Friday analysis of the unexpected hanging paradox

Question

The judge told me:

A1. You will be hanged on day X. (X is some day from Monday to Friday)

B1. You can't deduce what X is.

It's Friday morning and I'm still alive. My first deduction is (please tell me if it's not sound):

A2: I will be hanged today.

B2: I can't deduce if A2 is true or false.

In real life, there's no guarantee what the judge says is true, so I can't know if A2 is true or false. Therefore B2 is true. So there's no paradox here. (If I be hanged today, it will be a surprise.)

But let's assume that A1&B1 is true for sure. Since I've taken for granted that the judge is honest, I deduce that A2 is true, which means B2 is false. Therefore A2&B2 is false, which means A1&B1 is false. I conclude that the judge has made a statement which its truth leads to its falseness. Isn't this a variant of the liar's paradox?

Parallel thread in philosophy forum: https://philosophy.stackexchange.com/questions/50428/friday-analysis-of-the-unexpected-hanging-paradox

Answer 1

I see no parallel to the liar's paradox here. In particular, I don't see how the truth of $A2$ implies the falsity of $B2$, as it is perfectly possible for $A2$ and $B2$ to be both true: you get hanged, but you just don't know that you will be hanged.

Up to the millisecond before you get hanged you still don't know you will be hanged (who knows what strange thing can still happen that prevents you from actually being hanged) .. and of course once you're hanged you won't make any deductions of any kind any more!

So, the judge can easily speak the truth when uttering $A1 \land B1$.

Also, for it to be the liar's paradox, it also need to be the case that if the judge is lying, then the judge is speaking the truth. But that is also not the case: For example, if you don't get hanged .. you don't get hanged! So if the judge is lying ... the judge is lying.

Finally, I disagree with your suggestion that claims about events don't become true or false until they happen: If on Friday you will get hanged, then on Monday the statement that you will be hanged is true; you (and everybody else, despite their best intentions to hang you) just don't know it's true until you're actually hanged.

Answer 2

The problem you propose has an embedded time element to it tht you may not have considered, I'll rephrase to make my reasoning moreexplicit:

On a Sunday, the judge told me:

A1. It was decided by the competent authority that you will be hanged on day X. X is some day from Monday to Friday, which has already been selected.

B1. You can't deduce what X is. We are not informing you which day is X, nor at any time will you have enough information to deduce when is X, except when we come get to conduct you to your execution.

Assuming both statements are true, you deduce that you won't hang on a Friday, otherwise by Friday morning you would know X=Friday. Thus the range of X decreases from monday to thursday. But this reduces the range for X,it does not specify X. But tempted to follow this logic, you think that by Thursday morning you could know that X =Thursday, therefore Thursday is not an acceptable day to satisfy both conditions. You reduce the range again to Wednesday to but still don't know when $X$ is. The problem here, is that by Wednesday morning you haven't reached Thrusday morning, so you didn't reach the point in time when you could be 100% sure of your execution day. So If X=Thursday or X=Wednesday is undecidable at this point from your point of view. But X has already been picked. And it could be either. But being undecidable does not violate being non-deducible. The judge phrase can still be considered true.

More simply, the judge could have said: B1: You can't deduce now what X is. Off course you will on X.

This prevents the paradox immediately. The X=Friday thing seems impossible to happen if the judge sentences are true, thus the premise on your sentences (A2 and B2) is false.

Regarding your more philosophical questions, indeed I would say that no statement about the future is can be certain to be true or false, at any time whatever we say may fail to happen because some alien's death star lasers may finally reach Earth. Even for science related things, predicting the future is an art based on paradigms, that rise and fall in cycles. Yet, sticking to that lack of decisiveness for statements about the future leads to no practical logical conclusions, hence it becomes healthier to live with knowing that (some) statements about the future are "the best we have for now". And in the past, applying such logical has lead to better results in whatever was to be achieved than rejecting all statements Descartes-style.

For the second question, indeed the ideia of omniscience being held by a person could yield some paradoxes, but recall Kurt Gödel theorems: A logical system known to be consistent cannot be complete, and a complete logical system cannot be consistent. Furthermore, a system cannot demonstrate its own consistency.

So if all our science were consistent, there would always be sentences that cannot be proved, while even if all our knowledge was consistent you would never know it.

Answer 3

Regarding your first question: No, this isn't the liar's paradox. You're correct that if you have not been hanged by Friday morning the truth of A1 and B1 entails the truth of both A2 and B2, which are contradictory, so the judge's statement forces its own falsity. But that's not a paradox, that's a proof by contradiction! All you can conclude is it cannot be the case that the judge spoke truthfully and you will not have been hanged by Friday morning, or in other words provided the judge was truthful, you will be hanged on Thursday at the latest. This is where the paradox comes in; you can then repeat the argument, using Thursday as your "last possible day" instead of Friday, to conclude that you must be hung by Wednesday; and so on and so forth. But by eliminating each day, you're adding it back into the running as an option for execution - once you've deduced "I cannot be hanged on $X$ day", you've allowed (by statement B1) $X$ to be the day you're hanged.

Regarding your second question: This has nothing to do with time. As Mefitico pointed out in their answer, it's straightforward to rephrase by referring to the execution date rather than the execution - in other words, so make all of the statements be about a pre-determined execution date.

Regarding your third question: Of course it's possible to know whether a statement about the future is true. For example, in ten minutes, either the moon will be made of green cheese or it won't be. Statements like that, which are what's called deductively valid, are guaranteed to remain true no matter what happens (unless you're willing to accept that deductive reasoning might suddenly stop working). Mathematical statements are typically this kind of thing. The sort of statement that is technically questionable is one formed by inductive reasoning; things like "fire is hot", which is formed by repeated observation of a phenomenon which could conceivably change its properties tomorrow. I recommend the work of David Hume for a good introduction to the philosophical differences between deductive and inductive reasoning.

Regarding your last point about your third question: "Omniscience" isn't an issue unless you really wanted to ask "is it possible to know the truth of every statement in the future". This is a much bigger question, and far beyond the scope of MathSE.