Talk:Barbershop paradox

Why would Carroll have considered this a paradox?

I'm a little confused as to why Carroll considered this a paradox. In his notes he says
The paradox is a very real difficulty in the Theory of Hypotheticals....Are two Hypotheticals, of the forms If A then B and If A then not-B, compatible?
Of course they are! Why should (P > Q) and (P > ~Q) be contradictory? The contradiction of (P > Q) is ~(P > Q), which resolves to (P . ~Q), which is not the same as (P > ~Q) at all. So my question is: were the rules of logic different in Carroll's day? Was it not considered a standard law that from a falsity you can prove anything (F > R)? And if so, should we make it clear that he made the paradox under different logical rules? — Asbestos | Talk (RFC) 14:39, 19 May 2006 (UTC)

Asbestos,

My understanding is that when Carroll wrote his essay (1894) the notion of truth-functional material implication was not yet current in English logic; it was introduced mainly through the efforts of Bertrand Russell about a decade later. Truth-functional implication doesn't always line up very well with the intuitive use of "If-then" constructs so Carroll could not take for granted some of the features of material implication that are taken for granted by philosophers today. Note the questions that Carroll poses for logicians in his concluding "Note":

Can a Hypothetical, whose protasis is false, be regarded as legitimate?

Are two Hypotheticals, of the forms "If A then B" and "If A then not-B, compatible"?

What difference in meaning, if any, exists between the following Propositions?

# A, B, C, cannot be all true at once;

# If C and A are true, B is not true;

# If C is true, then, if A is true, B is not true;

# If A is true, then, if C is true, B is not true

If you think that material implication unproblematically expresses anything that you need to express by means of a conditional, then the "paradox" is simply obsolete: each of Carroll's questions has a definite and easy answer -- a hypothetical with a false protasis is regarded not only as legitimate, but as always true; two hypotheticals with the same antecedent but contradictory consequents are compatible so long as the antecedent is not true; and all of the listed propositions, if expressed truth-functionally (~(A . B . C), (C . A) > ~B, C > (A > ~B), A > (C > ~B)) are logically equivalent. And those answers provide a handy solution to the paradox proposed: Carr can be out as long as Allen is not also out, as per the answer to the second question.

If, on the other hand, you think that material implication fails to capture something interesting or important about conditional statements (which a lot of philosophers do think, for independent reasons), then Carroll's questions are likely to remain live ones for whatever notion of a hypothetical you think is uncaptured by material implication. Hope this helps. Radgeek 06:24, 21 May 2006 (UTC)

@Radgeek and Asbestos: I'm having a go at improving this article, based on the issues raised on the Talk page. Thanks Radgeek for your clear explanation about material implication -- in my comments in the section below I do briefly outline how this approach can help us understand the Barbershop problem better, but that I believe it is not the main issue at stake. Comments welcome. FrankP (talk) 20:18, 16 December 2019 (UTC)

"Paradox"?

I'm by no means an expert, so I can't be sure this wasn't covered by Radgeek, but I have a very sort of layman solution to the problem (which, most likely, does not go to the heart of the issue, but at least proves that a better wording of the problem is, in my opinion, necessary):

The conditions state that it is necessary that A) Someone is in the shop at any given time B) Allen can not walk to the shop (and be in the shop) without Brown

Based on the above conditions, it's possible to conclude that Allen never goes to the barber shop at all; IF he did, he would go with Brown. However, in this scenario, only Carr and Brown can switch off shifts (or work together) and none of the imposed conditions are false.

Even so, if we add the condition that C) Allen must work sometimes, there is nothing in the conditions that prevents all three barbers from being in the shop at some time, allowing Carr to switch off with Brown and Allen, who work always as a team. (the conditions state that the 3 barbers are not all always in the shop; that does not mean that they can not all 3 be together in the shop for some time to go on/off shifts)

Anyways, just a small contribution by an inquisitive layman. I can't guarantee that I'm right, and I'm fairly certain that this "solution" does not assess the actual problem proposed; however, it seems that for a philosophical paradox to exist, the conditions have to be spelled out precisely to the letter, and in this case, I believe there has been an ommission.

-Wasted —The preceding unsigned comment was added by 70.71.61.212 (talk) 07:29, 18 February 2007 (UTC).

