The Unexpected Hanging Problem

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In summary, the conversation discusses the unexpected hanging paradox and the question of its significance in philosophy. The participants analyze the paradox and its logical flaws, ultimately concluding that the premise of the problem is internally inconsistent. They also discuss the intuitive nature of the paradox and how it is often hidden in the formulation of the premise.
  • #1
Jamma
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Hi all. I'm sure a lot of you will have seen this (possibly a few will be sick of seeing it, I don't know) but I've only just seen it.

http://en.wikipedia.org/wiki/Unexpected_hanging_paradox

Is it not a bit embarrassing/degrading for philosophy to state that: "...have even led to it being called a "significant problem" for philosophy." ?

As far as I can make out, this isn't too difficult of a paradox to unravel.

If we look at the prisoner's logic (say by the point he has ruled out Thursday and Friday):

"I have ruled out Thursday and Friday, so if it is Wednesday and I haven't been hung yet, then I must be hung on Wednesday. But then I know I'll be hung on Wednesday, so I can't be hung then either."

or, more revealingly:

"By MY logic, I can't be hung on Thursday or Friday, so by MY logic, if it's Wednesday, I will definitely be hung, which wouldn't be surprising, so I won't be hung on Wednesday."

Is it not obvious where this paradox is coming from? We have a logical system but where we impose another rule which says that "if we conclude logically [I will not be hung on x-day] then [it is possible I will be hung on x-day]" (since it will then be a surprise).

It seems to me that the paradox is coming about from the fact that our logical system refers to its own conclusions in a non-trivial way, making it obviously non consistent. We have effectively set as a rule "any conclusion which we can make logically about when the prisoner cannot be hung must also be false".

It's be nice to get some other views on this paradox from people who have a better way with words than me! :D
 
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  • #2
Hmm, that is a nice way of looking at it. So in a sense we have both concluded that the whole premise of the problem in internally inconsistent (whether in equivalent ways, I'm not sure and am too tired at the moment to figure out).

I realize that my post looked a bit big-headed when I said "this isn't too difficult of a paradox to unravel", I didn't mean it like that, it's a really nice paradox, but I just can't see why it could ever possibly be described as "a "significant problem" for philosophy" when someone like me, who has little experience with dealing with these sort of philosophy questions (although, I do quite a bit of maths, which helps), can immediately pick out some of the obvious flaws with the argument. I hope you get what I mean.
 
  • #3
Suppose the guard says "you will be hanged Monday noon, but you will not know which day you will be hanged". What can he conclude from this? Absolutely nothing. The guard effectively posits "A and (not A)". Let this be case (0).

Suppose there are two days in question, say Monday and Tuesday. The guard says "You will be hanged either Monday or Tuesday noon, but you will not know which day you will be hanged". The guard can mean two things: (1): that before Monday the prisoner will not know which day he will be hanged, and (2): before Monday and after Monday noon, if he survives, he will not know which day he will be hanged.

If the guard chooses a day at random, (1) is perfectly possible. He can be hanged any day, he didn't know it before Monday. (2) is just [(1) and (0)] (where Monday is replaced by Tuesday in (0)). (0) is meaningless, so he cannot conclude anything from this. (2) is therefore also meaningless.

If there are three days in question, he can either mean (3): that the prisoner will not know before Monday which day he will be hanged, in which case a day chosen at random will do. He can similarly to (2) mean that (4): before Monday and on each day after noon he will not know which day he will be hanged. (4) is [(3) and (2)] (where Monday is replaced by Tuesday in (2)), but (2) didn't make any sense in the first place.

The same can be said for more days. It is simply a nonsensical premise.

What's puzzling is that it is intuitive that the prisoner can conclude that he will not be hanged. But the guard simply posits a meaningless premise, the prisoner cannot conclude anything from it. Case (0) does not imply that the prisoner will not be hanged, it doesn't imply anything at all (or vacuously it implies everything, have your pick, but he still cannot conclude anything).

EDIT: sorry, deleted it by mistake.
 
  • #4
Jamma said:
Hmm, that is a nice way of looking at it. So in a sense we have both concluded that the whole premise of the problem in internally inconsistent (whether in equivalent ways, I'm not sure and am too tired at the moment to figure out).

I realize that my post looked a bit big-headed when I said "this isn't too difficult of a paradox to unravel", I didn't mean it like that, it's a really nice paradox, but I just can't see why it could ever possibly be described as "a "significant problem" for philosophy" when someone like me, who has little experience with dealing with these sort of philosophy questions (although, I do quite a bit of maths, which helps), can immediately pick out some of the obvious flaws with the argument. I hope you get what I mean.

