 Mathematical foundations Logic of play as discussed here raises issues of how to interfaces local with global dynamics. This shows in particular with logical limit behavior, where observations and assertions are made repeatedly. Limit models of public announcement, as described earlier, can be ‘self-fulfilling’ or ’self-refuting’. In the first case, the property asserted becomes common knowledge among all agents, whereas in the second it eventually becomes false at the actual world. With soft update on plausibility models, a third option arises, namely, infinite oscillation, or even divergence (Baltag &amp; Smets 2009). To date, there is no general logical theory of these phenomena, but see van Benthem (2011) on the use of fixed-point logics for limit models, Miller and Moss (2005) on the high complexity of public announcement logic with finitely iterated announcements, and Klein and Rendsvig (2017) on limit behavior of product updates.