I'll first quote the problem from 'Computability and Logic' by Boolos, Burgess, Jeffrey.
9.3 Consider a language with a two-place predicate P and a one-place predicate F, and an interpretation in which the domain is the set of persons, the denotation of P is the relation of parent to child, and the denotation of F is the set of all female persons. What do the following amount to, in colloquial terms, under that interpretation?
(a) $\exists z\exists u\exists v(u \neq v \ \land \mathbf Puy\ \land \mathbf Pvy\ \land \mathbf Puz\ \land \mathbf Pvz\ \land \mathbf Pzx \ \land \lnot \mathbf Fy )$
I left out b, since it's quite similar to a. Now, I understand how to make a colloquial English sentence from a statement like this: $\forall a \forall b \ (\mathbf Pab \implies \mathbf \lnot \mathbf Pba)$. For example, this could be 'No one is a parent of his or her parent.' But in the problem above, x and y are unbound variables. How do I deal with this? Are they perhaps implicitly meant to be universally bounded? Thanks in advance.
If you have unbound (free) variables, then you should just refer to those variables.
That is, just like we define predicates by sayhing something like:
$P(x,y)$: '$x$ is a father of $y$'
we can likewise express complex formulas with free variables, e.g.
$\exists y(P(x,y) \land P(y,z))$: 'There is someone who has $x$ as a father, and who is a father of $z$'