The problem is that one somehow gets the impression that the guard is correct, even though the prisoner concludes that he is not correct. To take the one-day example, the prisoner actually just concludes from (A and not A) that A is false. Given A makes the assertion "true", while the prisoner is also correct. It's not all that astounding that one can arrive at this, when we are given something like (A and not A), even though it is hidden in the formulation of the premise.

Many paradoxes are of this sort. If there is a hidden contradiction somewhere in a premise, it is reasonable that we can arrive at conflicting conclusions.
 
  • #5
Yep, I agree with that summary. The internal inconsistencies arise from the use of the word "surprise", which should at least mean "I would be surprised if I got hung on a day that I logically concluded that I could not get hung on" or, in other words, "If I can't get hung on a certain day, then I can".
 
  • #6
But it's not about the word "surprise" at all! It's about the guards premise. In the base case he says that the prisoner will know which day and that he will not know which day he will be hanged (we simply define being surprised as being hanged if you didn't know before), and for more days the premise contain a contradiction of this sort. Being surprised and not knowing makes perfect sense and are not ambiguous. If you have concluded something logically and it does not happen you will of course be surprised, but then you haven't been able to conclude anything at all, since it didn't happen. You were mistaken, or the premise is itself a contradiction, which is the case here.

The point is that the prisoner cannot know he will not be hanged just because he is presented with a contradictory premise. The word "surprise" isn't ambiguous just because it is used in a contradictory statement.
 
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  • #7
Sorry, I didn't mean that the word surprise is ambiguous, I just meant that as soon as it is used, the judge is implying that any logical deduction which leads the prisoner to being certain the hanging can't be on an exact day immediately invalidates it. The word "surprise" is self-referencing to the logical system, because what is deemed as a surprise can be determined by the logical system itself.
 
  • #8
Jamma said:
Is it not obvious where this paradox is coming from? We have a logical system but where we impose another rule which says that "if we conclude logically [I will not be hung on x-day] then [it is possible I will be hung on x-day]" (since it will then be a surprise).

It seems to me that the paradox is coming about from the fact that our logical system refers to its own conclusions in a non-trivial way, making it obviously non consistent. We have effectively set as a rule "any conclusion which we can make logically about when the prisoner cannot be hung must also be false".

It's be nice to get some other views on this paradox from people who have a better way with words than me! :D



I would tend to agree with the above comments, as all human logic is self-referential and axiomatic. But i would not say that a good philosopher would find this surprizing or troublesome. Basically, all of reality and exstence are paradoxical and only a limited number of aspects of them seem logical to a chosen set of axioms we label 'logic'(and for which we find some kind of empirical confirmation, though at a deeper level, the 'logic' behind the 'empirical confirmation' would fail as well).

I would also be more interested in why certain aspects of reality succumb to a given set of axioms(though it's a different topic), than why, on the whole, reality and existence are grounded in paradoxes.
 
  • #9
Maui said:
I would also be more interested in why certain aspects of reality succumb to a given set of axioms(though it's a different topic), than why, on the whole, reality and existence are grounded in paradoxes.

I've also thought about these sorts of questions before, more on the lines of the question "why can we be sure that we can trust logic in real life?" which of course requires you to state what sort of axioms you are using.

However, I don't really know of any paradoxes that aren't explained by some reasonably obvious breach of logic or the addition of some axiom that is clearly troublesome, so you'd have to justify "Basically, all of reality and exstence are paradoxical"- they're only seem to be paradoxes when you add some extra (although, usually, seemingly intuitive) axioms.
 
  • #10
There is, of course, the obvious thing to point out here that we cannot logically show a given set of axioms to always give us factually correct statements, since the only way we could logically show that is by using the logical system itself (and this is assuming that there exists some notion of "factually correct" in reality).
 
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  • #11
Jamma said:
I've also thought about these sorts of questions before, more on the lines of the question "why can we be sure that we can trust logic in real life?" which of course requires you to state what sort of axioms you are using.

However, I don't really know of any paradoxes that aren't explained by some reasonably obvious breach of logic or the addition of some axiom that is clearly troublesome, so you'd have to justify "Basically, all of reality and exstence are paradoxical"- they're only seem to be paradoxes when you add some extra (although, usually, seemingly intuitive) axioms.


Bolding mine.

No, you don't have to add additional axioms for human logic, as we know it, to break down. We don't have an adequate(not talking about perfect!) understanding of any the concepts we take for granted in physics - from space, motion and matter to time and consciousness. It's all a superficial knowledge structure(though rather tall now really) based on a set of preferred axioms that works more than 99% of the time. It's only that the other <1% of the unanswerable questions concern the nature and integrity of the chosen axioms and seem to invalidate not just the attempts to answer the remaining bits, but pose a sizable risk of ruining all of the knowledge structure we've built to explain reality and existence. I would say that anyone seriously engaged in science would readily recognize that reality is more paradoxical and incomprehensible than the average Joe on the street would tend to believe. It's unexpected that we understand anything, is it not?
 
