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Interpolation and DefinabilityModal and Intuitionistic Logics$
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Dov M. Gabbay and Larisa Maksimova

Print publication date: 2005

Print ISBN-13: 9780198511748

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198511748.001.0001

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THE INTERPOLATION THEOREM IN INTUITIONISTIC PREDICATE CALCULUS

THE INTERPOLATION THEOREM IN INTUITIONISTIC PREDICATE CALCULUS

Chapter:
(p.103) 4 THE INTERPOLATION THEOREM IN INTUITIONISTIC PREDICATE CALCULUS
Source:
Interpolation and Definability
Author(s):

D.M. Gabbay

L. Maksimova

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198511748.003.0004

This chapter presents the equivalence of Craig's interpolation property to Robinson's joint consistency, and a proof of Lyndon's interpolation theorem for the classical predicate logic. It is proved that the general form of Robinson's consistency property (RCP) fails in the intuitionistic predicate logic HQ. A weaker form of RCP is equivalent to Craig's interpolation property (CIP) and holds in HQ, and a semantic proof is given. It is proved that in propositional intermediate logics the general form of RCP is equivalent to CIP. A derivation of Beth's property from CIP is given for the intuitionistic predicate logic. Kreisel's proof of validity of the Beth property for any propositional intermediate logic is also presented. It must be noted that there are intermediate predicate logics without Beth's property.

Keywords:   intuitionistic predicate logic, Craig's interpolation property, Lyndon's interpolation property, Robinson's joint consistency, Beth's property

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