We explore the idea of using a "possibilistic graphical model" as the basis for a world model that drives a dialog system. As a first step we have developed a system that uses text-based dialog to derive a model of the user's family relations. The system leverages its world model to infer relational triples, to learn to recover from upstream coreference resolution errors and ambiguities, and to learn context-dependent paraphrase models. We also explore some theoretical aspects of the underlying graphical model.