# Predicate symbol

*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:**

Relation symbol.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Relation_symbol&oldid=39424