GATE CS 2015 Set 2 Q.13 (CS Q.3): Propositional and First-Order Logic

Consider the following two statements.

$S_1$ : If a candidate is known to be corrupt, then he will not be elected.
$S_2$ : If a candidate is kind, he will be elected.

Which one of the following statements follows from $S_1$ and $S_2$ as per sound inference rules of logic?

If a person is known to be corrupt, he is kind

If a person is not known to be corrupt, he is not kind

If a person is kind, he is not known to be corrupt

If a person is not kind, he is not known to be corrupt

Official answer and solution

Solution

Formalize the statements in propositional logic and apply sound deductive inference rules (hypothetical syllogism and contraposition).

Define propositions:

The given premises translate to:

Take the contrapositive of $S_1$ :

\neg(\neg E) \implies \neg C \iff E \implies \neg C.

Chain with $S_2$ using hypothetical syllogism:

K \implies E \quad \text{and} \quad E \implies \neg C \implies K \implies \neg C.

In English: 'If a person is kind, he is not known to be corrupt.'

Final answer

Common trap

Affirming the consequent or assuming that not being corrupt implies kindness.

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