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Five Great Puzzles and Paradoxes to Tickle the Mind

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Paradoxes cause confusion at the roots of all our thinking. A paradox arises when we start with premises we believe to be obviously true, proceed by a process of reasoning that we also think incontrovertible, and end up with something we believe to be obviously false. Sometimes such thought experiments don’t end up in a full paradox – but puzzle the mind nonetheless.

We ponder over such problems because of a worry that they give us reason to doubt our most trusted source of knowledge – reason itself. At the very least – they are a fun way to baffle ourselves and confuse others at parties.

Here are five great puzzles/paradoxes to tickle the mind:

Newcomb’s Problem

Imagine you are playing a game – it works like this: a rich scientist has presented you with two opaque boxes labelled A and B. You can’t see what is inside of either of them. The scientist tells you that you can make one of the following two choices. You can either take box B, or you can take both boxes. These are your two options.

Now in box A he has placed a thousand dollars. But he doesn’t tell you what exactly is in box B. He tells you that he is a very great scientist that has developed a perfect model of human behaviour – and he can predict in advance any decision anyone is likely to make. He tells you that he has made a prediction as to which box you’re likely to take. And if he predicted that you would take both boxes, he left box B empty. If he predicted that you would take only box B – he placed a million dollars in that box. Of course he doesn’t tell you what prediction he has made.

He then asks you to make your choice. What do you do? Do you pick box B only? Or do you take both?

To make sure the puzzle cuts deeply enough – let’s rule one possibility of thought out. The scientist is as good as he says he is. It’s in the spirit of the puzzle to accept he can successfully make his prediction. Perhaps you have seen a hundred or so people play his game. All those that chose both boxes walked away with a thousand dollars only – but all those that chose box B only, walked away with one million dollars.

When confronted with this puzzle most people tend to say: that’s easy I’d do such and such. But the problem is that half the people you speak to will say they will take box B, and the other half say they’ll take both boxes. And each thinks the other is crazy.

There are two lines of equally compelling reasoning going on. On the one hand – you’re pretty confident that the scientist knows what you are going to do. As such, if you decide to take both boxes, the predictor would have known that and would have left box B empty. But if you decide to take only box B – then the predictor would have known that as well and would have placed a million dollars. Your decision to pick box B alone seems to count as evidence as to what the scientist has done. It seems obvious – pick box B only – and walk home with a million bucks.

But wait – says the other line of reasoning. The predictor has already made his choice. Whatever is there in those boxes is there. Irrespective of what the scientist has predicted – you stand to gain the most if you pick both boxes. Consider the two possibilities. Either he has predicted you would take both, or you wouldn’t. If he predicted you would take both, and you do, you get 1000 dollars. But in that scenario, if you pick box B only, you’ll get nothing. The other scenario, the scientist predicted that you would pick box B only. In this case, if you pick both boxes, you would get one million and one thousand dollars, but if you pick box B only, you get one million dollars only. So no matter what the scientist has predicted, you stand to gain the most by choosing both boxes.

It’s a cunning puzzle – and the philosophers are still arguing over it. But it is relatively recent. You can find the original paper by Robert Nozick in the reference below:

Nozick, Robert (1969), “Newcomb’s Problem and Two principles of Choice,” in Essays in Honor of Carl G. Hempel, ed. Nicholas Rescher, Synthese Library (Dordrecht, the Netherlands: D. Reidel), p 115.

The Liar Paradox


Consider the following sentence:

“This sentence is false.???

Now ask yourself if it is true. If it is true then what it actually says, that it is false, is true – so it’s false. If it is false, then what it actually says, that it is false, is false – so the sentence is true. So when it’s true, it’s false – and when false, true. This is a paradox of the first rank.

It’s a very old paradox, being attributed to an ancient greek fellow Eubulides of Miletus around 400 bc. Philosophers and logicians have been arguing about it ever since. Think about it – that’s thousands of years of argument.

Some obvious solutions are considered to be failures. For example – some say the problem arises because we assume that all sentences must be true or false. But that’s just not the way natural language works. There are plenty of sentences that are neither true nor false – and this sentence is one of them. But you can reframe the puzzle in the following way:

“This sentence is not true???

And the whole game starts up again.

Some have claimed the problem to be one about self reference. Sentences shouldn’t be allowed to refer to themselves. This solution is too strong – plenty of sentences refer to themselves meaningfully – without paradox. “This sentence is written in english??? – for example. But besides that – the solution is just wrong. Consider the following two sentences:

1) Sentence 2 is true
2) Sentence 1 is false

I’ll leave it to you to see why this causes all the same problems – but neither of these sentences refer to themselves – only to each other. Hence this version avoids that object.

The debate goes on and the literature is vast. They’ve been arguing over this one for centuries – so good luck.

The Unexpected Hanging

A prisoner who has commited a heinous crime is before a judge. The judge sentences the prisoner to death by hanging, but adds a cruel twist to the sentence (the prisoner’s crime is particularly heinous). The prisoner is to be hanged on one of the following seven days – but it must be a surprise which day it is. The prisoner is not allowed to know.

Returning to his cell the prisoner is a bit disturbed at the prospect of being hung without knowing when (presumably it worse when you don’t know what day you are to die) – and confides his fear to his lawyer. His lawyer tells him not to worry.

“Look,’ he says with a smile. “They can’t hang you at all now. The judge has made it a condition that you must be surprised. But think about it. If you make it to Saturday without being hung then Sunday is the last day they could do it. But then it wouldn’t be a surprise would it? So that makes Saturday the last day they could possibly hang you. But hang on – if Saturday is the last possible day then it also can’t be a surprise to hang you then. So that make Friday the last possible day – and so on back through all the days of the week. They can’t possibly hang without breaking the judge’s orders.???

The prisoner is comforted by this line of reasoning and stops worrying about the prospect of being hung at all. When suddenly on Wednesday, much to his great surprise, he is taken from his cell and hung.

This is a great puzzle, not so much because we have two different kinds of legitimate reasoning in conflict with one another, but because one strand of reasoning is shot down by reality itself! I’ll leave it to the reader to puzzle further on this one.

Hempel’s Paradox

Hempel’s paradox is a paradox of induction. Induction is method of reasoning we use to make generalisations about the world. Consider all the ravens you have seen in your life time. Hopefully they were all black. Now on the basis of seeing nothing but black ravens, it would be reasonable to generalise and claim that all ravens are black. This is a natural step – and it forms the basis to all our scientific reasoning.

Now consider the following statement: ‘All non-black things are not ravens’. This statement is logically equivalent to our generalisation. For if all ravens are black, then something which is not black can not be a raven. We could then go observe non-black things – and each time we saw that a non-black thing was not a raven, we would confirm that all ravens are black.
So it seems by observing a pink flamingo – it would confirm that all ravens are black.

But hold up! Seeing a pink flamingo would also confirm the statement: ‘All non-white things are not ravens’ and this is logically equivalent to ‘All raven’s are white.’ So it seems that the observation of a pink flamingo seems to confirm both that all ravens are white and that all ravens are black. But this is a contradiction!

Again I will leave it to the interested reader to go seek out the solution.

The Sorites Paradox

This is another ancient paradox bequeathed to us by those awfully intelligent Greek fellows.Imagine you have in front of you a heap of sand. It is unquestionably a heap of sand. It has more grains of sand in it than you would care to count. Now it seems fair to say that given any heap of sand, if you were to take one grain of sand away, then it would still be a heap of sand. And if you were to take another grain of sand away from the remaining heap, it would still remain a heap.

But what if you were remove some many grains of sand you were left with only one grain? Well it would no longer be a heap. But we have just proved that it must be! For the removal of one grain of sand from a heap, still leaves us with a heap. By repeated application, we must always be left with a heap. But at some point along the line it stops being a heap and our reasoning leads us astray. What has gone wrong?

This is a paradox that arises out of the vague use of our language. We know what a heap of sand is – but we do not know how to draw the line between a heap and a non-heap. Our use of language does not prescribe an answer.

Solutions to this paradox become very complicated very quickly – and again I’ll leave it to the interested reader to find his/her own solutions.

If you want to learn more about various logical and philosophical puzzles and paradoxes, I can’t recommend highly enough the book: Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge by William Poundstone. It’s a fantastic introduction to the world of paradox. He is comprehensive, yet explains everything in a clear and easy to understand language. If you’re up for the challenge, check him out today.

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96 Comments

  1. me wrote:

    Interesting about the box paradox. It reminds me of this newly proposed version of seeing things in QM. That is that information can travel backwards in time. In fact, their claim is that if you take two entangled electrons, spread them apart very far, and reveal one’s spin, that information travels back in time to the moment they were entangled to begin with. It would mean, among other things, that the information actually didn’t travel faster than light. Only in the other direction.

    For the box paradox, if we took this view for granted, only taking box B would be the only acceptable solution. No?

    Posted on 29-Mar-07 at 10:54 pm | Permalink
  2. admin wrote:

    Yes – if causation can happen backwards in time then taking box is the only choice. There are some respected philosophers out there that argue for backward causation – and those tend to be one boxers I believe.

    Posted on 29-Mar-07 at 11:13 pm | Permalink
  3. steve wrote:

    How does seeing a pink flamingo confirm the statement: ‘All non-white things are not ravens?’

