TITLE: What is the purpose of free variables in first order logic?
QUESTION [27 upvotes]: I understand the difference between free and bound variables, but what are free variables actually useful for? Can't you use quantifiers to express everything that you would want to express with both bound and free variables? I am a little frustrated that the ever-present possibility of free variables complicates introducing quantifiers into FOL statements, while I can't find any motivation for the free variables.  
Say you have a relation $R$, arity $2$. If you want to express that, for every x and for every y, $R(x,y)$, I guess you would write:
$$\forall_x\forall_yR(x,y)$$
Maybe you could also formulate this as:
$$R(x,y)$$
Since it seems that in order for this to be a valid proposition in FOL, you would have to capture all free variables with a universal quantifier. But this means that formulae with free variables are equivalent to closed formulae where these free variables have been captured by universal quantifiers, if you want to find out the validity of that formula. So what is the purpose of free variables when it comes to proving formulae in FOL?

REPLY [4 votes]: If one does not consider formulas with free variables, then one cannot write a proof by induction on the formation of formulas.  Every first-order formula is constructed from primitive formulas by either putting a quantifier in front of a formula with a free variable, thereby binding the variable, or applying operations like "and", "or", "not", "if...then...", etc., to other formulas.  In the inductive step in a proof by induction you show that if the thing you're trying to prove is true of two formulas $A$ and $B$ then it's true of $[A\text{ or }B]$ and it's true of $\exists x\ B$, and so on.
Łos's theorem says that if a first-order statement is true in "almost every" factor in an ultraproduct, then it's true in the ultraproduct.  Łos's theorem is proved by induction on the formation of first-order formulas.