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?


1 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 $A \vDash \psi$.


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