# Fuzzy sets as a basis for a theory of possibility

@article{Zadeh1999FuzzySA, title={Fuzzy sets as a basis for a theory of possibility}, author={L. Zadeh}, journal={Fuzzy Sets and Systems}, year={1999}, volume={100}, pages={9-34} }

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U = {u} which is characterized by its membership function μF, then a proposition of the form “X is F”, where X is a variable taking values in U, induces a possibility… Expand

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#### References

SHOWING 1-10 OF 44 REFERENCES

Calculus of fuzzy restrictions

- Mathematics
- 1996

ABSTRACT A fuzzy restriction may be visualized as an elastic constraint on the values that may be assigned to a variable. In terms of such restrictions, the meaning of a proposition of the form “x is… Expand

Similarity relations and fuzzy orderings

- Mathematics, Computer Science
- Inf. Sci.
- 1971

An extended version of Szpilrajn's theorem is proved and various properties of similarity relations and fuzzy orderings are investigated and, as an illustration, a fuzzy preordering is investigated which is reflexive and antisymmetric. Expand

A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges

- Mathematics
- 1972

Abstract A basic idea suggested in this paper is that a linguistic hedge such as very, more or less, much, essentially. slightly, etc. may be viewed as an operator which acts on the fuzzy set… Expand

Fuzzy sets and their applications to cognitive and decision processes

- Computer Science
- 1977

By providing a basis for a systematic approach to approximate reasoning, the theory of fuzzy sets may well have a substantial impact on scientific methodology in the years ahead, particularly in the realms of psychology, economics, law, medicine, decision analysis, information retrieval, and artificial intelligence. Expand

Probability measures of Fuzzy events

- Mathematics
- 1968

In probability theory [I], an event, A, is a member of a a-field, CY, of subsets of a sample space ~2. A probability measure, P, is a normed measure over a measurable space (Q, GY); that is, P is a… Expand

A fuzzy set approach to modifiers and vagueness in natural language.

- Psychology
- 1976

SUMMARY Recent developments in semantic theory, such as the work of Labov (1973) and Lakoff (1973), have brought into question the assumption that meanings are precise. It has been proposed that the… Expand

CONDITIONAL FUZZY MEASURES AND THEIR APPLICATIONS

- Mathematics
- 1975

Publisher Summary This chapter introduces a set function, that is, conditional fuzzy measure and a relation between a priori and a posteriori fuzzy measures. These are very useful for describing any… Expand

Applications of Fuzzy Sets to Systems Analysis

- Computer Science
- 1975

One of the objects of this book was to facilitate communication by bringing toge- ther different viewpoints and coloring them from a common viewpoint. Expand

Probability Theory I

- Computer Science, Mathematics
- 1963

These notes cover the basic definitions of discrete probability theory, and then present some results including Bayes' rule, inclusion-exclusion formula, Chebyshev's inequality, and the weak law of… Expand

Deductive verbal models of organizations

- Computer Science
- 1976

It is generally concluded that the present approach towards modelling the behavior of complex organizations is not without interesting potentialities, and that verbal models may, under certain circumstances, be superior to corresponding conventional simulation models. Expand