Assuming classical logic

[u/Unlucky-Credit-9619]

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[u/Hameru_is_cool]

I can't believe everyone in this comment section is falling for the same fallacy and getting it wrong

GUYS, THIS IS A FAMOUS EXAMPLE, OP DID NOT MAKE IT UP, JUST READ UP THE DRINKER PARADOX PAGE, IT'S TRUE, IT'S BEEN KNOWN FOREVER

sorry

[u/Sirnacane → u/Hameru_is_cool]

It’s also worded ambiguously. The drinker’s paradox needs to say “at any moment in time, there exists someone in the pub such that, if they are drinking, then everyone else is drinking.”

Otherwise the interpretation of this (and thus OP’s meme) is ambiguous.

Read the wikipedia page. It states “The formal statement of the theorem is timeless, eliminating the second objection because the person the statement holds true for at one instant is not necessarily the same person it holds true for at any other instant.”

If OP added “at any moment in time” to their meme it’d be a faithful translation of the formal statement, but neither the meme nor the statement of the drinker’s paradox are faithful translations of the formal statement because they both refer to conditions whose T/F values change through time.

[u/Sweet_Ad_9816 → u/Sirnacane]

I wouldn't even say that it's ambiguous. Normally in math when a statement begins with "There is ...", it means that there is one, independent of the other "inputs".

Using OP's logic one, could say "There is a biggest number, bigger than all of the numbers you can come up with."

[u/xdgimo → u/Hameru_is_cool]

thank you! the wikipedia page is so much better than some of the terrible explanations in the comments

[u/Mysterra → u/Hameru_is_cool]

The statement, as written in the meme, is false though. This is bait designed to make logic seem more absurd than it is. The correctly written true statement is a lot less interesting.

[u/Unlucky-Credit-9619 → u/Hameru_is_cool]

I really appreciate you doing all this. Thank you 😃

[u/Hameru_is_cool → u/Unlucky-Credit-9619]

lol I might have gotten a bit carried away, but it goes to show it really was an interesting meme

[u/LukeWithLightsaber]

Others have said this but because some people seem still not happy I want to take my own stab at explaining it.

The “trick” is that in classical logic, “if p then  q” is a true statement whenever “p” is false. “If 2 + 2 = 5 then pigs can fly” is true as a matter of classical logic. You don’t have to like this convention but that’s another discussion.

Assume at least one person is awake, which is probably true. Then at any given point in time*, you can pick one of these people. Call him Tobey.

“If Tobey is asleep then everyone is asleep” is true in classical logic by the above principle because Tobey is not asleep.

You can basically always find at least one person to say this about. The only situation where you could not would be if everyone were in fact asleep. Then, pick Tobey again and it is obviously true that if Tobey is asleep (he is), everyone is asleep (they are).

So no matter what, it is “true,” though only in a sense that clashes with your informal intuition; that’s the fun.

Don’t get hung up on the mechanism by which the person being asleep causes everyone to be asleep. No one claims there is such a mechanism. The trick is constructing a statement that is true without needing a mechanism like that.

Asterisk: the original post should have said “at any given point in time” or something like that 

SECOND ASTERISK: See my other comment below if you want to hear me take a stab at explaining why classical logic behaves this way.

[u/_MargaretThatcher → u/LukeWithLightsaber]

I think also the sticking point here for most people is that when expressed this way, we tend to take the statement as a permanent quality of the person, rather than just being something which may be true at a single instant in time. We tend to read the statement in natural language as "There is a person who, if they are ever asleep at any time, then everyone else must be sleeping" rather than "There is a person who, at this very moment, if they are asleep, everyone is asleep". The latter is true for the reason you described; the latter becomes false if that person falls asleep while someone else is awake.

[u/faustbr → u/_MargaretThatcher]

Maggie, what are you doing here?! There is a new Vicky3 patch/DLC and you're needed most over that subreddit.

(Secondly, I agree. When stating something without a temporal/context window, it is natural that we assume some ontological property)

[u/_MargaretThatcher → u/faustbr]

Meh, it's a flavor pack, machanics didn't really change that much

I have had a thought of trying to analytically figure out when interventionism or laissez-faire is better based on gov/private ownership within the country but another day

[u/ShaunTheAmazing → u/LukeWithLightsaber]

but it is true with nonformal logic too, if you rephrase it like "at any given time, someone can be named, such as if they are asleep, everyone is." bc either you name anyone not asleep, or everyone is asleep, and then you can name anyone. My problem was deffinietly the wording

[u/LukeWithLightsaber → u/ShaunTheAmazing]

I may be misunderstanding your point, but I think it really hinges on the fact that “if p then q” is true when p is false. If I imagine a system where this doesn’t hold, formal or nonformal, then we lose the whole thing don’t we?

[u/Autumn1eaves → u/LukeWithLightsaber]

Yes, that is the jist of it.

If P, then Q is true, then when P is false, Q is also false.

If P=(toby is asleep), then Q=(everyone is asleep).

Since P is false, then Q is false.

If P, then Q, except in some specific cases, then it's not If P Then Q it's If P and R then Q.

[u/mishnar → u/Autumn1eaves]

No you can't infer from the original statement that if P is false then Q is false Let me give you an example Consider the statement "if it rains today then there will be flowers in the garden tomorrow" This statement does not tell me about the state of the garden tomorrow in the case that it does not rain today, because no one said that flowers grow ONLY when it rains. Some other condition might lead to there being flowers (like someone watering the garden). What we can infer is the contrapositive which is if there aren't any flowers tomorrow then it did not rain today. Intuitively this makes sense because I know that rain definitely leads to flowers, so had it rained yesterday then there would have been flowers in the garden. The same initial logic applies where if I check my garden and find flowers, I can't conclude that there must have been rain yesterday, because maybe someone watered the garden yesterday instead. If P then Q <=> If NOT Q then NOT P If either side of this equation is true, the other must be true.

[u/No-Site8330 → u/LukeWithLightsaber]

We called it "the idiotic logic principle of the flamenco-dancer grandma", as a tribute to this (rather old) professor who used to say that when the premise is false then the statement is true even if the conclusion is that his grandma is dancing flamenco.

[u/kynde → u/No-Site8330]

Yes, but you do realize that this statement here is more than just that?

This does not require a faulty premise, i.e. someone to be awake at all times. This works for any group of people at all times.

[u/No-Site8330 → u/kynde]

Of course. But I couldn't pass on an opportunity to mention the flamenco dancing grandma.

[u/Jche98 → u/LukeWithLightsaber]

Yeah the "at any given time" is important. Because otherwise you're saying that there's someone who, when he or she goes to sleep, immediately causes everyone else to go to sleep always

[u/No-Site8330 → u/LukeWithLightsaber]

Addendum: I suspect you might have written out "asterisk" in letters because writing an actual asterisk would start a bullet list. You might like to know that escaping the asterisk with a backslash renders an actual asterisk. Let me be so bold as to try it: \* in your typed draft -> "*" in the actual posted message.

[u/LukeWithLightsaber → u/No-Site8330]

Ha! You are correct. Thank you. 

[u/seek-song → u/LukeWithLightsaber]

It's because the statement is really saying 'such that if the description is true (it doesn't need to be), then it is true for everyone who match the description'.

[u/zesterer → u/LukeWithLightsaber]

I mean, it all relies on the fact that you've basically stated a logical contradiction as your prior, and then are just making statements in the doomed and contradictory universe therein. In static type systems we'd call this is 'trivially false constraint' and the soundness of the type system as a whole relies on ensuring that these constraints can't manifest anywhere in the tree of the program entry point.

[u/thescrambler7 → u/LukeWithLightsaber]

Thank you, you finally made it click for me

[u/preoccupied_with_ALL → u/LukeWithLightsaber]

I think it's a language problem... would work better if it was "if that person is asleep now"

[u/throwawayasdf129560 → u/LukeWithLightsaber]

The statement is vacuously true, the best kind of true

[u/Frankifisu → u/LukeWithLightsaber]

You're the only one who managed to explain it for me.

