Essence and Definition by Abstraction

Abstract

We may define words. We may also define the things for which words stand. Definitions of words may be explicit or implicit, and may seek to report pre-existing synonymies, but they may instead be wholly or partly stipulative. Definition by abstraction seeks to define a term-forming operator by fixing the truth-conditions of identity-statements featuring terms formed by means of that operator. Such definitions are a species of implicit definition. They are typically at least partly stipulative. Definitions of things (real definitions) are typically conceived as statements about the essence of their definienda, and so not stipulative. There thus appears to be a clash between taking Hume's principle as an implicit, at least partly stipulative definition of the number operator and as a real definition of cardinal numbers. This chapter argues that this apparent tension can be resolved, and that resolving it shows how some modal knowledge can be a priori.