Second-Order Quantification

Abstract

In this chapter, I will deal with the problem of second-order quantification . I distinguish two senses of ‘second-order quantification’: (i) metaphysically second-order quantification, which occurs whenever we quantify into the position of individual variables of a domain that includes properties or sets and (ii) logically second-order quantification, i.e. cases in which we quantify into the position of variables for first-order predicates. Both kinds of second-order quantification may be used to establish the existence of universals and thus should be discussed. In the first part of Chap. 6, I deal with metaphysically second-order quantification and argue that any metaphysically second-order sentence that seems to refer to properties, e.g., ‘humility is a virtue’ and ‘red resembles orange more than blue’, has an ontologically fundamental first-order paraphrase . I develop and apply what I call the ‘method of grounded paraphrase’. In the second part, I discuss logically second-order quantification. I explain and defend the plausibility of two arguments of the ostrich against the line of reasoning that derives the existence of properties from logically second-order quantification. The first argument aims to cast doubts on the very intelligibility of quantification into the predicate position, the second aims to show that predicates can play an important semantic role even when they do not ‘stand for’ extra-linguistic entities like properties. In the end, I will maintain that even when one rejects these last two arguments, the priority nominalist may not be concerned about logically second-order predication, because all second-order truths are grounded in first-order truths.