Events and time in a finite and closed world

There are numerous occasions on which we need to reason about a finite number of events. And we often need to consider only those events which are given or which we perceive. These give rise to the Criteria of Finiteness and Closedness. Allen's logic provides a way of reasoning about events. In this paper I examine Allen and Hayes' axiomatisation of this logic, and develop two other axiomatisations based on the work by Russell and Thomason. I shall show that these three axiomatisations are weakly equivalent, and that only the last two meet the Criteria of Finiteness and Closedness (to different degrees). I shall then examine two ways of constructing instants of time in a finite and closed world, i.e. the Russell construction and the Thomason construction. I shall prove that these two constructions are equivalent under certain conditions.