Some Remarks on the Proof-Theory and the Semantics of Infinitary Modal Logic

Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 28)


We investigate the (multiagent) infinitary version \(\mathbf {K}_{\omega _1}\) of the propositional modal logic \(\mathbf {K}\), in which conjunctions and disjunctions over countably infinite sets of formulas are allowed. It is known that the natural infinitary extension \(\mathbf {LK}_{\omega _1}^{\Box }\) (here presented as a Tait-style calculus, \(\mathbf {TK}^{\sharp }_{\omega _1}\)) of the standard sequent calculus \(\mathbf {LK}_p^{\Box }\) for \(\mathbf {K}\) is incomplete with respect to Kripke semantics. It is also known that in order to axiomatize \(\mathbf {K}_{\omega _1}\) one has to add to \(\mathbf {LK}_{\omega _1}^{\Box }\) new initial sequents corresponding to the infinitary propositional counterpart \( BF _{\omega _1}\) of the Barcan Formula. We introduce a generalization of standard Kripke semantics, and prove that \(\mathbf {TK}^{\sharp }_{\omega _1}\) is sound and complete with respect to it. By the same proof strategy, we show that the stronger system \(\mathbf {TK}_{\omega _1}\), allowing countably infinite sequents, axiomatizes \(\mathbf {K}_{\omega _1}\), although it provably does not admit cut-elimination.


Modal logic Infinitary logic Kripke semantics Tait-style calculi Cut-elimination 



I wish to thank an anonymous referee for helpful comments and suggestions.


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Section of Philosophy, Department of Letters and PhilosophyUniversity of FlorenceFirenzeItaly

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