What does a conditional knowledge base entail?

This paper presents a purely logical approach to nonmonotonic reasoning. A conditional knowledge base, consisting of a set of conditional assertions of the type if ... then ..., represents the explicit defeasible knowledge an agent has about the way the world generally behaves. The set of all conditional assertions entailed by a conditional knowledge base is studied. This set does not grow monotonically with the knowledge base. This paper relies on both a proof-theoretic approach based on cumulative logic, as proposed by D. Gabbay in [Gabbay, 1985], and a semantical approach, as advocated by Y. Shoham in [Shoham, 1988] and [Shoham, 1987]. Two families of consequence relations are central to this work: preferential and rational consequence relations. Those families have been characterized in [Kraus et al., ] and [Lehmann and Magidor, ] as those consequence relations defined respectively by two families of models. Those models are sets of possible states equipped with a preference relation, some states being preferable, i.e. more natural, than others.