Peikoff Logic Homework on Hypothetical And Alternative Arguments
The following consists of my answers for one of the homework sets in Leonard Peikoff's Introduction to Logic course. The handout with the exercises is freely available on the Ayn Rand Institute's website for the course. These are from Page 7 on the handout. Note that I'm using the YouTube version of the lectures.
I used ✅ where I and Peikoff agreed and ❌ where I got a different answer and agreed with Peikoff's reasoning at the end of the day. For some questions I may use multiple emoji and the reason why should be clear upon reading my answer.
These problems deal with
- mixed-hypothetical syllogisms where one premise is a hypothetical (if-then) statement and one premise either affirms or denies part of that hypothetical statement.
- disjunctive syllogisms where one premise is a disjunct (either-or) and one premise either affirms or denies one of the disjuncts. Note that an assumption is made where we assume that statements are inclusive OR and not exclusive OR (XOR) unless exclusive OR is clearly specified.
See the links for more details on these syllogism types. This is fairly standard stuff so I won't recap it here.
Note that I use ¬ for negation, ∨ for inclusive-or, ⊂ or ➡ for implies (I switched part way through cuz I decided I liked ➡ better), and ∴ for therefore.
Note that Peikoff encourages writing shortened versions of the arguments when rewriting them. I have sometimes shortened them somewhat but not to the extent he suggests, since the context of his suggestion was for people who needed to write stuff out. That is just by way of indicating that my rewrites are not meant to be exact quotes but to capture the essence of the argument in a shorter and more standardized form.
Problem 1 ✅
To get to Boston by the time the speech started, he would have had to drive at least 60 mph on the average. Obviously, therefore, he did make the speech on time, because he always drives over 70 mph.
Informally Rewrite to Standardize
If he was in Boston by the time the speech started, then he drove at least 60 mph.
He always drives over 70mph.
Therefore, he made the speech on time.
Identify the Components
- Conclusion: Therefore, he made the speech on time.
- Hypothetical Premise: If he made the speech on time, then he drove at least 60 mph.
- Categorical Premise: He always drives [at least 60mph] (rewritten because we need to maintain the same language in the premises and we can treat over 70mph as equivalent to at least 60mph for the purposes of analyzing the syllogism).
Symbolize
| Statement | Symbol |
|---|---|
| If he made the speech on time, then he drove at least 60 mph. | P ⊂ Q |
| He always drives at least 60mph | Q |
| Therefore, he made the speech on time. | ∴ P |
Analysis
Invalid argument of the form affirming the consequent (or denying the antecedent if written differently as in "If he did not drive at least 60 mph, then he did not make the speech.".
Problem 2 ❌✅
Either the advocates of the Keynesian policies are ignorant of the facts of economics, or they are prejudiced in favor of statism. I know that these men are definitely prejudiced in favor of statism. It follows that at least they do know the facts of economics.
Informally Rewrite to Standardize
Either the advocates of Keynesian policies are ignorant of economics, or they are prejudiced towards statism.
They are prejudiced towards statism.
Therefore, they do know economics.
Identify the Components
- Conclusion: Therefore, they do know economics.
- Alternative Premise: Either the advocates of Keynesian policies are ignorant of economics, or they are prejudiced towards statism.
- Categorical Premise: They are prejudiced towards statism.
Symbolize ❌
(Erroneous): ❌
| Statement | Symbol |
|---|---|
| Either the advocates of Keynesian policies are ignorant of economics, or they are prejudiced towards statism. | P ∨ Q |
| They are prejudiced towards statism. | Q |
| Therefore, they do know economics. | ∴P |
Note: I initially made an error here in my symbolizing, and represented "ignorant of economics" and "know economics" with the same symbol. I actually managed to get the correct conclusion anyways regarding the validity of the argument, since, because of the form of the particular argument in question, the issue was that the categorical premise affirmed Q, which is invalid regardless of what's happening with P.
