Can P ↔ Q be true if P is true and Q is false?

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Multiple Choice

Can P ↔ Q be true if P is true and Q is false?

A biconditional says P ↔ Q is true exactly when P and Q share the same truth value. It’s true when both are true or both are false, and false when one is true and the other false. Here, P is true and Q is false, so they don’t match. The implication P → Q is false in this case (a true P with a false Q breaks the implication), while Q → P is true (a false Q makes that implication vacuously true). Since both directions must hold for the biconditional, the overall statement is false. So, it cannot be true when P is true and Q is false.

A biconditional says P ↔ Q is true exactly when P and Q share the same truth value. It’s true when both are true or both are false, and false when one is true and the other false. Here, P is true and Q is false, so they don’t match. The implication P → Q is false in this case (a true P with a false Q breaks the implication), while Q → P is true (a false Q makes that implication vacuously true). Since both directions must hold for the biconditional, the overall statement is false. So, it cannot be true when P is true and Q is false.

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