Logic

I think my logic really stinks here. How can we adequately express the idea that the sick guy A has to be in the shop at all times unless B is with him? Since it was stated all 3 are never there all at once (put potentially, 2 could be) it means that when A is out, C is in. It's not really a conflict to say 'if A is out' because why should he be? Of course, if the brothers had seen A out and about they should assume B was out too (and probably should have seen him with him) so it would be right to assume C was in. Man, Carrol is complex, I need to get some foundations and come back to these semantics. Tyciol 20:46, 7 March 2007 (UTC)

No, I think you've slipped up there -- you've said "it was stated all 3 are never there all at once", but that is not what is stated. The phrasing in the article is that "not all of them are always in the shop", which is different. It means there are times when not every one of the three is in, but it does not rule out some moment when all three are in at once.

If the article wording does not convey this clearly enough, I would be happy to change it. But I'm pretty sure that is what is meant. FrankP (talk) 17:22, 16 December 2019 (UTC)

Misunderstandings, plan for article

So from what I can tell the hypothesis referenced in the above two comments that someone must be in the shop at all times is misleading. The hypothesis listed in the article, "since someone must be in the shop for it to be open" is correct and the subtle distinction clears the air of all the switching off of shifts problems. Another point of confusion is in the actual article: it took a few minutes for me to realize that the fact "that Allen's recently been very ill" was not a hypothesis of the sort 'Allen cannot be the one manning the shop at the time of the story since he is ill.' I think it would be simpler to omit that sentence and just state that Allen never leaves the house without Brown, maybe playing up his nervousness and using the word 'fever' as Carroll did.

That brings me to propose major changes to this article. I'd like to add a section Simplifying Carroll's story, with terminology and a list of propositions like A - Allen is in. I think the two main points (that the contradiction that Joe says clinches his proof is between (~A > B) and (~A > ~B) and that this was Carroll's main dilemma, which has now been cleared up by the law of implication the Russell championed) are not made clear enough. Basically these changes won't affect the spirit of the current article, just incorporate the well written talk posts and make it easier to read for nonexperts (they shouldn't even have to read the original article linked to in the lede). - Callowschoolboy

The problem seems to be a misunderstanding the entire story! Lewis Carroll put forth a symbolic logic argument that is false in reality. That shouldn't be possible--symbolic logic should be perfect logic. Whether the theory of hypotheticals in use in 1894 was thusly flawed or whether Carroll misunderstood the application of the theory is a question better left for historical research, but this paradox doesn't pass muster by proper application of the theory by modern logicians.

Article needs to be scrapped and re-written. CAHeyden (talk) 19:31, 22 May 2010 (UTC)

@CAHeyden: I've responded to a call to review this article. I don't think it's necessarily as drastic as scrapping and rewriting but I'm sure it can use some clarification. Please see my comments below for some further historical context about this problem. FrankP (talk) 20:29, 16 December 2019 (UTC)

Further changes

I like the shorthand ya'll use here in the Talk, and it has historical support in Principia and all, but now that I've finally tracked down the more formal symbolism and how to implement it I think I'll update the article to use it.
- Callowschoolboy 17:22, 6 July 2007 (UTC)

Am I missing the paradox?

Ok so we have the rules set up. Here is where I'm hitting the problem and I think this is what Carroll was trying to prove. So far we have that If A is out, B must be in to have the shop open. It is then reasoned that if A is out than B is out because A won't leave without B. The paradox that I think is suggested is that the rules as given say that if A is out of the shop then B must be there and at the same time not be there. However that's not a paradox at all because B does not have to be out of the store simply because A is not there. There isn't a rule saying that A can't be left at home by himself. If C is not at the shop and neither is A, there is no reason that B can't be at the shop alone. H2P ^{(Yell at me for what I've done)} 07:10, 10 July 2007 (UTC)

Ok I see the problem. "If C is out" does NOT cause a paradox because there is nothing stopping A from being in. The paradox comes from "If C is out and If A is out" causing two solutions. The first rule states that in this instance, B must be in to run the shop. The second rule states that B must be out with A. B can't be both in and out and thus the paradox. However, there aren't any rules stating that when C is out, A must also be out. Actually, a new rule is formed by this paradox: If C is out A is in. H2P ^{(Yell at me for what I've done)} 08:17, 10 July 2007 (UTC)

