Identity is Irreducibly Relational

Abstract

This paper argues that identity is irreducibly relational: the statement A = A presupposes that A is defined, and definition requires distinction from a background. The thesis is developed at three levels: conceptual (definition requires distinction), formal (set-theoretic and type-theoretic foundations, including a 'Referential Set' formalization), and historical (Leibniz through Kripke). ZFC's extensionality axiom already makes identity relational for sets, and Homotopy Type Theory and Univalent Foundations treat identity as constituted by structural equivalence. The trajectory of foundational mathematics vindicates a relational conception of identity.