Modalities in non-classical variations of $S4$

Abstract

A classical result in modal logic states that $S4$ has $14$ modalities, that is, every sequence of negations, boxes, and diamonds is equivalent to one in a set of $14$ such sequences. We study analogous results for the non-classical analogues $CS4$, $IS4$, $GS4$, and $\mathsf{GS}{4}^{\mathsf{c}}$ of $S4$. First, we show that, while all these logics have finitely many $\{\square ,◊\}$- and $\{\neg ,\square \}$-modalities, the logic $CS4$ has infinitely many $\{\neg ,◊\}$-modalities. Second, we show that $IS4$ and $GS4$ have finitely many $\{\neg ,◊\}$-modalities, but they have infinitely many $\{\neg ,\square ,◊\}$-modalities. At last, we show that $\mathsf{GS}{4}^{\mathsf{c}}$ has finitely many $\{\neg ,\square ,◊\}$-modalities.