*Ken Satoh*

This paper presents a formal relationship for . probability theory and a class of nonmonotonic reasoning which we call laxy nonmonotonic reasoning. In lazy nonmonotonic reasoning, nonmonotonicity emerges only when new added knowledge is contradictory to the previous belief. In this paper, we consider nonmonotonic reasoning in terms of consequence relation. A consequence relation is a binary relation over formulas which expresses that a formula is derivable from another formula under inference rules of a considered system. A consequence relation which has lazy nonmonotonicity is called a rational consequence relation studied by Lehmann and Magidor (1988). We provide a probabilistic semantics which characterizes a rational consequence relation exactly. Then, we show a relationship between propositional circumscription and consequence relation, and apply this semantics to a consequence relation defined by propositional circumscription which has lazy nonmonotonicity.

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