All posts by Roman Roads

Truth Tree Catechism

Intermediate Logic, Logic

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.

I’m Just a Bill – Redux

Intermediate Logic, Logic

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; R = The bill is returned by the President within ten days; C = 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?

Must we do every unit of Intermediate Logic?

Intermediate Logic, 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.

Why Define?

Logic, Why learn logic?

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.

Help Solving Proofs

Intermediate Logic, Logic

If you are in Intermediate Logic and learning about proofs for the first time, or struggling through them again for the second or third time, here are some helpful suggestions for justifying steps in proofs, constructing proofs, or just getting better at proofs.

  1. Think about what a proof does. Recognize that the conclusion of a previous step becomes a proposition to use as a premise for a new step. Proofs are a series of connected arguments, conclusion of previous arguments becoming premises for new ones.
  2. If you are learning how to justify steps in proofs (that is, you are working on Exercise 14a:10-16, or 15a:1-6, or 16:11-18) and you are in the middle of a proof, ask yourself which steps you have not yet used. If you are trying to justify step 6, and the previous lines already used steps 1, 3 and 4, then you will probably use steps 2 and 5. Try reading them aloud, and listen for familiar patterns from the premises of the rules of inference.
  3. Rewrite the argument that you are trying to prove. This will help you more clearly see the premises you have and the conclusion you are aiming at. You also might recognize patterns for rules of inference and replacement that you need to use. Often, a proof is built around a single rule of inference or replacement, and the other steps are just needed to set the premises up. For example, if you read Exercise 17a problem 7 aloud, you might recognize the modus tollens. But it takes a couple of steps to set up the second premise of the modus tollens, and one step afterward to fix the double negation.
  4. In general, find the premises you have available to you (e.g. if you’re on step 5, the available premises are from steps 1-4), read them aloud, and listen for rule patterns. In fact, get used to the patterns of the rules by reading them aloud, using something other than p’s and q’s (e.g. for Disjunctive Syllogism say to yourself, “This or that, not this, therefore that.”
  5. See if later proofs use procedures from earlier proofs. Exercise 17a problem 9 is built around a hypothetical syllogism, but you need to modify the proposition is line 1 to turn the conjunction into a disjunction so that the middles match for the H. S. To do that, you follow the procedures you used in problems 3 and 4.
  6. If you’re stuck, consider whether the next step might use the rules of Addition or Absorption. These are the rules that are often difficult to see when you need to use them. This is why in Exercise 14a:10-16, five of the seven proofs use one of these two rules, and in Exercise 15a, half of them use one of these rules. You need the practice.
  7. Another hint for if you are stuck constructing a proof is to try writing down every possible conclusion you can make from the available premises, and see if any of them help.
  8. You may have struggling through the assignment, succeeded writing some proofs but needed to look at the answer key for others. That’s okay. But I would suggest that you then go back and do the assignment again without looking. Practice makes habit.
  9. If it is still hard for you, if you are still not quite getting it, don’t sweat it. Take your time. Go drink some coffee and come back. Don’t say to yourself, “Well, I don’t get this, but I’ll just go on to the next lesson.” No. The lessons build on each other. If you are worried about getting through the entire text, stop worrying about it. You don’t need to cover it all. Better to learn a small amount of material well then a large amount of material poorly.

Think about proofs like solving a puzzle, rather than thinking of it like homework. Make it a fun challenge.

Enjoy!

The ambiguous OR

Intermediate Logic, Logic

Logic is a symbolic language. It is also a very precise language, every term well defined and unambiguous. English, on the other hand, is a somewhat ambiguous language. The same word can have multiple meanings: a pen is a writing utensil and an enclosure for livestock.

One key term in logic is the disjunction “or”. In English, the word “or” has two meanings. The first is the inclusive or, which means basically “this, or that, or both.” If someone said, “Most Bible students read the King James or the NIV,” this statement is still true for a student who reads both the King James and the NIV. The “or” includes both possibilities.

The exclusive or basically means “this or that, but not both.” This is the sense used in this classic argument for the deity of Christ: “Jesus was either God or a bad man.” If Jesus was God, then He was not a bad man. If He was a bad man, then He was not God.

