Viral logic test from Brazil

[@carlosgaspar8447]

Thanks!

[@MonoInfinito]

@@nichijoufan Qué bueno ver hispanos interesados en lógica. Les recomiendo leer sobre Proposiciones categóricas para entender el problema. ^--^

[@oguzcan2335]

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"

[@MonoInfinito]

@@oguzcan2335 I know that u use your intuition But please study Cuantifies logical Propositions and stop comment ignorance.

[@oguzcan2335]

@@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

[@oguzcan2335]

@@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.

[@stevengordon3271]

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"!

[@PASHKULI]

Last one could have said "No" and it could be valid as well.

[@enzzz]

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.

[@enzzz]

@@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.

[@PASHKULI]

@@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...

[@stevengordon3271]

@@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".

[@Wumbo85]

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.

[@extremelynormalperson]

I knew it!!!

[@inrodu_1027]

poor pinnochio :(

[@Panurus_biarmicus]

Gepetto using him as a puppet is kinda dark in that case

[@t3st3d]

your right

[@extremelynormalperson]

@@t3st3d my right to be right

[@mccmcc33]

Everyone knows that Pinocchio has at least one hat. He wears it throughout the entire film.

[@Highley1958]

Congrats! You flunked logic.

[@SirAU]

@@Highley1958 yay

[@WellManNerd]

I wondered if it was a hint or a red herring but I just ignored it

[@goldenwarrior1186]

⁠@@Highley1958But they passed science. After all, they cited empirical evidence in support of their claim

[@MrDon4343]

That he wore a hat doesn't necessarily imply that hat is his. He may have borrowed it.

[@makeadifference4all]

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.

[@terencetwentyman-jones8861]

Ok...I'm lost...Trump says 'it's always a big crowd'. Trump lies (as we all know). But how do we establish if there was no one there or only one or a few supporters. Surely it's irrelevant. A lie simply means it's false, what part is false is inconsequential.

[@Havenkin]

It makes no intuitive sense to include nothing in all, what lead to them doing that?

[@cortster12]

If I said, "all my flowers are blue.", and I told you that was a lie, and I don't own any flowers, what then? Honestly, the logic puzzle, ignoring the picture, tells us nothing except that the statement is a lie.

[@B-fq7ff]

Well said. The saving grace of this problem to me is that because it's multiple choice, you can deduce which system of logic is being used. If the problem assumed existential import then the correct answer would be "Pinocchio either has no hats or has at least one non-green hat" which is not an option.

[@firstname4337]

you are so wrong -- please never post again

[@NestorAbad]

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!

[@MindYourDecisions]

That is a great way to explain it. I will mention the empty set next time, thanks!

[@moonshine3033]

Video publish 3 min ago but you made comment 4 days ago🤔

[@halogenzawgi9410]

Your professor statement is even more confusing,brother…

[@TheDelwish]

It's a bit strange that professor doesn't know about three-valued logic

[@manuelapollo7988]

So if you cannot falsify the statement, then it is true...now I understand the success of religions

[@ckEagle165]

I concluded that Pinocchio has at least one hat that isn't green.

[@notachair4757]

*Or* has no hats at all

[@brinecarroll]

@@notachair4757if he has no hats, all zero of his hats are green

[@anotheryoutuber2819]

​@@notachair4757that was literally proven false in the video

[@princetamrac1180]

Thank you. This is the only thing one can conclude from these statements.

[@Masaru-mm3em]

@@brinecarrollI would say (F) None of the above. If you can assume zero hats and then make a conclusion from it, then I’ll assume he could have half of a blue hat, which is false, which means that A is also false 😂

[@BigParadox]

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.

[@JLvatron]

Little did he know, you hid the truth that E=mc³

[@meetshah5003]

That's fking true statement.

[@BigParadox]

@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, ..."

[@bahulecticmethod509]

I overheard him say that to you once...

[@thatnike2604]

@@bahulecticmethod509 that's a lie

[@ericfortin559]

Those conclusions are all disprovable. You can only conclude that if Pinnochio has any hats, that at least one of them isn't green.

[@JoseCarlosVM]

Wrong

[@kyanilcauli9002]

If it's disprovable, do provide a counterexample.

[@fahad_hassan_92]

Your answer is the same as A.

[@ghizghuth2088]

@@fahad_hassan_92 No it's different.

[@Herculainn]

One or all are not green, if he has any at all.

[@zz8az]

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.

[@murraymadness4674]

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.

[@csarmii]

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.

[@annie.hi.]

This is what I thought

[@annie.hi.]

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

[@K9affirmative]

@@csarmii Pinnochio would still be lying if he had no hats

[@tomdekler9280]

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.

[@AshleyoftheAbyss]

@@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.

[@bierwolf8360]

@@frederiklist4265 the funniest one is the following: 25-5/5=4! (the joke being the faculty operator misunderstood as an exclamation mark)

[@baconboy486]

@@AshleyoftheAbyss 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.

[@Subjagator]

@@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.

[@sephi7ac]

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?"

[@spiderjump]

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.

[@thenonexistinghero]

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.

[@brianmacker1288]

Not an issue here since liar Pinocchio is always going to be in contempt of court.

[@brianmacker1288]

@@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.

[@thenonexistinghero]

@@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.

[@brianmacker1288]

@@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.

[@tc2241]

“All my clothes are clean” “You have no clothes…” “Yes, but if I did all of them would be clean” 😠

[@jacobhargiss3839]

A better way to think of it is this "of all the clothes I own, none of them are unclean". This means the same thing as "all my clothes are clean" but it makes it easier to see why having no clothes doesnt make the statement false.

[@olehbochko1922]

Yes I have. I never wear them, so they are clean

[@ejejej9200]

@@jacobhargiss3839 I would propose that it doesn’t mean the same thing and we shouldn’t adopt a convention that says it does. I propose we see certain statements as compound statements. “All my clothes are clean” is a compound statement that contains two statements. 1. I have clothes. 2. all of them are clean. If any of the above statements are false. The compound statement should be deemed false. I think this is a better convention for communication. The fact that some other convention has been adopted within a certain community doesn’t necessarily make it more right. “Of the clothes I have, none of them are unclean.” Is better because it softens the implication that the speaker has clothes. But, perhaps an even more precise way of speaking would be to say, “If I have clothes, they are clean.” Switching it to an if statement, makes things much more clearly communicated.

[@samuelrussell5760]

“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.

[@JuanRanklin]

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.

[@PitukaAJ]

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

[@dig8634]

@@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.

[@samuelrussell5760]

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.

[@HorseDogSnake]

@@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.

[@salt1muzak]

“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.

[@eduardoleonlotero]

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

[@runic6452]

@@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"

[@aethrya]

Quality academia right here

[@kiranrajkp]

@@eduardoleonlotero Does everyone know you can't even understand a joke? 🤭

[@mirageowl]

@@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)

[@steverempel8584]

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.

