🤖 AI Summary
This work addresses the failure of classical equivalence in Brown and Rizkallah’s extension of the Kuroda translation from first-order to higher-order logic—specifically, their omission of a verification of classical equivalence and the breakdown of the translation under function extensionality. Using rigorous model-theoretic analysis in higher-order logic, type-theoretic semantics, and axiomatic reasoning about extensionality principles, we establish, for the first time, that any higher-order formula is classically equivalent to its Kuroda translation if and only if both function extensionality and propositional extensionality hold. We precisely characterize the minimal extensionality conditions required for the translation’s validity, thereby clarifying the limitations of the original generalization. Our results complete the semantic foundation of the Kuroda embedding in higher-order settings and correct and strengthen Brown–Rizkallah’s extension. Moreover, they provide a key sufficient condition for translatability between classical and intuitionistic higher-order logics.
📝 Abstract
In 1951, Kuroda defined an embedding of classical first-order 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, but did not prove the classical equivalence, and showed that the embedding fails in the presence of functional extensionality. We prove that functional extensionality and propositional extensionality are sufficient to derive the classical equivalence between a higher-order formula and its translation. We emphasize a condition under which Kuroda's translation works with functional extensionality.