Talk:Drinker paradox

This is not a paradox!

This is not a paradox by any stretch of the imagination; it's a riddle where we're merely being confused due to the wording. The emphasized text shows where the problem exists:

Suppose, on the other hand, at least one person isn't drinking. For that particular person, it still can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking — because that person is, in fact, not drinking.

As you can see, there is an inconsistency in the logic. For us to make inquiry or assumptions about "that particular person" is fallacious when in fact there may be more than one person not drinking.

As simply as I can put it:

If in our pub there exists more than one non-drinker, we can not refer to that demographic(non-drinkers) as "that particular person". Thus the entire arguement that follows is inherently wrong. If no one objects, I suggest the article be removed or at least be put under a more appropriate category; perhaps riddles. --131.191.106.81 02:22, 16 September 2007 (UTC)

The first step of the proof, stating that either everybody drinks

or someone doesn't drink, is an application of the principle of
excluded middle, so the paradox is true only in classical logic
but false in intuitionistic logic;

I don't agree. This is valid in intuitionistic logic. The set of people in a bar is finite (!) and it should be decidable by direct observation whether someone is drinking or not^*. No appeal to the law of the excluded middle is needed.

The last step of the proof uses the notion of material
implication, according to which a hypothetical statement in the
form if A then B is always true if A is false.

This seems to be the essence of the paradox. It's about fallacies of relevance, not constructivism.

Paradoxes of this kind are used by the intuitionist school in the
philosophy of mathematics to argue against the the laws of
classical logic.

I'm curious about that. Does one of the references cited argue this way?

-Dan 13:49, 17 May 2006 (UTC)

* Even if not, it's not the essence of the paradox -- just change it to something directly decidable, like "in every non-empty hospital, there is a patient who, if he is declared dead, every patient is declared dead", or if you still object that a declaration of death might conceivably be ambiguous, you can't argue with "in every finite non-empty set of natural numbers there is a number which, if it is prime, all numbers in the set are prime."

It is true that excluded middle is not needed because the set of drinkers is finite, but the logical formula doesn't assume finiteness. Since finiteness is not formalizable in first order logic, the only way to prove the paradox is to use excluded middle.

(Anyway, I like your hospital example even better than the drinker. We could add it to the page, but the paradox is commonly known as the drinker paradox.)

About its relevance to intuitionism, maybe that's a silly part of the article. Some intuitionists argued in that way to me, but it is not to be found in any published work. So it probably must be deleted.

Eubulide 14:04, 17 May 2006 (UTC)

Said person may be only person in pub! —Preceding unsigned comment added by 70.139.185.216 (talk) 03:22, 27 December 2008 (UTC)

No, it's not the only way to prove it. We can still do without LEM.

• For a bar/hospital: every customer/patient is a human. The human population is less than one trillion (true now, almost certainly true in my lifetime, and can be increased as desired). "Less than one trillion" is formalizable in first-order logic.
• For prime numbers: we can talk about finite sequences of natural numbers in Heyting arithmetic just fine. I do believe the sentence "in every finite non-empty sequence of natural numbers there is a number which, if it is prime, all numbers in the sequence are prime" is expressible in HA and is a theorem!

I would never make this argument against LEM, and I consider myself a constructivist. Now I don't consider myself an intuitionist, still, Brouwer himself considered LEM to be something from our experience of finite space and time (and used improperly in discussing the infinite), so presumably he would not like this argument either. Please hit your intuitionist friends over the head! -Dan 16:09, 17 May 2006 (UTC)

It just occurred to me how your hospital interpretation actually shows the paradox to be essentially classic. Just change your interpretation a little: "In every hospital there is a patient such that, if he recovers, all the patients recover". So the doctors only need to cure that one patient! Of course, it is undecidable which one that patient is. Eubulide 14:21, 17 May 2006 (UTC)

A medical breakthrough! -Dan 16:09, 17 May 2006 (UTC)

Poor Wording

The article contains the phrase "either no one is drinking, or everyone is not drinking". These two mean exactly the same thing. Perhaps whoever wrote that intended to write "either no one is drinking, or not everyone is drinking, but at least one person is". —Preceding unsigned comment added by 69.171.176.43 (talk) 21:31, 2 December 2010 (UTC)

Infinite undecidable drunken health care

Speaking of Ralph Klein, is there a version of this which a) is classically (but not constructively) equivalent to the original statement that every non-empty hospital has a patient who, if cured, everyone is cured, and b) is constructively provable, even without assuming that the hospital is finite, or that death is decidable?

Answer: Yes. If, in a hospital, there isn't a patient who dies if any patient dies, then that hospital has no patients at all.

$[\not \exists p\in H:(\exists p'\in H:D(p'))\to D(p)]\to \not \exists p\in H$

Proof: exercise for the reader :-) -Dan 19:44, 18 May 2006 (UTC)

LME Redux

The paradox can be stated with less parentheses, i.e. (Ex) Drink(x) -> (x) Drink(x). Further, the argument seems to rely on the closed world assumption, and not on the excuded middle.

The Coq prover page uses sets and that goes beyond the scope of logic. So, in order to be certain, can somebody give a derivation of the paradox in classical logic, and if possible, in intuitionistic logic? Actually, I wonder if either is possible. If it is not possible, please correct the claims on the page. -- 129.247.247.238 16:37, 22 May 2006 (UTC)

I'm not sure that's a good way to write that, as it might be read $(\exists xDrink(x))\to (\forall xDrink(x))$ rather than $\exists x(Drink(x)\to (\forall xDrink(x)))$. As for a proof, alright:

SOLUTION TO QUESTION POSED ABOVE

In constructive free logic:

${\begin{matrix}{\mbox{1.}}&\exists x.\top &{\mbox{assumption}}\\{\mbox{2.}}&\neg \exists x.((\exists y.Dies(y))\to Dies(x))&{\mbox{assumption}}\\{\mbox{3.}}&\forall x.\neg ((\exists y.Dies(y))\to Dies(x))&{\mbox{2, constructive equivalent}}\\{\mbox{4.}}&\forall x.(\neg \neg \exists y.Dies(y))\wedge (\neg Dies(x))&{\mbox{3, constructive equivalent}}\\{\mbox{5.}}&(\forall x.\neg \neg \exists y.Dies(y))\wedge (\forall x.\neg Dies(x))&{\mbox{4, free logic equivalent}}\\{\mbox{6.}}&(\forall x.\neg \neg \exists y.Dies(y))\wedge (\neg \exists x.Dies(x))&{\mbox{5, constructive equivalent}}\\{\mbox{7.}}&(\neg \neg \exists y.Dies(y))\wedge (\neg \exists x.Dies(x))&{\mbox{1, 6, free logic inference}}\\{\mbox{8.}}&\bot &{\mbox{7,law of non-contradiction}}\\{\mbox{9.}}&\neg \exists x.\top &{\mbox{8, discharge assumption 1}}\\{\mbox{10.}}&(\neg \exists x.((\exists y.Dies(y))\to Dies(x)))\to \neg \exists x.\top &{\mbox{9, discharge assumption 2}}\\\end{matrix}}$

Which was my statement above. I trust you can take it from here to classical logic. Bonus points: where is the paradox of material implication (aka fallacy of relevance)? -Dan 20:17, 22 May 2006 (UTC)

I admit I am confused now. I have a straightforward proof using game semantics showing that (Ex) (Drink(x) -> (y) Drink(y)) is neither intuitionistically nor classically derivable:

  0                       (Ex) (Drink(x) -> (y) Drink(y))
  1  0?                   Drink(n) -> (y) Drink(y)
  2  1? Drink(n)          (y) Drink(y)
  3  2? m                 Drink(m)
  4  3?

Am I missing something badly? I think we have a case of hidden axioms which are needed to prove the proposition. -- 129.247.247.238 10:02, 23 May 2006 (UTC)

Yes, I was missing something -- there is a condition, in any non-empty-bar... This makes the domain non-empty and closes the domain of discourse, too. What is wrong, is that the formal version does not list these conditions formally.

The article should be modified accordingly. And the claim, that the proof relies on LME can be deleted, as shown by Dan. -- 129.247.247.238 14:15, 23 May 2006 (UTC)

I do not understand this. If $\forall x.Px\lor \exists x.\neg Px$, then $\exists x.\forall y.(Px\to Py)$ holds. Why must the bar be non-empty?

Nonintuitive

Shouldn't it be mentioned that while the original definition seems counter-intuitive, the contrapositive makes perfect sense (that is, if not (everybody drinks) then (someone doesn't drink))? Or am I wrong about that? MagiMaster

Contraposition is not an equivalence in intuitionistic logic. Precisely, $(p\rightarrow q)\rightarrow (\neg q\rightarrow \neg p)$ holds, but the reverse, $(\neg q\rightarrow \neg p)\rightarrow (p\rightarrow q)$, doesn't. Therefore, the drinker's principle only implies your preferred contrapositive version in all logics. I have no opinion on which version of the classical statement is more "intuitive" because I tend to find classical logic non-intuitive. — Kaustuv Chaudhuri 03:24, 16 July 2006 (UTC)

Actually, I don't think that's even the correct contrapositive. The implication occurs inside the existential quantifier. The contrapositive of $\exists x.(D(x)\to \forall y.D(y))$ should be $\exists x.(\neg (\forall y.D(y))\to \neg D(x))$, which is (non-constructively) equivalent to $\exists x.((\exists y.\neg D(y))\to \neg D(x))$. That is, "there exists someone who, if anyone doesn't drink, he also doesn't drink." 192.75.48.150 12:58, 17 July 2006 (UTC)

Hmm, I think the original comment was essentially correct. I interpreted the statement as talking about the body of the outer $\exists$. — Kaustuv Chaudhuri 13:47, 17 July 2006 (UTC)

Ah, never mind, I see what you're getting at. Yeah, $\neg (\forall y.\ D(y))\rightarrow \exists x.\ \neg D(x)$ is only implied by the contrapositive of the drinker principle. The original statement was imprecise because "someone doesn't drink" can be interpreted as an $\exists$. — Kaustuv Chaudhuri 14:20, 17 July 2006 (UTC)

Sorry, I forgot to sign my first comment. This is what I was thinking about when I wrote that though. I wasn't sure if it was right, because I don't remember exactly how negating qualifiers works. (Also, I don't know much about intuistionistic logic. I'm a computer scientist, so I studied classical logic.) MagiMaster 19:58, 17 July 2006 (UTC)

laymen

Could someone write out the reasoning in plain speech as opposed to latex? I'm not familiar with all the logic symbols, and furthermore, I prefer, as many probably do, the construction of natural language.

I hope this helps. 192.75.48.150 15:37, 14 August 2006 (UTC)

I guess not. We got a {{technical}} tag again. Please feel free to ask questions here. We don't bite. 192.75.48.150 21:06, 17 August 2006 (UTC)

(transcribed from WP:RfF --192.75.48.150 19:17, 28 August 2006 (UTC))

• It would be unwise for any newcomers to logic to leap straight in at this point, so the application of the technical tag is probably not appropriate. It is certainly a lot more accessible than certain other articles in this subject area. That said, there are a couple of points that could be cleared up:
  • In the proof section, I don't think it would be redundant to re-explain the proof (although I'm struggling to come up with alternate phrasing).
  • The proof diagram is scary - perhaps move it to its own subsection in the proof section, so readers can see that they don't need to be able to decipher that to understand the concept.
  • The reference to Goldbach's conjecture is awkward - you should make clear that it is unsolved, can not be solved like this because this reasoning is invalid, and introduce the section more clearly, something like "For instance, it would allow us to solve Goldbach's conjecture which has taxed mathematicians for hundreds of years...".
  • I'd always explain in English before using notations: the section "Material versus indicative conditional" in particular, would probably read better if the order of the sentences was changed and the notation explained.
  • In some cases you might like to point readers to appropriate connected themes explicitly rather than just a wikilink.
  • I also have a feeling that the title may draw an unintended audience, so some stress in the lead that it has nothing to do with pub culture or drinking games (or rather, overstressing the connection to first order logic) won't hurt.

Hope this helps. Yomangani^talk 11:03, 25 August 2006 (UTC)

I have simply removed the proof diagrams as in the present form they add no clarity to the article. I disagree with nearly all of the mumbo jumbo that was added as the logical issue seems fairly shallow to me, but I'll leave it be. — Kaustuv Chaudhuri 12:56, 3 September 2006 (UTC)

The paradox is less paradoxical and easier to understand when it is explained the following way: assume there is a person who would be the last to start drinking. If and when this person drinks, then everybody drinks.