Your first comment was dead on (not saying the second isn't but this whole thing is so confusing that I have to read everything several times ;). And I think in your second comment your basically saying that although the confusion between Allen being out of the shop and Allen being out of his house, "out and about" so to speak, also complicates the discussion about whether the "hypotheticals" that Uncle Jim cooks up are contradictory. I didn't help matters I guess when I stated Uncle Joe's second axiom as "Allen goes nowhere without Brown." I was trying to strengthen the narrative foundation for the logic question Carroll poses. You're right that even taking this strong statement it could be that Allen stays in one place while Brown goes elsewhere (eg A and B go together to the shop, then A stays in the shop while B galavants about town). Although Allen would be stranded, this would allow a narrative solution, rather than addressing the logical conundrum. In that sense it's not helpful to the reader to have this loophole in the story. Any suggestions how to close it? - Callowschoolboy 13:34, 11 July 2007 (UTC)

I would like to go back to something more like what was there before, in order to not skip steps. This is an extremely slippery subject and non-logicians might not easily see that (C ∨ A ∨ B) ∧ (C ∨ A ∨ ¬B) can be a true statement if Allen is in, nor would they seperate the issues we enumerated above from the underlying logic.
- Callowschoolboy 14:45, 11 July 2007 (UTC)

@HeirToPendragon and Callowschoolboy: H2P I agree you've hit the nail on the head there ("If C is out A is in"). I'm trying to clear up some of the confusions here, see new section below. Comments welcome. FrankP (talk) 20:33, 16 December 2019 (UTC)

Major point of confusion

I think the next step should be to identify the most important points (off the top of my head: LC wrote a story knowing that it was a paradox but intuitively shouldn't be with Jim as a straw man, underlying the story is a pure logic scenario, that scenario is not a paradox under modern logic which reconciles our intuition, etc).

Another list to make and emphasize (at the cost of any unimportant pieces of the article) would be points of confusion, closely related to that initial list. One of the big ones that just came up has to do with commonsense solutions such as "Why couldn't Allen be in, as long as Brown walks him to the shop?"

I started trying to summarize these sorts of objections, when I realized how big a problem my strong statement of Axiom 1 is. Using the result (C ∨ A ∨ B) ∧ (C ∨ A ∨ ¬B), which is really just (Axiom 1) ∧ (C ∨ A ∨ ¬B), I broke down the truth values of some possible commonsense solutions:

In the shop	Not in the shop	(A ∨ ¬B ∨ C)	Notes
Allen	Brown, Carr	(T ∨ ¬F ∨ F) ≡ T	B picked A up at A's house, they went to the shop together but B left at some point, A not compelled to go with him (argument against strong Axiom 2)
Brown	Allen, Carr	(F ∨ ¬T ∨ F) ≡ F	Again run into problem with the strong statement of Ax2, but what if Allen stays at his home, Brown goes in, and who knows what Carr does.
Allen, Carr	Brown	(T ∨ ¬F ∨ T) ≡ T	All this requires is that, as above, Brown escort Allen to the shop but not necessarily man the shop with him (and Carr in this case).

ALl of these assume that Allen does not live in the shop but we already know that "three barbers ... live and work in the shop."

Allen's house IS the shop so he can't be escorted there. Skrofler (talk) 19:41, 1 December 2011 (UTC)

Third case

What if both A and B are in the shop and C is out? I fail to see how this is a contradiction to the initial conditions. --86.127.22.71 12:30, 2 August 2007 (UTC)

Resolution

It would be nice if the article made it clearer why the paradox is not really a paradox. Rather than using paragraphs of symbols, we laymen would prefer an explanation in English. If the shop is open, then at least one of A, B and C must be in. A never goes anywhere without B, so it follows that if A is in then B must be in, and it follows that if A is out then B must be out. So the only possible outcomes from these conditions are (1) The shop is shut; (2) all three barbers are in; (3) C is in; (4) A and B are in. That's as complicated as it gets, as far as I'm concerned. In the story the shop is open, so that rules out (1) as an outcome. According to this, it's never going to happen that C is out and A is out, so "if C is out and A is out..." is a false premise. I have made it simple and not used any symbols, so does that mean I can't contribute to the debate? Brequinda 13:44, 27 August 2007 (UTC)

I agree with Brequinda's analysis. There are only 8 possibilities when you have 3 binary variables: 1 A.B.C -- all in 2 A.B.notC -- A.B are in 3 A.notB.C 4 notA.B.C 5 A.notB.notC 6 notA.B.notC 7 notA.notb.C -- neither A+B is in 8 notA.notB.notC -- none in, shop is closed

I can't understand why this posed a problem. It is either a simple problem in combinatorics or Boolean Algebra. Perhaps Boole's work hadn't percolated very far at that point? Diakron99 20:05, 13 November 2007 (UTC)

Yes, something seems wrong here

Either it is a misstatement of the conditions, or just plain wrong. It seems clear that Allen and Brown can be in the store, without Carr. So the assertion that Carr MUST be there seems false. Am I missing something? —Preceding unsigned comment added by Bigmac31 (talk • contribs) 20:39, 4 August 2008 (UTC)

I'm confused too.

