 The notion of ‘common knowledge’ that Lewis is working with here is not the standard modern notion. Lewis does not require that everyone know that everyone know etc., that all of these conditions hold. Rather, when Lewis says that it is common knowledge that p, he means that everyone has a reason to believe that p, and everyone has a reason to believe everyone has a reason to believe that p, and everyone has a reason to believe that everyone has a reason to believe everyone has a reason to believe that p, and so on. That people act on these reasons, or are known to act on these reasons, to form beliefs is unnecessary. And that the beliefs people would get if they acted on their reasons are true is also not part of the view. Hence it is necessary to specify truth as well as common belief in the definition.