Proof by Contraposition/Intuitionist Perspective

Proof by Contraposition: Intuitionist Perspective

Proof by Contraposition is often confused with Reductio ad Absurdum, which also starts with an assumption $\neg q$.

Reductio ad Absurdum itself is often confused with Proof by Contradiction.

Unlike Reductio ad Absurdum however, Proof by Contraposition can be a valid proof in intuitionistic logic, just like Proof by Contradiction.

Specifically, suppose:

$p \implies q$

is true.

Suppose furthermore that we have a proof of:

$\neg q$.

Then if we had a proof of $p$, it could be turned into a proof of $q$.

This would imply

$q \land \neg q$

which is impossible.

Therefore, it cannot be the case that $p$ is true.

That is:

$\neg p$

$\Box$

However, now suppose:

$\neg q \implies \neg p$

is true.

Suppose furthermore that we have a proof of $p$.

Then, if we were to find a proof of $\neg q$, it could be turned into a proof of $\neg p$.

This would imply:

$p \land \neg p$

which is impossible.

Thus it is not possible to prove $\neg q$.

Now intuitionistic logic does not accept the Law of Excluded Middle.

That is, from an intuitionist perspective, knowing that a statement $q$ is not false does not automatically allow us to deduce that $q$ is actually true.

Under this assumption, we only know:

$\neg \neg q$

$\Box$

Hence the rejection, by the intuitionist school of the Reductio ad Absurdum, but not the Proof by Contradiction.