The possibilities seem to be:

A, B and C

A, B and not C

C, not A and not B

But that's not what the page says atall. Stutley (talk) 09:53, 4 July 2009 (UTC)

Well, actually there are five possibilities:

A, B and C

A, B and not C

C, not A and not B

A, not B and C

A, not B and not C

The "paradox" arises only from the assumption that both C and A are out. Thus the assumption is proved to be wrong. That is at least A or C should be in. This statement can fully replace Axiom 1. Sharkb (talk) 09:32, 10 September 2009 (UTC)

I agree. The "paradox" only arises if one has strict assumptions that do not even make sense. It seems that Uncle Joe's original argument is that Carr MUST be in. However, this situation could arise: Brown and Allen are in. At the same time. Both cutting hair. So it's possible that, meanwhile, Carr could be out. Having a frikkin' beer... The problem is the sentence / assumption: "Suppose that Carr is out. If Carr is out, then if Allen is also out...." What is meant by "...if Allen is also out"? WE ALREADY KNOW THAT ALLEN CANNOT GO OUT WITHOUT BROWN. It should read: "Suppose that Carr is out. Therefore Allen and Brown are in" So Uncle Joe is wrong. It is easily possible for Carr to be out and for the shop to be OPEN AND RUNNING: ALLEN AND BROWN ARE IN, FOR CRYING OUT LOUD. What am I missing? Uncle Joe could easily arrive at the shop to find Brown and Allen there, cutting hair, talking, drinking coffee, yada yada. He would say: "Where's Carr? I prefer him", and Brown and Allen would say: "Charming. No-one's forcing you to come here with your ABC in-and-out bullshit paradoxes... go somewhere else, stubbly freak". CASE CLOSED. Oh, sorry for shouting, but this "paradox" was bugging the hell out of me. —Preceding unsigned comment added by 86.177.114.110 (talk) 00:27, 20 February 2010 (UTC)

I couldn't agree more to the above comments. Something is wrong here. I'm glad to see I'm not the only one seeing the apparent absurditiy of this so called paradox. The last unsigned comment was hitting the nail on the head, even more so in the sense of illustrating in written word my and possibly many others frustration with this article.

First let me say something. I love this kind of stuff. For many years I have enjoyed reading and solving many brain teasers, math problems, probability paradoxes etc. Im interested in and have read many logic paradoxes. I enjoy the challenge of wrapping my mind around what the authors are getting at. Usually the challenge is in just understanding why its a paradox in the first place. I feel for those who sometimes just dont get it. Some just can't let go of their stubborn common sense and see beyond the obvious. They refuse to open their mind and allow their thinking to be pushed into a mode we rarely achieve in day to day living. I never wanted to be or feel like one of those people, but I might now.

Because with this one, I just don't get it and I don't believe there is anything to "get"! The last sentence of this paradox is simply a false conclusion by any reasoning.

For the life of me I just can't see how the last sentence, "So, by contradiction, Carr must logically be in." is a logical conclusion. Given everything stated above it, the last sentence should and can only read:

"So by contradiction, if Carr is out, Allen & Brown(or just Allen) must be in."

The contradiction has nothing to do with the "feasability" (not sure of correct word to use here) of Uncle Joe's first hypothetheticle(if Carr is out). His second hypotheticle(if Allen is out) creates the contradiction and the two hypotheticles are "incompatible" because now Brown is required to be both in and out. So the contradiction demands that the SECOND hypotheticle cannot be. Not the FIRST. How possibly could Uncle Joe logically manage to leap back to his first... to conclude that his first hypothetical situation cannot be, just because he worked himself into contradiction further down the line? Its ridiculous. No sane reasonable person would think this way.

I mean seriously, I could do the same with anything. Here's my paradox:

Given: Ants are insects

If I am a human and if ants are reptiles...

oh wait, that's a contradiction.

