A Kuroda-style j-translation

Abstract

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 Gödel-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.

A General Glivenko–Gödel Theorem for Nuclei

“…This j is well-known to be a nucleus over ⊢ i [8,65], which we call the Glivenko nucleus. As stability (5) equals RAA, the strong extension ⊢ j i of intuitionistic logic ⊢ i is nothing but classical logic ⊢ c .…”

“…Now let jϕ ≡ ¬ϕ → ϕ. This j is a nucleus [8,65], which we call the Peirce nucleus, as it is a special case of the Peirce monad [27]. Over intuitionistic logic, it is easy to show that the Glivenko nucleus is equivalent to the Peirce nucleus, i.e., ¬¬ϕ ≈ i ¬ϕ → ϕ for every ϕ.…”

“…This j is a nucleus, in fact a closed nucleus [8,65]. We refer to this j as the Dragalin-Friedman nucleus.…”

“…Double negation over intuitionistic logic is a typical instance of a nucleus [4,37,43,50,65,74,75,8]. Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of the double negation of that formula [35].…”

“…Glivenko's theorem says that, in propositional logic, classical provability of a formula entails intuitionistic provability of the double negation of that formula [35]. This stood right at the beginning of the success story of negative translations, which have been put into the context of nuclei [8] or monads [27].…”