Philosophy Talk

A philosophy professor named Tim Sommers and I had an exchange at Three Quarks Daily about the Liar’s Paradox (the president wasn’t mentioned):

TS: There’s something wrong with the sentence, “This sentence is false.” Is it true or false? Well, if it’s true, then it’s false. But then if it’s false, it’s true. And so on. This is the simplest, most straightforward version of the “Liar’s Paradox”. It’s at least two thousand five hundred years old and well-known enough that you can buy the t-shirt on Amazon.com.

this-sentence-is-false

I’ve been thinking about the “Liar’s Paradox” lately, because I’m teaching an “Introduction to Philosophy” class on paradoxes (and writing a book) called “Life’s a Puzzle: Philosophy’s Greatest Paradoxes, Thought-Experiments, Counter-Intuitive Arguments, and Counter-Examples from AI to Zeno”. It starts with the “Liar’s Paradox” because it’s one of the oldest and most well-known, but also simplest and most daunting, of philosophical paradoxes. Some people think that while “puzzle” cases in philosophy are fun and showy, they are not where the real action is. I think every real philosophical puzzle is a window onto a mystery. And proposed solutions to that mystery are samples of the variety and possibilities of philosophy.

So, let’s start with this. Why is it called the “Liar’s Paradox”? Let’s go to the Christian Bible for that one, specifically, “St. Paul’s Letter to Titus” (Ch. 1, verses 12-14)

“They must be silenced, because they are disrupting whole households by teaching things they ought not to teach – and that for the sake of filthy lucre.12 One of Crete’s own prophets has said it: ‘Cretans are always liars, evil brutes, lazy gluttons.’13 This statement is true.14”

Verse 12 has philosophers dead to rights. We are disrupting whole households, teaching things we ought not to teach and – speaking for myself at least – it’s all about the filthy lucre (hence, the book). But verse 13 is what we want here. It has “Cretan’s own prophet” saying “Cretans are always liars.” Now, if that just means that all Cretans lie a lot, but not all the time, there’s no problem. But if it means that Cretans are always lying whenever they speak, given that this is asserted by a Cretan (read: liar), we have a paradox. This then is the primordial, liar’s version of the “Liar’s Paradox”. If that’s unclear you can simplify the liar’s version down to: “I am lying right now.”

By the way, “Crete’s own prophet” was “Epimenides of Crete”. Crete is the largest of the Greek islands, only 99 miles from the mainland, but (at that time) culturally distinct. It was the home of the Minoans. You might think you’ve never heard of the Minoans, but you have. They built a very famous labyrinth . . . 

What should we say about, “This sentence is false”? What if we said not every sentence has to be true or false and leave it at that? But “This sentence is not true” works just as well. Or rather works just as paradoxically. It’s true, if it’s not true. If it’s not true, it’s true. Even if we say not every sentence has to be true or false, we can’t have sentences that are both true and not true. That’s a straight-up contradiction. . . .

There’s . . . something called the Prior solution to the “Liar’s Paradox”, not because it’s prior to anything, but just because [it was formulated by] Arthur Prior. Prior says every sentence already implicitly implies its own truth. So, “This is fun”, really says “This sentence is true and this is fun.” Apply that to “This sentence is false” and you get “This sentence is true and this sentence is false”, which asserts a contradiction and so is just false now and not paradoxical. Voila! Problem solved.

Except, did you ever see the episode of “Rick and Morty” where Morty said, “What Rick just said is false”, and Rick said, “What Morty just said is true”? That’s the two-sentence version of the “Liar’s Paradox”. . . . Prior’s solution can’t help us here. (1) This sentence is true and the next sentence is false. (2) This sentence is true and the previous sentence is true. The paradox does not go away.

Maybe, what’s wrong with “This sentence is false” is that it’s self-referential. But, in general, it’s not a problem if a sentence is self-referential. Consider, “This sentence is six words long”. That’s just true. No problem. Or “This sentence is seven words long”. That’s just false. No problem. But “This sentence is false” is not just self-referential, it self-referentially assigns itself a truth value. So, let’s take a step back.

You have to admit, our ordinary, natural language is a mess. It’s imprecise and ambiguous and, most importantly, as any computer programmer will tell you, natural languages are over-flexible. They allow sentences to do things like self-referentially assign their own truth value. Computer languages can be thought of as artificial languages that replace natural languages in some contexts. Some universities now have departments of “logic and computation” that are no longer part of the philosophy department. This kind of thing happens all the time historically, by the way –  some part of philosophy morphs into a science. Four hundred years ago physics was “natural philosophy” and now they expect their own offices.

