in non-standard epistemic logics of knowing whether, knowing how, knowing what, knowing why and so on. The new epistemic modalities introduced in those logics all share, in their semantics, the general schema of $\exists x \Box \phi$, e.g., knowing how to achieve $\phi$ roughly means that there exists a way such that you know that it is a way to ensure that $\phi$. Moreover, the resulting logics are decidable. Inspired by those particular logics, in this work, we propose a very general and powerful framework based on quantifier-free predicate language extended by a new modality $\Box^x$, which packs exactly $\exists x \Box$ together. We show that the resulting language, though much more expressive, shares many good properties of the basic propositional modal logic over arbitrary models, such as finite-tree-model property and van Benthem-like characterization w.r.t.\ first-order modal logic. We axiomatize the logic over S5 frames with intuitive axioms to capture the interaction between $\Box^x$ and know-that operator in an epistemic setting.