I think there are numerous problems with the sentence theory of truth bearers. Here I will touch on one problem, that is, the problem of ambiguity. I start by assuming the sentence theory of truth bearers.
Consider the sentence:
S. It is logically possible that I exist and that I do not exist.
Is (S) true or false? I can’t tell because it is ambiguous. If you don’t see how it is ambiguous try deciding whether the predicate “It is logically possible” applies to only “I exist” or to both “I exist” and to “I do not exist”. Which is it? Logic helps us see the difference. We may formalize the two interpretations like this:
(Where “Ex” means x exists, “i” means I.)
We can translate these into english-ish:
1*. It is logically possible that I exist and it is logically possible that I do not exist.
2*. It is logically possible that (I exist and that I do not exist).
The first is true since my existence, anyone’s existence is a contingent matter (except contradictory entities). The second is false since it is not logically possible that I both exist and not exist (at the same time). That’s a contradiction. The problem is with deciding whether or not (S) is true or not. It can mean either (1) or (2), but which? It seems that there is no way in principle to tell whether (S) is true or not.
Both true and false
Another idea is to accept that it means both and simply say that (S) is both true and false. That doesn’t strike me as a good solution. It is basically giving up classical logic and accepting dialetheism.1
Neither true or false
One more plausible solution is to say that (S) is neither true or false; adopting this principle: All ambiguous sentences are neither true or false. The problem with this is that lots of sentences that we normally use are ambiguous, but maybe not in the context that they are used in. This is the best solution that I know of to the problem of ambiguity. Though it runs into methodological problems. When is a sentence ambiguous and when is it not?
1Maybe Priest would be happy about this? Another true contradiction discovered!