Preface to the Fourth Edition

My primary motivations for writing the fourth edition of this text have been to (1) reduce the length of the text, (2) correct the errata discovered since the publication of the third edition, and (3) introduce limited new material for the readers. The first motivation has been accomplished by eliminating some sections that are rarely taught in the classroom by various faculty using this text and by eliminating some sections that do not add to the utility of the text as a tool to learn basic fundamentals of the subject.

Since the first edition was published, in 1995, the technology of fuzzy set theory and its application to systems, using fuzzy logic, has moved rapidly. Developments in other theories such as possibility theory and evidence theory (both being elements of a larger collection of methods under the rubric "generalized information theories") have shed more light on the real virtues of fuzzy logic applications, and some developments in machine computation have made certain features of fuzzy logic much more useful than in the past. In fact, it would be fair to state that some developments in fuzzy systems are quite competitive with other, linear algebra-based methods in terms of computational speed and associated accuracy.

There is some new material which is included in the fourth edition to try to capture some of the newer developments; the keyword here is some because it would be impossible to summarize or illustrate even a small fraction of the new developments of the last six years since the third edition was published. As with any book containing technical material, the third edition contained errata that have been corrected in this fourth edition. As with the first three editions, a solutions manual for all problems in the fourth edition and software can be obtained by qualified instructors who visit www.wiley.com/go/ross/fuzzy4e and provide official documentation of their teaching status. In addition to the solutions manual, a directory of software will be made available to all student users and faculty of the text on this same website. Most of the software routines make use of the MATLAB platform, and most of the routines have been written by my students, except for the standard routines that exist as MATLAB functions.

As I discussed in the preface of the third edition, the axioms of a probability theory referred to as the excluded middle are again referred to in this edition as axioms-never as laws. The operations due to De Morgan are also not referred to as laws, but as principles because these principles do apply to some (not all) uncertainty theories (e.g., probability and fuzzy). The excluded middle axiom (and its dual, the axiom of contradiction) are not laws; Newton produced laws, Kepler produced laws, Darcy, Boyle, Ohm, Kirchhoff, Bernoulli, and many others too numerous to list here all developed laws. Laws are mathematical expressions describing the immutable realizations of nature. Definitions, theorems, and axioms collectively can describe a certain axiomatic foundation describing a particular kind of theory, and nothing more; in this case, the excluded middle and other axioms can be used to describe a probability theory. Hence, if a fuzzy set theory does not happen to be constrained by an excluded middle axiom, it is not a violation of some immutable law of nature like Newton's laws; fuzzy set theory simply does not happen to have an axiom of the excluded middle; it does not need, nor is constrained by, such an axiom. In fact, as early as 1905 the famous mathematician L. E. J. Brouwer defined this excluded middle axiom as a principle in his writings; he showed that the principle of the excluded middle was inappropriate in some logics, including his own, which he termed intuitionism. Brouwer observed that Aristotelian logic is only a part of mathematics, the special kind of mathematical thought obtained if one restricts oneself to relations of the whole and part. Brouwer had to specify in which sense the principles of logic could be considered "laws" because within his intuitionistic framework thought did not follow any rules, and, hence, "law" could no longer mean "rule" (see the detailed discussions on this in Chapters 5 and 13). In this regard, I continue to take on the cause advocated by Brouwer more than a century ago.

Also in this fourth edition, as in previous editions, I do not refer to "fuzzy measure theory" but instead describe it as "monotone measure theory"; the reader will see this in the title of Chapter 13. The former phrase still causes confusion when referring to fuzzy set theory; we hope to help in ending this confusion. In Chapter 13, in describing the monotone measure, m, I use the phrase describing this measure as a "basic evidence assignment (bea)," as opposed to the early use of the phrase "basic probability assignment (bpa)." Again, we attempt to avoid confusion with any of the terms typically used in probability theory.

As with the first three editions, this fourth edition is designed for the professional and academic audience interested primarily in applications of fuzzy logic in engineering and technology. I have found that the majority of students and practicing professionals are interested in the applications of fuzzy logic to their particular fields. Hence, the text is written for an audience primarily at the senior undergraduate and first-year graduate levels. With numerous examples throughout, this text is written to assist the learning process of a broad cross section of technical disciplines. It is primarily focused on applications, but each of the chapters begin with the rudimentary structure of the underlying mathematics required for a fundamental understanding of the methods illustrated.

Chapter 1 introduces the basic concept of fuzziness and distinguishes fuzzy uncertainty from other forms of uncertainty. It also introduces the fundamental idea of set membership, thereby laying the foundation for all material that follows, and presents membership functions as the format used for expressing set membership. The chapter summarizes a historical review of uncertainty theories and reviews the idea of "sets as points" in an n-dimensional Euclidean space as a graphical analog in understanding the relationship between classical (crisp) and fuzzy sets. A new section in the chapter addresses the intuition of propagating uncertainty by showing an example that compares the results of propagating probabilities on the one hand, or membership values on the other, through a simple nonlinear function. In this example there are some counterintuitive findings that readers will find both interesting and instructive.

Chapter 2 reviews classical set theory and develops the basic ideas of fuzzy sets. Operations, axioms, and properties of fuzzy sets are introduced by way of comparisons with the same entities for classical sets. Various normative measures to model fuzzy intersections (t-norms) and fuzzy unions (t-conorms) are summarized.

Chapter 3 develops the ideas of fuzzy relations as a means of mapping fuzziness from one universe to another. Various forms of the composition operation for relations are presented. Again, the epistemological approach in this chapter uses comparisons with classical relations in developing and illustrating fuzzy relations. Chapter 3 also illustrates methods to determine the numerical values contained within a specific class of fuzzy relations, called similarity relations. The section on a three-dimensional physical analogy of equivalence relations has been deleted.

Chapter 4 discusses the fuzzification of scalar variables and the defuzzification of membership functions. It introduces the basic features of a membership function and it discusses, briefly, the notion of interval-valued fuzzy sets. Defuzzification is necessary in dealing with the ubiquitous crisp (binary) world around us. The chapter details defuzzification of fuzzy sets and fuzzy relations into crisp sets and crisp relations, respectively, using lambda-cuts, and it describes a variety of methods to defuzzify membership functions into scalar values. Some of the defuzzification methods in the third edition have been deleted because they are seldom used in practice and because they are covered elsewhere in the literature. Examples of all methods are given in the chapter.

Chapter 5 introduces the precepts of fuzzy logic, again through a review of the relevant features of classical, or a propositional, logic. Various logical connectives and operations are illustrated. There is a thorough discussion of the various forms of the implication operation and the composition operation provided in this chapter. Three different inference methods, popular in the literature, are illustrated. Approximate reasoning, or reasoning under imprecise (fuzzy) information, is also introduced in this chapter. Basic IF-THEN rule structures are introduced and three graphical methods of inference are presented. The section on Natural Language has been shortened. A few more examples of the difficulties of using the axiom of the excluded middle are given in the summary of the chapter.

Chapter 6 provides several classical methods of developing membership functions, including methods that make use of the technologies of neural networks, genetic algorithms, and inductive reasoning.

Chapter 7 presents six...