Zeno's arrow and the unexpected hanging

[keithprosser2]

Zeno's arrow and the unexpected hanging (or exam) are examples of paradoxes that seem to be based on water-tight logic but lead to incorrect conclusions. Two things strike me - the first is what exactly is wrong with the arguments (if anything) and how sure can we be that many other apparently logic proofs are in fact invalid.

I have thought about this a fair bit but not come up with anything much - can anyone do better?

[Arising_uk]

Hmm.. I'm not much good at the Greeks logic but does not the existence of a paradox point to an error with the axioms? As logical deduction has always been able to deduce incorrect conclusions I thought, its why the axioms have to be true.

[Thundril]

We cannot have absolute certainty. But we do pretty well without it. Logic is one of our attempts to get as close as dammit to absolute certainty, and as such is does OK most of the time. The paradoxes are an example of where at some points in history we bump up against the limits of our understanding so far. Then we figure out other stuff that moves us on a bit. Eg calculus.

[keithprosser2]

To be concrete, are you familiar with the 'unexpected hanging/exam' paradox, Th or auk? How do you resolve it?

[Thundril]

To be concrete, are you familiar with the 'unexpected hanging/exam' paradox, Th or auk? How do you resolve it?

I'm more familiar with the 'interesting numbers' paradox, which I think is similar, (if rather less serious) in that it turns on a debatable defintion of a subjective mental condition.
With regard to the unexpected hanging paradox, my first attempt to resolve it would be this: If I choose a number from 1-6, and tell you it is one you will not expect, then as long as you are expecting a number in the range 1-6 my prediction is nonsensical; ie you will not be surprised unless it is a number outside the range. But suppose I demand that you ask 'Is it a 1?' and if it isn't a 1 then you will be required to ask 'Is it a 2?' and so on in strict order, then either you will be surprised to discover which number it is, or you won't, depending on whether you have formed a hunch.
So, if the judge tells the prisoner he will be hanged one day next week, but not on a day he expects, the apparent paradox is in the presentation of the 'sentence'.

[Arising_uk]

To be concrete, are you familiar with the 'unexpected hanging/exam' paradox, Th or auk? How do you resolve it?

Nope, I'm just taking a read of your link and the first paragraph puzzles me as I'd ask the judge what point the guard knocking on my door at noon when I'm being hung? Probably get myself hung there and then as a know-it-all philosopher. Back again later.

Okay,
Been a while since I've had to think this way but I sort of understand the logical approach as the word "surprise" is an issue. Think I prefer the reverse induction explanation but not completely clear on it, my take is that the prisoner can't infer from Wednesday as Friday and Thursday would reappear and its no good saying that "but I've thought of Friday as the base case" as I think for induction to work in the real world it actually has to be a case.

But in reality I'm not happy with higher logics so my understanding is this, the prisoner is making a mistake about the conjunction "hanging and surprise", in symbols (H ^ S). His approach is that he has deduced from the information that at least one day, Thursday, the conjunction will be false, i.e. ¬S( not S) means S is false, so (H ^ S) is false and he won't get hung as the judge will be breaking his word. He then uses his convoluted induction to cover all the previous days. But he's implicitly assuming that ¬S also means ¬H and this is not true, H can be true and the conjunction false. The best he can infer is that there is one day where he will get hung and not be surprised. That is, he's just wrong thinking that he's not going to get hung. Of course he can go to the gallows knowing that the judge was logically incorrect, although if he told him then I think the judge would agree and say "You will be hung and it will be a surprise, or you will be hung", happy now? Now take this idiot out and hang him! Surprise!!

[keithprosser2]

I would say the prisoner is right to infer he can't be hanged on Friday... if the judge had pre-decided Friday then the prisoner would know it was on Friday when the knock didn't come on Thursday. So if he alive on Thurday the sentence cannot be carried out as announced. IF. In fact, he could use that logic to realise that he must have been hanged either before or on Thursday to avoid that problem. If so, there is (as far as I can tell) no way to logically deduce which day it was he must have died.

To me the problem is the (hidden) assumption that the prisoner is still alive on Thursday. As I (think I) showed above that assumption not only could be false but must be false. Basing an argument on a false premise is never a good idea, although it probably the guy up - right up until the knock came on,say, Tuesday.

Of course he could be hanged on Friday as well. As he proved that he can't be hanged on that day, he will be most surprised (and disappointed) when he is.

[Arising_uk]

I would say the prisoner is right to infer he can't be hanged on Friday... if the judge had pre-decided Friday then the prisoner would know it was on Friday when the knock didn't come on Thursday. So if he alive on Thurday the sentence cannot be carried out as announced. IF. In fact, he could use that logic to realise that he must have been hanged either before or on Thursday to avoid that problem. If so, there is (as far as I can tell) no way to logically deduce which day it was he must have died.

