Viral logic test from Brazil
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- Опубликовано: 3 июн 2024
- The 17th annual Brazilian Olympiad featured an incredibly tricky logic puzzle that went viral on social media. Thanks to Guilherme who suggested and translated the problem from Portuguese to English!
Pinocchio problem discussions
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Pinocchio illustration
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See Bram28 explanation for vacuously true
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Wikipedia vacuous truth
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Wikipedia truth table
en.wikipedia.org/wiki/Truth_t...
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Thanks!
@@nichijoufan Qué bueno ver hispanos interesados en lógica. Les recomiendo leer sobre Proposiciones categóricas para entender el problema. ^--^
The answer is incorrect.
"has at least one hat" -> if he "has only one green hat" then "all my hats are green" becomes true but we know that he always lies.
The correct statement is "he has at least one hat that is not green"
@@oguzcan2335 I know that u use your intuition But please study Cuantifies logical Propositions and stop comment ignorance.
@@limaocalculista9539 The answer "has at least one hat" means he can have only one green hat, which is contrary to "all my hats are green" being a lie. Thats why the answer "has at least one hat" is incorrect. The correct answer is "he has at least one hat that is not green". And i'm not kidding
@@MonoInfinito I'm sure you didn't even understand what i'm talking about. And I don't expect you will realize that i'm right.
It's a trick question; Pinocchio always *lies* on the ground because he got in a car accident and is paralyzed from the neck down. He's just telling you all his hats are green.
I knew it!!!
poor pinnochio :(
Gepetto using him as a puppet is kinda dark in that case
your right
@@t3st3d my right to be right
My favorite logic joke: Three logicians walk into a bar. The bartender asks them if they all want a beer. The first logician says "I don't know". The second logician says "I don't know". The third logician enthusiastically says "yes"!
Last one could have said "No" and it could be valid as well.
But you know this actually a frequent occurrence, because such questions are very often asked from a group of people, so one person kind of has to take lead and guess whether everyone wants that or people have to offer their opinion without any order.
@@PASHKULI
Yeah, but only if they themselves didn't want it.
If the last person wanted a beer also, they would respond with "yes", because they would knew that first and second definitely wanted a beer, otherwise they would have said "no".
There's implication that others wanted it, because otherwise they would have said "no" and the statement would have been true, because only one needs to not want it.
@@enzzz Bartender asked "Would all three of you like a beer?" The correct question is "Who of you would like a beer?"
and then on...
@@enzzz Only makes it a better joke, at least for those who understand why logically only the last logician can say "yes", and only if all the logicians beforehand say "don't know".
I concluded that Pinocchio has at least one hat that isn't green.
Ah, yes!
*Or* has no hats at all
@@notachair4757if he has no hats, all zero of his hats are green
@@brinecarroll exactly
@@notachair4757that was literally proven false in the video
By that logic, saying my house has three floors is a true statement as long as I don't have a house
Thank you. I was mad from watching this video. The logic he/they are using is patently invalid and makes no logical sense in the real world. It ONLY makes sense in the realm of discrete mathematics where they are applying the P - > Q proposition. The presenter of this video "conveniently" leaves that fact out as in order to get the "correct" answer you MUST do it under the context of the P -> Q proposition, which was explained in the olympiad competition. Saying you own something when you don't in the real world is a lie, straight up, and you can even be charged with fraud and go to jail. For example, by saying it on banking paperwork or on federal documents.
@@resresres1 Math questions don't make real life sense most of the time. I mean, we don't usually see random people stop by the market to buy 10 boxes of pears, half with 8 and the other half with 12, and then calculating the probability of unripe pears per box and how many they'd get in the end.
@@AlineDreams then they shouldn't be asking the question in the form of a real life scenario because they'll only confuse people.
Ah, but what does "my house" mean? You can't point to it (either on the ground or on a map), tell us its address, or what its geographical coordinates are. I don't think you can avoid this clause meaning something like "there is a particular house for which the claim 'I own it' (or 'I live there') is true", which cannot be true unless there is such a house.
If, on the other hand, you said "all my houses have three floors", that formalizes to something like "of all the houses there are, if I own it then it has three floors", and this is not false if you do not own any of them: the issue of how many floors it has does not come up because there is no 'it'.
One thing that makes this unintuitive is that we use "if...then" ambiguously, sometimes - but not always - to mean "if and only if", but for logic to be consistent, we need to be clear whether that is what we mean.
Look up "quantification over the empty set" for more details.
@@ajayray4408 you are incorrect. Saying "all my houses have three floors" does not "formalize" or is even nearly the same statement as "of all the houses that exist, if I own it, then it has three floors". There is no if/then in the original statement, in fact, you can say the original statement already answered the if/then statement.
Everyone knows that Pinocchio has at least one hat. He wears it throughout the entire film.
Congrats!
You flunked logic.
@@Highley1958 yay
I wondered if it was a hint or a red herring but I just ignored it
@@Highley1958But they passed science. After all, they cited empirical evidence in support of their claim
That he wore a hat doesn't necessarily imply that hat is his. He may have borrowed it.
“Were you ashamed when you pooped your diaper? Yes or no only!” said Rodrick.
“Yes,” Greg said vacuously, for he had not actually pooped his diaper, yet had to answer Rodrick’s question within proper mathematical convention.
Wait I’m confused. If Greg said yes, it would’ve been that he was ashamed when he pooped his diaper, but he didn’t. Then what would happen if he said no, even though he was not ashamed when he pooped his diaper because he didn’t pooped his diaper at all. Hahah this is too confusing
@@eduardoleonlotero that's the whole trick, it's not supposed to be confusing, it's supposed to result in only one outcome, greg's humiliation. and btw it's from a book, "diary of a wimpy kid"
Quality academia right here
@@eduardoleonlotero Does everyone know you can't even understand a joke? 🤭
@@eduardoleonlotero If we interpret the statement as IF pooped your diaper THEN ashamed, the only way this can be false is if the first is true but the second statement is false. So the only time he would have to answer no is if he pooped his diaper but was not ashamed. (Look at a logic table for "if p then q" if you're still confused)
A better way to exain it at 5:39 is like this:
He has no hats
Hence "all hats are green" means "100% of the hats are green"
= "100% * 0 hats are green"
= "0 hats are green"
Which is true
This is much more convincing then the explanation he gave.
Doesn't this actually prove the opposite? If 0 hats are green, then his statement "all hats are green" is false, not true. Thus pinnichio can have 0 hats and still be lying, or he can have 1 or more non-green hats and still be lying. He can only tell the truth if he has atleast one hat.
@@xaelath7771that’s the entire point when you imagine an empty set of hats the claim is that mathematically whatever you say about the set is true in the sense that the set is empty so no-hats (as a category) is beautiful for example, nothing about this statement is false. no-hats are green etc it’s just an empty set it’s close to saying 0 hats are green, 0 hats are beautiful, subject (0 hats) are predicate(whatever) nothing is false about those statements (again mathematically)
@@xaelath7771 but you *want* pinocchio to be lying, that's the point of the question.
If statement A leads to statement B, then if B is true so must A, by necessity. Henceforth if "0 hats are green" is true, so must "all hats are green" since one leads to the other. I was trying to say that "all = 0" because all he has is 0 hats. So for him all his hats means 0 hats.
@@baraharonovich2926 But it's defintely ontological false. A non-existent hat doesn't have the property of colour, so the claim that it is green, or beautiful, or whatever, is false, not true. Else it would be true that the no-hat was green and blue, beautiful and ugly, X and not X. Wouldn't that violate the law of non-contradicton? But if all claims about empty sets are false, there is no contradiction.
The problem with this kind of question is words have to be given new definitions.
Exactly. This is almost diabolical.
He always lies
He claims to own hats = lie
He claims the hats he owns are all green= lie
Only logical conclusion is C.
@@dustking3569 Yes, because watching Destiny gives you more say over mathematicians in logic puzzles.
@@feelsdankman211 you have the green light my friend . I was completely wrong . He said explicitly "mathematical lie" not a lie in the traditional sense . Maybe I should watch less Destiny
@@dustking3569 on this basis, i.e., that "always lies" means lies about everything, which I agree with, the only conclusion you can come to is that some or none (the opposite of all) of someone else's (the opposite of my) non-hat possessions (the opposite of hats) might or might not be (the opposite of are) a colour other than green (the opposite of green). Which is pretty uninformative, and is exactly what you would expect from someone who lies about everything. Seems like this Pinocchio should have gone into politics.
Everytime I had lunch with Albert Einstein, he thanked me (without letting anyone else hear) for letting him take the credit for the theory of relativity.
Little did he know, you hid the truth that E=mc³
That's fking true statement.
@Caradoc
en.m.wikipedia.org/wiki/Theory_of_relativity
"The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, ..."
I overheard him say that to you once...
Relativity is very old older than galileo man its just comparision of 2things relative to each other
When I was in the university I remember that didn't understand why these kind of statements on the empty set were always true ("vacuously true").
