Soundness property, to my knowledge is the property that:
$\Gamma \vdash \varphi \implies \Gamma \vDash \varphi$
If $\varphi$ is provable (a syntactic consequence) from $\Gamma$ then $\varphi$ is also a semantic consequence of $\varphi$, which I believe is saying $\varphi$ is "true".
But what if, for example, $\varphi \in \Gamma$ and $\varphi = \bot$? It appears conceivable that some of the assumptions in $\Gamma$ are false, and then we might be able to prove things from it, but semantically they would be false.
It's possible I just have the definition of soundness wrong but how is this accounted for? We would normally say that the Hilbert system is both complete and sound but is this still the case even if we begin with a $\Gamma$ that contains some false premises? Or is it "sound only in certain cases"? How does this work?
If $\bot\in \Gamma,$ then we have $\Gamma\vdash \bot$ and $\Gamma \models \bot,$ so this is not in conflict with the soundness theorem. It is clear that $\Gamma \vdash \bot.$ The reason $\Gamma \models \bot$ is that, since $\bot\in \Gamma,$ there are no interpretations in which all the sentences in $\Gamma$ hold, i.e. no interpretations satisfying $\Gamma$. Hence, vacuously, $\bot$ holds in every interpretation satisfying $\Gamma$.