Past exam of the philosophy of science supplementary subject of the University of Oxford / 2015 / 2 / b / Solution

The raven paradox combines a plausible equivalence principle with an overly unrestricted account of positive-instance scientific confirmation. Let $R(x)$ and $B(x)$ mean that $x$ is a raven and black respectively. ThenA white shoe is a positive instance of the right-hand generalization. If every positive instance confirms its universal generalization, and logically equivalent statements receive the same evidential support, inspecting the shoe seems to confirm the original raven claim. There is no contradiction in the equivalence itself. The tension concerns the evidential rule and the intuition that an arbitrary shoe teaches us nothing about birds.^[ref]

The right response is to make background knowledge and sampling explicit. Under Bayesian confirmation, evidence increases support for $H$ precisely when its likelihood is greater under $H$ than under the alternative, assuming nondegenerate prior probabilities. A fact can therefore be compatible with $H$ yet provide no new support. The mere existence of a white shoe does not determine the needed likelihood ratio.^[ref]

Consider a deliberately simple finite model. There are $N$ objects, exactly $r$ ravens, and $M=N-r$ nonravens, with $M>r>1$. All nonravens are white. Under $H_0$ all ravens are black; under $H_1$ exactly one raven is white. Draw uniformly from the entire population. The event of drawing a white nonraven has probability $M/N$ under either hypothesis. Its Bayes factor is one and it leaves the prior odds unchanged. A black raven instead has likelihoods $r/N$ and $(r-1)/N$, so gives Bayes factor $r/(r-1)$ for $H_0$.

Now change the protocol: draw uniformly from the white objects, then discover that the selected object is a nonraven. This result has likelihoods $1$ and $M/(M+1)$. It gives the small positive Bayes factorFor $N=1000$ and $r=10$, this is $991/990$, whereas sampling a black raven from the full population gives $10/9$. The white nonraven can now help because the search could have revealed a white raven. The two experiments have different selection information, so their differing evidential values do not violate logical equivalence.

This model establishes possibilities, not a universal numerical law about shoes. Different alternatives, population information or selection procedures change the result. If the shoe's being a nonraven was already known, discovering that it is white need not add anything. Conversely, ruling out a potential counterexample can supply a small amount of information. Retain logical equivalence, but reject the unconditional instance-confirmation rule. Once the complete evidential situation is specified, neither zero support nor weak positive support is paradoxical, and neither amounts to proof of a universal law.