    It adds confirmation that all ravens are black and that all non-black things are not ravens.

    Is the reader supposed to use induction to inverse black into white?

    Someone please clarify.

    Posted on 29-Mar-07 at 11:15 pm | Permalink
  4. kathaclysm wrote:

    I think when choosing box A or box B, I would only pick B because the extra $1000 wouldn’t matter too much to me. Now, if box A had half a million, and box B had a million, that would make the dilemma a bit more interesting…

    And the issue with the heap of sand could easily be resolved by defining a quantity to the word heap. A heap of anything should consist of at least # of said objects. It’s not so much a paradox as poor language. But, that’s not to say this doesn’t have other, much more important implications. When does a ball of dough in the oven become bread? When does a fetus become a baby?

    Posted on 29-Mar-07 at 11:46 pm | Permalink
  5. james275 wrote:

    no need to clarify, steve – the author is wrong. and that’s not the only place he’s made an error in his reasoning. perhaps a better “logic puzzle” would be finding the other logical flaws in this post. you’ve already found one…

    Posted on 29-Mar-07 at 11:50 pm | Permalink
  6. admin wrote:

    Thanks for that James :)

    unfortunately I can’t take credit for any of the reasoning in these puzzles. They were all invented by those much smarter than myself…

    Please feel free to point out errors.

    For those wanting to stipulate a precise meaning of ‘heap’ – it’s unfortunately a no-go as a solution. The point is that we understand the word ‘heap’ and can pick out heaps without such a precisification. So offering a precisification doesn’t help us understand what it is about our use of the term that is causing all the problems.

    kathaclysm is right though when he implies that sorites paradoxes can be generated with a whole range of vague predicates.

    Posted on 29-Mar-07 at 11:58 pm | Permalink
  7. yourface wrote:

    A collection of comments regarding this post and other comments:
    The post is poorly written and the paradoxes all poorly worded, hence the confusion.
    Ship of Theseus is not included in the list.
    I resent the fact that I believed I would find anything interesting here.
    People who believe they have answers to paradoxical questions are boring.
    Writing well is always preceeded by reading well, not doing so lets Hemingway make you cry.
    Lastly, when discussing matters of logic, stuff histrionics up your ass.

    Posted on 30-Mar-07 at 12:12 am | Permalink
  8. admin wrote:

    The basic idea behind induction is that continued observation in support of a hypothesis leads to greater belief in that hypothesis. So if we are making the hypothesis all non-white things are non-ravens then anything you see which is non-white and also not a raven – then it supposedly confirms the hypothesis.

    If you have a problem with this – then you must also have a problem with non-black things that are also not ravens confirming: ‘all non-black things are non ravens.’ And well you might. It seems absurd to think that just by looking at all these things in front of me that have nothing to do with ravens I should somehow be confirming that they are all black.

    But if we had a genie that went and observed all the non-black things in the universe and reported back to us, we would be told that none of them were ravens. This is what underpins the intuition that ‘all ravens are black’ is equivalent to ‘all non-black things are not ravens’. But of course when the genie goes looking at all the non-white things – he finds some of them are ravens – disconfirming our nonwhite statement.

    So why should the contrapositive be equivalent only in the cases where the inductive hypothesis is true? What does this tell us?

    Posted on 30-Mar-07 at 12:13 am | Permalink
  9. Jeff wrote:

    Newcomb’s Problem is either stated incorrectly, or trivial. You said to assume that the scientist can perfectly anticipate your choice. The implication of this that the article misses is that the scientist basically becomes irrelevant: what you choose determines what you get. If you pick B, you get $1,000,000. If you pick both you get $1000. There’s no uncertainty: you’re immediately determining what you receive. Your statement, “Irrespective of what the scientist has predicted – you stand to gain the most if you pick both boxes,” implies a disconnection between the scientist’s prediction and your choice that the earlier statement, “It’s in the spirit of the puzzle to accept he can successfully make his prediction,” rules out. Even though your choice caused the scientist to arrange the money a certain way in the past, it did so so perfectly that the effect is equivalent to your choice arranging the money at the moment you make it. Given a perfect prediction, your choice determines the placement.

    Pick B.

    Posted on 30-Mar-07 at 12:14 am | Permalink
  10. admin wrote:

    I believe Nozick originally went for a 99 percent level of confidence on the predictors ability. Adjust the percentage to suit your intuitions.

    Posted on 30-Mar-07 at 12:17 am | Permalink
  11. The Sorites Paradox reminds me of the “what is the most I’d be willing to pay?” problem that I face every time I try to place a proxy bid on eBay.

    I try to simply bid the maximum I’d be willing to pay — but if I declare that to be $40, what if I lose because somebody bid $40.01?

    I’d always be willing to pay two cents more than any maximum bid I’ve previously set!

    Posted on 30-Mar-07 at 12:26 am | Permalink
  12. John wrote:

    does me fucking nut in donit

    Posted on 30-Mar-07 at 12:29 am | Permalink
  13. xeno wrote:

    1) The world doesn’t work this way – In the real world take both boxes unless the scientist can mess with them after you choose.

    2)Don’t believe everything you read. Uncertain is neither true or false.

    3)It takes more than 1 thing to be uncertain which. The argument breaks down when there are two days – a set of one is a special case.

    4)Things are not black or white, especially pink things.

    5)Heap is a description of randomly piled order not number, a heap if spread out would not be a heap. 1 thing (your pants) can be heaped on a floor.

    Posted on 30-Mar-07 at 12:53 am | Permalink
  14. You mentioned in the Liar Paradox:
    “Consider the following two sentences:
    1) Sentence 2 is false
    2) Sentence 1 is false
    I’ll leave it to you to see why this causes all the same problems”

    This isn’t quite like the original self-referencing sentence. Here is a possible solution: Sentence 1 is true, thus Sentence 2 is false, which leads to Sentence 1 is true (no contradiction).

    Since there is a double negation, it seems assuming either of the sentences to be either true or false won’t result in a contradition (as long as only an assumption regarding one of the sentences is made).

    Posted on 30-Mar-07 at 1:00 am | Permalink
  15. Steve wrote:

    These are pretty much all ridiculous in that they are not paradoxes at all. The only true paradox in this entire post is “This sentence is false.”

    A couple people have already pointed out why the box “paradox” is worded incorrectly.

    The second part of the Liar Paradox is not true at all, because either sentence one or sentence two can be exclusively false. I.E. if sentence one is false, then that means sentece two is true, and sentence two says that sentence one is in fact false, so no paradox at all.

    The entire logic in the hanging puzzle is flawed, because the idea that a particular day cannot be the day he’s hung only holds if it is the day before, not to mention the fact that there is no time-frame for a hanging decision specified. For instance, if he makes it to Saturday without being hung, he is home free because the only day left is Sunday and he’d know about that, right? No, because they could easily decide at 11:59pm on Saturday night, that he will be hung right then. Thus, he’ll be hung Saturday without knowing about it.

    The black raven paradox fails because of the laws of logic you learn the first day of any introductory philosophy class. Whenever you make a statement about all of something, the only way to prove or confirm it is through the method of exhaustion, meaning you have to test every other possibility (as the person who mentioned the hypothetical genie was getting at). Every single observation you make in favor of the statement does not confirm it as the author stated, it only supports it. And support does not prove anything.

    The Sorites paradox only hints at the subjectiveness and importance of context within our communication. I guess it is a bit of a paradox in the loosest definition of the word (though I think I like Jonathan Harford’s example of bidding on ebay a little more).

    To the author, you do write well, just perhaps not with enough understanding of the material at hand. You have to be very careful how you word things.

    Posted on 30-Mar-07 at 1:04 am | Permalink
  16. Bear wrote:

    The sentence “This sentence is true” is also a paradox. It could be true. But then it could just as well be false, since any false sentence which claims to be true is, um, false.

    Posted on 30-Mar-07 at 1:43 am | Permalink
  17. Michael Pohoreski wrote:

    Some of these puzzles for the trap of binary logic, instead of using tertiary logic. T, F, mu / indeterminate / n/a
    i.e.

    Q. Have you stopped beating your wife?
    A. Mu. I’m not married.

    Posted on 30-Mar-07 at 3:34 am | Permalink
  18. Alis wrote:

    Hi Daniel

    Neat presentation… Can I partially translate this in Turkish and duplicate your article for my students in our site?

    Sincerely

    Alis Sagiroglu

    Posted on 30-Mar-07 at 3:41 am | Permalink
  19. Mr Hector wrote:

    Contrary to the author’s assertion, there are at least two non-complicated solutions to the heap problem.

    1) Rather than viewing the term “heap” as binary description i,e. something is either a heap or it is not, define it as a fuzzy predicate e i.e. a collection of grains is heap-like to a certain extent. When there is only one grain the collection is clearly not a heap and as the number of grains increases the collection becomes more heap like.

    2) The collection of grain can only be considered to be a heap SO LONG as the effect of taking away one grain cannot be distinguished by an observer. As the number of grains is reduced, the impact of removing another grain becomes more significant and at some point, which will vary depending on the individuals powers of observation, it will start to be more useful to think of what is there as being a collection of individual grains rather than a continuous heap.

    Posted on 30-Mar-07 at 4:50 am | Permalink
  20. Ryan Devine wrote:

    Hempel’s Paradox is actually david hume’s pardox. Hume was the first person to find out about the problem with induction.

