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From: hardy@umnstat.stat.umn.edu (Michael Hardy)
Subject: Re: Fuzzy theory or probability theory?
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Date: Tue, 29 Nov 1994 02:08:47 GMT
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In article <3bdb9j$15jm@hearst.cac.psu.edu> caj@jerry.psu.edu writes:


>The only "axioms" for numerical representation of logic are
>Use 1 for TRUE
>Use 0 for FALSE
>which then allows us to use
>x*y to represent x and y   and
>x+y-x*y to represent x or y   (or any other similar functions).

	[snip]

>The numbers 0 and 1 were suggested quite arbitrarily by Boole so that he could
>write the logic AND and OR mathematically. Extending the concepts of TRUE and
>FALSE or set membership to continous measures is only attempting to allow for
>statements such as "His statement is somewhat TRUE."  And thus does not deserve
>a 1 but something smaller.  Since the statement is not FALSE it cannot be given
>a 0. Clearly, to translate this into a mathematical form, one must use a number
>between the two extremes (here, 0,1 because of Boole).

	[snip]

>0 through 1 are used in prob theory because it deals with fractions of trials
>(some  precentage of the trials yields outcome p). 0 through 1 are used in
>logic because they are the additive and multiplicative identities which makes
>mathematical formulas for Boolean Logic tables trivial for these numbers.


	This assumes that frequentism is the only interpretation of prob-
ability and ignores entirely the epistemic (or "Bayesian" interpretation).
The thesis of this post -- that the resemblance between logic and probability
theory is purely coincidental -- is entirely wrong.  Richard T. Cox's book
_Algebra_of_Probable_Inference_ has a good account of this, as does Edwin
T. Jaynes' forthcoming _Probability_Theory:_The_Logic_of_Science_, available
by anonymous ftp from bayes.wustl.edu.


	Mike Hardy


