  A very natural understanding of an implication A → B is reading it as expressing the inference rule which allows one to pass over from A to B. Licensing the step from A to B on the basis of A → B is exactly, what modus ponens says. And the deduction theorem can be viewed as the means of establishing a rule: Having shown that B can be deduced from A justifies the rule that from A we may pass over to B. A rule-based semantics of implication along such lines underlies several conceptions of proof-theoretic semantics, notably those by Lorenzen, von Kutschera and Schroeder-Heister.