Friday, October 3, 2014

Meta-comedy


While we’re on the subject of Steve Martin, consider the following passage from his memoir Born Standing Up.  Martin recounts the insight that played a key role in his novel approach to doing stand-up comedy:

In a college psychology class, I had read a treatise on comedy explaining that a laugh was formed when the storyteller created tension, then, with the punch line, released it... With conventional joke telling, there's a moment when the comedian delivers the punch line, and the audience knows it's the punch line, and their response ranges from polite to uproarious.  What bothered me about this formula was the nature of the laugh it inspired, a vocal acknowledgment that a joke had been told, like automatic applause at the end of a song...

These notions stayed with me for months, until they formed an idea that revolutionized my comic direction: What if there were no punch lines?  What if there were no indicators?  What if I created tension and never released it?  What if I headed for a climax, but all I delivered was an anticlimax?  What would the audience do with all that tension?  Theoretically, it would have to come out sometime.  But if I kept denying them the formality of a punch line, the audience would eventually pick their own place to laugh, essentially out of desperation.  This type of laugh seemed stronger to me, as they would be laughing at something they chose, rather than being told exactly when to laugh.

To test my idea, at my next appearance at the Ice House, I went onstage and began: “I’d like to open up with sort of a ‘funny comedy bit.’ This has really been a big one for me... it's the one that put me where I am today.  I'm sure most of you will recognize the title when I mention it; it's the Nose on Microphone routine [pause for imagined applause].  And it's always funny, no matter how many times you see it.”

I leaned in and placed my nose on the mike for a few long seconds.  Then I stopped and took several bows, saying, “Thank you very much.”  “That's it?” they thought.  Yes, that was it. The laugh came not then, but only after they realized I had already moved on to the next bit. (pp. 111-12)

Well, this kind of thing either works for you or it doesn’t.  No doubt Martin’s facial expressions and body language helped make it work on the occasions when it did.  But Martin evidently thought his unorthodox approach to relieving comic tension might play well on the printed page too.  A good example comes from his 1979 book Cruel Shoes.  The piece is titled “Sex Crazed Love Goddesses” and here it is in its entirety:

Little Billy Jackson had to go to the store for his mother to pick up some postage stamps.  When he got there, he found the stamp machine to be out of order, and decided to walk the extra three blocks to the post office.  On the way there, he passed a hardware store, a variety store and a lamp shop.  The line was short at the post office and he got the stamps quickly and returned home.  His dog, “Spider,” bounded out to greet him as his mom waved from the porch.  Billy’s mother was pleased at the job he did and congratulated him on having enough sense to go to the post office when he found the stamp machine broken.  Billy had a nice dessert that night and went to bed. (p. 55)

I know what you’re thinking, but the story is actually better in the book, because it runs to the bottom of the page and it isn’t clear until you turn the page that that was it

Well, again, this kind of thing either works for you or it doesn’t.  It got a laugh out of me but it probably helps that I’ve got a taste for the abstract and the absurd.  The joke will be either blindingly obvious to you or utterly opaque.  Either way, here it is: Even though you know it’s a gag piece in a Steve Martin book, the title “Sex Crazed Love Goddesses” cannot fail to raise in your mind the expectation that something salacious is to follow.  Hence as you read this utterly banal and irrelevant narrative about a kid buying stamps, etc., you feel sure that the story is going to shift gears at any moment.  Then it suddenly ends without having done so.  The comic “tension” Martin speaks of breaks precisely when it hits you that it’s never going to break, and that’s what gets the laugh.

In making a joke out of what we expect a joke to be or out of what we expect a story to be, Martin is doing something we might call meta-comedy.   Comedy itself and its conventions become the subject matter.  Notice how the “Nose on Microphone” bit can work only insofar as Martin gets his audience explicitly to think to themselves: “OK, here we all are watching a comedian, and we’re about to hear a really funny comedy bit.  Here it comes…”  That is the set-up of the joke, rather than something internal to the joke itself (“A priest and a rabbi walk into a bar…”), as we’d normally expect.  It is only when we become self-conscious about what we thought the joke would be and how it didn’t meet that expectation that the “payoff” can be delivered.  Normally we become “lost in” a joke, just as we become lost in the action of a movie or play and don’t constantly think to ourselves “These are actors, none of this really happened but they are trying to make it convincing” etc.  Martin’s joke works precisely by not letting us forget that “This is a stand-up comedian, and he is trying to make us laugh by telling us jokes,” like a movie or play that “breaks the fourth wall.” 

Similarly, in the case of the Cruel Shoes piece, the joke can work only insofar as we are not lost in the story, but instead start thinking about the conventions of story titles and how they relate to the content of a story: “Why would a story with a title like ‘Sex Crazed Love Goddesses’ be about something as mundane as a kid buying stamps?  Oh wait, that mismatch is the joke…”

Martin’s act during his stand-up days relied on this kind of thing to a very great extent, even if not entirely.  A big part of his shtick required that the audience have it at the forefront of their minds that this guy is a famous comedian who is here to entertain us.  (Consider this bit and this bit from The Tonight Show, as well as various clips of stand-up material you can find on YouTube.) 

Meta-comedy is essentially an instance of what, in a post from several years ago, I called “meta-art” -- art the theme of which is art itself, and the method of which involves a self-conscious stretching of art’s boundaries.  Martin was to stand-up comedy what Duchamp was to visual art, Schoenberg to classical music, and Ornette Coleman to jazz.  Meta-art, art gone self-conscious, is theory-driven in a way just-plain-old-art-without-the-“meta”-thank-you-very-much is not.  (It cannot be a coincidence that Martin was a philosophy major and has long been an art collector!)

This comparison of Martin’s stand-up comedy to other instances of meta-art prompts two reflections.  First, as I indicated in the post just linked to, while meta-art can be interesting, it can also be arid and repetitive and descend into self-parody.  Martin did not rely on meta-comedy entirely, and where he did the results do not always hold up well.  (Most of Cruel Shoes does not hold up well.  I’m not sure how well some of it held up in 1979.)  As he makes clear in Born Standing Up (which is a very good book), Martin was burned out by the early 80s.  The following passage is telling:

The act was still rocking, but audience disruptions, whoops and shouts, sometimes killed the timing of bits, violating my premise that every moment mattered.  The days of the heckler comebacks were over.  The audiences were so large that if someone was calling or signaling to me, only I and their immediate seatmates could hear them.  My timing was jarred, yet if I had responded to the heckler, the rest of the audience wouldn't have known what I was talking about.  Today I realize that I misunderstood what my last year of stand-up was about.  I had become a party host, presiding not over timing and ideas but over a celebratory bash of my own making.  If I had understood what was happening, I might have been happier, but I didn't.  I still thought I was doing comedy. (p. 185)

Martin does not put it this way, but it’s as if the meta-comedy had, without his realizing it, gone meta-meta.  People no longer showed up to hear meta-comedy anymore, let alone comedy.  They showed up to see the guy who was famous for doing the meta-comedy.  This couldn’t last, and Martin wisely made a transition into movies -- and, with them, a more conventional brand of comedy.

A second thought, though, is that it is quite remarkable how popular Martin’s stand-up then was given its often esoteric and “meta” character.  Meta-art is typically characterized by its lack of mass appeal.  Indeed, as Born Standing Up recounts, Martin’s act was by no means an overnight success.  But eventually it caught on in a big way.  Why?

One reason, of course, is that, as I have said, Martin’s stand-up comedy was not all of the surreal Cruel Shoes type.  It was a departure from the usual thing, but not a total departure.  (Thelonious Monk perhaps provides a better jazz analogy for Martin’s stand-up than Ornette Coleman does -- I compared Monk and Coleman in another earlier post.)  

A second reason is that the intentional absurdity of some meta-art, while a stumbling block to a popular audience in the case of visual arts, literature, and music, can have mass appeal in the case of meta-comedy because of its similarity to slapstick.  If you present the man on the street with Duchamp’s Fountain readymade as art or an Ornette Coleman piece as music, he will be offended by it.  But if you present it to him as comedy -- say, in a Three Stooges episode where the fellas are hired as musicians and start playing like Coleman, or try sculpture but produce only a urinal -- then he’ll probably love it.  Meta-comedy is just the next step.  “Sure it’s absurd, but then this is supposed to be comedy, so…” And Martin mixed old-fashioned slapstick in with his meta-comedy in any event (both onstage and via movies like The Jerk).  To a popular audience it all might have seemed more like Moe Howard than artistic modernism.

Finally, there is, possibly (especially in light of Martin’s “celebratory bash” remarks), what we might call the “Money for Nothing” factor.  Just as the average guy might both resent and admire the pop star for his ability to attract fame, wealth, and women with (so he assumes) little effort, so too might he be as drawn to, as annoyed by, a guy who acts goofy onstage for a couple of hours and gets tons of money for it.  The stand-up comic in a suit, chatting with Johnny Carson on TV, can have a sex appeal that the sullen and impoverished avant-garde painter or novelist does not.  Meta-comedy is not pretty, but boy its rewards are!

185 comments:

The Masked Chicken said...

Steve Martin wrote:

"These notions stayed with me for months, until they formed an idea that revolutionized my comic direction: What if there were no punch lines? What if there were no indicators? What if I created tension and never released it? What if I headed for a climax, but all I delivered was an anticlimax? What would the audience do with all that tension? Theoretically, it would have to come out sometime. But if I kept denying them the formality of a punch line, the audience would eventually pick their own place to laugh, essentially out of desperation. This type of laugh seemed stronger to me, as they would be laughing at something they chose, rather than being told exactly when to laugh."

Martin's theory is all wrong. His empirical results happen for a different reason. Suppressed tension exhausts the nervous system, if it is not released. It increases the level of cortisol in the blood, which can lead to anxiety and depression.

This type of non-dissipative humor Martin describes goes by the current common name of, "anti-humor." It is really popular among some teenage on-line posters. It is a type of encapsulated humor. It is a type of meta-process without being meta-humor, per se. There is a small literature about meta-humor, some of which I helped review for publication.

The linguist, Salvatore Attardo, has done the most in the field of analyzing meta-humor. Here, is a sample from an encyclopedia of humor studies in which I, also contributed some articles:

http://books.google.com/books?id=yjRzAwAAQBAJ&pg=PA418&lpg=PA418&dq=attardo+meta-humor&source=bl&ots=HkNYLiNicI&sig=QtWNetN99qGxCdjbhOl_hZ8Q5N0&hl=en&sa=X&ei=55QuVMSPAcGSyASAloC4Dw&ved=0CB0Q6AEwAA#v=onepage&q=attardo%20meta-humor&f=false

In fact, Attardo wrote a whole book on structural aspects of humor, which is available, on-line:

http://www.italiansubs.net/forum/televisione/doppiaggio-parliamone-qui/?action=dlattach;attach=212440

There isn't enough space in a single combox comment to explain what Martin stumbled onto. My principle areas of study in humor are neuro-mathematical modeling of humor processing and physiology and the logical mechanism of humor. We have some good theories and there is a (very) slow convergence among scholars towards a unified understanding of what humor is (at least, more and more, we are all starting to ask the same questions).

One could describe Martin's approach as, "silent absurdity." His approach is related to the non sequitor, but the non sequitur is silence in the face of expectations. This non-sequitor is a meta-element, which encapsulates the original joke text, but does not, strictly speaking, play off of any elements within the joke text, so the humor derives from a second, external joke process that does not reference the original material. Thus, it is a meta-process, but not meta-humor, per se, which would reference the original joke text in some way. A diagram would be nice, if I could insert one.

There is genuine meta-humor and it does follow what Prof. Feser describes, as far as it's aesthetics. It is self-referential and can be funny or not depending on how committed the listener is to the non-meta discourse (the more committed, the more the meta becomes more annoying than funny).

The Chicken

Glenn said...

The Masked Chicken,

There is genuine meta-humor and it does follow what Prof. Feser describes, as far as it's aesthetics. It is self-referential and can be funny or not depending on how committed the listener is to the non-meta discourse (the more committed, the more the meta becomes more annoying than funny).

We all know what goes on there, don't we? When a listener committed to the non-meta discourse finds the meta annoying, it's because there's no change in his mind.

The Masked Chicken said...

Dear Glenn,

Yes, that is one possibility. If one cannot see the meta as being connected to the primary-level discourse, it becomes mere interference. There is no unfolding in the joke processing. This is a form of something called a, "humor killer," which comedians learn to avoid - some sort of taboo that takes one right out of the imagination into reality. It is sort of like a comedian trying to tell you his jokes while pointing a gun to your head.

The Chicken

rank sophist said...

Very interesting article. It's probably worth adding that "meta-comedy" is still directed toward the same end as regular comedy: getting the audience to laugh. "Meta-art", so defined, changes the end of art to something besides aesthetic appreciation. (Not that I fully agree with Coleman's characterization as a meta-artist--even he has his moments of beauty, if you work for it.)

On a related note, Brandon recently made a blogpost that I found inspiring. Relevant excerpt:

If we strip away all irrelevancies, all purely arbitrary and irrational criteria, there is nothing about a work itself that can be assessed except these three things:

(1) Are the ends sought genuinely good?
(2) Are the means appropriate to the end?
(3) Are the means used in a way to achieve the end well?


A lot of so-called meta-art, like that of Damien Hirst, can be tossed out once you hit the first criterion. Too much of it isn't directed toward a genuinely good end. But good meta-comedy, like Martin's, will hit all three points: laughter is a genuine good; and the means he uses to bring it out are effective and suitable.

Tony said...

Chicken, thank you for your clarification. I think you are much more right than Steve Martin is about the matter. To me, "The Princess Bride" succeeds as humor precisely because it parodies the foolish elements of the "princess / adventure story" type while at the very same time being a fair princess / adventure story itself. Without the second part, the parodying would after 10 minutes just be a boring essay on dumb movie elements. But it isn't because it has enough reference back to the original type in its own sequence to keep on being funny. (I admit that there are some people who do not get anything from "The Princess Bride", so YMMV, but that seems to be true of all humor anyway. I take it as more telling that most intelligent people I know DO get it - not to mention the cast themselves, some of whom hurt themselves laughing so hard during filming.)

Thursday said...

Conceptual art only works when it's funny. That is all.

taylormweaver said...

This is, essentially, what Andy Kauffman did as well, right? I always found his "anti-humor" a bit more "meta" than Steve Martin, but I am not really as well-versed on the depth and breadth of Martin's work (though I do enjoy his stuff).

anti-nihilist said...

Given a genera, species and an individual, the 'elimination of the species' would produce an absurdity. The individual 'anti-joke' springs forth from the genera as amorphous matter. Without specific difference there is no natural end for the individual comedic display whereby it is traced back up to the genera (comedy), leaving the audience to abduce; "Well, gee, this must be comedy."

anonymist said...

"A second reason is that the intentional absurdity of some meta-art, while a stumbling block to a popular audience in the case of visual arts, literature, and music, can have mass appeal in the case of meta-comedy because of its similarity to slapstick. If you present the man on the street with Duchamp’s Fountain readymade as art or an Ornette Coleman piece as music, he will be offended by it. But if you present it to him as comedy -- say, in a Three Stooges episode where the fellas are hired as musicians and start playing like Coleman, or try sculpture but produce only a urinal -- then he’ll probably love it. Meta-comedy is just the next step. “Sure it’s absurd, but then this is supposed to be comedy, so…”"

There's probably an analogy with how "difficult" or avant-garde music has been a popular hit when presented to audiences as soundtrack<. (And not just modernist music: think of all the people who would never have sat through a Richard Strauss concert but were used to hearing very similar music thanks to mid-century Hollywood.) You could probably say the same about non-figurative techniques in art being presented as cartoon art.

Daniel said...

Completely off-topic but may I ask a question with regards to a point Ed raises in the context of the Third Way? It's not Third Way specific though.

David Gordon said...

This site gives a large number of examples of anti-jokes. http://anti-joke.com/

Anonymous said...

This reminds me of a joke:

"Have you ever seen the inside of Bruce Willis's house?"

"No"

"Well it is very nice."

Scott said...

Q: What do you sit on, sleep on, and brush your teeth with?

A: A chair, a bed, and a toothbrush, respectively.

jprev said...

Anti humour:
A man walks into a bar. He is an alcoholic, and his habit is ruining his family.

Glenn said...

It is nice to see that Mr. Attardo, author of a "book on structural aspects of humor" (for which book a link has been provided by The Masked Chicken above), has a sense of humor. That he has a sense of humor is evidence by the inclusion in his book of the following keen observation:

"This has to be the least funny book one has ever read" (Opening sentence of a review of Attardo 1994)

It is also nice to see that Mr. Attardo gives attention to the so-called "light bulb joke"--for which he provides a number of variants, and which now serves as an inspiration for:

An anti-anti-joke, i.e., a 'standard' joke, told in honor of Moe & Co. (since Moe Howard receives honorable mention in the OP):

How many Stooges does it take to screw in a light bulb?

Three. One to hold the light bulb, and two others to hold one leg each as they lift up the first Stooge and run in a circle.


- - - - -


Anyway...

What I gathered from the OP is that Martin had not offered what he intended to be taken as a formal theory of humor, but, rather, an accounting of how he had come to practice the style of comedy for which he became well-known--a style of comedy regarding which the OP's author has written, "[W]e might call meta-comedy."

Humor and comedy, though not infrequently conflated, are not necessarily one and the same. As Robert Mankoff puts it (here):

Strange as it may seem, there is actually a conflict between comedy and humor. All comedy has humor, but not all humor is comedy. Humor is the much broader category of anything that may make us laugh[.] Comedy is a form of professional entertainment, consisting of jokes and sketches intended to make people laugh. The humor in everyday life is often unintentional, consisting of harmless mistakes; the humor in comedy shouldn't be.

