Channels' Confirmation and Predictions' Confirmation: from the Medical Test to the Raven Paradox
This work addresses a foundational problem in philosophy of science and confirmation theory, offering a novel solution to a long-standing paradox, though it is incremental in the context of existing measures.
The paper tackles the Raven Paradox by deriving a new confirmation measure c* that supports the Nicod Criterion and undermines the Equivalence Condition, thereby eliminating the paradox. It compares existing measures like F and b*, showing they fail to resolve the paradox, and uses examples to illustrate these difficulties.
After long arguments between positivism and falsificationism, the verification of universal hypotheses was replaced with the confirmation of uncertain major premises. Unfortunately, Hemple discovered the Raven Paradox (RP). Then, Carnap used the logical probability increment as the confirmation measure. So far, many confirmation measures have been proposed. Measure F among them proposed by Kemeny and Oppenheim possesses symmetries and asymmetries proposed by Elles and Fitelson, monotonicity proposed by Greco et al., and normalizing property suggested by many researchers. Based on the semantic information theory, a measure b* similar to F is derived from the medical test. Like the likelihood ratio, b* and F can only indicate the quality of channels or the testing means instead of the quality of probability predictions. And, it is still not easy to use b*, F, or another measure to clarify the RP. For this reason, measure c* similar to the correct rate is derived. The c* has the simple form: (a-c)/max(a, c); it supports the Nicod Criterion and undermines the Equivalence Condition, and hence, can be used to eliminate the RP. Some examples are provided to show why it is difficult to use one of popular confirmation measures to eliminate the RP. Measure F, b*, and c* indicate that fewer counterexamples' existence is more essential than more positive examples' existence, and hence, are compatible with Popper's falsification thought.