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.

Logic and the World

SelfAwarePatterns is an excellent blog if you’re interested in science, philosophy and similar topics (which covers pretty much everything). Earlier this week, its author, a self-aware pattern named Michael Smith, wrote about the nature of logic. He quoted several brief definitions of logic, including one by Gottlob Frege (1848-1925), one of history’s greatest logicians. According to Frege, logic is “the science of the most general laws of truth”, to which Mike Smith responded:

Gottlob Frege’s definition seems closest to my own current personal intuition about it, namely that logic represents the most fundamental relationships in our universe. These relationships are so fundamental, that we can take them and extrapolate truths using them, and often we’ll be right.

After reading this, I began writing a comment but quickly saw that my comment was turning into a post of my own. And since I need to keep this blog going in order to continue raking in the big money, here it is: 

Whenever I try to understand what logic is and how it relates to the world, I end up thinking about the status of Aristotle’s three fundamental axioms of logic: the Law of Identity (A = A); the Law of Non-Contradiction (it is not the case that A and not A), and the Law of the Excluded Middle (either A or not A), where “A” represents a statement like “Snow is white” or “I’ve never made a single penny writing this blog”.

The Law of Identity seems to reflect how the world is without question, partly because it’s supremely uninformative. As Bishop Butler said: “Everything is what it is, and not another thing”. I’m not sure the Law of Identity states a fundamental relationship, since self-identity isn’t much of a relationship. There is only one party involved. But it seems undeniable that A equals A, whatever A happens to be.

The Law of Non-Contradiction seems to reflect how the world is too. It’s exceedingly hard to imagine how things could be otherwise in our universe or any other universe (e.g., “Vitamin C is ascorbic acid and yet it isn’t.”). Despite this difficulty, some enterprising logicians have accepted dialetheism: the view that the very same proposition can be both true and false. That seems plainly wrong. Can we step into the same river twice? Well, yes, we can (“It’s the mighty Mississippi”) and no, we can’t (“The Mississippi had different water in it yesterday”). But which answer is correct depends on what you mean by “same river”. It’s the same river it was yesterday in one sense, although it’s not the same in another sense.

How about a self-referential statement like “This statement is false”? To be fair, that’s the kind of sentence dialetheist logicians are interested in. If “this statement is false” is true, it’s false. But if it’s false, it’s true. That is certainly weird, but is the sentence in question really both true and false? I don’t think so. It seems to me that it’s a badly-formed sentence. Its apparent meaning contradicts our natural presumption as speakers of a language that speakers don’t undermine their own claims (i.e., give with one hand and take back with the other). In this case, it seems best to follow the doctor’s advice when the patient said “It hurts when I do this”. The doctor, of course, answered: “Don’t do that”. Or in this case, don’t say stuff like “This statement is false”. Just because we can put certain words together doesn’t make it a proper sentence.

Then there’s subatomic physics. Light is a field of waves and also a stream of particles! The evidence indicates that light acts as if it’s a wave in some cases and as if it’s a particle in others, but saying that it acts the same way at the same time makes no sense. To me anyway. It’s better in this case to infer that our everyday concepts of “wave” and “particle” aren’t adequate to describe the nature of light. But that doesn’t mean light is a counterexample to the Law of Non-Contradiction.

So far, so good for classical logic accurately representing the universe. Things get more complicated, however, when we consider the Law of the Excluded Middle. Personally, I don’t buy it at all. The idea is that every proposition is either true or false. Unless we define “proposition” as “a bearer of truth or falsity”, there are lots of propositions that aren’t clearly true or false. There are vague propositions, for example. Has George lost enough hair to be considered bald? What if he lost one more hair, or 500 more, or 50,000 more? Where is the line between being bald and being hairy? And there is the matter of probability. For example, according to the principle of quantum superposition, “a physical system – such as an electron – exists partly in all its particular theoretically possible states simultaneously”. Is an electron here or there? Most physicists think it’s a matter of probability. An electron could be here and it could be there, but it’s not definitely anywhere until it’s measured or otherwise interfered with.

Concerns about vagueness and probability have led to the creation of alternative logics. So-called “many-valued” logics reject the Law of the Excluded Middle. “Fuzzy” logic replaces it with a continuum of values, ranging from true to false and allowing points in between. We might instead reject the Law of Contradiction and accept that some well-formed declarative sentences, like “George is bald”, are both true and false. “Paraconsistent” logics do that. As Mike Smith pointed out in his post, there is even “quantum” logic, which tries to deal with the peculiar laws of quantum physics.

There is good reason, therefore, to believe that Aristotle’s three axioms are somewhat misleading if they’re taken as an attempt to state fundamental features of the world or even relationships between the world and language (or thought). We should agree that the Law of Identity applies to the world (in fact, it applies to every possible world). After that, we’re in a gray area. There is no denying that the world is what it is (as that annoying phrase “it is what it is” seems to call into question – after all, what isn’t what it is?). Furthermore, we learn logic by paying attention to the world and use logic to navigate the world, but logic, I think, is better understood as “the science of the laws of discursive thought” (James McCosh, 1811-1888) than as a general description of how things are.