Probabilistic entailment in the setting of coherence: The role of quasi conjunction and inclusion relation Original Research Article

In this paper, by adopting a coherence-based probabilistic approach to default reasoning, we focus the study on the logical operation of quasi conjunction and the Goodman–Nguyen inclusion relation for conditional events. We recall that quasi conjunction is a basic notion for defining consistency of conditional knowledge bases. By deepening some results given in a previous paper we show that, given any finite family of conditional events image and any nonempty subset image of image, the family image p-entails the quasi conjunction image; then, given any conditional event image, we analyze the equivalence between p-entailment of image from image and p-entailment of image from image, where image is some nonempty subset of image We also illustrate some alternative theorems related with p-consistency and p-entailment. Finally, we deepen the study of the connections between the notions of p-entailment and inclusion relation by introducing for a pair image the (possibly empty) class image of the subsets image of image such that image implies image. We show that the class image satisfies many properties; in particular image is additive and has a greatest element which can be determined by applying a suitable algorithm.