Viral logic test from Brazil

The 17th annual Brazilian Olympiad featured an incredibly tricky logic puzzle that went viral on social media. Thanks to Guilherme who suggested and translated the problem from Portuguese to English!

Pinocchio problem discussions

Pinocchio illustration

See Bram28 explanation for vacuously true

Wikipedia vacuous truth

Wikipedia truth table

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[stevengordon]

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"!

[Wumbo]

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.

[mccmcc]

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

[ckEagle]

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

[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.

[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!

[diueadslvrsl]

“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.

[diamondmemer]

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

[samuelrussell]

“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.

[Vgamer]

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.

[zzaz]

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.

[Krokodil]

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

[tomdekler]

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.

[steverempel]

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.

[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

[wittyjoker]

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

[rcnhsuailsnyfiue]

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!

[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).

[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.

[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.