We present a directly compositional and type-directed analysis of quantifier ambiguity, scope islands, wide-scope indefinites and inverse linking. It is based on Danvy and Filinski’s continuation hierarchy, with deterministic semantic composition rules that are uniquely determined by the formation rules of the overt syntax. We thus obtain a compositional, uniform and parsimonious treatment of quantifiers in subject, object, embedded-NP and embedded-clause positions without resorting to Logical Forms, Cooper storage, type-shifting and other ad hoc mechanisms. To safely combine the continuation hierarchy with quantification, we give a precise logical meaning to often used informal devices such as picking a variable and binding it off. Type inference determines variable names, banishing “unbound traces”. Quantifier ambiguity arises in our analysis solely because quantifier words are polysemous, or come in several strengths. The continuation hierarchy lets us assign strengths to quantifiers, which determines their scope. Indefinites and universals differ in their scoping behavior because their lexical entries are assigned different strengths. PPs and embedded clauses, like the main clause, delimit the scope of embedded quantifiers. Unlike the main clause, their limit extends only up to a certain hierarchy level, letting higher-level quantifiers escape and take wider scope. This interplay of strength and islands accounts for the complex quantifier scope phenomena. We present an economical “direct style”, or continuation hierarchy on-demand, in which quantifier-free lexical entries and phrases keep their simple, unlifted types.