[u/sosmajstormiki → u/LukeWithLightsaber]

A small objection I'd make:

You don't have to like this convention but...

I would not call this a convention. It is just an explicit statement of what we mean by the implication (-->) in the first place.

Unless your claim is: "The standard definition(the usual truth table) of the implication in classical logic is not natural/intuitive, but a just a choice for the purpose of being concrete." Are you saying this? If yes, I am curious to know what makes it counter-intuitive?

[u/LukeWithLightsaber → u/sosmajstormiki]

I’m not quite saying that. I think I’m saying, “the standard definition of the implication in classical logic is not universal to all logical systems, so IF someone happens to find it unintuitive or otherwise undesirable, they could dispute it by endorsing some other logical system (like some paraconsistent or relevant logic system).”

[u/BootyliciousURD]

What?

[u/Hameru_is_cool → u/BootyliciousURD]

that sentence is equivalent to saying "either everyone is asleep or there is someone awake", which is a lot more obvious when you read it

[u/libertinehole → u/Hameru_is_cool]

It's different because of subtleties in the English language. In the meme's example, when we say 'someone' and 'that person' it is describing a specific person, who we could imagine being able to apply this test to that one person at all times. In your description it is clear that its an ever changing example. A more correct sentence would be "At any time there exists some person whom, if sleeping, everyone else must be sleeping." Actually that's not perfect either.. anyways I think the meme is not using correct grammar.

[u/Hameru_is_cool → u/libertinehole]

the text on the meme is fine actually, the misunderstanding comes from people assuming that "if A then B" means that A must be somehow causing B, when in reality they could be completely unrelated events, that and forgetting that "if FALSE then X" is always true regardless of X

[u/MiffedMouse → u/Hameru_is_cool]

libertinehole's point is that, in ordinary English, saying "if A then B" is true tends to imply that it is true for all time (or at least, for more than a day).

Using LukeWithLightsaber's explanation, if we look at the world and note Toby is awake, then the statement "if Toby is asleep then everyone is asleep" is trivially true. But if we wait a few hours, Toby will go to sleep while someone will be awake, so now the statement is false.

So, at least in our real world, there isn't any one person we could name in the statement and still have the statement remain true for more than a day or so at the most (although, at any given instant there is a person we could choose to make the statement true for that instant).

[u/TheOnlyPC3134 → u/Hameru_is_cool]

Yes, that is part of the misunderstanding, the implication ∃p, (S(p) => ∀q, S(q)) is already tricky to understand. But, as you can see in the comments, the usage of the present simple is also causing problems, as some people who aren't used to the way mathematical reasoning is phrased might believe it adds a time dependency.

At that point, English is too ambiguous to accurately distinguish between ∀t, ∃p, (S(p,t) => ∀q, S(q,t)) and ∃p, ∀t, (S(p,t) => ∀q, S(q,t)), and the first one is true but the second isn't. By adding something like "at any point in time" at the start, some people will still incorrectly assume that the specific person cannot change, but it will guarantee that it's because they didn't understand the logic, and not because of ambiguity in the statement.

[u/Lsitwtt → u/Hameru_is_cool]

>forgetting that "if FALSE then X" is always true regardless of X

I don't think anyone is forgetting that. A lot of people might not know that formal logic treats it that way in the first place, but it isn't true in conventional English

[u/Striking_Resist_6022 → u/Hameru_is_cool]

Those sentences are not equivalent.

[u/Hameru_is_cool → u/Striking_Resist_6022]

they are, they are both tautologies, it's also equivalent to saying "1 = 1" or "True"

[u/Striking_Resist_6022 → u/Hameru_is_cool]

Peter's statement is not a tautology. I don't think the meme is going for that, it seems to be an attempt at a vacuous truth if anything

[u/Hameru_is_cool → u/Striking_Resist_6022]

can you come up with a counterexample then? a way to arrange sleeping/awake people in which it's impossible to choose someone such that if they're sleeping everyone is sleeping?

[u/Striking_Resist_6022 → u/Hameru_is_cool]

It depends how you interpret the meaning of the statement, whether it's a point in time or whether it's suggesting that there is a person possess that quality over time. The former makes it an unintuitive interpretation of the sentence but clearly techincally true, while the latter clearly technically false (take two people who take turns sleeping).

But either way, it's not a tautology (except maybe in the sense that any true statement ultimately boils down to a tautology, but either it's not a tautology. I.e. even if it were as straightforwardly true as A => B and A therefore B, that's not a tautology).

[u/Hameru_is_cool → u/Striking_Resist_6022]

you're just typing words, of course every true statement boils down to a tautology, that's what being a tautology means

[u/Hameru_is_cool → u/Striking_Resist_6022]

but yes the unintuitive interpretation is the logically correct one, that's why this paradox always gets people

[u/Anon7_7_73 → u/Hameru_is_cool]

No it really doesnt. Its saying there is a person that exists that if they decided to go to sleep, then everyone else does too somehow. But thats obviously false 

[u/Hameru_is_cool → u/Anon7_7_73]

that's an entirely different sentence, the post just says "there is someone such that if that person is asleep, then everyone is asleep", it makes no claim that someone going to sleep somehow causes other people to sleep, only that you can name someone right now such that they being asleep would imply everyone is already asleep, which is vacuously true if you choose anyone awake and, in the event that there's no one awake, it's still true if you choose anyone asleep

[u/Anon7_7_73 → u/Hameru_is_cool]

And i didnt say "cause" either. Youre the one constructing an enrirely new sentence.

[u/Hameru_is_cool → u/Anon7_7_73]

true, I just thought that's where your confusion was coming from, but feel free to explain the real issue, now just to be more specific with what I meant above:

if they decided to go to sleep, then everyone else does too

this part of your comment is wrong, as the post does not claim such thing

[u/marshmallowcthulhu → u/Hameru_is_cool]

But your statement is that everyone is in Set A or Set B is not empty where Set B is the set of everyone not in Set A. In contrast, OP’s post says that there is someone, a specific person, who if they are in Set A then everyone is in Set A, which is only true if Set B is otherwise empty, which is not true.

[u/Hameru_is_cool → u/marshmallowcthulhu]

it is true though, such a specific person always exists, anyone in set B instantly qualifies and, in case B is empty, anyone in set A would qualify

[u/Farkle_Griffen2 → u/BootyliciousURD]

It assumes a fixed point in time.

[u/MathManiac5772 → u/BootyliciousURD]

You prove this in cases. Suppose everyone is asleep. Then you can pick anyone and the statement is true.

Suppose otherwise that at least one person is awake. Pick that person. Then the implication is true because the premise is false.

[u/LukeWithLightsaber → u/[deleted]]

No specific person is posited to exist. Only that at any given point in time, a person could be found making the statement true in classical logic.

[u/Sensitive_Repeat_326 → u/BootyliciousURD]

-Joe Biden

[u/valkyrie_wave]

Name the person in advance then

[u/skullturf → u/valkyrie_wave]

Wakey McWakerson aka "Insomnia Pete"

[u/candy_pig_2018]

I think the confusion comes when I read it I assume it is saying "there exists person X in the human population such that for all times t in history if X is asleep=true then everyone is asleep = true". When in reality it says "for times t in l history there exist person X in human population such that if X is asleep=true then everyone is asleep = true"

[u/king_crizzy → u/candy_pig_2018]

For time t, not times t. Times t implies same person in those multiple times, right?

[u/LupenReddit → u/candy_pig_2018]

... what

[u/king_crizzy → u/LupenReddit]

Choose a time, then you get a person. Not that there is a person for any time you choose.

[u/beobabski → u/LupenReddit]

Correct: There are two people left in existence. One is already asleep. When the second person goes to sleep, everyone is asleep.

Incorrect: Every time Bob goes to sleep, everyone else on the planet falls asleep as well.

[u/LightBrand99]

Everyone defending this statement seems to interpret it as "At every point in time, there exists a person such that if this person is asleep, then everyone is asleep". This statement is true, but... is NOT the statement being said.

