GATE CS 2012 Q.1 (CS Q.1): Propositional and First-Order Logic

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?

Both $I_1$ and $I_2$ are correct inferences

$I_1$ is correct but $I_2$ is not a correct inference

$I_1$ is not correct but $I_2$ is a correct inference

Both $I_1$ and $I_2$ are not correct inferences

Official answer and solution

Solution

Identify the standard argument forms (modus tollens and denying the antecedent) to assess validity.

Let $p$ be "It rains" and $q$ be "The cricket match will not be played".

Inference $I_1$ :
- Premise 1: $p \rightarrow q$
- Premise 2: $\neg q$ ("The cricket match was played")
- Conclusion: $\neg p$ ("There was no rain")
  This is the standard rule of Modus Tollens: $(p \rightarrow q) \land \neg q \vdash \neg p$ . Since this argument form is valid, $I_1$ is a correct inference.
Inference $I_2$ :
- Premise 1: $p \rightarrow q$
- Premise 2: $\neg p$ ("It did not rain")
- Conclusion: $\neg q$ ("The cricket match was played")
  This argument commits the formal fallacy of denying the antecedent ( $(p \rightarrow q) \land \neg p \not\vdash \neg q$ ). The match might still be cancelled for reasons other than rain. Thus, $I_2$ is not a correct inference.

Therefore, $I_1$ is correct but $I_2$ is not a correct inference, which corresponds to option (B).

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

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