# Re: Fuzzy logic compared to probability

Andreas Poncet (poncet@isi.ee.ethz.ch)
Tue, 27 Feb 1996 18:17:20 +0100

> I am interested in a certain question concerning fuzzy logic
> and probability. I am trying to figure out whether there are
> theoretical reasons to prefer one to the other in an application
> concerning the computation of degrees of certainty.

Well, usually, in the papers about fuzzy logic, it is claimed that fuzzy
membership functions are not to be mixed up with probabilities.
But what kind of probabilities? The arguments used are typically that
a "membershipness" can be given for any single observation, whereas no
"probability" - in the sense of frequency - can be defined. Fine, but
what if we consider the Bayesian subjective view of probability, namely
as a degree of belief in (or a degree of relevance of) a given
proposition? Then it seems to me that a "membershipness" is in fact
a CONDITIONAL PROBABILITY.

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"

> It would appear that if the fuzzy logic concept of `degree of
> truth' is the same as the Bayesian `degree of belief,' then
> either fuzzy logic is the same as probability or else less
> powerful (as it would have to be inconsistent).

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