ABSTRACT
This chapter is devoted to a quantitative theory of possibility. We will use some facts about random sets to provide a firm foundation for possibility measures.
As discussed earlier, uncertainty arises in real-world problems in various forms, of which randomness is only one. In formulating the structure of a random experiment, such as games of chance, we list the collection of all possible outcomes of the experiment. It is on the concept of chance that we focus our attention. The theory of probability is first of all intended to provide an acceptable concept of quantitative chance. Despite the fact that events can be possible or probable, the lack of a quantitative theory of possibility has led us to focus only on probability. One of the contributions of probability theory to science is the rigorous quantification of the concept of chance, together with a solid theory of this quantification. This chapter is focused on the quantification of another type of uncertainty, namely possibility. But it is appropriate to review basic facts about probability theory, especially the concept of random sets, which will play an essential role in developing the foundation for other uncertainty measures, including belief functions in Chapter 10.
