On truth and satisfaction, again

Above, we have noted that sentences are either true or false relative to a structure, while open formulas (which are not sentences) are satisfied or not relative to a structure and an assignment. But we also saw that assignments are used in the calculations when we are dealing with quantified sentences. What gives?

In fact, there is no mistake here. How we bridge this apparent problem is by saying that a sentence (whether quantified or not) is true relative to a structure provided it is satisfied under every assignment (for that structure), no matter what. In other words, no matter what assignment you start out with, the sentence comes out satisfied under that assignment relative to the structure. This is trivially true for sentences without quantifiers, since there, assignments are completely superfluous. But as it turns out, the initial assignment is also completely superfluous when it comes to sentences with quantifiers.

We can return to our example $\exists x P(x)$ relative to the structure above. Is it the case that $\mathbf{S} \models P(x)$, regardless of the assignment? In fact, it is, since no matter which initial assignment we take, there will always be an $x$-variant assignment that sends $x$ to Tián Tián. In particular, for any initial assignment $\mathbf{a}$, we can just take $\mathbf{a}^x_\text{Tián Tián}$, which will satisfy $P(x)$ relative to $\mathbf{S}$. And the same goes for the universal quantified sentence $\forall x P(x)$ (and falsity).

Another way to think about this is to note that every assignment assigns a value to every variable, not just $x$ (or $y$, or whatever). But when it comes to a quantifier with a variable $x$, like $\exists x$ or $\forall x$, we are only interested in the possible values $x$ may take, which is where $x$-variant assignments come into the picture. But we can just disregard which values the other variables get, because it is irrelevant for our concerns. So again, the initial assignment drops out of the picture.

These considerations are why, as was briefly noted above, if we are considering a sentence $X$ relative to a structure $\mathbf{S}$, $\mathbf{S} \models X$ or $\mathbf{S} \not\models X$ (without specifying an assignment), and we can say that $X$ is either true in $\mathbf{S}$ or it is false in $\mathbf{S}$. Thus, we can say:

A formula $X$ is true in $\mathbf{S}$, written $\mathbf{S} \models X$, provided under every assignment $\mathbf{a}$, $\mathbf{S} \models X[\mathbf{a}]$. That is, provided under every assignment $\mathbf{a}$, $X$ is satisfied under $\mathbf{a}$ relative to $\mathbf{S}$.

And a formula $X$ is false in $\mathbf{S}$, written $\mathbf{S} \not \models X$, provided under every assignment $\mathbf{a}$, $\mathbf{S} \not\models X[\mathbf{a}]$. That is, provided under every assignment $\mathbf{a}$, $X$ is not satisfied under $\mathbf{a}$ relative to $\mathbf{S}$.