A short proof of Glivenko theorems for intermediate predicate logics

Archive for Mathematical Logic 52 (7-8):823-826 (2013)
  Copy   BIBTEX

Abstract

We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic

Other Versions

No versions found

Links

PhilArchive

    This entry is not archived by us. If you are the author and have permission from the publisher, we recommend that you archive it. Many publishers automatically grant permission to authors to archive pre-prints. By uploading a copy of your work, you will enable us to better index it, making it easier to find.

    Upload a copy of this work     Papers currently archived: 106,169

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.
On the proof theory of the intermediate logic MH.Jonathan P. Seldin - 1986 - Journal of Symbolic Logic 51 (3):626-647.
Embedding classical in minimal implicational logic.Hajime Ishihara & Helmut Schwichtenberg - 2016 - Mathematical Logic Quarterly 62 (1-2):94-101.
Completeness and incompleteness for intuitionistic logic.Charles Mccarty - 2008 - Journal of Symbolic Logic 73 (4):1315-1327.
A proof-search procedure for intuitionistic propositional logic.R. Alonderis - 2013 - Archive for Mathematical Logic 52 (7-8):759-778.
Conservation as Translation.Giulio Fellin & Peter Schuster - 2025 - Review of Symbolic Logic 18 (1):316-348.

Analytics

Added to PP
2013-11-23

Downloads
59 (#398,686)

6 months
11 (#332,542)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

Conservation as Translation.Giulio Fellin & Peter Schuster - 2025 - Review of Symbolic Logic 18 (1):316-348.
Decidable variables for constructive logics.Satoru Niki - 2020 - Mathematical Logic Quarterly 66 (4):484-493.

Add more citations

References found in this work

Introduction to metamathematics.Stephen Cole Kleene - unknown - Groningen: P. Noordhoff N.V..
Introduction to mathematical logic..Alonzo Church - 1944 - Princeton,: Princeton university press: London, H. Milford, Oxford university press. Edited by C. Truesdell.
Basic proof theory.A. S. Troelstra - 2000 - New York: Cambridge University Press. Edited by Helmut Schwichtenberg.
Applications of trees to intermediate logics.Dov M. Gabbay - 1972 - Journal of Symbolic Logic 37 (1):135-138.

View all 8 references / Add more references