This appeared on Daily Nous on 13 September 2022, just a couple days after the expression “the present King of England” changed(?) its meaning(?).
The “To φ or not to φ” comics feature on The Daily Nous is done by Tanya Kostocha , Assistant Professor of philosophy at Ashoka University. Russell’s theory of descriptions is long gone, but is still studied for the sake of understanding the variety of refutations and reformulations that succeeded it. Oh, and also for its well-remembered example, “The present King of France is bald” (uttered in 1905, when there was no current King of France).
Comics I Don't Understand 
This seems to be a version of the Sorites paradox, which is about vague terms like heaps and baldness … if one grain of wheat isn’t a heap, and one more doesn’t make a heap, and one more doesn’t and so on grain by grain until you find 1,000,000 grains aren’t a heap.
Similar with hairs on the head – is Charles bald or not? Where does baldness (or heapage) start? He has a lot of hair around his head but not much on top. In laypersons’ terms he is bald. The fact he is in fact now the present King of England is a bonus.
https://plato.stanford.edu/entries/sorites-paradox/
A different formulation of the paradox replaces the set of conditional premises with a universal generalization and proceeds by mathematical induction. Let ‘n’ be a variable ranging over the natural numbers and let ‘∀n(…n…)’ assert that every number n satisfies the condition …n…. Further, let us represent the claim ‘For any n, if αn is Φ then αn+1 is Φ’ as ‘∀n(Φαn→Φαn+1)
Mathematical Induction Sorites
Φα1
∀n(Φαn→Φαn+1)
∀n(Φαn)
For example, since a man with 1 hair on his head is bald, and since, for any number of hairs n, if a man with n hairs is bald then so is a man with n+1 hairs, every number n is such that a man with n hairs on his head is bald.
You lost me at “Russell”.
Powers, that would be Bertrand Russell, big name British philosopher of the Twentieth Century. He was important both in academic formal philosophy, and in what is now called public philosophy (opposition to the UK acquiring nuclear weapons, condemning the American war in Vietnam).
This cartoon is playing off an article where he asked if the statement “The present King of France is bald” would be counted True or False or Neither at a time when there was no King of France.
@Powers: My first LOL of the day. Thanks!
Is the Present King what they have in France instead of Santa Claus?
@Andréa, I guess that’s a Comment IDU?
Good one, @larK!
I have a few hairs growing on top. They began receding 60 years ago. But I’ve never admitted to myself that I’m bald. I’m balding.
@deety: Powers wrote, ‘You lost me at “Russell”‘, and I thought that was an LOL.
Because … ? Is there a current celebrity whose name begins with “Russell” and is in bad public regard?
The baldness paradox seems a bit far-fetched (I never liked Zeno’s paradoxes, either). I think I would have enjoyed this drawing a lot more if the caption had read “Ceci n’est pas un Roi“.
Thanks, @Dana K and @narmitaj!
I thought this was clever because of the timing. Just like “God save the Queen, oh sorry I mean King” and all the Silk baristas who were Q.C. becoming K.C. …. A couple weeks ago the sentence “The present King of England is bald” was problematic in much the way Russell’s version with “France” was. Then because of history marching on, and the promotion of a new King, that same sentence [though a different “utterance” since the time of utterance is different] becomes suddenly non-problematic and simply an empirical question.
But not without philosophical interest — as Narmitaj reminds us, settling what appears to be simply an empirical question may surprise us with problems. Some coming from Philosophy of Science — what are facts, what are measurements, when do we trust them — plus really old yet unsolved issues, such as a susceptibility to attack by a Heap argument (Sorites)!
While a person who is born, raised, and living in England may refer to Charles (III) as that person’s king, there exists no living person whose title is “King of England” as that position has not existed for hundreds of years. In 1905, there was no living person to refer to as the “King of France” in any truly monarch-ical way (but possibly as respect and deep declaration of loyalty).
The humor of this “comic” is simply the referral to the mentioned France quote and how the parts of the 2 statements differ in their true/false logic. After that, any discussion is intended to be philosophical; It is a delightful call to the members for discussion.
@Lark, that was a real day-brightener for me!
(My apologies for not getting up to find the dictionary today, I am often astonished anew to find there’s no spell-check active in the “Add a Comment” box” even though I’m in Chrome.)
@Deety: no, no, no . . . he and I were both already lost as to the meaning of the comic right when the name ‘Russell’ came up – neither of us realizing it was Bertrand Russell (although come to think on it, Russell Brand might fit your latter category, but that’s getting political, so never mind).
No spellcheck? Then . . . what/who has been checking my spelling in the comment box??
But Kilby, this is only secondarily about the baldness paradox. It’s primarily about the “failed referent” issue — and amusing because that issue “magically” evaporated, for this one particular instance of the very famous example.
BTW, for the curious, Russell’s answer was that “The present King of France is bald” is False. (Rather than True or No Truth Value.) This came from his analysis of the “definite description” as embodying an existence claim. This was stated very formally — these guys were basically inventing Symbolic Logic — but the informal analysis was that the sentence must be taken as “There is currently an entity which is a King of France, and there is no more than one such, and that [existent and unique] entity is bald.” Because the first clause in that conjunction is False, so is the conjunction. The “[existent and unique]” is an analysis of “the”.
This was unpalatable to many, including many who were basically his followers, and led to a rich literature. See https://plato.stanford.edu/entries/descriptions/ ]
@Kevin A, thanks for the correction on “King of England”. I was rather surprised to see what’s his name — okay, it’s William — referred to as Prince of Wales, already, as though it is automatic. I recall from The Crown that Charles at the time had a big investiture ceremony, and had to study his Welsh very hard to get thru it.
