116 CHAPTER 6 Compound Claims
For each of the following, write the contradictory of the claim. If it is an "or" claim,
identify the alternatives.
7. Inflation will go up or interest rates will go up.
8. Manuel can go everywhere in his wheelchair.
9. Maria or Lee will pick up Manuel after classes.
10. Neither Maria nor Lee has a bicycle.
11. You're either for me or against me.
12. You'd better stop smoking in here or else!
13. AIDS cannot be contracted by touching nor by breathing air in the same room as a
person infected with AIDS.
14. Maria will go shopping, but Manuel will cook.
15. Zoe (to Dick): Will you take the trash out or do I have to?
3. Reasoning with "or" claims
Often we can determine that an argument is valid or weak by looking at the role a
compound claim plays in it. For example,
Either there is a wheelchair ramp at the school dance, or Manuel stayed home.
But there isn't a wheelchair ramp at the school dance.
Therefore, Manuel stayed home.
The argument is valid: There's no possible way for the premises to be true and the
conclusion false at the same time.
SECTION A Consider the Alternatives 117
This is just one example of lots of arguments that have the same form and are
valid. In order to illustrate that form in a diagram, I'll use an arrow ( ) to stand
for "therefore," and the symbol "+" to indicate an additional premise.
This form of argument is sometimes called the disjunctive syllogism.
We also have the valid argument form: A or B, notB, therefore A.
Or there may be more than two alternatives:
Somebody's cat killed the bird that always sang outside. 1
Either it was Sarah's cat or the neighbor's cat or some stray. 2
Sarah says it wasn't her cat, 3 because hers was in all day. 4
My neighbor says her cat never leaves the house. 5
So it must have been a stray. 6
From 3 and 4 we get:
Sarah's cat didn't kill the bird, a
(Lowercase letters mark claims that are added to an argument.) And from 5 we get:
My neighbor's cat didn't kill the bird, b
With 2 rewritten as "Either Sarah's cat killed the bird, or the neighbor's cat killed
the bird, or some stray cat killed the bird," we now have:
A or B or C
not A, not B
Therefore C.
Sometimes we can only reduce the possibilities, not exclude all but one:
Either all criminals should be locked up forever, or we should put more
money into rehabilitating criminals, or we should accept that our
streets will never be safe, or we should have some system for
monitoring ex-convicts. 1 {this is all one claim)
We can't lock up all criminals forever, 2 because it would be too expensive. 3
We definitely won't accept that our streets will never be safe. 4
So either we should put more money into rehabilitating criminals, or we
should have some system for monitoring ex-convicts. 5
118 CHAPTER 6 Compound Claims
The argument is valid, because 2 and 4 eliminate some of the possibilities given
in 1. But even if 1 is true (it really lists all the possibilities), all we get from this argument is another "or" claim—we've reduced the possibilities.
A or B or C or D
not A, not C
Therefore, B or D.
Arguments like this are valid, too.
4. False dilemmas
Zoe has made a valid argument, but not a good one. She's posed a false
dilemma: "You're either going to have to stop smoking those nasty expensive cigars
or we'll have to get rid of Spot" is false. Dick could respond that Zoe could give up
talking to her mother long distance every day. Excluding possibilities is a valid form
of argument. But valid arguments need not be good. We get a bad argument when
the "or" claim doesn't list all the possibilities.
False dilemma A false dilemma is a bad use of excluding possibilities
where the "or" claim is false or implausible. Sometimes just the dubious
"or" claim itself is called a "false dilemma."
For example,
Society can choose high environmental quality but only at the cost of lower
tourism or more tourism and commercialization at the expense of the ecosystem,
but society must choose. It involves a tradeoff.
Robert Sexton, Exploring Economics
The alternatives are claimed to be mutually exclusive. But Costa Rica has created a
lot of tourism by preserving almost 50% of its land in parks. When you see a
versus -claim, think, "Is this a false dilemma?"
To avoid false dilemmas, we have to imagine other possibilities.
EXERCISES for Section A 119
Exercises for Section A
1. Give an "or" claim that you know is true, though you don't know which of the
alternatives is true.
2. a. State the form of valid arguments called "excluding possibilities."
b. Give two other forms of valid arguments that use "or" claims.
3. What is a false dilemma?
4. Give an example of a false dilemma you've used or which was used on you recently.
5. Why is using a false dilemma so good at making people do what you want them to do?
Is it a good way to convince?
6. Show that the argument about Manuel going to the dance on p. 116 is a false dilemma.
7. Sometimes a false dilemma is stated using an "if . . . then . . ." claim:
If you don't stop smoking, you're going to die.
(Either you stop smoking or you will die.)
Mommy, if you don't take me to the circus, then you don't really love me.
(Either you take me to the circus or you don't love me.)
