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.
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