Although the connection between natural language syntax and semantics has received serious attention in both linguistics and computational linguistics for the last several decades, it does not appear that it has yet been entirely satisfactorily identified. The present dissertation focuses on quantifier scope ambiguity in an attempt to identify such a connection. We show that there are some readings that are incorrectly allowed by the theories and that other readings that are available are allowed for the wrong reason. First, we distinguish referential NP interpretations from quantificational NP interpretations. Most traditional theories of scope do not, and they are shown to significantly overgenerate readings and/or miss a crucial generalization regarding quantificationally available readings. We present a hypothesis based on the notion of surface constituency to predict quantificationally available readings. The hypothesis is tested on core English constructions, including transitive verbs, dative alternation (ditransitive) verbs, attitude verbs, complex NPs containing prepositional phrases, possessives, and subject or non-subject Wh-relatives also with pied-piping and various coordinate structures. We argue that the scopings allowed under the hypothesis are the ones that are available. We then present a competence theory of quantifier scope, couched in a combinatory categorial grammar framework. The theory defines the connection between syntax and semantics in a precise way, utilizing the dual quantifier representation. We show theoretical predictions on the core English constructions, and verify that the theoretical predictions are consistent with the predictions made by the hypothesis and that there are further reasonable theoretically predicted readings. Finally, we describe an implementation of the theory in Prolog. The implemented system takes English sentences as ambiguous queries (regarding scope), generates logical forms that are associated with them, and evaluates those logical forms with respect to a predefined database of facts. The system also works as a proof-checker of the theory.