The rules I am considering are $\frac{\neg A, \ \Gamma \implies \Delta}{\Gamma \implies \Delta, \ A} (\neg L)$ and $\frac{\Gamma \implies \Delta, \ \neg A}{A, \ \Gamma \implies \Delta} (\neg R)$
I am trying to get my head around some of the sequent calculus rules, and while I think I understand most of them, I am struggling to apply any intuition to the negation rules shown above.
The intuition of looking at the left as a conjunction of literals and the right as a disjunction of literals seems to break down, and I am unclear how to explain these rules to myself.
What is a sensible way to view such rules and put some understanding on them?