    Posted on 30-Mar-07 at 5:14 am | Permalink
  21. admin wrote:

    Hi Alis,

    Please feel free to take what you like. :) I didn’t come up with these paradoxes so I don’t own em!

    I would love a back link though!

    Posted on 30-Mar-07 at 6:07 am | Permalink
  22. Kaarlo Mäki wrote:

    Seeing a pink flamingo will not confirm the statement: “All non-white things are not ravens” because we already know that black ravens exist and thus already know the statement to be false. Confirming things by induction is not logical proof, instead we could say that the likelihood of a given statement being correct increases by each observation.

    Posted on 30-Mar-07 at 6:30 am | Permalink
  23. Annie Mouse wrote:

    1) Newcomb’s Problem
    Assume the scientist will pay up as promised. If he was smart he’d predict every time that everyone would chose both boxes. That relieves him of his duty to pay $1 million. Essentially, the smart person would pick both boxes, thinking that “at least I’ll have $1000″!. If the person chose only box B that person is risking losing $1000 and $1 million. This is the paradox of risk. Do I play it safe and pick both boxes earning at least $1000 and possibly $1,001,000 or do I take a risk and chose only box B perhaps losing out on all? If the scientist performed his experiement as a true random prediction model, then still choosing both boxes every time grants the greater return on risk.

    2) Liar’s Paradox
    This sentence is false.
    This sentence is not true.
    A self-referencing sentence is acceptable. And this creates a paradox for certain. Mostly, however, it’s a paradox of language and not thought. When translated into thought we can easily see around this paradox by concluding that this sentence is irrelevant in that it contains no value logically. It in essence becomes an infinite logic loop.

    1) Sentence 2 is false.
    2) Sentence 1 is false.
    This pair of sentences requires that one statement be held in greater confidence than the other. Assuming sentence 1 to be true falsifies sentence 2 and thereby this “paradox” is solved. But what gives greater what to sentence 1 than sentence 2? Assume sentence 2 to be true and sentence 1 is false. Same scenario. The moment you assume both to be true, paradox. And again the only route around it is to assume greater weight to one of the sentences.

    3) The Unexpected Hanging
    I think the purpose of this was to show the falacy of certain logics. If the convicted were simply to expect his death everyday (and night) he would then not be hanged, by the logic of the Judge who insisted on him being surprized. If we take for granted that he is to be hanged within the week, a time constrait will set free the convict, free from death at least. The lawyer’s logic that if by Saturday the convict has not been hanged then indeed it would be no surprize to the convict and therefore would not be hanged, and if this logic were to regress through the days of the week, even without a time constraint (assume a year, ten years even, then regressing), then certainly it’s the same as assume everyday that the convict will behanged.

    4) Hemple’s Paradox
    Either this paradox is meant to expose a flaw or the flaw itself isn’t a flaw of logic but rather in language.

    Assume that all ravens are black. All ravens are black is the same as all non-black things are not ravens. Therefore, seeing a pink flamingo automatically rules it out as being a raven. However, by seeing this pink flamingo one cannot assume that all non-white things are not ravens. but it does point out the idea that author tries to grasp. When witnessing a pink flamingo one would say it’s not a raven, because it’s not black. We know that raven’s are black and not white. The pink flamingo is also not white, and by induction one could say all non-white things are not ravens (because ravens are black). Just the same one could say all non-red things are not ravens, again because ravens are black. Just the same, all pink things are not ravens, as ravens are not pink. But by the same logic that gives us all ravens are black and therefore all non-black things are not ravens can be reversed to say all non-black things are not ravens therefore ravens are black. Herein lies the paradox. If all non-white things are not ravens then all ravens are white…hmmm. Induction can be a very hazardous logic.

    5) Sorites’ Paradox
    A heap of sand is simply lots of gathered sand. A heap does not describe a quantity and shouldn’t. Numbers describe quantity and heap is not a number. It is, however, a linguistic point of reference to indicate lots of something. By removing a grain at a time eventually a logical person would agree that the heap gets smaller. Eventually the heap becomes a mound which becomes a pile which becomes handfuls and so on. The paradox here is about drawing lines and is meant to get one thinking about the language and logic mix. Socrates pointed out a lot that our language lacks a definitive structure for logic. We people tend to argue semantics. What one word means to me is different than to you; my heap, your mound. Here’s a paradox define speed, time, and distance without using those words to define the others. While it can be done, the general concept is very simple:

    Speed is the time it takes to move a distance.
    Time is the speed at which a distance is traversed.
    Distance is the observation of speed during a certain time.

    According to various sources speed is the obervation of movement. Movement represents a spacial relocation, distance.

    Enjoy!

    Annie Mouse

    Posted on 30-Mar-07 at 6:35 am | Permalink
  24. Adrian wrote:

    The problem with the “induction paradox” is that induction is defined wrongly here.

    Induction does not state that one piece of evidence proves something. Induction states that if you can prove that a statement is true for one thing, and then prove that it is true for all things smaller than X size, then you can work your way up to infinity, and it has been proved. For a simple case of induction:

    Show that 0 + 1 + 2 + 3 + 4 + … + n = (n(n+1))/2.
    Proof:
    Case n = 1: Let n be 1. This gives us the sequence 0 + 1 = 1(1+1)/2, or 1 = 2/2, which holds true.
    Inductive case: Assume that the original statement holds true for n – 1. This gives us 0 + 1 + 2 + … + n-1 = (n-1)(n-1+1)/2, which we know is true. To bring us to the next case, we add n:
    0 + 1 + 2 + … + n-1 + n = ((n-1)(n-1+1)/2) + n
    = ((n-1)n/2) + (2n/2)
    = ((n-1)n + 2n)/2
    = (n(n-1+2))/2
    = (n(n+1))/2
    So, we have shown that the statement holds true for one case (n=1), and that if a statement holds true for n-1, then it holds true for n, so we can extrapolate to say that the statement holds true for all cases above n = 1, and the proof is complete. QED.

    I’m not sure if this helps people (and the original poster) understand induction. Wikipedia might help (I stole the example from there, but did my own proof, I don’t know if it’s the same). But what it comes down to is that without the second step (show that if it holds for n-1, then it holds for n) the the induction has not been proved. This second step has not been performed with the pink flamingos (now that you have shown that there is one non-white bird that is not a raven, show that all non-white birds are not ravens, which cannot be done using induction). In fact, using the rules of logic you can easily disprove the statement:

    Show: not all non-white birds are not ravens.
    Proof by counter example: Consider the statement “all non-white birds are not ravens”. Let us consider this to be true. But here I have a bird which is a raven, and is clearly black. A black bird is a non-white bird, and thus the statement has been disproved, and thus the negative of the statement has been proved (as it is a binary statement). QED.

    There is no paradox, just a misunderstanding of induction in logic. If the OP has really found this paradox elsewhere, I’d like to know what is the original source material.

    Posted on 30-Mar-07 at 6:38 am | Permalink
  25. Kaarlo Mäki wrote:

    Adrian, you are talking about mathematical induction which is a different thing. What’s discussed here is the philosophical concept of inductive knowledge.

    Posted on 30-Mar-07 at 6:47 am | Permalink
  26. Emily wrote:

    As for Sorites’ paradox: The word ‘heap’ implies that there are multiple objects ‘heaped’ on top of each other, regardless of the size of said objects. So whether it is a heap of sand, or a heap of cars (for example), it is only a ‘heap’ if there are one or more objects on top of another one. So maybe what this depends on is not necessarily the number of grains in the ‘heap,’ but rather how they are arranged. So as long as there is more than one grain, it could be considered a heap…

    Then again, I’ve never heard anyone call two grains of sand a ‘heap.’

    Posted on 30-Mar-07 at 7:21 am | Permalink
  27. luc wrote:

    There is an error in Liar’s Paradox. The correct text should read:

    1) Sentence 2 is false
    2) Sentence 1 is true

    Also, Hempel’s Paradox is badly worded and confusing. One can find a much simpler explanation here: http://www.geocities.com/CapitolHill/Lobby/3022/hempel.html

    Posted on 30-Mar-07 at 7:59 am | Permalink
  28. admin wrote:

    It’s great to see so many comments – I’ll try to answer as many points as possible.

    There was indeed a typo in the liar paradox presentation as pointed out by Simon – that’s been fixed.

    Good to see a good two boxer in annie’s comment. To strengthen her case even more – imagine you had a friend who was allowed to look in both boxes before you made your decision and advise you what to do (your friend can’t tell you what’s in the boxes – they can only say take B or both). What do you think your friend would say?

    Annie also points out that language is not consistent across all users. This is true – but I don’t think that observation helps us in the case of the sorites. Faced with a real heap – we all agree that it is a heap. What we might differ on is the point at which it ceases to be a heap. Supervaluationists say – okay – let’s take all possible precisifications of the term heap – as given by various usages. If a particular heap is a heap across all precisifications then it is ‘super-true’ that it is a heap. But this solution runs into the problem of higher order vagueness.

    Kaarlo points out that induction doesn’t logically confirm with any kind of certainty – this is true, but the paradox doesn’t rely on that to get going. The pink flamingo doesn’t confirm ‘non white things are non ravens’ with any certainty. It supposedly just adds evidence – in the same way that getting burnt by fire a couple of times only confirms that it will burn you in the future (never proves it). But this is bad enough.