Then one professor told me something very simple that helped me understand:
"If you think that this statement on the empty set is not true, please find an element that doesn't meet the statement. You can't, can you? So it's true."
Thanks for sharing!
That is a great way to explain it. I will mention the empty set next time, thanks!
Video publish 3 min ago but you made comment 4 days ago🤔
Your professor statement is even more confusing,brother…
It's a bit strange that professor doesn't know about three-valued logic
So if you cannot falsify the statement, then it is true...now I understand the success of religions
You can go a step further. Not only must he own at least one hat, but he must specifically own at least one non-green hat.
Exactly. The statement can be written as “for all hats in Pinocchio’s possession, the statement ‘is green’ is true”. Simply negating that For All statement results in “There exists at least one hat in Pinocchio’s possession where the statement “is green” is false”. Naturally, the logic follows that Pinocchio must then own at least one hat.
I do not think the video does a good job explaining this problem, which is uncontroversial in modern formal logic. The underlying issue is "existential import." In older systems of logic, statements like "all my hats are green" are said to have 'existential import,' meaning that their truth requires that a hat exists. By contrast, in modern symbolic logic, these statements do not have existential import and are interpreted to mean, "If something is my hat, it is green," which is only falsified by finding a hat of mine that is not green. If I have no hats, then there is nothing falsifying the statement. Similarly, all my hats are yellow (go find a hat of mine that is not yellow if you want to falsify that statement). So, these "all" statements are true in an uninteresting way if the "all" ranges over the empty set.
“All my hats are green” can easily be interpreted to mean to contain the information that I have some hats. Certainly, if someone said that and I later learned they have no hats, I would consider them a liar. A better statement would have been, “Any hats I own are green.” That statement has the same logical meaning as the original if we assume the original doesn’t imply the ownership of hats. However, it lacks the ambiguity that makes this question disputed in the first place. In short, this isn’t really a logic question. It’s a language question, and language is often arbitrary.
This is so far the best explanation I've seen imo, cause honestly I did not understand at all how the video poster explained it.
This is the answer I agree with the most. Since this question's answer was made specifically to be solved with mathematical logic and not actual real-world applicable logic, the statement works. However, in a real setting it would depend entirely on how you interpret it. I wonder if in a differently structured language we wouldn't have this ambiguity issue
@@PitukaAJ But that's the thing. It is meant to test your knowledge of mathematical logic. It wouldn't be a good test question if it wasn't linguistically ambigious, because the skill you are supposed to learn is to set aside assumptions and follow only the logic defined by math. You are supposed to practice dismantling the statement to its pure logic formulation, and you can only practice doing that with statements not already formulated in a logical way.
But you can reasonably argue that the statement “All my hats are green,” means that I have hats and they are all green. Or you can argue that it just means that any hats I have are green and I may or may not have any hats at all. This is a linguistics dispute, not a logic dispute. We have to agree on the conversion of regular language into logically specific language before we can do the logic math. Any the reason this question is disputed is that people don’t agree. And no amount of logic will solve that because we disagree about what the English language sentence means.
@@samuelrussell5760 even if the sentence is interpreted as ‘I may or may not have any hats’, Pinocchio having no hats would not make his statement ‘all my hats are green’ false. That’s the point of this video. It is not a linguistics dispute.
Very interesting. It probably says more about me than the statements when the first thought I had to the question 'what can we conclude?' was "Pinoccio's nose just grew."
😆
I’m not reading any more comments! You won!
My conclusion was that it is true that Pinocchio only tells lies, and it is true that Pinocchio says "all his hats are green." What his hats colors are we don't know, but he sure does say they are green lol. Yours is more fun though
My first thought was "Pinoccio lied", then "oh wait" lmao
@@David-qj1mr exactly where my brain went too. And stopped 😀
There are two interpretation, both mathematically valid, of the English “All my hats are X” for some predicate X:
1) My hats are (as in they exist) and are all X.
2) My hats are or are not, but if they are, they are X.
The former could be interpreted to imply I have at least one hat or even strictly greater than one hat. Mathematicians or technically precise writers generally don’t write formal arguments without making it explicit whether the set could possibly be of size 0 or not.
You cannot make presumption on something that does not exist but if you say you have more than one when you don't, then you lie.
A mathematician will always use the second meaning. For example I can prove a statement about odd perfect numbers without knowing if they exist or not
@@yousauce7451 What about “My hats are in that closet.”? Would all mathematicians always assume the speaker might have no hats? Is that a truthful answer to the question “Where do you keep your hats?” if the responder had no hats? I’d imagine some mathematicians might say that that depends on what the English formally means.
That said, the intent of the problem in the video is easy to reverse-engineer because none of the other options make sense.
@@aroundandround Of course mathematicians are also humans, so if you would use that sentence in real life, then yes, we would assume that you have at least two hats. From a purely logical/mathematical perspective, if you would say "all my hats are in that closet" or "Every one of my hats is in that closet", then I would still see it possible that you have no hats. If you have no hats, then indeed it is true that each hat you have is in the closet. The statement is then said to be vacuously true. Even though it is true, it is void of any meaning.
The word 'all' then maybe has a bit of a different meaning than in normal use. The word 'or' is for example also used a bit differently in a mathematical/logical context. In regular speech, it is often used as an exclusive or, however a mathematician/logician would (/should) always use it inclusively (this or that does not exclude the possibility of both this and that being true).
I'll buy this logic when you successfully dereference a null pointer.
Yeah, vacuous truth can be confusing since we don’t usually refer to things we know don’t exist, but it makes more sense in terms of hypotheticals where we *aren’t sure* if they are.
For example, if an amusement park as the rule “All children must be accompanied by an adult” and a group of all adults shows up, are they violating the rule? No; there’s nobody the rule applies to, so nothing needs to be done.
Heck, pretty much any if statement follows this rule. If someone tells you to “bring and umbrella if it rains”, and it doesn’t rain, what do you need to do? Nothing; the request is only relevant if it rains, and otherwise it says nothing.
@KnakuanaRka The issue is that the statement in the video isn't an if statement, it's not if I have a hat, it is green.
@@algaeninja6806 In that case just replace the value of the amount. "All my hats" can be replaced by X. Then we have X are green.
How many hats does he have? We don't know but if we try to replace it with 0 hats we end up with.
0 hats are green.
And is that true? YES. There are 0 hats that are green so he would be telling the truth that contradicts the first rule about always telling lies
@@DajuSarIt could be interpreted as (X IS green), it could also be interpreted as ((X>1) AND (X IS green)). The question requires usage of either metalogic about the context in which you were provided the question or usage of linguistic conventions. The question is simply malformed.
The issue I feel is the same as with any math puzzle going viral.
People split into the camps of "math rules" and "conversation rules".
6+2x7=20, but in day-to-day life, you'll have to enunciate very carefully if you want to indicate order of operations, otherwise people will likely say 56.
By math rules, if I tell you all my cats have died in a fire, even if I didn't have any in the first place, that's called a "vacuous truth". By conversational rules I am a horrible lying excuse of a human being.
@@frederiklist4265 Well, not really. When most people say "6+2*7, they say it with an implicit comma (that is, six plus two, times seven). The parentheses cannot be stated outright, so most would interpret the way it was said to _mean_ that there's a parenthesis around the 6+2, even if there isn't. To get around this, you have to say "six, times two plus seven" if you want to make yourself clear, and while this arguably isn't enunciating 'very carefully', it's still a notable difference from the way that most people would say it.
TL;DR: Saying 6+2*7 out loud makes it sound like there's parenthesis around the 6+2 unless you put a pause in your sentence.
@@frederiklist4265 the funniest one is the following: 25-5/5=4! (the joke being the faculty operator misunderstood as an exclamation mark)
@@LilCharlet Bro, there is no need for that text in the brackets. Just say, "(6+2)*7" and then because 6+2 is contained in the brackets they solve the brackets first. Or, say "6+(2*7)" to make it easier for them.
@@baconboy486
I think you missed the original point.
Imagine some is speaking to you and specifically saying the words "what is six plus two times seven".
Obviously if you write an equation out then you can see any parenthesis, even if you write the words down you can see the punctuation such as a comma and a question mark etc.. but when spoken is just spoken casually the order of operations isn't always as clear as when written down. That was the point. I am going to assume you were talking about writing it down and not that they should instead be saying "what is open parenthesis six plus two closed parenthesis multiplied by seven?"
Just because there is maths in the problem, doesn't mean it is exclusively a maths problem, especially is phrased as a conversation or taken in the context of a spoken problem rather than a written one. This is often used as bad jokes such as "what is one plus one equals? Window." Or "what is one and one? Eleven." They aren't maths problems.
Conversationally, you wouldn't say it that way anyway. You'd state the problem as you desire it to be solved.
If you say 6+2×7, people will think (6+2)7. But if what you're after is 6+(2×7), then a normal person would day it as 2×7+6.
And the same for anything else. If I want to know what 12(5+15)/240 is, I'm going to say "Hey, what's 5+15×12÷240?"
