Deductive Nonmonotonic Inference Operations: Antitonic Representations

We provide a characterization of those nonmonotonic inference operations C for which C(X) may be described as the set of all logical consequences of X together with some set of additional assumptions S(X) that depends anti-monotonically on X (i.e., X is a subset of Y implies that S(Y) is a subset of S(X)). The operations represented are exactly characterized in terms of properties most of which have been studied in Freund-Lehmann(cs.AI/0202031). Similar characterizations of right-absorbing and cumulative operations are also provided. For cumulative operations, our results fit in closely with those of Freund. We then discuss extending finitary operations to infinitary operations in a canonical way and discuss co-compactness properties. Our results provide a satisfactory notion of pseudo-compactness, generalizing to deductive nonmonotonic operations the notion of compactness for monotonic operations. They also provide an alternative, more elegant and more general, proof of the existence of an infinitary deductive extension for any finitary deductive operation (Theorem 7.9 of Freund-Lehmann).