    Kaarlo is right about the mathematical induction case. Adrian you must be a mathematician! Mathematical induction is about as deductive as you get – so it’s a bit misleading they called it an inductive kind of proof (they probably did because it allows them to prove for infinite domains).

    Posted on 30-Mar-07 at 8:08 am | Permalink
  29. admin wrote:

    Forgot to mention – I’ll post references later tonight when I’m done at work.

    If anyone feels like digging me… the link is here:

    http://www.digg.com/general_sciences/Five_Great_Puzzles_and_Paradoxes_to_Tickle_the_Mind

    Posted on 30-Mar-07 at 8:14 am | Permalink
  30. Dan wrote:

    Dang! I thought I was being so clever perceiving a connection between Newcome’s problem and quantum entanglement! But “Me” said it first.

    To add a little detail, you can look at entanglement as forcing connections between events that should be isolated, relativistically-speaking. The canonical example is, two entangled photons have a quantum-mechanically enforced relationship between their respective spin states: if one is +, the other must be -, or vice versa. But they aren’t either one until someone does a measurement, at which point the unmeasured one INSTANTLY takes on the opposite value of the measured one.

    The “many worlds” interpretation says that there was no communication between the two, but instead the universe split in two, and one of the universes had the state (+, -) while the other had the state (-, +). When you make a measurement, all you find out is which universe “you” fell into. No universe had the state (+, +) and therefore you’ll never measure that.

    Similarly, if the rich scientist’s prediction is “entangled” with your choice, then there is no universe in which you can pick both boxes and get $1,001,000. So pick box B and be fully confident that it contains $1,000,000.

    Posted on 30-Mar-07 at 9:04 am | Permalink
  31. Kaarlo Mäki wrote:

    On Hempel’s Paradox:

    In my opinion, inductive reasoning makes the most sense if you think it in the terms of probability. To illustrate, let’s say I flip a coin. I would not be in the wrong in assuming that there’s a 50/50 chance of either heads or tails. Now, it is brought to my attention that the coin is weighed so that it will end tails up more often than not. You could say that a little evidence is added to the fact that the coin will land as tails the next time I flip it. But! If I have already flipped the coin and it landed heads, the new information makes no difference. The result was heads with a 100% probability and nothing will change it. The new information will not chance it towards a tails result in any manner.

    This is why I think there is no paradox with the ravens. If we knew nothing of ravens, then seeing a pink flamingo could indeed be said to add evidence to the statement that “All non-white things are not ravens” or “All ravens are white”. This would not be very interesting, since we know nothing of things called ravens, even that they exist – the logic could be used to add confidence to an infinite amount of equally uninteresting statements. But since in this case we already know that black ravens do exists, seeing a pink flamingo will not add any evidence to the whiteness of ravens, just as new information about the properties of the coin will not chance anything after the coin toss has already been performed.

    Posted on 30-Mar-07 at 6:59 pm | Permalink
  32. Kaarlo Mäki wrote:

    Thinking further, there’s a problem with the fact that in the language of logic any statement concerning an empty set is considered true. In the real (inductive) world statements about empty sets ie. nonexisting things are obviously not very interesting.

    Posted on 30-Mar-07 at 7:31 pm | Permalink
  33. Kaarlo Mäki wrote:

    Yet further reflection gives the following insight:

    Why does seeing a black raven add evidence to the fact that all ravens might be black? We could say that the raven we saw is a random sample from the set of all ravens. Every consecutive black raven we see reduces the odds of our streak of black ravens being just a random anomaly in the population of all ravens and increases the odds that all ravens indeed share the property of blackness.

    So then, does seeing a non-black thing such as a pink flamingo increase the odds of all ravens being black? If we were to exhaust the entire set of non-black things and found no ravens, we could conclude that ravens, if they exist, must be black. The set is obviously too big for us to do that so we consider the random sampling method. If we were able to somehow select a random specimen from the set of all non-black things and ended up with the said pink flamingo, then yes, it would indeed add evidence to the blackness of ravens. And each consecutive non-raven item selected this way would add more to the evidence.

    But in the real world, we simply cannot consider a pink flamingo to represent a random sample of all the non-black things (or non-white for that matter) in the world. We have no way of obtaining a sample that comes even close to that criteria. Whereas any raven we see could be said to be somewhat randomly selected from the set of all ravens (or atleast from the local population).

    Posted on 30-Mar-07 at 9:06 pm | Permalink
  34. Kaarlo Mäki wrote:

    The point being that if we purposefully go looking for pink flamingos and assume that to influence the color of ravens in any way we’re wrong. They can be only considered in the context of random sampling where there’s an equal possibility of getting a raven (should they exist) to that of any other object.

    Posted on 30-Mar-07 at 9:15 pm | Permalink
  35. admin wrote:

    Yes – It does seem as though the genie idea that can go and look at all non-black things is leading us astray. For us as finite creatures – we can’t do it and so we don’t allow the induction over the contrapositive.

    Posted on 31-Mar-07 at 10:09 am | Permalink
  36. Mikael wrote:

    Difficult is it?

    I take box B for one compelling reasons. If the prof. is true to his word he predicted my action and I get a mill. If not I can claim him for a scam and perhaps get my mill but certainly I get to take down an over-cocky prof. a notch and that’s worth more than 1 measly K.

    Alter the numbers and I might alter my decision because it’s true that what’s there is there – the laws of causality can’t be tampered with. Make the numbers 1 mill and 1 mill + 1 respectively and I will sure go for that safe mill + the possibility of another.

    Posted on 01-Apr-07 at 2:39 am | Permalink
  37. Anders Johnson wrote:

    As for Newcomb’s “box” problem…
    To choose both boxes A and B is simply illogical, according to the assumptions of the problem. If we are to assume that the scientist’s prediction was somehow automatically correct, as prescribed, then if one was to choose both boxes, the scientist would have left box B empty in his foreknowledge. Unless, by the latter solution, you violated the assumptions that were given earlier in the problem, choosing solely box B is the best “legal” answer.

    Posted on 01-Apr-07 at 8:09 am | Permalink
  38. admin wrote:

    A lot can turn on the wording here. To say that the scientist is good at his art should not be read as: ‘he WILL predict correctly by necessity”. I should have perhaps made this more clear.

    One way of strengthening the case for two boxing is by saying that your evidence for the skill of the scientist is inductive. You have observed many others take the test and he has been correct each time. This gives you strong evidence for his skill – but you can’t infer from this his infalliability.

    The question then boils down to this: is the fact that you would choose to one box evidence telling you that he has put 1 million dollars in box B? Can your own choices serve as evidence for anything? If it can serve as evidence then it is reasonable to take box B only.

    Causal decision theorists say that it is not reasonable to take only one box. Their argument is to the effect that your decisions should reflect the causal structure of the world – and your choice does not cause the scientists prediction (because we assume causality can’t work backward) – hence it can’t serve as evidence. Causal decision theory is currently the dominant school of philosophical thought on the issue. So it’s actually the two boxers that currently have the upper hand in the debate.

    Posted on 01-Apr-07 at 11:24 am | Permalink
  39. furu wrote:

    Hempel’s Paradox isn’t really a paradox as many have pointed out, but the reasons differ. Seeing a pink flamingo does not give any reason to make anyone say “all non-white things are not ravens”. That is simply as stupid a statement to make as “all non-existent things are not pencils.” Both statements are irrelevant and fallacious.
    One does not observe an object and make a statement about all other objects. The “all non-black things are not ravens” is speaking still only of ravens. The fact that it speaks of “non-black things” (other objects), is made irrelevant by the second “not”, which confirms that we are in fact only speaking of ravens, from which we are deriving the attribute we are describing.

    There is absolutely no point in Hempel’s Paradox except to perhaps illustrate how poor wording can create a paradox from a mistake.

    The Unexpected Hanging is also partly fallacious.
    By the phrase “much to his surprise”, it is deducted that the author is illustrating how the man was so secure in his fate as a free man, that he simply stopped expecting punishment, and when it came, he was surprised. The paradox is in the moron trusting the lawyer. “Let me help you friend, he’s not gonna kill you, don’t worry. Here’s why:…” “Wow, you’re really smart, here’s some cash.” then he dies.

    The reason for this obvious fallacy is that as the lawyers reasoning progresses, he negates his previous point. In saying that if not by Sunday then never, and if not ever as of Sunday, then not ever as of Saturday by the same reasoning, then the expectancy of surviving after Saturday creates a hole for surprise on Sunday AND that very Saturday.

    The only way to escape this is by not reasoning at all and living in fear constantly, expecting every noise to be your death, and when it comes you think “I knew it.” In which case, the king can neither confirm nor deny your foreknowledge and expectancy and will kill you anyway. Any continual muttering can be discounted as no actual foreknowledge or expectancy, but precaution which is irrelevant to the judge’s decree.

    Newcomb’s Problem is only a question of trust. He has succeeded in every previous test, but what if this test, he fails? Taking the premise of the problem that the man was accurate 100/100 so far, it stands to reason that winning the million a worthy gamble, in a test that was 100% successful so far. Taking the other box simply means you’re taking a 0/1 bet for no reason but blatant skepticism. It’s the amount difference that scares people.

