A Modern Introduction to Fuzzy Mathematics

5Possibility Theory

The English economist George Lennox Sharman Shackle [256] proposed that uncertainty is related to surprise. For example, when Jane says that an event can happen, then she will not be surprised if this event will actually happen. In general, when we say that an event can “happen”, we actually mean that it is a “possible” event. Since there are degrees of surprise, it is absolutely reasonable to talk about degrees of possibility. Thus, in a way, Shackle laid the theoretical basis for a theory of possibility. Indeed, Zadeh [314] used fuzzy sets to define a mathematical theory of possibility. Later on, others tried to describe possibilities using different tools that culminated to what is now known as possibility theory.

5.1 Fuzzy Restrictions and Possibility Theory

Although Zadeh was quite aware of the measure‐theoretic axiomatization of probability¹ (see Ref. [257] for a discussion of Kolmogorov's work), still he opted to define the notion of possibility in terms of fuzzy restrictions [309]. In Zadeh's own words, a fuzzy restriction is “a fuzzy relation which acts as an elastic constraint on the values that may be assigned to a variable.” Here the term “elastic” does not have any special meaning, and it just means that the constraint is not rigid. Let us first give an informal description of fuzzy restrictions.

Consider the fuzzy proposition Emma is young and let denote the value of variable Emma. The value is a whole number that belongs to the interval ...