Probability theory and fuzzy logic are the principal components of an array of methodologies
for dealing with problems in which uncertainty and imprecision play important roles.
In relation to probability theory, fuzzy logic is a new kid on the block. As such, it has been
and continues to be, though to a lesser degree, an object of controversy. The leitmotif of
Fuzzy Logic and Probability Applications: Bridging the Gap is that fuzzy logic and probability
theory are complementary rather than competitive. This is a thesis that I agree with
completely. However, in one respect my perspective is more radical. Briefly stated, I believe
that it is a fundamental error to base probability theory on bivalent logic. Moreover, it is my
conviction that eventually this view will gain wide acceptance. This will happen because
with the passage of time it will become increasingly obvious that there is a fundamental
conflict between bivalence and reality.

To write a foreword to a book that is aimed at bridging the gap between fuzzy logic
and probability theory is a challenge that is hard to meet. But a far greater challenge is to
produce a work that illuminates some of the most basic concepts in human cognition—the
concepts of randomness, probability, uncertainty, vagueness, possibility, imprecision, and
truth. The editors, authors, and publisher of Fuzzy Logic and Probability Applications have,
in my view, met this challenge.

In Fuzzy Logic and Probability Applications, in consonance with this book's central
theme, controversial issues relating to fuzzy logic and probability theory are treated with
objectivity, authority, and insight. Of particular interest and value is the incisive analysis
of the evolution of probability theory and fuzzy logic presented in Chapter 1. One of the
basic concepts discussed in this chapter is that of vagueness. In the authors' interpretation,
vagueness and fuzziness are almost synonymous. In my view, this is not the case. Basically,
vagueness relates to insufficient specificity, as in "I will be back sometime," whereas
fuzziness relates to unsharpness of boundaries, as in "I will be back in a few minutes." Thus
fuzziness is a property of both predicates and propositions, whereas vagueness is a property
of propositions but not of predicates. For example, "tall" is a fuzzy predicate, but "Robert
is tall" is fuzzy but not vague.

Complementarity of fuzzy logic and probability theory is rooted in the fact that probability
theory is concerned with partial certainty, whereas fuzzy logic is mainly concerned
with partial possibility and partial truth. A simple example is that if Robert is half-German,
then the proposition "Robert is German" may be viewed as half-true but not uncertain. On
the other hand, if it is possible that Robert is German, then the probability that he is German
may be 0.5.