In which the truth value of a statement can be any real number from 0 up to 1
Let us now move beyond the three-valued logics to the case of fuzzy logic, which offers up infinitely many truth values, uncountably many! In fuzzy logic, we allow the truth value of a statement to be any real number in the unit interval [0,1]—all the real numbers from 0 up to 1. In fuzzy logic we speak of a proposition or assertion holding with truth value 0.123 or with value 1/2 or value 1/π or with value 0 or 1 and so on. The motivating idea, of course, is that 0 represents false, 1 represents true, and the numbers between 0 and 1 represent shades or degrees of truth. And then we define the fuzzy logical operations on these values.
Such kinds of logic were studied from the 1920s by Łukasiewicz, Tarski, and Gödel and popularized as “fuzzy logic” in the 1960s by Zadeh. Fuzzy logic is now sometimes used in control theory and industrial applications, including household appliances—my rice cooker was advertised as using fuzzy logic in its control decision procedures. Perhaps the thermostat in your home uses fuzzy logic.
The fuzzy logic connectives
In fuzzy logic, we conceive of the fundamental logical operations as mathematical functions defined on the set of truth values, the real numbers from 0 to 1. One natural way to proceed, for example, is to define them as follows:
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