A liar who always lies says "All my hats are green."

Sum is the "sequence extension" of Plus.

    Sum([a, b, c]) = a + b + c -- basically our definition

    Sum([a, b, c]) = Sum([a, b]) + Sum([c]) -- distributive

    Sum([]) = 0 -- identity

    [0] Agg([a, b, c]) = a @ b @ c == ((a @ b) @ c) == (a @ (b @ c)) -- definition + associativity

    [1] Agg([a, b, c]) = Agg([a, b]) @ Agg([c]) -- distributive

    [2] Agg([]) = Identity(@)

So let's extend AND and OR to sequence operators. The extension of AND can be called "Every", and it operates on sets of booleans. In particular, Every([]) == Identity(AND). The Identity of AND is "true", so Every([]) == True. The extension of OR can be called "Any", and Any([]) == Identity(OR) == False.

This is the easiest way for me to remember the truth values of Every([]) and Any([]): they must be the identities of the corresponding boolean operators.

Any([Your name is Bob, you can fly]) == Any([Your name is Bob, you can fly] concat []) == Any([Your name is Bob]) OR Any([You can fly]) OR Any([]) == (Your name is Bob) OR (You can fly) OR (False). Any([]) == False.

It does not. All my unicorns fly. There is no assumption that I have a unicorn. There is an assumption, based on the claim but it is not a fact.

The puzzle also assumes that "my" implies there is some ownership (we'll take for granted "my" means "has" for simplicity), which is another quibble that unravels the whole thing.

E is correct. I don't see how A comes to be the accepted answer.

No, you're not.

Perhaps. But what if someone asks you "are all your hats green?" Then the interpretation is not so clear.

I've rarely encountered a case where it is isnt an extreme lack of self-awareness in the questioner -- eg., being extremely overfit to language/notation/etc. localised to their own area of expertise.

It’s why puzzlemaking a truly challenging and impressive skill. It must be fair, a challenge, and obvious in retrospect.

I think it’s why the twist in Bioshock worked so well for me: it was right there in plain sight numerous times, while other twists in mystery games/films elicit a groan.

Trying to return to your more naive understanding of logic not only helps understand the intent of the puzzle, it also makes the puzzle more fun.

Not to mention that a liar doesn't necessarily have to mean someone who tells a falsehood in every single statement. It could just mean someone who frequently tells falsehoods. Or, more deviously, someone who wants to cause maximum uncertainty in his listeners, in which case some mix of true and false statements would probably be the way to go.

Just like the barber paradox isn't literally Russell's paradox, but it made more people to look up the history of it and perhaps learned what Russell's paradox is. Hopefully 0.1% of them turn out to be mathematicians.

If we reframe it using C++ "std::all_of" function over an array of strings called "hats", and say that the following must be false (because he is a liar):

  std::all_of(hats.begin(), hats.end(), [](std::string hat) { return hat == "green"; })

A) The liar has at least one hat. Yes, because std::all_of returns "true" on an empty list

B) The liar has only one green hat. Unsure, because { "green", "green", "red" } is false, and { "green", "red" } is also false

C) The liar has no hats. No, it is the opposite of A

D) The liar has at least one green hat. Unsure, because { "green", "red" } is false and { "red" } is also false

E) The liar has no green hats. Unsure, for the same reason as in D

This is almost certainly not what the author will say, and it does have some tangential connections to computing so it's worth expanding on briefly.

In computing you often reduce problems to smaller problems, this can be done with recursion but it doesn't have to be, dynamic programming kind of is the reverse of it; whatever. When you do that though, you get these questions about really small collections, sets, lists, data structures.

Just for a quick concrete example so that we're not an abstract theory land, is a one-element list sorted? Is a zero element list sorted? Is [5, 5, 5] sorted ascending, descending, both, or neither? If you choose the wrong answer, then it means that your algorithm needs to be more complex, you don't trust zero-element lists to be sorted so your quicksort HAS to pivot on a median of 3, you don't trust one-element lists to be sorted so your quicksort HAS to pivot on a median of 5, or maybe when you see that you would recurse into a list of size 1 you generate a new median-pivot or something.

In mathematics, there is a convention which attempts to generalize this idea that an empty list is always sorted. In fact, an empty list is also always randomly sorted. It is always sorted descending, too. If it's an empty array of ints, all of those ints are greater than 1000—and they are also less than 50.

The mathematical convention is, in lay speak, “if you are talking about nothing, pretty much anything you say is going to be true unless it's gibberish.” If I said every int is green, well, synesthesia excepted ints don't really have colors, that's crap.

