Super Saturday Post (Yule Blog Challenge #1)

Table of Contents

1. Introduction

2. Yule Blog Prompt #1: A Success Story from 2021

3. Cheng's Art of Logic in an Illogical World, Chapter 3

4. Conclusion

Introduction

That's right -- a new blogging challenge is upon us.

Last year, Shelli, the founder of the Blaugust challenge, also started a Yule Blog challenge where we must attempt to blog twelve times during the winter break. This year, Shelli didn't do a Blaugust challenge, but she's making up for it in spades with the Yule Blog challenge.

http://statteacher.blogspot.com/2021/12/2021-mtbos12days-yule-blog-challenge.html

And so let's dive in. Even though Shelli tells us that we don't need to follow her prompts exactly, you know that I like to do it anyway -- I consider it part of the challenge. And since today is the first proper day of winter break, let's start with the first prompt.

Yule Blog Prompt #1: A Success Story from 2021

My most obvious success story is my getting this full-time teaching position in the first place. And so let me tell this story in more detail.

In fact, the story begins with my "success story from 2020" for last year's Yule Blog. I was hired to cover a long-term middle school position in Orange County, CA, where I taught Math 7 and Math 8. It lasted from the last week in September 2020 to the first week in January 2021. After that week, I spent the rest of the school year as a day-to-day sub, covering classes including middle and high school math.

I was hoping that with my experience as both a day-to-day and a long-term sub -- along with the letters of recommendation I'd gathered from my long-term school -- I'd have a stronger resume when it came to applying to full-time teaching positions. But as the summer stretched into August, I almost became disillusioned as I was still without a job.

Finally, a district in LA County hired me. I was assigned to a small magnet high school -- one that hadn't admitted any new students since the pandemic and thus has only juniors and seniors. I was hired on the first day of school, and my first day with the students was Day 2.

I'm one of only two math teachers at our school. I have mostly the senior classes -- four sections of Statistics and one section of AP Calculus AB. But as commonly happens, throughout August the seniors slowly realize that they already have all the math they need to graduate and start dropping my classes -- and this is at a school that's already tiny. In my first period Stats class, I was soon left with only a single student -- and he was eventually transferred to my fifth period Stats, leaving me with only four math classes rather than five.

The other two Stats classes are "Ethnostats." Notice that this class has nothing to do with Critical Race Theory or George Floyd -- in fact, most of the students are Hispanic, not black, and the district started to class well before Floyd was killed. Ethnostats is a one-year course -- the other general Stats class is only a one-semester class. In the second semester, I will be teaching Trigonometry instead of Stats.

In a certain episode of The Simpsons, Homer learns that the Chinese use the same word for crisis and opportunity -- and Homer pronounces it "crisitunity." Anyway, the pandemic has certainly been a crisitunity for me. Yes -- nearly a million Americans have died, and millions more have gotten sick, so it's a crisis. But for me, it's also an opportunity. Had COVID-19 never occurred, the long-term position last year would never have opened, and I wouldn't have been recommended for my current job. I'd still be stuck today as a day-to-day sub.

This job has allowed me to teach math to high school students, and while it has its ups and downs, I believe that I've been largely successful helping the students out. Therefore, this full-time teaching position is my biggest success of 2021.

Cheng's Art of Logic in an Illogical World, Chapter 3

There's actually an author whose books are highly relevant to some of the topics of Ethnostats -- Eugenia Cheng. She's written five books, and her third (on logic and race) and fourth (on gender) are the most relevant to Ethnostats. I read those books a few years ago when she first published them, and now I'm rereading (and blogging about) her books in order to get ideas for the Ethnostats course.

I covered the first two chapters of her logic book during Thanksgiving break. So we'll start our discussion of the book with Chapter 3.

Chapter 3 of Eugenia Cheng's The Art of Logic in an Illogical World, "The Directionality of Logic," begins almost the same way that Chapter 2 does:

"Eating chocolate makes me happy instantly. It has to be good chocolate, but it works without fail every time. Does being happy make me eat chocolate? That's a completely different question."

Here is Cheng's first example involving converses:

"In the example in the previous chapter, I argued that if you don't stand up for minorities who are being harassed then you are almost as bad as an outright bigot."

Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next three weeks and skip all posts that have the "Eugenia Cheng" label.

Cheng proceeds to tell us that if we turn this statement around, it becomes false:

"Perhaps you stand up for them in public but then quietly block their promotions or pay rises, turn them down for jobs, or refuse to vote for them."

The author's point is that we have an implication like this:

not standing up for harassed minorities => almost as bad as a bigot

but we can't just reverse the arrow to get this:

almost as bad as a bigot => not standing up for harassed minorities

Cheng's next example also reverses something from the previous chapter. The statement:

you have white privilege => you have privilege

is true basically by definition (a tautology), but:

you have privilege => you have white privilege

is false, since there are other types of privilege.

Here's her next example:

you are a woman => you have experienced sexism

you have experienced sexism => you are a woman

Cheng points out that the first statement is the premise of the Everyday Sexism project. She tells us that sometimes a man would tell her, "I've experienced sexism too," but technically, he'd be a counterexample to the second statement, not the first. The truth of one implication has nothing to do with the truth of the reverse implication.

Oh, and by the way, the author writes:

"The statement we get by turning the arrow around is called the converse of the original statement."

And this once again fits with yesterday's Lesson 2-4. The U of Chicago text defines the converse by writing that the converse of p => q is q => p.

Cheng's next example involves broccoli and ice cream. A parent might say to a child, "If you eat your broccoli you can have ice cream." A clever child might respond that perhaps eating other foods might lead to ice cream as well, since the parent said nothing at all about what no broccoli leads to:

"Here the precision might come across as pedantry to the adult, who might say exasperatedly 'You know what I mean," but the child is just seeking clarity and trying to find a loophole to avoid eating broccoli."

