Coercive Logic

Here’s a logic puzzle for you: Suppose I offer you a million dollars, in return for which you agree to answer a certain yes/no question. You can answer either truthfully or falsely as you desire. That’s it. Should you accept that offer? Solution below the fold.

Those of you reading this who enjoy logic puzzles are probably familiar with Raymond Smullyan. I was pretty young, eight or nine I think, when I first discovered his writing. Somehow I noticed his book What is the Name of This Book? sitting in a bookstore, and I persuaded my parents to buy it for me. The book opened with some very elementary puzzles I was able to appreciate even as a young child. But pretty soon I came to one of his most famous creations, the island of knights and knaves.

On this island the knights always tell the truth and the knaves always lie. The inhabitants of this island are in the habit of gathering in small groups and making various statements, from which you are to determine who among them is a knight and who is a knave. Smullyan intended these puzzles as a device for teaching basic ideas in propositional logic, and I use them for that purpose in my discrete mathematics classes. Anyway, at some point little kid me came across the following problem:

Suppose you meet three people, who we shall call A, B and C. You ask A, “How many knights are among you?” A mumbles an answer, but you cannot understand what he said. So you ask B to tell you what A said. B says, “A said that there is one knight among us.” At this point C interrupts and says, “Don’t listen to B! He’s lying.” Can you determine anything about what type B and C are?

At the time I was not able to make heads or tails of this. So I brought it to my father. He glanced at it, and with seemingly no effort at all he tossed off something like the following: “Well, let’s suppose B is a knight. Then C must be a knave, since he lied when he said that B had lied. Very well. Since B is a knight, then A really did say there was only one knight among them. But if A were a knight, then he would have been lying, since we know B is a knight and that would then make two knights. So A would have to be a knave. But then his statement would be true, since there really was only one knight among the three of them. So this is impossible as well. The only conclusion is that B must be a knave, and C is a knight.”

I was impressed. I have a clear memory of thinking, “I want to be able to do that.” With some fatherly guidance I soon got the hang of it, and pretty soon I was able to solve some of the easier problems. At that age I still wasn’t able to work out the more complex puzzles Smullyan devised, since I would quickly lose track of the analysis tree and I absolutely refused to write anything down. But I was hooked, and I have been a big fan of Smullyan ever since.

Anyway, knights and knaves are hardly his only invention. He also developed the idea of “Coercive Logic,” by which he meant logic that compels a person to do something he would not otherwise do. The puzzle from the start of this post is an especially ingenious example.

Of course, you should not accept my offer. If you did accept, and if you adhered strictly to the rules of the game, then you would find yourself paying me two million dollars. You see, the question I would have you answer is, “Will you either truthfully answer no this question, or falsely answer yes, or pay me two million dollars?”

We shall analyze the consequences of your various answers momentarily, but first we need to clarify one of the rules of propositional logic. In everyday speech, when we make a statement of the form “P or Q” there are two different meaning we might intend. Sometimes we mean, “Either P is true or Q is true or both are true.” Other times we mean, “Either P is true or Q is true, but it is not the case that both are true.” In logic, we always intend the first meaning. Thus, a statement that consists of a bunch of propositions connected by “or” is deemed to be true as soon as at least one of the individual propositions is true.

Now let’s consider my question. According to the rules of the game, you must answer either truthfully or falsely. That probably didn’t seem like much of a restriction, but it does rule out a big possibility. It prohibits you from answering in a manner that entails a contradiction.

Now, my question is asking whether at least one of these three alternatives holds:

  1. You will truthfully answer “no” to my question.
  2. You will falsely answer “yes” to my question.
  3. You will pay me two million dollars.

By answering “yes” to my question you are affirming that one of these three possibilities holds. Now suppose that your “yes” answer were false. In that case the second item would hold, meaning that you answered truthfully after all. So this leads to a contradiction, and you cannot falsely answer &yes&rdquo. Thus, if you are adhering to the rules, and answer in a way that is either true or false, then your yes answer must be true. But in this case the first two items clearly do not hold. That means the only way your yes answer could be true is if you pay me two million dollars.

What happens if you answer “no”? If that answer is true then you’re asserting that none of the three items will hold. But by giving a true answer of no the first item holds, and we have another contradiction. So you cannot truthfully answer “no.” That means your “no” answer must have been false, which is tantamount to asserting that at least one of the three items does hold. But which of the three? Certainly not the first two (since you neither truthfully answered “no” nor falsely answered “yes.”) What does that leave? Again, only the third item.

Thus, you can truthfully answer yes and pay me two million dollars, or you can falsely answer no and pay me. The only other alternatives entail logical contradictions, meaning you did not answer either truthfully or falsely in those cases.

Clever! Smullyan’s books are chock-full of similar puzzles, so if you found this amusing then I recommend his work to you.

Comments

  1. #1 Vince Whirlwind
    February 14, 2012

    You’ve asked me to answer Yes or No (and you’ve just explained that ‘or’ this context is not XOR, therefore the answer can be both Yes and No), and since you asked me three questions, therefore I will answer yes, I will truthfully answer no, no, I will not falsely answer yes, and no, I’m %#$%ed if I’m giving you $2,000,000.

    In other words, I don’t like your puzzle much.

