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

theguardian.com

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

#291

Post by Supermancho »

LudwigNagasena wrote: post #204
Supermancho wrote: post #183

Earlier quoted context omitted.

Yes

So if “all my hats” doesn’t imply that I have at least one hat, “some of my hats” doesn’t imply it either; otherwise we wouldn’t be able to derive “some” from “all”. Hence, “some of my hats are green” doesn’t imply that “at least one of my hats is green”. That’s a claim that contradicts both traditional formal logic interpretation and common sense English interpretation.

I think the same of your interpretation of some vs all. Some can contain all, just as it contains none. Both some/all imply, but do not assert existence. Claiming it tautologically defies logic is not compelling.

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

#292

Post by 8note »

jp57 wrote: post #131
FabHK wrote: post #95

SPOILER The statement translates to: ∀x ( IsAHatOfMine(x) => Green(x)) That's just equivalent to ∀x (~IsAHatOfMine(x) ∨ Green(x)) by the definition of implication (it's only false if the antecedent is true, and the conclusion false). The negation of that is (by repeated application of De Morgan's): ~∀x (~IsAHatOfMine(x) ∨ Green(x)) ∃x ~(~IsAHatOfMine(x) ∨ Green(x)) ∃x IsAHatOfMine(x) ∧ ~Green(x)) Thus, the liar has a…

Wait a second. 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.…

> A liar is someone who only says false statements.

we're using a made up definition of liar, who can only say things that are false. it's not part of the question for the liar to be able to tell a statement they arent certain is false

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

#293

Post by andrewla »

triclops200 wrote: post #246

The proposed solution misses the fact that you can't even derive that the liar has a hat because the subject of the predicate could be the lie. What if the liar had two green cars but any number or no hats? The lie is a lie then even if the statement is vacuous as there is too much ambiguity in the english language overall. Their hats could be any combination of colors or they could even be hatless if the lie was ove…

Even if that is the case -- if he has no hats, then the statement is technically true, and regardless of whether he intended to lie or not, he made a true statement.

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

#294

Post by njtransit »

scotty79 wrote: post #274
aftbit wrote: post #263

Earlier quoted context omitted.

It seems intuitively wrong that you can say "All X are Y" and yet that doesn't imply "At least one X exists".

That's how math defined what "all" means. You can be talking about all elements of empty set without implying it must have some. Basically it disentangled two unrelated concepts, that English language unduly mixes. Concept of every item having some quality and concepts of at least one item existing.

That how we define "all" logically but not linguistically. For instance, linguistically we would consider this inconsistent: "all 10-foot men have black hair AND all 10-foot men have blonde hair" yet it is true, logically, if there are no 10-foot tall men. The translation of the English word "all" should be something something like: |Q| > 0 ∧ ∀x∈Q, x is ...

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

#295

Post by quchen »

raincole wrote: post #219
mjburgess wrote: post #116

Earlier quoted context omitted.

This is generally the case for the vast majority of puzzles, and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency. 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.

My favorite example is Monty Hall problem. "Smart" people often use it as the evidence of how bad general people are at probability. It really isn't. The problem is usually given in this form: > Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, s…

Can you explain the mistake (of people taking the standardized tests)? I don't understand it from the message, and the Wikipedia article seems to detail exactly the "normal, smart" switching solution.

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

#296

Post by btilly »

Suppafly wrote: post #188
mjburgess wrote: post #116

Earlier quoted context omitted.

This is generally the case for the vast majority of puzzles, and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency. 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.

>and it equally drives me mad in those areas where academics set "puzzles" and conclude that people's inabilty to "solve" them is some cognitive deficiency. Does that actually happen in academia? It seems to mostly be a social media thing.

There is one notable example that I'm aware of, but it really is a cognitive deficiency.

The contrapositive is a rule that says that "A => B" is the same as "not B => not A". This is very confusing to people, and few can follow verbally why it works.

But here is a fun experiment. People are presented with a selection of envelopes, all face down, and are asked to verify the fact that, "All unstamped envelopes are small." They immediately begin turning over the large envelopes, then have trouble explaining their (correct) reasoning!

