First order logic can only detect formal logic fallacies. Informal logic fallacies like ad hominem, strawman, red herring, etc. are cast in language. They can't me defined and resolved mathematically. The model should be fine tuned with examples of these informal fallacies and counter-arguments to them. Even so it won't be able to detect them in all cases, but it will at least have some knowledge about them and how to reply to them. This knowledge could be further be refined with in context learning and other prompt engineering strategies.