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Glivenko theorems for substructural logics over FL

Published online by Cambridge University Press:  12 March 2014

Nikolaos Galatos  and
Hiroakira Ono
Nikolaos Galatos
Affiliation:
Japan Advanced Institute of Science and Technology School, of Information Science, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan, E-mail: galatos@jaist.ac.jp
Hiroakira Ono
Affiliation:
Japan Advanced Institute of Science and Technology School, of Information Science, 1-1 Asahidai, Nomi, Ishikawa, 923-1292, Japan, E-mail: ono@jaist.ac.jp
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Abstract

It is well known that classical propositional logic can be interpreted in intuitionistic prepositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part of the paper, we also discuss some extended forms of the Koltnogorov translation and we compare it to the Glivenko translation.

Keywords

Glivenko's theoremsubstructural logicinvolutivepointed residuated lattice
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1353 - 1384
Copyright
Copyright © Association for Symbolic Logic 2006

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References

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