Editing Raven paradox (section)

====Differing results of accepting the hypotheses====
Several commentators have observed that the propositions "All ravens are black" and "All non-black things are non-ravens" suggest different procedures for testing the hypotheses. E.g. Good writes:<ref name=Good1960/>

{{quote|As propositions the two statements are logically equivalent. But they have a different psychological effect on the experimenter. If he is asked to test whether all ravens are black he will look for a raven and then decide whether it is black. But if he is asked to test whether all non-black things are non-ravens he may look for a non-black object and then decide whether it is a raven.}}

More recently, it has been suggested that "All ravens are black" and "All non-black things are non-ravens" can have different effects when ''accepted''.<ref name=O'Flanagan2008>{{cite arXiv| author=Ruadhan O'Flanagan |date=Feb 2008 |title=Judgment |eprint=0712.4402 |class=math.PR }}</ref> The argument considers situations in which the total numbers or prevalences of ravens and black objects are unknown, but estimated. When the hypothesis "All ravens are black" is accepted, according to the argument, the estimated number of black objects increases, while the estimated number of ravens does not change.

It can be illustrated by considering the situation of two people who have identical information regarding ravens and black objects, and who have identical estimates of the numbers of ravens and black objects. For concreteness, suppose that there are 100 objects overall, and, according to the information available to the people involved, each object is just as likely to be a non-raven as it is to be a raven, and just as likely to be black as it is to be non-black:

<math display="block">P(Ra)=\frac{1}{2} \ \ \ \ \ \ \ \ P(Ba)=\frac{1}{2}</math>

and the propositions <math>Ra,\ Rb</math> are independent for different objects <math>a</math>, <math>b</math> and so on. Then the estimated number of ravens is 50; the estimated number of black things is 50; the estimated number of black ravens is 25, and the estimated number of non-black ravens (counterexamples to the hypotheses) is 25.

One of the people performs a statistical test (e.g. a [[type I and type II errors|Neyman-Pearson]] test or the comparison of the accumulated weight of evidence to a threshold) of the hypothesis that "All ravens are black", while the other tests the hypothesis that "All non-black objects
are non-ravens". For simplicity, suppose that the evidence used for the test has nothing to do with the collection of 100 objects dealt with here. If the first person accepts the hypothesis that "All ravens are black" then, according to the argument, about 50 objects whose colors were previously in doubt (the ravens) are now thought to be black, while nothing different is thought about the remaining objects (the non-ravens). Consequently, he should estimate the number of black ravens at 50, the number of black non-ravens at 25 and the number of non-black non-ravens at 25. By specifying these changes, this argument ''explicitly'' restricts the domain of "All ravens are black" to ravens.

On the other hand, if the second person accepts the hypothesis that "All non-black objects are non-ravens", then the approximately 50 non-black objects about which it was uncertain whether each was a raven, will be thought to be non-ravens. At the same time, nothing different will be thought about the approximately 50 remaining objects (the black objects). Consequently, he should estimate the number of black ravens at 25, the number of black non-ravens at 25 and the number of non-black non-ravens at 50. According to this argument, since the two people disagree about their estimates after they have accepted the different hypotheses, accepting "All ravens are black" is not equivalent to accepting "All non-black things are non-ravens"; accepting the former means estimating more things to be black, while accepting the latter involves estimating more things to be non-ravens. Correspondingly, the argument goes, the former requires as evidence ravens that turn out to be black and the latter requires non-black things that turn out to be non-ravens.<ref name=O'Flanagan2008/>