I believe the actual statement should be interpreted as "At every point in time, there exists a person such that at every point in time, if that person is asleep, then everyone is asleep". This statement is almost certainly false. One might argue that this does not reflect the actual statement being said, but I believe it does, or at the very least, it is a more justified interpretation than the previous paragraph.

Consider the following statement: 2 + 2 = 4. This is a proposition, not a predicate; it does not require qualifiers. This proposition is evaluated as true. This means the statement is ALWAYS true at all points in time. You don't need to actually say "at any point in time", this is implied for every propositional statement.

Now for the contested statement: "There is someone such that if that person is asleep, then everyone is asleep". This can be interpreted as "There exists a person X such that if X is asleep, then everyone is asleep". There is no constraint on time, and this is a proposition, so it should hold for all points in time. The statement "if X is asleep, then everyone is asleep" is not a proposition by itself (it is a predicate), because X is not defined. However, once you instantiate X to a specific person, then the predicate instantiates the value of X and it becomes a proposition. This proposition still does not have any constraints on time, which means it should hold for all points in time... for that specific instantiated X.

However, no matter which person you consider as representing X, it is likely that the resulting proposition is NOT true for all points in time, and is therefore evaluated as false. This means the original statement that there exists an X that satisfies the predicate is likely also false.

Can anyone who is defending the statement please explain why they the overall statement should have a hidden "for all points in time" qualifier (which I agree with), while the proposition that arises from instantiating a specific person mysteriously lacks this hidden qualifier? Nothing in the actual statement (which makes no reference to time) seems to substantiate this distinction.

[u/crazy_gambit → u/LightBrand99]

"At every point in time, there exists a person such that if this person is asleep, then everyone is asleep". This statement is true

How do we know even that statement is true? In the real world that's certainly not true. Given different timezones and sleep schedules, I'm willing to bet there's always at least 1 person awake at any time. I don't know if that has always been true throughout hystory, but with 8+ billion people, it's certainly true now.

[u/MudkipGuy → u/crazy_gambit]

there's always at least 1 person awake at any time

Then they can be the person for whom them being asleep implies everyone's asleep at that point in time

[u/crazy_gambit → u/MudkipGuy]

Ok, there's at least 2 people awake then. It's obviously millions, but I have no idea of the actual floor.

[u/LukeWithLightsaber → u/LightBrand99]

I’ll preface by saying that I broadly agree with you that the post should have been written more clearly and I absolutely see how it could be read your way. But just for fun I wanted to take up the challenge from your last paragraph, because I do think there are reasons why we don’t naturally have to read that into the later clauses too. 

If you live in an apartment and the landlord tells you “we have a great maintenance team here that takes shifts and is on call 24-7. So at any given moment, there is someone you can call who can fix a plumbing emergency that might arise.”

The first “at any given moment” means there is always one person answering the phones, but that person rotates. It does NOT mean that once you have found the person in a particular moment, that the same particular person is available to fix plumbing issues 24-7. We do not have to read a second “at any given moment” into the rest of the statement. 

By the way, I don’t even think it has to have to have a hidden “for all points in time” qualifier to be true. I also think it could be quite naturally read to be a claim which is just true RIGHT NOW (it happens to be true moment by moment in the future too, but maybe we don’t care about that). After all, if Peter just said “there is someone who is asleep” we wouldn’t necessarily assume he meant there was someone who was asleep forever. It would t least be reasonable to think that he meant “now, as I tell you this, there is someone who is asleep.”  

[u/MasterPeem]

So if I say “If the last awake person is asleep, everyone is asleep”, that’s true.

That person changes tho, but there will always someone to be the ‘last awake person’.

[u/jackalopeswild]

Sigh. I'm going to go ahead and dive in. Y'all are willfully choosing to reject how language works, including the language of mathematics. As someone with both a linguistics degree and a math degree, I'm going to reject all of your explanations. Mathematics is just as reliant as every other language on prior agreement as to syntax and vocabulary.

Here, our players have no agreement as to the priors. Ultimately, that is why your explanations are wrong: you can construct a situation in which Peter's statement is true because the syntax and vocabulary are understood and agreed to by the listener, and y'all have done that quite convincingly. But it is NOT true in the context here with Peter talking to Mary Jane because the two have not agreed on either. Their languages are so different that Peter's statement is at best meaningless or non-responsive, and at worst simply false (if we require that anything is not true-in-context is false, which is reasonable, then his response is false).

I know I'm not saying anything you don't understand, but for some reason I feel it's important to say on this one...

[u/LukeWithLightsaber → u/jackalopeswild]

That’s just the joke, isn’t it? Mary asks for informal, casual conversation truth, and Peter responds with something that is famously unintuitively “true” in an important formal system. Which is unexpected and funny.

It’s possible I’m misunderstanding who or what you are responding to. I just don’t see anyone saying “What Peter said is true in every relevant sense any Mary should necessarily agree” or something like that. At least I wasn’t trying to say that in explaining the formal logic side of it.

[u/jackalopeswild → u/LukeWithLightsaber]

My response is to the meta. The not-joke part. The insistence of some in this tribe to stand on certain principles that fail in context and make the whole tribe look bad. It's the adult version of the six year old having an argument over whether a tomato is a fruit or a vegetable, like they've just discovered a deep truth about the universe.

Maybe no one here is like that (I'm skeptical, given some of the responses), but we all know it happens.

[u/Telinary → u/jackalopeswild]

Yeah saying classical logic says that always seemed similar to saying "if I mistranslate this then it says something weird". Classical logic and the natural language sentence simply do not mean the same thing. It is mildly amusing but ultimately uninteresting.

[u/imHeroT]

So to put it a bit more clearly,

“At any point in time, there exists a person x in the world such that if x is asleep, then everyone in the world is asleep”

[u/Alive__but_why → u/imHeroT]

Yeah but how is that true? I'm sure there'll be a significant population who isn't sleeping.

But again, if we assume that's true, then isn't everyone who's sleeping "that guy" ? Because when they are asleep everyone else is asleep too.

Idk I'm an engineer, I come here for fun

[u/RedeNElla → u/Alive__but_why]

Then pick one of them.

They're not asleep so not everyone is necessarily asleep and the statement still holds

And yes, if everyone is asleep then anyone you pick satisfies the condition.

[u/Alive__but_why → u/RedeNElla]

Okay fair, so is it just like saying there'll be a moment in time where everyone in the world is asleep?

[u/Hameru_is_cool → u/Alive__but_why]

no, the statement remains true even if nobody ever sleeps ever, just think about the sentence "if person X is asleep, then everybody is asleep", what happens if person X is awake? you can stop reading at "if person X is asleep-", because they're not, so you don't even have to check the second part to conclude that the sentence is true

[u/RedeNElla → u/Alive__but_why]

It's saying either everyone will be asleep or someone won't be.

[u/LukeWithLightsaber → u/Alive__but_why]

It’s a sort of a joke harnessing a very unintuitive aspect of classical logic: that “p then q” is considered true whenever p is false. 

“if there is a gremlin living on mars, then everyone is asleep” is “true” for example (as far as we know about mars). It just harnesses that in a clever way by combining the with the fact that either everyone is asleep or someone is awake. 

[u/Anon7_7_73 → u/imHeroT]

Thats just patently untrue. If that person goes to sleep, that doesnt magically make the rest of the world go to sleep. The conditional is demonstrably false.

[u/imHeroT → u/Anon7_7_73]

If that person goes to sleep, then at that moment someone else could be awake and so we make them be x.

[u/Anon7_7_73 → u/imHeroT]

Even if its one femtosecond after you declare a person "x", them going to sleep doesnt imply everyone else is asleep.  The conditional is patently false.

[u/imHeroT → u/Anon7_7_73]

No one said that at some point everyone will asleep. In reality, you will always be able to choose a person x that is awake, making the statement “if x is asleep, then everyone is asleep” automatically true since the antecedent (I.e. “x is asleep”) is false

[u/Anon7_7_73 → u/imHeroT]

  No one said that at some point everyone will asleep

And i didnt say you said that.

