Abstract
We study Craig’s interpolation property in the extensions of Johansson’s minimal logic. We consider the Odintsov classification of J-logics according to their intuitionistic and negative companions which subdivides all logics into intervals. We prove that the lower endpoint of an interval has Craig interpolation property if and only if both its companions do so. We also establish the recognizability of the lower and upper endpoints which have Craig interpolation property, and find their semantic characterization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Johansson I., “Der Minimalkalkül, ein reduzierter intuitionistische Formalismus,” Compos. Math., 4, 119–136 (1937).
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).
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).
Maksimova L. L., “Craig’s interpolation theorem and amalgamable varieties,” Dokl. Akad. SSSR, 237, No. 6, 1281–1284 (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).
Maksimova L. L., “Decidability of the projective Beth property in varieties of Heyting algebras,” Algebra and Logic, 40, No. 3, 159–165 (2001).
Gabbay D. M. and Maksimova L., Interpolation and Definability: Modal and Intuitionistic Logics, Clarendon Press, Oxford (2005).
Maksimova L. L., “The decidability of Craig’s interpolation property in well-composed J-logics,” Siberian Math. J., 53, No. 5, 839–852 (2012).
Maksimova L. L., “The projective Beth property in well-composed logics,” Algebra and Logic, 52, No. 2, 116–136 (2013).
Maksimova L. L., “Decidability of the weak interpolation property over the minimal logic,” Algebra and Logic, 50, No. 2, 106–132 (2011).
Maksimova L. L., “A method of proving interpolation in paraconsistent extensions of the minimal logic,” Algebra and Logic, 46, No. 5, 341–353 (2007).
Maksimova L. L., “Negative equivalence over the minimal logic and interpolation,” Sib. Elektron. Mat. Izv., 11, 1–17 (2014).
Odintsov S., “Logic of classic refutability and class of extensions of minimal logic,” Logic Log. Philos., 9, 91–107 (2001).
Maksimova L. L., “Implicit definability in extensions of the minimal logic,” Logicheskie Issled., 8, 72–81 (2001).
Odintsov S. P., Constructive Negations and Paraconsistency, Springer-Verlag, Dordrecht (2008) (Ser. Trends in Logic; V. 26).
Maksimova L. L. and Yun V. F., “Recognizable logics,” Algebra and Logic, 54, No. 2 (2015).
Segerberg K., “Propositional logics related to Heyting’s and Johansson’s,” Theoria, 34, No. 1, 26–61 (1968).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2015 Maksimova L.L. and Yun V.F.
The authors were supported by the Russian Foundation for Basic Research (Grant 12-01-00168a) and the Presidential Grant Council for Government Support of Young Russian Scientists and the Leading Scientific Schools of the Russian Federation (Grant NSh-860.2014.1).
To Yuriĭ Leonidovich Ershov on the occasion of his jubilee.
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 3, pp. 600–616, May–June, 2015
Rights and permissions
About this article
Cite this article
Maksimova, L.L., Yun, V.F. Interpolation over the minimal logic and Odintsov intervals. Sib Math J 56, 476–489 (2015). https://doi.org/10.1134/S0037446615030118
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446615030118