For finite $\Sigma$, if $\Sigma \vdash A$ then $\Sigma \models A$

Question

In my book I have a theorem called "Soundness theorem" and it says:

For finite set $\Sigma$, if $\Sigma \vdash A$ then $\Sigma \models A$

Can someone tell me what the symbol $\vdash$ and the symbol $\models$ means?

Answer 1

The first is a syntatic notion, the second semantic:

• $\Gamma ⊢ \varphi$ means that there is a derivation $\mathcal{D}$ from hypothesis in $\Gamma$ and with conclusion $\varphi$.

• $\Gamma ⊨ \varphi$ means that any model $\mathcal{M}$ of $\Gamma$ makes $\varphi$ true.

Roughly, the soundness theorem says that the inference rules of logic system don't prove falsities.