To me the problem is the (hidden) assumption that the prisoner is still alive on Thursday. As I (think I) showed above that assumption not only could be false but must be false. Basing an argument on a false premise is never a good idea, although it probably the guy up - right up until the knock came on,say, Tuesday.

Of course he could be hanged on Friday as well. As he proved that he can't be hanged on that day, he will be most surprised (and disappointed) when he is.

I sort of follow you but I think it just a mistake to think he won't get hung on Friday. Because a conjunction, "hanging and surprised", is false because one of its conjuncts, "surprised", i.e. 'not surprised on Friday', which is correctly deduced, has no bearing on whether the other conjunct, "hanging or hung", is true or false. He's going to be hung, the most he can get from logic, I think, is whether he's surprised or not and a sense of superiority over the judge, given judges are trained in reasoning. I suppose he could argue that the conjunct is false because he won't be surprised at all, as the judge has said he's going to be hung on one day next week so he can assume that each day is the day and it'll be no surprise at noon. The surprise will be just after that when noon passes.

Me, I'm still stuck on why the guard is knocking on my door when I'm on the scaffold? Is this a lateral thinking exercise Your 'onour?

[Thundril]

Not being au fait with the rules of formal logical wrangling, dare I propose an attack from the rear?
Imagine trying to generate a new 'paradox'. What qualities would a 'paradox' necessarily have?
Without taking (dishonest) advantage of the ambiguities of language, how would one proceed? Zeno clearly supposed that things like 'moments' and 'points in space' actually existed. We know different. So we can't take refuge in pre-20th century concepts of time and space.
Tricky, isn't it?

[Impenitent]

"A judge tells a condemned prisoner that he will be hanged at noon on one weekday 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."

the judge never said that the method of execution would be by hanging...

-Imp

[Arising_uk]

"A judge tells a condemned prisoner that he will be hanged at noon on one weekday 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."

the judge never said that the method of execution would be by hanging...

-Imp

Then the prisoner is truly fucked and my logic has problems with "but that". But I can see that he could be stabbed first and still hung and that so far, on the prisoners reasoning, he appears logically wrong on all counts and the conjunction holds in this interpretation. So the paradox is how this is considered logic rather than linguistics, semantics or semiotics?

[Arising_uk]

Not being au fait with the rules of formal logical wrangling, dare I propose an attack from the rear? - Ooo! err! Missus!
Imagine trying to generate a new 'paradox'. What qualities would a 'paradox' necessarily have?
Without taking (dishonest) advantage of the ambiguities of language, how would one proceed? Zeno clearly supposed that things like 'moments' and 'points in space' actually existed. We know different. So we can't take refuge in pre-20th century concepts of time and space.
Tricky, isn't it?

The rules of the simplest formal logic, Propositional Logic, are simple Thundril, as they are based upon propositions and a combination of the logical operators "and" (^), "not" (¬), "or" (v), and one that is not necessarily needed but is handy when considering rational argumentation, "if...then..." (->). A proposition can be considered a sentence but its slightly different as a proposition is that thing that is described by many languages, "The cat is on the mat", "Le chat est sur la natte", "Die Katze ist auf der Matte", "Η γάτα είναι στο χαλί", "Il gatto è sulla stuoia", etc. So any proposition can be assigned a capital letter and by convention we start at P, so the above is P, another proposition would be Q, R, S, etc. What can you say truthfully say or symbolize about P other than "it is true" that the cat is sitting upon the mat, which is the proposition P. Well straight away you can say "its not the case ...", ¬P(the cat is not sitting upon the mat or as the logicians prefer "Its not the case that the cat is sitting upon the mat"), i.e. it is false that there is a cat sitting upon that mat. What is generally ignored I think is that you also get two other cases or relations for P, where P is true or false regardless of whether there is a cat sitting upon a mat, and they differ in that one is always true and the other always false. but you don't really see this until you deal with the operators that have two propositions attached, i.e. and, or and if...then. So say we have two propositions P, Q, what can we truthfully say about these? Well all the things that we could say about P are the things we can say about Q but there are some new things or relations, we can say about P,Q. We can say P and Q, (P ^ Q), P or Q, (P v Q)(although there are two types of this "or"), If P then Q (P->Q), although this is not strictly needed. Because its about propositions we can consider any of the last as singular propositions, R = (P ^ Q), etc, and as such we can use the "not" idea from just having P, i.e ¬(P ^ Q), ¬(P v Q), etc. Now this was really fun and in truth in philosophy what I'm saying is time-backward as in reality we started with what is now called Quantificational Logic, but what got interesting was when we realised that we can draw tables of truth-functions of the "and", "or", "not" and "if...then.." operators, so "^" is true when and only when both P and Q are true, any other time "^" is false, and its all based upon the actual state of P, Q. a rule of logic. There are the same kind of rule for "or" and "not", so "¬" is true only when the attached P is false and false when the attached P is true, on its own it has no meaning. "Or" has two meanings, but the standard or "inclusive" one is where (P V Q) is true when P is true or Q is true or when both are true, otherwise its false, i.e. one or both have to be true for it to be true. There's loads more but what made it very interesting was when we used the operators with the same proposition P, (P V ¬P) is true whatever the state of things. (P ^ ¬P) is false whatever the state of things, absolute truth and total contradiction, necessary and impossible. The 'laws' or boundaries of reason, or at least with certain words and propositions, that even 'god/s' can't fuck with.