  2. #2 Wonderist
    February 15, 2012

    Paradox! I don’t have the additional million dollars to give, so no answer can succeed! ;-)

  3. #3 MobiusKlein
    February 15, 2012

    What if I answer the question without evaluating the logic, or evaluating it incorrectly?
    Won’t the answer still be answered ‘falsely’ despite answering yes? Where does falseness actually lay, in my mind, your mind, or some freestanding logic.

  4. #4 hibob
    February 15, 2012

    “but it does rule out a big possibility. It prohibits you from answering in a manner that entails a contradiction.”

    Not so fast:

    1. “You can answer either truthfully or falsely as you desire. ”

    2. “when we make a statement of the form “P or Q” there are two different meaning we might intend. Sometimes we mean, “Either P is true or Q is true or both are true.&rdquo. Other times we mean, “Either P is true or Q is true, but it is not the case that both are true.” In logic, we always intend the first meaning. ”

    Ok, my answer is “No”. The definition of ‘or’ you stipulated (after the contract was made, very naughty BTW) allows me to answer either truthfully, falsely, or BOTH truthfully and falsely.

    So I choose both.

  5. #5 Who Cares
    February 15, 2012

    Took me a second to get this one.
    Then I realized it was just a variation on the Person A always tells the truth and Person B always lies logic.

  6. #6 Wow
    February 15, 2012

    “Will you either truthfully answer no this question, or falsely answer yes, or pay me two million dollars?”

    Me: “That’s not a yes/no question”.

    You *asked* for an answer to a yes/no question. That one has three propositions, therefore requiring either two binary answers or a single trinary (or higher) answer.

  7. #7 Richard Wein
    February 15, 2012

    Pedantic mode is now ON.

    “You can answer either truthfully or falsely as you desire.”

    No, I can’t. Given the question, my answer cannot be evaluated as true or false at the time I make it, since the future is uncertain and it is unknown whether I will subsequently pay you two million dollars. If I ever do pay you two million dollars, my answer will then be discovered to have been true (if “yes”) or false (if “no”).

    I think you are conflating two different meanings of “I will pay you two million dollars”. It can be an assertion about the future, in which case its truth value cannot yet be evaluated. Or it can be a promise to pay, in which case it is not an assertion of fact, it has no truth value and your propositional logic is not applicable.

  8. #8 Wow
    February 15, 2012

    “Or it can be a promise to pay, in which case it is not an assertion of fact”

    In which case you can get a million dollars and say you’ll pay. Just not actually pony up the money.

    Just ask bankers how to do that..!

  9. #9 Jason Rosenhouse
    February 15, 2012

    Richard –

    No, I can’t. Given the question, my answer cannot be evaluated as true or false at the time I make it, since the future is uncertain and it is unknown whether I will subsequently pay you two million dollars.

    But if you answer in a way whose truth value cannot be determined as true or false at the time you give your answer, then you have not held up your end of the bargain. The deal was that your answer be either true or false, not that you answer in a way that may or may not become true at some unspecified future time.

    So, as I said, you can truthfully answer yes and pay me, or you can falsely answer no and pay me. Any other response will be neither true nor false, and therefore unacceptable under the rules of the game.

  10. #10 Wow
    February 15, 2012

    “But if you answer in a way whose truth value cannot be determined as true or false at the time you give your answer, then you have not held up your end of the bargain”

    However, the fault for that is yours. Therefore, they have kept up their end of the bargain to the best of their possible ability.

    If you’d put “and you pay me one million dollars”, rather than two, then you could assert that this can be fulfilled immediately. As long as you have one million dollars to hand…

    Other alternatives are that you intend to, with all honesty. E.g. after 35 years. But in the meantime, either one dies and the result won’t happen precisely as stated.

    “you can truthfully answer yes and pay me, or you can falsely answer no and pay me”

    What about falsely answer yes? Or truthfully answer no and not pay you?

    Also, your query wasn’t that. It had three parts, and the second two were not definitely linked.

    Answer yes
    Answer no
    pay 2 million

    if the second two are necessarily linked by an “or”, why not the first two? Which makes it logically incoherent, or all three?

    Under the rules of the game, I can answer yes and get a million off you.

    Or answer no and get a million off you.

    Or pay two million to you and get a million off you, saying neither yes nor no.

  11. #11 Wow
    February 15, 2012

    “But if you answer in a way whose truth value cannot be determined as true or false at the time you give your answer, then you have not held up your end of the bargain”

    However, the fault for that is yours. Therefore, they have kept up their end of the bargain to the best of their possible ability.

    If you’d put “and you pay me one million dollars”, rather than two, then you could assert that this can be fulfilled immediately. As long as you have one million dollars to hand…

  12. #12 Wow
    February 15, 2012

    (cont…)

    Other alternatives are that you intend to, with all honesty. E.g. after 35 years. But in the meantime, either one dies and the result won’t happen precisely as stated.

    “you can truthfully answer yes and pay me, or you can falsely answer no and pay me”

    What about falsely answer yes? Or truthfully answer no and not pay you?

    Also, your query wasn’t that. It had three parts, and the second two were not definitely linked.

    Answer yes
    Answer no
    pay 2 million

    if the second two are necessarily linked by an “or”, why not the first two? Which makes it logically incoherent, or all three?

