Re: Request for information from a complete non-fuzzy person

WSiler (wsiler@aol.com)
Mon, 26 Oct 1998 03:13:27 +0100 (MET)

>Fuzzy logic is nothing more than a reformulation of statistical inference
>and probability theory.

Not true.

In the first place, while "fuzzy logic" is now a buzz word, we are really
referring to fuzzy systems theory. The basis is not fuzzy logic, but the theory
of fuzzy sets. This theory leads in turn to fuzzy logic, fuzzy numbers ...

So the basic mathematical concept which which we are dealing is neither
statistics nor probability, but set theory. In fact, fuzzy most (but not all)
fuzzy theoreticians make a point that what they are doing is not probability.

>What does bother me though is that there is no single formulation of
>fuzzy logic, based on some simple axioms. Unlike set theory and
>probability.

There is some truth to this, but the problem is inherent to multivalued logics.
It is generally agreed that multivalued logical operations should collapse to
classical logic for truth values of 0 and 1. However, this is not a very
restrictive requirement. For example, there is an infinity of AND operations
which collapse to the classical, ranging from the Zadeh logic a AND b =min(a,
b) through the probabilistic with independence a AND b = a*b to a AND b =
max(0, a+b-1). Usually, however, when one uses the term fuzzy logic, one refers
to the Zadeh operators, min for ANand max for OR.

>In fuzzy theory, none of the existing formulations correctly handle compound
>propositions in which the parts are not independent. For example, some
>fuzzy logic statements concerning (A and not A) appear spurious to me.

Not precisely true, although I agree that a strict use of almost any single
multivalued logic can produce counter-intuitive results.This is because
multivalued logics have trouble with the laws of excluded middle and
contradiction. However, we addressed this problem more than ten years ago. If
one wishes (as many do) to obey these laws, it can be done simply. The only
time the trouble arises is when one is combining the same proposition with
itself in a compound statement. If we combine a with a, then the correct logic
to use is the Zadeh max-min logic. I we combine a with NOT a, the correct logic
to use is the bounded sum-bounded difference logic. If we do this, excluded
middle and contradiction are obeyed.

>It could be argued that set theory/probability is a limiting special case of
>fuzzy logic. It could also be argued that probability exists 'inside' fuzzy
>theory, which appears to contradict the first statement.
>
I personally think that arguments of this type serve no purpose. We are arguing
about turf, not substance. All these theories can be useful, depending on the
context of the problem to be solved. I'm interested in solving problems and
developing methods to solve them, not about whether this includes that or vice
versa.

>What about the case where you need to unify information obtained from
>numerical and text data and the numerical data needs to be fuzzified?

This is an important problem. We are talking about reasoning with both numeric
and non-numeric data. We have paid a great deal of attention to this problem
over the last fifteen years, but most fuzzy theoreticians have ignored it. If
you are really interested in this and the theory behind it, I can send you a
preprint of a paper on exactly that topic. Meantime, if you visit my Web page,
you will see examples of fuzzy expert systems which deal with both numeric and
non-numeric data. Since you are in the UK, you might take a look at "Fril -
fuzzy and evidential reasoning in artificial intelligence" by Baldwin, Martin
and Pilsworth (Research Studies Press, 1995). FRIL is a fuzzy superset of
Prolog.

God knows this is not short, but I've attempted to address your major points.

William Siler

Odium ignorantem est odium infantem,
sed odium savantem est odium ferentem!

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