An interesting letter by Dr. Lotfi Zadeh on Determinism vs. Chance published on UAI. He also referneces his GTU paper for migration from random fuzzy sets to granule-valued distributions;

Dear Hung,

Thank you for your illuminating analyses of the Valentina

example,and your comments regarding the theory of random fuzzy sets.

Your high expertise in both probability theory and fuzzy logic is in

evidence.

Regarding the Bayesian approach outlined by Aleks Jakulin, I should

like to add the following. First, the conditioning information is

perception-based and not quantifiable. Specifically, how would wrinkles,

for example, be dealt with? Furthermore, if my perception is that

Valentina is young, how would the Bayesian approach apply? If a

subjective probability distribution is associated with young, what would

be the probability distribution associated with not young? We could

apply random sets to this example but fuzzy logic would be much simpler

to use.

Clearly, the power of probabilistic methods is enhanced when we move

from point-valued discrete probability distributions to set-valued

discrete probability distributions, that is, to random sets. The power

is enhanced further when we move from random sets to random fuzzy sets,

as you do. Please note that in my 1979 paper "Fuzzy Sets and Information

Granularity," Advances in Fuzzy Set Theory and Applications, M. Gupta,

R. Ragade and R. Yager (eds.), 3-18. Amsterdam: North-Holland Publishing

Co., 1979 (available upon request), I employed random fuzzy sets to

generalize the Dempster-Shafer framework. However, moving from random

sets to random fuzzy sets is not sufficient. What has to be done is

moving from random fuzzy sets to granule-valued distributions, as

described in my paper "Generalized Theory of Uncertainty

(GTU)--Principal Concepts and Ideas," in Computational Statistics & Data

Analysis 51, 15-46, 2006. Downloadable: http://www.sciencedirect.com/ or

available upon request. The concept of a granule is more general than

the concept of a fuzzy set. A granule is characterized by a generalized

constraint. In my view, this level of generalization is needed to

enhance the power of probability theory to a point where it can deal

with the examples given in my messages. Try the following: Most Swedes

are much taller than most Italians. What is the difference in the

average height of Swedes and the average height of Italians? A solution

is given in my GTU paper.

Theory of random fuzzy sets enriches probability theory but not to a

point where it can be said, as you do, that randomness and fuzziness can

coexist in the framework of probability theory. Many other problems

remain. One of them, as is pointed out in my JSPI paper, which is cited

in my previous message, is that in dealing with imprecise probabilities

we have to deal in addition with imprecise events, imprecise functions,

imprecise relations and other imprecise dependencies. More importantly,

a basic problem is that almost all concepts in probability theory are

bivalent, e.g, events A and B are either independent or not independent,

a process is either stationary or nonstationary, an event either occurs

or does not occur, with no shades of truth allowed. In reality, these

and other concepts are not bivalent--they are a matter of degree. It may

take a long time for this to happen, but I have no doubt that eventually

it will be recognized that bivalent logic is not the right kind of logic

to serve as a foundation for probability theory.

You have made and are continuing to make important contributions to

both probability theory and fuzzy logic, and building bridges between

them. Please continue to do so.

With my warm regards,

Lotfi

--

Lotfi A. Zadeh

Professor in the Graduate School

Director, Berkeley Initiative in Soft Computing (BISC)

----------------------------------------** Soft Computing Applications in Business**

*Final Call for book Chapters*

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(1). Soft Computing Applications in Business

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