On classical behavior of intuitionistic modalities
DOI:
https://doi.org/10.12775/LLP.2014.019Słowa kluczowe
intuitionistic logic, modal logics, intuitionistic modalities, compositions, possibility, necessity, un-necessity, impossibilityAbstrakt
We study connections between four types of modal operators – necessity, possibility, un-necessity and impossibility – over intuitionitstic logic in terms of compositions of these modal operators with intuitionistic negation. We investigate which basic compositions, i.e. compositions of the form ¬δ, δ¬ or ¬δ¬, yield modal operators of the same type over intuitionistic logic as over classical logic. We say that such compositions behave classically. We study which modal properties correspond to each basic compositions behaving classically over intuitionistic logic and also prove that KC constitutes the smallest superintuitionistic logic over which all basic compositions behave classically.Bibliografia
Blackburn P., and J. van Benthem, “Modal logic: A semantic perspective”, pages 1–84 in Handbook of Modal Logic. Volume 3, P. Blackburn, J. van Benthem, and F. Wolter (eds.), series “Studies in Logic and Practical Reasoning”, Elsevier Science Inc., New York, 2006. DOI: 10.1016/S1570-2464(07)80004-8
Božić, M., and K. Došen, “Models for normal intuitionistic modal logics”, Studia Logica 43 (1984): 217–245.
Cabalar, P., S.P. Odintsov, and D. Pearce, “Logical foundations of well-founded semantics”, pages 25–36 in Principles of Knowledge Representation and Reasoning: Proceedings of the 10th International Conference (KR2006), P. Doherty et al (eds.), AAAI Press, Menlo Park, California, 2006.
Došen, K., “Negative modal operators in intuitionistic logic”, Publication de l’Instutute Mathematique, Nouv. Ser., 35 (1984): 3–14.
Došen, K., “Models for stronger normal intuitionisticmodal logics”, Studia Logica 44 (1985): 39–70.
Drobyshevich, S.A., “A double negation operator in logic N*”, Vestnik NSU, Series: Math., mech. and informatics, 13, 4 (2013): 68–83.
Drobyshevich, S.A., “Composition of an intuitionistic negation and negative modalities as a necessity operator”, Algebra and Logic, 52, 3 (2013): 305–331. DOI: 10.1007/s10469-013-9235-8
Drobyshevich, S.A., and S.P. Odintsov, “Finite model property for negative modalities” (in Russian), Siberian Electronic Mathematical Reports, 10 (2013): 1–21.
Fischer-Servi, G., “On modal logics with an intuitionistic base”, Studia Logica 36 (1977): 141–149. DOI: 10.1007/BF02121259
Fischer-Servi, G., “Semantics for a class of intuitionistic modal calculi”, pages 59–72 in Italian Studies in the Philosophy of Science, M.L. Dalla Chiara (ed.), Reidel, Dordrecht, 1980.
Fischer-Servi, G., “Axiomatizations for some intuitionistic modal logics”, Rend. Sem. Mat. Univers. Polit., 42 (1984): 179–194.
Font, J., “Modaility and possibility in some intuitionistic modal logics”, Notre Dame Journal of Formal Logic 27, 4 (1986): 533–546.
Gabbay, D.M., and L.L. Maksimova, Interpolation and Definability: Modal and Intuitionistic Logics, Claeendon press, Oxford, 2005.
Maksimova, L.L., “Craig’s theorem in superintuitionistic logics and amalgamated varieties of pseudoboolean algebras”, Algebra and Logic, 16, 6 (1977): 427-455.
Mihajlova, M., “Reduction of modalities in several intuitionistic modal logics”, Comptes rendus de l’Académie Bulgaire des Sciences 33 (1980): 743–745.
Segerberg, K., “Propositional logics related to Heyting’s and Johansson’s”, Theoria, 34, 1 (1968): 26–61. DOI:10.1111/j.1755-2567.1968.tb00337.x
Sotirov, V., “Modal theories with intuitionistic logic”, pages 139–171 in Mathematical Logic, Proc. of the Conference Dedicated to the memory of A.A. Markov (1903–1979), Sofia, September 22-23, Bulgarian Acad. Of Sc., 1984.
Vakarelov, D., “Consistency, completeness and negation”, pages 328–363 in Paraconsistent Logics: Essays on the Inconsistent, G. Priest, R. Routley, and J. Norman (eds.), Filosophia, 1989.
Wolter, F., and M. Zakharaschev, “On the relation between intuitionistic and classical modal logics”, Algebra and Logic, 36, 2 (1997): 121–155. DOI: 10.1007/BF02672476
Wolter, F., and M. Zakharaschev, “Intuitionistic modal logics”, pages 227–238 in Logical Foundations of Mathematics, A. Cantini, E. Casari, and P. Minari (eds.), Synthese Library, Kluwer, 1999.
Yankov, V.A., “The calculus of weak law of excluded middle” (in Russian), Izv. Ak. Nauk SSSR Ser. Mat., 32, 5 (1968): 1044–1051.
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