"Fuzzy systems are excellent tools for representing heuristic, commonsense rules. Fuzzy inference methods apply these rules to data and infer a solution. Neural networks are very efficient at learning heuristics from data. They are 'good problem solvers' when past data are available. Both fuzzy systems and neural networks are universal approximators in a sense, that is, for a given continuous objective function there will be a fuzzy system and a neural network which approximate it to any degree of accuracy." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)
"Fuzzy set theory [...] is primarily concerned with quantifying and reasoning using natural language in which words can have ambiguous meanings. It is widely used in a variety of fields because of its simplicity and similarity to human reasoning." (Tzung-Pei Hong et al, "Genetic-Fuzzy Data Mining Techniques", 2009)
"Fuzzy systems are rule-based expert systems based on fuzzy rules and fuzzy inference. Fuzzy rules represent in a straightforward way 'commonsense' knowledge and skills, or knowledge that is subjective, ambiguous, vague, or contradictory."
"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)
"[Fuzzy logic p]rovides formal means for the representation of, and inference based on imprecisely specified premises and rules of inference; can be understood in different ways, basically as fuzzy logic in a narrow sense, being some type of multivalued logic, and fuzzy logic in a broad sense, being a way to formalize inference based on imprecisely specified premises and rules of inference. (Janusz Kacprzyk, "Foundations of Fuzzy Sets Theory", 2009)
"Granular computing is a general computation theory for using granules such as subsets, classes, objects, clusters, and elements of a universe to build an efficient computational model for complex applications with huge amounts of data, information, and knowledge. Granulation of an object a leads to a collection of granules, with a granule being a clump of points (objects) drawn together by indiscernibility, similarity, proximity, or functionality. In human reasoning and concept formulation, the granules and the values of their attributes are fuzzy rather than crisp. In this perspective, fuzzy information granulation may be viewed as a mode of generalization, which can be applied to any concept, method, or theory." (Salvatore Greco et al, "Granular Computing and Data Mining for Ordered Data: The Dominance-Based Rough Set Approach", 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)
"Unlike a conventional set, in a fuzzy set, a fuzzy membership function is used to define the degree of an element belonging to the set. Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth as defined by membership functions. Fuzzy logic contributes to the machinery of granular computing." (Zhengxin Chen, "Philosophical Foundation for Granular Computing", 2009)
"Unlike the classic set theory where a set is represented as an indicator function to specify if an object belongs or not to it, a fuzzy set is an extension of a classic set where a subset is represented as a membership function to characterize the degree that an object belongs to it. The indicator function of a classic set takes value of 1 or 0, whereas the membership function of a fuzzy set takes value between 1 and 0." (Jianchao Han & Nick Cerone, Principles and Perspectives of Granular Computing, 2009)
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