"If I eat chocolate then I am happy. Is that logical?"
Cheng asks several similar questions about what is or isn't logical -- how touching wood makes her feel better, and how it's often cheaper to fly to a suburb of London than the capital itself, even though the plane actually lands in London en route to the suburb. But then Cheng's next example is:
"If you are white, then you have white privilege. Are these things logical?"
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:
"The innocuous little word 'if' has a whole range of slightly different means. Some of them, but not all of them, encapsulate the most important building block of logical argument -- logical implication."
Again, I'm glad that our side-along reading of Cheng's book lines up with Chapter 2 of the U of Chicago text -- didn't we just read about "if" in Lessons 2-2 and 2-3? Anyway, the author shows us how important logic is to the study of mathematics:
"Logic is to mathematics as evidence is to science."
But Cheng provides us many uses of "if" in English that aren't logical. "If I eat chocolate then I am happy" is a personal taste, not logic. "If you eat your broccoli then you can have ice cream" is a bribe, not logic. "If you walk my dog for me then I'll pay you twenty quid" is an agreement, not logic. "If you are over 75 then you don't have to take your shoes off when going through airport security" is a rule, not logic:
"The difference is a bit blurry in real life, but we can try and find the distinction by thinking about examples."
The author tells us that the above examples are better described as "non-logical" or "alogical" rather than illogical. For examples of logical statements, Cheng returns to the contentious example:
- If you have white privilege then you have privilege.
- If you are white then you have white privilege.
- If you are white in Europe or the US then you have white privilege.
- If you are white in a place that has white privilege then you have white privilege.
Cheng points out that the last statement is logical because it's almost tautological (or pointless). The statements above depend on the definition of "white privilege," which she will discuss in more detail in a later chapter.
"If we continue to use normal everyday language we are doomed to have problems being completely logical because the words we use are not completely logically defined, but we can get close enough that to call it anything other than logical would, in my opinion, be pedantry rather than precision."
Cheng next writes about social services. Here she distinguishes between false negatives -- those who deserve help but don't get it -- and false positives -- those who don't deserve help but do get it:
- If you care more about false negatives than false positives you will believe in expanding social services.
- If you care more about false positives than false negatives you will believe in expanding social services.
I've thought about this myself (but never blogged about it, since such politics is usually off-topic for this blog), but I could never express myself as elegantly -- or logically -- as Cheng does here. And in fact, she compares social services to jetlag:
- If you're better at staying awake tired than falling asleep not tired you should deprive yourself of sleep in advance of crossing time zones.
- If you're better at falling asleep not tired than staying awake tired you should deprive yourself of sleep in advance of crossing time zones.
Cheng writes that these two ideas are similar. Staying awake when tired is a false positive, while falling asleep when not tired is a false negative. She writes that she herself is a false positive, so she stays awake before she travels across the pond.
The author proceeds to write about how logic leads to discovery:
"If a statement follows from pure logic then it has to be true, automatically. Saying it out loud doesn't exactly add new information, but it does add new insight."
To illustrate this, Cheng returns to her white privilege example again:
- If you are white then you have white privilege.
- If you have white privilege then you have privilege.
From these two statements we now have the implication "if you are white then you have privilege."
Now Cheng tells the remarkable story of Kyle MacDonald, an internet legend who ultimately traded a paper clip for a house. Here's how he did it:
paper clip -> pen shaped like a fish -> hand-sculpted doorknob -> camping stove -> 1000-watt generator -> "instant party" (beer keg with neon sign) -> snowmobile -> two-person trip to Yakh, British Columbia -> a large van -> a recording contract with Metalworks -> a room for a year in Phoenix, Arizona -> an afternoon with rock star Alice Cooper -> a part in a Corbin Bernsen film -> a house in Kipling, Saskatchewan
Of this story, the author writes:
"But I am really fascinated by this mental version of an optical illusion, where you can take tiny steps that don't seem too surprising, and get somewhere that is extremely surprising and a very long way from where you started. This is how logic works."
