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Prucnal proved, see [8] and [9], that Medvedev's logic ML of finite problems is structurally complete. So far, ML is the only known structurally complete intermediate logic with the disjunction property. Another proof of Prucnal's theorem was given by D. Skvortsov [10]. We refresh Prucnal's original argument and prove that all structurally complete intermediate logics with the disjunction property coincide on monadic formulas. The logic are also provided with appropriate Kripke frames.
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69--86
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- Institute of Mathematics, Silesian University, Bankowa 14, Katowice 40-007, wojtylak@ux2.math.us.edu.pl
Bibliografia
- [1] A. Diego, Les algebres de Hilbert, Gauthier-Villars Paris, 1966.
- [2] H. Friedman, One hundred and two problems in mathematical logic, Journal of Symbolic Logic, 40, 2 (1975), 113-129.
- [3] L.A. Levin, Some syntactical theorems on the Yu. T. Medvedev’s calculus of finite problems, Doklady Academii Nauk SSSR (Soviet Mathematics, Doklady), 185, 1 (1969), 32-33 (in Russian).
- [4] L.L. Maximova, On maximal intermediate logics with the disjunction property, Studia Logica, 45 (1986), 69-75
- [5] L.L. Maximova, D.P. Skvortsov, V.B. Sehtman, Impossibility of finite axiomatization of Medvedev’s logic of finite problems, Doklady Academii Nauk SSSR (Soviet Mathematics, Doklady), 245 (1979), 1051-1054 (in Russian).
- [6] P. Minari, A. Wroński, The property (HD) in intermediate logics. A partial solution of a problem of H. Ono, Reports on Mathematical Logic, 22 (1988), 21-25.
- [7] W.A. Pogorzelski, Structural completeness of the propositional calculus Bulletin de l’Academie Polonaise des Sciences, ser. math., astr. et phis., 19, 5 (1971), 349-351.
- [8] T. Prucnal, Structural completeness of Medvedev’s propositional calculus, Reports on Mathematical Logic, 6 (1976), 103-105.
- [9] T. Prucnal, On two problems of Harvey Friedman, Studia Logica, 38, 3 (1979), 247-262.
- [10] D.P. Skvortsov, On Prucnal’s theorem, in Logic at Work. Essays dedicated to the memory of Helena Rasiowa, ed. Orlowska E., A Springer-Verlag Company, Heidelberg (1999), 222-228.
- [11] M. Szatkowski, On fragments of Medvedev’s logic, Studia Logica 40, 1 (1981), 39-54.
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bwmeta1.element.baztech-article-BUJ1-0019-0089