Corrected: ✅
| Statement | Symbols |
|---|---|
| Either the advocates of Keynesian policies are ignorant of economics, or they are prejudiced towards statism. | ¬P ∨ Q |
| They are prejudiced towards statism. | Q |
| Therefore, they do know economics. | ∴P |
Analysis
Invalid. We know from the alternative premise that either one of two things is true. Because we are treating these as OR and not XOR statements, we can't make a definite claim about whether one alternant is true based on knowledge that the other is true.
Problem 3 ✅
Thomas Aquinas, Summa Theologica: “There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible.”
Informally Rewrite to Standardize
If something were to be its own efficient cause, it would be prior to itself.
Something being prior to itself is impossible.
Therefore, something being its own efficient cause is impossible.
Identify the Components
Conclusion: Therefore, something being its own efficient cause is impossible.
Hypothetical Premise: If something were to be its own efficient cause, it would be prior to itself.
Categorical Premise: Something being prior to itself is impossible.
Symbolize
| Statement | Symbol |
|---|---|
| If something were to be its own efficient cause, it would be prior to itself. | P⊂Q |
| Something being prior to itself is impossible. | ¬Q |
| Therefore, something being its own efficient cause is impossible. | ∴¬P |
Analysis
Valid example of denying the consequent. If you know that if P then Q, and you know that Q didn't happen, you can say that P didn't happen.
Problem 4 ❌
Only if you are eligible for marriage are you not a child. If you are a child, then you don’t wear large-size clothing. If you’re tall, you wear large-size clothing. Hence, if you’re not eligible for marriage, you’re not tall.
My original approach ❌
Informally Rewrite to Standardize
If you are a child, you are not eligible for marriage. (translation of "only if" into "if" statement)
If you are a child, you do not wear large-size clothing.
If you're tall, you wear large-size clothing.
If you're not eligible for marriage, you're not tall.
❌Symbolize
My original approach:
P = You are a child.
Q = You're eligible for marriage.
R = You wear large size clothing.
S = You're tall.
| Statement | Symbol | Contrapositive Form |
|---|---|---|
| If you are a child, you are not eligible for marriage. (translation of "only if" into "if" statement) | P⊂¬Q | Q⊂¬P |
| If you are a child, you do not wear large-size clothing. | P⊂¬R | R⊂¬P |
| If you're tall, you wear large-size clothing. | S⊂R | ¬R⊂¬S |
| If you're not eligible for marriage, you're not tall. | ∴¬Q⊂¬S | S⊂Q |
I thought something seemed wrong and was confused, gave up and listened to Peikoff's answer.
Post-Mortem
Originally, I skipped steps and tried to jump directly to an expression of the only statement.
Let's try going more carefully:
The original statement was
Only if you are eligible for marriage are you not a child.
We can say this is represented as:
Only Q ⊂ ¬P.
In the lectures Peikoff said you could translate an "only" statement a couple of different ways.
One was to negate both elements. In symbols this would be:
¬Q ⊂ ¬(¬P).
In English it would be something like: If you are not eligible for marriage, you are not a non-child.
A non-child would be an adult, and if you're not a non-child (adult), that means you are a child. So the simplified statement would be: If you are not eligible for marriage, then you are a child.
The symbolic representation follows this reasoning. If you have a double negation on a statement, you can just drop the negations. That would be: ¬Q ⊂ P.
You could also take the contrapositive, which would be ¬P ⊂ Q, and in English would be "If you are not a child, you are eligible for marriage."
