 Generalizations with exceptions illustrate the relationship between Popperian falsificationism and the learning-theoretic idea of reliable convergence to the truth. In some settings of inquiry, notably those involving universal generalizations, a naively Popperian “conjectures-and-refutations” approach of hanging on to conjectures until the evidence falsifies them does yield a reliable inductive method. In other problems, like the current example, it does not. Relying on falsifications is sometimes, but not always, the best way for inquiry to proceed. Learning theory has provided mathematical theorems that clarify the relationship between a conjectures-and-refutations approach and reliable inquiry. The details are discussed in  Section 3  (The Limits of Inquiry and the Complexity of Empirical Problems). Generally speaking methods that solve a learning problem with unfalsifiable hypotheses can be represented as employing a refined hypothesis space where the original hypotheses are replaced by strengthened hypotheses that  are  falsifiable.