One Connection between Standard Invariance Conditions on Modal Formulas and Generalized Quantifiers

Abstract

The language of standard propositional modal logic has one operator (□ or ♦), that can be thought of as being determined by the quantifiers ∀ or ∃, respectively: for example, a formula of the form □Φ is true at a point s just in case all the immediate successors of s verify Φ.This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of ∀ and ∃.