In a small town where everyone is clean-shaven, the barber shaves everybody who does not shave himself. Who shaves the barber?

http://content.answers.com/main/content/wp/en/6/6d/Rowlf.jpg

king tubby?

Please supply me with paradoxes?

How much bone marrow could Clarence Darrow eat if Clarence Darrow ate bone marrow?

When God sneezes, who is there to say "bless you"?

from wikipedia article on barber's paradox:

Chip Hop (rap) Artist MC Plus+ refers to the Barber's Paradox in his song "Man Vs Machine" from the album Chip Hop. He uses it to defeat his own fictional AI opponent, Max Flow, in a rap-battle.

Can God make a rock so big he can't lift it?

The paradox - http://en.wikipedia.org/wiki/Barber_paradox - specifies that the barber shaves both all and only those who don't shave themselves. Leaving out the "only" leaves open the possibility that the barber shaves himself. Anyway, can't there be a kid barber?

doood it's a chick

this is just like that doctor one

this is a really interesting thre..zzzzzzz

thanks for the contribution, chak

WHO SHAVES THE SHAVERS
WHO RAISES THE RAZORS

the barber cuts his own throat rather than deal with this kind of pointless paradoxical pish.

adagio for zings

^^^

ha ha!

The barber's son, who will some day take over the business.

tiki barber is a douchebag

BarberBot

RIP

http://www.riverwindgallery.com/STONE/barbaro%202.JPG

seriously - barber shaves himself. the hell

THE SURPRISE TEST

A teacher announced to his pupils that on exactly one of the days of the following school week (Monday through Friday) he would give them a test. But it would be a surprise test; on the evening before the test they would not know that the test would take place the next day. One of the brighter students in the class then argued that the teacher could never give them the test. "It can't be Friday," she said, "since in that case we'll expect it on Thurday evening. But then it can't be Thursday, since having already eliminated Friday we'll know Wednesday evening that it has to be Thursday. And by similar reasoning we can also eliminate Wednesday, Tuesday, and Monday. So there can't be a test!"

That there is my favourite paradox

if i was the teacher i'd send her to get gang-raped by the remedial kids

An autological word describes itself, e.g., ‘polysyllabic’ is polysllabic, ‘English’ is English, ‘noun’ is a noun, etc. A heterological word does not describe itself, e.g., ‘monosyllabic’ is not monosyllabic, ‘Chinese’ is not Chinese, ‘verb’ is not a verb, etc. Now for the riddle: Is ‘heterological’ heterological or autological? If ‘heterological’ is heterological, then since it describes itself, it is autological. But if ‘heterological’ is autological, then since it is a word that does not describe itself, it is heterological.

You do not believe this sentence.

i like stories

sorry, stupid simpsons joke. i like the heterological one.

The paradox thread.

Btw, the solution is simple, the barber is suffering fromalopecia universalis.

i can't make sense of that, ledge - they would only expect the test on Friday if they hadn't been given the test Mon-Thurs!

A teacher announced to his pupils that on exactly one of the days of the following school week (Monday through Friday) he would give them a test. But it would be a surprise test; on the evening before the test they would not know that the test would take place the next day. One of the brighter students in the class then argued that the teacher could never give them the test. "It can't be Friday," she said, "since in that case we'll expect it on Thurday evening. But then it can't be Thursday, since having already eliminated Friday we'll know Wednesday evening that it has to be Thursday. And by similar reasoning we can also eliminate Wednesday, Tuesday, and Monday. So there can't be a test!"

However, the teacher could answer, "Why yes, you are right!", and then give the test anyway next week, on whatever day he chooses, and the students would not guess it the day before.

(x-post)

Yeah, I thought it was a false paradox too, but couldn't pin the reason why. Thanks for the clarification, Tracer.

Back to the original question.

The barber's wife shaves the barber.

That's not possible, because the barber shaves everybody who doesn't shave himself.

Barber's wife is a her.

Alright, here's one that came from an actual SQL test (don't worry)

A database was set up to hold details of dogs.
Field 1 = sequential key field (integer number) (unique)
Field 2 = Key field of dog's father
Field 3 = Key field of dog's mother
Field 4 = Name of the dog.

How would you prevent a record being added that did not have values for father and mother?

I had a different question.

(have values for father and mother that actually exist in the table, I should clarify)

Barber's wife is a her.