Here's a summary of what we've figured out so far:
| Statement | Symbol |
|---|---|
| Only if you are eligible for marriage are you not a child. | Only Q ⊂ ¬P. |
| If you are not eligible for marriage, you are not a non-child. | ¬Q ⊂ ¬(¬P). |
| If you are not eligible for marriage, then you are a child. | ¬Q ⊂ P |
| If you are not a child, you are eligible for marriage. | ¬P ⊂ Q |
Anyways, from the starting point of ¬Q ⊂ P:
| Statement | Symbol | Contrapositive Form |
|---|---|---|
| If you are not eligible for marriage, then you are a child. | ¬Q ⊂ P | ¬P ⊂ Q |
| If you are a child, you do not wear large-size clothing. | P⊂¬R | R⊂¬P |
| If you're tall, you wear large-size clothing. | S⊂R | ¬R⊂¬S |
| Therefore, if you're not eligible for marriage, you're not tall. | ∴¬Q⊂¬S | S⊂Q |
So now you can see the flow. Just to be super clear, I'll say stuff in English.
If you are not eligible for marriage, then you are a child. (¬Q ⊂ P)
If you are a child, you do not wear large-size clothing. (P⊂¬R)
If you do not wear large-size clothing, you are not tall (contrapositive of third statement in above table: ¬R⊂¬S)
Therefore, if you're not eligible for marriage, you're not tall (∴¬Q⊂¬S )
This is a valid argument.
Peikoff's Approach
This is closer to how Peikoff did it (though I use different negation notation and write stuff out in full where he uses shorthand)
Only if you are eligible for marriage are you not a child.
If you are a child, then you don’t wear large-size clothing.
If you’re tall, you wear large-size clothing.
Conclusion: Hence, if you’re not eligible for marriage, you’re not tall.
P = You're eligible for marriage
Q = You're not a child.
R = You do not wear large size clothing.
S = You're tall.
| Statement | Symbol | Alternate Form |
|---|---|---|
| Only if you are eligible for marriage are you not a child. | Only P⊂Q | ¬P ⊂ ¬Q (translation of Only) |
| If you are a child, you do not wear large-size clothing. | ¬Q⊂R | |
| If you're tall, you wear large-size clothing. | S⊂¬R | R⊂¬S (contrapositive) |
| If you're not eligible for marriage, you're not tall. | ∴¬P⊂¬S |
I see how Peikoff's approach flows. You go:
¬P⊂¬Q
¬Q⊂R
R⊂¬S
and therefore
¬P⊂¬S .
Problem 5 ✅
Either you take a full lunch hour or you complete your letter today, but not both. You can’t, therefore, take a full lunch hour, because you must complete your letter today.
Informally Rewrite to Standardize
Either you take a full lunch hour or you complete your letter, but not both.
You must complete your letter today.
Therefore, you cannot take a full lunch hour.
Identify the Components
Conclusion: Therefore, you do not take a full lunch hour.
Strong alternative statement: Either you take a full lunch hour or you complete your letter, but not both.
Categorical statement: You complete your letter today.
Symbolize
| Statement | Symbol |
|---|---|
| Either you take a full lunch hour or you complete your letter today, but not both. | P∨∨ Q (strong or/XOR) |
| You complete your letter today. | Q |
| Therefore, you do not take a full lunch hour. | ∴¬P |
Analysis
I don't recall Peikoff dealing with exclusive alternative arguments much in the lecture. However, I think that this argument is valid. "But not both" means that one and only one of P or Q can be true. If we know that Q is true, we therefore know that P is not true.
Peikoff characterizes the "either" statement as a hypothetical statement in the lecture, but I don't actually understand that. It's not an if statement.
Update: A friend said that one way of treating the either-or statement as an "If" statement might be "If you take a lunch, your letter will be incomplete." That makes sense, though if that's what Peikoff had in mind it seems like a bit of a leap to expect people to just follow that.
