 for all moments of time \(t_0, t_1\), and \(t_2\). This means that the past relative to \(t_0\), PAST\((t_0)\), i.e., the set \(\{t \mid t \lt t_0\}\), is linear. In consequence, PAST\((t_0)\) will be a subset of any chronicle through \(t_0\). Much more on the mathematics of partially ordered sets and their use in temporal logic can be found in (van Benthem 1991, in particular in Part I). 