Definition of "$x$ occurs free as the $i$th symbol in a wff" In Enderton's logic book he defines the free variables of a wff recursively as


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*For atomic $\alpha$, $x$ occurs free in $\alpha$ iff $x$ occurs in (i.e., is a symbol of) $\alpha$.

*$x$ occurs free in $(¬\alpha)$ iff $x$ occurs free in $\alpha$.

*$x$ occurs free in $(\alpha → \beta)$ iff x occurs free in $\alpha$ or in $\beta$.

*$x$ occurs free in $\forall v_i \alpha$ iff $x$ occurs free in $\alpha$ and $x\neq v_i$ .


But then he has an exercise asking for "a precise definition of what it means for the variable $x$ to occur free as the $i$th symbol in the wff $\alpha$".
Is it just me, or is the definition trivial: $x$ occurs free as the $i$th symbol in $\alpha$ if $x$ occurs free in $\alpha$ and $x$ is the $i$th symbol in $\alpha$.
Are we not able to determine when something is the $i$th symbol? Maybe he wants a recursive definition, but such a definition would start with


*

*For atomic $\alpha$, $x$ occurs free as the $i$th symbol in $\alpha$ iff $x$ occurs as the $i$th symbol of $\alpha$.


But if we can determine the $i$th symbol of an atomic wff, shouldn't we be able to determine the $i$th symbol of any wff, or for that matter any string (just count). So isn't the recursive definition seemingly pointless? I'm a bit confused about the spirit of this exercise.

I've just realized my mistake. A wff can contain $x$ as both a free and bound variable. For example, in  $\alpha=(\forall x Fx\lor Gx)$, the last $x$ occurs free but $x$ as the 5th symbol is not free. Thus even though $x$ is free in $\alpha$ and $x$ is the 5th symbol, we cannot say "$x$ occurs free as the 5th symbol".
