If and are both true, clearly the implication is true . If the premise is false no conclusion can be drawn, but both being true and being false are consistent, so that the implication is true in both cases. Implication can also be expressed by other connectives, for example, is logically equivalent to . The Case of the Stupid Defense Attorney . Otherwise, is false.
Bill studies hard.
The biconditional may be viewed as a shorthand for a conjunction of two implications, as is logically equivalent to . A proposition is sufficient for if is a tautology. If is a tautology, then is logically equivalent to . If is a tautology, then it is true for all truth assignments. By the semantics of the biconditional, this means that and agree on every row of the truth table.