And of course, we do this all the time in Geometry classes -- we call it a "proof." Cheng writes about another example -- in his book The Power of Habit, Charles Duhigg argues that requiring a higher level of science achievement in teachers would eventually reduce birth defects in babies:
"It is an example of a masterful construction of a long chain of implications in normal life."
Now Cheng introduces the formal notation for implications. We've seen this notation a few days ago in Lesson 2-2 of the U of Chicago text -- "A => B" means "If A then B" or "A implies B." (Even though the notation appears in Lesson 2-2, I don't use it on a worksheet until tonight's lesson.)
She tells us that a proof is basically a whole series of implications strung together like this:
A => B
B => C
C => D
We can then conclude that A => D. Here are some longer chains of implications given by Cheng:
She tells us that a proof is basically a whole series of implications strung together like this:
A => B
B => C
C => D
We can then conclude that A => D. Here are some longer chains of implications given by Cheng:
- If you say women are inferior, that is insulting to women.
- If you think that "feminine" is an insulting way to describe a man, you are saying that women are inferior.
- Therefore if you think that "feminine" is an insulting way to describe a man, you are insulting women.
Here is another:
- If you don't stand up for minorities being harassed then you are letting bigotry flourish.
- If you let bigotry flourish you are complicit with bigotry.
- If you are complicit with something bad then you are almost as bad as it.
- Therefore if you don't stand up for minorities being harassed you are almost as bad as a bigot.
Cheng points out that whether A is true for a particular person has no effect on whether the entire implication A => B is true:
"'If you are a US citizen or permanent resident you are required to have health insurance' is true whether or not you are in fact a citizen or resident. The implication doesn't tell us whether or not someone needs health insurance; we only know they do if we already know they are a citizen or resident."
The author tells us that there are several things we must check before we attempt to start a proof:
- We should carefully define the concepts we're talking about.
- We should carefully state the assumptions we're making.
- We should carefully state exactly what we're going to prove, in an unambiguous way.
Here Cheng writes about something I've mentioned a few times on the blog -- the gender pay gap:
"For example, in arguments about why women earn less than men, on average, sometimes people assume that women don't care about earning money as much as men do."
But, as the author goes on to point out, she doesn't accept the implication that if women don't care about making money then they should be paid less for the same job. Cheng puts it succinctly and flatly: "I think that is exploitation."
Cheng warns us that arguments might break down due to problems of knowledge:
- Unstated assumptions, or using stated assumptions incorrectly.
- Incorrect definitions, or incorrect use of definitions.
Here is an example from the book: "...in arguments about clinical depression when people assume that depression is caused by circumstances and therefore there is no reason a successful person should be depressed."
Proofs also break down due to problems of logic:
- Gaps in the logic; leaping from one statement to another without justifying it, or leaving out too many steps in between.
- Incorrect inferences: this is when a logical step is made that is actually incorrect, where you say something follows from something else, but it doesn't.
- Handwaving: arriving at a conclusion without true use of logic, but by metaphorically waving your hands around enough that people think you are.
- Incorrect logic: there are many subtle ways that incorrect logic can get slipped into arguments as logical fallacies....
"An example of an incorrect inference is the view that 'scientists agree with each other, which shows there is a conspiracy.'"
Cheng repeats a story first told by Stephen Hawking -- an audience member insists that the world is standing on the back of a turtle, and in fact, "it's turtles all the way down." But we must be sure that our proofs start somewhere rather than stand on an infinite stack of turtles:
"Where does all this originate from? As I have mentioned, I think the process of working backwards is just like small children who ask 'Why?' repeatedly."
There are some questions that logic can't answer, like "Do I exist?" or "What is the meaning of life?":
"To be a moderately functioning adult, we have to stop asking those questions at some point -- not necessarily permanently, but at least for some part of every day."
And so in logic we must start with some basic assumptions -- which the author calls "axioms" -- with which everyone agrees beforehand. Another word for "axiom" is "postulate" -- and that's the word that appears in most Geometry texts, including Lesson 1-7 of the U of Chicago text.
