Propositional logic operates on whole propositions—that is, statements that are either true or false—and combines them using logical connectives. A truth table is a mathematical tool that determines the truth value of a compound proposition for every possible combination of truth values of its component statements, making explicit exactly when each connective yields a true or false result.
Conjunction (\(P \land Q\)) is true only when both P and Q are true. Disjunction (\(P \lor Q\)) is true when at least one of P or Q is true, and false only when both are false. Negation (\(\neg P\)) reverses the truth value of a proposition: if P is true, \(\neg P\) is false, and vice versa. The conditional or material implication (\(P \to Q\)) is false only when P is true and Q is false, and true in every other case. The biconditional (\(P \leftrightarrow Q\)) is true exactly when both propositions share the same truth value, whether both true or both false.
Compound statements can also be classified by their overall behavior. A tautology is true under every assignment of truth values to its variables, as with \(P \lor \neg P\). A contradiction is false under every assignment, as with \(P \land \neg P\). A contingency is neither a tautology nor a contradiction; its truth value depends on the values of its variables. These three categories cover every possible compound proposition in propositional logic.