I understand that $\bot \rightarrow A$ is a tautology, but I dont fully understand what it means that you can conclude anything after proving/assuming $\bot$.
For example:
Premise: $(A\rightarrow B)\rightarrow\bot$
Goal: $A\land(B\rightarrow\bot)$
You can prove this by considering the cases $B$ and $B\rightarrow\bot$.
So consider the case where $B$. Now, all conclusions that we make depend on the fact that $B$, right? So if we can conclude $A\rightarrow B$, we actually have $B\rightarrow (A\rightarrow B)$?
So since $B$, we can in fact conclude $A\rightarrow B$. And since we know that $(A\rightarrow B)\rightarrow\bot$, we can use modus ponens and conclude that $\bot$, so then i guess $(B\rightarrow\bot)$?
Here is the part I dont understand:
We proved $\bot$, which implies $A\land(B\rightarrow\bot)$, but I dont understand why.
Is it because $\bot\rightarrow A\land(B\rightarrow\bot)$ is a tautology? Therefore we can put anything on the righthand side, and since $\bot$, it must be that $A\land(B\rightarrow\bot)$
For example if you had $A$, you couldnt just say $A\rightarrow C$ or $A\rightarrow Y$, because those might not be tautologies.
I guess the concept of $\bot$ (false) being true is also making this confusing.
Given that $\bot \to A$ for any $A$ is a tautology, if you have $\bot$, concluding $A$ is simply an application of modus ponens on $\bot \to A$ and $\bot$.