nLab maximal consistent formulae

Definition

Given a normal modal logic, $\Lambda$, a set, $\Gamma$, of formulae is said to be $\Lambda$-consistent if $\neg(\Gamma\vdash_\Lambda \bot)$, i.e., $\bot$ is not deducible from $\Gamma$.

A set, $\Gamma$, of formulae is said to be $\Lambda$-maximal if it is consistent and, for any $\phi \in \mathcal{L}_\omega(n)$ either $\phi\in \Gamma$ or $\neg\phi\in \Gamma$.

Important

If $\Gamma$ is a $\Lambda$-maximal set of formulae, then within the Lindenbaum-Tarski algebra, $\mathfrak{A}^\Lambda_\omega$, the set $x_\Lambda = \{{\|\phi\|} \mid \phi \in \Lambda\}$ is an ultrafilter.

Canonical frame

Let $S^\Lambda_\omega = \{\Gamma \mid \Gamma \in \Lambda maximal\}$, then $\Gamma\leftrightarrow x_\Gamma$ is a bijection between $S^\Lambda_\omega$ and the set of ultrafilters of $\mathfrak{A}^\Lambda_\omega$.

This set forms the set of states / worlds for the canonical frame of $\Lambda$. The relations are given by

$$R_i \Gamma\Delta \;iff\; \Diamond_i\Delta \subseteq \Gamma.$$