If humor is one thing and comedy another, then it would seem to follow that meta-humor likely is one thing and meta-comedy likely another. And if meta-humor really is one thing and meta-comedy really another -- and if Martin had not offered a formal theory of humor, but instead an accounting of how he came to practice a particular style of comedy -- then we may have close at hand a reasonable explanation as to why the OP is entitled not as Meta-Humor, but as Meta-Comedy.

Robert Coble said...

How many programmers does it take to screw in a light bulb?

None: that's a hardware problem.

A nun has a habit, but it doesn't adversely affect her quality of life.

It is funny that all of the "serious" posts by Dr. Feser have myriad in-depth comments, yet a "humorous" post has so few. Perhaps the topic is too funny to be serious.

Scott said...

(This one is right on the border between humor and anti-humor, but it's one of my favorites.)

Three logicians walk into a bar. The bartender says, "Do all of you want beers?"

The first logician says, "I don't know."

The second logician says, "I don't know."

The third logician says, "Yes."

Glenn said...

Neat.

(Seems to be a variation of The Blind Prisoner, or vice versa. (Although the Blind Prisoner has more the character of a puzzle or problem, and the above, which is more concise, does come across as a joke.))

Scott said...

@Glenn:

Yeah, it's very much along the same lines as the Blind Prisoner riddle, and other similar puzzles like the one with the job interviewees some of whom had charcoal marks on their foreheads, and so forth.

The reason this one qualifies as a sort of meta-joke is that it starts out sounding like a joke ("Three such-and-suches walk into a bar"), but it turns out to be funny only to people who work it out as though it were a logic puzzle.

Scott said...

A priest, a minister, and a rabbi walk into a bar. The bartender says, "What is this, a joke?"

Glenn said...

Oh yeah? Well a termite walked into a bar and asked, "Is the bartender here?"


;)

Glenn said...

Perhaps the three-logicians joke can be expanded:

Three logicians walk into a bar. The bartender says, "Do all of you want beers?"

The first logician says, "I don't know."

The second logician says, "I don't know."

The third logician says, "Yes."

The bartender then says to the third logician, "That's what I like, decisiveness. Your drink is on the house. But your two buddies are indecisive, so they'll have to pay."

Scott said...

Heh. Well, the termite one definitely represents a turn toward the less meta, so…

A three-legged dog walks into a bar and says, "I'm lookin' fer the man who shot my paw."

Scott said...

"Perhaps the three-logicians joke can be expanded…"

Yes, that does make it a good deal less joke-like. ;-)

Timotheos said...

@ Scott

"A three-legged dog walks into a bar and says, "I'm lookin' fer the man who shot my paw."

Then a man with a really nice camera walks up and tells him, "I'll make sure to get those pictures of your paw out to him tomorrow"

Scott said...

Heh, nice.

All right, one more before bedtime:

A priest and a minister walk into a bar. The rabbi ducks.

Good night. ;-)

Glenn said...

>> "Perhaps the three-logicians joke can be expanded…"

> Yes, that does make it a good deal less joke-like. ;-)

I just received an email from the ABA (American Bartenders Association):

Dear Sir,

Scott is right that that was not a joke. It was, in fact, a disparagement.

The bartender in question had had a double ( a double shift, that is), and so perhaps was not as fleet of mind as he normally is.

Had the incident occurred on the first of his two shifts, or not later than the beginning of his second shift, he would have immediately recognized that the first two logicians had made their decisions on how to answer the bartender's question most quickly and quite effectively--which, of course, is the very definition of decisive.

Now that you have been made aware of the true circumstances of the matter, we trust that you will not be indecisive when considering our reasonable request that the disparagement be withdrawn.

Sincerely yours,
Etc., etc.

Maolsheachlann said...

The problem with all meta-humour and anti-humour is that it's ONE joke, repeated ad nauseum, whereas supposedly more formulaic and traditional humour is infinitely varied. The narrow gate leads to the broader way, always.

Maolsheachlann said...
This comment has been removed by the author.
Mr. Green said...

Scott: This one is right on the border between humor and anti-humor, but it's one of my favourites.

I suppose that means that it cancels itself out. Pity, I was looking forward to reading it, especially since I never heard that one before.

Yeah, it's very much along the same lines as the Blind Prisoner riddle, and other similar puzzles like the one with the job interviewees some of whom had charcoal marks on their foreheads, and so forth.

Perhaps the three-logicians joke can be expanded: 200 logicians walk into a bar, half with blue eyes and half with brown eyes…. (For various reasons, which goes without saying, at least if you are a logician who knows his metaphysics, the puzzle in this family of riddles of which I was reminded was this eye-coloured one (although despite the author’s claim, the given solution is in fact not correct — and interestingly, many people who rightly hit upon the flaw ended up being persuaded to change their minds… but I guess now we’re getting into meta-psychology).)

Of course, the dog-joke can be expanded too:

A four-legged dog walks into a photo shop and says, “I'm lookin' fer the man who shot my paw.”

…but that definitely make it a good deal less joke-like. I mean, how many people nowadays even know what a photo shop is??


Glenn: …he would have immediately recognized that the first two logicians had made their decisions on how to answer the bartender's question most quickly and quite effectively--which, of course, is the very definition of decisive.

Except, of course, the bartender did not ask, say, “Do you want to tell me your orders?” — and if the rimshot after the third logician said “Yes" hadn’t drowned him out, you would have heard the second logician saying,

“Shut up, I’m tying to decide whether I want a whiskey!”

…the joke of course being that the third logician did not pay attention to the setup and thought he was in a logic puzzle instead of a joke. Had he recognised the correct context, he would have known that he didn’t know whether the second logician didn’t know because he didn’t know what the third logician knew, or because he didn’t know what he himself knew. Or, to put it more simply, I could just say that the third logician didn’t know whether the second logician meta-didn’t-know or meta-meta-didn’t-know. At least, I could if I liked sticking “meta-“ in front of things, which really I’m not too keen on, because it seems to abuse the actual force of the Greek prefix based on a misunderstanding of “metaphysics”. And we all know how badly the average person misunderstands metaphysics. (Or, indeed, how well he misunderstands it.)


Scott: The reason this one qualifies as a sort of meta-joke is that it starts out sounding like a joke ("Three such-and-suches walk into a bar"), but it turns out to be funny only to people who work it out as though it were a logic puzzle.

Come to think of it, comedy is just logic solved anyway. (Which is why detective stories are comedies, for instance.) Of course, if “meta” humour means humour about humour, then is this really that? In gags like the clergymen walking into a bar (CLANG!), the twist comes in subverting one’s expectations about an established jocular form; but the joke is about clergymen and blunt-force trauma. Humour about humour would be something like:
Hey, have you noticed how so many comics are into “observational humour” these days? They’re basically letting other people write their material! Doesn’t anyone tell jokes any more?”


Scott: A priest, a minister, and a rabbi walk into a bar. The bartender says, "What is this, a joke?”
Scott: A priest and a minister walk into a bar. The rabbi ducks.

Well, apparently lawyers tell jokes. Now there’s a twist.

Daniel said...

I think so far Anonymous Bruce Willis one was best as it has just the right mix of potential punch-line and blandness. Having said that Comment deleted's entry gives it a run for its money though is probably too knowing and metareferential to fit into normal meta-entertainment (as is The Masked Chicken's).

Daniel said...

Not really meta-humour but one of my favorite shaggy-dog stories:

A man is running through the forest dodging under low-hanging branches and crawling along stream beds to avoid pursuit. Behind him follows a hideous beast horned and clawed and with eyes that shine in the darkness like patches of phosphorus. No matter what the ill-fated query does he cannot shake of his demonic assailant. At last he collapses in the lee of a great boulder and there awaits his fate. The creature breaks through the trees and stops in front of him. It extends a black clawed hand, lays it upon the man's shoulder and says: 'Tag'.

Scott said...

@Mr. Green:

"I suppose that means that it cancels itself out. Pity, I was looking forward to reading it, especially since I never heard that one before."

No one ever has. A shame, really, as I think it's quite good.

@Daniel:

"The creature breaks through the trees and stops in front of him. It extends a black clawed hand, lays it upon the man's shoulder and says: 'Tag'."

At which point the creature morphs into a human being and man turns into a horned, clawed, shiny-eyed beast—ecause, of course, now he's "It"…

Scott said...

@Daniel:

There's also a slightly more on-topic version that ends like this:

"The creature breaks through the trees and stops in front of him. It extends a black clawed hand, lays it upon the web developer's keyboard and types: 'Metatag'."

Scott said...

@Mr. Green:

(I hesitate to ask this question because it's off-topic and threatens to take us much further so.)

"[D]espite the author’s claim, the given solution is in fact not correct[.]"

Randall Munroe's explanation looks sound to me, at least insofar as we treat the puzzle as what it's intended to be: namely, a contrived problem of formal mathematical logic, common knowledge, nested hypotheticals, and so forth. What's the matter with it?

Scott said...

@Daniel:

"I think so far Anonymous Bruce Willis one was best as it has just the right mix of potential punch-line and blandness."

It also sets up a false expectation because it's based on a real joke. When I tell you that the original version involved Stevie Wonder, I'm sure you'll be able to work out the punchline.

(In fact, there's an arguably even better variant of it on the page to which David Gordon provided the URL. This one keeps Stevie Wonder and uses the "punchline" from the Bruce Willis version.)

Scott said...

@Mr. Green:

"Of course, if 'meta' humour means humour about humour, then is this really that?"

Hmm, arguably not.

Wouldn't that be funny?

Scott said...

@Mr. Green:

"Well, apparently lawyers tell jokes. Now there’s a twist."

A thousand lawyers walk into a bar association meeting. One says to the others, "Hey, what do you call us at the bottom of the ocean?"

(I'm sorry I didn't tell that better, but I'm out of practice.)

Daniel said...

A four-legged dog walks into a photo shop and says, “I'm lookin' fer the man who shot my paw.”

…but that definitely make it a good deal less joke-like. I mean, how many people nowadays even know what a photo shop is??


Well I just assumed both examples you give involved a Category Mistake. I mean how can a concrete corporeal substance walk into a program which only exists as such in relation to our interests?

Scott said...

@Daniel:

"[H]ow can a concrete corporeal substance walk into a program which only exists as such in relation to our interests?"

On the other hand, programs can be instantiated in corporeal substances, as when the Word became flesh.

Then again, a dyslexic Thomist would deny that Dog was a corporeal substance in the first place.

Tom said...

What, if I may ask, is the Blind Prisoner joke/riddle?

Scott said...

@Tom:

It's a logic puzzle. The basic setup is this (I omit irrelevant details):

The King offers to set free whichever of three prisoners can solve a riddle the soonest. "I have five hats," he says, "three of them white and two of them red. I will blindfold you and place one hat on each of your heads. Then I will remove the blindfold and ask each of in turn to tell me what color your hat is." [Insert other conditions as necessary in order to ensure that the prisoners can't speak to one another and that they'll get caught if they try to guess.]

The first prisoner doesn't know, and neither does the second. The third prisoner is blind, so when it's his turn, the King isn't even going to bother asking him—until the blind prisoner pipes up and says, "But, Your Majesty, I know what color my hat is."

The puzzle is: What color is his hat, and how does he know?

(In most versions, the first prisoner has two working eyes and the second has only one. That's one of the irrelevant details I omitted; it contributes nothing to the puzzle.)

Glenn said...

As indicated by Scott, there are variations to the problem (riddle, puzzle, etc.). One such variation, in which each prisoner sees fine, is:

A jailer tells three prisoners that from three white hats and two red hats, he will select three and put one hat on each of the prisoners' heads. Each prisoner is prevented from seeing what color hat is placed on his own head. They were brought together, and offered freedom if they could tell what color hat was on their own head. They were also condemned to death if they merely guessed and were wrong. The first two confessed that they could not tell, but the last one won his freedom. Show in all details how the last prisoner "deduced" the answer "white".


In an interesting variation on that variation, the prisoners stand in line, and the prisoners are lettered A through C from front to back. Prisoner A can't see any hats, prisoner B can see the hat worn by prisoner A but not the hat worn by prisoner C, and prisoner C can see the hats worn by prisoners A and B.

The jailer will ask each prisoner in turn if he knows what color hat he is wearing. If any prisoner answers correctly, all three will be set free; but if one prisoner answers incorrectly, all three will be condemned to death.

Jailer: "Prisoner C, what color is your hat?"

Prisoner C: "I don't know."

Jailer: "Prisoner B, what color is your hat?"

Prisoner B: "I don't know."

Jailer: "Okay, back in your cells."

Prisoner A: "Wait, you haven't given me a chance to answer."

Jailer: "You're at the head of the line, and can't see any hats."

Prisoner A: "Nonetheless, I know the color of my hat."

Jailer: "Really? Well, remember--if I ask and you answer incorrectly, all three of you will be condemned to death. Do you still want me to ask?"

Prisoner A: "Yes."

Jailer: "Very well, then. Prisoner A, what color is your hat?"

Prisoner A answers, his answer is correct, and all three prisoners are set free.

What color hat was worn by prisoner A, and how did he know?

Scott said...

@Glenn:

I especially like that latter variation, which I hadn't seen before. Its advantage is its bringing out clearly that it's only the color of the third prisoner's hat (A in this scenario) that matters to the reasoning of the second prisoner.

Scott said...

An afterthought: Do you suppose that that's why the most common version of the puzzle has the second prisoner being one-eyed? I've never seen a variant in which he couldn't see both of the other prisoners' hats, but perhaps in the original version, the idea was supposed to be that he could see only the third's.

David Gordon said...
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Scott said...

Speaking of prisoners, here's one I encountered in grad school about thirty years ago, from the physics professor in whose house I was rooming.

Three prisoners—A, B, and C—are scheduled for execution at dawn tomorrow, but the King decrees that one of them, chosen at random, will be set free.

The jailer brings this news to Prisoner A. A says to the jailer, "That's wonderful news, but at least one of my friends is still going to die tomorrow morning, so please tell me which one so that I can pray for his soul." The jailer says, "Well, I really shouldn't tell you this, but Prisoner B is definitely going to be executed."

After the jailer departs, A prays for B's soul (and throws in some prayers for C's as well, just in case it's A who is to be freed). But then he begins to reflect…

"Before I spoke to the jailer, I had just one chance in three of being the surviving prisoner. But now it seems that I have one chance in two, since I now know that either I or Prisoner C will be set free."

What's odd is that he could have reasoned the same way if the jailer had told him Prisoner C was to die, so he should have been able to reach the same conclusion without ever having asked the question at all. So how could a completely uninformative answer have altered the probability of A's death? Or is there an error in his reasoning?

(Hint, though perhaps not a universally helpful one: This is really a variant of the Monte Hall problem.)

Scott said...
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David Gordon said...

The solution to the puzzle is that A's hat is white. If A and B were wearing red hats, then C would know that his own hat is white, because there are only two red hats. But C doesn't know what color his hat is, so at least one of A or B must be wearing a white hat. If A were wearing a red hat, then when B is asked, he would know that his hat is white. If his hat were red, both A and B would have red hats, and C would have known that his hat was white. But he didn't know this. Thus, A's hat is white,as A is able to deduce once he knows the answers of C and B.

Scott said...

@David Gordon:

Well done; you've got it. All the other variants work similarly.

Scott said...
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Scott said...

And lest we wander too far off-topic…

Poland walks into a bar. The bartender says, "What can I get you?" Bronisław Komorowski says, "Nothing. We're here to change the light bulb."

Greg said...

The main problem with A's response is that it assumes B and C are rational.

Scott said...

@Greg:

Yeah, strictly speaking the premise that the prisoners are rational (to a sufficient degree, whatever that is) should be expressly stated.

Glenn said...

Scott,

An afterthought: Do you suppose that that's why the most common version of the puzzle has the second prisoner being one-eyed? I've never seen a variant in which he couldn't see both of the other prisoners' hats, but perhaps in the original version, the idea was supposed to be that he could see only the third's.

I don't rightly know; but the surmise works for me.

Glenn said...

Scott,

So how could a completely uninformative answer have altered the probability of A's death? Or is there an error in his reasoning?

a) While the probability of A's death was altered, it was altered by an informative answer rather than an uninformative one.

b) A's reasoning is correct--to the extent that he recognized the probability of his death had decreased.

c) A's reasoning is incorrect--in that the altered probability of his death actually is lower than he figured.

Glenn said...

David Gordon,

What Scott said--well done.

Glenn said...

Greg,

The main problem with A's response is that it assumes B and C are rational.

On the one hand, the objection is fair. OTOH, the assumption is standard for that type of problem.

- - - - -

The truth of the matter is that the jailer was just having fun--he was letting the prisoners go no matter what happened. ;)

Glenn said...

c) A's reasoning is incorrect--in that the altered probability of his death actually is lower than he figured.

I was thinking of the Monte Hall problem, so cancel that.

Scott said...

@Glenn:

Don't cancel (c) too hastily! As I mentioned, the problem is a variant of the Monte Hall problem, and that answer happens to be correct.

A's reasoning is flawed. If you run the numbers, you'll find that his chance of survival is still 1/3 (not 1/2), but C's is now 2/3—just as, in the Monte Hall problem, the chance that the prize is behind the door Monte didn't open is 2/3.

The key to the problem is to notice that if A is going to survive, the guard has a choice about what answer to give, just as Monte has a choice about which door to open if you've chosen the one with the prize.

