Abstract
We look into methods which make it possible to determine whether or not the modal logics under examination are residually finite w.r.t. admissible inference rules. A general condition is specified which states that modal logics over K4 are not residually finite w.r.t. admissibility. It is shown that all modal logics λ over K4 of width strictly more than 2 which have the co-covering property fail to be residually finite w.r.t. admissible inference rules; in particular, such are K4, GL, K4.1, K4.2, S4.1, S4.2, and GL.2. It is proved that all logics λ over S4 of width at most 2, which are not sublogics of three special table logics, possess the property of being residually finite w.r.t. admissibility. A number of open questions are set up.
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REFERENCES
M. A. Dummett and E. J. Lemmon, “Modal logics between S4 and S5,” Z. Math. Log. Grund. Math., 5, 250–264 (1959).
R. A. Bull, “That all extensions of S4.3 have the finite model property,” Z. Math. Log. Grund. Math., 12, No. 4, 341–344 (1966).
K. Fine, “The logics containing S4.3,” Z. Math. Log. Grund. Math., 17, No. 4, 371–376 (1971).
D. Gabbay and D. de Jongh, “A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property,” J. Symb. Log., 39, No. 1, 67–78 (1974).
D. Gabbay, “Selective filtration in modal logics,” Theoria, 30, 323–330 (1970).
D. Gabbay, “A general filtration method for modal logics,” J. Philos. Log., 1, 29–34 (1972).
K. Segerberg, “Decidability of S4.1,” Theoria, 34, 7–20 (1968).
K. Fine, “Logics containing K4, I,” J. Symb. Log., 39, No. 1, 31–42 (1974).
K. Fine, “Logics containing K4, II,” J. Symb. Log., 50, No. 3, 619–651 (1985).
V. V. Rybakov, “Decidable non-compact extensions of S4,” Algebra Logika, 17, No. 2, 210–219 (1978).
V. V. Rybakov, “A modal analog for Glivenko's theorem and its applications,” Notre Dame J. Formal Logic, 33, No. 2, 244–248 (1992).
M. Kracht, “Splittings and the finite model property,” J. Symb. Log., 58, No. 1, 139–157 (1993).
F. Wolter, “The finite model property in tense logic,” J. Symb. Log., 60, No. 3, 757–774 (1995).
V. V. Rybakov, “A criterion for admissibility of rules in the modal system S4 and intuitionistic logic,” Algebra Logika, 23, No. 5, 546–572 (1984).
V. V. Rybakov, “Problems of admissibility and substitution, logical equations and restricted theories of free algebras,” in Logic Methodology and Philosophy of Science VIII, Stud. Log. Found. Math., Vol. 126, North. Holland, Amsterdam (1989), pp. 121–139.
V. V. Rybakov, “Problems of substitution and admissibility in the modal system Grz and intuitionistic calculus,” Ann. Pure Appl. Log., 50, 71–106 (1990).
V. V. Rybakov, “Rules of inference with parameters for intuitionistic logic,” J. Symb. Log., 57, No. 3, 912–923 (1992).
V. V. Rybakov, “Criteria for admissibility of inference rules. Modal and intermediate logics with the branching property,” Stud. Log., 53, No. 2, 203–226 (1994).
V. V. Rybakov, Admissibility of Logical Inference Rules, Stud. Log. Found. Math., Vol. 136, Elsevier, Amsterdam (1997).
A. Chagrov and M. Zakharyaschev, Modal Logics, Oxford Logic Studies, Vol. 35, Clarendon, Oxford (1997).
M. Kracht, “Internal definability and completeness in modal logic,” Ph. D., Free University of Berlin, Berlin (1991).
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Rybakov, V.V., Kiyatkin, V.R. & Oner, T. Residual Finiteness for Admissible Inference Rules. Algebra and Logic 40, 334–347 (2001). https://doi.org/10.1023/A:1012557903153
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DOI: https://doi.org/10.1023/A:1012557903153