Logics of Strong Noncontingency

TL;DR

This paper introduces a novel modal logic with a strong noncontingency operator, analyzing its expressivity, frame definability, bisimulation, and axiomatization, expanding understanding of noncontingency modalities.

Contribution

It proposes the first logic with a strong noncontingency operator as the primitive modality, and provides its expressivity comparison, bisimulation, and axiomatization.

Findings

The logic is not a normal modal logic due to invalid distribution axiom.

The logic's expressivity is compared with standard modal and noncontingency logics.

A bisimulation notion is developed and used to characterize the logic within modal and first-order frameworks.

Abstract

Inspired by Hintikka's treatment of question embedding verbs in [8] and the variations of noncontingency operator, we propose a logic with strong noncontingency operator $\blacktriangle$ as the only primitive modality. A proposition is strongly noncontingent, if no matter whether it is true or false, it does it necessarily; otherwise, it is weakly contingent. This logic is not a normal modal logic, since $\blacktriangle(\phi\to\psi)\to(\blacktriangle\phi\to\blacktriangle\psi)$ is invalid. We compare the relative expressivity of this logic and other logics, such as standard modal logic, noncontingency logic, and logic of essence and accident, and investigate its frame definability. Apart from these results, we also propose a suitable notion of bisimulation for the logic of strong noncontingency, based on which we characterize this logic within modal logic and within first-order logic. We also axiomatize the logic of strong noncontingency over various frame classes. Our work is also related to the treatment of agreement operator in [10].