Kuroda's Translation for Higher-Order Logic

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.