While reminiscent of Zorn's Lemma, one need not delve into set theory at this point. Although the connection might be used in order to explain why the "paradox" need not arise in intuitionistic logic. -- Zz 13:13, 5 September 2006 (UTC)

Eh. I can see where you're headed with the last person to drink, but I'm not seeing the connection to Zorn's lemma. 192.75.48.150 17:27, 5 September 2006 (UTC)

The connection lies in the maximal element existing for every non-empty partially ordered set (e.g. the last one to drink). For finite sets the existence of such an element is obvious. For infinite sets, however, the argument presented might fail, unless the lemma guarantees the existence of this element. This echoes the objection listed for intuitionistic logic - for finite sets, the drinker paradox is provable (see above), for potentially infinite sets, this need not be so. -- Zz 02:09, 6 September 2006 (UTC)

Sorry Kaustuv, did you mean your only objection to the mumbo-jumbo is that it is unneeded, because the issue is shallow? Or was there something else? 192.75.48.150 17:27, 5 September 2006 (UTC)

has anyone come up with it yet?

if you apply these assumptions to a pub, where at least two people are NOT drinking, then the statement that everyone is drinking is false, because if he or she is drinking, there is still one more which might be not drinking :) !!! —The preceding unsigned comment was added by Katoa (talk • contribs) 04:16, 9 January 2007 (UTC).

== confusing or error?

"Suppose, on the other hand, at least one person isn't drinking. "

Right up to here.

"For any particular person,"

I think this "For any" is at least confusing if not wrong. _There exists_ a person for which "if that particular person is drinking, then everony else" holds true. But it doesnt hold true "For any" person.

"... it still can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking -- because that person is, in fact, not drinking."

---

The first comment above makes sense to me. If all but two people are drinking, either one could drink without the either drinking, correct? If this is the case then either one of them can drink without everyone drinking. Since everyone else was already drinking, obviously any of them can drink without everyone drinking. If any person in the room can drink at times when others aren't drinking, how can at be that there must be someone who only drinks when everyone is drinking?--Eloil 20:44, 17 April 2007 (UTC)

so much fuss for nothing

I am not a logician, but as an interested reader, this page appears to me as mind-bogglingly technical over a simple thing. That's fine with me, just as long as there is also a good initial paragraph with common-sense language that would explain that:

1. If everyone is drinking, then all the premises are true. The special drinker is drinking and the entire pub is also drinking;

2. If one person is not drinking, then none of the premises are true. We can deduce that such person must be the special drinker (though the paradox doesn't ask us that), because if it was anyone else, there shouldn't be anyone not drinking;

3. If there is more than one person not drinking, then none of the premises are true, and we can deduce that one of those people who are not drinking is the "special" drinker.

We can also say:

1. If the set has zero people, then the premises are irrelevant, but not necessarily untrue; 2. If the set has one person, then the premises are true: if he is drinking, everyone is drinking; 3. If the set has two people or more, we can infer who's the special drinker by going through the previous 3 points.

What's more than this about it? Sorry people, can't understand well. What's so special about it? I think that that is what this text lacks.

Luis Dias, 14-02-2007 —The preceding unsigned comment was added by 195.23.224.70 (talk) 13:35, 14 February 2007 (UTC).

I like the way you explain it. The technical details are obvious to the logically trained mind, which in turn proves that a training in formal logic can make things difficult. However, there is a reason why the paradox gets the attention of logicians... If no objections are raised, I will shorten the discussion in the article substantially and use your explanation instead. -- Zz 16:16, 24 April 2007 (UTC)

P.S.: Oh my, the explanation is already in the article. Nonetheless, I will shorten the technical details if no objections are raised.

How is the following true? "If there is more than one person not drinking, then none of the premises are true, and we can deduce that one of those people who are not drinking is the "special" drinker." I can understand that you may not be able to disprove this to be the case for each of the non-drinkers, but you wouldn't normally deduce something like that. Is there one person one earth who, if they're inside their mothers womb, all other people will be inside their mother's womb? Since there is more than one person not in their mother's womb can we deduce that this statement is correct? Constan69 (talk) 08:22, 12 February 2008 (UTC)

Why is this a paradox?

All it says is that there's always at least one person we're waiting on to take a drink in a given round, and when they decide whether they're taking a drink we'll know whether everyone's drinking that round or not. It doesn't seem paradoxical at all. 80.192.29.107 14:15, 25 August 2007 (UTC)

ISBN Wrong

The sole reference for this paradox (bookwise) seems to have the wrong ISBN, unless this book is contained within some other book. The ISBNs I get for the book "WHAT IS THE NAME OF THIS BOOK?" by Raymond Smullyan (Author) are (from Amazon):

(Paperback)

1. ISBN-10: 0671628321
2. ISBN-13: 978-0671628321

or What is the name of this book?: The riddle of Dracula and other logical puzzles (Unknown Binding)

1. ISBN-10: 0139550887
2. ISBN-13: 978-0139550881

Root⁴(one) 20:29, 30 January 2008 (UTC)

The 0140135111 isbn currently given in the article works fine in WorldCat and points to the 1990 Penguin reprint [1]. The actual problem is that book I've looked in, having ISBN 0-13-955088-7 by the way, says "(c) 1978". Easily confirmed in WordlCat [2] So I think the year and ISBN should be fixed. Actually, I've done that because I've added page numbers from my copy and they might not be the same in the Penguin reprint. Tijfo098 (talk) 20:47, 27 October 2012 (UTC)

It's merely a trick of language

One of the premises is "there is at least one person who is not drinking", but this is then sneakily changed (by implication) into "there is exactly one person who is not drinking", which is invalid since there could be five people, or thirty, who are not drinking. Therefore the entire problem is completely imaginary.

All it's saying is "if every person who is not drinking were to start, then everyone would be drinking", which is obvious.

86.63.16.62 (talk) 09:02, 1 May 2008 (UTC)

The unwritten implication (which I believe is the source of the intuitive confusion) is that there is one specific person who is special for the entire night at the pub. The "paradox" would make more intuitive sense if it were preceded with, "At any given moment in time..." BaruMonkey (talk) 15:27, 8 September 2009 (UTC)

Non-empty domain

This section is wrong, there has to be someone in the pub it clearly says so in the original statement "there is someone in the pub such that..." I suggest removing it. Also it isn't a paradox. —Preceding unsigned comment added by 122.106.16.105 (talk) 09:22, 2 September 2008 (UTC)

See: If there is noone in the pub, then there is not someone in the pub. Thus there is not someone in the pub which has any property you choose. So there is not anyone in the pub drinking or not drinking. There is no one which provides any implication on others in the pub. There is just no one.

