Understanding the Relationship Between Natural Language Predicates and First-Order Formulae Consider the class $C$ defined in terms of first-order formula $\phi(x,p_1, \ldots , p_n)$.  That is, $C = \{x : \phi(x, p_1, \ldots , p_n)\}$.  
Now this is all well and good, but I'm trying to understand how we can use the literal formulae of first-order logic to define classes that have meaning to us.
So here is a natural language predicate: $IsEven(x)$.
Now how could we define $C$ to be the set of all even numbers using only formulas of first-order logic?  I'm not sure how we could do this in first-order logic, given it would seem hard to define the set of even numbers using only the relation $\in$.
 A: There are two interpretations to your question. The first is, in the case of the integer for example, that we can represent the structure we care about using $\in$ directly. The second is when we can't, and we need to define a new language.
For the first we need to have a good definition for the integers (or our otherwise universe of discourse). For example, the finite von Neumann ordinals.
So we have a formula defining the set of finite ordinals, also dubbed as $\omega$. And we have a formula $\varphi(x,y,z)$ which says that $x,y,z$ are finite ordinals and $x+y=z$. Yes, we have a good sense of ordinal addition too, and we can easily restrict that to finite ordinals for this purpose. This formula would say something like "there exists a bijection from the disjoint union of $x$ and $y$, onto $z$" (and all three are finite ordinals).
Finally, $$\text{IsEven}(x)\stackrel{\rm def}{=}\exists y\varphi(y,y,x)$$
So $x$ is an even number if $x=y+y$ for some number.
Note that I don't dare to write this formula in explicit details down to the last $\in$. It's much easier to write it in parts. What is an ordinal, what is a finite ordinal, what is a bijection, a disjoint union, and so on. Those are a bit easier. Then you mix them all in a big jug...

For the second case, where we want to model something like "All green ducks are tasty", in which case there is no "natural" way of interpreting ducks, being green and being tasty as properties of sets.
In that case, normally, we would define a vocabulary, $D,G,T$ are unary predicates and so on. But now we want to do that with sets in the language of set theory. The idea is that we really just model the whole concept of logic as some sets, we don't care about which ones. We just care about their existence. This is exactly why replacement and separation axioms allow parameters. Then we can use any set we would like, and it's enough that there is one which satisfies our requirements.
So what is it that we do? Well, we pick any set which is of the right cardinality, and that would be the set of all symbols we are allowed to use. We would need $\land,\lor,\forall,\exists,\lnot,=,\ldots$ and any other symbol which may appear in our language (including variables!). That usually ends up as some infinite set whose exact contents depend on your preferences. Often we would require that the language will actually be composed of several disjoint sets: one for variables, one for "logical symbols" (connectives, etc.), one for relation symbols, another for function symbols, and last one for constant symbols. Note that we can do all sort of manipulations and end up with just relation symbols if we like.
Now, what is a formula in this language? It's just a finite sequence of symbols, but it has to be well-formed, so it's a finite sequence which was built inductively from the atomic formulas, using well-formed terms and whatnot. In the language of set theory we can describe these things, we can write a formula which tells us when a string of symbols from our vocabulary is a well-formed formula. So the set of well-formed formulas is a well-defined set (with parameters, of course, we need the parameters of the language) -- all thanks to separation.
As before, this is quite a difficult and long formula, but it's there. And it's much much easier to prove a more general theorem about the expressibility of inductive definitions, rather than to write them one by one. Note that there isn't a "unique way to do that", and this means that you just need to show that there is a way to do it. We hardly ever care about the exact algorithm or formulas which define the well-formed strings, or the sentences, or whatever.
So now the sentence we want to write "All green ducks are tasty" is a sequence which abides the rules of formations of well-formed formulas, and now we can represent this as a set -- because sequences are also sort of sets themselves.
So there is no concrete and obvious way of doing that, and it takes a lot of hard work to write these things by hand, and you have to develop a lot of theory first. But once you do, you know that you can and then we just skip to the part that we do.
In fact, we almost always ignore this part when we do mathematics. We assume that someone, somewhere, in some dark basement sat all night and verified this for us, and we just write things. Even more so when everything is finite (e.g. the set of relation symbols is finite, and so on), then we can directly manipulate strings on a piece of paper and we can ignore the fact that we can (if we like to) use set theory to model these things.
