Abstract

Cantor defined a set as the collection into a whole of determinate and well-distinguished objects, objects that can “exist together”. Although he recognized that not all objects can exist together in this sense, he did not provide a theory specifying when some objects can exist together. Without such a theory, his account of sets remains incomplete.

We argue that a version of plural logic, Critical Plural Logic, fits the bill. As a logic of co-existence, Critical Plural Logic captures the essence of the familiar “stage theory” of set formation but is far more general and avoids a gratuitous reification of stages. We also argue that this plural logic can be of great aid to Frege and logicism, supporting a large class of abstraction principles. Critical Plural Logic emerges as a compelling and fruitful theory to develop key ideas of Cantor and Frege.