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I never liked this type of puzzle. It is not formal logic but more about the idiosyncrasies and conventions of the English language. I put this puzzle on par wi

I never liked this type of puzzle. It is not formal logic but more about the idiosyncrasies and conventions of the English language. I put this puzzle on par with Agatha Christie’s murder mysteries. It requires a suspension of disbelief and logic to be believable. Someone who always lies means in the purest sense means you cannot trust anything they say. Even the word “hat” could mean they are talking about their pet cat that they like to carry on their head. What the author would probably say is “All my hats are green” means the liar is either lying about All or Green. Either all their hats are some other color or only one hat is green. This means you have to assume the liar has a hat. How do we know that? We only know that because of similar puzzles that came before. In other words this is not logic but more pattern recognition.

mjburgess 2y ago
This is generally the case for the vast majority of puzzles, and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency. I've rarely encountered a case where it is isnt an extreme lack of self-awareness in the questioner -- eg., being extremely overfit to language/notation/etc. localised to their own area of expertise.
- sillysaurusx 2y ago
  It’s not so much that an inability to solve them means you’re deficient, as it is that an ability to solve them means you’re capable in ways most people aren’t. My wife is great at puzzles, and it’s always humbling to watch her sail through them. I agree that puzzles alone shouldn’t e.g. determine whether you’re a good fit for a job, though. That’s one of the more annoying parts of software interviewing.
  - mjburgess 2y ago
    I think it should be the duty of the person asking the question to find a isomorphism to another question that the majority of people can answer. Eg., rather than asking about the risks of A,,B,C given various probabilities etc. ask them to make bets in a highly familiar environment with a resource that has uniform marginal utility to them... so eg., "suppose you were at home and your friend does..., how much of your time on a sunday would you bet to do... " You find that when "the very same question" is asked in highly familiar terms people get it right. Then this investigator-academic should ask: what features of their own puzzle induce the kind of mistakes they see? In my experience its often that people are far less socially incompetent than questioners, so put "interpretation & trust" priors on terms/presentations of questions that mean they don't parse the presentation into the problem the investigator has in mind. People who write puzzles tend to be the most bureaucratic sort of literalists who have profoundly eccentric modes of interpretation The problem that most people are solving is: "what do i say to make this person/question go away"
    - hinkley 2y ago
      > The problem that most people are solving is: "what do i say to make this person/question go away" reply This is it. Never attribute to incompetence that which can be readily attributed to apathy.
  - crazygringo 2y ago
    > as it is that an ability to solve them means you’re capable in ways most people aren’t. No, because the point is that they always hinge on an arbitrary distinction to give one answer. But if you made a different arbitrary distinction you'd get a different answer. And the arbitrary distinctions are, well, arbitrary. They reflect neither truth nor capability. Just whether you can read the questioner's mind as to what arbitrary and intentionally unstated assumptions they are making. No thanks.
- deleted 2y ago
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- stevenAthompson 2y ago
  Logic itself is a relatively new invention, and is symbolic itself. That is to say, logic is a map not the territory. That said, if someone can't fathom the most widely used symbolic languages humans use (math, logic, language, etc) they probably do have a cognitive deficit of some sort when compared to those who can.
  - bbor 2y ago
    I mean, 2400y at least… I guess you mean quantitative logical calculus notation? - the local Russell hater
    - hinkley 2y ago
      I guess sleeping through history class is also a mental deficit ;)
    - stevenAthompson 2y ago
      By "relatively recent" I meant in terms of human evolution. We've had it for around 2400 of the last 300,000 years or so. So less than 1% of our species existence. I used that phrasing to drive home the idea that logic is not some inherent aspect of nature, or even fundamental to the way humans perceive the world.
  - blueflow 2y ago
    Languages were made up by mankind at some point. They are not backed by the rest of physical reality. They can only be learned by induction, and there is no guarantee that people will get the same "version" of it. To speak in your analogy, people walk around with different maps of the same territory and realizing this is the self-awareness mjburgess is talking about.
  - lotharcable2 2y ago
    It is less 'logic is new'... it more about abstract thinking is a skill that is more useful in societies that have complex social arrangements. If you are a primitive farmer abstract thinking isn't really useful to you. Everything you deal with in your life, except religion, can almost entirely be dealt with absolutes with little in the way of abstractions. If it rains at the right time then you can have a good harvest. If the weather is bad then it sucks. If there is animals threatening your crops or herd you need to take steps to deal with them. There is a lot of logic in dealing with these things. You have to know the seasons, know the stars, know the dirt, etc. You have to understand the life cycle and manipulate the behavior and biology of plants and animals at the right stages in their lives. Things have a logical sequence and there are direct consequences that are predictable from events and your actions. Where as in modern society you have been conditioned to think in terms of hypothetical and abstractions through being exposed to testing your entire life. You first need to know how test questions work before you are able to answer them accurately. For a person who isn't exposed to this then the whole affair of asking hypotheticals and assuming imaginary situations with specific rules that don't actually apply to the present reality is very confusing. They don't even understand the question. So, of course, they are going to suck at answering them. And ultimately that is all IQ testing measures.. your ability to take tests.
- AtlasBarfed 2y ago
  The linguistics imo are pretty ill defined. "All" bring a common colloquial term doesn't have a strict set theory definition here. It is reasonable many people think zero hats is means the lie is in this very first word. A lot of people will consider "all" to implicitly mean 1 or more, while I think strict logicians will map colloquial all to 0 or more. All mat imply colloquially 2 or more as well, as why bother say "all" if you had one hat in the truthful sense "My hats" contrasts with the "has a hat" because having a hat in your possession that you could have borrowed does not confer ownership that the word "my" can imply. So great, a three letter word and a two letter word and we are knee deep in ambiguity. They could be wearing the hat to try to publicly locate the true owner who might say "hey I lost that hat at x". "Are green"... Green as in vegetable? Green as in the specific wavelength defined as green and not lime or some other named shade? Completely green dyed being undermined by a black spot or a pattern on the hat? Imo zero hats of ownership is a viable lie to the statement, as is having one red-green hat.
  - ewzimm 2y ago
    I like your way of phrasing this. Also, we can assume that this is not the only statement that the liar has ever told, so the context matters. If the preceding statement was "Every hat is my hat." Then we have a definition of "all" in the next sentence that works against the stated answer. So "we can't conclude anything" is really the answer because of undefined terms.
- Suppafly 2y ago
  >and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency. Does that actually happen in academia? It seems to mostly be a social media thing.
  - mjburgess 2y ago
    So frequently I'd say its the main case. Researchers are extremely poor at controlling for competing explanations, in many cases, strongly incentivsed not to. Suppose you're writing a paper what do you write: option A) Average People Cannot Understand Probaility!?!?!, option B) Inexperienced test takers with unfamiliar notation fail to grasp meaning of a novel question; option C) survey participants on technical questions often do not adopt a literal interpretation of question; D) etc. etc. In general researchers are extremely loath to, or poor at, recognising there's 101 alternative explanations for any research result using human participants and 99.99% of the time just publish the paper that says the experiment evidences their preferred conclusion.
    - whatshisface 2y ago
      "Democrats perform poorly on history tests designed by Republicans."
    - Suppafly 2y ago
      These sorts of puzzles aren't used by researchers though, so I'm not sure I follow the rest. They almost always seemed to be used by people trained in logical thinking to consider problems in different ways. They only seem get to the larger public when someone shares one on social media and people with no background in logic complain that they are poorly written or have no answer, since they don't realize that these sorts of puzzles presuppose some background in logic and familiarity with the setup.
  - btilly 2y ago