So by contradiction, I am not a human.

"Silly" you might say. "huh?", "uh... what?", "wtf" maybe, "nonsense" of course, but look, is it really so different from this paradox? I stated a given. Proposed a hypothetical. Fine so far. Followed by a second hypothetical which conflicted with the given. Concluded by contradiction the first hypothetical could not be. I followed the same formula as the Barbershop Paradox.

Shoot, if I were a famed respected writer or humorist or something, I would publish my dumb paradox of nonsense and 100 years from now there will be people like us sitting around having deep phylosophical disscussions about the brilliant hidden logic. On the flip side, if me personally or some other nobody wrote the "Barbershop Paradox" today, I would be laughed off the net and told countless ways how I don't know what I am talking about. (which will probably happen anyway here. its happened before elsewhere, hehe)

Now that I think about it, maybe the former is whats really going on here. I think LC was having a bit of fun with some people back then. He knew better but maybe he thought they wouldn't, and now he's getting us too. 71.81.66.222 (talk) 04:24, 9 March 2010 (UTC) didn't realize I was logged off when I signedRacerx11 (talk) 05:09, 9 March 2010 (UTC)

After rereading the entire article, it becomes apparent, for this to be clearly understood as a paradox, the article needs to specifically explain what relevant issues concerning the Theory of Hypotheticals were causing debate at the time it was written. Exactly what logic theory and what application of such theory, existing contemperary with LC would cause such a paradox and why? -Racerx11 (talk) 21:49, 16 March 2010 (UTC)

From a layman's point of view:

• As several people have pointed out, there really is no problem: Carr can be out, as long as Allen and Brown are in.

• Even if this was not the case (for example, because we postulate that "At all times, exactly one person must be in the shop"), it's not a paradox either: In this case, Carr is required to stay in the shop at all times. It's not more or less a paradox than to postulate "All sheep are white."

• Suppose we extend the set of rules even more to achieve a state where they can't all be reconciled with each other, I doubt that this constitutes a paradox in the stricter sense of the word -- rather, it would simply be an incoherent set of rules which are mutually incompatible, like a) "All sheep are white", and b) "You must be black to become a sheep".

I read Carroll's original text given by the link, and I think, Carroll wanted to arrive at something else: I surmise he wanted to question the validity of the statements ("Are these really valid sets of rules?"), or whether one could arrive at an incoherent system though building it up from individually sensible rules -- but I don't fully grasp what he was up to.

Someone more knowledgeable out there to fill in the gaps...? -- Syzygy (talk) 10:53, 25 August 2010 (UTC) (edited for sig)

The "simple explanation"

====The Simple Explanation of this "Paradox"====

The simple truth of the matter is that this is no paradox at all. Instead calling it a "paradox" is the result of an obvious error in logic. The reason being is that if Carr is out, then nothing is stopping both Allen and Brown from being in. What was shown is that if Carr is out, then for Brown to be out as well would be a contradiction, as that would mean Allen's absence as well, when there must be at least one person at the shop. Also it was shown that if Carr is out, then for Allen to be out as well would be another contradiction, for similar reasons. Then Carroll jumped to the conclusion that meant Carr must be in. But once more, nothing at all is stopping both Allen and Brown from being in the shop. Therefore all that has logically really be shown is that either Carr is in or Carr is out, in which case both Allen and Brown would be in.

I have take this from the article main page. Firstly, it's OR. (Please, no discussions about "it's so obviously logical, it don't need references".) Secondly, and more importantly, IMHO it is beside the point.

Obviously, the "paradox" at its face value can be resolved with no big problem. But if this was the case, nobody would seriously consider it a philosophical problem anymore. I feel that Carroll was up to something different, namely the inconsistencies created by a consistent set of rules, but I fail to graps what he exactly wanted to say. Someone more knowledgable would have to step in here. --Syzygy (talk) 08:51, 27 September 2010 (UTC)

But if this was the case, nobody would seriously consider it a philosophical problem anymore. -- That's like saying that, if Hitler wasn't onto something, there wouldn't be any neo-Nazis. People with "philosophical problems" with material implication end up in the same soup as Carroll ... suggesting that the problems they see are illusory. Much as people might feel uncomfortable with the assertion that "If squares are round then my mother is a tomato" is true, that's the only reasonable and consistent way to view it. -- 71.102.133.72 (talk) 07:11, 15 September 2014 (UTC)

Why this is not a logical paradox

Condition 1: There must be at least one person home as Joe and Jim clearly see that the shop is open.

