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A linguistic expression used to denote objects. For example, the expressions $1,0+1,\lim_{x\to0}(\sin x)/x$ are distinct terms denoting the same object. A term can contain free variables (parameters) (cf. Free variable), fixation of whose values uniquely defines some object according to the semantic laws of the language — the value of the term for the given values of its free variables. Thus, if $f$ is a variable with as values integrable real-valued functions, and $x$, $a$, $b$ are variables whose values are real numbers, then the expression $\int_a^bf(x)dx$ is a term with three parameters $a$, $b$, $f$, which denotes a well-defined real number for each set of values of the parameters ($x$ in this term is a bound variable). Syntactically, terms are characterized by the fact that they can be substituted for variables in other expressions of the language — terms or formulas, yielding new terms or formulas, respectively.

In a formalized language there exist formal rules, independent of the semantics of the language, for constructing terms and distinguishing free variables in them. In many-sorted languages there are also rules for determining the sorts of the terms which occur.



[a1] J.R. Schoenfield, "Mathematical logic" , Addison-Wesley (1967)
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This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article