In the current paper, we re-examine the concept of strong admissibility, as was originally introduced by Baroni and Giacomin. We examine the formal properties of strong admissibility, both in its extension-based and in its labelling-based form. Moreover, we show that strong admissibility plays a vital role in discussion-based proof procedures for grounded semantics. In particular it allows one to compare the performance of alternative dialectical proof procedures for grounded semantics, and obtain some remarkable differences between the Standard Grounded Game and the Grounded Persuasion Game.