Absolute and relative generality - Studd, JP (2013)
This thesis is concerned with the debate between absolutists and relativists about generality. Absolutists about quantification contend that we can quantify over absolutely everything; relativists deny this. The introduction motivates and elucidates the dispute. More familiar, restrictionist versions of relativism, according to which the range of quantifiers is always subject to restriction, are distinguished from the view defended in this thesis, an expansionist version of relativism, according to which the range of quantifiers is always open to expansion. The remainder of the thesis is split into three parts. Part I focuses on generality. Chapter 2 is concerned with the semantics of quantifiers. Unlike the restrictionist, the expansionist need not disagree with the absolutist about the semantics of quantifier domain restriction. It is argued that the threat of a certain form of semantic pessimism, used as an objection against restrictionism, also arises, in some cases, for absolutism, but is avoided by expansionism. Chapter 3 is primarily engaged in a defensive project, responding to a number of objections in the literature: the objection that the relativist is unable to coherently state her view, the objection that absolute generality is needed in logic and philosophy, and the objection that relativism is unable to accommodate ‘kind generalisations’. To meet these objections, suitable schematic and modal resources are introduced and relativism is given a precise formulation. Part II concerns issues in the philosophy of mathematics pertinent to the absolutism/relativism debate. Chapter 4 draws on the modal and schematic resources introduced in the previous chapter to regiment and generalise the key argument for relativism based on the set-theoretic paradoxes. Chapter 5 argues that relativism permits a natural motivation for Zermelo-Fraenkel set theory. A new, bi-modal axiomatisation of the iterative conception of set is presented. It is argued that such a theory improves on both its non-modal and modal rivals. Part III aims to meet a thus far unfulfilled explanatory burden facing expansionist relativism. The final chapter draws on principles from metasemantics to offer a positive account of how universes of discourse may be expanded, and assesses the prospects for a novel argument for relativism on this basis.