This paper explores the logic of vagueness, starting with the question of what the correct logic for vague languages should be. It then delves into the truth-conditions for vague languages and broader considerations of meaning and existence. The first half of the paper discusses and critiques an approach based on extended truth-tables, which fails to account for penumbral connection. An alternative framework is introduced, where truth-values are considered not only for actual sentences but also for their potential precise forms. This framework supports the claim that a vague sentence is true if and only if it is true for all ways of making it completely precise. The second half of the paper examines the consequences, complications, and comparisons of these approaches. It shows that the favored account leads to a classical logic for vague sentences and addresses objections to this position. The paper also explores higher-order vagueness, its impact on languages with definitely-operators or truth-predicates, and its relation to puzzles concerning priority and eliminability. The paper defines vagueness as a semantic notion, distinguishing it from generality, undecidability, and ambiguity. It uses artificial examples to illustrate these distinctions, such as different clauses for natural number predicates. The paper further characterizes vagueness in terms of extensional and intensional deficiency, with extensional vagueness referring to deficiency of extension and intensional vagueness to deficiency of intension. It also discusses the existence of truth-value gaps in vague sentences and the possibility of a vague sentence being both true and false.This paper explores the logic of vagueness, starting with the question of what the correct logic for vague languages should be. It then delves into the truth-conditions for vague languages and broader considerations of meaning and existence. The first half of the paper discusses and critiques an approach based on extended truth-tables, which fails to account for penumbral connection. An alternative framework is introduced, where truth-values are considered not only for actual sentences but also for their potential precise forms. This framework supports the claim that a vague sentence is true if and only if it is true for all ways of making it completely precise. The second half of the paper examines the consequences, complications, and comparisons of these approaches. It shows that the favored account leads to a classical logic for vague sentences and addresses objections to this position. The paper also explores higher-order vagueness, its impact on languages with definitely-operators or truth-predicates, and its relation to puzzles concerning priority and eliminability. The paper defines vagueness as a semantic notion, distinguishing it from generality, undecidability, and ambiguity. It uses artificial examples to illustrate these distinctions, such as different clauses for natural number predicates. The paper further characterizes vagueness in terms of extensional and intensional deficiency, with extensional vagueness referring to deficiency of extension and intensional vagueness to deficiency of intension. It also discusses the existence of truth-value gaps in vague sentences and the possibility of a vague sentence being both true and false.