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Abstract: In this paper, we propose a formalization of non-monotonic reasoning using a three-valued logic based on the strong definitions of Kleene. Non-monotonic reasoning is the rule rather than the exception when agents, human or machine, must act where information about the environment is uncertain or incomplete. Information about the environment is subject to change due to external causes, or may simply become outdated. This implies that inferences previously made may no longer hold and in turn must be retracted along with the revision of other information dependent on the retractions. We start by extending Kleene's three-valued logic with an ``external negation'' connective (-) where -A is true when A is false or unknown. In addition, a default operator D is added where DA is interpreted as ``A is true by default''. The addition of the default operator increases the expressivity of the language, where statements such as ``A is not a default'' are directly representable. The logic has an intuitive model theoretic semantics without any appeal to the use of a fixpoint semantics for the default operator. The semantics is based on the notion of preferential entailment, where a set of sentences Z, preferentially entails a sentence A, if and only if a preferred set of the models of Z are models of A. We also show that the logic belongs to the class of cumulative non-monotonic formalisms which are a subject of current interest.
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