Well, I might be too dense to grasp what you're saying, but I don't really understand it. I'm not sure, for starters, which axioms you're referring to.

It seems worth noting, however, that there is, and there should be, a distinction between formal logical fallacies and informal logical fallacies. The latter are what I have in mind when you use the term "social formal systems".

Mistaking ad hominem, or argument from authority, etc., for a formal fallacy is a huge mistake - the one OP seems to be referring to.

All I'm trying to say here to you is that there is no need to resort to formal systems theory, because the taxonomy of informal fallacies never was meant to be considered illogical in the same way that Affirming the Consequent (a formal fallacy)is.