A great example of how the correct answer can depend on what "rules" the question is asked under. This proof only works under the assumption that it is a mathematical lie that is being looked for, and is only useful within those rules. I find myself wanting to research vacuous truths now, to see if calling them "truths" is an arbitrary label or not.
I agree, the vacuously true statement is not what one can call true in any normal sense. Only within a specific definition of "true" does it make any sense, so essentially the question is misleading. I would say the bigger lie is when you say "all my hats" implies you have at least one hat in any normal sense.
It doesn't though. Answer B doesn't follow because it doesn't matter how many green hats he has, as long as he has a non-green hat he's lying. Answer C doesn't follow because again, there are ways for Pinocchio to be lying while having hats (say he has one red hat). Answer D doesn't follow because, again, the number of green hats he has is irrelevant. I don't even remember what answer E was.
And we know that answer A is true because for Pinocchio to be lying, he must have a non-green hat.
This is what I thought
It doesn’t make any kind of actual sense that “all my hats are green” is a truth if you have no hats. It can’t be true anymore than “all the phones in this room are turned off” is true. Neither are true
@@csarmii Pinnochio would still be lying if he had no hats
Great video! Also worth noticing: A + C cover every possible scenario, so if there is only one true answer it MUST be either A or C regardless of anything else!
Truth by uniqueness.
My deduction without knowing it was multiple answers:
Pinnocchio has at least one hat that is not green.
Not necessarily, just because the cases cover all possible scenarios does not mean that one of them must be the right answer, for example D and E also cover all possible cases, but neither of them are the true answer.
This is because we're not asking for trueness or falseness of the choices in a given scenario, we are instead asking which of these statements is always a lie, regardless of the scenario (and therefore logically follows the axioms set by the question no matter what). For both D and E there are cases where the statement could be the truth depending on the circumstances, therefore we can't conclusively determine that D or E is the right answer, it's dependent on the scenario in question. It would be true to say that in a given scenario where you know the number of hats Pinocchio has and their colors, those two statements will always be opposite to each other, but in this question, it does not mean that one of them must be the right answer.
@@zoeysalvesen8635 Hey Zoey thanks for your reply. I agree with you and certainly D + E also cover all possible scenarios, but there is a small caviat. Answers D and E cover all possibilities just like A + C, and therefore same principle applies: only one of D + E is the correct answer. We know that A + C cover all possible scenarios and deduce that C is the right answer. However, D + E cover all possible scenarios too, therefore E is the correct answer too. Notice:
C - Pinnocchio has no hats
E - Pinnocchio has no green hats
If Pinnocchio has no hats AT ALL, then certainly he also does not have any that are green. In fact, both C and E are correct answers, yet we stick with C, because it is more specific. (C means that Pinnochio has no hats, no green hats, no red hats, no blue hats, no any colour of hats).
The caviat is that even though both A + C = Ω and D + E = Ω , the subset C is contained within the subset E, giving as a more specific answer (in fact the answer C is as specific as it can be). Even though D + E also cover all scenarios, the sum of these sets only take the quantity into consideration, leaving the colour as constant (green). Note that "no green hats" also only considers green hats, since zero green hats is still a subset of green hats consideration. At the end of the day we just split the Ω omega set in different ways.
So while you are correct that D + E cover all possible scenarios too and I admit that blindly following what I said in the comment you replied to can be not precise enough, the principle still stands: If two subsets add up to Ω omega set, then one of the answers must be correct. Therefore when you said that "neither D or E are the correct answer" that is where the problem occurrs. Both C is correct (A+C=Ω) and E is correct (D+E=Ω), the question we just need to ask ourselves is, which of the 2 correct answers is more specific. And in This case size of set C is smaller that size of set E (set E fully contains subset C in itself), therefore C would be the preferred answer (as it gives us more information)
@GalaxyCimky I believe you are incorrect. For one thing A is the right answer, not C. The video proved C forms a vacuous truth with Pinocchio's statement, and therefore its opposite (in this case A) must be the correct answer. Because C is ALWAYS true no matter the circumstances Pinocchio could not have made the statement "All my hats are green" if he had no hats, because he always lies.
Now on to D + E. These do add up to an omega set as you said, but neither answer is correct and it is not because one answer is more specific than the other and therefore must be the most correct answer, it is because we can come up with examples for both D and E where Pinocchio is still lying but the statement can be either true or false. For example:
D - Pinocchio has at least one green hat
True Scenario: 1 green hat, 1 blue hat - Pinocchio's statement is still a lie
False Scenario: 0 green hats, 1 blue hat - Pinocchio's statement is still a lie
E - Pinocchio has no green hats
True Scenario: 0 green hats, 1 blue hat - Pinocchio's statement is still a lie
False Scenario: 1 green hat, 1 blue hat - Pinocchio's statement is still a lie
Because there are situations where D and E can both either be false or true based off of the colors of the hats Pinocchio has, we cannot conclusively determine which of these two options is correct. There is no way to do it for all scenarios, and like I said in my original message for a given scenario we 100% will have a definitive answer to whether those statements are true or false, and it would be true to say that exactly one of them will definitely be correct for that scenario because they form an omega set. However, given the set of all possible scenarios there will be some scenarios where D is true and E is false, and other scenarios where E is true and D is false. Therefore based on the premise of this question there is no logical way to definitively say that D or E is the correct answer even though they make up a full set of possible scenarios.
Conversely while A and C also form an omega set, in this case it IS possible to say one is definitively true in all scenarios, and therefore the other must be definitively false in all scenarios which means we can 100% say that the statement that is always false must be something we know to be true when talking about lying Pinocchio's statement.
It is not true to say that if a set of statements cover every possibility that one must be true, this is only true within a single specific scenario. For the set of all scenarios there could be inconsistencies within these statements, and because this question concerns itself with a scenario-less premise (i.e. we don't actually know the color or number of Pinocchio's hats) we cannot say definitively that a set of statements making up an omega set always will have a definitive piece of information that we can discern.
So me saying all my lamborghinis are green is not a lie?
A) vague amount
B) specific amount
C) specific amount
D) specific amount
E) specific amount
The number of times I used this strategy and succeeded really baffles me
Why is D) specific amount?
@@Grassmpl 0 is a specific amount!
@@dumbwaki5877 but D) is "at least one"
@@Grassmpl D) is also somewhat vague, but by specifying that one of them must be green, it becomes specific.
You could rewrite the sentence as "Pinocchio has a green hat," which is specific compared to "Pinocchio has a hat."
lol this is amazing
I came to the same conclusion a different way. I eliminated options B, D, and E for largely the same reasons. Then I looked at Pinocchio, who is wearing a hat, and concluded that he must have at least one hat.
Where does it say that is a picture of Pinocchio? ;)
@@kendraroth1276 An old colleague taught me a long time ago that assumption is the mother of all fuckups. Life has taught me he was correct. ;)
@@kendraroth1276 But did the question text talk about a picture at all? No. So the picture is not a part of the problem.
@ Helbore its common knowledge that this is Pinnochio in this picture, if i am not mistaken from the original book in which he is hanged at the end. I know another version in which he is burned but according to my italien teacher he was hanged and she also said this book gave her nightmares😉😉
It's A because if you don't own any hats, every hat you own could be green.
The opposite of all the hats Pinocchio has are green is Pinocchio has at least one non-green hat, which can’t happen when Pinocchio doesn’t have any hats at all.
I thought this was too easy so I was watching to see what I did wrong the whole time, only to be pleasantly surprised that I finally did one!
Likewise. Also pretty insightful in how so many "believers" use explicitly flawed thinking to make the types of vacuously true statements mentioned in the video, and then cling to them to the point of violence.
Same, immediately thought "at least one hat that's not green". If he had no hats at all that's just a "trick" question and not the clever kind.
Alternative title: Solve this viral test question, or you're going to Brazil
Then I would like to skip this question 😍
Dude of all fates. Brazil is the worst. But they...
i wanna double jump
I think both alternatives are better than staying where you are
Jokes on you, I'm a Brazilian
This is a rare case of a logic puzzle where the answer seems obvious at first but then when you dig deeper you find more depth than you expected until you eventually discover that you were actually right in the first place.
Yeah. Had a smoothbrain moment when I thought "Well duh he has at least one hat, it's right there on the picture!"
@@SpiralDownward I eliminated the picture from the puzzle when I addressed it. Logic is about premises and conclusion not empirical observation. And indeed the hat in the picture is green so then we leap to Pinocchio having more than one hat but it's really speculation. Focus on the given fact that is known and cannot be violated: Pinocchio always lies. Always. He makes a compound statement in the second premise. He states that he has hats and that they are all green. Is it then logical to falsify A by saying he has hats? In the puzzle I think not.
@@cre8tvedge the hat in the picture is yellow lol
I see you haven't done many logic puzzles.
If Pinocchio's nose always grows when he lies, how is that fella walking around gabbing about imaginary green hats. The very nature of Pinocchio is that he inherently has a flaw that makes his nose grow when he lies, so it's an activity he would otherwise avoid - so the question itself is a lie - why else choose him as the character in the question. Just my two cents.
