TITLE: When do free variables occur? Why allow them? What is the intuition behind them?
QUESTION [6 upvotes]: In the formula $\forall y P(x,y)$, $x$ is free and $y$ is bound.
Why would one write such a formula? 
Why are free variables allowed? 
What is the intuition for allowing free variables?

REPLY [6 votes]: Let's be clear. You don't need to countenance free variables to set up standard first order logic. [Thus it in fact isn't right to say e.g. "free variables are needed in any case to cope with some trivial syntactic matters, namely building up quantified formulas in an inductive fashion".]
As far as the syntax goes, we needn't allow wffs with free variables. We can have the rule that if $\varphi(a)$ is a wff containing one or more occurrences of the constant $a$, then so is $\forall \xi\varphi(\xi)$ and $\exists \xi \varphi(\xi)$, where $\xi$ is any variable and $\varphi(\xi)$ is of course the result of replacing each occurrence of $a$ in $\varphi(a)$ with $\xi$. Or to look at it another way, we can think of quantification not as a matter of just putting $\forall \xi$ or $\exists \xi$ in front of a wff with free variables, but rather as applying the single operator  $\forall \xi \ldots \xi \ldots $to the predicate $\varphi(\cdot)$.
As far as the semantics goes, the rule is that $\forall \xi\varphi(\xi)$ is true on an interpretation $\mathcal{I}$ just so long as every way of extending $\mathcal{I}$ to assign an interpretation to the new constant $a$ makes $\varphi(a)$ true -- and likewise $\exists \xi\varphi(\xi)$ is true on an interpretation $\mathcal{I}$ just so long there is some way of extending $\mathcal{I}$ to assign an interpretation to the new constant $a$ which makes $\varphi(a)$ true. (Careful! Note this is NOT a substitutional account of quantification: it is NOT the account that goes $\forall \xi\varphi(\xi)$ is true on  $\mathcal{I}$ just if $\varphi(a)$ is true on $\mathcal{I}$ for every constant $a$. Our account of the semantics talks about extensions of $\mathcal{I}$ by assigning references to new constants, which is in the same ball park as talking about extending $\mathcal{I}$ with assignments to variables).
So we can (and some authors do) treat both syntax and semantics without allowing wffs with free variables. Instead we can use "parameters", i.e. additional constants (whose interpretation is not initially fixed). I think there are principled reasons for preferring this approach, but don't need to argue that here: the present point is that it is certainly possible to do things this way.
And seeing that this is so should help understand what is going on when -- rather unfortunately perhaps -- logic texts (always tempted by Bauhaus austerity!) decide not to use new constants or parameters which are typographically distinguished from variables, but prefer (on the grounds of economy) to re-use the same letters both for 'real' variables tying quantifiers to slots in predicates and for 'free' variables serving the role of parameters. This doubling up of roles for your $x$s and $y$s isn't necessary as we have seen, even if it is probably the majority practice (and also reflects some informal mathematical practice). And it can help -- if this doubling seems confusing -- to first tell the syntactical and semantic stories the other way, distinguishing variables and parameters, and then see the doubled-up role for your $x$s and $y$s  as just a typographically economical trick.