On the proof theory of infinitary modal logic

The article deals with infinitary modal logic. We first discuss the difficulties related to the development of a satisfactory proof theory and then we show how to overcome these problems by introducing a labelled sequent calculus which is sound and complete with respect to Kripke semantics. We establish the structural properties of the system, namely admissibility of the structural rules and of the cut rule. Finally, we show how to embed common knowledge in the infinitary calculus and we discuss first-order extensions of infinitary modal logic

Infinitary Modal Logic and Generalized Kripke Semantics

This paper deals with the infinitary modal propositional logic Kω1, featuring countable disjunctions and conjunc- tions. It is known that the natural infinitary extension LKω1 (here presented as a Tait-style calculus, TK♯ω1 ) of the standard sequent calculus LKp 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 LKω1 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