[AndriyKunitsyn]

Okay, but how could the fact that these elements are in a partially ordered set, or whatever set, trump the basic law of logic, the law of identity, "a = a"?

Or the argument is that NaNs are not actually the values themselves, but the representations of the facts of different failures, and because we can't compare the facts, we shouldn't compare NaNs? Well, I guess one could say that numbers in general are also such incomplete representations; 2 is 2, I could get 2 by adding one crayon to another crayon, or by taking 10 crayons and removing 8 of them. That doesn't stop me from comparing these 2s.