Cheng concludes with a preview of the next chapter:
"The logic must all flow out of that starting point. In the next chapter we'll talk about the direction of that flow. Like time, logic has a direction and we must not try to violate it."
Lesson 2-4 of the U of Chicago text is called "Converses." (It appears as Lesson 2-3 in the modern edition of the text.)
This is what I wrote last year about today's lesson.
There is a little bit of politics near the end of this post, because I'd perceived one of the examples of fallacious reasoning (assuming that a statement and its converse are equivalent) as one often committed by Republicans. So I added a similar fallacy made by Democrats in order for this post to remain politically balanced -- the point being made that both parties are prone to making logical fallacies. But only the example from the text actually appears on the worksheet:
[2018 update: When I wrote this last year, I was worried posting too much about politics. But the examples I gave in this post are tame compared to what Cheng writes in Chapter 2 of her book! It all goes to show that the easiest examples of logical vs. illogical statements to find are political.]
Lesson 2-4 of the U of Chicago text deals with an important concept in mathematical logic -- converses. We know that every conditional statement has a converse, found by switching the hypothesis and conclusion of that conditional.
The conditional "if a pencil is in my right hand, then it is yellow" has a converse, namely -- "if a pencil is yellow, then it is in my right hand." The original conditional may be true -- suppose all the pencils in my right hand happen to be yellow -- but the converse is false, unless every single yellow pencil in the world happens to be in my right hand. A counterexample to the converse would be a yellow pencil that's on the teacher's desk, or in a student's backpack, or even in my left hand -- anywhere other than my right hand.
If converting statements into if-then form can be confusing for English learners, then writing their converses is even more so. Here's an example from the text:
-- Every one of my [Mrs. Wilson's] children shall receive ten percent of my estate.
Converting this into if-then form, it becomes:
-- If someone is Mrs. Wilson's child, then he or she shall receive ten percent of the estate.
Now if a student -- especially an English learner -- blindly switches the hypothesis and the conclusion, then the following sentence will result:
-- If he or she shall receive ten percent of the estate, then someone is Mrs. Wilson's child.
But this is how the book actually writes the converse:
-- If someone receives ten percent of the estate, then that person is Mrs. Wilson's child.
In particular, we must consider the grammatical use of nouns and pronouns. In English, we ordinarily give a noun first, and only then can we have a pronoun referring to that noun. Therefore the hypothesis usually contains a noun, and the conclusion usually contains a pronoun. (Notice that grammarians sometimes refer to the noun to which a pronoun refers as its antecedent -- and of course, the text refers to the hypothesis of a conditional as its antecedent. So the rule is, the antecedent must contain the antecedent.)
And so when we write the converse of a statement, the hypothesis must still contain the noun -- even though the new hypothesis may be the old conclusion that contained a pronoun. So the converse of another conditional from the book:
-- If a man has blue eyes, then he weighs over 150 lb.
is:
-- If a man weighs over 150 lb., then he has blue eyes.
Saying the converse so that it's grammatical may be natural to a native English speaker, but may be confusing to an English learner.
Now let's look at the questions to see which ones are viable exercises for my image upload. An interesting one is Question 9:
A, B, and C are collinear points.
p: AB + BC = AC
q: B is between A and C
and the students are directed to determine whether p=>q and q=>p are true or false. Now notice that the conditional q=>p is what this text calls the Betweenness Theorem (and what other books call the Segment Addition Postulate). But p=>q is the converse of the Betweenness Theorem -- and the whole point of this chapter is that just because a conditional is true, the converse need not be true. (Notice that many texts that call this the Segment Addition Postulate simply add another postulate stating that the converse is true.)
Now the U of Chicago text does present the converse of the Betweenness Theorem as a theorem, but the problem is that it appears in Lesson 1-9, which we skipped because we wanted to delay the Triangle Inequality until it can be proved.
Here, I'll discuss the proof of this converse. I think that it's important to show the proof on this blog because, as it turns out, many proofs of converses will following the same pattern. As it turns out, one way to prove the converse of a theorem is to combine the forward theorem with a uniqueness statement. After all, the truth of both a statement and its converse often imply uniqueness. Consider the following true statement:
-- Donald Trump is currently the President of the United States.
