Here |= means logical entailment, which is a semantic notion, while |- means provability, which is a syntactic, i.e. algorithmic notion. Completeness says that if A is true, then it can be proved eventually.
Given KB and A, it may be the case that KB |= A is not true and also KB |= ~A is also not true. Completeness says nothing about this case, where neither A nor ~A is entailed by KB, that is KB leaves the truth value of A unsettled.
Definition: The proof algorithm |- is refutation-complete if whenever KB |= A then KB union { ~A} |- false eventually, in finite time.
Theorem: Resolution is refutation-complete.
This theorem says that if KB entails A, then resolution will demonstrate this fact eventually. Again, nothing is said about the case where neither A nor ~A is entailed by KB.
Definition: The entailment relation |= is semi-decidable if a complete proof algorithm exists for it, while it is decidable if a proof algorithm exists that is complete and also always terminates in finite time.
Gödel's completeness theorem: First-order logic entailment |= is
semi-decidable.
Gödel's incompleteness theorem: First-order logic entailment
|= is not decidable.
Corollary: If A is not entailed by KB then resolution applied to KB union {~A} may fail to terminate.
Definition: A control strategy for resolution is complete if its use preserves refutation-completeness, i.e. if false can be proved, it can be proved while respecting the strategy.