Here is a correct implication process for their actions.

"All unstamped envelopes are small." => "unstamped envelope => small envelope" => "not small envelope => not unstamped" => "large envelope => stamped"

At which point it is easier to just check the large envelopes!

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

#297

Post by spencerflem »

tantalor wrote: post #277
LudwigNagasena wrote: post #264

Earlier quoted context omitted.

So you claim that in normal English “all” doesn’t imply “some”?

Does "when pigs fly" imply that some day pigs will be able to fly? No; people can understand impossibility when it is used rhetorically in every day speech. For example I might say, "all the honest politicians are doing a great job", which conveys my actual meaning, "all politicians are dishonest".

That stops working when its not obviously rhetorical.

Someone else in the thread mentioned: 'All my kids are in high school'. If you said this to a stranger with no other context, they will 100% think that you have kids. There is no possibility that you meant, 'I am asserting that in the set of my children, each element satisfies the property of being in high school'

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

#298

Post by LargeWu »

btilly wrote: post #287
LargeWu wrote: post #265

Earlier quoted context omitted.

The difference here is that the Monty Hall problem has an explanation that while counterintuitive, is statistically sound. You should always switch, because the probability you picked the correct door is locked in at the time you made the choice between 3 doors. It is 1/3 that you picked correctly, and 2/3 that you picked incorrectly. The counterintuitive part is that if you switch, you are effectively selecting all…

The problem here is that the usual explanation sneaks in multiple rarely stated assumptions. If Monty knows the door with the prize and is aiming for the game to continue, then you should switch. (This is the usual argument.) If Monty doesn't know where the prize is, then you learned nothing. (Monty's result was luck, and he can't impart information that he doesn't have.) If Monty knows where the prize is and wants y…

I think, canonically, Monty always knows where the prize is, and will always eliminate all doors except for one, and will never eliminate a door with the prize, and will always give you a choice to switch. There's no room for Monty's motivation.

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

#299

Post by gorfian_robot »

FabHK wrote: post #95

SPOILER The statement translates to: ∀x ( IsAHatOfMine(x) => Green(x)) That's just equivalent to ∀x (~IsAHatOfMine(x) ∨ Green(x)) by the definition of implication (it's only false if the antecedent is true, and the conclusion false). The negation of that is (by repeated application of De Morgan's): ~∀x (~IsAHatOfMine(x) ∨ Green(x)) ∃x ~(~IsAHatOfMine(x) ∨ Green(x)) ∃x IsAHatOfMine(x) ∧ ~Green(x)) Thus, the liar has a…

All my unicorns are green.

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

#300

Post by btilly »

raincole wrote: post #284
hansvm wrote: post #266

Earlier quoted context omitted.

Perhaps there's another formulation you meant to write, but every time I've seen it, it's been equivalent to the thing you just asked, and the only potentially ambiguous part of the question is "to your advantage". If you stay then you have a 1/3 chance of getting a car, and if you switch you have a 2/3 chance. Yes, it's also correct that you don't "know" the result, and you might prefer goats to cars (even then you…

I regret opening this can of worms called Monty Hall. But anyway I'll link the relevant section from Wikipedia: https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_... > The version of the Monty Hall problem published in Parade in 1990 did not specifically state that the host would always open another door , or always offer a choice to switch, or even never open the door revealing the car. However, Savant mad…

Just be glad that you didn't go for the 2 envelopes problem which was her next. Vos Savant was wrong about that one, and few people like the real answer.

The problem is that you're presented with two envelopes with 2 real numbers inside. You randomly select one, then look, and try to guess if you got the larger number. It doesn't seem like you can do better than even, but you can!

Unfortunately everyone hates the answer. Which is that you make up a random number and pretend it is the other one. Your odds of being right are

50% + (probability of choosing between the numbers) / 2

Which can always be strictly bigger than 50%. (Though possibly by only a little amount.)

Special case. If those numbers and yours were all independently randomly chosen from the same distribution, you'll be right 2/3 of the time.

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