Difference between revisions of "Predicate"
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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 | + | 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]]). | 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]]). |
Revision as of 14:59, 20 May 2013
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).
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) |
Predicate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predicate&oldid=29766