Negative Translations for Affine and Lukasiewicz Logic

Abstract: We investigate four well-known negative translations of classical logic into intuitionistic logic within a substructural setting. We find that in affine logic the translation schemes due to Kolmogorov and G\"odel both satisfy Troelstra's criteria for a negative translation. On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and vice versa. By contrast, in the setting of Lukasiewicz logic, we can prove a general result asserting that a wide class of formula translations including those of Kolmogorov, G\"odel, Gentzen and Glivenko not only satisfy Troelstra's criteria with respect to a natural intuitionistic fragment of Lukasiewicz logic but are all equivalent.