Precise reasoning about reasoning with fuzzy words

Comment by Tillers: legal scholars might avoid uttering a great deal of nonsense about imprecise legal concepts if they took the trouble to study fuzzy logic.

Lotfi A. Zadeh:

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Dear members of the BISC Group: 

    The concept of FL-generalization was introduced in my 2008 paper "Is there a need for fuzzy logic?" Information Sciences. The basic importance of FL-generalization has not been fully recognized as yet. For those who are not familiar with FL-generalization, a brief explanation is provided in the following.  

    In large measure, science -- including mathematics -- is based on the classical, Aristotelian, bivalent logic. Bivalent-logic-based science has achieved brilliant successes. But what is striking is that bivalent-logic-based science ignores a basic reality. In human cognition, almost all classes have unsharp (fuzzy) boundaries. Bivalent logic is not the right logic for dealing with such classes, nor is bivalent-logic-based probability theory. What is needed for this purpose is fuzzy set theory and, more broadly, fuzzy logic, FL. Informally, fuzzy logic is a system of reasoning and computation in which the objects of reasoning and computation are classes with unsharp (fuzzy) boundaries.  

    The point of departure in fuzzy set theory is a generalization of the concept of a set to the concept of a fuzzy set. A fuzzy set, A, in a space, U, is a graduated class of elements of U. Graduation involves association of each element, u, of U with its grade of membership in A. This very simple generalization has wide-ranging ramifications. 

    Let T be a bivalent-logic-based theory, formalism, algorithm, concept, etc. T is FL-generalized by adding to T the concept of a fuzzy set along with associated concepts and techniques. The result of FL-generalization is fuzzy T.Examples. Fuzzy arithmetic, fuzzy linear programming, fuzzy control, fuzzy stability, fuzzy support vector machine, fuzzy group theory, fuzzy topology, fuzzy convex set, fuzzy back-propagation algorithm, fuzzy probability, etc. T may be viewed as a special case of fuzzy T. FL-generalization is a matter of degree, reflecting the extent to which sets in T are replaced by fuzzy sets. In the limit, FL-generalization of T involves a shift in the foundations of T from bivalent logic to fuzzy logic. 

    What is gained by FL-generalization? There are two principal rationales. First, FL-generalization opens the door to construction of better models of reality. There is a fundamental conflict between two realities. In the world of human cognition, almost all concepts are classes with unsharp (fuzzy) boundaries, that is, are a matter of degree. In the world of science, almost all definitions are bivalent, with no degrees allowed. Here are a few examples. In economics, the official definition of recession is bivalent. Specifically, economy is in a state of recession if the GDP declined in two successive quarters. Realistically, recession is not a bivalent concept -- it is a matter of degree. In probability theory, stationarity is defined as a bivalent concept. Realistically, stationarity is a matter of degree. In stability theory, stability is defined as a bivalent concept. Realistically, stability is a matter of degree, and so on, and on and on. FL-generalization of definitions, serves an important purpose--replacement of bivalent definitions with fuzzy-logic-based definitions which are better models of reality. 

    The second rationale has a position of centrality in applications of fuzzy logic. Specifically, the second rationale involves an exploitation of tolerance for imprecision through replacement of numbers with precisiated words. A word is precisiated by representing it as a label of a fuzzy set which has a specified membership function. A striking example of exploitation of a tolerance for imprecision is the problem of stabilization of an inverted pendulum. The traditional approach starts with formulation of differential equations of motion, followed by application of stability theory. In the fuzzy-logic-based approach, a small number of very simple fuzzy if-then rules, with precisiated words in the antecedents and consequents, are employed to describe the dynamics of the inverted pendulum. This is the essence of what is called the Fuzzy Logic Gambit. Fuzzy Logic Gambit is an essential ingredient of the formalism of Computing with Words (CWW). Basically, CWW may be viewed as a progression from the use of numbers to the use of precisiated words. 

     In summary, FL-generalization may be viewed as an important instrument of generalization in which the point of departure is replacement of the concept of a set with the concept of a fuzzy set. In large measure, scientific progress is driven by a quest for better models of reality. What I see in my crystal ball is the following. In coming years, more and more theories, formalisms, algorithms and concepts will be FL-generalized. This is likely to be the case even in mathematics--a discipline in which the word "fuzzy" strikes a dissonant note. What should be recognized is that shifting foundations of a theory from bivalent logic to fuzzy logic need not involve a loss of rigor and precision. Example. Fuzzy topology is every bit as rigorous and precise as classical topology. Comments are welcome.

