This paper explores the logic of vagueness and the truth-conditions for vague language. It begins with the question of what the correct logic of vagueness is, leading to a broader consideration of meaning and existence. The first half of the paper presents key ideas. Section 1 critiques an approach based on extended truth-tables, which faces issues like penumbral connection. Section 2 introduces an alternative framework that accommodates penumbral connection by considering both actual and potential truth-values of sentences. Section 3 defends the favored account, which states that a vague sentence is true if it is true for all ways of making it precise. The second half discusses consequences and comparisons. Section 4 considers the logical consequences of different approaches, noting that the favored account leads to classical logic for vague sentences. Section 5 examines higher-order vagueness, its implications for truth-conditions, and its relation to puzzles about priority and eliminability. The paper also discusses vagueness as a semantic concept, distinguishing it from generality, undecidability, and ambiguity. It uses artificial examples to illustrate these distinctions. Vagueness is characterized as a deficiency of meaning, and it can apply to various expressions, including predicates, names, and sentence-operators. The paper distinguishes between extensional and intensional vagueness, with extensional vagueness involving borderline cases and intensional vagueness involving the possibility of such cases. It notes that 'bald' is extensionally vague but remains intensionally vague. The paper concludes by discussing the relationship between vagueness and truth-value gaps, noting that vague sentences are neither true nor false.This paper explores the logic of vagueness and the truth-conditions for vague language. It begins with the question of what the correct logic of vagueness is, leading to a broader consideration of meaning and existence. The first half of the paper presents key ideas. Section 1 critiques an approach based on extended truth-tables, which faces issues like penumbral connection. Section 2 introduces an alternative framework that accommodates penumbral connection by considering both actual and potential truth-values of sentences. Section 3 defends the favored account, which states that a vague sentence is true if it is true for all ways of making it precise. The second half discusses consequences and comparisons. Section 4 considers the logical consequences of different approaches, noting that the favored account leads to classical logic for vague sentences. Section 5 examines higher-order vagueness, its implications for truth-conditions, and its relation to puzzles about priority and eliminability. The paper also discusses vagueness as a semantic concept, distinguishing it from generality, undecidability, and ambiguity. It uses artificial examples to illustrate these distinctions. Vagueness is characterized as a deficiency of meaning, and it can apply to various expressions, including predicates, names, and sentence-operators. The paper distinguishes between extensional and intensional vagueness, with extensional vagueness involving borderline cases and intensional vagueness involving the possibility of such cases. It notes that 'bald' is extensionally vague but remains intensionally vague. The paper concludes by discussing the relationship between vagueness and truth-value gaps, noting that vague sentences are neither true nor false.