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It is not the case that Supervaluationism collapses under the 'higher-order vagueness' problem: the meta-language used to define 'precisifications' is itself vague.
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Reasons For
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Reason for
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1.
Meta-language vagueness doesn't collapse supervaluationism if we accept that higher-order vagueness is a general feature of language, not a special problem.
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2.
Supervaluationism only requires that *some* precisifications exist and are consistent, not that they be perfectly sharply defined at all orders.
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3.
The objection conflates vagueness in how we *describe* precisifications with vagueness in the precisifications *themselves*, which are mathematical constructs.
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Reasons Against
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Reason against
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1.
If precisifications are defined vaguely, they fail to provide determinate truth-conditions, undermining supervaluationism's core solution.
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2.
Higher-order vagueness is unavoidable: any attempt to sharpen 'precisification' via meta-language generates new vagueness at that level.
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3.
Supervaluationism claims vagueness is metalinguistic, not metaphysical, but vagueness in the meta-language contradicts this distinction.
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