Re: Godel's Theorem under Fuzzy Logic?

Dimitri Lisin (dima@wpi.edu)
Sun, 29 Mar 1998 00:43:37 +0100 (MET)

On Wed, 25 Mar 1998, Christian Borgelt wrote:

[...snip...]

> there are some difficulties involved. Due to the introduction of degrees
> of truth, we have to define carefully what we mean by ``proof'', by
> ``consistency'' and by ``completeness''. But I think that, if in doubt,
> we can always code the necessary interpretation into the G"odel formula
> F we construct in the system, for example, code F as saying intuitively
> ``I, formula, cannot be derived with a degree of truth of 1.''

[...snip...]

> With fuzzy logic, we face the additional complication that we have to
> fix our meta-interpretation of truth (looking from outside the system),
> in order to compare the result to the formal result in the system.

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?

> But all this, in the end, comes to no more than the question whether
> only a formal proof is proof and whether you, in your meta-reasoning,
> accept degrees of truth. This, obviously, cannot be decided by a
> formal system. You have to choose.

In general, does it even make sense to represent fuzzy logic as a formal
system, which is inherently crisp? Or maybe the rules of the formal
system itself should allow fuzzyness?

>
> Regards,
>
> Chris
>

Dimitri Lisin
dima@cs.wpi.edu
http://www.wpi.edu/~dima/