In the context of non-monotonic reasoning different kinds of consequence relations are defined for reasoning from (possibly) inconsistent information. Examples are consequence relations that are characterized in terms of maximal consistent subsets of the premise set. The *strong consequences* are those formulas that follow by Classical Logic from every maximal consistent subset. The *weak consequences* follow from some maximal consistent subset. The *free consequences* follow from the set of formulas that belong to every maximal consistent subset. In this paper the question is discussed which of these consequence relations should be applied in which reasoning context. First the concerns that are expressed in the literature with respect to the usefulness of the weak consequences are addressed. Then it is argued that making weak inferences is sensible for some application contexts, provided one has a (dynamic) proof theory for the corresponding consequence relation. Such a dynamic proof theory is what adaptive logics offer. Finally, all this is illustrated by means of a very simple adaptive logic reconstruction of the free, strong, and weak consequences