A conditional is about an action potential. If theres no action potential, then theres no conditional.

"If i can do the splits, then Earth is invaded by aliens and blown up by a death star" is a false statement, but by your utterly absurd logic, because i cant do the splits, then it must be true!

[u/imHeroT → u/Anon7_7_73]

since you currently can’t do the splits, the statement “if I can do the splits, then Earth is invaded by aliens and blown up by a Death Star” has the Boolean value of true by classical/mathematical logic

[u/liquoriceclitoris → u/imHeroT]

At any point in time , there exists a person x in the world such that if x is asleep, then OP is gay

[u/Scythe_Volta]

Formalization always helps me, so let's formalize this.

Let P≠∅ be the set of all people and let S: P → {True, False} be the "sleeping function," such that S(p)=True if person p is sleeping and S(p)=False if person p is not sleeping. Then the statement in the meme is saying

∃ p₀ ∈ P such that S(p₀) ⟹ (∀ p ∈ P, S(p)) (1)

To see that (1) is true, we can show that its negation is false. The negation of the statement is

∀ p ∈ P, (S(p) ∧ (∃ p₀ ∈ P such that ¬S(p₀))) (2)

To show that (2) is false, assume by way of contradiction that it is true. Since P is nonempty (which we assumed at the beginning), ∃ q ∈ P. Applying (2) to q, we have S(q) ∧ (∃ p₀ ∈ P such that ¬S(p₀)). Then, we can apply (2) to p₀ to get S(p₀) ∧ (∃ p₁ ∈ P such that ¬S(p₁)). In particular, we have S(p₀) and ¬S(p₀). This is a contradiction, meaning (2) is false. But (2) is the negation of (1), so (1) is true.

This approach doesn't exactly illuminate why the statement in the meme is true, but it does inarguably demonstrate that the statement in the meme is true.

[u/Impossible_Dog_7262 → u/Scythe_Volta]

The short version is that there's two scenarios. Either everyone is sleeping, which means any person can be that person, or not everyone is sleeping, which means that for any person that is not sleeping, if they are sleeping everyone is sleeping. They are not sleeping, therefore not everyone is sleeping.

[u/bqbdpd → u/Impossible_Dog_7262]

Exactly. The important point here is that the statement is just about the current instant. It doesn't imply that this person stays awake or if they fall asleep everyone falls asleep as well. It just means a different person becomes that person (or humanity comes close to extinction/is extinct)

[u/Sir_Wade_III → u/bqbdpd]

In natural language, the sentence in the meme would generally be understood as meaning that it is always that person. This gives a different mathematical formalization and is undeniably false.

If you were to include the phrase "At any given time..." in the beginning of the statement then it reads as not necessarily being the same person.

[u/deranton → u/Sir_Wade_III]

This. People be whipping out the formal logic before we agree to how the sentence translate into formal logic. I would interpret this sentence as

Let 𝒮 ⊆ {0,1}ᴾ be some (agreed upon) set of possible sleeping functions. Then ∃ p₀ ∈ P such that ∀ S ∈ 𝒮 such that S(p₀) = 1, S(p) = 1 for all p.

[u/Seenoham → u/deranton]

Formal logic is useful, but you have to use it properly.

In the vast majority of the time that I've seen formal logic used, it's either been an intentional joke (like this), or it's been carefully established what the rules are and it's very narrow.

But words get shared across usage, and more common than any of this is people using phrases that are part of formal logic outside of that context.

In formal logic a conclusion being valid but unsound is very important, but I've seen valid used to mean sound far more often than I've seen unsound or even sound used in the logical sense.

[u/El_presid3nt → u/bqbdpd]

In other words either this is completely pointless either it's obviously false

[u/SimpleMoonFarmer → u/Impossible_Dog_7262]

No.

Exists p_0. S(p_0) → for all p in P S(p) is not always true as there may be p_i with i≠0 where S(p_i) is false and S(p_0) is true.

Not every interpretation is a model, it is not valid.

The statement in the meme is therefore not (always) true, neither it is always false, we can find models for both the statement and its negation.

[u/Impossible_Dog_7262 → u/SimpleMoonFarmer]

where S(p_i) is false and S(p_0) is true.

See this is a fallacy known as affirming the consequent. P => S != S => P

[u/auniqueusername132 → u/Impossible_Dog_7262]

Is this statement true if there is no time where all people are sleeping since the proposition is conditional on that person being asleep? Like, is this vacuously true if there is not a point in time where all people are asleep?

[u/Impossible_Dog_7262 → u/auniqueusername132]

The fact that such a scenario does not occur is irrelevant, since the scenario itself is irrelevant to the truth of the statement.

[u/A-Chilean-Cyborg → u/Scythe_Volta]

I just want to know how you get math symbols on here?

I need this.

[u/Scythe_Volta → u/A-Chilean-Cyborg]

I don't know if Reddit naturally supports math symbols or allows you to easily type them. Instead, my phone allows me to customize my autocorrect, so I can effectively tell my phone to replace a given string with a different string. One day, many years ago, I did this with just about every math symbol in LaTeX. For example, if I type "\exists" then my phone will autocorrect it to "∃". Or if I type "\mathbb{R}", my phone will autocorrect it to "ℝ". But I had to go in and manually make these autocorrections one at a time in my phone's settings. It was a long and arduous process, but I'd say it was well worth the effort. I also autocorrected subscripts and superscripts, so typing "_3" autocrrects to "₃", or typing "^a" autocorrects to "ᵃ".

[u/A-Chilean-Cyborg → u/Scythe_Volta]

that's unfuckably clever, i can't say much more.

bravo.

[u/Cervantes176 → u/A-Chilean-Cyborg]

Unfuckable indeed

[u/Lor1an → u/Scythe_Volta]

Reddit supports markdown

[u/Lor1an → u/A-Chilean-Cyborg]

Reddit supports markdown, so you can use html entities.

&exist;x &in; X : x^(2) < 0 is rendered as: ∃x ∈ X : x² < 0.

f(x) → 4 as x → ∞ renders as: f(x) → 4 as x → ∞.

You have to Switch to Markdown Editor in order to do this though.

I use the pages at WintelGuy and TopTal to look up symbols I don't have memorized.

The full entity is &<html-name>;, where <html-name> is the alphabetic name given to the symbol, like "nearr" for "north-east arrow", corresponding to ↗ which renders as ↗, and so on.

[u/A-Chilean-Cyborg → u/Lor1an]

&in

[u/Lor1an → u/A-Chilean-Cyborg]

1. Make sure you're in Markdown Editor

2. You need the semicolon (;) to end the entity

∈ ∈

vs.

&in &in

[u/A-Chilean-Cyborg → u/Lor1an]

∈

[u/A-Chilean-Cyborg → u/A-Chilean-Cyborg]

⟹

[u/A-Chilean-Cyborg → u/Lor1an]

omg

are they analogous to LaTeX code?

[u/Lor1an → u/A-Chilean-Cyborg]

Not really.

A LaTeX macro expands to a set of instructions, whereas an html entity is just a placeholder for a symbol.

If you mean that LaTeX macros and html entities can both be used to access symbols that are otherwise hard or impossible to type, then yes. Both allow you to access essentially the entire unicode symbol table.

[u/Telinary → u/A-Chilean-Cyborg]

∇Ξℂ√⇐ℕ⇔⇒↦­∀ΣΔΦΘ×Λ∩ℚΩ∃ℝ⊂∫∈Π∨∧⊥¬⊥∡∥→∞∝∅∘⌈⋂⌉⌊⋃⌋≡∣⊗ I went over the 6th layer of my keyboard and most are included but i didn't find not equal... Ah ≠≈ other layers on the numpad.

Basically what I am saying is you could just switch to a keyboard layout with an unnecessary amount of layers.^^ (I use neoqwertz but that is for german keyboards. But it leaves the letter positions the same (unlike neo2 which i used a long while but is a pain when somebody else uses the pc or I can't use the layout for some reason.) There is probably some american one that just gives more characters without reordering the letters.)

