Formula vs. sentence
In first-order logic, statements about objects are written as strings of symbols with variables and quantifiers. A formula is every such string that is built by the rules of the language, such as $x > 3$ or $\forall x \, (x^2 \geq 0)$, and the variables in a formula can be free or bound by quantifiers. A sentence is a formula in which a quantifier binds every variable, so a sentence is true or false once its symbols are interpreted, without assigning values to variables.
sentence ⊂ formula
Definitions
Level:
Missing definition: formula_logic
Undergraduate (first-order logic)
Consider a formula of first-order logic, such as $\forall x \, (x^2 \geq 0)$ or $x > 3$. The formula is a sentence if it has no free variables: every variable in it is bound by a quantifier.
Examples
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$\exists x \, (x^2 = 2)$
$\exists x \, (x^2 = 2)$ is a sentence, because the quantifier $\exists x$ binds the only variable $x$.
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$\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$
$\forall x \, (x^2 \geq 0)$ is a sentence, because the quantifier binds its only variable $x$.
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The first-order sentence $\forall x \, (x^2 \geq 0) \land \neg \exists x \, (x^2 < 0)$, whose propositional form is $p \land \neg q$
$\forall x \, (x^2 \geq 0) \land \neg \exists x \, (x^2 < 0)$ is a sentence, because the quantifiers $\forall x$ and $\exists x$ bind every occurrence of $x$.
Non-examples
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“$7$ is prime”
“$7$ is prime” is not a sentence of first-order logic, because “$7$ is prime” is written in English, not as a string of symbols of a formal language.
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“$x > 3$”
“$x > 3$” is not a sentence, because $x$ is a free variable.
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$\exists y \, (y > x)$
$\exists y \, (y > x)$ is not a sentence, although $\exists y \, (y > x)$ has a quantifier: the quantifier binds $y$, but $x$ is free, so the truth of $\exists y \, (y > x)$ depends on the value of $x$ (for example, if $x$ and $y$ range over the natural numbers, then the formula is true for every $x$, and if $x$ and $y$ range over $\setdef{1, 2, 3}$, then the formula is false for $x = 3$).
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$p \land \neg q$
$p \land \neg q$ is not a sentence of first-order logic, because the variables $p$ and $q$ stand for whole statements, not for objects such as numbers, and $p \land \neg q$ has no quantifiers; a sentence such as $\forall x \, (x^2 \geq 0)$ has variables that stand for objects, and a quantifier binds each of those variables.
Properties
- Once its symbols are interpreted in a structure (such as $\reals$ with its usual operations), a sentence is true or false, so it expresses a proposition.
- A formula with free variables, such as $x > 3$, expresses a predicate. Quantifying every free variable turns it into a sentence.
Examples
| Example | formula_logic | sentence | Why |
|---|---|---|---|
| “$x > 3$” | ✓ | ✗ | “$x > 3$” is a formula but not a sentence, because its variable $x$ is free. |
| $\exists y \, (y > x)$ | ✓ | ✗ | $\exists y \, (y > x)$ shows that a formula with a quantifier can still fail to be a sentence: the quantifier binds $y$ but leaves $x$ free. |
| $\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$ | ✓ | ✓ | |
Missing example: x_squared_plus_1_as_an_expression | |||
Notes
- Quantifying every free variable of a formula gives a sentence: $\forall x \, (x > 3)$ and $\exists x \, (x > 3)$ are sentences.
- A formula with free variables expresses a predicate, and a sentence expresses a proposition once its symbols are interpreted.