All There Is: On the Semantics of Quantification over Absolutely Everything

Abstract

This thesis concerns the problem of providing a semantics for quantification over absolutely all there is. Chapter 2 argues against the common view that Frege understood his quantifiers in Begriffsschrift to range over all objects and discusses Michael Dummett's analysis of the inconsistent system of Grundgesetze, which generalises into his famous argument against absolute quantification from indefinite extensibility. Chapter 3 explores the possibility to adapt Tarski's first definition of truth to hold for sentences with absolute quantification. Taking the concept of logical consequence into account results in an argument for adopting a set-theory with an ill-founded membership relation as a metatheory. Chapter 4 reviews and deflates an influential argument due to Timothy Williamson against the coherence of absolute quantification. Chapter 5 discusses three important contemporary semantic theories for absolute quantification that tackle Williamson's argument in different ways. Chapter 6 challenges the widespread view that it is impossible to give a model-theoretic semantics for absolute quantification simply by providing such a semantics in NFUp. This semantic framework provides models with the universal class as domain. I show, furthermore, that the first-order logical consequence relation stays the same in this setting, by proving the completeness theorem for first-order logic in NFUp.