THE PARADOX OF CONFIRMATION (II)

The hypothesis, H, that all crows are black is the same as that all non-black things arc non-crows, and this is supported by the observation of a white shoe. The kernel of my argument was to nm*iAi*r a a by a contingency table with its rows labelled crow and shoe , and tts columns bock and white t and then to end an algebraic expression for the weight of evidence in favour of U in virtue of the evidence, £ , of the observation of a white shoe. (The use of the Stooge allows us to restrict our attention to a by 2 1 HHW *" y tables.) If the marginal totals of this table are known in advance, then E reaDy does support H. But I stated in Pan I mat this conrUition follows even if the marginal totals are not known in advance, but only have probability distributions. Here I made a mistake. In order to see this, let us suppose that we know in advance the total number of black shoes and the total number of white shoes that may be observed, each shoe being equally likely to be observed. Then it is a very simple matter to prove that die weight of evidence in favour of H provided by £ is tero, where ' weight of evidence' is defined, for rramplr. in Pan L Thus a * case of a hypothesis' does not necessarily support the hypothesis. (The example of a white raven shows that a case of a hypothesis can undermine it, but this example is irrelevant to Stooge-type experiments.) The question arises, what makes us imagine that a case of a hypothesis must support it? Light is shed on diis question by supposing an object to come gradually into view. Suppose that at some momrnt the object is seen to be white (event F). Then P(E| F. H)«= 1, where £ a the proposition that the object is a non-crow. But P(E | F. H) < 1. Hence W(H : E \ F) > o. For example, the observation of a white shoe supports H when the object is already known to be white. But W(H :F)<o under a wide class of assumptions, including those where one knows in advance the physical probability that die first object observed will be a black crow. (For on this assumption, the probability of seeing a white object is a litde smaller given H than given H. Thus it may easily happen that W(H :E.F), which is equal to W(H:F)+ W(H :E\F), may vanish although W(H:E\F)>o. Let us express the matter more generally. If we wish to test a hypothesis of the form that all A\ are B*s, we may go on sampling objects until we find an A, and then look to see if it i» a B, the observation being of Stoogian type. If it is a B, then the 1 1 . J. Good. Aajamul, i960,11, 145-140