Formal Logic

Question 1

Most serious students are happy students, and most serious students go to graduate school. Furthermore, all students who go to graduate school are overworked.

Which one of the following can be properly inferred from the statements above?

(A) Most overworked students are happy students.
(B) Some happy students are overworked.
(C) All overworked students are serious students.
(D) Some unhappy students go to graduate school.
(E) All serious students are overworked.

Answer 1

Answer = (B)

Most serious students are happy + most serious students go to grad school = at least SOME students are happy + go to grad school.
Given ALL grad students are overworked, that means at least SOME students are happy and overworked (since grad school is a sufficient condition for “overworked”)

(D) cannot be inferred since the prompt provides no info on unhappy students at all. Maybe unhappy students do not go to grad school. The only info we have is of the subset of ‘happy students’.

Domain: Students
Serious - M -> happy
Serious - M -> grad school
Grad school -> overworked
Therefore, Serious - M -> overworked (grad school)

Question 2

B swims immediately before C only if D does not swim immediately before E =

Answer 2

BC -> X DE

Question 3

G does not speak 4th unless Q speaks 2nd =

Answer 3

G4 -> Q2

Question 4

In an experiment, two-year-old boys and their fathers made pie dough together using rolling pins and other utensils. Each father-son pair used a rolling pin that was distinctively different from those used by the other father-son pairs, and each father repeated the phrase “rolling pin” each time his son used it. But when the children were asked to identify all of the rolling pins among a group of kitchen utensils that included several rolling pins, each child picked only the one that he had used.
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Which one of the following inferences is most supported by the information above?
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(A) The children did not grasp the function of a rolling pin.
(B) No two children understood the name “rolling pin” to apply to the same object.
(C) The children understood that all rolling pins have the same general shape.
(D) Each child was able to identify correctly only the utensils that he had used.
(E) The children were not able to distinguish the rolling pins they used from other rolling pins.

Answer 4

The question stem asks for “most supported.” (I Overlooked a lot of the qualifiers during my first attempt.)
Answer = (B)

X(A) - Not strongly supported. It seems possible that the children grasped the function because they were at least able to identify the rolling pin they used.

V(B) - We know no two children identified the same rolling pin and they each only identified the one they used. Hence, none of them thought the name “rolling pin” applied to anything beyond what they used. So this must be true.

X(C) - Goes against the stimulus. If this were true, then the children would all pick out more than one rolling pin.

X(D) - Out of scope. “Utensils” does not equal to “rolling pins.” We do not know anything about how the children handled other utensils.

X(E) - Goes against the stimulus. We know the children are able to identify the rolling pin they used.

Question 5

We can use the validity test to test whether something is a necessary assumption.

If it makes the argument valid, the assumption is necessary to the argument.

Is this correct?

Answer 5

No.
A necessary assumption is NOT required to make an argument valid.

If an option being true would make the argument look like it does NOT need additional support, then this option is SUFFICIENT.

If you assume an option is true and the argument looks like it checks out, then the option is BEYOND necessary; it qualifies as sufficient.

Question 6

Do necessary assumptions improve the validity of the argument?

Answer 6

NO.

When necessary assumptions are not true, the argument is factually untrue.

If the necessary assumption is true, the argument does not benefit from being more valid.

Question 7

As trees age, they grow rings. Therefore, counting the number of rings a tree has will tell us how old a tree is.

“Trees grow one ring per year” would be a necessary assumption. Yes or no?

Answer 7

No.

Knowing how old a tree is by the number of rings requires that the age of the tree be correlated to the number of rings in a constant rate and uniquely so.

A tree can grow one ring per year or per 4 years and it would still let us determine its age. Hence, “trees grow one ring per year” is NOT necessary.

Question 8

L attends the event unless K attends =

Answer 8

XL -> K

Question 9

G speaks 4th unless Q speaks 2nd =

Answer 9

XG4 -> Q2

Question 10

XA -> B =

Answer 10

At least one must be affirmative.
(when one negated condition guarantees the affirmative of another.)

This includes the possibility that both A and B are affirmative.

Question 11

A -> XB =

Answer 11

The two conditions are mutually exclusive.
(when one affirmative condition guarantees a negated one.)

Question 12

…not… unless… ;
EX: A does not speak 2nd unless B speaks 5th.

