Fuzzy Possibilities As Upper Previsions

In this paper we analyze, mainly in a finitary setting, the consistency properties of fuzzy possibilities, interpreting them as instances of upper previsions and applying the basic notions of avoiding sure loss and coherence from the theory of imprecise probabilities. It ensues that fuzzy possibilities always avoid sure loss, but satisfy the stronger coherence condition only in a special case. Their natural extension, i.e. their least–committal correction to a coherent upper prevision, is determined. The same analysis is then performed when min is replaced by a T–norm (or seminorm) in the definition of fuzzy possibility, showing that the consistency properties and also the natural extension remain the same. Some “closure” properties are also discussed, which are guaranteed to hold if the T–norm is continuous, and are satisfied by (ordinary) possibilities too.