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Current discussions of "Explainable AI" (XAI) do not much consider the role of abduction in explanatory reasoning (see Mueller, et al., 2018). It might be worthwhile to pursue this, to develop intelligent systems that allow for the observation and analysis of abductive reasoning and the assessment of abductive reasoning as a learnable skill. Abductive inference has been defined in many ways. For example, it has been defined as the achievement of insight. Most often abduction is taken as a single, punctuated act of syllogistic reasoning, like making a deductive or inductive inference from given premises. In contrast, the originator of the concept of abduction---the American scientist/philosopher Charles Sanders Peirce---regarded abduction as an exploratory activity. In this regard, Peirce's insights about reasoning align with conclusions from modern psychological research. Since abduction is often defined as "inferring the best explanation," the challenge of implementing abductive reasoning and the challenge of automating the explanation process are closely linked. We explore these linkages in this report. This analysis provides a theoretical framework for understanding what the XAI researchers are already doing, it explains why some XAI projects are succeeding (or might succeed), and it leads to design advice.