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Abductive reasoning is most easily understood through the analogy of a doctor diagnosing his patient’s illness. He gathers a hypothesis from the patient’s symptoms, or otherwise evidence that he deems factual, and from there, goes down the list of maladies and tries to assign the appropriate illness. This is as opposed to deductive or inductive reasoning.
More generally, abductive reasoning is the logical process where one chooses a hypothesis that would best fit the given facts.
In terms of archaeology, this type of reasoning was used Sir John Beazley. He would look at the finer details such as the nose and the eyes of Greek pottery then prescribe the entire piece to an artist according to similar characteristics from other pottery of the same artist.
Beazley used abductive reasoning to identify a vast number of pottery. He attributed the vases to their potters' hands and workshops. In using abductive reasoning, Beazley applied the Italian Renaissance approach of connoisseurship where many works are grouped together to identify an artist's style. Such style "cannot be grasped through the study of signatures, appraising the overall look of the work, or looking only at larger features such as composition or iconography" (Whitley 37).
Thus, Beazley uses abductive reasoning, examining many of the small details and then appropriating the piece's artist.
Biers, William R. The Archaeology of Greece. Ithaca: Cornell University Press, 1996.
Whitley, James. The Archaeology of Ancient Greece. New York: Cambridge University Press, 2007.