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  • #12
Maui said:
Basically, all of reality and exstence are paradoxical and only a limited number of aspects of them seem logical to a chosen set of axioms we label 'logic'(and for which we find some kind of empirical confirmation, though at a deeper level, the 'logic' behind the 'empirical confirmation' would fail as well).

I sort of agree here. But to put it more positively, logic leads to paradoxes because it choses to describe reality in terms of its limit states (and limits are "paradoxically" always just beyond where reality can actually reach).

To approach a limit is real. To arrive at the limit is then going to be unreal. That is why concepts like infinity or randomness are so "paradoxical", yet also so epistemically useful.

Furthermore, all such descriptions of limits are rooted in dichotomies. To have one direction (the limit state you mean to approach), you must have with equal definiteness its antithesis (a second opposing limit state which is the one you are leaving behind).

So randomness only has strong meaning as a condition if we also have the determined. The lack of one becomes a definite measure of the other. In the same way, the infinite is matched by the infinitesimal.

In the context of this brain teaser, the claim is "an event will take place randomly within this definite timeframe". So the source of the paradox is the fact that if the timeframe is allowed to shrink in practice, then the situation becomes so constrained that the event comes to seem determined rather than random.

The set-up statement is paradoxical because it demands and either/or response. Either the event is random, or it is determined. Those are the "logical alternatives". Yet reality in fact lies always between such absolute limiting conditions. So the event in fact goes from seeming relatively unconstrained (any time this week is good) to being highly constrained (today it has to happen now).

Constraints are global information. Clearly the issue here is the surprise of the prisoner, or the information he is allowed to accumulate as the week progresses and which thus steadily constrain his expectations. If the prisoner has no memory, or no way of telling which day it is, then there is no shift in knowledge.

So the set-up appears paradoxical - the process of the execution cannot be both random and determined, it has to be one or the other. And for the prisoner, it has to be random.

Yet the reality is that the execution is only globally constrained towards these polar choices by the information the prisoner has about the situation. And that state of information is easily changed.

Thus in the end, the brain teaser gets most of its charge from a sleight of hand confusion of an ontic reality (what the prison system is chosing to do - which is set a date presumably in advance that could equally well be any of five days) and an epistemic reality (how a prisoner reads the odds as information accumulates to constrain his knowledge about that already pre-determined "but random" choice).

The degree of surprise, or lack of globally constraining information, in fact can only be relative, even if the judge claims otherwise. For the prisoner, it can be relatively complete on the first day, relatively incomplete on the final.

If the judge's words are reframed to reflect this - "you will have a relative degree of uncertainty over the day of your death, and so your state of expectation will follow an asymptotic curve of probability over the course of a week" - then the paradox evaporates because the polar alternatives are being instead described more accurately in terms of dichotomous limits to a global process.
 
  • #13
Maui said:
But i would not say that a good philosopher would find this surprizing or troublesome. Basically, all of reality and exstence are paradoxical and only a limited number of aspects of them seem logical to a chosen set of axioms we label 'logic'(and for which we find some kind of empirical confirmation, though at a deeper level, the 'logic' behind the 'empirical confirmation' would fail as well).

I would also be more interested in why certain aspects of reality succumb to a given set of axioms(though it's a different topic), than why, on the whole, reality and existence are grounded in paradoxes.

What do you mean by that reality is paradoxical? A paradox is basically a contradiction, it has nothing to do with reality, just logic. What are you referring to when you say that aspects of reality "seem" logical? Logical statements can be statements about reality, but logical inference is just the utilization of certain logical rules which does not refer to reality, and has nothing to do with empirical inference. Our reasoning does not go further than what we can say from assumptions, and thus does not refer to anything but our assumptions.

In physics logic is constrained to models, which are mathematical in nature. So it is not the fault of logic nor mathematics that we fail to capture certain aspects of the world in physics, it is the failure of our models. The models can contradict each other, but this is not the same as reality contradicting itself (which doesn't make any sense anyway). This is grounds to try finding better models, not evidence of a paradoxical nature. We get paradoxes when we assume contradictory premises, not by observing nature.

I may have misunderstood you, you can probably clarify.
 
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  • #14
Hmm, I can't say that I totally agree, but I think that is because of my mathematical way of thinking, and I like to strip things down to the bare minimum.

