We provide a logical analysis of private international law, the body of law establishing when courts of a country should decide a case (jurisdiction) and what legal system they should apply to this purpose (choice of law). A formal model of the resulting interaction among multiple legal systems is proposed based on modular argumentation. It is argued that this model may be useful for understanding this rather esoteric, but increasingly important, domain of the law. Moreover, it might be useful for modelling the way in which interactions between heterogeneous agents, belonging to different and differently regulated virtual societies, can be governed without recourse to a central regulatory agency.