Editing Raven paradox (section)

====Role of background knowledge====
Much of the discussion of the paradox in general and the Bayesian approach in particular has centred on the relevance of background knowledge. Surprisingly, Maher<ref name=Maher1999/> shows that, for a large class of possible configurations of background knowledge, the observation of a non-black non-raven provides ''exactly the same'' amount of confirmation as the observation of a black raven. The configurations of background knowledge that he considers are those that are provided by a ''sample proposition'', namely a proposition that is a [[Logical conjunction|conjunction]] of atomic propositions, each of which ascribes a single predicate to a single individual, with no two atomic propositions involving the same individual. Thus, a proposition of the form "A is a black raven and B is a white shoe" can be considered a sample proposition by taking "black raven" and "white shoe" to be predicates.

Maher's proof appears to contradict the result of the Bayesian argument, which was that the observation of a non-black non-raven provides much less evidence than the observation of a black raven. The reason is that the background knowledge that Good and others use can not be expressed in the form of a sample proposition – in particular, variants of the standard Bayesian approach often suppose (as Good did in the argument quoted above) that the total numbers of ravens, non-black objects and/or the total number of objects, are known quantities. Maher comments that, "The reason we think there are more non-black things than ravens is because that has been true of the things we have observed to date. Evidence of this kind can be represented by a sample proposition. But ... given any sample proposition as background evidence, a non-black non-raven confirms A just as strongly as a black raven does ... Thus my analysis suggests that this response to the paradox [i.e. the Standard Bayesian one] cannot be correct."

Fitelson & Hawthorne<ref name=Fitelson&Hawthorne2006/> examined the conditions under which the observation of a non-black non-raven provides less evidence than the observation of a black raven. They show that, if <math>a</math> is an object selected at random, <math>Ba</math> is the proposition that the object is black, and <math>Ra</math> is the proposition that the object is a raven, then the condition:

<math display="block">\frac{P(\overline{Ba}|\overline{H})}{P(Ra|\overline{H})} \ - \ P(\overline{Ba}|Ra\overline{H}) \  \geq \ P(Ba|Ra\overline{H}) \frac{P(\overline{Ba}|H)}{P(Ra|H)}</math>

is sufficient for the observation of a non-black non-raven to provide less evidence than the observation of a black raven. Here, a line over a proposition indicates the logical negation of that proposition.

This condition does not tell us ''how large'' the difference in the evidence provided is, but a later calculation in the same paper shows that the weight of evidence provided by a black raven exceeds that provided by a non-black non-raven by about <math>-\log P(Ba|Ra\overline{H})</math>. This is equal to the amount of additional information (in bits, if the base of the logarithm is 2) that is provided when a raven of unknown color is discovered to be black, given the hypothesis that not all ravens are black.

Fitelson & Hawthorne<ref name=Fitelson&Hawthorne2006/> explain that:

{{quote|Under normal circumstances, <math>p=P(Ba|Ra\overline{H})</math> may be somewhere around 0.9 or 0.95; so <math>1/p</math> is somewhere around 1.11 or 1.05. Thus, it may appear that a single instance of a black raven does not yield much more support than would a non-black non-raven. However, under plausible conditions it can be shown that a sequence of <math>n</math> instances (i.e. of n black ravens, as compared to n non-black non-ravens) yields a ratio of likelihood ratios on the order of <math>(1/p)^n</math>, which blows up significantly for large <math>n</math>.}}

The authors point out that their analysis is completely consistent with the supposition that a non-black non-raven provides an extremely small amount of evidence although they do not attempt to prove it; they merely calculate the difference between the amount of evidence that a black raven provides and the amount of evidence that a non-black non-raven provides.