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This chapter is a general introduction to the book and an overview of its content. It describes the aims and scope of the book, and explains why a historical perspective is essential for achieving the aims. It introduces informally the key concepts involved, and the particular challenge fuzzy logic poses to the principle of bivalence in classical logic. It looks at the circumstances that led to the emergence of fuzzy logic in the academic community and as well as at the agendas of two main subareas of fuzzy logic, known as fuzzy logic in the narrow sense and fuzzy logic in the broad sense. The content of each of the subsequent chapters of the book is also briefly described.

The chapter examines the various propositional and predicate many-valued logics that were studied prior to the emergence of the concept of a fuzzy set in the mid-1960s, which led to the genesis of fuzzy logic in broad and narrow senses. Early ideas regarding formal systems of fuzzy logic allowed for deduction from partially true premises to partially true consequences, as suggested first by Goguen in the 1960s and further developed by Pavelka in the 1970s, and these ideas were developed from the 1990s onward. The systematic development of fuzzy logics based on t-norms and their residua, pursued under the leadership of Hájek in the 1990s, is discussed in some detail. An overview is presented of fuzzy logics that are not truth-functional, such as probabilistic, possibilistic and modal fuzzy logic. The chapter concludes by reviewing relevant additional issues, such as issues of computational complexity for fuzzy logic or higher-order fuzzy logics.

The high visibility of fuzzy logic and mathematics based on fuzzy logic is primarily due to their highly successful applications in many areas of human affairs. The aim of this chapter is to characterize the evolution and current status of these applications. Covered are applications in the various areas of engineering, natural, social and earth sciences, psychology, medicine, management and business, decision making, pattern recognition, image analysis, risk analysis, music, among others. In each of these areas, the significance of using fuzzy logic and the associated mathematics is critically assessed. It is argued that in some application areas, such as psychology or quantum mechanics, a cooperation between researchers in fuzzy logic and those in the application domain, leading possibly to new types of nonstandard fuzzy sets, is essential for further progress.

The chapter begins by introducing the important and useful distinction between the research agendas of fuzzy logic in the narrow and the broad senses. The chapter deals with the latter agenda, whose ultimate goal is to employ intuitive fuzzy set theory for emulating commonsense human reasoning in natural language and other unique capabilities of human beings. Restricting to standard fuzzy sets, whose membership degrees are real numbers in the unit interval [0,1], the chapter describes how this broad agenda has become increasingly specific via the gradual development of standard fuzzy set theory and the associated fuzzy logic. An overview of currently recognized nonstandard fuzzy sets, which open various new directions in fuzzy logic, is presented in the last section of this chapter.

This chapter first identifies the rare instances throughout the long history of classical logic when the principle of bivalence was challenged and shows that they all have been rather inconsequential. It then briefly examines the early research on many-valued logics during the first half the twentieth century, and describes in some detail circumstances that led to the emergence of fuzzy set theory and fuzzy logic in the mid-1960s. This is followed by characterizing the evolving attitudes toward fuzzy logic, especially within the academic community, and by a summary of major and well-documented debates between members of the emerging fuzzy logic community and opponents of fuzzy logic. Finally, the chapter describes how the supporting infrastructure for fuzzy logic evolved during the early and rather critical stage of development of fuzzy logic.

Mathematical reasoning is governed by the laws of classical logic, based on the principle of bivalence. With the acceptance of intermediate truth degrees, the situation changed substantially. This chapter begins with a characterization of mathematics based on fuzzy logic, an identification of principal issues of its development, and an outline of this development. It then examines the role of fuzzy logic in the narrow sense for developing mathematics based on fuzzy logic and the main approaches developed toward its foundations. Next, some selected areas of mathematics based on fuzzy logic are presented, such as the theory of sets and relations, algebra, topology, quantities and mathematical analysis, probability, and geometry. The chapter concludes by examining various semantic questions regarding fuzzy logic and mathematics based on it.

The aim of this chapter is to assess the significance of fuzzy logic based on its developments and its impact on mathematics and other areas of human affairs over its fifty-year history. For this purpose, the well-established concepts of a paradigm and a paradigm shift in science as well as their counterparts in mathematics and other areas of human affairs are employed as useful metaphors. Authors’ views regarding future prospects of fuzzy logic are also discussed. The prospects for a wider awareness of fuzzy logic depend on the development of specialist forums for discussion of its application, as well as on the availability of general-purpose textbooks. Fuzzy logic offers contributions to thinking about a large number of disciplines, and their acceptance will depend on how well the claims made for it materialize.

The aim of this paper is to provide a survey of issues regarding the problem of solving generalized fuzzy relational equations that are defined within a recently introduced framework of sup-preserving aggregation structures. Generalized fuzzy relational equations subsume the previously studied types of fuzzy relational equations, that is those based on either sup-t-norm or inf-residuum classes of compositions.

The principal aim of this paper is to bring the relatively little-known Hartley-like measure of uncertainty to the attention of readers of this journal. First, the reader is introduced to the classical Hartley measure of uncertainty, applicable to finite sets, and to the complementary Hartley-like measure of uncertainty, applicable to infinite sets. This is followed by an overview of some well-known applications of these measures to classical sets as well as standard fuzzy sets of possible alternatives. Applications of the Hartley-like measure to two types of non-standard fuzzy sets are then explored. This paper concludes with a discussion of two open research problems regarding the Hartley-like measure, solutions of which are essential for overcoming some practical limitations of this measure.

