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/sci/ - Science & Math

Apparently this is an "incredibly tricky logic puzzle that went viral on Brazilian social media". Are Brazilians retarded? This took me 1 minute.

It's obviously not B, D, or E, so A and C are left.

"All my hats are green" is a lie, so either "NOT all my hats are green" or "All my hats are NOT green" is true.

"NOT all my hats are green" implies he has green hats but not only green hats, "All my hats are NOT green" implies he has hats but none are green. In both cases he has hats and therefore C is incorrect and A is the answer.

Am I missing something?
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>>14593766
Pinocchio is a liar, so his claim must be negated. The negation of his claim is "I own a hat that is not green."
So A is true (note that Pinocchio's claim is not a lie if he owns zero hats).
B is indeterminate.
C is false.
D and E are indeterminate.
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>>14593766
Also if Brazilians are retarded than so are you. This is only a puzzle if you don't understand predicate calculus.
>"All my hats are green" is a lie, so either "NOT all my hats are green" or "All my hats are NOT green" is true.
This statement is true insofar as the disjunction of a true statement and a false statement is true. It also indicates that you don't understand predicate calculus.
>>
It's tricky because it assumes deductive principles, and the strictness of the use of "lie".

Abductively, Pinocchio can have no hats, always lie, and still say "All my hats are green", ands most people won't be treating lies as pure negation.
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>>14593766
None of that can be concluded.
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>>14594021
>None of that can be concluded.

BINGO!
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>>14593766
This is a question about English conventions more than math imo.

Assuming Plato never had children, what truth value would you assign to the sentence "All of Plato's children have dark hair"?

You could argue that it's nonsensical since "Plato's children" does not refer to anything.
You could argue that it false because the everyday usage of "All of" imply at least one thing of that type which isn't true in this case.
You could argue that it's vacuously true (if x is the child of Plato, then x does indeed have dark hair).

The convention in math is to consider it true so that it matches the universal quantifier: https://en.wikipedia.org/wiki/Universal_quantification.
So if Pinocchio had no hats, his sentence would be true, so we know he has at least one hat.

Once you know they mean the universal quantifier, then you also have this:
https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extension_to_predicate_and_modal_logic
NOT (all my hats are green) = there exists at least one of my hats that is NOT green
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>>14593933
>It's tricky because it assumes deductive principles
That's not a trick in itself, it's tricky because picrel is counterintuitive to many.
>the strictness of the use of "lie"
In what sense is it strict? The "always" quantifier?
>Abductively
Why would you prefer abductive reasoning when deductive reasoning is applicable to the problem at hand?
>most people won't be treating lies as pure negation
Most people won't encounter people who always lie in their day to day lives.
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Imagine listening to anything a compulsive liar puppet told you. Kill it with fire is the only correct answer. I don't debate demonic wooden puppets.
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>>14593801
>Pinocchio is a liar, so his claim must be negated.
No, his claim just doesn't have to be true, if he has no hats, "All my hats are green" is still a lie, you can't actually deduce the number of hats or colors just based on the fact he is a liar who said something about green hats.
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>>14593766
There's two statements here "all my hats" and "hats are green" I think. That needs to be negated and both must come out to true statements so as to not contradict each other. Pinochio isn't here to contradict, he's here to lie.

>Pinochio has hats
There's two option to this lie. He has no hats and he has one one hat.

>His hats are green
The option here is that his one hat is not green or his no hat is not green.

So the 4 options are

1) he has no hat, the one hat is not green
2) he has no hat, that no hat is not green
3) he has one hat, that one hat is not green
4) he has one hat, that hat is not green

The one that makes sense is 4) his one hat isn't green.
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>>14594131
>So if Pinocchio had no hats, his sentence would be true
No, all his hats are transparent color because they don't exist, they are not green like he claimed because he has no real hats.
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>>14593766
>Pinocchio always lies
Your mom asked to be first to sit on his face while he lies?
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>>14594430

4) and 2) would work fine.

He has no hats, and no green hat.
He has one hat, and no green hat
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>>14593816
>>14593801
>>14594131
Seconding these replies.
The majority of "puzzles" or "paradoxes" like these are more examples of bad communication than they are "puzzles". Which answer you choose depends on convention (aforementioned universal quantification), but as that convention is unstated there can be no strictly correct answer.

The true problem such "puzzles" demonstrate is the inability in the mind of their framers to realize that it is possible to interpret things other than assumed convention. This failure of imagination, or lack of honesty, should not be rewarded with the title of "a puzzle". It should be duly rewarded with the label "meaningless".
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>>14593801
>So A is true

totally false.

Nonce of the sentences can be said to be true.
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>>14593766
>>14594430
>>14594437
Also if we're choosing multiple answers, we'd choose A), C), E).
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>>14594427
>No, his claim just doesn't have to be true
No, he *always* lies. So it isn't sufficient to say that his claim need not be true. It in fact must be false, in every case.
>if he has no hats, "All my hats are green" is still a lie
No, it's not. "All my hats are green" is equivalent to "if a hat is mine, then it is green." A conditional statement of the form A=> B. As you can see in the truth table here: >>14594180
A=>B is always true if A is false.
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>>14594462
>"All my hats are green" is equivalent to "if a hat is mine, then it is green."
That is still false, he doesn't have a hat at all, so there is no green hat and if there were a green hat, he doesn't have it because he doesn't have a hat at all.
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>>14594487
ex false quodlibet
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>>14594509
*falso
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>>14594509
So if there was a green hat on the ground, you would think it would have to be true it belonged to some liar even though he doesn't own a hat at all?
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>>14594405
based and anti puppet pilled
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>>14594405
oy vey cool it with the antisemitism
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>>14593766
Well, it depends on the convention being used

According to the normal convention of formal logic, a syllogism is considered true if there are no counterexamples regardless of if there are any positive examples; so if he claims his hats are green when in fact he has no hats at all, it's considered a true statement, so for the claim to be a lie there must be at least one hat that must not be green.

Of course this runs entirely contrary to the normal conversational understanding of truth, where if you make a claim about your hats and you have no hats at all you're a liar; by this convention, we cannot be sure if he indeed has hats at least one of which must not be green, or if he has no hats at all and his statement is, as they say, cap

The difference between public opinion and the conventions of formal logic doesn't mean people are stupid, actually it means logicians are stupid for not accurately representing common usage of words.
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Unfortunately the language that is used to describe this scenario and context from it means that every single choice is pointless as nothing can be affirmed within this figurative system of thought. The questioner is a Brazilian with a low grasp of the English language, or low minded inability to actually present a legitimate question with an answer. Either these or that the question is purposely malicious and intended to be misinterpreted.

