Where do we stand on measures of uncertainty, ambiguity, fuzziness, and the like?

Abstract

It is argued in this paper that the theory of fuzzy sets involves at least four fundamentally different types of uncertainty. Each of these types requires a measure by which the degree of uncertainty of that type can be determined. Two main categories of uncertainty are connected with the terms ‘vagueness’ (or ‘fuzziness’) and ‘ambiguity’. In general, vagueness is associated with the difficulty of making sharp or precise distinctions in the world. Ambiguity, on the other hand, is associated with one-to-many relations, i.e., situations with two or more alternatives that are left unspecified. While the concept of a fuzzy set represents a basic mathematical framework for dealing with vagueness, the concept of a fuzzy measure is a general framework for dealing with ambiguity. Several classes of measures of vagueness, usually referred to as measures of fuzziness, have been proposed in the literature. Each class is based on some underlying conception of the degree of fuzziness. A general set of requirements for measures of fuzziness is formulated, followed by an overview of the measures proposed in the literature. Measures of ambiguity are discussed within the framework of plausibility and belief measures. Although it does not cover all fuzzy measures, this framework is sufficiently broad for most practical purposes, and represents a generalization of both probability theory and possibility theory. It is argued that three complementary measures of ambiguity should be employed. One of them is obtained by generalizing the Hartley measure of uncertainty; it measures the degree of nonspecificity in individual situations described by the various belief and plausibility measures. The other two are obtained by generalizing the well known Shannon measure of uncertainty; they measure the degree of dissonance and the degree of confusion in evidence, respectively. Basic mathematical properties of these measures are overviewed. It is also argued that each of the four types of uncertainty measures, which are fundamentally different from each other, can be used for measuring structural (syntactic) information in the same sense as the Hartley and Shannon measures have been used in this respect. As such, these measures are potentially powerful tools for dealing with systems problems such as systems modelling, analysis, simplification, or design.