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Fuzzy logic is controversial: it is widely accepted within the engineering and computer science communities, but generally rejected by mathematicians and statisticians because it seems to contradict the principle of bivalence. Critics also argue that it cannot be a superset of ordinary set theory since membership functions are defined in terms of conventional sets.
Fuzzy logic can be used to control household appliances such as washing machines (which sense load size and detergent concentration and adjust their wash cycles accordingly) and refrigerators.
A basic application might characterize subranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled.
In this image, cold, warm, and hot are functions mapping a temperature scale. A point on that scale has three "truth values" — one for each of the three functions. For the particular temperature shown, the three truth values could be interpreted as describing the temperature as, say, "fairly cold", "slightly warm", and "not at all hot".
Fuzzy logic has suffered many misconceptions, partly due to its name. "Fuzzy" often has negative connotations, either suggesting something cute or something imprecise; the latter sometimes causes people to equate "fuzzy logic" with "imprecise logic". However, fuzzy logic is not any less precise than any other form of logic: it is an organized and mathematical method of handling inherently imprecise concepts. The concept of "coldness" cannot be expressed in an equation, because although temperature is a quantity, "coldness" is not. However, people have an idea of what "cold" is, and agree that something cannot be "cold" at N degrees but "not cold" at N+1 degrees — a concept classical logic cannot easily handle due to the principle of bivalence.
Another common misconception is that fuzzy logic is a new way of expressing probability. However, Bart Kosko has shown that probability is a subset of fuzzy logic, as probability only handles one kind of uncertainty. He also proved a theorem demonstrating that Bayes' theorem can be derived from the concept of fuzzy subsethood . This should not by any means suggest that all those who study probability accept or even understand fuzzy logic, however: to many, fuzzy logic is still a curiosity.
Fuzzy logic is also sometimes said to be used only in AI, control systems, and/or expert systems (note that these fields can have significant overlap). These are by far the most common applications, but by no means the only possible: fuzzy logic can be applied in any situation requiring the handling of uncertainty.
Fuzzy logic has also been incorporated into some microcontrollers and microprocessors, for instance, the Motorola 68HC12.
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