Oops, I completely missed narmitaj’s comment and I had looked back several times (without refreshing) to see if anyone had touched on philosophy. Could his post have been delayed? (because I’m worried that I might need a trip to the doctor’s : – ))
And even though I CTRL-X’d and refreshed the comments before posting, Mitch4 got in before me. (thank goodness because it alerted me to the narmitaj post.)
Kevin (and anyone else wondering), yes the comment from Narmitaj was — for reasons unknown — held in Pending, but then released when noticed. As you probably know, WP keeps the original timestamp of such a comment, and then uses that in ordering them on a page.
But don’t worry, there was nothing about your comment that looks out of place for not mentioning narmitaj’s.
I forget what Ctrl-X does?
BTW when you said “After that, any discussion is intended to be philosophical; It is a delightful call to the members for discussion.” it left me unsure if you meant members here or at Daily Nous — the latter since it is a news outlet for academic philosophy and their “membership” moreso than ours may be predisposed to enjoy philosophical chatter.
Thanks, Andrea.
@MITCH4 (AND ANYONE ELSE WHO MIGHT BE INTERESTED): Line of succession now that Queen E II has died:
PREVIOUS:
William was created Prince of Wales in King Charles’ first speech.
Andrea, thanks for that informative video!
@ Mitch – Okay, in addition to to baldness paradox, I find the reference issue tiresome, especially after wasting an hour yesterday on a moronic function in Excel that instead of the simple “zero or one” answer(s) that I needed, produced “one” or a “Value Out of Range” error, forcing me to add superfluous layers of testing. In that vein, the pointer to “Current King of [location]” should have generated “[Nobody]“, so that the “IsBald{}” function would generate a “Missing Argument” error.
P.S. I suggested the modified “Trecherous Image” caption not just because I think it would have made for a funnier joke, but also because of the “#NotMyKing” protests that have been cropping up in Great Britain, and which have been quashed with an inordinate amount of force and a complete disregard toward the right to free expression.
P.P.S. Ctrl+X is “cut to clipboard“, the simplest method to pocket an already typed comment while performing a refresh to verify whether an intervening comment has rendered one’s own text obsolete.
Kilby say:
In that vein, the pointer to “Current King of [location]” should have generated “[Nobody]“, so that the “IsBald{}” function would generate a “Missing Argument” error.Yes, that is one approach! From section 5.1 of that Stanford Encyclopedia article,
Strawson (1950) objected that Russell’s theory is simply incorrect about the truth conditions of sentences like ‘The present king of France is bald’. According to Russell’s analysis, this sentence is false (since it contains an existence claim to the effect that there is a present king of France), but according to Strawson, this does not conform to our intuitions about the truth of an utterance of that sentence. In Strawson’s view, an utterance of the sentence in a world where there is no present king of France is neither true nor false; perhaps the sentence has a truth value gap, or perhaps it fails to express a determinate proposition (Strawson vacillated on this), but either way it does not appear to be false.[He also originated the phrasing of the problem as “failure of a presupposition” which basically led to the entire field of Pragmatics in Linguistics/Philosophy.]
But read on within that Section 5.1 for later dissents, and these examples:
Consider the following minimal pairs, where the examples marked with ‘#’ indicate ambivalence about assigning a truth value, and ‘F’ indicates it is more plausible to assign a value of falsehood.(34) a. # The present king of France is bald.
b. F The present king of France is a bald Nazi.
(35) a. # The present king of France is sitting in a chair.
b. F The present king of France is sitting that chair.
(36) a. # The present king of France read Anna Karenina.
b. F The present king of France wrote Anna Karenina.
(37) a. # The present king of France heard about Goldbach’s conjecture.
b. F The present king of France proved Goldbach’s conjecture.
Ctrl-X is the ‘cut’ of ‘cut-and-paste’ (in Windows, command-x on an Apple Mac). In an editable document window, it copies the selected text and deletes it from the document. I could use Ctrl-C (copy) to preserve my un-posted comment before refreshing the comments; I just like to know for sure that no glitch will cause it to posted.
I mostly meant the Daily Nous group and/or the Worlds loosely-knit philosophy fans , but I actually forgot that I knew (momentarily) the name of that site; perhaps because all my mind had wanted to do was try to think of more possibilities of what “nous” could be. (I had 7 years of French, so I feel bias may be blocking me.) (I believe the only ones are ‘news’ and ‘us’.)
FWIW, that’s how I saw it, too. What I would call a bi-lingual pun.
And I never was sure how to pronounce it!
Here’s Merriam-Webster:
b
2
ˈnau̇s chiefly British : COMMON SENSE, ALERTNESS
I don’t know why I find this concept of succession so fascinating: Is it ’cause I read so many British crime books and watch so many BritComs and BritCrime series? Is it ’cause the entire concept of a monarchy – no matter what country (and I come from a country where there is a very wealthy monarchy) is so anachronistic and ludicrous? Or because so much of world history derives from so-called legitimate and illegitimate monarchies and wars between them (and their interconnectedness with wars fought over/between religions)? All I know is that I am more interested in all this monarchical nonsense than is a friend of mine who was born in Scotland and has lived in England for much of her life ‘-)
I’m interested also in this #NotMyKing news . . . will have to look that up. Probably taken from the #NotMyPresident meme we’ve used in the US.
That last bit reminds me of the research that found that people will be more sure of the truth of a proposition when you add information, when statistically, adding another condition should make it less likely to be true. I can’t remember it exactly, but something like, “Bob is an accountant” people go, meh, maybe, but “Bob is an accountant who likes to collect buttons” and people go, yeah, that’s probably true, even though statistaclly you’d have to multiply the likelihood of either being true, making both being true more unlikely than either in isolation.