Condition 2: Allen is eccentric and cannot leave the house without Brown (He can't leave the house with just Carr, apparently).

Here are all the permutations and whether they are logically sound or not.

In the shop	Not in the shop	Logically sound or unsound	Reasoning
	Allen, Brown, Carr	Unsound	Violates condition 1
Allen	Brown, Carr*	Sound	Meets both conditions
Brown	Allen, Carr	Unsound	Violates condition 2
Carr	Allen, Brown	Sound	Meets both conditions
Allen, Brown	Carr*	Sound	Meets both conditions
Allen, Carr	Brown	Sound	Meets both conditions
Brown, Carr	Allen	Unsound	Violates condition 2
Allen, Brown, Carr		Sound	Meets both conditions

As one can see, there are two ways for the shop to be open without Carr being home [¬C ⇒ (A ∧ ¬B) ∨ (A ∧ B)], therefore, the logic that Uncle Joe used to assume that Carr must be home is erroneous, therefore there is no 'contradiction' and therefore no paradox. --HakuGaara (talk) 14:58, 29 June 2011 (UTC)

Exactly my point: The conditions can be met, and even if they couldn't, that wouldn't constitute a paradox any more than saying "You have to meet me on February 31st" -- it's simply a rule which can't be fulfilled.

Thus, as I said earlier, I feel we're all barking up the wrong tree. I don't think we outsmart the current set of pro philosophers so much that they would still a) consider this a true paradox and b) couldn't come up with the same conclusion to "solve" it. Hence IMHO the paradox lies somewhere else -- not in the fact that the barbershop couldn't be manned but... yes, this is the question: What really is the paradox? -- Syzygy (talk) 07:04, 4 July 2011 (UTC)

It's simple. This is NO paradox. You can simply have Allen and Brown together in the shop. — Preceding unsigned comment added by UltimateDragonMaster (talk • contribs) 20:02, 6 July 2011 (UTC)

You don't even need to have Allen and Brown together in the shop. Allen never leaves the shop without Brown. It doesn't say anything about Brown never leaving the shop without Allen. So either Carr is there, Allen is there, Carr and Allen are there or Allen and Brown are there. Brown is the only one who cannot be by himself, because that would require Allen to be out, which he is not keen on. So there is no paradox, just a flaw in logic. -- Jahkayhla (talk) 00:43, 31 January 2012 (UTC)

@Skrofler, Brequinda, Stutley, Sharkb, Racerx11, and Syzygy: All -- you are correct in your analysis of the problem, especially by looking exhaustively at the 8 possible cases, but that's a very modern approach (effectively a truth table). I've tried to fill out the historical context a bit (see below) as to why the problem has (or had) significance, and will try to improve the article accordingly. Comments welcome. FrankP (talk) 20:49, 16 December 2019 (UTC)

Deletion

For more than four years now, this talk page has pointed out why the topic of this article, as presented, is no paradox at all; and this can be easily seen by simply considering all seven options. It is, in fact, a rather obvious mistake: the fallacy of False dilemma, known since long before the 1890s. The only paradox I see in all of this is how anyone can rate this article as Mid-Importance: "The article covers a topic that has a strong but not vital role in the history of philosophy". As it stands it is simply a brain fart that fails the test of notability.

It is possible that Carroll was well aware of his mistake when writing the original article and that his point was something else entirely. If so, PLEASE TELL THE READER WHAT THE POINT IS. It's also possible that the Barbershop Paradox is commonly cited as a famous mistake akin to miasmas or the four classical elements, but if so PLEASE TELL THE READER WHY.

I appreceate that work has been put into this article, but to be honest, I think it lacks crucial parts to make it notable. -- Spearman (talk) 12:57, 14 February 2012 (UTC)

This article isn't about a paradox, it's about history of logic. The point is that it isn't a paradox but some people when logic was being developed thought it was a paradox. Later, more advanced techniques of logic were developed and showed easily it wasn't a paradox. This shows there was enough debate about the question in the early days for a significant logician to write a significant (more or less significant) paper on this (probably just to show it isn't a paradox.) RJFJR (talk) 19:26, 18 February 2012 (UTC)

In that case I think it needs to be reworked to reflect that. As it is, it presents a "problem", and proceeds to set it up and solve it formally, but the article doesn't really explain that he was using completely bogus logic until the last part; and when it does, it's done in a way that is not easily understandable to the non-logician (or at least not to me). The "bogus" part has nothing to do with later developments in logic; simple brute forcing of the problem can be done in your head. If it is, as you say, that this is such an important event in the history of logic, I think the article should be changed to focus on, or at least include, what (real) problems logicians struggled with at the time and what changes Carrols paper helped bring about.

