Predicate

A function whose values are statements about -tuples of objects forming the values of its arguments. For a predicate is called a "property" , for a "relation" ; propositions (cf. Proposition) may be regarded as zero-place predicates.

In order to specify an -place predicate one must indicate sets — the domains of variation of the object variables ; most often one considers the case . From the set-theoretical point of view a predicate is specified by a subset of the Cartesian product . Here is taken to mean "the ordered tuple a1…an belongs to M" . The syntactic specification of an -place predicate is realized by exhibiting a formula of a logico-mathematical language containing free variables. The notion of a predicate dates back to Aristotle; the apparatus for operating with statements containing predicates is developed in mathematical logic (cf. Logical calculus; Predicate calculus).

References

[a1]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1952) pp. Chapt. XIV
[a2]	P. Suppes, "Introduction to logic" , v. Nostrand (1957) pp. §9.8
[a3]	A. Grzegorczyk, "An outline of mathematical logic" , Reidel (1974)