i'm stuck for quite a while with those two questions. would appreciate your assistance with them:
1)$\varphi \models \psi$ then $\varphi$ is contradiction or $\psi$ is a tautology, or there exists elementary proposition in both $\varphi$ and $\psi$
2)show that if $\Gamma \:\cup \left\{\varphi \right\}⇒\psi$ and $\Gamma \:\cup \left\{\varphi \right\}⇒\lnot\psi$ then $\Gamma \models\lnot \varphi$
1)i don't see how $\varphi$ can be a contradiction, which means that $\varphi \models\psi$ must be an elementary proposition. however, i don't know how to prove or show that.
2)i don't understand the logic here. if the union gives that both $¬φ$ and $φ$, then it should be $Γ\models φ$. i don't understand that.
thank you very much for your help, maybe you can help me understand what i'm missing here and how to approach it correctly
Hint
I will try to explain the intuition here.
$\varphi \models \psi$ is trivially true if $\varphi$ is a contradiction or $\psi$ is a tautology. (I don’t understand your comment at all... of course $\varphi$ can be a contradiction, it is pretty much an arbitrary statement.) What you are to show is that if neither of these trivial cases hold, then $\varphi$ and $\psi$ must share a proposition. In other words for $\psi$ to be a nontrivial logical consequence of $\varphi,$ it must have something to do with $\varphi.$ This shouldn’t be hard to believe. To prove it you need to show that if the statements don’t share a proposition, $\varphi$ is not a contradiction and $\psi$ is not a tautology, you can find an interpretation where $\varphi$ is true and $\psi$ is false.
The premise is that $\Gamma \cup \{\varphi\} $ is inconsistent. This may be either because $\Gamma$ is inconsistent, or because $\Gamma$ implies $\lnot \varphi$ so adding $\varphi$ yields an inconsistency. Either way $\Gamma\models \lnot\varphi.$ By explosion in the first case, and obviously in the second.