Nonmonotonic Reasoning, Preferential Models and Cumulative Logics
  Many systems that exhibit nonmonotonic behavior have been described and
studied already in the literature. The general notion of nonmonotonic
reasoning, though, has almost always been described only negatively, by the
property it does not enjoy, i.e. monotonicity. We study here general patterns
of nonmonotonic reasoning and try to isolate properties that could help us map
the field of nonmonotonic reasoning by reference to positive properties. We
concentrate on a number of families of nonmonotonic consequence relations,
defined in the style of Gentzen. Both proof-theoretic and semantic points of
view are developed in parallel. The former point of view was pioneered by D.
Gabbay, while the latter has been advocated by Y. Shoham in. Five such families
are defined and characterized by representation theorems, relating the two
points of view. One of the families of interest, that of preferential
relations, turns out to have been studied by E. Adams. The "preferential"
models proposed here are a much stronger tool than Adams' probabilistic
semantics. The basic language used in this paper is that of propositional
logic. The extension of our results to first order predicate calculi and the
study of the computational complexity of the decision problems described in
this paper will be treated in another paper.