For example, I don't think that you will find any paradoxes in mathematics, yet it is all built upon the logic of set theory. Any paradoxes we seem to arrive at in reality, imo, always seem to stem from some earlier assumption/lack of understanding of a definition.

For example, you said that "all of reality and existence itself are paradoxical", yet for example, I could add the axiom "there is a thing called reality" to our list of axioms in set theory and not get any paradox. Clearly this is not satisfactory, all I've done is added an item, I'd need to add other axioms which characterise what reality is. If you did so tentatively, I'm sure you can get quite far before creating a paradox.

You may say that this approach is overly-cautious and just plain nit-picking. But I find it quite instructive. For example, there are philosophers (I think an example is William Lane Craig) who say things such as "God must exist" or "we can logically deduce that the universe must have had a beginning". However, I can IMMEDIATELY disprove such a statement by constructing a hypothetical universe which never began and never ends and nothing he can say can disprove that such a universe exists without invoking some sort of unfounded assumption.

I think that also a lot of problems arise from the fact that we often make claims about things yet we lack a precise definition of what it is we are talking about e.g. time, the self, consciousness, reality etc. and we end up inserting intuition about these things into the mix.
 
  • #15
Jarle said:
What do you mean by that reality is paradoxical? A paradox is basically a contradiction, it has nothing to do with reality, just logic.


Everything is a paradox. There is a contradiction underneath every model we have built to describe and explain reality. I would say that this is a direct consequence of nothing being what it seems. Hence, we have no working model of reality that pertains to established logic. The models that appear to work(have no internal contradictions, e.g. field theories) are far removed from common logic and have contradictions with the old models of reality(which were rather successful, but not perfect or complete).



What are you referring to when you say that aspects of reality "seem" logical? Logical statements can be statements about reality, but logical inference is just the utilization of certain logical rules which does not refer to reality, and has nothing to do with empirical inference. Our reasoning does not go further than what we can say from assumptions, and thus does not refer to anything but our assumptions.



I was saying that you shouldn't take anything for granted. The hardest thing to explain is the most obvious, which is paradoxical under closer examination.
 
  • #16
I wish this wasn't posted in the philosophy forum..

Maui said:
Everything is a paradox. There is a contradiction underneath every model we have built to describe and explain reality. I would say that this is a direct consequence of nothing being what it seems. Hence, we have no working model of reality that pertains to established logic. The models that appear to work(have no internal contradictions, e.g. field theories) are far removed from common logic and have contradictions with the old models of reality(which were rather successful, but not perfect or complete).

This does not make sense at all, no model is "far removed" from logic. Logic is used at all times and is a prerequisite for reasoning in the first place, and we certainly always reason about our models to determine what it predicts. What you are talking about is models failing to capture broad physical aspects of nature, but this is trivial and has nothing to do with "paradoxes of nature". Using a model does not mean we are assuming it encapsulates a broader aspect than strictly what it models, and we should at all times view a model as an empirical approximation to what we can measure.

Maui said:
I was saying that you shouldn't take anything for granted. The hardest thing to explain is the most obvious, which is paradoxical under closer examination.

This is not about taking anything for granted. In physics "assumptions" has a broader meaning than "taking things for granted". We may assume any model we want just to look at the consequences. If the predictions seem to coincide with the measurements we might adopt it as standard measure. Nothing has been taken for granted.
 
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  • #17
All (good) physicists accept that their models are not exact descriptions of reality, but I would hardly call that a paradox when they don't ever quite fit.

Let's put this another way, since you claim that even reality itself is a paradox, show it- using only the most basic rules of logic and some assumptions about reality that you like, find a contradiction/paradox. I assure you that any such paradox you reach will be due to some unwisely picked assumption. If, on the other hand, you say this is an unfair proposition, since we don't really know what reality is, then you have proved what I am trying to say from another angle- we often hit these paradoxes because we try to describe things for which we don't even know the definition.

There is something that I agree with you on, which is that our general preconceptions about what time, reality etc. often can lead to paradoxes, but I don't see this as anything other than naivety on our part.
 
  • #18
As I long ago thought about this paradox (by then the "headmaster/exam" version), I arrived at several different solutions. The one that satisfied me most was this:

The paradox presumes the execution must happen. But by saying the prisoner will be saved if he can reveal selected day in advance, they reveal he will not necessarily be hung. Therefore the prisoner cannot even argue last day Friday must be excluded, because that is the day they select if they will let him go.

It doesn´t help adding "they absolutely want him hung" because then Friday is not a possible day, in contradiction to declared conditions.

I mean the core of the paradox is that "selected day of execution" is not necessarily "day of execution".

