Logical connectives and truth tables

A proposition is a statement to which a truth value can be assigned: true or false, and nothing else. All of mathematical logic rests on this simple idea. From basic propositions, we build compound propositions using logical connectives, just as words form sentences.

Negation, written $\neg P$, reverses the truth value of $P$: if $P$ is true, $\neg P$ is false, and vice versa.

Conjunction, written $P \wedge Q$, is true only when both propositions are true at the same time. Disjunction, written $P \vee Q$, is true as soon as at least one of the two is: so it is not an "exclusive or" as in everyday language.

Implication, written $P \implies Q$, deserves special attention, because it often surprises beginners: it is false only when $P$ is true and $Q$ is false. In all other cases — including when $P$ is false — it is considered true. Remember this classic trap: an implication whose hypothesis is false is always declared true, whatever the intended conclusion.

Finally, equivalence, written $P \iff Q$, is true when $P$ and $Q$ share the same truth value: either both true, or both false. It is used to assert that two statements are logically identical, interchangeable in any proof.

To check these definitions without the slightest ambiguity, we build a truth table: a table listing all possible combinations of the values of $P$ and $Q$, together with the resulting value of the compound proposition. It is the ultimate reference tool in logic — it leaves no room for personal interpretation.

$P$	$Q$	$P \wedge Q$	$P \vee Q$	$P \implies Q$
T	T	T	T	T
T	F	F	T	F
F	T	F	T	T
F	F	F	F	T