1st Order, 2nd Order, What Next? Do We Really Need Third-Order Descriptions: A View from a Realistic (Granular) Viewpoint

To describe experts\u27 uncertainty in a knowledge-based system, we usually use numbers from the interval [0,1] (subjective probabilities, degrees of certainty, etc.). The most direct way to get these numbers is to ask the expert; however, the expert may not be 100\% certain what exactly number describes his uncertainty; so, we end up with a second-order uncertainty - a degree of certainty describing to what extent a given number d adequately describes the expert\u27s uncertainty about a given statement A. At first glance, it looks like we should not stop at this second order: the expert is probably as uncertain about his second-order degree as about his first-order one, so we need third order, fourth order descriptions, etc. In this paper, we show that from a realistic (granular) viewpoint, taking into consideration that in reality, an expert would best describe his degrees of certainty by a word from a finite set of words, it is sufficient to have a second-order description; from this viewpoint, higher order descriptions can be uniquely reconstructed from the second-order one, and in this sense, the second-order description is sufficient