Of course. But if you truly cannot model every true statement in any formally devised system, then you are by definition going to have to reject valid rules that your logic cannot verify if you intend your system to perfectly logical.
I believe that's right but only in a deductive setting, and as long as there's a requirement for soundness. Inductive and abductive logical inference are not sound and they are very useful for real-world decision-making. But that is a developing field and there are still many unknowns there.