slightly, quickly, and slowly. Other examples may be generated from a set of primary terms such as young or its antonym old or tall with its antonym short.
The first practical noteworthy applications of fuzzy logic and fuzzy set theory began to appear in the 1970s and 1980s. To effectively design modern everyday systems, one must be able to recognize, represent, interpret, and manipulate statistical and nonstatistical uncertainties. One should also learn to work with hedges and linguistic variables. One should use statistical models to capture and quantify random imprecision and fuzzy models to capture and to quantify nonrandom imprecision.
1.7 History Revisited – Early Mathematics
Fuzzy logic, with roots in early Greek philosophy, finds a wide variety of contemporary applications ranging from the manufacture of cement to the control of high‐speed trains, auto focus cameras, and potentially self‐driving automobiles. Yet, early mathematics began by emphasizing precision. The central theme in the philosophy of Aristotle and many others was the search for perfect numbers or golden ratios. Pythagoras and his followers kept the discovery of irrational numbers a secret. Their mere existence was also counter to many fundamental religious teachings of the time.
Later mathematicians continued the search for precision and were driven toward the goal of developing a concise theory of mathematics. One such effort was The Laws of Thought published by Stephan Korner in 1967 in the Encyclopedia of Philosophy. Korner's work included a contemporary version of The Law of Excluded Middle which stated that every proposition could only be TRUE or FALSE – there could be no in between. An earlier version of this law, proposed by Parmenides in approximately 400 BC, met with immediate and strong objections. Heraclitus, a fellow philosopher, countered that propositions could simultaneously be both TRUE and NOT TRUE. Plato, the student, made the same arguments to his teacher Socrates.
1.7.1 Foundations of Fuzzy Logic
Plato was among the first to attempt to quantify an alternative possible state of existence. He proposed the existence of a third region, beyond TRUE and FALSE, in which “opposites tumbled about.” Many modern philosophers such as Bertrand Russell, Kurt Gödel, G.W. Leibniz, and Hermann Lotze have supported Plato's early ideas.
The first formal steps away from classical logic were taken by the Polish mathematician Lukasiewicz (also the inventor of Reverse Polish Notation, RPN). He proposed a three‐valued logic in which the third value, called possible, was to be assigned a numeric value somewhere between TRUE and FALSE. Lukasiewicz also developed an entire set of notations and an axiomatic system for his logic. His intention was to derive modern mathematics.
In later works, he also explored four‐ and five‐valued logics before declaring that there was nothing to prevent the development of infinite‐valued logics. Donald Knuth proposed a similar three‐valued logic and suggested using the values of −1, 0, 1. The idea never received much support.
1.7.2 Fuzzy Logic and Approximate Reasoning
The birth of modern fuzzy logic is usually traced to the seminal paper Fuzzy Sets published in 1965 by Lotfi A. Zadeh. In his paper, Zadeh described the mathematics of fuzzy subsets and, by extension, the mathematics of fuzzy logic. The concept of the fuzzy event was introduced by Zadeh (1968) and has been used in various ways since early attempts to model inexact concepts were prevalent in human reasoning. The initial work led to the development of the branch of mathematics called fuzzy logic. This logic, actually a superset of classical binary‐valued logic, does not restrict set membership to absolutes (Yes or No) but tolerates varying degrees of membership.
Using these criteria, an element is assigned a grade of membership in a parent set. The domain of this attribute is the closed interval [0, 1]. If the grade of membership values is restricted to the two extrema, then fuzzy logic reduces to two‐valued or crisp logic.
In his work, Zadeh proposed that people often base their thinking and decisions on imprecise or nonnumerical information. He further believed that the membership of an element in a set need not be restricted to the values 0 and 1 (corresponding to FALSE and TRUE) but could easily be extended to include all real numbers in the interval 0.0–1.0 including the endpoints. He further felt that such a concept should not be considered in isolation but rather viewed as a methodology that moves from a discrete world to a continuous one. To augment such thinking, he proposed a collection of operations supporting his new logic.
Zadeh introduced his ideas as a new way of representing the vagueness common in everyday thinking and language. His fuzzy sets are a natural generalization or superset of classical sets or Boolean logic that are one of the basic structures underlying contemporary mathematics. Under Boolean algebra, a proposition takes a narrow view that a value is either completely true or completely false. In contrast, fuzzy logic introduces the concept of partial truth under which values are expressed anywhere within, and including, the two extremes of TRUE and FALSE.
Based on the idea of the fuzzy variable, Zadeh (1979) further proposed a theory of approximate reasoning. This theory postulates the notion of a possibility distribution on a linguistic variable. Using this concept, he was able to reason using vague concepts such as young, old, tall, or short. Zadeh also introduced the ideas of semantic equivalence and semantic entailment on the possibility distributions of linguistic propositions. Using these concepts, he was able to determine that a statement and its double negative are equivalent and that very small is more restrictive than small. Such conclusions derive from either the equality or containment of corresponding distributions.
Zadeh's theory is generally effective in reconciling ambiguous natural language expressions. The scope of the work was initially limited to laboratory sentences comparing hair color, age, or height between various people. Zadeh's work provided a good tool for future efforts, particularly in combination with or to enhance other forms of reasoning.
As often follows the introduction of a new concept or idea, questions arise: Why does that thing do this? Why doesn't it do that? Can you make it do another thing? An early criticism of Zadeh's fuzzy sets was: “Why can't your fuzzy set members have an uncertainty associated with them?” Zadeh eventually dealt with the issue by proposing more sophisticated kinds of fuzzy sets. New criteria evolved the original concept into numbered types of fuzzy subsets. His initial work became type‐1 fuzzy sets. Additional concepts grew from type‐2 fuzzy sets to ultimately type‐n in a 1976 paper to incorporate greater uncertainty into set membership. Naturally, if a type‐2 or higher set has no uncertainty in its members, it reduces to a type‐1. In this text, we will work primarily with type‐1 fuzzy subsets.
1.7.3 Non‐monotonic Reasoning
Non‐monotonic reasoning is an attempt to duplicate the human ability to reason with incomplete knowledge and to make default assumptions when insufficient evidence exists to empirically support a hypothesis. This proposed method of reasoning may be contrasted with monotonic reasoning in the following way.
A monotonic logic states that if a conclusion can be derived from a set of premises X, and if X is a subset of some larger set of premises Y, then x, a member of X, may also be derived from Y. This does not hold true for non‐monotonic logic since Y may contain statements that may prevent the earlier conclusion from being derived.
Consider the following example scenario: The objective is to cross a river, and at the edge of the river there is a row boat and a set of oars. Using monotonic logic, one can conclude that it is possible to cross the river by rowing the boat across. If the new information that the boat is painted red is
