Full Idea
If every semantically valid inference is proof-theoretically valid (so that |= entails |-), the proof-theory is said to be 'complete'.
Gist of Idea
A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity
Source
Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 1.1.7)
Book Reference
Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.2