Category: epistemology
Type: paradox
Associated thinker(s): Carl Hempel
Original publication or approximate period: Hempel, “Studies in the Logic of Confirmation” (Mind, 1945)
Difficulty: Intermediate
Themes: confirmation, induction, evidence
Related experiments: Goodman’s New Riddle of Induction (Grue); The Monty Hall Problem
Consider the hypothesis “All ravens are black.” A black raven is an instance of it. Now note that “All ravens are black” is logically equivalent to “All non-black things are non-ravens.” A white shoe is non-black and not a raven, so, by applying the same instance rule to the equivalent statement and then Hempel’s equivalence condition, it also counts as evidence confirming the raven hypothesis in Hempel’s qualitative account.
That seems absurd. A tidy indoor survey of shoes and teacups would seem to tell us something about ornithology. This is the familiar “indoor ornithology” worry, not a logical contradiction.
Why does it seem wrong that a white shoe confirms a claim about ravens, and which assumption—or account of confirmation—should we revise?
The standard derivation combines two commitments: Nicod’s instance principle, on which an observed instance supports a universal generalization, and Hempel’s equivalence condition, on which logically equivalent hypotheses receive the same confirmation. The raven hypothesis and its contrapositive are equivalent, so an observed non-black non-raven is treated as an instance of the latter and thereby as evidence for the former. This is a result within a particular qualitative account of confirmation, not a claim that every theory must assign the two observations equal evidential weight.
Background assumptions matter to other accounts of confirmation, especially probabilistic ones. How much an observation changes support can depend on the probability model, what is already known about the populations of ravens and non-black things, and how the object was selected. Those assumptions are disputed; logical equivalence alone does not establish a numerical degree of support.
Hempel’s response accepts that a non-black non-raven confirms the raven hypothesis under his account when confirmation is considered in isolation from additional empirical information. He explains much of the intuitive resistance by noting that people tacitly bring background knowledge to the example—for instance, that the object is already known to be a non-raven or that non-black things greatly outnumber ravens—while his formal comparison sets that information aside. This diagnosis does not make Hempel’s qualitative confirmation relation sensitive to all such background information, a limitation later discussions address.
Some Bayesian treatments also allow that a white shoe confirms the hypothesis, but argue that it may support it far less than a black raven. The result depends on a probability model and assumptions about the relative populations and how those populations relate to the hypothesis. Peter B. M. Vranas argues that a standard Bayesian solution relies on a disputed independence assumption; this is a criticism of that solution, not a settled rejection of Bayesian approaches generally.
I. J. Good’s counterexamples, as discussed by Jan Sprenger, show why an unrestricted rule that every positive instance confirms a universal hypothesis can fail under some background probability assignments. One constructed example makes observing a black raven favor a possibility in which the hypothesis is false. That is a deliberately artificial counterexample to a rule meant to hold under every circumstance, not a routine claim about ordinary raven observations.
The paradox does not show that shoes tell us anything useful about birds in practice, and it is not a logical contradiction. Hempel’s qualitative result should not be confused with the Bayesian claim that evidence can support a hypothesis by different degrees. Nor is the issue resolved simply by saying it is unscientific to survey shoes: how sampling and background knowledge affect support is part of the debate.
Versions with other generalizations, such as “All emeralds are green,” lead naturally to Goodman’s riddle about which generalizations evidence supports. This is a teaching connection, not a claim that the two puzzles have the same resolution.
Hempel’s 1945 primary text was consulted for the equivalence condition and the formal raven example; only relevant passages were read, not the complete paper. The Stanford Encyclopedia of Philosophy entry on confirmation and relevant passages of Jan Sprenger’s scholarly manuscript were also consulted. Sprenger reports Good’s counterexamples; Good’s original 1967 note was not consulted directly. Vranas’s publisher abstract was read, but not the full article. The Bayesian summary is schematic and does not reproduce formal derivations. The indoor-survey image and questions for reflection are editorial framing, not historical claims.
Research check: 2026-11-01. This is an educational explanation, not a claim that the paradox has one settled resolution or that the entry has received expert review.