Heritability is unary. Correlation is binary (I'm talking about arity, not domain). You shouldn't "wrong direction" on a unary relation, but I guess that's just another reason I shouldn't have put that in the form of an analogy. I see that you're taking "heritability(trait) X" as "causes(gene-variance,trait-variance) X". That's definitely not what I intended.

I certainly wasn't trying to convince anyone of "heritability is nonsense!". According to Wikipedia, it seems that narrow-sense heritability, with gene-environment correlation removed, would be a direct indication of "genetic variation causes phenotypic variation" (within a framework of simple linear combination of each gene, and environment). I don't know how to tell if someone has actually obtained this number properly, though.