A new version of an old modal incompleteness theorem

TL;DR

This paper demonstrates that a specific modal logic, previously known to be incomplete under Kripke and neighborhood semantics, is also incomplete under all classes of complete Boolean algebras with operators, making it completely incomplete.

Contribution

It proves the modal logic's incompleteness across all classes of complete Boolean algebras with operators, extending prior results to a broader semantic framework.

Findings

The logic is incomplete under Kripke semantics.

The logic is incomplete under neighborhood semantics.

The logic is completely incomplete across all classes of complete Boolean algebras.

Abstract

Thomason \cite{Thomason74} showed that a certain modal logic $\mathbf{L}\subset \mathbf{S4}$ is incomplete with respect to Kripke semantics. Later Gerson \cite{Gerson75} showed that $\mathbf{L}$ is also incomplete with respect to neighborhood semantics. In this paper we show that $\mathbf{L}$ is in fact incomplete with respect to any class of complete Boolean algebras with operators, i.e. that it is completely incomplete.