And if the execution day was selected by rolling a dice or something, of course no arguing had been possible.
 
  • #19
Jarle said:
I wish this wasn't posted in the philosophy forum..



This does not make sense at all, no model is "far removed" from logic.


Did you read what you quoted? Here it is again, in case you missed it:

"Hence, we have no working model of reality that pertains to established logic. The models that appear to work(have no internal contradictions, e.g. field theories) are far removed from common logic and have contradictions with the old models of reality(which were rather successful, but not perfect or complete)."



"Logic" as a system of thinking sometimes leads to nonsensical results. If we are cornered by a multitude of evidence leading in the same direction, we are forced to accept the seemingly nonsensical results as being correct.


Logic is used at all times and is a prerequisite for reasoning in the first place, and we certainly always reason about our models to determine what it predicts. What you are talking about is models failing to capture broad physical aspects of nature, but this is trivial and has nothing to do with "paradoxes of nature".


No, of course not. You are putting words in my mouth i didn't say or imply(no wonder you misunderstand). I never said or implied that our models were failing to capture broad physical aspects of nature but that the picture they paint is somewhat different than expected, i.e. it's paradoxical(incomprehensible). You can choose to believe and hope that this issue will be resolved, but it's all based on faith. The reality is that we have quite a lot of evidence that the deeper we probe, the more paradoxical everything appears to be and instead of a resolution, we are getting stuck deeper into the confusion.






This is not about taking anything for granted. In physics "assumptions" has a broader meaning than "taking things for granted". We may assume any model we want just to look at the consequences. If the predictions seem to coincide with the measurements we might adopt it as standard measure. Nothing has been taken for granted.

Assuming and taking for granted are different things. Look it up. I used "take for granted" as in "not questioning the obvious"(e.g. the physical nature of motion and how it happens)
 
  • #20
Jamma said:
\Let's put this another way, since you claim that even reality itself is a paradox, show it- using only the most basic rules of logic and some assumptions about reality that you like, find a contradiction/paradox.



Anything that is not fully explained is a paradox. Let's take the example of the previous post - how does physical motion take place? Does a pen i move with my hand stop existing at point X to reappear at point Y a plank time later and then at point Z etc. until it's finally where i intended it to be? Continuous motion would require that the pen would be elongated from X through Y to Z and exist in all points simultaneously. But that's not what happens in reality, so physcal motion must be quantized. Objects must cease to exist and reappear at plank times when in motion. Is that not a paradox? What is not a paradox?
Why do we teach our children that the universe is absolute, when we know for a fact that it's relative and quantum-mechanical? Because it makes reality and everything in it paradoxical and incomprehensible. But we do know better than that.
 
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  • #21
Maui said:
Did you read what you quoted? Here it is again, in case you missed it:

"Hence, we have no working model of reality that pertains to established logic. The models that appear to work(have no internal contradictions, e.g. field theories) are far removed from common logic and have contradictions with the old models of reality(which were rather successful, but not perfect or complete)."

Yes, I did, and it is exactly what you wrote: "The models that appear to work [...] are far removed from common logic..."

Maui said:
The reality is that we have quite a lot of evidence that the deeper we probe, the more paradoxical everything appears to be and instead of a resolution, we are getting stuck deeper into the confusion.

Being confused and having incomplete models which fails at certain places is not a paradox. My point is that your usage of the word paradox is entirely wrong, a paradox is something strictly contained within the logical realm, not in nature. No natural phenomenon can be paradoxical, only our assumptions about how they work (within the context of a model).


Maui said:
Assuming and taking for granted are different things. Look it up.

...And that is exactly what I was said. I talked about assumptions, but you brought up taking things for granted. I was trying to explain that these things are not the same, and that taking things for granted had nothing to do with assumptions in this context.
 
  • #22
Maui said:
Anything that is not fully explained is a paradox. Let's take the example of the previous post - how does physical motion take place? Does a pen i move with my hand stop existing at point X to reappear at point Y a plank time later and then at point Z etc. until it's finally where i intended it to be? Continuous motion would require that the pen would be elongated from X through Y to Z and exist in all points simultaneously. But that's not what happens in reality, so physcal motion must be quantized. Objects must cease to exist and reappear at plank times when in motion. Is that not a paradox? What is not a paradox?
Why do we teach our children that the universe is absolute, when we know for a fact that it's relative and quantum-mechanical? Because it makes reality and everything in it paradoxical and incomprehensible. But we do know better than that.

You seem to have proved what I was talking about. There are tonnes of assumptions that you make about reality lurking in there (to be honest, I don't quite understand your description). I don't see how continuous motion is paradoxical, you can, for example, make mathematical models of this sort of continuous motion which give no paradoxes (can't I say the pen is a subset of R^3 which moves according to some continuous rigid movement of the space? I know it's making very simple assumptions about reality, but so are you).
 