Scott said...

I should perhaps briefly explain the Monte Hall problem in case anyone isn't already familiar with it.

You're on the old TV game show "Let's Make A Deal," hosted by Monte Hall. He shows you three closed doors, behind one of which is a valuable prize, and tells you to choose one. You do so. He then opens one of the other two doors to show you that it doesn't hide the valuable prize, and then offers you a chance to switch your choice to the remaining closed door. Should you switch?

The answer is that you should. I think the easiest and simplest way to see why is this: there's only one chance in three that you chose the correct door to begin with, so there are two chances in three that Monte is in effect showing you the door you should have chosen.

Tony said...

"Before I spoke to the jailer, I had just one chance in three of being the surviving prisoner. But now it seems that I have one chance in two, since I now know that either I or Prisoner C will be set free."

What's odd is that he could have reasoned the same way if the jailer had told him Prisoner C was to die, so he should have been able to reach the same conclusion without ever having asked the question at all. So how could a completely uninformative answer have altered the probability of A's death? Or is there an error in his reasoning?

(Hint, though perhaps not a universally helpful one: This is really a variant of the Monte Hall problem.)


I once heard a math professor characterize the non-intuitive aspect of this result by saying "the correct statement is "before I spoke to the jailer, the probability that I was to be the one survivor was either 1 or 0 since the decision was already made, I just did not know which probability it was. After I spoke to him, the probability remained the same 1 or 0. Before the decision had been made, the probability I would be the survivor was correctly set at 1/3. The change in probability by the event of the decision went from 1/3 to 1 or from 1/3 to 0. The change in probability by the jailer telling me was a null set - there was no change.

To go on, before the decision had been made, the CONDITIONAL probability that I would survive provided B was certainly going to be killed, would be 1/2. After the decision was made, the condition ceased to be a hypothetical.

However, to speak of the probability that I would survive as if that probability were to be stated resting only on my knowledge and not the king's: I knew the conditional probability of 1/2 above before the king made his decision, and after the jailer told me that B would be killed, I knew the condition had ceased to be a hypothetical.

Does this sufficiently collar the un-intuitive part of this puzzle?

Scott said...

And @Glenn:

The parallel with the prisoner problem is probably clear to you already, but in case it's not…

Being the surviving prisoner = choosing the door with the prize

and

Being told which other prisoner will die = being shown which other door doesn't conceal the prize. (In either case it may be both, in which case the relevant "revealer" has a choice about what to reveal. But there are two chances in three that the decision will be determined for him.)

Of course Prisoner A isn't given an opportunity to switch, but otherwise the logic is the same.

Mr. Green said...

Scott: Randall Munroe's explanation looks sound to me, at least insofar as we treat the puzzle as what it's intended to be: namely, a contrived problem of formal mathematical logic, common knowledge, nested hypotheticals, and so forth. What's the matter with it?

As far as the basic induction, etc. that is the key to the puzzle goes, there’s no problem. However, at least as presented, what does the Guru communicate that was not previously known to anyone else? And if nothing is communicated, then nothing changes, and if nothing changes, then nothing happens (although it probably wouldn’t take much tweaking to get around this).

Overwhelmingly, there are 29 pages of comments there, including many that raise this issue. Someone helpfully posted a summary of it all, though the objection to this objection — in those comments and elsewhere that I’ve seen — misses the obvious point. For if I know something, then it doesn’t matter how I know it, or whether there is another way that I might not know it. Either I do or I don’t.


And in a vain (get it?!) attempt to stay on topic, I’ll add: what did one hundred logicians say to the other hundred logicians? — I don’t know, but I’m sure it made a hundred sense.

Glenn said...

Scott,

Wait a minute here; something isn’t kosher (or appears not to be). How can it be that my c) is correct and C's chance of survival is now 2/3? It must be that one or the other isn't right, yes?

According to the problem, (prisoner) A reasoned that this chance of survival went from 1/3 to 1/2. This means that his chance of death went from 2/3 to 1/2.

In my c), I said that "the altered probability of [A's] death actually is lower than he figured." If this is correct, i.e., if it is correct that the probability of A's death is lower than he figured, then, since he figured it was 1/2, the probability of his death must be less than 1/2. And if the probability of his death is less than 1/2, then the probability of his survival must be greater than 1/2.

But how can it be that the probability the A's survival is greater than 1/2 and C's chance of survival is greater than 1/2 (2/3)? In this case, the sum of the probabilities exceed 1. I'm quite fine with my c) not being correct, but not at all fine with the probabilities not properly footing.

What am I missing? Or am I being too literal, and when you said that my c) was correct, you meant that it is correct that the altered probability of someone's death is lower than 1/2? If so, that's fine by me.

Meanwhile, and so far as I know, I've got a decent grasp of the MH problem, and see, understand and can explain six ways from here to Sunday why the contestant should switch, and that his chance of winning when switching is 2/3. (I also concur that the explanation you offer quite likely is the simplest one going.)

The virtual switch in the prisoners problem, however, escapes me. I'll have to ponder the matter more.

(I get the analogical mapping (more or less); thanks for that. But I have yet to actually 'see' it; and I won't rest comfortably just 'knowing', I need to 'see' it. Oh well, I'll work on it.)

Glenn said...

Mr. Green,

However, at least as presented, what does the Guru communicate that was not previously known to anyone else?

As far as content is concerned, nothing. But how would anyone know whether a given day is day 1, day 17, or, say, day 94? Need a starting point for the inductions to commence, and the Guru's utterance constitutes that starting point.

(So I gather.)

Glenn said...

(I get the analogical mapping (more or less); thanks for that. But I have yet to actually 'see' it; and I won't rest comfortably just 'knowing', I need to 'see' it. Oh well, I'll work on it.)

Okay, I see it now.

(Pardon me while I talk to myself: Sheesh!)

Mr. Green said...

Glenn: Need a starting point for the inductions to commence, and the Guru's utterance constitutes that starting point.

I thought that too, but I talked myself out of it. With only a slight change, the riddle could be set up to work that way; and if so, then it would equally provide a non-arbitrary starting point for everyone, so they’d all end up knowing their eye-colours. However, as given, there is no motivation for the logicians to start counting (why pick that arbitrary non-arbitrary day as opposed to some other equally fixed starting point?) — they are not trying to find out their eye-colour, nor to avoid finding out.

One of the delightful elegancies of the original puzzle is that without any motivation one way or the other, the logic of the situation just naturally requires them to come to right conclusion on day n without any attempt on their part (other than their natural and immediate recognition of logical conclusions). And it works that way in other versions of the puzzle, such as those involving two or three mud-spattered children, or any other such variation. In those cases, there really is new information supplied, and so logical consequences follow that didn’t before. In the XKCD version, there is a sort of equilibrium that never gets disturbed, so I think the answer has to be that nothing happens.

Mr. Green said...

Mr. Green: "Of course, if 'meta' humour means humour about humour, then is this really that?”
Scott: Hmm, arguably not. Wouldn't that be funny?

It’s the Epimenides Skit: “This joke has no punchline.”

Actually, I think I was perhaps trying too hard to be pedantic: in the clerical quip, the story is about clergymen, but the joke is indeed about man-walks-into-bar jokes. Whereas my example is a story about comedy, but the joke is a standard play on the expected responsibilities of someone’s career. A better example would have been:

☞ What did one joke say to the other joke? — “These punch lines are really giving me a headache!”

Or, to run a good thing into the ground:

☞ What did one “meta” joke say to the other meta-joke? — “What did one joke say to the other joke?”

☞ What did one non-sequitur say to the other non-sequitur? — “I think so, Brain, but don’t camels spit a lot?”

☞ What did one absurdist joke say to the other absurdist joke? — ☃

☞ What did one post-modern joke say to the other post-modern joke? — “OK, so, like <something funny here>…”

☞ What did one anti-joke say to the other anti-joke? — Nothing, jokes can’t talk.


Of course, I could try to tie it all together with an ironic self-referential surrealistic reconstructed meta-parody, but that would be a bit over the top, don’t you think?

Mr. Green said...

Scott: On the other hand, programs can be instantiated in corporeal substances, as when the Word became flesh.

Well, now we’re getting into yet another branch of humour: Divine comedy. Of course, the Incarnation is a pun (…the lowest form of kenosis). And what is more self-referentially meta-redemptive than conquering death by death?

When does One equal Three?—Always. What did one Person of the Blessed Trinity say to the other Person of the Blessed Trinity?—Himself. What is always eaten, yet never consumed? Knock, knock.—It’s open unto you. My first is last and my last is first—where am I? How many sinners does it take to change?—Just one (out of 99), but he has to do it in good faith. Why did Christ cross Hades?—To get us to the other side.

Mr. Green said...

Daniel: Well I just assumed both examples you give involved a Category Mistake.

What did one category mistake say to the other category mistake?

“Well, that’s ironic!”

I mean how can a concrete corporeal substance walk into a program which only exists as such in relation to our interests?

By putting one foot in front of the other…

Mr. Green said...

Scott: A thousand score lawyers walk into a bar association meeting. One says to the others, "Hey, what do you call us at the bottom of the ocean?”

Well, I’d call it 20,000 legals under the sea.

Of course, I’m the type to forge ahead in these matters.

(I'm sorry I didn't tell that better, but I'm out of practice.)

http://instantrimshot.com

No one ever has. A shame, really, as I think it's quite good.

Or you think that’s what you think. But for all you know, you might think something else, pending mud in your eye from a local guru.

"The creature breaks through the trees and stops in front of him. It extends a black clawed hand, lays it upon the web developer's keyboard and types: '<Meta>’.”

I like that one. But then, I never met a tag I didn’t like.

Scott said...

@Glenn:

"What am I missing?"

Nothing, as it turns out. In reading your post I mixed up survival and death.

@Mr. Green:

"[A]t least as presented, what does the Guru communicate that was not previously known to anyone else?"

This is actually the key to understanding the entire puzzle.

I assume we're agreed that if there's just one blue-eyed islander, the Guru does communicate new information. The sole blue-eyed islander learns for the first time that there is a blue-eyed islander, and (because the announcement is made publicly) everyone else now knows that the blue-eyed islander knows this. (They couldn't have known it previously because none of them knew whether his own eye color was blue or not.)

If there are two blue-eyed islanders (say Adam and Bob), then of course everyone already knows there's at least one blue-eyed islander. But until the Guru speaks, Adam doesn't know that Bob knows this, and Bob doesn't know that Adam knows it. Now they do (and everyone else now knows that they know it), again because the announcement was made publicly.

If there are three (say Adam,Bob, and Chuck), then everyone knows that there's at least one blue-eyed islander and everyone also knows that everyone else knows it. But Adam doesn't know that Bob knows that Chuck knows it (and likewise for any other permutation of the three). Now they do know it, and the other islanders all know they know it.

In fact there are 100 blue-eyed islanders, but the principle is the same—just with 99 or 100 (depending on the eye color of the islander) now-knows-that-everybody-knows-es. That's pretty complicated for us ordinary mortals, but these islanders are perfect logicians!

The Guru turns the fact that there's a blue-eyed islander into what's technically called "common knowledge," which in the relevant philosophical parlance means something that not only everyone knows, but also everyone knows everyone knows, and everyone knows everyone knows everyone knows…

That's the new information the Guru provides. So you can feel safe in relying on the inductive argument.

Scott said...

@Glenn:

"Okay, I see it now."

Good, sounds like you've got it.

I like to put it pretty much the same way as I put the Monte Hall solution: there are two chances in three that A will die ("picked the wrong door"), and therefore there are two chances in three that the jailer ("Monte Hall") is in effect telling A which other prisoner will survive ("which door is correct").

I've inserted parenthetical comments in that version in order to make the parallelism clear (since perhaps not everyone who's interested has seen it yet), but of course I'd omit those in most contexts.

Scott said...

@Tony:

The point your math professor raised is certainly part of the issue my physics-prof friend thought the problem presented—namely, the nature of probability. Here's how he put it to me:

You have a coin. It's not a fair coin; it's biased toward either heads or tails, but you don't know which way or how much. What is the probability that you'll get heads on your first toss?

Most of us would (and initially I did) respond that without that information we couldn't know the probability. But Prof just smiled and said something like, "The probability is 1/2. Since you don't know anything about the direction or degree of the bias, you have no reason to tip the probability either way and your belief should be indifferent between the two alternatives."

The first answer just shows that we're thinking exclusively of what might be called "objective" probability. But Prof was a big fan (justifiably, in my opinion) of Richard Cox (and Bayesianism) and regarded probabilities as essentially degrees of subjective belief.

He thought the solution to the three-prisoner puzzle depended on that view (really a range of views) of probability, but I'm not sure he was right. Even if we wanted to say that once the decision was made, the objective probability that A would die was "really" either 0 or 1 and he just didn't know which, it seems to me that we could still consider his degrees of belief that it was one or the other and run the logic of the puzzle in just the same way—although we might not want to call those degrees of belief "probabilities." (It's even stated in the problem that A is told that the decision was made "at random"; on any relevant plausible understanding of that term, A's belief should initially be indifferent between the propositions that A, B, and C, respectively, will survive.)

At any rate, yes, I think that has something to do with the un- or counter-intuitive part of the puzzle, but I don't think that's all of it.

It's true, as you say, that the conditional probability of A's survival given that B will die was 1/2. But the really difficult bit in this puzzle is realizing that the relevant conditional probability is that of A's survival given that the jailer said B will die, which is a very different event.

The crucial insight is that the jailer is under no circumstances going to say that A will die (just as Monte Hall isn't going to open the door you chose); by the nature of A's question, the jailer's answer will be either "B" or "C." Once that's understood, it's a matter of using Bayes' Theorem to compute the probabilities that A and C, respectively, will die given that the jailer said B will die. These work out at 1/3 and 2/3, respectively.

Scott said...

Sorry, that last sentence should have read the probabilities that A and C, respectively, will survive.

Scott said...

@Mr. Green:

Instead of "So you can feel safe in relying on the inductive argument" I should have said that this new information is what sets off the induction. The way I put it made it sound as though you thought there was something wrong with the induction itself.

(By the way, Randall's presentation of the inductive argument for the solution If there are N blue-eyed islanders, they all leave on the Nth day isn't technically quite right, since he makes no explicit argument that the (K + 1)th case follows from the Kth but just walks us through the first few cases. But that argument isn't hard.)

Scott said...

Also, I wrote:

In fact there are 100 blue-eyed islanders, but the principle is the same—just with 99 or 100 (depending on the eye color of the islander) now-knows-that-everybody-knows-es.

I should make clear that this is the state of things before the Guru speaks. I've put this a bit misleadingly by including the word "now."

After the Guru speaks, everybody knows that everybody knows that…that everybody knows that there's a blue-eyed islander for as many everybody-knows-es as we care to include. That's not the case beforehand, and it's the change from one to the other that sets off the induction.

Scott said...

Three logicians walk into a bar. The bartender says, "Do all of you want beers?"

The first logician says, "Of course. Why else would we all walk into a bar?"

Glenn said...

Three logicians walk into a bar. The bartender says, "Do all of you want beers?"

The first logician says, "Of course. Why else would we all walk into a bar?"


Methinks the first logician may have jumped to contusions (as the saying goes). They all may have walked into a bar for the simple reason that not a one of them noticed it was there.

Glenn said...

Scott,

I like to put it pretty much the same way as I put the Monte Hall solution: there are two chances in three that A will die ("picked the wrong door"), and therefore there are two chances in three that the jailer ("Monte Hall") is in effect telling A which other prisoner will survive ("which door is correct").

I've inserted parenthetical comments in that version in order to make the parallelism clear (since perhaps not everyone who's interested has seen it yet), but of course I'd omit those in most contexts.


That does it just nicely. It tidily wraps things up, and brings matters back to St. Thomas.

(Well, it sort of, kind of (i.e., in a way) brings matters back to St. Thomas (at least ‘experientially’ speaking (functionality-wise, anyway)).)

1. Monte Hall Problem
2. Prisoners Problem
3. St. Thomas

1. Monte Hall Problem

Regardless of how the doors are actually labelled, we can label the door selected by the contestant as A, the door Monte opens as B, and the remaining door as C.

a. Use a line to separate door A from doors B and C:

A | B C

b. The chance that the lone prize is behind the door left of the line is 1/3, and the chance that the lone prize is behind a door right of the line is 2/3.

c. Monte opens door B to show that the prize isn’t there, so door B isn’t the door behind which the lone prize is, and door B can be (factored out):

A | (B) C

d. The chance that the lone prize is behind the door left of the line remains at 1/3, and the chance that the lone prize is behind a door right of the line remains at 2/3.

e. The contestant’s chance of winning the prize by standing pat (i.e., by sticking with door A, and not switching to door C) is 1/3, and his chance of winning the prize by switching to door C is 2/3.

2. Prisoners Problem

It is given in the Prisoners problem that the Prisoners are labelled A, B and C.

a. Use a line to separate Prisoner A from Prisoners B and C:

A | B C

b. The chance that the lone future survivor is left of the line is 1/3, and the chance that the lone future survivor is right of the line is 2/3.

c. "Prisoner B is definitely going to be executed", so he isn't the lone future survivor, and Prisoner B can be (factored out):

A | (B) C

d. The chance that the lone future survivor is left of the line remains at 1/3, and the chance that the lone future survivor is right of the line remains at 2/3.

e. A's chance of surviving is 1/3, and C's chance of surviving is 2/3.

