Predicate
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
Examples
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“$x > 3$”
High school
“$x > 3$” is a predicate because its truth depends on $x$.
Non-examples
Things that may be mistaken for examples.
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$\forall x \, (x^2 \geq 0)$, with $x$ ranging over $\reals$
Undergraduate
$\forall x \, (x^2 \geq 0)$ is not a predicate, because it has no free variables.
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$p \land \neg q$
Undergraduate
$p \land \neg q$ 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.
Notes
In grammar, the predicate of a sentence is the part that says something about the subject: in “$7$ is prime,” the subject is “$7$” and the predicate is “is prime.” Logic borrowed the word: the logical predicate “$x$ is prime” is the grammatical predicate with a variable in place of the subject. Let $P(x)$ be the predicate $x \text{ is prime}$. Then, the statement “$7$ is prime” is $P(7)$.
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).