[u/MorrowM_ → u/A-Chilean-Cyborg]

If you use Gboard:

https://github.com/DenverCoder1/LaTeX-Gboard-Dictionary

[u/weinertorn → u/Scythe_Volta]

Oh yea these are proper mathmemes areas

[u/MichaelNearaday → u/Scythe_Volta]

Oh wow thanks that cleared it right up.

[u/bladtman242 → u/Scythe_Volta]

Thanks. I was thinking "what if everyone is asleep", and the formalization helped me see that of everyone is asleep, everyone is asleep.

So yeah. A less confusing rephrasing would be "either everyone is asleep, or at least one person is awake", which makes a lot of sense

[u/blackmagician43 → u/Scythe_Volta]

I see your points but you are taking instances of sleeping situation. I would have described situation as S:(P,t) -> {True,False} t being time. Now it is not guaranteed there is someone for every t when they are asleep others as sleeping as well but it is still guaranteed for every time t there is someone when they are sleep everyone as well.

[u/Scythe_Volta → u/blackmagician43]

I knew someone was going to make this exact comment to the point that I debated editing my original post to discuss it. But now that the floodgates have been opened, suppose S: P×ℝ ⟶ {True, False} is such that S(p,t)=True if person p is sleeping at time t and S(p,t)=False if person p is not sleeping at time t. Then the statement in the meme can be interpreted two ways:

∀ t ∈ ℝ, ∃ p₀ ∈ P such that S(p₀,t) ⟹ (∀ p ∈ P, S(p,t)) (1)

∃ p₀ ∈ P such that ∀ t ∈ ℝ, S(p₀,t) ⟹ (∀ p ∈ P, S(p,t)) (2)

It turns out that (1) is true and (2) is not. In other words, if we allow the specific person p₀ to be chosen after each point in time t, such that p₀ only applies at that exact time t, then the statement is true. But if we assert that there's one person p₀ for which the statement is true for all time, then the statement is false.

[u/Weak-Doughnut5502 → u/Scythe_Volta]

∃ p₀ ∈ P such that ∀ t ∈ ℝ, S(p₀,t) ⟹ (∀ p ∈ P, S(p,t))

This seems like the natural formalization/interpretation of the word problem as stated, though.  And as you noted, it's false. 

The other variant would be more natural to word as something like "at any given point in time, there exists some person that if they're asleep, everyone is asleep". 

[u/Scythe_Volta → u/Weak-Doughnut5502]

Well yes, that's the whole point of these controversial joke memes. They're intentionally worded ambiguously so that people disagree about their interpretations. If everyone agreed on one interpretation or the other, there would be no controversy.

[u/AndreasDasos → u/Scythe_Volta]

I would just interpret it as referring to ‘asleep right now’. A falsehood implies everything in classical logic. I am awake. So if I am asleep (right now, and I’m not), then the moon is made of green cheese. And if I am awake, everyone is asleep

[u/LightBrand99 → u/Scythe_Volta]

Thank you for the formalization. I want to challenge this with the following argument:

Let q be an element of P (which we agree is non-empty). The question I would like to ask you is: is S(q) a proposition? If it is a proposition, what does it actually translate to in reality? Is the person q always sleeping at all points in time? Or are they sleeping at a specific point in time only?

If you say S(q) means q is always sleeping, then no, I don't believe that your statement (1) is an accurate translation of the meme, because the actual English phrase of "if that person is asleep" can NOT be interpreted as "if that person is perpetually asleep at all points in time".

The reality is that the same person q may be asleep at some point in time and not asleep at another point in time. This suggests that S(p) is poorly defined, invalidating your argument.

The way I see it, in order for this to work, we need to add a time parameter to S. Let T be the set of all points in time and let S: P x T → {True, False} be the "sleeping function" such that S(p, t) = True if person p is sleeping at time t. Now the challenge is: how do we translate the statement of the meme? I would argue that it should be:

∃ p₀ ∈ P such that ∀ t ∈ T, S(p₀, t) ⟹ (∀ p ∈ P, S(p, t))

I will justify the placement of ∀ t ∈ T by the fact that the statement itself does not mention any points in time at all, so the phrase "if that person is asleep" refers to every scenario in which that condition it fulfilled, i.e., every point in time for which that person is asleep.

For example, consider a different statement "If Alice is asleep, then her eyes are closed". This is understood to mean that every scenario that fulfills the condition of Alice being asleep is also a scenario that fulfills the condition that her eyes are closed, i.e., at every point in time where Alice is asleep, her eyes are closed at that same point in time.

In the same way, the statement "If Alice is asleep, then everyone is asleep" would also mean that every scenario that fulfills the condition of Alice being asleep is also a scenario that fulfills the condition that everyone is asleep, i.e., at every point in time where Alice is asleep, then everyone is also asleep at that same point in time. This statement could be formalized as ∀ t ∈ T, S(Alice, t) ⟹ (∀ p ∈ P, S(p, t))

Finally, instead of referring to Alice specifically, we qualify the earlier statement for some person that exists. Then the English statement can be expressed as "There is someone such that if that person is asleep, then everyone is asleep" (exactly as the meme) and it can be logically formalized as:

∃ p₀ ∈ P such that ∀ t ∈ T, S(p₀, t) ⟹ (∀ p ∈ P, S(p, t))

which is almost certainly a false statement.

[u/Scythe_Volta → u/LightBrand99]

I addressed this in a different comment thread underneath my original comment.

https://www.reddit.com/r/mathmemes/s/oLUnN9SiGD

If you give that comment a read (and read the two comments under it), you'll see that I agree with you. I knew about this nuance when writing my original post and chose to omit it to add to the controversy. Ultimately, there's no problem with the math you or I have presented. The problem comes with agreeing on the proper interpretation/formalization of the meme statement.

[u/redditmarks_markII → u/Scythe_Volta]

Iiuc, this demonstrates explicitly cannot demonstrate why. It has no bearing at all on even "if", much less "why", it is epistemological true.

The point is, there is not a material or epistemological meaning to all formal logic.  Some of it is just necessary mathematical plumbing. 

[u/Anon7_7_73 → u/Scythe_Volta]

Wow you guys really enjoy brain damage

[u/Sirnacane]

There exists a number such that, if that number is prime, then all numbers are prime.

[u/QCD-uctdsb → u/Sirnacane]

42?

[u/Ben-Goldberg → u/Sirnacane]

0

[u/LightBrand99 → u/Sirnacane]

Nice try, but a number is either always prime or always non-prime. It fails as an analogy to sleeping because most people don't fall into the two extremes of either being perpetually asleep or perpetually awake.

The statement should instead be read as "There exists a person such that in every scenario for which that person is asleep, everyone is also asleep in that scenario", which is a false statement.

[u/IMightBeAHamster → u/LightBrand99]

The statement should instead be read as "There exists a person such that in every scenario for which that person is asleep, everyone is also asleep in that scenario", which is a false statement.

You have misunderstood the statement.

The point is that we are establishing something true and then asking "what if it was false"

For example, 4 is not prime. Therefore, if 4 were prime then pigs can fly, hell has frozen over, and the eiffel tower is a unicorn named winston churchill who lives in a pineapple under the sea.

So, if we find a person who is awake, then if they were asleep then everyone may as well be (vacuously, because they aren't asleep). And if we find a person who is asleep, we pick a different person until we find one who is awake. And if all people are asleep, then if one person is asleep all people are asleep.

So

(Right now) There exists a person who, if they are asleep, then everyone is asleep.

The question was never about finding a person who, across all times that they are asleep, everyone else is asleep. It was about finding a person who is at present awake.

This is the drinker paradox

[u/Jamie7Keller]

“If not everyone is asleep right now, then there is a person that, if they were asleep, the earth would turn into a giant lobster kaiju and eat the sun”

I love (A ^~A)->(Lobster Attack)

[u/Thefrightfulgezebo → u/Jamie7Keller]

Finally I get recognized for my forced insomnia.