Anyway, in such an artificial language you might think there should be a kind of hierarchy of types of sentences. So, in computing, or in [the philosopher Alfred] Tarski’s logic, sentences can only assign truth values to sentences that are lower in the hierarchy. They are forbidden to assign truth-value to themselves or any of the sentences that outrank them.

“This sentence is false” is just nonsense, then, because it violates this rule. On this way of looking at it, the “Liar’s’ Paradox” is evidence that you need to enforce such a rule and/or develop an artificial language – or you get insoluble paradoxes.

So, how can we resolve the “Liar’s Paradox”. Beats me. I am hoping you will leave the solution in the comments section so I can go tell my class.

LF (me): I wonder what it would mean to “resolve” the Liar’s Paradox? Regarding the simple case, “this sentence is false”, would we have to show why it’s actually a regular true or false statement, or that it’s meaningless or vague or ungrammatical? There are lots of ways language can be used poorly. For example, some statements are rude. “This sentence is false” is paradoxical and not paradoxical in a good way if you’re having a normal conversation. If you’re teaching a course and want to discuss the issues the sentence raises about language, it’s paradoxical in an excellent way. But, in general, we don’t have to resolve a paradox like “this sentence is false”. We just have to avoid talking paradoxically (or rudely, falsely, ungrammatically, etc.), unless there’s a special reason to do otherwise.

TS: Thanks for the comment. I get what you are saying, I think. But still…what, if anything, is wrong with “This sentence is false”? It doesn’t appear to be, as you say, “meaningless or vague or ungrammatical” or even “rude”. Calling it “paradoxical” is just giving it a name. So, what would a resolution look like? Here are my thoughts on what might qualify. It’s true. Appearances aside, it’s just false. It’s ill-formed logically or as a speech-act. It’s meaningless, because… My hunch is that self-referentially assigning its own truth value violates some kind of “rule”, but I would like to see what that rule is better characterized – because I think it would be informative.  

[LF]: I will try to clarify. Presumably, to resolve a paradox means to show that it isn’t paradoxical. To resolve a problem is to make it go away, to no longer be problematic. Yet “This sentence is false” is clearly paradoxical, so it can’t be resolved or addressed in that way.

The next question is why is “this sentence is false” paradoxical? It’s paradoxical because it gives with one hand and takes away with the other. In ordinary, practical terms, declarative sentences are supposed to say something that’s true, yet “this sentence is false” simultaneously suggests that it’s not true. If the statement is meant to convey accurate information, it fails. It’s paradoxical. If it’s meant as entertainment, however, or as an example in a philosophy course, the fact that it’s paradoxical isn’t a problem; it’s a perfectly fine way to speak.

It’s the same situation with the two-sentence version of the paradox (“the next sentence is false”; “the previous sentence is true”). In practical terms, uttering these two declarative sentences suggests that each sentence is both true and not true (the principal difference being that each sentence refers to the other, and thereby indirectly to itself, not directly to itself, as in the one-sentence case). It’s another unresolvable paradox, a case in which language is being used poorly, if it’s meant in the usual way, to convey accurate information when uttering declarative sentences. The two-sentence case isn’t ungrammatical or meaningless in the usual sense, or rude or vague, other ways in which language can be spoken inappropriately unless there are overriding reasons to speak in those ways, for dramatic effect, for example.

I don’t think there’s a rule against “self-referentially assigning [a sentence’s] own truth value”. For example, everything I’m saying in this paragraph, including this sentence, is true. The relevant rule is “don’t make a declaration and indicate that it’s not true, unless you have a good reason for being paradoxical or ironic”. In similar fashion, “don’t tell somebody to do something and not do it, unless you have a good reason for contradicting yourself and don’t want it to be done” and “don’t ask somebody a question if you don’t want an answer, unless it’s a rhetorical question”. There are ordinary ways to speak that work well. Don’t mess with the formula unless you have a good reason to. I think that’s as deep as we need to go.

(Note: “Don’t mess with the formula” are the immortal words of Mike Love, addressed to Brian Wilson when things began to get strange around 1966.)

Unquote. (Prof. Sommers hadn’t responded to my response last time I looked.)

