22
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
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.
Alrighty I have more time so I will ramble about pragmatics for a bit. Hopefully it makes sense, but if it doesn't or if you are interested in going deeper, lmk.
Alrighty, so in linguistics there's a distinction between semantics and pragmatics. Semantics covers the propositions that are actually part of the utterance you make -- entailments, presuppositions, any actual info that is expressed by the actual words you utter in a particular combination. Pragmatics instead cares about meaning that's imparted to your utterances through context. In practice, the two are pretty inextricably linked and semanticists will argue over whether a particular phenomenon is pragmatic or not, but that's academics for you. Probably the most important text for pragmatics is Grice's Logic and Conversation. Gricean pragmatics isn't the only pragmatics, but it's foundational and introduced a lot of important stuff. It's 100% the first thing you'll cover in an intro to pragmatics, and it's what I'll cover here.
So Grice introduced what's known as the Cooperative Principle. The gist of that is that in most cases we care about as linguists, interlocutors are trying to cooperatively communicate information to each other. As a result, they (unconsciously) generally try to follow a few principles (which are now sometimes called "Gricean maxims"):
Your immediate reaction may be that people constantly violate the Gricean maxims. This is true! And there are a few reasons someone would do so. Sometimes they're just not following the cooperative principle and thus violate the maxims hoping you won't notice (for instance, if they're lying or trying to mislead the listener). Sometimes, instead, they're deliberately flouting the maxim for one of a variety of reasons (e.g. sarcasm, politeness, comedy). Usually in this case they expect the listener to notice and understand the actual meaning of what they're saying, since that's usually the source of the intended effect (be it humor or politeness or something else).
That said, by assuming that the speaker is trying their best to follow these maxims, listeners can gain added meaning beyond the surface meaning of what was said. This added meaning is called a "conversational implicature." For example, imagine I ask my friend "Are you going to Bob's party on Friday?" and their response is "I have to work." Technically, the utterance "I have to work" says nothing about Friday or Bob's party -- if someone said "I have to work" in a different context you wouldn't get anything about Bob or Friday -- so it doesn't make sense to include anything like that in the semantics of this utterance. But because we can usually safely assume the speaker is following the maxim of relevance, I can interpret the added meaning that my coworker has to work on Friday and that this means they'll miss Bob's party. This added meaning is called a conversational implicature, and they're CRAZY common in how we communicate with one another.
One key feature of conversational implicature is that it's cancelable. Even after the listener has interpreted the utterance and gained the information from the conversational implicature, the speaker can contradict that conversational implicature to make it clear that it's not true (for instance, speakers can do this if the implicature was unintended). For instance, my coworker could follow up "I have to work" by saying "I can get off work on Friday if I rearrange my schedule, though, so I'll be there," thus cancelling the implicature.
Conversational implicature is so common that it often seems to be part of the utterance's meaning to speakers. For instance, most semanticists analyze the understanding of the quantifier "some" to mean "some but not all" to be a conversational implicature. For instance if someone says "I ate some of the cookies," you can assume that if they ate all the cookies and are following the maxim of quantity, they'd say "all," since that would be more informative in most contexts. But it's still cancelable -- they can follow it up with "In fact, I ate all of them."
In the case of the green hats example, obviously saying "All my hats are green" is totally irrelevant if you don't own any hats, even if it's vacuously true. Some linguists have proposed that presuppositions (what I talked about in my earlier comment) are actually pragmatic in origin like this (though I don't really agree with that analysis, that perspective is described in the Stanford encyclopedia article I linked in my earlier comment).
It seems like "a liar who always lies" tells us that this is someone who doesn't follow Gricean maxims. We are being asked to assume they're saying something that's technically a lie. And there's the further implication that this is a logic puzzle.
Oh yeah, I don't think what I'm talking about in this comment is strictly related to the puzzle; I left it out of my initial comment because I think presupposition failure was a sufficient explanation for the difference between the formal logic-based answer and people's natural language intuitions. I went into conversational implicature here bc I got a comment interested in learning more, that's all.
The article I linked to is a great overview on presuppositions specifically! As for conversational implicature, I can leave another comment talking about it later when I have time (it's fundamental to my favorite subfield of linguistics so I can ramble on lol) but if you're interested in reading about it yourself in the meantime, Grice's "Logic and Conversation" is the foundation/introduction of the concept and is pretty approachable to read iirc. You can probably also find existing writeups online using keywords like "pragmatics," "conversational implicature" and "Gricean maxims" if you're interested in digging around yourself.
