 The Tarskian model theory for pure quantificational logic gives us exactly what we want. Since the axioms of pure quantificational logic are valid and the rules of inference preserve validity, all theorems of pure quantificational logic are valid.  (Soundness).[3]  One may in addition, prove that all valid formulas are, in fact, derivable from the axioms of pure quantificational logic. (Completeness). The entry on  classical logic  outlines a proof.