Summary
Infinitely many intermediate propositional logics with the disjunction property are defined, each logic being characterized both in terms of a finite axiomatization and in terms of a Kripke semantics with the finite model property. The completeness theorems are used to prove that any two logics are constructively incompatible. As a consequence, one deduces that there are infinitely many maximal intermediate propositional logics with the disjunction property.
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References
Anderson, J.G.: Superconstructive propositional calculi with extra schemes containing one variable. Z. Math. Logik Grundlagen Math.18, 113–130 (1972)
Bertoni, A., Mauri, G., Miglioli, P., Ornaghi, M.: Abstract data types and their extensions within a constructive logic. In: Semantics of data types (Lect. Notes Comput. Sci., vol. 173, pp. 177–195) Berlin Heidelberg New York: Springer 1984
Gabbay, D.M.: The decidability of the Kreisel-Putnam system. J. Symb. Logic35, 431–437 (1970)
Gabbay, D.M.: Semantical investigation in Heytings's intuitionistic logic. Reidel: Dordrecht 1981
Gabbay, D.M., De Jongh, D.H.J.: A sequence of decidable finitely axiomatizable intermediate logics with the disiunction property. J. Symb. Logic39, 67–78 (1974)
Jankov, V.A.: Constructing a sequence of strongly independent superintuitionistic propositional calculi. Sov. Math., Dokl.9, 806–807 (1968)
Kirk, R.E.: A result on propositional logics having the disjunction property. Notre Dame J. Formal Logic23, 71–74 (1982)
Kreisel, G., Putnam, H.: Eine Unableitbarkeitsbeweismethode für den intuitionistischen Aussagenkalkül. Arch. Math. Logik Grundlagenforsch.3, 74–78 (1957)
Levin, L.A.: Some syntactic theorems on the calculus of finite problems of Ju. T. Medvedev. Sov. Math., Dokl.10, 288–290 (1969)
Maksimova, L.L.: On maximal intermediate logics with the disjunction property. Stud. Logica45, 69–75 (1986)
Maksimova, L.L., Skvorkov, D.P., Sethman, V.B.: The impossibility of a finite axiomatization of Medvedev's logic of finitary problems. Sov. Math. Dokl.4, 180–183 (1979)
Medvedev, T.Ju.: Finite problems. Sov. Math., Dokl.,3, 227–230 (1962)
Meloni, G.C.: Modelli per le logiche intermedie KP e KP n . Atti degli incontri di Logica Mat. Siena. Scuola di Specializzazione in Logica Matematica — Dipartimento di Matematica —Università di Siena, 1984
Miglioli, P., Moscato, U., Ornaghi, M.: Constructive theories with abstract data types for program synthesis. In: Skordev, D.G. (ed.) Proc. Symp. Math. Logic Appl. Druzhba, Bulgaria: Plenum Press, pp. 293–302, 1987
Miglioli, P., Moscato, U., Ornaghi, M.: Semi-constructive formal systems and axiomatization of abstract data types. In: Tapsoft '89, Barcelona 1989. (Lect. Notes Comput. Sci., vol. 351, pp. 337–351) Berlin Heidelberg New York: Springer 1989
Miglioli, P., Moscato, U., Ornaghi, M., Quazza, S., Usberti, G.: Some results on intermediate constructive logics. Notre Dame J. Formal Logic30, 543–562 (1989)
Minari, P.: On the extension of intuitionistic propositional logic with Kreisel-Putnam's and Scott's schemes. Stud. Logica45, 55–68 (1986)
Skvorkov, D.P.: Logic of infinite problems and Kripke models on atomic semilattices of sets. Sov. Math., Dokl.20, 360–363 (1979)
Smorinski, C.A.: Applications of Kripke models. In: Troelstra, A.S. (ed.) Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. (Lect. Notes Math., vol. 344, pp. 324–391) Berlin Heidelberg New York: Springer 1973
Wronski, A.: Intermediate logics with the disjunction property. Rep. Math. Logic1, 35–51 (1973)
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Miglioli, P. An infinite class of maximal intermediate propositional logics with the disjunction property. Arch Math Logic 31, 415–432 (1992). https://doi.org/10.1007/BF01277484
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DOI: https://doi.org/10.1007/BF01277484