Is there a place for a third truth value?
There are two uses for a third truth value: one is the ontological trash bin; the other uncertainty. They are worlds apart.
The trashbin is filled with sentences like “The present king of France is bald” (Russell’s example), “Colorless green ideas sleep furiously ” (Chomsky’s), “The a book left,” “Did you stop beating your wife?” asked of someone who never beat his wife or doesn’t have a wife. Generally, these sentences fail in some presupposition: there is no king of France, nothing can be both green and not, a thing can’t be designated as the particular and not particular, you can’t spot beating your wife if you never did.
Dumping these sentences into a third category of neither true nor false seems harmless to me. It doesn’t have any adverse consequences for logic and has the advantage of dealing simply with presupposition failures. If such a sentence is neither true nor false, its negation with also be false. If there were an operator (like negation which takes a true sentence into the other truth value: false) that took true sentences into meaningless sentences, then logic would have to be restructured with far greater complexity. But since there is no such operator — there are too many ways for a presupposition to fail — there’s no reason to worry about this third value. It is logically inert.
The other strategy — using the third value for uncertainty — seems to me appealing in some ways, but has many adverse consequences. It levels necessary truth to mere truth. It’s also not clear whether the category of uncertainty means simply possibility or epistemic possibility. And it leaves the imagination bereft of conceptual fancy — possible worlds like ours, but different.
That’s not to say that there aren’t problems with conceptual possibilities — Nelson Goodman and David Lewis troubled at length about how such worlds can be consistently imagined. Bivalent modality in possibly worlds has a gross failure in that all necessary truths seem to have the same possible world meaning — true in all possible worlds. Possible worlds are too coarse-grained to distinguish those truths. (But leveling all necessary truths is not as bad as leveling all necessary truths and mere truths together.)
Looking at Kratzer’s lumping problem, or the ill fit of the material conditional with natural language, I get the impression that logic is in its infancy. Beginning with Frege there has been rapid innovation in logic. Unlike technology, which responds with increasing rapidity to a fiercely competitive market, logic hasn’t found its market value, so its progress will seem slow in comparison — unless someone can discover a logical structure that solves AI challenges better than the Aristotelean models.