                 Regards,

                 Lotfi

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Lotfi A. Zadeh
Professor Emeritus
Director, Berkeley Initiative in Soft Computing (BISC)

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Similarity and Fuzziness - and More

On the nature of and grounds for the relationship between fuzziness and similarity, see John F. Sowa What Is the Source of Fuzziness?
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Lotfi Zadeh

USA SCIENCE & ENGINEERING FESTIVAL – ROLE MODELS IN SCIENCE & ENGINEERING ACHIEVEMENT

Lofti Asker Zadeh

Lotfi Asker Zadeh -- Computer Scientist and Mathematician

The Father of the scientific concept,"Fuzzy Logic", as well as "Fuzzy Sets" and "Fuzzy Systems"

You've likely heard of the term "Fuzzy Logic", or "Fuzzy Mathematics". But despite what their names may imply, there is nothing inexact about these scientific concepts were formulated, says Lotfi Asker Zadeh, the famous UC Berkeley University mathematician and computer scientist who coined the names of these theories and spent a career advancing their applications.

Known as "The Father of Fuzzy Logic",Lotfi was born in 1921 in Baku, Soviet Azerbaijan to an Iranian father (a journalist) and a Russian mother (a pediatrician). In 1931, when Zadeh was ten years old, he moved with his family to Tehran, Iran where he was enrolled in Alborz College (an American-run Presbyterian school), where he was educated for the next eight years. In 1942, he graduated from the University of Tehran with a degree in electrical engineering and emigrated to the U.S. the following year, enrolling at the Massachusetts Institute of Technology (MIT).

Why He's Important: While a professor and scientist at the University of California, Berkeley, he published his seminal work on "Fuzzy Sets" in 1965 in which he detailed the mathematics of what he called the Fuzzy Set Theory. In 1973, he formally proposed his theory of "Fuzzy Logic", which, based on precise formulas, began allowing scientists, mathematicians and others to make accurate, realistic research data decisions when working in environments of incomplete information, uncertainty and imprecision. Fuzzy Logic does this by allowing for approximate values and inferences as well as for incomplete or ambiguous data (fuzzy data) -- instead of relying solely on crisp data (binary yes/no choices).

Other Achievements: Fuzzy Logic and his Fuzzy Set theory in general have been applied to numerous fields – from computer technology and control theory to artificial intelligence. Lotfi is also credited, along with John Ragazzini in 1952 with pioneering the development of the z-transform method in discrete time signal processing and analysis. These methods are now standard in digital signal processing, digital control, and other discrete-time systems used in industry and research. His latest work includes computing with words and perceptions.

Education: Lotfi received his Master's degree in electrical engineering from MIT in 1946, and his Ph.D in the same discipline from Columbia University where he taught for 10 years before joining UC Berkeley in 1959.

In His Own Words: Commenting how he came up with the name "Fuzzy Logic," he says: "I decided on the word 'fuzzy' because I felt it most accurately described what was going on in the theory. I could have chosen another term that would have been more 'respectable'. For instance, I had thought about using the word 'soft', but that really didn't describe accurately what I had in mind. Nor did 'unsharp', 'blurred', or 'elastic'. In the end, I couldn't think of anything more accurate so I settled on "'fuzzy'".

At age 92, Dr. Zadeh still remains active in his field. He serves as an editor of the International Journal of Computational Cognition, and his website at UC Berkeley lists him as Professor in the Graduate School of Computer Science. In addition, his Facebook page reports that he still goes to his office at UC Berkeley everyday whenever he is not out of town at national or international professional conferences.

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Lotfi Zadeh Wins Major Award

LOTFI A. ZADEH

The BBVA Foundation Frontiers of Knowledge Award in the Information and Communication Technologies (ICT) category has been granted in this fifth edition to the electrical engineer Lotfi A. Zadeh, “for the invention and development of fuzzy logic.” This “revolutionary” breakthrough, affirms the jury in its citation, has enabled machines to work with imprecise concepts, in the same way humans do, and thus secure more efficient results more aligned with reality. In the last fifty years, this methodology has generated over 50,000 patents in Japan and the U.S. alone. 

