Liar's Paradox Revisited

This is my second shot at the Liar's Paradox (and the Truth Teller), but I've made it a little tighter and simpler:

When we call a statement true or false, we are saying that we have compared that statement to a given fact or principle and either found corraboration or contradiction.
If we say, as in the Liar's Paradox, that "this very statement is false," we are limited by our terms to comparing that statement to the bare fact or principle of how we find a statement to be true or false.
But, since "this very statement is false" does not match or satisfy the very fact or principle of how we comparatively find a statement to be true or false, then that statement is in contradiction to the correct standard for finding truth or falsity, and thus it ultimately makes a true statement that it makes a false statement about its own truth value.
The fact that the statement calls itself false cannot supplant the correct process for finding truth or falsity, and so the statement is ultimately true.

Similarly, if we say, as in the Truth Teller, that "this very statement is true," this self-declared truth value does not match or satisfy the very fact or principle of how we find a statement to be true or false, and so the statement falsely asserts its own truth, and thus it ultimately makes a false statement.

If this is still too complex and ugly and unconvincing, please say so.

Very Respectfully,
Ray Donald Pratt

 

Log in or Sign up to hide all adverts.

unexpected hanging paradox

how do you resolve the unexpected hanging paradox?

a man is condemned to be hanged sometime before the year is out (its early in the year). the sheriff, wanting to make him suffer, tells him that when the day of his hanging comes that he the prisoner wont know it. the prisoner asks himself 'what is the very last day that the sheriff can hang me on without breaking his word'?


notice that even though this is clearly a paradox there is nothing in it that could be considered a self contradictory statement. everything in it is realistically possible.

 

Log in or Sign up to hide all adverts.

granpa,
really, any day. (I do know the whole, it can't be the last day because he would know and then it can't be the day before the last....and so on). but the 'won't know it' sounds like a category that we can use in logical formulas but it really isn't.

 

Log in or Sign up to hide all adverts.

logical formulas be danged. either he will or he wont know. its that simple. there is nothing unrealistic about it.

you are right though about one thing. there are certainly many days that he can be hanged on. but the question isnt how many days but rather which day is the last day?

 

Wow, I burned my last bag of tortilla chips getting enough sustained mental energy to work on the Liar's Paradox, and now this!

Let's see, if it's 12-31, the prisoner knows that the Sheriff goofed, so that's the day!

Very Respectfully,
Ray Donald Pratt

 

prisoner?
you aint still in jail are you?

i took a crack at the paradoxes and suffered brain damage
sorry

 

I've constructed a solution to the Liar's Paradox that makes use of the supposed paradox, and a solution to the Truth Teller that should have disposed of it in Aristotle's time:

When we call a statement true or false, we are saying that we have compared that statement to a given fact or principle and either found corroboration or contradiction. For example, if we say "All apples are cubes that glow in the dark," we can only judge that statement as being true or false by comparing the statement to what we know about apples.

In the Liar's Paradox, we say "This very statement is false." The alleged paradox is that if the statement is true, then it is false as it claims, but if it is false as it claims, then it has stated the truth and cannot be false, ad infinitum. To judge the truth or falsity of "This very statement is false," we must compare the statement not only to itself as explicitly required, but also to what we know about finding the falsity of any general statement. This is implied by the use of the term "false," much like the term "apple" would require us to compare a statement to what we know about apples. With the Liar's Paradox, the very fact of the supposed paradox proves that the statement cannot be definitively proven false. And as such, the statement is ultimately true because it admits that it falsely asserts that it is provably false.

Similarly, in the Truth Teller, we say "This very statement is true." Although there is no alleged paradox, the statement's self-reference to its own veracity is not sufficient evidence for its truth. The informal fallacy called petitio principii, or begging the question, occurs where a questioned fact is called in as proof of that fact, and such proofs are always illegitimate. However, we can go further here and say that the statement is definitely false because it falsely claimed that it was provably true.

Very Respectfully,
Ray Donald Pratt

 

So your argument is that it's true, because it isn't false?
You say that "This statement is false" is a true statement?

 

The two parts of the sherrif's promise are logically incompatible:
1 - You will be hanged next week
2 - The day of your hanging will be a surprise

The prisoner rationally concludes from the second statement that the first cannot be true.
So, the prisoner knows that the sheriff isn't completely rational, which makes all bets off. Since the prisoner can not be certain that they'll be hanged at all next week, any day of that week (even the last day) would be a surprise.

The sheriff's irrationality can be seen more clearly by reducing the scenario to one day:
"You will be hanged tomorrow, but only if it will be a surprise."
Its obviously not a surprise, so the prisoner knows that they won't be hanged tomorrow. When the sheriff comes in to hang him anyway, the prisoner is flummoxed - "But I can't be hanged today!" To which the sheriff responds, "Surprise!"

