Lesson 13-2: Negations (Day 132)

Lesson 13-2 of the U of Chicago text is called "Negations." In the modern Third Edition of the text, negations appear as part of Lesson 11-2 (which is the old Lessons 13-1 and 13-2 combined).

This is what I wrote last year about today's lesson:

For example, the statement:

All unicorns are white.

is actually true -- after all, we have never seen a unicorn that isn't white (precisely because there exists no unicorns at all, much less ones that aren't white). Another way of thinking about this is that there are zero unicorns in this world, and all zero of them are white! In if-then form this statement becomes:

If an animal is a unicorn, then it is white.

The hypothesis is false (since there are no unicorns), so the entire conditional is true. This statement has no counterexamples (unicorns that aren't white), and conditionals without counterexamples are normally called true.

The book then derives, from the statement 1=2, the statement 131=177. There is a famous example of a derivation of a false conclusion from a false hypothesis, often attributed to the British mathematician Bertrand Russell, about a hundred years ago. From the statement 1=2, Russell proved that he was the Pope:

The Pope and I are two, therefore the Pope and I are one.

that is, he used the the Substitution Property of Equality from the hypothesis 1=2.

In today's lesson, the U of Chicago text introduces the symbol not-p for the negation of p. In other texts, the notation ~p is used, but I have no reason to deviate from the U of Chicago here.

Before leaving this site, let me point out that this [Metamath] site gives yet a third way of writing the "not" symbol used in negations:

http://us.metamath.org/mpegif/wn.html

Today's post is fairly short so far. Last year, on the day I posted Lesson 13-2, most of my post was about my substitute teaching that day. I'd like to revisit that day briefly. Unfortunately, it was one of my toughest days as a sub.

Actually, today I want to write a little more about that first period class, which I didn't mention last year (yet I still remember all this a year later). I was supposed to play an audio file on the computer so that the students can listen and read the book together. The files for Part I and Part II were already opened up on the teacher's Mac, so all I had to do was press play. Each period had left off at a different location -- first period had already started Part II.

But Part I was already on the screen -- and the file had automatically maximized so that it took up the entire screen (which was mostly black, except for play/stop/volume buttons, and so on. I didn't know how to access Part II! No student in the class knew how to make Part II appear. I became so frustrated that I hit Command-Q to exit the file -- since that was the only way I knew to make anything other than the black screen with the Part I play/stop/volume buttons appear on the screen.

But unfortunately, a password was needed to access the files. The teacher had already opened up everything for me so that I wouldn't have needed her password. So once I closed the files, they were impossible for me to open on my own.

It wasn't until second period when another student knew how to access the files -- he used his own password to open them. It turned out that all I had to do was click near the top of the screen and the tabs for Part I and II would have appeared in first period -- but I hadn't known that. I suppose that I'm less familiar with Macs than with PC's, and this cost me dearly that day.

 The rest of the day was a domino effect. In later periods that day, I yelled -- all because of the frustration I'd felt starting in first period when I couldn't open the audio file. Since then, I now have a stress ball to squeeze so that I don't take my frustration out on the students (especially when that frustration comes from something that they have no control over, such as the audio file not playing).