A supervaluation is a function that assigns truth-values to open-statements depending on what the truth-values of the corresponding closed statements are.
An open statement is simply what results when a referring term in a closed statement is replaced with an empty or variable term. An example of a ‘closed statement’ is ‘Donald Trump is tall.’ A closed statement is simply a statement. An example of an ‘open statement’ is ‘x is tall.’ ‘Donald Trump is tall’ is an ‘instance’ or ‘completion’ of ‘x is tall.’
The stock of example of a supervaluation is the function that assigns the truth-value TRUE to ‘either Pegasus has wings or Pegasus does not have wings’ and that, more generally, assigns TRUE to all open-statements whose ‘completions’ are true and FALSE to ‘Pegasus has wings and Pegasus does not have wings.’ This supervaluation assigns TRUE to all open-statements whose completions are true and FALSE to all open-statements whose completions are false.
Here is the idea. First of all, ‘Pegasus has wings’ does not have a truth-value, since ‘Pegasus’, being an ‘empty’ term, does not refer to anything. So give or take a few irrelevant literary associations of ours, ‘Pegasus has wings’ is in the same category as ‘x has wings.’ Nonetheless, so it is argued, ‘either Pegasus has wings or Pegasus does not have wings’ in some way or other ‘deserves’ a truth-value or, at any rate, deserves some kind of a special epithet that indicates that it would be true if it didn’t have a referring term. Similarly, ‘Pegasus has wings and Pegasus does not have wings’, though neither true nor false, should be a given special epithet to indicate that, if only it didn’t contain an empty term, it would be false. And the supervaluation in question takes care of this problem by assigning TRUE to open-statements whose completions are all true and FALSE to open statements are all false.
The concept of a supervaluation is due to Bas van Fraassen (1966).
What is the purpose of supervalutions? They supposedly help us deal with puzzles relating to vagueness and other semantic indeterminacies. Somebody with one hair (of the appropriate thickness etc.) is bald; somebody with a billion hairs is not bald; somebody with 749 hairs is vaguely bald and vaguely not bald. So there doesn’t seem to be a definite cutoff line between bald and not bald or event between not bald and vaguely bald (or even between vaguely bald and vaguely vaguely bald; etc.). Supervaluations have been used to ‘solve’ this problem by consolidating statements with questionable truth-values into ‘super-statements’ with determinate truth-values, thereby restoring (supposedly) the definite cut-off lines required by Classical Logic.
And, as we have already seen, supervaluations are used to deal with the problems for classical logic supposedly created by ‘Pegasus has wings’ and ‘Fred Flintstone snores’, and the like.
Does supervaluationism solve these problems? No it doesn’t. It is just a clumsy and apparatus-heavy way of articulating what we already know, namely that these quasi-statements are statement-like in some ways and not statement-like in others. As for the puzzles in question, they can be solved intuitively and cleanly, without supervaluations or any other such encumbrances.