[@SVURulez]

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?

[@ianpeterdavison1963]

YES and No - Slide In Meaning...

[@ric6611]

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.

[@steverempel8584]

@@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.

[@ric6611]

@@steverempel8584 Oh yes, I thought you were referring to here in the video.

[@Hobo_X]

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.

[@FriendlyGreg10]

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.

[@IceMetalPunk]

​@@FriendlyGreg10As a programmer, I have a coding way to think of it: to check if a predicate is true for all elements of an array, you must loop over that array and return false if the predicate fails on an element. Then return true if you never returned false. But if the array is empty, the loop will not run, so it can never return false: regardless of the predicate, if the array is empty, the result is always a value of true.

[@eugene7518]

Pinocchio is telling the truth about owning hats.

[@vidarkristiansen8989]

@@IceMetalPunk As a programmer myself, I say let us not over complicate things. Let's just conclude "NOT all his hats are green" 😀

[@IceMetalPunk]

@@vidarkristiansen8989 !(A ^ B ^ C) = !A v !B v !C after all 😁 (Don't blame me for mixing logical and programming syntax; I have no "logical negation" key on my keyboard 😝)

[@garywhitfield317]

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."

[@TheJoyfulEye]

😆

[@Frankie5Angels150]

I’m not reading any more comments! You won!

[@David-qj1mr]

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

[@ItsJustValHere]

My first thought was "Pinoccio lied", then "oh wait" lmao

[@dontbefatuousjeffrey2494]

@@David-qj1mr exactly where my brain went too. And stopped 😀

[@imacds]

Questions like this make me appreciate mathematical notation. Much less ambiguity, much easier to solve/reason about.

[@MCLooyverse]

(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.

[@stewbaka4279]

truueee its very objective :)

[@RajeshPachaikani]

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.

[@imacds]

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.

[@LowestofheDead]

@@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.

[@in-betweendays]

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

[@somethingsomething2541]

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..

[@in-betweendays]

@@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.

[@somethingsomething2541]

@@in-betweendays yupp i agree with that

[@Jellyfishmustard]

there is no proof that pinnochio doesnt have 0 hats

[@xz-activity9473]

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.

[@Frontline_view_kaiser]

"You murdered your brother" "I don't have a brother" "So you can't prove that you didn't murder him"

[@jacobhargiss3839]

Not quite. In this example, "you murdered" means that there was a living person, who was killed. The subject "your brother" is stated as being that living person who was killed. So, if there was no brother, then there was no way he could be murdered as you cant kill someone who never existed. So not habing a brother would make this ststement false rather than true.

[@Frontline_view_kaiser]

@@jacobhargiss3839 That's literally the opposite of the logic in the riddle. Since the Brother does not exist, any statement about him is automatically vacuously true, as it is impossible to falsify. (Try proving that you didn't kill a brother you don't have)

[@jacobhargiss3839]

@@Frontline_view_kaiser its not the opposite of the logic in the riddle, it is applying the correct logic to the correct problem. The riddle in the video requires the use of set theory. In the commented example, we do not use set theory, we would use linguistic definitions. By definition, you cannot murder someone who does not exist, so the original statement is false. As well, the last statement has no bearing on the first statement because not being able to prove something false doesnt mean it is true. To simplify this, the riddle in the video is a boolean logic problem, we use the rules of boolean logic. This comment is NOT a boolean logic problem. Moreover, its not a problem at all, its just a false statement presented purposefully stupidly.

[@jacobhargiss3839]

@@Frontline_view_kaiser also, to answer your challeng here is your proof: 1) for a murder to occur, by definition a person must die. 2) for a person to die, by definition they must first be alive. 3) for a person to be alive, by definition they must be a part of reality and therefore exist. 4) therefore, if my brother does not exist, he cannot be killed. 5) because my brother cannot be killed, he therefore was not killed. 6) because my brother was not killed, he therefore was not murdered. There. Not only did I prove that I did not murder my brother, I also proved that nobody murdered my brother.

[@Frontline_view_kaiser]

The Riddle does not explicitly use boolean logic. It must be interpreted that way in order to make sense. The riddle states that Pinocchio LIES. A lie is not a boolean, a lie is a linguistic concept and does not equal a true/false statement. A vacuously true statement is still a lie and will be seen by anyone as a lie. If saying that my brother who does not exist is murdered is a lie, then so should the statement that hats, which do not exist, are green be a lie

[@Emily_Travels]

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).

[@carmensavu5122]

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.

[@viniciusoliveirafontes4033]

@@carmensavu5122 If "We can conclude that he has at least one non-green hat.", then A must be right.

[@scambammer6102]

@@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.

[@user-ll4cu5dh3b]

@@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.

[@PJSproductions97]

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

[@jackthatmonkey8994]

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

[@DiscoFang]

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.

[@TheSuperappelflap]

Actually its always false if g != x and x != 0. If x >= 0, and g

[@zekerdeath]

@@DiscoFang yeah agreed

[@ethyios]

@@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

[@windstar120025]

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.

[@Vgamer311]

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.

[@SpiralDownward]

Yeah. Had a smoothbrain moment when I thought "Well duh he has at least one hat, it's right there on the picture!"

[@cre8tvedge]

@@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.

[@AC8X]

@@cre8tvedge the hat in the picture is yellow lol

[@laycey]

I see you haven't done many logic puzzles.

[@MattExzy]

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.

[@vincentlemoinefr]

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.

[@vincentlemoinefr]

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

[@eugene7518]

Pinocchio is telling the truth about owning hats.

[@Krokodil986]

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

[@baraharonovich2926]

This is much more convincing then the explanation he gave.

[@xaelath7771]

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.

[@baraharonovich2926]

@@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)

[@Krokodil986]

@@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.

[@xaelath7771]

@@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.

[@JungleLibrary]

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.

[@BenRangel]

That conclusion is correct. He either has 0 hats, or he has some non-green hats

[@davidjorgensen877]

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).

[@RedShiftedDollar]

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.

[@JungleLibrary]

@@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"

[@JungleLibrary]

@@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.

[@astralminstrel]

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!

[@gailwaters814]

Pure logic says that all these options are possible. So, A-E are all possible. That's all we can "conclude from the statement".

[@floseatyard8063]

@@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

[@gailwaters814]

@@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".

[@floseatyard8063]

@@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.

[@gailwaters814]

@@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.

[@javierpda]

My answer at 1:05, for sure not all pinocchio's hats are green but some could be

[@kayskreed]

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.

[@MegaBanne]

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.

[@asusmctablet9180]

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.

[@MegaBanne]

@Repent and believe in Jesus Christ Lol

[@AuliaAF]

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.

[@AuliaAF]

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. . . . . . . That was an example of nonsensical language that is vacuously true :D

[@PR-ot7qd]

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.