The problem described in this section is that in first order logic it's different than in normal logic: The domain must not be empty.

Just think about the two sentences in human language and it should be clear.

--88.217.21.134 (talk) 14:52, 6 October 2011 (UTC)

The domain is non empty because it is in classical predicate logic and not a free logic. See the very first statement in the lead and there is a bit about this in the article. The statement about the pub should probably be made a bit tighter to conform with the logic like 'when the pub isn't empty then there is someone in it who is not drinking if not everyone is drinking'. Of course then it sounds blindingly obvious Dmcq (talk) 15:15, 6 October 2011 (UTC)

The analysis of this is simply wrong - there is no paradox

I can't believe that so many people have failed to see the fallacy here.

Suppose that two persons A and B aren’t drinking, and another person C is drinking.

It is wrong to say that if A is drinking, then everyone in the pub is drinking.

It is wrong to say that if B is drinking, then everyone in the pub is drinking.

It is wrong to say that if C is drinking, then everyone in the pub is drinking.

So the assertion that: there is someone in the pub such that, if he is drinking, everyone in the pub is drinking. is simply wrong - there is no paradox here at all. —Preceding unsigned comment added by 81.141.218.244 (talk) 20:43, 7 December 2010 (UTC)

Ok, but the thing is the first two statements aren't really wrong because A and B are in fact NOT drinking. You may say but if either A or B takes a drink then each statement regarding them specifically becomes false. BUT WAIT, aha! The other steatement then remains true! And I can say it was THAT person who was the special person all along. In other words, in your example of three people:

1)if A is drinking, then everyone in the pub is drinking.

2)if B is drinking, then everyone in the pub is drinking.

3)if C is drinking, then everyone in the pub is drinking.

C is drinking, so statement 3 is false.

If A starts drinking then statement 1 is false, but statement 2 is still true.

If B starts drinking then statement 2 is false, but statement 1 is still true.

If neither A and B take a drink then both statements 1 and 2 are true because they are in fact NOT drinking.

So no matter what anyone does there will always be at least one person where its true if he is drinking then everyone is drinking. Racerx11 (talk) 19:06, 12 December 2010 (UTC)

You can't win an arguement by concocting the most tanlged sentence in the world. I myself think this is not a paradox because it doesn't fit the basic criteria: If one person is drinking, everyone is drinking has absolutely no contradiction and is infact a positive within a positive, so the first major criteria of a paradox--contradiction--has not been met. The second major criteria is that at least one parameter somewhere in the statement has to be absolutely impossible. Let's see: Drinking is possible, Coincidence is possible, it is possible, albeit rare, for a coincidence of everybody in a bar to drink at once, there is no law of physics that denies that, even several times consecutively. The fact could also be that the person who is drinking is a well-respected figure, and that everbody has made it their duty to drink exactly the same time he or she drinks.

Now let's compare this to a real paradox, the Irresistible force paradox. An immovable object is impossible, an irresistable force is impossible, a force meeting an object is possible. When combined, it creates a statement with two outcomes, both of which seem true and perfectly legitimate, but they contradict eachother and therefore something in the equation is absolutely impossible, be it one, or two, or even a hundred parameters, all it takes is one.

Furthermore, a paradox is usually created to question something somebody believes and render it illogical, like time travel. The grandfather paradox has at least three major parameters: Being born, being killed, and time travel. If you kill your grandfather in the past, you couldn't have been born to kill him, if you haven't been born, you didn't kill him. Something in this equation is false or impossible by the laws of physics. Let's test this against the criteria: The contradiction - Killing your grandfather in the past Vs. Being born to be able to do it, The impossibility(ies): Being born is possible, Being killed is possible, therefore time travel must be impossible, and the fact that we've never accomplished this feat is more proof than needed: Time travel is impossible and it has been proved by the grand father paradox.

So the question is: What was this "Paradox" made to disprove the possibility of? That coincidence is possible? We all know it is possible since we all have experienced it. There is no impossible parameter and there is no apparent contradiction, therefore there is no paradox. two negatives can make a negative (-1, -1, --> -2)(carrots can't be genuinely rainbow, rainbows can't taste like carrots, they don't nullify eachother), two positives make a positive ( 1, 1, --> 2) (carrots are orange, orange is the natural color of a carrot), a real paradox has a positive statement ( 1), and a negative one (-1), they both make each other impossible by nullifying eachother to a logical state of zero (0) or impossibility ( 1 being omnipotence, both cases are -1, both statments make a 0) (If god makes a stone he cannot lift, it is a -1 to his 1, if god can not make that stone, then there is a limit to his power, another -1). Again we return to this drinker paradox. the act of drinking is a 1, the happening of coincidence is 1, and consecutive coincidence, there is nothing stopping it from being a 1. In all cases, no 0 is produced.

By the way, there is probably no actual mathematical or logical gauging of logical possibility with 1s and -1s, but this is what I chose to demonstrate my point as clearly as possible. Seriously think about this, if this is somehow a paradox, then it is very abnormal. I want you to prove beyond a doubt with simple logic that this is indeed a paradox, then I might agree. But for now, this is anything but a paradox. Prove me wrong. 168.103.126.103 (talk) 19:38, 8 December 2011 (UTC)

Another Point I would like to make: This paradox would make sense if the statement were "If one is drinking, no one in the pub is drinking." Because at least then there is a contradiction--if one man is drinking, then you can't say that zero people are drinking because there is still that one person who is (1 =/= 0). But seeing as the statement is infact the exact opposite of that, there is no contradiction and therefore no paradox. And if you do decide to challenge my logic, do not use math equations, if you use math equations of any form (including using A and B to represent people and situations) you automatically lose the debate (to truly win a debate, you must convince your opponent beyond a reasonable doubt, your view) because in my eyes, you could easily be making up nonsense--in other words: debate in English, not in Spahish, not in German, not in Long, ugly quantum theory snippets, but in English. Long calculus equations didn't convince me in Buffalo⁸ that Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo is a grammatically correct sentence, and they won't convince me here that this is a paradox. 168.103.126.103 (talk) 19:50, 8 December 2011 (UTC)

Racerx11 says:

If A starts drinking then statement 1 is false, but statement 2 is still true.

