During conversation, people often make assumptions or suppositions that are not explicitly stated. Failure to identify these suppositions may lead to mis-communication. In this paper, we describe a procedure that postulates such suppositions in the context of the discourse interpretation mechanism of BIAS – a Bayesian Interactive Argumentation System. When a belief mentioned in a user’s discourse differs from that obtained in BIAS’ user model, our procedure searches for suppositions that explain this belief, preferring suppositions that depart minimally from the beliefs in the user model. Once a set of suppositions has been selected, it can be presented to the user for validation. Our procedure was evaluated by means of a web-based trial. Our results show that the assumptions posited by BIAS are considered sensible by our trial subjects.