Skip to main content
Log in

Interpolation in non-classical logics

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

We discuss the interpolation property on some important families of non classical logics, such as intuitionistic, modal, fuzzy, and linear logics. A special paragraph is devoted to a generalization of the interpolation property, uniform interpolation.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Areces C., Blackburn P., Marx M. (2003) Repairing the interpolation theorem in quantified modal logic. Annals of Pure and Applied Logics 123: 287–299

    Article  Google Scholar 

  • Areces, C., & de Rijke, M. (1998). Interpolation and bisimulation in temporal logic. In Proceedings of WoLLIC’98. Workshop of Logic, Language, Information and Computation (pp. 15–21). Sao Paulo, Brazil, July 1998, IME/USP.

  • Baaz M., Veith H. (1999) Intrepolation in fuzzy logic. Archive for Mathematical Logic 38: 461–489

    Article  Google Scholar 

  • Bilkova, M. (2006). Intrepolation in modal logic. PhD thesis, Charles University in Prague.

  • D’Agostino G., Hollenberg M. (2000) Logical questions concerning the μ-calculus: Interpolation, Lyndon, and Łoś-Tarski. Journal of Symbolic Logic 65: 310–332

    Article  Google Scholar 

  • Dyckhoff R. (1992) Contraction-free sequent calculi for Intuitionistic Logic. Journal of Symbolic Logic 57: 795–807

    Article  Google Scholar 

  • Fine K. (1979) Failure of the interpolation lemma in quantified modal logic. Journal of Symbolic Logic 44(2): 201–206

    Article  Google Scholar 

  • Gabbay, D. M. (1969). Semantic proof of Craig Interpolation Theorem for intuitionistic logics and extensions II. In Logic Colloquium ’69 (Proc, Summer School and Colloq. Manchester) (pp. 403–410). Amsterdam: North Holland.

  • Gabbay D.M., Maksimova L.L. (2005) Interpolation and definability. Clarendon Press, Oxford

    Google Scholar 

  • Ghilardi, S., Luts, C., Wolter, F., & Zakharyaschev, M. (2006). Conservative extensions in modal logic. In Proceedings of Advances in Modal Logic (AiML 06) (pp.187–207). College Publications.

  • Ghilardi S., Zawadowski M. (1995) Undefinability of propositional quantifiers in the modal system S 4. Studia Logica 55: 259–271

    Article  Google Scholar 

  • Ghilardi, S., & Zawadowski, M. (2002). Sheaves, games, and model completions trends in logic. Kluwer.

  • Henkin L. (1963) An extension of the Craig–Lyndon Interpolation Theorem. Journal of Symbolic Logic 28: 201–216

    Article  Google Scholar 

  • Krystek P.S., Zachorowski S. (1977) Reports on Mathematical Logic 9: 39–40

    Google Scholar 

  • Maksimova L.L. (1977) Craig’s Theorem in superintuitionistic logics and amalgamated varieties of pseudoboolean algebras. Algebra Logika 16: 643–681

    Google Scholar 

  • Maksimova L.L. (1991) Absence of interpolation and of Beth’s property in temporal logics with “the next” operation. Siberian Mathematical Journal 32(6): 109–113

    Google Scholar 

  • Marx, M. (1998). Interpolation in modal logic. In AMAST’98, LNCS 1548/1998 (pp. 154–163).

  • Pitts A. (1992) On an interpretation of second order quantification in first order intuitionistic propositional logic. Journal of Symbolic Logic 57(1): 33–52

    Article  Google Scholar 

  • Priest, G. (2001). An introduction to non-classical logic. Cambridge University Press.

  • Rautenberg W. (1983) Modal Tableau calculi and interpolation. Journal of Philosophical Logic 12: 403–423

    Article  Google Scholar 

  • Renardelde Lavalette G.R. (1989) Interpolation in fragments of intuitionistic propositional logics. Journal of Symbolic Logic 54(4): 1419–1429

    Article  Google Scholar 

  • Roorda D. (1994) Interpolation in fragments of classical linear logic. Journal of Symbolic Logic 52(2): 419–444

    Google Scholar 

  • Schutte K. (1962) Der Interpolationssatz der der intuitionistischen Prädikatenlogik (German). Mathematische Annalen 148: 192–200

    Article  Google Scholar 

  • Shavrukov, V. Yu. (1994). Adventures in diagonalizable algebras. PhD thesis, University of Amsterdam. ILLC Dissertation Series 1994-7.

  • ten Cate, B. (2005). Model theory for extended modal languages. PhD thesis, University of Amsterdam. ILLC Dissertation Series DS-2005-01.

  • Visser, A. (1996). Bisimulations, model description and propositional quantifiers. Logic Group Preprint Series, Utrecht University, no. 161.

  • Visser, A. (2002). Löb’s logic meets the μ-calculus. Processes, terms and cycles: Steps on the road to infinity, LNCS 3838/2005 (pp. 14–25).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Giovanna D’Agostino.

Additional information

Supported by PRIN project 2006/2007 ‘Large-scale development of certified mathematical proofs’.

Rights and permissions

Reprints and permissions

About this article

Cite this article

D’Agostino, G. Interpolation in non-classical logics. Synthese 164, 421–435 (2008). https://doi.org/10.1007/s11229-008-9359-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11229-008-9359-x

Keywords

Navigation