If P is false, what can be concluded about P -> Q?

• A. P -> Q is false
• B. P -> Q cannot be determined
• C. P -> Q is both true and false
• D. P -> Q is true

Answer: (D)

In logic, an implication P → Q is true whenever the antecedent P is false. This happens because the statement is claiming that Q follows from P; if P doesn’t occur, there’s no requirement on Q, so the implication isn’t falsified by Q’s truth value. Formally, P → Q is equivalent to ¬P ∨ Q; if P is false, ¬P is true, and the whole expression is true regardless of Q. So when P is false, the implication is true, even if Q is false. For example, if P is “it is raining” (false in this situation) and Q is “the sidewalk is wet,” the statement “If it is raining, the sidewalk is wet” is considered true because the premise didn’t hold.