The only way that this question can work at all is if the reader purposely tries to invent reading between the lines of (E), and accepting that the answer implies that Pinocchio can still have one green hat. Meaning that Pinocchio does not have multiple green hats but he could very well have one singular green hat. It is the implication that there is a continuation after the arrival of an answer. So (E) to me is the only way to salvage this likely purposeful time waster, by still choosing one of the following five answers to participate in the spirit of giving the benefit of doubt to the question itself. Though as a whole, this has been an awful experience, and whoever did make this question should be immediately exited from the gene pool for being so low minded.
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>>14594535
I think that you already assumed that the liar in question doesn't own a hat. So if you further assume that he also does own a hat, then you can prove anything you want. I'm not making that second assumption, I'm noting the consequences of that second assumption.
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>>14593766
The only thing that can be cleanly determined is that if Pinocchio has any hats then not all of them are green
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>>14593766
What I concluded
Possibility 1:
Pinocchio has no hats
Possibility 2:
If Pinocchio has one hat, it's not green
Possibility 3:
If he has several hats atleast one of them is not green. Could be some, could be all.
Possibility 4: Pinocchio has only (a) green hat(s) in his possession but it(they're) actually owned by someone else and Pinocchio is being a real cheeky cunt.

This is honestly a dumb fucking logic puzzle. Anyone could construe different scenarios which make Pinocchios statement entirely untrue.
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>>14594663
>>14594673
The key is to note that Pinocchio is neither saying that he owns hats, nor that he doesn't. He's strictly making a conditional claim: "If I own a hat, then it is green." And because every claim he makes is a lie, that conditional claim must be false.

The only way to disprove a conditional claim is for the claim's hypothesis to be true while its conclusion is false. i.e. if A is true and B is false, then "A implies B" is false.

Now again, Pinocchio made a conditional claim. And because all his claims are lies, his conditional claim must be disproven. The A must be true and the B must be false. The A is "I own a hat" and the B is "it's green." Thus, he MUST own a non-green hat in order for his claim to be a lie.
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>>14594444
waste of digits, see>>14594703
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>>14593766
>All my hats are green
That could be a lie for all of the following:

>I have no hats
>I have one green hat and one red hat
>I have no green hats

A - Can't be proven, he may have no hats.
B - Can't be proven, he may have no green hats or multiple green hats.
C - Can't be proven, he could have hats that aren't green.
D - Can't be proven, he may have no green hats.
E - Can't be proven, he could have multiple green hats.

Ultimately, the question is ambiguous. Does "always lie" mean every portion of the statement is a lie, or does just one portion of the statement being a lie fulfill the criteria since it's still a false statement?
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>>14594703
>He's strictly making a conditional claim: "If I own a hat, then it is green."
No anon he's speaking in plurals, he's claiming that if he owns any hats they are all green.
You seem to have misread it.
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>>14595011
>he's speaking in plurals
Yeah that's a universal quantifier. All it does in this case is transform "if I own a hat, then it's green" to "for any hat, if I own it, then it's green." Where it does matter is that when you negate a universally-quantified statement, you get an existentially-quantified statement. i.e. "There exists a hat such that I own it and it isn't green." Which as you'll note also confirms option A.
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>>14593766
Either he has no hats, or he has a hat that isn't green.
None of those options in the image can be stated with certainty.
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>>14595097
>"for any hat, if I own it, then it's green."
He's not claiming to possibly own a hat, he's claiming to own a hat. He could either be lying about having more than one hat, about owning any hats, or about the color of the hats. He can't be lying about all three, because if he owns no hats, then color isn't involved, in which case he has a hat, which could indicate he was truthful about having a hat, which contradicts that he always lies.
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>>14593801
Pinocchio can't lie unless he actually has a hat, but that doesn't mean it has to be materialised. The hat must be there for him to stipulate in his sentence, either materially (>0) or merely conceptually (0). Otherwise there would be nothing to base the subject of the sentence on to begin with in order to lie.

BUT, you made the mistake of assuming that a lie must be disproven with material evidence here, when he can make a lie without having a hat at all.
Lies are not just established here on the basis of him having a hat. We would generally call having nothing and then stipulating that you have something a "lie" too.

He must have "greater than or equal to" zero hats that are not green.

Either "Pinocchio has a hat that not green or has no hat" in common english or potentially "Pinocchio does not necessarily have a hat that is green" (which is rather oddly vague or even a misused style of wording, but still a way to word it - albeit badly and in an ambiguously wishy washy way. It works only because of the word "does" alone I think. "May not necessarily have" does not negate the phrase enough, though neither does "does not necessarily have" really in my opinion.).
Nobody says "greater than or equal to zero hats that are not green" in common tongue.

>>14594021 is correct regardless.
But I just wanted to point out the phrase "does not necessarily" which is strange as a phrase to me.
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>>14594438
isnt human language itself based on some universal grammar/logic ? otherwise it would be impossible to convey unambiguous information. like even if some languages are SVO and others are SOV , you still have subjects, objects and verbs
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>>14595287
>isnt human language itself based on some universal grammar/logic ?
Absolutely not.
Have you ever tried to learn Japanese.
The logic is very esoteric with some of their interpretations of their characters and words.
Hell, you try describing "moe" to an english person who doesn't know animu and it's basically impossible. There's just too much water.
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>>14595242
>He's not claiming to possibly own a hat, he's claiming to own a hat
No, he's not. He's made a claim about every member of the set of all hats he owns, but he hasn't claimed that that's a non-empty set.
>He could either be lying about
There's no "either" to it, he's only made a single claim. If A, then B. That's one proposition.
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>>14595293
>>14595293
logic is logic. knowledge can be esoteric. yeah if you dont have the necessary information/background you cant understand this logic but it still has to be there, otherwise these constructions couldnt be passed on from person to person
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>>14595301
To be fair, the english language is lacking array A[0] which is required in order to describe this situation of having "0 or more" hats better.
Well played C and C++.
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>>14595301
>If A, then B. That's one proposition.
No, it's A and B.
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>>14595311
If you don't know a single logical imperative or any way of using one, then it becomes logic that is esoteric to you.
This is why we prove something, then reject it with evidence stating otherwise later and appending the logic later that we communicate and agree upon.