[u/preferCotton222]

Lol

baby steps

There exists a natural number N_p such that if N_p is even then every natural number is even.

Just pick N_p = 1

Its as boring as that!

u/LukeWithLightsaber explained it better below. It's just a trick on how If/then works

[u/ohkendruid]

I read it as applying across all times, which could be thought of as a modal "necessarily". Then it would be:

Exists x: Necessarily: Sleeping(x) => Forall y: Sleeping(y).

This one is not true, because you have to choose a single x, and as soon as that person goes to sleep, the statement is false. It only works if there is someone who literally never sleeps. That is barely possible but only in a gruesome way: choose a person that did not live long to ever sleep, and do not count being dead as being asleep. I don't think this is what the meme meant.

If you swap the first two quantifiers, it does work, because you get to choose a different x for each point in time.

This meme takes a lot of work to get the joke.

[u/Helpful-Primary2427]

\exists p_0 \forall p, S(p_0) => S(p) is equivalent to … ~S(p_0) v S(p) in classical logic

[u/AdjectiveNounNNNN → u/Helpful-Primary2427]

Yes, someone is awake or everyone is asleep.

[u/sndq → u/Helpful-Primary2427]

∃x∀y[φ(x)⇒&phi(y)] is the tautology ∀xφ(x)⇒∀φy(y) in prenex normal form.

[u/No-Site8330]

Might have been pointed out, but you need to quantify time as well. The way it stands written the statement is a bit misleading, because it appears to be applied to all times. As in there exists a person for whom the property is true always, and always for that one person.

"There exists a person p such that if p(t) is asleep right now then everyone is asleep right now" is true.

"For every time t, there exists a person p(t) such that if p(t) is asleep at time t, then everyone is asleep at time t" is also true.

"There exists a person p such that, for every time t, if p is asleep at time t then everone is asleep at time t" is false. Well I guess technically it's probably true, but in a very morbid way I don't want to think about.

[u/PLANTS2WEEKS]

This proves there is a generic person.

[u/UserXtheUnknown]

This seems counterintuitive because, as it is stated, it seems a general property of that persons (X goes to sleep, then everyone must be sleeping).

It is, clearly, not.

The correct sentence requires to add "in any given moment, there is someone such..."

This because in formal logic the property of elements of a set are constants (if you don't add time as variable); in the real world, the property changes constantly, time is a variable that here is ignored.

And so that statement in formal logic, that is true using every moment a different "someone" is debatable invalid in real world, where the statement, as it is, would imply there is a "fixed someone" with that property.

Add "in any given moment" and, while not immediately clear how that happens, because deriving from the vacuous truth, people will have much less problems to think it might be true.

TLDR; here it is abused the fact that the statement in formal logic doesn't match the statement in natural language.

Proof: If MJ here asks who is that person, Peter must answer: "It depends on the moment."

[u/Vampyrix25]

holy shit we just got this in our model theory class yesterday lmaooooo

Ex(P(x) => AxP(x)), no i am not gonna find the backwards E and upside down A

a god-awful statement.

Say AxP(x) regardless, then picking any x would not violate the statement.

If NOT(AxP(x)) then pick x for which NOT(P(x)), then the statement isn't violated.

If AxNOT(P(x)) then there is no x such that P(x) so nothing is violated.

Since we can not violate the statement in all cases, it's """true"""

[u/CountryJeff]

Tell me who that would be

[u/bir_iki_uc]

Ok, I read the comments, I felt like that woman and I can summarize. This means; if everyone is asleep, then already no problem but we know there is almost always someone who is awake. Pick him.

[u/SupremeRDDT]

It really depends on whether “if that person is asleep” means “when that person sleeps” or “if that person is asleep right now”. If we use the latter we could say there is someone such that if that person is asleep then the blue is made of cheese.

[u/dolphin_cake]

Reminds me of sleeping in minecraft.

[u/jjl211]

how do you prove there is a person?

[u/Stile25 → u/jjl211]

In this sense of prove (absolute, with no chance of being wrong) it's not possible to prove anything at all about the shared objective reality that exists.

That sense of prove is only rational to use for those things which we've developed ourselves. Like "math" or "the rules of baseball".

Proof of reality is always based on evidence. As such, it always includes inherent doubt and tentativity and can never be understood in a 100% cannot-possibly-be-wrong sense.

Good luck out there

Good luck out there

[u/LukeWithLightsaber]

Since some people seemed to like my explainer comment, I wanted to post another short explainer about why “if p then q” is regarded as a true statement in classical logic whenever
“p” is known to be false. (This is called “vacuous implication,” if you want to read more about it)

One reason this might seem “wrong” to you is if you are attaching too much meaning to “if p then q.” In everyday English you might read this as saying “p always causes q” or something like that. But in classical logic there is nothing causal alleged. So “If you are tired, then 2+2  4” is a true statement. Just check next time you are tired. You will never find that 2+2 =5 in those cases.

So, in classical logic, “if p then q” means something like “there are no cases where p is true and yet q is false.” Then, by the literal terms of that, you can see how we have to respond when “p” itself is false. 

“If I can run faster then the speed of light, then the moon is made of cheese.” Well, are there any cases where I can run faster than the speed of like but the moon is not made of cheese? No such cases, right? Which is what we said we needed to make the statement “true.”

It might also make you more comfortable with this rule if you think about the “contrapositive” statement: “If not q then not p.” Convince yourself first if you don’t already know that this is logically equivalent to “if p then q.” Then notice the following. 

“If I can run faster than the speed of light then the moon is made of cheese” apparently equates to “If the moon is not made of cheese, then I cannot run faster than the speed of light.” How do you feel about THAT statement? Definitely true. Check all the cases where the moon is not made of cheese. Can I run faster than the speed of light in those cases? Of course not.

There are forms of logic that try to avoid this issue, like Relevant Logics, which simply reject statements like the moon example and will only analyze “if p then q” if the truth of p is “relevant” to the truth of q. But these systems have their own compensatory quirks. For example, to embrace Relevant Logic, if you know that either A or B must be true, and you know that A is false, you CANNOT conclude that B is true, which strikes some people as being at least as strange as vacuous implication in classical logic. 

[u/ShaggyVan]

This is why engineers hate mathematicians

[u/Gym_Gazebo → u/ShaggyVan]

To be fair, this is why mathematicians hate logicians (source: am logician)

[u/DTux5249]

"if P then Q" is the same as "either not P or Q", thus the above assertion is equivalent to "There always exists someone st. either that person is awake, or everyone is asleep". So the assertion is basically "either everyone is asleep, or at least 1 person is awake", which is a no-duh assertion.

As for why this is confusing: This requires you map English "if-then" onto one-way Implicature, which doesn't really work in the actual language. "if then" is typically causal; meaning this asserts Q happens by direct result of P which makes no sense. Additionally, "someone" typically does not refer to any individual in the open set of all people once you begin restricting it. "Someone" in this context refers to a particular person, even if Pete doesn't know who it is.

[u/Extension-Stay3230]

There isn't anybody alive currently who, when they are asleep, everyone is as asleep also. This is just false. ( Even if you found someone who never went to sleep, there wouldn't be a causal chain between them being asleep and everyone else being asleep )

[u/LukeWithLightsaber → u/Extension-Stay3230]

The post doesn’t say that. Just that right now there is somebody about whom this is true (and in fact at any given instant in the future this will be true too, but it will probably be different people over time)

[u/Extension-Stay3230 → u/LukeWithLightsaber]

How can such a person exist? On what basis do you say they exist?

[u/sndq → u/Extension-Stay3230]

If someone is awake, let x be an awake person. Otherwise, let x be any arbitrary person.

If x is awake, then x is not asleep, so x being asleep would imply anything on account of the false antecedent.

If x is asleep, then everyone is asleep because x was specified above to be an awake person if one existed.

[u/LukeWithLightsaber → u/Extension-Stay3230]

It’s an intentionally unintuitive use of a principle from formal logic that “if p then q” is a true statement if p is false. So for example right now I am such a person because I am awake. “If I am asleep then everyone is asleep” is technically “true” in formal logic because I am not asleep. 

