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While you may find appeals to arguments from the regression to the mean to be "horrendously bad", I can only report that, so far as I have been able to make out, the logical legitimacy of such arguments is pretty much taken for granted among the disputants on both sides of the IQ nature/nurture controversy.

The first link you point to, which seems most directly to address the issue of regression to the mean, in turn points to papers which were written about 30 years ago or more, without, it seems, anyone in the dispute taking them seriously.

Don't you think that that would suggest that there's something deficient in the argument that use of regression to the mean in this context is a logical fallacy?

Here's the basic problem with claiming that regression to the mean in the context of, say, human traits is simply some mathematical artifact: it does nothing to explain WHY there should be a regression to the mean.

Yes, not only do the average IQs (or heights) of children regress to the mean from the average IQs of their parents; the opposite is also true -- the average IQs (or heights) of parents regress to the mean from the average IQs of their children. Does that mean that there is no causal relation established by regression to the mean effects? No, absolutely not. It only establishes that the direction of a causal arrow can't be determined from the fact of regression to the mean alone. But we know the direction of that arrow, if the cause is genetic (or environmental, presumably): it goes from parents to children, not the other way around. When we understand this, we can also explain why we see regression to the mean in the other direction as well; the same underlying set of causes are working, though, again, the direction of the causal arrow is opposite.

The fact of regression to the mean strongly argues that there is SOME underlying causal mechanism (be it genetic or environmental or a combination) that explains that fact. Why is it that the children of high IQ parents regress partly to the mean, but not all the way?

Regression to the mean in traits in both directions, from children to parents and vice versa, can be explained by luck -- those parents or children who have greater IQs or greater heights are, on average, luckier than average; they are, in particular, luckier than their own children or parents, respectively. But what are they luckier AT? What have they received more of? If one says, genes that increase the trait in question, then a perfectly coherent explanation emerges. One might say that they've received a better environment -- but that becomes a very difficult explanation in the case of IQ, since typically quite the opposite seems to be true (parents with high IQs have on average greater incomes and generally should establish a better environment for their children than they themselves experienced.)

In short, the existence of regression to the mean in the expression of traits across generations presents an important fact -- one that one might not a priori expect. Something must explain that fact. Do you seriously think that that explanatory problem simply goes away by declaring that appeals to regression to the mean constitute a "logical fallacy"?