"A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint." (Lotfi A Zadeh, "Fuzzy Sets", 1965)
"The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply denned criteria of class membership rather than the presence of random variables." (Lotfi A Zadeh, "Fuzzy Sets", 1965)
"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)
"As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics." (Lotfi A Zadeh, 1973)
"The closer one looks at a real-world problem, the fuzzier becomes its solution." (Lotfi A Zadeh, 1973)
"[Fuzzy logic is] a logic whose distinguishing features are (1) fuzzy truth-values expressed in linguistic terms, e. g., true, very true, more or less true, or somewhat true, false, nor very true and not very false, etc.; (2) imprecise truth tables; and (3) rules of inference whose validity is relative to a context rather than exact." (Lotfi A Zadeh, "Fuzzy logic and approximate reasoning", 1975)
"[...] much of the information on which human decisions are based is possibilistic rather than probabilistic in nature, and the intrinsic fuzziness of natural languages - which is a logical consequence of the necessity to express information in a summarized form - is, in the main, possibilistic in origin." (Lotfi A Zadeh, "Fuzzy Sets as the Basis for a Theory of Possibility", Fuzzy Sets and Systems, 1978)
"Fuzziness, then, is a concomitant of complexity. This implies that as the complexity of a task, or of a system for performing that task, exceeds a certain threshold, the system must necessarily become fuzzy in nature. Thus, with the rapid increase in the complexity of the information processing tasks which the computers are called upon to perform, we are reaching a point where computers will have to be designed for processing of information in fuzzy form. In fact, it is the capability to manipulate fuzzy concepts that distinguishes human intelligence from the machine intelligence of current generation computers. Without such capability we cannot build machines that can summarize written text, translate well from one natural language to another, or perform many other tasks that humans can do with ease because of their ability to manipulate fuzzy concepts." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)
"In essence, a basic assumption in classical mathematics is that a concept must admit of precise definition which partitions the class of all objects into two classes: (i) those objects which are instances of the concept; and (ii) those which are not, with no borderline cases allowed. For example, a function is either continuous or discontinuous; it cannot be continuous to a degree. Similarly, a matrix is either symmetric or not symmetric; it cannot be somewhat symmetric or more or asymmetric or symmetric to a degree." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)
"It is important to observe that there is an intimate connection between fuzziness and complexity. Thus, a basic characteristic of the human brain, a characteristic shared in varying degrees with all information processing systems, is its limited capacity to handle classes of high cardinality, that is, classes having a large number of members. Consequently, when we are presented with a class of very high cardinality, we tend to group its elements together into subclasses in such a way as to reduce the complexity of the information processing task involved. When a point is reached where the cardinality of the class of subclasses exceeds the information handling capacity of the human brain, the boundaries of the subclasses are forced to become imprecise and fuzziness becomes a manifestation of this imprecision." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic", 1989)
"As systems became more varied and more complex, we find that no single methodology suffices to deal with them. This is particularly true of what may be called information intelligent systems - systems which form the core of modern technology. To conceive, design, analyze and use such systems we frequently have to employ the totality of tools that are available. Among such tools are the techniques centered on fuzzy logic, neurocomputing, evolutionary computing, probabilistic computing and related methodologies. It is this conclusion that formed the genesis of the concept of soft computing." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)
"[...] fuzzy logic [FL] has many distinct facets - facets which overlap and have unsharp boundaries. Among these facets there are four that stand out in importance. They are (i) the logical facet; (ii) the set-theoretic facet: (iii) the relational facet, and (iv) the epistemic facet. […] The logical facet of FL, FL/L, is a logical system or, more accurately, a collection of logical systems which includes as a special case both two-valued and multiple-valued systems. […] The set-theoretic facet of FL, FL/S, is concerned with classes or sets whose boundaries are not sharply defined. […] The relational facet of FL, FL/R, is concerned in the main with representation and manipulation of imprecisely defined functions and relations. […] The epistemic facet of FL, FL/E, is linked to its logical facet and is centered on applications of FL to knowledge representation, information systems, fuzzy databases and the theories of possibility imprecise probabilities." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)
"In science, it is a long-standing tradition to deal with perceptions by converting them into measurements. But what is becoming increasingly evident is that, to a much greater extent than is generally recognized, conversion of perceptions into measurements is infeasible, unrealistic or counter-productive. With the vast computational power at our command, what is becoming feasible is a counter-traditional move from measurements to perceptions. […] To be able to compute with perceptions it is necessary to have a means of representing their meaning in a way that lends itself to computation." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)
"A concept which has a position of centrality in fuzzy logic is that of a fuzzy set. Informally, a fuzzy set is a class with a fuzzy boundary, implying a gradual transition from membership to nonmembership. A fuzzy set is precisiated through graduation, that is, through association with a scale of grades of membership. Thus, membership in a fuzzy set is a matter of degree. Importantly, in fuzzy logic everything is or is allowed to be graduated, that is, be a matter of degree. Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, equivalence, similarity, proximity or functionality. Graduation and granulation form the core of fuzzy logic. Graduated granulation is the basis for the concept of a linguistic variable – a variable whose values are words rather than numbers. The concept of a linguistic variable is employed in almost all applications of fuzzy logic." (Lofti A Zadeh, "Fuzzy Logic", 2009)
"In essence, logic is concerned with formalization of reasoning. Correspondently, fuzzy logic is concerned with formalization of fuzzy reasoning, with the understanding that precise reasoning is a special case of fuzzy reasoning." (Lofti A Zadeh, "Fuzzy Logic", 2009
"Science deals not with reality but with models of reality. In large measure, scientific progress is driven by a quest for better models of reality. In the real world, imprecision, uncertainty and complexity have a pervasive presence. In this setting, construction of better models of reality requires a better understanding of how to deal effectively with imprecision, uncertainty and complexity. To a significant degree, development of fuzzy logic has been, and continues to be, motivated by this need." (Lofti A Zadeh, "Fuzzy Logic", 2009)
"As complexity rises, precise statements lose meaning and meaningful statements lose precision." (Lotfi A Zadeh)
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