You missed such a great opportunity for irony. You had all the setup you needed: "You're using argument from authority wrong--and I studied Philosophy."

You even got the "I hope I don't have to explain the problem with that" thing in--totally no argument there, and you could next have pointed to your credentials again. But, alas, you then started to explain things.

What a waste! Common advice for writers is "show, don't tell", you know!