Weak Nonmonotonic Probabilistic Logics

Abstract

We present an approach where probabilistic logic is combined with default reasoning from conditional knowledge bases in Kraus et al.'s System P, Pearl's System Z, and Lehmann's lexicographic entailment. The resulting probabilistic generalizations of default reasoning from conditional knowledge bases allow for handling in a uniform framework strict logical knowledge, default logical knowledge, as well as purely probabilistic knowledge. Interestingly, probabilistic entailment in System P coincides with probabilistic entailment under g-coherence from imprecise probability assessments. We then analyze the semantic and nonmonotonic properties of the new formalisms. It turns out that they all are proper generalizations of their classical counterparts and have similar properties as them. In particular, they all satisfy the rationality postulates of System P and some Direct Inference property. Moreover, probabilistic entailment in System Z and probabilistic lexicographic entailment both satisfy the property of Rational Monotonicity and some Irrelevance property, while probabilistic entailment in System P does not. We also analyze the relationships between the new formalisms. Here, probabilistic entailment in System P is weaker than probabilistic entailment in System Z, which in turn is weaker than probabilistic lexicographic entailment. Moreover, they all are weaker than entailment in probabilistic logic where default sentences are interpreted as strict sentences. Under natural conditions, probabilistic entailment in System Z and lexicographic entailment even coincide with such entailment in probabilistic logic, while probabilistic entailment in System P does not. Finally, we also present algorithms for reasoning under probabilistic entailment in System Z and probabilistic lexicographic entailment, and we give a precise picture of its complexity.