This collection of papers celebrates 50 years since the publication of Zadeh’s seminal paper creating the field of fuzzy sets and fuzzy logic [1]. This is certainly not an introductory book on the field of fuzzy logic, as it presupposes quite a lot of knowledge on fuzzy set theory, mathematics, and computer science. In fact, the book contains 31 papers mostly on recent results of the applications of fuzzy logic to a wide variety of fields, witnessing the success of the area in penetrating and contributing to the development of those areas.

The book opens with a paper authored by Lotfi Zadeh, as a deserved recognition to the father of the field. But Zadeh, still active and pushing the frontiers of fuzzy logic, instead of looking back on the developments of the past 50 years, proposes a new way of pursuing a theory of truth and meaning. His paper, “Toward a Restriction-Centered Theory of Truth and Meaning (RCT),” builds on 50 years of fuzzy set and fuzzy logic research to propose a theory of meaning based on the notion of “precisation of meaning.” In the tradition of fuzzy logic, the theory departs from the analysis of vague terms in natural language, proposes and analyzes a concept of restriction, and veers into logic and computational aspects of RCT.

A summing up of the field is presented in the 12th paper, by another well-known developer of the field and author of a classical introductory book [2], George Klir. Klir’s paper sets out to compare what was expected from the area, and what actual developments occurred in the past 50 years. In fact, he sets out to describe the discord between expectations and development, concentrating mainly on the applications of fuzzy logic, defined in a broad sense of “degrees of truth,” claims that the discrepancy between the expected and the actual lie in the extreme success of the fuzzy approach, going well beyond what was initially imagined. One such success story is the development of fuzzy controllers, which did not count among the reasons for introducing the notion of a fuzzy set, but became one of its first successes.

The other papers in the collection cover a huge variety of topics. In mathematics, we see papers on fuzzy measures and fuzzy integrals, and on fuzzy Markov chains and other fuzzified versions of probabilistic models: the latter is particularly interesting, as it indicates not an antagonism between the probabilistic and fuzzy approaches, but a complementary stance, at least from the part of the practitioners of the fuzzy approach. In artificial intelligence, there are papers on machine learning, human and machine intelligence, natural language processing, cognition, geospatial representation, and vision. In computer science, there are papers covering fuzzy versions of information retrieval, databases, the web, and support vector machines, among others. Other areas are also covered, such as econometrics and science.

Conspicuously absent from the book are papers that consider the logical aspect and the philosophical challenges of fuzzy logic. On the logical aspect, Zadeh’s initial paper was perhaps the only one to mention those topics with any relevance, while most others only mention it tangentially, stressing the fact that the editors’ choice for this celebratory book lies mainly on the applications, not on the foundations, of fuzzy logic. On the philosophical side, there is a paper by Rudolf Seising on aspects of fuzzy sets in science and philosophy, but the book ignores the disputes that surrounded the creation and the acceptance of the area as a respectable topic of research. These disputes were fueled by the claim made by Zadeh and colleagues about the “inadequacy” of probability theory and purely logical approaches to deal with the vagueness and imprecision of terms that abound in natural languages and, thus, are an integral part of human reasoning. The fact that this claim was not posed as the unsuitability of probability theory, but rather as the inadequacy of the probabilistic approach, has generated an ongoing debate on the philosophical stand of fuzzy logic, with recent and poignant claims on the inadequacy of the original claim itself [3]. It seems that, 50 years later, the practitioners of fuzzy logic have, consciously or not, abandoned the original inadequacy claim, finding a motivation for fuzzy sets and logic not on the inadequacy of other approaches, but on the wide number of areas in which it can be applied.

The final paper, by the three editors, is the only one to present an introductory overview of the theory of fuzzy sets and fuzzy logic. As a collection of papers with no fixed order, the last one is a good place for a novice to start, serving almost as an appendix for a quick run over the concepts mentioned in the rest of the book.