Re: Godel's Theorem under Fuzzy Logic?

Stan Rice (
Sat, 28 Mar 1998 22:48:39 +0100 (MET)

This (below) is hardly a mathematical point, but is it not true?--
Fuzzy gradations, no matter how fine the steps involved (let alone
only 10 steps,) come down in the end to binary distinctions. I.e. in
the end either "this degree" does or does not apply, is or is not
adequate to these criteria, is or is not alowed to trigger an action.
In other words, there is no distinction that is not bivalent, because
any distinction whatever is bivalent.
Can anyone show otherwise?

In other words, fuzzy is bivalent as long as it admits of distinction.
A more profound question is whether the consciousness in which
all distinctions appear actually supports them in the manner that
we imagine. Penrose seems right to me.
Cheers, Stan R
-------------------------- wrote:
> Can Godel's Incompleteness Theorem be extended to formal systems based on
> fuzzy logic?
> This is a crucial question, IMO, with regard to Artificial Intelligence,
> because as I understand it, Roger Penrose's whole argument in his book
> "Shadows of the Mind" that convincingly attempts to show that AI is
> in principle unachievable on any algorithmic computer, assumes that the
> posited machine intelligence thinks in terms of either-or (Aristotlean) logic
> rather than fuzzy logic. Surely fuzzy logic would invalidate Godel's Theorem
> and hence his whole argument? - therefore AI might in fact be possible after
> all.
> However, I wonder whether Godel's Theorem can be extended to formal systems
> based on fuzzy logic. In this case, Penrose might be correct after all!
> I am only a first-year mathematics student, so please try to keep your replies
> as simple as possible.
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