Everything else is “trivially true.” More formally, any predicate of the form {for all elements in S, this is true of that element} is taken to be “trivially true” when S is an empty set. It is something like the code,

    let agg = true
    for (const element of mylist) {
      agg = agg || f(element)
    }

So that's the convention and besides making base cases much easier, one reason to do this is that the negation of any “for all Xs in the set, Y” is always perfectly specified as “there exists an X in the set such that not-Y.” The other convention you'd have to negate as “Either the set is empty or (...)”.

So the claim is that the perfect Liar is lying according to mathematical rules of Truth and falsehood, and according to those rules, this statement “all of my hats are green” can only be false if there exists at least one hat belonging to the liar which is not green. If there are no hats then the statement was trivially true.

It reminds me of the math "puzzles" on Twitter which go:

1 shoe + 1 shoe = 2

2 shoes + 2 shoes = 4

3 shoes + 2 shoes = ???

And the answer isn't 5 because a) we're not counting shoes and b) the shoe laces were different colors. There's nothing clever, it just teaches you to be hyper cynical and question every little detail which isn't relevant to either Math or the real world.

The solution article also discusses vacuous truths ("all my hats are green" is true if you have no hats, vacuously). Truth has a definition in formal logic too, but we've gotten far enough that this problem is already solvable and unambiguous.

the one I hate the most is the Monty Hall Problem.

Which is another listener’s bias: is lying by omission a lie?

All my horses are unicorns. I don’t have any horses, nor unicorns. So it’s true but also not.

The article mentions a maths exam, and of course the answer to the hats puzzle is very straightforward if you convert to statements about sets and logic in the way the teacher expects.

But converting ambiguous language into logical statements by taking everything extremely literally is the opposite of a useful life lesson, so I think it's a terrible exam question.

Well yea, that's the fun part! Logic puzzles without interaction with language are dull affairs indeed, basically just computation.

When you’re writing functions that work on lists or sets, you need to decide what to do about statements that are vacuously true. Call it out in the documentation? Put in a special case? Assert that the set is non-empty? Or maybe even create a new datatype to make the corner case unrepresentable. These design choices have side-effects, making a function easier or harder to use, more or less error-prone.

Recognizing vacuous statements is pattern recognition, but it’s sometimes useful pattern recognition, and for children, learning about it might even be fun.

In js if I write:

  const allHatsGreen = hats.every(hat => hat.color === 'green');

  const matchAllConditions = conditions.every(condition => condition.matches(item))

I'm not well versed, in functional programming, but in Ocaml (of F#, hehe) it would be

  let rec all_hats_green = function
    | [] -> true
    | hat::rest -> hat.color = "green" && all_hats_green rest

1. A deceit is an attempt to make somebody believe something false.

2. A falsehood is a statement which is false.

3. A lie is a deceitful falsehood.

Regardless of whether they miscommunicated about hats vs cats, and regardless of whether they were being deceitful in the process, for the statement to be a lie it would also have to be a falsehood, implying they have at least one hat which is not green.

Yes, these problems require suspension of disbelief, especially given the shorthand "he's a liar" always meaning that the person is often deceitful (usually with other negative implications), but the problem statement being that the "person always lies" is pretty clear and doesn't require special pattern recognition or other mental gymnastics, and it's not that different from the suspension of disbelief you invoke when playing a game of chess and not literally sending a knight to murderously dethrone your opposition.

Unfortunately, there are some obvious discrepancies. My favorite is that "can't" really means something closer to "not can".

This can be demonstrated with a close analysis of the statement, "I can't not do that." To get our usual understanding of the sentence we need to parse it as, "I (not can) (not do that)." And then turn that into, "I must not (not do that)." And now cancel the double negative to get, "I must do that."

Suppose that you try to parse it as, "I can not not do that." You quickly get, "I can do that." Which is not at all what that sentence actually means.

We know this is occurring on a math exam, so it must be a logic puzzle. The exam-taker would've probably known this refers to a first-order logic question, and have been taught a certain way of translating sentences like this into first-order logic.

This is true of all word puzzles that they need to be mapped to math in a way that assumes knowledge about the world and context.

In fact, I would extend this to arguments as well: Let's be clear up front what we mean by important words and go from there. Too often people ending up arguing about definitions, but in roundabout ways.

If the liar says, "All ten-foot tall men have brown hair," we cannot conclude that there must exist a ten-foot tall man.

EDIT: I'll clarify to say I wasn't taking issue with the derivation, but rather with the translation of the English statement into first-order predicate logic. No non-logician would conclude that there must be a ten-foot tall man if "All ten-foot-tall men have brown hair" is false. But since we can derive it from the translated logic statement, then there must be a problem with the translation.

In normal discourse people don't accept vacuous truths like that as meaningfully true. Rather I think people would interpret such a statement as a kind of hypothetical: "If there were a ten-foot tall man, he would have brown hair."