Anyways, said friend illustrated the point via the following truth table (the right-most table column represents a disjunction of an expression representing XOR, ((P ∨Q)&¬(P&Q)), and an expression representing the possibility of doing neither P nor Q. Notice that the P ➡¬Q, which represents "If you take a lunch, you will not complete your letter today" and the expression representing XOR + neither P or Q have the same truth table.
| P | Q | P ➡¬Q | ((P ∨Q)&¬(P&Q))∨(¬P&¬Q) |
|---|---|---|---|
| T | T | F | F |
| T | F | T | T |
| F | T | T | T |
| F | F | T | T |
Problem 6 ✅
In today’s world, a man becomes an altruist unless he is an independent thinker. I can only conclude that John must be opposed to altruism, because no one is more intellectually independent than he.
Informally Rewrite to Standardize
If John is not independent thinker, then altruist.
John is intellectually independent.
Therefore, John is opposed to altruism.
Identify the Components
Conclusion: Therefore, John is opposed to altruism.
Hypothetical Statement: If John is not independent thinker, then altruist.
Categorical Statement: John is independent thinker.
Symbolize
| Statement | Symbol | Contrapositive |
|---|---|---|
| If John is not independent thinker, then altruist. | ¬P➡Q | ¬Q ➡P |
| John is independent thinker. | P | |
| Therefore, John is opposed to altruism. | ∴¬Q |
Analysis
This is invalid. It Denies the antecedent.
(Peikoff made "John is not independent thinker" P instead of ¬P but otherwise agreed).
Problem 7 ✅
If you are rich, then you are happy. If you are not rich, then you haven’t worked hard. Unless you are happy, you need a psychiatrist. Hence, if you have worked hard, you do not need a psychiatrist.
Informally Rewrite to Standardize
If you are rich, then you are happy.
If you are not rich, then you haven't worked hard.
If you are not happy, you need a psychiatrist.
Therefore, if you worked hard, you do not need a psychiatrist.
Identify the Components
Hypothetical Statement: If you are rich, then you are happy.
Hypothetical Statement: If you are not rich, then you haven't worked hard.
Hypothetical Statement: If you are not happy, you need a psychiatrist.
Conclusion: Therefore, if you worked hard, you do not need a psychiatrist.
Symbolize
| Statement | Symbol | Contrapositive | Weak Alternative |
|---|---|---|---|
| If you are rich, then you are happy. | P➡Q | ¬Q ➡ ¬P | |
| If you are not rich, then you haven't worked hard. | ¬ P➡¬R | R ➡ P | |
| If you are not happy, you need a psychiatrist. | ¬Q ➡S | ¬S➡Q | S ∨ Q |
| Therefore, if you worked hard, you do not need a psychiatrist. | ∴R➡¬S |
Analysis
R ➡ P and P➡Q so R ➡Q but I don't see a way to go from Q to ¬S. Invalid.
Problem 8 ✅
There are two possible explanations of the American Constitution: (a) The Founding Fathers were philosophically committed to the principle of individual rights; (b) The Founding Fathers were selfishly interested in protecting their own private property. There is no doubt that this latter is definitely true. Therefore, the Founding Fathers were not philosophically committed to the principle of individual rights.
Informally Rewrite to Standardize
Either Founding Fathers were committed to rights, or they were selfish.
They were selfish.
Therefore, they were not committed to rights.
Note that "but not both" is specified in the first statement, and thus we assume that we are using the inclusive or.
Identify the Components
Weak Alternative Statement: Either Founding Fathers were committed to rights, or they were selfish.
Categorical Statement: They were selfish.
Conclusion: Therefore, they were not committed to rights.
Symbolize
| Statement | Symbol |
|---|---|
| Either Founding Fathers were committed to rights, or they were selfish. | P∨Q |
| They were selfish. | Q |
| Therefore, they were not committed to rights. | ∴¬P |
Analysis
This is "fallacy of weak alternation"; specifically, it affirms a disjunct.
Problem 9 ✅
Only were A a Q could G be a T. Thus G is not a T, since A is not a Q.
Informally Rewrite to Standardize
Only if A were Q could G be a T.
A is not a Q.
Therefore, G is not a T.
Translating the only statement: Only if A were Q could G be a T.
