 At world w a formula \(\varphi\) is commonly known among the agents if and only if \(\varphi\) is the case in every world (the “for all” in C’s semantic interpretation) that can be reached by any finite non-zero sequence of transitions in \(R_E\) (the fact that \(R_C\) is the transitive closure of \(R_E)\). In other words, \(\varphi\) is commonly known among the agents if and only if everybody knows \(\varphi\) (any sequence of length 1), everybody knows that everybody knows \(\varphi\) (any sequence of length 2), and so  on.[5]