Help needed in task involving structures and models

Question

The task is as follows:
Let $\Sigma$ be a set of closed formulas such that for any closed formula $\phi$, either $\Sigma\models\phi$ or $\Sigma \models \neg\phi$.
Now let structure $A$ be a model of the set $\Sigma$. Show that for any closed formula $\psi$,

$$A\models \psi \iff \Sigma \models \psi $$

(Closed formula means here a formula such that it's variables are bound with quantifier)

I know that since structure $A$ is model of a $\Sigma=\{\phi_1,\phi_2\dots\}$, it follows that $\Sigma\models\psi$, since all formulas are sharing the same model. Not sure whether this is correct or formal enough.

Also don't know how to prove it to the other direction.

Any tips?

Answer 1

For the first part, we have to use the definition of $Σ \vDash \psi$:

$\text { for every structure } A, \text { if } A \vDash \phi_i, \text { for every } \phi_i \in \Sigma, \text { then } A \vDash \psi$.

Thus, we have that $\Sigma \vDash \psi$ implies $A \vDash \psi$, for a structure $A$ that is a model of $\Sigma$.

For the other part: assume $Σ \nvDash \psi$.

This means $Σ⊨¬ψ$, by property of $Σ$, and this implies that in every structure $A$ that is a model of $Σ$ we have that: $A⊨¬ψ$, contradicting the fact that $A \vDash \psi$.