Actuality in Propositional Modal Logic

Abstract

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood mod- els). Specifically, we prove that for every formula φ in the propositional modal language with A, there is a formula ψ not containing A such that φ and ψ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality operator in the actuality extension of any normal propositional modal logic and of any "classical" modal logic. As an application, we provide an alternative proof of a result of Williamson's to the effect that the compound operator A behaves, in any normal logic between T and S5, like the simple necessity operator in S5.