2. Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems
Timothy J. Ross, W.J. Parkinson · Society for Industrial and Applied Mathematics eBooks · 2002
2.1 Introduction Making decisions about processes that contain nonrandom uncertainty, such as the uncertainty in natural language, with the use of classical theories has been shown to be less than perfect. Lotfi Zadeh suggested that set membership is the key to decision-making when faced with linguistic and nonrandom uncertainty. In fact, Dr. Zadeh stated in his seminal paper of 1965: “The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables.” Suppose we are interested in the height of people. We can easily assess whether someone is over 6 feet tall. In a binary sense, this person either is or is not, based on the accuracy, or imprecision, of our measuring device. For example, if “tall” is a set defined as heights equal to or greater than 6 feet, a computer would not recognize an individual of height 5′11.999″ as being a member of the set “tall.” But how do we assess the uncertainty in the following question: Is the person nearly 6 feet tall? The uncertainty in this case is due to the vagueness, or ambiguity, of the adjective nearly. A 5′11″ person clearly could be a member of the set “nearly 6 feet tall” people. In the first situation, the uncertainty of whether a person's height, which is unknown, is 6 feet or not is binary; it either is or is not, and we can produce a probability assessment of that prospect based on height data from many people. But the uncertainty of whether a person is nearly 6 feet tall is nonrandom.