3. St. Thomas

o [T]o understand is simply to apprehend intelligible truth: and to reason is to advance from one thing understood to another, so as to know an intelligible truth. And therefore angels, who according to their nature, possess perfect knowledge of intelligible truth, have no need to advance from one thing to another; but apprehend the truth simply and without mental discussion, as Dionysius says (Div. Nom. vii). -- ST I Q 79 A 8

Dionysius does indeed say:

o [T]he intelligible and intelligent powers of the Angelic Minds [do not collect] their knowledge of God in partial fragments or from partial activities of Sensation or of discursive Reason... but rather...they perceive the spiritual truths of Divine things in a single immaterial and spiritual intuition... [On the other hand,] human souls possess Reason, whereby they turn with a discursive motion round about the Truth of things[.] -- Chapter VII (Concerning "Wisdom," "Mind," "Reason," "Truth," "Faith"), #2

Scott said...

@Glenn:

Nicely done.

And your quotation from St. Thomas also suggests an alternative way of formulating the problem of the blue-eyed islanders: they could be angels, who don't reason in steps but immediately apprehend all of the everybody-knows-that-everybody-knows-es.

But they'd still have to wait the requisite number of days!

(I'm also not sure what property an angel might have that would be unknown to the angel him/her/itself but known to all the other angels.)

Glenn said...

...your quotation from St. Thomas also suggests an alternative way of formulating the problem of the blue-eyed islanders: they could be angels, who don't reason in steps but immediately apprehend all of the everybody-knows-that-everybody-knows-es.

!

Glenn said...

(I'm also not sure what property an angel might have that would be unknown to the angel him/her/itself but known to all the other angels.)

This makes for an interesting question.

(For we human, it does.)

Glenn said...

(s/b "...we humans...")

Mr. Green said...

Scott: In fact there are 100 blue-eyed islanders, but the principle is the same—just with 99 or 100 (depending on the eye color of the islander) now-knows-that-everybody-knows-es. That's pretty complicated for us ordinary mortals, but these islanders are perfect logicians!

Indeed, they wouldn’t even use induction — instantly seeing any logical result, they would directly understand the n=100 case as quickly and easily as we can see it for n=1 or 2. So the induction itself is really a red herring; it’s just a shortcut we use because we don’t want to work out n=100 manually. (Part of understanding the solution is understanding how to figure out general cases without working them through manually, but it’s not particularly relevant to what the people in the puzzle are doing.)

If you read various defences of the solution, a lot of people seem to talk about “running the induction” as if it were something that was actually happening on the island — almost as though we started out with one person present there, then another physically came along, and another…. The induction itself is purely abstract; that discursive process needs an initial case to “set it going” in our heads, but once we’ve established the pattern, we just need something to apply it to. And what it applies to is a situation where seeing blue eyes is common knowledge. In a case where n=1 or 2, the only way for that to be common knowledge happens to be for someone else to point it out. In the case where everyone is surrounded by dozens and dozens of blue eyes, it’s almost self-evident. A simple deduction shows that common knowledge in this case applies to seeing n-3 sets of blue eyes, and so 100-3 is clearly large enough to satisfy “seeing at least one pair of blue eyes”.

In other words, our induced principle is: If there are n people with certain-coloured eyes, and if it is common knowledge that someone has that colour of eyes, and if there are “rounds” where anyone’s figuring out his eye-colour since the previous round is made common knowledge, then everyone with the colour will deduce it on round n.
For n≤4, that condition is not satisfied unless somebody makes it commonly known that there are any blue/etc. eyes. For n=100, it’s obvious; and so the blues — and browns — will all know on day 100. At least they will if the rounds kick in. That’s why many people (at one time including myself) argued that the guru’s statement just provided a fixed starting point (and maybe a fixed colour to be counted, except that that isn’t necessary, assuming everyone can see who leaves, and thus know how many people of which eye-colour leave on any day).

However, someone’s statement of already-known information is not going to matter — unless, say, we add a new rule about everyone’s actively trying to figure out his eye-colour. In the n≤4 cases, the outside statement, by providing the necessary common knowledge, also effects a change of state, and things naturally count down from there — it is the first act in the per se chain from which the following conclusions are activated. When there is no change in knowledge, the chain remains purely potential and no deduction actually happens. We could of course stipulate that everyone arrives on the island on the same day, and I think should be sufficient — seeing each for the first time would be a change in knowledge, and the rest would follow. Then everyone in a sub-population > 4 would leave on their corresponding day n, and anyone in a group with fewer than 4 of the same eye-colour would be left at the mercy of outside help.

Mr. Green said...

I was not entirely satisfied with the functional level of verbosity incorporated in the lexical content of my previous reply vis-à-vis the clarificatory power of said formerly noted response. So let me try again:

In a population of 100, person A may see only 99 sets of blue eyes (since, if he is blue-eyed, he cannot see his own eyes). Thus for all he knows, person B may see only 98 (since if A is not blue-eyed he won’t see A, and B cannot see himself). And person C may see only 97 blues — that is, as far as A knows that B knows. Now we could continue this series of A knows that B knows that … Z may see 0 blues — we could if it were not that we know Z does in fact see blue eyes. But that knowledge isn’t special to us as outside the puzzle; nobody has special knowledge, so all the people are symmetrical with respect to each other. So if A knows that B knows that C sees 97 blues, then A knows that anyone else knows that any-other-one else sees 97 blues (or more). And that’s as far as A needs to go. By symmetry, everyone knows that everyone knows that everyone sees some (many!) blue eyes. We don’t care about going all the way down to Z — we can short circuit it because of the surplus of blue eyes.

Now from this it is clear why external assistance is needed for small values of n: the highest value that can be safely assumed is n-3, and at least two people are needed to start deducing who will leave when, so n-3 ≥ 2 means n must be at least 5. And when people work out examples, they manually go through n=1, 2, 3 … maybe 4. No one notices that at n=5 and up we have a qualitatively different scenario. For “large” n, “someone has blue eyes” is automatically common knowledge, and thus the guru’s statement changes nothing, and nobody leaves. (Or possibly everyone leaves, depending on the other details.)

Scott said...

@Mr. Green:

"For 'large' n, 'someone has blue eyes' is automatically common knowledge, and thus the guru's statement changes nothing, and nobody leaves. (Or possibly everyone leaves, depending on the other details.)"

This isn't right. If you mean "common knowledge" in its technical sense, then that statement is false; it's not the case (even for large n) that the Guru's statement is something everyone knows everyone knows everyone knows…. And if you mean it in any other sense, the statement is true, but then it's also false that the Guru's statement adds no information and changes nothing.

In order to bring out this important point more clearly, let's go back to the case in which there's only one blue-eyed islander (Adam). In this case, of course, Adam doesn't know his own eye color and sees no other blue-eyed islanders, so he doesn't know that there's at least one blue-eyed islander. So it's obvious that the Guru's announcement conveys new information to him (and in fact would have done so if the Guru had just spoken privately to each of the islanders, or even just to Adam).

But just as importantly, because the Guru's announcement was made to all the islanders at once and they all know that the rest of them heard it, the announcement also conveys information to everyone else—in particular, that Adam now knows that there's at least one blue-eyed islander. (And everyone knows that, and everyone knows everyone knows that, and… ).

Previously, no one else could tell whether or not Adam knew there was at least one blue-eyed islander. A brown-eyed islander (Zeke) doesn't know his own eye color, so for all he knows, Adam does see a blue-eyed islander (namely Zeke himself)—or maybe not; there's no way for Zeke to be sure.

Obviously, in this case it's this new information that sets the calendar running. Every islander except Adam is now waiting to see what Adam will do at the end of the first day: if Adam leaves (each other islander reasons), then he didn't see any other blue-eyed islanders, and if he doesn't leave, then I must have blue eyes. The reason none of the other islanders starts this process any earlier is that until now, none of them knew that Adam knew there was a blue-eyed islander.

The important point is that the Guru's announcement conveyed new information even to the 199 islanders who already knew its content. It did so because it was made publicly. Had the Guru spoken privately to Adam, Adam would still have left the island at the end of the first day, but none of the other islanders would have been waiting to see whether that happened, and when it did happen, none of them would have known why.

[cont'd]

Scott said...

[cont'd]

Now consider the case of two blue-eyed islanders (Adam and Bob). In this case, each islander can see at least one blue-eyed islander (and everyone except Adam and Bob sees two), so nothing happens if the Guru makes her announcement privately to each islander alone. Adam and Bob each think, "Yes, I know that; what I don't know is whether there are one or two"; everyone else thinks, "Yes, I know that; what I don't know is whether there are two or three." But neither Adam nor Bob (nor anyone else) prepares to leave the island.

Everything changes when the Guru makes her announcement publicly. Here's why:

(1) Before the Guru's announcement:

(a) Adam knows that there's at least one blue-eyed person on the island (Bob). But not knowing his own eye color, Adam doesn't know whether Bob knows this; since Bob is the only blue-eyed islander Adam sees, he knows that whether Bob sees any blue-eyed islanders depends solely on Adam's own eye color. (Likewise, Bob doesn't know whether Adam knows it.)

(b) The other islanders don't know that significant fact about the state of Adam's knowledge. They don't know, that is, whether Adam knows whether Bob knows there are any blue-eyed islanders. Any such islander (Zeke) knows that Adam sees Bob's blue eyes (and thus knows that Adam knows there's at least one blue-eyed islander), but Zeke doesn't know what color Zeke's eyes are. If Zeke's eyes are blue, then Adam knows that Bob sees Zeke's blue eyes, and therefore Adam does know that Bob knows there's a blue-eyed islander (Zeke); if not, not. But there's no way for Zeke to tell. (Likewise, no one else knows whether Bob knows whether Adam knows there are any blue-eyed islanders.)

(2) After the Guru's announcement:

Everybody knows that there's a blue-eyed islander, that everybody knows there's a blue-eyed islander, that everybody knows everybody knows there's a blue-eyed islander, ….

(a) So now Adam does know that Bob knows there's at least one blue-eyed Islander, and Bob knows that Adam knows it.

(b) And all the other islanders know that Adam and Bob know those things.

Again the calendar starts to run. Adam waits to see whether Bob leaves at the end of the first day; if Bob doesn't, then Adam will know that Adam has blue eyes. Bob also waits to see whether Adam leaves. Neither does in fact leave, so both of them leave at the end of the next day.

Meanwhile, everybody else knows neither Adam nor Bob will leave at the end of the first day (since each of them knows there are two blue-eyed islanders); they're waiting to see what Adam and Bob do at the end of the second day. Zeke reasons that if Adam and Bob don't leave at the end of Day 2, Zeke must have blue eyes; every other non-blue-eyed islander reasons in the same way. And as with Adam and Bob, this calendar starts to run precisely because the Guru's public announcement conveyed new information to them.

The case of 100 blue-eyed islanders (or other "large n") is no different in principle from the case of one or two, on the premise that the islanders are perfect logicians (as we are not, so it's not a good idea to use our ordinary reasoning as a model). In each case the Guru's statement conveys new information to every islander, not just because of its content, but because it's made publicly. (From what I've already said, I think you can see that there's a sound inductive argument to this effect.) And the new information is about the state of the other islanders' knowledge, meta-knowledge, meta-meta-knowledge….

Scott said...

[cont'd]

"And that's as far as A needs to go."

I think that's the point at which you're losing me. This is surely not as far as A needs to go (or will go) as a perfect logician (who doesn't need any special motivation to "figure out" his eye color). There's a world of difference between that state of knowledge and the state of "common knowledge" in the technical sense of that term (and feel free to substitute some other term if you like; I'm not happy with it myself).

Scott said...

One last point:

"No one notices that at n=5 and up we have a qualitatively different scenario."

That's because this isn't true if the islanders are assumed to be perfect reasoners. For n=6, for example, the scenario is that before the Guru's announcement, Adam doesn't know that Bob knows that Chuck knows that Doug knows that Ed knows that Fred knows that there's at least one blue-eyed islander, nor does Zeke know whether Adam knows this; whereas after the Guru's announcement, everybody knows these things and more. That is not qualitatively different from the scenarios for lower values of n.

The Masked Chicken said...

For a long time (okay, a short time, but it seemed longer), humor was thought to require paradox. The linguist, Victor Raskin, was able to argue that humorous narratives have discernible truth content - one can ask questions about the joke scenario that can, actually, be answered. I was able to use this to show that humor does not use paradox, but something that looks like a paradox.

The biggest single problem in current theories of humor is that no one has been able to define incongruity in any rigorous way. It was (is) believed that the two unfolding texts or situations at the punchline are incongruous and the processing of the two incongruous texts initiates the humor response.

This turns out to be an over-simplification. I have a way to define incongruity in a rigorous mathematical way, up to the problem of universals (which, fortunately, does not have to be solved for the theory to work) so that one can assign an actual measure for each object in a joke scenario as to how incongruous it is. The theory is a bit complicated and uses lattice theory, but produces pretty pictures of concept lattices.

It would take a while to explain things, but humor turns out, it seems, to be a statistical concept. It plays off of the probability assessments of congruity of the elements in the joke scenario. I am in the process of writing up the results, which I have already presented at humor conferences (which turn out to be not that humorous), for publication.

The Chicken

The Masked Chicken said...

[cont.]
There is a smoking gun test for the theory, but current detection thresholds are just a little too slow to see it on fMRI. I have a suspicion that magenetoencephelographs have the speed needed, if anyone out there should happen to have one.

There is a lot of cool logic involved in humor. Unfortunately, there is a nomenclature problem in that the Raskin/Attardo General Theory of Verbal Humor (GTVH) uses the term, "logical mechanism," differently than I do. They see it as the logical relationship between the two unfolded joke texts at the punchline. I see it as the actual mechanism of logic that lets the joke happen. We have no idea how to make a proper classification of the logical mechanisms of humor in the Raskin/Attardo sense. The logical mechanism of humor, in my sense, is pretty understandable in terms of modal logic (ah, see where the connection to probability creeps even into the logic).

This work goes back to the late 1990's and I have some of it published in peer-reviewed journals, but I want the bigger picture. I have been trying to tie everything together - the neural processing, the logic, the physiological responses. Due to recent work in a number of fields it is now possible, more or less to develop a truly comprehensive theory of humor, at least insofar as the material aspects are concerned, so the ten or so papers on the different topics that I have been working on and presenting, hopefully, will get published, soon. Unlike most theories of humor, this one is actually scientifically testable and makes predictions. Oh, did I mention the pretty computer simulations?

It's not like one can just make up a theory and call it science. The science is incredibly complex and I have had to solve a lot of technical problems along the way, like how to represent probabilities in a lattice or formal concept map and build a computer model of synaptic responses in the pre-frontal cortext during humor processing and figuring out how to use counterfactual access points to travel between the two different possible worlds that the unfolded joke texts activate. You didn't think humor would be that simple, now, did you?

Beyond the nuts-and-bolts, there is also the metaphysics and even moral implications of humor to be done - areas where almost no one is doing any work.

Anyway, beyond my scribblings, humor turns out to be a very complex and interesting field of study with lots of unanswered questions. If only they would pay us, sigh. A million dollars for depression research and none for humor. There is something wrong with that picture.

The Chicken

Tony said...

For n=6

Scott, for any scenario where there are more than 3 blue eyed persons, EVERY islander knows that EVERY person, including every blue eyed person, can see 3 people with blue eyes. Therefore, every person KNOWS that every person knows that _everyone_ knows there are at least 2 blue eyed persons. Therefore Adam knows that Bob knows that Chuck knows that Doug knows that Ed knows that Fred knows there are at least 2 blue-eyed persons. This remains true also at N=7, at N=8, etc.

Brandon said...

Holy moly, I've been out of commission for a while and this comments thread became very complex!

Masked Chicken,

That's very interesting. Incongruity accounts of humor have been around for a while -- they were developed at length by the Scottish common sense philosophers, for instance, and some of their work (Beattie's, in particular) have had a small but traceable influence on research into humor. But giving a serious account of incongruity itself, rather than leaving it as an intuitive primitive, would be a fairly significant advance.

Mr. Green said...

Scott: The case of 100 [is no different] from the case of one or two

We agree that it’s no different in some relevant ways, but I showed it was different in at least one other relevant way. Where do you think my argument went wrong? It’s true our ordinary reasoning is subject to flaws, but on the other hand, no amount of fancy logical deduction can make a truth into a falsehood. Not even an obvious truth.

(Yes, I meant the technical sense (which is apparently contrasted to “mutual knowledge”, which seems to me to be backwards, but I don’t have any better suggestions).)

the Guru's statement conveys new information to every islander, not just because of its content

It has to be about the content too. If the guru came out and said, “Hey, nice weather we’re having!” that obviously wouldn’t help in any of the cases. Nor does the guru say anything about meta-knowledge. (She doesn’t, for example, say “I see someone who sees someone who sees ….”) Whatever knowledge anyone gets from her statement is deduced from it, not obtained directly. And if it can be deduced from “someone has blue eyes”, then it doesn’t matter who says it. Anyone who knows there are any blue eyes out there will deduce it. And everyone sees blue eyes, so everyone will deduce it. And everyone knows that everyone else sees blue eyes, so everyone knows that everyone else knows it.

you can see that there's a sound inductive argument to this effect

Sure, but it doesn’t show what I think you want it to show. You can show that for ∀x, x²≥0; but it doesn’t follow that x﹤0 unless I square it first. Sometimes x is already ﹥ 0 by itself. Similarly, the guru’s announcement is sufficient to say that everyone knows etc.; but when n﹥4, it’s redundant, like squaring a positive number to get a positive.

