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2. MEMBERSHIP FUNCTIONS

Since Zadeh (1965) introduced fuzzy sets, the main difficulties have been with the meaning and measurement of membership functions as well as their extraction, modification, and adaptation to dynamically changing conditions.

2.1. MEANING OF MEMBERSHIP

In particular, lack of a consensus on the meaning of membership functions has created some confusion. This confusion is neither bizarre nor unsound. However, this cloud of confusion has already been diffused with rigorous semantics and practical elicitation methods for membership functions (Bilgic and Türkos¸en,1996).

In general, there are various interpretations as to where fuzziness might arise from. Depending on the interpretation of fuzziness one subscribes to, the meaning attached to the membership function changes. It is the objective of this subsection to review very briefly the various interpretations of membership functions.

We first start with the formal (i.e., mathematical) definition of a membership function. A fuzzy (sub)set, say F, has a membership function mF, defined as a function from a well-defined universe (the referential set), X, into the unit interval as: µF: X --> [0,1] (Zadeh, 1965).

Thus, the vague predicate "temperature (x = 35°C) is high (H)" for a summer day is represented by a number in the unit interval, µH(x) [element of] [0,1]. There are several possible answers to the question, "What does it mean to say [element of]H(x) = 0.7?":

Likelihood view: 70% of a given population declares that the temperature value of 35°C is high.
Random set view: 70% of a given population describes "high" to be in an interval containing the given temperture value, 35°C.
Similarity view: the given temperature value, 35°C, is away from the prototypical temperature value, say 45°C, which is considered truly "high" to the degree 0.3 (a normalized distance).
Utility view: 0.7 is the utility of asserting that the given temperature value, 35°C, is high.
Measurement view: When compared to other temperatures, the given temperature value, 35°C, is higher than some and this fact can be encoded as 0.7 on some scale.

It is important to realize that each of these interpretations is a hypotheses about where fuzziness arises from and each interpretation suggests a calculus for manipulating membership functions. The calculus of fuzzy sets as described by Zadeh (1965; 1973) and his followers is sometimes appropriate as the calculus of fuzziness but sometimes inappropriate, depending on the interpretation.

2.2. GRADE OF MEMBERSHIP

When someone is introduced to the fuzzy set theory, the concept of the grade of membership sounds fairly intuitive since this is just an extension of a well-known concept. That is, one extends the notion of membership in a set to "a grade of membership" in a set. However, this extension is quite a bit more demanding: "How can a grade of membership be measured?" This question has been considered in the context of many-valued sets by many people from different disciplines.

Although more than 2000 years ago, Aristotle commented on an "indeterminate membership value," the interest in formal aspects of many-valued sets started in early 1900s (McCall and Ajdukiewicz, 1967; Rosser and Turquette, 1977). But the meaning of multiple membership values has not been explained satisfactorily. For some, this is sufficient to discard many-valued sets altogether (Kneale, 1962; French, 1984). On the other hand, the intellectual curiosity never lets go of the subject (Scott, 1976).

Part of the confusion arises from the fact that in two-valued theory, both the membership assignments, i.e., {0,1}, to a set and the truth value assignments {F,T} to a proposition are taken to be the same without a loss of generality. But this creates a problem in infinite-valued theory. As explained recently (Türks¸en,1966), the graded membership assignments in a fuzzy set must be separated and distinguished from the truthood assignments given to a fuzzy proposition.

In general and in particular, anyone who is to use fuzzy sets must answer the following three questions:

  1. What does graded membership mean?
  2. How is it measured?
  3. What operations are meaningful to perform on it?

To answer the first question, one has to subscribe to a certain view of fuzziness. Basically, there has been two trends in the interpretations of fuzziness: those who think that fuzziness is subjective as opposed to objective, and those who think that fuzziness stems from an individual's use of words in a language as opposed to a group of people's use of words (or individual sensor readings versus a group of sensor readings, etc.).

Both the likelihood and the random set views of the membership function implicitly assume that there is more than one evaluator or experiments are repeated. Therefore, if one thinks of membership functions as "meaning representation," one comes close to the claim that "meaning is essentially objective" and fuzziness arises from inconsistency or errors in measurement. On the other hand, during the initial phases of the development of fuzzy sets, it has been widely accepted that membership functions are subjective and context dependent (Zadeh, 1965; 1975). The similarity and utility views of the membership function differ from the others in their espousing a subjective interpretation. The measurement view is applicable to both the subjective and objective views in the sense that the problem can be defined in both ways, depending on the observer(s) who is (are) making the comparison. The comparisons can be results of subjective evaluations or results of "precise" (or idealized) measurements (Bilgic and Türks¸en,1996).


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