Abstract
We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic.
Similar content being viewed by others
References
Anglberger, A.J.J., Gratzl, N., Roy, O. (2015). Obligation, free choice, and the logic of weakest permissions. The Review of Symbolic Logic, 8(4), 807–827.
Arisaka, R., Das, A., Straßburger, L. (2015). On nested sequents for constructive modal logics. Logical Methods in Computer Science, 11.
Askounis, D., Koutras, C.D., Zikos, Y. (2012). Knowledge means “all”, belief means “most”. In Logics in artificial intelligence (pp. 41–53): Springer.
Bellin, G., de Paiva, V., Ritter, E. (2001). Extended Curry-Howard correspondence for a basic constructive modal logic. In Proceedings of Methods for Modalities.
Bierman, G.M., & de Paiva, V. (2000). On an intuitionistic modal logic. Studia Logica, 65(3), 383–416.
Chellas, B.F. (1980). Modal logic: an introduction. Cambridge University Press.
Collinson, M.J., Hilken, B.P., Rydeheard, D.E. (1999). Semantics and proof theory of an intuitionistic modal sequent calculus. University of Manchester, Department of Computer Science, Technical Report Series.
Dalmonte, T., Olivetti, N., Negri, S. (2018). Non-normal modal logics: bi-neighbourhood semantics and its labelled calculi. In Proceedings of AiML 12 (pp. 159–178).
Fairtlough, M., & Mendler, M. (1997). Propositional lax logic. Information and Computation, 137(1), 1–33.
Fischer Servi, G. (1977). On modal logic with an intuitionistic base. Studia Logica, 36(3), 141–149.
Fischer Servi, G. (1980). Semantics for a class of intuitionistic modal calculi. In Italian studies in the philosophy of science (pp. 59–72): Springer.
Fitch, F.B. (1948). Intuitionistic modal logic with quantifiers. Portugaliae Mathematica, 7(2), 113–118.
Gabbay, D.M., Kurucz, A., Wolter, F., Zakharyaschev, M. (2003). Many-dimensional modal logics: theory and applications. In Studies in logic and the foundations of mathematics, Vol. 148: North Holland Publishing Company.
Galmiche, D., & Salhi, Y. (2018). Tree-sequent calculi and decision procedures for intuitionistic modal logics. Journal of Logic and Computation, 28(5), 967–989.
Galmiche, D., & Salhi, Y. (2010). Label-free natural deduction systems for intuitionistic and classical modal logics. Journal of Applied Non-Classical Logics, 20 (4), 373–421.
Gilbert, D.R., & Maffezioli, P. (2015). Modular sequent calculi for classical modal logics. Studia Logica, 103(1), 175–217.
Goldblatt, R.I. (1981). Grothendieck topology as geometric modality. Mathematical Logic Quarterly, 27(31-35), 495–529.
Hansson, S.O., & et al. (2013). The varieties of permission. In Gabbay, D. (Ed.) Handbook of deontic logic and normative systemsn (pp. 195–240): College Publications.
Heilala, S., & Pientka, B. (2007). Bidirectional decision procedures for the intuitionistic propositional modal logic IS4. In Proceedings of CADE.
Iemhoff, R. (2019). Uniform interpolation and the existence of sequent calculi. Annals of Pure and Applied Logic.
Indrzejczak, A. (2005). Sequent calculi for monotonic modal logics. Bulletin of the Section of logic, 34(3), 151–164.
Indrzejczak, A. (2011). Admissibility of cut in congruent modal logics. Logic and Logical Philosophy, 20(3), 189–203.
Kojima, K. (2012). Relational and neighborhood semantics for intuitionistic modal logic. Reports on Mathematical Logic, 47, 87–113.
Lavendhomme, R., & Lucas, T. (2000). Sequent calculi and decision procedures for weak modal systems. Studia Logica, 66, 121–145.
Lellmann, B., & Pimentel, E. (2019). Modularisation of sequent calculi for normal and non-normal modalities. ACM Transactions on Computational Logic (TOCL), 20 (2), 7.
Negri, S. (2017). Proof theory for non-normal modal logics: the neighbourhood formalism and basic results. IfColog Journal of Logics and their Applications, 4, 1241–1286.
Orlandelli, E. (2019). Sequent calculi and interpolation for non-normal logics. arXiv:1903.11342.
Marin, S., & Straßburger, L. (2014). Label-free modular systems for classical and intuitionistic modal logics. In Proceedings of AiML 10.
McNamara, P. (2006). Deontic logic. In Gabbay, & Woods (Eds.) Handbook of the history of logic, (Vol. 7 pp. 197–288): Elsevier.
Mendler, M., & de Paiva, V. (2005). Constructive CK for contexts. In Proceedings of CONTEXT05. Stanford.
Mendler, M., & Scheele, S. (2011). Cut-free Gentzen calculus for multimodal CK. Information and Computation, 209(12), 1465–1490.
Montague, R. (1970). Universal grammar. Theoria, 36(3), 373–398.
Pacuit, E. (2017). Neighborhood semantics for modal logic. Springer.
Scott, D. (1970). Advice in modal logic. In Lambert, K (Ed.) Philosophical problems in logic (pp. 143–173): D. Reidel Publishing Company.
Simpson, A.K. (1994). The proof theory and semantics of intuitionistic modal logic. PhD thesis, School of Informatics, University of Edinburgh.
Straßburger, L. (2013). Cut elimination in nested sequents for intuitionistic modal logics. In International conference on foundations of software science and computational structures (pp. 209–224): Springer.
Stewart, C., de Paiva, V., Alechina, N. (2018). Intuitionistic modal logic: a 15-year retrospective. Journal of Logic and Computation, 28(5), 873–882.
Troelstra, A.S., & Schwichtenberg, H. (2000). Basic proof theory. Cambridge University Press.
Wijesekera, D. (1990). Constructive modal logics I. Annals of Pure and Applied Logic, 50, 271–301.
Wijesekera, D., & Nerode, A. (2005). Tableaux for constructive concurrent dynamic logic. Annals of Pure and Applied Logic, 135(1-3), 1–72.
Witczak, T. (2018). Simple example of weak modal logic based on intuitionistic core. arXiv:1806.09443.
Wolter, F., & Zakharyaschev, M. (1999). Intuitionistic modal logic. In Logic and foundations of mathematics (pp. 227–238): Springer.
von Wright, G.H. (1963). Norm and action: a logical enquiry. Routledge.
Acknowledgments
We are grateful to the anonymous reviewers for their insightful remarks, suggestions, and constructive criticisms that helped us to improve a first version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was partially supported by the Project TICAMORE ANR-16-CE91-0002-01.
Rights and permissions
About this article
Cite this article
Dalmonte, T., Grellois, C. & Olivetti, N. Intuitionistic Non-normal Modal Logics: A General Framework. J Philos Logic 49, 833–882 (2020). https://doi.org/10.1007/s10992-019-09539-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-019-09539-3