Questions: Modus Ponens and Modus Tollens

Question 1

You know: 'If it rained, the ground is wet' (R → W). You observe that the ground IS wet. A classmate concludes 'Therefore it rained.' What logical error did they commit?

• A. No error — this is a valid application of modus ponens
• B. No error — this is a valid application of modus tollens
• C. Affirming the consequent — an invalid inference; the ground could be wet for reasons other than rain
• D. Denying the antecedent — correctly ruling out rain by observing the ground is wet

Answer: (C)

The implication R → W only says: whenever it rains, the ground will be wet. It does not say the ground is wet ONLY because of rain — sprinklers, flooding, or a spilled hose could also cause wetness. Concluding R from W alone is the invalid form 'affirming the consequent.' Modus ponens (valid) would reason: it rained (R is true) → the ground is wet. Modus tollens (valid) would reason: the ground is NOT wet (¬W) → it did not rain (¬R).

Question 2

From the theorem 'If f is differentiable at a point, then f is continuous there' (D → C), and the fact that function g is NOT continuous at x = 0, what can you validly conclude?

• A. g is differentiable at x = 0 (by modus ponens)
• B. g is not differentiable at x = 0 (by modus tollens)
• C. No conclusion is possible — the theorem only tells us what happens when differentiability holds
• D. g might be differentiable at x = 0, since non-continuity does not affect differentiability

Answer: (B)

This is modus tollens: we have D → C and ¬C (not continuous), therefore ¬D (not differentiable). Since the implication guarantees continuity whenever differentiability holds, discontinuity guarantees non-differentiability. Modus tollens is equivalent to applying modus ponens to the contrapositive: ¬C → ¬D, and since ¬C is true, ¬D follows. Options C and D are wrong because the contrapositive gives us definitive information.

Question 3

Modus tollens is logically equivalent to applying modus ponens to the contrapositive of the original implication.

• T. True
• F. False

Answer: True

The contrapositive of P → Q is ¬Q → ¬P, which is logically equivalent to P → Q (same truth table). Modus tollens says: given P → Q and ¬Q, conclude ¬P. This is the same as: given ¬Q → ¬P (the contrapositive) and ¬Q (the antecedent), conclude ¬P by modus ponens. The two forms are interchangeable, which is why proof by contrapositive works — you set up the contrapositive and then apply modus ponens forward.

Question 4

If P → Q is true and Q is true, then P is expected to be true. This valid inference form is called modus ponens.

• T. True
• F. False

Answer: False

This is NOT modus ponens — it is 'affirming the consequent,' which is an invalid inference. Modus ponens says: if P → Q is true AND P is true, then Q must be true. The premise that triggers the conclusion is the antecedent (P), not the consequent (Q). Affirming the consequent feels persuasive — if Q happened, maybe P caused it — but the implication only promises Q given P, not P given Q. A function can be continuous without being differentiable, so continuity alone tells you nothing about differentiability.

Question 5

Give a concrete example of 'affirming the consequent' where the premises are true but the conclusion is false, and explain why the inference fails.

Model answer: Example: 'If a number is divisible by 4, then it is even' (P → Q). The number 6 is even (Q is true). Conclusion by affirming the consequent: 6 is divisible by 4 (P). But 6 ÷ 4 = 1.5 — this is false. The inference fails because the implication only guarantees Q when P holds; it does not say P is the only way to get Q. Many numbers are even without being divisible by 4. The implication creates a one-directional logical channel: P flows to Q, but Q does not flow back to P.

Affirming the consequent confuses the implication P → Q with the biconditional P ↔ Q (which would go both ways). In mathematics, many theorems are one-directional implications, not 'if and only if' statements. Recognizing this asymmetry is essential for rigorous reasoning and is one of the central lessons of formal logic.