What does this matter? She can't shave the barber, because according to the paradox, is someone doesn't shave himself, the only one who can shave him is the barber.

the barber actually lives in Fort Collins and commutes to the small town every day

he is also just 6 years old

they would only expect the test on Friday if they hadn't been given the test Mon-Thurs!

The idea of the surprise is that they can't know they're getting the test until they actually arrive in the classroom on that day - but if they aren't given the test on Thursday then they immediately know they'll be getting it the next day - hence no surprise. So the surprise test can *never* be on the Friday, so Thursday is the last possible day for the test. But then if they don't get it on Weds they immediately know they're getting it tomorrow... etc etc.

> doood it's a chick
> this is just like that doctor one

both these have been printed in the guardian weekend magazine in the last two months as part of their brainteaser questions (along with the one where Peter is the killer and the police can instantly identify him despite being in a group of people) and the answer is always the same - 'dood, it's a chick'. it is the brainteaser equivalent of Stick.

I mean I see your point... I think that is part of the nature of the paradox, that the test simultaneously can and can't be given on the Friday - or any other day. xpost

There's a difference between paradoxes and trick questions tho. You could easily word the barber paradox to avoid the "it's a chick" get-out. I think logical paradoxes are much more interesting than "he was a dwarf so he could only reach the button for the fifth floor" games.

ledge i don't get that teacher one. why can't the teacher give it on monday, tue, wed or thur. either day would surely count as a valid surprise day ?

what am i missing, obv.

If it can't be Friday, because as the last day it would not be a surprise, then Thursday becomes the last day - which the test can't fall on, because it won't be a surprise - and so on in a regression until all days are ruled out. But we intuitively feel like the surprise test can be given, so there must be a fault in the logic. But what is it?

so Thursday is the last possible day for the test. But then if they don't get it on Weds they immediately know they're getting it tomorrow... etc etc.

why would the kids think this?

i'd hold it on the monday and surprise the shit out of em.

gabbneb in being a chauvinist pig SHOCKER.

U.N. OUT OF MY UNTERUS

OK, I didn't read the link (or closely read gabbneb's early post.)

many xposts

I disagree. I think that 'the' means that the set has one member in the context of discourse. In a town where there are two equally well known barbers, the sentence would not make sense unless one had already been mentioned. It would also not make sense if the town does not have a barber.

This is-the-barber-a-girl stuff is why it works much better as "there is a set of all sets that are members of themselves. There is another set of all sets that are not members of themselves. Which set is the second set a member of?"

Also, Grimly, this paradox came about in a little offhand comment from Russell to Frege. But it actually blew a fatal hole in what had until that point been a proof of mathematics based solely on logic, and sent Russell on a massive multi-year digression as he constructed the theory of types in a attempt to get around what he'd initially seen as a small stumbling block. It's not pointless!

As I see it, both are false paradoxes. The Barber's Paradox is false cuz it uses language to describe an impossible situation. By saying that the barber shaves all and only those who don't shave themselves, we're creating a situation that is logically incompatible with the idea that everyone - including the barber - is shaved bald. (Barring trick answers like "someone from out of town", or "the barber is congenitally bald", of course.) It's not really a paradox, it's just a pair of mutually negating premises.

The Student's Paradox is false, but in a somewhat more complex manner. Here, the phrasing of the puzzle tricks us into overlooking a crucial flaw in the student's logic. The following variation illustrates this:
Teacher says, "I'm going to throw this here rubber ball through a small hole into that there sealed box. The box has five lidded compartments, labeled according to the order in which I will open and reveal them to you: Monday, Tuesday, Wednesday, Thursday and Friday. Due to the construction of the box, once thrown, the ball MUST land in one of those five compartments. Until the box is opened, however, you will not know which compartment the ball has landed in."

The student might pose the same objection: "Well, we know the ball can't land in the Friday box, since once you've opened all the other boxes, we'll know it's gotta be there, and you said we wouldn't know. And it can't land in Thursday, because when you open the Wednesday box, we'd know it's got to be Thursday, given that we've already ruled out Friday. And so on. Basically, the ball can't land anwhere without us knowing exactly where it is, cuz no matter where it lands, that's the only place left it could land."

Phrased this way, the error is obvious. The student has overlooked the fact that we can't rule out any given box (day for the test) until all the other boxes have been opened (days have elapsed). That "until" is the crucial bit. On Sunday night, nothing has been ruled out, not even Friday. The box is closed. The teacher's decision is unknown to all. The cat is neither alive nor dead. Sure, once we've gone through some number of days, we can rule them out and select from the remaining, but prior to that, we know nothing and can eliminate nothing.

