Short answers, pulled from the story.
Fuzzy logic is a form of many-valued logic in which a variable's truth value can be any real number between 0 and 1, rather than only 0 or 1 as in Boolean logic. It is used to represent partial truth, ranging between completely true and completely false.
The term fuzzy logic was introduced in 1965 when mathematician Lotfi Zadeh proposed fuzzy set theory. Basic infinite-valued logic behind the idea had already been studied since the 1920s by Łukasiewicz and Tarski, and Zadeh continued developing the concept with Joseph Goguen through the 1960s and 1970s.
Fuzzy logic and probability both use a scale from 0 to 1, but they model different things. Fuzzy logic represents vagueness through degrees of truth, while probability represents ignorance about whether an event will occur.
Fuzzy logic has controlled the Sendai Subway 1000 series train system, anti-lock braking systems, helicopter flight aids, automobile fuel efficiency systems, single-button washing machines, and vacuum cleaner power controls. It has also been used for handwriting recognition in Sony pocket computers and early earthquake recognition through Japan's Institute of Seismology.
The Mamdani system fuzzifies inputs, executes rules, and then defuzzifies the output into a single crisp value. The Takagi-Sugeno-Kang (TSK) system instead folds defuzzification into the rules themselves, using constant or linear polynomial functions as outputs, which makes it more computationally efficient but less intuitive than Mamdani.
When analyzed closely, neural networks and fuzzy logic share the same underlying logic, since a neural network reaches a decision by weighting and combining inputs rather than following either-or steps. Researchers in the 1980s debated whether decision tree learning or neural networks were the better approach to machine learning.