This comment section will understand vacuous truth when pigs fly.
I see what you did here
so we can conclude that not every hat he has is green.
Just below this in my feed is a meme about how far a squirrel has to fall to die, with the answer "0 feet, as squirrels have been known to die without falling". Same energy.
1. What precisely is a meme? 2. Why is your squirrel thing one? 3. Why is every single image, video, text or now just a meme?
@@dunnedigby4957read selfish gene by richard dawking (only the first or so chapter are necessary). I wrote a comment but mid writting it on the phone it got deleted.
Resumed form is meme is culture under natural selection, almost all if not all culture is under natural selection by the people. so the above comment is a meme by definition.
i chose A, but i thought about it differently:
if pinocchio always lies, then
1) Not all of his hats are green
2) None of his hats are green / All of his hats aren’t green
that would mean he has to have at least one hat, which might or not be green. solved this in a linguistic way more than mathematical though. im brazilian btw, didnt take the exam but i remember seeing this all over the internet a few months ago lol
This is not linguistic at all, if in the statement the word "all" is a lie then it could mean anything like "none my hats are green" thus making answer that none of his hats are green.. you in no way shape of form can come to th "correct" conclusion by linguistic simply because thats not how it works(you just got lucky(.. its a maths question and cant be solved otherwise.. if u apply actual logic this question will have no answers.. there is another case where u could say what if he lied about the "hat" part.. example- "all my shirts are green"..he was lying about the fact that the green things he has are hats but they are actually shirts.. oh wow see that dosent mean he has atleast one hat..
@@somethingsomething2541 by reading my comment again i think i might’ve expressed it wrongly - regardless, even if it is a math question, i think there’s still a linguistic undertone to it.
the second sentence is a lie, so you’re supposed to negate the “all”. therefore: “at least one hat isn’t green” (if one of them is a different color, saying that all are the same is a lie) -> option A.
i get what you mean and i know you can’t solve it *completely* by using language, but it’s part of the process.
@@in-betweendays yupp i agree with that
there is no proof that pinnochio doesnt have 0 hats
The reason that Pinnochio has to have one hat tho, lies in the meaningless truth, i.e. If there are no hats in the room, then we have to assume that the fact that "All the hats in the room are green" is true, we can apply the same thing to pinnochio owning a hat, Pinnochio says "All the hats I own are green" If he owns no hats, then we have to assume that all the hats he owns are green because its a meaningless truth, but Pinnochio cannot speak any kind of truth, because he always lies, therefore in order for him to be able to lie about that statement, we have to assume he owns at least one hat.
You actually cannot conclude anything about Pinocchio's hats. He SAYS all his hats are green. But that is not true. If he had said he had no hats we could conclude he had at least one hat. As it is his hat possession status is indeterminate. Schroedinger's Hats.
This is literally a contextual statement, theres no right answer
The answer to this problem is different depending on how you define the word "lie." With a more human, and real life definition of the word lie, you can't say that any of these options are true. If you say all your hats are green, and you have no hats, that's misleading enough to be considered a lie in the real world.
These problems that go viral and are discussed always have some ambiguity like that.
The definition of "lie" in the context of a logic puzzle like this is pretty obvious to anyone with common sense. Why would you deliberately choose to interpret it as a trick question when there is a clear logical solution?
YES and No - Slide In Meaning...
I think that's why it was stated this was a problem in a math olympiad. If you didn't consider the mathematical, rigid definition, it's kind of on you.
@@ric6611 I guess if you are training on logic puzzles, and come across this question it's pretty easy, to know the right interpretation. But when you just post this question on social media, and try to answer it honestly with no biases, then the ambiguity shows up.
So you need the bias that comes with studying and understanding logical theory for this question to become unambiguous basically.
@@steverempel8584 Oh yes, I thought you were referring to here in the video.
If someone testified in court, when he told the bank to get a loan “ all my business are profitable “ when he in fact had no businesses , and insists his statement is vacuously true … the judge is going to add the charge of contempt of court.
Pretty much. There's no true answer to this puzzle, the data to solve which one of the statements is true just isn't there.
Not an issue here since liar Pinocchio is always going to be in contempt of court.
@@thenonexistinghero I am a credentialed and professional logician. There is a true answer to the question. However it is not one of the multiple choice answers.
The answer is:
"We know Pinocchio either has no hats or at least one hat that is not green." That is he could be lying about having hats and their color, or just lying about their color but we know he is lying.
@@brianmacker1288 That's not one of the provided answers. And it is also not a single answer, but one that combines multiple answers.
Anyhow, that being said... the discussion is about 1 out of those 5 answers being the right one. And the issue is that there quite simply isn't enough data to deduce which one of the five shown answers is the real one. And the 'logic' used to prove which one of those answers is true is not logical at all.
@@thenonexistinghero I know it is not one of thr provided answers, Duh. Because all the provided answers are entirely wrong. Every one of them is false.
Nor does the correct answer "combine multiple answers". The question is what we know. The statement "Pinocchio has no hats" is not an answer to that question. Nor is "Pinocchio has at least one non-green hat" an answer.
My answer is the single and only correct answer as to what is known.
As I stated elsewhere I am a credential and professional logician. My answer is the correct one. It is not using the "or" operator to combine two correct answers in this case.
Really interesting! I loved how you went thinking through the problem
You could really just use the same logic for all four wrong options; propose a scenario in which the statement is false, but the answer could still apply. For choice C, if Pinocchio had no hats, all of his hats (which don’t exist) could theoretically be green, as his “hats” are all a figment of his imagination. If the statement *could* be true in any way, then it’s not the answer.
My only problem with the question is the use of the word "lie", since that can be used for misleading but not necessarly false statements. The premise should be that pinochio always tells false statements, and by simple negation we would conclude A.
@@mrdkaaa I know he addressed it, I am just refering to the question, not the video, it's still bad wording since it's being used outside the context in which it was created for, which was the Math olympiad.
For me they are the same thing. Can you come up with an example where a statement is a lie and not false or vice-versa?
@@pedrotraposo all my ducks have a green neck. How many ducks do I have?
@@PR-ot7qd I dont know. I dont get it.
@@pedrotraposo I do not have ducks, which makes my statement misleading, ergo, a lie. However, if you see in a purely logical perspective, 0 ducks have 0 green necks, making my statement true, not false.
I was wondering how we can even figure from Pinocchio's statement whether he has any hats at all - imagining an option (F) which were 'We cannot know whether Pinocchio has any hats" - but understandably within the math/logic framework the statement implies he must have at least one hat so as to not make a vacuous true statement.
All it says is he has no green hats, he could have a blue one, an orange one, it doesn’t specify.
@@petermello55 my bad I forgot there was a real option E. I meant a sixth option
I got A but for a less “good” reason - the sentence structure. The way the sentence is built is that what Pinocchio is lying about is the colour of his hats, so therefore saying he has no hats is wrong. I don’t think this logic would hold up under inspection, but perhaps because it was written in translationese that’s what I got from it.
I just thought that if the question was trying to get us to think about if Pinocchio even owned hats, then suddenly the grammar of the sentence gets very shonky and isn’t how anyone would say or write it.
As he explained in the structure, the problem is that if he has no hats, then any statement about what hats he made would still be vacuously true, because there would be no hat that exists to falsify the statement. He has to have at least one hat in order to falsify the statement and make it a lie.
@@KryptikM3 Isn't that overthinking the solution though? His reasoning for ruling out option D also applies to option C. If Pinochio has 2 blue hats then the statement by P that he is lying is accurate as required by the problem. However, Option C...P has no hats is NOT always True if P has two blue hats. Therefore C is not correct. One can come to the correct answer of A without knowing what "vacuously true" statements are.
I disagree with that conclusion. If I had no cars, and I say "all my cars are green" I would be lying, only because of the "all my cars" part. Just my opinion.
What the video explain is that if I say something about an object I don't have, it's always true. I could say all my cars are planes... Even if I don't have cars this would be true
the picture provided shows pinocchio wearing a green hat so d is correct although some may say the hat shown is closer to yellow, so he owns a yellow hat and any number (including 0) of different coloured hats potentially including green ones. These answers assume that he is in fact the owner of the hat on his head - if he is borrowing it then he owns any number (including 0) of hats and the number of these hats that are green ranges between 0 and the total of owned hats minus 1 as long as he has more than 1 hat. If we don't take the image into consideration then he must have at least one hat as per the video. If we interpret "always lies" in a way that includes setting a false premise instead of merely making incorrect statements then i think "all my hats are green" is false if he owns no hats but if we take "lying" to mean making false statements he must have at least 1. Very cool puzzle
I’m a computer programmer and picked option A after treating the problem like a negation statement. By assuming Pinnocchio NEVER lies, then Pinnocchio would truthfully say “NOT all my hats are green”. The only compatible option with that statement was A. Great puzzle!
wait, doesn't D also fit within this logic? Since not all his hats are green, at least one is green, no?
When Pinocchio says "my hats" he is claiming to own hats, but everything he says is a lie, so he mustn't own any hats, otherwise his claim to own hats would be true which would contradict the statement that he always lies.
