Archive for Mathematical Logic 58 (5-6):627-634 (2019)

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.
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DOI 10.1007/s00153-018-0656-x
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Set Theory and the Continuum Hypothesis.Kenneth Kunen - 1966 - Journal of Symbolic Logic 35 (4):591-592.
Forcing in Proof Theory.Jeremy Avigad - 2004 - Bulletin of Symbolic Logic 10 (3):305-333.
The Peirce Translation.Martín Escardó & Paulo Oliva - 2012 - Annals of Pure and Applied Logic 163 (6):681-692.
Basic Subtoposes of the Effective Topos.Sori Lee & Jaap van Oosten - 2013 - Annals of Pure and Applied Logic 164 (9):866-883.

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