The devil is there, but the logic is just a mere compliment we think for it at the time. We will always come up with new ones. But regardless the devil is there demanding your compliments and that you lick her butthole.
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>>14595301
>There's no "either" to it, he's only made a single claim.
>All
>My
>Hats
>Green
One statement, but four points that need proven. There's no "if" involved. The question is written so that it's impossible to solve unless you add criteria to the question that doesn't exist, which is when the arguments start.
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>>14595281
>you made the mistake of assuming
No.
>We would generally call having nothing and then stipulating that you have something a "lie" too.
He never stipulated that he has a hat. He only stipulated that if he does have a hat, it's a green hat.
>He must have "greater than or equal to" zero hats that are not green.
No, sorry. If he has zero hats, then his conditional proposition isn't false, contradicting the assumption that his statement is a lie. QED.
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>>14595335
>He never stipulated that he has a hat.
"All my hats are green" is stipulating he has multiple hats AND that they are green.

>He only stipulated that if he does have a hat, it's a green hat.
The question is not "If I have hats, they're all green." That question is much more concise, there would be no debate, and this thread wouldn't exist.
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>>14595318
No Anon. The truth value of A and of B are independent of the truth value of (A => B), though of course the reverse doesn't hold.
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>>14595329
>One statement, but four points that need proven.
That's not how propositional logic works. The formula is whatever it's specified to be. All, My, Hats, and Green are what they are, there's no proving them. All is a quantifier, hats is a variable, my and green are functions. When combined in the manner specified, they yield a conditional proposition. The logic of the problem further specifies that that conditional proposition is false. A conditional proposition's falsity necessarily implies the truth of it's hypothesis AND the falsity of its conclusion.
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>>14595337
>"All my hats are green" is stipulating he has multiple hats AND that they are green.
No, it isn't. I have zero hats. All of them are green. Are you suggesting that zero isn't a number?
>there would be no debate
There isn't room for debate. There's room for people who have studied logic to give the correct answer, and people who haven't to learn something. This isn't even a difficult problem. Any math major who's made it halfway through their third year can tell you that it's practically trivial.
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>>14595406
>No, it isn't. I have zero hats. All of them are green. Are you suggesting that zero isn't a number?
I'm arguing C is not the only possible conclusion. He could have 50 red hats and one green hat.
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>>14595487
C isn't a possible conclusion at all. If he has no hats, he isn't lying.
>He could have 50 red hats and one green hat
correct, which is covered by conclusion A, like I said all the way at the beginning. >>14593801
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>>14595500
>If he has no hats, he isn't lying.
It's a lie because he says he has hats.
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>>14595511
>he says he has hats
Okay, let's try it like this. He says he has x hats, and they're all green, right? Where does he say that x can't equal zero?
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>>14595517
If I tell you something exists that doesn't exist, do you consider that a lie?
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>>14595518
He doesn't claim that something exists. That would be an existentially quantified claim. He makes a universally quantified claim, which is literally the opposite of an existentially quantified claim.

https://en.m.wikipedia.org/wiki/Universal_quantification
https://en.m.wikipedia.org/wiki/Existential_quantification
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>>14595529
Okay, so if I told you all of my charity donations are huge, when in fact I've made zero donations, you'd consider that to be an honest statement?
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>>14595532
>honest
No, because a statement can be true and still misleading.
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>>14595535
Well the donation amount cannot be huge because there was never a donation. That seems to be a lie, not just misleading.
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>>14594405
Thank you /pol/. truly our greatest ally
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>>14595559
But you never actually said there were donations. You merely described an attribute of any and all donations you made. It's misleading to allow someone to mistakenly infer from that description that the set of all your donations is non-empty, but it's still a true statement nonetheless.
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>>14595570
>You merely described an attribute
I lied about the attribute. Whether the donations exist or don't exist, they cannot be huge. It's like dividing by zero.
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>>14595577
No, it's not like dividing by zero. Dividing by zero is undefined. The logic for this situation is very well defined, it's merely counterintuitive.
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>>14593766

All of you are faggots who failed to learn simple deductive logic in class.

There are two certain truths in this problem:

1. Everything Pinocchio says is a lie
2. Pinocchio said "All my hats are green."

This means, that Pinocchio's statement, "All my hats are green," is a certain lie.

This is abstracted away in to the negation of the declaration that all objects of a certain set (Pinocchio's hats) have a certain property (the color green).

This means that he never stated how many hats he actually has, and that it can be a set of any size of zero, where he doesn't own any hat, or an infinite set of hats. It doesn't matter, because he never stated the size of the set.

This means, for however many hats that Pinocchio owns (n), he can have between 0 and n-1 green hats and still be lying.

Immediately we can rule out A, because that would exclude a set of size 0.

We can rule out B, because having a single green hat in a set of one green hat would mean that all of his hats are green, therefore he didn't lie.

We can rule out C, because that would exclude sets of hats that are 1 or greater.

This leaves option D: He has at lease one green hat and option E, he has 0 green hats, which are both applicable under different circumstances.

For hat sets of size 2 or greater, option D can apply, because he can still lie about it.

For any hat set of any size, including a size of 0, option E can apply, because none of the hats in his set of hats would be green.

I would choose E: Pinocchio has no green hats because it is the only option among the five provided that holds true for all possible sets of hats that Pinocchio could own, from a set of 0 to a set of infinity.
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>>14595579
Apparently what I'm describing is informally called a vacuous truth.
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>>14595583
>All of you are faggots who failed to learn simple deductive logic in class
lol. lmao.
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>>14595583
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>>14595593
Nobody can prove that he has at least one hat because he never made a statement about how many hats he has.

He can have zero hats (less than the supposed "at least one"), say to the world that all of his hats are green, and he would be lying about it, therefore fulfilling both truths while invalidating the truth of A.
>>
e
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>>14595588
They're not the only curve balls that logic will throw at you, either. Common sense, it isn't. I truly believe that everyone owes it to themselves to at least study classic syllogistic logic, if not symbolic logic. If for no other reason than because it's good practice to avoid being misled in a similar manner.
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>>14595596
>say to the world that all of his hats are green
In the real world it's a lie, but it's a vacuous truth in mathematics.

Vacuously true statements

A statement is vacuously true if the premise is false or cannot be achieved. For example, consider a room with no mobile phones. If someone says, “All mobile phones in the room are turned OFF,” then that would technically be a true statement. After all, there are no mobile phones in the room, so one could say all of the ones in the room are turned off. But for the same reason, one could also speak a vacuously true statement, “All mobile phones in the room are turned ON.”

Both statements are technically true, but they are meaningless because there are actually no mobile phones in the room.

Let’s apply the principle to the Pinocchio problem.

(C) Pinocchio has no hats.

If Pinocchio has no hats, then any statement he makes about the hats is vacuously true. Thus the statement, “All my hats are green” is technically a true statement. But we know Pinocchio always lies, so this is an impossibility.