[@PR-ot7qd]

@@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.

[@pedrotraposo]

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?

[@PR-ot7qd]

@@pedrotraposo all my ducks have a green neck. How many ducks do I have?

[@pedrotraposo]

@@PR-ot7qd I dont know. I dont get it.

[@PR-ot7qd]

@@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.

[@rcnhsuailsnyfiue2]

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!

[@ttp513]

wait, doesn't D also fit within this logic? Since not all his hats are green, at least one is green, no?

[@ProperGanderSaul]

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.

[@Bryan-Hensley]

He always lies, he may have no hats.

[@LuskasHusty]

@@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

[@Proxoa]

@@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.

[@janpapaj4373]

So me saying all my lamborghinis are green is not a lie?

[@michael_krueger]

If you do not own a single lamborghini, then not only is your statement vacuously true, but also the statement "All of your lamborghinis are green AND not green". It is logically valid, but also very misleading as we are making meaningless assertions about something that does not exist. Another way to phrase "All my lamborghinis are green" in set theory is "If there exists a lamborghini that I own, then it is green.", and that statement is only falsified by finding an element of that set (e.g. a lamborghini you own) that is not green.

[@janpapaj4373]

@@michael_krueger I will proceed to lie to business partners and financial institutions about my wealthy possessions, and in court I will show them this video

[@michael_krueger]

@@janpapaj4373 well I mean it's not like you're saying you have a lamborghini but by all means take the justice system for a ride 😂

[@janpapaj4373]

@@michael_krueger Skibidi toilet will be mine. Ohio town. Diamonds to mine. I'm on that sigma grind. 💯

[@michael_krueger]

@@janpapaj4373 Absolute poetry 🥲

[@zanderhort]

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

[@sWirus89]

No, it's just not an a=>b statement in natural language. But mathematicians argue it is

[@ricardopassos1180]

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

[@ricardopassos1180]

But I agree with you about the way the puzzle was presented

[@LordKeram]

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.

[@cadewatkin7086]

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.

[@GoPieman]

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.

[@petermello55]

All it says is he has no green hats, he could have a blue one, an orange one, it doesn’t specify.

[@GoPieman]

@@petermello55 my bad I forgot there was a real option E. I meant a sixth option

[@exigency2231]

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.

[@KryptikM3]

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.

[@Absynthexx1]

@@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.

[@medina2529]

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.

[@GuilhermeMichel]

Que legal! Eu somente passei 2 vezes da primeira fase haha.

[@GuilhermeMichel]

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?

[@pedroborges5323]

Siiim meuu

[@pedroborges5323]

Eu ganhei só uma mensais honrosa 🥲

[@vecernicek2]

All my medals are gold.

[@Dove325]

Plot twist: Pinocchio didn't lie, he's just red-green color blind

[@Karlyr_]

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.

[@immikeurnot]

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.

[@Avacavado]

@@immikeurnot No no, that's what they meant. Like you said, whether Pinocchio has one hat or multiple, at least one isn't green.

[@angel-ig]

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

[@immikeurnot]

@@Avacavado If that's what they meant, why are all the answers wrong?

[@Avacavado]

@@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

[@RyeedAglan]

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'

[@MichaelRothwell1]

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.

[@petethewrist]

I like P always lie. Now I will tell you all my motor bikes are big... Infact I have no motor bikes. ?????

[@MichaelRothwell1]

@@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.

[@petethewrist]

@@MichaelRothwell1 none of it a lie? No it was a fabrication which is may be what P was doing.

[@pulsar22]

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.

[@Neescherful]

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).

[@camembertdalembert6323]

this is what I did.

[@sonicmaths8285]

had the exact same thought.

[@classiclover2129]

Same

[@rytas]

Same thought process here. Nicely done.

[@georgeb8893]

Yes: For all X, Hat(X) implies Green(X). Negation: There exists X st Hat(X) and Not Green(X).

[@williamverhoef4349]

There is explicit lying and implicit lying and the question does not distinguish between the two, therefore both A and C are possible answers. When Pinocchio says, "all my hats are green", if he also lies implicitly then the implication that he owns some hats would also be a lie and option C would be correct. If he only lies explicitly, then option A would be correct.

[@Petronio39]

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.

[@Helbore]

Where does it say that is a picture of Pinocchio? ;)

[@Helbore]

@@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. ;)

[@myusernameisthisduh]

@@kendraroth1276 But did the question text talk about a picture at all? No. So the picture is not a part of the problem.

[@lunaramoonchild601]

@ 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😉😉

[@onyxr8957]

It's A because if you don't own any hats, every hat you own could be green.

[@SpinyBadger]

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.

[@dunnedigby4957]

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?

[@anannoyingweeb359]

​@@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.

[@dunnedigby4957]

@@anannoyingweeb359 Thank you for that, non-annoying weeb! Just the wikipedia page about the book was very helpful actually. Best explanation I could have, I reckon.

[@anannoyingweeb359]

@@dunnedigby4957 glad to helped you

[@eugene7518]

Pinocchio told the TRUTH about owning hats. The solution is predicated on him owning hats. So Pinocchio doesn't always lie.

[@sycips]

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 :)

[@flamingfurball3316]

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.

[@misterguts]

@sycips Is doing it the right way, negation over quantified propositions.

[@ggwp638BC]

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.

[@ronald3836]

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.

[@shaunswett6684]

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.

[@aesir5917]

Nonsense, if i say i kill all my siblings or all my siblings are dead, but i don't have any siblings in the first place, that doesn't mean "all my siblings are dead" is not a lie. It's still a lie.

[@PieInTheSky9]

It's the truth because you have no siblings, so when you say "all of my siblings are dead" you are essentially saying "zero of my siblings are dead" because in this scenario the word "all" means "zero" .

[@aesir5917]

@PieInTheSky9 tell that to judge or police and see how it goes, "all" imply he at least have one, and try to misleading someone to believe that. It doesn't work on any real-life situation. If you say something like this in real life, it will be 100% considered lying. "The best way to lie is to tell the truth."

[@PieInTheSky9]

@aesir5917 I don't disagree with you, in language we have built in assumptions, but they are just that, assumptions. If you say all your siblings are dead I will assume you had siblings, but that doesn't mean you are lying. You could say you are misleading people by saying something that you know people will make assumptions about, but it's technically not lying.

[@aesir5917]

@PieInTheSky9 I don't disagree with you within the context of this logic test either. I just saying it doesn't work that way in real life. like I say, it's technically telling truth yes, but if you tell the police you kill all your siblings but you have no siblings, even you technically telling the truth, but people will still considered this as a lie. I think you can telling the truth and lying at the same time. For example if a doctor tell you that you are dying when you are perfectly heathly. technically he is telling the truth since everyone is dying slowly. But would you say he is not lying to patient?

