If seeing a black raven confirms "all ravens are black", then seeing a white shoe confirms "all non-black...

Kevin B. Korb, "Infinitely Many Resolutions of Hempel's Paradox",
Dept. of Computer Science, Monash University, Clayton, Victoria,
Australia, November 29, 1993.  [ Editor: published in R. Fagin
(ed.), Theoretical Aspects of Reasoning about Knowledge: Proceedings
of the Fifth Conference (TARK 1994), Morgan Kaufmann, 1994,
pp. 138-149. ]

Abstract:

   What sorts of observations could confirm the universal hypothesis
   that all ravens are black?  Carl Hempel proposed a number of simple
   and plausible principles which had the odd ("paradoxical") result
   that not only do observations of black ravens confirm that
   hypothesis, but so too do observations of yellow suns, green seas
   and white shoes.  Hempel's response to his own paradox was to call
   it a psychological illusion--i.e., white shoes do indeed confirm
   that all ravens are black.  Karl Popper on the other hand needed no
   response: he claimed that no observation can confirm any general
   statement--there is no such thing as confirmation theory.  Instead,
   we should be looking for severe tests of our theories, strong
   attempts to falsify them.  Bayesian philosophers have (in a loose
   sense) followed the Popperian analysis of Hempel's paradox (while
   retaining confirmation theory): they have usually judged that
   observing a white shoe in a shoe store does not qualify as a severe
   test of the hypothesis and so, while providing Bayesian
   confirmation, does so to only a minute degree.  This rationalizes
   our common intuition of non-confirmation.

   All of these responses to the paradox are demonstrably
   wrong--granting an ordinary Bayesian measure of confirmation.  A
   proper Bayesian analysis reveals that observations of white shoes
   may provide the raven hypothesis any degree of confirmation
   whatsoever.

-- Kevin B. Korb

[ Editor: summary.  Korb measures the support e gives h by the
likelihood ratio lambda(e|h) = P(e|h)/P(e|not-h), which exceeds 1
exactly when the posterior probability of h exceeds the prior.  His
point is that this ratio depends on the protocol by which the
observation was obtained (e.g. picking a raven and finding it black
versus picking a black thing and finding it a raven) and on the space
of alternative hypotheses, not just on the recorded observation.  In
a small worked example (a world of ravens and shoes; under h, 10 black
ravens, under not-h, 5 black and 5 white ones; 10^6 black and 10^9
white shoes either way), a white shoe drawn at random from all objects
gives lambda = 1, no impact at all; a black raven gives lambda = 2
when drawn from the ravens and 2 - epsilon when drawn from the black
things (agreeing with Horwich against Howson and Urbach that the
protocol matters, but not because of "falsifiability"); and a protocol
of drawing from the white things gives a white shoe 1 + epsilon, which
he offers as the honest form of the usual Bayesian "minute degree"
answer.  With other protocols and hypothesis spaces any likelihood
ratio can be obtained, hence the title.  A note shows the same protocol-dependence for a coin: with e =
"equal numbers of heads and tails", h = bias 2/3 against fairness,
flipping twice gives lambda = 8/9, flipping four times gives
(8/9)^2, and flipping until heads equal tails gives 2/3.  He concludes
that positive instances commonly but not necessarily confirm, so that
it is Nicod's criterion, not Hempel's equivalence condition, that
fails, and that "We should well and truly forget about positive
instance confirmation: it is an epiphenomenon of Bayesian
confirmation." ]

[ Editor: the full paper was in the TODO collection as
hempel.tarkpaper.ps.gz. ]