Abstract

The main purpose in this dissertation is to show how certain modal-semantic considerations can be used to make sense of the subject of impredicativity. A secondary purpose is to rebut in a more direct manner the charge of vicious circularity. ;In Chapter 1, I examine Russell's early idealist work in the foundations of geometry. Although Russell increasingly disassociated himself from this work, as indeed from Kant and Hegel, an examination of Russell's idealist foundations can shed light on Russell's later ban on impredicativity. Russell's idealist metaphysical views make extensive appeal to modal notions such as essentiality and presupposition. It was largely his change in attitude toward just these modal notions that lead him to reject idealism and adopt in its place logical atomism and an analytic philosophical methodology. The modal account of impredicativity I give in Chapter 3 will rely chiefly on modal notions Russell rejected when he abandoned his idealist philosophy. Thus the purpose of the first chapter is largely historical: to sketch Russell's views regarding essentiality and ontological presupposition as they were applied in foundations of mathematics. ;Chapter 2 concerns Poincare. I present Poincare's views in the foundations of arithmetic and geometry prior to his rejection of impredicativity in 1906. I then try to highlight certain tensions in his thought which the rejection of impredicativity created. These tensions arise from Poincare's use of Kant's claim that mathematical knowledge is based upon synthetic a priori intuition. The principles Poincare held such intuition to justify require, for their proof, the use of impredicative definitions or the postulation of impredicative objects. Poincare took his ban on impredicativity to show that explicit proofs of these principles were not possible, and that therefore these principles presupposed a role for synthetic a priori intuition. I argue that this conclusion is misguided, and that Poincare does not successfully avoid impredicativity in the foundations. ;In Chapter 3 I discuss Russell's ramified type theory and argue first that Russell's motivations for introducing this theory can be expressed as certain modal prejudices Russell held. I then extend the modal notions used to express Russell's motivations to define a notion of mutual presupposition or reciprocal ontological dependency, which can be seen to constitute the impredicativity of objects in the context of ramified type theory