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.
Definitions
Level:
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.
Non-examples
-
“$x > 3$”
“$x > 3$” is not a term, because “$x > 3$” makes a claim about $x$ instead of naming an object.
-
$\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$
$\forall x \, (x^2 \geq 0)$ 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.
Missing definition: formula_logic
Examples
| Example | term | formula_logic | Why |
|---|---|---|---|
Missing example: x_squared_plus_1_as_an_expression | |||
| “$x > 3$” | ✗ | ✓ | |
| $\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$ | ✗ | ✓ | |
Notes
- The algebraic distinction between an expression and an equation is close to the distinction between a term and a formula: $3x + 2$ is a term, and the equation $3x + 2 = 8$ is a formula.