In order to properly characterize the content of this book, it is important toclarify first the intended meaning of its title Fuzzy Logic. This clarification is needed since the term “fuzzy logic,” as currently used in the literature, is viewed either in a narrow sense or in a broad sense. In the narrow sense, fuzzy logic is viewed as an area devoted to the formal development, in a unified way, of the various logical systems of many-valued logic. It is concerned with formalizing syntactic aspects (based on the notion of proof) and semantic aspects (based on the notion of truth) of the various logical calculi. In order to be acceptable, each of these logical calculi must be sound (provability implies truth) and complete

(truth implies provability). The most representative publication of fuzzy logic in this sense is, in my opinion, the classic book by Peter HajekAfter examining the content of this book, it is easy to conclude that its title, Fuzzy Logic, refers to fuzzy logic in the broad sense. This is consistent, by and large, with the usual meaning of the term “fuzzy logic” in the literature. Indeed, most papers and books that use the term “fuzzy logic” in their titles or in their lists of keywords are, in fact, dealing with issues of fuzzy logic in the broad sense. Literature devoted to fuzzy logic in the narrow sense, which is lately referred to as mathematical fuzzy logic or formal fuzzy logic, is only a small fraction of the overall literature on fuzzy logic, and it is primarily literature shared by a relatively small community of researchers working in this area.

After reading this book and comparing the various issues discussed in its chapters with the simple idea of a set with unsharp boundaries suggested in 1965 by Zadeh, one can hardly fail to recognize that the idea, regardless of or, perhaps, due to its simplicity, has proven to be an extremely profound idea. Considering the huge body of theoretical results and the fascinating spectrum of applications that are described in this book, which all emerged over the period of about forty years from such a simple idea, one can hardly be left unimpressed. Let me mention just some of the most impressive developments in this area that are well captured in this book.

After reading this book and comparing the various issues discussed in its chapters with the simple idea of a set with unsharp boundaries suggested in 1965 by Zadeh, one can hardly fail to recognize that the idea, regardless of or, perhaps, due to its simplicity, has proven to be an extremely profound idea. Considering the huge body of theoretical results and the fascinating spectrum of applications that are described in this book, which all emerged over the period of about forty years from such a simple idea, one can hardly be left unimpressed. Let me mention just some of the most impressive developments in this area that are well captured in this book.

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