Re: Fuzzy logic compared to probability

Bo Yuan (yuan@binghamton.edu)
Tue, 5 Mar 1996 18:42:58 +0100


In article <4goljv$cbv@geraldo.cc.utexas.edu>, bill@clyde.as.utexas.edu
(William H. Jefferys) wrote:

> In article <4gn6re$r5u@watt.electriciti.com>,
> Fred A Watkins <fwatkins@hyperlogic.com> wrote:
> #hrubin@b.stat.purdue.edu (Herman Rubin) wrote:
> #
> #>>I am aware that proponents of this Bayesian probability theory
> #>>claim that it is the only consistent generalization of binary
> #>>logic (according to Cox's theorems). Where does this leave
> #>>fuzzy logic?
> #
> #Alone. Cox's Theorem needs twice-differentiable operators; fuzzy uses
> #min and max, which do not meet the requirement.
>
> Cox's Theorem [Cox, R.T. (1946), "Probability, Frequency, and Reasonable
> Expectation," _American Journal of Physics_ 14, pp. 1-13] used
> differentiability; however, this assumption is not neccessary [Aczel, J.
> (1966), _Lectures on Functional Equations and Their Applications_, New
> York: Academic Press.]
>
> I expect that Cox, who was a physicist, rediscovered something that was
> already well-known. Jaynes notes in the draft of his book that I have
> that the key equation was known to Abel (1826).

If you read Cox's proof carefully, you can find that Cox did not prove
anything. He just provided another way to solve a so-called associative
functional equation under some conditions. The flaw of his proof is that
he made an assumption, which is "if b|a is one such measure (a measure of
the reasonable credibility of the proposition b when the proposition a is
known to be true) then an arbitrary function f(b|a) will also be a
measure."
This assumption is obiviously false, because if f is a strict
decreasing function, then the measure of credibility is reversed. If the
original measure has a property such as b|ac|a if c implies b, then the
new measure will have the property f(b|a)f(c|a) if c implies b, which
contradicts the ordering of probability measure.

Bo Yuan, ABD
Center for Intelligent Systems and
Dept. of Systems Science and Industrial Engineering
Binghamton University-SUNY
Binghamton, New York 13902-6000
E-mail: yuan@binghamton.edu