Adam doesn't know that Bob knows that Chuck knows that Doug knows that Ed knows that Fred knows that there's at least one

Sure he does: with 6 blue-eyes, Adam sees 5 of them. He knows Bob can’t see himself, and—on the assumption that Adam is brown-eyed—Bob won’t see Adam as blue either. But Adam knows that Chuck, Doug, Ed, & Fred are blue, so he knows that Bob sees 4 blues. And— in Adam’s mind—Bob might think Chuck can see only Doug, Ed, & Fred. Which is as far as we need to go. Adam sees (more than) two blues, and he knows that Bob sees more than two blues, and he knows that Bob knows that Chuck sees more than two too. But—to the hypothetical Bob—Doug & Ed & Fred are interchangeable with Chuck. From hypo-Bob’s perspective, he knows they equally can see three other blues. So hypo-Bob knows that everyone else is going to play the counting game. And to Adam, Chuck through Fred are interchangeable with Bob. So Adam knows that everyone else sees enough blue eyes to play the counting game; and that everyone else knows that everyone will play.

Or look at it from the bottom up: could Fred think nobody has blue eyes? Each person knows there are four blues besides himself and Fred—of course Fred sees someone blue! Fred isn’t seen but unable to see the others. So Adam knows everybody knows Fred sees blue. If he starts imagining that Bob might imagine some sequence that ends up with Fred not seeing any blue, then Bob has gone off his rocker. So any perfect logician will immediately rule out those impossible paths from the chain of possible meta-metas.

The chain isn’t illogical by itself; it simply fails to take into account all the information that is available. (I think the problem is that considering what Adam knows that Bob knows … is too specific. We ought to be dealing with disjunctions: Adam knows that (Bob|Chuck|Doug|Ed|Fred) knows that ((Chuck|Doug|Ed|Fred) | (Bob|Doug|Ed|Fred) | …) etc. Well, you could write it out properly, but it’s easier to do it the way you did and stick a “& ∑(blues) ≥ 5” on the end.)


And if you still aren’t convinced, I have an induction too: n islanders will leave in n days without any announcement. Just start it at n=5, and you will find it works fine.

Tony said...

Anyway, beyond my scribblings, humor turns out to be a very complex and interesting field of study with lots of unanswered questions. If only they would pay us, sigh. A million dollars for depression research and none for humor. There is something wrong with that picture.

Chicken: of course, people don't usually kill themselves over an excess of humor.

Actually, quite the reverse. Have you thought about running a study to measure physiological differences of humor in depressed people versus non-depressed people? You might get some money there, because it might actually lead to a new treatment for depression! Dr: "Yes, Mrs. Jones, your prescription is to read 3 xkcd comic strips per day..." Boy, the pharmaceutical companies would get mad.

The Masked Chicken said...

"Chicken: of course, people don't usually kill themselves over an excess of humor. "

Well, strictly speaking, an extended tragic state is called depression and an extended humor state is called mania. Normally, tragic and humor responses damp down, relatively quickly, but, under certain circumstances (existence of an external energy source in the brain: biochemical or physiological) - the responses don't damp down and one is stuck in that state. With two energy sources, one can get an oscillation between the two states, which we call bipolar disorder.

As for studying laughter in depressed people, those studies have been done for geriatric patients and stroke victims.

The Chicken

Scott said...

One last round on the blue-eyed islanders puzzle and then I'll drop the subject. It really is an interesting (and difficult!) puzzle, though, and thanks to Mr. Green and Tony for their discussion of it. (Every time I think it through again in reply to one of them, I learn something more about it.)

We're agreed, I take it, that the inductive argument is sound; the disagreement is about why, and indeed whether, the Guru's statement starts the calendar running on the number of days until the blue-eyed islanders leave. The claim I'm defending is that the Guru's statement turns the knowledge that there's at least one blue-eyed islander into what's technically called "common knowledge," and the main question I want to address here is why that knowledge isn't "common knowledge" in the first place.

Let's continue with the case n=6, where (as usual) Adam, Bob, Chuck, Doug, Ed, and Fred are the six blue-eyed islanders in arbitrary order. In good Scholastic fashion I'll state the objection first: Everybody on the island sees at least five blue-eyed islanders, and everybody on the island knows that, so how could it possibly not be the case that Adam knows Bob knows Chuck knows Doug knows Ed knows Fred knows there's at least one blue-eyed islander?

Tony and Mr. Green have each put this objection nicely. Here's Tony:

[F]or any scenario where there are more than 3 blue eyed persons, EVERY islander knows that EVERY person, including every blue eyed person, can see 3 people with blue eyes. Therefore, every person KNOWS that every person knows that _everyone_ knows there are at least 2 blue eyed persons. Therefore Adam knows that Bob knows that Chuck knows that Doug knows that Ed knows that Fred knows there are at least 2 blue-eyed persons.

And here's Mr. Green:

[C]ould Fred think nobody has blue eyes? Each person knows there are four blues besides himself and Fred—of course Fred sees someone blue! Fred isn’t seen but unable to see the others. So Adam knows everybody knows Fred sees blue. If he starts imagining that Bob might imagine some sequence that ends up with Fred not seeing any blue, then Bob has gone off his rocker.

[cont'd]

Scott said...

[cont'd]

However, there's a problem, and it's hidden inside a string of nested hypotheticals.

Suppose Adam* is entertaining the proposition that Bob knows Chuck knows Doug knows Ed knows Fred knows there's a blue-eyed islander. Who, from Adam's point of view, could this islander be?

It obviously can't be Fred himself; Fred doesn't know his own eye color, and Adam knows that. Just as obviously, it can't be Adam; Adam is the one entertaining the proposition, and he doesn't know his own eye color either.

What about Bob? Adam knows Bob's eyes are blue, and so does Fred. But wait—Bob is also one the islanders in the nested hypothetical. Bob doesn't know his own eye color, and Adam knows Bob doesn't know it. Never mind what real Fred sees; in order for the hypothetical Fred in this proposition to see Bob's blue eyes, Adam would have to think Bob knew his own eye color. Sure, Adam himself can know that Fred sees Bob's blue eyes, but he can't suppose that Bob knows this. So from Adam's point of view, in order for Bob to know the relevant sub-proposition, the blue-eyed islander Fred sees would have to be Chuck, Doug, or Ed.

But for the same reasons, it can't be any of them either. Chuck is also one of the islanders in the nested hypothetical, and (still from Adam's point of view) in order for Bob to know that Chuck knows the relevant sub-proposition, the blue-eyed islander Fred sees would have to be Doug or Ed. And so forth.

So Adam can't possibly know what, on the face of it, it seems he must know. As a perfect logician, he has to acknowledge that he doesn't know that Bob knows Chuck knows Doug knows Ed knows Fred knows there's a blue-eyed islander.

----

* I'm using Adam here for convenience, but of course the other islanders don't know their own eye colors either and similar arguments apply to a brown-eyed Zeke.

[cont'd]

Scott said...

[cont'd]

It's for this very reason that we can't infer Adam knows Bob knows Chuck knows Doug knows Ed knows Fred knows there's at least one blue-eyed islander from Adam knows Fred (or Bob, or anyone else) sees at least four blue-eyed islanders. The problem just discussed arises each time we add another "knows" step. The best Adam can do is know that Bob knows that Chuck sees at least three blue-eyed islanders. If he goes beyond three, he's implicitly assumed either that he himself knows his own eye color or that he thinks someone else does.

We therefore have to decrement the number of hypothetically-known blue-eyed islanders at each step. That means that when we get to Fred, the number is zero. That's probably the most counterintuitive bit: even though it's obviously not zero for the real Fred, it's zero for the hypothetical Fred at issue in the propositions Adam is entertaining! But that's enough to show that before the Guru speaks, There is at least one blue-eyed islander doesn't qualify as "common knowledge" in the technical sense. (And it's surely no stranger than the first step of the inductive argument, in which the islanders entertain the possibility that there's exactly one blue-eyed islander even though every single one of them knows it's false.) However tempting it is to think otherwise, we can't just keep adding Islander So-and-so-knows-es without limit.

What happens when the Guru speaks, then? At that point, everyone hears the Guru's statement and everyone knows everyone else hears it, and so all of a sudden the number of hypothetically-known blue-eyed islanders in the nested hypotheticals tapers down not to zero, but to one. (That's why the content of the Guru's statement matters even though it's something all the islanders already know.) Weird and over-technical as it may sound, for an island full of perfect logicians that's sufficient to start the calendar running.

Now, Mr. Green also thinks that nothing is necessary to start the calendar running—that is, that on whatever day the conditions of the puzzle take effect, the calendar will start to run whether the Guru speaks or not. I'd be interested in seeing his inductive argument to that effect, although in order to keep the thread from going too much farther off-track I won't post a reply (at least not a lengthy one) if he chooses to elaborate.

However, I suspect he'll turn out to be solving a different problem. I might agree, for example, that the calendar would also start to run if each of the islanders had an exhaustive list of all the eye colors on the island and knew all the others had it too; I would certainly agree that in this puzzle, the calendar would have started to run if the Guru had instead said, "Here is an exhaustive list of all the eye colors of which I can see at least one: blue, brown." (In that case, of course, the 100 brown-eyed islanders would also leave on the 100th day.) But there's no such premise in this puzzle. Here, no islander knows that he has one of a specified range of eye colors and each islander knows that none of the other islanders know it either, so nobody has any reason to start any deductions.

Step2 said...

Beyond the nuts-and-bolts, there is also the metaphysics and even moral implications of humor to be done - areas where almost no one is doing any work.

What are you talking about?

Scott said...

Postscript. In checking back and rereading my previous three-part post, I see that I haven't explicitly addressed Mr. Green's question:

We agree that [the case n=100 is] no different [from the case n=1 or 2] in some relevant ways, but I showed it was different in at least one other relevant way. Where do you think my argument went wrong?

Mr. Green is right in one important respect; I was wrong that the case n=100 doesn't differ from the case n= 1 or 2 in any way that's relevant to Mr. Green's own claim. It does, because in neither of the latter cases does every islander know that every islander sees at least one blue-eyed islander. What I should have said is that it doesn't differ in principle from the case n=3, the first case in which every islander does know that.

I do, however, continue to insist that the case n=1 or 2 doesn't differ from the case n=100 in any way that's relevant to the "common knowledge" problem. Merely having every islander know that every islander sees at least one blue-eyed islander isn't sufficient to make There's at least one blue-eyed islander "common knowledge," for the reasons I've already explained at length.

As for where the argument goes wrong: as I've already implied, there's nothing wrong with Adam's considering the hypothetical proposition that Fred doesn't know there's at least one blue-eyed islander even if Adam knows it's not true. Mr. Green says, "If [Adam] starts imagining that Bob might imagine some sequence that ends up with Fred not seeing any blue, then Bob[?] has gone off his rocker." On the contrary, if that happens, Adam has a reductio ad absurdum argument against the scenario he began by imagining—namely, that Adam himself knows Bob knows Chuck knows Doug knows Ed knows Fred knows there's at least one blue-eyed islander.

My own argument can be taken to show that Adam can't know (and can argue to himself that he doesn't know) that proposition, precisely because his knowing it would entail the absurd consequence that Fred both did and didn't know that there was a blue-eyed islander. Surely it's permissible for Adam to consider the possibility that he knows X for just long enough to see that if he did, he would also "know" something else that's demonstrably untrue!

Jeremy Taylor said...

Brandon,

Yes, I'd certainly be interested in exploring the metaphysical place of humour, especially from a traditional perspective.

Christianity seems to completely skirt over humour and laughter, although certain European folk lore traditions, like that of Till Eulenspiegel, have sought to make up for this.

Some religious traditions have included a role for humour and whimsy. Loki has something of a trickster role in Norse mythology and Taoism has a humourous and whimsical side (which sets it apart from the other two great Chinese religions, Confucianism and Buddhism).

Glenn said...

Insofar as the Catholic brand of Christianity, if it may be put that way, is concerned:

As the absence of an official Catholic Book of Cooking ought not to be taken as an indication that eating food is considered anathema, so too the absence of an official Catholic Book of Humor ought not to be taken as an indication that laughter and/or humor is considered anathema.

Indeed,

a) though immoderate laughter constitutes a venial sin, laughter not immoderate is "not opposed to the love of God and neighbor" (CCC 1856); and,

b) the "Catholic wisdom of the people...affirms the dignity of every person as a child of God, establishes a basic fraternity, teaches people to encounter nature and understand work, [and] provides reasons for joy and humor[,] even in the midst of a very hard life. (CCC 1676)

Tony said...

Never mind what real Fred sees; in order for the hypothetical Fred in this proposition to see Bob's blue eyes, Adam would have to think Bob knew his own eye color. Sure, Adam himself can know that Fred sees Bob's blue eyes, but he can't suppose that Bob knows this.

No, I don't think this does what you claim.

Since every person sees 3 blue-eyed persons, every person knows that every other person sees 2 blue-eyed persons. And given that, everyone knows that every other person knows that there is AT LEAST 1 blue-eyed person.

It is unnecessary for the "common knowledge" needed for the puzzle that every person know that "A knows that B knows that C knows that..." N-1 knows that there is a blue-eyed person N as a single ordered chain. It is sufficient for the solution that every person knows that every other person knows that there is a blue-eyed person, and this is known without identifying a SPECIFIC blue-eyed person, nor a single specific chain that includes every person "knowing that X1 knows that X2..." in a specific order. Your comprehensive and ordered "A knows that B knows that..." is unnecessarily restrictive. A knows that every person can see 3 blue-eyed people, so every person knows that every person knows there are at least 2 blue-eyed persons is a given and it doesn't matter that there be a single specific ordered chain of "A knows that B knows that..." which runs through the whole list in the same order for everyone.

It is sufficient that any person G be able to identify 2 other blue-eyed people that both he and any possible h can see to know that h knows there are at least 2 blue-eyed people, and HE CAN NAME 2 SUCH PEOPLE, for each possible h from A to N. They (the two blue-eyes G and h can both see) simply don't include G himself or that particular h.

And since G can do this for any other person h, G and h together can do also between G and h and any other k, regarding seeing ONE blue-eyed person in common: G and h can know that k knows G and h all see some blue-eyed person in common with him, and so they all know there is a blue-eyed person and that each other knows it.

Since every person from A to N knows he can do this for every OTHER person from A to N, then he knows that every OTHER person from A to N can do this with him included, so every person from A to N knows that every other person knows there is a blue-eyed person.

Scott said...

@Tony:

"No, I don't think this does what you claim."

Sure it does; it shows that the proposition in question isn't "common knowledge" in the technical sense*, which is exactly what I claimed (in the second paragraph of Part I of III) to be showing.

Your reply says in essence that no such "common knowledge" is actually necessary in order to start the calendar running; it's sufficient that everyone knows everyone knows there's at least one blue-eyed person. (And of course you needn't persuade me that everyone knows everyone knows there's at least one; as I've already acknowledged, that's obviously true for all n≥3.)

But that's not enough to make the base proposition "common knowledge." Whether any such knowledge is needed or not is a separate question.

----

* …,which, again, is as follows: a proposition qualifies as "common knowledge" for a group iff everyone in the group knows it, everyone knows everyone knows it, everyone knows everyone knows everyone knows it, and so on without limit.

The Masked Chicken said...

Dear Step2,

I know Manoff, at least collegially, and this is a far cry from either metaphysics or morality. He has some good ideas, but as I have said, there is no good in-depth treatment of either of these two subjects.

As for benign violation, we've known about this aspect of humor for years - heck, slipping on a banana peel is a prototypical example: it is funny only if no one is seriously injured. The author has re-discovered this and published an experiment to confirm parts of it. It seems to be good science and a useful study, but it does not, really, advance theory. He describes an observation - that humor occurs via benign violation, but doesn't really get at the mechanics in any satisfying way. He fails to prove that this is a necessary condition of humor, showing, instead that it is sufficient, in some cases. Indeed, absurdist humor cannot be explained easily by this theory.

If he had attended some research conferences on the subject, he would have known that I and others were doing work on this back in the 1990's, but, since I am not a psychologist, my presentation was on the unfolded states in humor and what is allowed and what isn't. The concept of humor killers, which is a limiting case of benign violation, has been known for years. The whole idea can be traced back, in embryonic form, to Kant. We often see many people who think they have discovered a new theory of humor, but aren't up on the work of others and begun touting their work as something new and important. It is a problem that we have had to address on the humor research bulletin board more than once. Raskin got so frustrated that he published a resource book for people entering the field.

I am not against the author of benign violation theory, but he has not, to my knowledge, presented at any conferences nor is widely known in the humor research community, although, obviously, I don't know everyone. We already know that humor requires these conditions. Robert Provine from the University of Maryland at Baltimore, has included something like this in his evolution-based theory of laughter.

Sorry, I sound so grouchy. If I thought benign violation theory were, "it," as far as theories go, I would be shouting it to the rooftops. It, like many modern theories of humor, encompasses some aspects of humor, but not is still not either a proper scientific nor general theory.

The benign violation theory, also, addresses nothing about the actual moral aspects of humor. This is not a study of the moral implications of humor, even though it is billed as one. It is a study, properly speaking, of the aesthetics of humor involving a few moral cases.

I do know the literature, more or less, as I have been researching the morality of humor for years and, if I live long enough, plan to write a book on it. The metaphysical aspects are even less studied and needs the attention of an expert.

As for the Catholic Church's appreciation of humor, it is profound, but, outside of a few passing commentaries, such as St. Thomas's treatment of mirth, there is very little in the way of systematic treatment.

The Chicken

Scott said...