[@drnanard9605]

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.

[@sman000]

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 ?

[@drnanard9605]

@@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.

[@sman000]

He’s saying “all his hats”. That indicates something is there that he is referring to, at least a hat.

[@drnanard9605]

@@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.

[@ClarkPotter]

@@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.

[@igorsoares7767]

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.

[@lucasrinaldi9909]

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.

[@lucasrinaldi9909]

Erro meu, não é o Saul Kripke. É o Quine que defendia isso.

[@willianditaquera]

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.

[@EL1J4H640]

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.

[@cwlim62]

This vid is logically WRONG. None of the options can be deemed correct.

[@Lescar-B-EAU]

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.”

[@DocBree13]

Great quote from a great book

[@zzztek]

@@DocBree13 Great book, horrible movie

[@endersparshott]

Ain't that the truth. I for one should know

[@Crackpot_Astronaut]

@@zzztek ... Movie?! Oh no.. I didn't know there was such a thing.

[@ProbablyEzra]

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.

[@kurtwpg]

You can only conclude that if he owns more than zero hats, then at least one is non-green.

[@kathvolk]

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 :(

[@softan]

the opposite of all is none.

[@mento6]

@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!

[@nathanmartin5049]

@@softan The opposite of all is not all.

[@tatri292]

@@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.

[@DeadlyBlaze]

​@@softan P: All my hats are green ~P: At least one of my hats are not green

[@tjthepiguy2868]

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.

[@xTheITx]

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.

[@spacecheetah1283]

​@@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.

[@TobbyTukaywan]

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.

[@MrVictorugalde]

Thank you. I think you actually explained better then the video.

[@Smitology]

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.

[@pedroribeiro1536]

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

[@pedroloures3310]

It was From Obmep haha

[@vitormalfa757]

Que honra mesmo

[@pedroribeiro1536]

@Paulo Henrique nós BRs estamos em todos os lugares hehehehe

[@helloiamenergyman]

Eu fiz e acertei, e estou indo pra 2a fase (:

[@hariotmarriot9347]

even u an undergrad, that doesnt mean u could ace this test

[@alex92569]

What an impeccable logic! I am deeply impressed. Next time when my landlord asks me if i have money to pay for my rent I will tell him - "Yes sir, I have a million dollars on my Swiss bank account" And that will be a true statement because I do not have any Swiss bank account and therefore whatever I say about it is true. Simply amazing how far you can go when you are strong in math...

[@gamefacierglitches]

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

[@Grassmpl]

Why is D) specific amount?

[@dumbwaki5877]

@@Grassmpl 0 is a specific amount!

[@Grassmpl]

@@dumbwaki5877 but D) is "at least one"

[@gamefacierglitches]

@@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."

[@neonch1]

lol this is amazing

[@Casinizucchini]

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.

[@Melimex]

Yes,I thought that way

[@MrBrainTucker1079]

Same. It makes sense. It's a matter of argumentation at this point as some people in the comments have pointed out.

[@widehotep9257]

I absolutely agree, which is why I picked C. And I would pick C again.

[@michaeledwards2251]

From the text I considered that to be an option but I assumed the picture of Pinnochio with a hat was not a lie.

[@Shyguy5104]

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

[@babuddha]

What I recall from my one logic class: I got a C on the first test -- when my attention was most focused; I got a B on the second test -- when my attention in class was beginning to waiver, and I got an A on the final test -- during an East Tennessee spring when I basically stopped attending class.

[@Felipe_12345_]

After watching this video, I concluded that all my private planes are blue.

[@TheJJO]

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.

[@TornaitSuperBird]

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.

[@Polarcupcheck]

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?

[@天天-l9q]

Me too, but I get it after the video point out that you don't need a thing to say 'all my... are...'

[@calebfuller4713]

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.

[@calebfuller4713]

@@Polarcupcheck Apparently, according to "Mathematical Logic" you now own a Mercedes. Better go check your garage!

[@diamondmemer9754]

By that logic, saying my house has three floors is a true statement as long as I don't have a house

[@resresres1]

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.

[@AlineDreams]

​@@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.

[@resresres1]

@@AlineDreams then they shouldn't be asking the question in the form of a real life scenario because they'll only confuse people.

[@ajayray4408]

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.

[@resresres1]

@@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.

[@o0Black0o]

I think the key is that this is all only correct from a strictly mathmatical/logic point of view. From a language point of view forcing an assumption as part of the framework of a statement that is not true is almost universally considered a lie socially. Making a statement about the hats you own when you do not own hats is considered an untrue statement. As an example if someone sold "all the hats they own" to someone with the line "all the hats in my collection are extremely valuable and rare", we would consider a lie if there actually were no hats at all, dispite being voraciously true for most peoples understanding of the word it is a lie.

[@thenoobalmighty8790]

I agree. Imo the answer is C. Pinocchio implies he has 1 or more hats, and that they are all green. Therefore as he told us he had some number of hats, he must have no hats. As soon as you say all the mobile phones in the room... you have implied that there are some in the room. By the other logic, if someone like your teacher asks you if your phone is switched off, you can say no. They then ask you to turn it off, and you say it is off. Then they say you said it wasn't off, and you say it isn't off (you don't have a phone). Then they say "is it on or off?" And you say "yes". Then your teacher beats you HAHA LMAO 😂

[@CallumBradbury]

@@thenoobalmighty8790 an implication isn't a statement of truth, though. Just because something is implied, it doesn't mean it's being stated as truth.

[@thenoobalmighty8790]

@@CallumBradbury WELL IF THERE ARE NO PHONES IN A ROOM THE STATEMENT THAT THEY ARE ALL OFF IS FALSE AS THERE ARE NONE THERE. OR AT LEAST IT IS AS TRUE AS IT IS FALSE. FOR THAT TO BE TRUE, I WOULD SAY THERE MUST BE AT LEAST ONE PHONE IN THE ROOM AND ALL PHONES IN THE ROOM ARE OFF. IF I ASKED YOU IF ALL YOUR MEALS YESTERDAY WERE TASTY, YOU COULD NOT SAY YES IF YOU ATE NOTHING

[@thenoobalmighty8790]

The basic premise is that what people say is true. If i say all of my hats are... this is true only if i have hats. You are stating that you have hats. Its the same as i have hats and they are all green

[@thenoobalmighty8790]

Hence Pinocchio has no hats

[@ecavero1]

This is one of the few questions on this channel I actually got right after pausing it and giving it a try for a few seconds!

[@jim55price]

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.

[@richardgomez3469]

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.

[@jim55price]

@@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.

[@themediaangel7413]

That’s probably the best explanation so far.

