Berkson's paradox is a special case of Simpson's for the two subgroups selected and non-selected.
The difference is that Berkson's paradox involves selecting the subgroup a posteriori and in a particular way, Simpson's paradox assumes a selection a priori.
I like how you put paradox in quotes. I also annoys me when people call these things paradoxes. They're more properly called counter-intuitive phenomena. I wonder if there's a single-word name for that.
So I actually checked and... turns out you're right.
According to Wikipedia, "paradox" can either mean "logically self-contradictory statement" or a "statement that runs contrary to one's expectation". I always thought that it meant the former only.
These two concepts should really really have separate words.
Berkson's paradox is a special case of Simpson's for the two subgroups selected and non-selected.
The difference is that Berkson's paradox involves selecting the subgroup a posteriori and in a particular way, Simpson's paradox assumes a selection a priori.