Another post discussed what is commonly called the Paradox of the Unexpected Hanging. The conclusion was not that the judge might have been lying, even though he turned out not to be, and that the lawyer's argument is somehow flawed for not taking that possibility into account. The conclusion was that the judge most definitely was lying (or, used synonymously here, mistaken).
This conclusion follows from a common type of mathematical argument: Consider a set of statements and deduce from them, possibly in combination with other undisputed premises, a contradiction (a pair of statements which are logically contradictory). Then it follows that at least one of the original statements must be false.
That is what happened in the paradox. From the judge's statements, the lawyer proved—quite rightly—that the prisoner cannot be hanged. From the same statements, the hangman proved—equally rightly—that the prisoner can be hanged. It follows that at least one of the judge's statements was false, even if it is not determined which one. So the judge is a liar.
This sort of reasoning is not something which only mathematicians do in abstruse articles in obscure journals. It is something normal people do in everyday life all the time.
For example, consider a lawyer cross-examining a witness:
You say that on the date and time in question you were in Times Square. But a minute ago you said that half an hour later you were on the Champs-Élysées. You are a liar!
Opposing counsel may stand up and say:
Objection! My learned friend has introduced absolutely no evidence about where the witness was on the day in question. So it is quite possible that he was in Times Square when he says he was. It is also quite possible that he was on the Champs-Élysées when he says he was. Hence, the outrageous claim that he is lying must be struck from the record and the jury instructed to disregard it.
Does any reader dispute that this objection would be invalid?
Or consider another real-world example. A lothario is confronted by one of his inamoratas:
You tell me you love only me. But you tell her that you love only her! You are a liar!
Would anybody consider the following to be a valid response?
Now dear, you have not shown that either of my professions of love is untrue. So take back your accusation that I am liar!
