Richard L Epstein
Page 21
Here are some examples of claims and their contradictories:
Claim Contradictory
All dogs bark. Some dogs don't bark.
Some dogs bark. No dogs bark.
Some dogs don't bark. All dogs bark.
No women are philosophers. Some women are philosophers.
Every Mexican likes vodka. Some Mexicans don't like vodka.
Some Russians like chile. No Russian likes chile.
Some whales eat fish. Not even one whale eats fish.
162 CHAPTER 8 General Claims
The contradictory of "Only S are P" can be made in two ways:
Not every P is S.
Some P are not S.
So "Only postal employees deliver mail" is contradicted by "Some people who
deliver mail are not postal employees." If we want to say that just exactly postal
employees and no one else delivers U.S. mail, we should say that. Or we can say:
All postal employees and only postal employees deliver U.S. mail.
Contradictory Either some postal employees don't deliver U.S. mail,
or some people who deliver U.S. mail aren't postal employees.
Because there are so many ways we can make general claims, it's hard to give
set formulas for contradictories. With some practice you ought to be able to use your
common sense to get the contradictory right. As an aid, here is a rough guide:
Claim
Contradictory
All —
Some are not —
Not every —
Some —
N o -
AH are not —
Not even one —
Some are not —
All are —
No —
Some are —
Only S are P
Some P are not S
Not every P is S
Exercises for Section A
1. Give two other ways to say "All dogs eat meat."
2. Give two other ways to say "Some cats can swim."
3. Give two other ways to say "All computers are powered by electricity."
4. Give two other ways to say "Some state governors are women."
5. Give another way to say "Only birds fly."
6. Give two other ways to say "No police officer is under 18 years old."
7. Give another way to say "Everything that's a dog is a domestic canine, and everything
that's a domestic canine is a dog."
8. Give two other ways to say "No pig can fly."
SECTION B Some Valid and Invalid Forms 163
9. Judging from your experience, which of the following claims are true? Be prepared to
defend your answer.
a. Only dogs bark.
b. All blondes are dumb.
c. Some textbooks are designed to fall apart after one semester.
d. Crest toothpaste is not for sale in all stores.
e. Some English professors are women.
f. Dictionaries are the only way to learn the meaning of new words.
g. No student can register for this course after the first week of classes.
10. For each of the following, give a contradictory claim.
a. All students like to study.
b. No women are construction workers.
c. Every CEO of a Fortune 500 company is a man.
d. This exam will be given in all of the sections of critical thinking.
e. No exam is suitable for all students.
f. Some exams don't really test a student's knowledge.
g. Not all drunk drivers get in accidents,
h. Donkeys eat carrots.
i. Only the good die young.
j. All teachers and only teachers are allowed to grade exams.
k. Nothing both barks and meows.
1. Tom will start every football game if he's not suspended.
m. If some football player is a vegetarian, then his coach will hate him.
n. All decisions about abortion should be left to the woman and her doctor.
o. The Lone Ranger was the only cowboy to have a friend called "Tonto."
11. There are general claims about time, too. Give a contradictory for each of the following:
a. Dr. E always gives an exam when he is irritated with his students.
b. It never rains in Seattle in July.
c. Sometimes Spot will not chase Puff.
d. Only during the winter are there flocks of birds along the river.
B. Some Valid and Invalid Forms
Recall the first argument in this chapter:
All good teachers give fair exams. Professor Zzzyzzx gives fair exams.
So Professor Zzzyzzx is a good teacher.
We saw that it's weak: Professor Zzzyzzx could be among the bad teachers who give
fair exams. Here's a diagram that summarizes the discussion:
164 CHAPTER 8 General Claims
This argument sounds good because it's similar to a valid form of argument.
Schematically, where "a" stands for the name of someone or something:
Valid: All dogs bark.
Weak: All dogs bark.
Ralph is a dog.
Ralph barks.
So Ralph barks.
So Ralph is a dog.
The argument on the right is overlooking possibilities. One way to be something that
barks is to be a dog, but there may be other ways (seals and foxes).
The diagram on the previous page is an example of a way to check whether
certain kinds of arguments that use general claims are valid.
Checking for validity with diagrams
• A collection is represented by an enclosed area.
• If one area is entirely within another, then everything in the one
collection is also in the other.
• If one area overlaps another, then there is something that is
common to both collections.
• If two areas do not overlap, then there is nothing common
to both collections.
• An "a" or a dot in an area marks that a particular object is in
that collection.
• Draw the areas to represent the premises as true while trying to
represent the conclusion as false. If you can, then the argument
is invalid. If there's no way to represent the premises as true
and the conclusion as false, the argument is valid.
