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“K23798” — 2015/2/2 —

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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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