Abstract
We investigate an enrichment of the propositional modal languageℒ with a “universal” modality ▪ having semanticsx ⊧ ▪ϕ iff āy(y ⊧ ϕ), 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(
) ofℒ, where
is an additional modality with the semanticsx ⊧
ϕ iff āy(y⊧ x → y ⊧ϕ). 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.) fromℒ to ℒc are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.
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Gargov, G., Goranko, V. Modal logic with names. J Philos Logic 22, 607–636 (1993). https://doi.org/10.1007/BF01054038
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DOI: https://doi.org/10.1007/BF01054038