Category Archives: Intermediate Logic

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?
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

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)  ✔




/        \

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

|                               V

|                               O

      /        \

A            ~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.


Help Solving Proofs

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.


The ambiguous OR

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.”

Quick negation rules

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.