Modal logic with names

Abstract

We investigate an enrichment of the propositional modal language ℒ with a "universal" modality {black small square} having semantics x {models} {black small square}φ{symbol} iff āy(y {models} φ{symbol}), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒc proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ() of ℒ, where is an additional modality with the semantics x {models}φ{symbol} iff āy(y{models} x → y {models}φ{symbol}). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒc. Strong completeness of the normal ℒc-logics is proved with respect to models in which all worlds are named. Every ℒc-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from ℒ to ℒcare discussed. Finally, further perspectives for names in multimodal environment are briefly sketched. © 1993 Kluwer Academic Publishers.