If P→Q is true and P is false, what can be concluded about Q?

• A. Q must be true
• B. Q must be false
• C. Q is true if P is true
• D. Q may be either true or false

Answer: (D)

An implication is true when the antecedent is false, regardless of the consequent. If P→Q is true and P is false, the truth of Q isn’t constrained by these premises. Because P→Q is logically equivalent to ¬P ∨ Q, having ¬P true makes the entire statement true no matter what Q is. This is known as a vacuous truth: the implication holds simply because the antecedent doesn’t occur. So Q could be true or false. The other options would require Q to have a specific truth value, which isn’t dictated by P→Q being true with P false.