Intuitionistic logic- proof.

Question

$$ A \implies \neg \neg A$$

A proof of $\neg \neg A $ is a proof of $\neg A \implies \bot$. Assume $p : A $ and $q : \neg A$. Then $q(p): \bot$. What is $q(p)$. How to understand it?

Answer 1

The notation $p: A$ means that $p$ is a proof of the formula $A$. This proof may have hypotheses; $q(p)$ means that in the proof $q$ we have plugged in the proof $p$ to prove the hypothesis used in $q$.

A proof of $\neg A$ proves the proposition $A \rightarrow \bot$. It will consist in assuming $A$ as a hypothesis and from this eventually concluding $\bot$.

So the proof $q(p)$ consists of the above proof but with the added hypothesis of $A$. We now have that $A$ and that $A \rightarrow \bot$ and therefore conclude $\bot$. So $q(p)$ is a proof of $\bot$.