The concept of a linguistic variable and its application to approximate reasoning-III

In the classical approach to probability theory, an event, A, is defined as a member of a o-field,,d, of subsets of a sample space ft. Thus, ifP is a normed measure over a measurable space (a,_~#), the probabibty of A is defined as P(A), the measure of.4, and is a number in the interval [0, I]. There are many real-world problems in which one or more of the basic assumptions which are implicit in the above definition are violated. First, the event, A, is frequently ill-defined, as in the question, “What is the probability that it will be a warm day tomorrow ?” In this instance, the event warm day is a fuzzy event in the sense that there is no sharp dividing line between its occurrence and nonoccurrence. As shown in [48], such an event may be characterized as a fuzzy subset, A, of the sample space Q, with tiA , the membership function of A, being a measurable function. Second, even if A is a well-defined nonfuzzy event, its probability, P(A), may be ill-defined. For example, in response to the question, “What is the probability that the Dow Jones average of stock prices will be higher in a month from now?” it would be patently unreasonable to give an unequivocal numerical answer, e.g., 0.7. In this instance, a vague response like “quite probable,” would be much more commensurate with our lack of understanding of the dynamics of stock prices, and hence a more realistic-if less precisecharacterization of the probability in question.