SECTION B Some Valid and Invalid Forms 165
For example, we can use diagrams to check whether the following is valid:
All dogs bark. Everything that barks is a mammal.
So all dogs are mammals.
We first draw the diagram to represent the premises as true.
The "dogs" area is completely inside the
"things that bark" area: All dogs bark.
The "things that bark" area is completely
inside the "mammals" area: All things that
bark are mammals.
So the "dogs" area ends up being inside the "mammals" area. There's no way it
couldn't be. That represents that all dogs are mammals. So if we represent the
premises as true, we are forced to represent the conclusion as true. The argument
is valid, reasoning in a chain with "all."
Compare that to a similar argument:
Some kangaroos are tame. Some creatures that are tame
live in New Zealand. So some kangaroos live in New Zealand.
Is the argument valid? What do we need to have in a diagram?
The "kangaroos" area must overlap the
kangaroos
"tame" area: Some kangaroos are tame.
The "tame" area must overlap the
"New Zealand" area: Some creatures
that are tame live in New Zealand
We were able to draw the diagram to represent both premises as true, yet there's
/> no overlap between the "kangaroos" area and the "New Zealand" area, so the
conclusion is false: It's possible that no kangaroos live in New Zealand. Thus, the
argument is invalid. Even though its conclusion is true (there are some kangaroos in
zoos there), it's weak.
166 CHAPTER 8 General Claims
Here's an argument with "no":
All dogs bark. No professor is a dog.
So no professor barks.
How do we check if this is valid? We do what we've always done: Look for all the
possible ways that the premises could be true. Only now we can use diagrams to
represent those possibilities. We know that the "dogs" area must be entirely within
the "things that bark" area (All dogs bark). So we just have to figure out where to
put the "professors" area. We know that there must be no overlap of the "professors"
area and the "dogs" area (No professor is a dog). Here are three possibilities:
These (schematically) represent all the ways the premises could be true. Yet in both
(2) and (3) the conclusion is represented as false. It's possible for there to be a
professor who barks, even though he (she?) isn't a dog. The argument is invalid.
It mimics a valid form of argument.
Drawing diagrams to check validity is just another way to look for possibilities
that make the premises true and the conclusion false. The method works for some
arguments that use general claims, but not for all. Even the simple argument about
dogs that like cats with which we began the chapter can't be analyzed using diagrams
this way. You'll have to think your way through all possible ways the premises
could be true when you do some of the exercises.
EXERCISES for Section B 167
Here is how Lee and Maria have been using diagrams to check for validity.
The Workbook has two more examples of Lee and Maria using this method.
Exercises for Section B
Which of the argument forms in Exercises 1-6 are valid? Justify your answer. Then give an
argument of that form.
1. All S are P.
4. Only S are P.
No Q is S.
a is S.
So some Q aren't P.
So a is P.
2. All S are P.
5. Some S aren't P.
a is S.
So no Pare S.
So a is P.
3. Some S are P.
6. All S are P.
All P are Q.
No Q is P.
So some S are Q.
So no Q is S.
168 CHAPTER 8 General Claims
Exercises 7-14 are simple examples for you to develop some skill in analyzing general
claims. For each, select the claim that makes the argument valid—you're not asked to judge
whether the claim is plausible, just whether it makes the argument valid.
7. All turtles can swim. So turtles eat fish.
a. Anything that eats fish swims.
b. Fish swim and are eaten by things that swim.
c. Anything that swims eats fish.
d. None of the above.
8. Anyone who plagiarizes is cheating. So Ralph plagiarizes.
a. Ralph wrote three critical thinking essays in two days.
b. Ralph cheated last week.
c. Both (a) and (b).
d. None of the above.
9. Pigs are mammals. So pigs eat apples.
a. Anything that eats apples is a mammal.
b. Pigs don't eat meat.
c. Anything that is a mammal eats apples.
d. None of the above.
10. All professional dancers cannot hold a day job. So no lawyer is a professional dancer.
a. Lawyers don't usually like to dance.
b. Dancers aren't interested in making money.
c. Being a lawyer is a day job.
d. Professional dancers can't write essays.
e. None of the above.
11. Every voter must have a legal residence. So no sex-offender has a legal residence.
a. No sex-offender is a voter.
b. No sex-offender can register to vote.
c. If you're a sex-offender, then no one will want to live near you.
d. None of the above.
12. Some cats chase songbirds. So some songbirds are eaten by cats.
a. Some cats catch songbirds.
b. Some things that chase songbirds eat them.
c. Some songbirds attack cats.
d. None of the above.
13. Every dog chases cats. So Spot chases Puff.
a. Spot is a dog.
b. Puff is a cat.
c. Puff irritates Spot.
d. Both (a) and (b).
e. None of the above.
