A footnote. Let's say that a logical system $S$ allows truth-value gaps if for some $S$-sentences (closed wffs without free variables) and some $S$-valuations, a sentence comes out neither true nor false. And a logical system $S$ allows truth-value gluts if for some $S$-sentences and some $S$-valuations, a sentence comes out both true and false.
As other answers have noted, the text-book classical version of first-order logic doesn't allow truth-value gluts. Nor does it allow truth-value gaps. But there are other first-order logics.
It is worth saying, then, that there are perfectly sensible first-order logics (logics for regimenting first-order quantificational inference) which allow gaps. For suppose we drop the requirement that every term denotes (after all, we may use names in ordinary language which, by misadventure, fail to denote: and in maths we do use partial functions so that $\varphi_e(n)$ can fail to denote). Then it is natural to say that an atomic sentence $Ft$ lacks a truth-value when $t$ fails to denote. And we can develop free versions of first-order logic (versions free of the assumption that all terms denote) based on this natural idea. See, for an accessible review, http://plato.stanford.edu/entries/logic-free/
You can also -- perhaps more surprisingly -- develop systems that allow truth-value gluts. Obviously if these are to be non-trivial, we'll have to drop the explosion rule that from $A$ and $\neg A$ we can infer any $C$. But we might want to drop explosion anyway. Why should we allow gluts? Well, it is one way to go with the paradoxes. Add a truth-predicate $T$ to a theory of arithmetic, lay down some plausible rules governing it, and (oh dear!) you find that -- because you can do some Gödelian self-reference-via-coding -- you can construct a liar sentence L where $T(L)$ comes out both true and false. What to do? Some have gone for biting the bullet and said, yep, some special sentences are both true and false, and we can assert this without the contagion of apparent paradox spreading everywhere. And (as I said, perhaps surprisingly) this idea can be developed into a coherent logical system. I'm not recommending it, but it can be done! For an accessible review see http://plato.stanford.edu/entries/dialetheism/ and http://plato.stanford.edu/entries/logic-paraconsistent/