Abstract. This paper revisits the results of Barendregt and Ghilezan [3] and generalizes them for classical logic. Instead of λ-calculus, we use here λµ-calculus as the basic term calculus. We consider two extensionally equivalent type assignment systems for λµ-calculus, one corresponding to classical natural deduction, and the other to classical sequent calculus. Their relations and normalisation properties are investigated. As a consequence a short proof of Cut elimination theorem is obtained.

We present a judgemental formulation of natural deduction for classical logic, similar in spirit to Wadler’s dual calculus, but founded on the logical judgements A true and A false; proof-by-contradiction, which puts these two judgements in opposition, lies at the heart of our system. We then show directly a normalization property for this system by a purely syntactic structural induction. 1

Abstract not found

received..., revised..., accepted.... Combinatory logic shows that bound variables can be eliminated without loss of expressiveness. It has applications both in the foundations of mathematics and in the implementation of functional programming languages. The original combinatory calculus corresponds to minimal implicative logic written in a system “à la Hilbert”. We present in this paper a combinatory logic which corresponds to propositional classical logic. This system is equivalent to the system