Rules of Inference

Rules of Inference are the building blocks for constructing valid arguments: templates that say which conclusions can be drawn from a set of premises.

You could check an Argument (Logic) with a truth table, but that gets laborious fast (8 propositional variables needs $2^8$ rows). Rules of inference give a simpler way to prove an argument is a Valid Argument, and every rule can be proved using a Tautology.

Common rules include Modus ponens ($p \rightarrow q$ and $p$, therefore $q$), modus tollens ($p \rightarrow q$ and $\neg q$, therefore $\neg p$), simplification, addition, hypothetical syllogism, disjunctive syllogism and resolution. Predicate logic adds rules for quantifiers, like universal instantiation and existential generalisation.

See Week 8 - Predicate Logic B.