The property that a logic has when the only truth-values are "True" and "False" and every well-formed formula must be one or the other is called "Bivalence". There are a number of ways to deny bivalence.
Two of the most popular options are to recognize the existence of truth-value "gaps" or truth-value "gluts" (or both).
Accepting gaps, as the name would suggest, is to accept that there are some statements which are neither true nor false--- they are indeterminate. You could treat indeterminacy as a third truth value, or as simply the lack of truth or falsity. As far as I know there isn't much of a philosophical reason to preference one approach or the other. The difference corresponds to a choice of semantic apparatus. If you take the assignment of truth-values to statements to be functional then you treat indeterminacy as a third truth-value. If the assignment of truth-values to statements is relational then you can avoid positing a third truth-value by simply having some statements relate to neither truth nor falsity.
The issues are much the same for truth-value gluts except that accepting gluts means that you accept some statements are both true and false. Dialetheism is the philosophy most commonly associated with the existence of truth-value gluts.
For further information of these issues and exploration of technical systems of both sorts I recommend Graham Priest's An Introduction to Non-Classical Logic. The book presents these systems in a tableuax style of proof, which combines sytactic and semantic proof. It has the disadvantage of being somewhat unnatural (it doesn't resemble normal reasoning as much as, say, a natural deduction system) but it is rather easy to get accustomed to.
EDIT: Since I can't comment yet, I'll note here that as the other answerer indicates, it is unclear whether or not Aristotle accepted that there were truth-value gaps (he almost certainly didn't accept gluts). One of the areas where it seemed that he accepted truth-value gaps is in his discussion of future contingents. You can find discussion of this issue in his De Interpretatione.
EDIT 2: I just noticed that you were specifically asking for authors, so here are some:
Graham Priest for Dialetheism/Truth-Value Gluts
L.E.J. Brouwer and other intuitionists for Truth-Value Gaps
Petr Hajek on Fuzzy Logic (i.e., infinitely many valued logic)
Saul Kripke on his theory of truth and the semantic paradoxes (gaps)
Hartry Field on the semantic paradoxes (gaps)
Those are the big figures I can think of off the top of my head.