  Transitivity is not the only property which we might want to require of the frame \(\langle W, R\rangle\) if \(R\) is to be read ‘earlier than’ and \(W\) is a set of moments. One condition (which is only mildly controversial) is that there is no last moment of time, i.e. that for every world \(w\) there is some world \(v\) such that \(wRv\). This condition on frames is called seriality. Seriality corresponds to the axiom \((D): \Box A\rightarrow \Diamond A\), in the same way that transitivity corresponds to (4). A \(\mathbf{D}\)-model is a \(\bK\)-model with a serial frame. From the concept of a \(\mathbf{D}\)-model the corresponding notion of \(\mathbf{D}\)-validity can be defined just as we did in the case of 4-validity. As you probably guessed, the system that is adequate with respect to \(\mathbf{D}\)-validity is \(\mathbf{KD}\), or \(\bK\) plus \((D)\). Not only that, but the system \(\mathbf{KD4}\) (that is \(\bK\) plus (4) and \((D))\) is adequate with respect to \(\mathbf{D4}\)-validity, where a \(\mathbf{D4}\)-model is one where \(\langle W, R\rangle\) is both serial and transitive.