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UNIFORM INTERPOLATION IN SUBSTRUCTURAL LOGICS

Published online by Cambridge University Press:  27 May 2014

MAJID ALIZADEH ,
FARZANEH DERAKHSHAN  and
HIROAKIRA ONO
MAJID ALIZADEH*
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran
FARZANEH DERAKHSHAN*
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran
HIROAKIRA ONO*
Affiliation:
Japan Advanced Institute of Science and Technology
*
*SCHOOL OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE, COLLEGE OF SCIENCE, UNIVERSITY OF TEHRAN, P.O. BOX 14155-6455, TEHRAN, IRAN E-mail: [email protected]
†SCHOOL OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE, COLLEGE OF SCIENCE, UNIVERSITY OF TEHRAN, P.O. BOX 14155-6455, TEHRAN, IRAN E-mail: [email protected]
‡JAPAN ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY, NOMI, ISHIKAWA, 923-1292, JAPAN E-mail: [email protected]
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Abstract

Uniform interpolation property of a given logic is a stronger form of Craig’s interpolation property where both pre-interpolant and post-interpolant always exist uniformly for any provable implication in the logic. It is known that there exist logics, e.g., modal propositional logic S4, which have Craig’s interpolation property but do not have uniform interpolation property. The situation is even worse for predicate logics, as classical predicate logic does not have uniform interpolation property as pointed out by L. Henkin.

In this paper, uniform interpolation property of basic substructural logics is studied by applying the proof-theoretic method introduced by A. Pitts (Pitts, 1992). It is shown that uniform interpolation property holds even for their predicate extensions, as long as they can be formalized by sequent calculi without contraction rules. For instance, uniform interpolation property of full Lambek predicate calculus, i.e., the substructural logic without any structural rule, and of both linear and affine predicate logics without exponentials are proved.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 3 , September 2014 , pp. 455 - 483
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

BIBLIOGRAPHY

Bílková, M. (2007). Uniform interpolation and propositional quantifiers in modal logics. Studia Logica, 85, 1–31.CrossRefGoogle Scholar
D’Agostino, G. (2008). Interpolation in non-classical logics. Synthese, 164, 421–435.Google Scholar
Dardžaniá, G. K. (1977). Intuitionistic system without contraction. Bulletin of the Section of Logic, 6, 2–8.Google Scholar
Došen, K. (1988). Sequent systems and groupoid models I. Studia Logica, 47, 353–385.CrossRefGoogle Scholar
Dyckhoff, R. (1992). Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 78, 795–807.Google Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Vol. 151, Studies in Logic and the Foundations of Mathematics. Amsterdam: Elsevier, 532 pp.Google Scholar
Ghilardi, S., & Zawadowski, M. (1995a). A sheaf representation and duality for finitely presented Heyting algebras. Journal of Symbolic Logic, 60, 911–939.Google Scholar
Ghilardi, S., & Zawadowski, M. (1995b). Undefinability of propositional quantifiers in the modal system S4. Studia Logica, 55, 259–271.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science, 50, 1–102.CrossRefGoogle Scholar
Grišin, V. N. (1982). Predicate and set-theoretical calculi based on logic without the contraction rule. Mathematical USSR Izvestiya, 18, 41–50.Google Scholar
Henkin, L. (1963). An extension of the Craig-Lyndon interpolation theorem. Journal of Symbolic Logic, 28, 201–216.CrossRefGoogle Scholar
Heuerding, A. (1998). Sequent calculi for proof search in some modal logics. Ph. D. thesis, University of Bern.Google Scholar
Hudelmaier, J. (1989). Bounds for cut elimination in intuitionistic propositional logic. PhD Thesis, University of Tübingen.Google Scholar
Idziak, P. M. (1984). Lattice operations in BCK-algebras. PhD Thesis, Jagiellonian University, Kraków, Poland.Google Scholar
Kanger, S. (1957). Provability in Logic, Stockholm Studies in Philosophy 1. Stockholm: Almqvist & Wiksell.Google Scholar
Kiriyama, E., & Ono, H. (1991). The contraction rule and decision problems for logics without structural rules. Studia Logica, 50, 299–319.Google Scholar
Komori, Y. (1986). Predicate logics without the structure rules. Studia Logica, 45,393–404.Google Scholar
Lambek, J. (1958). The mathematics of sentence structures. American Mathematical Monthly, 65, 154–170.Google Scholar
Maehara, S. (1960). On the interpolation theorem of Craig (Japanese). Sugaku, 12, 235–237.Google Scholar
Montagna, F. (2012). ∆-core fuzzy logics with propositional quantifiers, quantifier elimination and uniform Craig interpolation. Studia Logica, 100, 289–317.CrossRefGoogle Scholar
Ono, H. (1990). Structural rules and a logical hierarchy. In Petkov, P. P., editor. Mathematical Logic. London: Plenum, pp. 95–104.CrossRefGoogle Scholar
Ono, H. (1998). Proof-theoretic methods for nonclassical logic – An introduction. In Takahashi, M., Okada, M., and Dezani-Ciancaglini, M., editors, Theories of Types and Proofs, MSJ Memoirs 2. Tokyo: Mathematical Society of Japan, pp. 207–254.CrossRefGoogle Scholar
Ono, H., & Komori, Y. (1985). Logics without the contraction rule. Journal of Symbolic Logic, 50, 169–201.Google Scholar
Pitts, A. M. (1992). On an interpretation of second order quantification in first order intuitionistic propositional logic. Journal of Symbolic Logic, 57, 33–52.Google Scholar
Tamura, S. (1974). On a decision procedure for free lo-algebraic systems. Technical report of Mathematics, Yamaguchi University 9.Google Scholar
Troelstra, A. S. (1992). Lectures on Linear Logic, Vol. 29, Lecture Notes. Stanford: CSLI.Google Scholar
van Benthem, J. (2008). The many faces of interpolation. Synthese, 164, 451–460.Google Scholar
Visser, A. (1996). Uniform interpolation and layered bisimulation. In Hájek, P., editor. Lecture Notes in Logic 6 “Gödel ’96: Logical foundations of mathematics, computer science and physics — Kurt Gödel’s legacy”. Berlin: Springer, pp. 139–164.Google Scholar
Wang, H. (1963). A Survey of Mathematical Logic, Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland.Google Scholar