This is a question about the meaning of a mathematical sentence.
Let $\mathcal M$ be a structure with universe $M$. Let $a_1,...,a_m \in M$. Let $\phi(x_1,...,x_m,y_1,...,y_n)$ be a quantifier-free formula. What does this mean:
$\mathcal M \vDash (\forall y_1)...(\forall y_n) \phi(a_1,...a_m)$ $(*)$
Here is the definition of satisfaction,
$ \mathcal M \vDash \psi$ if every sequence $(a_1,a_2...)$ from $M$ satisfy $\psi$.
Also
$ \mathcal M \vDash \forall(x_i)\psi$ if every sequence that differs from $(a_n)$ in at most the $i^{th}$ component satisfy $\psi$.
I have two question actually.
1) Are $(a_1,...,a_m)$ the free variables in $\phi$? What if $\phi$ is a sentence?
2) Can you give a definition to $(*)$? What I mean by definition is, "$(*)$ happens if this and this and this happens."