The notation p ⊨ q is read ``p entails q''; it means that q holds in every model in which p holds.
The notation p ⊢_{m} q means that q can be derived from p by some proof mechanism m.
A proof mechanism m is sound if p ⊢_{m} q &rarr p ⊨ q.
A proof mechanism m is complete if p ⊨ q &rarr p ⊢_{m} q.
Resolution for predicate calculus is:
We generally are not willing to give up soundness, since we want our conclusions to be valid. We might be willing to give up completeness: if a sound proof procedure will prove the theorem we want, that is enough.