Sets, Properties, and Unrestricted Quantification

Call a quantifier unrestricted if it ranges over absolutely all things: not just over all physical things or all things relevant to some particular utterance or discourse but over absolutely everything there is. Prima facie, unrestricted quantification seems to be perfectly coherent. For such quantification appears to be involved in a variety of claims that all normal human beings are capable of understanding. For instance, some basic logical and mathematical truths appear to involve unrestricted quantification, such as the truth that absolutely everything is self-identical and the truth that the empty set has absolutely no members. Various metaphysical views too appear to involve unrestricted quantification, such as the physicalist view that absolutely everything is physical. However, the set-theoretic and semantic paradoxes have been used to challenge the coherence of unrestricted quantification. It has been argued that, whenever we form a conception of a certain range of quantification, this conception can be used to define further objects not in this range, thus establishing that the quantification wasn’t unrestricted after all.1 This paper has two main goals. My first goal is to point out some problems with the most promising defense of unrestricted quantification developed to date. My second goal is to develop a better defense. The most promising defense of unrestricted quantification developed to date makes use of a hierarchy of types (Section 3). I show that there are some important semantic insights that type-theorists cannot express in full generality (Section 4). I argue that this problem is analogous to those faced by philosophers who deny the coherence of unrestricted quantification. My alternative defense of unrestricted quantification is based on a sharp distinction between sets and properties (Section 5). Sets are combinatorial entities, individuated by reference to their elements. Properties are intensional entities, individuated