If B starts drinking then statement 2 is false, but statement 1 is still true.

So Racerx11’s own analysis demonstrates that the statement 2 does not have a definitive true/false value that is independent of the scenario - it is dependent on the actual scenario of who is/is not drinking.

And if the true/false value of the statements depends on the scenario, what I pointed out is correct. In the scenario where A and B aren’t drinking, and another person C is drinking, the statements 1), 2) and 3) are incorrect. There is no paradox. —Preceding unsigned comment added by 81.141.218.231 (talk) 11:19, 27 December 2010 (UTC)

I don't think this is a paradox, it seems to rely on this little sentance not being noticed by the listener, it's more of a riddle:

"Suppose, on the other hand, at least one person isn't drinking. For that particular person, it still can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking"

In the first sentance it says "at least one" but the second sentance it says "for that particular", which means just one person.

Still I have just read the "heap paradox" and I think that the definition of paradox except for in the case of the sci-fi staple "temporal paradox", is ill-defined. DarkShroom (talk) 21:45, 1 January 2011 (UTC)

81.141.218.231 I agree with your reasonings, but I'm playing devil's advocate for this being a paradox. Since in your example you are stating if a person (A,B or C) is drinking... What if I insist for sake of argument, that they do each in fact start drinking? What if I insist that you demonstratively prove the statement is false by having each person, one by one, in any order, actually take a drink? When you get to last person you will not have proven the statement false. In fact you will have confirmed that there is at least one person where if he is drinking then everyone is drinking. Namely, it would be the last person to take a drink. To me this is paradoxical. Racerx11 (talk) 04:11, 25 January 2011 (UTC)

Racerx, that's not a paradox, that is what's called "irony". The fact that using this guy as a litmus test to find out whether or not everyone is drinking because he waits until that very moment and then starts drinking to make it so everyone is drinking is irony. Trying to stop a building from exploding and then ultimately causing the explosion because of stupid rash actions is irony, most sitcoms base their situations on irony, going back in time and killing your grandfather is a paradox. Seriously ¬_¬ 168.103.126.103 (talk) 20:55, 8 December 2011 (UTC)

delete please

jesus christ, can somebody delete this? i'm too lazy to figure how such is done, but this entirely unsourced article is one incomprehensible and futile attempt to prove how something that is obviously not a paradox, is a paradox.

this article is complete and utter nonsense that doesnt meet almost any of the criteria of wikipedia. please delete this crap· Lygophile has spoken 23:51, 7 February 2011 (UTC)

You could nominate the article for deletion. I believe it pretty clearly is a paradox: one would not expect it to be the case that "there is such a person that if he is drinking, everybody is drinking", although that is truly the case. It highlights a curiosity of formal logic that contradicts everyday language (the nature of "If P, then Q"), and so something that seems nonsensical is technically correct. So, while I do feel that this is in fact a paradox, the article is also poorly written. The introduction to the paradox and description of it make sense, but everything after that is quite awful. Chris3145 (talk) 00:01, 12 February 2011 (UTC)

Inaccurate statements regarding theorems of classical predicate logic

Several contributors to this talk page have disputed the contents of the article. The article does not cite any references to support its claims, other than referencing a book which features the so-called ‘paradox’, so that there are no peer reviewed references to support the claims of the article. In addition, one contributor, Chris3145, has simply deleted notifications that noted the lack of references, instead of actually providing authoritative references.

This article asserts that $\exists x.[\neg Drinking(x)\vee \forall y.Drinking(y)]$ (which is asserted to be a a representation of the English sentence ‘There is someone in the pub such that, if he is drinking, everyone in the pub is drinking’) is a theorem of classical predicate logic.

This is factually incorrect. In classical predicate logic, every proposition has a fixed truth value, and every proposition MUST be either true or false, and REMAIN so. Which presents no difficulties when dealing with arithmetic for example. But in the real world, which is the case with drinkers and pubs, the case of whether a person is or is not drinking is subject to change over time and does not have a fixed truth value.

If $\exists x.[\neg Drinking(x)\vee \forall y.Drinking(y)]$ were a true statement of classical predicate logic, then it must always be true, and it must be true either because some person in a given pub is not drinking, or because everybody is drinking. Obviously, for some pub situations, not everyone will be drinking. That means that for every such pub situation where not everybody is drinking, since $\exists x.[\neg Drinking(x)\vee \forall y.Drinking(y)]$ must nonetheless remain true, the truth of it requires that there always is some person "$\ x$" in any given pub who never drinks, and never has done, and never will. This is obviously not the case for every pub situation. Hence in the real world, where $\ Drinking(x)$ does not have a fixed truth value, the statement $\exists x.[\neg Drinking(x)\vee \forall y.Drinking(y)]$ cannot have a fixed truth value of 'true'.

Note that $\exists x.[\neg A\vee B]$ means ‘There exists an x, such that A or B.’ , which in classical predicate logic is perfectly equivalent to stating, ‘There exists an x, such that if A, then B.’, which is given as $\exists x.[A\rightarrow B]$. It does not follow that the same equivalence applies in natural languages such as English that deal with things that do not have fixed truth values. Ironically, the article notes that the properties of properties of classical predicate logic do not always agree with ordinary language.

Perhaps it is time that this article is nominated for deletion?

Jamesrmeyer (talk) 12:48, 17 February 2011 (UTC)

You write: This article asserts that $\exists x.[\neg Drinking(x)\vee \forall y.Drinking(y)]$ (which is asserted to be a representation of the English sentence ‘There is someone in the pub such that, if he is drinking, everyone in the pub is drinking’) is a theorem of classical predicate logic. Yes, it is a theorem of classical predicate logic. With all respect, your struggle with shows why it is also labeled as a paradox. -- Zz (talk) 16:17, 25 March 2011 (UTC)

Why do some people find this so difficult? The simple facts of the matter are this. If it is a theorem of classical logic, then:

1. it is provable within that logic, and so
   
2. for any given $\ x$, $\ Drinking(x)$ must have a fixed truth value, since all truth values in predicate logic are fixed as either true or false.