    An equivalent problem, to demonstrate the structure of Newcomb’s, is this:
    A scientist has machine that surges 20 million volts of electricity. It grants the subject immortality, but has to be calibrated perfectly. Any slight error in the machine’s operation or calculations will instantly kill the subject. It has been tested on over 100 patients 150 years ago, all of which are still the same age in appearance, including the scientist. If you opt out of the procedure, you win $10,000. If you stay, you become immortal. This is assuming that the subject would want immortality anyways.

    The Pile problem was properly addressed. There simply needs to be a definition of a pile. Another user posted an ignored suggestion the makes the most logical sense: At the point where our brain decides that it would be more beneficial to regard the grains of sand as individual grains, instead of as a single pile, it stops being a pile.

    You should all read the other posts to stop being redundant.

    Posted on 01-Apr-07 at 4:30 pm | Permalink
  40. casey wrote:

    furu, how do you know the suggestion you cited was ignored–simply because no one rehashed it (as you did)? But to do so would be redundant and would imply, as you point out, that the post was, after all, ignored. Now that’s a paradox! Or is it?

    Posted on 02-Apr-07 at 7:12 am | Permalink
  41. admin wrote:

    I really wasn’t sure what furu meant by being redundant.

    While furu finds the puzzles easy to solve, still I appreciate the contribution. I’m not convinced by most of the solutions. But I’ll only address the point about the newcomb problem – because this has come up a number of times.

    Many are arguing that you should one box if the scientist is an infallible predictor – a god like predictor. I shouldn’t have placed so much emphasis on his infalliability because it causes people to miss the point. Say you observe him 999/1000 times. Adjust the figures to suit your intuitions – so as to generate the problem proper. What the problem is trying to get you to look at is the issue of whether or not your own choice can serve as evidence as to which prediction the scientist made. So it not so much about trust – as about the basis on which decisions should be made.

    Usually we have no problem because the evidence – qua evidence – usually matches the causal structure of the world. But this example is specifically setup so that it doesn’t. That’s the essence of the problem.

    Posted on 02-Apr-07 at 7:37 am | Permalink
  42. Joe wrote:

    All of these paradoxes can be solved by applying computer science problem solving.

    Posted on 02-Apr-07 at 11:09 am | Permalink
  43. Faloo Faloo wrote:

    Here’s a paradox by the size of logic in any introductory philosophy class.

    These are not ravens. The paradoxical questions are obviously too big, for one cannot assume causality who can’t serve, as many others take two boxers that are the second part of ravens, even without knowing about it. These are black ravens, then the wording can successfully make his infalliability. Proof by not a pink flamingo is to exhaust the same, and all ravens are arguing to pay $1 million. However, by defining a number of the fact, only one of the grains of sentences requires that surges 20 million dollars in time to be calibrated perfectly.

    Essentially, the days of context within the effect is defined wrongly here.

    Every single observation of all non-black things called ravens, again causes the scientist’s prediction (because we already pointed out on him being a certain time). Their argument breaks down an infallible predictor – a floor.

    Yet further reflection gives us reason to think of probability. I’d be black? Contrary to simply stop being black? No, because it’s true for n = 1(1+1)/2, or you should confirm it will regress through the collection, not to prove that black ravens are not ravens. But, that’s not the state (-, +).

    You could be somewhat randomly selecting this view for logic. Similarly, if you do so I would not be included in assuming either true is, um, false.

    Thus, he’ll be either true or false.

    Someone please clarify.

    Posted on 02-Apr-07 at 10:00 pm | Permalink
  44. john wrote:

    steve said,

    March 29, 2007 at 11:15 pm

    How does seeing a pink flamingo confirm the statement: ‘All non-white things are not ravens?’

    It adds confirmation that all ravens are black and that all non-black things are not ravens.

    Is the reader supposed to use induction to inverse black into white?

    Someone please clarify.

    This seems pretty logical to me – “all non-white things are not ravens”, so a pink, purple or mauve thing certainly could be a raven. If ‘a’ not numeric.. well doh.

    Posted on 03-Apr-07 at 5:29 am | Permalink
  45. john wrote:

    I mean you don’t need binary, hex or any other conclusion, “non-white” = “not raven”, so “other than white” = ‘raven”.

    Posted on 03-Apr-07 at 5:35 am | Permalink
  46. casey wrote:

    faroo faroo:

    clarify flamingo heap logic. n number of scientists black non-binary. 1 million observation induction for raven grain. Now do you get it?

    Posted on 03-Apr-07 at 6:11 am | Permalink
  47. feo wrote:

    Hempel’s paradox is not a paradox at all, it’s simply a good illustration of why the Scientific Method is so important. “All non-black things are not ravens” is a hypothesis, and hypotheses like this can *never* be proven, only disproven. Seeing a pink flamingo doesn’t confirm anything, it only discounts things. So, since seeing one doesn’t discount either “all non-black things are not ravens” or “all non-white things are not ravens”, it gives you no information (about these two hypotheses). Seeing a black raven, however, does disprove “all non-white things are not ravens.” Taking observations as proof of a hypothesis goes directly against the scientific method, and this is exactly why. Further discussion about this “paradox” is gratuitous.

    Posted on 03-Apr-07 at 10:02 am | Permalink
  48. furu wrote:

    To Admin: Newcomb’s problem
    I still fail to see the terrible paradox. Indeed if the scientist could predict through -I assume- psychoanalytical means, then it would have to take into account that very doubt. Any person who was put to the test would have the same doubts, suggesting it is part of the psychological process that the scientist understands enough to make accurate predictions about.. I don’t see how the causal nature of the choice fails to put the weight on trust in his abilities. Only someone who doubts, even a little, his abilities would go for both boxes. Are we somehow not supposed to assume that the basis of the choice was what the scientist analyzed?

    Posted on 05-Apr-07 at 2:02 pm | Permalink
  49. admin wrote:

    Hi Furu,

    Technically it’s not a paradox… we’re not led into a contradiction of sorts – it’s just that there are two lines of reasoning that urge us to make opposite decisions.

    the evidential decision theorist says – even at 99 percent odds… or 98, or 97 (down to your own betting preferences) that the predictor is right – then those odds are still good enough to one box it. A little bit of doubt itself isn’t enough to cause you to two box…

    But there does seem to be some implication that in choosing to two box you are claiming that you lie somewhere outside the normal sample of people who two boxed and got a thousand dollars. But then again – if you two boxed and got 1000 dollars you would believe that that was the most you could have possibly gotten.

    The causal issue confuses me too – it’s part of the debate that I haven’t quite gotten my head around. The idea is – as I understand it – is that our decisions should match the causal structure of the world. But I’ve never quite been able to fully suss the reasoning.

    I’ll look into it and do another post on it – cause I need to understand this stuff better.

    Posted on 06-Apr-07 at 10:02 am | Permalink
  50. John McNellis wrote:

    How do we know “anything”
    when we can’t “know” anything?

    see “The Nature of Awareness” at http://www.anglingtek.com

    Posted on 09-Apr-07 at 11:49 pm | Permalink
  51. Charlie wrote:

    John McNellis:

    We don’t ‘know’ anything because we can’t ‘know’ anything, but it doesn’t matter. Knowledge is not necessary, only a reliably high predictability. We don’t ‘know’ that all ravens are black, but we have enough anecdotal evidence supporting the theory that they are all black that we routinely assume it to be true, even if we have certain knowledge of the existence of anomalous non-black ravens [albinos, etc.]. The statement is supported by such an overwhelming proportion of observations that the anomalies are simply ignored, or noted as anomalous, or explained away by better precision in the use of language, as: “With very limited exceptions, all the ravens ever recorded as being observed have been black.”

    Posted on 10-Apr-07 at 12:47 am | Permalink
  52. Charlie wrote:

    I should add that my observation does nothing to solve any of the paradoxes; it simply observes that in the ‘real’ world, such paradoxes often don’t apply, or are irrelevant to our ability to live, work, reason or otherwise make our way in it. They are very useful in training us to observe carefully and use language precisely in the [futile] attempt to be consistently accurate.

    Posted on 10-Apr-07 at 12:53 am | Permalink
  53. admin wrote:

    I think John’s statement was more a bit of a plug for his site than a direct comment on the problems. It’s interesting enough site though – and it has a nice photo of a boat on the front… so I’ll let it stay.

    Charlie: – I wonder about your second comment. One of the things I often complain to philosophical colleagues and my supervisor is in finding some real pragmatic consequences to the various problems we consider. No one has been able to give me a good answer yet – so you might be right.

    Posted on 10-Apr-07 at 11:19 am | Permalink
  54. [This was just pure entertainment for me, so if reading more strong opinions on the paradoxes as you stated them would disturb your professional work, feel free to ignore my comment.]

    Newcomb’s Problem

    Logical answer: Take only box B if you have 33% or more confidence that the predictor is infallible. Otherwise take both boxes and make more money on average.

    Intuitive answer: Offer to take only box A or nothing at all. The scientist has not said what he would put in box B if he predicted that for your choice, so it might be something troublesome enough to make up for the $1000 in box A.

