Classical Proofs via Basic Logic

Cut-elimination, besides being an important tool in proof-theory, plays a central role in the proofs-as-programs paradigm. In recent years this approach has been extended to classical logic (cf. Girard 1991, Parigot 1991, and recently Danos Joinet Schellinx 1997). This paper introduces a new sequent calculus for (propositional) classical logic, indicated by ? C. Both, the calculus and the cut-elimination procedure for ? C extend those for basic logic (Sambin Battilotti Faggian 1997). Two new structural rules are introduced, namely transfer and separation. As in basic logic, the cut rule has two forms, corresponding to substitution on the left and on the right, resulting in a tighter control over the cut. The control over the structural rules, achieved once they are kept distinct from the operational rules, results in a ne control over the form of the derivations. These features of ? C beneet both the proof search and the cut-elimination process. In relation to the framework of basic logic, a remarkable result is that the extensions of basic logic (the ones that are "symmetric") are obtained by means of structural rules. Also, and in agreement with the spirit of uniformity propound in Sambin 97], the procedure given here provides technical tools that allow us to treat cut-elimination for all such logics in a modular way. Cut-elimination, besides being an important tool in proof-theory, plays a central role in the proofs-as-programs paradigm. As a way of computing with proofs, cut-elimination depends on the system in which the proofs are written. As remarked by Jean-Yves Girard in Girard 1991, p.256], cut-elimination for classical logic is a \very clumsy" process, \full of very bad critical pairs" and where we have \not the slightest way of control". Nonetheless, in the last few years the proofs-as-programs paradigm has been extended to classical logic, and new computational systems have been presented (cf. Girard 1991],,Parigot 1991], and recently Danos Joinet Schellinx 1997]), improving the cut-elimination process of Gentzen's calculus LK. As pointed out by Danos, Joinet and Schellinx, the interest in the extraction of programs from classical proofs lies in the potential access to other proofs, and thus to other, and possibly more eecient, algorithms. This paper introduces a new sequent calculus for propositional classical logic, indicated by ? C. It arises in the framework of basic logic ((Sambin Battilotti ? This work was written while the author was a visitor at Imperial College, London and was …