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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."

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  • $\begingroup$ You wrote "$a_1,...,a_m \in M$", so $(a_1,...,a_m)$ are not variable at all $\endgroup$ – Holo Dec 22 '18 at 20:35
  • $\begingroup$ The definition of (*) is that for any interpretation, $σ$, for $\cal M$, we have $σ((\forall y_1)...(\forall y_n) \phi(a_1,...a_m))$ $\endgroup$ – Holo Dec 22 '18 at 20:36
  • $\begingroup$ See D.Marker, Model Theory, page 11 for the definition. $\endgroup$ – Mauro ALLEGRANZA Dec 23 '18 at 11:03

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