We can write this as a true conditional:
-- If a person is Donald Trump, then he is currently the President of the United States.
The converse of this conditional:
-- If a person is currently the President of the United States, then he is Donald Trump.
This converse is clearly true as well. By claiming the truth of both the conditional and its converse, we're making a uniqueness statement -- namely that Trump is the only person who is currently the President of the United States.
So let's prove the converse of the Betweenness Theorem. The converse is written as:
-- If A, B, and C are distinct points and AB + BC = AC, then B is onAC.
(I explained why segment AC has a strikethrough back in Lesson 1-8.)
Proof:
Let's let AB = x and BC = y, so that AC = x + y. To do this, we begin with a line and mark off three points on it so that B is between A and C, with AB = x and BC = y. This is possible by the Point-Line-Plane Postulate (the Ruler Postulate). By the forward Betweenness Theorem, AC = x + y.
Now we want to show uniqueness -- that is, that B is the only point that is exactly x units from A and y units from C. We let D be another point that is x units from A, other than B. By the Ruler Postulate again, D can't lie onAC -- it can only be on the same line but the opposite side of A (so that A is between D and C), or else off the line entirely (so that ACD is a triangle).
In the former case, the forward Betweenness Theorem gives us AC + AD = DC. Then the Substitution and Property of Equality give us DC = (x + y) + x or 2x + y, which isn't equal to y (unless 2x = 0 or x = 0, making A and D the same point when they're supposed to be distinct).
In the latter case, with ACD a triangle, we use the Triangle Inequality to derive AD + DC >AC. Then the Substitution Property gives us x + CD > x + y, and then the Subtraction Property of Inequality gives us the statement CD > y, so DC still is not equal to y.
So B is the only point that makes BC equal to y -- and it lies onAC. QED
(The explanation in the book is similar, but this is more formal.) Later on, we're able to use this trick to prove converses of other theorems. So the converse of the Pythagorean Theorem will be proved using the forward Pythagorean Theorem plus a uniqueness statement -- and that statement is SSS, which tells us that there is at most one unique triangle with side lengths a, b, and c up to isometry. And, more importantly, the converse to the what the text calls the Corresponding Angles Postulate (the forward postulate is a Parallel Test, while the converse is a Parallel Consequence) can be proved using forward postulate plus a uniqueness statement. The uniqueness statement turns out to be the uniqueness of a line through a point parallel to a line -- in other words, Playfair's Parallel Postulate. This explains why the forward statement can be proved without Playfair, yet the converse requires it.
In the end, I decided to throw out Question 9 -- I don't want to confuse students by having a statement whose converse requires the Triangle Inequality that we skipped (and the students would likely just assume that the converse is true without proving it) -- and used Question 10 instead.
But now we get to Question 13 -- and here's where the controversy begins. I like to include examples in this lesson that aren't mathematical -- doing so might engage students who are turned off by doing nothing but lifeless math the entire period. This chapter on mathematical logic naturally lends itself to using example outside of mathematics. But the problem is that Question 13 is highly political. Written in conditional form, Question 13 is:
-- If a country has communist, then it has socialized medicine.
and its converse is:
-- If a country has socialized medicine, then it is communist.
[2018 update: Here I snip out the long discussion -- once again, if I were too worried about what I'm including in this post, then I would have skipped Cheng's book altogether.]
In the end, I will include Question 13 on the worksheet. But I left a space so that if a teacher feels that the question is politically slanted, then he or she could add another question to balance it. For example, that teacher can add a fallacious argument often made by Democrats, for example:
In the review section, I'd have loved to include Question 15, a review of the last lesson on programming (and of course changed it to TI-BASIC). But in deference to those classes that skipped the lesson because not every student has a graphing calculator, I've thrown it out and included only questions from the fully covered Lessons 2-2, 1-8, and 1-6.
And so in logic we must start with some basic assumptions -- which the author calls "axioms" -- with which everyone agrees beforehand. Another word for "axiom" is "postulate" -- and that's the word that appears in most Geometry texts, including Lesson 1-7 of the U of Chicago text.