"5.123 If a god creates a world in which certain propositions are true, then by that very act he also creates a world in which all the propositions that follow from them come true. And similarly he could not create a world in which the proposition 'p' was true without creating all its objects." Tractatus Logico-Philosophicus, L. Wittgenstein. (I also think this book has the best introduction and preface in philosophy)
Hope this starts an interest in formal logics.

With respect to generating new paradoxes, to much for me but if I was to take on the task then the bit in the wiki link about the logical argument and self-referring statements is where I'd start but I don't really think there is a paradox here as I agree it a misuse or misunderstanding of inductive and propositional logic.

[keithprosser2]

I really wish auk would write 'hanged' rather than hung... but that is beside the point. My argument against a Friday hanging is that I am interpreting 'surprised' to mean that the prisoner would not know ahead of time, and more over that 'know' includes 'logically infer'.

The fact is that a minute after noon on Thursday - if he is alive at all - the prisoner can start to loudly announce that he knows the day of his execution must be Friday (if at all), negating the Judge's clear statement that he would not know the day before it came.

@Thundril, the interesting thing about this 'paradox' is that the prisoner's argument that the Judges sentence cannot be carried out as announced seems rock solid, but it is clearly false because he could be hanged on, say, Wednesday in complete accord with the Judge's sentence. I don't think it is a paradox as such, just a real pain to spot the error in the prisoner's reasoning. Other (more genuine) paradoxes like Russell's paradox aren't like that...

[Arising_uk]

I really wish auk would write 'hanged' rather than hung... but that is beside the point. My argument against a Friday hanging is that I am interpreting 'surprised' to mean that the prisoner would not know ahead of time, and more over that 'know' includes 'logically infer'.

The fact is that a minute after noon on Thursday - if he is alive at all - the prisoner can start to loudly announce that he knows the day of his execution must be Friday (if at all), negating the Judge's clear statement that he would not know the day before it came.

I agree but what he can't infer from knowing that he won't be surprised is that he won't be hung. Just because one part of the conjunction is false implies in no way the other can't be true even if the conjunction is false.

[Mike Strand]

Not sure this helps, but the Wikipedia statement of the paradox says the prisoner concludes that "he will escape from the hanging". But the judge's statement only implies that the prisoner will be surprised, leaving open his being hanged without being surprised.

The statement of the paradox should say that the prisoner concludes he won't be surprised about the day he'll be hanged.

This said, maybe I'll try to analyze this in more depth, to see why the prisoner can still be surprised (unless not hanged by Thursday - but even in this case, the prisoner would be surprised, or at least unable to predict, not being hung by Thursday, so ...). It appears to involve conditional events (can X happen if Y has already occurred), as well as semantics.

Incidentally, this reminds me of an interesting play on words: If a man is alive and strung up on a rope and this makes him die, he's been "hanged". If he is already dead before being strung up, he's been "hung". Like a dead goose being hung in the smokehouse.

This led me to wonder: If the man is alive before the bad rope experience, is he "stranged" (or "stringed") up on the rope to be hanged?

I'll be danged! (Not "I'll be dung") -- English is a wonderful language.

Still working on the paradox, but isn't it interesting how "human" the prisoner's reasoning is? After all, he knows he's going to be hung. First, he jumps to the unwarranted conclusion that unless he's surprised about the day, he won't be hung at all. Then he gets into a routine of "logic" that eliminates the possibility of being surprised. Trust a desperate man to come up with "reasoning" like this.

It would be interesting, however, explicitly to show the flaws in this reasoning, even with my restatement of the paradox to stipulate that he concludes that he won't be surprised about the day of his hanging.