Journal of Symbolic Logic

Glivenko theorems for substructural logics over FL

Nikolaos Galatos and Hiroakira Ono

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It is well known that classical propositional logic can be interpreted in intuitionistic propositional 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 Kolmogorov translation and we compare it to the Glivenko translation.

Article information

J. Symbolic Logic, Volume 71, Issue 4 (2006), 1353-1384.

First available in Project Euclid: 20 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary: 06F05, Secondary: 08B15, 03B47, 03G10, 03B05, 03B20

Glivenko’s theorem, substructural logic, involutive, pointed residuated lattice


Galatos, Nikolaos; Ono, Hiroakira. Glivenko theorems for substructural logics over FL. J. Symbolic Logic 71 (2006), no. 4, 1353--1384. doi:10.2178/jsl/1164060460.

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