I have some question about how to use completeness theorem, I need to show the follow:
Let $\Sigma$ be a consistent set of propositions and $\phi, \psi$ formulas. Show that
- If $\Sigma\not\vdash \phi$ then $\Sigma\vdash\neg \phi$
- If $\Sigma\not\vdash \phi$ then $\Sigma\not\vdash[\phi\lor (\psi\land \phi)]$
I think that by completeness theorem this is equivalent to the following:
Let $\Sigma$ be a satisfiable set of propositions and $\phi, \psi$ formulas. Show that
- If $\Sigma\not\models \phi$ then $\Sigma\models\neg \phi$
- If $\Sigma\not\models \phi$ then $\Sigma\not\models[\phi\lor (\psi\land \phi)]$.
My attempt: The latter is easy to prove,
If $\Sigma\not\models \phi$, then there exists a model $\mathcal{A}$ such that $\mathcal{A}\models \Sigma$ but $\mathcal{A}\not\models \phi$, then $\mathcal{A}\models \neg \phi$, then $\Sigma\models \neg \phi.$
If $\Sigma \not\models\phi$ then there exists a models $\mathcal{A}$ such that $\mathcal{A}\models\Sigma$ but $\mathcal{A}\not\models \phi$ then $\mathcal{A\not\models \phi\land\psi}$, then $\mathcal{A}\not\models \phi\lor (\phi\land\psi)$ and then $\Sigma\not\models [\phi\lor (\psi\land \phi)]$
I appreciate if you tell me if I have any errors, or if I'm right, any help is welcome.