What is a fuzzy set, and what distinction (per Zadeh) motivates fuzzy logic over probability?

Answer

A fuzzy set assigns each element a degree of membership m_F(s) ∈ [0,1] instead of a binary in/out. It models vagueness (e.g. "tall"), which is distinct from randomness — probability handles randomness, fuzzy logic handles vagueness.

Degrees of membership violate exclusive membership and the excluded middle; an element can partly belong to a set and its complement at once.