Nonmonotonicity, Fuzziness, and Multi-Values

Abstract

The use of fuzzy sets to represent extensions of predicates has as a consequence that the truth of a predicate belongs to the interval [0,1]. In this case the underlying logic is a many-valued one in which the law of excluded middle does not hold. This is due to the presence of a truth-value which expresses ignorance about whether an object has a property or not without rejecting the possibility that it might have this property. This is exactly the type of knowledge used in non-monotonic reasoning systems which allows a fact to be asserted as true by default. We capitalize on the natural existence of such a truth value when interpreting fuzzy predicates and propose a formalization of a fuzzy non-monotonic logic. We start by extending fuzzy logic with two connectives M and L where Ma reads as “it may be the case that α is true” and Lα reads as “it is the case that ± is true. In addition, a default operator D is added where D± is interpreted as “a is true by default”. 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. The logic also belongs to the class of cumulative non-monotonic formalisms which are a subject of current interest.