Consider this theory. To utter a sentence is to offer a description of a proposition. Sometimes, whether contingently or necessarily, no proposition meets the description. That is a case of nonsense. Sometimes more than one proposition meets the description. That is a case of vagueness (ambiguity is a case where the sentence's words fail to determine the proposition, but the sentence as a whole does—in agreement with how one might read Frege on tense, I take the context to be a part of the sentence or, in his terminology, the "expression"). Subintensional vagueness is when the propositions that meet the description necessarily have the same truth value. Intensional vagueness is when they possibly have different truth values. Extensional vagueness is when they actually have different truth values. I suspect that subintensional vagueness occurs all over the place.
How do these descriptions work? I think in a recursive way. For instance, the sentence "F(a)" describes a proposition that applies Fness to a. What does it mean to say that a proposition "applies" Fness to a? Well, we can say this: necessarily, a proposition that applies Fness to a is true if and only if a exists and has Fness. But this is not sufficient to determine the concept of application—after all, there are many propositions that satisfy this description, such as the proposition that God knows that a has Fness and the proposition that a has Fness or 2+2=5. We can lay down a few more conditions on application. A proposition that applies Fness to a is not going to be truthfunctionally complex if F is not truthfunctionally complex. (But truthfunctional complexity needs to be defined, and it is hard to do that.) The functor application from property-object pairs to propositions is natural. Maybe something can be said about explanatory priority.
But, on the present theory, there is little reason to think that filling out the above story will narrow down the notion of application, or of Fness for that matter, so far as to ensure that there is only one proposition that meets the description given by the sentence "F(a)". Similar points apply to truthfunctionally complex sentences. Thus, "s and u" describes a proposition which the output of the conjunction functor as applied to the propositions described by "s" and by "u". But we cannot give an unambiguous definition of the conjunction functor. So subintensional vagueness is everywhere.
Now we have a story to tell about the classic liar sentence: "This is a falsehood." Such a sentences attempts to describe a proposition that denies truth to itself. But there is no reason to suppose that there is such a proposition, and indeed there is a very good reason to suppose that there isn't—namely, that the supposition that there is such a proposition immediately leads to a logical contradiction (what better argument could one ask for?). (One might try Goedel numbering and diagonalization. But Goedel numbering works for sentences, not propositions, and because of subintensional vagueness, the correspondence between sentences and propositions is not one-to-one.)
The contingent liar is tougher, but perhaps some generalization of the solution will work. (It might help if some propositions are contingent beings.)