It has been known for six years that the restriction of Girard's polymorphic system F to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each β-reduction step of the full intuitionistic propositional calculus translates into one or more βη-reduction steps in the restricted Girard system. As a consequence, we obtain a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to η-conversions.

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that 'one tends to think that natural deduction should be modified to correct such atrocities.' We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions.

It is known that the β-conversions of the full intuitionistic propositional calculus (IPC) translate into βη-conversions of the atomic polymorphic calculus F at . Since F at enjoys the property of strong normalization for βη-conversions, an alternative proof of strong normalization for IPC considering β-conversions can be derived. In the present paper we improve the previous result by analyzing the translation of the η-conversions of the latter calculus into a technical variant of the former system (the atomic polymorphic calculus F ∧ at ). In fact, from the strong normalization of F ∧ at we can derive the strong normalization of the full intuitionistic propositional calculus considering all the standard (β and η) conversions.

Several proof translations of classical mathematics into intuitionistic (or even minimal) mathematics have been proposed in the literature over the past century. These are normally referred to as negative translations or doublenegation translations. Amongst those, the most commonly cited are translations due to Kolmogorov, Gödel, Gentzen, Kuroda and Krivine (in chronological order). In this paper we propose a framework for explaining how these different translations are related to each other. More precisely, we define a notion of a (modular) simplification starting from Kolmogorov translation, which leads to a partial order between different negative translations. In this derived ordering, Kuroda, Krivine and Gödel-Gentzen are minimal elements. A new minimal translation is introduced.

It is known that there is a sound and faithful translation of the full intuitionistic propositional calculus into the atomic polymorphic system Fat, a predicative calculus with only two connectives: the conditional and the second-order universal quantifier. The faithfulness of the embedding was established quite recently via a model-theoretic argument based in Kripke structures. In this paper we present a purely proof-theoretic proof of faithfulness. As an application, we give a purely proof-theoretic proof of the disjunction property of the intuitionistic propositional logic in which commuting conversions are not needed.