Second, with all due deference to the right of specialists to develop specialized vocabulary, construing “related” to exclude such pairs is a piss-poor choice* that invites confusion, since it conflicts violently with the common meanings of the word ("1. Being connected; associated. 2. Connected by kinship, common origin, or marriage.” AHD, emphasis mine.) Saying father and Vater can be called related but mill and muileann cannot seems like saying that the Wilton brothers are not related to each other but Washoe the chimp and I are.
Third, springing the specialist’s restricted sense into the discussion in the way you did is like telling a programmer who has remarked that he has a lot of work to do that he’s wrong because “work” is force times distance and working at a keyboard involves only small forces acting over small distances. Or like Dr. Doowop’s objection to “delusion” here, except that it’s plausible that he was unaware of the non-specialist sense of “delusion”; “related” is too common a word for you to be unaware of the common senses.
Oh, I quite agree, and in general I try to use commonly understood vocabulary rather than obscure specialist terms (and “phonetic respellings” rather than IPA, which is gibberish to most nonspecialists). My problem here is that I didn’t realize my use of “cognate” was so counterintuitive to most people; now that I know, I will try to avoid it.
The case of father is one where we can state that it is true with extreme confidence, similar to that with which a physicist talks about relativity, but this is not the case with all PIE roots.
Of course not, that’s why I picked a root for which it can be stated with confidence. I’m not trying to get into a detailed discussion of the ins and outs of comparative reconstruction, I’m trying to establish a very basic principle, the difference between inheritance and borrowing. And Dr. T, with all due respect, the connection between mill and muileann is not “somewhere in between.” The fact that both words have undergone phonetic change since they were borrowed is neither here nor there; the important fact is that they were borrowed in the first instance. No amount of phonetic change or analogical reconstruction can change that fact. Neither word is inherited, and that’s what I’m trying to get firmly entered into people’s consciousness.