How does the supervaluationalist defend his/her theory of truth since the correspondence theory of truth seems to presuppose bivalence? It would seem then that the only truth is Super-Truth. And, the only false is Super-False. So, while correspondence holds for each precification, indeterminacy still reigns in the actual world. So, how can there be correspondence to truth for boundaryline cases?
Bivalence and supertruth
Yes, clearly a supervaluationist makes a distinction between the truth of a particular precisification and the supertruth of a statement true for all possible precisifications, which on the face of it could imply multiple truth values rather than the pure true/false dichotomy of bivalence.
If we take for example the statement "Andrew loves maths", this is clearly not an example of a supertruth because we can make this statement false or true by making it precise in different ways, whereas "Andrew is a banana or is not a banana" is supertrue since its truth is not affected by how it is made precise.
However, you can interpret a supertruth as a collection of true precisification rather than as something which is "more" true than an individual precisification - there's no need to introduce a multi-valued logic to use this concept.
[Conversely, you could also think of supervaluation as using the set of true precisification of a statement to determine a partial order on statements (assuming you fix a particular language), and the maximum element is the supertrue one, and the minimum element is the superfalse one. This can be viewed as consistent with a boolean algebra with more than two truth values, but it's not necessary to interpret it this way.]
Some logicians would require that any statement be completely disambiguated before assigning a truth value to it, so could argue that none of these had a truth value, which links nicely with my next point about borderline cases and bivalence.
Borderline cases and bivalence
The supervaluationist claims that borderline cases have neither truth nor falsehood, precisely because by definition they are intrinsically undecidable. You can argue that this is inconsistent with bivalence by saying that you now have three truth values: true, false and indeterminate, but a supervaluationist could counter that indeterminacy was exactly an absence of a truth value, not a third truth value.
Correspondence and bivalence
Correspondence may indeed seem to presuppose bivalence. Classical logic definitely presupposed bivalence, but nevertheless it was found that more generally any boolean algebra would do.
The correspondence between a statement and reality can be seen as a relation in the mathematical sense, with a pair of statement and fact being in the relation or not in a binary yes/no way, but there's no inherent problem with marrying correspondence to a non-bivalent logic.
For example, we could assign a number between zero and one to indicate the extent to which a statement corresponds to a fact in reality; it's possible to argue that bivalence is not implied by correspondence, regardless of whether any protagonists have seen the two as part of a single theory of truth.
Indeterminacy, borderline cases and correspondence
There's a difference between unclear to a given person at a particular time and intrinsically inquiry-resistant. (Interestingly, quantum mechanics asserts that some things are unknowable, and Godel's incompleteness theorem implies that some things are undecidable, but note that neither is the result of multiply-interpretable statements.)
Indeed, as you note, in practice many things aren't known at the time, and many things an individual such as you or I "know" are merely learned and classed as fact rather than determined as fact. It's even possible to argue that nothing can be known, but this contradicts neither correspondence nor supervaluation. It's consistent to assert that the correspondence relation exists without demonstrating or knowing any of it.
In fact, borderline cases could be described as the cases where a statement cannot be classified as either in or out of the correspondence between true statements and facts, because of its inherent vagueness of language. This is an entirely different issue to whether a given non-vague statement is knowable as true or false.
The fact that it doesn't make sense to say whether "yellow < 7" or not doesn't spoil the fact that we can compare any two real numbers with "<". Similarly, a supervaluationist rejects some statements as truth-valueless but needn't feel that this spoils correspondence theory. To quote Dr Vassili Corbas (very much out of context): "It's worse than wrong, it's meaningless!". A supervaluationist may take this view of borderline cases - they're external to the theory rather than contradictory to it.
- A supervaluationist could happily accept that they know few of correspondences with fact in the actual world (they are as yet indeterminate in practice), without admitting them as borderline cases (inherently indeterminate).
- They could happily see the existence of inherently indeterminate statements not as a rejection of correspondence theory, but rather as an externality to the correspondence relation itself.
- They could retain bivalence by asserting that lack of a truth value is by definition not itself a truth value and also asserting that a supertruth is a collection of truths rather than a greater truth.
- Alternatively they could reject bivalence and retain correspondence by using multiple truth values associating statements to facts.
Finally, it's worth noting that you can see supervaluation as your favourite valid, purely formal logical system without asserting anything whatsoever about its applicability to life!