He always lies, he may have no hats.
@@JackyPup The negation of "All my hats are green" is "At least one of my hats is not green". The only way he can have at least one hat that is not green is by having at least one hat, so A
@@ProperGanderSaul I agree with you, one step further though. It aren't his hats to begin with, as he said MY, so you can't even say anything about pinocchio to begin with. as he is lying about the hats being his.
I saw this problem as a mathematical logic problem.
The negation of "All of my hats are green" is "There exists a hat of mine such that it is not green." Thus, the phrase "There exists a hat of mine" implies that Pinocchio has at least one hat.
Perhaps you can clarify my confusion: Shouldn't answer A then qualify that not only does Pinocchio have at least one hat, but that necessarily at least one of those hats isn't green. Statement A is incomplete because it includes the possibility of the hat or hats that he owns being all green.
@@xTheITx Statement A indeed isn't complete, but it doesn't need to be. The question isn't about concluding everything possible, it's giving a set of statements and asking which must be true. The only thing you can conclude is that Pinocchio has at least one non-green hat; the only statement that must be true because of that is A.
In my opinion, I view "All of my hats are green" as meaning "The number of green hats I have (G) is equal to the total number of hats I have (H)" or "G = H". Thus, the negation would be "G < H".
So, if he had 0 hats, "G = H" would be true since he has no hats in total, and by extension also has no green hats (G and H are both 0). This statement can't be true, however, since we know he always lies. So, he cannot have 0 hats, meaning he must have at least 1, making A the only conclusion we can be 100% sure of.
Thank you. I think you actually explained better then the video.
This is because of the mathematical edge case in which "for all" statements are true if the universe of discourse is empty. Because "for all" really means there does not exist any counter example, which is true.
It's like, mathematically, the statement "all my iphones are red" is true because I don't own any iphones, even if it does not make sense in english.
1:10 at this point it's interesting because i feel like the answer should be "Pinocchio has at least one non-green hat" but that isn't one of the options
I thought the same thing but in different wording. “Not all of Pinocchio’s hats are green.”
Well, if he ALWAYS lies, then all parts of the statement "All my hats are green" are lies, then it means he never wears hats.
Funny, I'm an English teacher, so I approached this problem linguistically. I also ended up with answer A, by ticking off answers based on conversational maxims and exploring deep structure vs. surface structure. Though if this were a question on a linguistics test, you would still be awarded points for any of the answers as long as you can argue to which maxim the answer belongs (by explaining as to how you interpreted the deep structure).
I'm a research linguist, and my first thought was none of the answers. We can conclude that he has at least one non-green hat. I can see why A is the "right" answer, but I am also of the opinion that natural language is too complex for this type of logical reasoning to apply properly. A statement like "all my hats are green" when you own no hats is considered true in logic, but I think that is forced, at best. In natural language the determiner "all", just like "the" comes with a presupposition of existence, in and of itself. So the sentence "all my hats are green" is actually "I have (at least too) hats and they are all green", and if "I have hats" is false", "I have hats and they are all green" is also false.
@@carmensavu5122 If "We can conclude that he has at least one non-green hat.", then A must be right.
@@viniciusoliveirafontes4033 there is no reason to conclude that. We were told he is a liar. You shouldn't assume that he is telling the truth about having any hats.
@@carmensavu5122 Well, even then, the statement wouldn't necessarily be false or a lie. If Pinnochio was a green hat seller, sold all his hats, then claimed "all my hats are green," then just by the hats mere non-existence doesn't guarantee the statement to be false, logically or linguistically.
This is sort of how I came to my answer, and I think my reasoning actually reflects the "vacuously true" mathematical answer as well. Since the sentence doesn't become a statement of a fact until "are green" is tacked onto "all my hats," I elected to ignore the word "All" as a word he could be lying about
Questions like this make me appreciate mathematical notation. Much less ambiguity, much easier to solve/reason about.
(forall hat of Hats . isGreen hat) = false => (!forall hat of Hats . isGreen hat) => exists hat of Hats . !isGreen hat
Pardon my writing on a phone, I can't get to nice symbols.
truueee its very objective :)
The question is to partly test the verbal aptitude of the candidates, otherwise they could have given the mathematical notation which will be solved easily by most candidates who prepared for the test.
Yeah. I mean that trying to solve it in words is very confusing, at least to me. I think the concept of vacuous truth violates grice's maxims, lol.
While if you translate the words into a math notation of your choice like set theory or formal logic then the answer is quite simple and straightforward to derive.
@@imacds You're the first person I've seen to talk about Grice's Maxims online. They're so invaluable but not so well-known.
But a isn't necessarily right either. As with b d and e, there are situations, where Pinocchio might not be lying.If he has 3 green hats, he has at least one hat and all his hats are green. A is correct, but so is Pinocchio's statement.. so the only fitting answer isn't an option. It would have to be: Pinocchio has at least one hat, that isn't green.
The first half isn't enough to be the right solution..
Me: i need a loan!
Bank: what is your credit score?
Me: 1000
Bank: you don't have a credit score. But that statement is true, so here is your loan! Have a good day
This just proves to me that mathematics are fundamentally divorced from reality
it's actually LITERALLY THE OPPOSITE.
It proves to me exactly why nobody likes or enjoys having conversations with mathematicians.
That makes no sense lmfao
They are too much into reality while your daily interactions are with the shadows of the reality they work with.
@@grimendancehall Okay, can i give you 1.23 negative dollars?
Without the multiple choice I said outloud : "the only thing we can conclude is that pinochio has at least 1 hat that isn't green." And somehow got confused by the multiple choices.
And you're wrong. The only thing we can conclude is that if Pinocchio has only one hat, it isn't green, but if he has more than one hat, at least one isn't green.
The multiple choices are all incorrect.
@@immikeurnot No no, that's what they meant. Like you said, whether Pinocchio has one hat or multiple, at least one isn't green.
Exactly! If you know propositional logic, you know the negative of "for all" is "there exists" (followed by the negative of the condition). As the sentence "For all hats H, H is green" is false, it must be true that "There exists a hat H such that H is not green", which is exactly what you claimed
@@yes1570 If that's what they meant, why are all the answers wrong?
@@immikeurnot No, the right answer is A, which would still match with the statement that Pinocchio has at least one not green hat. It’s in the video. OP is just saying they got confused by the multiple choice even though they knew the answer
An infinite number of mathematicians walk into a bar…
... and say: "You can count (on) us." Is that a lie? 😅
Keep the math to the math. Making a problem after translating a problem makes the new problem as much of a linguistic problem as it is mathematical. You can see this because the original problem in Portuguese is much more direct in its wording.
You don't know about logic in discrete math yet then. This can easily be written in an equation.
"You can see this because the original problem in Portuguese is much more direct in its wording" Not sure if you're parroting someone else, but no it's not. It's literally translated as directly as possible, I'm a native brazilian portuguese speaker and there is absolutely nothing that I would change in this translation. It's as precise as it can possibly be, and there is no real difference in meaning or interpretation.
Just showed the beginning to a friend, so we could solve this together, and he went "The opposite of 'all' is 'at least' ". After this he just went from the logic and solve the problem in 10 seconds. He has a math degree, and i forgot about this for a sec. Not funny :(
the opposite of all is none.
@softan Think of it this way, the opposite of ‘at least’ is ‘at most’, so ya basically ‘all’. Didn’t make sense to me at first either!
@@softan The opposite of all is not all.
@@softan How do you prove that something isn't always true? By finding a single counterexample. You don't have to show that it is never true.
@@softan
P: All my hats are green
~P: At least one of my hats are not green
Very odd indeed, but interesting nonetheless. The language itself leaves room for interpretation and it becomes evident that there is a discrepancy between pure logic/math and the world in an empirical sense.
Here the problem is mostly just that 0 is treated as something.
When it is defined as the absence of something.
If you multiply 5 with nothing is it still 5 or is it 0?
It is just mathematical semantics when used in math.
The only field of math where 0 actually has a use is Boolean algebra.
In Boolean algebra there is only 1 and 0.
It is used to understand and build computers from scratch.
In Boolean algebra 1+1=1 (since 2 does not exist).
"A+B" is the mathematical equation for an OR gate.
The truth table he showed is pretty much Boolean algebra.
He just replace 0 with false and 1 with true.
Yeah not only that but "vacuously true" doesn't exist in some modern philosophical logics, which are a priori to math. In some logics, you can say "all my hats are green" when there are 0 hats is neither true nor false. If Pinocchio only says false things then he can never say a thing that's neither true nor false.
@Repent and believe in Jesus Christ
Lol
Language and math have similarity, though. Both are based on consensus. For example, "square root is always non-negative" is based on consensus instead of absolute truth or something. The difference is that language is based on applicable habit of communication while math is based on consistency of the rules.
If I were you, I would study all languages, try to understand the logic behind the structures, start dancing on white house dinner table, and then turn into alien piranha.
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That was an example of nonsensical language that is vacuously true :D
(F) Pinocchio is color blind.