Anyone have links to what I’m talking about?
@Andréa – The reason I thought that there was no spell-check was my typing two words I was sure couldn’t be spelled the way I spelled them. The only one I remember is “debuted”. I also tried to misspell a word or two and they weren’t getting highlighted. (I could have misspelled them into words I didn’t know existed.)
I am glad that I had enough conviction to at least finish typing “debuted” before it became so unfamiliar- feeling that I decided to change the sentence.
Whoops, fast moving discussion! I meant, the last bit of Mitch’s quoting section 5.1 of that Stanford Encyclopedia article…
larK I don’t have links but names to look up would be Kahneman and Tversky
@larK, I think it’s in the same ballpark as that “added info” study, but only example 34 is really close to that — the others are more like, we have additional basis for knowing this claim must be false regardless of the success or failure of the referring-expression. Thus in 35 (which I think has a typo, missing “in” in b.) we can see that nobody at all is sitting in that chair; in 36 we know who wrote Anna Karenina and he wasn’t in any way the King of France; in 37 we know that nobody has proven Goldbach’s conjecture.
@Mitch4 – re: ‘nous’ WOW!.. I took a look in the Oxford Dictionary of English. I can’t find any memory in my mind about knowing that word. I feel there must have been several movies (British) where I missed it. (hard to ignore it in a book; perhaps easy to forget)
@Mitch: exactly; only the first example tingled my reminder-senses.
@Mitch4: Charles was named as Prince of Wales in 1958, but they didn’t hold his investiture until 1969. The investiture ceremony is what he studied Welsh for.
While William has been named Prince of Wales already, it is not known if he will have an investiture, nor when that might take place if it does.
The discussion reminds me a bit of the Brazilian Olympiad Pinocchio’s Hats logic puzzle.
Assume that both of the following sentences are true:
• Pinocchio always lies;
• Pinocchio says, “All my hats are green.”
We can conclude from these two sentences that:
(A) Pinocchio has at least one hat.
(B) Pinocchio has only one green hat.
(C) Pinocchio has no hats.
(D) Pinocchio has at least one green hat.
(E) Pinocchio has no green hats.
In case it’s not clear, only one answer is correct.
LOTS of money the British taxpayers are putting out – funeral, coronation, investiture. Does the family itself pay for ANY of this??
Yes, they do; the British tax payer pays close to nothing to the royal family (you can argue about extra police support and army and navy support). Remember, the king WAS the State, all money was his; one of the Georges agreed to give most of it to a fund the government could administer so he didn’t have to bother with paying for things like parliament, but that money remains the royals money. They get paid out of that fund (about 25% currently, but that is rather higher than normal because they are renovating some castles). The monarch also has two duchies they still own outright, that provide about a £20 million a year income — the queen chose to voluntarily pay taxes on this, so that’s a boon to the tax payer. (Remember, the king is the State, so when you pay taxes to the state, you were paying them to the king — so who does a king pay taxes to? Himself?)
Brian, I saw “hat” and thought it was going to be one of that family of puzzles involving a group of people and rules about who can see the others’ hats, and deducing one’s own hat color based on “A didn’t announce a solution but he would have if he was seeing two white hats so he must be seeing …” .
But this is a good riddle of a different flavor. The nature of hats is not a factor as with the indirect-reasoning puzzles.
I briefly seized on (C) but of course that is wrong, and based on a too-quick misreading. The right answer has to be (A). This turns on the modern principle of a universal being satisfied vacuously, but still counted true. At one time an All implied an Exists, but not any more.
BTW have you at some point been a fan of Raymond Smullyan?
In one of Martin Gardner’s books, ther was a comment that most “lying” puzzles and paradoxes depend on an unrealistic (far too logical) understanding of the art of lying. If “all my hats are green” is to be interpreted as a “logical” falsehood, then it could mean “not all of my hats are green” or it could mean “all of my hats are not green” (hence the determination that “a” is correct: he has at least one hat, since the latter interpretation does not rule out “all=one”). However, if the lie is an “artful” or “denotational” lie, then the statement might mean “all of my ‘hats’ are white wigs“, in which case both C and E would be correct. The condition “only one answer is correct” could then be interpreted as ruling out the art of lying.
Vaguely relevant: the queue to see the Queen lying in state has reached capacity (about five miles long) and so the ability to join the end of it has been paused for six hours.
https://www.bbc.co.uk/news/live/uk-62921700
The question is (without researching it): before people are again allowed to join the end of the queue, what are they doing now? Are they milling about randomly at a distance, constantly being shooed away by officers if they get too close to the end? Or are they actually queueing unofficially to join the end of the queue – in which case, aren’t they still actually queueing to see the Queen in some way?
Prospect of queue to join queue as Queen mourners told capacity reached
Government request to stay away from Southwark Park ignored as hundreds file in
https://www.theguardian.com/uk-news/2022/sep/16/queen-mourners-queue-southwark-park-capacity-reached-london
Similarly, if the true statement is “all of my hats were green yesterday”, then they still might be green today, or not, in which case we can’t conclude any of A-E at all and any of them might or might not be true, depending on whether Pinocchio was visited by a hat thief overnight, and if so whether he’s a victim or a fence.
A more subtle variation on that is if what he means is that the hats are green now but subject to change, since that is not the expected interpretation of “are” or the one we’re using in points A-E, and in that case all of A-E would be false.
Somewhat relatedly, also, deducing anything at all from false statements tends to rely on the excluded middle, and that’s not necessarily sustainable here. In particular, whether a hat is green or not is probably vulnerable to a sorites argument. Similarly, if Pinocchio is out to make trouble, he might have acquired a green hat using a Turing-complete contract such that “mine” is undecidable.
