# [10] What is the relationship between fuzzy truth values and probabilities?

```Date: 21-NOV-94

This question has to be answered in two ways: first, how does fuzzy
theory differ from probability theory mathematically, and second, how
does it differ in interpretation and application.

At the mathematical level, fuzzy values are commonly misunderstood to be
probabilities, or fuzzy logic is interpreted as some new way of handling
probabilities.  But this is not the case.  A minimum requirement of
probabilities is ADDITIVITY, that is that they must add together to one, or
the integral of their density curves must be one.

But this does not hold in general with membership grades.  And while
membership grades can be determined with probability densities in mind (see
[11]), there are other methods as well which have nothing to do with
frequencies or probabilities.

Because of this, fuzzy researchers have gone to great pains to distance
themselves from probability. But in so doing, many of them have lost track
of another point, which is that the converse DOES hold: all probability
distributions are fuzzy sets! As fuzzy sets and logic generalize Boolean
sets and logic, they also generalize probability.

In fact, from a mathematical perspective, fuzzy sets and probability exist
as parts of a greater Generalized Information Theory which includes many
formalisms for representing uncertainty (including random sets,
Demster-Shafer evidence theory, probability intervals, possibility theory,
general fuzzy measures, interval analysis, etc.).  Furthermore, one can
also talk about random fuzzy events and fuzzy random events.  This whole
issue is beyond the scope of this FAQ, so please refer to the following
articles, or the textbook by Klir and Folger (see [16]).

Semantically, the distinction between fuzzy logic and probability theory
has to do with the difference between the notions of probability and a
degree of membership.  Probability statements are about the likelihoods of
outcomes: an event either occurs or does not, and you can bet on it.  But
with fuzziness, one cannot say unequivocally whether an event occured or
not, and instead you are trying to model the EXTENT to which an event
occured. This issue is treated well in the swamp water example used by
James Bezdek of the University of West Florida (Bezdek, James C, "Fuzzy
Models --- What Are They, and Why?", IEEE Transactions on Fuzzy Systems,
1:1, pp.  1-6).

Delgado, M., and Moral, S., "On the Concept of Possibility-Probability
Consistency", Fuzzy Sets and Systems 21:311-318, 1987.

Dempster, A.P., "Upper and Lower Probabilities Induced by a Multivalued
Mapping", Annals of Math. Stat. 38:325-339, 1967.

Henkind, Steven J., and Harrison, Malcolm C., "Analysis of Four
Uncertainty Calculi", IEEE Trans. Man Sys. Cyb. 18(5)700-714, 1988.

Kamp`e de, F'eriet J., "Interpretation of Membership Functions of Fuzzy
Sets in Terms of Plausibility and Belief", in Fuzzy Information and
Decision Process, M.M. Gupta and E. Sanchez (editors), pages 93-98,
North-Holland, Amsterdam, 1982.

Klir, George, "Is There More to Uncertainty than Some Probability
Theorists Would Have Us Believe?", Int. J. Gen. Sys. 15(4):347-378, 1989.

Klir, George, "Generalized Information Theory", Fuzzy Sets and Systems
40:127-142, 1991.

Klir, George, "Probabilistic vs. Possibilistic Conceptualization of
Uncertainty", in Analysis and Management of Uncertainty, B.M. Ayyub et.
al. (editors), pages 13-25, Elsevier, 1992.

Klir, George, and Parviz, Behvad, "Probability-Possibility
Transformations: A Comparison", Int. J. Gen. Sys. 21(1):291-310, 1992.

Kosko, B., "Fuzziness vs. Probability", Int. J. Gen. Sys.
17(2-3):211-240, 1990.

Puri, M.L., and Ralescu, D.A., "Fuzzy Random Variables", J. Math.
Analysis and Applications, 114:409-422, 1986.

Shafer, Glen, "A Mathematical Theory of Evidence", Princeton University,
Princeton, 1976.

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