View source for Predicate

A function whose values are statements about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743501.png" />-tuples of objects forming the values of its arguments. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743502.png" /> a predicate is called a  "property" , for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743503.png" /> a  "relation" ; propositions (cf. [[Proposition|Proposition]]) may be regarded as zero-place predicates.

In order to specify an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743504.png" />-place predicate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743505.png" /> one must indicate sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743506.png" /> — the domains of variation of the object variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743507.png" />; most often one considers the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743508.png" />. From the set-theoretical point of view a predicate is specified by a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p0743509.png" /> of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p07435010.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p07435011.png" /> is taken to mean  "the ordered tuple a1…an belongs to M" . The syntactic specification of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p07435012.png" />-place predicate is realized by exhibiting a formula of a logico-mathematical language containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074350/p07435013.png" /> 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|Logical calculus]]; [[Predicate calculus|Predicate calculus]]).



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.C. Kleene,   "Introduction to metamathematics" , North-Holland  (1952)  pp. Chapt. XIV</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Suppes,   "Introduction to logic" , v. Nostrand  (1957)  pp. §9.8</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Grzegorczyk,   "An outline of mathematical logic" , Reidel  (1974)</TD></TR></table>