EXERCISES for Section B 169
14. Manuel is sweating. So he must be hot.
a. Manuel sweats when he is hot.
b. Anyone who is hot sweats.
c. Only Manuel sweats when he is hot.
d. Only people who are hot sweat.
e. None of the above.
Which of Exercises 15-32 are valid arguments? You're not asked to determine whether the
argument is good, only whether it is valid. Check by doing one of the following:
• Give a possible way in which the premises could be true and the conclusion false
to show it's invalid.
• Draw a diagram.
• Point out that the argument is in one of the forms we have studied.
• Explain in your own words why it's valid.
15. Not every student attends lectures. Lee is a student. So Lee doesn't attend lectures.
16. No professor subscribes to Rolling Stone magazine. Maria is not a professor.
So Maria subscribes to Rolling Stone magazine.
17. No professor subscribes to Rolling Stone magazine. Lou subscribes to Rolling Stone
magazine. So Lou is not a professor.
18. Some dogs bite postal workers. Some postal workers bite dogs. So some dogs and
postal workers bite each other.
19. Everyone who is anxious to learn works hard. Dr. E's students work hard. So Dr. E's
students are anxious to learn.
20. All CEOs of Fortune 500 companies earn more than $400,000. Ralph earns more than
$400,000. So Ralph is a CEO of a Fortune 500 company.
21. All students who are serious take critical thinking in their freshman year. No one who
smokes marijuana every week is a serious student. So no one who smokes marijuana
every week takes critical thinking his or her freshman year.
22. No student who cheats is honest. Some dishonest people are found out. So some
students who cheat are found out.
23. Only ducks quack. George is a duck. So George quacks.
24. Everyone who likes ducks likes quackers. Dick likes ducks. Dick likes cheese.
So Dick likes cheese and quackers.
25. No dogcatcher is kind. Anyone who is kind loves dogs. So no dogcatcher loves dogs.
26. Some things that grunt are hogs. Some hogs are good to eat. So some things that grunt
are good to eat.
27. Dogs are loyal. Dogs are friendly. Anything that is friendly and loyal makes a great pet.
Hence, dogs are great pets.
170 CHAPTER 8 General Claims
28. Every newspaper Dr. E reads is published by an American publisher. All newspapers
published by an American publisher are biased against Muslims. So Dr. E reads only
newspapers that are biased against Muslims.
29. Some paraplegics can't play basketball. Belinda is a paraplegic. So Belinda can't
play basketball.
30. Every dog loves its master. Dr. E has a dog. So Dr. E is loved
.
31. Only janitors have access to this building after midnight. Paul is a janitor. So Paul has
access to the building after midnight.
32. All mammals have both a heart and a liver. The fossil remains of this animal show that
it had a heart and a liver. So it must have been a mammal.
33. Arguing backwards with "all" and arguing backwards with conditionals are related.
We can rewrite:
All dogs bark. If anything is a dog, then it barks.
Ralph barks. as Ralph barks.
So Ralph is a dog. So Ralph is a dog.
Rewrite the following claims as conditionals:
a. All cats cough hair balls.
b. Every donkey eats hay.
c. Everything that's made of chocolate is good to eat.
d. Ducks like water.
C. Between One and All
1. Precise generalities
There are a lot of quantities between one and all. For example,
7 2 % of all students who take critical thinking from Dr. E think he's the
best teacher they've ever had. Harry took Dr. E ' s critical thinking course
last year. So Harry thinks Dr. E is the best teacher he's ever had.
This is not valid. Where does it land on the strong-weak scale? We can say exactly:
There's a 2 8 % chance the premises could be true and the conclusion false, which is
not strong. If the percentages are very high or very low, though, we can get a strong
argument, assuming we know nothing more about the people or things involved:
9 5 % plus-or-minus 2% of all cat owners have cat-induced allergies.
Dr. E ' s ex-wife has a cat. So Dr. E ' s ex-wife has cat-induced allergies.
Only 4 of the 123 students who take Dr. E ' s classes failed his final exam.
Mary Ellen took Dr. E ' s class. So Mary Ellen passed Dr. E ' s final exam.
SECTION C Between One and All 171
2. Vague generalities
There are a lot of ways we talk about all or a part of a collection without specifying a
precise number:
All dogs bark.
Almost all dogs bark.
Many students at this school will vote.
Most dogs bark.
A lot of students at this school will vote.
Some students study hard.
A few students study hard.
Very few students dislike Dr. E.
Despite the ambiguity of the words "all" and "some," we can analyze whether
arguments using those are valid. We have enough precision.
The rest of these quantity words are too vague to figure in valid arguments.
Most of them are too vague even to be used in a claim. How could we tell if