A Boolean-valued function (sometimes called a
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, ...
or a
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
) is a
function of the type f : X → B, where X is an arbitrary
set and where B is a
Boolean domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
, i.e. a generic two-element set, (for example B = ), whose elements are interpreted as
logical value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or ''false'').
Computing
In some prog ...
s, for example, 0 =
false
False or falsehood may refer to:
* False (logic), the negation of truth in classical logic
*Lie or falsehood, a type of deception in the form of an untruthful statement
* false (Unix), a Unix command
* ''False'' (album), a 1992 album by Gorefest
* ...
and 1 =
true
True most commonly refers to truth, the state of being in congruence with fact or reality.
True may also refer to:
Places
* True, West Virginia, an unincorporated community in the United States
* True, Wisconsin, a town in the United States
* ...
, i.e., a single
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...
of
information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, ...
.
In the
formal science
Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, gam ...
s,
mathematics,
mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
,
statistics, and their applied disciplines, a Boolean-valued function may also be referred to as a characteristic function,
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
, 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
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
, a truth predicate is a predicate on the
sentences of a
formal language
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
The alphabet of a formal language consists of s ...
, 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
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or ''false'').
Computing
In some prog ...
.
See also
*
Bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...
*
Boolean data type
In computer science, the Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted ''true'' and ''false'') which is intended to represent the two truth values of logic and Boolean algebra. It is named ...
*
Boolean algebra (logic)
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
*
Boolean domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
*
Boolean logic
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
*
Propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
*
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...
*
Logic minimization
Logic optimization is a process of finding an equivalent representation of the specified logic circuit under one or more specified constraints. This process is a part of a logic synthesis applied in digital electronics and integrated circuit d ...
*
Indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x ...
*
Predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, ...
*
Proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
*
Finitary boolean function
In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values.
In standard mathematics, an operati ...
*
Boolean function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function ...
{{colend
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. 3rd edition, McGraw–Hill, 2010.
*
Korfhage, Robert R. (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
*
Mathematical Society of Japan
The Mathematical Society of Japan (MSJ, ja, 日本数学会) is a learned society for mathematics in Japan.
In 1877, the organization was established as the ''Tokyo Sugaku Kaisha'' and was the first academic society in Japan. It was re-organized ...
, ''Encyclopedic Dictionary of Mathematics'', 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM.
*
Minsky, Marvin L., and
Papert, Seymour, A. (1988), ''
Perceptrons
In machine learning, the perceptron (or McCulloch-Pitts neuron) is an algorithm for supervised classification, supervised learning of binary classification, binary classifiers. A binary classifier is a function which can decide whether or not an ...
, An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. Revised, 1972. Expanded edition, 1988.
Boolean algebra