Predicate Logic: Truth in a Structure Under a Variable Assignment

Question

Let $A$ be some predicate logic structure, let $a$ be any variable assignment (over $A$), and let $\Phi$ be a sentence of predicate logic.

My instructor has said that we should not hold the following:

(1) $\vert\forall v\Phi\vert$ is true in $A$ iff, for every $a$, $\vert\Phi\vert$ is true in $A$ under $a$.

Instead, we should hold:

(2) $\vert\forall v\Phi\vert$ is true in $A$ iff, for every $a$, $\vert\forall v\Phi\vert$ is true in $A$ under $a$.

He mentioned that it has something to do with sentences that contain multiple quantifers, e.g., $\forall x\exists yRxy$.

I can't figure out why (1) doesn't do the job. What's wrong with taking, for instance, $\vert\forall x\exists yRxy\vert$ to be true in $A$ iff, for every $a$, $\vert\exists yRxy\vert$ is true in $A$ under $a$? This seems to be the intuitive thing to do, no?

Answer 1

"$|\forall v \phi|$ is true in $A$" expresses truth in a structure without reference to a particular variable assignment. Since the semantics defined inductively on the structure of the formulas is always defined relative to assignments ("$|\phi|$ is true in $A$ under $a$ iff ..."), one has to define what global truth in a structure (without reference to a partiuclar $a$) means. The definition of that is just truth under all assignments:

(a) $|\phi|$ is true in $A$ iff it is true in $A$ under every variable assignment $a$.

Note that this definition applies to arbitrary formulas $\phi$, not just quantified ones.

After decomposing the meaning of global truth in a structure, one can expand the usual semantics clauses defined relative to assignments as usual:

(b)
(b-i) $|P(t_1, ..., t_n)|$ is true in a structure $A$ under variable assignment $a$ iff $\langle |t_1|, ..., |t_n| \rangle \in \mathcal{I}(P)$
$\vdots$
(b-vii) $|\forall v \phi$| is true in $A$ under $a$ iff for every variable assignment $a'$, $|\phi|$ is true in $A$ under $a'$

So the first step when evaluating $|\forall v \phi|$ in $A$ is to unpack the definition of "global" truth, which yields your (2):

1. $|\forall v \phi|$ is true in $A$ iff for every $a$, $|\forall v \phi|$ is true in $A$ under $a$ (by rule (a))

Then, once one has the $a$ back in place, one can start to unpack the universal quantifer by rule (b) and run the following rule on every assignment $a$:

2. $|\forall v \phi|$ is true in $A$ under $a$ iff for every $a'$, $|\phi|$ is true in $A$ under $a'$ (by rule (b-vii))

(1) would be wrong for the reason that one would skip the definition of truth in a structure $A$ and jump directly to the evaluation of the universal quantifier under some assignment $a$ which is, at that point, not yet given, but only enters trough the implicit quantification over all assgignments triggered by "true in $A$".