(G) Pinochio has a green hat that identified as being red
The college class on logic I had 10 years ago still left something useful in me lol.
Approaching the question logically rather than mathematically, I thought the only information you can glean is "if Pinnochio has any hats, at least one is not green", but I didn't know about vaccuously true statements, so thanks for explaining.
That conclusion is correct. He either has 0 hats, or he has some non-green hats
I'd never heard of a "vacuously true" statement, but I deduced A) to be the correct answer because C) is the logical equivalent of dividing by zero. For example, if he has 3 hats and 2 are green, you can express the proportion of green hats as 2/3. But if he has zero hats, then the proportion of green hats is 0/0. Since division by zero is undefined, claiming that all hats out of zero are green is neither true nor false, it's simply mathematically illogical. Therefore, the only logically True answer is A).
If Pinocchio is truly speaking about hats then he is telling the truth that the subject of his sentence is hats. So if he ALWAYS lies, he cannot be speaking about hats at all. Therefore none of the answers are correct.
@@RedShiftedDollar I don't know if I can agree with that. A lie is saying "I didn't eat your icecream" when you did, not saying "I didn't eat your icecream" when you are asked "where is your work assignment"
@@davidjorgensen877 I like your reasoning, but you're assuming that one of the answers is correct (not a bad assumption) whereas I was looking at just the statement. It shouldn't make a difference which approach you take on a well written question, but in this case we come to different conclusions.
The idea that saying “all my hats are green” is true when you have no hats irks me. If I was cooking dinner and said all of the burgers are cooked medium well, but there were no burgers, I’ve just lied to someone. It feels like there’s a disconnect between the logic/mathematic argument and the human side, which makes the logic puzzle kind of contrived or mean spirited to be presented as a little verbal puzzle rather than a mathematics question. I’m not sure that being able differentiate the last two answers shows any form of cleverness other than a skill check on if someone has been educated with a mathematics degree
No, it's just not an a=>b statement in natural language. But mathematicians argue it is
I also found it very confusing. The trick for me was to think like this: the fact is that there are no burguers; that's a fact, you can't deny that. But then you say the burguers are cooked medium well, it is a truth statement in its own. The second statement is not linked to the first statement and because of that it is true. Both statements are separated, they're not linked. Now, if you said "there are no burguers AND they're cooked medium well" it would be a false statement because both statements are linked to each other and since each negates the other, it becomes a false statement.
Truth table for AND:
T T = T
T F = F
F T = F
F F = F
But I agree with you about the way the puzzle was presented
I agree with you, the assignment of this task is unclear. That's why in most mathematical Olympiads people avoid these sort of assignments and opt to express similar ideas in mathematical terms.
It definitely can feel frustrating that the answer relies on a technicality, because generally when we communicate with each other, we tend to follow certain rules, like not sharing more information than necessary, and only sharing relevant information. But if you don’t have any hats, and were to say “all my hats are green” seems to violate the rules we generally use to communicate.
I think another way to analyze the “all my hats are green” is to think of it like this:
If you wanted to check that all of someone’s hats were green, you would look at the first one, and if it wasn’t green, you would stop and conclude some hats are not green. Otherwise you continue and look at the next hat and repeat. If you reach the end, and every hat that you have checked is green, then all hats are green.
If there are 0 hats to start, then every single hat that you have checked is green, thus all hats are green.
This is correct in paper, but it's a flat lie anywhere else.
"All the money you just won can be use to cure your mother :D!!!!!"
"All you have is happiness"
Literally anything can be turn this misleading way
Pinnochio shaves all barbers who do not shave themselves. Is Pinnochio a barber?
I thought this way; the negation of 'all my hats are green' is 'I have at least one hat that is not green,' which is naturally a subset of the case 'I have at least one hat'
This is absolutely correct. It's surprising that Presh doesn't give this argument or indeed give any explanation of why the answer "I have at least one hat" is correct.
I like P always lie. Now I will tell you all my motor bikes are big... Infact I have no motor bikes. ?????
@@petethewrist you didn't lie, assuming you have no motorbikes.
For "all my motorbikes are big" to be a lie, you would need to have at least one motorbike that is not big, which you don't. So the statement is true.
Similarly it is true if you say "all my motorbikes are small". For it to be a lie, you would need to have at least one motorbike that is not small, which you don't.
I hope this is clear.
@@MichaelRothwell1 none of it a lie? No it was a fabrication which is may be what P was doing.
Incorrect. The phrase could be broken down into two statements I have a some hats and they are all green.
So either he has no hats or at least one hat is not green to make it a false statement.
If you are a computer programmer, you will understand how to translate that into a code and you'll know why is also a possible situation and why is not a unique solution.
Another way to look at this that I find more intuitive : we tend to assume that "all" means "at least one". But it also can refer to zero. If you have zero hat, then all of your hats means "zero". Therefore, zero hats are green, which is true. Therefore, Pinocchio can't be lying. He MUST have at leat one non-green hat for the statement to be false.
Fascinating.
If everything he states is false, wouldn’t “all my hats” in of itself be false. There is either nothing or something(like bianary 1 0).. if he’s saying there is something “all hats”.. or even one hat is something, then there must be nothing, regardless of color ?
@@sman000 I'm not sure I understand what you're saying, but "all" doesn't necessarily mean "something". "All" of zero is equal to zero, therefore "all" can be nothing.
He's saying every hat he possesses is green, but he doesn't possess any, therefore it's true. All of zero is zero.
He’s saying “all his hats”. That indicates something is there that he is referring to, at least a hat.
@@sman000 Again, if he has zero hats, then "all of his hats" is literally zero. You're falling in the same trap I explicitely warned about in my initial comment : that we tend to assume "all" means "at least one", but that isn't the case. "All" and "every" do not, in logic, infer number. All of zero is zero. All of 1 is 1. All of 1000 is 1000. The meaning of "all" is determined by the number it's associated with.
If you have zero hats, then zero of your hats are green. Therefore ALL of your ZERO hats are green.
@@sman000 All that matters for the given condition to be correct, "that he always lies," is that each statement in itself is false. Therefore you can't break the first part apart like that because it's possible that all his hats are not green, or, that he has at least one hat that is not green.
answering before finishing the video, none; because the true answer is Pinocchio has less green hats than the total amount of hats
The best way to read these statements is put NOT at the the start, but in a programming sense, not in the sense of natural English.
"NOT all my hats are green."
This is different to "Not all OF my hats are green."
I solved this by reducing "all my" to a number : "0 hats are green." If Pinocchio has 0 hats, this is a true statement; ergo, Pinocchio must have at least 1 hat.
However Pinocchio can have exactly 1 green hat under option A making it a true statement. the only true answer would be that Pinocchio has at least 1 non-green hat.
@@richardgomez3469 Understand that the issue isn't what CAN be the case, but rather what MUST be the case, given the two introductory sentences which, for the sake of the riddle, also MUST be true. It is child's play to construct specific instances where one or more of options A-E are true; excepting option A, however, it is logically impossible to show that any of the rest of them MUST be true. Again, if Pinocchio has 0 hats, then "All my hats are green" is TRUE, so Pinocchio must NOT have 0 hats. // Additionally, please note also that your "solution" isn't one of the listed options, but is rather a meaningless tautology directly inferable from the necessary truth of option A.
That’s probably the best explanation so far.
@@themediaangel7413 Thank you. I tries. :)
Ohhhhhh that makes sense
That reminds me of a dialogue in Ender’s Game, when colonel Graff asks Valentine to write a letter to her brother Ender. She had written him numerous times before, but unbeknownst to her Graff had never forwarded any of her letters.
G- “I want you to write a letter.”
V- “What good does that do? Ender never answered a single letter I sent.”
Graff sighed. “He answered every letter he got.”
It took only a second for her to understand. “You really stink.”
Great quote from a great book
@@DocBree13 Great book, horrible movie
Ain't that the truth. I for one should know
@@zzztek
... Movie?! Oh no..
I didn't know there was such a thing.
A thing to note here is that she couldn't determine whether A) he got the letters and she didn't receive the answers or B) if he simply didn't get the letters.
He saw a man with binoculars
1. Man had binoculars
2. The man who he witnessed had binoculars
I like to think of option c as this: we can say that “all” of his hats is equal to the number of hats he has, so if he had 5 hats the statement “all my hats are green” is “5 of my hats are green”. If Pinocchio had no hats, then the statement becomes “0 of my hats are green”. Now, if he had no hats, then this is true, since none of his hats are green since he has no hats, and since he always lies, then we have a contradiction.
I'm a Bronze Medallist of the OBMEP, so it's awesome to see one of its tricky questions here. Look for more, there are many cool ones.
Que legal! Eu somente passei 2 vezes da primeira fase haha.
Nessa pergunta eu acertei porque eu pensei, "ele não iria falar com tanta especificidade de algo que ele não tem, se ele não tivesse ele somente ia dizer que ele tem", faz sentido?
Siiim meuu
Eu ganhei só uma mensais honrosa 🥲
All my medals are gold.
