 The publication of “The Relation of the Principles of Logic to the Foundations of Geometry” (1905) and “An Extension of the Algebra of Logic” (1913b) were significant works of pure mathematics but were also intended to illustrate invariants across transformations of all sorts. Burch (2011) has left open the question of how Royce’s work in mathematical logic is related to his general philosophical project, but has argued that Royce nevertheless made a significant contribution to mathematical logic by developing the concept of a Boolean Ring and an algebra identical to the system devised at about the same time by Russian logician, Ivan Ivanovich Zhegalkin, renowned as “the ‘father’ of the Russian school of logic” (231). Central to the Boolean Ring operations and to the larger System \(\Sigma\) is Royce’s development of the operation of symmetric difference (discussed elsewhere as the \(O\)-relation), which is also viewed as one of Royce’s major contributions by later reviewers, Burton and Leblanc (1952). While Burch suggests the possibility of a full exposition of System \(\Sigma\), apart from Royce’s own there is currently no such treatment available. Burch also discusses Royce’s work in philosophical logic in the introduction to a paper by Royce addressing Russell’s paradox as presented in Russell’s 1903 The Principles of Logic. This paper, Burch argues, both demonstrates Royce’s familiarity with developments in modern logic at the time and presents an alternative resolution of the paradox similar to Frege’s response while also offering a non-foundational set theory (1987). Other important assessments of Royce’s work in mathematics and logic include early reviews by Sheffer (1908), De Laguna (1906), and C. I. Lewis (1916), and more recent assessments in papers by Crouch (2011) and Scanlan (2011).