 This result rules out the possibility of a reduction theorem for \(\PALC\): we cannot find a public announcement-free equivalent of every \eqref{PAL+C}-formula. This led van Benthem, van Eijck, and Kooi (2006) to develop a common knowledge-like operator for which a reduction theorem does hold. The result is the binary relativized common knowledge operator \([B*](F|G)\), which is read, “F is common knowledge among group B relative to the information that G is true”. The language \eqref{RCK} of relativized common knowledge is given by the following grammar: 