UGCNET Questions

For a conditional statement p → q, the contrapositive ¬q → ¬p is logically equivalent to the original. Here p is “the number is divisible by 4” and q is “the number is even.” The contrapositive is “if a number is not even, then it is not divisible by 4.” Both the original statement and its contrapositive are true or false together in all possible cases, so they are logically equivalent.

Option A:
Option A reverses the direction of the implication to q → p, which is the converse. The truth of the original statement does not guarantee the truth of its converse; being even does not ensure divisibility by 4.

Option B:
Option B states “If a number is not divisible by 4, then it is not even,” which is the inverse. The inverse is not logically equivalent to the original because many numbers are not divisible by 4 but are still even.

Option C:
Option C correctly expresses the contrapositive ¬q → ¬p: if a number is not even, then it is not divisible by 4. This preserves the logical content of the original implication and is therefore equivalent to it.

Option D:
Option D, “If a number is not divisible by 4, then it is even,” contradicts the original pattern and misstates the relationship between being even and being divisible by 4.

Modus ponens is the basic valid argument form that says: from “If P then Q” and “P,” we may validly infer “Q.” It affirms the antecedent and concludes the consequent. The pattern “If P then Q; P; therefore Q” guarantees that if the premises are true, the conclusion must be true. This is one of the most frequently used inference rules in formal and informal reasoning.

Option A:
Option A presents the correct modus ponens form, with the conditional and affirmation of the antecedent leading to the consequent. It preserves validity in all interpretations.

Option B:
Option B is the fallacy of denying the antecedent: from “If P then Q” and “not P” it draws “not Q,” which is not logically guaranteed, because Q might still be true for other reasons.

Option C:
Option C is the fallacy of affirming the consequent: from “If P then Q” and “Q” it infers “P,” which is also not always valid.

Option D:
Option D is modus tollens, a different but still valid form, which infers “not P” from “If P then Q” and “not Q,” but it is not modus ponens.