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.
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
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
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
Dyckhoff R. (1992) Contraction-free sequent calculi for Intuitionistic Logic. Journal of Symbolic Logic 57: 795–807
Fine K. (1979) Failure of the interpolation lemma in quantified modal logic. Journal of Symbolic Logic 44(2): 201–206
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
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
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
Krystek P.S., Zachorowski S. (1977) Reports on Mathematical Logic 9: 39–40
Maksimova L.L. (1977) Craig’s Theorem in superintuitionistic logics and amalgamated varieties of pseudoboolean algebras. Algebra Logika 16: 643–681
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
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
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
Renardelde Lavalette G.R. (1989) Interpolation in fragments of intuitionistic propositional logics. Journal of Symbolic Logic 54(4): 1419–1429
Roorda D. (1994) Interpolation in fragments of classical linear logic. Journal of Symbolic Logic 52(2): 419–444
Schutte K. (1962) Der Interpolationssatz der der intuitionistischen Prädikatenlogik (German). Mathematische Annalen 148: 192–200
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by PRIN project 2006/2007 ‘Large-scale development of certified mathematical proofs’.
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-008-9359-x