Quasi Conjunction and Inclusion Relation in Probabilistic Default Reasoning

We study in the setting of probabilistic default reasoning under coherence the quasi conjunction, which is a basic notion for defining consistency of conditional knowledge bases, and the Goodman & Nguyen inclusion relation for conditional events. We deepen two results given in a previous paper: the first result concerns p-entailment from a finite family of conditional events to the quasi conjunction , for each nonempty subset of ; the second result analyzes the equivalence between p-entailment from and p-entailment from , where is some nonempty subset of . We also characterize p-entailment by some alternative theorems. Finally, we deepen the connections between p-entailment and inclusion relation, by introducing for a pair the class of the subsets of such that implies E|H. This class is additive and has a greatest element which can be determined by applying a suitable algorithm. © 2011 Springer-Verlag Berlin Heidelberg.