# Tackling More Difficult Proofs

Mr. Nance,

In our logic studies, my son and I wrestle to work through the proofs, generally together. When we get stuck, really stuck, we go to the answer key, cover the answer, and move through the proof step-by-step until we find where we veered off-track. Then we use that one step to get us back where we need to be; and then, hopefully, we finish the proof. My question is, is this a reasonable approach? Continue reading Tackling More Difficult Proofs

# Shorter Truth Tables for Equivalence

Mr. Nance,

I have a question Lesson 11 of Intermediate Logic. I have all of the right answers to the exercises, but I noticed that on a few of the questions, I wrote more lines in my answers because I thought I had to be exhaustive in my efforts to find no contradiction. A specific example would be #5. I agree that there is only one way for the conditional to be false. But, there are multiple ways the conditional can be true. Why didn’t you try lines with I as true and C false, or C false and I true? Am I misinterpreting the fourth instruction from p.70 to “switch the assigned truth values and try again”? Is this directive leveled at each whole proposition, or at each constant? (I hope this is making sense). Continue reading Shorter Truth Tables for Equivalence

# Improving in Proofs

If you are studying Intermediate Logic, Unit Two, and you are having trouble writing formal proofs of validity (especially if you are in either Lesson 15 or 17), here are my last two suggestions (after reading THIS and THIS):

1.  Work together with someone who can help you. If you do not have a study partner, shoot me a question on my Facebook logic page. I would be happy to answer specific questions about solving any of the proofs. Or give me a call during work hours. My phone number is on my personal Facebook page. I’m serious.
2. If you and a friend finally get through the proofs in the exercise after a lot of struggle and effort together, then do this. Take a break, go have lunch. Then return to the exercise, and re-do it, looking back at the answers if you need to. Repeat until you can complete all of the proofs without looking at the answers.

Logicians hate this trick because it lets you solve proofs without effort!

Sorry. No magic pill. No “Logicians hate this trick because it lets you solve proofs without effort!” Just hard work and practice.

# Rules for Proofs

Two types of rules can be used to justify steps in formal proofs: rules of inference and rules of replacement. In order to use these properly, you should understand the differences between them.

The main difference is that rules of inference are forms of valid arguments (that’s why they have a therefore ∴  symbol), but rules of replacement are forms of equivalent propositions (which is why they have the equivalence sign  ≡  between the two parts).  This fundamental distinction is the cause of all other differences in how they are applied in proofs. Continue reading Rules for Proofs

# An exhortation to teachers regarding formal proofs

Formal proofs are hard, like many other things worth learning!
In this video, I talk through the difficulties of formal proofs of validity, and why it’s worth enduring the hardship to learn them.

# A “100 Cupboards” Story

My students and I are having a lot of fun in Good Books I . We finished The Horse and His Boy by C. S. Lewis, and are now reading N.D. Wilson’s 100 Cupboards. Later we will read The Hobbit by J.R.R. Tolkien, and Watership Down by Richard Adams. One way to appreciate a story is to enter into it, trying the think the author’s thoughts after him. So I thought you might enjoy the writing assignments I am giving my students after each book:

The Horse and His Boy
Write a story about an adventure from your own life, and make your story imitate the style of Aravis’ story in chapter 3. The story should be true, though you may embellish it slightly.

100 Cupboards
In chapter 12, Henry and Richard have been to Tempore, Carnassus, Badon Hill, and perhaps other places we are not told about. I want you to write a short story about Henry and Richard going through another cupboard looking for Henrietta. Make your story consistent with the rest of the story, the characters, and the cryptic description of the place from Grandfather’s journal.

The Hobbit
After the Battle of the Five Armies we are told that, on his return home, Bilbo had many adventures (the wild was, after all, still the wild), but he was never in any real danger because the orcs were scattered or destroyed, and he was with Beorn and Gandalf most of the way. Write a story about an adventure Bilbo had on his trip home.

Watership Down
Throughout this classic, the rabbits tell stories about their folk hero, El-ahrairah: “The Blessing of El-ahrairah,” “The King’s Lettuce,” “The Trial of El-ahrairah,” “The Black Rabbit of Inle,” and “Rowsby Woof and the Fairy Wogdog.” Write another tale of El-ahrairah.

# Re: Formal Proofs

Formal proofs of validity are challenging. Unlike truth tables (longer and shorter), completing formal proofs is not merely a question of following all the steps correctly; they require some creativity. Consequently, students may have more difficulty solving them. But some students enjoy the challenge of figuring out how to prove the conclusion. It is very much like solving puzzles, and can be an enjoyable challenge. This is how the instructor should present them. I have found that many of my students over the years have risen to the challenge, done exceptionally well with formal proofs, and enjoyed them.

Formal proofs of validity give students practice thinking in a straight line. The process teaches them how to connect premises in a proper way in order to reach the desired conclusion. For example, consider this argument:

If we want to send a manned mission to Mars then it must be either funded by taxpayers or privately funded. We want to send a manned mission to Mars and other planets. It should not be funded by taxpayers. Therefore a Mars program must be privately funded.

How do I get to that conclusion? The argument can be symbolized as follows:

M ⊃ (T v P)    M • O    ~T    ∴ P

This can be shown to be valid by truth table, but how to we prove the conclusion by connecting the premises? In the video below, I work through the proof, showing how to connect the premises using the rules of inference to reach the desired conclusion.

Trouble with video? YouTube version HERE.

# Dilemmas in Stories

Great stories often owe their greatness in part to dilemmas that confront the protagonist, who must make some difficult choice. Below, I have summarized several example dilemmas from stories I love. As you read through them, try to figure out which method is (or could be) used to escape the dilemma in the story: going between the horns, grasping the horns, or rebutting the horns with a counter-dilemma.

The Odyssey
If Odysseus sails close to the rocks then he will lose some men to Scylla, but if he sails close to the whirlpool then he will lose his entire ship to Charybdis. He must either sail close to the rocks or close to the whirlpool. Thus he will either lose some of his men to Scylla or lose the entire ship to Charybdis.

The Aeneid
If Aeneas stays in Carthage then he will not fulfill his destiny to found Rome, and if he flees to Italy then he will lose the pleasures of a kingdom. He will either stay or flee, therefore he will either lose Rome or lose Carthage.

The Fellowship of the Ring
If Frodo goes to Mordor alone, then he will likely fail in his quest, but if he goes to Mordor with the fellowship then he endangers his friends. He will either go alone or with the fellowship. Therefore he will either endanger his friends or he will likely fail in his quest.

The Lion, the Witch, and the Wardrobe
If the Narnians release the traitor Edmund to the Witch then he will be killed, and if they do not let the Witch have him as her rightful kill for treachery then Narnia will perish in fire and water. The Narnians must either release Edmund, or not let the Witch have her rightful kill. Therefore either Edmund will be killed, or Narnia will perish.