    There is one notable example that I'm aware of, but it really is a cognitive deficiency. The contrapositive is a rule that says that "A => B" is the same as "not B => not A". This is very confusing to people, and few can follow verbally why it works. But here is a fun experiment. People are presented with a selection of envelopes, all face down, and are asked to verify the fact that, "All unstamped envelopes are small." They immediately begin turning over the large envelopes, then have trouble explaining their (correct) reasoning! Here is a correct implication process for their actions. "All unstamped envelopes are small." => "unstamped envelope => small envelope" => "not small envelope => not unstamped" => "large envelope => stamped" At which point it is easier to just check the large envelopes!
- hinkley 2y ago
  It works better for everybody when these puzzles are told as inside jokes instead.
- raincole 2y ago
  My favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability. It really isn't. The problem is usually given in this form: > Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? [1] The only correct ansewr to this question is "I don't know." People who answer yes are just taking way too many standardized tests. [1]: https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem
  - copypasterepeat 2y ago
    I guess I don't understand why you think "I don't know" is the only correct answer. It's clearly not, which is the point of the problem in the first place. It's a bit hard to grok, but once you do, it's clear what the right answer is.
    - raincole 2y ago
      "I don't know" is the only correct anwser because the original problem isn't properly worded. You need to (at least) add this statement to the original question: > The host must open a door in any situation, and you know this rule. Because "host" in daily language is a human being with agency to choose whether to not open a door. Without this statement, the original problem is actually a game theory problem with two players. The host might be playing a strategy where he only opens the door when he knows you picked the car at the first place, or any other strategy. The above statement removes the ambiguity and that's what the original author meant (but failed to put in words). (If you're interesting in this, the wikipedia page linked above actually contains a quite extensive discussion in history over it)
      - dooglius 2y ago
        I don't think improper wording makes it an example of the type of puzzle GP is complaining about, because the counter-intuitive aspect holds up when everything is clarified.
- afiori 2y ago
  In my opinion a lot of these puzzles are about empathy toward the examiner as they generally ask variations of "what would I answer to this question?". In the case of a logician and the properties of the elements of the empty set the frame of mind of the examiner is probaly going to be about using algebraic logical connectives. For another nice example I can quote [0] via [1] > Luria: All bears are white where there is always snow. In Novaya Zemlya there is always snow. What color are the bears there? > Peasant: I have seen only black bears and I do not talk of what I have not seen. > Luria: What what do my words imply? > Peasant: If a person has not been there he can not say anything on the basis of words. If a man was 60 or 80 and had seen a white bear there and told me about it, he could be believed. This is a more extreme case, but in my opinion it is the same phenomenon of being asked to take external things as true and work on them. [0] https://languagelog.ldc.upenn.edu/nll/?p=481 https://languagelog.ldc.upenn.edu/nll/?p=481 [1] https://www.astralcodexten.com/p/somewhat-contra-marcus-on-ai-scaling?utm_source=publication-search https://www.astralcodexten.com/p/somewhat-contra-marcus-on-a...
Waterluvian 2y ago
Agreed. I bin this with other “rules lawyering,” which usually feels unfair, uninteresting, and unsatisfying. It’s why puzzlemaking a truly challenging and impressive skill. It must be fair, a challenge, and obvious in retrospect. I think it’s why the twist in Bioshock worked so well for me: it was right there in plain sight numerous times, while other twists in mystery games/films elicit a groan.
paxys 2y ago
These puzzles are entirely formal logic. Now you may not like or understand the intricacies of the logic/math and how it interacts with the English language, but the rules, and thus the solutions, are pretty objective and not open to interpretation.
- 38 2y ago
  You aren't reading the same article. It's obviously ambiguous
  - paxys 2y ago
    And yet everyone here has come up with the right answer.
- SirMaster 2y ago
  Then the puzzle shouldn't use the word lie, because to lie can mean to be deceptive, and saying all your hats are green when you have no hats is clearly deceptive and thus can be considered a lie.
  - mr_mitm 2y ago
    It's not deceptive, it's simply true in formal logic. I've been thought in linear algebra 101: all statements about the elements of the empty set are true. That's the core of this puzzle, and contrary what OP claimed, it's a matter of logic, not language.
    - SirMaster 2y ago
      But the question is being presented to humans via language and without any formal logic training. If it's a formal logic problem it should be presented in formal logic terms (presumably that only those with formal logic training would even understand) so then there is no possibility of language ambiguity. I'm not trying to be argumentative, but this is just how this come across to me, and I am pretty confident most humans would consider it a lie which I think would count for something given the place and the way in which the question was presented.
dotancohen 2y ago
Only people with formal training in logic, or those who work with formal definitions of logic, have trouble with this aspect of the puzzles. Trying to return to your more naive understanding of logic not only helps understand the intent of the puzzle, it also makes the puzzle more fun.
- eblume 2y ago
  I'm genuinely perplexed by this, how is the puzzle even a puzzle if you don't judge it by formal logic?
  - gosub100 2y ago
    It's still useful for young people or those who have never stopped to consider it. I think it's a good introduction to the warts of a context sensitive grammar
  - dotancohen 2y ago
    Because in everyday life we do not formally define ideas. Just for example I play a game with my children where we tickle each other when we see a yellow vehicle. Is a yellow tractor a vehicle for purpose of the game? How about a yellow baby cart. How about a billboard with a photograph of a yellow baby's toy car? We honestly have more fun discussing what is a vehicle for purpose of the game than we do tickling. Especially when it becomes clear that a yellow baby carriage is a vehicle only if it is daddy to be tickled... Honestly a good lesson to learn early that life is not fair and that rules are subjective.
    - LudwigNagasena 2y ago
      Reminds me of "The rule says, “No vehicles in the park”", https://news.ycombinator.com/item?id=36453856 https://news.ycombinator.com/item?id=36453856
- garciasn 2y ago
  > it also makes the puzzle more fun. No; it makes the puzzle illogical and thus about as much fun, for those who prefer logical outcomes, as discussing religion or politics.
copypasterepeat 2y ago
Agreed. Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement. It could just mean someone who frequently tells falsehoods. Or, more deviously, someone who wants to cause maximum uncertainty in his listeners, in which case some mix of true and false statements would probably be the way to go.
- fortran77 2y ago
  In the world of logic puzzles, it is understood that a "liar" never makes a true statement, and that "vacuously true" statements are false.
  - dingaling 2y ago
    Which seems to be an untenable premise. "I never make a true statement" says the liar.
    - latexr 2y ago
      > "I never make a true statement" says the liar. I’ve never seen one of this puzzles phrasing it like that. With good reason, that would break the puzzle.
  - JadeNB 2y ago
    > In the world of logic puzzles, it is understood that a "liar" never makes a true statement, and that "vacuously true" statements are false. I agree with the first stipulation, but, outside of Lewis Carroll's puzzles, I think that the second isn't true. For example, I believe you'll find just the opposite in Raymond Smullyan's books.
- pleurotus 2y ago
  > Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement. FTA: >Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic. A liar is someone who only says false statements. I think it's pretty clear how on definitions
raincole 2y ago
Puzzles like this are to make more people interested in formal logic. The "gateway drug" of formal logic. Just like the barber paradox isn't literally Russell's paradox, but it made more people to look up the history of it and perhaps learned what Russell's paradox is. Hopefully 0.1% of them turn out to be mathematicians.
- dleeftink 2y ago
  Why not hopefully more? Less? Is mathematics this inaccessible to the other 99.9%?
  - Retric 2y ago
    Another way of considering that statement is diminishing returns on the utility of mathematicians. Some is good, but more isn’t necessarily better.
    - dleeftink 2y ago
      And less isn't necessarily better either. But as we further mathematics, large cohorts are falling behind. I welcome puzzles, but I also think we need to shed some of the exclusionary aspects of mathematics/compsci, the brunt of which are far too known: inaccesible formalisms, leetcode, competitive grants/hackathons, steep admission requirements, code bounties, interview puzzle rounds, etc. Some necessary and organised in good faith I'm sure, but I hope we can move past the implicit assumption that 'maths/code isn't for everyone' and self-select based on that, as that doesn't further the cause.
  - raincole 2y ago
