Glivenko theorems and negative translations in substructural predicate logics

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

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 QFLe. It is shown that there exists the weakest logic over QFLe 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 QFLe 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.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 92,931

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Glivenko Theorems for Substructural Logics over FL.Nikolaos Galatos & Hiroakira Ono - 2006 - Journal of Symbolic Logic 71 (4):1353 - 1384.
Modal translations in substructural logics.Kosta Došen - 1992 - Journal of Philosophical Logic 21 (3):283 - 336.
Substructural Logics.Greg Restall - forthcoming - Stanford Encyclopedia of Philosophy.
Metacompleteness of Substructural Logics.Takahiro Seki - 2012 - Studia Logica 100 (6):1175-1199.
Kripke semantics for modal substructural logics.Norihiro Kamide - 2002 - Journal of Logic, Language and Information 11 (4):453-470.
Normal modal substructural logics with strong negation.Norihiro Kamide - 2003 - Journal of Philosophical Logic 32 (6):589-612.
Deduction theorems for weak implicational logics.M. W. Bunder - 1982 - Studia Logica 41 (2-3):95 - 108.

Analytics

Added to PP
2013-10-27

Downloads
56 (#292,952)

6 months
9 (#356,042)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Hiroakira Ono
Japan Advanced Institute of Science and Technology

References found in this work

Introduction to metamathematics.Stephen Cole Kleene - 1952 - Groningen: P. Noordhoff N.V..
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.

View all 7 references / Add more references