Undermining and confirmation
Next, undermining. Suppose we are testing a model H according to which the probability that a certain type of coin toss results in heads is 1/2. On some accounts of physical probability, including frequency accounts and "best system" accounts, the truth of H is incompatible with the hypothesis that all tosses of the relevant type in fact result in heads. So we get a counterexample to simple formulations of the Principal Principle: on the assumption that H is true, we know that the outcomes can't be all-heads, even though H assigns positive probability to all-heads. In such a case, we say that all-heads is undermining for H.
Underminingness comes in degrees. All-heads is strongly undermining insofar as it directly refutes H. By contrast, the hypothesis that 80 percent of the first ten million coin tosses result in heads may be logically compatible with H, even on best system (or frequency) accounts, but it still undermines H. Thau, Hall, and Lewis 1994 showed how underminingness can be quantified, and why even quite ordinary evidence is often very slightly undermining.
How does undermining affect confirmation? Well, if we receive some evidence that is logically incompatible with a hypothesis, then obviously the hypothesis is refuted. So in undermining cases we must take care not to conflate Cr(E/H) in Bayes' Theorem with the probability that H assigns to E. In other words, we must take care not reason by "direct inference".
So undermining-type counterexamples to the simple Principal Principle do not pose a serious challenge to Bayesian Confirmation Theory. However, they do contradict some popular ideas about the likelihoods Cr(E/H) in Bayesian Confirmation Theory. For example, in this recent handbook article on BCT, James Hawthorne claims that the "likelihoods represent the empirical content of hypotheses: what hypotheses say about the observationally accessible parts of the world" (p.202). He takes this to explain why there can't be rational disagreement about likelihoods. However, if we allow for the possibility of undermining, then the likelihoods can't be interpreted as what the relevant hypotheses say about the world: H may say that all-heads has probability 0.000001, yet Cr(all-heads/H) = 0.
Undermining also presents a challenge to one of the main rivals to Bayesian Confirmation Theory: likelihoodism. Likelihoodists argue that all there is to the confirmation of a probabilistic theory H by E is somehow captured by the probability assigned to E by H. In undermining cases that is clearly false. The fact that H is refuted by all-heads cannot be retrieved from the (non-zero) probability that H assigns to all-heads.