Term (logic)
Undergraduate (first-order logic)
Consider a first-order language, with variables, constant symbols, and function symbols. A term is a string of symbols built by the rules of the language that names an object, such as $x$, $3$, or $x^2 + 1$. Terms are built in two steps:
- every variable and every constant symbol is a term, and
- if $f$ is a function symbol with $n$ inputs and $t_1, \dots, t_n$ are terms, then $f(t_1, \dots, t_n)$ is a term.
Examples
Non-examples
Things that may be mistaken for examples.
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“$x > 3$”
High school
“$x > 3$” is not a term, because “$x > 3$” makes a claim about $x$ instead of naming an object.
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$\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$
Undergraduate
$\forall x \, (x^2 \geq 0)$ is not a term, because $\forall x \, (x^2 \geq 0)$ makes a claim, which is true, instead of naming an object.
Properties
- A term has a value, such as a number, once its variables are given values, but a term has no truth value.
- A relation symbol applied to terms, such as $x^2 + 1 > 3$, is a formula, not a term.
Notes
In algebra, a “term” is one of the summands of a sum, such as $3x^2$ in $3x^2 - 5x + 2$; that meaning is different.
Distinctions involving this term
- Term vs. formula: In logic, the expressions of a formal language, which are the well-formed strings of symbols, come in two kinds. A term, such as $x^2 + 1$, names an object, so a term has a value, such as a number. A formula, such as $x^2 + 1 > 3$, makes a claim about objects, so a formula has a truth value. Applying a relation symbol, such as $>$ or $=$, to terms gives a formula.