What is an axiom schema for minimalist propositional logic that characterizes negation ($\lnot$), but, does so without introducing the $\bot$ symbol to the language?
I thought that maybe $$\dfrac{X, X \to \bot}{\bot}$$ could be replaced with $A \to X \to \lnot X \to \lnot A$ , but that doesn't seem to work for really small expressions like trying to prove $(B \to \lnot \lnot B)$....or maybe it does and I am missing an obvious proof?