Therefore, Pinocchio cannot have no hats; he must have at least one hat. We can eliminate answer choice (C) and the only correct answer is (A) Pinocchio has at least one hat.
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>>14595596
Precisely so. In fact, he never said he has any hats at all. All he said is that every member of the set of all his hats has the property of being green. That statement isn't a lie (false) if the set is empty.
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>>14595605
I still don't get it entirely, honestly. I'll let it sink in and come back to it.
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>>14595608
You have state that any declaration of properties regarding a set is vacuously true. Saying "all phones in the room are off" is just as true as "all phones in the room are on," if all phones in the room are off, then saying that they are all on is a lie, and visa versa.

Similarly with his hats, they could all be green, or they may not be.

I can argue that he is in fact presenting the opposite of the vacuous truth that "None of my hats or green" or "Not all of my hats are green," therefore making the statement "All of my hats are green" a vacuous lie in the same vein that it is a vacuous truth.

You cannot determine if he is telling a vacuous truth because the negation of a vacuous truth. A negative zero is still zero.
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>>14595629
Not the Anon you replied to but
>You have state that any declaration of properties regarding a set is vacuously true
Only if the set is empty.
>Saying "all phones in the room are off" is just as true as "all phones in the room are on," if all phones in the room are off, then saying that they are all on is a lie, and visa versa.
Vacuous truth is only entailed if there aren't any phones in the room. Again, we're specifically talking about empty sets.
>I can argue that he is in fact presenting the opposite of the vacuous truth that "None of my hats or green" or "Not all of my hats are green," therefore making the statement "All of my hats are green" a vacuous lie in the same vein that it is a vacuous truth.
You can argue anything you want if you arbitrarily change the parameters of the problem. But then you're no longer talking about the same situation as everyone else.
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>>14595641
>Only if the set is empty
I intended to type such, but my mind outpaced my fingers.

To clarify what I was saying, when it comes to empty sets, he can say whatever he wants, because any property he assigns to a vacuous set is both a vacuous truth and a vacuous lie. Rather, it's truthfulness is only defined by the context in which it's interpreted.

You can say it's a vacuous truth, and I can say it's a vacuous lie, and neither of us are wrong and both of us are wrong.
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>>14595641
Or, if you want to be more autistic about it, a vacuous truth only exists because it's antecedent is false.

The antecedent "All my hats" with a set of zero hats is false, there simply him saying "All my hats" is a lie, regardless of what property he assigns to them.
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>>14595583
Wrong because that's nonsense linguistically and this isn't a mere logical deduction problem.
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>>14595654
>when it comes to empty sets, he can say whatever he wants
correct
>because any property he assigns to a vacuous set is both a vacuous truth and a vacuous lie
No, it's just true. If you try to make it false, then you end up with a system of logic in which the conditional operation is indistinguishable from the conjunction operation.
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>>14595608
Bullshit, Pin is a liar so how can you trust any statement, about how many hats he has, at all?
He could just be bullshitting and he actually has no hats.

I bet he'll shiv us when he goes to show us his hat collection.

Don't trust them puppets, they're full of shit.
Let one in, you let them all in. We need a puppet wall to keep them out of America.
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>>14595671
>a vacuous truth only exists because it's antecedent is false
correct
>simply him saying "All my hats" is a lie
No, because he never claimed that the antecedent is true. He only claimed that if the antecedent is true, it necessarily follows that the consequent is true.

The only way that statement can be false is if the antecedent is true yet the consequent is false. So, in order to ensure that Pinocchio's statement is a lie, it's necessary that the antecedent be true.
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>>14595681
If you have to assume something to make a truth, it's not a truth.
Why is that called a truth? That makes no sense.

Unless you somehow obtain empirical evidence to suggest otherwise, it's simply not a truth. It's a hypothesis.
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>>14595692
>If you have to assume something to make a truth
It's not an assumption, it's a deduction. It necessarily follows from the stated specification that Pinocchio always lies. It's the only possible permutation of truth values consistent with all specified parameters.
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>>14595698
>It's not an assumption, it's a deduction.
I don't care what you call it. It's not a truth and it won't establish a truth without evidence.

God this would never stick in court.
You have to conclude, which infers establishing a solid truth. And the only truth there is here is that "pinocchio lies". That means his logical statement is useless.

Why do we teach this in science? It's fucking nonsense. At most, you can "spin" something there. But you are "spinning" in the sense that you are "making up crap".
Unless you somehow have the power to say things in red, of course.
>>
>>14593766
you're missing having had any basic human interaction with the opposite sex
>>
>>14595672
"Nonsense linguistically?" If you can't understand the problem or the solution I provided, then, I don't know, go to /lit/ and take some reading lessons.

>>14595674
>>14595681
Because the antecedent of the hats is false, you cannot use it as a way to infer a meaningful statement, because it has no meaning.

Besides, E can still be used to describe if Pinocchio had a set of zero hats, if he has no hats, then he can't possibly have any green hats anyways.
>>
>>14595715
see >>14595707
You can't assume anything from the statement pinhead says. You can't even assume the context of anything in the statement of "All my hats are green" is going to be remotely relevant to any conclusion you make and you cannot make any decisions on it. It is a "junk statement".

How do you know he's even talking about his own hats or "hats" at all if all he says is garbage?
Of course, that's not what most people interpret from that statement right?
We assume it has relevance to the context in the streets.
But realistically, calling him a liar means his statement is not worth anything as a statement to make a decision upon.
>>
>>14595733
If you're going to argue that nothing he said be trusted in a legal sense because he's a liar, then maybe it would be better if you lurked over on the political board rather than the science & math board.

This problem was posted on /sci/ because it is intended to be a problem with a solution to be deduced utilizing some form of math or logic.

Or maybe OP is the pinhead we can't trust that you're talking about and this entire thread is just autists having a brain aneurysm over a faggot that lies.
>>
>>14595707
>God this would never stick in court.
Actually, math and philosophy majors rank at the top of the LSATs.
>it won't establish a truth without evidence
Okay, so there's these things called axioms and these other things called rules of inference.

Axioms are basic statements of truth. They're not proven, they're the basics that you have to assume in order to prove anything. Things like "A is A" or "if A is B and B is C, then A is C".