Answer 12

The affirmative (the condition minus the “not”) of the first condition is the sufficient condition
In this case, A2 -> B5

Question 13

L does not attend unless K attends =

Answer 13

L -> K

Question 14

No cats like being walked =

Answer 14

If it’s a cat, it does NOT like being walked.

Question 15

No student goes to the gym =

Answer 15

If student -> No gym

Question 16

Does “unless” = “only if” ?

Answer 16

No. They are quite the opposite.

“Unless” signifies a necessary condition while “only if” signifies a sufficient one.

The rule of negating the first condition only applies to necessary conditions.

Question 17

L is selected only if K is also selected =

Answer 17

L -> K

Question 18

L does not perform 3rd unless K performs 1st =

Answer 18

L3 -> K1

Question 19

H goes 2nd unless B goes 5th =

Answer 19

X H2 -> B5

Question 20

If jays are in the forest, then mockingbirds are not. =

Answer 20

J -> XM (Mutually exclusive)

Question 21

If roses are not in the bouquet, then lilies are. =

Answer 21

XR -> L (The bouquet must at least include R or L, but it does not exclude it having both)

Question 22

Sufficient and necessary assumptions:

If a necessary assumption is true, does it increase the validity of the argument?

Answer 22

NO.

If a necessary assumption is true, it should not make an argument more persuasive.
But every single necessary assumption is required.

EX: “I am the best current player.”
Necessary: “I am alive” and “I know what a game is.”
Sufficient: “I’ve won against every other player.”

Question 23

Sufficient and necessary assumptions:

Descriptive differences between sufficient and necessary assumptions?
(What level of inferences can we make with necessary v. sufficient assumptions?)

Answer 23

A necessary assumption would never be air-tight nor would it be very strong in stance.

EX:
“Knowing strategy is relevant to being the best general in the world.” = necessary assumption

“If you know strategy, then you must be the best general in the world.” = sufficient assumption

Question 24

Sufficient and necessary assumptions:

What are necessary assumptions testing for in questions that ask us to find the necessary assumption?

Answer 24

Necessary assumptions test whether we have conflated two different concepts to bear an air-tight correlation.

EX: You beat out the fiercest competitor in the race so you must have won the race.

A necessary assumption would be that “beating out the fiercest competitor” means “you won the race.” But nothing in the stimulus actually guarantees that the two ideas have an air-tight link.

Question 25

Question Stem:

“Which one of the following is an assumption the argument requires in order for its conclusion to be properly drawn?”

Answer 25

Necessary Assumption

Question 26

Sufficient and necessary assumptions:

What are some red flags in the options when it comes to questions asking for necessary assumptions?

Answer 26

A necessary assumption will likely not include fixed parameters or details (A necessary assumption might be that snakes molt at a constant rate ; it would NOT be necessary for a snake to molt oncec per year. We can still calculate a snake’s age by its molt even if it molts 5 times per year.)

Remember necessity has less of a burden than you might think.
For example, we can use the molt of snake skin to determine its age.
A sufficient assumption will likely include details (EX: snakes molt once per year).

Question 27

Sufficient and necessary assumptions:

Do necessary assumptions improve the validity of the argument?

Answer 27

NO.

When necessary assumptions are NOT true, the argument is factually untrue.
EX: I am the best swimmer in the world because I can hold my breath for 60 seconds.
- Necessary: I know how to swim.
- Sufficient: Simply being able to hold your breath for 60 seconds would automatically make one the best swimmer in the world.

If the necessary assumption is true, the argument does not benefit from being more valid.

Question 28

Sufficient and necessary assumptions:

Necessary assumptions test whether the reader has conflated what?

Answer 28

Necessary assumptions test whether the reader have conflated the level of proof afforded by necessary assumptions with something affords higher level of proof/support.
In other words, these questions test whether the reader conflated sufficient assumptions with necessary ones.

EX: “Nectar contains more calories so birds must take less time consuming nectar to get the same amount of calories as they would seeds.”

Why is the concept of calories linked to the concept of consumption time? Maybe bird anatomy makes it really hard for birds to consume nectar.

Question 29

Sufficient and necessary assumptions:

What is the logical function of proving a necessary assumption true?
(In other words, what does that do for the support structure of an argument?)

Answer 29

Necessary assumptions, when we stipulate them to be true, essentially rule out alternative hypothesis.

EX: If we loosened regulations, banks would loan more.

An alternative hypothesis would be that banks loaned less since there was already less cash flow to begin with, which would mean regulations were irrelevant.

