Any proof of the Completeness Theorem consists always of two parts.

First we have show that all formulas that have a proof are tautologies. This implication is also called a Soundness Theorem, or soundness part of the Completeness Theorem.

The second implication says: if a formula is a tautology then it has a proof. This alone is often called a Completeness Theorem. In our case, we call it a completeness part of the Completeness Theorem.

— Cse371, Math371, LOGIC, Fall 2011

— Professor Anita Wasilewska

2012.09.14 Friday ACHK