Rules of inference are formulas for drawing conclusions. They're also not proven, they have to be assumed. Things like
>if A, then B
>A
>therefore B

You are not arguing against anything that needs to be proven. You are arguing against the basic rules that we assume in order to prove things. You are by all means free to do this. It may even be that such a logic has already been developed - I don't know much about paraconsistent logics and other more esoteric things of that nature. But at the level of logic that's used to define things like any kind of mathematics you've ever used, the rules are as I've said.
>>
>>14595293
means fire
>>
>>14595389
Anon if an illiterate peasant who always lies, told you that "all of his degrees are firsts" is it likely that he even has a degree?
You cannot assume from the statement that the speaker possess anything in the first place.
>>
>>14595850
*But you also can't assume that he possesses nothing.
>>
It seems easy.
We know that Pinocchio always lies
so when he says all my hats are green - it's a lie
which means
>not all his hats are green

This means he has
1) at least one hat that is not green
2) he has no green hats at all

However Pinocchio mentioned the green color, why?
Does this mean he has at least one green hat?
Or he just picked the color randomly?

The only thing we can conclude is that Pinocchio has at least one hat.

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>>14595854
Here anon, all of the bridges that I own are blue. If you want I can sell you one of them, very cheap only tree fiddy.
>>
>>14593766
>Am I missing something?
That there are many ways to lie.
The puzzle has no correct answer, because in either of those cases what he said would be a lie.
>>
>>14595865
There is a difference between a statement and a hyperbolic sarcastic statement which is OBVIOSLY a lie.
Unfortunately most people can't tell one from another which is why i'm banned from tinder and bumble for violent threats.
>>
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>>14593766
> logic puzzle
>Dude what if this impossible thing is true ?
>Logic wins again !

This is literally useless and of no value whatsoever in the real world there are no people who only tell lies 24/7 this is literally a useless skill.

>It's obviously not B, D, or E, so A and C are left.
You expect me to even take this garbage seriously ?
>>
>>14595892
>there are no people who only tell lies 24/7
you would be surprised.
People do tell lies 24/7. People actually lie to themselves and don't even realize.
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>>14595893
Cope harder. Logic belongs in the trash.
>>
>>14595895
I am already in the trash, might as well use logic
>>
>>14595898
You are dumber then the dumbest village idiot.
Even rednecks can understand that
>What practical use is this even?
Is the first thing you should think about.
>>
>>14595899
It's entertainment, there is no practical use for it.
Just like other 99% of activities humans do.
>>
>>14595902
>It's entertainment,
Then extremely dumb entertainment. With the possibility of making you even more stupid.
>>
>>14595903
Every form of entertainment is dumb man. There is no goal in it besides fun you're having in the process.
Even science is entertainment and a waste of time.
>>
>>14595904
>Even science is
>>14595887
>>14595887
>>14595887
>>
>>14595905
I've argued before that Karl Popper's falsifiability criterion is absolute cancer and must go.
The only reason why it exists is because Popper was a pozzed kike on a payroll and turning science into business was his goal.
>>
>>14595899
Logic is extremely useful. The fact that OP's example is contrived and stupid and doesn't work due to the imprecision of natural language is not a condemnation of logic.
That's like saying calculus is useless if a textbook uses a stupid, contrived example to illustrate it;
>>
>>14595913
>I've argued before that Karl Popper's falsifiability criterion is absolute cancer and must go.
Based.
What do you think about the theory that science dies in the 1960s and now it is all academics creaking out nonsense paper after nonsense paper to fulfill their minimum publishing criteria ?

>falsifiability criterion
Another thing that makes no sense whatsoever.
>>
>>14595919
>Logic is extremely useful
It is actually not and logic self contradicts itself AKA logic proves that logic is useless.

>That's like saying calculus is useless if a textbook uses a stupid, contrived example to illustrate it;
Only you can show that 2kg + 2kg = 4 kg etc you know real things.

Can you show the same for logic ?

>That's like saying calculus is useless if a textbook uses a stupid, contrived example to illustrate it;
Good thing I am a mathematical fictionalist you have a world of trouble if you are not one.

>to the imprecision of natural language
Then WTF is it referring to I wait.
>>
>>14595937
Poppers thinking makes no sense to me whatsoever. Every time I try to understand him I feel like vomiting.

He was either insane schizophrenic with a cooked brain or just deliberately evil scumbag. It’s just DARK, man.
How the his bullshit is so influential I have no idea.

> What do you think about the theory that science dies in the 1960s and now it is all academics creaking out nonsense paper after nonsense paper to fulfill their minimum publishing criteria ?
What do you mean theory? It’s the objective truth, everyone knows about it, nobody talks about cause if you ask too many questions on the conference you might cause someone to lose funding.
It’s just how it is, people do useless and pointless research for money.
>>
>>14595943
>What do you mean theory?
I mean it is a theory lots of things are theories to know if it was different we need to basically time travel before 1960.
>>
>>14595943
>He was either insane schizophrenic
Tell us more.
>>
>>14595950
You follow someone’s thoughts and realize the thing goes so deeply out of touch with reality it makes you feel bad physically.
You feel kinda disconnected from the real world and stuck inside you own mind and then the entire thing turns into unintelligible chaos where you can’t tell correct from incorrect, right from wrong and it’s just bullshit.
If that’s not what insane mind looks like I don’t know what it is.

It doesn’t happen when I try to understand Richard Feynman or Socrates for instance. Cause these people are fine. Hell, Einsteins logic makes sense somewhat.
>>
>>14593766
E, obviously. If Pinocchio has any hats they are not green.
>>
>>14595960
What made you the most offended in his writings ?
>>
>>14595583
This seems to be the correct logic. The only thing required to make the statement a lie is that Pinocchio owns no green hats. Whether or not he owns any other hats at all is completely irrelevant. If he owns no hats it's a lie, if he owns a million red hats it's a lie. The statement's truth hinges entirely on the color, not the quantity.
>>
>>14595583
brainlet
>>
>>14595968
I don't like that his philosophy is basically elitarism or fascism.
Aka
>fuck the unenlightened, no need to educate them, and help them, actually let's turn them all into cattle and abuse them forever
>>
>>14593766
r/confidentlyincorrect
>>
>>14595871
lol
>>
ITT: brain trust that can't figure out a child's logic problem stomps their feet and pouts "logic bad!"
>>
>>14593766
The picture of Pinocchio has him wearing a yellow hat. A is the obvious answer.
>>
>>14593766
>Are Brazilians retarded?
Um... oh dear... anons who should break the news to OP?
>>
>>14595892
>This is literally useless and of no value whatsoever in the real world there are no people who only tell lies 24/7 this is literally a useless skill.
The skill part is using logic (and stretching the brain in the process), it doesn't have to lies and truths, rather, it's a process for evaluating a situation or equation to form a valid conclusion.
>>
>>14593801
The negation of a "for all" quantifier is "there exists" hence the negation of All of Pinocchio's hats are green is Some of Pinocchio's hats are not green. So none of the given are implied
>>
>>14593766
You can't infer any of those...