    Under the rules of the game, I can answer yes and get a million off you.

    Or answer no and get a million off you.

    Or pay two million to you and get a million off you, saying neither yes nor no.

  13. #13 eric
    February 15, 2012

    Without looking at the other posts or below the fold…
    the only two combinations that work are knight-knave-knight, and knave-knave-Knight. Let K = knight and V = knave.

    KKK fails because B and C could not contradict each other.
    KKV fails because A would not then say there’s one knight.
    KVK works.
    VKV fails because liar A would not say there’s one knight.
    VVK works.
    VVV fails because then C told the truth about B lying.

    Thus, B is a knave and C is a knight. We don’t know whether (1) A is a knight who mumbled “two,” or (2) A is a knave who mumbled “two” or “three.”

    Hmmm…interestingly, we also know that A could not have mumbled “one.”

    Anyway, that’s my guess. Hope I’m not wrong.

  14. #14 Jason Rosenhouse
    February 15, 2012

    Wow –

    However, the fault for that is yours. Therefore, they have kept up their end of the bargain to the best of their possible ability.

    Not so! The fault for that is with the person who agreed to play the game. If the person answers yes and pays the two million dollars, then we can see that he answered truthfully. If he answers no and pays the two million dollars, then he answered falsely. So it is clearly possible for the person playing the game to adhere to the rules. But if he does anything other than the two options I just spelled out he will not have answered truthfully or falsely, and therefore will not have adhered to the rules of the game.

    Of course, the person may not have two million dollars, or upon hearing the question he might decide he doesn’t want to play anymore. But perhaps he should have thought of that before agreeing to play!

  15. #15 Wow
    February 15, 2012

    “The fault for that is with the person who agreed to play the game”

    Nope.

    No question was asked until AFTER the game started.

    You’re like Origin who claims that you must have agreed to the EULA since you bought the game!
    :-P

    “But perhaps he should have thought of that before agreeing to play!”

    Perhaps you should say what the game was before asking him to agree!

  16. #16 Jason Rosenhouse
    February 15, 2012

    Wow –

    Perhaps you should say what the game was before asking him to agree!

    I don’t see how the rules of the game could have been spelled out more clearly. I will ask you a yes/no question. You will answer either truthfully or falsely. I will pay you a million dollars for your trouble.

    What’s not clear about that?

  17. #17 Sammi C
    February 15, 2012

    No, I don’t buy this as anything other than an illustration of the tedium of the trivialities of propositional logic.

    First, it conflates language and fact. Reality establishes facts, not word games.

    Second, the commenter who points out that the truth value of the answer cannot be established at the point of utterance is heading in the right direction. This is a performative statement, as in “I promise to…” or “I intend to…” not a true/false statement. If you want to use logic analysis, you need at least some form of modal logic. In fact, with pedant mode on, “will you…” means “do you intend to…?”, it’s a statement of intent, not of logical necessity.

    Third, what do you mean by:

    Thus, you can truthfully answer yes and pay me two million dollars, or you can falsely answer no and pay me. The only other alternatives entail logical contradictions, meaning you did not answer either truthfully or falsely in those cases.

    Rubbish! the “mark” could sing “oh for the wings of a dove” and push a custard pie into the face of the annoying questioner before skipping off into the sunset. Does that “logical contradiction” (in your eyes) lead to a breakdown of the structure of the universe? No. Logical contradictions are usually an indication of the failure of logical analysis, not a problem with reality.

    Fourth, these puzzles seduce one into a profound mis-understanding of the purpose and nature of language. Real language is almost never about true/false statements, it is about influencing other people, it is a social act. What even a scientific paper? Yes – one could argue that the primary purpose of most scientific papers is to promote the authors’ status in the scientific hierarchy and their job prospects, and only secondarily to communicate information. That is one reason why puzzle-setters have to come up with absurdities like absolutely reliable truth-tellers and liars, when none such exist; it is an admission that this sort of logic won’t wash with realistic people and realistic language, even if it works brilliantly in designing binary electronics.

  18. #18 Wow
    February 15, 2012

    “I don’t see how the rules of the game could have been spelled out more clearly.”

    The bit about “you will have to pay me two million”.

    If your rules are purely:

    “I offer you a million dollars, in return for which you agree to answer a certain yes/no question. You can answer either truthfully or falsely as you desire.”

    There’s nothing there about agreeing to pay you.

  19. #19 Wow
    February 15, 2012

    You see, unlike the knight/knave problem, this one isn’t intending to make a logical proposition, it’s intended to show off how you can “win” a logical puzzle.

    Just like the other logical “puzzle” you had a while ago.

    The knight/knave leads you to thinking about what you’ve been told and how you can use inference to find the answer (though as eric has shown, the answer may not be a single answer, but a reduction in the possible answers).

    This one, however, isn’t trying to teach you how to use inference to expand the available knowledge into an answer. It’s made so that you can’t answer. It’s just mental willy-waving, showing off how you trapped someone else.

    In this case, the only winning move is not to play. Unless that was your intent, your proposition has holes you placed in there you can drive a barn through.

