Compound truth tables do more than compute values row by row. They allow us to decide whether a statement is always true, always false, or true only in some cases. This leads to tautologies, contradictions, and valid arguments. In this lesson, tautologies, contradictions, and valid arguments will be tested by final columns of truth tables.

A compound statement is called a \textbf{tautology} if it is true for every possible truth-value assignment of its component statements. Thus, the final column of its truth table contains only $1$'s.