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.
Verbs like "to stop" are also presupposition triggers (and profs love "stopped smoking" as an example in classes lol so that brings me back). Can you elaborate on what you mean when you say these two examples seem like they should be qualitatively different? It's hard for me to tell what you specifically mean there.
It is worth knowing that while formal semantics uses symbolic logic, it does include a lot of stuff that you probably wouldn't learn in a course on symbolic logic, since it's specific to modeling natural language. In my case, my first/only exposure to symbolic logic was in this context, so we're definitely coming at this from opposite backgrounds!
Well, it's also been a couple of decades since my linguistics courses as well! They are why I used "stopped smoking" because it is the classic example, from what I recall.
I think that the qualitative difference has to do with what the presupposition trigger is saying and the problems that arise when those triggers are untrue, but when I try to express what I think the results are, it all comes back to the use of the null set in the argument, which just seems circular.
If you say "I stopped smoking" then the presupposition trigger implies that you used to smoke. If you have never smoked, then that's a conflict with a state based action. The lack of truth makes the statement nonsense because it is the opposite of what has happened. From the reading you suggested, I would parse this as a "semantic catastrophe", because it renders the statement to be nonsense because it cannot evaluate to true.
If you say "My hats are all green" then the presupposition trigger implies that you have hats. If you have no hats, then the presupposition is false, but the falseness of this doesn't actually cause a conflict, because of the understanding of the null set; "no hats" being green doesn't cause the same semantic catastrophe as the smoking example.
But that seems like it's begging the question pretty hard; the truthiness of the green hat statement is different because it uses the null set, and the statement about why it's different is because it uses the null set.
Ultimately I don't think there's really a difference when it comes to the natural language meanings there, at least not in respect to both being cases of presupposition failure. I think both end up being parsed as "semantic catastrophes" (love that word choice) in most speakers' minds -- if you utter either in an inappropriate context, the listener is going to interpret it as either an uncooperative discourse participant (which I think is ultimately what people saying the speaker is "lying" are getting at) or as them having incorrect information about truth value of the presupposition ("ah, I guess they must own a hat after all that I didn't know about").
I don't think that the way speakers interpret this utterance allows for an empty set to be evaluated the way you describe. I think you could get a "truthy" evaluation of a vacuous truth by wording the sentence differently to avoid the presupposition trigger, by saying something like "Any hats I own are green." I think this semantically corresponds better in natural language to the formal logic interpretation the original puzzle is talking about. But in natural language, hopefully you at least feel some level of intuitive difference between "All my hats are green" and "Any hats I own are green," even without parsing it out formally?
The latter is (I think) the source of the famous (for semanticists) "Hey wait a minute!" test to distinguish presuppositions from assertions. If I say "I stopped smoking", it'd be weird to respond with "Hey, wait a minute! I didn't know you stopped smoking!" whereas it'd be pretty reasonable to respond with "Hey, wait a minute! I didn't know you smoked!" I think this type of response to presupposition failure is pretty common in this type of "semantic catastrophe" -- it wouldn't be weird to respond to "All my kids are blonde" with "Hey, wait a minute! I didn't know you had kids!" Same with the green hats example (though I think in practice most people would be less likely to challenge this presupposition failure in practice because most people don't care about how many hats someone else owns).
I do see the difference between "any hats I own are green" and "all my hats are green". I do still think there is a difference between "I have stopped smoking" and "all my hats are green" though; there must be because when you dive into them "all my hats are green" is vacuously true, and "I have stopped smoking" is not; if they were not different, we wouldn't have discovered the differences in their purely logical contexts.
I think what I'm grasping towards is often connoted in natural language as "I'm technically correct - the best kind of correct!" If I explain this puzzle to someone and I say "it's technically correct that all of my no hats are green" they will understand. My source is experiential; my son did the puzzle the other day, and easily got to "it's A or C", and hit a bit of a wall. I asked him "what colour are no hats?" and he got the rest of the way there. Now there's probably some other linguistic and genetic things at play - as my son, he's probably more inclined towards logic puzzles and math and such, both via nature and nurture - but I think that if I can get a 10 year old to quickly arrive at the answer, a lot of other people should be able to.