  20. #20 Karl
    February 15, 2012

    Anyone who’s read Douglas Hofstadter knows “mu” is an acceptable answer to a yes-or-no question.

  21. #21 Richard Wein
    February 15, 2012

    Jason,

    But if you answer in a way whose truth value cannot be determined as true or false at the time you give your answer, then you have not held up your end of the bargain.

    You missed my point. If this is the bargain, then I cannot possibly uphold my side, since I cannot answer in a way that can be true or false at the time I give it. My answer will only become true or false later, if and when I pay you the two million dollars.

    Now perhaps (contrary to what you wrote here) we can interpret the bargain as requiring me to act in such a way that my answer eventually becomes true. In that case, I can only uphold my side of the bargain by answering yes or no and then paying you two million dollars. But your wording doesn’t seem quite to be saying that. As your own interpretation here suggests, it sounds like the bargain requires me to give an answer that is true or false at the time I give it.

    P.S. Above I gave two possible interpretations of the sentence, “I will pay you two million dollars”. Perhaps a third possible interpretation is that I am reporting my state of mind: I currently have the intention to pay you two million dollars. That interpretation raises difficult philosophical questions about the nature of intentions and the degree of certainty we can have about our own intentions. But let’s say, for the sake of argument, that at the moment I give my answer I have the intention to pay you two million dollars. In that case, on this interpretation, I can give an answer that is true or false at the time I give it, since the truth about my intention is already fixed. But in that case I have already fulfilled my side of the bargain as soon as I reply, even if I subsequently change my mind or am incapable of paying you two million dollars.

  22. #22 Jason Rosenhouse
    February 15, 2012

    Wow –

    There’s nothing there about agreeing to pay you.

    But there is something there about answering truthfully or falsely. And it is impossible to answer my question truthfully or falsely without paying me two million dollars.

    Richard –

    You missed my point. If this is the bargain, then I cannot possibly uphold my side, since I cannot answer in a way that can be true or false at the time I give it. My answer will only become true or false later, if and when I pay you the two million dollars.

    Of course, you could pay me the money and answer either yes or no at the same time. But if you really want to be that picky about what it means for the answer to be true or false at the time you give it, then let’s simply put a time limit on how long you get to (a) give me an answer and (b) take whatever other actions you are going to take. At the end of that time limit, I will make a determination as to whether you answered truthfully or falsely. Would that satisfy you?

    Perhaps a third possible interpretation is that I am reporting my state of mind: I currently have the intention to pay you two million dollars.

    Adding a time limit would seem to rule out this interpretation. If we add the rule that you have, say, one minute in which to answer and take whatever other actions you are going to take, then my question is clearly addressing your actions after one minute from the time I finished the question. Your intentions at the instant you answered yes or no would then be irrelevant.

  23. #23 Wow
    February 15, 2012

    “But there is something there about answering truthfully or falsely.”

    Yup, and I’ll do that.

    What I won’t do is pay you because that wasn’t part of the game. At the very least, the PC gaming industry tells you how much the game costs, even if they neuter it by making it “limited install”, they don’t ask you buy it (FREE!!) then, when opening up the game, present you with another bill for several grand.

    Sorry, your game is simple to pass.

    “Yes”

    Honestly answered, now where’s my million.

  24. #24 taylormattd
    February 15, 2012

    “Will you either truthfully answer no this question, or falsely answer yes, or pay me two million dollars?”

    That is not a “yes/no” question.

  25. #25 Wow
    February 15, 2012

    Aye, that was the other problem (out of three?) I pointed out.

    It doesn’t work at education, doesn’t work as a gotcha.

  26. #26 Richard Wein
    February 15, 2012

    Jason,

    Thinking this over some more, I realise that there’s an issue with the time frame of the question and answer. Suppose you ask, “Will X occur?”, and I reply, “Yes, X will occur”. In that case I’m definitely putting the occurrence of X at a time subsequent to my reply. If, on the other hand, I simply answer, “Yes”, it might be argued that this is consistent with X occurring any time after your question. Consider this sequence:

    1. You ask, “Will X occur?”
    2. X occurs.
    3. I answer “Yes”.

    Was my answer true? It’s hard to say. If we accept that it’s true, then by the same token I suppose I could truthfully answer “Yes” to your original question by paying you two million dollars between the times of your question and my reply (or simultaneously with my reply). But it seems to make the problem rather unsatisfying if you have to rely on this interpretation.

    But if you really want to be that picky about what it means for the answer to be true or false at the time you give it, then let’s simply put a time limit on how long you get to (a) give me an answer and (b) take whatever other actions you are going to take. At the end of that time limit, I will make a determination as to whether you answered truthfully or falsely. Would that satisfy you?

    Well, you’re still using the phrase “answered truthfully or falsely”, which you yourself seemed to think requires that the answer be true or false at the time it’s given. Also, the determination you are going to make is not between true and false. It’s between true/false and neither. Your wording can be interpreted either way, but is likely to be taken the wrong way. Perhaps that was deliberate, as the aim is to fool me into accepting a bad deal. Anyway, a clearer way to state these terms is as follows: “You agree to answer Yes or No, but you will only be considered to have fulfilled your side of the bargain if at time T your answer turns out to be true or false.”

