The question doesn't go beyond the title.
And I don't mean logics that merely just don't have it as a primitive rule - I'm interested in logic where you can't actually use it.
I've searched around and looked at the more exotic logics that I know, but all use modus ponens. Are there logic that do have implication but go without that rule?
But maybe I'm confused and if I take away that tool to syntactically go from knowing $P$ to knowing $Q$, then there isn't anything to $\to$ left.