Section 1 introduces basic concepts and terminology. Section 2 delimits the paper topic from related but distinct issues. Section 3 presents the relevant notion of higher-order vagueness and five basic logical principles that together characterize this notion; it sets out the major problem faced by the received positions on higher-order vagueness: that they cannot preserve “robust” higher-order vagueness without being caught in the paradoxes of higher-order vagueness. Sections 4 to 10 present an alternative theory of higher-order vagueness that maintains robustness without falling foul of the paradoxes. Section 4 introduces an unclarity operator that covers the borderlinehood of vague expressions, and its interpretation. The operator is an unclear-whether operator governed by a logic modeled on non-contingency logics. The interpretation is based on the idea that whether some a is borderline F depends on whether a relevantly competent, rational, informed speaker (crisp) could tell whether a is F. The notion of a crisp is obtained by stripping ordinary speakers of all relevant epistemic handicaps. crispness is indexed to the scope of the unclarity operator. The relevant notion of telling-whether is governed by a margin-for-error principle. It is shown why, if borderline crisps exist, they cannot be criterial for borderlinehood. Section 5 introduces a criterion for distinguishing the unclarity characteristic for borderlinehood (“radical unclarity”) from other kinds of unclarity covered by the unclarity operator with the given interpretation. This criterion is absolute epistemic inaccessibility. It forces upon us absolute agnosticism about borderline cases. Section 6 collects the basic axioms of a logic of higher-order borderline vagueness, including an axiom that embodies the element of radical unclarity characteristic for borderlinehood. This principle (the uu-Principle) says that, when it is unclear whether Fa, it is unclear whether it is unclear whether Fa. An informal proof is given that the notion of radical unclarity, together with the interpretation of the unclarity operator, entails the uu-Principle. Section 7 explicates why unclear, or hidden, clear cases are incompatible with the interpretation of the unclarity operator. A further axiom is introduced to exclude such cases. This axiom entails the principle that if something is clear, it’s clear that it’s clear (the cc-Principle). Section 8 presents the resulting logical system, which combines homogeneity of all borderline cases of a vague expression relative to a context with infinite orders of unclarity. Section 9 shows how both robustness and tolerance are preserved in the system. Section 10 considers what semantics and logic — if any — hold for the borderline zones and what happens to penumbral connections. Section 11 demonstrates that the theory of absolute agnosticism developed in Sections 4–10 is immune to the various paradoxes of higher-order vagueness that have been put forward by Wright, Fara, Shapiro and Williamson. In an appendix (i. e. Section 12) various questions about the existence of crisps are answered.