One other thought. I’m enjoying my news vacation during these troubled and troubling times, even though news leaks through. One of the results of avoiding the news is that you’re living in a different context. It occurred to me that, if I ran across the phrase “Fump Truck” during a really thorough news vacation (or if FBI Director James Comey had followed the rules and kept his mouth shut in 2016), I’d probably think it was a variation on “Dump Truck”. It wouldn’t suggest an entirely different phrase (similar to, but not “Dump Trump”).

Irrationality: A History of the Dark Side of Reason by Justin E. H. Smith

Smith teaches philosophy at the University of Paris. This book is more of a survey than a history. He has chapters on irrationality as it relates to logic, nature, dreams, art, myth, pseudoscience, humor, the internet and death. He occasionally discusses the fact that the American president is a dangerous buffoon. (Or, as a New York Times columnist put it so well: “The most powerful country in the world is being run by a sundowning demagogue whose oceanic ignorance is matched only by his gargantuan ego”. Or, as a Washington Post columnist concluded: “If he can’t argue that he has delivered prosperity, all that remains is the single most repugnant human being to ever sit in the Oval Office, befouling everything he touches”. But back to the book.)

Smith argues that the difference between rationality and irrationality often depends on one’s perspective. For example, is it rational or irrational to concern ourselves so much with the future when we’re all going to die anyway? It depends on what our goals are. His principal thesis is that “irrationality is as potentially harmful as it is humanly ineradicable, and that efforts to eradicate it are themselves supremely irrational” [287].

It’s an interesting book, but I’d say the first part of that last sentence didn’t need to be proved (irrationality is harmful, both actually and potentially, and will never be eliminated) — and that Smith fails to prove the second part (that efforts to eliminate irrationality are supremely irrational).

This is from his chapter on death and one of the best passages in the book:

… what lifted the soldier out of his foxhole was not his faculty of reason, but rather something deeper, something we share with the animals, which the Greeks called “thumos” and which is sometimes translated as “spiritedness”. It is a faculty that moves the body without any need for deliberation. It is like something that propels us when we are driven by desire, when we dive into a mosh pit or into bed with someone we don’t quite trust. It is something to which we are more prone when we are drunk, or enraged, or enlivened by the solidarity and community of a chanting crowd.

These manifestations of irrationality, it should be clear, are, as he saying goes, beyond good and evil. Life would be unlivable if they were suppressed entirely. But to what precise extent should they be tolerated or, perhaps, encouraged? It will do no good to say flatly that they should be tolerated “in a reasonable balance” or “in moderation”. For the ideal of moderation is one that is derived from reason, and it is manifestly unfair to allow reason to determine what share it should itself have in human life in a competition between it and unreason. So if we can neither eliminate unreason, nor decide on a precise amount of it that will be ideal for human thriving, we will probably just have to accept that this will always remain a matter of contention, that human beings will always be failing or declining to act on the basis of rational calculation of expected outcomes, and that onlookers, critics and gossipers will always disagree as to whether their actions are worthy of blame or praise.

The speeder and the duelist and the others seem guilty of no failure to correctly infer from what they already know, in order to make decisions that maximize their own interests. Rather, in these cases, there is a rejection of the conception of life that it must be a maximization of one’s own long-term interests in order to be a life worth living [263-264].

And Another Thing (Not About Politics for the Most Part)

What follows may be the least important of the 522 posts I’ve written since beginning this blog. That’s saying a lot, I know, but here it is anyway:

One thing I learned from years of studying philosophy is the distinction between naming, mentioning or referring to a word and using it. One half of that distinction should be perfectly clear. We all use words when we write or speak them. For example, I used either nine or ten words in that last sentence, depending on how you want to count the word I used twice.

Brief tangent: If the same word is used twice in a sentence, should it be counted twice since it appears twice or should it be counted once, since it’s the same word both times? The answer to that question has to do with another distinction (the one between token and type). That distinction isn’t the subject of this post, however, so let’s keep going. 

As I was saying, it’s clear that we all know how to use a word. But how do you name it? It’s very simple, actually: you put quotation marks around it. For example, in order to refer to or mention the word “cat”, I put quotation marks around the three letters that spell it. Doing so allows me to ask you, for example, whether you can spell “cat”. If I asked you to spell the word cat, without the quotation marks, it wouldn’t mean the same thing at all. (The best I can figure is that it would mean to relieve the cat that’s good with words by taking its place.)