(In the traditional interpretation, btw, we know that the hat he must have isn’t green, because the negation of “for all h in hats, Mine(h) implies Green(h)” is “exists h in hats, Mine(h) and not Green(h)”. Or “for all h, Hat(h) implies Mine(h) implies Green(h) ” becomes “exists h, Hat(h) and Mine(h) and not Green(h)”.)
Related to the green hats is the “confirming instance” paradox. If you claim “all hats are green” and go out to observe hats, every green hat you find strengthens your position … until you find a hat that is not green, of course.
Now if you set out to prove “All crows are black,” you can go out counting crows. But at the end of the day, you might have seen only a few crows. Instead, consider that the statement “All crows are black” is equivalent to “all things that are not black are not crows.” Now you just look around: there’s a green hat! That’s not black, and it’s not a crow. You can find thousands of confirming instances just looking around your room.
Mark in Boston, I heard about that under the name Hempel’s Paradox or the Ravens Paradox. I don’t remember learning any good answers to it!
Though it’s not really a logical paradox. It’s more a trick-argument, for the position “verification as increasing probability is hoooey”.
Or else “you can’t just go around swapping in contrapositives”
The book by Martin Gardner† that I mentioned in commenting about the “art of lying” also contains an excellent description of Hempel’s paradox. MiB’s description is similar enough that I wonder whether he has read the same book.
@ Mitch – According to Gardner, there are several possible reasons that the logical reversal seems to defy “common sense”. For instance, MiB’s “green hat” example is a potential confirmation for “all crows are black“, but it also could be used to confirm “all crows are white“. In addition, serious problems arise when the set of objects is effectively unlimited (both “crows” and “objects in the room” would be exceedingly difficult to enumerate). Gardner provides a counter example with a more limited set, namely “all the typists employed by a (large) company”, and examines the intersection of the subsets “redhead” and “married” (it’s an old book, the implicit sexism was “normal” back when it was written).
P.S. † “The Scientific American Book of Mathematical Puzzles and Diversions” (1959) — I received my copy (of the 3rd printing) as a gift in 1985. The discussion of Hempel’s paradox starts on page 52, the “lying” puzzle is on page 25, but the best part are the solutions (and a hilarious letter from a reader) on pages 28-32.
Ah yes, I followed his column in the magazine for a long time, and also enjoyed many of the collections in book form.
I see that there is now “Martin Gardner’s Mathematical Games: The Entire Collection of his Scientific American Columns”. This link is to an e-book listing: https://bookstore.ams.org/gardner-set . This doesn’t seem to be a traditional printed book. Amazon is featuring a CDR, of all things!
But I see they have reissued some of the collections as books. “Hexaflexagons and Other Mathematical Diversions: The First ‘Scientific American’ Book of Puzzles and Games Paperback – September 15, 1988” That date is certainly not the original publication — I made some hexaflexagons that my family took delight in playing with while sitting in the Florida room watching tv, in a house that we lived in in the first half of the sixties. And I made a Science Fair poster project on “Topology” based on a couple chapters (or not-yet-collected magazine columns by Gardner.
I was reminded that Gardner spent some time at the University of Chicago, whose neighborhood I live in. I don’t remember the name of a character he assumed for writing a mx of genuine number theory and amusing crank numerology. But this Dr. oh oh oh it was Dr Matrix! This Dr. Matrix once remarked on what look like unlikely-by-chance or supernatural-influenced street addresses; was Bantam Books at 666 Fifth Avenue a satanic enterprise, for instance.? Then Gardner in propria persona mentions 1234 East 56th Street in Chicago and some of the notable occupants of that building. (Such as the Institute of General Semantics, which still exists but not at that address.) Then I realized I walked by that spot very regularly, and I checked that the street address remained. Though it had been converted from offices to a Co-op or Condo building, with large apartments very suitable for renting to a group of like a half-dozen students.
At some point I discovered that I had started reading Martin Gardner shortly after I learned to read. This was in the 1950’s, and my parents bought me a subscription to Humpty Dumpty’s Magazine for Children. “Humpty Dumpty sat on a wall. Humpty Dumpty had a great fall. All the King’s horses and all the King’s men Couldn’t put Humpty together again. But an American doctor with scissors and glue Put Humpty together as good as new. And now he is back on the scene Busily editing this magazine.”
Supposedly Humpty Dumpty was the editor, but Gardner was one of the writers and editors. He wrote The Adventures of Humpty Dumpty Junior, a little egg, Humpty Senior’s son. The magazine was full of little puzzles and tricks and clever cut-outs.
I read everything by Martin Gardner that I could get my hands on. And made lots of hexaflexagons.
Andréa – My 94yo mom who was who pointed me to towards being an anglophile, from her interest in Queen Elizabeth from when mom and she were young, cannot understand that the royal line extends beyond Prince William’s children. I keep trying to point out to her that even Princess Margret and her descendants still stand in line for the throne (though with very little chance of getting it).
I tried to explain using the movie “King Ralph” with John Goodman and Peter O’Toole that the line extends beyond the immediate families. Which I have to do by explaining the plot to her as she has never seen it. Basically – the entire main royal family – including Princess Margaret and family and beyond – have been killed at an event (explosion I think). They need the next nearest relative who happens to be an American named Ralph – beer drinking, bowling, blue collar fellow played by Goodman. O’Toole is the servant tasked with getting him “up to snuff” as the new King.
(I won’t give away the rest of the story unless it is asked for.)
“He wrote The Adventures of Humpty Dumpty Junior, a little egg, Humpty Senior’s son.”
