Answer:

1. If he was indeed Cosian his belief would be true, and therefore he would have to be lucid in order to have held such a belief. If he was Tyrosian his belief would be false, and once again, he would have to be lucid in order to have held such a belief. Therefore, we cannot say for sure whether he was Cosian or Tyrosian, but we do know that he was lucid.
2. In similar fashion to the above case, if he was indeed delirious then his belief would be true, and he would have to be Tyrosian, as a Cosian cannot believe true things while he is delirious. If he was in fact lucid, his false belief implies that he is Tyrosian. Hence, we cannot tell whether he is delirious or not, but we can tell that he is Tyrosian.
3. We have seen above that a man who believes he is delirious must be Tyrosian, and it can easily be shown similarly that a man who believes he is lucid must be Cosian. Hence when he believes he is Tyrosian, it is plain to see that he must be delirious.
4. The proposition "If I am a lucid Cosian then I am the Ubar" must itself be true or false. It can, in fact, only be false if Bortus is a lucid Cosian and is not the Ubar. But if Bortus is a lucid Cosian then the proposition must be true, or else Bortus could not have thought it. Hence the proposition is true. This means that Bortus must be either a lucid Cosian (and hence the Ubar) or a delirious Tyrosian (in which case we have no idea whether or not he is the Ubar).

Now by similar logic, Kortus's belief can only be false if Kortus is not the Ubar and is either a lucid Cosian or a delirious Tyrosian. But if he were either a lucid Cosian or a delirious Tyrosian, his belief would have to be true, as he could not have believed it otherwise. Hence his belief is true and so he must be either a lucid Cosian or a delirious Tyrosian to have believed it. In either case, by his proven-true belief, he is the Ubar. Hence Bortus is not (as there is only one Ubar) and so Bortus must be a delirious Tyrosian. Hence Kortus is also a delirious Tyrosian.

Sadly, it is of course extremely unlikely that the poor Ubar could do anything useful with the knowledge...

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