    Either way, I think this interpretation largely undermines the original point of the exercise. It seems to me the original idea was to trick me into committing myself to paying two million dollars, under the pretence that all I need do is answer a question. And the trick is based on conflating multiple meanings of “will”. Even with your wording (but more so with mine) it becomes clearer that fulfilling my side of the bargain may involve me doing something more than just answering a question, and so I’m going to be more suspicious about accepting the deal, especially if I feel I would be committing myself to doing my best to fulfill the deal.

    I would suggest seeing the problem as an illustration of how we can be fooled by ambiguity and conflation of meanings, rather than seeing it as a case of “coercive logic”.

  27. #27 Owlmirror
    February 15, 2012

    So, as I said, you can truthfully answer yes and pay me, or you can falsely answer no and pay me. Any other response will be neither true nor false, and therefore unacceptable under the rules of the game.

    It occurs to me that I could back up a level and say that I was responding falsely when I agreed to the rules that lead to the trick question.

  28. #28 Calli Arcale
    February 15, 2012

    I tend to enjoy logic puzzles, but this one isn’t a logic puzzle. It’s a trap — offer a person a million bucks in exchange for answering a yes/no question and then furnishing a question which cannot reasonably be answered in that form.

    There are quite a few puzzles in which you have a limited number of questions to ask a sage or perhaps a group of people, and the challenge is to work out how to obtain all the necessary information within those parameters. This is like trying to cheat and jam two questions (or in this case three, though I’d allow the first two as a single question since they are opposites) into one by writing the first question as a run-on sentence. It’s not a puzzle; it’s a trick question.

  29. #29 Hal
    February 15, 2012

    Suppose I answer:
    (1/sqrt(2))|Yes> + (1/sqrt(2))|No>

  30. #30 Composer99
    February 15, 2012

    I think the issue is that the puzzle re: yes/no/$2M is not a yes/no question in the colloquial sense. It’s a propositional logic puzzle.

    Let A = You will truthfully answer ‘no’ to this question
    Let B = You will falsely answer ‘yes’ to this question
    Let C = You will pay me (or, well, Jason) $2M.

    The question is not IMO actually a question. It’s a statement of propositional logic:

    A or B or C == TRUE

    If memory serves, by the rules of propositional logic, if any one of A,B,C is TRUE, then the entire statement (joined as it is solely by ‘or’ connectives) is TRUE.

    It’s possible I need to add in parentheses to make the order of operations clearer, but because the only connectives are ‘or’, it doesn’t change the final outcome.

  31. #31 Brandon
    February 15, 2012

    The core problem, it seems to me, is that the question you ask isn’t what anyone would consider a yes/no question, at least not in general parlance. I think that’d pretty much be my reply: “I’m still waiting for the yes/no question so I can have my million dollars…”.

  32. #32 mandas
    February 15, 2012

    I will of course, take your challenge, and I will keep the million dollars and pay you nothing. Let me explain.

    You said you would offer to pay me a million dollars in return for me agreeing to answering this yes/no question, “….Will you either truthfully answer no this question, or falsely answer yes, or pay me two million dollars?…’

    There is nothing in that question which compels me to pay you two million dollars. You gave me three options – say ‘yes’, say ‘no’, or pay you the money. If I answer yes or no, then I have fulfilled my end of the agreement and I don’t have to give you any money. It is only if I stay silent that I need to pay.

    And besides, your question is a nonsense. You say I need to truthfully answer no or falsely answer yes, but there is nothing to indicate truthfully or otherwise to what.

  33. #33 Richard Wein
    February 16, 2012

    I think I have a way of wording the question which removes the problems I raised. Those problems were caused by the fact the question was asking about the future. Here’s my suggested wording:

    “Is any of the following the case: your answer to this question is a true ‘no’, or your answer to this question is a false ‘yes’, or you have given me two million dollars by the time you answer?”

    With this question I can truthfully answer “yes” or falsely answer “no”, provided I have given you two million dollars by the time I answer.

    I still think you need to change your phrase, “You can answer either truthfully or falsely as you desire.” My ability to answer truthfully or falsely is not simply a matter of my desire. It depends on my ability to pay two million dollars. Also, that phrase does not strictly require me to answer truthfully or falsely; it just offers me those options. Since your trick relies on holding me strictly to our agreement, I think you have to be strict throughout.

    The problem for you is that you want to fool me into thinking that I only have to give an answer, when in reality you want to bind me into doing whatever it takes to make that answer true or false. But you must play fair and word it in such a way that it’s suitably binding without being untrue. I suggest: “You must answer ‘yes’ or ‘no’, but your answer doesn’t have to be true, as long as you do your best to answer truthfully or falsely.” This last bit does give a slight hint that there’s something fishy going on, but I can’t see a way to be binding and truthful without giving some hint of what you’re up to.

  34. #34 Ian Kemmish
    February 16, 2012

    1) Neither your father nor you explain why you discount the possibility that B sincerely but mistakenly believed what he told you.

    2) That’s not a yes or no question. Pay up, or I’ll call that cop over there.

  35. #35 Sam C
    February 16, 2012

    Shorter summary of some (not all!!) of the objections:

    At the appropriate time, did you think “will you marry me?” was a true/false question?

