WHAT IS FUZZY LOGIC?

Is it the paradox of the century?

How can fuzzy thinking be logical? However difficult this may be to accept,Kosko makes it clear that it has come to be a reality and a new successful way to dealwith uncertainty. In particular, we wish to have a computer be capable of reasoning withvague statements that have no probabilistic meaning. This is what we are preparingto illustrate. There is no complete agreement within the fuzzy community on a definitivedefinition for fuzzy logic. However it seems only reasonable to follow a definition accordingto L. Zadeh when he said in a talk he gave at the National Science Foundationthat:

* "In a Narrow sense, fuzzy logic is a logical system that aims at a formalization of approximatereasoning. As such, it is firmly established in multiple valued logic, butits agenda is quite different from that of the traditional Lukasiewicz's logic becausefuzzy logic as logic of approximate reasoning is not part of the traditional multi-valuedlogic. Although originally there was much antagonism, the scientific community hasremarkably changed its attitude. For example, Zadeh has kept a close count, and washappy to announce at a BISC seminar in September 1999, that there were 3240 papers,between 1995 and 1998, containing fuzzy in its title cited in Mathematical Reviews incontrast of only 521 in 1993.

* In a Broad sense, Zadeh often adds, fuzzy logic is almost synonym with fuzzy set theorywhich, as the name suggests, is basically the theory of "classes without boundaries".

Thus, it is highly capable to handle ambiguities and vagueness.

Finally, most of us ingeniously propose that fuzzy logic is an extension of multiple-valuedlogic to the continuum case. With the sharp distinction that in the multiple-valuedcase we are limited to the use of rational numbers within the unit interval, while in thecontinuum case we may use any real number within the unit interval".

Thornber makes a clear case for this claim. In [12], with the introduction of thelinguistic variable, Zadeh makes it even more clear how the tool of fuzzy numbers allowsthe representation and manipulation of the meaning of vague concepts whose value is notprobabilistic, and certainly not stochastic. Why insist that only probability can handle uncertaintywhen we have long recognized different forms of uncertainty. For further explanation,read Novak, see references [11,12]. Different kinds of uncertainty should be treatedand studied accordingly.

EXAMPLE 1. Suppose we say: "Wanted: a cheap house in a good neighborhood with anexcellent school system".

The vagueness, that is absolutely linguistic, of the above statement, an example due tomy good friend Mike Smith, is without a doubt, unsuitable to any attempt at a probabilisticapproach, and keep its original meaning.

EXAMPLE 2. Suppose we say "Matthew is young".

A probabilistic expression of the above would change the statement into somethinglike:

"The probability that Matthew is young is 0.9".

Note the implicit law of the excluded middle: Matthew is either young or he is not.What happens is that we only have a 90 % likelihood of being right in knowing what he isThus, we could have said, using probabilistic language:

"There is only a 90% chance that Matthew is young".

In a fuzzy set context, we would say:

"The membership grade of Matthew within the set of young people is 0.9"or that, by using any of the permissible fuzzy quantifiers:

"Matthew is more or less, [orsomewhat] young".

The semantic is clearly distinct. Again a strikingly distinct semantic that includes quantifiersof a new kind. In other words, the distinction is in the use, in its meaning, thus in thesemantic. We shall often have to return to this distinction that is fundamental to removethe antagonism and the confusion often existing between the idea of probability values anddegrees of membership because most often they end up to be from the unit interval.

In a fuzzy context, we have a continuum of possible valuations which means that mathematicallywe can use anything within the unit interval. If we were able to say that the valueof the truth is either zero or one, then we do not hesitate, and we have a crisp case. It is whenwe are unsure that the truth-value is neither 0 or 1, maybe it is somewhere in between, thenwe may believe that it is a likely event, then we have a probability, when it is not this caseeither then we have a fuzzy case and thus we cannot force a crispness or probability modelingapproach. Thus we must interpolate. Note the frequent mention of the term Interpolativein the quote below. It turns out that these matters lend themselves to the use of fuzzynumbers. Finally, we conclude this introductory section by quoting a concise viewpoint asexpressed by Zadeh at his NSF talk in May 1993. Namely, he said:

"The basic concepts of fuzzy logic are:

* The concept of linguistic variable, that is, a variable whose values are words rather thannumbers;

* The concept of canonical form which represents information as an elastic constrainton a variable;

* The concept of Interpolative reasoning which makes it possible to reason with incompleteinformation.

* Interpolative reasoning plays a key role in human cognition and lies at the basis ofpattern classification, qualitative reasoning, system identification, system modelingand neural network modeling. In the context of fuzzy logic, a concept which is centralto Interpolative reasoning is that a collection of fuzzy if-then rules which serveto provide an approximate characterization of the input–output relation of partiallyspecified system approximation. To fill the gaps in knowledge, a number of differentarchitectures may be employed, prominent among which are disjunctively combinedfuzzy if-then rules with a defuzzifier and the Takagi-Sugeno-Kang fuzzy if-then ruleswith a convex aggregator. The role-played by Interpolative reasoning in the implicationof fuzzy logic. The importance of fuzzy logic derives from the fact that almostall of human reasoning is approximate in nature. In fact, it is the ability to recognizedistorted speech, decipher sloppy handwriting, and, more generally, make rational decisionsin an environment of uncertainty and imprecision". Note that the membershipfunction is not subject to any restrictions other than to be continuous on the intervalof interest.

* EXAMPLE. To emphasize the...