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Combining Monotone and Normal Modal Logic in Nested Sequents – with Countermodels

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

Abstract

We introduce nested sequent calculi for bimodal monotone modal logic, aka. Brown’s ability logic, a natural combination of non-normal monotone modal logic M and normal modal logic K. The calculus generalises in a natural way previously existing calculi for both mentioned logics, has syntactical cut elimination, and can be used to construct countermodels in the neighbourhood semantics. We then consider some extensions of interest for deontic logic. An implementation is also available.

Supported by WWTF project MA16-28.

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References

  1. Brown, M.A.: On the logic of ability. J. Philos. Log. 17(1), 1–26 (1988)

    Article  MathSciNet  Google Scholar 

  2. Brown, M.A.: Action and ability. J. Philos. Log. 19(1), 95–114 (1990)

    Article  MathSciNet  Google Scholar 

  3. Brünnler, K.: Deep sequent systems for modal logic. Arch. Math. Log. 48, 551–577 (2009)

    Article  MathSciNet  Google Scholar 

  4. Chellas, B.F.: Modal Logic. Cambridge University Press, Cambridge (1980)

    Book  Google Scholar 

  5. Ciabattoni, A., Gulisano, F., Lellmann, B.: Resolving conflicting obligations in Mīmāmsā: a sequent-based approach. In: Broersen, J., Condoravdi, C., Nair, S., Pigozzi, G. (eds.) DEON 2018 Proceedings, pp. 91–109. College Publications (2018)

    Google Scholar 

  6. Dalmonte, T., Olivetti, N., Negri, S.: Non-normal modal logics: bi-neighbourhood semantics and its labelled calculi. In: Bezanishvili, G., D’Agostino, G., Metcalfe, G., Studer, T. (eds.) Advances in Modal Logic 12, pp. 159–178. College Publications (2018)

    Google Scholar 

  7. Hansen, H.H.: Monotonic modal logics. University of Amsterdam, ILLC, Masters (2003)

    Google Scholar 

  8. Indrzejczak, A.: Sequent calculi for monotonic modal logics. Bull. Sect. Log. 34, 151–164 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Indrzejczak, A.: Labelled tableau calculi for weak modal logics. Bull. Sect. Log. 36(3/4), 159–171 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Kashima, R.: Cut-free sequent calculi for some tense logics. Studia Logica 53(1), 119–135 (1994)

    Article  MathSciNet  Google Scholar 

  11. Lavendhomme, R., Lucas, T.: Sequent calculi and decision procedures for weak modal systems. Studia Logica 65, 121–145 (2000)

    Article  MathSciNet  Google Scholar 

  12. Lellmann, B., Pattinson, D.: Constructing cut free sequent systems with context restrictions based on classical or intuitionistic logic. In: Lodaya, K. (ed.) ICLA 2013. LNCS, vol. 7750, pp. 148–160. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36039-8_14

    Chapter  MATH  Google Scholar 

  13. Lellmann, B., Pimentel, E.: Proof search in nested sequent calculi. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 558–574. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48899-7_39

    Chapter  Google Scholar 

  14. Lellmann, B., Pimentel, E.: Modularisation of sequent calculi for normal and non-normal modalities. ACM Trans. Comput. Logic 20(2), 7:1–7:46 (2019)

    Article  MathSciNet  Google Scholar 

  15. Marin, S., Straßburger, L.: Label-free modular systems for classical and intuitionistic modal logics. In: Goré, R., Kooi, B.P., Kurucz, A. (eds.) Advances in Modal Logic 10, pp. 387–406. College Publications (2014)

    Google Scholar 

  16. Negri, S.: Proof theory for non-normal modal logics. IfCoLog J. Log. Their Appl. 4(4), 1241–1286 (2017)

    Google Scholar 

  17. Pacuit, E.: Neighbourhood Semantics for Modal Logic. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-67149-9

    Book  MATH  Google Scholar 

  18. Poggiolesi, F.: The method of tree-hypersequents for modal propositional logic. In: Makinson, D., Malinkowski, J., Wansing, H. (eds.) Towards Mathematical Philosophy. Trends In Logic, vol. 28, pp. 31–51. Springer, Dordrecht (2009). https://doi.org/10.1007/978-1-4020-9084-4_3

    Chapter  Google Scholar 

  19. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge Tracts In Theoretical Computer Science, vol. 43, 2nd edn. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. TU Wien, Vienna, Austria

    Björn Lellmann

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  1. Björn Lellmann
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Correspondence to Björn Lellmann .

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Editors and Affiliations

  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

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Lellmann, B. (2019). Combining Monotone and Normal Modal Logic in Nested Sequents – with Countermodels. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_12

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  • DOI: https://doi.org/10.1007/978-3-030-29026-9_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-29025-2

  • Online ISBN: 978-3-030-29026-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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