# 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.

Enjoy!

# 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.”

# 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:

# 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.

# The Biblical Biconditional

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”?

# 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.

# If/Then Truth Table

One of the difficulties new students of symbolic logic must overcome is understanding the defining truth table for the conditional, the “if/then” logical operator. The defining truth table tells us what the truth value of the proposition is, given the truth value of its component parts. For the conditional, it looks like this:

p    q     p ⊃ q
T    T         T
T    F         F
F    T         T
F    F         T

One way to defend this is to look at real-life conditional propositions with known truth values, for which we also know the truth value of the component parts. We will take our examples from the Bible.

The first row of the defining truth table states that a conditional with a true antecedent and a true consequent is true. In Genesis 44:26, Judah says about Benjamin, “If our youngest brother is with us, then we will go down.” The antecedent “Our youngest brother is with us” is true, and the consequent, “We will go down” was also true. We also know this is a true statement; Judah is speaking truthfully. There is the first row: If true then true is true.

The second row says a conditional with a true antecedent and a false consequent is false. In Judges 16:7 Samson says to Delilah, “If they bind me with seven fresh bowstrings, not yet dried, then I shall become weak, and be like any other man.” In verse 8 they bind him with seven fresh bowstrings (i.e. the antecedent is true) , but in verse 9 he breaks them easily (the consequent is false). So in verse 10, Delilah recognizes that Samson had lied to her; that is, she knows the conditional was false. Thus, if true then false is false.

The third row says a conditional with a false antecedent and a true consequent should be considered true. In Genesis 24:41, the servant quotes Abraham: “You will be clear from this oath when you arrive among my family; for if they will not give her to you, then you will be released from my oath.” The antecedent of the conditional (they will not give her to you) is false (they do give her), but the consequent (you will be released from my oath) is true (given the statement in verse 41 prior to the conditional). And we know the conditional was true; Abraham was speaking the truth. So if false then true is true. (See also Genesis 34:17).

Finally, the fourth row says conditionals with false antecedents and false consequents are also true. Here is one example: “If I find in Sodom fifty righteous within the city, then I will spare all the place for their sakes” (Genesis 18:26). The conditional is true, since it is God speaking. But note that the antecedent (God finds in Sodom fifty righteous people) is false, and the consequent (God spares the city) is also false. (See also Genesis 42:38). Thus, if false then false is true.

We see that it is not difficult to find examples of if/then statements in the Bible that support the traditional defining truth table for the conditional logical operator.

# Minecraft Logic

So apparently, creating digital logic circuits on the game Minecraft redstone is a thing.

I was recently sent some screen-captures of the answer to Exercise 34, problem 4. You can create the circuit in the game, and it will give you the outputs for the various inputs. It appears to use an SPST switch for the inputs.

Anyone else out there use the Minecraft game for their Digital Logic studies?

# 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.

# Logic Video Session Info

If you are using the Introductory or Intermediate Logic videos to teach your students, you may want to know the duration of the sessions. That information is now available on this printable document: Video Session durations.

Here are some quick facts:

Introductory Logic Videos
Total duration 10 hrs, 52 min, 33 sec.
Average session 13 min, 7 sec. (excluding the test and optional sessions)
Longest session 33 min, 21 sec.
Shortest session 5 min, 33 sec.
Total number of sessions 45.

Intermediate Logic Videos
Total duration 12 hrs, 30 min, 42 sec.
Average session 14 min, 14 sec. (excluding the test sessions)
Longest session 49 min, 54 sec.
Shortest session 4 min, 35 sec.
Total number of sessions 51.

You’re welcome.