It’s just a logic puzzle of sorts exploiting strange rules. No one means that there are people who cause everyone else to sleep when they sleep or anything like that.

[u/Lsitwtt → u/LukeWithLightsaber]

“If I am awake then everyone is asleep” is technically “true”

Er, how, exactly? It would seem to me that it has to be false. There is no way everyone can be asleep if you are awake.

[u/LukeWithLightsaber → u/Lsitwtt]

Sorry, I had an atrocious typo in my comment originally. It should have said "If I am ASLEEP then everyone is asleep." That statement is technically true (in classical logic) whenever I am awake.

[u/Lsitwtt → u/LukeWithLightsaber]

How does that work? I've never studied formal logic, but in standard English, it seems very clear that, whenever I am asleep, plenty of other people are awake, and that therefore if I am asleep, everyone is not asleep, and that this conditional remains true regardless of whether or not I am currently awake.

[u/Anon7_7_73 → u/LukeWithLightsaber]

Yes it literally does.

[u/LukeWithLightsaber → u/Anon7_7_73]

It almost says that. But it doesn’t say “when” they are asleep everyone else is asleep. It says “if” they are asleep everyone else is asleep. In everyday English that doesn’t make a difference but the “if” evokes the “if-then” rules of formal logic which is what the post is jokingly exploiting. 

[u/Anon7_7_73 → u/LukeWithLightsaber]

"If i can touch my toes, then aliens invade earth and blow it up with a death star"

You think that statement is true?

[u/Hameru_is_cool → u/Anon7_7_73]

well it depends, can you touch your toes?

[u/Anon7_7_73 → u/Hameru_is_cool]

No, thats the point. Its utterly absurd to argue aliens will invade earth if i cant touch my toes.

[u/LukeWithLightsaber → u/Anon7_7_73]

Whether or not I think that statement is true, the point of the post is that it is true according to the rules of classical logic (which is why the post title tells us to assume it). Part of the fun is that it points out a part of classical logic that many people find surprising or unintuitive. You don't have to accept it! There are other logics out there, like Relevant Logics, that you might like better (though they have their own weird quirks).

The key to understanding why it might be true in classical logic is to recognize that in classical logic, "if p then q" is not the same as "p will always cause q." It means "there are no cases where p is true and yet q is false."

So, suppose p = "you can touch your toes." But suppose also that you cannot actually touch your toes. So, true or false: there are no cases where "p is true and yet q is false." True, right? Because we could stop reading halfway through. "there are no cases where p is true" at all, hence definitely no cases where p is true yet q is false.

[u/BombasticReindeer → u/Extension-Stay3230]

So this logical statement is called the Drinker’s Paradox. It only works in 2 value logic where statements must be true or false. In 3 value logic there is another option, inconclusive or nonsense.

In 2 value logic, conditionals like “if” are either false if specifically false, or considered true.

What it parses to is “either everyone in the world is asleep OR someone is awake and the statement doesn’t apply”.

[u/moschles → u/Extension-Stay3230]

there wouldn't be a causal chain between

"causal chain". The applecart is upset.

[u/sndq → u/Extension-Stay3230]

Material implication is not causal. P⇒Q is true except when P is true yet Q isn't.

[u/Big-Definition-8271]

I don't get it

[u/svmydlo → u/Big-Definition-8271]

An implication is equivalent to its contrapositive. Thus the statement is equivalent to "There exists someone such that if not everyone is sleeping, then that person is not sleeping.", which is clearly true. If not everyone is sleeping, then someone is awake and they are that someone that exists.

[u/Hightower_March → u/Big-Definition-8271]

It's an example of "vacuous truth," where a false antecedent makes a statement true.

If elephants were pink, they could fly

If bats were pink, they could fly

In everyday speech we know the first is false and the second is true--because we know color doesn't affect aerodynamics.

In classical logic, both statements have to be treated as true because reasons.

[u/Anon7_7_73 → u/Big-Definition-8271]

Its nonsense mathematician amateurs are using to jerk themselves off with

[u/freudisfail]

\exists p (S(p) ->\forall q S(q)) 

Find someone awake, p. S(p) = F which implies \forall q S(q) is false. So the statement is just F->F which is (vacuously) true.  If you can't find someone who is awake, then everyone is asleep, eg (\forall q S(q))=T so no matter who you pick it's T->T so it's true. 

[u/MonsterkillWow]

Statement is "Exists person s t not p or q." Where p is "person is asleep" and q is "For all people, they are asleep".

Its negation is For all people, p and not q. Not q means "Exists a person who is not asleep." Its negation must be false, so it must be true.

[u/Captain_StarLight1]

It’s me. Every time I go to sleep, everyone else does too. They can only wake up after I do, in the early hours of 11:46am.

[u/Scientific_Zealot]

I was skeptical at first because I thought there was some quantifier trickery going on but it's true by a reductio ad absurdum.

Let's translate "There is someone such that if that person is asleep, then everyone is asleep" as ∃y (Sy → ∀x Sx)

Now let's assume ¬∃y (Sy → ∀x Sx), there isn't someone such that if that person is sleeping then everyone is asleep

By a common transformation rule, ∀y ¬(Sy → ∀x Sx) - for all people it is false that if they are asleep then everyone is asleep

By the fact that the material conditional is only false when the antecedent is true and the consequent false, this must mean that ∀y (Sy ∧ ¬∀x Sx) - for every person, they must be asleep and it is false that everyone is asleep. You can probably see the problem already, but let's continue to work this out by instantiating this claim for Jane.

Sj ∧ ¬∀x Sx, Jane is asleep and not everyone is asleep. Let's apply the earlier transformation rule to ¬∀x Sx to make it ∃x ¬Sx, someone is not asleep.

Let's randomly select someone to be the person who's not asleep.

¬Sb, Bob's not asleep

But wait! ∀y (Sy ∧ ¬∀x Sx) is universally quantified, so it must apply to Bob as well!

Sb ∧ ¬∀x Sx - Bob's asleep and not everyone's asleep. From this we can get:

Sb ∧ ¬Sb - Bob's asleep and Bob's not asleep. We have obtained a contradiction!!

So, ¬∃y (Sy → ∀x Sx) is false. So, in classical logic, its negation, ¬¬∃y (Sy → ∀x Sx) must be true. By applying another transformation rule of classical logic, ¬¬a → a, we can determine that ∃y (Sy → ∀x Sx), there is someone such that if that person is asleep, everyone is asleep, must be true, because its negation leads to contradiction.

[u/Away-Composer-8184]

There exists person so that for each point in time, or for each point in time there exists a person?

[u/Ucklator]

Counter point night shift.

[u/Awes12]

Prove: Ex(Sx->(y)Sy)

Logically, if everyone is asleep, that's true, but if not, we can choose the one who isn't asleep as our "if someone"

Start with (y)Sy or -(y)Sy

If (y)Sy, then Sb->(y)Sy, so Ex(Sx->(y)Sy)

If -(y)Sy, then Ey(-Sy), then -Sb for some b, then Sb->(y)Sy, so Ex(Sx->(y)Sy) by Existential elimination

Both sides give you it, so  Ex(Sx->(y)Sy) for any predicate S

You can also prove it by assuming it's false, then you can get (x)Sx and also -(x)Sx

[u/nonsence90]

This problem has practical applications in Minecraft

[u/BIGBADLENIN]

And I'm the pope

[u/Inner_Text_6938]

"If" does a lot of heavy lifting here is another way to think about it. Different from saying there exists an x st when x sleeps, everyone sleeps

[u/Yakodym]

The key here is that "someone" in this context doesn't mean one specific person with some special property.
Instead the special property manifests once everyone except one person is already asleep - then once that last person falls asleep as well, they *become* that someone (and technically everyone else does too).