  • #23
It's an important thing to note that a model is not an assumption of reality, but an assumption of measurements. We don't "assume" that the pen is moving in a continuous curve in your example, but we are assuming this will give a sufficiently approximate mathematical representation of what measurements we hypothetically could do.

Not only that; continuity is a mathematical notion, not a physical one. It doesn't even make sense to say that the pen physically is in continuous motion. We can say that we can mathematically represent its motion as continuous, and that this model will be sufficient for the type of measurements we will do make. The paradoxes which may appear will in this context be purely a consequence of our choice of mathematical model. In this sense we can only make paradoxes-we can only force upon a paradox in natural phenomena, but we can't find any paradoxes.
 
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  • #24
Jamma said:
You seem to have proved what I was talking about. There are tonnes of assumptions that you make about reality lurking in there (to be honest, I don't quite understand your description). I don't see how continuous motion is paradoxical,


Not quite what i said. I was saying that physical motion is paradoxical and continuous motion impossible(an idea we get from casually observing moving objects and concluding motion must be continuous, because it looks so). The closest idea to continuos motion would be if the object in motion was spacetime itself, but this renders the idea of motion as we know it useless and needs a number of additional assumptions. This is the "motion" that comes out of GR and the the so-called "blockworld" universe.



you can, for example, make mathematical models of this sort of continuous motion which give no paradoxes (can't I say the pen is a subset of R^3 which moves according to some continuous rigid movement of the space? I know it's making very simple assumptions about reality, but so are you).


Perturbation methods, renormalization, state vector reduction all deal with attempting to bridge the gap between continuous and discrete. I don't think anyone knows of a way to introduce continuous motion in physics, now that all fundamental concepts are quantized. But i don't hold a strong opinion that atoms are entirely real(as real as a teapot for example), so from my probably naive position, physical motion is a series of measurements and observations. But it's still a paradox, like all else in reality when subjected to more rigor.
 
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  • #25
Jarle said:
The paradoxes which may appear will in this context be purely a consequence of our choice of mathematical model. In this sense we can only make paradoxes-we can only force upon a paradox in natural phenomena, but we can't find any paradoxes.


I understand what you are pointing out but what does it change if i believed that nature wasn't in any way paradoxical but our logic(method of inquiry) was flawed and/or insufficient(for the time being)? If all efforts fail to resolve a paradox, one is equally entitled to call nature paradoxical as they are to call their method of inquiry, model or logic "paradox raising", as there is no guarantee that we will ever understand reality. So how would we tell which position is right? i am fully aware that i am making the usual assumption(axiom) in the sciences, that human logic is something special and the universe is intelligible.
 
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  • #26
Maui said:
I understand what you are pointing out but what does it change if i believed that nature wasn't in any way paradoxical but our logic(method of inquiry) was flawed and/or insufficient(for the time being)?

But this is the point, our logic isn't flawed, it's the assumptions about reality which is flawed. As I said, we never get these paradoxes unless we make some sort of assumption about reality.

We never get paradoxes in mathematics. Is it just a coincidence that when we start applying logic to (our understanding of) reality that we start to acquire these paradoxes?
 
  • #27
Maui said:
I understand what you are pointing out but what does it change if i believed that nature wasn't in any way paradoxical but our logic(method of inquiry) was flawed and/or insufficient(for the time being)? If all efforts fail to resolve a paradox, one is equally entitled to call nature paradoxical as they are to call their method of inquiry, model or logic "paradox raising", as there is no guarantee that we will ever understand reality.

As Jamma said, it is not the logic which is flawed if we arrive at something paradoxical, but the model in question. If a model turns out to be flawed or contradictory to others, we may at no point say that we have found out something about nature-that it somehow therefore must be paradoxical. It is simply a matter of failed attempts to explain/model.

Maui said:
So how would we tell which position is right? i am fully aware that i am making the usual assumption(axiom) in the sciences, that human logic is something special and the universe is intelligible.

We don't have to assume that the universe is intelligible, that is something we discover when we see that natural laws and models seem to describe it pretty well, so the universe is intelligible in this trivial sense. Maybe you meant something else. What do you mean by that human logic is "special"?
 
  • #28
Jarle said:
As Jamma said, it is not the logic which is flawed if we arrive at something paradoxical, but the model in question. If a model turns out to be flawed or contradictory to others, we may at no point say that we have found out something about nature-that it somehow therefore must be paradoxical. It is simply a matter of failed attempts to explain/model.

