David Lewis articulated minimal constraints on a formal theory of natural language semantics that have been widely adopted by subsequent theorists: compositionality and sentence truth in a given context. In the process, Lewis distinguished between the compositional semantic value of an expression and its propositional content relative to a context. This dissertation consists of a series of essays in which I address several questions that arise from this distinction, including how we should understand semantic values, how we should understand propositional content, and how we should understand the relation between them. Related to this, I explore and address a number of interesting and unresolved methodological issues that arise in relation to context-sensitivity, and provide an account of the role of speaker intentions in a formal theory of natural language semantics. Additionally, I provide a detailed analysis of the role of context in a theory of natural language semantics and its connection to various aspects of language use and communication. I also motivate coherence with syntactic structure (in the tradition of generative grammar) as an additional constraint on a formal theory of natural language semantics and assess its import for how we theorize about tense and modality and issues related to the syntax-semantics interface, including covert structure and logical form. In broad strokes, this dissertation addresses issues concerning the aims, scope and criteria of a theory of natural language semantics. I approach these issues from the perspective of generative grammar, a theoretical framework that aims to characterize our understanding of natural language independent of its use. These essays help to clarify what should be expected of a formal theory of natural language semantics and its contribution to theories of speech acts and communication.