On Császár's condition in nonmonotonic reasoning

Császár’s condition is a well-known property introduced about 50 years ago in the axiomatic theory of conditional probability. In recent years such condition has been reconsidered by some authors, who have studied its role in the coherence-based approach to conditional probability. In this paper we consider the probabilistic entailment of a conditional knowledge base by another one. We represent Loop rule in a generalized way and, using Császár’s condition, we give a simple probabilistic interpretation of it. Then, exploiting the rules Cautious Monotonicity and Cut, we obtain some related results on p-entailment by the knowledge base associated with Loop rule. We also determine the best probability bounds for the quasi-conjunction of two conditional events and we give a probabilistic semantics for the QAND rule. Finally, we reconsider our results in the setting of conditional objects.