    The Liar Paradox

    It’s a nonsense sentence. If all sentences have to be classified true or false, it would be better to classify it false and reserve the word true for sentences that don’t negate the truth when classified as true. That results in there being a sentence that is false and yet all it says is that it is false, so why isn’t it true? Because it’s nonsense and can’t be true. The lesson is that if every sentence that isn’t true is false, then there are some sentences that are false while they also say they are false, which is a little “truthiness,” but not enough to make them true.

    The variation with two sentences referring to each other is still reflexive, taken as a whole. Pure reflexiveness, lack of reference to some outside fact or truth, is what makes the sentences nonsense, therefore not true, therefore if all sentences not true are false, false.

    “This sentence is true.” is also nonsense. You seem to have a choice whether to classify it as true or false, which becomes a self-fulfilling classification. Therefore, it is nonsense, and false, in the sense not really true.

    The Unexpected Hanging

    Instead of believing a lawyer, stupid prisoner, the prisoner should have announced that he expects the hanging, if it is to occur, on a certain day with absolute certainty and he will not say which day. Then the prisoner would not be hanged, at least not in any country where executioners are cautious enough not to take a one out of seven chance of hanging a prisoner against judicial orders.

    Hempel’s Paradox

    I don’t get it. Why IS a raven like a writing desk?

    The Sorites Paradox

    It’s a heap of sand if there are one or more grains resting on top of others. You can make it not a heap by spreading the grains out without removing any. You can keep it a heap, removing grains until there are only three left if you leave one grain resting on two others. If you left one grain resting on only one other grain, that would be a stack, not a heap.

    That’s not presissification. It’s practical sense and the dictionary backs it up. Trying to find a philosophical conundrum in the shadings of use of a word while ruling out investigating the circumstances of the actual use and meaning of it is the philosophical error that causes the paradox.

    I see xeno and Emily above have already given some similar opinions about the meaning of the word heap.

    ALL solutions of real-world problems in words involve “the problem of higher order vagueness.” That’s why most people are right not to trust using symbolic or formal logic for solving real world problems and not to use it even if they can. Problems stated and to be answered in words, when they involve the real world and not just symbol manipulation to entertain a judge of symbol manipulation, always include the problem of whether the words really apply and constantly checking that they do still apply to reality despite logical or grammatical manipulations.

    Posted on 10-Apr-07 at 3:39 pm | Permalink
  55. admin wrote:

    Good post sonny,

    I also wonder a lot about just how much logic is going to be able to help us answer these questions – or whether it’s through logical thinking that they become questions at all.

    I think in the case of the sorites paradox though – assuming your definition of heap is correct – (i.e. you’re effectively saying that it’s not a vague predicate at all), then there are still plenty of predicates out there that are vague and could be used to generate sorites. I doubt the dictionary will save us in all cases.

    But I really do like your last comment – what is the relationship between language, the real world on the one hand – and logical, mathematical systems we use to represent them on the other? It’s a big question. :)

    Posted on 11-Apr-07 at 8:28 am | Permalink
  56. Chris wrote:

    The only thing of which the convict can be certain is that he will be hanged this week. He knows his death is imminent and a distinction between weekdays is an unnecessary division of time: he could simply ready himself in each passnig moment. Thus, he can never be surprised and likewise never hanged. It doesn’t need to be complicated further, and it is in this way that he could avoid surprise and spare his neck. Of course, the judge would not even consider this and would still schedule his hanging on Wednesday. “Surprise, your booked on Wednesday.” The problem not a paradox but a natural conclusion: the Judge’s order is simply absurd.

    Posted on 11-Apr-07 at 2:56 pm | Permalink
  57. Tibor Alfred Richter wrote:

    Alfred said on April 12+1 2007 \ Nisaan 25 5767 at 10:00 am

    A wonderful (or wonderfool?) page of spiritual entertainment, indeed.

    Thanks The Eternal. I really enjoyed that.

    Posted on 13-Apr-07 at 4:01 pm | Permalink
  58. Dennis wrote:

    The paradox of the boxes comes down to a question of free will. If the universe is deterministic, this paradox becomes meaningless since no choice can be made. If free will exists, then the scientist cannot predict with 100% accuracy and so the choice is a matter of statistics as any good investor could tell you.

    The liar paradox reminds of divide by zero. Dividing by zero has different results in different situations, but in general is has a result of no meaning–as Sonny said. So perhaps the paradox rises from lack of this idea. Rather than classifying things as true or false, we should classify them as true, false, or meaningless.

    The Sorites paradox is one that touches everyone’s life. We are all either adults or children, but when should we make the distinction? Under different legal jurisdictions, we get different answers (20 years old where I live), but the same reasoning could be applied. A child is 1 year old. Add one second to its life and it is obviously still a child. At what point does it become an adult. The only reasonable answer seems to be that you can’t break these ideas down in this way and when you try to get different interpretations depending on who does the breaking down because language is relative. By asking relative questions about it, you can always get the answer you want. This is a common trick used in moral reasoning which leads to “the slippery slope”.

    Try this experiment instead. Make a heap of sand. Show it to 100 people an ask each person if they think it could be described as a heap. Then, take away 1 grain of sand and ask 100 different people if they think it could be a heap. Repeat until sand is gone. Now the relativity is taken away from the individual observer and you can formulate some definition of what consitutes a heap (such as the limit at which no one will agree it is a heap or the point at which fewer than 50% of people agree that it is a heap).

    Posted on 13-Apr-07 at 4:53 pm | Permalink
  59. LIZ wrote:

    Hi I would just like to mention that Ravens are not all black. If a Raven was hit by a car on the side of the road, it is still considered a raven right? But there would be blood and guts which are red. Also, I would consider a raven’s beak more greyish. And The eyes have white in them. Plus there are different kinds of ravens with various different colours.

    We would still consider a raven to be black, like we would consider a person with white skin but brown eyes and blond hair to be white. But that does not mean the raven is only black, thus the statement “all non black things are not ravens” is not a logical statement. As stated earlier, a dead raven may be red but it is still a raven

    Posted on 14-Apr-07 at 2:00 am | Permalink
  60. avignon wrote:

    Ravens are like writing desks because they both produce flat notes.

    Posted on 14-Apr-07 at 3:01 pm | Permalink
  61. anon wrote:

    Newcomb’s Problem. This puzzle illuminates the value of faith applied in “real” world situations. That is, if you have faith that the person who gives you the choice of boxes is looking out for your best interests and wants you to get the money, you can signal that belief on your part by a deed of faith, i.e. by taking box B. You can ponder the possible existence of the ability to change past events by our present actions and think about what mechanisms may or may not be involved. But if you take the person making the offer at their word and have faith that the promised result will obtain regardless of whether you understand the mechanism involved, you guarantee for yourself the optimal outcome.

    I believe that faith in the idea that others are looking out for your welfare on the part of all participants would lead to the optimal result in the Prisoner’s Dilemma as well. That is, if each suspect remains silent in the faith that the other is looking out for his best interests, the total amount of jail time served by the two will be optimal.

    With respect to the implications of quantum theory for philosophy in general, I would guess that much of philosophy falls apart without the presupposition that time is linear and flows in only one direction. The future will see a convergence and mutual enrichment of science and religion. Physicists, theologians and philosophers are doing themselves a grave disservice by not studying the Seth material presented in Jane Roberts’ books. The existence of depraved occult practices and of hoax and fabrication should not keep serious minds from considering the validy of what Seth had to say.

    Posted on 15-Apr-07 at 3:16 am | Permalink
  62. Spencer wrote:

    For the Sorite’s Paradox:

    A heap is no longer a heap when there are 37 grains of sand remaining.

    Posted on 18-Apr-07 at 6:04 am | Permalink
  63. tom m wrote:

    most of these are so obvious as to not even be interesting.

    Posted on 18-Apr-07 at 12:23 pm | Permalink
  64. andrew wrote:

    newcomb’s paradox as you presented it is absurd. you stipulated as part of the riddle that the scientist always predicts correctly. therefore, the entire line of reasoning as to why you should pick both boxes is wrong. you need to make up your mind whether the scientist always predicts right or not. a riddle that contradicts itself is not a paradox, it’s just poorly constructed.

    Posted on 21-Apr-07 at 11:56 am | Permalink
  65. Peter wrote:

    As to the ‘this sentence is false’ quandary: This isn’t much of a problem at all if you
    a) eliminate self reference (as noted) into (for example)
    The next sentence is correct.
    The preceeding sentence is false.

    The solution is obviously that noone said that BOTH sentences must be lies (or truth). Someone who utters these sentences is not confined to speak either truth or lie (he may do both or neither).

    Another solution is that there are states other than ‘right/true’ or ‘wrong/false’. Don’t believe me? Pick up a book on the Heisenberg Uncertainty Principle or superposition of probabilities. Only human minds insist on truth/lie distinctions. Sorry, but the universe does not work that way. Uttering such sentences is doomed to produce paradox – since they are, ab initio, fraught with invalid assumptions.

