Propositional logic supplies a set of rules of inference—reliable patterns for deriving conclusions from premises. Modus ponens reasons from \(P \to Q\) and P to Q. Modus tollens reasons from \(P \to Q\) and \(\neg Q\) to \(\neg P\). Hypothetical syllogism chains conditionals: \(P \to Q\) and \(Q \to R\) together yield \(P \to R\). Disjunctive syllogism resolves alternatives: \(P \lor Q\) and \(\neg P\) together yield Q. These patterns preserve validity, while faulty counterparts such as affirming the consequent (from \(P \to Q\) and Q inferring P) or denying the antecedent (from \(P \to Q\) and \(\neg P\) inferring \(\neg Q\)) do not.
Several classical laws govern how propositions behave. The law of excluded middle holds that for any proposition P, either P or \(\neg P\) is true; there is no middle ground, expressed as \(P \lor \neg P\). The law of non-contradiction states that a proposition cannot be both true and false at the same time and in the same sense, expressed as \(\neg(P \land \neg P)\). The law of identity holds that every object is identical to itself, formalized as \(P \equiv P\).
De Morgan's laws describe how negation distributes over conjunction and disjunction: \(\neg(P \land Q) \equiv (\neg P \lor \neg Q)\) and \(\neg(P \lor Q) \equiv (\neg P \land \neg Q)\). Working with conditionals requires care, since the contrapositive of \(P \to Q\), namely \(\neg Q \to \neg P\), is logically equivalent to the original, but the converse (\(Q \to P\)) and the inverse (\(\neg P \to \neg Q\)) are not. More generally, two statements are logically equivalent when they share the same truth value in every possible scenario, as with \(P \to Q\) and \(\neg P \lor Q\).