I know a bunch of facts about weak and strong completeness in many valued logic, that there is strong completeness for the finite mv logic, and that for the infinite ones you can either only have weak completeness or strong finitary completeness. My question is about what the fact that it is weak or strong implies. I know that strong completeness works for any arbitrary set of premises, but I'm unclear on what exactly makes the weak completeness weak.
Strong completeness of a calculus establishes its suffciency for capturing the logical consequence; namely, whenever a sentence follows logically from a set of hypothesis, there is a proof of this sentence in the calculus. While the weak completeness says that we have proofs for all validities.