The discussion clarifies the distinction between proof by contraposition and proof by contradiction in the context of the theorem p → q. Proof by contraposition involves assuming ¬q to derive ¬p, providing a clear logical path. In contrast, proof by contradiction starts with the assumption that p ∧ ¬q is true, leading to a contradiction without a defined outcome. The equivalence of these methods holds in classical first-order predicate logic, but may differ in intuitionist logic and others that reject certain axioms. Ultimately, proof by contradiction is critiqued for its lack of clarity and practical utility in problem-solving.