People, of course, have names like “Peter” and “Susan”. See how I just gave you the names of those fictional people by using quotation marks? In effect, I gave you the names of their names. If I’d said Peter knows Susan, I’d have simply used their names. I’m sure you get the idea.

My next point: We can also name, mention or refer to a sentence. How? Again, simply by putting quotation marks around it. For example, “This is a sentence” is a sentence. In that last sentence, I referred to the sentence “This is a sentence” in the same way I referred to the word “cat” above.

Okay, I’m now getting close to the point I wanted to make when I began this post several hours ago. When I was in school, I was taught a certain way to use quotation marks. It’s a rule that appears in a list Google presented when I asked their search engine how to use quotation marks correctly:

Rule 4. Periods and commas ALWAYS go inside quotation marks.

Examples:
The sign said, “Walk.” Then it said, “Don’t Walk,” then, “Walk,” all within thirty seconds.
He yelled, “Hurry up.”

If you follow this rule, you’ll write things like this:

I can spell “cat.” 

Instead of:

I can spell “cat”.

But the name of the word “cat” isn’t “cat.”. So what’s that period doing inside the quotation marks instead of being at the end of the sentence where it obviously belongs?

Likewise, if you follow Rule 4, you’ll write sentences like this:

This is a sentence: “This is a sentence.”

Instead of:

This is a sentence: “This is a sentence”.

Putting the period inside the quotation marks would be okay if you wanted to emphasize how declarative sentences are supposed to end with a period, but if that’s what you were doing, you’d need another period outside the quotation marks to mark the end of the whole sentence, like this:

This is a sentence: “This is a sentence.”.

Now take a look at the next rule in the list:

Rule 5a. The placement of question marks with quotation marks follows logic. If a question is within the quoted material, a question mark should be placed inside the quotation marks.

Examples:
She asked, “Will you still be my friend?”

The question Will you still be my friend? is part of the quotation.

But wait a second. We’re supposed to follow logic with respect to question marks, but not when it comes to periods? What this apparently means is that the question below should be written this way:

Can you spell “cat”?

Not this way:

Can you spell “cat?”

Which makes sense, because “cat?” is not a good way to refer to the word “cat”, in the same way “cat.” isn’t a good way either.

Sometimes you have an idea and wonder why you’ve never heard that idea before. Why hasn’t someone else concluded that Rule 4 is dumb? Why are we all taught to use logic when using quotation marks with question marks but not with periods? Was there a problem with typesetting many years ago that gave rise to Rule 4? Was it a solution to a perceived problem or simply someone like Ben Franklin or Noah Webster who had a few too many and thought Rule 4 would be a good idea? 

I decided somewhere along the line that I would ignore Rule 4 on this blog and in my other written communications. So far nobody has complained or, so far as I know, even noticed. 

But imagine my surprise and relief when I found this today at the website of Capital Community College of Hartford, Connecticut: 

Here is one simple rule to remember:

In the United States, periods and commas go inside quotation marks regardless of logic. 

In the United Kingdom, Canada, and islands under the influence of British education, punctuation around quotation marks is more apt to follow logic. In American style, then, you would write: My favorite poem is Robert Frost’s “Design.” But in England you would write: My favorite poem is Robert Frost’s “Design”.

Capital Community even offer an explanation of sorts for this discrepancy:

There are peculiar typographical reasons why the period and comma go inside the quotation mark in the United States. The following explanation comes from [a link that no longer works]: “In the days when printing used raised bits of metal, “.” and “,” were the most delicate, and were in danger of damage … if they had a [double quote] on one side and a blank space on the other. Hence the convention arose of always using ‘.”‘ and ‘,”‘ rather than ‘”.’ and ‘”,’, regardless of logic.”

This seems to be an argument to return to something more logical, but there is little impetus to do so within the United States.

That last sentence sums up the situation perfectly. There is little impetus to be more logical here in the United States. It’s not the entire English-speaking world that’s doing it wrong, it’s just us. And we’ll keep doing it wrong because, well, most of our Constitution is more than 200 years old and much of it is out-of-date. Why shouldn’t our punctuation be old and out-of-date too?

So, here we are. I’ve explained the situation and assured you that I’m committed to breaking Rule 4 at every opportunity. I call on the rest of America to do the same. Sometimes we have to stand up for what’s right, even if it means risking reprisals from illogical, conservative elements among us.

But, I know, I know, you probably think you’ve wasted the past several minutes reading this post. Maybe you’d rather have been reading about Trump and Congress and France and Baton Rouge.

