The prisoner concludes that the judge’s instructions were impossible, which is true. Their conclusion was that there’s an additional possibility: that they don’

The prisoner concludes that the judge’s instructions were impossible, which is true. Their conclusion was that there’s an additional possibility: that they don’t get hung at all. Which is also true. Their mistake is believing that this new possibility will come to pass, instead of realizing that the new possibility means that the judge’s instructions aren’t impossible after all. So in the end, the judge was telling the truth, and the prisoner was mistaken, and then dead.