If you can't remember what you wanted to say, it's not important.
(Either you remember what you want to say or it's not important.)
Give two examples of false dilemmas stated using "if . . . then . . .".
Trade with a classmate to rewrite them as "or" claims.
8. A particular form of false dilemma is the perfectionist dilemma, which assumes:
Either the situation will be perfect if we do this, or we shouldn't do it.
{All or nothing at all.)
— I'm voting for raising property taxes to pay for improvements to the schools.
— Don't be a fool. No matter how much money they pour into the schools,
they'll never be first-rate.
a. Give the unstated premise that shows that this argument is a false dilemma.
b. Give an example of a perfectionist dilemma you've heard or read.
Evaluate Exercises 9-13 by answering the following:
Argument? (yes or no)
Conclusion (if unstated, add it):
Premises:
Additional premises needed (if none, say so):
Classify (with the additional premises): valid strong weak
Good argument? (choose one—if it's a false dilemma, say so)
• It's good (passes the three tests).
• It's valid or strong, but you don't know if the premises are true,
so you can't say if it's good or bad.
• It's bad because it's unrepairable (state which of the reasons apply).
120 CHAPTER 6 Compound Claims
9. Tom: Look, either you'll vote for the Republican or the Democratic candidate
for president.
Lee: No way I'll vote for the Democrat.
Tom: So you' 11 vote for the Repu
blican.
10. Lee: Manuel and Tom went to the basketball game if they didn't go to the library.
Maria: I know they're not at the library because I was just there.
Lee: So they must have gone to the basketball game.
11. Tom: Both Lee and I think they should allow logging on Cedar Mountain. You do,
too—don't you, Dick?
Dick: Actually, no, . . .
Tom: I didn't know you were one of those environmentalist freaks.
12. Dick: Somebody knocked over our neighbor's trash can last night. Either our neighbor
hit it with her car when she backed out again, or a raccoon got into it, or Spot
knocked it over.
Zoe: Our neighbor didn't hit it with her car, because she hasn't been out of her house
since last Tuesday.
Dick: It wasn't a raccoon, because Spot didn't bark last night.
Zoe: Spot! Bad dog! Stay out of the trash!
13. Zoe: We should get rid of Spot. He keeps chewing on everything in the house.
Dick: But why does that mean we should get rid of him?
Zoe: Because either we train him to stop chewing or we get rid of him. And we
haven't been able to train him.
Dick: But I love Spot. We can just make him live outdoors.
(Evaluate what Zoe says as an argument. Consider Dick's answer in doing so.)
B. Conditionals
1. Conditionals and their contradictories
Suppose your instructor says to you:
If you do well on the final exam, then I'll give you an A in this course.
This is one claim. If it shows up in an argument, we don't say one premise is
"You do well on the final exam" and another is "I'll give you an A in this course."
Rather if you do well, then your instructor will give you an A in this course. There is no promise to give you an A, only a conditional promise. If you do poorly on the
final exam, your instructor is not obligated to give you an A.
Sometimes "then" is left unsaid, or the order of the two parts is reversed:
a. If Dick loves Zoe, he will give her an engagement ring.
b. I'll meet you at the cafeteria if they're not serving beef stroganoff.
And sometimes neither "if" nor "then" is used, yet it's clear the claim makes sense as an "if. . . then . . ." claim. For example,
SECTION B Conditionals 121
c. Bring me an ice cream cone and I'll be happy.
Conditional claim A claim is a conditional if it can be rewritten as an
"if. . . then . . ." claim that must have the same truth-value.
In a conditional (rewritten as) "If A, then B " , the claim A is the
antecedent, and the claim B is the consequent.
In (a), the antecedent is "Dick loves Zoe," and the consequent is "He will give
her an engagement ring."
In (b), though the order is reversed, it is the part that follows "if" that is the
antecedent, "They're not serving beef stroganoff," and the consequent is "I'll meet
you at the cafeteria."
In (c), the antecedent is "Bring me an ice cream cone," and the consequent is
"I'll be happy."
How do we form the contradictory of a conditional? Yesterday Manuel said,
"If Maria called in sick today, then Lee had to go to work." To decide whether this
is true, we ask whether Lee was obligated to work if Maria called in sick. He wasn't:
She called in sick and he didn't have to go to work.
Contradictory of a conditional If A, then B has contradictory A but not B.
Zoe: I'm so worried. Spot got out of the yard. If he got out of the yard,
then the dogcatcher got him, I'm sure.
Suzy: Don't worry. I saw Spot. He got out of the yard, but the dogcatcher
didn't get him.
122 CHAPTER 6 Compound Claims
The contradictory of a conditional is not another conditional.
Sometimes, when we reason about how the world might be, we use a
conditional with a false antecedent:
If cats had no fur, they would not give people allergies.
