this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as

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. 1

LAMA- Équipe de logique, Université de Savoie, F-73376 Le Bourget du Lac, France In this paper, we introduce the λµ ∧ ∨- call-by-value calculus and we give a proof of the Church-Rosser property of this system. This proof is an adaptation of that of Andou (2003) which uses an extended parallel reduction method and complete development. Keywords: Call-by-value, Church-Rosser, Propositional classical logic, Parallel reduction, Complete development 1

Confluency property of the call-by-value λµ ∧ ∨-calculus