    Maybe "hopefully" isn't the right word. My point is that not everyone who understands barber paradox (in plain english) has to understand formal logic, and not everyone who understands formal logic has to become a mathematician. However I still believe the existing of the plain english puzzle is a net positive for humanity's collective mathematical comprehension.
gweinberg 2y ago
Right. Some people will claim "all my hats are green" is equivalent to "I have no hats which are not green", so he ,must have one hat which is not green. But other people would argue that teh original statement also implies you have at least one hat, so the statement can still be a lie if he has no hats at all. Lewis Carrol explicitly endorses the second view in his book "The Game of Logic". Good luck trying to argue logic with Lewis Carrol.
jkaptur 2y ago
The end of the article notes "A liar is someone who only says false statements." I agree that this is quite different from the colloquial definition of a liar, someone who mixes truth and lies in order to mislead.
- efdee 2y ago
  The beginning of the article quite clearly states "A liar who always lies", though.
gosub100 2y ago
Any time I hear the Rick Roll song (Rick Astley - Never Gonna Give You Up) I always notice the ambiguity in the lyrics "Never Gonna run around AND desert you" and ponder how his proclamation would still be valid provided he only committed one of those transgressions individually.
GuB-42 2y ago
That's the reason natural languages are not very well suited to formal logic, and one of the reasons programmers use programming languages. If we reframe it using C++ "std::all_of" function over an array of strings called "hats", and say that the following must be false (because he is a liar): std::all_of(hats.begin(), hats.end(), [](std::string hat) { return hat == "green"; }) Then we can answer the questions without ambiguity: A) The liar has at least one hat. Yes, because std::all_of returns "true" on an empty list B) The liar has only one green hat. Unsure, because { "green", "green", "red" } is false, and { "green", "red" } is also false C) The liar has no hats. No, it is the opposite of A D) The liar has at least one green hat. Unsure, because { "green", "red" } is false and { "red" } is also false E) The liar has no green hats. Unsure, for the same reason as in D
- gpderetta 2y ago
  The question becomes why all_of returns true for an empty list. A better example would use std::accumulate and an 'and' function argument. The original STL documentation [1] described the function argument as having to model the Monoid[2] concept and have an identity value. The identity value for the 'and' Monoid is 'true'. [1] https://www.boost.org/sgi/stl/MonoidOperation.html https://www.boost.org/sgi/stl/MonoidOperation.html [2] note: Monoid, not Monad, and yes, category theory left its mark in C++ as well.
  - JadeNB 2y ago
    > The question becomes why all_of returns true for an empty list. Because it is, if you'll forgive my Haskell-ese, the only implementation that means that `all_of $ l1 ++ l2 == (all_of l1) && (all_of l2)` for all lists `l1` and `l2`, including empty ones.
    - gpderetta 2y ago
      yes of course, I was trying (and failed?) to draw a connection from all_of to accumulate (i.e. fold).
      - 2y ago
crdrost 2y ago
> What the author would probably say is “All my hats are green” means the liar is either lying about All or Green. Either all their hats are some other color or only one hat is green. This means you have to assume the liar has a hat. How do we know that? This is almost certainly not what the author will say, and it does have some tangential connections to computing so it's worth expanding on briefly. In computing you often reduce problems to smaller problems, this can be done with recursion but it doesn't have to be, dynamic programming kind of is the reverse of it; whatever. When you do that though, you get these questions about really small collections, sets, lists, data structures. Just for a quick concrete example so that we're not an abstract theory land, is a one-element list sorted? Is a zero element list sorted? Is [5, 5, 5] sorted ascending, descending, both, or neither? If you choose the wrong answer, then it means that your algorithm needs to be more complex, you don't trust zero-element lists to be sorted so your quicksort HAS to pivot on a median of 3, you don't trust one-element lists to be sorted so your quicksort HAS to pivot on a median of 5, or maybe when you see that you would recurse into a list of size 1 you generate a new median-pivot or something. In mathematics, there is a convention which attempts to generalize this idea that an empty list is always sorted. In fact, an empty list is also always randomly sorted. It is always sorted descending, too. If it's an empty array of ints, all of those ints are greater than 1000—and they are also less than 50. The mathematical convention is, in lay speak, “if you are talking about nothing, pretty much anything you say is going to be true unless it's gibberish.” If I said every int is green, well, synesthesia excepted ints don't really have colors, that's crap. Everything else is “trivially true.” More formally, any predicate of the form {for all elements in S, this is true of that element} is taken to be “trivially true” when S is an empty set. It is something like the code, let agg = true for (const element of mylist) { agg = agg || f(element) } where no matter what f is, if mylist was empty, agg is true. (“Trivial” here is sometimes replaced with the word “vacuous” because “trivially” is also a common English word meaning “easily,” so something “trivially true” might also by normal English rules mean “is easily seen to be true, is easily proven to be true, we can debunk the opposite with a 5 second look at the Wikipedia page,” etc.) So that's the convention and besides making base cases much easier, one reason to do this is that the negation of any “for all Xs in the set, Y” is always perfectly specified as “there exists an X in the set such that not-Y.” The other convention you'd have to negate as “Either the set is empty or (...)”. So the claim is that the perfect Liar is lying according to mathematical rules of Truth and falsehood, and according to those rules, this statement “all of my hats are green” can only be false if there exists at least one hat belonging to the liar which is not green. If there are no hats then the statement was trivially true.
bingohbangoh 2y ago
I agree. It reminds me of the math "puzzles" on Twitter which go: 1 shoe + 1 shoe = 2 2 shoes + 2 shoes = 4 3 shoes + 2 shoes = ??? And the answer isn't 5 because a) we're not counting shoes and b) the shoe laces were different colors. There's nothing clever, it just teaches you to be hyper cynical and question every little detail which isn't relevant to either Math or the real world.
- ninalanyon 2y ago
  I have a little monograph written many decades ago on Dimensional Analysis. Since reading it, not quite so many decades ago, I simply dismiss puzzles of this sort because the two sides of the equations are dimensionally incongruent. This means that I have to try to guess the state of mind of the questioner rather than solve a logic problem. It's a handy stance because I'm no good at either solving logic problems or getting inside other people's heads! Another on that really irritates me is the kind that presents a series of integers and asks which integer comes next. Any integer will do, you just have to fit the appropriate polynomial.
  - hinkley 2y ago
    I expect people in different branches of statistics or physics would potentially come up with different answers based on what sorts of series appear in their realm of expertise.
  - chromaton 2y ago
    > Another on that really irritates me is the kind that presents a series of integers and asks which integer comes next. Any integer will do, you just have to fit the appropriate polynomial. This one bugs me to no end because it's part of the standard elementary school curriculum, for example here: https://byjus.com/maths/patterns-questions/ https://byjus.com/maths/patterns-questions/ But surely someone with a strong imagination could come up with a pattern to fit any number as the next in the sequence. I doubt most elementary educators even grasp the issue.
  - bingohbangoh 2y ago
    > This means that I have to try to guess the state of mind of the questioner rather than solve a logic problem That is an excellent way to put it! It explains why it appeals to non-math people. They are (usually) better at these more verbal-based games ime.
grog454 2y ago
The article clearly defines someone who always lies to mean they only make false statements. It's not hard or ambiguous to determine various scenarios in which "all my hats are green" is false. The solution article also discusses vacuous truths ("all my hats are green" is true if you have no hats, vacuously). Truth has a definition in formal logic too, but we've gotten far enough that this problem is already solvable and unambiguous.
gryn 2y ago
yeah, you can add most "paradoxes" to that list. the one I hate the most is the Monty Hall Problem.
- fargle 2y ago
  monty hall is not a true paradox. it could be classed as "veridical" (truthful) paradox, or i've seen it called a pseudo-paradox. but you realize that it's unambiguously mathematically true without gimicky word-play or puns or other logic puzzle trickery. it's just simply an un-intuitive result. most statistics really is. curious: why the hate for the monty-hall problem?
  - naniwaduni 2y ago
    The claimed Monty Hall result becomes untrue with very subtle changes to the premises. For example, if the host blindly chooses randomly between remaining doors, the resulting distribution is 1/3 contestant door choice correct (switch loses), 1/3 contestant and host both incorrect (switch wins), 1/3 contestant incorrect host correct (no switch can be offered).
  - gryn 2y ago