It's not clear to me if first-order predicate logic is really equipped to even handle reasoning about these cases of "a liar who always lies." Such a situation seems to be intrinsically higher-order. If a liar states a hypothetical, what does that mean, exactly?

My interpretation of the negation of the statement is, "If there were a ten-foot-tall man, he would not necessarily have brown hair." This doesn't imply the existence of any ten-foot-tall man.

That being said, I would argue that, "All my hats are green." has different meaning than "I may or may not own a hat. Any hat that I own is green".

The use of 'all' and the plural of 'hat' implies that the author has multiple hats.

Perhaps that means I'm wrong.

The liar either has zero hats or some amount of hats. The only thing we know for certain is that if they do have hats, there is at least one non Green hat.

So if the liar speaks of "all my hats" while having none, that is deceptive. I would consider it a lie.

Personally, I'd change the original wording (from the quoted italic section) to "someone who only tells lies" if I were the author. It's probably specific to me, but I'm always thrown by phrases like that because it seems like it's trying to differentiate in some way from someone else; surely there isn't anyone who "always lies" who _isn't_ a liar, so why say that? It's distracting to me in the same way as if someone said "the speech-capable human being who speaks only in lies". Normally it wouldn't bother me, but because puzzles like this often seem to be used to try to illustrate some smug point about how bad people are at logic, phrasing things in an unnecessarily confusing way just makes it seem even more smug (see https://xkcd.com/169/).

Not sure why so many people got it wrong maybe ESL is at play here

He may also be claiming that the hats are ecologically friendly.

Are we also to assume that if all the hats were each mainly green on the outside but had brown linings, they are, or are not, each said to be "all green"?!

I need a pint.

This is so much so that if you use the 'vacuous all', people will suspect that you have ulterior motives, and are being deliberately obtuse to hide them.

I wonder whether, if we attempted to make explicit all these tacit rules and conventions, we would end up with a consistent logic, and I believe that this looseness of natural languages was the main motivation for formalizing logic, from the enlightenment onward.

That makes A) a good candidate.

But E) can also be true — and in fact both A) and E) can both be correct.

>Note: this question was originally set in a maths exam, so the answer assumes some basic assumptions about formal logic.

You could have a similar situation in pure logic or math - if I were to say "the largest prime number is odd", is that false? Or something else entirely? (This is what Hofstadter calls mu, from a related concept in Zen.)

the statement is ambiguous when i have no hats, therefore, i must have at least a hat. weve seen the case so often as to know its true without thinking about it

By contrast, if you want to play in Math space, you need to be rigorous, and provide a Math-quality conversion.

If you refuse, what this reveals about the one posing the problem is that they themselves labor under the delusion that you can apply math rules to English, that there is exactly one and only one such mapping, that everyone should know and agree upon that mapping, and that is so true that you are justified in playing "gotcha!" games with people who don't know this nonexistent mapping.

It doesn't paint a terribly flattering picture of such people, in my opinion.

In the meantime, the rest of us should meditate a bit on "The Only Way To Win Is Not To Play The Game", because spiraling down trying to figure out the exact nature of the aforementioned nonexistent mapping is just a waste of time. There isn't one, so arguing it is just a waste of time and emotional energy. Pick a lane up front. They're both fine, but anything that functions by always choosing whichever lane maximizes the "gotcha!" in the moment is not worth spending time on.

If they always lie, then statement one is false - they lied about having hats.

If a person doesn’t have any hats, is the statement ”All my hats are hats” false?

We have a set of items X1, X2, X3, ... etc.

Each of X1, X2, X3 differently matches conditions A, B, C, D, etc.

Think of "All" as: A AND B AND C AND D AND...

Each term we add we further restricts the result-set.

Thus if we start with lots of conditions, as we remove conditions, we increase the matching results, until we remove all conditions, and "All([])" matches everything, i.e. is vacuously true.

Likewise, think of "Any" as: A OR B OR C OR D.

Here, as we increase the number of conditions, we increase how many things match, thus "Any" on an empty set returns nothing, i.e. is vacously false.

The statement "All my hats are green" would map to

    things.every(thing => thing.type != 'hat' || thing.color == 'green')

The negated form would then be

    things.some(thing => thing.type == 'hat' && thing.color != 'green')

And, as usual, the xkcd is fantastic.

6÷2(1+2) is written in a deliberately confusing fashion. This is a simple math olympiad-style question. All universal quantifications on the empty set are true, for the same reason that the implication A -> B is true when A is false regardless of B. It cannot be any other way. Precedence of infix operators on the other hand is completely arbitrary, we settled on multiplication before addition because otherwise it would be a pain to write polynomials.

o1 gave A as the answer, and it even mentioned "vacuous truth"