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It can be proved by proof by contradiction. hence it is just a variant of it?

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- #1

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It can be proved by proof by contradiction. hence it is just a variant of it?

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CRGreathouse

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Proof by contradiction: Assume p and ¬q; prove ⊥.

Proof by contraposition: Assume ¬q; prove ¬p.

A proof by contraposition can be turned into a proof by contradiction by adding the step "p ∧ ¬p ⇒ ⊥". But proofs by contradiction don't need to prove ¬p as an intermediate step; they can prove contradiction in other ways. So proof by contradiction is stronger.

On the other hand, the more constrained proof by contraposition is considered a constructive technique, while proof by contradiction is always considered nonconstructive. So if you have a choice use contraposition.

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HallsofIvy

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However, "proof by contradiction" more general. You start with "if not Q" but only need to arrive at two statements that contradict one another. I have seen proofs in which the two contraditory statements have no apparent connection with the hypothesis, P.

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