nLab
maximal consistent formulae

Λ\Lambda-maximal consistent formulae

Definition

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

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

Important

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

Canonical frame

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

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

R iΓΔiff iΔΓ.R_i \Gamma\Delta \;iff\; \Diamond_i\Delta \subseteq \Gamma.

Revised on February 24, 2014 13:17:25 by Toby Bartels (64.89.53.235)