Fuzzy Logic and Mathematics ─ A Historical Perspective

The term "fuzzy logic" (FL), as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic: the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediary truth values between true and false are allowed, which are interpreted as degrees of truth. This opens a new way of thinking--thinking in terms of degrees rather than absolutes. For example, it led to the definition of a new category of sets, referred to as fuzzy sets, in which membership is a matter of degree.

The book examines the genesis and development of two research programs within FL, known as fuzzy logic in the narrow sense (FLN) and fuzzy logic in the broad sense (FLB), whose agendas are very different. The agenda of FLN is the development of propositional, predicate, and other fuzzy-logic calculi that are sound and complete in a similar sense as in classical logic. The agenda of FLB is to explore the rejection of the rigid principle of bivalence for emulating commonsense human reasoning in natural language and other unique capabilities of human beings. The book also examines principles for developing mathematics based on fuzzy logic and surveys established and prospective applications of fuzzy logic and mathematics based on fuzzy logic in various areas of human affairs. The principal aim of the book is to assess the significance of FL for these various areas and, especially, for mathematics. For this purpose, the well-known notion of paradigm shifts in science and its more recently discussed counterparts in mathematics and other areas, such as engineering or medicine, are employed.

Radim Belohlavek is Professor of Computer Science and Head of Department of Computer Science at the Palacky University, Olomouc in the Czech Republic. He obtained PhD in Computer Science, PhD in Mathematics, and DSc in Mathematical and Physical Sciences. He is the author of over 150 papers in peer reviewed journals and conferences on various aspects of uncertainty, logic, and discrete mathematics.

Joseph W. Dauben is Distinguished Professor of History and the History of Science at the City University of New York. A graduate of Claremont McKenna College (A.B. '66) and Harvard University (A.M., Ph.D. '72), Professor Dauben is the author of biographies of Georg Cantor and Abraham Robinson. He has been a member of the Institute for Advanced Study (Princeton) and Clare Hall (Cambridge), and the recipient of a Guggenheim Fellowship and a Senior ACLS Fellowship. In January of 2012 he received the Albert Leon Whiteman Memorial Prize for History of Mathematics, conferred by the American Mathematical Society in recognition of a career of outstanding contributions to the history of mathematics.

George J. Klir spent most of his academic career (1969-2008) with the State University of New York at Binghamton where he was appointed Distinguished Professor of Systems Science in 1978. He published well over 300 research papers and 31 scholarly books. He received numerous awards and honors including 6 honorary doctoral degrees.