Abstract
We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness syntactically fails due to the hierarchical structure of nested sequents. Consequently, we only provide the semantic proof of soundness. The countermodel construction used to prove completeness required a completely novel approach to deal with two independent sources of non-termination in the proof search present in the case of transitive and Euclidean logics.
Article PDF
Similar content being viewed by others
References
von Plato, J.: Saved from the Cellar: Gerhard Gentzen’s Shorthand Notes on Logic and Foundations of Mathematics. Sources and Studies in the History of Mathematics and Physical Sciences. Springer (2017). https://doi.org/10.1007/978-3-319-42120-9
Gentzen, G.: Untersuchungen über das logische Schließen. I. Mathematische Zeitschrift 39(1), 176–210 (1935). https://doi.org/10.1007/BF01201353
Maehara, S.: Eine Darstellung der intuitionistischen Logik in der klassischen. Nagoya Math. J. 7, 45–64 (1954). https://doi.org/10.1017/S0027763000018055
Dyckhoff, R.: Intuitionistic decision procedures since Gentzen. In: Advances in Proof Theory, volume 28 of Progress in Computer Science and Applied Logic, pp. 245–267. Springer (2016). https://doi.org/10.1007/978-3-319-29198-7_6
Fitting, M.C.: Intuitionistic Logic, Model Theory and Forcing, volume 54 of Studies in Logic and the Foundations of Mathematics. North-Holland (1969). http://www.sciencedirect.com/science/bookseries/0049237X/54
Beth, E.W.: The Foundations of Mathematics: A Study in the Philosophy of Science, volume 25 of Studies in Logic and the Foundations of Mathematics. Harper & Row (1959). http://www.sciencedirect.com/science/bookseries/0049237X/25
Curry, H. B.: Foundations of Mathematical Logic. Dover, Dover edition, 1977. First edition published by McGraw–Hill in (1963)
Egly, U., Schmitt, S.: On intuitionistic proof transformations, their complexity, and application to constructive program synthesis. Fundam. Infor. 39(1,2), 59–83 (1999). https://doi.org/10.3233/FI-1999-391204
Simpson, A.K.: The Proof Theory and Semantics of Intuitionistic Modal Logic. Ph.D. thesis, University of Edinburgh (1994). http://hdl.handle.net/1842/407
Brünnler, K.: Deep sequent systems for modal logic. In: Governatori, G., Hodkinson, I., Venema, Y. (eds.) Advances in Modal Logic, Volume 6, pp. 107–119. College Publications (2006). http://www.aiml.net/volumes/volume6/Bruennler.ps
Brünnler, K.: Deep sequent systems for modal logic. Arch. Math. Log. 48(6), 551–577 (2009). https://doi.org/10.1007/s00153-009-0137-3
Poggiolesi, F.: The method of tree-hypersequents for modal propositional logic. In: Makinson, D., Malinowski, J., Wansing, H. (eds.), Towards Mathematical Philosophy, volume 28 of Trends in Logic, pp 31–51. Springer (2009). https://doi.org/10.1007/978-1-4020-9084-4_3
Straßburger, L.: Cut elimination in nested sequents for intuitionistic modal logics. In.: Pfenning, F. (ed.), Foundations of Software Science and Computation Structures, 16th International Conference, FOSSACS 2013, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2013, Rome, Italy, March 16–24, 2013, Proceedings, volume 7794 of Lecture Notes in Computer Science, pp. 209–224. Springer (2013). https://doi.org/10.1007/978-3-642-37075-5_14
Fitting, M.: Prefixed tableaus and nested sequents. Ann. Pure Appl. Logic 163(3), 291–313 (2012). https://doi.org/10.1016/j.apal.2011.09.004
Marin, S., Straßburger, L.: Label-free modular systems for classical and intuitionistic modal logics. In: Goré, R., Kooi, B., Kurucz, A. (eds.) Advances in Modal Logic, Volume 10, pp. 387–406. College Publications (2014). http://www.aiml.net/volumes/volume10/Marin-Strassburger.pdf
