I mean some additional comments may be helpful.
> In a message dated 5/16/01 8:50:08 PM Central Daylight Time,
> firstname.lastname@example.org 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
> (it is somewhere on the Web, for sure) and you will understand
> (since you seemingly have read Ruspini's reply already).
> 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
> 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".
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
> 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.
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
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
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
> 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.
But, let me repeat: it is the heart of the matter.
> 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.
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.
Prof. Siegfried Gottwald
Institut fuer Logik und Wissenschaftstheorie
phone: (0341) 97 35770/71
fax: (0341) 97 35798
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