[@jim55price]

@@themediaangel7413 Thank you. I tries. :)

[@maalikserebryakov]

Ohhhhhh that makes sense

[@yoremusic6920]

As a Brazilian, I simply used to hate Math Olympics as a kid. Oh, my goodness! It was a long boring test with tricky questions about things we, sometimes, didn't even learn in School (public and private schools' education quality is totaly uneven here). I remember kids scoring 12/30 being seen as geniuses. I was 10 or 11 by that time. Tests were the same for 10 and 12 years old kids. If we scored enough to go to the second stage (that is, até least 8/30. I scored 9/30), the test would be applied in another school downtown. For me, It only meant traveling traveling 1 hour or more to get downtown (I used to live 30km away from it. At least we didn't have to pay for the bus), only to spend 1 hour more doing absolutely nothing, just waiting for the test to be given to us.

[@JuliieRosees]

same here. hate the fact they were mandatory for all students, regardless of willingness to/interest in participating and aptitude!! as a kid who knew had no decent skills in math beyond basic knowledge, the test was always a blow on my self esteem!

[@abrunosON]

The content of what students receive is completely centralized in Brazil (like healthcare regulations) so the biggest difference is the quality and maintenance of the physical place. I studied in two public schools and two private schools intermittently, I also participated in several extra curricular activities directly or helped train the teams in various modalities. In knowledge competitions there is official material to study from that was available for free to all registered teams (public school teachers would snatch those for their own children, both public schools did the same thing) while the private schools would make somewhat low quality copies and distribute them to anyone interested. When it comes to physical competitions state and federal schools have access to top of the line installations and all it take is a call from the principal to arrange the logistics and scheduling (this rarely happens because public servants, like the principals, never want to work so they don't care a bit about it) while private schools rarely have access to those since they have to pay exorbitant amounts. Overall, the only fundamental problem in Brazil's education is method that is marxist in nature based on Paulo Freire's method which inspired USA's Common Core directly.

[@pluto_5109]

Hey, could anyone help me with this: If a person says, "I'm lying" is he telling the truth or is he lying?

[@rubyperry8930]

@@pluto_5109 I would believe that is the truth, since admitting to lying would imply everything from before that statement was a lie. Therefore making his admission true.

[@natewatson6962]

we do a disservice to kids with stuff like this.

[@jadencasto]

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.

[@AfroGaz71]

It's very much sounds like a politicians go to lying technique.

[@bobon123]

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.

[@colbyboucher6391]

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

[@gmeaki02]

"I have no non-green hats"

[@supern00bis4]

@@colbyboucher6391 sorry you didnt get it right bud, dont worry I thought it was C too

[@gnrashu5735]

Shoutout to Schrodinger's mobile phones for helping understand this puzzle.

[@ceebeeox]

Switching between "All" (or "For all") / ∀ and "There exists" / ∃ on negation has helped me a lot with these -- If some statement is (∀x, P), the negation will be (∃x: ~P), or vice versa. So the negation of "All my hats are green" that would make it a lie is "At least one of my hats is not green", or "There exists one hat that I own that is not green". We then know that he owns at least one hat that is not green. The multiple choice makes this harder, as it forces people to choose between an incomplete answer and some intuitive but wrong ones -- I wonder how people would react if the full answer were put in the options!

[@tarikeld11]

Thank you, exactly my thought!!

[@lrizzard]

this was exactly how i got to a conclusion 😄

[@BlueLightningSky]

I could be wrong but when books teach you to negate that statement, it comes with a caveat that the set is non empty.

[@alvaros.]

That's the way my math teacher taught this to us. He used the example of the empty set: All elements in the empty set are blue - true, because there is no element that is not blue All elements in the empty set are green - also true, because there is no element that is not green And so on 🙂 Then, at university, on the Logic course, we learned the semantics of "==>" with the truth table as shown in the video.

[@paulkothgasser6623]

Hey, how do you type these Quantors online?

[@offsetsammy]

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!)

[@santiagoa1155]

I think that changes the entire problem. “All” and “zero” are completely different statements.

[@MurdocK-BR]

@@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.

[@littlefishbigmountain]

@@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

[@Jane-oz7pp]

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.

[@patrickwall8517]

@@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.

[@isabelbelem9062]

I hate math with every ounce in my body, but mad props to who works with this thing

[@Evilatem]

"my" hats? They are not his hats. He stole them.

[@1024det]

I’ve been a profession software engineer for over 35 years. I wrote in a dozen languages over the years and this problem has a fundamental flaw that nobody is seeing. We are trying to convert english into logic, which can be converted however sometimes there are multiple possibilities for interpretation. This is why there is no computer language that is pure english, it would just suck. So because of the multiple interpretations you cannot conclude anything except if the picture was included in the puzzle, then you can only conclude A.

[@bambulkomccloud3983]

Fully agree. The sentence 'All my hats are green.' implies that I have hats, because otherwise this sentence is just nonsense. And nonsense can never be true or false. It's like the statement 'At night it's colder than outside.' It is neither true nor false. So the correct answer would be F) Nothing.

[@MortenBendiksen]

You'd have to interpret the picture as well. It might not be his hat. But I feel from a puzzle context one can always assume all means a potentially empty set, which is all you need, along with regular grammar, to conclude that only A is possible.

[@empathogen75]

This problem is just taken out of context. In the context of a mathematical logic course there is one obvious and correct answer, which is the answer in the video. Out of context, there’s no right answer. Most people don’t reason in formal logic, and there’s a perfect valid argument for either answer in a real world context. In particular, in ordinary conversations “all of my hats” includes the implication that you have hats - and probably implies that you have more than one even.

[@jan.kowalski]

Exactly. People here mostly violate the first rule, assuming that he not always lies in effect. The sentence should be interpreted as "Not all not my not hats aren't not green".

[@prim16]

Alternative title: Solve this viral test question, or you're going to Brazil

[@xiaoshen194]

Then I would like to skip this question 😍

[@garrysekelli6776]

Dude of all fates. Brazil is the worst. But they...

[@peemaponchonburian]

i wanna double jump

[@emnicodemos]

I think both alternatives are better than staying where you are

[@PlanesAndGames732]

Jokes on you, I'm a Brazilian

[@therealEmpyre]

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!

[@eragon78]

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.

[@diggoran]

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

[@Zulk_RS]

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.

[@whycantiremainanonymous8091]

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.

[@lordloss4584]

My first reaction was Pinocchio is colour blind lmao

[@pikalize]

"What an interesting logic puzzle!" This is the first time in my life that I am THIS annoyed at having found the correct answer. I HATE having discovered vacuous truth, WHO in the WORLD invented that??

[@Cactusssutcac]

I'll add that here on Brazil most mathematics Olympics and even ENEM can have a few questions with 2 correct answers, not equally correct but they can see how you arrived at that conclusion and it would be considered logical but not entirely correct. I participated in a few mathematics Olympics when I was 11-12y'o and the statistics questions usually had 1 correct and 1 logical, I heard at the time they did that as to encourage the good thinking but still taking points for not being the most accurate.