Humpty Dumpty had a son? And were no chickens involved, or did they just not speak of same as too facts of lifey for the magazine’s little readers? (“Which came first, the egg or the egg?”)
Maybe you can show her/print out this chart . . .
Shrug: Yes, Humpty Dumpty, an egg, was married to Mrs. Dumpty, another egg, and they had a son, also an egg. How exactly an egg produced an egg was never explained, but that makes sense because it was a children’s magazine, and children’s magazines didn’t give a lot of detail as to where babies come from. I bet though that if they did provide a backstory, a stork would be involved. Which makes sense if you think about it, a stork delivering an egg to a couple of eggs.
Meryl A: Long before “King Ralph”, in 1969 Richard Lester made “The Bed Sitting Room.” War has been declared on Britain, and massive atomic bomb attacks on London result in the shortest war in history. Much of the population is gone, but the remainder continue to carry on as if nothing has happened, living in the bombed-out shells of their apartments, and taking the escalator to the train in the morning, falling off at the top because the rest of the train station is gone.
But even though the Royal Family has been killed along with all the rest of the upper class, the line of succession still works because it includes everybody. The new ruler is Mrs. Ethel Shroake of 393A High Street, Leytonstone.
The words of the National Anthem have been changed to “God save Mrs. Ethel Shroake, Long live Mrs. Ethel Shroake, God save Mrs. Ethel Shroake of 393A High Street, Leytonstone”.
“. . . in 1969 Richard Lester . . .”
He of ‘Hard Day’s Night’ fame . . . and I was just looking at Leo McKern in
‘Rumpole of the Bailey’ . . . feel like I’m in a time machine.
Not just “A Hard Day’s Night”, but also “Superman II” and “Superman III” (the latter with Richard Pryor), and “A Funny Thing Happened On The Way To The Forum” with Zero Mostel and Buster Keaton. And the version of “The Three Musketeers” that had Oliver Reed, Richard Chamberlain, Raquel Welch, Michael York, Frank Finlay and Christopher Lee.
Oliver Reed – I remember him in a PBS (Educational TV or something like that it was known then) bio of Dante Gabriel Rossetti.
So in a circuitous way, I was brought back to our logic discussion today, and how common sense understandings often get in the way of more rigorous logical understanding. I was trying to remember a point made on the TED Radio Hour yesterday, where the speaker used the mnemonic “Preacher, Prosecutor, Politician”, and I couldn’t remember the “Prosecutor” (I knew it was a lawyer of some type, but I couldn’t think of any that began with “p”), but my brain did throw up “Poets, priests, and politicians” …who all have words to thank for their positions; so when their eloquence escapes you; their logic ties you up and Rapes you: the doo doo doo, the da da da, is all I have to say to you! The Doo doo doo; the Da, da, da, they’re meaningless, and all that’s true!
So, they’re meaningless, and all that’s true. Which is to say, nothing is true.
But what does it mean, logically, to say a meaningless phrase is the only thing that’s true? According to Pinocchio’s Hat from above, if Pinocchio only lies, and he has no hats, he cannot say that all his hats a green, because that statement is neither true nor false, and Pinocchio can only make false statements. Colloquially, we’d say this kind of statement is meaningless. Yet Sting assigns it the value “true”, to thus imply that if it is true, then nothing can be true, and we all understand that — even though the logicians were arguing about how to classify those statements, going from they must be false, to they must be out of bounds, and changing the implications depending on where it fell. But we all understand Sting’s sentiments exactly, I think.
OK, granted, Sting is talking about Truth with a capital “t”, which is a different beast from what the logicians are dealing with.
Andréa pointed out in email that this week in Wayno’s blog he brings up Bertrand Russell, and uses this picture as his “pipe pic of the week”:
This week’s blog entry (with his comments on this week’s Bizarro weekday cartoons) is linked here.
As I mentioned somewhere above, you can knock out two of the Pinocchio statements and be left with “no hats” and “one hat”. Those are contradictory statements, so they can’t both be true. For me, since the “one hat” statement causes no problems or ambiguities that I can see (can’t be falsified), I considered it true. That then means that the “no hats” statement must be false. The reasons why aren’t even that important to me.
When I took Graph Theory during my MSCS program, I was a bit unprepared for the number of proofs of various sorts we had to do†. One on an exam was a true or false about the number of nodes for a given graph, and gave a specific number. Some people falsified the statement through counter-example, by showing there was a higher number nodes that fit the criteria.
I started off that way, but then went and asked the professor if it meant exactly X nodes. I figured he’d probably say to use the information given on the exam, but he said “no, it doesn’t meant that”. So I got it right.
The people that got it wrong were unhappy, including one guy who complained loudly in class and demanded the professor admit that he was wrong. That didn’t work.
† I hadn’t done a proof by induction in over 15 years, back in some of the math classes for my BS Physics. We did a lot of them in that class.
Brian, I don’t think that’s the intended answer or really the right approach to the Pinocchio’s Hats puzzle.
But first, let’s agree that this is not done under an earlier age’s understanding of the truth conditions of an “all” statement. If it is of the form “all x’s are P” and it so happens that there are no x’s, the modern understanding is that any such statement counts as TRUE. Not “ill-defined”, not “lacking truth value”; just (vacuously) TRUE. And that holds regardless of our discomfort over noticing that, for instance, “all x’s are not-P” is also TRUE when there are no x’s at all.
Don’t bother arguing that this is not the best system. That rejection may be correct (and Europe’s logicians thought so for a long time), but it’s the only way quantified logic can really work. It’s fine if you disagree, but it’s not something we can settle. And the puzzle was clearly constructed under the vacuous-truth system.
