Identity is Irreducibly Relational

Abstract

The law of identity (A=A) is not foundational but derivative. It presupposes that A is defined, and definition requires distinction from a background. Formalizing this via a 'Referential Set' R(A), we prove that identity implies R(A) is not empty. We demonstrate that Homotopy Type Theory (HoTT) and the Univalence Axiom vindicate this view by treating identity as structural equivalence rather than primitive property.