A necessary assumption assumes that such alternative hypothesis do not exist.

Question 30

Job opportunities in the tech industry are expanding. Companies that specialize in developing software and technology solutions are relatively limited in number, while the number of individuals seeking jobs in this industry is rapidly growing.

Is the above an argument?

Answer 30

No, this is not an argument. Neither claim supports the other. These are just two claims asserted to be true without support.

Question 31

All libraries and bookstores are intellectual places. Most well-stocked intellectual places showcase a wide range of books on various subjects. But if an intellectual place is disorganized, it is not well-stocked.

Is the above an argument?

Answer 31

Not an argument.

None of the statements necessarily provide support for one another.

In other words, there is no common link shared between the statements.

Question 32

What is a necessary function of the “support” of an argument?

Answer 32

It MUST increase the likelihood/truth of the statement.

Question 33

Population growth increases the demand for housing. Construction companies capable of building new houses are relatively few, while individuals in need of housing are many.

Is this an argument?

Answer 33

Not an argument.

These statements never said demand for housing is increasing.

Neither claim supports the other. These are just two claims asserted to be true without support.

Question 34

What is an assumption? What impact does it have?

Answer 34

An assumption is the implicit link between premise and conclusion. An assumption can support, hurt, or not have any impact on an argument.

Question 35

Can an argument be strong and still contain assumptions?

Answer 35

Yes.

If an argument is strong, it would either contain fewer assumptions or have very reasonable assumptions - notice an argument can contain assumptions and still be strong.

Question 36

Assumptions, when assumed to be true, tend to be equally valid given they are all implicit.

True or false?

Answer 36

False.

Assumptions are NOT equal. Some assumptions may be more reasonable than others = they would provide stronger support for the argument they’re in.

Example: “Miracles are a rare sight” would be an assumption that’s more reasonable than “miracles are commonplace” even though we cannot consider either statement to be objectively true.

Question 37

Where on the spectrum would the assumptions and support of a strong argument be?

Answer 37

Assumptions - Little to none. If there are, they would be quite reasonable.

Support - Valid inferences. Must be true.

Question 38

Where on the spectrum would the assumptions and support of a weak argument be?

Answer 38

Assumptions - Many unreasonable assumptions.

Support - Unsupported. Could be either true or false.

Question 39

Addressing any assumption contained in an argument would increase the strength of the support.

True or false?

Answer 39

False.

Any argument would have an infinite number of assumptions.

Unless one proves the truth of a SUFFICIENT assumption, the level of increased strength would never be air-tight.

On the other hand, proving the truth of an unreasonable assumption would do very little to increase the level of support.

Question 40

What parts of the excerpt below is context, not argument?

Jazz music has existed for over a century and is hugely influential across the world. So, it seems unlikely there would be much left to learn about what makes it resonate with people. Yet, musicologists have recently discovered a new rhythm that seems to generate a powerful sense of excitement. Jazz musicians in New Orleans have been observed to use this unique rhythm to electrify their performances.

Answer 40

Context: Jazz music has existed for over a century and is hugely influential across the world. So, it seems unlikely there would be much left to learn about what makes it resonate with people.

Conclusion: Yet, musicologists have recently discovered a new rhythm that seems to generate a powerful sense of excitement.

Premise: Jazz musicians in New Orleans have been observed to use this unique rhythm to electrify their performances.

Question 41

Separate the concession point, the premise, and the conclusion:

Despite heavy usage of antibiotics in hospitals over the past few decades, there hasn’t been an exponential surge in antibiotic-resistant bacteria. Furthermore, even if the use of antibiotics doubled for the next few decades, it would have little impact on creating more resistant strains. Consequently, the fears of a looming antibiotic resistance crisis seem overblown.

Answer 41

Concession point: Despite heavy usage of antibiotics in hospitals over the past few decades…

Premise: …there hasn’t been an exponential surge in antibiotic-resistant bacteria.

Premise (imbedded Concession): Furthermore, even if the use of antibiotics doubled for the next few decades, it would have little impact on creating more resistant strains.

Conclusion: Consequently, the fears of a looming antibiotic resistance crisis seem overblown.

Question 42

“Assumptions” and the “support” of an argument are separate issues.

True or false?

Answer 42

False.

A weak support and a damaging assumption are not necessarily two different things

“The support of this argument is weak” CAN equal to “the assumptions in this argument are unreasonable”

Question 43

What are the three steps to parsing out comparative grammar?

