Abstract
We consider versions of the interpolation property stronger than the Craig interpolation property and prove the Lyndon interpolation property for the Grzegorczyk logic and some of its extensions. We also establish the Lyndon interpolation property for most extensions of the intuitionistic logic with Craig interpolation property. For all modal logics over the Grzegorczyk logic as well as for all superintuitionistic logics, the uniform interpolation property is equivalent to Craig’s property.
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References
Craig W., “Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory,” J. Symbolic Logic, 22, No. 3, 269–285 (1957).
Model-Theoretic Logics, J. Barwise and S. Feferman (eds.), Springer-Verlag, New York (1985).
Gabbay D. M. and Maksimova L., Interpolation and Definability. Modal and Intuitionistic Logics, Oxford Univ. Press, Oxford (2005).
Van Benthem J., “The many faces of interpolation,” Synthese, 164, No. 3, 451–460 (2008).
Lyndon R., “An interpolation theorem in the predicate calculus,” Pacific J. Math., 9, 129–142 (1959).
Pitts A. M., “On an interpretation of second order quantification in first order intuitionistic propositional logic,” J. Symbolic Logic, 57, No. 1, 33–52 (1992).
Ghilardi S., “An algebraic theory of normal forms,” Ann. Pure Appl. Logic, 71, 189–245 (1995).
Shavrukov V. Yu., Subalgebras of Diagonalizable Algebras of Theories Containing Arithmetics, Thesis Doct. Phylosophy (Mathematics), Polska Academia Nauk, Math. Inst., Warszawa (1993).
D’Agostino G. and Hollenberg M., “Logical questions concerning the µ-calculus: interpolation, Lyndon and Łoś-Tarski,” J. Symbolic Logic, 65, No. 1, 310–332 (2000).
Ghilardi S. and Zawadowski M., “A sheaf representation and duality for finitely generated Heyting algebras,” J. Symbolic Logic, 60, No. 3, 911–939 (1995).
Maksimova L. L., “The Lyndon interpolation theorem in modal logics,” in: Mathematical Logic and Algorithm Theory [in Russian], Nauka, Novosibirsk, 1982, 2, pp. 45–55.
Boolos G., “On systems of modal logic with provability interpretations,” Theoria, 46, 7–18 (1980).
Visser A., “Uniform interpolation and layered bisimulation,” in: Goedel’96, Springer-Verlag, Berlin, 1996, pp. 139–164.
Shamkanov D. S., “Interpolation properties of the provability logics GL and GLP,” in: Algorithmic Problems of Algebra and Logic [in Russian], MAIK, Moscow, 2011, 274, pp. 329–342.
Segerberg K., An Essay in Classical Modal Logic, Uppsala Univ., Uppsala (1971).
Maksimova L. L., “Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-Boolean algebras,” Algebra and Logic, 16, No. 6, 427–455 (1977).
Rasiowa H. and Sikorski R., The Mathematics of Metamathematics, Panstwowe Wydawnitstwo Naukowe, Warszawa (1963).
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Original Russian Text Copyright © 2014 Maksimova L.L.
The author was supported by the Russian Foundation for Basic Research (Grant 12-01-00168a) and the Program “Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 2.1.1.10726).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 1, pp. 147–156, January–February, 2014.
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Maksimova, L.L. The Lyndon property and uniform interpolation over the Grzegorczyk logic. Sib Math J 55, 118–124 (2014). https://doi.org/10.1134/S0037446614010145
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DOI: https://doi.org/10.1134/S0037446614010145