  36. #36 Michael
    February 16, 2012

    I can simply decide not to answer, void our agreement, forfeit the million, and punch you in the face. Rationally, you would be better off reducing the 2 million to just shy of 1 million.

  37. #37 Jason Rosenhouse
    February 16, 2012

    For those of you claiming that the question in the post is not yes/no, perhaps you can explain to me what answer other than yes/no you would give to it. Of course it’s a yes/no question. It’s not the sort of question remotely like what someone would typically ask in normal conversation, but that’s part of the point. If you do come up with an answer other than yes/no, then I will tell you in advance that my reply will be, “You didn’t answer my question.”

    Richard –

    We can clarify the instructions to say explicitly that you are not permitted to answer in a way that entails a contradiction. I think I would also build the time limit into the question: “Will you truthfully answer no to this question or falsely answer yes or pay me two million dollars ten minutes from now?” With those changes, though, I don’t see any ambiguity.

    If I ask you simply, “Will you pay me two million dollars ten minutes from now?” and you answer yes, then at that moment we cannot determine whether your answer is true or false. But we can certainly wait ten minutes, see if you pay me the money, and then make a determination as to whether your answer was true or false. If after ten minutes you don’t pay me then your answer was false. Is there anything ambiguous about that? I don’t see any issue here about your intentions or beliefs at the time you answered my question.

    Now, if I ask you a question about the future that has three propositions connected by or’s, then it seems clear what needs to occur to say that you answered the question truthfully or falsely. We wait until sufficient time has passed so that all three propositions in the question can be tested. With regard to the the question I asked in this case, that interval is ten minutes. After ten minutes I can make a determination as to whether your answer was true or false.

    It seems to me that if you answer either yes or no without paying me two million dollars then your answer entails a contradiction. That means that the only way you can answer my question non-contradictorily is to pay me the two million.

    Even without building a time limit into the question, I would think the logic of the situation does coerce you into doing something you would not otherwise have done. It coerces you into taking on an obligation. You can defer that obligation indefinitely, leaving me unable to determine for quite some time whether you have answered truthfully or falsely, but you have taken on an obligation nevertheless. Your natural death can serve as a time limit on how long I have to wait to determine if you have answered in a way that adheres to the rules of the game. Thus if you answer either yes or no and then die without paying me, I can say that you did not follow the rules, since you answered in a way that entails a contradiction.

  38. #38 Wow
    February 16, 2012

    “For those of you claiming that the question in the post is not yes/no, perhaps you can explain to me what answer other than yes/no you would give to it.”

    Well, absent that we’ve already said that there are three statements there, only two of which are mutually exclusive, therefore answerable with a binary response…

    You: Will you do A, B or C?

    Can’t be answered with yes or no.

  39. #39 g
    February 16, 2012

    It may be enlightening to consider the following variant of the trap question. Suppose the person who’s just given you $1M in exchange for your undertaking to answer a yes/no question asks instead simply “Will you either truthfully answer no or falsely answer yes to this question?”. That is, the same as before, but without the bit about $2M.

    Then the same logic that apparently leads to the conclusion that you have to give him $2M instead leads unconditionally to a contradiction. So, on the face of it, you simply can’t answer his question.

    And yet, of course, you can. Nothing prevents you uttering the word “yes” or “no”, whichever you happen to prefer.

    Perhaps neither answer can rightly be said to be either “true” or “false”, in the context of the question you’ve been asked. And what’s different about the $2M version is that he’s kindly provided you with a way to answer with an actual truth value, on condition that you pay him $2M. This is logically not much different from the version where he asks you the version without the $2M, then says “aha, I see you’ve got yourself into an impossible position and simply cannot fulfil your promise. Tell you what: I’ll release you from the obligation if you pay me $2M”.

    Is the Bad Guy really entitled to your $2M? Only in so far as he would be if he’d done that. Or if his question had been “Given that you’re now about to pay me $2M, is the weather today nice or not?” (Which, just like the original question, is one that can only really be *answered* if you accept the presupposition in it, and which the mark probably didn’t anticipate.)

  40. #40 Wow
    February 16, 2012

    “And what’s different about the $2M version is that he’s kindly provided you with a way to answer with an actual truth value”

    Nope, that’s not what’s being given.

    What’s being given is a chance for Jason to show of how s.m.r.t he is by building a trap.

    Without the $2m bit, it becomes “will you answer yes truthfully or no falsely” and that shows a logical trap. Damned if you do, damned if you don’t.

    WITH the $2m, it becomes a trinary and therefore there is no binary answer. Will you do A or B or C? Yes. Yes what? No? No what? Which option was the yes or no to?

    Or simpler from the Schoolyard: “Are you going to give me your money or do I smash your face in?”

  41. #41 g
    February 16, 2012

    Wow, I think maybe you’re misinterpreting the question. It’s not “which of these three things are you going to do?” but “are you going to do one of these three things?”. Is it the case that you will do A or B or C, or not?

  42. #42 Wow
    February 16, 2012

    No, it IS which of those three things are you going to do. There are three things there. Say yes, say no, pay 2 million.

    Now, if the game was:

    You must answer yes or no, and whatever your answer I’ll pay you a million, but if you do not give an answer, you pay me a million

    THEN that would be only two things asked AND would have been fully informed before agreement.

