1st order, 2nd order, what next? We do not really need third-order descriptions: a view from a realistic (granular) viewpoint

Abstract

To describe experts' 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 an expert; however, the expert may not be 100% certain what exact 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's 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.