Predicate (mathematical logic)

From Wikipedia, the free encyclopedia

(Redirected from Predicate (mathematics))

For other meanings of "predicate", see Predicate.

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 {x | P(x)}, is just a collection of all the objects for which P is sensible and true.

For instance, {x | x is a positive integer less than 4} is the set {1,2,3}.

Thus, an element of {x | P(x)} is an object t for which the statement P(t) is true. Such a sentence P(x) is called a Predicate. P(x) is also called a propositional function, because each choice of x produces a proposition P(x) that is either true or false.

In formal semantics a predicate is an expression of the semantic type of sets. An equivalent formulation is that they are thought of as indicator functions of sets, i.e. functions from an entity to a truth value.

In first-order logic, a predicate can take the role as either a property 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, a predicate is either a relation or the boolean-valued function that amounts to the characteristic function or the indicator function of such a relation.

A function P: X→ {true, false} is called a predicate on X. When P is a predicate on X, we sometimes say P is a property of X.