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Contents 1 Why would Carroll have considered this a paradox? 2 "Paradox"? 3 Logic 4 Misunderstandings, plan for article 5 Further changes 6 Am I missing the paradox? 7 Major point of confusion 8 Third case 9 Resolution

[edit] 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)

[edit] "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).

[edit] 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)

[edit] 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

[edit] 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)

[edit] 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)

[edit] 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).

[edit] 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)

[edit] 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)