Boolean-valued function

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A boolean-valued function is a function of the type $$f : X \to \mathbb{B},$$ where $$X\!$$ is an arbitrary set and where $$\mathbb{B}$$ is a boolean domain.

In the formal sciences &mdash; mathematics, mathematical logic, statistics &mdash; 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 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.

Examples
A binary sequence is a boolean-valued function $$f : \mathbb{N}^+ \to \mathbb{B}$$, where $$\mathbb{N}^+ = \{ 1, 2, 3, \ldots \},$$. In other words, $$f\!$$ is an infinite sequence of 0's and 1's.

A binary sequence of length $$k\!$$ is a boolean-valued function $$f : [k] \to \mathbb{B}$$, where $$[k] = \{ 1, 2, \ldots k \}.$$

Focal nodes

 * Inquiry Live
 * Logic Live

Peer nodes

 * Boolean-Valued Function @ InterSciWiki
 * Boolean-Valued Function @ Subject Wikis
 * Boolean-Valued Function @ Wikiversity
 * Boolean-Valued Function @ Wikiversity Beta

Logical operators

 * Exclusive disjunction
 * Logical conjunction
 * Logical disjunction
 * Logical equality


 * Logical implication
 * Logical NAND
 * Logical NNOR
 * Negation

Related topics

 * Ampheck
 * Boolean domain
 * Boolean function
 * Boolean-valued function
 * Differential logic


 * Logical graph
 * Minimal negation operator
 * Multigrade operator
 * Parametric operator
 * Peirce's law


 * Propositional calculus
 * Sole sufficient operator
 * Truth table
 * Universe of discourse
 * Zeroth order logic

Relational concepts

 * Continuous predicate
 * Hypostatic abstraction
 * Logic of relatives
 * Logical matrix


 * Relation
 * Relation composition
 * Relation construction
 * Relation reduction


 * Relation theory
 * Relative term
 * Sign relation
 * Triadic relation

Information, Inquiry

 * Inquiry
 * Dynamics of inquiry


 * Semeiotic
 * Logic of information


 * Descriptive science
 * Normative science


 * Pragmatic maxim
 * Truth theory

Related articles

 * Cactus Language
 * Futures Of Logical Graphs
 * Propositional Equation Reasoning Systems


 * Differential Logic : Introduction
 * Differential Propositional Calculus
 * Differential Logic and Dynamic Systems


 * Prospects for Inquiry Driven Systems
 * Introduction to Inquiry Driven Systems
 * Inquiry Driven Systems : Inquiry Into Inquiry

Document history
Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.


 * Boolean-Valued Function, InterSciWiki
 * Boolean-Valued Function, PlanetMath
 * Boolean-Valued Function, Wikinfo
 * Boolean-Valued Function, Wikiversity
 * Boolean-Valued Function, Wikiversity Beta
 * Boolean-Valued Function, Wikipedia