[dvt]

This is actually wrong.. There's an old joke in logic: sound, complete, consistent -- pick two :)

I think you're misunderstanding what soundness is. Soundness means that a proof in that logic implies a (semantic) entailment in that logic. That is:

  Γ ⊢ A ⇒ Γ ⊨ A

It's a very specific characteristic.

[mafribe]

Sorry, I mean inconsistent where I wrote sound.