Chaudhuri, K., Marin, S., Straßburger, L.: Modular focused proof systems for intuitionistic modal logics. In: Kesner, D., Pientka, B. (eds.) 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016), vol 52, pp. 16:1–16:8. Schloss Dagstuhl (2016). https://doi.org/10.4230/LIPIcs.FSCD.2016.16
Galmiche, D., Salhi, Y.: Label-free natural deduction systems for intuitionistic and classical modal logics. J. Appl. Non Class. Log. 20(4), 373–421 (2010). https://doi.org/10.3166/jancl.20.373-421
Mendler, M., Scheele, S.: Cut-free Gentzen calculus for multimodal CK. Inf. Comput. 209(12), 1465–1490 (2011). https://doi.org/10.1016/j.ic.2011.10.003
Došen, K.: Higher-level sequent-systems for intuitionistic modal logic. Publications de l’Institut Mathématique (N.S.), 39(53):3–12 (1986). http://www.komunikacija.org.rs/komunikacija/casopisi/publication/53/d001/show_download
Fischer Servi, G.: Axiomatizations for some intuitionistic modal logics. Rendiconti del Seminario Matematico, Università e Politecnico di Torino, 42(3):179–194 (1984). http://www.seminariomatematico.unito.it/rendiconti/cartaceo/42-3.html
Plotkin, G., Stirling, C.: A framework for intuitionistic modal logics. In: Halpern, J. (ed.) Theoretical Aspects of Reasoning About Knowledge, pp. 399–406 (1986). Extended abstract. http://www.tark.org/proceedings/tark_mar19_86/p399-plotkin.pdf
Fitch, F.B.: Intuitionistic modal logic with quantifiers. Port. Math. 7(2):113–118 (1948). http://purl.pt/2174
Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Dover, Dover edition, 2006. Originally published as a Ph.D. thesis by Almquist & Wiksell in (1965)
Bierman, G.M., de Paiva, V.C.V.: On an intuitionistic modal logic. Stud. Log. 65(3), 383–416 (2000). https://doi.org/10.1023/A:1005291931660
Pfenning, F., Davies, R.: A judgmental reconstruction of modal logic. Math. Struct. Comput. Sci. 11(4), 511–540 (2001). https://doi.org/10.1017/S0960129501003322
Alechina, N., Mendler, M., de Paiva, V., Ritter, E.: Categorical and Kripke semantics for constructive S4 modal logic. In: Fribourg, L. (ed.) Computer Science Logic, 15th International Workshop, CSL 2001, 10th Annual Conference of the EACSL, Paris, France, September 10–13, 2001, Proceedings, volume 2142 of Lecture Notes in Computer Science, pp. 292–307. Springer (2001). https://doi.org/10.1007/3-540-44802-0_21
Arisaka, R., Das, A., Straßburger, L.: On nested sequents for constructive modal logic. Log. Methods Comput. Sci. 11(3:7), pp. 1–33 (2015). https://doi.org/10.2168/LMCS-11(3:7)2015
Garson, J.: Modal logic. In: Zalta, E.N. (ed.), The Stanford Encyclopedia of Philosophy. Stanford University, Spring 2016 edition (2016). https://plato.stanford.edu/archives/spr2016/entries/logic-modal/
Ewald, W.B.: Intuitionistic tense and modal logic. J. Symb. Log. 51(1), 166–179 (1986). https://doi.org/10.2307/2273953
Marin, S.: Modal proof theory through a focused telescope. Ph.D. thesis, Université Paris-Saclay & École Polytechnique (2018)
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). We would like to thank the anonymous reviewer for useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the ANR/FWF project “FISP” and by the PHC/BMWFW Amadeus project “Analytic Calculi for Modal Logics”. The first author was supported by the Austrian Science Fund (FWF) Grants P25417-G15 and S11405 (RiSE).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kuznets, R., Straßburger, L. Maehara-style modal nested calculi. Arch. Math. Logic 58, 359–385 (2019). https://doi.org/10.1007/s00153-018-0636-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-018-0636-1
Keywords
- Proof theory
- Sequent calculus
- Nested sequents
- Modal logic
- Intuitionistic logic
- Cut elimination
- Multiple conclusion
- Intuitionistic modal logic