The puzzle attempts the trick us by slyly dropping the crucial phrase, "if we get through [day x] without having yet been tested." All of these seeming elminations only work once the week has begun to pass. Thus they mean nothing at the time the student raises them.

It's not really a paradox, it's just a pair of mutually negating premises.

UH

Your phrasing of the objection doesn't reflect the paradox properly, because it doesn't incorporate the teacher's will. The teacher isn't putting the ball in the box via a random mechanism, but is doing it deliberately. And the rejection is not "it can't land in" but "you can't put it in"

None of the boxes have to be opened to rule out Friday, either. On Sunday night you can say "it can't be Friday" purely within the terms of "you won't know what box it is in". And then the cascade begins.

xpossst

Wouldn't it be more logical to conclude that the test would happen on a random day but wouldn't be a surprise, than to assume that there wouldn't be a test at all?

The teacher screwed up the surprise part by telling them all about it anyway.

re John Justen:

Touche. I mean that it isn't a substantial paradox. After all, we can easily create sets of mutually negating premises til our faces turn blue. All butterflies are insects. Tom is a butterfly. Tom is not an insect. But there isn't any puzzle in that, and it doesn't seem to call into question the nature of logic itself.

The Barber's Paradox does exactly the same thing as the Tom/butterfly example, albeit in a more complex manner. It first creates a set that must include the barber ("everyone is shaved bald"). It then creates a set that cannot include the barber ("no one shaves themselves"). Finally, it simply insists that the barber must belong to both sets ("the barber shaves everyone"). That's the trick, the flaw: the final statement hides a craftily unstated clarificatory stipulation: "the barber shaves everyone including himself."

Put another way, the "paradox" works like this:
1) All things are members of set X.
2) No thing is a member of set Y.
3) Thing A is a member of both sets X and Y.

How is that a paradox? How is that anything other than a set of exclusionary premises? If that's a "paradox", this is a paradox:

1) 1+1 equals 2
2) 1+1 does not equal 2
3) What does 1+1 equal?

And the rejection is not "it can't land in" but "you can't put it in."

The distinction you're making between random and willful selection has nothing to do with the logic puzzle itself. After all, the teacher could well make his selection randomly. The puzzle has nothing to say on that point. If the teacher picks Friday, sure, at the end of the day Thursday it becomes clear that test must then fall on Friday. But prior to that, nothing is known. That's the crucial point that the phrasing of the "paradox" leaves out. Nothing becomes true until something is known, and on Sunday night, nothing is known.

For what it's worth, that's the essence of Shroedinger's Cat (sp?) - a paradox that essentially inverts this one.

That's not the flaw in the paradox -- it's actually what the paradox was trying to highlight. The point is that there are things our intuitions say should be possible, but when we try to use them in certain ways, they turn out to be logically impossible. The "is heterological heterological" one is probably a better phrasing. xpost

No, the teacher couldn't make his selection randomly. Because if he did that there would be nothing to prevent it being on Friday! The thing that prevents it being on Friday is that the teacher is trying to make it a 100% surprise, which means they can't set the test for Friday. And so on.

ie on Sunday night it is known that there is no way there can be a completely surprise test this coming Friday.

(Assuming the test is guaranteed to be held in the coming week. Like someone said, making the test uncertain avoids the paradox)

Bob OTM about Barber paradox. I think I might have actually done a presentation in Gr 12 math about the Russell paradox now that I look at it.

The flaw in the surprise paradox is that you *can* have the test on the Friday. The student who was sure you couldn't have it then will be surprised to be set it.

thanks for ruining this thread with logic guys.

Fuck! You're right, stet. I fooled myself by paying too little attention to the exact language of the Student's Paradox. I allowed myself to gloss over the teacher's insistence that "on the night before" the students will not know the test is set for the following day. Shit. Fuckin brain.

Nonetheless, what I'm talking about still holds true, I think. I'm probably wrong, but lemme give it a shot...

The basic premises of the puzzle are these (given both the teacher's rules and a basic model of the progress of days):

1a) Test (T) must have a Date Value (DV)
1b) Student (S) must also have a DV
1c) Possible DVTs = 1, 2, 3, 4, or 5
1d) DVS = 0, then 1, then 2, then 3, then 4, until DVS = DVT-1 *
1e) If only one valid DVT remains when DVS = DVT-1, DVT = Known (K)
1f) DVT != K

* Not necessarily going through all steps, but ascending until DVS = DVT-1.

