Fuzzy Sets and Systems 50 (1992) 279-292 279 North-Holland On the specificity of a possibility distribution Ronald R. Yager Machine Intelligence Institute, lona College, New Rochelle, NY, USA Received January 1991 Revised August 1991 Abstract: The specificity of a possibility distribution measures the degree to which the distribution allows one and only one element as its manifestation. As such it is a measure of amount of uncertainty or information. We investigate a number of issues related to specificity measures. We discuss the connection between the specificity of a possibility distribution and the entropy of a probability distribution. We describe unifying view for constructing specificity measures. We look at the relationship of the specificity of a distribution and its negation. We consider the case where the base set is continuous. Keywords: Specificity; uncertainty measure; possibility distribution; information; cardinality; fuzzy sets. 1. Introduction Assume V is a variable taking its value in the set X. A proposition in the theory of approxi- mate reasoning is a statement of the form V isA where A is a fuzzy subset of X. As suggested by Zadeh [16] such a proposition induces a possibility distribution/7 defined on the space X. In particular for each x ~ X, H(x) = A(x), where A(x) is the membership grade of x in A. In this environment /7(x) indicates the degree to which x is a possible value of V. The concept of a possibility distribution is in the spirit of a probability distribution. These two kinds of distributions both provide information about a variable whose actual value is uncertain. In Correspondence to: Prof. R.R. Yager, Machine Intellig- ence Institute, Iona College, New Rochelle, NY 10801, USA. studying these kinds of distributions one is concerned with the amount of information provided by a particular possibility or probability distribution. In the framework of probability theory the Shannon entropy is commonly used to measure the amount of information contained in a probability distribution. In possibility theory the measure of specificity can be used in an analogous manner. The concept of specificity was originally introduced by Yager [6-14] to measure the degree to which a fuzzy subset contains one and only one element. This measure can also be used to indicate the degree to Which a possibility distribution allows one and only one element as possible. Specificity provides an important measure of the amount of information contained in a possibility distribution. We recall the basic properties of a measure of specificity. Assume/7 is a possibility distribution defined on the finite set X, /7:X---~ [0, 1]. The specificity of/1, denoted Sp(/7), is a value lying in the unit interval such that (1) Sp(/7) = 1 iff there exists one and only one x* ~ X with I-l(x*) = 1 and/7(x) = 0 for all other xq:x*. (2) If 1-I(x) = 0 for all x then Sp(/7) = 0. (3) If /71 and /72 are normal I and /71(x)>~ II2(x) for all x then Sp(/7,) ~< Sp(/-/2). (4) If H1 and /72 are normal and crisp 2 possibility distributions then if card(//0 ~< card(/72) (where card(II) = ~i l-l(xi)) ~We recall a possibility distribution is normal if there exists at least one element with possibility one. 2A possibility distribution H is crisp if H(x) is either 1 or 0. 0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved