Inference to the Best Explanation in the Catch-22: How much autonomy for Mill’s method of difference?

In his seminal "Inference to the Best Explanation", Peter Lipton adopted a causal view of explanation and a broadly Millian view of how causal knowledge is obtained. This made his account vulnerable to critics who charged that Inference to the Best Explanation is merely a dressed-up version of Mill's methods, which in the critics' view do the real inductive work. Lipton advanced two arguments to protect Inference to the Best Explanation against this line of criticism: the problem of multiple differences and the problem of inferred differences. Lipton claimed that these two problems show Mill's method of difference to be largely unworkable unless it is embedded in an explanationist framework. Here I consider both arguments as well as the best Millian defense against them. Since the existing Millian defense is only partially successful, I will develop a new and improved account. As an integral part of the argument, I show that my solutions to the problems of multiple and inferred differences offer new insight into Lipton's main case study: Ignaz Semmelweis's discovery of the cause of childbed fever. I conclude that the method of difference can overcome Lipton's challenges outside an explanationist framework