Talk:Barbershop paradox

message: 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)
thread_title: "Paradox"?

message: 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)
replies: text: 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)
thread_title:  Misunderstandings, plan for article 

message: 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)
thread_title:  Further changes 

message: 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)
thread_title:  Am I missing the paradox? 

message: 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)
thread_title:  Major point of confusion 

message: 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)
thread_title:  Third case 

message: 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)
thread_title:  Layman's terms? 

message: 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)
thread_title:  Nonsense 

message: 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: 
: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)
thread_title:  Hoping to clear this up 

message: 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)
thread_title:  What you talkin about Willis