We give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction.

Abstract. The symmetric λµ-calculus is the λµ-calculus introduced by Parigot in which the reduction rule µ ′ , which is the symmetric of µ, is added. We give arithmetical proofs of some strong normalization results for this calculus. We show (this is a new result) that the µµ ′ -reduction is strongly normalizing for the un-typed calculus. We also show the strong normalization of the βµµ ′ -reduction for the typed calculus: this was already known but the previous proofs use candidates of reducibility where the interpretation of a type was defined as the fix point of some increasing operator and thus, were highly non arithmetical.

In this paper we consider a type system with a universal type ω where any term (whether open or closed, β-normalising or not) has type ω. We provide this type system with a realisability semantics where an atomic type is interpreted as the set of λ-terms saturated by a certain relation. The variation of the saturation relation gives a number of interpretations to each type. We show the soundness and completeness of our semantics and that for different notions of saturation (based on weak head reduction and normal β-reduction) we obtain the same interpretation for types. Since the presence of ω prevents typability and realisability from coinciding and creates extra difficulties in characterizing the interpretation of a type, we define a class U + of the so-called positive types (where ω can only occur at specific positions). We show that if a term inhabits a positive type, then this term is β-normalisable and reduces to a closed term. In other words, positive types can be used to represent abstract data types. The completeness theorem for U + becomes interesting indeed since it establishes a perfect equivalence between typable terms and terms that inhabit a type. In other words, typability and realisability coincide on U + . We give a number of examples to explain the intuition behind the definition of U + and to show that this class cannot be extended while keeping its desired properties.

We give in this paper a short semantical proof of the strong normalization for full propositional classical natural deduction. This proof is an adaptation of reducibility candidates introduced by J.-Y. Girard and simplified to the classical case by M. Parigot.