 

The prisoner rationally concludes from the second statement that the first cannot be true.

ydu cant be serious. in my version i specifically said that it was the beginning of the year and the prisoner had to be hanged sometime that year. i did this to make it obvious that in fact he can be hanged. the sheriff has only to choose a day at random from that whole year. how could the prisoner know when a randomly selected day arrives?

 

But taking the sheriff's words literally, the day is not randomly selected. If it's going to be a surprise, then the probability that it will be the last day is zero. It logically follows that it can't be the second last day, either. Or the third, fourth... or any day at all! So the sheriff's statements are not logically consistent.

If the prisoner is as smart as the story says, he'll understand this, and realise that there is, in fact, a small chance that he won't be hanged until the last day of the year. It's the small chance that the sheriff is mistaken that in practice allows the sheriff's prediction to come true.

 

Not clearly a paradox, the Sheriff could decide to dose him with luadnum and hang him while incoherant. Thus the day of the hanging the prisoner would not know it.

 

why do people make this so diffecult. clearly the sheriff has only to choose a day at random and the prisoner cannot possibly know when it arrives. so we know for a fact that there are some days on which the sheriff can hang the prisoner and still keep his word. the question the prisoner asks himself though is 'what is the last day the sheriff can hang me on and still keep his word'. clearly if the prisoner could deduce what that day was then the sheriff would choose some other day. so the conclusion is that the prisoner cannot know what the last day is. it is unknowable from his point of view.

 

Unless he randomly chooses the last day of the year. And he can't choose the second last day either... unless you implicitly allow the possibility that he might have chosen the last day. That's the whole point of the "paradox".

You resolve the paradox by seeing that the sheriff can choose the second last day, because the prisoner doesn't know for sure if the sheriff was just yanking his chain.

You've made this harder to see by choosing a large number of days. With this many days, it's hard to see the paradox to start with. The usual form of the makes it only a week. Reducing the days even further makes it even more obvious.

 

if he chooses a day at random somewhere in the first six months of the year then how exactly would the prisoner know when it arrived that he was going to be hanged that day? he cant. so this proves that there are some days that he can be hanged on therefore the sheriffs statements are consistent. we know that there is at least one day that he cant be hanged on (the very last day of the year) so we know that there must be a last day that the sheriff can hang him on and still keep his word. yet if the prisoner can deduce which day that is then clearly the sheriff would choose some other day. hence if, hypothetically speaking, the prisoner knew which day was the last day then it wouldnt be the last day. it is unknowable from the prisoners point of view.

the question is not 'what is the last day the sheriff can hang him on'? the question is what day will the prisoner conclude is the last day he can be hanged on'? that is totally different. from the prisoners point of view the question is unanswerable.

 

Like I said, the sheriff's statements are consistent only by allowing the possibility that he could be wrong.

How do you address the usual form of the paradox, which covers only a few days:

A judge tells a condemned prisoner that he will be hanged at noon on one day in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that if the hanging were on Friday then it would not be a surprise, since he would know by Thursday night that he was to be hanged the following day, as it would be the only day left. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the hanging cannot be on Thursday either, because that day would also not be a surprise. On Wednesday night he would know that, with two days left (one of which he already knows cannot be execution day), the hanging should be expected on the following day.

By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner's door at noon on Wednesday — an utter surprise to him. Everything the judge said has come true.​

Where is the prisoner's mistake?

 

where is he wrong?

why do you want to change the number of days? that doesnt add anything.

i have already stated my conclusion. why do you want to hear it again? didnt you read what i said? there are days he can be hanged on and there is at least one day he cant be hanged on but the prisoner himself cant know the last day he can be hanged on. it is unknowable from his point of view. clearly it is absurd to follow the line of reasoning in your example for a full year. the sheriff has only to choose a day at random from the first half of the year.

 

I don't want your conclusion again, because it is inadequate. All you have concluded so far is that yes, the sheriff/judge can effect a surprise hanging. And yes, we all agree on that. The problem is in identifying the mistake in the prisoner's logic.

Reducing the number of days clarifies the problem.

Clearly the prisoner is wrong, because he logically concluded that the hanging could not occur. The puzzle is that his logic seems valid.

So, can you point out the mistake in the prisoner's logic?

 

i dont see how anything you said is different from what i have been saying. my guess is that the last day can be solved to within one of two days. if day one is the last day then day 2 is the last day but if day 2 is the last day then day 1 is the last day. its that second part that is missing from your solution.

 

Did you read the bit in blue? The prisoner deduces that he can't be hung on any day.
Where did he make a mistake?