Bimodal logics with contingency and accident

TL;DR

This paper introduces a bimodal logic framework for contingency and accident, exploring their interaction, model-theoretic properties, and extending to dynamic public announcement logic with applications to Moore sentences.

Contribution

It is the first to unify contingency and accident in a bimodal logic, providing axiomatizations, completeness proofs, and dynamic extensions.

Findings

Logic captures interaction between contingency and accident

Complete axiomatizations over various frame classes

Dynamic extension with public announcements and Moore sentences

Abstract

Contingency and accident are two important notions in philosophy and philosophical logic. Their meanings are so close that they are mixed sometimes, in both everyday discourse and academic research. This indicates that it is necessary to study them in a unified framework. However, there has been no logical research on them together. In this paper, we propose a language of a bimodal logic with these two concepts, investigate its model-theoretical properties such as expressivity and frame definability. We axiomatize this logic over various classes of frames, whose completeness proofs are shown with the help of a crucial schema. The interactions between contingency and accident can sharpen our understanding of both notions. Then we extend the logic to a dynamic case: public announcements. By finding the required reduction axioms, we obtain a complete axiomatization, which gives us a good application to Moore sentences.