Sheffer’s Criticism of Royce’s Theory of Order

Henry Sheffer in his 1908 Ph.D. thesis disproves the claim of his teacher Josiah Royce that he had shown how to define an asymmetric order relation in terms of a symmetric relation of incompatibility in his 1905 ∑-system. The paper examines the nature of Royce’s “mistake” in the context of the evolving conception of mathematical definition in that period. It then goes on to consider the significance of defining the asymmetric in terms of the symmetric for Royce. Part of this was connected with a logicist project of defining mathematical concepts in terms of logical concepts.