Abduction is a pattern of inference in which an agent seeks an explanation for an observation or report. Iterated abduction is a variety of abduction in which evidence is acquired and explained over time. The long-term goal is to maintain highly plausible consistent explanations for as much of the evidence as possible. Some reports, at the time of acquisition, may be inconsistent with the agent's present beliefs, so some beliefs must be contracted in order to find an explanation of the new reports. Existing work in iterated belief revision only addresses how to maintain consistent beliefs in light of inconsistent observations; whether or not existing beliefs serve as explanations is not considered. What is needed to meet the goal of iterated abduction is a means of seeking new explanations for old evidence when previously-accepted explanations are contracted. We develop a logical formalism for this process as well as a computational implementation.