On the probability of a fuzzy event, with application to nonmonotonic reasoning

The idea of a probability measure for fuzzy events was first studied by Zadeh in 1968. The approach taken there was to adapt the conventional notion of an event as a member of a σ-field of subsets of a sample space Ω to the case where the subsets are fuzzy subsets of Ω. The present work takes a different perspective and employs the concept of a linguistic variable, where the latter was introduced by Zadeh in 1975. First the notion of a fuzzy event is defined. Then follows a definition of the notion of a probability for such an event. It is shown how these notions can be used to provide a semantics for a logic that expresses syllogistic arguments such as `Most Swedes are tall; Helge is a Swede; therefore it is likely that Helge is tall´, where tall is a fuzzy predicate. This entails providing a semantics also for fuzzy quantifiers (most, many, few, etc.) and fuzzy likelihood modifiers (likely, uncertain, unlikely, etc.). Finally, it is outlined how this can be applied to nonmonotonic default reasoning with exceptions.