Boolean—Valued and Boolean—Like Processed Belief Functions

Abstract

The reasons for which it may seem useful to reconsider the Dempster—Shafer model of uncertainty quantification and processing from the point of view of possible non-numerical quantification of occurring uncertainty degrees can be divided into two groups: why to refuse the numerical real-valued degrees, and why to choose just this or that set of values and structure over this set as an adequate alternative to the original numerical evaluation. First, there are some general arguments in favour of the claim that structures over sets of abstract objects of non-numerical nature can be sometimes more close to the spaces of uncertain events and structures over them than the space of real numbers with all the riches of notions, relations and operations over these numbers (over-specification of the degrees of uncertainty by real numbers, these degrees need not be dichotomic, a danger of an ontological shift from structures over real numbers to structures over uncertainties, and so on). A more detailed discussion in this direction can be found in Drossos (1990) and Novák (1989), as far as fuzzy sets are concerned, in Bundy (1985) for set-valued probability measures, and in Kramosil (1989,1991) for applications of such probabilities in uncertain data processing expert (knowledge) systems; we shall not repeat this discussion here and refer to these sources.