    Yes, I didn't mean it as it being a paradox just the problem I hate the most. I should've worded that better. The reason I hate it is that it's a example of how to lie and mislead using statistics and that the only reason it exist is that a content creator in the print media wanted to give an edgy true answer to farm engagement, and now as a consequence many introductory statistics course make students suffer for the same reasons. The assumptions made to reach that answer are not made explicit and it changes the response. And teachers mess it up a lot of the time which lead to a lead of head-scratching (or sometime just leave under-specified on purpose). It was the right answer to a question that wasn't asked. the host opens the door before giving the choice and the door he's choosing isn't random. Wikipedia explains this better than I would be here's the part I'm talking about: > In Morgan _et al four university professors published an article in _The American Statistician_ claiming that Savant gave the correct advice but the wrong argument. They believed the question asked for the chance of the car behind door 2 _given_ the player's initial choice of door 1 and the game host opening door 3, and they showed this chance was anything between 1/2 and 1 depending on the host's decision process given the choice. Only when the decision is completely randomized is the chance 2/3 . [Monty Hall problem - Wikipedia](https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem) The game theoretic explanation (in the same page) as to why you should switch is more convincing and less click-baity though without needing to give a specific probability value or assume the host strategy.
shever73 2y ago
This is an Alex Bellos puzzle, which I’ve learned through experience to ignore. The terms are always so loosely defined that any solution you come up with is a gamble at best.
hinkley 2y ago
> Either all their hats are some other color or only one hat is green. This means you have to assume the liar has a hat. How do we know that? Which is another listener’s bias: is lying by omission a lie? All my horses are unicorns. I don’t have any horses, nor unicorns. So it’s true but also not.
missinglugnut 2y ago
There's an early xkcd for this sort of riddle: https://xkcd.com/169/ https://xkcd.com/169/ The article mentions a maths exam, and of course the answer to the hats puzzle is very straightforward if you convert to statements about sets and logic in the way the teacher expects. But converting ambiguous language into logical statements by taking everything extremely literally is the opposite of a useful life lesson, so I think it's a terrible exam question.
whatshisface 2y ago
The article said that the problem was based in formal logic and appeared on a "maths test." That means it isn't ambiguous question about the English language.
- from-nibly 2y ago
  No that doesn't mean anything. Tests and books can be wrong.
  - jldugger 2y ago
    Well, part of the coursework, and probably the hardest part, is about translating natural language problems into formal logic statements, ie: "The liar says All my hats are green" becomes "¬∀x|x∈hat,OWNS(x) (GREEN(x))" or similar. And then from there you can also translate the five provided answers and try to find a contradiction.
  - deleted 2y ago
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- seanhunter 2y ago
  Yes. It isn’t ambiguous and is entirely solvable using old school formal logic of the “All Cretans are liars” sort. If the liar owns no hats the statement “All my hats are green” would be true. Under the parameters of the question it must be a lie and therefore cannot be true. So the liar owns at least one hat which is not green. They may own additional hats which can be of any colour. People who are saying “if they are a liar they might not even be talking about hats” are somewhat missing the point: 1) Whether or not they are talking about hats they have made a statement about hats and under the “rules of the game” of formal logic it must be untrue because we are given they are a liar. That’s enough to answer the question. 2) The “rules of the game” are about predicates and properties and inference. The language is plenty precise enough to convey both the question and to deduce the solution.
  - stickfigure 2y ago
    I think there's a sort of divide-by-zero problem here. Does an empty set of hats have a color? You could arbitrarily define "all my hats are green" for the empty set as either true or false as part of a consistent logical system. There isn't enough information in the question to know whether we should pick one or the other, though there's probably a colloquial preference for true.
idrios 2y ago
If a stranger told me "All of my cars are red" and then I learned that they didn't own a car, I wouldn't call them clever I'd call them a liar.
smohare 2y ago
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PittleyDunkin 2y ago
> It is not formal logic but more about the idiosyncrasies and conventions of the English language Well yea, that's the fun part! Logic puzzles without interaction with language are dull affairs indeed, basically just computation.
skybrian 2y ago
Yes, this is an artificial example and it uses ambiguous language. But it doesn’t really matter whether you got it right or whether the puzzle is fair. Knowing about “vacuously true” statements is useful because that’s a corner case where a bug can happen. (Or it could be a loophole in a legal document.) When you’re writing functions that work on lists or sets, you need to decide what to do about statements that are vacuously true. Call it out in the documentation? Put in a special case? Assert that the set is non-empty? Or maybe even create a new datatype to make the corner case unrepresentable. These design choices have side-effects, making a function easier or harder to use, more or less error-prone. Recognizing vacuous statements is pattern recognition, but it’s sometimes useful pattern recognition, and for children, learning about it might even be fun.
dllthomas 2y ago
I think the puzzle is better if the person "only speaks falsehoods" than if they "always lie" - depending on context, a true statement can be a lie.
- aftbit 2y ago
  This is called out in the article: >Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic. A liar is someone who only says false statements.
  - dllthomas 2y ago
    Yeah, my comment was just a suggestion that that wording be imported into the question, rather than living in a clarification.
torginus 2y ago
if you look at it as a programmer, vacuously true makes sense. In js if I write: const allHatsGreen = hats.every(hat => hat.color === 'green'); it will be true if hats is empty. Same thing for C# Linq. Imo this makes sense, because if I write: const matchAllConditions = conditions.every(condition => condition.matches(item)) In some condition matcher (lets say I want to check rules before an user is allowed to post), the correct behavior is the result to be true, when the array is empty. I'm not well versed, in functional programming, but in Ocaml (of F#, hehe) it would be let rec all_hats_green = function | [] -> true | hat::rest -> hat.color = "green" && all_hats_green rest so in a recursive implementation, this is the behavior that makes sense.
hansvm 2y ago
I don't think it's that idiosyncratic? Everywhere I've lived: 1. A deceit is an attempt to make somebody believe something false. 2. A falsehood is a statement which is false. 3. A lie is a deceitful falsehood. Regardless of whether they miscommunicated about hats vs cats, and regardless of whether they were being deceitful in the process, for the statement to be a lie it would also have to be a falsehood, implying they have at least one hat which is not green. Yes, these problems require suspension of disbelief, especially given the shorthand "he's a liar" always meaning that the person is often deceitful (usually with other negative implications), but the problem statement being that the "person always lies" is pretty clear and doesn't require special pattern recognition or other mental gymnastics, and it's not that different from the suspension of disbelief you invoke when playing a game of chess and not literally sending a knight to murderously dethrone your opposition.
btilly 2y ago
What you're saying can only be true to the extent that natural language doesn't look like formal logic. Unfortunately, there are some obvious discrepancies. My favorite is that "can't" really means something closer to "not can". This can be demonstrated with a close analysis of the statement, "I can't not do that." To get our usual understanding of the sentence we need to parse it as, "I (not can) (not do that)." And then turn that into, "I must not (not do that)." And now cancel the double negative to get, "I must do that." Suppose that you try to parse it as, "I can not not do that." You quickly get, "I can do that." Which is not at all what that sentence actually means.
golly_ned 2y ago
Most english sentences are semantically ambiguous and can often be parsed in hundreds of different ways. And it's true that there is an element of reading comprehension. But a reasonable reader won't assume the thing about the cat. We know this is occurring on a math exam, so it must be a logic puzzle. The exam-taker would've probably known this refers to a first-order logic question, and have been taught a certain way of translating sentences like this into first-order logic. This is true of all word puzzles that they need to be mapped to math in a way that assumes knowledge about the world and context.
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Nifty3929 2y ago
Yes - the puzzle should not revolved around semantic ambiguities like this. Let's be clear about the terms of the puzzle, agree on definitions, and then argue from there. In fact, I would extend this to arguments as well: Let's be clear up front what we mean by important words and go from there. Too often people ending up arguing about definitions, but in roundabout ways.

