Wednesday, November 19, 2014

Review for Chapter 6 Test (Day 67)

Last night I tutored my geometry student. He was learning from Section 2-4 of the Glencoe text, which is titled "Deductive Reasoning." Even though Chapter 2 of both the Glencoe and U of Chicago texts cover logic, the worksheets that I covered back in the second chapter are very different from what the student learned today. In fact, the closest counterpart in the U of Chicago text is actually Section 13-1, "The Logic of Making Conclusions."

Both Section 2-4 of the Glencoe text and Section 13-1 of the U of Chicago text mention two laws that are important to logical thinking. Both texts call the first law the "Law of Detachment," which states that from p and p=>q, conclude q. This law was mentioned in my geometry text 20 years ago, except it was in the back rather than in the main text, and called by its Latin name, modus ponens.

The second law has different names in the two texts. The Glencoe text calls it the "Law of Syllogism," and states that it is sort of like a transitive property for logic. Therefore, the U of Chicago calls it the "Law of Transitivity." From p=>q and q=>r, conclude p=>r. But two British logicians, Bertrand Russell and A.N. Whitehead, used the name "Syllogism" about a hundred years ago.

Even though there are many laws in logic, these two are by far the most important. In fact, the Metamath website -- a site that seeks to write two-column proofs for many math theorems, calls the Law of Syllogism the most commonly used assertion, followed by Detachment (modus ponens):

My student had a little trouble at first distinguishing the two, but by the end of our session he was getting the hang of it. I can see why Glencoe would place this before beginning proofs. I'm not sure why the U of Chicago waits until Chapter 13 to present this information. That one may want to delay indirect proofs until Chapter 13 is understandable, but I don't know why this lesson is in Chapter 13 while converses -- a related logical concept -- appear in Chapter 2. I had no worksheet of mine to show my student, since I'm nowhere near writing a worksheet for Lesson 13-1 yet.

Instead of Chapter 13, let's think about Chapter 6 instead. We have just about finished Chapter 6, and so let's get ready for a test. The following worksheet is intended as a review for the upcoming test on the material of Chapter 6. As usual, I base my problems on the SPUR section of the U of Chicago text, and this review worksheet may contain repeats from the Chapter 6 quiz.

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