You can&#x27;t have the same axioms with different rules of inference. The rules of inference are axioms.<p>By the axioms being the same, you mean the ink-shapes making up the symbols on a piece of paper being the same.<p>Not the actual meaning behind them.<p>Edit: Since you added more content to this reply later on, let me respond. In this case I was talking about specifically a formal system where a=0 makes sense and is expressible, but to keep it short and concise I didn&#x27;t choose to write out all the details. So the rebuttal is moot.