Quite the contrary. A pretty simple reductio proof demonstrates that more housing lowers prices. All you need is to imagine a market in which there is one more apartment available than there are people to rent it.
I actually regret making that comment. We should be suspicious of logic-based arguments about topics for which there is actual evidence. We already know what it looks like when supply outpaces demand, because there are housing markets all over the rust belt where that's the case.