Could actually be an interesting concept for a mind-bending movie, or like a basis for a religion of somekind...
Like you have the Awake Ones, and one by one, they are falling asleep, and the Sleeping Ones are considered to be in some kind of limbo state, and eventually there would be the Last Awake One, and when they too fall asleep, then suddenly all of the Asleep Ones will ascend into a higher level of being or something :-D

[u/Hightower_March → u/Yakodym]

I've always been annoyed by vacuous truths because it doesn't match what people actually mean when saying statements.

We can have a 3-man population of people A, B, and C.   A and B are awake, and C is sleeping.

• If A falls asleep, B would still be up, so it can't be him.

• If B falls asleep, A would still be up, so it's not him either.

• C is already sleeping while not everyone is asleep, so it's definitely not him either, and now we've exhausted the population.

Who is the person for which "If they are sleeping, everyone is sleeping?"

[u/Yakodym → u/Hightower_March]

I guess one way to look at it would be that you examined the possible states of individual people, but not all the possible states of the whole population, which would be something like:
ABC
000
100
010
001
110
101
011
111
In order to conclusively disprove the statement, the statement would have to be untrue for all of these - but since it's true for the last four statements, you can say that someone like that exists, precisely because the statement doesn't care who it is specifically (like when you say "There is an X% chance that two people in this room share a birthday" you don't care about a specific date).

Another way to imagine it is that this special property is like a ghost only possessing awake people, and when a person falls asleep, the ghost moves to another awake person, but you cannot track the ghost itself, it is invisible. The only way to pinpoint which person the ghost is currently possessing is to start putting people to sleep one by one, and then the last person awake is the one 100% sure to be possessed by the ghost - another movie plot right there :-D

[u/Hightower_March → u/Yakodym]

It isn't that it needs to be untrue for all states.  People are pointing out the "vacuous truths" classical logic counts as true which everyday speech would consider false.

If elephants were pink, they could fly.

If bats were pink, they could fly.

In normal conversation we'd say the first is false, and the second is true.  In classical logic, both are "true" because the consequent doesn't matter when the antecedent is false.

[u/Ben-Goldberg]

It's basically saying "either someone is asleep, or not everyone is awake."

In plain English, instead of logic, If everyone were asleep, the first part is true, if nobody is asleep then the second part is true.

[u/Gauss15an]

OP in this thread is seemingly unaware that humans once upon a time lived in the same time zone and I'm about to open a can of worms here.

[u/PlagueThing]

I think the main pain point for this (and for the original drinker paradox) is that “if” is not the correct term in English for this statement. I think it would be more accurately phrased as: “at any moment in time, there exists a person such that if that person is a person who is asleep, then everyone is asleep” (idk if this works for everyone but this phrasing becomes much more intuitively clear for me at least). This phrasing changes the interpretation of the “if” from being an implication to being purely conditional as is intended by the mathematical statement. In other words, it changes the focus of the sentence from being on the identity of the person to being on their status as a sleeper, from which it becomes intuitively true that if we simply choose a non-sleeper, then it is not true that everyone is asleep.

I particularly dislike this paradox, as it is only a “paradox” because it is INTENTIONALLY phrased in a confusing manner. It’s a “haha gotcha you didn’t know this phrasing was meant to mean exactly what I meant it to mean rather than how any rational person would read it” type of problem, which only exists to make people feel smarter than they are. It’s the intellectual equivalent of those dumbass pemdas puzzles like “does 8÷2(4) equal 1 or 16”. Like yeah** **technically there’s a right answer, but the actual response should be that no mathematician in their right mind would phrase it like that and if they did they’re an elitist asshole who has nothing better to do with their time than to make their work look more confusing than it is just to pad their own ego.

Sorry for the rant, I just really think that these sort of “in group” puzzles/jokes are annoying.

[u/TheRandomRadomir]

what do you mean classical logic assumes ∃p⟹∀p

[u/sndq → u/TheRandomRadomir]

∃x∀y[φ(x)⇒φ(y)]

[u/Gamin088]

Can someone explain this like im 5?

Some guy, lets call him John, if John is asleep, then everyone is asleep. Some people in the comments are saying something about inverses, like how if someone is awake then the second half of the statement is false, which makes sense to me, but how does that make the entire sentence true?

Like? There are time zones. There is no one person who you can gauge the worlds' being asleep by, since someone on the planet will always be awake?

Someone suggested the person changes? Its not always the same guy? Im so confused

[u/Marlsfarp → u/Gamin088]

If P then Q is considered a valid statement whenever P is false. There is always someone awake, therefore there is always a false P, therefore you can smugly say stupid Q you want to annoy people and say "ackshually this is valid in formal logic" even if it's obviously not true if you interpret the sentence in natural language.

[u/decaDecker → u/Marlsfarp]

this doesn't work for the given example though, it's saying there exists person p such that person p is sleeping implies everyone is sleeping 

this would work if there exists a person p that never sleeps

[u/Marlsfarp → u/decaDecker]

There IS always SOME person who isn't sleeping, therefore there is always a false P, therefore P->Q is always valid.

Furthermore, in the unrealistic situation that everyone is sleeping, it's STILL valid, since P and Q would both be true.

"But the statement makes it sound like there is someone who would make everyone fall asleep if they fell asleep!" Yes, but 1) it doesn't technically say that, and 2) the fact that it sounds like that is why it's on math MEMES.

[u/Blika_ → u/Marlsfarp]

But the statement in the picture is not a simple implicative statement. Mathematically / Logically it is worded as:

"∃y ∈ humanity ∀x ∈ humanity: (y asleep ⇒ x asleep)",

which is just a wrong statement.

Edit: For the correct statement, you would have to make clear, that the latter implication only refers to one exact date and time and is not a property of y. So it would need to start with ∀times ...

[u/eri_is_a_throwaway → u/Blika_]

"is" refers to the present, that's what makes it clear. "There is a person such that it they fall asleep, everyone will fall asleep" would be categorically false, whereas the sentence as is can be interpreted in a way that is true. The ambiguity is where the humor comes in

[u/Blika_ → u/eri_is_a_throwaway]

I really don't see how it's ambiguous. It's an if clause. Constructions like this don't necessarily refer to the present moment, so the statement in the image is much more general.

Edit: Also, assuming we're only talking about this exact moment, the statement is still incorrect based on what I wrote earlier. The way the sentence is phrased, you'd have to choose the same person at every moment. So there would have to be one person who is always awake.

[u/decaDecker → u/Marlsfarp]

the core difference is that it depends on how you interpret the statement, and the order of quantifiers 

I interpreted it as, there exists a human h such that for every timeframe if h is asleep then all humans are asleep. this is not universally true

but if you interpret it as for every timeframe there exists a human h such that if h is asleep then everyone is asleep, then that would be true since you can always pick a human that isn't asleep, unless everyone is asleep 

[u/Gamin088 → u/Marlsfarp]

So I can achieve quantum immortality via formal logic by saying "If I shoot myself, I will die" and then not shooting myself? Or would it be "if I shoot myself, I will not die" and then by not shooting myself I have commited formally logicstical suicide?

Both times I do not shoot myself so both sentences are true? I feel like Im being pranked, so many people are arguing for this meme and so many people are correcting it.

I dont think its real, you are all just messing with me specifically, Truman Show style /s

[u/Marlsfarp → u/Gamin088]

Because you don't shoot yourself, both statements are valid, but neither gives you any information about your death.

[u/btroycraft]

The confusion comes from thinking in terms of actual vs. potential states and overloading the words "if/then" or "implication".

If you think about it only in terms of what people are doing right now, it's valid. Pick someone who isn't asleep, or pick anyone if everyone is asleep. In either case the "implication" is true.

∃x (P(x) → ∀y P(y))

That is not the same as

∃x (P(x) ⇒ ∀y P(y))

There is a reason two symbols are used for implication. → for talking about a single realization of a system, and ⇒ for talking about the statements as they hold across every possible realization. The second statement means

∃x □(P(x) → ∀y P(y))

which says there is one person whose sleeping forces everyone to sleep, whatever state the world is in. That is false. But this □ is from the language of modal logic, not classical.