Beyoncé Logic
Implicitly, Alex was arguing, “If you are an independent, then you have a mind of your own.”
From which she concludes, “Conversely, if you are not an independent,” then you do not have a mind of your own.
Alex, I think, is making both a mistake in English usage and a mistake in logic.
Her mistake in usage is that she should have said “inversely” instead of “conversely.” The converse of “If p, then q” is “If q, then p.” But the last frame concerns an inverse: “If not p, then not q.” An interesting empirical study would look to see how often newspapers or academics misuse these adverbs (I’m sure I have).