  \(\mathbf{S4}\) is the system that results from adding (4) to \(M\). Similarly \(\mathbf{S5}\) is \(M\) plus (5). In \(\mathbf{S4}\), the sentence \(\Box \Box A\) is equivalent to \(\Box A\). As a result, any string of boxes may be replaced by a single box, and the same goes for strings of diamonds. This amounts to the idea that iteration of the modal operators is superfluous. Saying that \(A\) is necessarily necessary is considered a uselessly long-winded way of saying that \(A\) is necessary. The system \(\mathbf{S5}\) has even stronger principles for simplifying strings of modal operators. In \(\mathbf{S4}\), a string of operators of the same kind can be replaced for that operator; in \(\mathbf{S5}\), strings containing both boxes and diamonds are equivalent to the last operator in the string. So, for example, saying that it is possible that \(A\) is necessary is the same as saying that \(A\) is necessary. A summary of these features of \(\mathbf{S4}\) and \(\mathbf{S5}\) follows.