And just to be clear: the general result, which I've shown for n=6 but which works as I've described for any number of blue-eyed islanders, is that before the Guru speaks,

Islander₁ knows that Islander₂ knows… that there's at least one blue-eyed islander

is false for any string of So-and-so knows-es that includes all of the blue-eyed islanders, even though everybody knows everybody knows there's at least one blue-eyed islander. The former simply doesn't follow from the latter.

I have not shown, or claimed to show, that this knowledge (or any other item of "common knowledge") is necessary to start the calendar running on the mass exodus of blue-eyed islanders (although I think that's true as well).

Glenn said...

Oh boy; I have received another email from the ABA (America Bartenders Association):

Dear Sir,

We have yet to see evidence of your having honored our reasonable request that the disparagement of the bartender be withdrawn. As you do not seem like the sort of fellow who is given to indecisiveness, we have no alternative but to conclude that you have decisively set yourself against honoring said request.

It is true that the first and second logicians were not indecisive, i.e., were not saying by their respective 'I-don't-knows', "Golly gee, I don't know if I want a drink", but were decisive, i.e., were saying, "I know I want a drink, but I don't know if each of my two companions also want a drink, so can only answer whether we all want a drink by saying, 'I don't know'."

However, your having chosen to make a joke of the bartender's thinking -- in the face of having just witnessed two fine examples of decisiveness -- that he had the wherewithal to recognize indecisiveness, serves as indication that Plato was right: some people will laugh at the self-ignorance of others.

We think such behavior not to be a virtue, but a vice.

Shame on you.

Sincerely yours,
Etc., etc.


Some people have no sense of humor; oh well, what can one do?

Write back, that's what:


Dear Etc., etc.,

You allude to the Superiority Theory of Humor, but I think it has no place in the matter. The Superiority Theory is but one of several theories of humor. Another theory of humor is the Incongruity Theory, and I think it is this theory which has a place in the matter.

The joke in question was not intended to encourage laughter at the bartender, but at the incongruity of something decisive being treated as if it were indecisive. It's just as if someone were to say, "Two fish are in a tank, and one says to the other, 'Do you know how to drive this?'" The joke is not intended to enourage laughter at fish, but at the incongruity of a fish tank being treated as if it were a military tank.

Sincerly yours,
Glenn

Glenn said...

The Masked Chicken,

I'm going to spend some time reading at the first link you provided above, as your writing on and treatment of the subject makes it interesting. Thank you.

Tony said...

Scott, well, OK, under that definition of common knowledge, fine. But I would maintain both that (a) such common knowledge is NOT necessary for the solution, and (b) that such a definition is not necessarily a good measure of what we normally mean by "common knowledge". For instance, take the principle of non-contradiction. It is common knowledge in the sense that every sane person holds it. (For, every sane person attempts to do things like argue their position, which would be nonsensical without holding the principle.) In that sense, everyone "knows it", and in that sense it is common. But some foolish people imagine that it would be possible to be sane and not hold it as always and everywhere true, so given that there are these foolish people, it is perhaps not true that everyone knows that everyone knows it is true. That is, the foolish people attempt to think it not to be true in all conditions: even though they do not SUCCEED in so thinking, they imagine that they think this, and thus there are some people who are unaware that they know the principle. (Likewise people who do know the principle but have never thought about it formally.)

Or, take Euclid's "common notions" such as "the whole is greater than the part." Self-evident axiomatic truths are true because they have this quality that as soon as you understand the terms and their juxtaposition in a statement, the truth of the statement is immediate (does not need any further support or basis for the mind to adhere to it as certainly true). They are common knowledge to everyone who has thought about the matter, because you cannot fail to have the knowledge of in the proposition as soon as you understand the proposition. But some people don't understand that some things are self-evident (they think that all propositions must require the support of prior propositions to be held with certainty). So although the common notions are commonly known, it is not true that they are known to be common knowledge to the entirety of the same set that knows them.

But surely there is a reasonable sense of "common knowledge" that needs only that everyone knows it, and not that "everyone knows that everyone knows that everyone knows..." it.

Scott said...

@Tony:

"Scott, well, OK, under that definition of common knowledge, fine. But I would maintain both that (a) such common knowledge is NOT necessary for the solution, and (b) that such a definition is not necessarily a good measure of what we normally mean by 'common knowledge'."

I agree with you entirely about (b), and I even mentioned a few posts ago that I don't care for the term myself. (I'd prefer something like "recursively common knowledge," say.) Unfortunately I wasn't consulted when the term came into vogue three or four decades ago, so I've tried to be very careful to be consistent in specifying that I'm using the term with its technical rather than its normal meaning.

As for (a), it's surely obvious that some sort of "common knowledge" (scare quotes here and below indicate the technical sense) is needed in order to start the countdown. In fact, it's easy to state a necessary and sufficient condition: that the proposition Islander so-and-so (it doesn't matter which) is beginning the countdown be "common knowledge." (The sufficiency is obvious: if one perfect-logician islander starts the countdown, s/he must know that all the rest are starting as well. The necessity is almost as obvious, though I don't intend the following as a full proof: I, an arbitrary islander, won't start counting down unless I know that all the other islanders are doing so as well, in particular the arbitrarily-chosen Adam; I know Adam won't start unless he knows that Bob is starting; I know Adam knows that Bob won't start unless Chuck is starting; …)

Any other necessary and sufficient condition will therefore be found to be equivalent to that one—which is one reason why any other proposition that sets off the countdown must itself be "common knowledge."

Mr. Green said...

Tony: But I would maintain both that (a) such common knowledge is NOT necessary for the solution, and (b) that such a definition is not necessarily a good measure of what we normally mean by "common knowledge”.

Yes, that’s the “technical” definition referred to above. And I agree, and Scott mentioned likewise, that it’s a poor use of the term. What we normally call “common knowledge” has apparently ended up being referred to in the technical jargon as “mutual knowledge”, which frankly has got it backwards. But if I get started on complaining about bad jargon I’ll never stop… (or as far as you know, I’ll never stop. Or as far as I know you know I know I’ll never………).


Scott: I have not shown, or claimed to show, that this knowledge (or any other item of "common knowledge") is necessary to start the calendar running on the mass exodus of blue-eyed islanders (although I think that's true as well).

I had managed to assure myself that it wasn’t necessary, but actually you have almost convinced me otherwise. The inductive proof I alluded to is simply that if it works without the guru for n islanders, then of course it works for n+1. The catch is proving a base case: clearly it doesn’t work for n≤4, but I need to write it all out again to check where I thought that could change for n≥5 and somehow short-circuit the requirement.

Scott said...

(Oops, that should be "Bob won't start unless he knows Chuck is starting." I'm sure everyone understood that, but in this context it's pretty important.)

Scott said...

@Mr. Green:

"I had managed to assure myself that it wasn’t necessary, but actually you have almost convinced me otherwise."

In that case my reply to Tony just above yours might do the trick.

Scott said...

@Glenn:

I'm sorry to hear that the ABA is giving you so much trouble. Would you like me to put in a word for you if I'm ever again at the bar?

Anonymous said...

Any other necessary and sufficient condition will therefore be found to be equivalent to that one—which is one reason why any other proposition that sets off the countdown must itself be "common knowledge."

I would claim that the countdown exists inherently as part of the givens of the situation, just as all of the persons know "A will leave the island only if he knows his eye color".

if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay.

The countdown is included in the scenario from the first day the scenario exists, which is the first day the ferry runs, which the first day the logicians have subject matter on which to logicize (i.e. that they know the conditions)...etc. If they all deduce all logical conclusions instantly, then they all start deducing everything that can be deduced from the condition "a ferry stops at the island. Any Islanders..."

Glenn said...

Scott,

I'm sorry to hear that the ABA is giving you so much trouble. Would you like me to put in a word for you if I'm ever again at the bar?

The offer is tantalizaing, and I'm tempted to accept. But I notice the kind of word that might be put in for me has yet to be specified, so I'm also hestitant to accept. I do assume the word would be a kind word, but when dealing with a lawyer, one cannot be too careful.

;)

Scott said...

@Anon:

"I would claim that the countdown exists inherently as part of the givens of the situation[.]"

In the sense you appear to mean, I agree. But I'm talking about the countdown's starting to run; that is, this or that day's being the one from which each islander starts to count If the n blue-eyed islanders that I can see leave on the nth day from today, then I know I don't have blue eyes, but if they don't leave, then I know I do have blue eyes and I have to leave with them on the (n+1)st day.

I don't think you mean to say that this day is just the day the puzzle's condition's take effect and that teh Guru's statement adds nothing; I'm taking you to mean that the possibility of a countdown is implied by the setup even if something else is required to set it running. If I'm misreading you, let me know.

@Glenn:

Very statute—even lawyerly. As it happens, the word I had planned to put in for you was zymurgy. Let me know what you think.

Scott said...

("Conditions." "The." Yeesh. The scary part is that I did proofread.)

Scott said...

(And oh, boy—"statute" s/b "astute." That was no typo; it was a thinko.)

Glenn said...

@Glenn,

...As it happens, the word I had planned to put in for you was zymurgy. Let me know what you think.


I think it's a good thing I: a) didn't act on my assumption; and, b) refrained from accepting the offer.

;)

Scott said...

@Glenn:

Oops, I guess I should have been less hasty myself. [quickly heads down to bar to retrieve word]

Glenn said...

No, no; quite alright. Had to run earlier, so didn't type up a complete response. Left out was,

"Nonetheless, the word-choice is excellent."

Scott said...

Ah, good; glad to hear it. I must say, I do like the word zymurgy and (particularly in view of its meaning) I don't think it will do you any harm to have bartenders associate it with you.

Shall I put in a good number for you too? I was thinking of e.

The Masked Chicken said...

Glenn wrote:

"You allude to the Superiority Theory of Humor, but I think it has no place in the matter. The Superiority Theory is but one of several theories of humor. Another theory of humor is the Incongruity Theory, and I think it is this theory which has a place in the matter."

In 2011 I presented a paper at the International Society for Humor Studies arguing that Superiority Theory, Incongruity Theory, Arousal Theory, Discharge Theory, Pleasure Theory, etc. of humor are part of a single, unified process in brain processing. Metaphysically, I am not sure that we know what humor is. The best we can do is analyze what humor does. We can see its tracks in how the brain responds.

We can see that the ventro-lateral pre-frontal cortex is active in processing the, "scripts," of the two humorous unfoldings of the incongruities, but we can also see that the dorsolateral pre-frontal cortex is active in processing social hierarchy. We can also see that the pleasure center is active. What I showed was that there are neural connections between all of these parts of the brain, so they form a circuit, in parallel. When humor is tripped, all of these regions are tripped, simultaneously. Thus, perceptions of incongruity, superiority, and pleasure are created at the same time and flow from the same event. As in any parallel circuit, one can isolate each individual component and notice a separate change in superiority or incongruity or pleasure. Without knowing that these elements were connected in the brain (something we could not have known until 2009, when the social hierarchy processing section was isolated), different people isolated different aspects of the humor response. My idea is that it is one single thing that triggers all of these responses. Because of the reciprocal nature of the connections, tripping anyone of the outbound elements, superiority, incongruity, or pleasure, can also trip all of the other elements and cause the overall humor response.

[cont.]

The Masked Chicken said...

I believe that the respiratory center in the brainstem is also tied in, so that when the signal is sent, it also modulates respiration and vocalization and this leads to laughter. I created a mathematical model that was printed in the early 2000's showing how laughter respiration might come about. The computer model matches, almost exactly, the measured esophageal pressure response. In fact, I almost didn't publish the model for a long time because there was a dip at the bottom of the curve that I thought did not exist, but Willi Ruch, the premier humor physiologist in the world, was giving a literature review about 15 years ago and put up a slide showing the measured pressure response in the esophagus that some Italian researchers had measure in the 1960's of which I was totally unaware. It was virtually identical to my computer model, including the dip at the bottom of the laughter curve. When I showed him this, he asked if I had published it and that led to the paper, with a direct comparison of the empirical and calculated laughter curves included in the paper.

Since then, I have made models of the pre-frontal cortex reflecting the best evidence we have about how humor works and the switching between the two different scripts (incongruities) shows up. If we can just see that oscillation (about 5 Hz) in real-time, then we will have the first handle on an actual emotional process, at least in a material sense. Humor, itself, as I say, is a metaphysical action, which we do not understand at that level, yet.

So, Incongruity and Superiority are both different ways of describing what goes on in a humorous action. It all depends on what one wants to focus on.

The big stumbling block, for years, has been a real definition of incongruity. That, now, seems to be possible.

So, here is something for you to puzzle over. How do the following two statements differ:

Did you hear about the guy who fell into a vat of gum at work? The boss chewed him out...

Did you hear about the guy who fell into a vat of gum at work? The boss yelled at him...

The Chicken

P. S., some of this material has been presented at conferences, but not published, yet. Don't publish ahead of me :) I want to tighten up the papers a bit more.

Step2 said...

Dear Groucho MC,

Indeed, absurdist humor cannot be explained easily by this theory.

Absurdist humor hinges on the same type of violation of expectations much of humor already uses, without any pretense of being logical or plausible. Besides, benign violation theory covers the basics of tickling, which is a behavior found in other primate species, and explains its peculiarities better than any other theory. Related to that it also explains why the same joke can be funny for one person and offensive to another, how it has to subjectively fit within the overlap of benign and violation categories. The objection that benign violations only establish a sufficient rather than necessary condition is just as valid against incongruity theory.

I believe that the respiratory center in the brainstem is also tied in, so that when the signal is sent, it also modulates respiration and vocalization and this leads to laughter.

Speaking of humor killers…

In regard to its moral implications, humor has its evolutionary roots in tickling and play, it explores novelty in meaning and expectations, it functions socially as a temporary distraction and a way to think outside convention. In short, for those of a religious mindset, it might be a version of Calvinball.

Jeremy Taylor said...

Glenn,

Indeed, but the point is that humour has no significant place in the Christian dogmatic and imaginal world. Christianity shares this trait with several of the great religions of the world. But Taoism and some mythologies and folklore do hold a noteworthy place for humour. That is not to say humour doesn't feature in the former traditions at all, but simply it is not accorded a noteworthy place.

There is a famous Chinese story about the three great sages of Chinese religion, Confucius, Lao Tzu, and Gautama Buddha tasting a pot of vinegar, representing life. For Confucius life is sour, because man's laws do not accord with those of nature; for the Buddha, life is bitter, because of illusion and attachment; for Lao Tzu, though, life is sweet as the universe begins and has its being in the Tao.

I'm not criticising, or praising, any particular religious tradition. I think, as the allegory alluded to probably is meant to suggest, that Confucius, Buddha, and Lao Tzu all express perspectives here that represent something of the truth, in various ways. I think it is interesting background for the topic of the metaphysical, mythological, and spiritual place of humour and whimsy and light-heartedness.

Jeremy Taylor said...

An interesting related topic is the holy fool, whom Shakespeare made such good use of.

Glenn said...

Scott,

I must say, I do like the word zymurgy and (particularly in view of its meaning) I don't think it will do you any harm to have bartenders associate it with you.

Alas, ere they could associate the word with me, they'd first have to have an association of their own with me.

Shall I put in a good number for you too? I was thinking of e.

I'll take the positive side of that, and respond by saying that it's good to know I (somehow) move your mind to consider something transcendental. ;)

Glenn said...

The Masked Chicken,

So, here is something for you to puzzle over. How do the following two statements differ:

Did you hear about the guy who fell into a vat of gum at work? The boss chewed him out...

Did you hear about the guy who fell into a vat of gum at work? The boss yelled at him...


The first statement is funny, whereas the second statement is not funny.

The meaning of the second statement is literal-based, and that's it--no secondary meaning arises, is fashioned or comes into play to compete with it.

The first statement, on the other hand, has two meanings, a primary meaning and a secondary meaning.

The primary meaning is idiom-based, and the secondary meaning arises, not concomitantly but subsequently, to compete with it. (It might seem that the secondary meaning arises concomitantly, but, and at the risk of being wrong, I would bet it really doesn’t (not if the time slices are made small enough, that is).) This secondary meaning is literal-based, and forms as a consequence of 'gum', already passed by and left behind, being retrieved and associated with 'chew'.

(Ironically, the arising of a secondary meaning drags one’s attention down from the ‘higher’ non-literal meaning, to the ‘lower’ literal meaning.)

I would guess that, in sequence:

a) there is a kind of rapid flip-flopping between the two meanings, the idiom-based primary meaning and literal-based secondary meaning;

b) some unspecified something is ratcheted up -- some metaphorical ‘charge’ (or, given your earlier remarks, possibly an actual charge of some kind) -- as a result of the rapid flip-flopping between the two meanings; until,

c) some threshold is reached; whereupon,

d) boom!--it's 'funny'.

And whether when d) there is simply a recognition that “hey that’s funny”, or an actual burst of laughter, of whatever strength, would depend on other factors.

Glenn said...

Jeremy,

Although I was somewhat defensive earlier, I do see your point.

Also, I like the Chinese story you related. I have read a little, a very little, about the "undifferentiated aesthetic continuum", and it would seem that that story catches Confucius and Gautama Buddha in the act of differentiating.

The Masked Chicken said...

Dear Step2,

Thanks for your reply.

Benign violation, as I keep pointing out, is already in the literature. It is contained in Provine's book in embryonic form, but we already knew about this aspect of humor years ago. This is another researcher putting a new name to an old concept.

To respond to some points you raise:

"Absurdist humor hinges on the same type of violation of expectations much of humor already uses, without any pretense of being logical or plausible."

Yes, it uses violation of expectation, but it is a mata-trigger as it does not exist on the object level of discourse, but refers to the discourse, itself. We already know this.

