The Logic of Classes of the No-Class Theory

Abstract

In developing modern logic, Frege aims to substantiate the logicist view of arithmetic. In this view, arithmetic is reducible to logic: one can define all arithmetical notions using only logical notions, and prove all arithmetic truths using only logical truths together with definitions of arithmetical notions. A key component of his attempt to show this is his analysis of numbers as extensions of ‘concepts’ or properties (Frege, 1884: 80).¹ Russell, who shares Frege’s logicism, accepts the same analysis of number. In his early analysis of mathematics, he writes:

... a number is nothing but a class of similar classes: this definition allows the deduction of all the usual properties of numbers ... and is the only one (so far as I know) which is possible in terms of the fundamental concepts of general logic. (POM: 116)²

Like Frege, Russell identifies classes as extensions and considers the notion of extension a logical notion because it pertains to the logic or semantics of predicates.