Also, noting that “only one answer is correct” should not be taken as a statement to be factored into the logical evaluations — it is just a reassuring hint from the puzzle setter not to get wrapped up in multiple guesses.
Okay, then.
1) Pin always lies. [Any statement he makes we must consider FALSE]
2) Pin says “All hats that belong to Pin are green”.
Suppose Pin has no hats at all. Then condition (2) is TRUE — no, I mean the statement quoted in condition 2 is TRUE. This contradicts condition 1. So we rule out this supposition, that Pin has no hats.
That eliminates answer (C), which was exactly that supposition.
And it shows the truth of (A), which is just the direct negation of that.
If we were using the “only one correct answer” principle we would be done, since we have found one correct answer, A.
But can we, without using “only one correct answer” as a presupposition, rule out B, D, and E?
B cannot be concluded from the premises, because nothing rules out Pin having multiple green hats. Show by counterexample: say he has 3 green hats and none of other colors. This falsifies B, so it cannot be said to logically follow.
It is left to the reader to do the same move for D and E.
When I saw that you were describing a puzzle with colored hats, I thought it was going to be the one I have known and liked the longest, requiring getting into what one character can conclude about other characters’ reasoning.
This may have been from Smullyan, via Gardner when he was first introducing him to his readership in the SciAm column. Or maybe I saw or heard it elsewhere — I think it’s the one I’ve known the longest. I think the original version may have been explained with forehead patches instead of hats; but hats make it easier to set up the right who-can-see-whose-marking conditions.
We also have to make some agreements about all the characters being competent but not super reasoners. Also the oldest versions I recall just said “They are all silent for a while — Then person A announces that she has solved it. What is her answer?” I have introduced the story variant that “The puzzlemaster asks C if they can solve it, and after just a minute C says no. Then the puzzlemster asks B, and after a moment B also says no. Then the puzzlemaster asks C, who does solve it. What is C’s correct solution?”
The puzzlemaster brings in contestants A, B, and C and locks the door. Puzzlemaster shows the contestants that there are a supply of five hats, three white and two black. The contestants now are lined up facing forward in a queue, with A at the back, B in the middle, and C at the front.
The puzzlemaster now puts one hat from the supply on each contestant’s head and puts away the leftover hats. None of the contestants can see their own hat. A can see what color hats B and C are wearing; B can see what C is wearing (but not A); C cannot see any hat.
The puzzlemaster makes these conditions explicit to the contestants, and tells them the goal for each is to determine the color of their own hat.
Now the puzzlemaster asks A “Can you figure out the color of your own hat?”. A answers “No”.
Then the puzzlemaster asks B “Can you figure out the color of your own hat?”. B answers “No”.
Then the puzzlemaster asks C “Can you figure out the color of your own hat?”. C answers “Yes, it has to be ____”. And the puzzlemaster verifies that this is correct.
What was C’s correct conclusion, and the reasoning that got them there?
You’re in maze of twisty little passages, all alike. Eventually you come upon the two doors from “The Lady or the Tiger”; obviously you don’t want to go through the door with the tiger behind it. There are two guards here, one of whom always tells the truth, and the other of whom always lies, but you do not know which is which. You may ask exactly one question and only one question to either one of the guards. What one question do you ask?
(@Mitch: I like you hat puzzle, and I’m confident I know the answer, but I won’t post it yet…)
@larK I won’t try to get it precisely, but as an attempt schematically at your liar puzzle, you should ask either of the guards a question based around “If I were to ask the OTHER guard xxxx, what would he say?”
@ larK – XYZZY !!
Mentioning the truth conditions of an “all” statement reminds me of a very basic difference between mathematical logic and everyday logic. Mathematically, “a or b” is true if a is true or if b is true or if both are true. But when Mom says “You may have pie or cake for dessert” she means you can’t have pie AND cake. I guess she should have said “You may have pie XOR cake.”
Likewise “If pigs have wings, then green is blue” is taken as a nonsensical statement, even though logically it is true because pigs never have wings.
The hat puzzle makes me think of a puzzle I read a long time ago. There was a certain far off city where every resident had a very logical mind. Marriage was taken very seriously; infidelity was considered as one of the worst offenses.
Yet there were 30 unfaithful married women in the city. If a woman was unfaithful, everybody but the woman’s own husband knew she was unfaithful. People were really good at spreading gossip but even better at keeping secrets.
The king of the city had had enough of this infidelity. He declared that he knew there were unfaithful wives and decreed that for the next 30 nights, and those 30 nights only, a man who knew his wife was unfaithful had license to kill his wife in the night, but only on the night of the first day he found out about her infidelity.
29 nights went by with no deaths. Then on the 30th night, 30 unfaithful wives were killed by their husbands. Why did that happen?
If there were only one unfaithful wife, her husband would think “I thought there were no unfaithful wives but the King says there are more than zero. The husband who doesn’t know his wife is unfaithful must be me,” and he would kill his wife on the first night.
But if there were two unfaithful wives, Jones would say “There is only one unfaithful wife and that is Smith’s, and so Smith will kill his wife on the first night,” and Smith would say the same about Jones. But after the night passes with no killing, Smith and Jones will each say “Uh-oh, there are two of us” and kill their wives on the second night.
And so on.
@ larK – In addition to Danny’s “What would the other guy say?” solution, Gardener’s book offers these options: (1) “If I were to ask you whether this door leads to the [lady], would you say ‘yes’?” and (2) “Of the two statements ‘you are a liar’ and ‘this door leads to the [lady]’, is one and only one of them true?”
