Predicate

A function whose values are statements about $n$-tuples of objects forming the values of its arguments. For $n=1$ a predicate is called a "property", for $n>1$ a "relation"; propositions (cf. Proposition) may be regarded as zero-place predicates.

In order to specify an $n$-place predicate $P(x_1,\dots,x_n)$ one must indicate sets $D_1,\dots,D_n$ — the domains of variation of the object variables $x_1,\dots,x_n$; most often one considers the case $D_1=\dots=D_n$. From the set-theoretical point of view a predicate is specified by a subset $M$ of the Cartesian product $D_1\times\dots\times D_n$. Here $P(a_1,\dots,a_n)$ is taken to mean "the ordered tuple $(a_1,\dots,a_n)$ belongs to $M$". The syntactic specification of an $n$-place predicate is realized by exhibiting a formula of a logico-mathematical language containing $n$ 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)