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

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

• 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