The readers’ letter rejects all of these “logical” solutions: “It is a sad commentary on the rise of logic that it leads to the decay of the art of lying…. the proposed solution [implies]… that liars can be made dupes of their own principles,… whenever lying takes the form of slavish adherence to arbitrary rules. No [true liar] could be expected to display the scrupulous consistency required … nor would any liar capable of such acumen be so easily outwitted.”
“We therefore propose as the most general solution the following question or its moral equivalent: ‘Did you know that [the lady] is serving free beer?’“
Mark in Boston, the “infidelity” puzzle strikes me as a version of the “Unexpected Hanging” or as Martin Gardner recast it,”The Unexpected Egg”.
If anyone’s inclined to disagree about empty “for all” propositions being true, the reason to prefer this interpretation is that it’s the limit as the set becomes empty.
This follows from “for all” being the transitive closure of “and” (that is, “for all X, P(X)” can be expanded to “X1 and X2 and X3 and …”, at least for finite sets).
So if you have a bag of statements X, all the statements in X are true if you take one (X1) out and it’s true, and all the rest of the statements in X are true. So when you have one statement (Xn) left, the condition is that it’s true and all none of the statements left in the bag are true. So you want to treat the empty bag as containing all true statements so that this case doesn’t need to become a special case.
For “there exists”, which is the transitive closure of “or”, it’s the other way around; that is, “there exists a statement in the bag such that XYZ” is false when the bag is empty. This is both necessary to maintain standard equivalences and also fortunately consistent with the general understanding of what existence means.
The reason it may seemsweird is that in natural language when we say things like “all hats are green” we normally actually mean something like “some hats exist, and in general any hat you find will be green… though there might be exceptions if there are a lot of hats”. We don’t speak in first-order logic and it’s not natural to try.
Andréa – Thank you. She understands through William’s children, but not that the rest of the family, cousins, etc follow beyond that – she says that they are too far away in the line (including Edward and Anne are too far away). Best also sometimes to just agree with her these days. I am sure we will discuss it again on another weekend.
Here is a solution to the black/white hats puzzle I posed previously. (Not to be confused with the green hats puzzle, about satisfaction of an empty “for all”.) With luck, you could get to it by the link https://cidu.info/2022/09/15/russell-on-denoting/#comment-118077 . Or by slogging thru the comments in the “Russell on Denoting?” thread to the timestamp “SEPTEMBER 24, 2022 AT 7:11 PM”. All of this is trying to avoid a big paste job here…
So, taking the conditions and question as read, let’s proceed by steps.
STEP 1.
What could Contestant A be seeing, consistent with his declaration that he cannot conclude the color of his own hat? Well, one thing we can observe is that A cannot be seeing two Black hats (one on B and one on C) — for if he saw that, and knows there are only two black hats in the original supply, he must have a white hat.
STEP 2.
Now consider what B (middle position) could be seeing on C’s head, and what could be concluded from that. Suppose B saw a Black hat on C. Then B couold frame a hypothtical —
“Suppose I have on a Black hat. Then A would be seeing my Black hat and C’s Black hat, and by Step 1 A would have concluded that he has White hat. But he does not announce that conclusion. So the supposition is wrong, and I do not have Black. So I can respond that I have concluded I have a White hat.”
STEP 3
But B did not have that response. So the failed hypothetical was “Suppose B saw a Black hat on C.”. Now C can follow this reasoning, and conclude that his own hat is White.
It’s like the US Presidential succession. Even though some are far down the line, they are still in line. Someone in the line goes into seclusion as the “designated survivor” during the State of the Union speech.
Brian, I don’t think that’s the intended answer or really the right approach to the Pinocchio’s Hats puzzle.
I missed this first part back when, and only noticed it while going through email. I think I didn’t clearly state that I meant the “at least one hat” and not “exactly one hat” and did not provide a letter answer, so you probably didn’t see that we actually agreed.
If the vice president of the United States dies, who’s president?
If the plane crashes right on the border of Czech Republic and Slovakia, where are the survivors buried?
@ Mitch – Usually in their own home towns, but it may take years (or even decades) for that to happen.
P.S. This isn’t the right opportunity for the classic question about a rooster laying an egg exactly on the apex of a roof (“Will it roll down to the east or to the west?”), so instead I will repeat the following linguistic riddle:
“What comes between ‘fear‘ and ‘sex‘?“
How many legs does a cow have, if you call her tail a leg?
Ah! Now we’re getting somewhere! So how many green hats may Pinocchio claim owning when you call having no hats ‘true’?
Shrug offers: “How many legs does a cow have, if you call her tail a leg?”
Phrased with dog instead of cow, this was attributed to Abe Lincoln, with the answer “Four. Calling a tail a leg doesn’t make it one.”
Speaking of quantification (as we have been elsewhere in this same thread), there is potential for confusing understanding of the quantifiers in another saying attributed to Lincoln: “You can fool some of the people all the time, and you can fool all of the people some of the time; but you cannot fool all the people all the time.” Both the first and second parts have their own ambiguity of quantifier scoping. But it makes no real difference to the point.
Kilby, you almost fell for the deceptive wording! The survivors do not need to be buried! (Not yet anyway, as you notes.)
For your addendum riddle, I can’t think of an English word that would match “fünf”; maybe just “fun”?
On the Lincoln quantification ambiguities:
Clause 1, You can fool some of the people all of the time
Repr 1
(∃x∊People)(∀t∊Times)[CanFoolAtTime(you,x,t)]
Repr 2
(∀t∊Times)(∃x∊People)[CanFoolAtTime(you,x,t)]
Cl 1 Repr 1 is like “Some people are so gullible, they can always be fooled”
Cl 1 Repr 2 is like “There are always going to be some people you can fool at the moment”
Clause 2, You can fool all of the people some of the time
Repr 1
(∀x∊People)(∃t∊Times)[CanFoolAtTime(you,x,t)]
Repr 2
(∃t∊Times)(∀x∊People)[CanFoolAtTime(you,x,t)]
Cl 2 Repr 1 is like “Everybody is susceptible to getting fooled, nobody is immune”
Cl 2 Repr 2 is like “Once in a while you can score a total 100% fooled-them triumph!”
