There have been already two questions asked about "for that matter":
I read them carefully and the following dictionary entries:
- "for that matter" in MacMillan Dictionary
- "for that matter" in Cambridge Dictionary
- "for that matter" in Oxford Living Dictionaries
The meaning explanations seem to contradict.
One group of explanations point out that "for that matter" is used to introduce a generalization or extension:
He doesn’t like young women, or any women for that matter.
Ming's never been to Spain, or to any European country for that matter.
He didn't eat the cucumbers, or any food at all for that matter.
I didn't complain about the food. I have nothing but praise for the staff for their service, for that matter.
Whereas the other group of explanations just say "used to indicate that a subject, though mentioned second, is as relevant as the first":
I am not sure what value it adds to determining public, or for that matter private, policy.
I don't like bread and rice, or porridge for that matter.
So, is "for that matter" not only for generalization purposes but to add any kind of information to a statement? Even for correction of something what has been said (as "public policy" vs. "private policy" in one of the examples above)?