Does $p$ follow from $p \land \neg p$?

Question

Ex contradictio states that you can follow anything from a contradiction. But is it possible to follow one of the conjuncts itself?

$$ (p \land \neg p) \to p $$

Is that valid?

Answer 1

Since what you are asking about is trivially true, I suspect that you are worried about something else. In a proof by contradiction, once you reach a contradiction you don't move on from the contradiction itself, deducing additional consequences of it (which include both $p$ and $\neg p$, so not very helpful in deciding between the two). Instead, you step back and reject the assumption that got you to that point in the first place.

Answer 2

In general, $p$ follows from $p\land q$ for any $q$. The specific case $q=\lnot p$ is no different.

Answer 3

Yes, we can deduce either conjunct. Indeed, the reason a contradiction implies anything is because by combining $\neg p$ with the consequence $p\lor q$ of $p$ we get $q$.