Soundness Theorem

Soundness Theorem

If $\Sigma$ has a model then $\Sigma$ is consistent.

Soundness Theorem

If $\Sigma$ is a set of formulas and $\phi$ is a formula(all in the same language $\mathcal{L}$), then $\Sigma ⊢\phi$, implies $\Sigma \models \phi$

Remark

Together with Completeness Theorem Version 2, we get $\Sigma$ has a model iff $\Sigma$ is consistent, moreover, with $\Sigma$ has no model $⟺\Sigma \models \perp$, we get $\Sigma \models \perp ⟺\Sigma ⊢\perp .$ $\Sigma \models \phi ⟺\Sigma ⊢\phi .$