Extension (predicate logic) explained

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\{ (x _{1,...,x} _{n)} \mid \Phi(x _{1,...,x} _{n)} \}.

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