Fuzzy Logic III

"Another direction of research is fuzzy systems. This will greatly increase the use of mathematics from the inanimate to the animate. In the past, mathematics has been used for the analysis of physical systems. With fuzzy systems and computer simulation we can study many processes in the social sciences." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"In the real world, none of these assumptions are uniformly valid. Often people want to know why mathematics and computers cannot be used to handle the meaningful problems of society, as opposed, let us say, to the moon boondoggle and high energy-high cost physics. The answer lies in the fact that we don't know how to describe the complex systems of society involving people, we don't understand cause and effect, which is to say the consequences of decisions, and we don't even know how to make our objectives reasonably precise. None of the requirements of classical science are met. Gradually, a new methodology for dealing with these 'fuzzy' problems is being developed, but the path is not easy." (Richard E Bellman, "Eye of the Hurricane: An Autobiography", 1984)

"One advantage of the use of fuzzy models is the fact that their complexity can be gradually increased as more information is gathered. This increase in complexity can be done automatically or manually by a careful commission of the new operating point." (Jairo Espinosa et al, "Fuzzy Logic, Identification and Predictive Control", 2005)

"Each fuzzy set is uniquely defined by a membership function. […] There are two approaches to determining a membership function. The first approach is to use the knowledge of human experts. Because fuzzy sets are often used to formulate human knowledge, membership functions represent a part of human knowledge. Usually, this approach can only give a rough formula of the membership function and fine-tuning is required. The second approach is to use data collected from various sensors to determine the membership function. Specifically, we first specify the structure of membership function and then fine-tune the parameters of membership function based on the data." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)

"Logic is the study of methods and principles of reasoning, where reasoning means obtaining new propositions from existing propositions. In classical logic, propositions are required to be either true or false; that is, the truth value of a proposition is either 0 or 1. Fuzzy logic generalizes classical two-value logic by allowing the truth values of a proposition to be any numbers in [0, 1]. This generalization allows us to perform fuzzy reasoning, also called approximate reasoning; that is, deducing imprecise conclusions (fuzzy propositions) from a collection of imprecise premises (fuzzy propositions). In this section, we first introduce some basic concepts and principles in classical logic and then study their generalizations to fuzzy logic." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)

"Fuzzy logic is an application area of fuzzy set theory dealing with uncertainty in reasoning. It utilizes concepts, principles, and methods developed within fuzzy set theory for formulating various forms of sound approximate reasoning. Fuzzy logic allows for set membership values to range (inclusively) between 0 and 1, and in its linguistic form, imprecise concepts like 'slightly', 'quite' and 'very'. Specifically, it allows partial membership in a set." (Larbi Esmahi et al,  Adaptive Neuro-Fuzzy Systems, 2009)

"Like classical logic, fuzzy logic uses formulas to formally represent statements about the world. Given an appropriate semantic structure (such as an evaluation of propositional symbols in the case of propositional logic, or a relational structure in the case of predicate logic), a truth degree of formula ϕ is denoted by ||ϕ||. It is significant that the truth degree ||ϕ|| of ϕ may in general be any element of the set of truth degrees. That is, formulas in fuzzy logic are true to degrees , not just true or false as in the case of classical logic." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"Nevertheless, the use of fuzzy logic is supported by at least the following three arguments. First, fuzzy logic is rooted in the intuitively appealing idea that the truths of propositions used by humans are a matter of degree. An important consequence is that the basic principles and concepts of fuzzy logic are easily understood. Second, fuzzy logic has led to many successful applications, including many commercial products, in which the crucial part relies on representing and dealing with statements in natural language that involve vague terms. Third, fuzzy logic is a proper generalization of classical logic, follows an agenda similar to that of classical logic, and has already been highly developed. An important consequence is that fuzzy logic extends the rich realm of applications of classical logic to applications in which the bivalent character of classical logic is a limiting factor." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"The principal idea employed by fuzzy logic is to allow for a partially ordered scale of truth-values, called also truth degrees, which contains the values representing false and true , but also some additional, intermediary truth degrees. That is, the set {0,1} of truth-values of classical logic, where 0 and 1 represent false and true , respectively, is replaced in fuzzy logic by a partially ordered scale of truth degrees with the smallest degree being 0 and the largest one being 1." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"We use the term fuzzy logic to refer to all aspects of representing and manipulating knowledge that employ intermediary truth-values. This general, commonsense meaning of the term fuzzy logic encompasses, in particular, fuzzy sets, fuzzy relations, and formal deductive systems that admit intermediary truth-values, as well as the various methods based on them." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)