Fuzzy Set Theory and Chemistry

As chemists, we often employ inexact terms, such as fast/slow, strong/weak, concentrated/dilute, hot/cold, etc., yet we are confident that these words suffice to carry the desired message, unburdened by superfluous precision, within the context of the report. Although quantification of these terms by an appropriate measurement is always possible, it is often unnecessary, as in the specification of color. Indeed, it is generally recognized that terms such as the above play a vital role in the efficient 

A formal classification theory for such terms, which are neither very exact nor very inexact, was introduced by Zadeh, who, among others, saw the need for generalizing the theory of sets to admit the possibility of unsharp boundaries between classes of imprecisely defined objects. Whereas the characteristic function can only have a value of 0 or 1 (in harmony with the Aristotelian logic of the excluded middle), membership in a fuzzy set is a matter of degree. This degree (or grade) of membership (ix) may have any value in the continuum of the real interval [0,1] in the special case where the value of /x is restricted to 0 or 1, the degree of membership becomes the characteristic function. 

The mapping of T from the subdomain of temperatures onto the interval [0, 1] is the membership or compatibility function fx T). The form of this function is subject to some relatively nonarbitrary constraints to be consistent with one s primitive notions of hotness, fx T) should increase monotonically and smoothly with T, and it should be roughly sigmoidal in shape. However, since a whole family of curves will fit this description, the choice of numerical values for the parameters of fi(T) will to a considerable extent be an arbitrary one. Therefore, the threshold criteria, which are given by fi(T), are themselves fuzzy. 

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