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

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

## Abstract

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.

## Keywords

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

## References

1. 1.
K. Brünnler, Deep sequent systems for modal logic. Arch. Math. Logic 48, 551–577 (2009)
2. 2.
E. Calardo, A. Rotolo, Variants of multi-relational semantics for propositional non-normal modal logics. J. Appl. Non-Classical Logics 24, 293–320 (2014)
3. 3.
R. Fagin, J.Y. Halpern, Y. Moses, M.Y. Vardi, Reasoning About Knowledge (The MIT Press, Cambridge, 1995)
4. 4.
S. Feferman, Lectures on proof theory, in Proceedings of the Summer School in Logic, Leeds 1967, vol. 70, Lecture Notes in Mathematics, ed. by M.H. Löb (Springer, Berlin, 1967), pp. 1–107
5. 5.
L. Goble, Multiplex semantics for deontic logic. Nordic J. Philos. Logic 5, 113–134 (2000)
6. 6.
N. Kamide, Embedding linear-time temporal logic into infinitary logic: application to cut-elimination for multi-agent infinitary epistemic linear-time temporal logic, in Computational Logics in Multi-Agent Systems, vol. 5405, Lecture Notes in Artificial Intelligence, ed. by M. Fisher, F. Sadri, M. Thielscher (Springer, Berlin, 2009), pp. 57–76
7. 7.
E.G.K. Lopez-Escobar, An interpolation theorem for denumerably long formulas. Fundam. Math. 57, 253–272 (1965)
8. 8.
E.G.K. Lopez-Escobar, Remarks on an infinitary language with constructive formulas. J. Symb. Logic 32, 305–318 (1967)
9. 9.
P. Minari, Labeled sequent calculi for modal logics and implicit contractions. Arch. Math. Logic 52, 881–907 (2013)
10. 10.
S. Negri, Proof analysis in modal logic. J. Philos. Logic 34, 507–544 (2005)
11. 11.
M. Ohnishi, K. Matsumoto, Gentzen method in modal calculi. Osaka Math. J. 9, 113–130 (1957)
12. 12.
S. Radev, Infinitary propositional normal modal logic. Studia Logica 46, 291–309 (1987)
13. 13.
P. Schotch, R. Jennings, Non-kripkean deontic logic, in New Studies in Deontic Logic, ed. by R. Hilpinen (Reidel, Dordrecht, 1981), pp. 149–162
14. 14.
W.W. Tait, Normal derivability in classical logic, in The Syntax and Semantics of Infinitary Languages, vol. 72, Lecture Notes in Mathematics, ed. by J. Barwise (Springer, Berlin, 1968), pp. 204–236
15. 15.
Y. Tanaka, Kripke completeness of infinitary predicate multimodal logics. Notre Dame J. Formal Logic 40, 327–339 (1999)
16. 16.
Y. Tanaka, Cut-elimination theorems for some infinitary modal logics. Math. Logic Q. 47, 326–340 (2001)
17. 17.
Y. Tanaka, H. Ono, Rasiowa-Sikorski lemma and Kripke completeness of predicate and infinitary modal logics, in Advances in Modal Logic 98, vol. 2, ed. by M. Zakharyaschev, K. Segerberg, M. de Rijk, H. Wansing (CSLI Publications, Stanford, 2000), pp. 419–437Google Scholar