Global constitutionalists argue that the international legal order can only be meaningfully construed as having an objective, value-based purport. There is, however, something hybrid about the constitutionalist argument, as constitutionalists espouse a normative agenda whilst at the same time setting out to ground their approach in positive international law. It is contended that to avoid both this foundational problem and the charge of utopianism, and as a rejoinder to positivistic arguments for the denial of objective purport, constitutionalists are forced to reason along indirect, transcendental lines. Thus, constitutionalists are to be construed as avouching global values as necessary conditions for making sense of existing international legal practice, rather than merely invoking direct, positivistic evidence and/or mere normative arguments to ground their position. Moreover, it is submitted, first, that global constitutionalists would do better by adopting a less objectivist stance as regards global values, as on the ideal-agent theory of value. Second, it is argued that even though there might be room for so-called constitutionalist ‘mindsets’, these fall short of establishing the objective purport of the international legal order. Third, d'Aspremont's positivistic argument contra objective purport is construed as (also) an argument to the effect that the rules and architecture of the international legal order only warrant the existence of Hobbesian interests as necessary conditions for making sense of it. The constitutionalist case for objective purport, then, hinges on the issue of whether constitutionalism is necessitated by considerations as regards the intelligibility of international legal argument, by explanatory desiderata regarding trends in international law-making, and as a viable response to the problems posed by fragmentation, deformalization, and international legal scepticism.