Symbolic logic deals with the ambiguous “or” this way. The logical operator OR is taken in the inclusive sense. “A or B” is true if A is true, B is true, or both A and B are true. To represent the exclusive or, we use the compound proposition “A or B, but not both A and B.”

King’s Grand Style

Rhetoric

It has been maintained that Martin Luther King Jr. was the last American orator to use the grand level of style appropriately. In  Fitting Words Classical Rhetoric, the grand level defined as that “in which the stylistic devices are intended to be dramatic, apparent, and impressive. Its purpose is not only to inform the mind and persuade the will, but to grip the emotions and heart. It is most appropriate for speeches delivered on formal occasions.”

Anyone who has listened to (or at least read) some of his speeches – especially his most famous “I Have a Dream” – is aware that MLK uses stylistic devices in a dramatic and impressive way, a way that can grip the mind and heart of his hearers.  Here are some quotes from my text which shows his skill in using the grand level of style.

Perhaps no modern orator mastered rhythm as well as Martin Luther King, Jr. Consider the rhythm of these sentences, taken from his powerful speech “I Have a Dream”:

In the process of gaining our rightful place, we must not be guilty of wrongful deeds.

And so, we’ve come to cash this check, a check that will give us upon demand the riches of freedom and the security of justice.

I have a dream that my four little children will one day live in a nation where they will not be judged by the color of their skin but by the content of their character.

These beautifully framed words are memorable not only for what they say, but for how they were said. Listen for the rhythm by reading them aloud or, better yet, find a recording of Dr. King’s original delivery of the speech. (Fitting Words, page 279)

A study of the most powerful speeches will reveal that many of them end with a longer, highly-coordinated sentence, a sentence that climaxes in a short, powerful word. A few examples…

And when this happens, and when we allow freedom to ring, when we let it ring from every village and every hamlet, from every state and every city, we will be able to speed up that day when all of God’s children, black men and white men, Jews and Gentiles, Protestants and Catholics, will be able to join hands and sing in the words of the old negro spiritual, ‘Free at last, free at last; Thank God Almighty, we are free at last.’ (Martin Luther King, Jr., “I Have a Dream.”)

(Fitting Words, page 281f)

Much more could be said. We could identify dozens of figures of speech which he uses masterfully, even including a subtle use of rhyme. The orations of Martin Luther King Jr. are well worth studying by the student of formal rhetoric.

Quick negation rules

Intermediate Logic, Logic

Here are some quick rules to help you symbolize propositions that use negation:

Not both p and q  =  ~(p ⋅ q)
Either not p or not q  =  ~p v ~q
Both not p and not q  =  ~p ⋅ ~q
Neither p nor q  =  ~(p v q)

Truth tables can be used to show that the first two proposition forms are equivalent, and the last two forms are equivalent. The meaning of the sentences also help to show this.

The Biblical Biconditional

Intermediate Logic, Logic

Symbolic logic has five standard logical operators, each of which has a standard translation in English:

negation is “not”
conjunction is “and”
disjunction is “or”
conditional is “if/then”
biconditional is “if and only if”

While the translations of the first four logical operators are frequent in English, the phrase “if and only if” is used very infrequently, and then only occasionally among mathematicians, philosophers, and lawyers.

For instance, while it is easy to find hundreds of nots, ands, ors, and if/thens in the Bible, the phrase “if and only if” is completely absent. However, for those who look carefully, biconditional reasoning is used several times in scripture. Keeping in mind that p if and only if q means if p then q and if q then p — and remembering other equivalences we have learned — the following verses all reflect biconditional reasoning:

Genesis 43:4-5, “If you will send our brother along with us, we will go down… But if you will not send him, we will not go down.”

2 Kings 7:4, “If they spare us, we live; if they kill us, then we die.”

John 6:53-54, “Unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Whoever eats my flesh and drinks my blood has eternal life.”

The first example could be translated as “We will go down if and only if you send our brother with us.” The second, “We will live if and only if they spare us.” The third, “You have eternal life if and only if you eat Christ’s flesh and drink His blood.”

Can you find any other biblical examples of statements that could be translated with the “if and only if”?