Part of the issue is that you are not subject to the principle of explosion. You can assert contradictory things without also asserting that 2+2=3, so you can be confident that you will never tell anyone that 2+2=3 without being confident that you will never contradict yourself. Formal systems using classical logic can't do this: if they prove any contradiction at all, they also prove that 2+2=3, so proving that they don't prove 2+2=3 is exactly the same thing as proving that they are perfectly consistent, which they can't consistently do.