"Besides, benign violation theory covers the basics of tickling, which is a behavior found in other primate species, and explains its peculiarities better than any other theory."

The neurobiology of tickling has been known since the late 1990's. See, for example,

http://www.nature.com/neuro/journal/v1/n7/abs/nn1198_635.html

I, myself, developed a model of ape laughter that matches the empirical measurements, but the article is behind a paywall.

"Related to that it also explains why the same joke can be funny for one person and offensive to another, how it has to subjectively fit within the overlap of benign and violation categories."

This is already understood under human variations. It is a statistical phenomenon. It forms the Bayesian a priori distribution function that can be used to calculate humor responses.

"The objection that benign violations only establish a sufficient rather than necessary condition is just as valid against incongruity theory."

Benign violation is a form of incongruity: an action is both benign and violation, hence incongruous. Again, the psychologist should have presented his work at conferences in order to get feedback. There are examples of benign violation that are not funny. An example would be waking up feeling like a spider were crawling over you, only to find out that someone is dragging a plastic spider over your arm. Most people would be angry in this situation, not a used.

It is possible to make incongruity theory, benign violation theory, superiority theory, etc. into necessary conditions for humor, but we have lacked clear definitions of incongruity up until now to determine why some incongruities are funny and why dome are not. I claim, or at least my paper will, that we can know this. It is a problem of a statistical corridor where the incongruity is not too much, but not too little. It varies from person to person in a sliding fashion and is specific for each given word. We don't have a dictionary of how probable one attribute is to be found with another. That could take a century to map out, but once done, one can identify the trigger range for incongruity, benign violation, etc.

The Masked Chicken said...

Sorry for the spelling error. IPad error correction went bezerk and I couldn't re-enter the editor to change things, even though I had not hit publish. Maybe sub iPad thing?

Should be:

Most people would be angry in this situation, not amused.

The Chicken

The Masked Chicken said...

Finally, there is a lot of free reference material available, on-line, should anyone wish to delve into humor in more detail

The Chicken

Tony said...

In the sense you appear to mean, I agree. But I'm talking about the countdown's starting to run; that is, this or that day's being the one from which each islander starts to count...

I don't think you mean to say that this day is just the day the puzzle's condition's take effect and that teh Guru's statement adds nothing; I'm taking you to mean that the possibility of a countdown is implied by the setup even if something else is required to set it running. If I'm misreading you, let me know.


Scott: I was the "Anon" who posed that the countdown is included in the scenario from the first day the scenario exists.

And no, I am NOT saying merely that the scenario sets up the possibility of a countdown. I am saying that the scenario having a beginning in actuality implies an automatic starting point of the countdown and the Guru's statement is unnecessary. You either have to intentionally structure the scenario to have always existed in the past (with all the problems that has, re St. Thomas' First Way), or it has a beginning, and if it has a beginning in itself there is no need for anyone to announce a beginning. I won't countenance a theory of the scenario having always existed in the past without someone explaining how that could be in itself and avoiding all the problems that entails. And proposing that "well that's what the puzzle assumes" requires claiming that such assumptions are not of themselves contradictory or impossible in order for the puzzle to be sound.

Scott said...

@Tony:

"And no, I am NOT saying merely that the scenario sets up the possibility of a countdown. I am saying that the scenario having a beginning in actuality implies an automatic starting point of the countdown and the Guru's statement is unnecessary."

Thanks for the clarification.

Well, I've already pointed out that the countdown whenn* anyone knows that someone (in fact everyone) else is doing so, so unless you disagree with that, you must have some reason why you think the initial setup informs everyone of this. I don't know what that reason is, but I do think it will have to involve "common knowledge" in the technical sense.

The islanders do already have some "common knowledge" in the setup itself, so I'm certainly open to the possibility that such an argument will work. Perhaps your suggestion is that The conditions of the puzzle have taken effect today constitutes such "common knowledge." If so, I'd want to know how the "common knowledge" that Some islander is starting the countdown follows from it, but I wouldn't rule out in advance that it could.

But I've already explained (and you've agreed) that There's at least one blue-eyed islander doesn't qualify as "common knowledge," so if something along those lines is to start the countdown, then there must be something wrong with my brief and enthymematic demonstration that some kind of "common knowledge" is required at all.

I'll also be interested to hear what Mr. Green thinks when he's done reconsidering the question himself.

----

* By analogy with "iff." ;-)

** For example, it's easy to show that All the islanders are perfect logicians is "common knowledge," and the puzzle seems to assume (and, I think, should have either stated or given us enough information to infer) that The Guru speaks truthfully is too.

Scott said...

"…that the countdown begins whenn…"

Scott said...

@Glenn:

"[I]t's good to know I (somehow) move your mind to consider something transcendental. ;)"

Oh, there's clearly something transcendental at the very heart of Glenn.

Something similar would be true if your name were Glπnn.

Paul Amrhein said...

@Scott, Glenn and other puzzlers.

It's strikes me that you might enjoy almost anything by Ray Smullyan. His most famous book is probably *Forever Undecided* on Godel's theorem. He discusses logic and higher math in terms of Knights and Knaves and things like that.

Glenn said...

Scott,

Something similar would be true if your name were Glπnn.

Mmmm, that apple pie a la mode was good.

Little known fact: any pion worthy of the name would commit seppuku before daring to refer to itself as meson.

Glenn said...

Paul,

@Scott, Glenn and other puzzlers.

It's strikes me that you might enjoy almost anything by Ray Smullyan.


That is a good recommendation, and, yes, I have had the pleasure. Although it has been a while, quite a while.

(But in following memory traces back into the past, I arrive at the conclusion that I once had been particularly fond of the retrograde analyses in his The Chess Mysteries of Sherlock Holmes.)

Glenn said...

(Oh darn. 'twould have been better had it been: An oft overlooked fact...)

Glenn said...

Back to those blue-eyed islanders...

It is given in the problem that they live on an island, and it has been rightly pointed out here that it cannot be that they have always been there.

Were they born there? All on the same day? Were they brought there? All at once? Or on different days? Or were some born there, and others brought there? Etc., etc.

Let it be supposed that they all arrived at exactly 11:45:01pm on the same day.

Do they all begin counting on the day of their arrival? Why? Why start counting on a day that has less than 15 minutes remaining? Why not start counting on the first full day after their arrival?

Since it is given in the problem that, aside from their being able to see the eyes of others there is no communication amongst them, how do they manage to start (i.e., what precipitates their starting) to count on that partial day of their arrival (rather than on the first full day afterwards), or on the first full day after their arrival (rather than on the partial day before)?

Glenn said...

Jeremy,

Hmm; interesting.

I had commented that "...it would seem that that story catches Confucius and Gautama Buddha in the act of differentiating."

I hadn't previously encountered the story, but have just now poked around to see if I could find out something about it.

It turns out that the story is, at least allegedly, based on a painting--a Taoist painting.

If true, then little wonder now that Lao Tzu is kept from the 'bad light' in which Confucius and Gautama Buddha are portrayed in the story.

There may be some irony there, however.

And whether there is, depends on the extent to which the Taoist painter had willfully differentiated, in the aforementioned manner, Lao Tzu from Confucius and Gautama Buddha.

Scott said...

@Paul Amrhein:

Yep, longtime Smullyan fan here—since the 1970s, in fact. I heartily second Glenn's approval of your excellent recommendation, and of course I therefore third the recommendation itself.

I suppose anyone who approves of my approval of Glenn's approval of your recommendation would therefore be willing to fourth it.

Scott said...

(However, it wouldn't be valid to infer, from the facts that Glenn approves and I approve of his approval, that therefore this guy approves.)

Scott said...

@Glenn:

"Since it is given in the problem that, aside from their being able to see the eyes of others there is no communication amongst them, how do they manage to start (i.e., what precipitates their starting) to count on that partial day of their arrival (rather than on the first full day afterwards), or on the first full day after their arrival (rather than on the partial day before)?"

Wouldn't the time remaining until the first ferry departs constitute the first "day" on the island for logical purposes? If one leaves in fifteen minutes, the first "day" is fifteen minutes long; if one doesn't, then nobody even can leave until the following midnight. It doesn't even matter whether they know ahead of time that it's coming; if they don't, they can always just wait fifteen minutes and see.

(Mind you, I still don't think they have any reason to start their Countdown to Blue-Eyed Exodus on that day! But the time until the first ferry departure would count as the first "day" that the conditions of the puzzle obtain.)

Glenn said...

Scott,

If the day the ferry departs constitutes the first day, then, since the ferry arrives at midnight, and midnight marks the beginning of a new day, it would be the first full day after their arrival at the island which marks the first day.

Possible. And logical.

But why would it be illogical to count as the first day the day on which they had all arrived?

And, hmm, even if it is the first full day after their arrival that the counting begins, then wouldn’t it be true that they all actually leave on the 101stt day?

(It would take 100 days for each to figure with certainty what his eye-color is, true. But the ferry won’t arrive until midnight. And midnight marks the beginning of a new day. So, it really would be the 101st day on which they’d all leave.)

Glenn said...

(Oops. It takes only 99 days for everyone to figure his own eye-color. So, yeah, it's on the 100th day that they leave.)

Scott said...

@Glenn:

For logical purposes, once the conditions of the puzzle take effect, each "day" is just the period until the next ferry departs. If those conditions take effect upon their arrival and the first ferry departs fifteen minutes thereafter, then the first logical "day" is just the fifteen-minute period from 11:45 to 12:00.

I don't think the logical structure of the puzzle is altered by having the ferry arrive at random intervals, as long as each arrival is recursively common knowledge* among the islanders. (In fact even that condition may be too strong; at first look, it seems to suffice that every islander knows every islander knows when a ferry arrives. But I haven't thought that through fully.)

I don't suppose it makes much difference whether we regard the departure of a ferry as the end of one such interval or the beginning of the next one; the 100 blue-eyed islanders leave at the end of the 100th interval or, equivalently, at the beginning of the 101st. With that in mind, I think it's fairly clear in the original version of the puzzle that the exodus takes place at the end of the 100th day.

----

* I think I'll provisionally start using that term just to avoid any possible further confusion with the ordinary meaning of "common knowledge."

Scott said...

Oops, you posted while I was composing my reply and I missed it. Yes, now that you mention it, I think it may come out as the 100th day whether we take midnight as the end of one day or the beginning of another.

Glenn said...

(However, it wouldn't be valid to infer, from the facts that Glenn approves and I approve of his approval, that therefore this guy approves.)

Scott is right. Additionally, not only would the inference be invalid, it would be incorrect. For with Clarice Starling on your side, who needs Raymond Smullyan? -- Mr. Glenn

Scott said...

Also, even though my first name is John (Scott is my middle name), it wouldn't be valid to infer that this man approves.

For that matter, even if Jesus Himself approved and I approved (as of course I would) of His approval, it wouldn't be valid to infer that this man would have approved.

Glenn said...

Also, even though my first name is John (Scott is my middle name), it wouldn't be valid to infer that this man approves.

"On the motion of whether the inference is valid, what say ye?"

"Nay."

For that matter, even if Jesus Himself approved and I approved (as of course I would) of His approval, it wouldn't be valid to infer that this man would have approved.

I think you may have stumbled on this one:

"On the motion of whether the inference is valid, what say ye?"

"Raymond that."

Glenn said...

(Since it, or something like it, is bound to happen, I may as well be the one to do it:

(@Glenn:

(I think you may have stumbled on this one:

(I think you may have stumbled in thinking I may have stumbled. There were two motions, not one. You quoted the first motion correctly, but the second motion incorrectly, i.e., the first motion had to do with whether the inference was valid, but the second motion had to do with whether the inference was invalid.)

Step2 said...

Jeremy,
But Taoism and some mythologies and folklore do hold a noteworthy place for humour.

Most humor is irreverent and thus creates a dissonance when combined with reverence or sacramental awe. However there is a minor story in the Talmud where the spirit of Elijah declares a heavenly reward for jesters.

Since you mentioned the Bard of Avon (half price if you place your order in sonnet form), I'll submit a relevant quote - "Jesters do oft prove prophets."

Step2 said...

Dear Masked Chicken,

I must thank you for your reply that finally solves the ancient mystery, why did the chicken cross the road? To find a statistical corridor that will take a century to map out. I had always imagined it was because the road egged him on but realize now how scrambled up that was, possibly even cheesy.

Yes, it uses violation of expectation, but it is a mata-trigger as it does not exist on the object level of discourse, but refers to the discourse, itself.

I don't understand what its meta-character is supposed to subtract from its violation status. It seems to me like your process (joke diagrams, lattice models, Bayesian functions) is preventing you from treating humor holistically as a dynamic and integrated behavior. This type of analysis incurs the cost of ignoring invention and learning which can quickly change the distribution function, even within the joke itself as part of the setup.

Most people would be angry in this situation, not amused.

I wonder why people feel angry when they were involuntarily experimented on. After the initial shock and relief wears off it probably would make for an amusing story in retrospect.

Scott said...

@Step2:

"…the Bard of Avon (half price if you place your order in sonnet form)…"

To Amazon

I'd like to place an order for a book
Of plays by William Shakespeare for the stage.
It seems to be in stock, for when I look
It says so on the volume's product page.

I hope that you will have it quickly shipped,
For, as a member of your program Prime,
I'm bound to feel hard done by, even gypped,
If you don't get it here in two days' time.

Please use the credit card I indicate
(The one with these four final digits on it)
And sure, a half-price discount would be great!
That's why I've phrased my order as a sonnet.

Now that it's done, here comes the hardest part:
How do I put this in my shopping cart?

Glenn said...

Step2,

It seems to me like your process (joke diagrams, lattice models, Bayesian functions) is preventing you from treating humor holistically as a dynamic and integrated behavior.

a) WIP.

Work-in-progress.

b) "Before there's an integrated whole, individual things need to isolated and worked on (mapped out, developed, shaped up, refined, etc.). Then they all get bundled into a neat package.

"I want the bigger picture. I have been trying to tie everything together - the neural processing, the logic, the physiological responses. Due to recent work in a number of fields it is now possible, more or less to develop a truly comprehensive theory of humor, at least insofar as the material aspects are concerned, so the ten or so papers on the different topics that I have been working on and presenting, hopefully, will get published, soon...

"It's not like one can just make up a theory and call it science. The science is incredibly complex and I have had to solve a lot of technical problems along the way...

"Beyond the nuts-and-bolts, there is also the metaphysics and even moral implications of humor to be done..."

c) WIP.

Work-in-progress.

- - - - -

Or would you propose that a mandatory 7-year prison term be imposed on any adult not yet 40 by the time he reaches his 33th birthday?

I hope not.

;)

Glenn said...

33th?...

Scott said...

Well, it's nth for n=33…

Glenn said...

You're a face-saver; thank you.

(I was going to use n, but nrd didn't look right.)

Glenn said...

Step2,

My earlier comment wasn't to say that there isn't a sense in which your point is valid.

However, Tzvi Freeman asks a good question, and, in responding to the question, asks another good question--to which he adds the following parenthetical comment: "I'm thinking of the parable of the blind men and the elephant, all with a complete and true understanding of their part of the elephant".

I had thought of that parable when The Masked Chicken wrote, "My idea is that it is one single thing that triggers all of these responses[.]"

As there are several theories of "elephant" (Pillar Theory, Rope Theory, Thick Branch Theory, Hand Fan Theory, etc.), so there are several theories of "humor" (Superiority Theory, Incongruity Theory, Arousal Theory, Discharge Theory, Pleasure Theory, etc.)

After seeing where the 'jesters' link led to, I thought, "Hmm, I wonder if there's any mention of the blind men and the elephant..." A brief search on the site returned favorable results.

Although it isn't required that anyone be interested in what the "elephant" might be really like, some people are. I myself am, and applaud the work of similarly interested others who, unlike me, have what it takes to make strides in finding out.

Scott said...

Q: When is it time to go to the dentist?

A: When it's nth-hurty for n=2.

Mr. Green said...

Jeremy Taylor: Christianity seems to completely skirt over humour and laughter

It’s “over”, but not so much skirting as transcending: there is humour in the Bible and lives of the saints, but of course the focus is on joy. “Humour” is only joy writ small, so I would not especially expect it to figure prominently in dogmatic pronouncements. Since Christianity lets the world be what it is — good (though with evil in it) — vinegar to be sour, wine to be sweet — humour occupies its natural place, and is appropriately studied on a natural level.

Mr. Green said...

Tony: You either have to intentionally structure the scenario to have always existed in the past (with all the problems that has, re St. Thomas' First Way)

Since Thomas thought it impossible to prove the world had a beginning, I’m not sure why that’s necessarily a problem. Humans “instantly” seeing all logical conclusions is surely more problematic; except of course the idea is to describe an abstract mathematical structure, so it doesn’t matter so much whether the additional details make sense, as they’re there only to aid our imaginations (or perhaps to confuse them by tempting us to apply common sense to a logical abstraction that could never be instantiated that way…).


Glenn: But why would it be illogical to count as the first day the day on which they had all arrived?

I concur with Scott; all that matters is that we have discrete periods so that it’s possible to identify unambiguously whether someone has left or not. And speaking of misdirections, the puzzle is slightly misleading when it says there is no communication between the islanders: leaving — or in this case, the curious incident of the logician in the nighttime — is a form of communication. Connoting that it isn’t perhaps is a way to draw people’s attention away from considering its possible role in the solution. It would work just as well, if more obviously, to have anyone who figures out his eye-colour smack his forehead and exclaim, "Of course, my eyes are blue!"


