In this paper, we consider a tractable sub-class of propositional linear time temporal logic, and provide a complete clausal resolution calculus for it. The fragment is important as it can be used to represent simple B?uchi automata. We also show that, just as the emptiness check for a B?uchi automaton is tractable, the complexity of deciding unsatisfiability, via resolution, of our logic is polynomial (rather than exponential). Consequently, a B?uchi automaton can be represented within our logic, and its emptiness can be tractably decided via deductive methods. This may have a significant impact upon approaches to verification, since techniques such as model checking inherently depend on the ability to check emptiness of an appropriate B?uchi automaton. Thus, we also discuss how such a logic might form the basis for practical deductive temporal verification.