In response, the student is saying this:

2a) If DVT = 5, DVT will eventually = 4
2b) When DVS = 4, DVT can only = 5
2c) When DVS = 4, DVT = K
2d) DVT != 5

The student works back from there in the same manner, eliminating all the other DVT values, and QED!

But hold on, step 2 doesn't quite work. 2d) is overstated. We seem to have learned that Friday is impossible (and realistically speaking we have), but from the standpoint of pure logic, we've only said this:

2d) When DVS = 4, DVT != 5

That's the crucial bit. Given the internal logic of the problem, the conclusion that the test cannot occur on Friday only becomes true on Thursday night. In the real world, we don't see this as a problem, because we're used to the forward march of time, we know that Thursday will come around eventually, and we easily grasp how this rules out Friday as a possibility. But while looking at this as an exercise in pure logic, we have to stay focused on the limited, circumstantial nature of our finding.

If we go on to say, as the student does, that DVT = 4 (test on Wednesday night), we can learn about that circumstance:

3a) If DVT = 4, DVS will eventually = 3
3b) When DVS = 3, DVT could possibly = 4 or 5
3c) When DVS = 3, DVT != K

What we can't do is to cross-apply what we learned when we assumed that DVS = 4. We want to do this. Our core assumptions tell us that we should be free to apply what we learned when DVS = 4 to the situation of DVS = 3. But given our premises, that's illegal. Just as we only exist at one point in time at any given time, DVS can only hold one value at a time. And what holds true when DVS = 4 does not necessarily hold true when DVS = 3.

This brings up another problem with the puzzle. It's vastly more complex than it seems on the surface. It's not just about the logic of time, it's about the logic of cognition, of knowing. To straighten it out, we have to parse the logic that allows the extention of knowing from one point in time to another. And that's more complex than the relatively straightforward nature of the puzzle suggests.

Let's say the teacher sets the test for Tuesday. Do the students know on Sunday night that the test will be on Tuesday? No. Nor do they know it on Monday night. They might think they can deduce something about the possibility of the test occurring on any given day, but all they can say for certain is that it won't be on Friday, that it seems it shouldn't therefore be on Thursday, and that Wednesday is maybe beginning to be a gray area, maybe not...

Why? Why does our inbuilt assumption that we can apply what we learn in one temporal circumstance to other circumstances fail? Why does it work for Friday and even Thursday (in the real world, anyway, if not the logic model I built), but break down beyond that?

Good questions. I dunno. And maybe I'm missing something important here, but it's late, and I'm not seeing things clearly anymore.

hioly fuck

GODDAM! 2a) contains a typo! Should read:

2a) If DVT = 5, DVTS will eventually = 4

I'm sure you're just as relieved as fuck to hear that.

I think what you're attempting is the "leaky inductive argument":

http://en.wikipedia.org/wiki/Unexpected_hanging_paradox

(I like this a little more than the Stanford link, I think. The "additivity of surprise" seems to make sense to me right now but I don't know how well I could explain it.)

Thanks for the leg up, Sundar. Leaky Inductive (a phrase I wasn't familiar with) seems to describe what I'm objecting to here. Kinda. More or less.

I was thinking about this again last night and I'm not really satisfied with the epistemic blind spot solution. And I don't agree with Bob saying "the conclusion that the test cannot occur on Friday only becomes true on Thursday night." I think the test can *never* be held on the Friday and the students know this, but up until Thursday night the students don't know if they are going to get a proper surprise test, or if the teacher is going to break his promise and give them a test they are expecting.

It's clearer if you reduce it down to two days. On Friday the teacher says "I'm going to give you a surprise test next Monday or Tuesday". The test can only be a surprise on Monday, but it will still be a surprise, because of the possibility that the teacher will break his promise.

To be honest, the phrase "I'm going to give you a surprise test" actually negates the surprise immediately.

I am going to give you a surprise beating.

j/k

In a small town where everyone is clean-shaven, the barber shaves everybody who does not shave himself. Who shaves the barber?

So everyone is clean-shaven. Some people shave themselves. Some people are shaved by the barber. The barber shaves himself. I can't see any paradox there at all, or even a puzzle. The question might as well be "There are two houses in town: a red one and a yellow one. Everyone lives in a house. Everyone who doesn't live in the red house lives in the yellow one. The barber doesn't live in the yellow one. Where on earth does he live????"

he waxes !

