In contrast to the below post, I do not think that the vagueness of an object or a predicate is a result of some vagueness about the real world.
For example, we can consider "Mars" as being a non-vague entity, as "Mars" clearly refers to a finite entity. However, things like "Mount Everest," etc. are considered "vague" entities, as there is no definitive Mount Everest (as TW mentions in the first paragraph).
In this sense, the problem seems to be with language, as we use the term "Object" or "Extension," or "Predicate" to apply to two distinctly different categories of things, both vague and non-vague, when they have distinctly different properties and differing ontologies. We just tend to apply vague predicates and treat their truth values the same way we would non-vague predicates, even though they are very different things and it is not clear we should do so.
To illustrate, we can consider a vague predicate versus a non-vague predicate. Let's consider the two predicates, "is dry," and, "has 5 water molecules." Both descriptors refer to the exact same planet at the same time, though one is vague and the other not. The state of affairs of the world is not vague, as it can be described by the latter definite description, and follows all of the usual, "Either Mars has 5 water molecules or it does not have 5 water molecules" and the like. It is only when we take the vague predicate, "is dry" and try to apply it to the world as we would a definite description, that things start to become complicated.
So (I think) what I am saying is that in the case of the predicate "is dry" The only reason the sentence, "Mars is always dry or not dry," is difficult to ascribe a truth value to is not because the world is vague, just that it is our general tendency to apply descriptors and concieve of things in ways that are useful for practical reasons (just as the term "Mount Everest" serves practical purposes) rather than encoded with definitive meanings. It's not the world that is the problem -- the world is as it is, with finite properties -- but rather it is our way of thinking/talking about the world that runs us into the difficulties that we encounter with the "Original Question." Or that's probably what TW was saying, I'm not sure anymore.
I think I'm on board with this "thought and talk" method of "serious" philosophy, but only because of footnote #14, and the sentence that preceeds it. Mainly because I don't want to admit to having a short attention span.