Proposition
Elementary
A proposition is a statement that is either true or false.
Examples
Worked examples
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$\exists x \, (x^2 = 2)$
: True if $x$ ranges over $\reals$; false if $x$ ranges over the rational numbers
Undergraduate
$\exists x \, (x^2 = 2)$ 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.
Examples
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“$7$ is prime”
High school
“$7$ is prime” is a proposition because it is true.
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“$7$ is prime and $4$ is not prime”
Undergraduate
“$7$ 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 $\reals$
Undergraduate
With $x$ ranging over $\reals$, $\forall x \, (x^2 \geq 0)$ is a proposition, because it is true.
Non-examples
Things that may be mistaken for examples.
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“$x > 3$”
High school
“$x > 3$” is not a proposition, because it is not true or false until $x$ is given.
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$p \land \neg q$
Undergraduate
$p \land \neg q$ is not a proposition, because it is not true or false by itself—the truth value depends on the truth values of $p$ and $q$.
Notes
In mathematical writing, “Proposition” also labels a proven result, usually less important than a theorem. That usage is unrelated. “Proposition” is also different from “preposition,” a word from grammar, such as “in,” “on,” or “of,” that relates a noun to another word.
Distinctions involving this term
- Proposition vs. predicate: In logic, statements are classified by whether they include enough information to determine whether they are true or false. A proposition is a statement that is true or false, such as “$7 > 3$.” A predicate is a statement with variables, such as “$x > 3$,” that becomes a proposition once its variable $x$ is given a value, such as “$7 > 3$” (true) or bound by quantifiers, such as “$\forall x,\, x > 3$” (false).
- Proposition vs. propositional formula: In logic, statements are combined with logical connectives such as “and” ($\land$), “or” ($\lor$), and “not” ($\neg$). A proposition is a particular statement that is true or false, such as “$7$ is prime.” A propositional formula is a pattern, such as $p \land \neg q$, built from propositional variables that stand for propositions, and the formula has a truth value only once each variable is assigned a truth value. Substituting a proposition for each variable in a propositional formula gives a proposition.
- 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.