Kuroda's Translation for Higher-Order Logic

Thomas Traversié

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arXiv · 2404.19503

Kuroda's Translation for Higher-Order Logic

Thomas Traversié

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Abstract

Kuroda's translation embeds first-order classical logic into intuitionistic logic, such that a formula and its translation are equivalent in classical logic. Recently, Brown and Rizkallah extended this translation to higher-order logic. However, they showed that the translation fails in the presence of functional extensionality, and they did not prove the classical equivalence between a formula and its translation. In this paper, we emphasize different conditions under which Kuroda's translation works in the presence of functional extensionality, including the double-negation shift. We show that the classical equivalence between a formula and its translation does not necessarily hold in higher-order logic. However, it is sufficient to assume both functional extensionality and propositional extensionality.

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Thomas Traversié. 2024-04-30. Kuroda's Translation for Higher-Order Logic. https://arxiv.org/abs/2404.19503

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