Cheng concludes with a preview of the next chapter:
"The logic must all flow out of that starting point. In the next chapter we'll talk about the direction of that flow. Like time, logic has a direction and we must not try to violate it."
Lesson 2-4 of the U of Chicago text is called "Converses." (It appears as Lesson 2-3 in the modern edition of the text.)
This is what I wrote last year about today's lesson.
There is a little bit of politics near the end of this post, because I'd perceived one of the examples of fallacious reasoning (assuming that a statement and its converse are equivalent) as one often committed by Republicans. So I added a similar fallacy made by Democrats in order for this post to remain politically balanced -- the point being made that both parties are prone to making logical fallacies. But only the example from the text actually appears on the worksheet:
[2018 update: When I wrote this last year, I was worried posting too much about politics. But the examples I gave in this post are tame compared to what Cheng writes in Chapter 2 of her book! It all goes to show that the easiest examples of logical vs. illogical statements to find are political.]
Lesson 2-4 of the U of Chicago text deals with an important concept in mathematical logic -- converses. We know that every conditional statement has a converse, found by switching the hypothesis and conclusion of that conditional.
The conditional "if a pencil is in my right hand, then it is yellow" has a converse, namely -- "if a pencil is yellow, then it is in my right hand." The original conditional may be true -- suppose all the pencils in my right hand happen to be yellow -- but the converse is false, unless every single yellow pencil in the world happens to be in my right hand. A counterexample to the converse would be a yellow pencil that's on the teacher's desk, or in a student's backpack, or even in my left hand -- anywhere other than my right hand.
If converting statements into if-then form can be confusing for English learners, then writing their converses is even more so. Here's an example from the text:
-- Every one of my [Mrs. Wilson's] children shall receive ten percent of my estate.
Converting this into if-then form, it becomes:
-- If someone is Mrs. Wilson's child, then he or she shall receive ten percent of the estate.
Now if a student -- especially an English learner -- blindly switches the hypothesis and the conclusion, then the following sentence will result:
-- If he or she shall receive ten percent of the estate, then someone is Mrs. Wilson's child.
But this is how the book actually writes the converse:
-- If someone receives ten percent of the estate, then that person is Mrs. Wilson's child.
In particular, we must consider the grammatical use of nouns and pronouns. In English, we ordinarily give a noun first, and only then can we have a pronoun referring to that noun. Therefore the hypothesis usually contains a noun, and the conclusion usually contains a pronoun. (Notice that grammarians sometimes refer to the noun to which a pronoun refers as its antecedent -- and of course, the text refers to the hypothesis of a conditional as its antecedent. So the rule is, the antecedent must contain the antecedent.)
And so when we write the converse of a statement, the hypothesis must still contain the noun -- even though the new hypothesis may be the old conclusion that contained a pronoun. So the converse of another conditional from the book:
-- If a man has blue eyes, then he weighs over 150 lb.
is:
-- If a man weighs over 150 lb., then he has blue eyes.
Saying the converse so that it's grammatical may be natural to a native English speaker, but may be confusing to an English learner.
Now let's look at the questions to see which ones are viable exercises for my image upload. An interesting one is Question 9:
A, B, and C are collinear points.
p: AB + BC = AC
q: B is between A and C
and the students are directed to determine whether p=>q and q=>p are true or false. Now notice that the conditional q=>p is what this text calls the Betweenness Theorem (and what other books call the Segment Addition Postulate). But p=>q is the converse of the Betweenness Theorem -- and the whole point of this chapter is that just because a conditional is true, the converse need not be true. (Notice that many texts that call this the Segment Addition Postulate simply add another postulate stating that the converse is true.)
Now the U of Chicago text does present the converse of the Betweenness Theorem as a theorem, but the problem is that it appears in Lesson 1-9, which we skipped because we wanted to delay the Triangle Inequality until it can be proved.
