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 $c_{a_1}, \dots, c_{a_n}$?
Best Answer
There is a subtle difference between constant symbols in a language and the elements of a structure.
Let's start with a language, the language of groups. So we have a binary function symbol $m(x,y)$ for multiplication and a constant $e$ for the identity element. Any structure in this language should assign an element to this constant (and an operation to the function symbol). For example, in the additive group $\mathbb{Z}$ the element $0$ is assigned to $e$, but in the multiplicative group $\mathbb{Q} - \{0\}$ the element $1$ is assigned.
Now jump to satisfaction of sentences. If $\mathcal{M}$ is an $\mathcal{L}$-structure and $\varphi(x_1, \ldots, x_n)$ is an $\mathcal{L}$-formula, then of course, it would not really make sense to write $\mathcal{M} \models \varphi(x_1, \ldots, x_n)$ (see also footnote). In our example of the language of groups we could take the formula $m(x_1, x_2) = e$, what would $\mathcal{M} \models m(x_1, x_2) = e$ mean? Of course, this makes sense once we plug in elements from $\mathcal{M}$ for the free variables $x_1$ and $x_2$.
For example, for the multiplicative group $\mathbb{Q}$, asking whether or not $\mathbb{Q} \models m(1/3, 3) = e$ makes perfect sense. But we have to replace the symbols in our language by their interpretations first to give a final answer to this question. Doing that we arrive at "$1/3 \cdot 3 = 1$", which is true, so indeed $\mathbb{Q} \models m(1/3, 3) = e$. Note that "$m(1/3, 3) = e$" is no longer a formula (or sentence) in our language, so it will generally not make sense in any other structure (e.g. it does not make sense in $\mathbb{Z}$). But when replacing free variables by elements, and replacing all symbols by their interpretations, we can turn a formula into a statement that can be true or false in our structure. This is what $\mathbb{Q} \models m(1/3, 3) = e$ means, or more generally $\mathcal{M} \models \varphi(a_1, \ldots, a_n)$.
What often happens in model theory, is that we wish to fix some elements of a structure as parameters and act as if they were in the language in the first place. Formally what happens is this. Let $\mathcal{M}$ be an $\mathcal{L}$-structure.
Now comes the magic of this construction, which tells us why $c_a$ and $a$ are often used interchangeably. Even though they technically are different things! So read carefully where everything lives.
What the above shows is that, viewing $\mathcal{M}$ as an $\mathcal{L}_\mathcal{M}$-structure, we have $$ \mathcal{M} \models \varphi(c_{a_1}, \ldots, c_{a_n}) \quad \Longleftrightarrow \quad \mathcal{M} \models \varphi(a_1, \ldots, a_n). $$ So the subtle difference between elements and constant symbols disappears in this way. Which is why many authors will not distinguish between them.
Footnote: some authors use $\mathcal{M} \models \varphi(x_1, \ldots, x_n)$ as an abbreviation for $\mathcal{M} \models \forall x_1 \ldots x_n \varphi(x_1, \ldots, x_n)$, which does make sense since then there are no longer free variables.