I think your comment is more or less grasping at the difference between formal logic and natural language. In a formal model of natural language, "my hats are green" simply isn't a vacuous truth anymore because the presupposition is an inherent part of its meaning. But in discussions of formal logic, which deliberately differ from natural language use by defining a number of terms more rigidly, it is a vacuous truth. I think the difference between the formal logic and the natural language (even when the natural language is modeled formally) is to blame for your differing judgments of these two examples -- you're not actually modeling "stopped smoking" in formal logic and set theory with or without the included presupposition, but you are with the "all my green hats" sentence (understandably, because it would be a complicated job to model the meaning of "stopped" without doing a lot of annoying formal semantics anyway).
I think the reason many people struggle with the formal logic of vacuous truths (as evidenced by the other people in this topic trying to define why they don't think vacuous truths are really true) is because they're unintuitive when there's a mismatch between formal logic and natural language interpretations of the same statement like this. You can get someone to interpret it correctly if you teach them formal logic and/or set theory, but that's different from it being the intuitive meaning of the actual natural language. I think, without you trying to explain the formal logic vacuous truth to them, your ten-year-old is going to have a "Hey wait a minute"-type reaction because of that. This doesn't make it impossible to learn the formal logic of vacuous truths but it is why they're unintuitive to learn and why people are not going to interpret natural language utterances in a way that's compatible with them. Part of the reason for having presuppositions like this in formal semantics is precisely because without including them, this behavior doesn't make sense for what should be vacuous truths like this.
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."
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.
Absence of evidence is not evidence of absence.
Take the statement mentioned in this thread “I have read every book on this shelf” for an empty bookshelf. As stated elsewhere, this is vacuously true. Now invert the statement “I have not read every book on this shelf”. Because of the inversion this should be considered vacuously false. But looking at the statement in isolation, it is vacuously true. Vacuousness does not, to me, imply truthiness or falseness.
The statement “all my hats are green” for a person with no hats can be argued to be vacuously false or vacuously true. So that statement could be said by someone who always lies.
I have no training in formal logic, but it seems to me that the formal discipline of logic may have become disconnected from how logic is actually used in the real world.
The breakdown you discovered is because that is not the inverted statement. Because in English we associate that "I have not read" as the entire inner part. The correct inversion is: "It is not true that I have read every book on the shelf", and indeed that is false. Your statement is "All the books on the shelf are ones I have not read", and indeed that is true.
The correct inversion - "It is not true that I have read every book on the shelf" - is false because it requires some counterexample. There must be some book that I have not read, and there are no books so this must be false. Written as a plain statement - "I have not read any of the books on this shelf".
There's this interplay between "not-every P" becoming "any not-P" and "not-any P" becoming "every not-P". These are De Morgan's Laws.
The trick when applying the logic to English is carefully identifying the subject of each "not", and carefully identifying the synonyms "all", "every", "each", "none", "and", etc. distinct from the synonyms "any", "some", "exists", "one", "or", etc. I'm sure there are other similarl words I'm not thinking of here. That first group are all conjunctive (and-like) operations, and the second group are all disjunctive (or-like). De Morgan's laws just state that when you move a negation in or out of some operation like this, you have to also switch between conjunction and disjunction.
For example "none are P" is equivalent to "all of them are not P" is equivalent to "there does not exist one that is P".
The not-any-P stuff gave me some flashbacks to my discrete math class. I loved that class, but it’s been a long time and I was never very good at the logic parts.
All of this just agrees with my statement that formal logic doesn’t align with how laypeople use language. Those deMorgan laws (I plan on reading those after I type this; thanks for the link) may be the correct way to invert a statement in formal logic. And I am not proposing that the field of logic needs to change anything. But they should understand that formal logic doesn’t align with language perfectly. I think that most laypeople would agree that my inversion is much closer to what they would expect than a deMorgan law abiding inversion.
The original question was asked in English, not the language of logic. Since the question is in English, I think that the conclusion that the liar owns zero or more hats is a valid conclusion. If the question were written in the language of logic, which is much more precise and has less room for ambiguity, than I would wholeheartedly agree with the given answer.
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?
Based on my one discrete math class, the way I think about it is anything you say about an empty set is like multiplying by zero. You can throw in whatever numbers you want, but as soon as you multiply the whole thing by zero it's just zero. You can say whatever modifiers you want about an empty set, but they all become meaningless because empty is empty.