The real problem for this article is that I'm not the first to say this - Asbestos, as the first of several, pointed it out in 2006. When readers today are left with the same questions that he was all those years ago, not only is the article badly structured and lacking in crucial information, but I'm starting to think that the people whose task it is to show the rest of us why this is notable are either not willing or not able to do so. -- Spearman (talk) 19:58, 3 March 2012 (UTC)

@Spearman and RJFJR: I think you're right that it's more about the historical development of logic than being a truly challenging (paradoxical) logical problem. I'm going to try to improve the article, see below. FrankP (talk) 20:52, 16 December 2019 (UTC)

Layman's terms?

As far as I can see, these are the possible scenarios that do not contradict the rules:

At least one person is in the shop; and, If A is out, B is out

Shop: A, B, C Out:

Shop:A, B Out: C

Shop: A, C Out: B

Shop: A Out: B, C

Shop: C Out: A, B

It is very easy to see that in two of these scenarios, C is out. So where is the paradox? In reference to what someone said before, it says nowhere that they cannot all be in the shop at once. Which also means if there was ever a problem with shifts, C could go to the shop from wherever he was, followed by A and B going to the shop from wherever they are (note that someone is of course already in the shop), followed by any of the above scenarios occurring by having a person or people leave as necessary. I cannot find how this is a paradox even slightly. It seems to be based on the assumption that A and B are NOT together in the shop, which is nowhere in the rules. — Preceding unsigned comment added by 74.15.25.97 (talk) 06:54, 9 March 2012 (UTC)

Nonsense

It's obvious that ¬C => A ∧ B (¬C => A ∨ B and A <=> B), so based on the hypothesis ¬C, saying ¬A is a contradiction by itself and can't be used in the implication. Or, in simpler terms, if C is out, then both A and B are in (because they can't be separate), so if we suppose that C is out, it is illogical to suppose that A could be out. This article should either be deleted, or it should be clearly stated that his paradox is false. 184.144.193.166 (talk) 03:09, 11 November 2014 (UTC)

Hoping to clear this up

Hello all, I've followed a link to here from Project Mathematics. There's obviously been a bit of frustration over this article, but I do think I can see a way forward. So I'm first going to address some of the important points made above, and explain how I see the problem, before (boldly) having a go at the article to see if I can improve its clarity and coverage.

• In vs Out: Before coming to the knottier logical stuff, let's just clear up one small confusion. Some commenters have assumed that Allen or Brown go out to the shop to work, or one has to pick the other up at his house, or whatever. As noted by some replies, this is not what the problem states. The barbers live and work at the shop. If you have access to the original paper you'll see there is no doubt about this. In the narrative the states of being "in", "at home" and on shaving duty are all used interchangeably.

• Paradox?: My next point is about whether or not this is an actual paradox. In the lede of that article you will see that, "Informally, the term paradox is often used to describe a counter-intuitive result." It is probably in that sense that we should take it. In fact, Carroll presented this problem in a succession of versions (see next item), and did not always describe it as a paradox, but also under other titles such as A logical puzzle and A disputed point in logic. He never used the exact phrase The Barbershop Paradox, but that is how it has come to be known.

• Historical context: Commenters have rightly asked for the puzzle to be given some context, especially as to why it might be important, ultimately asking whether it merits an article at all. I am strongly of the opinion that it does, and hope to show you why. I rely on the following source which I will add to the article when I edit it: Carroll, Lewis (1977). Bartley, William Warren (ed.). Symbolic Logic, Parts I and II. Harvester Press. ISBN 0855279842.

Bartley tracked down, in various collections, never-published galley proofs of the unfinished second volume of Carroll's Symbolic Logic, which had long been thought to be lost. He has also collected together correspondence and privately printed papers by Carroll which together tell the story of the debates around this problem and other developments within the study of logic in this period. Carroll would habitually print and circulate challenging logical puzzles to various acquaintances, and in particular he had a long-running antagonism with his Oxford colleague, the Wykeham Professor of Logic John Cook Wilson. Wilson is here represented by the character of Uncle Joe, who attempts to prove that Carr must always be in the shop.

