A common concern in temporal reasoning frameworks, discussed in detail by Vila and others [VilaVila1994,van Benthamvan Bentham1983], is the divided instant problem. This is the problem that is apparent when considering what happens at the moment of transition from, say, truth to falsity of a propositional variable. The question that must be addressed is whether the proposition is true, false, undefined or inconsistently both true and false at the instant of transition. Clearly the last of these possibilities is undesirable. The solution we adopt is a combination of the pragmatic and the philosophically principled. The pragmatic element is that we choose to model actions as instantaneous transitions with effects beginning at the instant of application. Thus, the actions mark the end-points of intervals of persistence of state which are closed on the left and open on the right. This ensures that the intervals nest together without inconsistency and the truth values of propositions are always defined. The same half-open-half-closed solution is adopted elsewhere. For example, Shanahan shanahantutorial observes that a similar approach is used in the event calculus, although there the intervals are closed on the right. Although the two choices are effectively equivalent, we slightly prefer the closed-on-the-left choice since this allows the validation of a plan to conclude with the state at the point of execution of its final action, making the determination of the temporal span of the plan unambiguous.
From a philosophical point of view the truth value of the proposition at the instant of application of an action cannot be exploited by any other action, by virtue both of the no moving targets rule and our position, outlined above, that a valid plan cannot depend on precise synchronisation of actions. This forces actions that require a proposition as a precondition to sit at the open end of a half-open interval in which the proposition holds.