# Re: Thomas' Fuzziness and Probability

From: Herman Rubin (hrubin@odds.stat.purdue.edu)
Date: Sat Aug 18 2001 - 08:33:15 MET DST

• Next message: Joe Pfeiffer: "Re: Thomas' Fuzziness and Probability"

S. F. Thomas <sfrthomas@yahoo.com> wrote:
>I see your difficulty. You think that if A is a fuzzy term, and its
>membership function is denoted simply by a, let's say, then the
>one-minus rule of negation gives the membership function of NOT A as
>1-a. Hence the "middle" is included, so to speak, and LEM and LC
>should fail, as indeed it obviously does if the min-max rules are then
>applied. For we have A AND NOT A being modeled in the meta-language as
>min(a,1-a), which gives us the well-known middle with a peak at 0.5
>(assuming of course that a has its max at 1, its min at 0, and there

>Now let's try another rule of conjunction, in particular the
>Lukasiewicz bounded-sum rule, for which we have for two membership
>functions a and b, and their corresponding terms A and B,

> mu[A AND B] = a AND b = max(0, a+b-1).

>In the particular case where B is NOT A, and b=1-a, we have under this
>rule

> a AND b = max(0,a+1-a-1) = 0 everywhere

>and in accordance with the law of contradiction, the term A AND NOT A
>is rendered as the comstant absurdity whose membership value is
>everywhere 0. LC is upheld.

And what if B is A? Do we not want A AND A to be A?
The rule fails! ANY rule for computing truth values
for compound propositions from those simple ones is
not going to do all that is desired if the truth value
system is not a Boolean algebra. BTW, rather large
Boolean algebras as truth values are ways of obtaining
some of the highly counterintuitive results of set theory.

Probability measures exist on Boolean algebras, but the
sloppy current language of considering probabilities as
measures of truth (which they are) does not make them
truth values.

Any consistent scheme for allocating odds for some bets
and conditional bets can be extended to a probability
measure. If fuzziness is consistent, it can be as well.

```--
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
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