In this paper we describe a method for efficient argument-based inquiry. In this method, an agent creates arguments for and against a particular topic by matching argumentation rules with observations gathered by querying the environment. To avoid making superfluous queries, the agent needs to determine if the acceptability status of the topic can change given more information. We define a notion of stability, where a structured argumentation setup is stable if no new arguments can be added, or if adding new arguments will not change the status of the topic. Because determining stability requires hypothesizing over all future argumentation setups, which is computationally very expensive, we define a less complex approximation algorithm and show that this is a sound approximation of stability. Finally, we show how stability (or our approximation of it) can be used in determining an optimal inquiry policy, and discuss how this policy can be used to, for example, determine a strategy in an argument-based inquiry dialogue.