Archive for Mathematical Logic 51 (7-8):695-707 (2012)

Hiroakira Ono
Japan Advanced Institute of Science and Technology
Along the same line as that in Ono (Ann Pure Appl Logic 161:246–250, 2009), a proof-theoretic approach to Glivenko theorems is developed here for substructural predicate logics relative not only to classical predicate logic but also to arbitrary involutive substructural predicate logics over intuitionistic linear predicate logic without exponentials QFL e . It is shown that there exists the weakest logic over QFL e among substructural predicate logics for which the Glivenko theorem holds. Negative translations of substructural predicate logics are studied by using the same approach. First, a negative translation, called extended Kuroda translation is introduced. Then a translation result of an arbitrary involutive substructural predicate logics over QFL e is shown, and the existence of the weakest logic is proved among such logics for which the extended Kuroda translation works. They are obtained by a slight modification of the proof of the Glivenko theorem. Relations of our extended Kuroda translation with other standard negative translations will be discussed. Lastly, algebraic aspects of these results will be mentioned briefly. In this way, a clear and comprehensive understanding of Glivenko theorems and negative translations will be obtained from a substructural viewpoint
Keywords Glivenko’s theorem  Negative translations  Double negation shift  Substructural predicate logics  Proof-theoretic methods
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DOI 10.1007/s00153-012-0293-8
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References found in this work BETA

Shoenfield is Gödel After Krivine.Thomas Streicher & Ulrich Kohlenbach - 2007 - Mathematical Logic Quarterly 53 (2):176-179.
Glivenko Theorems for Substructural Logics Over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - Journal of Symbolic Logic 71 (4):1353 - 1384.
Glivenko Theorems Revisited.Hiroakira Ono - 2009 - Annals of Pure and Applied Logic 161 (2):246-250.
Applications of Trees to Intermediate Logics.Dov M. Gabbay - 1972 - Journal of Symbolic Logic 37 (1):135-138.

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