# Re: Do we have proofs that human think in a fuzzy (logic) way?

From: P.G.Hamer (pgh@nortelnetworks.com)
Date: Sun Jan 21 2001 - 11:14:39 MET

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Vilem Novak wrote:

> > Moreover, Cox's Theorem [Cox, R.T.: ``Probability, frequency, and
> > reasonable expectation,'' Am. J. Phys. v.14, pp.1--13 (1946)] proves
> > conclusively that =ALL= possible approximate reasoning methods other than
> > Bayesian Probability Theory fail to be logically self-consistent in some
> > way or other --- and that includes fuzzy logic. Hence, if any method of
> > approximate reasoning is to be preferred, it would have to be Bayesian
> > reasoning. (This is =NOT= to say that _humans_ use Bayesian reasoning,
> > but only that _if_ we wish our machines to think in a way compatible with
> > logic, Bayesian reasoning has been mathematically proved to be the =ONLY=
> > approximate reasoning method that is capable of doing so. Any other method
> > --- including Fuzzy Logic --- will necessarily yield logical paradoxes
> > and inconsistencies under certain circumstances.)

I rather thought it required the combinatory rules of statistics, rather than
specifically those of Bayesian stats.

Can I emphasis that there are problems with the internal consistency
of many formal systems. While a total absence of inconsistency would
be nice,the most that can often be achieved is that the inconsistencies
don't lead to significant problems very often. So even if Cox is
absolutely right, it doesn't prove that the resultant logical paradoxes
got.

> ??????????????? - this is very strange and clear nonsense.

A strange response. Do you know of this mathematical work and have
objections to its assumptions or reasoning? Or are you objecting to the
apparently rational conclusion thar fuzzy logic cannot be entirely logically
self-consistent?

Or is it simply a knee-jerk reaction to something that could imply that
fuzzy logic is less than perfect.

Can you say why? I get the impression that these books are going to
say that the authors think that fuzzy logic is neat, and can address
significant problems. Which is irrelevant to the formal truth of Cox's
theorem.

> A Treatise on Many-Valued Logics.
> Research Studies Press Ltd., Baldock, Herfordshire, UK (to
> appear).
>
> H\'ajek, P.: Metamathematics of fuzzy logic. Kluwer, Dordrecht
> 1998.
>
> Nov\'ak, V., Perfilieva I. and J. Mo\v{c}ko\v{r}:
> Mathematical Principles of Fuzzy Logic. Kluwer, Boston 1999.
>
> Turuen, E.: Mathematics Behind Fuzzy Logic. Springer, Heidelberg
> 1999.

If any of them specifically address the truth or the implications of Cox's
theorem can you say which (and where).

Peter

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