Identity is Irreducibly Relational: A Critique of Primitive Identity from ZFC to Homotopy Type Theory

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), and historical (Leibniz through Kripke).

Key Arguments

• Against primitive identity: The apparent primitiveness in first-order logic reflects a genuine conceptual difficulty
• ZFC already relational: The extensionality axiom makes identity relational for sets
• HoTT vindication: Homotopy Type Theory and Univalent Foundations treat identity as constituted by structural equivalence
• Trajectory argument: The development of foundational mathematics vindicates a relational conception of identity

Links

• ASCRI: systems.ac/4/DAI-2603
• Research Lab: Dissensus AI

Version update (February 2026): Expanded literature review with additional citations and substantive engagement with recent scholarship.