# Completeness

Definition. A Proof Theory is **complete** if whenever $P_{1},P_{2},…,P_{n}⊨Q$ (truth/valid) then $P_{1},P_{2},…,P_{n}⊢Q$ (proof).

Reminder from Logic: $∣−$ -> Proof Theory, pronounced “proves” $∣=$ -> Semantics, pronounced “entails” / “valid” / “semantic entailment”

### Related

I am still confused at how these two are related. https://math.stackexchange.com/questions/3536805/what-is-true-about-a-proof-system-that-is-complete-but-not-sound

If a proof system is unsound and complete, then all the semantically valid inferences can be proven, but in addition it proves some sequents that are not actually valid.