zbMATH — the first resource for mathematics

On logics with coimplication. (English) Zbl 0976.03020
Summary: This paper investigates (modal) extensions of Heyting-Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gödel-embedding of intuitionistic logic into S4, it is shown that all (modal) extensions of Heyting-Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok-Esakia-Theorem is proved for this embedding.

MSC:
 03B45 Modal logic (including the logic of norms) 03B20 Subsystems of classical logic (including intuitionistic logic)
Full Text:
References:
 [1] Balbes, R. and Dwinger, Ph., Distributive Lattices, University of Missouri Press, 1974. [2] van Benthem, J., The Logic of Time, Reidel, Dordrecht, 1983. · Zbl 0508.03008 [3] van Benthem, J., Temporal logic. In: Gabbay, Hogger, and Robinson (eds), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 4, 1995, pp. 241–350. [4] Blok, W., Varieties of Interior Algebras, Dissertation, University of Amsterdam, 1976. [5] Blok, W. and Köhler, P., Algebraic semantics for quasi-classical modal logics, Journal of Symbolic Logic 48 (1983), 941–964. · Zbl 0562.03011 [6] Basic tense logic. In: D. Gabbay and F. Guenthner (eds), Handbook of Philosophical Logic, Vol. 2, 1984, pp. 89–133. · Zbl 0875.03046 [7] Bosic, M. and Došen, K., Models for normal intuitionistic modal logics, Studia Logica 43 (1984), 217–245. · Zbl 0634.03014 [8] Chagrov, A. V. and Zakharyaschev, M. V., Modal companions of intermediate propositional logics, Studia Logica 51 (1992), 49–82. · Zbl 0766.03015 [9] Chagrov, A. V. and Zakharyaschev, M. V., Modal and superintuitionistic logics, Oxford University Press, 1996. LOGISEG6.tex; 19/03/1998; 13:42; v.7; p.33 · Zbl 0856.03017 [10] Dummett, M. and Lemmon, E., Modal logics between S4 and S5, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 5 (1959), 250–264. · Zbl 0178.30801 [11] Esakia, L., On Varieties of Grzegorczyk Algebras, in Studies in Nonclassical Logics and Set Theory, Moscow, Nauka, 1979, pp. 257–287. · Zbl 0435.03012 [12] Fine, K., Logics containing K4, Part I, Journal of Symbolic Logic 39 (1974), 229–237. · Zbl 0287.02010 [13] Fine, K., Logics containing K4, Part II, Journal of Symbolic Logic 50 (1985), 619–651. · Zbl 0574.03008 [14] Fine, K. and Schurz, G., Transfer theorems for stratified modal logics, in Proceedings of the Arthur Prior Memorial Conference, Christchurch, New Zealand, 1991. · Zbl 0919.03015 [15] Fischer Servi, G., Semantics for a class of intuitionistic modal calculi. In: M. L. Dalla Chiara (ed.), Italian Studies in the Philosophy of Science, Reidel, Dordrecht, 1980, pp. 59–72. [16] Fischer Servi, G., Axiomatizations for some Intuitionistic Modal Logics, Rend. Sem. Mat. Univers. Polit. 42 (1984), 179–194. · Zbl 0592.03011 [17] Font, J., Modality and possibility in some intuitionistic modal logics, Notre Dame Journal of Formal Logic 27 (1986), 533–546. · Zbl 0638.03017 [18] Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines mathematischen Kolloquiums 6 (1933), 39–40. · Zbl 0007.19303 [19] Goldblatt, R., Metamathematics of Modal Logic, Reports on Mathematical Logic 6 (1976), 41–78, 7 (1976), 21–52. · Zbl 0356.02016 [20] Goldblatt, R., Logics of Time and Computation, Number 7 in CSLI Lecture Notes, CSLI, 1987. [21] Grzegorczyk, A., A philosophically plausible formal interpretation of intuitionistic logic, Indag. Math. 26, 596–601. · Zbl 0131.00701 [22] Köhler, R., A subdirectly irreducible double Heyting algebra which is not simple, Algebra Universalis 10 (1980), 189–194. · Zbl 0431.06015 [23] Kracht, M., Even more on the lattice of tense logics, Arch. Math. Logic 31 (1992), 243–257. · Zbl 0787.03013 [24] Kracht, M. and Wolter, F., Properties of independently axiomatizable bimodal logics, Journal of Symbolic Logic 56 (1991), 1469–1485. · Zbl 0743.03013 [25] Kripke, S., A semantical analysis of intuitionistic logic I. In: J. Crossley and M. Dummett (eds), Formal Systems and Recursive Functions, North-Holland, Amsterdam, 1965, pp. 92–129. · Zbl 0137.00702 [26] Makkai, M. and Reyes, G., Completeness results for intuitionistic and modal logics in a categorical setting, Annals of Pure ans Applied Logic 72 (1995), 25–101. · Zbl 0830.03036 [27] Maksimova, L. and Rybakov, V., Lattices of modal logics, Algebra and Logic 13 (1974), 105–122. · Zbl 0315.02027 [28] Ono, H., On some intuitionistic modal logics, Publ. Kyoto University 13 (1977), 687–722. · Zbl 0373.02026 [29] Rauszer, C., Semi-Boolean algebras and their applications to intuitionistic logic with dual operators, Fund. Math. 83 (1974), 219–249. · Zbl 0298.02064 [30] Rauszer, C., A formalization of propositional calculus of H-B logic, Studia Logica 33 (1974). · Zbl 0289.02015 [31] Rauszer, C., An algebraic and Kripke-style approach to a certain extension of intuitionistic logic, Dissertationes Mathematicae, vol. CLXVII, Warszawa, 1980. · Zbl 0442.03024 [32] Rautenberg, W., Klassische und Nichtklassische Aussagenlogik, Wiesbaden, 1979. · Zbl 0424.03007 [33] Segerberg, K., An Essay in Classical Modal Logic, Uppsala, 1971. · Zbl 0311.02028 [34] Segerberg, K., That every extension of S4.3 is normal. In: S. Kanger (ed.), Proceedings of the Third Scandinavian Logic Symposium, Amsterdam, 1976, pp. 194–196. [35] Thomason, S. K., Semantic analysis of tense logics, Journal of Symbolic Logic 37 (1972), 150–158. · Zbl 0238.02027 [36] Troelstra, A. and van Dalen, D., Constructivism in Mathematics, vol. I, North-Holland, Amsterdam, 1988. · Zbl 0661.03047 [37] Wojcicki, R., Theory of Logical Calculi, Dordrecht, 1988. [38] Wolter, F., The finite model property in tense logic, Journal of Symbolic Logic 60 (1995), 757–774. · Zbl 0836.03015 [39] Wolter, F., Superintuitionistic companions of classical modal logics, Studia Logica 58 (1997), 229–259. · Zbl 0952.03020 [40] Wolter, F., Completeness and decidability of tense logics closely related to logics containing K4, Journal of Symbolic Logic 62 (1997), 131–158. · Zbl 0893.03005 [41] Wolter, F. and Zakharyaschev, M., Intuitionistic modal logics as fragments of classical bimodal logics, in logic at work, Essays in honour of H. Rasiowa, forthcoming. · Zbl 0922.03023 [42] Wolter, F. and Zakharyaschev, M., On the relation between intuitionistic and classical modal logics, to appear in Algebra and Logic, 1996. [43] Zakharyaschev, M., Canonical Formulas for K4, Part II, to appear in Journal of Symbolic Logic, 1996. · Zbl 0884.03014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.