Talk:Proof by Contraposition

The standard technique for this on $\mathsf{Pr} \infty \mathsf{fWiki}$ is usually to set such a proof up as a Proof by Contradiction.

That is:

"To be proved: $p \implies q$."

"Let $p$ be assumed.

"Aiming for a contradiction, suppose $\neg q$.

After some working:

"Hence $\neg p$.

"But this contradicts $p$.

"Hence, by Proof by Contradiction, $q$.

"Hence the result."

--prime mover (talk) 16:38, 13 September 2018 (EDT)

I see. Thank you for keeping the page up anyway :-) I believe it is important pedagogically to distinguish between these two types of proofs. In a proof by contradiction, you begin with an assumption and end up with a nonsense statement (a logical contradiction). In a proof by contraposition, you only end up with $\neg p$.

As for "keeping the page up" -- it is very rare that we actually delete pages from $\mathsf{Pr} \infty \mathsf{fWiki}$. --prime mover (talk) 03:25, 14 September 2018 (EDT)

Now of course, if you include the assumption $p$ at the start (as above), the conclusion $\neg p$ is indeed nonsense; but often the assumption $p$ is not actually used inside the proof, and an astute reader could point out that the contradiction only arises because we insisted on assuming $p$ at the start, as otherwise there would be no problem.

So I think the goals of these two types of proofs are quite different, and I think it is good to point out the underlying principle (the rule of transposition). If you don't mind, I may edit some proofs to refer to the page Proof by Contraposition instead. Robsiegen (talk) 03:09, 14 September 2018 (EDT)

As I founder on logic, I will defer to you -- I find the details tiresome, and have difficulty with the philosophical implications. :-) Feel free to go ahead, but try and keep a rigorously consistent style, with a view to going forward and building a template for oft-repeated boilerplate. --prime mover (talk) 03:24, 14 September 2018 (EDT)