View source for Philosophy:Extension (predicate logic)

{{short description|Set of tuples in mathematical logic that satisfy a predicate}}
The '''extension''' of a [[Predicate (mathematical logic)|predicate]]{{spaced ndash}}a truth-valued [[Function (mathematics)|function]]{{spaced ndash}}is the [[Set (mathematics)|set]] of [[Tuple|tuple]]s of values that, used as arguments, satisfy the predicate. Such a set of tuples is a [[Relation (mathematics)|relation]].

==Examples==
For example, the statement "''d2'' is the weekday following ''d1''" can be seen as a truth function associating to each tuple (''d2'', ''d1'') the value ''true'' or ''false''. The extension of this truth function is, by convention, the set of all such tuples associated with the value ''true'', i.e.

 {(Monday, Sunday),
  (Tuesday, Monday),
  (Wednesday, Tuesday),
  (Thursday, Wednesday),
  (Friday, Thursday),
  (Saturday, Friday),
  (Sunday, Saturday)}

By examining this extension we can conclude that "Tuesday is the weekday following Saturday" (for example) is false.

Using [[Set-builder notation|set-builder notation]], the extension of the [[Arity|''n''-ary]] predicate <math>\Phi</math> can be written as
:<math>\{ (x_1,...,x_n) \mid \Phi(x_1,...,x_n) \}\,.</math>

==Relationship with characteristic function==
If the values 0 and 1 in the range of a [[Indicator function|characteristic function]] are identified with the values false and true, respectively{{spaced ndash}}making the characteristic function a predicate{{spaced ndash}}, then for all relations ''R'' and predicates <math>\Phi</math> the following two statements are equivalent:
*<math>\Phi</math> is the characteristic function of ''R''
*''R'' is the extension of <math>\Phi</math>

== See also ==
* Extensional logic
* Extensional set
* [[Extensionality]]
* [[Philosophy:Intension|Intension]]

==References==
* [https://ncatlab.org/nlab/show/extension+%28semantics%29 extension (semantics)] in [[NLab|nLab]]

[[Category:Predicate logic]]



{{Sourceattribution|Extension (predicate logic)}}