Below is a conversation I recently had with my (imaginary) friend ;-)
"Is there a way of looking at fuzzy theory in the context of traditional
mathematics?"
Set theory and logic are both binary branches of mathematics. In set theory
everything is either in the set, or not in the set. Similarly, in logic, a
statement is either true or false. Meanwhile probability takes a real
number between 0 and 1 and is therefore continuous. The only sensible way
of combining these is to make traditional set theory/logic a special
limiting case of fuzzy set theory/fuzzy logic.
"How can we best interpret fuzzy theory using traditional mathematics 'from
within'?"
The fuzzy membership function could be interpreted as a probability density
function (pdf). The fuzzy set then becomes a set with a probability
distribution attached and the degree of membership of the set becomes a
probability.
"But I've been told that I mustn't confuse probability with degree of
membership. In fuzzy theory if we say that a man is tall with a degree of
membership of 0.8, we don't mean that he is tall 80% of the time."
Yes, but how can we justify using the figure 0.8? Where did it come from?
We could ask a random sample of people to define their interpretation of
(say) short, average and tall. If the man is 6' tall and it transpires that
80% of the sample consider this to be tall, we could also argue that if we
ask a person at random, the probability that they consider 6' as belonging
to the set 'tall' is 0.8.
"Usually the shapes and ranges of the fuzzy membership functions are defined
by a human expert - how can this involve statistics?"
The human expert has built up his knowledge from experience of past events,
which is in effect, statistical inference.
"So fuzzy theory relies on human reasoning and if there were no people,
there would be no fuzzy theory?"
Correct.
"What's inherently wrong with fuzzy theory?"
Fuzzy theory forces continuous data into discrete sets, which necessarily
effects a loss of information.
"I think I can see the real advantage of fuzzy set theory. In traditional
set theory if a man grew by just 1/4 inch he could fall into a completely
different set. Fuzzy set theory would overcome such a clear-cut, abrupt
change with a gradual smoothing effect."
Fuzzy logic is solving a problem it created in the first place. In
traditional maths, for this type of problem we would not try to split up a
set of real numbers into subsets in the first place. We already have far
more information contained in the actual height of a man, say 6' 0" than we
would if we said that he was in the set 'tall'. Note that 6' 0" also
conveys far more information than 'a member of the fuzzy set 'tall' with a
degree of membership of 0.8', and that it even requires less storage space
in terms of memory.
"But I insist that we do incorporate 'tallness', because dealing with such
vague concepts is the whole point of fuzzy theory."
No problem. There must exist a least height such that over half of a random
sample consider this to be tall. Any height above this will be considered
to be tall. Depending on the context, you may read 'relevant expert' rather
than 'random sample'.
"But then we lose information on how ambiguous 'tall' is."
Not if we also record from our sample the distribution of the limits of the
sets.
"Fuzzy classification evidently violates Aristotle's law of the excluded
middle."
Yes, this is inherent in any multivalued logical system - in the sense that
you can't have it both ways (except in such systems, where of course you
can!)
Martin Sewell
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