I was also torn between answer A and C. I'm not familiar with "mathematically true/false" statements. Thanks for making this kind of logic game accessable!
Pure logic says that all these options are possible. So, A-E are all possible. That's all we can "conclude from the statement".
@@gailwaters814 but if he says all my hats are green he's lying about having hats in the first place so he has no hats and he doesn't have any green ones either. Easy solution, it's C and E
@@floseatyard8063 Nope, because once he says "all" it means that he can either have no hats or a large number of hats of which some are green, or none, etc. So all options are possible because he used the word "all".
@@gailwaters814 do you not remember the puzzle said pinnochio always lies? If he said all my hats are green he would be lying about having hats and about how all his hats are green so its C and E.
@@floseatyard8063 Yes, but a lie could mean either A B C D or E. Each one of those would be the result of a lie.
I'm brazilian, and i have done this test in my school back in the day, and the logic i used was: the statement is not based on whether or not he has a hat, but rather the color of a hat. It would be like saying "there is a plane in the sky", the statement is not whether there is a sky, but whether there is a plane in it... I don't know if it made much sense, but I got it right on the day of the test (Sorry if I made any grammar mistakes)
Logically, it's not feasible that someone would always lie, so the narrator is unreliable, therefore I can't conclude anything.
The way I solved this, is by remembering that a logical statement is false if and only if the negation is true. The negation of the statement "For all X, Y is true" is "There exists at least one X for which Y is not true". The negation of the statement "All my hats are green" is "I have at least one hat that's not green". Therefore the answer is quite clear, it can't be (C).
this is what I did.
had the exact same thought.
Same
Same thought process here. Nicely done.
Yes: For all X, Hat(X) implies Green(X). Negation: There exists X st Hat(X) and Not Green(X).
Looking from a non mathematical standpoint, one that would be applied in normal conversation. If somebody were to say “All my hats are green” when in fact they have no hats, that would be lying. Because it implies the possession of hats which if he were to have none, he would be lying.
Yes,I thought that way
Same. It makes sense. It's a matter of argumentation at this point as some people in the comments have pointed out.
I absolutely agree, which is why I picked C. And I would pick C again.
From the text I considered that to be an option but I assumed the picture of Pinnochio with a hat was not a lie.
actually no if they have no hats and said all their hats are green it could be taken that if they actually had a hat it would be green
The only thing you can infer is that Pinocchio has an inderminate number of hats, which could be zero or not, and that if that number is positive then one hat at least is not green. Therefore none of the statements are correct.
I thought it was "at least one of Pinocchio's hats is not green"
What an honor as a Brazilian to see this problem being discussed here hehehe. Unfortunately I couldn't take this Olympiad test since I'm already an undergrad, but I loved it
It was From Obmep haha
Que honra mesmo
@Paulo Henrique nós BRs estamos em todos os lugares hehehehe
Eu fiz e acertei, e estou indo pra 2a fase (:
even u an undergrad, that doesnt mean u could ace this test
It's hard to wrap my brain around "c" being incorrect, as in that case the lie isn't about the hats being green, the lie is about ownership of hats in the first place.
Apparently the deal lies within admission of having a quantity of something must mean that the admittant must have at least one of something, if that made any sense.
Basically, if I say "all of my cats are calicos", then the logic in this case dictates that I have at least one cat. Even if you didn't know I was lying or otherwise, you'd still assume I have at least one cat. Especially if you weren't told I was lying beforehand.
If I say, all my Mercedes are red. I own no Mercedes. Therefore, I can't have at least one red one. How do I have at least one red one?
Me too, but I get it after the video point out that you don't need a thing to say 'all my... are...'
I get why they derive the answer from a mathematical point of view, but from a linguistics point of view, I agree with what you say. He can be lying about owning any hats at all.
@@Polarcupcheck Apparently, according to "Mathematical Logic" you now own a Mercedes. Better go check your garage!
Love how these videos teach me how to think!
Cut ties with Pinocchio, he's a liar.
Pinocchio: "There is one correct answer."
Pinocchio: "It is assumed to use vacuous logic"
if its a Mathematiacal Problem, then its not a Logic Problem. Also it says what can you conclude for the two sentences. You cannot conclude that pinocchio has at least one hat, because he doesnt tell the truth. He simply can have no hats despite the picture because he could lie about the hats too. none of the answers are correct, if we use pure logic. And this is also the problem with liars in the real world!
@@crashoverwrite5196 No, A and C are left over because of the reasons stated, C is eliminated simply because if he says "all my hats are green" and he possesses no hats, then he didn't lie, all the hats in his posession are indeed green. Going by both logic and mathematics, A is the only possible answer.
@@crashoverwrite5196 logic is literally a branch of discrete mathematics.
@@olivermatthews8110 Sure but not the full range of the physical world. Mathematical logic isnt always useable for our world.
@@emriys1334 We cannot conclude C because he could have at least one hat wich isnt green! But we also cannot conclude A because he could have no hats!!! Maybe mathematical logical but not in our realm by logic. If you have no hats you cant be right that every of your hats are green, because there is no hat so its a lie.
The sentence p says: " all my hats are Green" is true because he said it. But he tells a lie! Logic at its finest.
I think this explanation makes sense and is correct when this question is understood to be from a math/logic perspective. But from a real world perspective, if someone said all of their hats are green, and I found out they had no hats, I would say they were lying in their statement.
It's very much sounds like a politicians go to lying technique.
I would not say they were _lying._ It was clearly a misleading statement, aimed to purposefully confuse you. It is a dishonest statement. But it is not technically false. Information meant to mislead you but technically true is very different from lying: most advertisement and political communication is based on falsely represent reality without lying.
If I were to say "No girl I slept with complained about my performance", and I were a virgin, I would not be lying: I would be surely misleading the audience, but it would be technically true - the best kind of true.
Yep, artificially twisting a natural-language question into a truth table for the sake of getting a clean answer is a very... mathematician thing to do
"I have no non-green hats"
@@colbyboucher6391 sorry you didnt get it right bud, dont worry I thought it was C too
I feel like the correct answer would be: "pinocchio has at least one hat, which is not green"
I thought all his hats were green but he was colourblind so he thought his hats were a different colour
The brazilian channel Victorelius made a very good video answering this question. Just remember that the negation of a total affirmative is a partial negative (many people make the mistake of thinking that the negation of a total affirmative is a total negative). That is, the negation of "All my hats are green" is "At least one hat of mine is not green". Therefore, we conclude that Pinocchio has at least one hat (one hat that is not green: it could be one green hat and one red hat, just one red hat, etc.)
He also points out the misleading in the question statement: lying is not the same thing as expressing falsehood. E.g., I can think, for some reason, that a pencil is white and lie saying that it is black. However, the pencil is actually black. So I lied but I spoke the truth.
Para Saul Kripke, essa resposta não seria tão óbvia.
Ele dizia que tudo que predicamos, assumimos a existência (mesmo sem usar quantificadores existenciais).
Logo, a afirmação de Pinocchio seria mais ou menos assim: X (chapéu que é meu) existe, tal que, para todo X, X é verde.
Erro meu, não é o Saul Kripke. É o Quine que defendia isso.
Eu que não estudei nada disso entendi que pra considerar uma afirmação de negação,ou vc aceita como total negação,ou tem algo que afirma a negação. Se ele diz que todos os chapéus dele é verde, como não sabemos a quantia de chapéu, não tem como ele não ter um pelo menos. Pois ai não teria como ele mentir sobre usando uma afirmação,pois seria redundante.
Mano, eu nunca vou entender negação como matéria. Parece uma perda de tempo ficar rachando a cabeça com uma pergunta que pode ter N respostas.
This vid is logically WRONG. None of the options can be deemed correct.
The statement was actually "For all hats I have, the hat is green". When negating the statement you get "There exists a hat for which the hat is not green". Not only can you say pinnochio has a hat, but you can also say that it's not green
Negating statements is fun. For all swaps with there exists and there are also rules for what happens if you negate logical operators. I missed a small introduction of logical operators in the video but it was fun to watch :)
I agree with this. If pinocchio had no hats it would be vacuously true that none of pinocchio's hats were green, and from a mathematical standpoint he wouldn't be lying.
@sycips Is doing it the right way, negation over quantified propositions.
The statement on the actual quizz is "Todos os meus chapéus são verdes" which directly translates to "All my hats are green". This line can basically be translated word for word and work in both english and portuguese.
He may also have a hat that is green.
But I agree, before seeing the answer you expect "P has at least one hat which is not green". After then seeing answer (a), you still expect to find the more complete statement among (b)-(e), but it is not there.
Never studied logic, but that explanation makes a lot more sense to me than the concept of vacuous truth. My answer was, if he has any hats, at least one of them is not green, before the choices came up.
Before finishing the video which is the way to do these, I conclude that you can't trust anything Pinocchio says. There is no solution based upon what he says.
Could Pinocchio know future lotto numbers by going through each digit and watching his nose react?
