Re: Godel's Theorem under Fuzzy Logic?

Dimitri Lisin (dima@wpi.edu)
Sun, 29 Mar 1998 00:51:55 +0100 (MET)

On 25 Mar 1998, Stephan Lehmke wrote:

> In article <Pine.OSF.3.96.980325104226.20804A-100000@scooter.wpi.edu>,
> Dimitri Lisin <dima@wpi.edu> writes:
> >
> > But by fixing our meta-interpretation of truth (e.g. F is true, if it can
> > be drived with a gegree of 1) aren't we killing the very idea of
> > fuzzyness?
>
> Christian's argument was only about whether or not you could construct
> a Goedel formula in a system of "fuzzy logic". In most systems of fuzzy
> logic I know, you _can_ express crispness. This does not make it impossible
> to express fuzziness as well (but it's not needed for Goedel's proof).

What I am trying to understand, though, is whether it even make sense to
talk about "incompleteness" or, especially, "consistency" of a system in
which a statement or a formula is both true and false to some degree.

Christian mentioned that choosing appropriate definitions for the
key concepts, such as "proof", "consistency", and "completeness" presents
a difficulty. It seems to me that by defining these concepts crisply we
are creating a crisp "wrapper" around fuzzy logic, essentially
defuzzyfying it. Certainly, fuzzy logic can express crispness, as a
special case. However, if we restrict such general and fundamental
concepts as "truth", "proof", "completeness", and "consistency" to crisp
definitions then we arrive back to two-valued logic.

IMHO, to be *consistant* with the fundamental ideas of fuzzyness, the
formal system of fuzzy logic should be complete to some degree,
and consistant to some degree. We can define what it means to be
"complete enough" and "consistent enough" for our purposes, and then
Godel's theorem will hold to some degree... :)

Also, didn't Kosko deal with the classic self-referencing contradictions
such as the barber, who shaves everyone who doesn't shave himsel?
I, personally, find this very hard to dygest, but didn't he
declare that these statements are both true and false to the
degree of .5 and place them in the middle of his hypercube?
Well, maybe he was not the one who first thought of this, but that's where
I first saw it. :)
Wouldn't Godel's formula also be in the middle of the hypercube?

> When you go down to the level of bits, of course _every_ system running
> on a computer (or written down on paper in a fixed symbolic language)
> is "inherently crisp", even fuzzy controllers and the like.

What a pity... :) All we can get is *simulated fuzzyness*... :)
Thanx for the URL.

Dima