He could have one red hat. Many hats but none green. Or no hats. Or many hats but only some green. Can't be determined at all...
>>
>>14596139
If a dude says "all my hats are green" and he has no hats, you'd say that's truthful. Motherfucker, he doesn't even HAVE a hat, let alone a green one.
>>
>>14596501
A situation who will literally never happen because you get others using reverse psychology on you.

So outside of magic characters who magically can only tell lies .. how the fuck is this useful ?
>>
>>14596585
"Hey bro I paid our rent all in \$100 bills."
>Has not even paid the rent
"Guess you were telling truth bro."

>>
>>14593766
>question is tricky bc it hinges on some sort of semantic ambiguity
into the trash
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>>14596610
>into the trash
See>>14595892
>>
He obviously must have at least one non-green hat because he’s wearing a yellow one in the picture faggots
>>
>>14596580
>A situation who will literally never happen because you get others using reverse psychology on you.
Anon, you need logic for everything, whether it's investing, networking, mathematics, science, engineering...
>>
>>14596139
In normal conversation, the statement "All my hats are green" is not interpreted as being equivalent to "If there exists a hat such that it belongs to me, that hat is green.". "All my hats are green" means "I have several hats and they are all green".
>>
>>14596651
No, it really is the former.
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>>14596651
In a normal conversation, your conversational partner wouldn't be a wooden doll that is able to speak and tells only lies.
>>
>>14596585
How do you know what colour his non-existent hats are? He says they're green, and you can't check, because they don't exist. You should take his word for it.
>>
>>14596649
>Anon, you need logic for everything
Stop making up shit.
>Anon, you need ROMAN CATHOLICISM for everything, whether it's investing, networking, mathematics, science, engineering...
>>
>>14596648
The picture of Pinocchio serves no purpose except to show him with a hat on. All of the answers are inconclusive originally, but knowing from the specifically given information that he’s wearing a yellow hat (A) becomes conclusive. There is no reason to illustrate what he looks like otherwise, as it wouldn’t matter
>>
>>14596661
>The correct answer to this problem is what I said it is.
> what I said it is.
Behold the power of logic !
>>
>>14596664
>In a normal conversation, your conversational partner wouldn't be a wooden doll that is able to speak and tells only lies.
See: >>14595892

Forget the wood part there are no people who only tell lies 24/7 and even if there are you can not prove that they will not start telling the truth in the future.
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>>14596672
I'm sorry you're too dumb to tell when you're being made fun of.
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>>14596687
>Logic is so universal
>THE CORRECT ANSWER IS WHAT I SAY IT IS !
>BECAUSE IT IS !

And you are not a dogmatic religion how exactly ? Remember the same shit was done by Roman Catholicism and Buddhism AKA think what the right answer is and if you tell me anything that is not 100% what I think it is I beat you with a stick and tell you to think more we repeat this until you get the right answer.

If it is only a collection of whatever you think is the right answer it is not universal.
>>
>>14596689
I'll stay, someone has to make fun of the hysterical brainlets.
>>
>>14596680
Now of course you may be thinking, “B-But it says we can conclude FROM THESE TWO SENTENCES!! It specifies!!!” But then why supply the fucking image with the hat? Is that not, then, what he looks like standing in front of us telling us this? Is this not extra contextual information that should be taken into account?
>>
>>14596705
But what if the hat is green and the image was digitally altered (by Pinocchio himself, to throw you off guard) in order to trick the reader?
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>>14596711
Because the image was supplied by the Brazilian Mathematical Olympiad of Public Schools and not Pinocchio
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>>14593766
If he has no hats, is his sentence a lie?
I mean, if he has no hats, then all his hats are green, blue, ... any damn colour he chooses. All true statements, right?
But let's say you ask him a question, "what colour are your hats", and he has no hats, an honest person would say "I have no hats", whereas a dishonest person would say "all my hats are green".
From mathtistic point of view, the "honest" statement is actually incorrect and is not answering the question correctly, while the "dishonest" statement is logtistically true.
>>
>>14596754
Dozens, huh? No worries, buddy, you'll learn to count when you get to kindergarten.
>>
>>14596775
Yes, you should go back to studying, dear.
>>
>>14593766
I couldn't even get past the first two lines.
I dont get it.
We are asked to assume that Pino always lies. Therefore no statement he makes can be true. Yet we are then asked to assume that his second statement "All my hats are green" is true. Which is a outright contradiction of the first assumption.
How does that make any sense? Can someone explain it to a brainlet like me?
>>
>>14596792
>Yet we are then asked to assume that his second statement "All my hats are green" is true.
No, you're asked to assume that the statement "he said x" is true.
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>>14596515
>The negation of a "for all" quantifier is "there exists"
correct
>hence the negation of All of Pinocchio's hats are green is Some of Pinocchio's hats are not green
correct
>So none of the given are implied
incorrect, if he has at least one non-green hat then he must have at least one hat, so A is true.
>>
>>14596656

If someone told you "all my hats are green", and he doesn't even have a fucking hat, in what world do you think he was therefore telling the truth? All his hats are devoid of color because he doesn't even have one. They don't exist. Lmao.

I'm gonna start telling women all my Ferraris are blue.
>I don't even own a Ferrari.
It's okay though, when they see I just have a FIAT they'll realize I was telling the truth.

>PS
Know how I know he has at least one green hat? Because he's wearing one in the fuckin' picture.
>>
>>14595892
>This is literally useless and of no value whatsoever in the real world
Actually, this is what makes the magic box you're reading these words on work. All the wizards who work on these kinds of magic spells have to learn logic, but that doesn't concern you. Just leave the magic to the experts.
>>
>>14596925
>Know how I know he has at least one green hat? Because he's wearing one in the fuckin' picture.
That's a yellow hat.
>>
>>14593766
Was this question originally even asked in English? There might be nuance in Portugese that is going to be lost in translation. What defines a "lie" (and dishonesty, etc) isn't identical in all languages. The answer in English is either A or none of them, dependent on whether dishonestly implying that you have hats at all counts as a lie.
>>
>>14596651
Doi. What sort of autismal fuck starts doing math to decode the guy's message? Words and language are used to convey a message. The use of that phrase implicitly claims he owns hats. Any English speaker would interpret it as such.