Answer 43

Step 1. What are the two things/subjects that we’re comparing? (Identify A vs B)

Step 2. What is the quality or the characteristic that is being compared (re A vs B)?
- Can think of this as the context that makes A and B relevant to each other
- Ex: “People are more scared of strangers than they are of monsters.” Here “strangers” and “monsters” are relevant to each other because people are scared of both of them and that is being compared.

Step 3. Identify the “winner” (Does A or B come out on top in the comparison?)

Question 44

How to find the core structure of sentences with heavy modifiers?

Answer 44

Focus on the object-predicate. In other words, focus on the “that”.

Example:
The cats discovered that the food dish replenishes itself whenever the pedal at the bottom of the dish is pressed
^ the above sentence can be understood as “the cats discovered that.” “That” equates to the rest of the details that supplements but does not change the primary message of the sentence.

Question 45

What is being compared?
What is the quality being compared?

A complicated hotel security system costs more in customer goodwill than it saves in losses by theft.

Answer 45

Step 1. Identify A vs. B.
A: cost in customer goodwill
vs.
B: saves in losses by theft

Step 2. Identify what we’re comparing
For a complicated hotel security system, which one is more?

Step 3. Identify the “winner.”
A, cost in customer goodwill is more.

Question 46

Explain why modifiers serve as the sufficient condition and why the predicates serve as the necessary conditions?

Answer 46

The modifiers (Ex: Kingdoms “whose economies rely predominantly on trade”) set the conditions for where the rest of the conditional will always apply; hence, they serve as the sufficient condition.

The predicates are triggered when the sufficient condition is true; so they serve as the necessary conditions.

Question 47

Formal argument #4, “some before all.” Some A are B. All B are C. Therefore, some A are C.

All arguments that instantiate this form are valid. We can substitute any concept with A, B, and C and the argument will always be valid.

Premise: A ←s→ B → C
________________
Conclusion: A ←s→ C

Answer 47

As we know SOME of the A set has membership in the B set (A ←s→ B); we also know EVERY member of the B set simultaneously has membership in the C set (B → C)

This means any member of the A set that has membership in the B set would also have membership in the C set by virtue of their membership in the B set.
Therefore, SOME of the A set also has membership in the C set.

Question 48

A → B
A —m→ C
________
B ←s→ C

From the lesson on the relationships between the quantifiers, we know that the “all” arrow implies the “most” arrow which means that “A → B” implies “A —m→ B.”

Answer 48

When we see “A → B”, we can automatically assume it means:
“A —m→ B” and “A —s→ B”
Therefore, we can see it as:
A → B
A —m→ B
A —s→ B
A —m→ C
________
B ←s→ C

We know:
A —m→ B
A —m→ C
________
B ←s→ C

If ALL of the A set has membership in the B set, we can assume the status “MOST of the A set has membership in the B set” and “SOME the A set has membership in the B set” must be true as well. (Since “most” and “some” are covered under “all”.)

Question 49

Remember three rules:
1. Sufficient satisfied, necessary must be as well. (Conditional argument)
2. Necessary failed, sufficient must fail as well. (Contrapositive argument)
3. Merge together the same symbol to create a chain. (Chaining conditionals)

Question 50

Remember three traps:
1. Sufficient failed yields no information about the necessary.
2. Necessary satisfied yields no information about the sufficient.
3. Do not confuse sufficiency for necessity.

Question 51

Only roses are red. Only thorny things are red. There’s a red thing in the flower garden.

“Some roses are thorny.”
Can we make the above inference?

Answer 51

Yes.

Rose —-> Red
Thorny —-> Red

We can also make the below inferences:
1. There’s a rose in the flower garden.
2. There’s a thorny thing in the flower garden.
3. Some roses are thorny.

Note: Two most relationships imply a “some” relationship.
Although there isn’t a “most” relationship here, remember that “all” implies “most”.

Question 52

Most classically trained opera singers can recite the lyrics to Musetta’s Waltz and most people who have not received such training cannot. Therefore, Anna, who can recite the lyrics to Musetta’s Waltz, was classically trained.

Is the statement valid?

Answer 52

No.
Don’t read unidirectional arrows backwards. A —m→ B does not imply B —m→ A because the A set could be tiny in comparison to the B set.