@ Mitch – The answer can be delivered in either language, so both “Fünf” or “Five” would be acceptable. It depends on the audience.
Those puzzlers who are still reading this thread might be interested in the website “Gathering 4 Gardner”, which was mentioned today in the Post’s “Style Invitational”.
So I’m working my way through Gardner’s book mentioned earlier in this thread. The first thing that strikes me is the level of intellectual discourse for a popular, general population book! (I have a copy from 1959, the 11th reprint.) I feel like a dunce! He introduces a problem with a casual, of course everyone knows this, and states a problem that I feel thrilled I can generally solve, though I’m sweating, but that is just for introduction! Then he gets to the real problem, and generally it requires a clever insight just to see that a really clever insight is required to actually solve the problem. Any time there’s a question involving how few steps needed to solve the problem (how may weighings, how many questions asked), you can be sure that the eventual answer will be 1, if not zero — an extreme and extremely unlikely answer looking at the problem. (Yeah, it looks like I’d have to weigh at least 5 of those stacks of coins, but I guess if I’m clever, I could reduce that to 4, and if I’m really clever, I see a way to reduce it to 3 weighings… but the solution actually only involves one weighing…) I’d like to say I was going along at a C- level, but honestly, I’m below 50%, so I’m flunking royally!
The other thing that strikes me is how many famous names are casually mentioned as having contributed to or worked on a particular problem; a lot of them possibly even before they were generally famous, but others who might have already been famous, but messed around on this silly, trivial (and I can’t solve) thing in their earlier days. R. P. Feynman, A. Turing, John Nash, Claude Shannon are just a few off the top of my head I remember coming up.
larK, which Gardner book do you mean? I think there were several mentioned. If the one you’re reading is not the first in the series, there may be an explanation for the “everybody knows this one” effect, if he is referring to one he treated in an earlier column / collected in an earlier volume.
You indicated earlier that you generally knew how to approach the hat puzzle I posted. Did you see my solution? And does it fit with what your approach was to be?
Here is one of those coin-weighing problems; and an answer. Is this among the ones he presents early?
You are given a bunch of commemorative coins — actually there are 27 of them. [Yes, the choice of that number is a hint.] You are told that due to a manufacturing accident, one of them is heavier than all the rest, and your task is to identify that incorrect one. You have a two-pan balance which only shows if one side or the other is heavier, but does not provide actual weights. How many comparison-weighings will you need to identify the one incorrect coin?
Answer: Divide them into three piles of nine coins each. Place two of the piles on the balance pans, nine on either side. If the balance swings, take the heavier bunch into the next weighing; if the two sides are in balance, take the bunch which was not weighed to move into the next step.
Now you have nine coins, including the one heavier one. For your second weighing, divide the nine into three groups of three, and weigh two of them against each other. As before, if one side swings down, keep that group for the next step; if the two sides balance, keep the unweighed group of this step to go into the next step.
Now you have three coins. For the third weighing, put one coin on each pan. If they don’t match you have found the heavy coin on the side that swings down; if they match, the unweighed coin from this weighing is the heavy one from the original group.
So: three weighings.
This might be one of those that comes up as “you all know how to do this” before complicating the picture; do you agree?
I don’t know what the one-weighing problem you saw was, but here’s one!
You have ten bags of coins, ten coins in each bag. You are told all the coins are supposed to weigh 100 grams, but due to a manufacturing error all 10 coins in one of the bags were made to weigh 110 grams each. You have a weighing scale (not a balance) with a big dial and numerical readout. Or it can be digital, fine! Now how to identify the bag with the heavy coins?
One weighing will suffice.
@Mitch: The Scientific American Book of Mathematical Puzzles and Diversions by Martin Gardner. Kilby mentioned it on the previous page of comments. It has the second coin weighing problem you give, that I was referring to, where as you note, the solution can be done in one weighing, though the problem as presented just innocuously asks how many weighings do you need to do, leading you to expect, with ten piles of coins, something on the order of 5 weighings, and if you’re clever you might be able to reduce that, but you don’t really expect the answer to be “one”.
As an example of the “introductory” puzzle, in the chapter entitled “Nine Problems”, the first is introduced with the following: A guy walks a mile due south, a mile due east, and then a mile due north, and arrives at the same spot he started, and then he shoots a bear — what color was the bear? It is assumed everyone can figure this out (it doesn’t sound like it was previously given puzzle — it sounds like it’s just trivial). Then he goes on from that answer to the actual puzzle…
SPOILER
(So walking the way he did to arrive where he started, he must be on the North Pole, so the bear must be white, being a polar bear.) The actual puzzle is: is there any other spot on earth from which you can walk due south for a mile, due east for a mile, due north for a mile, and arrive at the same spot you started? (There are actually an infinite number of them, but they are particularly confined, and you need to discover and describe where on Earth this is…)
The hat problem I had the answer “white” for the logic you gave (no, really, I did!).
Ah yes, “the man walks a mile south…” puzzle — one of the all time greats!
For the southern circumpolar set of solutions, the story can no longer involve a polar bear!
In the book, Gardner points out that the single weighing solution will still work even if there are eleven bags (or stacks) of ten coins each.
larK – The President is President if the Vice President dies.