Obama’s relatives Did not Vote for McCain
Only about 3 percent of the small ellipse is left of the line to represent the premise that 3 percent of Obama’s relatives voted for McCain. The area that lies within both ellipses represents the people who are both Republicans from California and also relatives of Obama. About 42 percent of that area is left of the line to represent the premise that 42 percent of Republicans from California who were relatives of Obama voted for McCain. The whole dia- gram now shows how all of these premises can be true, even though they lead to conflicting conclusions.
This series of arguments illustrates in a clear way what we earlier called the defeasibility of inductive inferences: A strong inductive argument can be made weak by adding further information to the premises. Given that Mar- vin is a Republican from California, we seemed to have good reason to think that he voted for McCain. But when we added to this the additional piece of information that he was a relative of Obama, the original argument lost most of its force. And new information could produce another reversal. Suppose we discover that Marvin, though a relative of Obama, actively campaigned for McCain. Just about everyone who actively campaigns for a candidate votes for that candidate, so it seems that we again have good reason for thinking that Marvin voted for McCain.
It is clear, then, that the way we select our reference classes will affect the strength of a statistical application. The general idea is that we should define our reference classes in a way that brings all relevant evidence to bear on the subject. But this raises difficulties. It is not always obvious which factors are relevant and which are not. In our example, party affilia- tion is relevant to how people voted in the 2008 election; shoe size presum- ably is not. Whether gender is significant, and, if so, how significant, is a matter for further statistical research.