A Kuroda-style j-translation

In topos theory it is well-known that any nucleus j gives rise to a translation of intuitionistic logic into itself in a way which generalises the Goedel-Gentzen negative translation. Here we show that there exists a similar j-translation 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. The development is entirely syntactic and no knowledge of topos theory is required to read this small note.