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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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INTERPOLATION IN NORMAL EXTENSIONS OF THE MODAL LOGIC S4

INTERPOLATION IN NORMAL EXTENSIONS OF THE MODAL LOGIC S4

Chapter:
(p.225) 8 INTERPOLATION IN NORMAL EXTENSIONS OF THE MODAL LOGIC S4
Source:
Interpolation and Definability
Author(s):

D.M. Gabbay

L. Maksimova

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198511748.003.0008

It appears that the behaviour of interpolation over the modal S4 logic is similar to interpolation in superintuitionistic logics. It is shown that all extensions of S4 with interpolation property for deducibility IPD are modal companions of superintuitionistic logics with CIP, but there is an intermediate logic with CIP that has no modal companions with IPD. On the other hand, all modal companions of intermediate logics with CIP have a weaker version of interpolation, which is CIP restricted to those implications, where all occurrences of variables are preceded by necessity symbol. It is proved that there are no more than forty-nine modal logics with IPD in the family of normal extensions of S4; in this list there are twelve logics that have IPD but do not have CIP. All the forty-nine logics are finitely axiomatizable and have the finite model property. It is proved that IPD and CIP problems are deducible over S4, and amalgamation and superamalgamation are base-decidable in varieties of closure algebras.

Keywords:   Craig's interpolation, interpolation property for deducibility, modal companion, well-representable logic

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