Classically trained opera singers number in the thousands. People who are not so trained number in the billions. Granted the truth of both premises, it could still be that there are thousands of classically trained opera singers who can recite and millions of people not so trained who also can recite. Why is it more likely that Anna, who can recite, should be in the first set? In fact, it seems more likely that she would be in the second set.

Question 53

Qualifiers will change the logic of any chained condition:
All before Most

if the all arrow shows up first in the chain and then you see the most arrow, there are no valid conclusions to be drawn via the chain.

A —m→ B → C yields valid conclusions via the chain.
A → B —m→ C yields no valid conclusions via the chain.

Question 54

A ←s→ B → C yields valid conclusions via the chain.

A → B ←s→ C yields no valid conclusions via the chain.

Answer 54

From the lesson on the relationships between the quantifiers, you know that the “most” arrow implies the “some” arrow which means that “A —m→ B” implies “A ←s→ B.” Given that the previous argument is invalid and we can transform this argument into the previous one, then this one must be invalid as well.

Another way you can see this is to think about how strong your premises need to be to support a conclusion. B —m→ C is stronger than B ←s→ C. If the stronger premise can’t even support the conclusion, then of course the weaker premise cannot either.

This argument also looks similar to a valid formal argument where a some arrow precedes the all arrow. In that argument, we can draw a valid conclusion via the chain.

Question 55

A —m→ B —m→ C yields NO valid conclusions via the chain.

A —m→ B and A —m→ C yield a valid conclusion.

It means there is a non-zero, (at least one) overlap between A and C.

Question 56

A ←s→ B ←s→ C yields no valid conclusions via the chain.

Question 57

Causal logic is not formal. It’s informal. That means arguments that use causal logic will never be valid. Their premises, even when true, will never guarantee the truth of their conclusions.

But, that doesn’t mean causal arguments are necessarily weak. In fact, causal arguments can be very strong. The fact that they can never be valid merely sets a ceiling on how strong the support relationship can be.

Question 58

Why does Presumption of Truth matter?

What impact does it functionally have on the options/questions?

Answer 58

It impacts, and thus matters to the logic, in that it changes the direction of the basis of proof.

This direction is decided by the question stem (i.e., question type).

Ex: “If we assume the below options to be true, which option most weakens the above statement?”
^In this case, the options enjoy Presumption of Truth; the prompt statement does not.
The options are then used to question the logic of the prompt.

Ex: “Which of the below options is most strongly supported by the statements above?”
^In this case, the prompt statement enjoys Presumption of Truth.
We then check the options against the prompt statement, using the prompt statement as the evidence/proof.

Question 59

Be rigorous.
For Weaken Questions.

The correct answer must fulfill the task of directly weakening the relationship between the premise and the conclusion in the prompt.

What are key indicators of whether an option is on task?

Answer 59

The identifiers (domain) of the option must DIRECTLY address the same level/identifier (domain) of the prompt.

For example, the prompt says ‘a dog barked when it was given ten steaks but did not did not bark when it was given one steak.’

An option that says ‘dogs only bark when they are given their favorite food’ would be less relevant as the difference in identifiers lies in the QUANTITY of the food given.

Question 60

Negate the statement ‘all cats are quiet’.

Answer 60

Negation means the statements must contradict the original statement. In other words, they must be mutually exclusive.

Answer:
NOT all cats are quiet
OR
SOME cats are NOT quiet

Question 61

Negate the statement ‘some mammals are aquatic.’

Answer 61

Negation means the statements must contradict the original statement. In other words, they must be mutually exclusive.

Answer:
It is NOT the case that some mammals are aquatic
OR
NO mammals are aquatic
OR
ALL mammals are NOT aquatic

Question 62

Negate the statement ‘most cats come out at night.’

Answer 62

Note that ‘negation’ is NOT the same as ‘opposition.’

‘A minority of cats come out at night’ would not be a negation of the statement.

The only forms of negation would be:
It’s NOT the case that most cats come out at night
OR
Anywhere from none to exactly half of all cats come out at night

Question 63

Every planetary society will be endangered by impacts from space.

Translate this to formal logic.

Answer 63

Planetary society -> Endangered by impacts from space
X Endangered by impacts from space -> X Planetary society

Question 64

Only Italian plumbers can fly while wearing raccoon suits.

Translate this to formal logic.

Answer 64

Fly in raccoon suit -> Italian plumber
X Italian plumber -> X Fly in raccoon suit