The present thesis is an inquiry into the metasemantics of natural languages, with a particular focus on the philosophical motivations for countenancing degreed formal frameworks for both psychosemantics and truth-conditional semantics. Chapter 1 sets out to offer a bird's eye view of our overall research project and the key questions that we set out to address. Chapter 2 provides a self-contained overview of the main empirical findings in the cognitive science of concepts and categorisation. This scientific background is offered in light of the fact that most variants of psychologically-informed semantics see our network of concepts as providing the raw materials on which lexical and sentential meanings supervene. Consequently, the metaphysical study of internalistically-construed meanings and the empirical study of our mental categories are overlapping research projects. Chapter 3 closely investigates a selection of species of conceptual semantics, together with reasons for adopting or disavowing them. We note that our ultimate aim is not to defend these perspectives on the study of meaning, but to argue that the project of making them formally precise naturally invites the adoption of degreed mathematical frameworks (e.g. probabilistic or fuzzy). In Chapter 4, we switch to the orthodox framework of truth-conditional semantics, and we present the limitations of a philosophical position that we call "classicism about vagueness". In the process, we come up with an empirical hypothesis for the psychological pull of the inductive soritical premiss and we make an original objection against the epistemicist position, based on computability theory. Chapter 5 makes a different case for the adoption of degreed semantic frameworks, based on their (quasi-)superior treatments of the paradoxes of vagueness. Hence, the adoption of tools that allow for graded membership are well-motivated under both semantic internalism and semantic externalism. At the end of this chapter, we defend an unexplored view of vagueness that we call "practical fuzzicism". Chapter 6, viz. the final chapter, is a metamathematical enquiry into both the fuzzy model-theoretic semantics and the fuzzy Davidsonian semantics for formal languages of type-free truth in which precise truth-predications can be expressed
