A black raven
The paradox of Hempel (sometimes called the ravenparadox) is a paradox by German philosopher Carl Gustav Hempel in the 40's to illustrate a problem where inductive logic contradicts intuition. The problem
Hempel gives an example of the induction principle: the statement that all ravens are black. A sample is being taken and one million ravens are checked and they are all black. After each observation, the belief in the statement that "all ravens are black" will increase slightly. The induction principle seems reasonable here. Non-black non-raven
Now the problem is coming. The statement "all ravens are black" is logically equivalent to the statement "all non-black things are non-raven". If we notice a red apple, that is consistent with this second statement. A red apple is a non-black thing, and it's a non-raven. Thus, by the induction principle, one can say that observing a red apple increases the belief in the statement that all ravens are black. This goes against the intuition.
In fact, there is still a chance that a white raven will be found, because even this case can be tested one million times (like an apple one million times from a tree), "infinite" remains one million times of occurrence of a white raven (in which the apple does not fall out of the tree). The argument that all non-black things are also non-raven are therefore subject to sufficient inductive experience. The moment one white raven is discovered, which is highly unlikely, due to the inductive experience of one million black ravens, this "certain knowledge" collapses. Decision: Knowledge remains uncertain, because one has to leave for induction, instead of deduction.
However, the story is not finished. In 1967, British mathematician Jack Good wrote an article describing Hempel's paradox. Hempel is based on the intuitive premise "A case of an hypothesis supports the hypothesis". However, this is not correct.
Example: Take world 1 with a million birds. Hundreds of these are ravens, all blacks. World 2 has one million ravens of which there are one hundred and one thousand black and the rest is white. One now observes a bird, a black raven. In which world are you? Application of the Bayes statement with equal a priori for the two hypotheses teaches that one can assume 1000 by 1 in world 2. The fact that a black raven was observed can now clearly not be construed as a support for the hypothesis that all ravens are black. This shows that Hempel's premise is in general untrue. Hempel's reasoning comes to loose, the paradox disappears.
The crux of the example is that the assessment of a hypothesis must always be done in competition with other hypotheses that are (also) arranged to determine the probability of the observations.
wiki
