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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|"relation"]] ; propositions (cf. [[Proposition|Proposition]]) may be regarded as zero-place predicates.
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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|"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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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|Logical calculus]]; [[Predicate calculus|Predicate calculus]]).
  
  

Latest revision as of 15:00, 30 December 2018

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).


Comments

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)
How to Cite This Entry:
Predicate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predicate&oldid=29767
This article was adapted from an original article by S.Yu. Maslov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article