All posts by Roman Roads

Relating Terms from Birmingham Jail

One practical method of organizing arguments is to identify relationships between terms. Terms may be related as different parts of a whole (including different steps in a process) or as different species of a genus. In his “Letter from Birmingham Jail” Martin Luther King Jr. uses both methods of relating terms to organize and clarify his arguments. Continue reading Relating Terms from Birmingham Jail

Limitations of Logic

Limitations of Logic

“Without the aid of trained emotions the intellect is powerless against the animal organism… In battle it is not syllogisms that will keep the reluctant nerves and muscles to their post in the third hour of bombardment. The crudest sentimentalism (such as Gaius and Titius would wince at) about a flag or a country or a regiment will be of more use. We were told it all long ago by Plato. As the king governs by his executive, so Reason in man must rule the mere appetites by means of the ‘spirited element.’ The head rules the belly through the chest.” — C. S. Lewis, The Abolition of Man

Truth Tree Catechism

Q: What is a truth tree?
A: A truth tree is a diagram that shows a set of compound propositions decomposed into literals following standard decomposition rules.

Q: What is a literal?
A: A simple proposition symbolized as a constant or variable, or the negation of the same.

Q: What does it mean to decompose a compound proposition?
A: It means to show the components that must be true for the decomposed proposition to be true. A fully decomposed proposition is broken down into literals.

Q: Why do some compound propositions branch when decomposed?
A: 
The branching shows that there is more than one way for the proposition to be true.

Q: What does consistency mean?
A: Consistent propositions can all be true at the same time.

Q: How does the truth tree show consistency?
A: If the propositions in the set are fully decomposed into literals on at least one branch without contradiction, the propositions are consistent.

Q: What does it mean to recover the truth values?
A: It means to show the truth values of the component propositions that make every proposition in the given set true.

Q: What does SM mean?
A: It stands for Set Member; a label for a proposition in the given set.

Q: What is the meaning of the number and the symbols at the end of a row?
A: It is the justification for the decomposition, showing the number of the compound proposition that is decomposed, and the abbreviation of the rule used to decompose it.

Q: What is the meaning of the Ο at the bottom of a truth tree branch?
A: It designates an open branch, meaning that there are no contradictions on that branch.

Q: What is the meaning of the numbers separated by an Χ at the bottom of a branch?
A: The X designates a closed branch; the numbers are the line numbers of the propositions that contradict on that branch.

Q: What is the benefit of using truth trees?
A: Truth trees do the same things as truth tables — showing consistency, equivalence, validity, etc. — but in a visual way. They are a tool used in higher-level logic.

A Logical Declaration

Have you ever read through the Declaration of Independence and thought to yourself, “What a clear presentation of reasoning. Why can’t our leaders today argue in such a straightforward way?” If you read the Declaration carefully, you can discern this pattern of reasoning:

If P then Q
P
Therefore Q.

This pattern of reasoning is called modus ponens, and the three parts of the Declaration can be understood in this way.

     If  any form of government becomes destructive of our unalienable rights of life, liberty, and the pursuit of happiness, then it is the right of the people to alter or to abolish it, and to institute new government.

     The government of the king of Great Britain has shown itself to be destructive of these rights (as shown in a long list of grievances).

Therefore, these United Colonies are, and of right ought to be free and independent States.

One wishes that such clear reasoning would once again rule in the hearts and minds of our nation, its people and its rulers.

 

I’m Just a Bill – Redux

How does a bill become a law (from the U.S. Constitution, Article 1, section 7)? My answer to this question (putting our propositional logic tools to use) is:

(H • S • P) • {[A ∨ (~R • ~C)] ∨ (V • O)}

Given: H = The bill passed the House; S = The bill passed the Senate; P = The bill is presented to the President; A = The President approves and signs the bill; = The bill is returned by the President within ten days; = The Congress by adjournment prevent the return of the bill; V = The President vetoes the bill (i.e. he returns it with objections to congress); O = The veto is overridden (i.e. the bill is reconsidered and approved by two-thirds majority of both houses).

Truth tree decomposition is a rarely used tool, but this is one time we want to pull it out of the toolbox. Decomposing that compound proposition shows remarkably clearly the three paths that a bill can take in becoming a law:

(H • S • P) • {[A ∨ (~R • ~C)] ∨ (V • O)}  ✔

H • S • P  ✔

[A ∨ (~R • ~C)] ∨ (V • O)  ✔

H

S

P

/        \

A ∨ (~R • ~C)  ✔              V • O  ✔     

|                               V

|                               O

      /        \

A            ~R • ~C  ✔

~R

~C

Now, wasn’t that fun?

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.

The Adventures of Tom Sawyer
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?

Must we do every unit of Intermediate Logic?

A common question for new parents, teachers, or tutors going into Intermediate Logic:

“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.

It is common for teachers to skip either truth trees or digital logic. In fact, only the best classes successfully complete 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.

Rules for Guessing

Shorter truth tables can help us find if an argument is valid, or a set of propositions are consistent, or if two propositions are equivalent. However, when completing a shorter truth table, we must sometimes guess a truth value for a variable. This occurs when there are no “forced” truth values — that is, when there exists more than one way to complete the current truth value for every remaining proposition.

Here are two rules to keep in mind when you must guess a truth value:

  1. If guessing allows you to complete the shorter truth table without contradiction, then stop; your question is answered. Either you have shown the argument is invalid, or the given propositions are consistent, or the two propositions are not equivalent.
  2. If the guess leads to an unavoidable contradiction, then you must guess the opposite truth value for that variable and continue, because the contradiction just might be showing that your guess was wrong.

Take a look at this post for a flowchart for guessing with validity.

 

Why Define?

We define terms in order to make their meaning understood, of course. But we might ask, what does understanding the meaning of a term give us? Let’s dig a little deeper into this question.

There are at least six purposes for defining terms.

  1. To show relationships between terms. A good example is Aristotle, in his Rhetoric, defining indignation as “a feeling of pain caused by the sight of undeserved good fortune” and pity as “a feeling of pain cause by the sight of undeserved bad fortune.” This helps us to see how these terms are related, and that a just man would tend to feel both indignation and pity, because he has an understanding of what is deserved.
  2. To remove ambiguity. An ambiguous word has more than one meaning, like the word bar. “A tall man walked into a bar. He said ‘ouch!'” We define terms in order to deal with equivocations like this, making it clear which meaning we have in mind when we use the word.
  3. To reduce vagueness. A vague word is unclear as to its extent. The instructions for constructing a model, instead of saying “For adults,” will give a more precise definition for adult, e.g. “For people 18-years old and up.” Such definitions will usually not be found in dictionaries,  applying only to particular situations.
  4. To increase vocabulary. Much teaching consists in giving students new meanings for new terms, thus enlarging their vocabulary and adding to their storehouse of knowledge. Thus, when we learn the definitions of ambiguous and vague, our personal vocabulary has increased, and we can begin to understand how these words relate.
  5. To explain theoretically. Often a definition will be theoretical in a way that accepting the definition implies that one is buying into a particular theory. If you agreed that light is “visible electromagnetic waves,” this is the same as accepting the wave theory of light (as opposed to the particle theory).
  6. To influence attitudes.  Sometimes we give a definition for a term in order to make people feel a certain way, either good or bad, about the thing being defined. When you father calls the television “a one-eyed brain sucker,” he is trying to influence your attitude about watching it.

There are no doubt additional purposes for defining terms, but this list of six perhaps goes a little ways toward expanding our understanding.