• The central issue: The earliest form of the problem arose from correspondence between Carroll and Cook Wilson, and was presented by Carroll in different forms before settling on the Barbershop narrative that was published in Mind in 1894. The heart of the issue was Wilson's inability to correctly negate a conditional. With hindsight it's easy for us to say how thoroughly wrong he was (hope that's NPOV enough?!). But (i) at the time, especially in Oxford, modern logical methods were poorly understood and (ii) Carroll did everything he could to obfuscate the issue, perhaps to trap the eminent Professor (there is support for this view in Carroll's journals) or perhaps to make a point about the difficulties of capturing natural language in logic (WP:OR).

Suppose we wish to negate a simple conditional statement, like "If A then B". Anyone who has studied logic will know that to say "If A then not B" is the wrong answer. "If you are French, you are a great painter" is untrue. But we don't express its opposite by claiming, "If you are French, you are not a great painter". Cook Wilson's reply to Carroll's problem relied on this exact fallacy.

Here's how Cook Wilson's argument actually went:

• We are told that (X) "If Allen is out, Brown is also out" (to keep him company)
• We also know (Y) "If Carr is out, then if Allen is out, Brown must be in" (so that there is someone to mind the shop)
• But (Y) is equivalent to "If Carr is out, then not (X)" (reasoning fallaciously as to the negation of a conditional)
• And we know (X) is true, so by reductio ad absurdum Carr is definitely in.

• Counter-intuitive: But this is not just about one logic professor's incompetence. Another way to look at the structure of the problem is to see that, supposing Carr is out, we have to simultaneously hold true that "if Allen is out, Brown is out" and also "if Allen is out, Brown is in", seemingly incompatible statements. We resolve the apparent contradiction when we realise that Carr being out only presents a problem if Allen tries to go out, whereas if he stays in, we avoid any difficulty. We conclude correctly, "if Carr is out then Allen is in". Again, this is easier to see with modern logical methods, but certainly presented problems for Carroll's contemporaries, especially those unfamiliar with the work of Boole, de Morgan and others.

• Material implication: One final point relates to the connection with material implication, which is a technical term referring to the mainstream interpretation (today) of how conditionals should work in mathematical logic. The conditional ${\text{A}}\Rightarrow {\text{B}}$ ("A implies B") is False only when A is True yet B is False. The conditional is True otherwise -- firstly in the intuitive case when A is True and B is consequently True, but also whenever A is False, no matter what value B has. This formulation works nicely in maths, but it leaves natural language behind, because sentences like "If I have three arms then the moon is made of green cheese" have to be accounted as true, whereas for most people they are patent nonsense.

There are other philosophical approaches to implication, such as causal implication, where a causal connection is required between antecedent and consequent before regarding the conditional statement as true. While the Barbershop Paradox is easily resolved by applying material implication, it should be noted that this is not where the problem lies. The arguments concerned will might just as well if the implications are considered causally -- Brown has to accompany Allen for a cause (Allen's nervousness), and the three barbers cannot leave at the same time because the shop may not be left unattended. The real issues are as described above, correctly negating conditionals and realising the compatibility of ${\text{A}}\Rightarrow {\text{B}}$ and ${\text{A}}\Rightarrow \neg {\text{B}}$ so long as A is False.

FrankP (talk) 19:20, 16 December 2019 (UTC)

What you talkin about Willis

The claim that it is universally true that, if Allen is not in then Brown is not in, is incorrect! No constraint was placed on Brown, only Allen. Allen can't go anywhere without Brown also going. But Brown can go somewhere without Allen going. Brown can go to the shop without Allen. But Allen can't go to the shop without Brown. So the possibilities are:

1. Allen and Brown go to the shop and they are both in.

2. Brown goes to the shop while Allen stays home and only Brown is in the shop.

3. Neither Brown or Allen go to the shop and neither is in.

So, has the fourth possibility, Allen is in while brown is out, been eliminated and is not possible?

No it hasn't because it's more complex and further things can happen.

Allen and Brown can both go to the shop. Since Brown is not constrained he can then leave the shop without Allen. Now only Allen is in the shop.

 — Preceding unsigned comment added by 2603:3024:204:B00:E0E1:5AAD:4A60:8B50 (talk) 17:11, 6 March 2020 (UTC)