Notation for free variables in a wf $\mathscr B$

So I am having some confusion about the notation that is used in "Introduction to Mathmatical Logic" by Elliot Mendelson for indicating that some of the variables are free in a wf $$\mathscr B$$.[Below exercise 2.5]

We shall often indicate that some of the variables $$x_{i_1},...,x_{i_k}$$ are free variables in a wf $$\mathscr B$$ by writing $$\mathscr B$$ as $$\mathscr B(x_{i_1},...,x_{i_k})$$ .This does not mean that $$\mathscr B$$ contains these variables as free variables , nor does it mean that $$\mathscr B$$ does not contain other free variables.

In order for a variable to be a free variable in $$\mathscr B$$ , it has to have an free occurrence in the wf $$\mathscr B$$ atleast once.So if a variable $$x$$ doesn't exist in a wf $$\mathscr B$$ , then it is not a free variable in $$\mathscr B$$.
But , the above passage allows us to indicate these "non existent variables" as free variables in $$\mathscr B$$ by using the notation $$\mathscr B(x_{i_1},...,x_{i_k})$$. Is this right to do?

There is another confusion of mine regarding non-Existent variables. Lets say $$y$$ is a variable that doesn't exist in $$\mathscr B$$ .Then because $$\mathscr B$$ doesn't have $$y$$ as a variable , so we can write $$\mathscr B \equiv \forall y\mathscr B$$ according to this passage.

Notice that $$\mathscr B$$ need not contain the variable $$y$$. In that case, we understand (($$\forall y)\mathscr B$$) to mean the same thing as $$\mathscr B$$

In this case , $$y$$ is bound in $$\forall y \mathscr B$$ .And because $$\mathscr B \equiv \forall y\mathscr B$$ , we can say that $$y$$ is bound in $$\mathscr B$$.Is this faulty?

• It is only a convention. See also van Dalen – Mauro ALLEGRANZA Feb 23 at 7:35
• If it is disturbing for you, you can imagine $B':=B\land(x_1=x_1\land x_2=x_2\land\dots)$. – Berci Feb 23 at 8:18
• Again, the issue is the difference between a formula of the language: $A_1^2(x_1,x_2)$ and a schematic expression of the meta-language. Having said that, see page 49: "An occurrence of a variable $x$ is said to be bound in a wf $\mathscr B$ if either it is the occurrence of $x$ in a quantifier “$(∀x)$” in $\mathscr B$ or it lies within the scope of a quantifier “$(∀x)$” in $\mathscr B$. Otherwise, the occurrence is said to be free in $\mathscr B$." Thus, if var $x_1$ does not occur in the formula it is neither free nor bound. – Mauro ALLEGRANZA Feb 23 at 10:13
• Mendelson's convention is rather loose but suggestive: $\mathcal{B}(x_{i_1}, \ldots, x_{i_k})$ is just an indicator that some of the $x_{i_j}$ may appear free in $\mathcal{B}$. This allows him to write $\mathcal{B}(t_1, \ldots, t_k)$ for the result of substituting a list of terms $t_j$ for the $x_{i_j}$, which makes it reasonably clear what is substituted for what, but doesn't imply that all the $x_{ij}$ actually appear free. (So he can describe substitution of $1$ for $x$ and $2$ for $y$ in $x = x \land z = 3$, even though $y$ doesn't appear free in the formula, while $z$ does.) – Rob Arthan Feb 23 at 11:31
• If $y$ is not free in $\mathcal{B}$, then $\forall y \mathcal{B}$ and $\mathcal{B}$ have the same meaning (they are logically equivalent) but they are not syntactically identical. You can write $\mathcal{B} \equiv \forall y \mathcal{B}$ (if you are using $\equiv$ for logical equivalence, which I don't think Mendelson does), but it doesn't mean that the two formulas have the same syntactic properties and whether or not a variables occurs bound in a formula is a syntactic property. – Rob Arthan Feb 23 at 12:38