He absolutely gave her something to use against him by being sarcastic in a public forum, but I think he made a rational decision that an interesting dialogue in which he could be called snide would catch much more attention than the dull one in which he makes a polite, logically airtight case and receives a shorter reply full of nothing much.

Edit: Oh, I was going to add: and I now know a lot more about Armstrong than I would otherwise, namely, that her argumentative approach is deceitful and based on manipulating her audience's moral feelings.