Not so as logic is our model of causality. It derives from our experience of the world and the general way things seem to work.

As a model, it can embed assumptions that are not "true" - but which are effective and good enough within their limits of use.

As I've pointed out, a key assumption of standard logic is that reality is either/or - Aristotle's law of the excluded middle. And this can lead to logical paradox.

Other suggested models of logic - such as Peirce's vagueness-based approach - avoid this kind of paradox by finding a way to say "both". The middle is not excluded ontically. Or at least, it is accepted that this exclusion is itself a process that has to happen operationally. It is not something that just brutely exists.

In as much as Maui has a point here, it is right that QM conflicts with standard either/or thinking. But it then seems to fit with a logic of vagueness.

So yes, nature itself cannot be paradoxical. That is only something that happens in our models of causality.

But it can also be fixed in those models. As it only crops up because the models have become too simplistic.
 
  • #29
I often dealt with the problem like this.
I add that The prisoner was required to predict (anticipate) the day of hanging (of course before the hanging) by writing down in a paper.
So,lets go with this rule. If he isn't hung until Thursday night then he will write down "Friday" in the paper. He will be saved.
If he isn't hung until Wednesday night then he will have two options to write, Thursday Or Friday. If he write one, the judges will hang him on the other day. The judges has authority to read the paper which is equivalent to ability to read his mind in the original problem. the ability to read the mind is required because how else can we decide whether the hanging was unexpected or not?
So, I see that he could be hung any-day except Friday. But if he has already written down his prediction (make up his mind) before Thursday, then he can even be hung on Friday. I think we have to impose the rule that he can't change his mind, because the prisoner can easily avoid hanging by always thinking that today is hanging day.
 
  • #30
apeiron said:
Not so as logic is our model of causality. It derives from our experience of the world and the general way things seem to work.

As a model, it can embed assumptions that are not "true" - but which are effective and good enough within their limits of use.

As I've pointed out, a key assumption of standard logic is that reality is either/or - Aristotle's law of the excluded middle. And this can lead to logical paradox.

I will concede this one, we can't actually prove that our logic is an absolute which always derives truths if fed with other truths. I was going to mention this, but didn't bother. However, I have never seen such a process happen. Also, I've never seen a paradox derived simply from assuming the law of the excluded middle, can you please give an example?

This was my point, I wasn't putting our logic on a pedestal, but I was demonstrating that it is at least very effective and never gives paradoxes, and that all of these "paradoxes" that are derived are nothing to do with our basic system of logic, but always some later assumption.
 
  • #31
Jamma said:
Also, I've never seen a paradox derived simply from assuming the law of the excluded middle, can you please give an example?

I did in my initial post. But a trivial standard example is...

An example might be to affirm or deny the statement "John is in the room" when John is standing precisely halfway through the doorway. It is reasonable (by human thinking) to both affirm and deny it ("well, he is, but he isn't"), and it is also reasonable to say that he is neither ("he's halfway in the room, which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.
http://en.wikipedia.org/wiki/Paradox#Logical_paradox
 
  • #32
But again, that doesn't seem to be a problem with the law of the excluded middle to me, more a problem with the definition of "being in a room". If we defined "being in a room" to mean you have to totally within the room, then no paradox occurs, the same if only a bit of you has to be in the room. There is no precise statement of what "being in the room" is, and this is emphasised by the statement adding in "by human thinking".
 
  • #33
in physics boundary problems, we define three different cases. Inside the room, outside the room, and the transient case. This can be seen in more abstract mathematical concepts too, like convolution.

But it's still important to recognize that while your whole arm is inside the room, your hand can't be outside the room at the same time. In quantum, we can talk about a single particle being completely inside and outside the room at the same time, but that's a completely different story.
 
  • #34
The point is not that we haven't defined what it means to physically be in the room, but that we haven't decided that "being in the room" is a logical proposition at all. If you can say p and (not p) you are not dealing with standard logic, but a different logic altogether which will have a different and non-overlapping use, though potentially useful.

Logic is not about physical states of affairs, the proposition "being in the room" does not logically refer to the physical state of "being in the room" (in whatever way it is defined). But we can treat it logically if we decide that it shall be a logical proposition, which for any useful purpose ought to have a well-defined physical counterpart. It is important not to confuse any statement about physical states of affairs with the logical counterpart which may or may not have any use depending on how we treat the statement as a statement of physics. In the case of "being in the room"-if we are inclined to say "both"-we are simultaneously deciding that we are not stating a logical proposition (in standard logic).
 
