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From: ST401843@BROWNVM.BITNET
Newsgroups: comp.ai.digest
Subject: Re:Fuzzy Logic vs. Probability Theory
Message-ID: <8802210846.AA02229@ucbvax.Berkeley.EDU>
Date: 20 Feb 88 17:27:36 GMT
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Here is my two bits about fuzzy logic: Richard Golden writes:

  >...Rational  selection in this case meaning consistency with the classic
  >deductive/symbolic logic - boolean algebra...

But here's the rub! In Boolean Algebra you have clear cut True or False
truth values.In Crisp Set Theory (to a subclass of which theory Boolean Algebra
is isomorphic) you have clear cut "x belongs A" relationships. Write this as
B(x,A)=1 , where x is an element, A is a set and B(.,A) takes values  (for
"belongs" and 0 for "not belongs". In other words B(.,A) is the "belongingness"
function of A. In Fuzzy Set Theory on the other hand, B(.,A)  can take any
value between 0 and 1. There are no clear cut  answers to questions such as:
"does a person 5 ft. 10 in. tall belong to the set of tall persons?".

Cisp Set Theory is what we traditionally call Set Theory. It is a subset of
Fuzzy Set Theory, in that in CST B can take only the extreme values 0 and 1.

Of course in all of FST we use the reasoning, syllogisms etc. of the Boolean
Algebra. I don't know if anyone does Fuzzy Mathematics. In FM, things such
as reasoning by Reductio ad Absurdum would not be valid. Why? Well, in
RaA we usually want to prove something for x, call it P(x). And we begin by
saying: "assume NOT P(x)..." But in FST P(x) and NOT P(x) are just two of
uncountably many possibilities. But maybe ther is a Fuzzy generalization
to RaA. Does anyone know?

Anyways, I think FST is still an alternative way to reasoning about
uncertain events, different from Probability Theory. In fact there has been
work done on Fuzzy Probability, postulating fuzzy probability measures. I did
not have time to check before I send  this, but I suppose they would drop
things like the countable additivity hypothesis. And classical PT is by
no means the only way to reason about uncertain events- see von Mises   and
deFinneti, among others.

One last observation. I do not want to make much of it, but is it not
remarkable that FST, an alternative to the Aristotelian logic, was invented
by Zadeh, an Indian?




                  Thanasis Kehagias