Thursday, September 18, 2014

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:

Berkeley Initiative in Soft Computing (BISC)

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.


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