Extension (predicate logic)

The extension of a predicatea truth-valued functionis the set of tuples of values that, used as arguments, satisfy the predicate. Such a set of tuples is a 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.

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

Using set-builder notation, the extension of the ''n''-ary predicate $\Phi$ can be written as
:$\\,.$

Relationship with characteristic function

If the values 0 and 1 in the range of a characteristic function are identified with the values false and true, respectivelymaking the characteristic function a predicate, then for all relations ''R'' and predicates $\Phi$ the following two statements are equivalent:
*$\Phi$ is the characteristic function of ''R''
*''R'' is the extension of $\Phi$

See also

* Extensional logic
* Extensional set
* Extensionality
* Intension

References

extension (semantics)
in nLab

Predicate logic

{{mathlogic-stub