A Kuroda-style j-translation

Abstract

A nucleus is an operation on the collection of truth values which, like double negation in intuitionistic logic, is monotone, inflationary, idempotent and commutes with conjunction. Any nucleus determines a proof-theoretic translation of intuitionistic logic into itself by applying it to atomic formulas, disjunctions and existentially quantified subformulas, as in the Gödel–Gentzen negative translation. Here we show that there exists a similar translation of intuitionistic logic into itself which is more in the spirit of Kuroda’s negative translation. The key is to apply the nucleus not only to the entire formula and universally quantified subformulas, but to conclusions of implications as well.