    Posted on 23-Apr-07 at 8:12 am | Permalink
  66. Evan wrote:

    I have recently begun to get the impression that much of the Greek-based logic is far to digital to be applicable to real-life scenarios. For example, the the grain of sand problem would define one grain of sand as a heap so long as it came from a heap that had it’s grains of sand removed on at a time. This seems to me to be too digital, if you know what I mean. Either it is something or it isn’t something, whereas I find the human mind to act in a much more analog fashion. A heap has a varying definition within our own minds that could be based on the size of the original pile of sand, or some odd standard that arose in our minds through some kind of societal reinforcement of the concept of a heap. As the author stated, our language does not define specifically what a heap is. This is why I have found this kind of Greek logic to be inapplicable to real-life scenarios. It is too digital or mathematical for a realm of approximations and varying definitions. Perhaps we need to develop some kind of Quantum Physics-based logic… Or something like that. Anyway! I love the puzzles. I love having that kind of thought stimulated.

    Evan

    Posted on 24-Apr-07 at 1:13 pm | Permalink
  67. admin wrote:

    Hi Evan,

    There are plenty of people who agree with your intuitions. Although they try to cash them out in different ways.

    One approach is to use fuzzy logic to account for these sorts of intuitions.

    If your keen – you can check out my post:

    http://danielhaggard.com/18/the-fuzzy-view-of-vagueness/

    Althought it might be a little tedious if you aren’t from a philosophy/logic background (and my writing probably isn’t the greatest on the topic).

    Posted on 24-Apr-07 at 1:29 pm | Permalink
  68. Tim Gane wrote:

    Placed in reality, a heap of sand no longer remains a heap as soon as the quantity is definable by the human eye (which is obviously variable) As it is sand, this would mean the heap would be tiny before it is considered a non-heap.

    As soon as the quantity of sand becomes definable, i.e. the point where taking away grains of sand suggests there “are fewer grains of sand” rather than simply “less sand” then the heap is not a heap but a collection of sand grains.

    The sand itself transforms from mass to individuality when the human mind can discern all grains as individual.

    PROFESSOR SCIENCE would tell us that this point is a maximum of 16 grains. Why? Because the human brain does not (cannot?) discern individual items greater than 16. (If I were to ask you to imagine 17 cats, you would merely imagine a grouping of cats, but your mind could feasibly imagine up to 16 individual cats, if you used all your will and might)

    So a heap ceases to be a heap once you take the 17th grain of sand remaining away.

    “Now, if you’ll excuse me, I have a heap of 17 cats to imagine…”

    Posted on 24-Apr-07 at 4:46 pm | Permalink
  69. max wrote:

    Amazing! What an amusing way to waist a few minutes in the office. And It’s true. Our language is not precise enough to express what our mind can come up with.
    I like it! you can get computers confused with some of these paradoxes. Programmers call it a dead lock! Check out the “Dining Philosophers Problem”

    a lovely day this page is bookmarked!

    greetz Max

    Posted on 25-Apr-07 at 9:26 pm | Permalink
  70. AttemptingReason wrote:

    I’m sure some on else has mentioned this. Hempel’s paradox isn’t very hard. The perceived error stems not from the logic, but from the input. Observing a pink flamingo can’t tell you anything about the color of ravens, only observing ravens can do that. Also, the phrasing of the paradox said that observing the flamingo ‘confirmed’ both of the statements; I would suggest that it ‘did not disprove’ either of them.

    Posted on 26-Apr-07 at 9:56 am | Permalink
  71. Starspawn wrote:

    You all must be collage students or grads. Can’t really believe you waste your time on this stuff. Does it really matter in the face of what’s going on in the world. No wonder God doesn’t like philosophers, you waste your time debating instead of doing!!!

    Posted on 26-Apr-07 at 11:17 am | Permalink
  72. admin wrote:

    Hi Starspawn,

    There are a number of responses I could make to you Starspawn:

    First I could ask you to give a positive account of what does matter in your view. In general I encourage people to be constructive in their criticism. Fair enough that you think we’re doing something pointless – but until you provide at least some measure of an argument for this view – you haven’t said anything at all.

    Secondly – I might ask why you think this sort of exercise is in some way an inferior sort of practice to say, watching tv, movies, playing sport, going for a walk, playing chess … etc and all the other recreational activities in which people enjoy. Are all these things a waste of time too? Are you similarly against them?

    Thirdly, philosophy in a general sense tries to make sense of the foundations of thinking and understanding. You might like to think of this as the clarification of all doing. But I can understand why the great abstractedness of it all might make it difficult to understand why this might be important.

    Posted on 26-Apr-07 at 2:41 pm | Permalink
  73. bah wrote:

    These are all very real paradoxes, in that none of them can be resolved with traditional (2-valued) logic. The best one is the unexpected hanging, nobody’s even come close to a decent answer (anywhere).

    Posted on 27-Apr-07 at 7:08 am | Permalink
  74. starspawn wrote:

    Dear admin,
    What I think is important is we need people to go out and feed the hungry, house the cold, be a friend to a stranger, do one anonymous good act a day, in other words, to help others without recompense. Now I’m not a student or grad, but I think, and so I have been told, that my thought process is on a very abstract plane. And I just think that if more people did instead of just thinking about it, this world would be a better home for us all. Thank you.

    Posted on 28-Apr-07 at 10:28 am | Permalink
  75. admin wrote:

    Hi Starspawn,

    Good reply. I guess I just feel that thinking through problems like these is at least as worthy as watching a game of football – yet I don’t see people turning up to football games and telling all the people there to stop wasting time and go feed the poor.

    I believe in balance – action without thought is a very dangerous thing. So we have to have both. And just because a few of us enjoy thinking through a few puzzles on a blog (perhaps after a hard days work feeding the poor for some of us) – doesn’t mean this balance is in anyway being disrupted.

    It’s just a bit of fun

    Posted on 28-Apr-07 at 12:00 pm | Permalink
  76. Starspawn wrote:

    Dear admin,
    Good response, thank you. and I always wondered what infinity squared minus one is. Perhaps you can help. And what about inertialess drive for space craft, think it’ll ever happen, and is there a good way to shield spacecraft from cosmic rays as it approaches the speed of light? Just askin’.

    Posted on 29-Apr-07 at 6:48 am | Permalink
  77. moeshroom wrote:

    Actually none of these paradoxes were meant to be taken at face value anyway. The people that proposed these questions were likely proving a point through questioning the students basis for individual thought in the first place. the meaning lies not in the paradox but rather in the lesson that may be incurred through examination of the problem itself.
    Newcomb’s problem is not about boxes and prediction. Instead the problem arises when thinking upon the nature of trust of one’s self and others. If you trust that the scientist always predicts correctly and trust yourself in this conclusion, then the clear answer is to pick B, knowing without doubt tht he knew you would pick that and planned accordingly. The converse to this situation is also true. It is also a perplexity about the nature of risk and greed. If you are so inclined to take both boxes then you inevitably get only the thousand dollars, however; if you are to risk losing a thousand dollars for the adventure of seeking the undiscovered, you have the opportunity to gain far more. The lesson is far simpler than the question would imply – “Nothing ventured, nothing gained.”
    The Liar is not about the sentence either. The lesson is only gained when one realizes that something may be both true and false at the same time suggestively symbolizing the duality and perspectivity of our subjective life. In addition, the sentence is incomplete grammatically.
    The Hanging is a simple excersize in expecting the unexpected. Also the lawyes logic is in error because they add an extra “if/then” exploration into the question, which negates the impression already given by the judge. Thus the logic added by the advocate provides the truth needed to execute the court order by throwing the convicted prisoner into confidence that they will not be hanged.
    Hempel’s or the induction paradox is answered simply because it approaches the concept of objective versus subjective statements. To induce something one must rely on one’s own experience which may easily be faulty. The real question is to question one’s own conclusions;) rather than accept an incomplete version of perspective.
    Sorites’ does indeed rely on use of an incomplete linguistics model but at the same time, using modern scientific knowledge one may eliminate this particular term’s paradox. A heap is a relative quantity of particles or objects – so by that definition even a single grain may be considered a heap of sand, being that is a conglomeration of molecules of sillicon dioxide.
    Anyway… none of these so called paradox’s are relevant to themselves as such. Somebody was trying to postulate about the broader psychological and sociological aspects of the human endeavor and indeed life itself.

    Posted on 02-May-07 at 4:00 pm | Permalink
  78. harsha wrote:

    ha! some pure fun,no matter what others have stated,i really enjoyed all the paradoxes,really good fun. GR8 work dude;-)

    Posted on 03-May-07 at 1:46 am | Permalink
  79. Kev wrote:

    Further to the liars paradox, consider the sentence “I am a compulsive liar.”

    Always makes me smile when I try and figure that one out in my head.

    If I am a compulsive liar, then the statement is true, which means I can not be a compulsive liar. if I am not a compulsive liar, then I lied when I said that sentence so therefore I may be a compulsive liar.

    Posted on 03-May-07 at 7:52 am | Permalink
  80. admin wrote:

    Thanks Harsha :)

    Stay tuned for some more puzzles in the future.

    Kev – if by ‘compulsive’ you mean that everything you say is a lie, then I think this is just a case where the sentence is self-refuting – but not actually paradoxical.

    It could have the form:

    (Vx)(Sx -> Lx)

    Which could be translated as For every x – if x is a statement, then x is a lie. Or in more natural english – every statement is a lie. It is itself a statement – so if it is is true, then it implies that it is a lie – but this itself implies that it is false. but if the statement is false, then it should imply Ex(Sx.~Lx) which says at least one statement is not a lie. But it seems to me that it doesn’t imply that THAT statement is the one which is the true statement. So in this case – there is no reason to conclude that the statement is true (when originally thought of as false). So if it is true, it is false, and if it is false, it is just false. So it must be false.

