Questions: The Contrapositive and Logical Equivalence

Question 1

Someone argues: 'If you exercise regularly, you will be healthy. John is healthy. Therefore, John exercises regularly.' What is wrong with this argument?

• A. Nothing — the argument correctly applies the contrapositive of the original conditional
• B. It commits the fallacy of affirming the consequent, confusing a conditional with its converse
• C. The premise is false — exercise does not guarantee health
• D. It incorrectly applies the inverse rather than the contrapositive

Answer: (B)

The argument takes 'If P then Q' (if exercise, then health) and 'Q is true' (John is healthy) and concludes 'P is true' (John exercises). This is affirming the consequent — a fallacy. It treats the conditional as if it were its converse ('if healthy, then exercises'), which is not equivalent. The correct inference from 'Q is true' would be nothing useful about P. The contrapositive would allow: 'If John is not healthy, then John does not exercise regularly' — which is valid. This is one of the most common logical errors in everyday reasoning.

Question 2

Which of the following statements is logically equivalent to 'If it rains, the ground gets wet'?

• A. If the ground is wet, then it rained (converse)
• B. If it does not rain, then the ground is not wet (inverse)
• C. If the ground is not wet, then it did not rain (contrapositive)
• D. The ground being wet causes rain to fall (causal reversal)

Answer: (C)

The contrapositive of 'If P then Q' is 'If not-Q then not-P,' and these are logically equivalent — they have identical truth conditions. 'If the ground is not wet, then it did not rain' is the contrapositive and is logically equivalent to the original. The converse (option A) and inverse (option B) are not equivalent: a sprinkler could wet the ground without rain, making the converse false in cases where the original is true. Only the contrapositive preserves the full logical content.

Question 3

The converse of a conditional ('If Q then P') is logically equivalent to the original conditional ('If P then Q').

• T. True
• F. False

Answer: False

The converse is NOT logically equivalent to the original — this is the contrapositive's key contrast. 'If it rains, the ground is wet' does not mean 'if the ground is wet, it rained' (a sprinkler could be the cause). Confusing a conditional with its converse is one of the most common logical errors in ordinary reasoning and argumentation. What IS equivalent to the original is the contrapositive ('if not-Q then not-P'). The converse is instead equivalent to the inverse ('if not-P then not-Q').

Question 4

Modus tollens ('If P then Q; not-Q; therefore not-P') derives its validity from the logical equivalence between a conditional and its contrapositive.

• T. True
• F. False

Answer: True

Modus tollens is valid because 'If P then Q' is logically equivalent to 'If not-Q then not-P' (the contrapositive). Applying modus ponens to the contrapositive gives: 'If not-Q then not-P; not-Q is true; therefore not-P.' This is exactly modus tollens on the original. So modus tollens reduces to modus ponens plus contrapositive equivalence — a satisfying logical explanation of why the inference works.

Question 5

Explain why the contrapositive of a conditional is logically equivalent to the original, while the converse is not. Use a concrete example to illustrate the difference.

Model answer: A conditional 'If P then Q' is false in exactly one case: P is true and Q is false. The contrapositive 'If not-Q then not-P' is false when not-Q is true and not-P is false — i.e., when Q is false and P is true. Same falsity condition, same truth table: they are logically equivalent. The converse 'If Q then P' is false when Q is true and P is false — a completely different condition. Concrete example: 'If it is a dog, then it is a mammal' (P→Q). Contrapositive: 'If it is not a mammal, then it is not a dog' — logically equivalent, same relationship. Converse: 'If it is a mammal, then it is a dog' — clearly false (cats are mammals). The converse introduces an entirely new claim.

The key is truth conditions. Two statements are logically equivalent when they are false (and true) in exactly the same circumstances. The contrapositive achieves this by negating both P and Q and flipping the direction — the flip and the negations together preserve the original falsity condition. The converse only flips without negating, which changes which cases make it false.