Generalized Possibility and Necessity Measures on Fields of Sets

Abstract. We give a generalization of possibility andnecessity measures: their domains are extended towardsfields of sets, and their codomains towards arbitrarycomplete lattices. In this way, these measures can beassociated with ( Q;fl )-fuzzy sets, where ( Q;fl ) is atleast a poset. An important inconsistency problem, in-tricately linked with this association, is solved. It isargued that order lies at the basis of a mathematicaldescription of vagueness and linguistic uncertainty. Theresults obtained here allow us to mathematically repre-sent and manipulate linguistic uncertainty in the pres-ence of incomparability.Keywords: confidence relation, fuzzy set, linguisticinformation, necessity measure, possibility measure. 1. Introduction The notion of a possibility measure was first intro-duced by Zadeh in 1978 [14], as a mathematicaldescription of the (linguistic) information conveyedby vague propositions such as, for instance, ‘Johnis old’. Zadeh’s course of reasoning can be brieflysummarized as follows. With the predicate ‘old’, heassociates a fuzzy set on the universe