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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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SUPERINTUITIONISTIC LOGICS AND NORMAL EXTENSIONS OF THE MODAL LOGICS S4

SUPERINTUITIONISTIC LOGICS AND NORMAL EXTENSIONS OF THE MODAL LOGICS S4

Chapter:
(p.61) 3 SUPERINTUITIONISTIC LOGICS AND NORMAL EXTENSIONS OF THE MODAL LOGICS S4
Source:
Interpolation and Definability
Author(s):

D.M. Gabbay

L. Maksimova

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198511748.003.0003

It is well known that the intuitionistic logic Int may be translated into S4. This translation was introduced by Gödel to give an interpretation of Int in the classical logic with an additional provability operator. This chapter pays special attention to extensions of the intuitionistic logic and of the modal logic S4. An algebraic semantics and Kripke semantics for these logics are presented in more detail. The main interrelations of the family of superintuitionistic logics and of the family of normal extensions of S4 induced by Gödel's translation are explained. In particular, any intermediate logic L has an infinite family of its modal companions over S4, which have L as their superintuitionistic fragment.

Keywords:   translation, pseudoboolean algebra, topoboolean algebra, characteristic formula, frame

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