Instantiation vs substitution in Smulyan's First order Logic

Question

I am learning logic by reading R.M. Smullyan's First Order logic. I have included images of the relevant pages below.

On page 44 he defines: a formula $A$ is closed if for every variable $x$ and every parameter $a$, $A_{a}^{x}=A$ (i.e. no variable has a free occurrence in $A$).

Then on page 47 he considers the set $E^{\cup}$ of all closed $\cup$-formulas and states that for every $A$ in $E^{\cup}$ and every variable $x$

1. $\nu$ is a Boolean valuation of $E^{\cup}$.
2. (a) $(\forall x)A$ is true under $\nu$ iff for every $k\in\cup$, $A_{k}^{x}$ is true under $\nu$.

This is what confuses me: Since $A\in E^{\cup}$ then $A$ is closed, which means that no variables occur freely in $A$, i.e. $A_{k}^{x}=A$. Also, since $A$ is closed $(\forall x)A$ is simply $A$. From this (a) appears to be: $A$ is true under $\nu$ iff $A$ is true under $\nu$.

On the other hand, (a) intuitively seems to mean that $(\forall x)A$ is true under $\nu$ when every instance of it in $\cup$ is true under $\nu$. If $x$ occurs freely in $A$ then (a) seems to coincide with this meaning. But it is stated clearly that $A$ is closed.

For example, take $A:=\forall x Rxx$, with $\cup=\mathbb{N}$ and $R:=\leq$.

What is the proper way to understand this? The expression $A_{a}^{x}$ seems to sometimes mean substitution of free variable for element of $\cup$ while others instantiation in $\cup$. Thank you.

Pages 43-47

Answer 1

Denoting by $A$, Smullyan talks about a formula in a generic sense (atomic/compound, open/closed, parametric/non-parametric). So, for example, $A(a/x)$ —this is more widespread and preferable than Smullyan's notation— denotes the resultant formula obtained by substituting $a$ for every free occurrence of $x$ in $A$. Thus, let $A$ be the closed formula $$(0<a<1)\rightarrow (a^{2}<a)$$ for some defined parameter $a$. Take notice that this is nothing but, with an abuse of notation for illustration, $$(0<[a/x]<1)\rightarrow ([a/x]^{2}<[a/x])$$

Hence, $A = A(a/x)$.

The same idea applies to the case parameters replaced by individual constants. In other words, Smullyan goes back and forth between two types exemplified by the following statements:

1. Socrates is mortal.
2. $\ldots$ is mortal, where '$\ldots$' is substituted by 'Socrates'.

Originally, the term sentence is used as an alternative for closed formula, because it does not contain any '$\ldots$' just as it is in natural language.

As a side note, quantifying over closed formulas, called vacuous quantification, is legitimate in the majority of logical systems.