17
votes
A liar who always lies says “All my hats are green.”
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- Title
- Can you solve it? That Sally Rooney hat puzzle
- Authors
- Alex Bellos
- Published
- Dec 9 2024
- Word count
- 282 words
The technical answer is that the liar having no hats would make the statement vacuously true, so they must have at least one hat. There's insufficient information to infer anything more.
Philosophically, I don't consider vacuous truths to actually be truths (or at least speaking vacuous truths doesn't preclude you from being a liar), it just makes predicate logic work out neatly. Sort of like how it can be convenient in some instances to say that dividing by zero results in infinity, when it's really just undefined.
This. It requires the external framework to make it work that way. I prefer lateral thinking puzzles (even when they make me groan) to formal logic
The way I think about it is that vacuous truths are a corner case that programmers and possibly lawyers need to aware of. It's an ambiguity: if you get an empty set, what do you do?
Often, ambiguity is fine since it never comes up or you can figure out what someone meant to say from context, but in a computer program it can result in bugs when an answer isn't what you expected.
Logic puzzles like this one are a fun introduction for kids and anyone else who hadn't thought about vacuous truths before.
They're not really "fun" if you don't have the pre-existing background in formal logic. Because they don't really make sense without that. The "one tells the truth, the other lies, you can ask one question" sort of logic puzzles are much more fun because you don't need an educational grounding, especially for kids.
Also rebuses and riddles are great for kids, once they can grasp the twist of the joke. Just beware, my younger brother discovered riddles and jokes and was just young enough that the answer to every one was "Teenage Mutant Ninja Turtles."
Why did the chicken cross the road?
Teenage mutant ninja turtles.
It made him laugh every time but I might have threatened to throw him down a sewer grate a few times as a kid. Just a few times.
Can you share any good lateral thinking puzzles? I wanted to introduce them to my daughter but went I went searching most of the ones I found were either outlandish — "gotcha" solutions that no one would be able to deduce, like some old adventure game puzzles — or highly macabre. Usually both. Hoping to find some fair stumpers that aren't about "what gruesome way did this guy die?"
You could try listening to Tom Scott's Lateral podcast, which presents six or so lateral thinking puzzles an episode. Some of them fall a bit into gotcha, but you do get to hear the panelists trying to explore different means of thinking for each of them.
I was going to suggest the same thing, there are also some collections of the questions/transcripts out there! But I've definitely come across other sorts of puzzles too, I'll have to go hunting. @balooga
They also recently released a book with Lateral questions, made to help people play the game as they do on the podcast.
Here's a collection of riddles I had a lot of fun with about 20 years ago.
https://www.ocf.berkeley.edu/~wwu/riddles/easy.shtml
Thanks for prompting me to go look for this. It brought back a lot of memories.
When I was a kid, there was a game called MindTrap that was a lot of lateral thinking puzzles on a cassette tape with a supporting booklet. I can't say how many are/would be the gruesome ones you're talking about... maybe check the age on the box if it still exists! For me, I know there are lateral thinking puzzles that have "solutions", but to me a correct solution is one that is cogent given the situation and objects/people in play. The point is to consider all the evidence and come up with a logical solution - whether it matches the solution listed or not is one thing, the important piece is that the solution uses all of the factors in play and shows an ability to think creatively and extrapolate information to form a hypothesis. That's the fun part!
When I was a kid I also had a subscription to a monthly book called Puzzlemania - a bunch of logical and thinking puzzles that are 100% age-appropriate for kids. I looked them up and they're still around, although they only ship to the US.
Oh I had Puzzlemania as a kid too! Published by Highlights, right? Those were great!
I'll look up MindTrap, that sounds like something that might have been adapted for the digital age and could be right up my alley.
I did go hunting a bit last night for more lateral puzzles and damn, you're right a lot of them are about death or implied death/kidnapping etc.
Weird
I find this really interesting, and I am asking this earnestly and in good faith. How can you consider a vacuous truth to be a lie? To me, their lack of meaning is based on the fact of their truth; the fact that they are meaningless relies on the fact that they are true, and the antipodal statement is also true, like the bookshelf example in the solution; if they were lies, they would be vacuously false.
As someone with a background in formal semantics, which is all about modelling the meaning of natural language formally, I'm so excited this came up. There are a couple ways to approach this on a linguistic level that still align with formal logic.
