**Previous message:**Ashish Pandey: "Re: Fuzzy systems in the process industries"**Maybe in reply to:**Vilem Novak: "fuzzy proofs and law of excluded middle"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]

I mean some additional comments may be helpful.

*> In a message dated 5/16/01 8:50:08 PM Central Daylight Time,
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*> ulrich.bodenhofer@scch.at writes:
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*>
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*> >>Don't believe anybody who calls that a crisis! Elkan's statement was simply
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*> a dull error. He, more or less, assumed that fuzzy logic obeys all laws of
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*> Boolean algebras (which is not true) and proved, under these FALSE
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*> assumptions, that it collapses into Boolean logic. Read the original Elkan
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*> paper
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*> (it is somewhere on the Web, for sure) and you will understand
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*> (since you seemingly have read Ruspini's reply already).
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*>
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*> The question arises whether something which is not a Boolean algebra
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*> may be considered as a concept of logic. Nowadays, the accepted opinion
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*> is yes! I would not like to go into detail, but to recommend the following
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*> books:
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*> >>
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*>
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*> By "accepted opinion", I presume you mean accepted by most fuzzy
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*> mathematicians. Certainly there are many others, including most AIers, who do
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*> not accept the opinion that Elkan has simply committed a "dull error".
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*>
*

Sorry, but I think that Bodenhofer was right with his evaluation.

However, the reason may be completely another one as given by him.

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

Well, it is just this what is not true. What Elkan discussed was

neither the law of excluded middle, nor the law of non-contradiction

at all.

Let me concentrate on the law of excluded middle, for the law of

non-contradiction the situation is completely similar.

The law of excluded middle is a metalogical statement, related to

classical logic, saying that a proposition has to be true or has to

be false. By the very approach toward fuzzy sets, as well as by the

very approach toward many-valued logics, this law has to fail there.

And this is completely trivial: it is just the heart of the matter

in fuzzy and many-valued topics that the law of excluded middle has

to fail there, because one accepts degrees in between "true" and

"false". To make a paper out of this obvious fact is really

astonishing. (And giving an award to such a paper even more.)

However, there is something more complicated here. And this comes

from the fact that the FORMULA $p \lor \neg p$ in some (weak) sense

CODES (or: represents) this law of excluded middle inside CLASSICAL

PROPOSITIONAL CALCULUS.

That this representation is not a faithful one in any situation

becomes clear just from the fact that this formula $p \lor \neg p$

may become a logical truth in a suitable system of many-valued logic,

if one reads the connectives $\lor$ and $\neg$ in a suitable way:

e.g. if they are the Lukasiewicz arithmetical disjunction, and the

Lukasiewicz negation.

*>
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*> Many fuzzy mathematicians assert that this failure is a virtue. After having
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*> been involved in creating fuzzy expert systems and a fuzzy expert system
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*> shell for over 15 years, I can not accept that this failure is a good thing.
*

But, let me repeat: it is the heart of the matter.

*> In some circumstances it produces highly counter-intuitive results. For
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*> example, if "~2" is a triangular fuzzy two, then the intersection of "~2 and
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*> NOT ~2 is bimodal, and the union "~2 OR NOT ~2" has two notches in it.
*

Here, however, the OR is the max-disjunction. If one considers also

here the Lukasiewicz arithmetic disjunction, together with the

1-..-negation (i.e. Lukasiewicz negation), "~2 OR NOT ~2" becomes

the universal fuzzy set (over the intended universe of discourse) -

and this is not at all counterintuitive (for me). So the core problem

seems to be how to choose the (generalized) connectives: particularly

in cases in which there are more than only one candidate in the

generalized (fuzzy or many-valued) setting.

Siegfried Gottwald

------------------------

Prof. Siegfried Gottwald

Universitaet Leipzig

Institut fuer Logik und Wissenschaftstheorie

Burgstrasse 21

D-04109 Leipzig/Germany

email: gottwald@rz.uni-leipzig.de

phone: (0341) 97 35770/71

fax: (0341) 97 35798

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