And if $\ Drinking(x)$ has such fixed truth values, then the statement $\exists x.[\neg Drinking(x)\vee \forall y.Drinking(y)]$ bears no relationship to the matter of drinkers in a pub, since there are persons who may be drinking at one time, and not drinking at another time, so there is not a fixed truth value of $\ Drinking(x)$ for some $\ x$.

You can't have it both ways. Either:

1. there is a theorem of logic which is simply $\exists x.[\neg A(x)\vee \forall y.A(y)]$, where $\ A$ has no relationship to persons drinking, or
   
2. there is an expression $\exists x.[\neg Drinking(x)\vee \forall y.Drinking(y)]$ where $\ Drinking(x)$ indicates that a person $\ x$ is drinking, which means that the expression is not an expression of predicate logic.

-- Jamesrmeyer (talk) 19:46, 26 March 2011 (UTC)

Let me repeat: it is a theorem of classical predicate calculus. You do not understand. That happens with paradoxes. I could try to show the mistakes of your reasoning, but that is not the point of wikipedia. And the experience with discussions on the net is that they mostly waste time. If it is really important to you, we can try by mail. -- Zz (talk) 21:28, 27 March 2011 (UTC)

The previous contributor is not addressing the issue. Instead of simply repeating over and over bland assertions such as, 'This is a theorem of predicate logic', and 'You do not understand', if anyone thinks that there is an error in what I have said, why don't they try and point out what they think that is. Otherwise, yes, this is a waste of time. --Jamesrmeyer (talk) 10:54, 28 March 2011 (UTC)

There is a proof of $\exists x.[\neg D(x)\vee \forall y.D(y)]$ in the very reference whose relevance you question below. Sławomir Biały (talk) 01:37, 29 March 2011 (UTC)

Why don't you read what is written. I'm not disputing that the formal statement is true, I'm disputing its application to a real world scenario where, unlike a formal system, true statements at one point in time can be false statements at another point in time, and vice-versa.--Jamesrmeyer (talk) 20:57, 24 November 2019 (UTC)

Inaccurate transcription into English

It seems to me that the source of the paradox is the fact that the logical formula is not transcribed properly into English. It should be "there is a person such that if we assume that 'he is drinking' then everybody is drinking". Tkuvho (talk) 04:54, 27 March 2011 (UTC)

I am not a native speaker, but I doubt the translation is correct. It is not about what we assume, it is about him drinking. -- Zz (talk) 21:30, 27 March 2011 (UTC)

Relevance of reference?

Can anyone explain what is the relevance of this recently added reference?

Coscoy, Yann; Kahn, Gilles; Théry, Laurent (1995), Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, pp. 109-123, doi:10.1007/BFb0014048, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.36.8114&rep=rep1&type=pdf .

If no-one can suggest any relevance, I think it should be deleted --Jamesrmeyer (talk) 11:03, 28 March 2011 (UTC)

They seem to mention the paradox as an example. I agree it's pretty tangential. — Carl (CBM · talk) 16:53, 28 March 2011 (UTC)

The reference is relevant because it includes a proof of the paradox, which should dispel the above concern that $\exists x.[\neg D(x)\vee \forall y.D(y)]$ is not a theorem of classical predicate logic. Obviously, better sources are desirable, but I think this one should stay in the mean time. James, do you have any sources you would like to add? Sławomir Biały (talk) 01:41, 29 March 2011 (UTC)

Fitch-style calculus proof

.2 | |__________ ~Ex(Px -> VyPy) . New Subproof
.3 | | |________ Pa. . . . . . . . New Subproof
.4 | | | |_____b . . . . . . . . . new variable for universal introduction
.5 | | | | |____ ~Pb . . . . . . . New Subproof
.6 | | | | | |__ Pb. . . . . . . . New Subproof
.7 | | | | | | . _|_ . . . . . . . 5,6 _|_ Introduction
.8 | | | | | | . VyPy. . . . . . . 7 _|_ Elimination
.. | | | | | <-------------------- end subproof
.9 | | | | | . . Pb -> VyPy. . . . 6-8 -> Introduction
10 | | | | | . . Ex(Px -> VyPy). . 9 Existentional Introduction
11 | | | | | . . _|_ . . . . . . . 2,10 _|_ Introduction
.. | | | | <---------------------- end subproof
12 | | | | . . . ~~Pb. . . . . . . 5-11 ~ Introduction
13 | | | | . . . Pb. . . . . . . . 12 ~~ Elimination
.. | | | <------------------------ end subproof
14 | | | . . . . VyPy. . . . . . . 4-13 Universal Introduction
.. | | <-------------------------- end subproof
15 | | . . . . . Pa -> VyPy. . . . 3-14 -> Introduction
16 | | . . . . . Ex(Px -> VyPy). . 15 Existentional Introduction
17 | | . . . . . _|_ . . . . . . . 2,16 _|_ Introduction
.. | <---------------------------- end subproof
18 | . . . . . . ~~Ex(Px -> VyPy). 2-17 ~ Introduction
19 | . . . . . . Ex(Px -> VyPy). . 18 ~~ Elimination

What is the point of this?First we would need it in a reliable source. Secondly the proof would have to be notable in itself as Wikipedia does not include proofs, it is not a textbook. Dmcq (talk) 12:15, 24 May 2014 (UTC)

Yet another proof

Somebody left an incomplete proof in game semantics. This is a complete version:

		$O$		$P$
1.				$\exists x.(Drinks(x)\to \forall y.Drinks(y))$
2.	$1?$		$2!$	$Drinks(m)\to \forall y.Drinks(y)$
3.	$2?$	$Drinks(m)$	$3!$	$\forall y.Drinks(y)$
4.	$3?$	$n$	$2!$	$Drinks(n)\to \forall y.Drinks(y)$
5.	$4?$	$Drinks(n)$	$4!$	$Drinks(n)$

At stage 4, the proponent goes back to stage 2 in order to defend against that attack again. For classical logic, that is allowed. In intuitionistic logic, more background is needed. -- Zz (talk) 11:49, 29 March 2011 (UTC)

I don't think this is interesting because the winning strategy for the verifier is to pick a non-drinker if it exists or a drinker otherwise, which is really just a restatement of the simple proof. Tijfo098 (talk) 17:24, 28 October 2012 (UTC)

explanation of paradox?

