# Re: Thomas' Fuzziness and Probability

From: Stephan Lehmke (Stephan.Lehmke@cs.uni-dortmund.de)
Date: Fri Aug 17 2001 - 12:15:33 MET DST

• Next message: S. F. Thomas: "Re: Thomas' Fuzziness and Probability"

I'm afraid I'll need some more explanations to understand your
arguments.

In article <9lem1i\$3nr4@odds.stat.purdue.edu>, Herman Rubin writes:
> In article <9l818o\$36a\$4@fbi-news.cs.uni-dortmund.de>,
> Stephan Lehmke <Stephan.Lehmke@cs.uni-dortmund.de> wrote:
>>In article <23af61c2.0108122107.6f0e7aab@posting.google.com>, Robert Dodier writes:
>>
>>> The difficulty is that rules of the kind applied in fuzzy logic
>>> ignore relations between the elements of a compound proposition.
>
>>What about two-valued logic?
>
> It does not give any problems, looked at carefully.

You mean that if Robert's example were formalized in two-valued logic,
the problem he mentions would not occur? Could you give such a
formalization?

> Probability is NOT a truth value system, and does not
> pretend to be so.

I agree completely.

> In a truth value system, the truth
> of a compound proposition follows from that of the
> simple propositions involved.

That goes without saying.

> Probabilities have to
> be exist for all compounds simultaneously.

I'm not sure this gets to the point. In a truth value system, it is
also the case that an interpretation fixes all truth values
simultaneously.

In both cases, there are atomic or `basic' independent entities the
truth value and probability, respectively, of which can be fixed
independently.

Truth values and probabilities of more complex entities depending on
the atomic ones have to be calculated dependent on the atomic
entities. The only difference is that the truth value of a compound
statement can be calculated from the truth value of its substatements
(whether these are compound or atomic) while for probability, you have
to go all the way down to atomic entities.

> If one
> starts with the fuzzy idea and adds this, probability
> is what will result.

I don't understand. Extensionality (aka truth-functionality) is an
essential part of "the fuzzy idea" (which started out as fuzzy set
theory, remember). I don't see how you can add something to this and
arrive at probability.

regards
Stephan

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