On hearing of the award, Zadeh remarked that it meant a lot to him for several reasons: “First, because fuzzy logic has been somewhat controversial. Some people have greeted it with enthusiasm but others have been skeptical. It also has a special significance for me because I am a great admirer of Spain and the Spanish people. I’d therefore like to take this opportunity to express my deep appreciation to all those who were involved in my receiving this award, particularly Luis Magdalena and Enric Trillas of the European Centre for Soft Computing in Mieres, who were among those putting forward my nomination."

Classical logic, based on class membership, imposes that an element should strictly belong ir not belong to a clearly demarcated set, like for instance the set of even numbers. But reality is a lot more complex. Hence we have groups, classes and sets whose boundaries are blurred, like that of “good basketball players.” To belong to this set, a basketball player must “be tall” and “shoot well”, but these concepts are imprecise. A binary system would specify, for example, that “be tall” equates to “measure more than 185 cm” and discard all players below this height, regardless of their shooting prowess. But fuzzy logic, like a human coach, would find room in the set of good players for one who measures 184 cm but is an excellent shooter. In this sense, what fuzzy logic does is bridge the gap between classical logic and the real world.

This indeed is what Zadeh was seeking when the began the research that led him to fuzzy logic: “As an engineer, I was always convinced that mathematics held the answers to almost any problem, but I also realized that classical mathematics was constrained by its inability to tolerate imprecision.” To get over this shortcoming, Zadeh turned to the human model: “We humans have a remarkable capability to reason and make decisions in an environment of uncertainty and incompleteness of information (…). The principal objective of fuzzy logic is the formalization of this capability.”

Human beings intuitively apply fuzzy logic to their decisions, juggling imprecise data and weighing up each relevant element. Zadeh’s contribution was to apply such logic to the decision-making processes of systems and computers, so they cease to operate as mere calculating machines and become capable of evaluating degrees and shades of reality and deciding accordingly in an autonomous or semi-autonomous fashion (with little or no human intervention). 

According to the jury, the contributions of Lotfi A. Zadeh (Baku, Azerbaijan, 1921) have been “enthusiastically adopted by industry, where thousands of engineers have designed a whole plethora of complex and intelligent systems (…)."

But Zadeh’s work has also changed the face of numerous industrial processes, where it has simplified design, providing more efficient products that are easier to use and more tractable to change, while bringing down production costs. 

A seminal paper

En 1965, Lotfi Zadeh articulated fuzzy sets for the first time in a paper that would come to be among the most cited of the 20th century, with over 35,000 mentions. And the next step from there was the development of fuzzy logic, a brilliant contribution to extending the frontiers of knowledge. Indeed Zadeh is defined in the jury’s citation as the founder of “a new field of research which has proved powerful in many application domains.”

The controversy around fuzzy logic began with the name: “The word fuzzy has a pejorative connotation in English, and this turned out to be a handicap when it came to gaining the acceptance of the scientific community. But it was the word that came closest to what I had in mind. In Asia, however, they don’t have problems with the word fuzzy, so they were more receptive to my work. They also have a culture that accepts shades of grey, as opposed to the western – Cartesian – tradition where everything is either black or white.”

This was perhaps the reason, he speculates, that one of the earliest applications of his concept was the automated subway system in the Japanese city of Sendai. 

Fuzzy logic opened the door to machine understanding of such imprecise instructions as “brake smoothly” or “refrigerate until the air is cool,” which would be instantly understood by any human being acquainted with the system, but are utterly impenetrable for a conventional computer program. The conceptual shift was so abrupt that Zadeh initially had to face the skepticism of many scientist colleagues, until the success of the practical applications of his theory dissipated all such doubts. 

Zadeh’s work has enabled us to communicate with machines through an increasingly natural, human language. 

The laureate, still working at the age of 91, sees this as the most promising research avenue in the fuzzy logic field, and hopes to author some further advance that will connect computers and systems more closely with natural language.   

International jury

The jury in this category was chaired by George Gottlob, Professor of Computer Science at the University of Oxford (United Kingdom), with Ramón López de Mántaras, Director of the Artificial Intelligence Research Institute of the Spanish National Research Council (CSIC) acting as secretary. Remaining members were Oussama Khatib, Professor in the Artificial Intelligence Laboratory in the Computer Sciences Department of Stanford University (United States), Rudolf Kruse, Head of the Department of Knowledge Processing and Language Engineering at Otto-von-Guerike-Universität Magdeburg (Germany), Mateo Varelo, Director of the Barcelona Supercomputing Center (Spain) and Joos Vandewalle, Head of the SDC Division in the Department of Electrical Engineering at the Katholieke Universiteit Leuven (Belgium).

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