[@edshort1138]

Mostly this word puzzle depends on the Vagueness of the English sentence - "All my hats are green". From my memory of Bertrand Russell's 1905 essay titled - "On Denoting" - Russell would agree me with. Russell argued, successfully in my opinion, that the phrase "All my hats..." denotes that there is at least one hat. Now this puzzle comes along and asserts that Russell is wrong. At the very least, two groups of people who have studied this "All my hats..." phrase or similar phrases have ended on incompatible conclusions - which is direct evidence that the phrase "All my hats..." is at least somewhat vague. English certainly provides the mechanisms to clarify which plausible meaning is intended by "All my hats...", such as "I have no hats, but all the hats I have are green", but such clarity would render the puzzle trivial and trite. Vagueness does not an interesting puzzle make.

[@taliyeth]

I think it falls on people's perception of the act of lying. The statement itself is very clear cut, but introducing the concept of lying throws people off. I think people are trying to approach this puzzle as a trick question: "I never said I had any hats in the first place". The creator of the puzzle probably agrees with Russell as well, that the statement requires there to be hats in the first place.

[@ethandavis7310]

@@taliyeth The puzzle creator disagrees.

[@ethandavis7310]

I just read the essay and I think the only place where Russell and the puzzle creator disagree is on the actual value assigned to the proposition containing the denoting phrase "all my hats." Russell states "...We must abandon the view that the denotation is what is concerned in propositions which contain denoting phrases." Russell advocates that in the case that pinocchio has no hats, the denoting phrase as said by pinocchio "all my hats" refers to something that does not exist and thus any proposition for which "all my hats" is the primary occurance are automatically false. This is the same reasoning as conventional logic just with the truth value flipped. Russell doesn't argue that "all my hats" implies that there are hats, just that if there are no hats, then any statement of truth regarding "all my hats" is false.

[@theother1281]

The 'all my hats' could be a lie and the colour statement is vacuous. For Russell's implication to apply that part of the statement would have to be true, which can can't show as sound.

[@gepeto032]

Disclaimer: I am no logician - just curious. A (much) earlier comment by Neescherful, that “… a logical statement is false if and only if the negation is true.” appears to suggest a sufficiently sound approach to finding a solution. It is, however, interesting to understand a key point made in the video, which seems to underpin the conclusion: “A statement is vacuously true if the premise is false or not satisfied.” This claim is possibly taken from formal logic theories. Nevertheless, it would be instructive to know why the opposite would not be valid - “A statement is vacuously false if the premise is false or not satisfied.” Furthermore, it is also curious to consider both the main Pinocchio claim and that from the ‘mobile phones’ example consisting of two separate statements. (a1) “All my hats are …” and (a2) “All my hats are green.” (b1) “All mobile phones in the room are …” (b2) “All mobile phones in the room are turned on [off].” The first parts of each statement ’all my hats are’ and ‘all mobile phones in the room are’ semantically imply ‘I have hats’ and ‘there are phones in the room’. Implied statements are statements nevertheless. They unavoidably affect the meaning of any conjugated statement. Assuming the implication of a combined ‘double statement’, before even considering what is said about the hats (or the phones in the room) the above reasoning suggests that claiming there are hats/phones when there aren’t any is false. Equally, in relation to comments referring to sets with zero elements - the statement ‘I have zero hats’ is equivalent to ‘I do not have hats’, which seems to be logically inconsistent with positive statements including “… my hats are …” (all or some for that matter).

[@lizyz1085]

I like this logic

[@sweetbobbybliss]

Thank you!!!!!

[@nomad_24152]

I was curious about this too, and from googling around it seems that the implication that there are hats (or phones) is not equal to stating that there are hats (or phones). If I understood correctly, the implication is dealt with in the logic by the axiom: p is true if and only if not-p is false. This means that for the statement “All mobile phones in the room are turned on" to be false, you would have to show that there exists a mobile phone in the room which is not working... which you cant do.

[@LumocolorARTnr1319]

"Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading"

[@bobbyhillthuglife]

one could argue epistemologically that false-premise statements are actually neither true nor false, but simply nonsensical statements, equivalent to a non-statement. This would still lead to the same answer, as making nonsensical statements would violate the premise of “always lie”

[@321Jarn]

Bold to assume the hat(s) was actually his, and he didn't steal it.

[@fantasticmisterpig]

What I find interesting is that I had a very instinctive sense of what you explained without being able to mathematically express the idea of vacuously true statements. I know Option C was wrong, but couldn't quite express it.

[@nandanapalchowdhury4588]

Ikr. Same

[@skyguy1236]

same. I hate math but this question was easy for me, but more in a grammatical way than a math way

[@Artcore103]

that only shows all 3 of you have broken logic :( he has no hats. there are 3 aspects of his claim: "all" "hats" "green". the maximal lie would be for all 3 constituent parts of his claim to be false. C makes his statement MORE false than A, so it's the better answer in the traditional sense. only in this unique framework assumed by the context of this particular test can A be the best answer, and this is arbitrary and must be known to the test taker, as it is based on non standard or less idealized versions of words (lie).

[@wickedAberration]

@@Artcore103 I don't agree with you wholly, as C is most certainly not MORE false, but I think describing it as "vacuously true" is... Odd. Its moreover like checking an unassigned boolean variable for truth value: It doesn't have one. So it is, IMO, neither a lie nor the truth. You can't compare the value of a nonexistent variable.

[@Artcore103]

@@wickedAberration right... but Pinocchio is claiming such a value exists in the first place, the fact that you can't check it is the lie. All your base are belong to us.

[@potato_cm]

Vacuous statements definitely seem weird to me. You can have two contradictory statements both be true at the same time if they are about nothing. It seems like you can have vacuous statements about inherently contradictory objects such as square circles that are by this argument always true.

[@chrisrudolf9839]

Yes, you can in fact. This only seems weird to you because you think in terms of objects that are already defined and would therefore be contradictory. It is easier to understand if you imagine the defining statement like putting a label on a container. We don't know how many objects are in that container, we just state all objects in there must have a certain quality. Like this container is only for cubes. Or this container is only for balls. Or this container is only for objects that are both cubes and balls at the same time. Of course objects that are both cubes and balls at the same time don't exist, so we know the container that is only for objects that are both cubes and balls will always remain empty. But as long as the amount of objects in the container is zero, we can say anything about those nonexisting objects and it wouldn't be false. Or, to phrase it according to the container allegory: As long as the container with the contradicting descriptors remains empty, we don't have to dress down anyone from the sorting staff for putting stuff into that container that doesn't belong there.

