Sentence vs. proposition
In logic, a statement can be studied as a string of symbols or as a claim that is true or false. A sentence is a formula of first-order logic, a string of symbols in which a quantifier binds every variable. A proposition is a claim that is true or false. A sentence expresses a proposition only once its symbols are interpreted, for example by choosing the set that the variables range over, and the same sentence can express different propositions under different interpretations.
Definitions
Level:
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.
Elementary
A proposition is a statement that is either true or false.
Examples
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“$7$ is prime”
“$7$ is prime” is a proposition because it is true.
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$\exists x \, (x^2 = 2)$
: True if $x$ ranges over $\reals$; false if $x$ ranges over the rational numbers
$\exists x \, (x^2 = 2)$ expresses a proposition only once the set that $x$ ranges over is chosen: over $\reals$, the number $\sqrt{2}$ satisfies $x^2 = 2$, and over the rational numbers, no number does.
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“$7$ is prime and $4$ is not prime”
“$7$ is prime and $4$ is not prime” is a proposition, because the statement is true: $7$ is prime, and $4$ is not prime.
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$\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$
With $x$ ranging over $\reals$, $\forall x \, (x^2 \geq 0)$ is a proposition, because it is true.
Non-examples
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“$x > 3$”
“$x > 3$” is not a proposition, because it is not true or false until $x$ is given.
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$p \land \neg q$
$p \land \neg q$ is not a proposition, because it is not true or false by itself—the truth value depends on the truth values of $p$ and $q$.
Examples
| Example | sentence | proposition | Why |
|---|---|---|---|
| $\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$ | ✓ | ✓ | |
| $\exists x \, (x^2 = 2)$ | ✓ | True if $x$ ranges over $\reals$; false if $x$ ranges over the rational numbers | The sentence $\exists x \, (x^2 = 2)$ expresses a true proposition when $x$ ranges over the real numbers and a false proposition when $x$ ranges over the rational numbers. |
| “$7$ is prime” | ✗ | ✓ | “$7$ is prime” is a proposition stated in English, not a sentence of a formal language, although a sentence of first-order logic can express the same proposition. |
| “$x > 3$” | ✗ | ✗ |