NM3 — A three-valued cumulative non-monotonic formalism

Abstract

In this paper, we propose a formalization of non-monotonic reasoning using a three-valued logic based on the strong definitions of Kleene. We start by extending Kleene's three-valued logic with an “external negation” connective where ∼ α is true when α is false or unknown. In addition, a default operator D is added where Dα is interpreted as “α is true by default”. The addition of the default operator increases the expressivity of the language, where statements such as “α 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 Γ preferentially entails a sentence α, if and only if a preferred set of the models of Γ are models of α. We also show that the logic belongs to the class of cumulative non-monotonic formalisms which are a subject of current interest.