Well, I agree with Auto in spirit about A = A, but I again want to suggest that the Misesian-Mengerian way of phrasing it is more elucidating: that it simply inheres in the approach we take to thinking about A that it is inconceivable (nonsense, meaningless) to write or say “A != A” as a claimed truth. No one ever really believes something that could be accurately described as a contradiction in that sense, because it is impossible to think or believe.
Sure, it is possible to think the words “A and not-A,” but there is no experience that corresponds to those words. Those words don’t refer to anything. They don’t communicate anything that the speaker is experiencing or has ever experienced, hence they are gibberish in the plainest sense.
Therefore, since we know for absolute certain that if someone claims something of the form “A != A” the words must be a miscommunication, I say why not call it a miscommunication rather than say they broke a law? That way they can explain what they did mean, or else they just might realize that they were just spinning meaningless words that didn’t mean anything, even to them (this is actually extremely common). The case for this is elaborated here.