Kuroda’s Translation for Higher-order Logic
Year: 2026
Type: article
Abstract: Kuroda’s translation embeds first-order classical logic into first-order intuitionistic logic. It satisfies the soundness property—if a judgment is provable in classical logic, then its translation is provable in intuitionistic logic—as well as the characterization property—a formula and its translation are equivalent in classical logic—and the reverse translation property—if the translation of a judgment is provable in intuitionistic logic then the original judgment is provable in classical logic. Brown and Rizkallah extended Kuroda’s translation to higher-order logic. However, they showed that the soundness property fails in the presence of functional extensionality, and they neither proved the characterization property nor the reverse translation property. In this article, 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 characterization property and the reverse translation property do not necessarily hold in higher-order logic. Nevertheless, we prove that it is sufficient to assume both functional extensionality and propositional extensionality.
Source: Journal of Symbolic Logic
Author: THOMAS TRAVERSIÉ
Institutions: Centre National de la Recherche Scientifique, Institut national de recherche en sciences et technologies du numérique, Université Paris-Saclay, École Normale Supérieure - PSL, CentraleSupélec, Mathématiques et Informatique pour la Complexité et les Systèmes, DEDUCTEAM: Deduction modulo, interopérabilité et démonstration automatique
Language: English
Topic: Advanced Algebra and Logic
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W4396244392