E.g. "next week lotto numbers are as follows: first digit 0 (nose grows)". Repeat the process until his nose doesn't grow and repeat for each digit. Obviously Pinocchio would have to be very specific about which lotto game and date he's talking about, and he would have to phrase it as he's making a claim not guessing.
What do you think?
you can only lie about numbers from the past.
Also: If C were correct, that would automatically make E correct as well (No hats means also no green hats)
Since this is a test question with only one answer, an answer choice that makes another one true cannot be correct
Same goes with B and D-if he has one green hat he also has at LEAST one green hat, and therefore B cannot be the answer as this would also make D true.
No green hats may mean he has other hats. C) is specifically refuting his truth claim that he has any hats.
Yeah, I know. What I'm saying is that if he has no hats, he can't have green hats. This means that for C to be correct, E would have to be correct. We can't have two correct answers
My knee-jerk reaction was "None of the above". I eliminated B, D and E just like you did, but I also eliminated both A and C, thinking that the statement had no information about the number of hats. You have convinced me that we can indeed conclude that he has at least one hat. Well done!
well, C cant be true no matter what without even using the logic in the video. Imagine the case where Pinocchio has 1 blue hat. This would make his statement of "All my hats are green" a false statement, but it would also mean C is not forced. There can exist a situation where pinocchio's statement is false without C being true. Same way you proved it couldnt be B,D or E.
So the only possible answer that could be correct was A. It was either A or "none of the above". Now you still have to do the logic in the video to show A is indeed the correct choice, but you dont need that logic to prove C false.
On the assumption that we are talking about “mathematical lies” where a liar never tells vacuous truths. I think a real life liar would love to tell vacuous truths because they can also be interpreted as lies that you can’t disprove! :P
My reaction was "Pinnchio has at least one non-green hat". But then I went with answer A because C just felt wrong and B, D, E were eliminated because those are wrong.
Your knee-jerk reaction isn't necessarily wrong. Famously, there were decades of arguments around whether Russell's example, "The present King of France is bald" does or does not imply that there exists at present a King of France. At some point, the experts agreed to disagree (or, in other words, you can set your axioms one way or the other). The same goes for "All my hats are green". You can have a system where this implies "I have at least one hat", and another where it doesn't.
My first reaction was Pinocchio is colour blind lmao
If he always lies and says he has hats, then he can't have any hats.
As a reward for this nice video I will donate to the channel all of the camels, jets and castles that I have.
I find it helps to substitute the word “all” for “zero” when testing the statement against an empty set. E.g. “all of my hats are green” = “zero of my hats are green” when Pinocchio owns zero hats. The statement is technically correct (the best kind of correct!)
I think that changes the entire problem. “All” and “zero” are completely different statements.
@@santiagoa1155 Generally, yes. However, say N is the number of hats Pinocchio has and N = 0, then all of Pinocchio's hats (N) is equal to zero.
@@santiagoa1155
He’s not saying for the entire problem. Just in the case where it’s “against an empty set”. I.e., when all=zero anyway, like in answer choice C
That isn't the answer to the question, though, because the second true statement is about Pinocchio making a claim, a claim which is known to be false. If the statement of him having only green hats was not already known to be false, then sure, but it is false, that's the entire premise.
If you render his statement technically true, then you negate the first premise of the question, meaning you're answering an entirely different question.
@@Jane-oz7pp The statement says all his hats are green. From a logical standpoint that means that he has some or at least one hat. What you can conclude since he always lies is that not all of the hats are green.
Thanks for explaining the concept of a vacuously true statement. I tried to explain to myself why I found answer A to be correct, though I only selected answer A after you talked about mathematical falsehoods
My explanation would be that this situation can be represented by x^2 = g*x
Where x is the amount of hats pinocchio owns (x>=0) and g is the amount of hats he owns that are green (g 0, the statement is always false
Too bad it appears arbitrary
Except A makes Pinocchio's statement vacuous too. Pinocchio uses a plural, meaning a situation where he only has one hat "...at least one hat" it makes his statement vacuous, therefore true.
Actually its always false if g != x and x != 0. If x >= 0, and g
@@DiscoFang yeah agreed
@@DiscoFang actually no. When Pinocchio says 'all my hats are green' he is implying 'i have hats' AND 'all my hats are green'. This question is about mathematics logic. The correct part in the answer is that when you have P and Q and you negate both, you have a true answer, but if you negate only one of them, you have a false. What 'pinocchio always lies' means is that 'pinocchio's statements are false' and the only answer provided that makes it true is P and not Q
Unfortunatly Logic debunks most of the statement. Basicaly "A statement is Vacuously true if the premise is false or not satisfied" is in itself a BS statement and False by nature, as exemplified by the word Vacuously, which means empty, or that the truth itself is only ever true because the statement alone says it is, not because it actualy is. The given example ignores the understanding that the Phones being ON or OFF is areflection of a fact of the statement, aka the phones CANNOT be EITHER ON/OFF because NO phone IN the room is in the state of being ON/OFF, which checks a factual piece of information.
I think the worst aspect of this question is that whe can only "conclude" letter c because there is no option in which "there is at least one hat which isn't green".
This question should be rephrased. "What affirmative is one viable conclusion, between the following?"
What I find fascinating is that the question and Pinocchio's statement both imply things that are not actually stated, and those things are at odds with each other.
By giving the reader 5 choices as to what you can conclude, the question IMPLIES that at least one of those choices is correct-- that one of the options presented is something you can deduce just from the statements given.
However, Pinocchio's statement IMPLIES that he owns at least one hat. It may still be logically true if he owns no hats, but it would then carry a false implication, so from a linguistic angle, it could still be considered a "lie."
If we take the unstated implications of Pinocchio's statement into account-- that he owns at least one hat, and would be lying if he did not-- then the unstated implications of the question itself are broken. We cannot reach any of the given conclusions if owning no hats counts for making Pinocchio's statement a lie. Similarly, if we take the unstated implications of the question-- that at least one of the answers is correct-- then the unstated implications of Pinocchio's statement must be broken. For the question to make linguistic sense, one of the two implications MUST be false, and it's easy to see why people would argue over which one.
Now I'm imagining a version of Pinocchio where he misleads people by telling vacuously true statements.
"Somebody stole money from my purse. Pinocchio, did you see anyone steal from my purse?"
"Well, all the money Giorgio stole from you was in $100 bills."
"That can't be true, because I never have any $100 bills in my purse anyway. We're in Italy, we use Euros here."
@@yurenchu "Oh, my mistake. I mean €10 notes. I got the number of 0s and the type of currency wrong."
"So your nose doesn't grow when you accidentally tell a lie?"
"...That certainly would appear to be the case."
@@LimeGreenTeknii What would happen if Pinocchio makes a paradoxical self-referential statement? If he says "I'm lying" does his nose fall off?
This tickled me
@@gdclemo "This statement will make my nose grow longer." - Pinocchio the curse-breaker.
I expected you to take the symbolic logic route, but I felt you left out a key premise. The universally quantified statement “All my hats are green” is equivalent to the conditional statement “If I have a hat, then it is green.” This would more directly tie the second statement to the truth table.
But even more so, if the second statement is false, then its antecedent (“I have a hat”) must be true, and its consequent (“It is green”) must be false, making a stronger connection to the truth table ad a means for explaining the solution.
So, if he says “All of my hats are green” and it’s false, then it must be the case that he has a hat and it is not green.
that makes no sense, why do you assume that the antecedent must be true regardless? why do you assume the falsehood only applies to the quality of the hats rather than the existence of the hats?
Well, to be fair, he didn't need to since none of the answer choices included both conclusions. But yes, the negation of the universal statement is another way to approach this problem and you'll still arrive at the same answer
Edit: My bad. I didn't realize that you weren't really talking about the universal negation at all. But yeah, the video mainly talked about how C is a vacuously true statement (why C is incorrect). This way, people wouldn't be wondering why C doesn't work as well
@@jdavi6241 I'm not really trained in this field, but I feel that if you don't have a hat, you can't have a green one. So if you have no hats you have no green hats but if you have a hat then it could be green. You can't have the situation in which you having a green hat and not having any hat coincide
@@jdavi6241 The antecedent must be true to consider whether P -> Q is a false statement or not. If the antecedent is false, then just as the video explained, you have meaningless true statements since there will be no premises to consider. Hence, the antecedent has to be true in all false P -> Q statements
@@thesidecharacter6499 Why would the statements be "meaninglessly true" rather than false? If the antecedent is false then wouldnt it be the case that consequent is automatically false rather than automatically true? If I have no hats, then I have no green hats. So In that case, if I then say I have a green hat, it's not vacuously true, its just false since there is no hat to be green in the first place.
if P is false why is Q then automatically true rather than also inheriting the quality of being false?
You are working on the premise that factors in these statements are either true or false but there is no allowance for a statement being irrelevant (neither true or false). Pinocchio makes two statements, 1) that he has one or more hats and 2)they are green. Your chosen answer assumes that the first statement is true and only the second statement is false. I don't believe that there is enough information to allow us to choose any one of the answers.