$p \land \neg q$

Terms

Term	This example	Why
propositional logic	A formula of propositional logic	$p \land \neg q$ is a formula of propositional logic, because $p \land \neg q$ combines the propositional variables $p$ and $q$ with the connectives $\land$ and $\neg$.
first order logic	Becomes a first-order sentence once $p$ and $q$ are replaced by sentences, such as $\forall x \, (x^2 \geq 0) \land \neg \exists x \, (x^2 < 0)$	$p \land \neg q$ describes how first-order sentences combine, because replacing $p$ and $q$ by first-order sentences gives a first-order sentence.
propositional formula	✓	$p \land \neg q$ is a propositional formula because its truth depends on the truth values of $p$ and $q$.
proposition	✗	$p \land \neg q$ is not a proposition, because it is not true or false by itself—the truth value depends on the truth values of $p$ and $q$.
predicate	✗	$p \land \neg q$ 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.
sentence	✗	$p \land \neg q$ 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.
formula logic	✗	$p \land \neg q$ is not a formula of first-order logic, because its variables $p$ and $q$ stand for whole statements, and a first-order formula is built from relation symbols applied to terms that name objects.

Distinctions using this example

All examples · All distinctions · All definitions