 We understand the notion of an inference relation in a broad way. In essence, an inference relation maps sets of sentences to sets of sentences such that this mapping is guided by some idea of truth preservation: if all members of a given set of premises are true, then all sentences inferable from this set must be true as well. Moreover, there are inference relations that are guided by a weaker requirement: if all members of a given set of premises are believed, then it is rational to believe a certain set of conclusions. Nonmononotonic logics and formal theories of belief revision define inference relations in this sense. We shall say more about such inference systems in  Section 2.2.  Suffice it to say for now that an inference relation understood in our broad way may be defined syntactically in terms of relations of derivability, or semantically in terms of possible worlds and model-theoretic interpretations, or both ways.