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).
Definitions
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 $q$.
Consider a true/false statement with a variable in it, such as “$x$ is even.” The statement is a predicate: it becomes true or false once a value is chosen for the variable.
Consider a set $X$ of objects. A predicate on $X$ is a function $P$ that assigns to each $x \in X$ a proposition $P(x)$; for example, the predicate “$x$ is prime” assigns to $7$ the proposition “$7$ is prime.” Each proposition is either true or false, so the predicate $P$ determines the function
\[P : X \to \setdef{\text{true}, \text{false}}\]that sends each $x$ to the truth value of $P(x)$. Logic books usually identify the predicate with this function.
Examples
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“$x > 3$”
“$x > 3$” is a predicate because its truth depends on $x$.
Non-examples
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$\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$
$\forall x \, (x^2 \geq 0)$ is not a predicate, because it has no free variables.
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$p \land \neg q$
$p \land \neg q$ is not a predicate, because the variables of a predicate stand for objects, such as numbers, sets, or points, which a statement can be about, but $p$ and $q$ stand for propositions, which are statements that are true or false. For example, in the predicate “$x > 3$,” the variable $x$ stands for a number, and the number $5$ is neither true nor false; in $p \land \neg q$, the variable $p$ stands for a statement such as “$7$ is prime,” which is true.
Examples
| Example | proposition | predicate | Why |
|---|---|---|---|
| “$7$ is prime” | ✓ | – | |
| “$x > 3$” | ✗ | ✓ | “$x > 3$” is a predicate, and giving $x$ the value $5$ turns the predicate into the proposition “$5 > 3$,” which is true. |
| $\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$ | ✓ | ✗ | Binding the variable of the predicate “$x^2 \geq 0$” with the quantifier $\forall x$ turns the predicate into a proposition. |
Notes
- An $n$-place predicate and an $n$-ary relation carry the same information: the relation is the set of tuples that satisfy the predicate (its extension).