No. This is not the case. Just because something is a fallacy doesn't make its negation true. Thus for example (P->Q) -> (Q->P) is a fallacy. But ~((P->Q) ->(Q->P) ) is not a theorem of first order logic. So even if I throw (P->Q) -> (Q->P) as an additional axiom in I can't get a general explosion in first order logic. Contradictions lead to explosion, but fallacies do not necessarily do so.
So even if I throw (P->Q) -> (Q->P) as an additional axiom in I can't get a general explosion in first order logic.
Sure you can.
Edit:
(P->Q) -> (Q->P) is not a fallacy. ∀P,Q: (P->Q) -> (Q->P) is a fallacy, and its negation is ∃P,Q: (P->Q)^~(Q->P) which is indeed a theorem in first order logic.
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