Rejecting a Null-Hypothesis under Uncertainty

Alexandre Borovik has a nice post on the fallacy of null hypothesis rejection. A brief overview of his post.

Typically:

If the null hypothesis is correct, then this datum cannot occur.
It has, however, occurred.
Therefore, the null hypothesis is false.

Thus, for example,

If a person is a Martian, then he is not a member of Congress.
This person is a member of Congress.
Therefore, he is not a Martian.

So far, so good. Now, let throw in some uncertainty.

If the null hypothesis is correct, then these data are highly unlikely.
These data have occurred.
Therefore, the null hypothesis is highly unlikely.

While the above may seem reasonable, and is in fact quite widely misused even in academic literature, the following example highlights the inherent error.

If a person is an American, then he is highly unlikely to be a Congressman
This person is a member of Congress.
Therefore, it is highly unlikely that he is an American.