Fuzzy Set Theory - And Its Applications, Third Edition is a textbook for courses in fuzzy set theory. It can also be used as an introduction to the subject. The character of a textbook is balanced with the dynamic nature of the research in the field by including many useful references to develop a deeper understanding among interested readers. The book updates the research agenda (which has witnessed profound and startling advances since its inception some 30 years ago) with chapters on possibility theory, fuzzy logic and approximate reasoning, expert systems, fuzzy control, fuzzy data analysis, decision making and fuzzy set models in operations research. All chapters have been updated. Exercises are included.

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Fuzzy Set Theory - and Its Applications

A review of fuzzy set aggregation connectives

Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions

On a general class of fuzzy connectives

PRUF—an acronym for Possibilistic Relational Universal Fuzzy—is a meaning representation language for natural languages which departs from the conventional approaches to the theory of meaning in several important respects. First, a basic assumption underlying PRUF is that the imprecision that is intrinsic in natural languages is, for the most part, possibilistic rather than probabilistic in nature. Thus, a proposition such as “Richard is tall” translates in PRUF into a possibility distribution of the variable Height (Richard), which associates with each value of the variable a number in the interval [0,11 representing the possibility that Height (Richard) could assume the value in question. More generally, a proposition, p , translates into a procedure, P, which returns a possibility distribution, Π p , with P and Π p representing, respectively, the meaning of p and the information conveyed by p . In this sense, the concept of a possibility distribution replaces that of truth as a foundation for the representation of meaning in natural languages. Second, the logic underlying PRUF is not a two-valued or multivalued logic, but a fuzzy logic, FL, in which the truth-values are linguistic, that is, are of the form true, not true, very true, more or less true, not very true , etc., with each such truth-value representing a fuzzy subset of the unit interval. The truth-value of a proposition is defined as its compatibility with a reference proposition, so that given two propositions p and r, one can compute the truth of p relative to r . Third, the quantifiers in PRUF—like the truth-values—are allowed to be linguistic, i.e. may be expressed as most, many, few, some, not very many, almost all, etc. Based on the concept of the cardinality of a fuzzy set, such quantifiers are given a concrete interpretation which makes it possible to translate into PRUF propositions exemplified by “Many tall men are much taller than most men,” “All tall women are blonde is not very true,” etc. The translation rules in PRUF are of four basic types: Type I—pertaining to modification; Type II—pertaining to composition; Type III—pertaining to quantification; and Type IV—pertaining to qualification and, in particular, to truth qualification, probability qualification and possibility qualification. The concepts of semantic equivalence and semantic entailment in PRUF provide a basis for question-answering and inference from fuzzy premises. In addition to serving as a foundation for approximate reasoning, PRUF may be employed as a language for the representation of imprecise knowledge and as a means of precisiation of fuzzy propositions expressed in a natural language.

PRUF—a meaning representation language for natural languages

A new form of logic is described, originally developed for the formalization of physical theories, the essential feature being a “fuzzification” of the concept of a proposition A proposition is not regarded as being necessarily true or false; it is defined not via truth conditions but in terms of a definite commitment that is assumed by the speaker In the case of an atomic proposition the commitment amounts to a bet on the outcome of some agreed test; for a compound proposition it leads to a dialogue between the speaker and an opponent The resulting logic corresponds closely to the infinitevalued logic Ł α, of Łukasiewicz In fact, the approach provides a dialogue interpretation of Ł ∞ and leads to a convenient method for establishing logical identities Set theory is then developed, not by taking set as a primitive concept but by assuming each set A is determined by a property P characteristic of its members: A = {x:P(x)}  When this is expressed formally the result can be read in two ways according to whether the underlying logic is classical logic or Ł ∞ (with the above interpretation) If the propositions P(x) are classical we get ordinary sets; if they are propositions in the new “fuzzy” sense we get fuzzy sets ( f -sets) The situation is illustrated by a number of definitions and theorems involving simple operations on f -sets Lastly, the notion of a convex f-set is defined, and a simple theorem is stated and proved using Ł ∞ and the dialogue method of proof All statements and proofs are expressed in terms of the new logic In particular, use of the quantitative notion of “grade of membership” in a fuzzy set is entirely avoided

Robin Giles

Papers

Łukasiewicz logic and fuzzy set theory

A computer program for fuzzy reasoning

A non-classical logic for physics

The concept of grade of membership

A formal system for fuzzy reasoning