  In retrospect, we can identify two sorts of approaches to nonmonotonic logic: those based on preference and those based on conflict. Theories of the first sort (like circumscription) involve a relatively straightforward modification of the ordinary model-theoretic definition of logical consequence that takes into account a preference relation over models. Theories of the second sort (like default logic) involve a more radical rethinking of logical ideas. The possibility of multiple extensions—different possible coherent, inferentially complete conclusion sets that can be drawn from a single set of premises—means that we have to think of logical consequence not as a function taking a set of axioms into its logical closure, but as a relation between a set of axioms and alternative logical closures. Since logical consequence is so fundamental, this represents a major theoretical departure. With multiple extensions, we can still retrieve a consequence relation between a theory and a formula in various ways, the simplest being to say that DT nonmonotonically implies C if C is a member of every extension of DT. Still, the conflict-based account of consequence provides a much richer underlying structure than the preferential one.