These people would make awful homicide detectives...
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>>14596941
Well it certainly ain't "honesty"... The thing is you still can't actually infer any statement for certain, since there are multiple possibilities for that lie... Maybe he has one hat and it's red, maybe he has hundreds and they're all blue. Both of those render his statement a lie... So both would be possible... You can't say any of the options are certainly correct.
>>
>>14596942
>What sort of autismal fuck starts doing math to decode the guy's message?
The kind that considers logic to be math and not philosophy.
>>
>>14596976
It's a linguistic "puzzle"... And even if having zero hats would render the bullshit statement true (though I'd say the imaginary zero hat can't have any color since only things that exist have color), there are still multiple possible lies on that list, so the question doesn't even work.

AKA it's ESL riddles.
>>
>>14596991
Oh I see you could infer A... It's still ESL bullshit.
>>
>>14593801
Fpbp, this is a basic question about formal logic. Anybody who has ever learned it already understands that phrasing this type of logic in English leads to confusion, this problem is mentioned in basically every textbook about the subject. There is a reason that symbols for these logical statements exist. Rephrase the question in notation for mathematical logic and there is no ambiguity, which it is obvious you are supposed to do because the pic states this question is from a math contest.
>>
>>14596996
Actually Anon, this is that "critical thinking" thing you sometimes hear about. Here's another example:
If this is ESL bullshit, then only ESL students would have difficulty with it.
Therefore, if this is ESL bullshit, you're an ESL student.

Note that this doesn't imply that you're an ESL student. It implies that you're either an ESL student or else you're wrong.
>>
>>14596676
So you believe mathematics and engineering can be learned without logic?
>>
>>14597093
It's ESL to think no hats would make the statement true... I just gave the "benefit of the doubt" and assumed it true and thought the question would still fail, as I forgot the list of answers...

There is no native English speaker who would interpret it that way. And how does a non existent hat even have a color? Only material things can even possess a color.

It's a total fail because it's apparently meant to be a math problem, yet is written as a linguistical riddle...
>>
>>14596927
>Actually, this is what makes the magic box you're reading these words on work
I'm a programmer you retard.
Tell me what questions about hats from magic wooden men are used to make a processor.
Aha Aha I see.
You have no fucken idea !

This is not a XOR operation or anything not that you bit flip like this anyway or that explaining a nested negation is difficult you fagot.

>Terminate loop when game is not not not working.

Literally watching this video is more enriching to your intelligence then anything about your retarded logic puzzles.
>>
>>14596942
>These people would make awful homicide detectives...
>>14596955
>Well it certainly ain't

>homicide detectives...
>homicide detectives...
>homicide detectives...
And here is the thing magic humans who only tell lies do not exist IRL and no homicide detective assumes this. Literally learning about real life homicide detectives or what they are learning is more important then this crap.

It is literally the definition of vapid shit.

The scenario is shit and everyone instinctively knows this it involves actors who can not exist.

>Muh computers
No fucken computer will give you
>I only tell lies
>I have no green hats
You retarded dumb fuck !
>>
>>14597116
>can be learned without logic?
YES.

Prove logic is even useful or what it even is.
>>
>>14593766
>t. 86 iq
>>
>>14597257
So when you're given an algorithm, and you're then given a question that requires a tweak of the algorithm, how do you figure out what adjustments to make? Magic?
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>>14593766
Fucking retards it's obviously C, it says that Pinocchio always lies and if he has any hats then that means his assertion that he even has any hats is true and thus he doesn't always lie which contradicts the rules of the puzzle.
>>
The idea that there might not be any hats is nothing but overthinking.
It’s an equivalent of
>a random person asks you for a cigarette
>you don’t smoke
>assume that he thinks you smoke
>assume that you look like you smoke
>assume that you look like shit deadbeat loser
>assume it was an insult
>get mad at person and call him a faggot in response
>the guy goes like “what the fuck”

It’s thinking too much where thinking is not required and looking for pitfalls where none exist. Very midwit trait.
>>
>>14597737
It's not overthinking, it's the only possible answer according to the rules.
>>
>>14593766
(((Pinocchio)))
>>
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>>14597676
Mind blown
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>>14597800
It’s paranoia.
The answer is simple. Not hard.
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>>14593801
>Pinocchio's claim is not a lie if he owns zero hats).
What planet do you live on? Retard, he made up the colour for these "hats". Negate yourself.

It is C. Nor the hats and colour were true. The whole fucking sentence is a lie, every word of it.
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>>14597820
You're so paranoid that you think the rules of the puzzle are fucking lies.
>>
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>>14597826
>>
>>14597826
This is probably the best thread I've ever read. All the ESLs from planet poppadom unable to decipher basic English comprehension.

That son of a bitch has one hat and it's yellow. He's wearing it in the photo. I'm bout to beat his ass.
>>
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Look at this cocky motherfucker, lying about his fucking hats.
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>>14597833
It’s simpler than you make it up to be
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>>14597849
You have no argument, you just pretend you do.
>>
>>14597850
My argument is that anything beyond surface level obvious in this puzzle is meaningless paranoia
>>
>>14597853
Then you're retarded because C is the surface level obvious answer.
>>
>>14597861
When people ask you what a chair is they don’t ask you for it’s chemical composition, you fucking autist schizophrenic, lmao. Kill yourself by repeatedly stabbing yourself with a fork in the eye, don’t forget to jump of a cliff in the process, subhuman.
>>
>>14597241
liar
>>
>>14597873
All my statements are lies.
All my beatings of you will be quick and painless.
>>
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>>14597873
>strawman
>projection
>extreme anger
The truth has a toxic effect on you. NPCs will be NPCs I suppose.
>>
>>14597601
Give example.
Also show where magic characters who will only tell lies start to talk about hats show up.

>how do you figure out what adjustments to make? Magic?
Thinking (you get a bootstrapping error if you argue otherwise).