Here, I'll discuss the proof of this converse. I think that it's important to show the proof on this blog because, as it turns out, many proofs of converses will following the same pattern. As it turns out, one way to prove the converse of a theorem is to combine the forward theorem with a uniqueness statement. After all, the truth of both a statement and its converse often imply uniqueness. Consider the following true statement:
-- Donald Trump is currently the President of the United States.
We can write this as a true conditional:
-- If a person is Donald Trump, then he is currently the President of the United States.
The converse of this conditional:
-- If a person is currently the President of the United States, then he is Donald Trump.
This converse is clearly true as well. By claiming the truth of both the conditional and its converse, we're making a uniqueness statement -- namely that Trump is the only person who is currently the President of the United States.
So let's prove the converse of the Betweenness Theorem. The converse is written as:
-- If A, B, and C are distinct points and AB + BC = AC, then B is on
(I explained why segment AC has a strikethrough back in Lesson 1-8.)
Proof:
Let's let AB = x and BC = y, so that AC = x + y. To do this, we begin with a line and mark off three points on it so that B is between A and C, with AB = x and BC = y. This is possible by the Point-Line-Plane Postulate (the Ruler Postulate). By the forward Betweenness Theorem, AC = x + y.
Now we want to show uniqueness -- that is, that B is the only point that is exactly x units from A and y units from C. We let D be another point that is x units from A, other than B. By the Ruler Postulate again, D can't lie on
In the former case, the forward Betweenness Theorem gives us AC + AD = DC. Then the Substitution and Property of Equality give us DC = (x + y) + x or 2x + y, which isn't equal to y (unless 2x = 0 or x = 0, making A and D the same point when they're supposed to be distinct).
In the latter case, with ACD a triangle, we use the Triangle Inequality to derive AD + DC >AC. Then the Substitution Property gives us x + CD > x + y, and then the Subtraction Property of Inequality gives us the statement CD > y, so DC still is not equal to y.
So B is the only point that makes BC equal to y -- and it lies on
(The explanation in the book is similar, but this is more formal.) Later on, we're able to use this trick to prove converses of other theorems. So the converse of the Pythagorean Theorem will be proved using the forward Pythagorean Theorem plus a uniqueness statement -- and that statement is SSS, which tells us that there is at most one unique triangle with side lengths a, b, and c up to isometry. And, more importantly, the converse to the what the text calls the Corresponding Angles Postulate (the forward postulate is a Parallel Test, while the converse is a Parallel Consequence) can be proved using forward postulate plus a uniqueness statement. The uniqueness statement turns out to be the uniqueness of a line through a point parallel to a line -- in other words, Playfair's Parallel Postulate. This explains why the forward statement can be proved without Playfair, yet the converse requires it.
In the end, I decided to throw out Question 9 -- I don't want to confuse students by having a statement whose converse requires the Triangle Inequality that we skipped (and the students would likely just assume that the converse is true without proving it) -- and used Question 10 instead.
But now we get to Question 13 -- and here's where the controversy begins. I like to include examples in this lesson that aren't mathematical -- doing so might engage students who are turned off by doing nothing but lifeless math the entire period. This chapter on mathematical logic naturally lends itself to using example outside of mathematics. But the problem is that Question 13 is highly political. Written in conditional form, Question 13 is:
-- If a country has communist, then it has socialized medicine.
and its converse is:
-- If a country has socialized medicine, then it is communist.
[2018 update: Here I snip out the long discussion -- once again, if I were too worried about what I'm including in this post, then I would have skipped Cheng's book altogether.]
In the end, I will include Question 13 on the worksheet. But I left a space so that if a teacher feels that the question is politically slanted, then he or she could add another question to balance it. For example, that teacher can add a fallacious argument often made by Democrats, for example:
- If a white person is racist, then he or she opposes Obama.
- If a white person opposes Obama, then he or she is racist.
In the review section, I'd have loved to include Question 15, a review of the last lesson on programming (and of course changed it to TI-BASIC). But in deference to those classes that skipped the lesson because not every student has a graphing calculator, I've thrown it out and included only questions from the fully covered Lessons 2-2, 1-8, and 1-6.
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