## Friday, January 25, 2013

### An Important Message about Fuzzy Logic

The recent award of a major scientific prize to Lotfi Zadeh provoked many admirers from around the world to send along their congratulations on a discussion list devoted to fuzzy logic and soft computing. In so doing, these well-wishers also made a variety of comments about fuzzy logic. Professor Zadeh eventually responded with a substantive comment. (See below.) His comment is important and enlightening in several different ways. Although I will let his comment speak for itself, I do wish to emphasize the astonishing reach of fuzzy logic that Zadeh highlights. It is also worth mentioning that the use of fuzzy logic is apparently accelerating rather than plateauing. There is thus reason to think and to hope that an increasing number of legal scholars (in addition to luminaries such as Kevin Clermont and Lothar Phillips) will decide to use fuzzy logic to explore reasoning about and in law. It is high time that they do so!

Dear members of the BISC Group:

The BBVA Award has rekindled discussions and debates regarding what fuzzy logic is and what it has to offer. The discussions and debates brought to the surface many misconceptions and misunderstandings. A major source of misunderstanding is rooted in the fact that fuzzy logic has two different meanings -- fuzzy logic in a narrow sense, and fuzzy logic in a wide sense. Informally, narrow-sense fuzzy logic is a logical system which is a generalization of multivalued logic. An important example of narrow-sense fuzzy logic is fuzzy modal logic. In multivalued logic, truth is a matter of degree. A very important distinguishing feature of fuzzy logic is that in fuzzy logic everything isor is allowed to be, a matter of degree. Furthermore, the degrees are allowed to be fuzzy. W
ide-sense fuzzy logic, call it FL, is much more than a logical system. Informally, FL is a precise system of reasoning and computation in which the objects of reasoning and computation are classes with unsharp (fuzzy) boundaries. The centerpiece of fuzzy logic is the concept of a fuzzy set. More generally, FL may be a system of such systems. Today, the term fuzzy logic, FL, is used preponderantly in its widsense. This is the sense in which the term fuzzy logic is used in the sequel. It is important to note that when we talk about the impact of fuzzy logic, we are talking about the impact of FL. Intellectually, narrow-sense fuzzy logic is an important part of FL, but volume-wise it is a very small part. In fact, most applications of fuzzy logic involve no logic in its traditional sense.

What is not widely recognized within the scientific community and the general public, is that fuzzy logic has become a vast enterprise.There are over 280,000 papers in the literature with fuzzy in title. There are 25 journals with fuzzy in title. There are close to 25,000 fuzzy-logic-related patents issued or applied for in the United States and Japan. There is a long list of applications ranging from digital cameras to fraud detection systems. Particularly worthy of note, on one end, is the fuzzy logic subway system in Sendai, a city of over 1 million in Japan. On the other end, numerically, is Omron's 120 million fuzzy logic blood pressure meters.

Most, but not all of the constituents of fuzzy logic are what are called FL-generalizations
of traditional, bivalent-logic-based systems of reasoning and computation. Examples. Fuzzy arithmetic, fuzzy cluster analysis, fuzzy differential equations, fuzzy control, fuzzy linear programming, etc. FL-generalization of a theory or a formalism, T, involves introduction into T of the concept of a fuzzy set, followed by addition of related concepts and techniques. FL-generalization may be applied to any field, any theory, any system, any formalism and any algorithm. The  fundamental importance of FL-generalization derives from the fact that in the real world almost all classes have unsharp (fuzzy) boundaries. As a consequence, FL-generalization opens the door to construction of better models of reality.

It is of interest to observe that the impact of FL-generalization is growing in visibility and importance in mathematics -- a field in which precision plays a quintessential role. We see a growing number of papers with fuzzy in title imany branches of mathematics, among them topology, algebra, differential equations, group theory, set theory, and functional analysis.
What may come as a surprise to many is that Math.Sci.Net database lists over 22,383 papers with fuzzy in title. I did not anticipate that this will happen when I wrote my first paper on fuzzy sets. My expectation was that the concept of a fuzzy set would find its main applications in the realm of soft, human-centered sciences.

When it comes to practical application of fuzzy logic, there is a major source of misunderstanding. Fundamentally, fuzzy logic is aimed at precisiation of what is imprecise. But in many of its applications fuzzy logic is used, paradoxically to imprecisiate what is precise.
In such applications, there is a tolerance for imprecision, which is exploited through the use of fuzzy logic. Precisiation carries a cost. Imprecisiation reduces cost and enhances tractability. This is what I call the Fuzzy Logic Gambit. What is important to note is  that precision has two different meanings: precision in value and precision in meaning. In the Fuzzy Logic Gambit what is sacrificed is precision in value, but not precision in meaning. More concretely, in the Fuzzy Logic Gambit imprecisiation in value is followed by precisiation in meaning. An example is Yamakawa's inverted pendulum. In this case, differential equations are replaced by fuzzy if-then rules in which words are used in place of numbers. What is precisiated is the meaning of words.

Some critics have been saying that fuzzy logic is a passing fad. This assessment of fuzzy logic fails to recognize that the world we live in is, in large measure, a world of fuzzy classes, and that science has much to gain from shifting its foundation from classicalAristotelian logic to fuzzy logic. Comments are welcome.

Regards to all,

Lotfi

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