Re: Fuzzy logic compared to probability

Fred A Watkins (fwatkins@hyperlogic.com)
Fri, 1 Mar 1996 22:27:10 +0100


yuan@binghamton.edu (Bo Yuan) wrote:

>In article <4gh66g$r8@elna.ethz.ch>, poncet@isi.ee.ethz.ch 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"

>A is not a proposition in the two-valued logic that is the foundation of
>probability theory or Bayesian theory.

Well now, it's most convenient to say "... not a proposition". But
that doesn't end the matter, because "the man is tall" is a sentence
that does have meaning. So you still have to come up with a truth
value for it. If you so not, then you limit your theory. It sounds
like you want "fuzzy logic" to be some sort of axiomatic thing.
Axiomatics is nice, but limiting.

Axiomatics is what saved mathematics from Russell's Paradox. Before it
appeared one could say "set of all sets"; not one can't. If we refuse
to allow paradox (i.e., insist on consistency) we have no choice but
to simply refuse to utter paradoxes. Even then it's not clear whether
our efforts have been successful.

You are right that "the man is tall" isn't a proposition in bivalent
logic. This is because it is *fuzzy*. And it's a good idea to remember
that the words "Probability" and "Bayesian Interpretation" are NOT
etched into the sky.

>> 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
>> theory guarantees consistent answers.
>>

>Fuzzy logic (in its narrow sense) and probability focus on different
>things. Fuzzy logic (in its narrow sense) or fuzzy set theory deals with
>problems at set theory or logic level, while probability theory or
>Bayesian theory deals with problems that are based on set theory or logic.
>If you want to compare the two, you need to consider probabilities of
>fuzzy sets (fuzzy events) or belief of fuzzy sets (fuzzy events).

Interesting. "Fuzzy set theory deals with problems at set theory or
logic level", and: "probability theory or Bayesian theory deals with
problems that are based on set theory or logic". What do you intend
the difference ("at" vs "based on") to be? And assuming you can come
up with a satisfactory clarification to that, how do you know that
there *are* probabilities (beliefs, if you're a Bayesian) of fuzzy
sets? You are surely aware that not every set admits a probability
measure. And, not everyone who thinks himself an expert on
"probability" would equate that idea with "belief", no matter what the
Bayesians say.

>Your example "the man is tall" is not a legal proposition in the
>two-valued logic. Hence, the classical Bayesian inference cannot be
>applied. You need a Bayesain inference for fuzzy sets (or events).

What was being offered is an instance of random Boolean set, as you
asked for in the first place. The notation is supposed to answer the
question "What is the probability that "the man is tall", *given* that
his height is x?" In fact it's identical to the notion that the input
fuzzy sets in an additive fuzzy system reflect conditional
probabilities.

Agreed, the original author didn't spell everything out in gross
detail, but the intent was clear enough. Viewing input activations as
conditional probabilities dates back to 1992 at least (by Kosko). It
follows from this idea that centroid defuzzification computes the
conditional mean of the output value, given the input data. Pedantic
mutterings about admissability of propositions won't get it in this
case.

Fred Watkins
fwatkins@hyperlogic.com