[auggierose]

It is a matter of presentation and what people are used to. In particular, in abstraction logic, in the simplest (and most general!) case, you only have these proof rules: Axiom, Assume, AddAnte, BindAnte, FreeAnte, InferAnte. If we just focus on this case, we can choose simpler names for these rules: Axiom, Assume, Add, Bind, Free, Infer. That's it. That is all there is to proof.