Scott: What I should have said is that it doesn't differ in principle from the case n=3, the first case in which every islander does know that.

At n=3, everyone can see someone with blue eyes, but doesn’t know that everyone can see blue. At n=5+, everyone knows that everyone can see at least 2, so that’s a difference. The question is whether this difference is relevant. It certainly seems to be, because once we know nobody can be fooled into not counting by not seeing anyone else who is blue, then it’s possible for everyone to start counting, with the inevitable result. For small n, we know that to rule out someone’s being fooled in this way, we need the guru’s statement. Though it is clearly necessary in these cases, and clearly sufficient to kick off the countdown in all other cases, that does not show it’s necessary in all cases. That requires its own argument, such as another induction: if n people can’t start the countdown, then n+1 cannot; because person n+1 sees only n other blues, and thus has to consider that we have the n situation, which by assumption does not work. Ergo, etc.

So how can we square that with my (and Tony’s) claim I just finished making, that for large enough n, it is possible to count down? I think what it comes down to is that actually the scenario is subtly different: you can figure it out for 5 and up, but that assumes that the islanders have some motivation to start counting whenever it becomes possible. And the original puzzle is supposed to follow from immediate logical possibilities, not from anyone’s trying to figure it out. That is, it is not that the islanders were all trying to figure out their eye colours in any way possible, but simply couldn’t until the guru came along; rather, with that information, the logical state of affairs changes to a different one (because of the recursively common knowledge), such that it is impossible to consistently refrain from counting down. Simply having five our more people does nothing to change the logical configuration, so something else has to happen that is sufficient to motivate the islanders to actually begin figuring it out. For example, the guru might say, “Free paper hats for everyone who figures out his eye-colour”, and that would be enough. At least, I think that is the correct distinction… emphasising again that we are talking not about real people but about an abstract pattern.

Mr. Green said...

Maolsheachlann: The problem with all meta-humour and anti-humour is that it's ONE joke, repeated ad nauseum, whereas supposedly more formulaic and traditional humour is infinitely varied.

I suppose it depends upon the details. A twist on a joke is per se no worse than twist on anything else, as far as whether it is humorous (or not). The sort of absurdist or non-sequitur “joke” where the audience is set up to get something (funny) and gets nothing instead cannot really be repeated much and still work, though. I do think that humour requires some substance — well, at least a form — the more cleverness or wit it involves, the better it will be (again suggesting the connection between comedy and puzzles). Even things like “nonsense verse” are not (when done well) truly nonsensical.

Anonymist: There's probably an analogy with how "difficult" or avant-garde music has been a popular hit when presented to audiences as soundtrack

And I think this is a related point: a piece of music may be good as a soundtrack, but bad as a stand-alone composition because it does not have enough substance to stand on its own. As one part of a film, it needs only to support, and be supported by, all the other components; like any disembodied part, it just won’t work in isolation.


Glenn: 33th?…

Hey! What’th wrong with thaying “33th”?!?

Glenn said...

Scott,

Q: When is it time to go to the dentist?

A: When it's
nth-hurty for n=2.

Histories of appointments past
Removal that wasn't fast
Rummaging in a canal
Eruption of a howl

Son ov a n! That hurty-gurty, man!


Mr. Green,

Glenn: 33th?…

Hey! What’th wrong with thaying “33th”?!?


Now that you mention it, nording. Nording at all.


Yours truly (not to mention inaccurately),

I had thought of that parable when The Masked Chicken wrote,...

As much as it pains me to do so, I must acknowledge that I am not psychic. 'twas when I read what The Masked Chicken wrote that the thought had occurred to me.

The Masked Chicken said...

Dear Step2,

You wrote:

"I must thank you for your reply that finally solves the ancient mystery, why did the chicken cross the road? To find a statistical corridor that will take a century to map out. I had always imagined it was because the road egged him on but realize now how scrambled up that was, possibly even cheesy."

Thank, you, thank, you. One has to look far and wide for such carefully prepared sarcasm.

"I don't understand what its meta-character is supposed to subtract from its violation status. It seems to me like your process (joke diagrams, lattice models, Bayesian functions) is preventing you from treating humor holistically as a dynamic and integrated behavior. This type of analysis incurs the cost of ignoring invention and learning which can quickly change the distribution function, even within the joke itself as part of the setup."

I don't think you understand. The Bayesian approach allows, specifically, for the real-time modification of the a priori distribution as the joke progresses. Also, the neural linkages I mentioned, above, is the integrated behavior.

"I wonder why people feel angry when they were involuntarily experimented on. After the initial shock and relief wears off it probably would make for an amusing story in retrospect."

Maybe, maybe not. The anger comes from a defensive position about the original spider.

As for benign violation, compare that to Kant's classic definition of humor and laughter, from the Critique of Judgment (if memory serves), to see the parallel:

"Laughter is an action arising from a strained expectation (violation) suddenly being reduced to a nothing (benign)."

As I have said, nothing new.

The Chicken

Scott said...

@Mr. Green:

"At n=3, everyone can see someone with blue eyes, but doesn’t know that everyone can see blue."

Sure they do. Every islander sees at least two blue-eyed islanders and knows that they can see each other…

"At n=5+, everyone knows that everyone can see at least 2, so that’s a difference."

…and for n≥5, everyone knows that everyone can see at least three (in general, at least n-2). So I still don't see what's supposed to happen for n=5 that doesn't happen for n=3.

(Your earlier remarks seemed to imply that two additional blue-eyed islanders are needed so that they can be the two islanders that everybody knows will start the countdown. But I don't see (a) why these two alleged countdown-starters can't be two of the three blue-eyed islanders, nor (b) why they have to be blue-eyed at all.)

"So how can we square that with my (and Tony’s) claim…that for large enough n, it is possible to count down?"

I don't think we can.

"[Y]ou can figure it out for 5 and up, but that assumes that the islanders have some motivation to start counting whenever it becomes possible."

But it doesn't. They're perfect logicians; whenever the countdown becomes possible, they'll all know it (and grasp all of the logical implications) no matter what their motivations are.

For that matter, the puzzle doesn't even assume anyone wants to leave the island; maybe they'd all rather stay. In that case why would they ever have "motivation" to start counting?

Scott said...

@Mr. Green:

"For example, the guru might say, 'Free paper hats for everyone who figures out his eye-colour', and that would be enough."

But it's already recursively common knowledge among the islanders that Everyone who figures out his eye color leaves the island. If that's not sufficient to start a countdown, why is Everyone who figures out his eye color leaves the island and gets a free paper hat?

Scott said...

@Mr. Green:

"[O]nce we know nobody can be fooled into not counting by not seeing anyone else who is blue, then it’s possible for everyone to start counting, with the inevitable result.…Though [the Guru's statement] is clearly necessary [for small n], and clearly sufficient to kick off the countdown in all other cases, that does not show it’s necessary in all cases. That requires its own argument[.]"

As far as I can see, your inductive argument is valid, and n=1 obviously works as a base case. What do you think is wrong with it?*

I've given a different argument: A necessary and sufficient condition for the starting of the countdown (for any eye color and any corresponding value of n) is that it be recursively common knowledge (henceforth RCK) that some islander is starting it—again, because islander A won't start without knowing that B will start, and knows B won't start unless B knows C will start, and knows B knows C won't start unless C knows D will start, etc., and it's completely arbitrary which islanders we call A, B, C, and so forth. (That's easily shown to be equivalent to its being recursively common knowledge that all the islanders are starting. If it's RCK that all of them are starting, then of course it's RCK that at least one is doing so; and of course if it's RCK that any one islander is starting, it's RCK that he believes all the others are doing so and that, being a perfect logician, he's correct in this belief.) This is why the very heart of the puzzle is figuring out what everybody thinks everybody else thinks everybody else thinks everybody else thinks…

It should be pretty clear that for any value of n, some other condition must be met in order to generate this RCK; in order for these perfect logicians to start popping out of the recursion stack, they need a limit on the depth of the recursion. It should also be pretty clear (though I haven't actually given an argument for this lemma) that this condition must involve new information, and that this information must itself be RCK. And I've given a very detailed argument showing that Everyone knows everyone knows there's at least one blue-eyed islander doesn't qualify.

----

* Or do you think it's correct but still consistent with your claim that the countdown can get started for sufficiently large n?

Scott said...

And just to recap and tie a couple of things together for the sake of clarity:

For n=3, everybody knows everybody sees at least one blue-eyed islander. In particular, if A, B, and C are the blue-eyed islanders (in arbitrary order), then A knows B sees at least one blue-eyed islander (C). What A doesn't know is whether B has enough information to start the countdown, because A doesn't know whether B knows that C sees a blue-eyed islander. (A doesn't know his own eye color, so he has to assume it's possible that B and C are the only two blue-eyed islanders.) So A doesn't start the countdown because he can't be sure B will do so: from A's point of view, B may not have enough information to be sure C will start it.

This argument can be extended to any positive integer n (and in particular nothing significant changes for n=5); we just have to keep adding knows-es. For n=5, for example, A can't be sure B will start the countdown because he can't tell whether B knows C knows D knows E knows F will start it.

Note well that A is not entertaining the proposition F doesn't see any blue-eyed islanders; he knows better than that. Nor is he entertaining the proposition E doesn't know F sees any blue-eyed islanders; he knows better than that too. For that matter—and this is the really important bit, so pay close attention—he knows that C knows D knows E knows F sees a blue-eyed islander. But he doesn't know whether B has that same knowledge, because for all A knows, B may see one fewer blue-eyed islander than A does. And that's why A can't be sure B will start the countdown: because A can't be sure that B knows F will start it.

There's no short-circuiting this part. It's perfectly obvious (I hope) that no one will start counting down without knowing that everyone else is doing so at the same time, without knowing everyone else knows everyone else is doing so, and so forth. If so, it should also be obvious that no one will start counting down unless the same thing is true of There's at least one blue-eyed islander, because that proposition's being recursively common knowledge (RCK) is logically equivalent to its being RCK that Some islander is (equivalently, all islanders are) starting the countdown.

Scott said...

(Oops, I wrote "For n=5" and then considered the case of six blue-eyed islanders! Ah, well; hindsight is always better than threesight.)

Tony said...

Note well that A is not entertaining the proposition F doesn't see any blue-eyed islanders; he knows better than that.

The Guru doesn't tell A or F anything they didn't know.

In the case of N=1, the guru's statement (can we shorten this to GS?) tells A the blue-eyed person (shorten this to BEP?) something he didn't know. Thus the first step of the inductive proof relies on the fact that GS (right on the surface) provides something that A the BEP didn't know, neither by direct observation nor by deduction from observable facts.

In the case of N=2, A and B, A saw a BEP and B saw one, but A did not know if B saw one or not. Because A did not know if B saw a BEP, A didn't know if B knew that N>0. GS added to A's knowledge of B's knowledge: PRECISELY BECAUSE the highest possible number of BEPs was (so far as A or B could tell directly) 2 but the lowest number possible was 1, it was possible for each to think that the other might not see a BEP (or otherwise know there was one), i.e. each could see a possibility of the other unable to be sure there is at least one BEP. In that way GS added to both A and B's knowledge of each other's information: each now knew that the other knew there was a BEP. Or, to put it another way, through GS, each BEP knew that everyone knew there was a BEP.

However, in these 2 cases GS was about N or N-1 blue-eyed persons. And, so for case N=1 and for N=2, GS supplies for information regarding either N or N-1 that CANNOT be achieved by direct observation or simple (single-step) deduction, with a statement REFERRING TO either N or N-1. It is precisely this facet that makes GS new information.

In both cases a BEP sees N-1 BEPs. As a result he knows that the number of BEPs is at most N, one more than he sees. When he can establish by logic that the minimum has changed to N, he knows he is a BEP and that gets him off the island. For N=1, the GS itself changes his knowledge that the number of BEPs is one (up from a possible 0). For N=2, it changes every BEPs knowledge from the 1 BEP I see might not know there is a BEP (might think N=0, to a new datum: nobody else can still think that N=0.

Knowledge of _when_ to get off the island relates to knowing beforehand that there is at a minimum N-1 BEPs (which he can see) and using logic and passage of time to establish a new datum, that there are N, which N must include you because you were the one you didn't know about when you knew that there were at least N-1 but you were unsure if there were N.

Any passage of N days must definitively establish that there are more than N-1 BEPs. But any BEP can see N-1 of them. Therefore any passage of N days tells them there are exactly N BEPs and that he is a BEP.

The guru always speaks of 1 BEP but it is knowledge that there is at least N-1 that is critical. So the cases of N=1 and N=2 are special cases where GS provides new information, the number he speaks of falls in N or N-1. And the newness of the information creates a timing point but the timing point is not critical, the knowledge is.

When N>2, the GS doesn't provide new information and does nothing for any timing.

Tony said...

In any situation where N > 2, GS is not about N or N-1 blue-eyed persons. And so for these cases, GS DOES NOT increase anyone's information about anything: even before he says anything, everyone knows THAT everyone can see a blue-eyed person. The reason the induction result cannot be achieved for the 100th night without the guru is that the induction process REGARDS two cases where the guru's statement is needed to provide information that cannot be achieved without it, N=1 and N=2, not because the guru's statement provides new information to anyone when N=100, for it doesn't.

When N>2, the logicians choosing to "start" when the guru makes his statement is completely arbitrary, they could just as well have said "this tells me (and everyone else) nothing new so I cannot not act on it." For N=3, every BEP already knew that every other BEP knew there was at least 1 BEP. Any passage of 3 days would tell all 3 BEPs that N=3 (since they each can see 2 BEPs and they all know that possible -for anyone to believe the minimum for N - is 1), they each know that the MAXIMUM number anyone can think N is, is 3: for A regarding C's knowledge, he can be unsure of C's knowledge of A and of C, so the largest gap between "least possible" and "greatest possible" is always 2 - and everyone knows that. So when N>2 the GS is insignificant for their knowledge. Any passage of N-1 days tells them what N is. The GS is only needed for N=1 and 2, but it is needed for them, and the induction proof needs N=1 and N=2.

Scott said...

@Tony:

I'll try to be brief here so as not to repeat myself, but I'm afraid a little repetition is inevitable.

"The Guru doesn't tell A or F anything they didn't know.…When N>2, the GS doesn't provide new information and does nothing for any timing."

You're right that GS in and of itself doesn't tell anyone anything he doesn't already know. If the Guru had made GS to any of the BEPs (or for that matter any islander) privately, he would have thought, "Yeah, I already know that. So what?"

But the Guru's making the statement publicly does give them new information about the state of everyone else's knowledge (namely that everyone on the island has sufficient information to start the countdown, because Everybody heard GS is recursively common knowledge), and I've been at some pains to explain why (including why the islanders didn't have that information before the Guru spoke).

For example, for n=6, with the BEPs arbitrarily designated as A, B, C, D, E, and F, I've shown that A does know the proposition C knows D knows E knows F sees a blue-eyed islander, but doesn't know that B knows it too. I take it it's obvious that A does know this after hearing GS publicly. Is it your claim that this isn't new information? Or are you acknowledging that it's new but saying it isn't relevant to the start of the countdown?

"Any passage of N days must definitively establish that there are more than N-1 BEPs."

In other words, the countdown should have started to run on the day the conditions of the puzzle began to obtain. I know you and Mr. Green think this, but I still haven't seen a successful argument for it.

"For N=3, every BEP already knew that every other BEP knew there was at least 1 BEP."

Of course. But I've already shown in detail why this knowledge isn't sufficient to start a countdown.

Daniel said...

Hmm this touches on something which came to mind when I first discovered the Argument from Queerness. Is the Principle of Economy normative? If so it's hard to see how Mackie's argument can fail to undercut itself (likewise it would seem impossible to state the principle without reference to mathematical objects).

All this talk of Wittgenstein is amusing since it neglects to mention his later views towards the inadequacy of scientific as opposed to Ordinary Language. Not that even that matters particularly since Plain Language Philosophy is about as dead as Logical Positivism. If anything Analytical Thomists are too keen on Wittgenstein.

Daniel said...

Oh dear, wrong blog entry. My apologies.

Scott said...

@Daniel:

No worries. Your posting it here is meta-funny.

Mr. Green said...

Scott: Sure they do. Every islander sees at least two blue-eyed islanders and knows that they can see each other…

Sorry, there was supposed to be another “…knows that…” in that claim. Anyway, for 5 and up, it’s possible for everyone to know that everyone knows that everyone sees two (or more) blue eyes, so it’s “possible” for everyone to count in a way that isn’t possible for 4 or under. So what’s wrong with the induction I offered last time? Well, I can argue that now it’s “legitimate” for everyone to start counting (yeah, you know where this is going…), but the problem is that nobody knows whether he’s in the n case or the n+1 case. And obviously, that applies no matter how large an n we want to pick; so sure, everyone could count, but it’s never possible to know whether everyone else is counting with you. Which just is my previous induction — and, really, is more or less equivalent to the recursive argument you gave since the start, just looking through the other end of the telescope. (Hey, at least I got some exercise taking the scenic(?!) route to get here!)

Scott said...

@Mr. Green:

"[S]ure, everyone could count, but it’s never possible to know whether everyone else is counting with you."

Exactly, and therefore no one will ever actually start counting, because the count is logically useless unless everyone is doing it at the same time (and known to be doing so, etc.). But you're right, I think, that there's a fairly clear sense in which it's possible to start the count; it's like a supersaturated solution waiting for a grain of salt to be dropped in so that it can instantly crystallize. The Guru's statement provides it.

I like the scenic route, especially for this puzzle. Every time I have a chance to discuss it with someone, I end up understanding at least a little more about it.