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Difference between revisions of "Predicate symbol"

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''predicate letter''
 
''predicate letter''
  
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 (or $n$-ary) predicate symbol, then the following rule should be among the syntactic rules for forming expressions in the formalized language:  "If $t_1,\ldots,t_n$ are terms, then $P(t_1,\ldots,t_n)$ is a formula". Thus, predicate symbols are syntactically used to form formulas, and semantically denote predicates.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.L. Ershov,  E.A. Palyutin,  "Mathematical logic" , Moscow  (1970)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.L. Ershov,  E.A. Palyutin,  "Mathematical logic" , Moscow  (1970)  (In Russian)</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
A predicate symbol is also called a relation symbol.
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A predicate symbol is also called a ''relation symbol''.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Yu.I. Manin,  "A course in mathematical logic" , Springer  (1977)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  Yu.I. Manin,  "A course in mathematical logic" , Springer  (1977)  (Translated from Russian)</TD></TR>
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</table>

Latest revision as of 13:16, 14 February 2020

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 (or $n$-ary) predicate symbol, then the following rule should be among the syntactic rules for forming expressions in the formalized language: "If $t_1,\ldots,t_n$ are terms, then $P(t_1,\ldots,t_n)$ 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=16839
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article