On Various Negative Translations

Gilda Ferreira
Paulo Oliva

Several proof translations of classical mathematics into intuitionistic mathematics have been proposed in the literature over the past century. These are normally referred to as negative translations or double-negation translations. Among those, the most commonly cited are translations due to Kolmogorov, Godel, 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 and Krivine are minimal elements. Two new minimal translations are introduced, with Godel and Gentzen translations sitting in between Kolmogorov and one of these new translations.

In Steffen van Bakel, Stefano Berardi and Ulrich Berger: Proceedings Third International Workshop on Classical Logic and Computation (CL&C 2010), Brno, Czech Republic, 21-22 August 2010, Electronic Proceedings in Theoretical Computer Science 47, pp. 21–33.
Published: 27th January 2011.

ArXived at: http://dx.doi.org/10.4204/EPTCS.47.4 bibtex PDF

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