    You can’t be a compulsive liar, while telling everyone about it!

    Posted on 03-May-07 at 9:17 pm | Permalink
  81. Perfect!

    -Steven Burda
    http://www.linkedin.com/in/burda

    Posted on 06-May-07 at 7:09 am | Permalink
  82. I had never heard of the Newcomb problem until I read this post. And here I was, thinking I knew most of the major philosophical brain-teasers!

    Posted on 07-May-07 at 1:55 pm | Permalink
  83. Barry McCawkiner wrote:

    The real puzzle/paradox is why these are referred to as “great.” These are fairly weak… but maybe I’m just being an asshole. Or maybe they really do suck. Your choice!

    Posted on 19-May-07 at 3:51 am | Permalink
  84. Christian Burnham wrote:

    Unexpected hanging solution

    The prisoner has to have some way of communicating the fact that he knows he is to be hanged on a particular day. If he is right, then they will let him go.
    Let’s say he has a bell which he can ring on the day he thinks he is to be hanged.
    By inductive reasoning he would ring the bell on Monday, then again on Tuesday etc. etc.
    Obviously it would be to his advantage to ring the bell every day — he has to be right eventually.
    But for this paradox to mean anything we have to impose a new rule– he can only ring the bell once. It would be unfair to let him ring it every day.
    Thus he rings the bell on Monday. However it happens that he’s not killed until the very last day. On the last day he knows for sure he will be killed, but there’s nothing he can do about it. The bell has already been rung once, and so he is hanged.

    Posted on 03-Jul-07 at 6:29 pm | Permalink
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    Posted on 31-Jul-07 at 11:57 pm | Permalink
  86. Revelation 23 wrote:

    A bit late to the party (just going through older posts here, for lack of anything else to do at the moment), I’m going to take a crack at a few of these. I might take a different approach than others have (and open up some new problems at the same time), but is there a right way and a wrong way to this?

    Newcomb’s Problem

    Let us stick to how it’s originally stated and not give the scientist’s predictions any margin of error; he is ALWAYS correct. Does he determine the outcome, or does the outcome determine the prediction? Not unlike the chicken or egg question, although certainly with less flavor.

    Given that he’s always right, you’re walking away with at least $1,000, provided you play by the rules and not choose to take only box A instead. You’re not going to lose. But, one million dollars is more than a thousand and so long as you trust that the scientist will keep his word, take box B and walk away with the million; don’t question the amount each box could hold. If you want a million dollars, that’s what you’re walking away with.

    So long as you’re dealing with a person who knows the difference between the contents of the boxes and/or their value or importance, the million dollar box is the choice. Logic doesn’t need to be dragged through all of this. Take the million dollars and be done with it.

    Take away the money and its effect on people and you have a harder choice. Box A has a cube. Box B has a sphere. Now what?

    The difference between the two doesn’t have the same meaning for us that the difference between one thousand dollars and one million dollars does. There is no ‘greater than’ involved. They aren’t the same thing, but as far as we are concerned, they are on equal standing. It’s just a matter of simply choosing one, not taking what we want.

    Now you get to what I think the problem is and what the situation should be (just my opinion, mind you).

    By using money, you’re playing off something we’ve been taught, not things we know.

    For the sake of argument, you had someone who was blank slate, never taught, never had contact with anyone else. They are completely oblivious to everything we do, everyhing we know. Forget the problems one might encounter presenting this game to them, let’s just say they understand that taking one box grants them one thing, taking both gets them something different.

    While they could probably tell the difference between one million things and one thousand things, to stick to the original amounts presented here, and might also be able to understand a concept of “more” (don’t assume either way). that doesn’t mean they’d make the choice the same way we do. Take one, get one thing. Take two, get another thing instead. Here’s where the scientist’s game could get interesting. Maybe, but it could very well be the thing that proves that he can do what he can, dealing with someone who doesn’t know what the two prizes are or what they mean.

    We’ve put value and importance on things. But, give a monkey two checks, each for a different value. Is the monkey going to know the difference? No. Monkeys don’t know about money and in general aren’t taught anything about math or values. It’s just a piece of paper with ink on it. While monkeys and chimps may be able to grasp some of these things like we are, it’s not something that’s of use to them. They are intelligent, but in a different way than us. To us, the check represents money, a value, something we can use. To them, the check is just apiece of paper with ink on it. The check matters to us, because we equate it to having a value. The monkey has no idea what it is or what it’s for.

    With the cube or the sphere, even if you had never seen one before, had no clue what we called them, and didn’t know what it was that made a cube a cube and made a sphere a sphere, you should still be able to know there’s a difference. There’s no meaning to them and no reason to have instead of the other. It’s just something in a box.

    Just like a checks given to a monkey, and I don’t mean the boss’ inept relatives.

    There’s no incentive, but you have to decide.

    It’s a level playing ground now. Make your choice.

    Imagine the [added] chaos if you could choose between A, B or both boxes!

    The Unexpected Hanging

    The prisoner is hanged because he thinks that he won’t be. Forget the lawyer who gave him bad advice, send him to the gallows first.

    I don’t think it matters how many days he’s survived, he’s going to be surprised when they hang him without telling him about it before hand, no matter how prepared he is.

    “They’re coming to take me away, ha-ha!”

    Sorry, I had to do it…

    “I know it, so they can’t take me.”

    We’re taking you anyway. Surprise. Wasn’t this fun?

    If you want to go by days, the seventh day is the perfect day to do it, unless the day is the only thing that matters (as in only being able to do it at one point in the day, something we aren’t told). Why?

    “I just survived trough six days, they can’t do it today! I’m in the clear.”

    If he really believes this, he won’t expect it to happen. But we don’t really know how the prisoner is really thinking about this, other than he doesn’t seem to be worried. For all we know, armed with this last possible day logic, he could completely disregard the first day and be hanged then. Why it happened Wednesday, I don’t know. The judge could have already made up his mind and decided to not tell the prisoner, instead giving him something to occupy his mind.

    Surprise, you’re dead.

    The Sorites Paradox

    The heap of sand isn’t a heap anymore once you’re down to the last grain. Forget how many items the average human can tell apart under normal conditions. Forget that the word heap is used. It could be pile, it could be mass. Heck, you might even be able to get away with collection. And let’s toss out a ‘what if’ that briefly came to mind. Disregarding how large or small this ‘heap’ is, the grains of sand don’t have to be touching, just within a defined area. You can have two grains of sand left and it’s still a collection of sand. Not a large collection, but it’s still more than one.

    Like my take on Newcomb’s Problem, this comes down to our own, often confusing, way of thinking about things. Where’s the threshold? Are we looking to find a point where the “heap” looks different than it did before? If so, that’s not going to happen if you’re taking sand away grain by grain AND are paying constant attention to all the sand from start to finish. If you get up and leave for a while, then return, you might see a difference, more so if you’re just an observer. It’s the same with watching grass grow. You aren’t going to notice any progress as you’re watching it. You’ll only notice if you pay attention to it at different times.

    Maybe when you return, you wouldn’t consider it a heap anymore. But that’s open to interpretation.

    Just as grass doesn’t grow fast enough for us to see it happening, taking sand away grain by grain isn’t enough of a change to notice, so long as it remains a single grain each time. If we leave or look away long enough, then return our attention to the sand, it’s possible we’d see a difference. You could say the same about the grass. It all depends on how much time has passed and what the rate of growth ( the grass) or de-heaping (the sand, and yes, I just made that up) is during that time.

    If you can rememeber with 100% accuracy how far along the grass was or how exaclty how much was left in the heap when you stopped paying attention, and knew the exact figures for the current time, then you’d notice the difference, not just know that there should be one.

    One grain.

    Okay, my head hurts now. I think I’m done for a while.

    Posted on 16-Jan-08 at 1:14 am | Permalink
  87. Revelation 23 wrote:

    Okay, I lied, and I’m telling the truth about that.

    One more thing comes to mind regarding the heap of sand.

    No matter what, when you have only one grain of sand, it’s not a heap of sand anymore. Me, I think one is the answer (and it’s the loneliest number). Maybe two is still a heap, maybe it’s above ten, or a hundred, or a thousand, or…

    But one is never a heap of sand.

    Posted on 16-Jan-08 at 1:30 am | Permalink
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  93. brian wrote:

    in Newcomb’s problem, it would always be more profitable to take both boxes.

    if ‘A and B’ is predicted and ‘A and B’ is chosen, you would have $1000.
    if ‘A and B’ is predicted and ‘B’ is chosen, you would have $0.
    if ‘B’ is predicted and ‘A and B’ is chosen, you would have $1001000.
    if ‘B’ is predicted and ‘B’ is chosen, you would have $1000000.

    Posted on 18-Jun-10 at 1:03 pm | Permalink
  94. Rex wrote:

    On hempel’s paradox:

    this is no paradox at all. it can be assumed that all ravens are black, however black does not define the raven. it is merely a common condition of the being of the raven. if i for instance caught a raven and presented you the bird, you would confirm that it is indeed a raven, and if i painted that bird before you blue, it does not make him a blue jay. therefore the statement all things that are not black are not ravens is not a logical equivalent

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