Which statement is logically equivalent to the conditional 'If P, then Q'?

• A. If P, then not Q.
• B. If not P, then Q.
• C. Q implies P.
• D. If not Q, then not P.

Answer: (D)

The main idea here is the contrapositive of a conditional statement, which preserves truth values. For a statement like “If P, then Q,” the opposite direction with negations—“If not Q, then not P”—has the same truth conditions as the original.

Think of a concrete example: If it is raining, the sidewalk is wet. The contrapositive would be, If the sidewalk is not wet, then it is not raining. Whenever the original statement is true, the contrapositive is also true, and whenever the original is false, the contrapositive is false as well. That symmetry is what makes them logically equivalent.

The other forms don’t line up in truth conditions with the original. Saying, “If P, then not Q” changes the outcome when P is true and Q is true, so it isn’t equivalent. Saying, “If not P, then Q” shifts the focus to cases where P is false, which also doesn’t match the original’s truth pattern. Reversing the order to “Q implies P” flips the direction entirely, which changes when the statement is true or false. Thus, the form with the negated Q leading to a negated P is the one that matches the original in all cases.