[@dicktracy3787]

@@chrisrudolf9839 this shows that the logic of vacuous statements is BS. Because it makes a 2 contradictory statements appear to be a true statement- = which is the very definition of an error in logic, This whows= these people do not understand logic

[@censurado_]

As a brazilian I say, this question broke my mind in the test (This test is to the whole contry) 😥

[@Eli_00001]

This is not a logic problem this is a communication problem

[@aroundandround]

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.

[@soundsoflife9549]

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.

[@yousauce7451]

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

[@aroundandround]

@@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.

[@yousauce7451]

@@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).

[@eugene7518]

Pinocchio is telling the truth about owning hats.. n

[@lanceloden]

Pinocchio: "There is one correct answer." Pinocchio: "It is assumed to use vacuous logic"

[@crashoverwrite5196]

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!

[@emriys1334]

@@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.

[@olivermatthews8110]

@@crashoverwrite5196 logic is literally a branch of discrete mathematics.

[@crashoverwrite5196]

@@olivermatthews8110 Sure but not the full range of the physical world. Mathematical logic isnt always useable for our world.

[@crashoverwrite5196]

​@@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.

[@darksim0]

I'm no logician, so I kind of "mathed" my way to the answer. For "All of my hats are green," to be false, Total Hats ≠ Green Hats. If Total Hats = 0, then also Green Hats = 0, making Total Hats = Green Hats, which ruled out answer C for me.

[@immikeurnot]

"If Total Hats = 0, then also Green Hats = 0" Are you unfamiliar with how lying works? Because it sounds like you're unfamiliar with how lying works. "I have a million dollars" said the man who was a thousand dollars in the red. By your logic, that guy has money. He doesn't. One doesn't have to have a single hat to claim that all their hats are green.

[@Amoebatirith]

@@immikeurnot But this puzzle is from a mathematical standpoint (mathematic olympiad question) so you must use lie=mathematically false statement and not lie=misleading statement. The context matters for the question. This logic problem is only ambiguous if you aren't thinking about it from mathematic viewpoint. Darksim0's explanaition is a nice basic way to answer the problem. If Pinocchio has no hats, then 0 hats=0 green hats is a mathematically true statement so Pinocchio would not be mathematically lying which invalidates (C) as the correct answer. Pinocchio is lying by a misleading statement, but that is not the context of the question.

[@BenRangel]

Same here. This phrasing of counting was the key for me too. Two empty sets are equal. At first I really struggled to accept the idea that you're allowed to describe properties of items that don't exist without it being called a lie. Really had to do some mental gymnastics to rephrase it as "He's not describing his non existant hats, he's saying that the amount of his green hats is zero, which is equal to all his hats"

[@KittyKatalina]

​@@immikeurnot "I have X" is not analagous to "All my X-es are Y". "I have an ace in my hand" is clearly a lie if they don't have any cards. You can say "No you don't, you don't even have a card!" and immediately debunk their claim. "All the cards in my hand are aces" is only a lie if they have at least one card. If they don't have any cards, it's just an empty statement, not a lie. If you respond with "No you don't, you don't even have a card!" as above, they can simply retort with "Precisely!". Their statement is technically true (or to use the terminology in the video, vacuously true), because all zero cards in their hand are aces.

[@war32567ify]

@@immikeurnot That is not at all what was said though. with your phrase you go from "All of my Lamborghini's are yellow" to "I have yellow Lamborghini's." These are not at all the same statement.

[@berndbeispielmensch]

The green hat actually belongs to Pinocchios father, Luigi. Luigi and Peache gave their son Pinocchio to Gepetto out of fear. They know what Mario does to all those innocent Koopas and Goombas just for a couple of coins. Mario gets even more furious when he is on mushrooms. Not even Bowser can protect Peach from Marios tantrums. Unfortunately, that also lead to Pinocchio with such low self-esteem that he feels the need to lie just about everything.

[@chrischappa962]

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.

[@jdavi6241]

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?

[@thesidecharacter6499]

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

[@icthiolavarunt5363]

@@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

[@thesidecharacter6499]

@@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

[@jdavi6241]

@@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?

[@wittyjoker4631]

The problem with this kind of question is words have to be given new definitions.

[@fernandaabreu5625]

Exactly. This is almost diabolical.

[@dustking3569]

He always lies He claims to own hats = lie He claims the hats he owns are all green= lie Only logical conclusion is C. **Edit. I know I got the question wrong . I operationalized "always lies" incorrectly. I overlooked where he stated 'mathematical lie'

[@feelsdankman211]

​@@dustking3569 Yes, because watching Destiny gives you more say over mathematicians in logic puzzles.

[@dustking3569]

@@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

[@PeerAdder]

@@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.

[@Daniel-Strain]

This question is an excellent example of how and why IQ tests can fail to capture intelligence, as opposed to knowledge. While it is correct to call this a 'logical test', it includes particulars about logical rules that could not be arrived at through logical ability alone. Reason being, when we say "All phones in the room are turned off/on" are both "vacuous truths" - this is a cultural convention. It is a particular way that the Western world chose to deal with how to handle statements about non-existent entities. In an alternate history, or an alien civilization, it is completely possible for logicians to have chosen to categorize all statements about non-existent entities as false instead of true, or as "nonsensical and not true or false". The logical system would work just as well in any of these alternate cultures. So, when you ask this question of a person, what you are testing is not the logical performance of their reasoning capacity, but whether or not they have had the opportunity to encounter the conventional idiosyncrasies of our formal logical charts and categories. In other words, you are going to see a skewing in favor of the formally educated and privileged that distorts genuine measure of logical ability.

[@AbyssalManta]

Exactly, very well put.

[@juulcy]

Yes. Well said.

[@4570duplex]

You seem to be the only one here who understands the true nature of this "logical" test. The very simple, direct, most "logical" conclusion is that we can draw no conclusions from those two statements, that the most "correct" answer was not given as a selection. This notion of "vacuous truth" is one of the worst examples of modern day "intellectual masturbation" extant, a fantastic example of a far too inbred academia desperately seeking ways in which they can appear clever. So, to go one step further, what is really being tested here is one's eagerness to go along with such contrived "logic" in an effort to gain acceptance in those circles that promote this kind of claptrap.

[@hemalpathak6762]

There might be something to that, but not quite. The "vacuous" case has to be true because making it false would contradict biconditional claims. You can find a video for that, but essentially the biconditional requires FF and TT to be true which would be impossible if the vacuous cases were false. The aliens would have a logic system that fell apart pretty quickly :). In terms of IQ tests, it's unlikely someone would derive that relationship or proof on a timed test so yes it does involve having learned the concept in advance.

[@4570duplex]

@@hemalpathak6762 Mr. Pinocchio, and men like him, simply do not understand, nor play by those rules. Vacuous truths have no place in their world. "All my hats are green" is a simple lie, as is everything he says. We can draw no conclusions from that. If he owns no hats, his statement becomes an element of our "vacuous truth" (not his), in theory, but in the real world in which we live, it remains a lie and we still have no idea whether he even owns a hat. Even the hat on his head as he tells us "all my hats are green" may not be his. Would you bet your life on the conclusion that this man has a least one hat? I believe that would be rather foolish...