For this specific utterance, I think the issue is that the meaning of a possessive noun phrase like "my hats" is not just picking out the set of hats that are mine regardless of whether that set is empty. Possessives are what we call presupposition triggers, a large class of semantic elements impart the utterance with one or more presuppositions. This article does a good job going over the characteristics of a presupposition and a variety of linguistic theories about them, but for the purpose of this comment we can simply acknowledge that the phrase "my hats" triggers the presupposition that "There are hats that I own." While there are some contexts where presuppositions may be cancelled, none are at play here.
Thus, ∀x(hat'(x) ∧ mine'(x) ⇒ green'(x)) simply isn't an accurate representation of the propositional content of the natural language utterance "All my hats are green," because it does not account for the presuppositions triggered by the possessive noun phrase. When an accurate formal representation of that utterance reflects that presupposition, it evaluates to false, which reflects how said utterance is generally understood.
There are other similar utterances where presuppositions are not responsible for the difference between the formal logic representation and the way an utterance is interpreted by the speaker, and it's instead what's called conversational implicature at play. But since they're not relevant to the utterance in question, I'll not make my comment any longer by including that. If anyone's interested, though, I can elaborate later.
I would definitely be interested if you have the chance to explain or point me to some resources! I have absolutely no background but find this sort of thing endlessly fascinating.
Great comment, I appreciate it! I think that there is still an issue specific to vacuous truths though, but it's been a couple of decades since my symbolic logic courses, so please bear with me.
Consider both of these phrases and their presupposition triggers:
All my hats are green / I own at least one hat and all my hats are green.
I have stopped smoking. / I used to smoke, but now I do not.
These seem as if they should be qualitatively different, assuming I own no hats, and have never smoked.
I mean, it's all just semantics anyways. But I would consider it neither a truth nor a lie. It would be like if the person who always lies were to read a book aloud.
My issue with vacuous truths is that you can say things that are both true but contradict each other. For instance, that each purple elephant has five legs and each purple elephant has three legs.
In standard logic, instead of contradicting each other, these statements imply that there are no purple elephants. If you add the constraint that that there is at least one purple elephant then you get a contradiction, so you can use proof-by-contradiction to show that there are no purple elephants.
I believe constructive logic works differently.
I'm definitely out of my element here, and admittedly I do know what you're getting at, but if each purple elephant does indeed have five legs, then each purple elephant would indeed have three legs, you're just not counting two of them in that instance.
It's like that old Mitch Hedberg bit. "I used to do drugs. I still do, but I used to, too."
Sorry, I meant exactly 5 and exactly 3, which would both be vacuously true.
No worries, I was just goofin'!
It's not really any different than saying that the empty sum is 0, or that the empty product is 1. What else could it be but the identity element? 0 is the value that does nothing in a sum, and 1 is the value that does nothing in a product. If there are no operands, nothing is being done, so it must be that value.
It's just that, in boolean algebra, for the "all" operation the identity element is True. It does nothing in an "all" operation but defer to the other statements. If there are no other statements, it's the only option that agrees with potential other statements.
Say I just got some new hats. If I say "all my hats are green", we want it to mean the same thing as "all my old hats are green AND all my new hats are green". If I have no old hats, then "all my old hats are green" is a vacuous statement - the thing must defer entirely to the statement about new hats. The only way to do this with "AND" is for the vacuous statement to be true.
Similarly, for the "any" operation, the identity element is False. It is not true that "Any of my hats are green" if I have no hats.
I don't understand how it can be considered a vacuous truth to have no hats. I understand the logic of the other examples given, like "I have read all the books on my shelf", while having no books on the shelf, because the status of what you have read can be nothing through something. So it can be possible to have read all the books on the shelf and not have any books on the shelf, because it means you read nothing, which is a possible status. The status of nothing cannot be the color green however. The status of having no hats cannot be that all your hats are green or that they have any properties at all. Sure maybe if you get into some kind of quantum level or something you can argue about states of existence but without that context mentioned that wouldn't be an obvious interpretation of what is otherwise plain language to me.
Not sure about that... Consider the statement "I have no green hats."
The hats I don't have don't exist. Can I ascribe a color to those non-existent hats?
I guess you could make the argument that "some" green hats are a real thing, and I just don't have any of them. But is the statement "I have no antigravity hats" any different, just because there's no such thing as an antigravity hat?
This puzzle is pretty easy for people who are familiar with logic, but maybe you’ll find it amusing?
I love Alex Bellos! I have a couple of his books, and they are quite different but both delightful. This one would be right at home in "The Language Lovers Puzzle Book" which is a good read.
Here are some videos featuring the author from Numberphile, with my favourite probably being Lewis Carroll's Pillow Problem.
I prefer the one that Pinnochio says "my nose will grow now"