Or maybe in retrospect it was good to get away from that for a bit? If you’ve missed it, here’s a long article about Trump’s even longer history of lies and exaggerations. The reporter says it’s hard to find a project Trump worked on that didn’t involve dishonesty or misinformation of some sort. I couldn’t stand to read the whole thing, but maybe you can. (By the way, there’s a poll out that says a majority of registered voters think Trump is more honest and trustworthy than Hillary Clinton. Amazing. This is what years of propaganda and intense ideology will do.)

And in case you want to submit a question to one of the speakers at the Republican Convention, which starts tomorrow, there’s a form you can fill out here. Posing as a fictitious reporter, I asked to interview a member of the Trump family on the topic: “Is Donald Trump a psychopath or merely a dangerous con man?” Still waiting for a reply.

What Should Sleeping Beauty Say, Logically Speaking? (Part 2)

In yesterday’s post, I described the so-called “Sleeping Beauty Problem”. Mathematicians and philosophers have been debating this problem for the past 15 years (most recently in response to an article at the Quanta Magazine site). What should Sleeping Beauty say when she wakes up on Monday or Tuesday and is asked the following question about a coin flip that happened on Sunday: “What are the odds that the coin we flipped on Sunday came up heads?”.

To review: Sleeping Beauty doesn’t know what day it is when she wakes up or whether she’s already woken up during the experiment. But she is told that if the coin flip on Sunday came up heads, she’s only being awakened on Monday. If, however, the coin came up tails, she’s being awakened on both Monday and Tuesday. “Halfers” think the odds that the coin came up heads is 1/2. “Thirders” think the answer is 1/3.

Here’s the answer Pradeep Mutalik, the author of the Quanta article, gave in response to a whole lot of comments:

This is the crux of the idea: that there are two propositions in the Sleeping Beauty problem:

1) the probability of heads when the fair coin is tossed, which is obviously 1/2. This is the proposition modeled by halfers.
2) the probability of heads encountered later, which thanks to the experimental conditions that are asymmetric between heads and tails, is 1/3. This is the proposition modeled by thirders.

… the same problem statement can mean tossed if understood one way and encountered if interpreted another way.

Once you accept that there are two valid interpretations, the solution of the Sleeping Beauty problem can be expressed in one sentence:

A fair coin is tossed: The probability of it landing heads when tossed is ½; the probability of Sleeping Beauty encountering heads later, thanks to the asymmetric experimental conditions, is 1/3.

And that, in a nutshell, is all there is to it.

Mr. Mutalik also added this note in response to my own comment on his article, in which I came out in support of the 1/3 answer:

As in my reply to Rich, your answer is correct for the heads “encountered” probability. The heads “tossed” probability is, and remains ½ until SB learns the actual result of the coin toss.

The point of all this is to show that there is no incompatibility in the thirder and halfer arguments. The same person can simultaneously be both a halfer and a thirder.

As a new supporter of the thirder position, I’m not convinced by Mr. Mutalik’s response. As I wrote in the previous post, it’s clear that the odds of getting heads or tails when a coin is flipped are always 1/2 heads and 1/2 tails. That’s how coin flips work (putting aside the extremely rare occasions when they land on their edge and don’t fall either way). 

But the question presented to Sleeping Beauty is: “What are the odds that the coin flip came up heads?” Since she knows how the experiment is being conducted when she’s asked this question, shouldn’t she conclude that the odds are 1/3, not 1/2? After all, she knows that a coin that came up tails means this could be either Monday or Tuesday. If it came up heads, it can only be Monday. So the odds that the coin came up tails is twice as likely as the odds that it came up heads (i.e., 2/3 vs. 1/3).

It seems to me that asking Sleeping Beauty to ignore the information she’s been given about the experiment is asking her to give an answer she knows is incorrect. What a question means depends on its context. If the context in this case were simply: “Please explain how coin flips work. In particular, what are the odds that a fair, properly-executed coin flip will come up heads instead of tails, such as the coin flip we executed on Sunday?” In that context, the correct answer is obviously 1/2. Everybody should be a halfer when asked how coin flips work.

But that isn’t the context in which Sleeping Beauty is being asked to give an answer. She’s not being asked to explain how coin flips work in general. She understands that she’s involved in a bizarre, possibly unethical (the memory-destroying drug!) experiment and that the question she’s being asked is part of that experiment. She’s being asked what the odds are given her current situation. That’s why I think it’s more reasonable for Sleeping Beauty to be a thirder than a halfer. She should answer “1/3”. It’s twice as likely that the coin came up tails!

