GATE CS 2012 Q1 — Propositional and First Order Logic

Let us define the propositional variables:

• $p$: It rains.
• $q$: The cricket match is played.

Analysis of $I_1$:

• Premise 1: $p \rightarrow \neg q$
• Premise 2: $q$
• Inference: $\neg p$

By the rule of Modus Tollens (or contrapositive reasoning), since $p \rightarrow \neg q$ is logically equivalent to $q \rightarrow \neg p$, the presence of $q$ logically guarantees $\neg p$. Thus, $I_1$ is a correct inference.

Analysis of $I_2$:

• Premise 1: $p \rightarrow \neg q$
• Premise 2: $\neg p$
• Inference: $q$

This is a classic logical fallacy known as denying the antecedent. Knowing that it did not rain ($\neg p$) does not allow us to conclude whether the match was played or not (it could have been cancelled for other reasons). Thus, $I_2$ is not correct.

Hence, $I_1$ is correct but $I_2$ is not a correct inference.