mjburgess 2y ago

This is generally the case for the vast majority of puzzles, and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency. I've rarely encountered a case where it is isnt an extreme lack of self-awareness in the questioner -- eg., being extremely overfit to language/notation/etc. localised to their own area of expertise.

sillysaurusx 2y ago
It’s not so much that an inability to solve them means you’re deficient, as it is that an ability to solve them means you’re capable in ways most people aren’t. My wife is great at puzzles, and it’s always humbling to watch her sail through them. I agree that puzzles alone shouldn’t e.g. determine whether you’re a good fit for a job, though. That’s one of the more annoying parts of software interviewing.
- mjburgess 2y ago
  I think it should be the duty of the person asking the question to find a isomorphism to another question that the majority of people can answer. Eg., rather than asking about the risks of A,,B,C given various probabilities etc. ask them to make bets in a highly familiar environment with a resource that has uniform marginal utility to them... so eg., "suppose you were at home and your friend does..., how much of your time on a sunday would you bet to do... " You find that when "the very same question" is asked in highly familiar terms people get it right. Then this investigator-academic should ask: what features of their own puzzle induce the kind of mistakes they see? In my experience its often that people are far less socially incompetent than questioners, so put "interpretation & trust" priors on terms/presentations of questions that mean they don't parse the presentation into the problem the investigator has in mind. People who write puzzles tend to be the most bureaucratic sort of literalists who have profoundly eccentric modes of interpretation The problem that most people are solving is: "what do i say to make this person/question go away"
  - hinkley 2y ago
    > The problem that most people are solving is: "what do i say to make this person/question go away" reply This is it. Never attribute to incompetence that which can be readily attributed to apathy.
- crazygringo 2y ago
  > as it is that an ability to solve them means you’re capable in ways most people aren’t. No, because the point is that they always hinge on an arbitrary distinction to give one answer. But if you made a different arbitrary distinction you'd get a different answer. And the arbitrary distinctions are, well, arbitrary. They reflect neither truth nor capability. Just whether you can read the questioner's mind as to what arbitrary and intentionally unstated assumptions they are making. No thanks.
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stevenAthompson 2y ago
Logic itself is a relatively new invention, and is symbolic itself. That is to say, logic is a map not the territory. That said, if someone can't fathom the most widely used symbolic languages humans use (math, logic, language, etc) they probably do have a cognitive deficit of some sort when compared to those who can.
- bbor 2y ago
  I mean, 2400y at least… I guess you mean quantitative logical calculus notation? - the local Russell hater
  - hinkley 2y ago
    I guess sleeping through history class is also a mental deficit ;)
  - stevenAthompson 2y ago
    By "relatively recent" I meant in terms of human evolution. We've had it for around 2400 of the last 300,000 years or so. So less than 1% of our species existence. I used that phrasing to drive home the idea that logic is not some inherent aspect of nature, or even fundamental to the way humans perceive the world.
- blueflow 2y ago
  Languages were made up by mankind at some point. They are not backed by the rest of physical reality. They can only be learned by induction, and there is no guarantee that people will get the same "version" of it. To speak in your analogy, people walk around with different maps of the same territory and realizing this is the self-awareness mjburgess is talking about.
- lotharcable2 2y ago
  It is less 'logic is new'... it more about abstract thinking is a skill that is more useful in societies that have complex social arrangements. If you are a primitive farmer abstract thinking isn't really useful to you. Everything you deal with in your life, except religion, can almost entirely be dealt with absolutes with little in the way of abstractions. If it rains at the right time then you can have a good harvest. If the weather is bad then it sucks. If there is animals threatening your crops or herd you need to take steps to deal with them. There is a lot of logic in dealing with these things. You have to know the seasons, know the stars, know the dirt, etc. You have to understand the life cycle and manipulate the behavior and biology of plants and animals at the right stages in their lives. Things have a logical sequence and there are direct consequences that are predictable from events and your actions. Where as in modern society you have been conditioned to think in terms of hypothetical and abstractions through being exposed to testing your entire life. You first need to know how test questions work before you are able to answer them accurately. For a person who isn't exposed to this then the whole affair of asking hypotheticals and assuming imaginary situations with specific rules that don't actually apply to the present reality is very confusing. They don't even understand the question. So, of course, they are going to suck at answering them. And ultimately that is all IQ testing measures.. your ability to take tests.
AtlasBarfed 2y ago
The linguistics imo are pretty ill defined. "All" bring a common colloquial term doesn't have a strict set theory definition here. It is reasonable many people think zero hats is means the lie is in this very first word. A lot of people will consider "all" to implicitly mean 1 or more, while I think strict logicians will map colloquial all to 0 or more. All mat imply colloquially 2 or more as well, as why bother say "all" if you had one hat in the truthful sense "My hats" contrasts with the "has a hat" because having a hat in your possession that you could have borrowed does not confer ownership that the word "my" can imply. So great, a three letter word and a two letter word and we are knee deep in ambiguity. They could be wearing the hat to try to publicly locate the true owner who might say "hey I lost that hat at x". "Are green"... Green as in vegetable? Green as in the specific wavelength defined as green and not lime or some other named shade? Completely green dyed being undermined by a black spot or a pattern on the hat? Imo zero hats of ownership is a viable lie to the statement, as is having one red-green hat.
- ewzimm 2y ago
  I like your way of phrasing this. Also, we can assume that this is not the only statement that the liar has ever told, so the context matters. If the preceding statement was "Every hat is my hat." Then we have a definition of "all" in the next sentence that works against the stated answer. So "we can't conclude anything" is really the answer because of undefined terms.
Suppafly 2y ago
>and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency. Does that actually happen in academia? It seems to mostly be a social media thing.
- mjburgess 2y ago
  So frequently I'd say its the main case. Researchers are extremely poor at controlling for competing explanations, in many cases, strongly incentivsed not to. Suppose you're writing a paper what do you write: option A) Average People Cannot Understand Probaility!?!?!, option B) Inexperienced test takers with unfamiliar notation fail to grasp meaning of a novel question; option C) survey participants on technical questions often do not adopt a literal interpretation of question; D) etc. etc. In general researchers are extremely loath to, or poor at, recognising there's 101 alternative explanations for any research result using human participants and 99.99% of the time just publish the paper that says the experiment evidences their preferred conclusion.
  - whatshisface 2y ago
    "Democrats perform poorly on history tests designed by Republicans."
  - Suppafly 2y ago
    These sorts of puzzles aren't used by researchers though, so I'm not sure I follow the rest. They almost always seemed to be used by people trained in logical thinking to consider problems in different ways. They only seem get to the larger public when someone shares one on social media and people with no background in logic complain that they are poorly written or have no answer, since they don't realize that these sorts of puzzles presuppose some background in logic and familiarity with the setup.
- btilly 2y ago
  There is one notable example that I'm aware of, but it really is a cognitive deficiency. The contrapositive is a rule that says that "A => B" is the same as "not B => not A". This is very confusing to people, and few can follow verbally why it works. But here is a fun experiment. People are presented with a selection of envelopes, all face down, and are asked to verify the fact that, "All unstamped envelopes are small." They immediately begin turning over the large envelopes, then have trouble explaining their (correct) reasoning! Here is a correct implication process for their actions. "All unstamped envelopes are small." => "unstamped envelope => small envelope" => "not small envelope => not unstamped" => "large envelope => stamped" At which point it is easier to just check the large envelopes!
hinkley 2y ago
It works better for everybody when these puzzles are told as inside jokes instead.
raincole 2y ago
My favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability. It really isn't. The problem is usually given in this form: > Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? [1] The only correct ansewr to this question is "I don't know." People who answer yes are just taking way too many standardized tests. [1]: https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem
- copypasterepeat 2y ago
  I guess I don't understand why you think "I don't know" is the only correct answer. It's clearly not, which is the point of the problem in the first place. It's a bit hard to grok, but once you do, it's clear what the right answer is.
  - raincole 2y ago
    "I don't know" is the only correct anwser because the original problem isn't properly worded. You need to (at least) add this statement to the original question: > The host must open a door in any situation, and you know this rule. Because "host" in daily language is a human being with agency to choose whether to not open a door. Without this statement, the original problem is actually a game theory problem with two players. The host might be playing a strategy where he only opens the door when he knows you picked the car at the first place, or any other strategy. The above statement removes the ambiguity and that's what the original author meant (but failed to put in words). (If you're interesting in this, the wikipedia page linked above actually contains a quite extensive discussion in history over it)
    - dooglius 2y ago
      I don't think improper wording makes it an example of the type of puzzle GP is complaining about, because the counter-intuitive aspect holds up when everything is clarified.
afiori 2y ago
In my opinion a lot of these puzzles are about empathy toward the examiner as they generally ask variations of "what would I answer to this question?". In the case of a logician and the properties of the elements of the empty set the frame of mind of the examiner is probaly going to be about using algebraic logical connectives. For another nice example I can quote [0] via [1] > Luria: All bears are white where there is always snow. In Novaya Zemlya there is always snow. What color are the bears there? > Peasant: I have seen only black bears and I do not talk of what I have not seen. > Luria: What what do my words imply? > Peasant: If a person has not been there he can not say anything on the basis of words. If a man was 60 or 80 and had seen a white bear there and told me about it, he could be believed. This is a more extreme case, but in my opinion it is the same phenomenon of being asked to take external things as true and work on them. [0] https://languagelog.ldc.upenn.edu/nll/?p=481 https://languagelog.ldc.upenn.edu/nll/?p=481 [1] https://www.astralcodexten.com/p/somewhat-contra-marcus-on-ai-scaling?utm_source=publication-search https://www.astralcodexten.com/p/somewhat-contra-marcus-on-a...

