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Chapter 1
Fuzzy Maximal Subsets
One paradigmatic change in science and mathematics this century concerns
the concept of uncertainty. It was once thought science should strive for cer-
tainty and that uncertainty was regarded as unscientific. An alternate view
now believes that uncertainty is essential to science. The publication of Lotfi
A. Zadeh’s seminal paper, [23], introducing the concept of fuzzy set theory
is an important work in the modern concept of uncertainty. Zadeh’s paper
challenges both Aristotelian two-valued logic and probability theory as being
the sole agent for uncertainty. Given a universal set X, a fuzzy subset µ of X
is a function of X into the closed interval [0, 1]. For an element x in X, the
statement that x is in µ is not necessarily true or false, but may be true to
some degree. This degree is given by µ(x).
It is stated by Zadeh in [23] that in the realm of soft sciences, sciences in
which imprecision, uncertainty, incompleteness of information and partiality of
truth lie at the center rather than the periphery, crisp-set-based mathematics
is not adequate as a modeling language. It is the inadequacy of crisp-set-
based mathematics as a modeling language that underlies the paucity of solid
theories in soft sciences. See also [4]. To overcome these inadequacies, we make
use of fuzzy set theory.
By Zadeh’s argument [23], we would expect to see considerable interest
in fuzzy logic in the social sciences. However, we do not. Why is that so?
One reason relates to the division in these disciplines between empirical and
theoretical research. The two are often disconnected, with theoretical findings
either being ignored by empiricists or those same conclusions having no rea-
sonable empirical referent. This is particularly evident in political science and
is the major impetus behind the National Science Foundation’s (NSF) initia-
tive to bring the two into more meaningful dialog [9]. Labeled the Empirical
Implications of Theoretical Models (EITM) program, the effort has arguably
yet to have made any notable impact in the discipline. Thus explicitly formal
approaches, to which fuzzy logic might contribute, have a minimal impact on
the empirical work that dominates the discipline’s journals.
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