    Additionally, it places post-hoc restriction. Why can’t I say no truthfully? “agree to answer a certain yes/no question. You can answer either truthfully or falsely as you desire.” doesn’t say I’m not allowed to answer “no” truthfully if I desire, nor “yes” falsely if I wish.

    Your alternative challenge, should you decide to accept, is to prove that there was only two options there, rather than three. So far, all you’ve done is say there isn’t a third.

    But we can see them right there. They’re separated by “or”.

    So you’ll have to show that the second AND ONLY THE SECOND or is not like the other and doesn’t indicate a choice.

  43. #43 g
    February 16, 2012

    Wow, I suggest just assuming a less ambiguous wording of the question — say, “Is it the case that, in response to this question, you will either answer ‘no’ truthfully, or answer ‘yes’ falsely, or give me $2M?”.

  44. #44 Joel
    February 16, 2012

    The correct answer is “no”.

    And by that I mean the answer to the knights and knaves question, of course.

  45. #45 Wow
    February 17, 2012

    “Will you either truthfully answer no this question, or falsely answer yes, or pay me two million dollars?”

    Yes (truthfully): Yes, I will falsely agree to pay you 2 million.
    Yes (falsely): Yes, I will agree to pay you 2 million and I falsely answered yes.
    No (truthfully): No, I will not pay you 2 million and I truthfully answered no.
    No (falsely): No, I will truthfully answer no to this question

    And joel, according to Cohen the Barbarian, the answer to the question is to put your sword up against the first “knight’s” throat and ask him to repeat his statement.

  46. #46 Wow
    February 17, 2012

    “Is it the case that, in response to this question, you will either answer ‘no’ truthfully, or answer ‘yes’ falsely, or give me $2M?”.

    Look, if you want to insist, despite all logic, that it’s all one question, since “giving you 2million” is not a “yes or no” answer, I can’t do that either.

  47. #47 Lenoxus
    February 17, 2012

    I think I’ve created a simpler, perhaps clearer version. Firstly, you have the mark agree to answer the question completely truthfully, and with either a “yes” or a “no”, nothing more or less. He will possibly expect the question to be of an embarrassing nature or something, but he’s not worried; he’s got nothing to hide. To be nice (and limit the amount of legal trouble for yourself), you promise to give the money only after he fulfills his end, and you do not try to put him into a bind whereby he literally has to answer your question “or else”, whatever that could possibly mean.

    Then you ask the following question: “Is one or more of the following two statements true: (a) your answer to this question is (/was/will be) ‘no’; (b) you will be (/are/were) handing me $2 mill while answering this question?”

    His only option is to say “yes” while handing you the money. If he does anything else, that amounts to his saying “The bargain’s off.”

    Does anyone see problems with this version? In my opinion, tense shouldn’t be an issue; it’s not a crucial aspect of language, and the whole thing can probably be communicated in a way that eliminates tense problems.

  48. #48 Wow
    February 17, 2012

    “Does anyone see problems with this version?”

    Might as well cut to the chase and say “Are you going to say ‘no’?”. At least that one is only one question unquestionably.

    The question posed by Jason and g have tried are all basically making your choices for you, and you are NOT free to say yes or no.

    And yours still has “you have to do more than just say yes or no” in it.

    And even mine is bad because it DOESN’T HAVE a yes or no answer. By design.

    It’s broken, really. Let it die.

  49. #49 g
    February 17, 2012

    Wow, no one is claiming that giving the Bad Guy the $2M is an *answer* to the question. The claim is that in order to be consistent you have to (1) give a particular answer and (2) give him the $2M. The latter isn’t part of your answer; it’s just allegedly something you have to do in order to make your answer be either true or false.

    The idea that these questions are “basically making your choices for you, and you are not free to say yes or no” is the whole point of them; it’s why the term “coercive logic” is being used.

  50. #50 Lenoxus
    February 17, 2012

    I think it’s fair to say that part of why this works is that the original bargain required the person to answer “either truthfully or falsely”. Then, if it is phrased properly, the question “forces” the answerer to either do whatever it is you want them to do (thus “answering truthfully or falsely”), or to drop out.

    Meanwhile, from both a legal and logical perspective, I’d say it’s problematic to suppose that the answerer is somehow “forced” to give you the money. He just has to give it if he wants to “receive” your money, though of course he won’t. Otherwise, what is the “alternative” to his answering in a logically consistent manner… that he goes to jail? Would any decent courtroom actually enforce such a thing? Of course not.

    All things considered, this trick is perhaps the more sophisticated, academic version of “Dumbassaywha?” ;)

  51. #51 lenoxuss
    February 17, 2012

    Wowbagger:

    Might as well cut to the chase and say “Are you going to say ‘no’?”. At least that one is only one question unquestionably.

    Are all the statements in the Bible true? I would consider that a single question, and not tens of thousands packed into one, but whatevs.

  52. #52 Lenoxus
    February 17, 2012

    Oops! I seem to have sleepily referred to Wow as Wowbagger; my apologies!

    signed, Lenoxus or lenoxuss, can’t seem to keep even my own name straight

  53. #53 matt
    February 17, 2012

    “Suppose I offer you a million dollars, in return for which you agree to answer a certain yes/no question [truthfully or falsely]”

    Apparently all I have agreed to do is answer. There is nothing in your statement about taking any other action, including paying you anything. Paying money is not answering a question.

