Glivenko theorems and negative translations in substructural predicate logics

Abstract

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.