Siberian Mathematical Journal

, Volume 56, Issue 3, pp 476–489 | Cite as

Interpolation over the minimal logic and Odintsov intervals

  • L. L. MaksimovaEmail author
  • V. F. Yun


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.


Johansson minimal logic Craig interpolation property recognizability Odintsov interval 


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  1. 1.
    Johansson I., “Der Minimalkalkül, ein reduzierter intuitionistische Formalismus,” Compos. Math., 4, 119–136 (1937).Google Scholar
  2. 2.
    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).MathSciNetCrossRefGoogle Scholar
  3. 3.
    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).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Maksimova L. L., “Craig’s interpolation theorem and amalgamable varieties,” Dokl. Akad. SSSR, 237, No. 6, 1281–1284 (1977).MathSciNetGoogle Scholar
  5. 5.
    Maksimova L. L., “Implicit definability in positive logics,” Algebra and Logic, 42, No. 1, 37–53 (2003).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Maksimova L. L., “Interpolation and definability in extensions of the minimal logic,” Algebra and Logic, 44, No. 6, 407–421 (2005).MathSciNetCrossRefGoogle Scholar
  7. 7.
    Maksimova L. L., “Decidability of the projective Beth property in varieties of Heyting algebras,” Algebra and Logic, 40, No. 3, 159–165 (2001).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gabbay D. M. and Maksimova L., Interpolation and Definability: Modal and Intuitionistic Logics, Clarendon Press, Oxford (2005).CrossRefGoogle Scholar
  9. 9.
    Maksimova L. L., “The decidability of Craig’s interpolation property in well-composed J-logics,” Siberian Math. J., 53, No. 5, 839–852 (2012).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Maksimova L. L., “The projective Beth property in well-composed logics,” Algebra and Logic, 52, No. 2, 116–136 (2013).zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Maksimova L. L., “Decidability of the weak interpolation property over the minimal logic,” Algebra and Logic, 50, No. 2, 106–132 (2011).zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Maksimova L. L., “A method of proving interpolation in paraconsistent extensions of the minimal logic,” Algebra and Logic, 46, No. 5, 341–353 (2007).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Maksimova L. L., “Negative equivalence over the minimal logic and interpolation,” Sib. Elektron. Mat. Izv., 11, 1–17 (2014).Google Scholar
  14. 14.
    Odintsov S., “Logic of classic refutability and class of extensions of minimal logic,” Logic Log. Philos., 9, 91–107 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Maksimova L. L., “Implicit definability in extensions of the minimal logic,” Logicheskie Issled., 8, 72–81 (2001).MathSciNetGoogle Scholar
  16. 16.
    Odintsov S. P., Constructive Negations and Paraconsistency, Springer-Verlag, Dordrecht (2008) (Ser. Trends in Logic; V. 26).zbMATHCrossRefGoogle Scholar
  17. 17.
    Maksimova L. L. and Yun V. F., “Recognizable logics,” Algebra and Logic, 54, No. 2 (2015).Google Scholar
  18. 18.
    Segerberg K., “Propositional logics related to Heyting’s and Johansson’s,” Theoria, 34, No. 1, 26–61 (1968).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

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