  The supervaluational response is more interesting, both because it engages directly with the intuitions behind this argument, and because two of its leading proponents (Vann McGee and Brian McLaughlin, in their 2001) have responded directly to this argument using the supervaluational framework. Roughly (and for more detail see the section on supervaluations in the  entry on  vagueness)  supervaluationists say that whenever some terms are vague, there are ways of making them more precise consistent with our intuitions on how the terms behave. So, to use a classic case, ‘heap’ is vague, which to the supervaluationist means that there are some piles of sand that are neither determinately heaps nor determinately non-heaps, and a sentence saying that that object is a heap is neither determinately true nor determinately false. However, there are many ways to extend the meaning of ‘heap’ so it becomes precise. Each of these ways of making it precise is called a precisification. A precisification is admissible iff every sentence that is determinately true (false) in English is true (false) in the precisification. So if a is determinately a heap, b is determinately not a heap and c is neither determinately a heap nor determinately not a heap, then every precisification must make ‘a is a heap’ true and ‘b is a heap’ false, but some make ‘c is a heap’ true and others make it false. To a first approximation, to be admissible a precisification must assign all the determinate heaps to the extension of ‘heap’ and assign none of the determinate non-heaps to its extension, but it is free to assign or not assign things in the ‘penumbra’ between these groups to the extension of ‘heap’. But this is not quite right. If d is a little larger than c, but still not determinately a heap, then the sentence “If c is a heap so is d” is intuitively true. As it is often put, following Kit Fine (1975), a precisification must respect ‘penumbral connections’ between the borderline cases. If d has a better case for being a heap than c, then a precisification cannot make c a heap but not d. These penumbral connections play a crucial role in the supervaluationist solution to the Problem of the Many. Finally, a sentence is determinately true iff it is true on all admissible precisifications, determinately false iff it is false on all admissible precisifications.