by experiment. In neither case is the possibility of serendipitous discovery excluded.
Discovery, Carbonell proposes, is a three‐step process begun by hypothesis formation. The hypothesis may be either data driven, as is the case for observation, or theory driven as with experimentation. The initial step is followed by a refinement process in which partial theories are merged and boundary conditions are established. Finally, what has been learned and created is extended to new instances.
Clearly, any theory or model of human (or machine) learning must include aspects of each member of the established taxonomy. Today, no comprehensive and unified theory of human learning exists. Only partial theories that attempt to explain portions of the whole of human learning have been developed. Looking back over each of the ideas, we can take away the notion that learning and problem solving are effectively used interchangeably.
1.5 Crisp and Fuzzy Logic
As we move forward, we will explore, study, and learn two forms of logic and reasoning called crisp logic or reasoning and fuzzy logic or reasoning. The two graphical diagrams shown in Figure 1.1 suggest the difference between these two forms.
Figure 1.1 Crisp and fuzzy circles.
On the left is a crisp, precise circle and on the right is one in which the shape is less precise but can still be viewed in many contexts as a kind of circle.
1.6 Starting to Think Fuzzy
Over the years, fuzzy logic has been found to be extremely beneficial and useful to people involved in research and development in numerous fields including engineers, computer software developers, mathematicians, medical researchers, and natural scientists. As we begin, with all those people involved, we raise the question: What is fuzzy?
Originally, the word fuzz described the soft feathers that cover baby chicks. In English, the word means indistinct, imprecise, blurred, not focused, or not sharp. In French, the word is flou and in Japanese, it is pronounced “aimai.” In academic or technical worlds, the word fuzz or fuzzy is used in an attempt to describe the sense of ambiguity, imprecision, or vagueness often associated with human concepts.
Revisiting an earlier example, trying to teach someone to drive a car is a typical example of real‐life fuzzy teaching and fuzzy learning. As the student approaches a red light or intersection, what do you tell him or her? Do you say, “Begin to brake 25 m or 75 ft from the intersection?” Probably not. More likely, we would say something more like “Apply the brakes soon” or “Start to slow down in a little bit.” The first case is clearly too precise to be implemented or executed by the driver. How can one determine exactly when one is 25 m or 75 ft from an intersection? Streets and roads generally do not have clearly visible and accurate millimeter‐ or inch‐embedded gradations. The second vague instruction is the kind of expression that is common in everyday language.
Children learn to understand and to manipulate fuzzy instructions at an early age. They quite easily understand phrases such as “Go to bed about 10:00.” Perhaps with children, they understand too well. They are adept at turning such a fuzzy expression into one that is very precise. At 9:56, determined to stay up longer, they declare, “It's not 10:00 yet.”
In daily life, we find that there are two kinds of imprecision: statistical and nonstatistical. Statistical imprecision is that which arises from such events as the outcome of a coin toss or card game. Nonstatistical imprecision, on the other hand, is that which we find in instructions such as “Begin to apply the brakes soon.” This latter type of imprecision is called fuzzy, and qualifiers such as very, quickly, slowly or others on such expressions are called hedges in the fuzzy world.
Another important concept to grasp is the linguistic form of variables. Linguistic variables are variables with more qualitative rather than numerical values, comprising words or phrases in a natural or potentially an artificial language. That is, whether simple or complex, such variables are linguistic rather than numeric. Simple examples of such variables are very, 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