Abstract
Analogs of Robinson’s theorem on joint consistency are found which are equivalent to the weak interpolation property (WIP) in extensions of Johansson’s minimal logic J. Although all propositional superintuitionistic logics possess this property, there are J-logics without WIP. It is proved that the problem of the validity of WIP in J-logics can be reduced to the same problem over the logic Gl obtained from J by adding the tertium non datur. Some algebraic criteria for validity of WIP over J and Gl are found.
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References
Robinson A., “A result on consistency and its application to the theory of definition,” Indag. Math., 18, No. 1, 47–58 (1956).
Barwise J. and Feferman S., eds., Model-Theoretic Logics, Springer-Verlag, New York (1985).
Craig W., “Three uses of Herbrand-Gentzen theorem in relating model theory,” J. Symbolic Logic, 22, No. 3, 269–285 (1957).
Gabbay D. M. and Maksimova L., Interpolation and Definability. Modal and Intuitionistic Logics, Oxford Univ. Press, Oxford (2005) (Oxford Logic Guides, 46; Oxford Sci. Publ.).
Schütte K., “Der Interpolationssatz der intuitionistischen Prädikatenlogik,” Math. Ann., Bd 148, 192–200 (1962).
Gabbay D. M., Semantical Investigations in Heyting’s Intuitionistic Logic, D. Reidel Publ. Co., Dordrecht (1981).
Johansson I., “Der Minimalkalkül, ein reduzierter intuitionistische Formalismus,” Compos. Math., 4, 119–136 (1937).
Segerberg K., “Propositional logics related to Heyting’s and Johansson’s,” Theoria, 34, No. 1, 26–61 (1968).
Maksimova L., “Interpolation and joint consistency,” in: We Will Show Them! Essays in Honour of Dov Gabbay. V. 2, S. Artemov, H. Barringer, A. d’Avila Garcez, L. Lamb and J. Woods, eds., King’s College Publ., London, 2005, pp. 293–305.
Maksimova L. L., “Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudoboolean algebras,” Algebra i Logika, 16, No. 6, 643–681 (1977).
Maksimova L. L., “Implicit definability in positive logics,” Algebra and Logic, 42, No. 1, 37–53 (2003).
Maksimova L. L., “Interpolation and definability in extensions of the minimal logic,” Algebra and Logic, 44, No. 6, 407–421 (2005).
Odintsov S. P., Constructive Negations and Paraconsistency, Springer-Verlag, Dordrecht (2008).
Rasiowa H. and Sikorski R., The Mathematics of Metamathematics, Panstwowe Wydawnitstwo Naukowe, Warszawa (1963).
Maksimova L. L., “Intuitionistic logic and implicit definability,” Ann. Pure Appl. Logic, 105,No. 1–3, 83–102 (2000).
Maksimova L. L., “Weak form of interpolation in equational logic,” Algebra and Logic, 47, No. 1, 56–64 (2008).
Maltsev A. I., Algebraic Systems [in Russian], Nauka, Moscow (1970).
Odintsov S. P., “Logic of classical refutability and class of extensions of minimal logic,” Logic Log. Philos., 9, 91–107 (2001).
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Original Russian Text Copyright © 2010 Maksimova L. L.
The author was supported by the Russian Foundation for Basic Research (Grant 09-01-00090a), the Leading Scientific Schools of the Russian Federation (Grant NSh-3606.2010.1), and the Russian Federal Agency for Education (Grant 2.1.1.419)
To Yuriĭ Leonidovich Ershov.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 3, pp. 604–619, May–June, 2010.
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Maksimova, L.L. Joint consistency in extensions of the minimal logic. Sib Math J 51, 479–490 (2010). https://doi.org/10.1007/s11202-010-0050-3
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DOI: https://doi.org/10.1007/s11202-010-0050-3