Translation 1: If G were T, A could be Q.
Translation 2: If A were not Q, G would not be T.
Let's analyze both variations, starting with translation 1:
First Argument Variation
Identify the Components
Hypothetical statement: If G were T, A could be Q.
Categorical Statement: A is not a Q.
Conclusion: Therefore, G is not a T.
Symbolize
| Statement | Symbol |
|---|---|
| If G is T, A is Q. | P➡Q |
| A is not a Q. | ¬Q |
| Therefore, G is not a T. | ∴¬P |
Analysis
This is denying the consequent, and is a valid argument.
Second Argument Variation
Now let's do the second translation of the only statement, keeping the mapping of the symbols the same as in our first variation.
Identify the Components
Hypothetical statement: If A were not Q, G would not be T.
Categorical Statement: A is not a Q.
Conclusion: Therefore, G is not a T.
Symbolize
| Statement | Symbol |
|---|---|
| If A is not a Q, G is not a T. | ¬Q➡ ¬P |
| A is not a Q. | ¬Q |
| Therefore, G is not a T. | ∴¬P |
Analysis
This variation is affirming the antecedent, and is valid.
Problem 10 ✅
Given the premises: If you study, you learn the course material; only if you graduate with honors and get a well-paying job have you done well in the course; you will be truly happy provided that you graduate with honors and get a well-paying job; if you don’t do well in the course, you haven’t learned the course material. Can one validly conclude: If you study, you will be truly happy?
Informally Rewrite to Standardize
If you study, you learn the course material.
Only if you graduate with honors and get a well-paying job have you done well in the course.
If you graduate with honors and get a well-paying job, then you will be happy.
If you don’t do well in the course, you haven’t learned the course material.
Conclusion: If you study, you will be truly happy?
The only statement can be written as either:
- If you do not graduate with honors and get a well-paying job, you did not do well in the course.
- If you did well in the course, you will graduate with honors and get a well-paying job.
Let's use the first one since we have another "didn't do well in the course". Now are premises are:
If you study, you learn the course material.
If you do not graduate with honors and get a well-paying job, you did not do well in the course.
If you graduate with honors and get a well-paying job, then you will be happy.
If you don’t do well in the course, you haven’t learned the course material.
Conclusion: If you study, you will be truly happy?
Identify the Components
Hypothetical Statement: If you study, you learn the course material.
Hypothetical Statement: If you do not graduate with honors and get a well-paying job, you did not do well in the course.
Hypothetical statement: If you graduate with honors and get a well-paying job, then you will be happy.
Hypothetical Statement: If you don’t do well in the course, you haven’t learned the course material.
Conclusion: If you study, you will be truly happy.
Symbolize
| Statement | Symbolic | Contrapositive Statement | Contrapositive Symbol |
|---|---|---|---|
| If you study, you learn the course material. | P➡Q | If you did not learn the course material, you did not study. | ¬Q➡¬P |
| If you do not graduate with honors and get a well-paying job, you did not do well in the course. | ¬R➡¬S | If you did well in the course, you will graduate with honors and get a well-paying job. | S ➡R |
| If you graduate with honors and get a well-paying job, then you will be happy. | R➡T | If you are not happy, then you did not graduate with honors and get a well-paying job. | ¬T➡¬R |
| If you don’t do well in the course, you haven’t learned the course material. | ¬S➡¬Q | If you learned the course material, you did well in the course. | Q ➡S |
| If you study, you will be truly happy. | ∴P➡T |
Analysis
By taking the contrapositive of our statements, we can build a lengthy chain of reasoning:
If you study, you learn the course material.
If you learned the course material, you did well in the course.
If you did well in the course, you will graduate with honors and get a well-paying job.
If you graduate with honors and get a well-paying job, then you will be happy.
Or symbolically:
P ➡Q
Q ➡S
S ➡R
R ➡T
The chain is valid.