Skip to main content
SpringerLink
Log in

The decidability of craig’s interpolation property in well-composed J-logics

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Under study are the extensions of Johansson’s minimal logic J. We find sufficient conditions for the finite approximability of J-logics in dependence on the form of their axioms. Using these conditions, we prove the decidability of Craig’s interpolation property (CIP) in well-composed J-logics. Previously all J-logics with weak interpolation property (WIP) were described and the decidability of WIP over J was proved. Also we establish the decidability of the problem of amalgamability of well-composed varieties of J-algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Johansson I., “Der Minimalkalkül, ein reduzierter intuitionistische Formalismus,” Compos. Math., 4, 119–136 (1937).

    MathSciNet  Google Scholar 

  2. 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).

    Article  MATH  MathSciNet  Google Scholar 

  3. Model-Theoretic Logics, J. Barwise and S. Feferman (eds.), Springer-Verlag, New York (1985).

    Google Scholar 

  4. Gabbay D. M. and Maksimova L., Interpolation and Definability. Modal and Intuitionistic Logics, Clarendon Press, Oxford (2005).

    Book  MATH  Google Scholar 

  5. Schütte K., “Der Interpolationssatz der intuitionistischen Prädikatenlogik,” Math. Ann., Bd 148, 192–200 (1962).

    Article  MATH  Google Scholar 

  6. Gabbay D. M., Semantical Investigations in Heyting’s Intuitionistic Logic, D. Reidel Publ. Co., Dordrecht (1981).

    Google Scholar 

  7. Beth E. W., “On Padoa’s method in the theory of definitions,” Indag. Math., 15, No. 4, 330–339 (1953).

    MathSciNet  Google Scholar 

  8. 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).

    Article  MathSciNet  Google Scholar 

  9. Maksimova L. L., “Intuitionistic logic and implicit definability,” Ann. Pure Appl. Logic, 105, No. 1–3, 83–102 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  10. Maksimova L. L., “Decidability of the projective Beth property in varieties of Heyting algebras,” Algebra and Logic, 40, No. 3, 159–165 (2001).

    Article  MathSciNet  Google Scholar 

  11. Maksimova L. L., “Implicit definability in positive logics,” Algebra and Logic, 42, No. 1, 37–53 (2003).

    Article  MathSciNet  Google Scholar 

  12. Maksimova L., “Problem of restricted interpolation in superintuitionistic and some modal logics,” Log. J. IGPL, 18, No. 3, 367–380 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  13. Maksimova L. L., “The projective Beth property and interpolation in positive and related logics,” Algebra and Logic, 45, No. 1, 49–66 (2006).

    Article  MathSciNet  Google Scholar 

  14. Maksimova L. L., “Decidability of the weak interpolation property over the minimal logic,” Algebra and Logic, 50, No. 2, 106–132 (2011).

    Article  MathSciNet  Google Scholar 

  15. 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.

    Google Scholar 

  16. Maksimova L. L., “Joint consistency in extensions of the minimal logic,” Siberian Math. J., 51, No. 3, 479–490 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  17. Maksimova L., “Interpolation and definability over the logic Gl,” Studia Logica, 99, No. 1, 249–267 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  18. Maksimova L. L., “Interpolation and definability in extensions of the minimal logic,” Algebra and Logic, 44, No. 6, 407–421 (2005).

    Article  MathSciNet  Google Scholar 

  19. Maksimova L. L., “Interpolation and the projective Beth property in well-composed J-logics,” Algebra and Logic, 51, No. 2, 163–184 (2012).

    Article  Google Scholar 

  20. Maksimova L. L., “The projective Beth properties in modal and superintuitionistic logics,” Algebra and Logic, 38, No. 3, 171–180 (1999).

    Article  MathSciNet  Google Scholar 

  21. Kreisel G., “Explicit definability in intuitionistic logic,” J. Symbolic Logic, 25, 389–390 (1960).

    Article  Google Scholar 

  22. Maksimova L., “Restricted interpolation in modal logics,” in: Advances in Modal Logics, Kings’s College London Publ., London, 2003, V. 4, pp. 297–312.

    Google Scholar 

  23. Mal’cev A. I., Algebraic Systems, Springer-Verlag and Akademie-Verlag, Berlin, Heidelberg, and New York (1973).

    Book  Google Scholar 

  24. Odintsov S. P., “Logic of classical refutability and class of extensions of minimal logic,” Logic Log. Philos., 9, 91–107 (2001).

    MATH  MathSciNet  Google Scholar 

  25. Miura S., “A remark on the intersection of two logics,” Nagoya Math. J., 26, No. 2, 167–171 (1966).

    MATH  MathSciNet  Google Scholar 

  26. McKinsey J. C. C. and Tarski A., “Some theorems about the sequential calculi of Lewis and Heyting,” J. Symbolic Logic, 13, No. 1, 1–15 (1948).

    Article  MATH  MathSciNet  Google Scholar 

  27. McKay C. G., “The decidability of certain intermediate propositional logics,” J. Symbolic Logic, 33, 258–264 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  28. Kuznetsov A. V. and Gerchiu V. Ya., “On the superintuitionistic logic and residual finiteness,” Dokl. Akad. Nauk SSSR, 195, No. 5, 1029–1032 (1970).

    MathSciNet  Google Scholar 

  29. Diego A., Sur les algebras de Hilbert, Gauthier-Villars, Paris (1966).

    Google Scholar 

  30. Stukacheva M. V., “The disjunction property in the class of paraconsistent extensions of minimal logic,” Algebra and Logic, 43, No. 2, 132–141 (2004).

    Article  MathSciNet  Google Scholar 

  31. Maksimova L. L., “A weak form of interpolation in equational logic,” Algebra and Logic, 47, No. 1, 56–64 (2008).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    L. L. Maksimova

Authors
  1. L. L. Maksimova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. L. Maksimova.

Additional information

The author was supported by the Russian Foundation for Basic Research (Grant 09-01-00090-a), 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).

Original Russian Text Copyright © 2012 Maksimova L.L.

__________

Novosibirsk. Translated from Sibirskiı Matematicheskiı Zhurnal, Vol. 53, No. 5, pp. 1048–1064, September–October, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maksimova, L.L. The decidability of craig’s interpolation property in well-composed J-logics. Sib Math J 53, 839–852 (2012). https://doi.org/10.1134/S0037446612050096

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446612050096

Keywords

Search

Navigation