| Summary: | <p>Towards probabilistic formalisms for resolving local inconsistencies under model-theoretic probabilistic entailment, we present probabilistic generalizations of Pearl's entailment in System <em>Z</em> and Lehmann's lexicographic entailment. We then analyze the nonmonotonic and semantic properties of the new notions of entailment. In particular, we show that they satisfy the rationality postulates of System <em>P</em> and the property of Rational Monotonicity. Moreover, we show that model-theoretic probabilistic entailment is stronger than the new notion of lexicographic entailment, which in turn is stronger than the new notion of entailment in System <em>Z</em>. As an important feature of the new notions of entailment in System <em>Z</em> and lexicographic entailment, we show that they coincide with model-theoretic probabilistic entailment whenever there are no local inconsistencies. We also show that the new notions of entailment in System <em>Z</em> and lexicographic entailment are proper generalizations of their classical counterparts. Finally, we present algorithms for reasoning under the new formalisms, and we give a precise picture of its computational complexity.</p>
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