Questions: Modus Ponens and Modus Tollens

Question 1

Consider this argument: 'If an animal is a mammal, it is warm-blooded. This animal is warm-blooded. Therefore it is a mammal.' Is this argument valid?

• A. Valid — it follows the form of modus ponens
• B. Invalid — it commits the fallacy of affirming the consequent
• C. Valid — it follows the form of modus tollens
• D. Invalid — it commits the fallacy of denying the antecedent

Answer: (B)

This is affirming the consequent — an invalid form. The argument knows 'mammal → warm-blooded' and 'warm-blooded,' and concludes 'mammal.' But Q's truth does not guarantee P's truth: birds and reptiles are also warm-blooded without being mammals. Valid modus ponens would require asserting the antecedent (mammal) to conclude the consequent. The error is treating a one-way conditional as if it ran in both directions.

Question 2

Consider: 'If a number is divisible by 4, it is divisible by 2. The number 6 is not divisible by 4. Therefore 6 is not divisible by 2.' What is wrong with this argument?

• A. Nothing — this is a valid application of modus tollens
• B. The second premise is false — 6 is divisible by 4
• C. This is denying the antecedent, an invalid form — the conclusion does not follow
• D. This is affirming the consequent, an invalid form

Answer: (C)

This is denying the antecedent: we know 'P → Q' and 'not-P,' and incorrectly conclude 'not-Q.' The form is invalid. Indeed, 6 IS divisible by 2, showing directly that the argument fails. Valid modus tollens would start from 'not-Q' (not divisible by 2) to conclude 'not-P' (not divisible by 4). The antecedent (divisible by 4) was denied, not the consequent.

Question 3

Modus tollens is valid because if the truth of P guarantees the truth of Q, then the falsity of Q guarantees the falsity of P.

• T. True
• F. False

Answer: True

This captures the logic exactly. If 'If P then Q' holds, then P and ¬Q cannot both be true simultaneously — if P were true, Q would have to be true. So knowing ¬Q, P must be false. The form (1) P → Q, (2) ¬Q, therefore (3) ¬P is deductively valid. Modus tollens is the logical engine behind scientific falsification: if a theory predicts Q and Q is observed to be false, something in the theoretical premises must be false.

Question 4

Modus tollens and affirming the consequent are both valid argument forms — they both start from the same conditional and information about Q.

• T. True
• F. False

Answer: False

Only modus tollens is valid. Modus tollens asserts ¬Q (denying the consequent) and validly concludes ¬P. Affirming the consequent asserts Q and invalidly concludes P — this is a formal fallacy. The two forms do start from the same conditional, but differ crucially in what the second premise asserts: denying Q is valid, affirming Q is not.

Question 5

Explain in your own words why 'affirming the consequent' is an invalid argument form. Use a concrete example to illustrate.

Model answer: A conditional 'If P then Q' is a one-way claim: P guarantees Q, but Q does not guarantee P, because Q might be true for other reasons. Example: 'If it rains, the street is wet. The street is wet. Therefore it rained.' The street could be wet because a pipe burst — rain is not the only way streets get wet.

The fallacy treats 'If P then Q' as a biconditional ('P if and only if Q'), which would allow inference in both directions. But a conditional only guarantees the forward direction. Recognizing this asymmetry is the key to correctly analyzing any conditional argument: always ask whether the second premise asserts the antecedent (P) or the consequent (Q), and whether it affirms or denies it.