Waterluvian 2y ago

Agreed. I bin this with other “rules lawyering,” which usually feels unfair, uninteresting, and unsatisfying. It’s why puzzlemaking a truly challenging and impressive skill. It must be fair, a challenge, and obvious in retrospect. I think it’s why the twist in Bioshock worked so well for me: it was right there in plain sight numerous times, while other twists in mystery games/films elicit a groan.

paxys 2y ago

These puzzles are entirely formal logic. Now you may not like or understand the intricacies of the logic/math and how it interacts with the English language, but the rules, and thus the solutions, are pretty objective and not open to interpretation.

38 2y ago
You aren't reading the same article. It's obviously ambiguous
- paxys 2y ago
  And yet everyone here has come up with the right answer.
SirMaster 2y ago
Then the puzzle shouldn't use the word lie, because to lie can mean to be deceptive, and saying all your hats are green when you have no hats is clearly deceptive and thus can be considered a lie.
- mr_mitm 2y ago
  It's not deceptive, it's simply true in formal logic. I've been thought in linear algebra 101: all statements about the elements of the empty set are true. That's the core of this puzzle, and contrary what OP claimed, it's a matter of logic, not language.
  - SirMaster 2y ago
    But the question is being presented to humans via language and without any formal logic training. If it's a formal logic problem it should be presented in formal logic terms (presumably that only those with formal logic training would even understand) so then there is no possibility of language ambiguity. I'm not trying to be argumentative, but this is just how this come across to me, and I am pretty confident most humans would consider it a lie which I think would count for something given the place and the way in which the question was presented.

dotancohen 2y ago

Only people with formal training in logic, or those who work with formal definitions of logic, have trouble with this aspect of the puzzles. Trying to return to your more naive understanding of logic not only helps understand the intent of the puzzle, it also makes the puzzle more fun.

eblume 2y ago
I'm genuinely perplexed by this, how is the puzzle even a puzzle if you don't judge it by formal logic?
- gosub100 2y ago
  It's still useful for young people or those who have never stopped to consider it. I think it's a good introduction to the warts of a context sensitive grammar
- dotancohen 2y ago
  Because in everyday life we do not formally define ideas. Just for example I play a game with my children where we tickle each other when we see a yellow vehicle. Is a yellow tractor a vehicle for purpose of the game? How about a yellow baby cart. How about a billboard with a photograph of a yellow baby's toy car? We honestly have more fun discussing what is a vehicle for purpose of the game than we do tickling. Especially when it becomes clear that a yellow baby carriage is a vehicle only if it is daddy to be tickled... Honestly a good lesson to learn early that life is not fair and that rules are subjective.
  - LudwigNagasena 2y ago
    Reminds me of "The rule says, “No vehicles in the park”", https://news.ycombinator.com/item?id=36453856 https://news.ycombinator.com/item?id=36453856
garciasn 2y ago
> it also makes the puzzle more fun. No; it makes the puzzle illogical and thus about as much fun, for those who prefer logical outcomes, as discussing religion or politics.

copypasterepeat 2y ago

Agreed. Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement. It could just mean someone who frequently tells falsehoods. Or, more deviously, someone who wants to cause maximum uncertainty in his listeners, in which case some mix of true and false statements would probably be the way to go.

fortran77 2y ago
In the world of logic puzzles, it is understood that a "liar" never makes a true statement, and that "vacuously true" statements are false.
- dingaling 2y ago
  Which seems to be an untenable premise. "I never make a true statement" says the liar.
  - latexr 2y ago
    > "I never make a true statement" says the liar. I’ve never seen one of this puzzles phrasing it like that. With good reason, that would break the puzzle.
- JadeNB 2y ago
  > In the world of logic puzzles, it is understood that a "liar" never makes a true statement, and that "vacuously true" statements are false. I agree with the first stipulation, but, outside of Lewis Carroll's puzzles, I think that the second isn't true. For example, I believe you'll find just the opposite in Raymond Smullyan's books.
pleurotus 2y ago
> Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement. FTA: >Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic. A liar is someone who only says false statements. I think it's pretty clear how on definitions

raincole 2y ago

Puzzles like this are to make more people interested in formal logic. The "gateway drug" of formal logic. Just like the barber paradox isn't literally Russell's paradox, but it made more people to look up the history of it and perhaps learned what Russell's paradox is. Hopefully 0.1% of them turn out to be mathematicians.

dleeftink 2y ago
Why not hopefully more? Less? Is mathematics this inaccessible to the other 99.9%?
- Retric 2y ago
  Another way of considering that statement is diminishing returns on the utility of mathematicians. Some is good, but more isn’t necessarily better.
  - dleeftink 2y ago
    And less isn't necessarily better either. But as we further mathematics, large cohorts are falling behind. I welcome puzzles, but I also think we need to shed some of the exclusionary aspects of mathematics/compsci, the brunt of which are far too known: inaccesible formalisms, leetcode, competitive grants/hackathons, steep admission requirements, code bounties, interview puzzle rounds, etc. Some necessary and organised in good faith I'm sure, but I hope we can move past the implicit assumption that 'maths/code isn't for everyone' and self-select based on that, as that doesn't further the cause.
- raincole 2y ago
  Maybe "hopefully" isn't the right word. My point is that not everyone who understands barber paradox (in plain english) has to understand formal logic, and not everyone who understands formal logic has to become a mathematician. However I still believe the existing of the plain english puzzle is a net positive for humanity's collective mathematical comprehension.

gweinberg 2y ago

Right. Some people will claim "all my hats are green" is equivalent to "I have no hats which are not green", so he ,must have one hat which is not green. But other people would argue that teh original statement also implies you have at least one hat, so the statement can still be a lie if he has no hats at all. Lewis Carrol explicitly endorses the second view in his book "The Game of Logic". Good luck trying to argue logic with Lewis Carrol.

jkaptur 2y ago

The end of the article notes "A liar is someone who only says false statements." I agree that this is quite different from the colloquial definition of a liar, someone who mixes truth and lies in order to mislead.

efdee 2y ago
The beginning of the article quite clearly states "A liar who always lies", though.

gosub100 2y ago

Any time I hear the Rick Roll song (Rick Astley - Never Gonna Give You Up) I always notice the ambiguity in the lyrics "Never Gonna run around AND desert you" and ponder how his proclamation would still be valid provided he only committed one of those transgressions individually.

