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CONSERVATION AS TRANSLATION

Part of: Algebraic logic Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  30 January 2025

GIULIO FELLIN Open the ORCID record for GIULIO FELLIN [Opens in a new window]  and
PETER SCHUSTER Open the ORCID record for PETER SCHUSTER [Opens in a new window]
GIULIO FELLIN*
Affiliation:
DIPARTIMENTO DI INFORMATICA UNIVERSITÀ DEGLI STUDI DI VERONA STRADA LE GRAZIE 15 37134 VERONA ITALY E-mail: peter.schuster@univr.it
PETER SCHUSTER
Affiliation:
DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE UNIVERSITÀ DEGLI STUDI DI BRESCIA VIA BRANZE 38 25123 BRESCIA ITALY E-mail: giulio.fellin@unibs.it
*
e-mail: giulio.fellin@univr.it
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Abstract

Glivenko’s theorem says that classical provability of a propositional formula entails intuitionistic provability of the double negation of that formula. This stood right at the beginning of the success story of negative translations, indeed mainly designed for converting classically derivable formulae into intuitionistically derivable ones. We now generalise this approach: simultaneously from double negation to an arbitrary nucleus; from provability in a calculus to an inductively generated abstract consequence relation; and from propositional logic to any set of objects whatsoever. In particular, we give sharp criteria for the generalisation of classical logic to be a conservative extension of the one of intuitionistic logic with double negation.

Keywords

Glivenkonucleusentailment relationnegative translationdouble negation

MSC classification

Primary: 03F50: Metamathematics of constructive systems
Secondary: 03B55: Intermediate logics 03G27: Abstract algebraic logic
Type
Research Article
Information
The Review of Symbolic Logic , First View , pp. 1 - 33
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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Footnotes

This note grew out from a chapter of the first author’s doctoral thesis [33], and is a revised and extended version of a conference paper [35]. As compared to the latter, the main amendments include: the study of j-homogeneous functions and j-translations; a more informative statement of the Gödel conservation theorem; the new Kolmogorov conservation theorem and its corollary, the Kuroda conservation theorem; and an application to lax logic.

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