Good questions, Walid.

But first of all, the contraposition of “All mathematicians are smart” is “All non-smart (people) are non-mathematicians” which is not quite the same as your H1'.

what does the Bayesian analysis say about someone who is a mathematician AND a teacher, or a mathematician AND a singer?

Both of those people fall into the category of mathematicians, and are outside the group of non-mathematicians, so if they are smart, they are evidence for “All mathematicians are smart,” but would not be evidence against the statement that “All non-mathematicians are not smart”

Also, I would like to point out that I see no reason why “natural kinds” should be relevant but not “artificial kinds”. I am taking “natural kinds” to means something like explained here. At that link they give an example of “footwear under $100” as an artifical kind. The raven paradox could also apply to this category, even though it is artificial. Consider the following statement:

All footwear under $100 is black.

Does the fact that this category mean that we can’t gather evidence about it?

Finally, my intuition is that Bayesian analysis actually does take natural kinds into consideration. If something is a natural kind, it seems like we might have priors about it. If not, it may be more difficult to identify our priors or to find a representative sample.

For example, consider the following statements:

H1: All extra-terrestrial life is not carbon based

H1': No carbon based things are extra-terrestrial life

Suppose I pick up 50 things. 30 are carbon based, 20 are not carbon based. None are extra-terrestrial life. Does that confirm H1? Not directly, since nothing in my sample is extra-terrestrial life. Does it confirm H1'? I have a sample of 20 carbon based things, and none are extra-terrestrial life. If this were a representative sample of carbon based things, I would estimate that the percentage of carbon based things that are extra-terrestrial life is approximately 0.

However, I have some other additional relevant information, which Bayesian analysis encourages me to use. Namely, I believe that

H2: All extra-terrestrial life exists far away from Earth; or

H2': All things close to Earth are not extra-terrestrial life

My sample was all collected from Earth, so while it may be representative of things close to Earth, I have no reason to believe it is representative of things far away from Earth.

Perhaps this is all you mean by your statement about natural kinds. We can’t use Bayesian inference if we don’t have a representative sample…and if something is not a natural kind it can’t have a representative sample. Is that the point you are trying to make?