Whether she should be allowed to go back to sleep after giving the correct answer, so that she can later be awakened by a handsome young prince, or whether she should receive appropriate medical care and reacquire a normal sleeping pattern, is an ethical question, not a logical one, and beyond the scope of this post.

What Should Sleeping Beauty Say, Logically Speaking?

Imagine that Sleeping Beauty agrees to take part in an experiment. First, she’s going to be given a special memory-loss drug that will make her forget what day it is. Then she’ll go to sleep. While she’s asleep, the people running the experiment will flip a coin. If the coin comes up heads, they’ll wake her up on Monday and ask her a question. If it comes up tails, they’ll wake her up on Monday, ask her the same question, give her the drug, let her fall asleep again, and then wake her up on Tuesday, asking the same question as before.

Because of the drug, however, she won’t know what day it is when she wakes up. Furthermore, she won’t know whether this is the first or second time she’s woken up. Each day will be a brand new experience for her. But each time Sleeping Beauty wakes up during the experiment, she’s told about the coin flip and the rule that says heads will mean she will wake up on Monday, and tails will mean she will wake up Monday and Tuesday. 

Now here’s the problem: Each time Sleeping Beauty wakes up and is told the rules of the experiment, she’s asked this question: “What are the odds that the coin flip came up heads?” That is, what are the odds that the coin came up heads and this is Monday (as opposed to tails and this is either Monday or Tuesday)? What should Sleeping Beauty say?

I came across this problem at the Quanta Magazine site yesterday. You may be surprised to learn that experts have been arguing about its solution for years:

The famous Sleeping Beauty problem has polarized communities of mathematicians — probability theorists, decision theorists and philosophers — for over 15 years…. This simple mathematical problem has generated an unusually heated debate. The entrenched arguments between those who answer “one-half” (the camp called “halfers”) and those who say “one-third” (the “thirders”) put political debates to shame… Halfers and thirders tend to remain firmly rooted in their view of the Sleeping Beauty problem. Both camps can certainly do the math, so what makes them butt heads in vain? Is the problem underspecified? Is it ambiguous?

So, should Sleeping Beauty say there is a 1/2 chance that the coin came up heads, since there’s always a 1/2 chance that coin flips come up heads and a 1/2 chance that they come up tails? Should Sleeping Beauty be a halfer?

Or should she be a thirder and say there is only a 1/3 chance it was heads? After all, if it’s Monday, she was awakened because the coin came up heads or tails. If it’s Tuesday, she’s awake because the coin came up tails. That means it’s twice as likely she’s awake because the coin came up tails. There’s a 2/3 chance the coin came up tails and a 1/3 chance it came up heads. It sounds like Sleeping Beauty should be a thirder.

But not so fast! There was only that one coin flip on Sunday and we all know that coins have a 1/2 chance of coming up heads! Maybe she should be a halfer?

The Quanta article is fairly long and delves into why halfers and thirders give the answers they do, as well as why they often resist changing their minds. The comments that follow the article are even longer and include mathematical formulas. I didn’t read all the comments, and I’ve never studied probability, but I left my own comment anyway:

I wasn’t a halfer or a thirder before reading the article. Now I’m a committed thirder.

If you were to simply ask Sleeping Beauty whether a fair coin toss came up heads, she should say the odds were 1/2. Without any other information, that’s the rational answer. But you’re asking what odds Sleeping Beauty should assign, given the additional information she’s been given about the experiment. Since today could be either Monday or Tuesday (as far as she knows), it’s more likely that the coin came up tails. The fully-informed, rational answer she should give is 1/3.

So I think Sleeping Beauty should say this each morning: “Given how coin flips work, the odds are 1/2 that heads came up. But given how coin flips work and given what you’ve told me about this peculiar experiment, the odds are only 1/3 that heads came up. I can easily flip back and forth between the halfer and thirder positions, but why should I ignore the additional information you’ve given me? Taking into account what I  know about the experiment, I must conclude that the odds are 1/3 that heads came up. If that makes me a thirder, so be it. Now where’s my prince?”

But she shouldn’t take my word for it. She should make up her own mind.

The author of the article, Pradeep Mutalik, responded to my comment and everyone else’s last night. I’ll post his response tomorrow in case you haven’t already rushed to the Quanta site to read it.