Rule Underdetermination and the Non-Existence of Sharp Boundaries in Vague Predicates

This paper argues that the extension of vague predicates is neither uniquely determined by structural facts nor governed by any single privileged semantic rule. Through a set of interconnected counterexamples, I show that a single physical configuration can support multiple incompatible yet equally reasonable classification rules. This shifts the problem of vagueness from first-order boundary indeterminacy to second-order rule underdetermination. I further develop a closure argument showing that no rule-selection mechanism can provide a non-circular and sufficient basis for fixing a unique classification. Consequently, the hypothesis of a unique sharp boundary cannot be sustained—not merely epistemically, but metaphysically. I then argue that this underdetermination undermines multi-valued logic,“neither true nor false” approaches, weak negation, and epistemicism.