Chapter 14 Review (Day 139)

It is still spring break in my new district, and I'm currently enjoying it. I spend my day today doing things that I normally can't do when school is in session. Many of those things involve being away from my computer.

Thus today's a day when I really don't want to blog. Yet I'm scheduled to -- since the blog calendar is based on a district where it isn't spring break.

Don't worry -- I'm still going to post as scheduled. Fortunately, today's Chapter is mostly about logic, and I can actually just link to Metamath today. I'll just add Hofstadter's cute names to each of his main rules of logic. Thus all I have to do is quote a few lines from the Dialogue.

Dialogue 7 of Douglas Hofstadter's Godel, Escher, Bach is called "Chromatic Fantasy, and Feud." It begins as follows:

"Having had a splendid dip in the pond, the Tortoise is just crawling out and shaking himself dry, when who but Achilles walks by."

Tortoise: Ho there, Achilles. I was just thinking of you as I splashed around in the pond.
Achilles: Isn't that curious? I was just thinking of you, too, while I meandered through the meadows. They're so green at this time of year ...

...

Tortoise: Thank you. I just went swimming and washed off several layers of dirt which had accumulated last century. Now you can see how green my shell is.
Achilles: Such a good healthy shell, it's nice to see it shining in the sun
Tortoise: Green? It's not green.
Achilles: Well, didn't you just tell me your shell was green?
Tortoise: I did.
Achilles: Then we agree: it is green.
Tortoise: No, it isn't green.

...

Tortoise: I never said such a thing; but I wish I had.
Achilles: You would have liked to say that?
Tortoise: Not a bit. I regret saying it, and disagree wholeheartedly with it.
Achilles: That certainly contradicts what you said before!

...

Tortoise: Well, you didn't give ONE sentence. You gave TWO.
Achilles: Yes -- two sentences that contradict each other!

...

Tortoise: For instance, you'd grant that "Politicians lie" is true, wouldn't you?
Achilles: Who could deny it?
Tortoise: Good. Likewise, "Cast iron sinks" is a valid utterance, isn't it?
Achilles: Indubitably.
Tortoise: Then, putting them together, we get "Politicians lie in cast-iron sinks." Now that's not the case, is it?

...

Achilles: You should have used the word "and," not "in."
Tortoise: I should have? You mean, if YOU'D had YOUR way, I should have.

...

Tortoise: Today's exchange may have served a little to right your course. Good day, Achilles.
Achilles: Good Day, Mr. T.

Chapter 7 of Douglas Hofstadter's Godel, Escher, Bach is called "The Propositional Calculus." It begins as follows:

"The preceding Dialogue is reminiscent of the Two-Part Invention by Lewis Carroll. In both, the Tortoise refuses to use normal, ordinary words in the normal, ordinary way -- or at least he refuses to do so when it is not to his advantages to do so."

Here are the symbols of this formal system -- Proposition Calculus:

< > P Q R ^ v -> ~ [ ]

(Note: -> means "implies." Hofstadter uses an upside-down C, but this doesn't show well in ASCII.)

The argument between the Tortoise and Achilles is over the Rule of Joining:

Rule of Joining: If x and y are theorems of the system, then so is the string <x^y> (read "x and y").

Here are some other rules:

Atoms: P, Q, R are called atoms. New atoms are formed by adding primes to the right of old atoms -- thus R', Q", P'".

Formation Rules: If x and y are well-formed, then the following four strings are also well-formed:

(1) ~x ("not x")
(2) <x^y> ("x and y")
(3) <xvy> ("x or y")
(4) <x->y> ("x implies y")

These are called wff's in Metamath. For example, here's a link to Formation Rule (3) above:

http://us.metamath.org/mpeuni/wa.html

Rule of Separation: If <x^y> is a theorem, then both x and y are theorems.

Double-Tilde Rule: The string '~~' can be deleted from any theorem.
http://us.metamath.org/mpeuni/notnotri.html

A few minutes after I typed this, I saw the following commercial:

"They're not, not Pringles."

According to the Double-Tilde Rule, this is the same as saying "They're Pringles."

Fantasy Rule: If y can be derived when x is assumed to be a theorem, then <x->y> is a theorem.
Carry-Over Rule: Inside a fantasy, any theorem from the "reality" one-level higher can be brought in and used.

(The fantasy ones are a bit hard to explain, but we consider "pushing" and "popping" from the Dialogue from two days ago. The fantasy begins with [ and ends with ].)

The converse of the fantasy rule is the Rule of Detachment. This appears in the U of Chicago text, right in Lesson 13-1. (It's also the focus of the original Lewis Carroll Achilles/Tortoise Dialogue, and called "Modus Ponens" in Metamath.):

Law of Detachment: If x and <x->y> are both theorems, then y is a theorem.
http://us.metamath.org/mpeuni/ax-mp.html

The following is in Lesson 13-2 of the U of Chicago text. (It is called Transposition in Metamath.):

Contrapositive Rule: <x->y> and <~y->~x> are interchangeable.
http://us.metamath.org/mpeuni/ax-3.html

De Morgan's Rule <~x^~y> and ~<xvy> are interchangeable.
http://us.metamath.org/mpeuni/ioran.html

Switcheroo Rule: <xvy> and <~x->y> are interchangeable.
http://us.metamath.org/mpegif/ori.html
http://us.metamath.org/mpegif/orri.html

According to Hofstadter, this rule is named for an Albanian railroad engineer, Q. q. Switcheroo. The author ends the chapter by assuring us that the Propositional Calculus is complete and consistent:

"So if ever an incompleteness or inconsistency is uncovered, one can be sure that it will be the fault of the larger system, and not of its subsystem which is the Propositional Calculus."

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

Find x.

Almost all the necessary givens are in an unlabeled diagram, so let me fill it all in:

A, B, C, D all lie on a circle. AC is a diameter, Angle BCA = 73, Angle BDC = x.

Once again, this is a job of the Inscribed Angle Theorem:

Angle BCA = 73
Arc AB = 146
Arc BC = 34 (since Arc AB + BC = semicircle ABC)
Arc BDC = 17

Therefore the desired value of x is 17 -- and of course, today's date is the seventeenth.

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

Let me at least post the answers to the review worksheet.

1a. 16x
1b. 16xsqrt(2)
2a. 7sqrt(2)
2b. 7
2c. 7sqrt(3)
2d. 14
3. 12/13
4. 5/12
5. 0.406
6. 1
7. 1/2
8. 1
9-10. There is an error in the way I wrote this questions last year. Teachers may fix the error, and then the answers will vary.
11. AC/AB
12. BD, BC
13. DAB, DCA
14. Draw a vector the same direction as AB, but pointing in the opposite direction.
15. (-2, -9)
16. about 9 yards
17-20. Last year I delayed this activity, but this year I posted it yesterday. Thus my original intent with these questions has been restored.