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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: majidalizadeh@ut.ac.ir
†SCHOOL OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE, COLLEGE OF SCIENCE, UNIVERSITY OF TEHRAN, P.O. BOX 14155-6455, TEHRAN, IRAN E-mail: f.derakhshan@ut.ac.ir
‡JAPAN ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY, NOMI, ISHIKAWA, 923-1292, JAPAN E-mail: ono@jaist.ac.jp
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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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