That's just in the abstract though. Once you start translating it to real life and ordinary language these things don't apply. If I've gone apple picking three times and picked 8 apples each time, it's nonsense to say I then went apple picking zero times and have a result of zero total apples picked. If I tell a story about how I took my dog to the park, the dog ate some chocolate it found on the ground so I had to take it to the vet, and also I don't own a dog and there was never any dog to begin with... that's not so much false as it is nonsense.
I think if we can accept that multiplying by zero is a thing that is useful in abstract, it's not too far off to accept that vacuous truths are useful in abstract as well. I don't feel that this particular riddle is terribly good as a riddle since there is not a good plain language interpretation, but as mentioned elsewhere it might be an ok intro to some logic concepts.
Sure, because you're saying that you did go picking [a fourth time], but you didn't.
Say you went picking Apples on Tuesday, Wednesday, and Thursday. It is correct to say "I picked no apples on the weekend". Or, more obtusely but still equally correct: "My weekend picking trips yielded no apples".
@Grumble4681 this is the strategy that makes vacuous statements most sensible to me: take the overall set you're considering - books on a shelf, or hats, or apple-picking-trips - and figure out some way to divide it in two parts. Logically, you want the entire "all of S" statement to be the same as "all of (one part of S) AND all of (the other part of S)". If one of those parts is empty - weekend trips in this example - then that part must be True. If it were false, you'd get "all of S AND False" which is a contradiction.
This suffers from the same problem as all the other objections in this thread. If you told this to anyone other than a logician in conversation, they would assume you went on an apple picking trip. If you didn't, it's a nonsense statement that might make people think you lied to them, just like my dog example where there is a story with many details about a dog that never existed. It may be correct in logic, but it's not correct in conversational language.
My point with the apple example is that once you have some number of apples, you can't really multiply them by zero in real life. Once you start describing things, you can't go back and say there was no thing to begin with. But in math 24 * 0 = 0, and in logic you can truthfully call things any color you want if they don't exist. These concepts might be real and useful, but they don't always have a good real-life mapping, which is what is happening here.
I did say it was the more obtuse option. "I didn't pick any over the weekend" is a much more natural way to phrase the same idea.
I suppose if I'm going to be arguing about all this I should say the real point I'm trying to make.
I've seen this sentiment many times in this thread:
Generally the sentiment seems to be that logicians say vacuous truths are true, but they're really not, they're nonsense, and if you believe they are true then it's only because some formal education told you so.
I object to this. Vacuous truths are the only sensible way to handle an "every" statement about nothing. And vacuous falsehoods are the only sensible way to handle an "any" statement about nothing. Even in natural language.
The problem is that when you say "None of my weekend picking trips yielded apples." is that there is a presupposition that the speaker is only talking about relevant things, so they shouldn't bring up weekend trips if there were none. One must acknowledge this assumption if they are to have a consistent view of reality.
In the sentiment above, there's this strange refusal to acknowledge the presupposition. If you acknowledge it, then there is no logical contradiction! Don't reject the details that formal logic is so concerned about, then complain that formal logic is inconsistent with reality!
Besides, people violate these presuppositions all the time. Especially advertisers and politicians, it seems. It's important to be aware of what assumptions you're making about the conversation before you make conclusions based on the person's words. Do you assume they speak in good faith? Do you assume they cite real sources? These things, whether you acknowledge them or not, have bearing on whether your conclusions are sensible or not. Think critically. Listen critically.
There's also statistics in general, where you'll hear about "a study", and make assumptions about the population used for the study. Those assumptions may or may not be true, and have bearing on how strong a conclusion should be made on the study. This relationship between the size and validity of a population and the confidence you can have in a result is closely related to De Morgan's laws and vacuous statements. You always arrive at this kind of relationship from so many different fields and so many different approaches, I must believe vacuous truths are a fundamental property of the universe in general - including language - not just formal logic.
Here's a more concise way to phrase why I'm so amped up about this - a rejection of vacuous truth comes across as a rejection of critical thinking. It feels like "gateway anti-intellectualism" and I don't like it.
Sorry, I don't understand what you're getting at here. If you acknowledge the presupposition, then the given answer to the OP riddle is incorrect (he must have hats if you take "my hats" as a statement that you are not talking about nothing). If you're saying that we cannot take a presupposition as part of a statement in conversation, I would disagree with you, since the use of presuppositions is a huge part of how we communicate. So, could you please clarify this point?
