Consider the following logical inferences.
I_1: If it rains then the cricket match will not be played. The cricket match was played. Inference: There was no rain.
I_2: If it rains then the cricket match will not be played. It did not rain. Inference: The cricket match was played.
Which of the following is TRUE?
A. Both I_1 and I_2 are correct inferences
B. I_1 is correct but I_2 is not a correct inference
C. I_1 is not correct but I_2 is a correct inference
D. Both I_1 and I_2 are not correct inferences
GATE 2012 · Discrete Mathematics · Logical Reasoning · medium
Answer: I_1 is a correct inference (Modus Tollens); I_2 is not correct (denying the antecedent fallacy). Answer: B.
Evaluate I_1 using Modus Tollens: Premise: Rain -> NOT match (P -> Q). Given: match was played, i.e., NOT (NOT match) = match played means NOT Q is false... Restating: let P = rains, Q = match not played. Given Q is false (match WAS played = NOT Q). By Modus Tollens from (P -> Q) and NOT Q: conclude NOT P (no rain). This is VALID.
Evaluate I_2 for fallacy of denying the antecedent: I_2 uses: P -> Q and NOT P, concludes NOT Q. This is NOT Modus Tollens. It is the 'denying the antecedent' fallacy. P -> Q does NOT imply NOT P -> NOT Q. Counterexample: match could still be cancelled due to bad pitch (Q is true even when P is false). I_2 is INVALID.