GuB-42 2y ago

That's the reason natural languages are not very well suited to formal logic, and one of the reasons programmers use programming languages. If we reframe it using C++ "std::all_of" function over an array of strings called "hats", and say that the following must be false (because he is a liar): std::all_of(hats.begin(), hats.end(), [](std::string hat) { return hat == "green"; }) Then we can answer the questions without ambiguity: A) The liar has at least one hat. Yes, because std::all_of returns "true" on an empty list B) The liar has only one green hat. Unsure, because { "green", "green", "red" } is false, and { "green", "red" } is also false C) The liar has no hats. No, it is the opposite of A D) The liar has at least one green hat. Unsure, because { "green", "red" } is false and { "red" } is also false E) The liar has no green hats. Unsure, for the same reason as in D

gpderetta 2y ago
The question becomes why all_of returns true for an empty list. A better example would use std::accumulate and an 'and' function argument. The original STL documentation [1] described the function argument as having to model the Monoid[2] concept and have an identity value. The identity value for the 'and' Monoid is 'true'. [1] https://www.boost.org/sgi/stl/MonoidOperation.html https://www.boost.org/sgi/stl/MonoidOperation.html [2] note: Monoid, not Monad, and yes, category theory left its mark in C++ as well.
- JadeNB 2y ago
  > The question becomes why all_of returns true for an empty list. Because it is, if you'll forgive my Haskell-ese, the only implementation that means that `all_of $ l1 ++ l2 == (all_of l1) && (all_of l2)` for all lists `l1` and `l2`, including empty ones.
  - gpderetta 2y ago
    yes of course, I was trying (and failed?) to draw a connection from all_of to accumulate (i.e. fold).
    - 2y ago

crdrost 2y ago

> What the author would probably say is “All my hats are green” means the liar is either lying about All or Green. Either all their hats are some other color or only one hat is green. This means you have to assume the liar has a hat. How do we know that? This is almost certainly not what the author will say, and it does have some tangential connections to computing so it's worth expanding on briefly. In computing you often reduce problems to smaller problems, this can be done with recursion but it doesn't have to be, dynamic programming kind of is the reverse of it; whatever. When you do that though, you get these questions about really small collections, sets, lists, data structures. Just for a quick concrete example so that we're not an abstract theory land, is a one-element list sorted? Is a zero element list sorted? Is [5, 5, 5] sorted ascending, descending, both, or neither? If you choose the wrong answer, then it means that your algorithm needs to be more complex, you don't trust zero-element lists to be sorted so your quicksort HAS to pivot on a median of 3, you don't trust one-element lists to be sorted so your quicksort HAS to pivot on a median of 5, or maybe when you see that you would recurse into a list of size 1 you generate a new median-pivot or something. In mathematics, there is a convention which attempts to generalize this idea that an empty list is always sorted. In fact, an empty list is also always randomly sorted. It is always sorted descending, too. If it's an empty array of ints, all of those ints are greater than 1000—and they are also less than 50. The mathematical convention is, in lay speak, “if you are talking about nothing, pretty much anything you say is going to be true unless it's gibberish.” If I said every int is green, well, synesthesia excepted ints don't really have colors, that's crap. Everything else is “trivially true.” More formally, any predicate of the form {for all elements in S, this is true of that element} is taken to be “trivially true” when S is an empty set. It is something like the code, let agg = true for (const element of mylist) { agg = agg || f(element) } where no matter what f is, if mylist was empty, agg is true. (“Trivial” here is sometimes replaced with the word “vacuous” because “trivially” is also a common English word meaning “easily,” so something “trivially true” might also by normal English rules mean “is easily seen to be true, is easily proven to be true, we can debunk the opposite with a 5 second look at the Wikipedia page,” etc.) So that's the convention and besides making base cases much easier, one reason to do this is that the negation of any “for all Xs in the set, Y” is always perfectly specified as “there exists an X in the set such that not-Y.” The other convention you'd have to negate as “Either the set is empty or (...)”. So the claim is that the perfect Liar is lying according to mathematical rules of Truth and falsehood, and according to those rules, this statement “all of my hats are green” can only be false if there exists at least one hat belonging to the liar which is not green. If there are no hats then the statement was trivially true.

bingohbangoh 2y ago

I agree. It reminds me of the math "puzzles" on Twitter which go: 1 shoe + 1 shoe = 2 2 shoes + 2 shoes = 4 3 shoes + 2 shoes = ??? And the answer isn't 5 because a) we're not counting shoes and b) the shoe laces were different colors. There's nothing clever, it just teaches you to be hyper cynical and question every little detail which isn't relevant to either Math or the real world.

ninalanyon 2y ago
I have a little monograph written many decades ago on Dimensional Analysis. Since reading it, not quite so many decades ago, I simply dismiss puzzles of this sort because the two sides of the equations are dimensionally incongruent. This means that I have to try to guess the state of mind of the questioner rather than solve a logic problem. It's a handy stance because I'm no good at either solving logic problems or getting inside other people's heads! Another on that really irritates me is the kind that presents a series of integers and asks which integer comes next. Any integer will do, you just have to fit the appropriate polynomial.
- hinkley 2y ago
  I expect people in different branches of statistics or physics would potentially come up with different answers based on what sorts of series appear in their realm of expertise.
- chromaton 2y ago
  > Another on that really irritates me is the kind that presents a series of integers and asks which integer comes next. Any integer will do, you just have to fit the appropriate polynomial. This one bugs me to no end because it's part of the standard elementary school curriculum, for example here: https://byjus.com/maths/patterns-questions/ https://byjus.com/maths/patterns-questions/ But surely someone with a strong imagination could come up with a pattern to fit any number as the next in the sequence. I doubt most elementary educators even grasp the issue.
- bingohbangoh 2y ago
  > This means that I have to try to guess the state of mind of the questioner rather than solve a logic problem That is an excellent way to put it! It explains why it appeals to non-math people. They are (usually) better at these more verbal-based games ime.

grog454 2y ago

The article clearly defines someone who always lies to mean they only make false statements. It's not hard or ambiguous to determine various scenarios in which "all my hats are green" is false. The solution article also discusses vacuous truths ("all my hats are green" is true if you have no hats, vacuously). Truth has a definition in formal logic too, but we've gotten far enough that this problem is already solvable and unambiguous.

gryn 2y ago

yeah, you can add most "paradoxes" to that list. the one I hate the most is the Monty Hall Problem.

