Abstract
In a previous paper [21] all extensions of Johansson’s minimal logic J with the weak interpolation property WIP were described. It was proved that WIP is decidable over J. It turned out that the weak interpolation problem in extensions of J is reducible to the same problem over a logic Gl, which arises from J by adding tertium non datur.
In this paper we consider extensions of the logic Gl. We prove that only finitely many logics over Gl have the Craig interpolation property CIP, the restricted interpolation property IPR or the projective Beth property PBP. The full list of Gl-logics with the mentioned properties is found, and their description is given. We note that IPR and PBP are equivalent over Gl. It is proved that CIP, IPR and PBP are decidable over the logic Gl.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barwise, J., Feferman, S (eds): Model-Theoretic Logics. Springer-Verlag, New York (1985)
Craig W.: ‘Three uses of Herbrand-Gentzen theorem in relating model theory and proof theory’. J. Symbolic Logic 22, 269–285 (1957)
Gabbay D.M.: Semantical Investigations in Heyting’s Intuitionistic Logic. D. Reidel Publ. Co., Dordrecht (1981)
Gabbay D.M., Maksimova L.: Interpolation and Definability: Modal and Intuitionistic Logics. Clarendon Press, Oxford (2005)
Johansson I.: ‘Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus’. Compositio Mathematica 4, 119–136 (1937)
Jonsson B.: ‘Algebras whose congruence lattices are distributive’. Mathematica Scandinavica 21, 110–121 (1967)
Kreisel G.: ‘Explicit definability in intuitionistic logic’. J. Symbolic Logic 25, 389–390 (1960)
Maksimova L.L.: ‘Craig’s Theorem in Superintuitionistic Logics and Amalgamable Varieties of Pseudo-Boolean Algebras’. Algebra and Logic 16, 427–455 (1977)
Maksimova L.: ‘Intuitionistic Logic and Implicit Definability’. Annals of Pure and Applied Logic 105, 83–102 (2000)
Maksimova L.L.: ‘Decidability of projective Beth’s property in varieties of Heyting algebras’. Algebra and Logic 40, 159–165 (2001)
Maksimova L.L.: ‘Implicit definability in positive logics’. Algebra and Logic 42, 37–53 (2003)
Maksimova L.: ‘Restricted interpolation in modal logics’. In: Balbiani, P., Suzuki, N.-Y., Wolter, F., Zakharyaschev, M (eds) Advances in Modal Logics, Volume 4, pp. 297–312. King’s College London Publications, London (2003)
Maksimova L.L.: ‘Interpolation and definability in extensions of the minimal logic’. Algebra and Logic 44, 407–421 (2005)
Maksimova L.: ‘Interpolation and joint consistency’. In: Artemov, S., Barringer, H., d’Avila Garcez, A., Lamb, L., Woods, J (eds) We Will Show Them! Essays in Honour of Dov Gabbay Volume 2., pp. 293–305. College Publications, London (2005)
Maksimova L.L.: ‘Projective Beth property and interpolation in positive and related logics’. Algebra and Logic 45, 49–66 (2006)
Maksimova L.L.: ‘A method of proving interpolation in paraconsistent extensions of the minimal logic’. Algebra and Logic 46, 341–353 (2007)
Maksimova L.L.: ‘A weak form of interpolation in equational logic’. Algebra and Logic 47, 56–64 (2008)
Maksimova L.: ‘Problem of restricted interpolation in superintuitionistic and some modal logics’. Logic Journal of IGPL 18, 367–380 (2010)
Maksimova L.: ‘Weak interpolation in extensions of minimal logic’. In: Feferman, S., Sieg, W (eds) Proofs, Categories and Computations: Essays in honor of Grigori Mints, pp. 159–170. College Publications, London (2010)
Maksimova L.L.: ‘Joint consistency in extensions of the minimal logic’. Siberian Mathematical Journal 51, 479–490 (2010)
Maksimova L.L.: ‘Decidability of weak interpolation property over the minimal logic’. Algebra and Logic 50(2), 106–132 (2011)
Maltsev A.I.: Algebraic systems. Nauka, Moscow (1970)
McKinsey J., Tarski A.: ‘Some theorems about the sententional calculi of Lewis and Heyting’. J. of Symbolic Logic 13, 1–15 (1948)
Miura S.: ‘A remark on the intersection of two logics’. Nagoya Math. Journal 26, 167–171 (1966)
Odintsov S.: ‘Logic of classical refutability and class of extensions of minimal logic’. Logic and Logical Philosophy 9, 91–107 (2001)
Odintsov S.: Constructive negations and paraconsistency. Springer, Dordrecht (2008)
Rasiowa H., Sikorski R.: The Mathematics of Metamathematics. PWN, Warszawa (1963)
Robinson A.: ‘A result on consistency and its application to the theory of definition’. Indagationes Mathematicae 18, 47–58 (1956)
Schütte K.: ‘Der Interpolationsatz der intuitionistischen Prädikatenlogik’. Mathematische Annalen 148, 192–200 (1962)
Segerberg K.: ‘Propositional logics related to Heyting’s and Johansson’s’. Theoria 34, 26–61 (1968)
Stukacheva M.V.: ‘On disjunction property in the class of paraconsistent extensions of the minimal logic’. Algebra and Logic 43, 235–252 (2004)
Wójcicki R.: Theory of Logical Calculi: Basic Theory of Consequence Operation. Kluwer Academic Publishers, Dordrecht (1988)
Wójcicki R.: ‘Some remarks on the consequence operation in sentential logics’. Fundamenta Mathematicae 68, 269–279 (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
To Professor Ryszard Wójcicki on his 80th Birthday
Special issue in honor of Ryszard Wójcicki on the occasion of his 80th birthday
Edited by J. Czelakowski, W. Dziobiak, and J. Malinowski
Rights and permissions
About this article
Cite this article
Maksimova, L. Interpolation and Definability over the Logic Gl. Stud Logica 99, 249 (2011). https://doi.org/10.1007/s11225-011-9351-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11225-011-9351-1