Quasi Conjunction and Inclusion Relation in Probabilistic Default Reasoning

Abstract

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 $\mathcal{F}$ of conditional events to the quasi conjunction $\mathcal{C}(\mathcal{S})$, for each nonempty subset $\mathcal{S}$ of $\mathcal{F}$; the second result analyzes the equivalence between p-entailment from $\mathcal{F}$ and p-entailment from $\mathcal{C}(\mathcal{S})$, where $\mathcal{S}$ is some nonempty subset of $\mathcal{F}$. 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 $(\mathcal{F},E|H)$ the class of the subsets $\mathcal{S}$ of $\mathcal{F}$ such that $\mathcal{C}(\mathcal{S})$ implies E|H. This class is additive and has a greatest element which can be determined by applying a suitable algorithm.