"You will not know, 10 minutes before I administer the beating, that it's going to happen, but it will be sometime between now and 19:00 on Friday"

NBS: the barber shaves those and only those who do not shave themselves.

and he lives in a red house

A football manager has two formations, 4-4-2 and 4-5-1. 4-4-2 has seen his team play well, 4-5-1 has seen them do little other than lose. Which formation does he play in an all-important Euro 2008 qualifier?

Well, ten minutes before kick-off, they will not know etc...

surely the fact that "the barber" and "himself" are eventually one and the same negates the paradox anyway?

the student one is much, much more interesting.

still: all this reminds me why i like to avoid logical philosophy like the plague :)

I've seen that argument in a couple of guises, Bob -- didn't Ayer use it as well? It doesn't totally convince me, because I think there are ways of rephrasing the question which stump it (like Ayer says, you can set it to start on Wednesday and there are circumstances in which the student *will* know).

The rebuttal I really like goes something like this (this might be Quine's and is a bit like Tuomas's, though it pisses logicians off:
1. Teacher says there absolutely will be a test.
2. But student's logic shows there can't be a test. So there "can be no test" he says.
3. This is admitting the possibility that there will be no test, which rules out the idea that it "cannot happen on Friday" and the whole chain never starts
4. Then the teacher says "but there will be a test"
5. The fact that a test arrived on a day the student showed it couldn't, comes as a surprise.

"But it couldn't happen on a Friday, because we would know, so [...] ergo that means it can't happen *any* day, but he said it *will* happen, so when will it happen? Argh I dunno I guess it will be a surprise."

"if a man says something in forest, and there is no woman around rto hear him, is he still wrong?"

The barber is shaved by Kirk Cameron

Brothers and sisters I have none
But this man's father is my father's son

-- Abbott, Thursday, 22 November 2007 00:16 (15 hours ago) Bookmark Link

It's Alex Ferguson!

classic :)

Stet, the thing I don't like about that solution is I'm inclined to pare everything down to a set of hard, clear, non-assumptive premises and work through from there. I suspect the semblance of paradox in the initial puzzle arises simply from unclear, self-contradictory assumptive premises on some level or another. But because the puzzle is much more complex than it first appears, we don't recognize this.

By simply accepting the paradox without examining it further, and then saying that the paradox is in turn negated (and even trumped) by the teacher's continued insistence that, "no, really, there WILL be a test," I think we're taking an easy way out.

That solution does work in the real world, and is of course the right answer in that sense. But in approaching logic puzzles, I'm not much concerned with real world right answers. I want to investigate the formal logic and only the formal logic. The hope is that when we recognize and correctly formulate all valid, applicable premises and then precisely describe the ensuing logic chains, we'll likewise describe what goes on in the real world.

but this is just a logical limeric, there is no solution or end result. It's just a clever paradox. It stumped me at first but now I understand it, we don't need to hammer at it to find anything further surely.

i guess thats actually what you're saying though isn't it. ignore me.

Where does the paradox arise, though, Ste? Is it language dependent? Does it arise from the fact that we're intermingling incompatible ways of thinking about time and knowing? Is it just a simple trick question that depends on our assuming something that isn't necessarily true in this case? Or does it reveal some gap in the way we habitually describe the world?

Okay, after a shower and a little more thought... Stet's solution boils down to this unstated premise in the original puzzle: The teacher could be lying (or mistaken) about any or all of his rules. Basically, this premise insists that all the other premises are unreliable. This keeps the students in suspense about anything they might logically deduce, no matter what. In the real world, this is doubtless true, and maybe that's the only solution we can come to here. Maybe the apparent paradox arises simply from the nature of "surprise", and its dependence on doubt and unpredictability - the unknowability of the future and other limits on human knowledge.

That works, but I'm not inclined to give up so easily on the logic as logic, given that all the premises ARE reliable. Still working things over...

OK, here's my final offering.

THE PARADOX IS THE TEST!

They did not know, the previous evening, that he was going to set the test.

They had from now until some time next week to work it out.

That's it.

the paradox is the 'surprise' surely ?

Told you logicians hate that answer! It can be formalised though. It's simply that the premise it all rests on -- that "there can be no surprise test on Friday" -- is never true, because if it were true then it would automatically be false again (as it then has to allow for the possibility of no test).

And the only way to repair it then is to define surprise so tightly in terms of itself that you're getting self-reflexive, and basically just re-doing the barber's paradox!

nabisco otm

the barber is a WOMAN

the barber is a robot.