This article is based on my presentation at a symposium organised by Binghamton University (SUNY) on the occasion of my retirement. The purpose of the article is to pay tribute to three great scholars, Antonin Svoboda, W. Ross Ashby and Lotfi A. Zadeh, who have decisively influenced my ideas and research work pertaining to the emergence of systems science and to the study of the key role of uncertainty-based information in dealing with systems problems.

An overview of those W. Ross Ashby's ideas that contributed to the emergence of systems science is presented in this paper from my personal perspective. Ashby's visible influence on three research areas in systems science that I have pursued for many years is discussed in more detail.

In a recent paper, a functional AM (ambiguity measure) is introduced and an attempt is made to show that this functional qualifies as a measure of total aggregated uncertainty in the Dempster-Shafer theory. We show that this attempt fails due to a particular error in the proof of one of the principal theorems in the paper. Some additional remarks are made regarding recent research pertaining to the subject of the discussed paper.

The problem of defuzzification is examined in this paper from a broader perspective as a special way of dealing with the general problem of retranslation. The paper includes an overview of different formulations of the problem of defuzzification, as well as an overview of methods that have been suggested in the literature for dealing with the problem. Our own approach to defuzzification, which is described in the paper in more details, is based on relevant measures of uncertainty-based information. This paper is a companion to our recent paper that addresses the general problem of retranslation in computing with perceptions (Martin and Klir 2006).

This paper deals with one problem that needs to be addressed in the emerging field known under the name “computing with perceptions”. It is the problem of describing, approximately, a given fuzzy set in natural language. This problem has lately been referred to as the problem of retranslation. Two approaches to dealing with the retranslation problem are discussed in the paper, one based on a pre-defined set of linguistic terms and the associated fuzzy sets, and one based on approximating the given fuzzy set by another fuzzy set that is easier to describe in natural language. The retranslation problem is discussed in terms of two criteria—validity and informativeness.

We present a new approach to measure uncertainty/information applicable to theories based on convex sets of probability distributions, also called credal sets. A definition of a total disaggregated uncertainty measure on credal sets is proposed in this paper motivated by recent outcomes. This definition is based on the upper and lower values of Shannon's entropy for a credal set. We justify the use of the proposed total uncertainty measure and the parts into which it is divided: the maximum difference of entropies, which can be used as a non-specificity measure (imprecision), and the minimum of entropy, which represents a measure of conflict (contradiction).

This article deals with the claims that “a standard version of fuzzy logic collapses mathematically to two-valued logic” made by Charles Elkan in two papers [Proc 11th National Conf on AI, Menlo Park, CA: AAAI Press, 1993, pp 698–703; IEEE Expert 1994;9:3–8]. Although Elkan's effort to trivialize fuzzy logic has been questioned by numerous authors, our aim is to examine in detail his formal arguments and make some new observations. We present alternative, considerably simpler proofs of Elkan's theorems and use these proofs to argue that Elkan's claims are unwarranted. © 2007 Wiley Periodicals, Inc. Int J Int Syst 22: 203–207, 2007.

Various types of integrals with respect to signed fuzzy measures on finite sets with cardinality n can be presented as corresponding rules for partitioning the integrand. The partition can be expressed as an n-dimensional vector, whereas the signed fuzzy measure is also an n-dimensional vector. Thus, the integration value is the inner product of these two vectors. Two pairs of extremes, the Lebesgue-like integral versus the Choquet integral and the upper integral versus the lower integral, are discussed in detail. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1073–1092, 2006.

The paper gives an overview of applying fuzzy measures and relevant nonlinear integrals in data mining, discussed in five application areas: set function identification, nonlinear multiregression, nonlinear classification, networks, and fuzzy data analysis. In these areas, fuzzy measures allow us to describe interactions among feature attributes towards a certain target (objective attribute), while nonlinear integrals serve as aggregation tools to combine information from feature attributes. Values of fuzzy measures in these applications are unknown and are optimally determined via a soft computing technique based on given data.

A general way of representing incomplete information is to use closed and convex sets of probability distributions, which are also called credal sets. Each credal set is associated with uncertainty, whose amount is quantified by an appropriate uncertainty measure. One of the requisite properties of uncertainty measures is the property of additivity, which is associated with the concept of independence. For credal sets, the concept of independence is not unique. This means that different definitions of independence lead to different definitions of additivity for uncertainty measures. In this paper, we compare the various definitions of independence, but our principal aim is to analyze those definitions that are employed in the most significant uncertainty measures established in the literature for credal sets.

The principal purpose of this paper is to present a comprehensive overview of generalized information theory (GIT): a research program whose objective is to develop a broad treatment of uncertainty-based information, not restricted to classical notions of uncertainty. After a brief overview of classical information theories, a broad framework for formalizing uncertainty and the associated uncertainty-based information of a great spectrum of conceivable types is sketched. The various theories of imprecise probabilities that have already been developed within this framework are then surveyed, focusing primarily on some important unifying principles applying to all these theories. This is followed by introducing two higher levels of the theories of imprecise probabilities: (i) the level of measuring the amount of relevant uncertainty (predictive, retrodictive, prescriptive, diagnostic, etc.) in any situation formalizable in each given theory, and (ii) the level of some methodological principles of uncertainty, which are contingent upon the capability to measure uncertainty and the associated uncertainty-based information. Various issues regarding both the measurement of uncertainty and the uncertainty principles are discussed. Again, the focus is on unifying principles applicable to all the theories. Finally, the current status of GIT is assessed and future research in the area is discussed.