A cut-free sequent calculus for the bi-intuitionistic logic 2Int

Abstract

Note that the rules for ∧L a , ∨L c , → L c and L a could also be given in the form of two rules, each with only one active formula A or B, as it is for example done in Gentzen's original calculus for the left conjunction rule. We need this single rule formulation, however, in order to get the invertibility of these rules (cf. Lemma 3.3.1 below), which is important for the proof of admissibility of contraction. As said above, the structural rules do not have to be taken as primitive in the calculus but can be shown to be admissible. We want to consider rules for weakening, contraction and cut. Due to the dual nature of the calculus, we need two rules for each of these rules: