Knowing That, Knowing What, and Public Communication: Public Announcement Logic with Kv Operators

In his seminal work [Plaza, 1989], Plaza proposed the public announcement logic (PAL), which is considered as the pilot logic in the field of dynamic epistemic logic. In the same paper, Plaza also introduced an interesting “know-value” operator Kv and listed a few valid formulas of PAL+Kv. However, it is unknown that whether these formulas, on top of the axioms for PAL, completely axiomatize PAL+Kv. In this paper, we first give a negative answer to this open problem. Moreover, we generalize the Kv operator and show that in the setting of PAL, replacing the Kv operator with its generalized version does not increase the expressive power of the resulting logic. This suggests that we can simply use the more flexible generalization instead of the original PAL+Kv. As the main result, we give a complete proof system for PAL plus the generalized operator based on a complete axiomatization of epistemic logic with the same operator in the single-agent setting.