Probability Applications

Foreword
Writing this foreword after reading Lotfi Zadeh's foreword takes me back more than
two decades. When Lotfi was first developing fuzzy logic, we talked about it quite a lot.
My most vivid memories of those conversations are about the times when Lotfi presented
his ideas at Stanford to my seminar on the foundations of probability. We had some great
arguments, with lots of initial misunderstandings, especially on my part. I belonged, in those
days, to the school of thought that held that anything scientifically serious could be said
within classical logic and probability theory. There were, of course, disagreements about
the nature of probability, but not really about the formal properties of probability. The main
exception to this last claim was the controversy as to whether a probability measure should
be only finitely additive or also countably additive. De Finetti was the leading advocate of
the finitely additive viewpoint.
But new ideas and generalizations of probability theory had already been brewing on
several fronts. In the early pages of the present book the authors mention the Dempster-
Shafer theory of evidence, which essentially requires a generalization to upper and lower
probability measures. Moreover, already in the early 1960s, Jack Good, along with de Finetti,
a prominent Bayesian, had proposed the use of upper and lower probabilities from a different
perspective. In fact, by the late 1970s, just as fuzzy logic was blossoming, many varieties
of upper and lower probabilities were being cultivated. Often the ideas could be organized
around the concept of a Choquet capacity. For example, Mario Zanotti and I showed that
when a pair of upper and lower probability measures is a capacity of infinite order, there
exists a probability space and a random relation on this space that generates the pair. On the
other hand, in a theory of approximate measurement I introduced at about the same time, the
standard pair of upper and lower probability measures is not even a capacity of order two.
This is just a sample of the possibilities. But the developing interest in such generalizations
of standard probability theory helped create a receptive atmosphere for fuzzy logic.
To continue on this last point, I also want to emphasize that I am impressed by the
much greater variety of real applications of fuzzy logic that have been developed than is
the case for upper and lower probabilities. I was not surprised by the substantial chapters
on aircraft and auto reliability in the present volume, for about six years ago in Germany I
attended a conference on uncertainty organized by some smart and sophisticated engineers.
They filled my ears with their complaints about the inadequacy of probability theory to
provide appropriate methods to analyze an endless array of structural problems generated
by reliability and control problems in all parts of engineering. I am not suggesting I fully
understand what the final outcome of this direction of work will be, but I am confident that
the vigor of the debate, and even more the depth of the new applications of fuzzy logic,
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