Model theory substitution notation Let $\mathcal M$ be a structure in language $\mathcal L$ and let expanded language $\mathcal L_\mathcal M$ be the language with added constants $c_a$ for $a \in \mathcal M$.
I thought I understood notation but I keep finding myself questioning if my understanding is actually accurate. We write $\phi(a_1, \dots, a_n)$ as shorthand notation for the substitutions $\phi[x_1 / c_{a_1}, \dots, x_n / c_{a_n}]$.  Now if we write $\mathcal M \models \phi (a_1, \dots, a_n)$, what does this precisely mean? It was my understanding that for satisfaction, ex. $\mathcal M \models \psi$, $\psi$ is a sentence so all variables are bound and none can be substituted. Does this mean before the substitution $\phi$ has free variables $x_1, \dots, x_n$ and after the substition $\phi(a_1, \dots, a_n)$ is a sentence?
On a related note, what does $\phi(c_{a_1}, \dots, c_{a_n})$ in the expanded language mean? Is it the same result as $\phi(a_1, \dots, a_n)$, that is a sentence with constants $c_{a_1}, \dots, c_{a_n}$?


Does this mean before the substitution $\phi$ has free variables $x_1, x_2, \ldots, x_n$ and after the substitution $\phi(a_1, \ldots, a_n)$ is a sentence?

Yes, that's precisely what it means.

On a related note, what does $\phi(c_{a_1}, \dots, c_{a_n})$ in the expanded language mean? Is it just indicating that $\phi$ has constants $c_{a_1}, \dots, c_{a_n}$ ?

That notation is kind of confusing.  I would have assumed that $a_i$ are constants themselves, not indices to constants.
