# Re: fuzzy proofs and law of excluded middle

From: WSiler@aol.com
Date: Sat Jun 23 2001 - 07:39:33 MET DST

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In a message dated 5/16/01 8:50:08 PM Central Daylight Time,
ulrich.bodenhofer@scch.at writes:

>>Don't believe anybody who calls that a crisis! Elkan's statement was simply
a dull error. He, more or less, assumed that fuzzy logic obeys all laws of
Boolean algebras (which is not true) and proved, under these FALSE
assumptions, that it collapses into Boolean logic. Read the original Elkan
paper
(it is somewhere on the Web, for sure) and you will understand

The question arises whether something which is not a Boolean algebra
may be considered as a concept of logic. Nowadays, the accepted opinion
is yes! I would not like to go into detail, but to recommend the following
books:
>>

By "accepted opinion", I presume you mean accepted by most fuzzy
mathematicians. Certainly there are many others, including most AIers, who do
not accept the opinion that Elkan has simply committed a "dull error".

To restate the stituation in less euphemistic terms, Elkan showed (without
explicitly so stating) that fuzzy logic fails to obey the laws of excluded
middle and non-contradiction. This is not just a matter of "obeys all laws of
Boolean algebras"; it is a matter of not obeying laws of logic which have
been accepted for a couple of thousand years. This is not a "dull error", but
an annoyiing statement of fact.

Many fuzzy mathematicians assert that this failure is a virtue. After having
been involved in creating fuzzy expert systems and a fuzzy expert system
shell for over 15 years, I can not accept that this failure is a good thing.
In some circumstances it produces highly counter-intuitive results. For
example, if "~2" is a triangular fuzzy two, then the intersection of "~2 and
NOT ~2 is bimodal, and the union "~2 OR NOT ~2" has two notches in it.

It is, however, quite possible to define multivalued logics in a way which
does obey both excluded middle and contradiction. Jim Buckley and I have a
couple of fairly recent papers in Fuzzy Sets and Systems which defines such a
family of logics. If the Zadehian max-min logic is taken as a default, then
Elkan's proof crumbles.

William Siler

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