# The meaning of $\mathcal M \vDash (\forall y_1)…(\forall y_n) \phi(a_1,…a_m)$

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."

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