# Probabilistic Lexicographic Entailment under Variable−Strength Inheritance with Overriding

*Thomas Lukasiewicz*

### Abstract

In previous work, I have presented approaches to nonmonotonic probabilistic reasoning, which is a probabilistic generalization of default reasoning from conditional knowledge bases. In this paper, I continue this exciting line of research. I present a new probabilistic generalization of Lehmann's lexicographic entailment, called lex_lambda-entailment, which is parameterized through a value lambda in [0,1] that describes the strength of the inheritance of purely probabilistic knowledge. Roughly, the new notion of entailment is obtained from logical entailment in model-theoretic probabilistic logic by adding (i) the inheritance of purely probabilistic knowledge of strength lambda, and (ii) a mechanism for resolving inconsistencies due to the inheritance of logical and purely probabilistic knowledge. I also explore the semantic properties of lex_lambda-entailment.