Is there an adverb for circularly using the function/method you are discussing?

For instance, John Mill argues in his essay "On Liberty" about how you must address controversy of opinion in order to become more certain of your truth, and in the structure of his essay he brings up counter points and defeats them in order to prove his truth.

Something of the essence of reciprocal, circular, reusing, using what you’ve established

Adverb form like mistakenly, or correspondingly

"Mill proves his argument by _____ addressing counter points to prove his truth."

Perhaps there isn’t a word for this?