I looked through the material at the main page and this talkpage and did not find the following explanation: let's list everybody in the bar in a master list in such a way that all drinkers come before all non-drinkers (other than this condition the order is arbitrary within the set of all drinkers, and similarly for all non-drinkers). Then if we choose the last person on the list, we can affirm convincingly that if he is drinking then everybody is drinking. Doesn't seem that paradoxical, either, when you put it that way. Any comments? Tkuvho (talk) 16:04, 29 March 2011 (UTC)

That is a good introduction why it is not so counter-intuitive. But we do not even need a master list connected with a kind of determinism. The last person to drink (and that can change in different scenarios) is the one who, if he drinks, then everyone drinks. If there is not going to be a last person, because too few are drinking anyway, the statement is true in the first place. -- Zz (talk) 09:08, 30 March 2011 (UTC)

This only works if there are finitely many people in the pub, but the theorem is true for any (nonempty) domain. Sławomir Biały (talk) 11:43, 30 March 2011 (UTC)

Just make sure the ordering has a last element, by pulling someone from the final ω chain if necessary. — Carl (CBM · talk) 11:45, 30 March 2011 (UTC)

Ok, well if we're going to use the axiom of choice anyway, may as well do it properly: by well-ordering, the set of all nondrinkers on the has a least element. :-D Sławomir Biały (talk) 12:11, 30 March 2011 (UTC)

It is only paradoxical in that 'person' might be taken to mean a particular person who doesn't change identity. It's like saying there's a woman giving birth every five minutes (or however long it is) in Britain. As to the idea of ordering - any explanations should be based on something that can be cited. Dmcq (talk) 09:26, 30 March 2011 (UTC)

BTW I see it is about one and a third babies born every minute in Britain, I should have just remembered the phrase "There's one born every minute". :) Dmcq (talk) 11:58, 30 March 2011 (UTC)

Is there a consensus that this is helpful in explaining the paradox? While I have no evidence that this is in the sources, if everyone agrees that this is "common knowledge" perhaps we can include it in the page anyway. Tkuvho (talk) 12:22, 30 March 2011 (UTC)

I think I'd be against it without a source that was somewhat along those lines. Dmcq (talk) 14:21, 30 March 2011 (UTC)

The main point of the "paradox" is never mentioned

The article is written in such a way that most readers can't understand what is going on. This is a contrived problem based on two nonintuitive aspects of formal logic:

• The logical statement is considering only one configuration D of the pub! In other words, the statement assumes that D is fixed, whatever its values may be. Of course, in "real life" that is completely uninteresting. We generally don't make statements "if ... then ..." about single instances! When we read There is someone in the pub such that, if he is drinking, everyone in the pub is drinking., then in "real life" we assume that this statement applies to all possible states of the pub. (In which case, the statement is false because that person would have to keep changing to ensure that she is a nondrinker.) However, the formal logical statement presented assumes that D is fixed, and so it is only considering a single state of the pub, "the actual state". What this means is that the English utterance is being badly modeled. The logical statement should include "for all D..." for it to model what we mean. It does not! And this has been done intentionally by the creator of the paradox for the sake of confusing people. It's a problem of intentionally poor modeling.
• This is compounded by the nonintuitive use of "if X then Y" by which, in formal logic, the statement is always true with one exception: when X is true but Y is false. This means that any "nondrinker" will establish the truth that "if X is not drinking, then everybody is drinking". But this is secondary. And note that we can't be sure that any particular "someone" is not drinking! So this logic is only true when we can choose our "someone" to be a nondrinker. This is not true, in general, if we are considering all configurations of the pub, as normally one would assume.

The point is that you can have logical statements that don't apply to real life, and then you can pretend that those logical statements are good models.

The above two issues are what's relevant here. The other discussion on "excluded middle", "nonempty pubs" is peripheral and greatly adds to the confusion.

I'm doing a study of paradoxes and the article was so confusing that I spent much more time than I expected to. It should be deleted or explained for what it is, a contrivance, intentionally confusing. It's miseducation and makes Wikipedia look bad. AndriusKulikauskas (talk) 19:37, 29 January 2012 (UTC)

Non-emptiness of the pub has actually a crucial importance, because in an empty domain any ∃-formula will evaluate to false, a method of inference used in the proof will not be applicable and the theorem therefore will not hold. Incnis Mrsi (talk) 21:25, 29 January 2012 (UTC)

You think that's what makes this paradox interesting? or a paradox? or worthy of an article? or mention in an article? You surely have references! In general, is there any other reference to the Drinker paradox beside the creator's book? — Preceding unsigned comment added by AndriusKulikauskas (talk • contribs) 21:35, 29 January 2012 (UTC)

Surely, it worths a small article or at least a section somewhere in the Wikipedia. What else references do you want: mentioning and proof in the paper of Coscoy, Kahn and Théry is not enough? If you feel that Drinker paradox#Excluded middle is off-topical (and probably even WP:OR), then go forth and delete it. Incnis Mrsi (talk) 22:06, 29 January 2012 (UTC)

I've removed most of the excluded middle section as I think it was probably OR, I've put in a cite which actually mentions this though. The business about the empty set though seems to be referenced a number of times. Dmcq (talk) 01:27, 30 January 2012 (UTC)

The "Clausification in Coq" paper says that proof by resolution uses the Excluded Middle, the Axiom of Choice (AC), and non-emptiness of the domain, but that by using an obscure result of Kleene, proof without AC is possible. I can't say I've grokked the last part yet. It refers for further reading on that to a paper I've not read yet: "G. Dowek. Automated theorem proving in type theory. In Course notes for the 2nd International Summer School in Logic for Computer Science, University of Chambery, France, 1994", and which (unusually for a CS paper) is nowhere on the interwebs, although Dowek lists it here. Tijfo098 (talk) 12:32, 27 October 2012 (UTC)

obviously a riddle

it is a riddle, for he is the only man in said pub. therefore if he is drinking everyone in the pub is drinking. — Preceding unsigned comment added by 201.200.37.126 (talk) 03:04, 26 June 2012 (UTC)

Do we have any articles on the related notions?

Escardo and Oliva have an interesting result, showing that the "Drinker paradox" (as they call it) is perhaps more important than a mere amusement.