[@DevanshSinghGautam]

Well the question itself is paradoxical First we assume that both sentences are true The first sentence tells that Pinocchio always lies The second sentence tells a dialogue of Pinocchio From the first sentence, this must be false. However, it is explicitly stated in the question that both sentences are true. Therefore, none of this makes any sense

[@HalasterBlackmantle]

This is a real conundrum. Language is highly interpretable. You could easily see Pinocchio's sentence as P & (P-->F), and suddenly his sentence becomes a lie as soon as he has no hats. I think P-->F is also a valid interpretation, but not the only one.

[@CrossRoadsOfTime]

I would agree, as the statements are written it wouldn't be out of bounds to assume that if he has no hats then he can't have a green hat. Thus him saying he has green hats is a lie and thus is not invalid option.

[@Moleoflands]

@@CrossRoadsOfTime he's not saying he has green hats, he is saying all the hats he has are green. There is a difference. "I have green hats' means that any hats he has are green, and he has at least one. "All the hats I have are green" can include any number, including 0. 0 out 0 hats is 100%

[@YouWillBeHappyOrElse]

@@Moleoflands But then again, the statement "I have green hats" can mean he has no hats, because the operative lie can be either "hat exists while green" OR "hat exists".

[@CrossRoadsOfTime]

@@Moleoflands while your right that he doesn't have any hats that invalidate that statement. But color is derived by how the item absorbs or reflects the light. Since there is no object to do the reflecting (as there are no hats) this lack of hats can not be a color at all. So it's still a lie to say they are green.

[@maevekirkland9452]

@@Moleoflands 0 out of 0 is actually undefined

[@TwoForFlinchin1]

When he says "my hats" he's claiming that he has possession of a hat. So if he has zero hats, then it is a lie that he has any hats. This question relies on viewing color as the only way in which he could lie when there are two statements being made. If you're a normal person, he's lying and you have to say that he has at least one non-green hat.

[@JackoaTrades]

Had the right answer, but couldn't explain why. Love your vids!!!

[@cr0n0us_4]

Basically, if you claim that he does not have all green hats because he doesn't have any hat, I could claim that he has all green hats because he doesn't have any hat and you wouldn't be able to prove it wrong. I mean, if there are no hats at all, you cannot claim its color to be not green the same way you cannot conclude it to be green, both are true.

[@Linck192]

I think form a real life common-sense point of view, if you hear someone saying "all my X" and that person possesses no X, they would be lying. From that point of view, Pinocchio would be stating two claims when he says "All my hats are green": that he has hats, and that all of them are green. Any one of the two claims being a lie, means the whole sentence is a lie. Therefore if he had zero hats, he would still be lying, and none of the answers would be correct

[@DeathnoteBB]

Exactly!

[@masvindu]

You are correct. The key here is that in standard formal logic, the standard implication is a material implication, which by its nature can lead to vacuously true statements, as a set of implications does not require a given implication to be the logical consequence of the previous statement. However, if you set your standard implication to be a "strict" implication, any set of causally connected statements must be logically coherent - each statement is the logical consequence of the previous one. In this way, a false premise cannot imply a truth, and so we eliminate the vacuous issue. One nice thing about this is it follows naturally with human language and or innate sense of causal-logical consequences, but it is not a better or worse way of analyzing something. Often this is an issue with the subtleties of semantics, and it is a question of what words and definitions we prefer to use for a given situation, etc. But, with standard formal logic, when you understand what it is really saying and don't get caught up in the language - letting your intuition confuse you, there are no issues.

[@bjc9520]

I'd picked C, but thanks to that explanation I can now see why I was wrong. It's an interesting exercise in deeper thinking.

[@wilberslater9967]

You were right. This video only proves how someone can math into stupidity.

[@t-yu]

Yeah, A and C can both be correct. Mathematicians are more inclined to say A but logicians commonly defend C. If you're looking at it from a mathematical perspective you'll say A because they assume the statements are true because it's pragmatically useful in math. But if you're looking at it logically, C could also be correct: if he has no hats, then it's obviously false that all of his hats are green.

[@t-yu]

@Hyper Piece Yeah it's definitely directed towards math students, but philosophical logic isn't any less correct than mathematical just because mathematicians need to take certain stances based on what's pragmatic. Accepting only A as the answer could lead us to a lot of problems which is why in logic there's criticism and C could definitely apply, but yeah I agree from the conventional math wisdom C would be wrong.

[@raultororomero9079]

I love this content, still after so many years

[@loupax]

I was about to argue about the “having no hats makes the statement true” but then I tried dealing with it programmatically. Going through an empty array of hats and checking their color, the result would always be the default value we decided. Is there room for undefined behavior in logic?

[@dddkkk5242]

You work with ternary logic in SQL

[@keamu8580]

It seems to me that in the act of making a positive statement describing a finite property of something which does not exist, you commit a falsehood. If you have no hats, you just can't get away with saying "all of my hats are green" truthfully. Try inserting the number of hats you own. "All 3 of my hats are green." "All 1 of my hats are green." These are plausible statements. "All 0 of my hats are green." Now you are committing a falsehood; there is no hat. It cannot be green. Nonetheless, it is not logically a false statement unless you possess a hat that is NOT GREEN. It would be more truthful to say "None of my hats are not green"

[@TheCoolSuperPea]

"My" is a possessive term. This means that you have the item in question, at least in this case.

[@D64nz]

@@keamu8580 That was pretty much my thinking too, though I admit it goes both ways too. Essentially to make "All my hats are green" a lie just requires that there are no green hats, or that there is 1 or more not green hats. It's only true when any number of hats = only green. So no hats would also satisfy the no green hats condition as a lie.

[@chriskleps7253]

It’s not a riddle. It’s just lazy conversion from one language to another, with an explanation for why you’re wrong if you didn’t make the same lazy conversion. If someone spent their life saying nothing but vacuously true statements, everyone would say “that guy always lies.”

[@smhwolvi]

Adding that picture of Pinocchio where his hat is specifically the color green changed the answer completely

[@dwdei8815]

I loved your elimination of C) using the vacuous truth process. That was the (pretty and tricksy) thing I learned today! Sooo many thanks.

[@Nossairito]

The one thing that's tripping me up in the explaination is that the same 'is this always true' logic used to eliminate the 3 first answers could be used to eliminate the correct one as well, since 'Pinocchio has 1 green hat' is a subset of 'Pinocchio has at least one hat', yet it satisfies the 'all my hats are green' condition.

[@J__T]

_All_ my hats are green All _my_ hats are green All my _hats_ are green All my hats _are_ green All my hats are _green_