On algebraic and topological semantics of the modal logic of common knowledge S4CI

TL;DR

This paper explores algebraic and topological semantics for the modal logic S4CI, establishing strong completeness results and a Stone-type representation theorem for certain algebraic structures.

Contribution

It introduces a comprehensive semantic framework for S4CI, including infinitary extensions and a characterization of completable algebras with a representation theorem.

Findings

Strong completeness in local semantics

Strong completeness in global semantics with infinitary extensions

Stone-type representation theorem for S4CI-algebras

Abstract

We investigate algebraic and topological semantics of the modal logic S4CI and obtain strong completeness of the given system in the case of local semantic consequence relations. In addition, we consider an extension of the logic S4CI with certain infinitary derivations and establish strong completeness results for the obtained system in the case of global semantic consequence relations. Furthermore, we identify the class of completable S4CI-algebras and obtain for them a Stone-type representation theorem.