Predicate (logic)

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Predicate (logic)

Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common. The notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) written , is just a collection of all the objects for which P is sensible and true.

For instance, is the set .

Thus, an element of is an object t for which the statement P(t) is true.

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Encyclopedia

Semantics

Semantics is the study of meaning in communication. The word is derived from the Greek language word s??a?t???? , "significant", from s??a??? , "to signify, to indicate" and that from s??a , "sign, mark, token"....
a predicate is an expression of the semantic type

Type theory

In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general....
of set

Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
s. An equivalent formulation is that they are thought of as indicator function

Indicator function

In mathematics, an indicator function or a characteristic function is a Function defined on a Set that indicates membership of an element in a subset of ....
s of sets, i.e. function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s from an entity

Entity

An entity is something that has a distinct, separate existence, though it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities....
to a truth value.

In first-order logic

First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
, a predicate can take the role as either a property

Property (philosophy)

In modern philosophy, mathematics, and logic, a property is an attribute of an Object ; thus a red object is said to have the property of redness....
or a relation between entities.

The following explanation is from, http://www.cs.odu.edu/~toida/nerzic/content/logic/pred_logic/predicate/pred_intro.html

To cope with deficiencies of propositional logic we introduce two new features: predicates and quantifiers. A predicate is a verb phrase template that describes a property of objects, or a relationship among objects represented by the variables.

For example, the sentences "The car Jane is driving is blue", "The sky is blue", and "The cover of this book is blue" come from the template "is blue" by placing an appropriate noun/noun phrase in front of it. The phrase "is blue" is a predicate and it describes the property of being blue. Predicates are often given a name. For example any of "is_blue", "Blue" or "B" can be used to represent the predicate "is blue" among others. If we adopt B as the name for the predicate "is_blue", sentences that assert an object is blue can be represented as "B(x)", where x represents an arbitrary object. B(x) reads as "x is blue".

Similarly the sentences "Mary gives the book to John", "Jane gives a loaf of bread to Mary", and "John gives a lecture to Mary" are obtained by substituting an appropriate object for variables x, y, and z in the sentence "x gives y to z". The template "... gives ... to ..." is a predicate and it describes a relationship among three objects. This predicate can be represented by Give( x, y, z ) or G( x, y, z ), for example.

Note: The sentence "Mary gives the book to John" can also be represented by another predicate such as "gives a book to". Thus if we use B( x, y ) to denote this predicate, "Mary gives the book to John" becomes B( Mary, John ). In that case, the other sentences, "Jane gives a loaf of bread to Mary", and "John gives a lecture to Mary", must be expressed with other predicates.

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a predicate is either a relation

Relation (mathematics)

In mathematics , a relation is a property that assigns truth values to combinations of k first-order logic. Typically, the property describes a possible connection between the components of a k-tuple....
or the boolean-valued function

Boolean-valued function

A boolean-valued function, in some usages a Predicate_ or a Proposition, is a function of the type f : X ? B, where X is an arbitrary Set and where B is a boolean domain....
that amounts to the characteristic function

Characteristic function

In mathematics, characteristic function can refer to any of several distinct concepts:* The most common and universal usage is as a synonym for indicator function, that is the function* The characteristic state function in statistical mechanics....
or the indicator function

Indicator function

In mathematics, an indicator function or a characteristic function is a Function defined on a Set that indicates membership of an element in a subset of ....
of such a relation.

A function P: X? is called a predicate on X. When P is a predicate on X, we sometimes say P is a property

Property

Property is any physical or virtual entity that is ownership by an individual or jointly by a group of individuals. An owner of property has the right to consumption, sell, Renting, mortgage, transfer and exchange his or her property....
of X.

Predicate (logic)

See also

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