fargle 2y ago
monty hall is not a true paradox. it could be classed as "veridical" (truthful) paradox, or i've seen it called a pseudo-paradox. but you realize that it's unambiguously mathematically true without gimicky word-play or puns or other logic puzzle trickery. it's just simply an un-intuitive result. most statistics really is. curious: why the hate for the monty-hall problem?
- naniwaduni 2y ago
  The claimed Monty Hall result becomes untrue with very subtle changes to the premises. For example, if the host blindly chooses randomly between remaining doors, the resulting distribution is 1/3 contestant door choice correct (switch loses), 1/3 contestant and host both incorrect (switch wins), 1/3 contestant incorrect host correct (no switch can be offered).
- gryn 2y ago
  Yes, I didn't mean it as it being a paradox just the problem I hate the most. I should've worded that better. The reason I hate it is that it's a example of how to lie and mislead using statistics and that the only reason it exist is that a content creator in the print media wanted to give an edgy true answer to farm engagement, and now as a consequence many introductory statistics course make students suffer for the same reasons. The assumptions made to reach that answer are not made explicit and it changes the response. And teachers mess it up a lot of the time which lead to a lead of head-scratching (or sometime just leave under-specified on purpose). It was the right answer to a question that wasn't asked. the host opens the door before giving the choice and the door he's choosing isn't random. Wikipedia explains this better than I would be here's the part I'm talking about: > In Morgan _et al four university professors published an article in _The American Statistician_ claiming that Savant gave the correct advice but the wrong argument. They believed the question asked for the chance of the car behind door 2 _given_ the player's initial choice of door 1 and the game host opening door 3, and they showed this chance was anything between 1/2 and 1 depending on the host's decision process given the choice. Only when the decision is completely randomized is the chance 2/3 . [Monty Hall problem - Wikipedia](https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem) The game theoretic explanation (in the same page) as to why you should switch is more convincing and less click-baity though without needing to give a specific probability value or assume the host strategy.

shever73 2y ago

This is an Alex Bellos puzzle, which I’ve learned through experience to ignore. The terms are always so loosely defined that any solution you come up with is a gamble at best.

hinkley 2y ago

> Either all their hats are some other color or only one hat is green. This means you have to assume the liar has a hat. How do we know that? Which is another listener’s bias: is lying by omission a lie? All my horses are unicorns. I don’t have any horses, nor unicorns. So it’s true but also not.

missinglugnut 2y ago

There's an early xkcd for this sort of riddle: https://xkcd.com/169/ https://xkcd.com/169/ The article mentions a maths exam, and of course the answer to the hats puzzle is very straightforward if you convert to statements about sets and logic in the way the teacher expects. But converting ambiguous language into logical statements by taking everything extremely literally is the opposite of a useful life lesson, so I think it's a terrible exam question.

whatshisface 2y ago

The article said that the problem was based in formal logic and appeared on a "maths test." That means it isn't ambiguous question about the English language.

from-nibly 2y ago
No that doesn't mean anything. Tests and books can be wrong.
- jldugger 2y ago
  Well, part of the coursework, and probably the hardest part, is about translating natural language problems into formal logic statements, ie: "The liar says All my hats are green" becomes "¬∀x|x∈hat,OWNS(x) (GREEN(x))" or similar. And then from there you can also translate the five provided answers and try to find a contradiction.
- deleted 2y ago
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seanhunter 2y ago
Yes. It isn’t ambiguous and is entirely solvable using old school formal logic of the “All Cretans are liars” sort. If the liar owns no hats the statement “All my hats are green” would be true. Under the parameters of the question it must be a lie and therefore cannot be true. So the liar owns at least one hat which is not green. They may own additional hats which can be of any colour. People who are saying “if they are a liar they might not even be talking about hats” are somewhat missing the point: 1) Whether or not they are talking about hats they have made a statement about hats and under the “rules of the game” of formal logic it must be untrue because we are given they are a liar. That’s enough to answer the question. 2) The “rules of the game” are about predicates and properties and inference. The language is plenty precise enough to convey both the question and to deduce the solution.
- stickfigure 2y ago
  I think there's a sort of divide-by-zero problem here. Does an empty set of hats have a color? You could arbitrarily define "all my hats are green" for the empty set as either true or false as part of a consistent logical system. There isn't enough information in the question to know whether we should pick one or the other, though there's probably a colloquial preference for true.

idrios 2y ago

If a stranger told me "All of my cars are red" and then I learned that they didn't own a car, I wouldn't call them clever I'd call them a liar.

smohare 2y ago

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PittleyDunkin 2y ago

> It is not formal logic but more about the idiosyncrasies and conventions of the English language Well yea, that's the fun part! Logic puzzles without interaction with language are dull affairs indeed, basically just computation.

skybrian 2y ago

Yes, this is an artificial example and it uses ambiguous language. But it doesn’t really matter whether you got it right or whether the puzzle is fair. Knowing about “vacuously true” statements is useful because that’s a corner case where a bug can happen. (Or it could be a loophole in a legal document.) When you’re writing functions that work on lists or sets, you need to decide what to do about statements that are vacuously true. Call it out in the documentation? Put in a special case? Assert that the set is non-empty? Or maybe even create a new datatype to make the corner case unrepresentable. These design choices have side-effects, making a function easier or harder to use, more or less error-prone. Recognizing vacuous statements is pattern recognition, but it’s sometimes useful pattern recognition, and for children, learning about it might even be fun.

dllthomas 2y ago

I think the puzzle is better if the person "only speaks falsehoods" than if they "always lie" - depending on context, a true statement can be a lie.

aftbit 2y ago
This is called out in the article: >Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic. A liar is someone who only says false statements.
- dllthomas 2y ago
  Yeah, my comment was just a suggestion that that wording be imported into the question, rather than living in a clarification.

torginus 2y ago

if you look at it as a programmer, vacuously true makes sense. In js if I write: const allHatsGreen = hats.every(hat => hat.color === 'green'); it will be true if hats is empty. Same thing for C# Linq. Imo this makes sense, because if I write: const matchAllConditions = conditions.every(condition => condition.matches(item)) In some condition matcher (lets say I want to check rules before an user is allowed to post), the correct behavior is the result to be true, when the array is empty. I'm not well versed, in functional programming, but in Ocaml (of F#, hehe) it would be let rec all_hats_green = function | [] -> true | hat::rest -> hat.color = "green" && all_hats_green rest so in a recursive implementation, this is the behavior that makes sense.

hansvm 2y ago

I don't think it's that idiosyncratic? Everywhere I've lived: 1. A deceit is an attempt to make somebody believe something false. 2. A falsehood is a statement which is false. 3. A lie is a deceitful falsehood. Regardless of whether they miscommunicated about hats vs cats, and regardless of whether they were being deceitful in the process, for the statement to be a lie it would also have to be a falsehood, implying they have at least one hat which is not green. Yes, these problems require suspension of disbelief, especially given the shorthand "he's a liar" always meaning that the person is often deceitful (usually with other negative implications), but the problem statement being that the "person always lies" is pretty clear and doesn't require special pattern recognition or other mental gymnastics, and it's not that different from the suspension of disbelief you invoke when playing a game of chess and not literally sending a knight to murderously dethrone your opposition.

btilly 2y ago

What you're saying can only be true to the extent that natural language doesn't look like formal logic. Unfortunately, there are some obvious discrepancies. My favorite is that "can't" really means something closer to "not can". This can be demonstrated with a close analysis of the statement, "I can't not do that." To get our usual understanding of the sentence we need to parse it as, "I (not can) (not do that)." And then turn that into, "I must not (not do that)." And now cancel the double negative to get, "I must do that." Suppose that you try to parse it as, "I can not not do that." You quickly get, "I can do that." Which is not at all what that sentence actually means.

golly_ned 2y ago

Most english sentences are semantically ambiguous and can often be parsed in hundreds of different ways. And it's true that there is an element of reading comprehension. But a reasonable reader won't assume the thing about the cat. We know this is occurring on a math exam, so it must be a logic puzzle. The exam-taker would've probably known this refers to a first-order logic question, and have been taught a certain way of translating sentences like this into first-order logic. This is true of all word puzzles that they need to be mapped to math in a way that assumes knowledge about the world and context.

deleted 2y ago

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Nifty3929 2y ago

Yes - the puzzle should not revolved around semantic ambiguities like this. Let's be clear about the terms of the puzzle, agree on definitions, and then argue from there. In fact, I would extend this to arguments as well: Let's be clear up front what we mean by important words and go from there. Too often people ending up arguing about definitions, but in roundabout ways.