Extension (predicate logic)

The extension of a predicate is the set of true propositions that can be formed by substituting a term for each of its free variables.

For example, consider the predicate "d2 is the weekday following d1". Its extension is the set

{Monday is the weekday following Sunday, Tuesday is the weekday following Monday, Wednesday is the weekday following Tuesday, Thursday is the weekday following Wednesday, Friday is the weekday following Thursday, Saturday is the weekday following Friday, Sunday is the weekday following Saturday}

By examining this extension we can conclude, under the Closed World Assumption and the principle of bivalence, that "Tuesday is the weekday following Saturday" (for example) is false.

Note that some predicates have different extensions in different situations. For example, that of "a is the mother of b" changes over time, whenever somebody is born. However, that of "x < y" (in the domain of numbers) can be safely assumed to have the same extension in all situations--in particular, at all times.