You mention violating presuppositions, and I think most people would say that abuse of presuppositions is a form of lying. For example, if I say "our wall on the border, it's keeping immigrants out, and it's a wonderful wall," when there is no wall, that's known in common parlance as lying.
I wouldn't say that these concepts are similar to making unfounded assumptions on studies unless some headline or article has added descriptions of things that don't exist, which I've said is considered lying. I feel that linking this to a rejection of critical thinking is not an idea that I can agree with. I say that this is more a semantic argument on whether something would be called a lie or not rather than a rejection of any logical concept.
Because in this thread I see people saying that (conversational) "All my hats are green" is not equivalent to (formal) "All my hats are green" and therefore formal logic is something to be disregarded.
It is true that these are not the same! (formal) "All the contents of this statement exist and I am not lying and all my hats are green" is clearly different from (formal) "All my hats are green". You need to acknowledge that the former has these additional assumptions, and a critical thinker should try to be aware of assumptions in any situation.
Not considering the assumptions at all and declaring formal logic to be inconsistent with language is a straw-man argument.
Hardly. I'm saying you should be aware of those assumptions, and avoid drawing conclusions which do not use reasonable assumptions. If you find some assumption to be unreasonable, you should adjust your conclusions accordingly. It is not possible to do this if you are not aware of the assumptions at play.
Most of the time it is reasonable to assume the other speaker is conversing in good faith. If you determine that not to be the case, you need to examine their words more closely. Especially if health or finance or policy are on the line.
This is a fair take. And, I must also concede that if you're not familiar with the definition of "A person who never lies" and similar from formal logic, and instead apply the layman's definition, that's confusing.
Despite that, though, there is a difference between arguing over the definition of a lie and saying something is inconsistent with reality. Vacuous truths (if relevant assumptions are considered) are the only way for language or any of formal logic to be consistent with reality.
I'd also point out that your example, with presuppositions explicitly stated, is in fact false according to formal logic. "[I am arguing in good faith, and] our wall on the border [exists, and] it's keeping immigrants out, and it's a wonderful wall". The last two parts are vacuously true, but the first two parts are directly false.
The same arguments around logical statements about empty sets can extend to making sweeping extrapolations from nearly-empty datasets. My real point bringing this up - and re-reading my message I don't think I communicated this well at all, I apologize - my point is that the general principal behind vacuous truth shows up in other contexts too so can hardly be dismissed as inconsistent.
I should probably emphasize that I'm not trying to call anyone here anti-intellectual. The word "gateway" in my other comment is doing a lot of work. The general sentiment that formal logic should be disregarded or does not apply to natural language is what I don't like, and feels like has potential to foster anti-intellectualism.
If you object to the hats you don’t have being green, you should also object to describing the things you don’t have as “hats,” because that’s also a property, and you want nonexistent things to have no properties.
But from a computer programming point of view, these are properties of the search query, not the things that you’re searching through. You can search for hats or green things or books or your car keys, and it doesn’t assume that the things you’re searching through have these properties. That’s what you’re trying to find out!
The result of the search tells you something about the things you searched. If you searched for your car keys on a shelf, not finding any tells you that none of the things on the shelf are your car keys, but they might be other keys or other car keys, or not keys at all. Or the shelf might be empty. There are many possibilities.
These aren’t really properties of nonexistent things, though. They are non-properties of the collection of things on the shelf. It’s like when food has labels like “fat free” or “gluten free.” Supplying answers to questions you didn’t have is a marketing trick. You could just as well say that it’s not purple or not an elephant. In a game of twenty questions, there’s no limit to the questions you can answer “no” to if the person asking the questions is bad at guessing.
The trickiness of the statement “all my hats are green” comes from the question of what assertions you’re making about your possessions. The standard logic approach assumes that it’s asserting that a particular search would come up empty, and nothing else. But sparkbet brought up presuppositions and I think that’s the right way to think about it. What presuppositions are there in this sentence?
(When writing code to do a search, the presuppositions would show up as types or preconditions.)
Logically them having one hat makes sense.
But if I was to face this problem in real life, both A and C would be possible choices due to vacuous truths essentially being nonsense.
This question boils down to, which of these statements is more of a lie than the others. So I don’t find it very interesting at all.
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"