On the Relation Between Various Negative Translations

Abstract

Several proof translations of classical mathematics into intuitionistic (or even minimal) mathematics have been proposed in the literature over the past century. These are normally referred to as negative translations or doublenegation translations. Amongst those, the most commonly cited are translations due to Kolmogorov, Gödel, Gentzen, Kuroda and Krivine (in chronological order). In this paper we propose a framework for explaining how these different translations are related to each other. More precisely, we define a notion of a (modular) simplification starting from Kolmogorov translation, which leads to a partial order between different negative translations. In this derived ordering, Kuroda, Krivine and Gödel-Gentzen are minimal elements. A new minimal translation is introduced.

Double Negation Semantics for Generalisations of Heyting Algebras

“…Hence, by (DNS2), (A N 1 → A) N 2 is intuitionistically valid. With a further assumption that these translations are modular (see [11]), we also have (…”

“…Schemes for translating classical logic into intuitionistic logic have been studied since the 1920s and are important for understanding the computational content of classical logic. These so-called negative translations or double negation translations such as those proposed by Kolmogorov, Gödel, Gentzen and Glivenko are generally presented as syntactic translations and are studied by mainly syntactic methods (e.g., see [9,11]). In this paper we use an algebraic framework for investigating proposed double negation translations.…”

Subframization and stabilization for superintuitionistic logics

“…Remark 6.2. The translation τ is a version of the Gödel-Gentzen translation (see, e.g., [18]). It has been pointed out by Aczel Recall that by Notation 5.4, given an S-frame (F, G), we always assume that F = (X, ≤) and G = (S, ≤).…”

Postponement of $$\mathsf {raa}$$ raa and Glivenko’s Theorem, Revisited

“…A consequence of this fact, together with Remark 13.1 below, is that the translation A → ¬¬A m (resp. A → ¬¬A j ) is modular negative according to [8]. Besides, the modular negative translations A → ¬¬A m and A → ¬¬A m ′ (resp.…”

“…Besides, the modular negative translations A → ¬¬A m and A → ¬¬A m ′ (resp. A → ¬¬A j and A → ¬¬A j ′ ) are the same according to [8,Definition 2], in the sense that they are interderivable with respect to minimal logic. 21 However, quite interestingly, they have a very different behavior with respect to the postponement of raa: only the negative translation A → ¬¬A m (resp.…”