A new version of an old modal incompleteness theorem

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.