Theorem 1. The following are equivalent for any inhabited set X:

1. X is searchable.
2. X is boolean Dubuc-Penon compact.
3. X satisfies the boolean drinker paradox.

4. X satisfies the principle of omniscience.

But we don't seem to have any articles on the related notions: searchable set, Dubuc-Penon compactness, or principle of omniscience although we have Limited principle of omniscience. Tijfo098 (talk) 16:48, 27 October 2012 (UTC)

Beginners - read this first!

This is a ridiculously long article which, in spite of Wikipedia:Be bold, I do not feel bold enough to edit and yet which, in spite of its great length, and the length of its talk page, does not at any time explain the so-called "paradox" in simple terms.

The "paradox" arises because correlation does not imply causation. So when the "paradox" claims that "it can't be wrong to say that if that particular person is drinking, then everyone in the pub is drinking" it can be wrong - simply put, it would be more accurate to say "if that particular person is drinking, and everyone in the pub is drinking". Remove the requirement that any given drinker "causes" the rest of the pub to drink, and the "paradox" disappears. 87.194.209.163 (talk) 04:39, 22 November 2012 (UTC)

No that's not it. An easier way to understand it is to rephrase it as: "there exists a person such that either: (1) he is not drinking [period] or (2) he is drinking and everyone else is drinking." That way it seems trivially true. Tijfo098 (talk) 04:56, 22 November 2012 (UTC)

I think we should be open to the possibility that there are multiple ways of understanding this paradox. Personally, I think it makes more sense as a "paradox" when spoken in natural language, and I don't really like the inclusion of logic-type in the lede (defined as the "actual theorem".) I fear that these symbols will turn off the average reader, and I don't think they particularly convey the fun paradox that applies to the statement in question. More broadly, I don't agree with the interpretation currently presented that the theorem is the "actual" theorem behind the statement. groupuscule (talk) 15:05, 23 November 2012 (UTC)

Unfortunately all the sources discussing this topic are discussing it as a theorem in FOL. You're welcome to try find other sources, but I rather doubt you'll turn up any—I've look for such sources myself, unsuccessfully. As for the rest of your proposal, see WP:OR. Tijfo098 (talk) 06:26, 24 November 2012 (UTC)

Could you express it like this? "There is at least one person in a pub. Everyone in the pub is either drinking or not drinking. If the last non-drinker takes a drink, everyone in the pub will be drinking." 203.217.150.76 (talk) 03:54, 6 March 2014 (UTC)

Quantifier scope problem

It seems to me that the apparent paradox arises because of a difficulty in expressing the scope of the existential quantifier. The theorem states:

$\exists x\in P.\ [D(x)\rightarrow \forall y\in P.\ D(y)].\,$

and this is a true theorem, because as explained in the article it depends on the fact that a material implication is trivially true if either the antecedent is false or the consequent is true. As such, it is somewhat misleading to translate it into English using if/then because material implication is only a crude approximation to natural language conditionals. The problem is that when the reader hears the English text There is someone in the pub such that, if he is drinking, everyone in the pub is drinking he mistakes it for

$\exists x\in P.\ D(x)\rightarrow \forall y\in P.\ D(y).\,$

Note the lack of brackets, which means that the scope of the existential quantifier is restricted to the antecedent only. This is quite different and certainly not a theorem. One might try to express it as If anyone in the pub is drinking then everyone is drinking. The paradox therefore arises because natural languages such as English are rather poor at being able to express quantifier scope (and quantifier order) unambiguously. Indeed, this is the very reason why it is helpful to use predicate logic, because it permits a degree of precision of expression that is difficult to capture otherwise. This is also why, despite the requests above for a non-mathematical statement of the paradox, this cannot be done. Dezaxa (talk) 20:48, 9 June 2015 (UTC)

The set P is irrelevant. This

$\exists x\ [D(x)\rightarrow \forall y\ D(y)].\,$

is a validity of FOL. — Preceding unsigned comment added by 81.155.219.165 (talk) 21:57, 8 October 2018 (UTC)

Trivial resolution if there exists even a single non-drinker

Smullyan's Original Drinker Principle:

"[T]here exists someone such that whenever he (or she) drinks, everybody drinks."

--R. Smullyan, "What is the name of this book?" p. 209

We begin by proving, for arbitrary predicate D and proposition Q:

$\exists x:[D(x)\implies Q]$

We make the very reasonable assumption that somewhere there exists even a single non-drinker. Then we would have:

$\exists x:\neg D(x)$.

Applying existential specification, we can infer:

$\neg D(a)$

For any proposition $Q$ whatsoever, be it true or false, we also then have:

$\neg D(a)\lor Q$.

Applying the definition of logical implication, we have:

$D(a)\implies Q$.

Generalizing, we conclude:

$\exists x:[D(x)\implies Q]$

Note again that $Q$ need not even be true. It could be false or contradictory. Clearly, such constructs have very limited application in the "real world." Here, anything goes.

Example 1

As in Smullyan's original "Drinker Principle," we could set:

$Q~\equiv ~\forall y:D(y)$

Here, Q contradicts our initial assumption!

As in Smullyan's original "Drinker Principle," we would have:

$\exists x:[D(x)\implies \forall y:D(y)]$

Example 2

Alternatively, as in other popular versions of DP which explicitly mention a pub, we could set:

$Q~\equiv ~\forall y:[P(y)\implies D(y)]$

Where "$P(y)$" means y is in the pub.

Then we would have:

$\exists x:[D(x)\implies \forall y:[P(y)\implies D(y)]]$

---

A Set-theoretic Variation of DP

If we assume the non-existence of a universal set, then every set must exclude something. Thus, similar to the above argument, for any set $D$ and proposition $Q$, we must have:

$\exists x:[x\in D\implies Q]$

--Danchristensen (talk) 17:27, 17 April 2024 (UTC)

Simply because the formal statement is timeless does not mean it can apply to the real world

In the current article the 2nd paragraph includes a claim unsupported by any reference: "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."

There are obvious difficulties with a claim that the formal statement applies to the real physical world for an instant of time. One is that according to the general theory of relativity, there is no such thing as a single instant that applies over a region of space. Another is defining when a person is drinking and when they are not. How many molecules of the drink have to pass the lips of a person before they are considered to be drinking? What precise line on the constantly moving lips is to be the determining line? What are the exact locations of the moving drink molecules that exhibit quantum behavior? The notion of the formal statement being true in the real world at an instant in time is ludicrous. --Jamesrmeyer (talk) 20:51, 24 November 2019 (UTC)