# Re: Fuzzy logic compared to probability

Tom Whalen (dscthw@gsusgi2.Gsu.EDU)
Fri, 1 Mar 1996 20:55:44 +0100

On Tue, 27 Feb 1996, Andreas Poncet wrote:

> As an example: instead of saying, e.g., about a specific man, that
> his height x has "tall"-membership, say, 0.8,
> one could express this equivalently as P(A|x) = 0.8,
> with the proposition A defined as
> A = "the man is tall"

This would seem to imply one of the following two scenarios:

(1)
There exist three ordinary sets: a set of unambiguously tall heights, a
set of heights that are unambiguously not-tall, and a set of heights 80%
of whose members are also members of the tall set and the remaining 20%
are all members on the not-tall set. Our knowledge about the man's
height gaurantees that it is in the thord set but gives no further clue
about which of the first two it belongs to.

(2)
There exists a population of observers 80& of whom will say without
equivocation "the man is tall!" and 20% of whom will say without
equivocation "The man is NOT tall!"

A fuzzy interpretation would be that the proposition "he is tall" conforms
to our overall knowledge better than any statement with truth value .79
but not so well as any statement with truth value .81.

> Fuzzy logic couldn't convinced me yet, for there seems to be much "ad
> hockery", as in the choice of the "tent" form, e.g., of membership
> functions. Why such a form? How can it incorporate prior information?
> (Probability theory decomposes the problem of specifying P(A|x) in two
> steps: the prior P(A) and the likelihood p(x|A).)

Let M = the truth value of the statement that the assumptioms of
probability theory are justifiable in a given situation. Then the truth
value of the statement that probability theory is optimal in that
situation can be brought arbitratily close to M by judicious choice ot the
criterion for optimality, but it cannot exceed M.

> Furthermore, how to consistently specify JOINT membership functions of DEPENDENT
> characteristics (such as "tall"-"heavy")? Again, using probability