                       Excerpt from Computer Design
                                April 1992
         
     
                    The Seven Noble Truths Of Fuzzy Logic
                                by Earl Cox
     
                                
                                TRUTH ONE
                 There Is Nothing Fuzzy About Fuzzy Logic
     
     The idea that fuzzy logic is fuzzy or intrinsically imprecise is
     one of the most commonly expressed fables in the fuzzy logic
     mythos. This wide-spread belief comes in two flavors, the first 
     holds that fuzzy logic violates common sense and the well proven
     laws of logic, and the second, perhaps inspired by its name,
     holds that fuzzy systems produce answers that are somehow
     ad-hoc, fuzzy, or vague. The feeling persists that fuzzy logic
     systems somehow, through their handling of imprecise and
     approximate concepts, produce results that are approximations of
     the answer we would get if we had access to a model that worked
     on hard facts and crisp information.  Nothing could be further
     from fact.  
     
     There is nothing fuzzy about fuzzy logic, Fuzzy Sets differ from
     classical or crisp sets in that they allow partial or gradual
     degrees of membership.  We can see the difference easily by
     looking at the difference between a conventional (or "crisp")
     set and a fuzzy set.  Thus someone 34 years, eleven months, and
     twenty eight days old is not middle aged.  In the Fuzzy 
     representation, however, we see that as a person grows older he
     or she acquires a partial membership in the set of Middle Aged
     people, with total membership at forty years old.
     
     But there is nothing ambiguous about the fuzzy set itself.  If 
     we know a value from the domain, say an age of 35 years old,
     then we can find its exact and unambiguous membership In the
     set, say 82%.  This precision at the set level allows us to write
     fuzzy rules at a rather high level of abstraction.  Thus we can
     say, if age is middle-aged, then weight is usually quite heavy;
     and means that, to the degree that the individual's age is
     considered middle aged, their weight should be considered
     somewhat heavy.  A weight estimating function, following this
     (very simple) rule might infer a weight from age through the
     following fuzzy implication process.
     
     
     Much of the discomfort with fuzzy logic stems from the implicit 
     assumption that a single ``right'' logical system exists and to
     the degree that another system deviates from this right and
     correct logic it is in error.  This ``correct'' logic, of
     course, is Aristotelian or Boolean logic.  But as a logic of
     continuous and partial memberships, Fuzzy Logic has a deep and
     impressive pedigree.  Using the metaphor of the river, Heraclitus
     aptly points out that a continuous reasoning system more
     correctly maps nature's logical ambiguities.  From his dictum
     that all is flux, nothing is stationary, he devcloped a 
     rudimentary multi-valued logic two hundred years before
     Aristotle.  Recently, Bart Kosko, one of the most profound
     thinkers in fuzzy logic, has shown that Boolean logic is, in
     fact, a special case of fuzzy logic.
     
                                 TRUTH TWO
                 Fuzzy Logic Is Different from Probability
     
     The difference between probability and fuzzy logic is clear when
     we consider the underlying concept that each attempts to model. 
     Probability is concerned with the undecidability in the outcome
     of clearly defined and randomly occurring events, while fuzzy
     logic is concerned with the ambiguity or undecidability inherent
     in the description of the event itself.  Fuzziness is often
     expressed as ambiguity rather than imprecision or uncertainty
     and remains a characteristic of perception as well as concept.
     
                                TRUTH THREE
                   Designing the Fuzzy Sets is very asy
     
     Not only are fuzzy sets easy to conceptualize and represent, but
     they reflect, in a general "one-to-one" mapping, the way experts
     actually think about a problem.  Experts can quickly sketch out
     the approximate shape of a fuzzy set.  Later, after we have run
     the model or examined the process, the precise characteristics
     of the fuzzy vocabulary can be adjusted if necessary.
     
                                TRUTH FOUR
                  Fuzzy Systems are Stable, Easily Tuned,
                    and can be conventionally Validated
     
     Creating fuzzy sets and building a fuzzy system is faster and 
     quicker than conventional knowledge-based systems using "crisp" 
     constructs. These fuzzy systems routinely show a one or two
     order of magnitude reduction in rules since fuzzy logic
     simultaneously handles all the interlocking degrees of freedom. 
     Fuzzy systems are very robust since the over-lapping of the
     fuzzy regions, representing the continuous domain of each
     control and solution variable, contributes to a well-behaved and
     predictable system operation.  These systems are validated in
     the same manner as conventional system.  The tuning of fuzzy
     systems, however, is usually much simpler since there are fewer 
     rules; representation if visually centered around fuzzy sets,
     and operations act simultaneously on the output areas.
     
                                TRUTH FIVE
                     Fuzzy Systems are Different From
                   and Complementary to Neural Networks
     
     There is a close relationship between fuzzy logic and neural 
     systems.  A fuzzy system attempts to find a region that
     represents the space defined by the intersection, union, or
     complement of the fuzzy control variables.  This has analogies
     to both neural network classifiers and linear programming
     models.  Yet fuzzy systems approach the problem differently with
     a deeper and more robust epistemology.  In a fuzzy system, the
     classification and bounding process is much more open to the
     developer and user with capabilities for explanations, rule and
     fuzzy set calibration, performance measurements, and controls
     over the way the solution is ultimately derived.
     
                                 TRUTH SIX
             Fuzzy logic "ain't just process control anymore"
     
     Historically we have come to view fuzzy logic as a process 
     control and signal analysis technique, but fuzzy logic is really
     a way of logically representing and analyzing information,
     independent of particular applications.  The information
     management field in particular has, until recently, ignored
     fuzzy logic, delaying its introduction into expert system and
     decision support technology.  Recently, however, new types of
     knowledge base construction tools have emerged.  Such tools will
     make it easier for experts who are not computer experts to
     intuitively represent and manipulate information.
     
                                TRUTH SEVEN
           Fuzzy Logic is a Representation and Reasoning Process
     
     Not the "Magic Bullet" for all AI's current problems - Fuzzy
     Logic is a tool for representing imprecise, ambiguous, and vague
     information.  Its power lies in its ability to perform meaningful
     and reasonable operations on concepts that are outside the
     definitions available in conventional Boolean logic.  We have
     used fuzzy logic in such applications as project management,
     product pricing models, health care provider fraud detection,
     sales forecasting, market share demographic analysis, criminal
     identification, capital budgeting, and company acquisition
     analysis.  Although fuzzy logic is a powerful and versatile tool,
     it is not a solution to all problems.  Nevertheless, it opens the
     door for the modeling of problems that have generally been
     extremely difficult or intractable.
     Earl Cox, CEO
     Metus Systems     
     White Plains, NY
     (914) 238-0647

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