How to semantically show this statment?
$$\varphi \models (\forall x)\varphi$$ I'm very new to first order logic and I don't even know if this statment holds or not. What does the symbol $\models$ mean in this case? I have read that quantifiers do not change truth value (if there are no free variables) but I am not sure what exactly this statment says.
The symbol $\vDash$ means entailment or Logical consequence.
The proof relies on the details of the semantics specifications.
According to one approach (see: Dirk van Dalen, Logic and Structure, Springer (5th ed. 2013), page 67) we have that the definition of meaning and truth value is limited to sentences, i.e. "closed" formulas.
For open one, the following convention is adopted:
$\mathfrak A \vDash \varphi$ iff $\mathfrak A \vDash \text{Cl}(\varphi)$,
where $\text{Cl}(\varphi)$ is the universal closure of $\varphi$.
Having said that, we have that $\Gamma \vDash \psi \text { iff (if } \mathfrak A \vDash \Gamma \text {, then } \mathfrak A \vDash \psi)$, where $\Gamma$ and $\psi$ are sentences.
If so, the proof of $\varphi \vDash (\forall x) \varphi$ is trivial, because it amounts to:
if $\mathfrak A \vDash \text {Cl}(\varphi)$, then $\mathfrak A \vDash(\forall x) \varphi$.
But if there are no free variables, then $\text {Cl}(\varphi)=\varphi$.
