So what is it then? A mathematical approach to the problem aiming to minimize the total effort of the system reveals that I should not put down the toilet seat to avoid the useless effort of putting down the seat only to lift it back up in the event the males of the household go twice in a row. So conveniently, minimization of effort leads us to believe that I shouldn't put it down.
Then again, if effort minimization is the absolute goal, I should just always go with the toilet seat down, and we wouldn't want that would we? Cleanliness is key.
So I am not such a big fan of public bathrooms. You never know what the last guy had on their hands when they pulled the flusher handle or even the faucet at the sink. So I try to avoid contact. Using your shoed feet is a big help here. This includes, probably more than anything else the toilet seat. Women may not be as privvy to the joy of lifting the seat to discover the untold splatter whether fresh or dried. Needless to say, it's never a positive moment in my day.
And that's what I think this is all about. If I put the toilet seat down, a woman doesn't have to encounter arguably - and hopefully - the least sanitary surface in the house. And that is a simple enough luxury to make demands.
But, I have never heard this sentiment expressed directly. Is this it? If so, I think I can put it down. But I am going to do it like I do it in the public bathroom, with my shoe. ;) See, we can all win.