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Difference between revisions of "Predicate symbol"

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''predicate letter''
 
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A notation for some concrete predicate or relation. E.g., the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074370/p0743701.png" /> often denotes the order relation on the real numbers; it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074370/p0743702.png" />-place predicate. In the formal structure of a language, the symbols denoting predicates must be used, in a well-defined way, for constructing expressions of the language. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074370/p0743703.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074370/p0743704.png" />-place predicate symbol, then the following rule should be among the syntactic rules for forming expressions in the formalized language:  "If t1…tn are terms, then Pt1…tn is a formula" . Thus, predicate symbols are syntactically used to form formulas, and semantically denote predicates.
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A notation for some concrete predicate or relation. E.g., the symbol $\leq$ often denotes the order relation on the real numbers; it is a $2$-place predicate. In the formal structure of a language, the symbols denoting predicates must be used, in a well-defined way, for constructing expressions of the language. In particular, if $P$ is an $n$-place predicate symbol, then the following rule should be among the syntactic rules for forming expressions in the formalized language:  "If t1…tn are terms, then Pt1…tn is a formula" . Thus, predicate symbols are syntactically used to form formulas, and semantically denote predicates.
  
 
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Revision as of 20:36, 11 April 2014

predicate letter

A notation for some concrete predicate or relation. E.g., the symbol $\leq$ often denotes the order relation on the real numbers; it is a $2$-place predicate. In the formal structure of a language, the symbols denoting predicates must be used, in a well-defined way, for constructing expressions of the language. In particular, if $P$ is an $n$-place predicate symbol, then the following rule should be among the syntactic rules for forming expressions in the formalized language: "If t1…tn are terms, then Pt1…tn is a formula" . Thus, predicate symbols are syntactically used to form formulas, and semantically denote predicates.

References

[1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
[2] Yu.L. Ershov, E.A. Palyutin, "Mathematical logic" , Moscow (1970) (In Russian)


Comments

A predicate symbol is also called a relation symbol.

References

[a1] Yu.I. Manin, "A course in mathematical logic" , Springer (1977) (Translated from Russian)
How to Cite This Entry:
Predicate symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predicate_symbol&oldid=31562
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article