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  • #35
Jamma said:
But again, that doesn't seem to be a problem with the law of the excluded middle to me, more a problem with the definition of "being in a room". If we defined "being in a room" to mean you have to totally within the room, then no paradox occurs, the same if only a bit of you has to be in the room. There is no precise statement of what "being in the room" is, and this is emphasised by the statement adding in "by human thinking".

You are making my point that paradox can be avoided if you accept that middles don't come excluded. It is an action that has to be performed. And so paradox arises if you have an axiom system where middles do come ready-excluded.

You say this kind of intelligent softening of the formal logic is no big deal, just a pragmatic exercise. I agree, of course. But then much better is also to put that insight itself on a formal basis as "a logic". Which is what a Peircean vagueness approach would be about.

More recently, we've had fuzzy logic and paraconsistent logic. I actually think Peirce's work remains far more radical. But here is discussion that is more orthodox.

This contribution deals with developments in the history
of philosophy, logic and mathematics before and
when fuzzy logic began. Even though the term
“fuzzy” was introduced by Lotfi Zadeh in 1964/65 it
should be noted that older concepts of “vagueness”
and “haziness” have been discussed in philosophy,
logic, mathematics, applied sciences, and medicine.
This paper delineates some specific paths through the
history of the use of these “loose concepts” in science.
The theory of fuzzy sets is a proper framework for
“loose concepts”, that connote the nonexistence of
sharp boundaries.

http://www-bisc.cs.berkeley.edu/BISCSE2005/Abstracts_Proceeding/Saturday/SA3/Rudi_Seising.pdf [Broken]

Jarle said:
Logic is not about physical states of affairs, the proposition "being in the room" does not logically refer to the physical state of "being in the room" (in whatever way it is defined).

No quarrels with your position here. Clearly, what I am interested in are the non-standard approaches that would be formal solutions to the familiar formal paradoxes. And this interest seems justified by QM as well as the developmental perspective that underpins biology and the modelling of life/mind.
 
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<h2>1. What is "The Unexpected Hanging Problem"?</h2><p>The Unexpected Hanging Problem is a famous paradox in mathematics and logic that involves a prisoner sentenced to death by hanging, and a judge who presents the prisoner with an unexpected twist to the execution date.</p><h2>2. What is the paradox in "The Unexpected Hanging Problem"?</h2><p>The paradox lies in the fact that the prisoner is unable to logically determine the date of their execution, even though they are given new information about it. This leads to a contradiction, as the prisoner's reasoning leads to both the execution being expected and unexpected at the same time.</p><h2>3. How is "The Unexpected Hanging Problem" relevant in science?</h2><p>The Unexpected Hanging Problem is relevant in science as it highlights the limitations of logical reasoning and the potential for paradoxes to arise in seemingly simple situations. It also demonstrates the importance of considering all possible scenarios and information when making decisions and drawing conclusions.</p><h2>4. Is there a solution to "The Unexpected Hanging Problem"?</h2><p>There is no definitive solution to the Unexpected Hanging Problem, as it is a paradox that challenges traditional logic and reasoning. However, there have been various proposed solutions and explanations, each with their own flaws and limitations.</p><h2>5. What can we learn from "The Unexpected Hanging Problem"?</h2><p>The Unexpected Hanging Problem teaches us to be critical thinkers and to question our assumptions and reasoning. It also highlights the complexity of decision-making and the potential for unexpected outcomes, even in seemingly straightforward situations.</p>

1. What is "The Unexpected Hanging Problem"?

The Unexpected Hanging Problem is a famous paradox in mathematics and logic that involves a prisoner sentenced to death by hanging, and a judge who presents the prisoner with an unexpected twist to the execution date.

2. What is the paradox in "The Unexpected Hanging Problem"?

The paradox lies in the fact that the prisoner is unable to logically determine the date of their execution, even though they are given new information about it. This leads to a contradiction, as the prisoner's reasoning leads to both the execution being expected and unexpected at the same time.

3. How is "The Unexpected Hanging Problem" relevant in science?

The Unexpected Hanging Problem is relevant in science as it highlights the limitations of logical reasoning and the potential for paradoxes to arise in seemingly simple situations. It also demonstrates the importance of considering all possible scenarios and information when making decisions and drawing conclusions.

4. Is there a solution to "The Unexpected Hanging Problem"?

There is no definitive solution to the Unexpected Hanging Problem, as it is a paradox that challenges traditional logic and reasoning. However, there have been various proposed solutions and explanations, each with their own flaws and limitations.

5. What can we learn from "The Unexpected Hanging Problem"?

The Unexpected Hanging Problem teaches us to be critical thinkers and to question our assumptions and reasoning. It also highlights the complexity of decision-making and the potential for unexpected outcomes, even in seemingly straightforward situations.

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