In this paper I analyse the reason behind the following well settled belief in the international arbitration community: that arbitration exists thanks to the state and its infrastructures. This conceptual error has a bigger practical impact than we might think of at first glance, but has no real basis. The two erroneous starting points that could explain the reason behind this erroneous and well extended belief are: (i) the idea that law and justice are of a public nature; (ii) the unresolved practical problems posed by the lack of imperium of the arbitrators. I analyse why neither of these points has any real basis. Only if this initial conceptual error is dismantled will it be possible to build an appropriate legal theory of international arbitration.