Extension (predicate logic)

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The extension of a predicate is the set of assignments of values to its arguments that make it true.

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, 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.