(Professor Lotfi A. Zadeh)

In science—and especially in mathematics—it is a universal practice to express definitions in a language based on bivalent logic. Thus, if C is a concept, then under its definition every object, u, is either an instance of C or it is not, with no shades of gray allowed. This deep-seated tradition—which is rooted in the principle of the excluded middle—is in conflict with reality. Furthermore, it rules out the possibility of graceful degradation, leading to counterintuitive conclusions in the spirit of the ancient Greek sorites paradox.

In fuzzy logic—in contrast to bivalent logic—everything is, or is allowed to be, a matter of degree. This is well known, but what is new is the possibility of employing the recently developed fuzzy-logic-based language precisiated natural language (PNL) as a concept-definition language to formulate definitions of concepts of the form “approximate X,” where X is a crisply defined bivalent-logic-based concept. For example, if X is the concept of a linear system, then “approximate X” would be a system that is approximately linear.

The machinery of PNL provides a basis for a far-reaching project aimed at associating with every—or almost every—crisply defined concept X a PNL-based definition of “approximate X,” with the understanding that “approximate X” is a fuzzy concept in the sense that every object x is associated with the degree to which x fits X, with the degree taking values in the unit interval or a partially ordered set. A crisp definition of “approximate X” is not acceptable because it would have the same problems as the crisp definition of X.

As a simple example, consider the concept of a linear system. Under the usual definition of linearity, no physical system is linear. On the other hand, every physical system may be viewed as being approximately linear to a degree. The question is: How can the degree be defined?

More concretely, assume that I want to get a linear amplifier, A, and that the deviation from linearity of A is described by the total harmonic distortion, h, as a function of power output, P. For a given h(P), then, the degree of linearity may be defined in the language of fuzzy if-then rules – a language which is a sublanguage of PNL. In effect, such a definition would associate with h(P) its grade of membership in the fuzzy set of distortion/power functions which are acceptable for my purposes. What is important to note is that the definition would be local, or, equivalently, context-dependent, in the sense of being tied to a particular application. What we see is that the standard, crisp, definition of linearity is global (universal, context-independent, objective), whereas the definition of approximate linearity is local (context-dependent, subjective). This is a basic difference between a crisp definition of X and PNL-based definition of “approximate X.” In effect, the loss of universality is the price which has to be paid to define a concept, C, in a way that enhances its rapport with reality.

In principle, with every crisply defined X we can associate a PNL-based definition of “approximate X.” Among the basic concepts for which this can be done are the concepts of stability, optimality, stationarity and statistical independence. But a really intriguing possibility is to formulate a PNL-based definition of “approximate theorem.” It is conceivable that in many realistic settings informative assertions about “approximate X” would of necessity have the form of “approximate theorems,” rather than theorems in the usual sense. This is one of the many basic issues which arise when we cross into the uncharted territory of approximate concepts defined via PNL.

A simple example of “approximate theorem” is an approximate version of Fermat’s theorem. More specifically, assume that the equality x^{n} + y^{n} = z^{n} is replaced with approximate equality. Furthermore, assume that x, y, z are restricted to lie in the interval [I,N]. For a given n, the error, e(n), is defined as the minimum of a normalized value of | x^{n} + y^{n} - z^{n} | over all allowable values of x, y, z. Observing the sequence {e(n)}, n = 3,4,…, we may form perceptions described as, say, “for almost all n the error is small;” or “the average error is small;” or whatever appears to have a high degree of truth. Such perceptions, which in effect are summaries of the behavior of e(n) as a function of n, may qualify to be called “approximate Fermat’s theorems.” It should be noted that in number theory there is a sizeable literature on approximate Diophantine equations. There are many deep theorems in this literature, all of which are theorems in the usual sense.

In a sense, an approximate theorem may be viewed as a description of a perception. The concept of a fuzzy theorem was mentioned in my 1975 paper “The Concept of a Linguistic Variable and its Application to Approximate Reasoning.” What was missing at the time was the concept of PNL.

* Professor in the Graduate School and director, Berkeley initiative in Soft Computing (BISC), Computer Science Division and the Electronics Research Laboratory, Department of EECS, Univeristy of California, Berkeley, CA 94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712; E-Mail: zadeh@cs.berkeley.edu. Research supported in part by ONR Grant N00014-00-1-0621, ONR Contract N00014-99-C-0298, NASA Contract NCC2-1006, NASA Grant NAC2-117, ONR Grant N00014-96-1-0556, ONR Grant FDN0014991035, ARO Grant DAAH 04-961-0341 and the BISC Program of UC Berkeley.

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