## Journal of Symbolic Logic

### Glivenko theorems for substructural logics over FL

#### Abstract

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

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

Dates
First available in Project Euclid: 20 November 2006

https://projecteuclid.org/euclid.jsl/1164060460

Digital Object Identifier
doi:10.2178/jsl/1164060460

Mathematical Reviews number (MathSciNet)
MR2275864

Zentralblatt MATH identifier
1109.03016

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

#### Citation

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. https://projecteuclid.org/euclid.jsl/1164060460

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