So I was reading about resolutions to the Sorites paradox, and I got most of them, but I didn't understand the one labelled supervaluationism. Would someone explain it to me in very basic terms please? Are there any objections to this resolution? Thanks in advance.
You deliver a large and complex topic ! The following extract from an article by Matti Eklund (2011) could scarcely be fully satisfying but it does (a) relate sorites and supervaluationism and (b) start a discussion of difficulties:
Vagueness, as discussed in the philosophical literature, is the phenomenon that paradigmatically rears its head in the sorites paradox, one prominent version of which is:
One grain of sand does not make a heap.
For any n, if n grains of sand do not make a heap, then n + 1 grains of sand do not make a heap.
So, ten billion grains of sand do not make a heap.
It is common ground that the different versions of the sorites paradox arise because of vagueness in a key expression, in this case 'heap'. One central concern in the literature on vagueness is to find a solution to the sorites paradox.
Vagueness also gives rise to borderline cases. Since 'heap' is a vague ex- pression, 'heap' also gives rise to borderline cases: there are some possible entities such that they are neither clearly heaps nor clearly non-heaps, but instead are borderline heaps. Borderline cases can seem intuitively to present counter-examples to bivalence and the law of excluded middle (LEM). If Harry is a borderline case of baldness, 'Harry is bald' can seem neither true nor false, whence it presents doubts concerning bivalence. Similarly, one can find the corresponding instance of LEM, 'Either Harry is bald or Harry is not bald', suspect.
There are three main theories that involve dtures from bivalence, and two of them involve rejection of LEM as well. Standard three-valued logic invokes the idea of a third truth-value or truth-value gaps; fuzzy logic invokes the idea of continuum-many truth-values intermediate between truth and falsity. Both these views preserve truth-functionality. Traditional supervaluationism involves rejection of bivalence, but adherence to LEM. The idea is that if an expression is vague there is a range of precisifications associated with it, where, roughly, precisifications are ways the expression could be made more precise consistently with the settled facts. Given the notion of precisification, one can talk about what sentences are true under all precisifications, what sentences are false under all precisifications, sentences have different truth-values under different precisifications. Traditional supervaluationism identifies supertruth as truth under all precisifications, and correspondingly for falsity. There are then sentences which are neither true nor false, and so bivalence is rejected. But each precisification is perfectly classical, and each classical logical truth, including all instances of LEM, comes out true under all precisifications. Supervaluationism rejects truth-functionality.
These views rejecting bivalence themselves face well-known problems. Some of these problems are specific to the view in question. Others are more general. One general problem is this. It seems quite implausible that the use of a vague predicate should determine a cut-off in a sorites series: a pair of cases such that one is classed as true and the other as false. But these views do not avoid unwanted cut-offs of their own. Three-valued logic and supervaluationism just replace the true/false cut-off with two new ones: the true/neither cut-off and the neither/false cut-off.
(Matti Eklund, 'Recent Work on Vagueness', Analysis, Vol. 71, No. 2 (APRIL 2011), pp. 352-363: 352-3.)
Further difficulties are exposed and examined as the article continues. Nicholas K. Jones, 'Williams on Suoervaluationism and Logical Revisionism', The Journal of Philosophy, Vol. 108, No. 11 (November 2011), pp. 633-641, is also worth looking into if you have not come across it.