Can the "$\forall x\in X $" be moved in this statement? "$\Gamma$ satisfiable $\implies \exists v:v(\alpha)=1 \forall \alpha \in \Gamma$".

Question

Can the "$\forall x\in X $" be moved in this statement? "$\Gamma$ satisfiable $\implies \exists v:v(\alpha)=1 \forall \alpha \in \Gamma$".

I mean, is this the same than to write "$\Gamma$ satisfiable $\implies \exists v: \forall \alpha \in \Gamma\text{ }v(\alpha)=1$"?

Answer 1

In a formal language, we have to follow the syntactical rules; thus a f-o formula must be written (as usual) with quantifiers in front of the sub-formula which is their scope.

When we mix symbols and natural language, as in your case, where you are expressing a meta-mathematical fact :

"if a set of formulae is satisfiable, then there exists a valuation that satisfy all formulae in that set"

we have more freedom, but in any case we have to improve readibility without introducing ambiguities.

Thus, if we want to use : $∃v:v(α)=1, ∀α \in Γ$, we have to be aware that someone may read it as : $∀α \in Γ, ∃v:v(α)=1$.