# Impressive Logic Students

Those who have taught a subject for many years have occasionally had the blessing of teaching a student who naturally has (or has developed) an intuitive insight into the subject.  I was blessed this way by two of my logic students this morning!

I was teaching Intermediate Logic, Lesson 34, “Converting Truth Tables into Digital Logic Circuits.”  We were going over the exercise, problem 3, for which the final answer in the text is the unsimplified proposition (~A • ~B • C) v (A • ~B • C). One of my students said, “I know that B has to be false and C has to be true, and it doesn’t matter about A. So that would simplify to ~B • C.” When I asked him if he has been reading ahead to future lessons, he said no, “I can just see it by looking at the proposition.” I was very impressed. He just saw future lessons on simplification techniques, without being taught them.

But the surprises for the morning were not yet over. Continue reading Impressive Logic Students

# Applying Intermediate Logic

The most recent edition of Intermediate Logic has two new sections, Unit 4 and Unit 5. I’ve included these new units in the text because I wanted to answer the important question, “What are some practical applications of the tools that we are learning about, i.e. truth tables, formal proofs, and truth trees?” I believe that the applications of these tools in the new units will deepen and solidify student understanding of the concepts that, up to that point, have been largely theoretical.

Unit 4 (Lesson 28) gives logic students the opportunity to analyze chains of reasoning. The arguments to be analyzed are taken from Boethius’s The Consolation of Philosophy, the Apostle Paul’s argument proving the general resurrection of the dead in I Corinthians 15, and a section on angelic will from Augustine’s City of God. I work through this last one in full on the DVD.

An exercise not in the text that may be beneficial would be to have students write their own chains of reasoning, arguing for a conclusion of their choosing, in imitation of these authors. Their arguments must include at least one NOT, AND, and OR, and two IF/THENs. Tell them to include a truth-table or truth-tree analysis of their own chain of reasoning.

Unit 5 (Lessons 29-40) teaches students how to apply what they have learned to the fascinating topic of Digital Logic. Do not be intimidated by the 0’s, 1’s, and new symbols. It’s just the same old true, false, and logical operators that they have already learned about presented in a new way. Students often find this a fun application of what they have learned. It helps them to understand the electronic world around them, and it shows that the tools that they have learned apply not only to philosophy and theology, but to digital clocks and iPhones!

# Do these truth tree branches close?

Mr. Nance,

I have a student asking me if this would be a valid way of completing this truth tree for consistency. She thinks since she already found inconsistency in the branches, that she doesn’t have to do line 3 (per lesson 24). I’m thinking that this thought process only applies if she finds it consistent, not inconsistent. She’s also asking if it matters the order they are done in (I told her non-branching first, then branching, but that I didn’t think the order mattered if it was all branching that was left). Please help me give her direction! Continue reading Do these truth tree branches close?

# Logic & the Resurrection

Intermediate Logic Unit 4 teaches how to apply the tools we have learned in logic to real-life arguments. One such argument is contained in 1 Cor. 15:12-20, in which Paul argues that Christ has been raised from the dead, and as such He is the firstfruits of the general resurrection to come.

There are many points to Paul’s argument, but the main one is from verses 13, 16, and 20:

“If there is no resurrection of the dead, then Christ is not risen…But now Christ is risen from the dead, and has become the firstfruits of those who have fallen asleep.”

This argument can be symbolized as follows (C = Christ is risen, R = There is a resurrection of the dead):

~R ⊃ ~C    C     ∴   R

You can use the tools of truth table, truth tree, or formal proof to demonstrate that Paul’s argument is valid.

There are several other arguments in 1 Cor. 15:12-20, but the others leave premises assumed, so they take additional effort to analyze. But it is a beneficial exercise to work through them. Look at Exercise 28b.

Have a blessed Good Friday!

# Where does the CPA come from?

Mr. Nance,

I’m stumped on Logic lesson 18 #5. We got same answers as answer key until line 7…I can see from line 8 why line 7 is important, but how did we deduce a consequent that was not the original consequent of line 1 (from which we assumed the antecedent in line 3)?

Hope that makes sense! Continue reading Where does the CPA come from?

# 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

# 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.

If Tom Sawyer confesses that Injun Joe killed Dr. Robinson, then Injun Joe will kill him. If he doesn’t confess, then Muff Potter will be falsely accused. He will either confess or he won’t. Hence, either Injun Joe will kill him, or Muff Potter will be falsely accused.

Watership Down
If Hazel and his rabbits again ask the Efrafans for some does then they will be imprisoned. If they try to fight the Efrafans then they will lose. They either ask them or fight them. Therefore they will either be imprisoned or defeated in battle.

The Princess Bride
If Westley and Buttercup enter the Fire Swamp then they will be killed by flame, quicksand, or R.O.U.S. If they do not enter the Fire Swamp then they will be captured by Humperdinck. They enter the Fire Swamp or they do not, so they will either be killed or captured.

Harry Potter and the Sorcerer’s Stone
If Harry seeks the Sorcerer’s Stone then he will be expelled, but if he does not seek the Stone then Voldemort will return. Harry will either seek the Sorcerer’s Stone or he will not, so he will either be expelled or Voldemort will return.

Can you think of dilemmas that the protagonists face in other stories you have read?

# Shorter Truth Table Summary

Here is a chart that summarizes the shorter truth table methods for validity, consistency, and equivalence, taught in Intermediate Logic Lessons 8 through 11.

# Shorter truth table flow

If you are going through Intermediate Logic Lesson 9 and want some help understanding shorter truth tables for validity, here is a flowchart that shows the decision path:

# Must we do every unit?

Over my 25 years of teaching logic, I have often been asked this question:

“Intermediate Logic is a challenging course, especially trying to complete it all in one semester. Is each unit equally important, or can I skip something if I can’t fit it all in?”

The short answer is “You don’t have to do it all.” Unit 1 on Truth Tables is foundational to propositional logic, as is Unit 2 on Formal Proofs. Both of these are essential and must be completed by every student. Unit 3 teaches the Truth Tree method. A truth tree is another tool that does the same job as a truth table: determining consistency, equivalence, validity, etc. Some people like truth trees more than truth tables, since they are more visual. But Unit 3 could be considered an optional unit. Unit 4 covers Applying the Tools to Arguments. This is where the rubber meets the road for propositional logic, showing how to apply what has been learned up to this point to real-life reasoning. Consequently, Unit 4 should be completed by every student. Note that if you skip Unit 3, one question in Unit 4 will have to be skipped (namely, Exercise 28c #1). Unit 5 on Digital Logic – the logic of electronic devices – is entirely optional. Like Unit 4, this unit covers a real-life application of the tools of propositional logic, but one that is more scientific (though ubiquitous in this age of computers and smart phones). Though optional, many students find that they really enjoy digital logic.

As a teacher I have sometimes skipped either truth trees or digital logic. In fact, only with my best classes have I taught both Unit 3 and Unit 5. The Teacher Edition of the Intermediate Logic text includes two different schedules, one for completing every unit, and another for skipping Unit 5.

For answers to more FAQs, take a look HERE.