$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}}$ I am trying to derive some rules without the use of the $\bot$ symbol.
First I want to describe how I am defining certain inference rules:
Negation Introduction: $\{(a\to b), (a \to \lnot b) \} \vdash \lnot a$:
$$\fitch{} {\fitch{a} {\vdots \\b} \\a\to b\\ {\fitch{a} {\vdots \\\lnot b}} \\ a \to \lnot b \\ \lnot a}$$
Now presume that we take the law of excluded middle $a \lor \lnot a$ as an axiom and use it (with negation introduction and disjunction elimination) to prove the double-negation rule:
$$\fitch{\lnot \lnot a}{ a \lor \lnot a \\ \fitch{a}{ a } \\ a \to a \\ \fitch{\lnot a}{ \vdots \\ a } \\ \lnot a \to a \\ a \text{ (by or-elim using LEM and the two implications)} }$$
I can't quite figure out how to fill in the dotted section. Normally we'd just restate $\lnot \lnot a$ and note the contradiction with $\lnot a$, state the contradiction $\bot$, and then use ex falso $\bot \to a$ to invoke $a$ and finish the proof.
But without the $\bot$ symbol, is this impossible to do? I don't know how to prove ex falso in the first place without using $\bot$ or double negation elimination which is the very thing I am trying to prove.