Having expressed validity through logical implication, we now replace long truth-table verification with standard valid argument forms. Direct rules of inference are reusable patterns: once a rule is known to be valid, it may be applied whenever the same logical form appears. A common mistake is to remember only the name of a rule but not its exact pattern. In this lesson, direct rules of inference begin with Modus Ponens, the Law of the Syllogism, and Modus Tollens.

A \textbf{rule of inference} is a valid argument form that allows a conclusion to be deduced from given premises. If $$\begin{equation*} (p_1\land p_2\land\cdots\land p_n)\to q \end{equation*}$$ is a tautology, then the corresponding argument $$\begin{equation*} p_1,\ p_2,\ldots,p_n\ \therefore\ q \end{equation*}$$ is a valid rule of inference.