A General Glivenko-G\"odel Theorem for Nuclei

TL;DR

This paper generalizes Glivenko's theorem to arbitrary nuclei and abstract consequence relations, providing a broad conservation result applicable across various logical systems and including first-order logic.

Contribution

It extends Glivenko's theorem from double negation to general nuclei and from propositional logic to arbitrary objects, with precise criteria for validity.

Findings

Provides a generalized conservation theorem for nuclei

Includes G"odel's counterpart for first-order logic

Introduces a form of the deduction theorem for positive logic

Abstract

Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of double negation of that formula. We generalise Glivenko's theorem 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. The resulting conservation theorem comes with precise criteria for its validity, which allow us to instantly include G\"odel's counterpart for first-order predicate logic of Glivenko's theorem. The open nucleus gives us a form of the deduction theorem for positive logic, and the closed nucleus prompts a variant of the reduction from intuitionistic to minimal logic going back to Johansson.