The problem
Take the set of “tall people.” A classical (crisp) set needs a boundary: taller than 180 cm belongs, 179 cm does not. But nobody actually reasons that way: a person who is 179 cm is still “tall” — just a little less so.
Theory
Fuzzy Logic
In classical mathematics, an element either belongs to a set or it doesn't. Zadeh's idea: in the real world, that is almost never true.
The membership function
In “Fuzzy Sets,” published in June 1965 in Information and Control, Zadeh replaced 0/1 membership with a measure on the interval [0, 1]. His own definition — the very formula carved into his tombstone in Baku — reads:
A = { (x, μA(x)) | x ∈ X }
Here X is the set of all possible elements, and μA(x) is the membership function giving the degree to which x belongs to A: 0 means no membership at all, 1 means full membership.
A simple example: the triangular function
In practice a triangular membership function is common — membership equals 1 at the centre (a) and falls to zero at the edges (a distance b away):
μ(x) = 0.60
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Set operations
Zadeh also extended the classical operations of union, intersection and complement to fuzzy sets — using the max and min functions:
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1973: the move to fuzzy logic
In January 1973 Zadeh published “Outline of a New Approach to the Analysis of Complex Systems and Decision Processes” in IEEE Transactions on Systems, Man, and Cybernetics — turning fuzzy set theory into a full formal system of logic.
In 1975 he introduced the “linguistic variable”: a variable whose values are words, not numbers. The variable “temperature,” for instance, can be expressed as {cold, cool, warm, hot} — each one a fuzzy set with its own membership function.
Computing with words and Z-numbers
In later years Zadeh developed “computing with words” — computers working directly with the imprecise expressions of natural language. Late in his career he proposed “Z-numbers”: Z = (A, B), where A describes the value itself and B how much confidence there is in that value.
Where it's used
Fuzzy control systems run in washing machines, rice cookers, air conditioners, camera autofocus and car transmissions. Automatic subway control in Sendai, Japan, has used fuzzy control since 1987. The method also underlies fuzzy databases, pattern recognition and neuro-fuzzy AI systems.
The 1965 paper had been cited more than 122,743 times as of February 2021 — one of the most-cited works in the history of science.
