When one wants to draw non-trivial inferences from an inconsistent belief base, a very  natural approach is to take advantage of the maximal consistent subsets of the base. But few inference relations from maximal consistent subsets exist. In this paper we point out new such relations based on selection of some of the maximal consistent subsets, leading thus to inference relations with a stronger inferential power. The selection process must obey some principles to ensure that it leads to an inference relation which is rational. We define a general class of monotonic selection relations for comparing maximal consistent sets. And we show that it corresponds to the class of rational inference relations.