Re: Godel's Theorem under Fuzzy Logic?

Stephen Paul King (spking1@mindspring.com)
Mon, 20 Apr 1998 18:13:13 +0200 (MET DST)

Dimitri Lisin <dima@wpi.edu> wrote
>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.
>o
Is it neccessary to use universal qualifiers? Could we not
just be satisfied with 'bounded' truths? The idea here is that it is
impossible in practice to verify any universal statement, like "all
crows are black." If we just stick to to finite and constructible
subsets when we are talking about statements we might avoid these
difficulties. The fuzzyness would then be understood as the measure of
unverifiability (from within the statement).

>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... :)
>
Could it be possible to use some labeling or other device to
define the consistency, proof and completeness terms as contingent to
a given subset of the hypercube I^n? This would be like saying that
subset X, having such and such coordinates in I^n, is 'f-consistent'
with respect to to some other subset of I^n. The difficult task is to
find a way to embed ladeling of this kind in I^n.

>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?
>
It would seems that any "liar's paradox" would be situated in
the center of the fuzzy hypercube, e.g. have max. fuzzy entropy.

>> 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
>
>
Just as student,

Stephen Paul King