>how do you figure out what adjustments to make? Magic?
I think neural networks debunked your crap.
>>
>>14593766
The question seems to hang on whether "All my hats are green" would be a false statement if Pinocchio has no hats. I think it would be, since Pinocchio would be implying that he has hats where he has none. Therefore, none of the choices can be concluded.
>>
>>14593766
He either has no hats or ( he owns one non-green hat and possibly more hats which may be any color ). Therefore it's a troll puzzle, because this choice isn't listed. Most of these puzzles are trolls, written by math students as a joke.
>>
>>14594021
I thought it was this clear. Is there something I'm not getting?
>>
>>
>>14598423
Based actual native English speaker.
>>
>>14594462
the definition of a lie isn't strictly a statement that's untrue, a statement that's intended to deceive is also by definition a lie
>>
>>14598903
>He either has no hats or ( he owns one non-green hat and possibly more hats which may be any color ). Therefore it's a troll puzzle, because this choice isn't listed. Most of these puzzles are trolls, written by math students as a joke
Only correct and intelligent comment on this thread/
>>
>>14593766
For one thing, in Brazilian logic A AND NOT A isn't a contradiction.
>>
(F) We can't know whether this liar has any hats, but if he does, then at least one of them isn't green.
>>
(F) If Pinocchio has hats, 1 hat can't be green.
>>
(F) If hats > 0, green hats < hats
>>
>>14593766
He is part owner of half a hat which itself has some green pin stripes but is otherwise iridescent-beetle colored
>>
>>14593766
it was instantly clear to me after reading all possible answers that none could be concluded. But since it was "a tricky question", I rechecked it over and over again, still thinking none could be concluded. But if I had to pick one, it would be A, because the question seems to be indirectly implying that he had at least one hat he was referring to while lying. I still think this question is wrong though, which is a testament to Brazilians being retarded, considered it was selected to the national math Olympiad.
>>
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>>14593766

since he always lies, he has also learned to be a credible liar.
he must be wearing a green hat while making the statement.

>>
>>14600490
bingo. the pictures shows him wearing a green hat too. are the test writers also compulsive liars?
>>
>>14600518
What makes you think the hat is his?
>>
>>14600529
while its in his possession, it is his hat
>>
>>14600328
>Brazilian logic
Soooooo, bullshit, essentially?
>>
>>14600529
Because he's a custom made marionette. No normal hat would fit his head, but as we see this one is snug. It's reasonable therefore to conclude it's his barring some really bizarre circumstance (he has a twin puppet he stole it from).
>>
>>14594703
>because every claim he makes is a lie, that conditional claim must be false.
it doesn't say it must be false, it says it must be a lie, which is not necessarily false
>>
>>14593766
The correct answer is that none of these can be deduced.

But if the question absolutely required you to pick one, then it must be C because then both sentences are true. Unless we are getting into the matter of English semantics where lying is not the same as something being false, in which case this entire question is absolutely retarded and on the same level of bullshit as stuff like Monty Hall and assorted "trick maths questions".
>>
>>14593766
Both sentences are true.
Pinocio is a lier.
All his hats are green.

Thus he has
B: only one hat

The lie was the quantity stated by him, not the colour
>>
>>14598396
>Thinking (you get a bootstrapping error if you argue otherwise).
Aka logic?
>>
>>14599956
>>14600918
>The most widely accepted definition of lying is the following: “A lie is a statement made by one who does not believe it with the intention that someone else shall be led to believe it” (Isenberg 1973, 248)
https://plato.stanford.edu/entries/lying-definition/
It's not necessary that a lie be false, but it is necessary that the agent telling the believe it to be false.
>>
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>>14593766
Oh a logic puzzle with no practical use? Sorry bud, busy with something productive
>>
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>>14593766
Pinocchio is a metaphor for Jews who wish to become human but no matter what they do, they can't help but lie and expose their true nature as symbolized by the growing of Pinocchio's nose

so the answer is "(C) Pinocchio has no hats." since he would need to buy them
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>>14602477
again not necessarily, the language isn't that exact, the core of a "lie" seems to be in the deliberate deception, not the untruth
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>>14602669
Can you cite an example? It should be literal, done by an agent rather than an inanimate object (not e.g. "the mirror never lies"), and unqualified (not e.g. "lie by omission").
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>>14602692
i mean, i am trilingual but two of the languages i am fluent in are irrelevant so from experience there is not much reason to bring them up as they would just make this conversation awkward, however even in normal english a "lie" is defined as primarily deception
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It is E: Pinocchio has no green hats.

If everything he says is a lie and he is claiming not only that he has a hat but has many hats and they are all green, then he can't have a green hat and may not even have a hat. Since you can know the color of the hat is not green but you can't know or even if he has a hat, the only statement that could be true is that he has no green hats, which itself could mean not only that his hats are not green but that he may not have any hats at all, but does not negate that he may have a hat.
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>>14602705
>even in normal english a "lie" is defined as primarily deception
It's not though.
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>>14602705
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>>14602705
>>14602737
Note how in that second definition, that sort of matches with your claimed meaning, it's an inanimate object that's lying.
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someone post more puzzles
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>>14593766
None of these options are guaranteed. The person who wrote the problem is a midwit.
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>>14593801
You seem to have missed the important phrase "we can conclude." We can conclude A, but (as you write) not B, C, D or E.
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>>14594131
>>14594432
>>14594438
If you wanted to be really clever, you should note that this is from a mathematical olympiad and in that context the correct interpretation is

~(∀x, PH(x) -> G(x))
which is equivalent to
∃x ~(PH(x) -> G(x))
and
∃x PH(x) ^ ~G(x)
which implies
∃x PH(x)
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>>14595592
lol, your mom is a bone vac
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>>14593766
I conclude that Pinocchio said what he said and that what was said was a lie.
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>>14604391
If the participants haven't seen formal quantifiers yet, then I'd argue finding/guessing this interpretation (which I agree is the correct interpretation given the context) is 99% of the problem.

If the participants have already seen formal quantifiers, then this question is trivial and wouldn't be on a contest.
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>>14604423
That's a good point.
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>>14604423
I agree that this question doesn't really fit on a contest. But even if the students have had enough exposure to elementary logic to translate the problem as >>14604391 did, I wouldn't consider the inference steps trivial. For example, neither $\neg\forall x\,\varphi\vdash\exists x\,\neg\varphi$, nor [math]\neg(\varphi\to\psi)\vdash\varphi\wedge\neg\psi[\math] hold intuitionistically.
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>>14604801
> doesn't hold intuitionistically
I completly agree, that's why I say that 99% of the work is figuring out the intended interpretation, which in this case is classical first order logic.

If someone knows enough to differenciate between classical and intuistionistic logic and has seen enough math to know that classical first order logic is the "default" logic in math, I feel they should at least know the De Morgan laws for quantifiers which makes the inference for this question trivial.

But I think this question is intended for people who have never seen logic in a formal way before. I think the author assumed that first order logic is "obviously" the one and only "correct" interpretation. I don't think people realize the amount of subjective reading comprehension that needs to happen before the objective math can start.
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>>14604943
>I don't think people realize the amount of subjective reading comprehension that needs to happen before the objective math can start.
I've seen this quite often. It's mildly infuriating if you're teaching some kid or being asked for advice and all you can tell them is: "The question is worded poorly, it was probably meant like this."

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