  54. #54 Wow
    February 18, 2012

    “I think it’s fair to say that part of why this works is that the original bargain required the person to answer “either truthfully or falsely”.”

    I think it’s fair to say that’s wrong:

    “you agree to answer a certain yes/no question. You can answer either truthfully or falsely as you desire.”

    It doesn’t work because it requires you to truthfully of falsely answer yes or no but then gives you a question that can’t be answered yes or no. Handing over cash isn’t a yes or no. And there are three questions.

    If it were one question, then the answer I could give would be “Yes”.

    The ONLY way you can make that an “I WIN!” answer is if you make it apply to a part of the question you choose.

    You might as well say what is the difference between a duck’s legs? Or are you going to answer this question “no”?

    Neither can be answered with a yes or no.

    Therefore the deal is invalid.

  55. #55 Wow
    February 18, 2012

    “Wow, no one is claiming that giving the Bad Guy the $2M is an *answer* to the question.”

    Then it’s not part of the question. Therefore doesn’t matter.

    The question is now “Will you falsely answer yes or truthfully answer no”.

    But then it’s not “answer truthfully or falsely as you desire”.

    Additionally, since I didn’t agree to that coda, I don’t have to pay. I only agreed to:

    “answer a certain yes/no question.”

  56. #56 Michael Kremer
    February 18, 2012

    @Jason #37:

    Actually this question is at least remotely like something that would be asked in normal conversation, and the answer to such a question is typically not “yes” or “no.” Consider:

    Will you either come with me to the party, stay at home, or go out to dinner by yourself? An answer to this would be “I will go out to dinner by myself.” That is, a question in the form “Will you either do x, y or z” is colloquially understood as asking the respondent to choose between doing x, y, or z.

    So, similarly, an answer to your question might be “I will truthfully answer yes to this question.” That would be a false answer, but one which does not commit me to giving you two million dollars.

    I understand that that is not an answer to the question you intended to pose. But I’d say your formulation was ambigous at best. And it’s that ambiguity which is causing a lot of the dissension above.

    But if we rephrase your question as follows

    “Is it the case that one of the following is true: you will answer this question truthfully, you will answer this question falsely, or you will give me two million dollars?” then we have a yes/no question, which is the one you intended. Then you can get your argument going.

  57. #57 Michael Kremer
    February 18, 2012

    I will add that only a logician making a joke would answer “Will you either come with me to the party, stay at home, or go out to dinner by yourself?” with “yes”. Even “no” is a strange answer, though “none of those” does make sense (as refusing the choice).

  58. #58 Lenoxus
    February 18, 2012

    The point of the original question (if properly phrased) is that giving the money, though not at all “part of the answer”, is a necessary event for the answer to have a consistent truth value. Suppose you just said “no”. Would your answer be true, or false? It can’t be either.

    Of course, with this particular question, I can think of a couple more exceptions that may work. For example, what about answering in a foreign language? You could falsely say “Sí”, or truthfully say “Nein”. And how do we analyze an answer of “Maybe”? It seems that the full situation would have to be done with a contract beforehand, specifying “Yes or no only”. (Which would be silly insofar as he’s going to decline anyway.)

    Finally, suppose the “deal” were simplified such that the mark (foolishly) pledges to “truthfully answer yes” to your question, and then the question is “Did you give me $2 million before answering this?” Apart from tense issues, is there any dispute that the mark is obligated to give you $2 million if he wants your $1 million? In my humble opinion, he can’t possibly argue that doing anything else constitutes truthfully answering yes.

    The fact that you didn’t tell him beforehand that holding up his end would require some further action on his part is irrelevant; you never promised that he would only have to answer your question. Nor are you obligated to give him $1 million if his circumstances are such that he can’t fulfill his end. Why would it work that way? Suppose I promise to give you $1 million if you spin straw into gold. If you point out that this is impossible for you, am I still obligated to give you the money? The Randi prize thankfully doesn’t work that way!

  59. #59 CherryBombSim
    February 20, 2012

    To me, the proposition seems like a complicated way of saying “I’ll bet you a million dollars that I can ask you a yes/no question that you cannot answer without creating a contradiction.” I think it is logically equivalent, and easier to reason through. In either case, collecting on the bet might be problematic, but that’s not the point of the puzzle.

  60. #60 Wow
    February 22, 2012

    “is that giving the money, though not at all “part of the answer”, is a necessary event for the answer to have a consistent truth value.”

    Except that it is only so because you’ve precluded answers and included restrictions you had pre-arranged were not there.

    And, rather than be a “necessary event for the answer to have a consistent truth value” (whatever the heck THAT means, I’ll take it as meaning “a conclusion to the problem”), then you might as well skip to the chase and say “give me two million”. Therefore, the only “consistent truth value” here is to give me two million.

    This is also why it is a separate question (question since it says “or will you…”, otherwise it’s merely a demand).

  61. #61 Shecky R
    February 25, 2012

    Similarly also originally from Smullyan:

    http://richardelwes.co.uk/2011/06/06/an-idiotic-paradox/

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