Estimating a probability distribution corresponding to the negation of a property

Abstract

Suppose that we know the probability distribution corresponding to some property A. How can we then describe the distribution describing the negation of this property? In this paper, we show that, in general, we need additional information to uniquely determine the desired probability distribution. If we do not have this additional information and we still need to select one of the possible distributions for the negation, which one should we choose? We show that reasonable arguments lead to a formula proposed by Yager (IEEE Trans Fuzzy Syst 23(5):1899–1902, 2014). However, if we apply Yager’s formula to negation, we do not get the original distribution back. We show how to modify Yager’s formula so that the distribution for “not not A” will be the same as the original distribution for the property A.