Abstract 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 inference relations based on selection of some 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 subsets and show that it corresponds to the class of rational inference relations