OK to kick off, here's the little paragraph I had to put together with my topic for my qualifying exams next semester, to give some sense of the general scope of the project.
General topic: philosophy of logic, with a focus on dialetheism and the semantic and set-theoretic paradoxes
The semantic paradoxes, like the Liar and it's multi-sentence equivalents, are the best-known cases of prima facie true contradictions. The set-theoretic paradoxes seem to constitute equally good evidence for dialetheism. An artificial, purely formal solution is much more widely accepted in the set-theoretic than in the semantic case, but this sort of move seems equally arbitrary in either case. Explosion-based arguments against dialetheism are radically question-begging for familiar reasons, and at best demonstrate the existence of too many true contradictions, as opposed to none at all. In fact, this begins to look like an innate structural imbalance in the argument, since any argument either for or against true contradictions must take place either within a logical context that tolerates contradictions or within one that rules them out. A dialetheist can argue for true contradictions without begging the question by generating them within contexts in which they are supposed to be ruled out, but it is hard to see what sort of parallel move within enemy territory would be possible for those on the anti-dialetheist side. Arch-dialetheist Graham Priest also identifies candidates for the status of true contradictions in the philosophy of law (in which citizens can have real, but contradictory, legal rights and obligations) and in the metaphysics of change. Given the scope and rigor of the Priest's case and the unsatisfactory nature of many of the existing attempts to defuse the semantic and set-theoretic paradoxes which provide the best evidence for dialetheism, there should definitely be enough material here for a dissertation-length argument against true contradictions.
After Turkey-murdering-pumpkin-pie-and-whitewashing-genocide-day*, we'll start off with a quick look at how *not* to argue against the possibility of true contradictions.
*sometimes alternatively referred to as "Thanksgiving"