Infinitary Modal Logic and Generalized Kripke Semantics

Abstract

This paper deals with the infinitary modal propositional logic Kω1, featuring countable disjunctions and conjunc- tions. It is known that the natural infinitary extension LK􏰀ω1 (here presented as a Tait-style calculus, TK♯ω1 ) of the standard sequent calculus LK􏰀p for the propositional modal logic K is incomplete w.r. to Kripke semantics. It is also known that in order to axiomatize Kω1 one has to add to LK􏰀ω1 new initial sequents corresponding to the infinitary propositional counterpart BFω1 of the Barcan- formula. We introduce a generalization of Kripke seman- tics, and prove that TK♯ω1 is sound and complete w.r. to this generalized semantics. By the same proof strategy, we show that the stronger system TKω1 , allowing countably infinite sequents, axiomatizes Kω1, although it provably doesn’t admit cut-elimination.