If you eat your broccoli you can have ice cream.

broccoli => ice cream

You can eat ice cream only if you eat broccoli.

ice cream => broccoli

The first statement tells us that broccoli is sufficient for ice cream, but the second statement tells us that broccoli is necessary for ice cream. To combine both, we should actually say:

You can have ice cream if and only if you eat your broccoli.

"The trouble is that only a pedantic mathematician would bother saying that, so we grow up with the vague sense that 'only if' means the same as 'if and only if.'"

For Cheng's next example, imagine that you're trying to catch a group of bank robbers and you know the whole gang was white men. So you know:

If someone you encounter is in that gang then they are a white man.

This is equivalent to:

Someone you encounter can only be in that gang if they are a white man.

The converse is:

If someone you encounter is a white man then they are in the gang.

But this is false -- being a white man is necessary for being in the gang, but not sufficient. Using arrows we have:

True: gang => white

False: white => gang

Cheng tells us that arrows make the implications easier to understand, especially since the word "if" can appear anywhere in an English sentence:

You can have ice cream if you eat your broccoli.

If you eat your broccoli you can have ice cream.

are logically equivalent. But using arrows, the implications are more obvious:

The converse of A => B is B => A.

The equivalent of A => B is B <= A.

At this point the author draws a chart of all the converse and equivalent implications of A and B. And speaking of charts, Cheng now moves on to Venn diagrams:

"Math gets its power from being abstract, this is, removed from the real world of objects and things we can touch."

And she quotes Tristan Needham in his book Visual Complex Analysis:

"While it often takes more imagination and effort to find a picture than to do a calculation, the picture will always reward you by taking you nearer to the Truth."

But of course, such pictures are difficult for bloggers like me, since I'm bound to what I can represent here in ASCII. And so I'll merely describe some of Cheng's Venn diagrams. (Again, if you really want to see the diagrams, just get Cheng's book.)

If you are from England then you are from the UK.

So we draw a circle representing England inside a circle representing the UK, which in turn is inside a box representing the world, the universal set. Cheng draws similar diagrams for "If you have white privilege then you have privilege," and "If you are a US citizen then you can live in the US legally."

For the general case we have this: A => B, a circle representing A inside a circle representing B, which in turn is inside a box representing the universal set.

Cheng also draws a traditional Venn diagram where "can live in the US legally" and "US citizens" are interlocking circles inside the universal set "people." But she asserts that this diagram, while mathematically correct, is misleading. The subset of US citizens who can't live in the US legally is, in fact, empty.

The author includes eight different ways of writing A => B in words:

• A implies B. B is implied by A.
• If A then B. B if A.
• A is a sufficient condition for B. B is a necessary condition for A.
• A is true only if B is true. Only if B is true is A true.

Cheng gives a chilling example here. In 2017, a Georgia cop reassured a white woman that "we only shoot black people." The following are equivalent:

We shoot you only if you are black.

Only if you are black do we shoot you.

you are black <= we shoot you

we shoot you => you are black

If we shoot you then you must be black.

The author continues with converse errors:

"Converse errors occur when someone makes the mistake of thinking the converse of a statement is equivalent to the statement."

Cheng provides the following four examples to show that a statement and its converse aren't necessarily equivalent:

1. If you are a US citizen then you can legally live in the US. (True statement, false converse -- you could be a permanent resident or have a visa.)
2. If you have a university degree then you are false. (False statement, false converse -- this often arises in the traditionalists' debates about who should and shouldn't go to college.)
3. If you have experienced prejudice then you are a woman. (False statement, true converse -- Cheng already mentioned Everyday Sexism. The statement is false because men experience prejudice too, and Cheng also mentions non-binary people.)
4. If you support Obamacare then you support the Affordable Care Act. (True statement, true converse -- Obamacare = ACA, yet some people doubt the converse.)

We can sum up those conclusions in this table:

                    original  converse

statement 1  true        false

statement 2  false       false

statement 3  false       true

statement 4  true        true

Cheng writes a little more about logical equivalence. Many people commit the fallacy of false equivalence, as in the degree/intelligence example. They occasionally commit the fallacy of false inequivalence, as in Obamacare/ACA above:

"Logically those are the same, but emotionally they are different to some people, who feel fine supporting something with the calm and compassionate name 'Affordable Care Act,' yet can't bear the idea of supporting something referring to Obama."

Now I wish this blog to be politically neutral. But unfortunately, the problem is that this is a Common Core blog, and Common Core is itself politically charged. Once again, it was a Democratic administration that passed Common Core, and so once again, many Republicans oppose the standards (indeed, one nickname for the standards is "Obamacore"). One argument against Common Core is that it promotes slanted viewpoints that favor the Democrats.

The author also provides a list of eight ways to say that two things are equivalent:

• A is true if and only if B is true. B is true if and only if A is true.
• A is a necessarily and sufficient condition for B. B is a necessarily and sufficient condition for A.
• A is logically equivalent to B. B is logically equivalent to A.
• If A is true B is true, and if A is false B is false. If B is true A is true, and if B is false A is false. 

Cheng concludes with a preview of the following chapter:

"We'll come back to this in the next chapter, in which we are going to explore what it means for things to be false."

Conclusion

Twelve -- that's a lot of blog entries! But last year, my Orange County had a two-week winter break. On the other hand, my LA County district (which isn't LAUSD, but just like the huge district) will take three weeks off for the holidays. Last year I made eight Yule Blog posts in two weeks. So solving the proportion, that means that I should make twelve posts in three weeks.

So this year I have no excuse -- I'm going for it. Expect a dozen posts from me during winter break.