A boolean-valued function, in some usages a predicate or a proposition, is a function of the type $f\; :\; X\; \backslash to\; \backslash mathbb\{B\}$, where $X$ is an arbitrary set, where $\backslash mathbb\{B\}$ is a generic 2-element set, typically $\backslash mathbb\{B\}\; =\; \backslash left\; \backslash \{\; 0,\; 1\; \backslash right\; \backslash \}$, and where the latter is frequently interpreted for logical applications as $\backslash mathbb\{B\}\; =\; \backslash left\; \backslash \{\; false,\; true\; \backslash right\; \backslash \}$.

In the formal sciences, mathematics, mathematical logic, statistics, and their applied disciplines, a boolean-valued function may also be referred to as a characteristic function, indicator function, predicate, or proposition. In all of these uses it is understood that the various terms refer to a mathematical object and not the corresponding semiotic sign or syntactic expression.

In formal semantic theories of truth, a truth predicate is a predicate on the sentences of a formal language, interpreted for logic, that formalizes the intuitive concept that is normally expressed by saying that a sentence is true. A truth predicate may have additional domains beyond the formal language domain, if that is what is required to determine a final truth value.

References

• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.

• Kohavi, Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.

• Korfhage, Robert R. (1974), Discrete Computational Structures, Academic Press, New York, NY.

• Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).

See also

• Boolean domain

Equivalent concepts

• Characteristic function
• Indicator function
• Predicate, in some senses.
• Proposition, in some senses.

Related concepts

• Finitary boolean function

Logic | Mathematics | Mathematical logic | 布尔值函数