We could form the contradictory as for any conditional. But more commonly we use
words like "although" or "even if":
Even if cats had no fur, they would still give people allergies.
"Even i f " does not make a conditional. "Even i f " is used in much the same way as
"although" or "despite that."
Exercises for Section B.l
1. a. What is a conditional?
b. Is a conditional a compound claim?
2. Make a conditional promise to your instructor that you believe you can keep.
3. What is the antecedent of a conditional?
4. What is a contradictory of a claim?
5. Make up five examples of conditional claims that don't use the word " i f " or don't use the word "then." At least one should have the consequent first and antecedent last.
Exchange with a classmate to identify the antecedents and consequents.
6. How do you say the contradictory of "If A, then B"?
7. a. Give a contradictory of:
(*) If Suzy studies hard, then she'll pass Dr. E's class.
Show that each of (b)-(d) is not a contradictory of (*) by giving for each a possibility
where both it and (*) could be true or both of them could be false at the same time.
b. If Suzy doesn't study hard, then she'll pass Dr. E's class.
c. If Suzy doesn't study hard, then she won't pass Dr. E's class.
d. If Suzy studies hard, then she won't pass Dr. E's class.
EXERCISES for Section B. 1 123
8. Make up two conditionals and two "or" claims. Exchange them with a classmate to
write the contradictories.
Here are two examples of Tom's work on conditionals.
Getting an A in critical thinking means that you studied hard.
Conditional? (yes or no) Yes.
Antecedent: You get an A in critical thinking.
Consequent: You studied hard.
Contradictory: You got an A in critical thinking, but you didn't study hard,
(or Even though you got an A in critical thinking, you didn't study hard.)
G oodwork.
Spot loves Dick because Dick plays with him.
Conditional? (yes or no) No.
Antecedent: Spot loves Dick. No
Consequent: Dick plays with him. No
Contradictory: Spot loves Dick but Dick doesn't play with him.. No
You're right, it's not a conditional; the word "Because" tells you it's an argument.
But if it's not a conditional, then there is no antecedent and no consequent. And
there can't be a contradictory of an argument.
For each exercise below, answer the following. Remember that even though it might not be
a conditional, it could still have a contradictory.
Conditional! (yes or no)
Antecedent:
Consequent:
Contradictory:
9. If Maria goes shopping, then Manuel will cook.
10. Lee will take care of Spot next weekend if Dick will help him with his English exam.
11. If you don't apologize, I'll never talk to you again.
12. Flo's mother won't go to the movie if she can't get someone to watch Flo.
13. Loving someone means you never throw dishes at them.
14. Since 2 times 2 is 4, and 2 times 4 is 8,1 should be ahead $8, not $7.
15. Get me some cake mix at the store and I'll bake a cake.
16. Tuna is good for you even though they say you shouldn't eat it more than once per week.
17. Tom: Being late for football practice will make the coach really mad.
18. If it's really true that if Dick takes Spot for a walk Dick will do the d
ishes, then Dick
won't take Spot for a walk.
124 CHAPTER 6 Compound Claims
19. If Manuel went to the basketball game, then he either got a ride with Maria or he left
early to wheel himself over there.
20. When there's a raccoon in the yard, you can be sure that Spot will bark.
21. Lee didn't go to the lecture because he knew Maria would take notes.
2. Necessary and sufficient conditions
We say that two claims are equivalent if each is true exactly when the other is. For
example, the following are equivalent:
If interest rates go down, then unemployment will go down.
If unemployment doesn't go down, then interest rates won't go down.
Contrapositive The contrapositive of If A, then B is If not B, then not A.
A claim and its contrapositive are equivalent.
Sometimes it's easier to understand a conditional via its contrapositive:
If you get a speeding ticket, then a policeman stopped you.
Contrapositive: If a policeman didn't stop you, then you didn't get a ticket.
Conditionals are crucial for understanding what we mean by necessary or
sufficient conditions. For example, what's necessary for getting a driver's license?
Well, you've got to pass the driving exam. That is, if you don't pass the driving
exam, you won't get a driver's license. There's no way you'll get a driver's license
if you don't pass the driver's exam.
What's sufficient for getting money at the bank? Well, cashing a check there
will do. That is, if you cash a check at the bank, then you'll get money at the bank.
A is necessary for B means If not A, then not B is true.
A is sufficient for B means If A, then B is true.
For example, passing an eye test is necessary but not sufficient for getting a
driver's license. This is the same as saying: "If you don't pass an eye test, you can't
get a driver's license" is true, but "If you pass an eye test, then you get a driver's
license" is false.
But lots of times we get confused. Here's what Lee and Manuel were saying
last week:
EXERCISES for Section B.2 125
Manuel: It's just wrong that Betty didn't make the basketball team.
Lee: Yeah. I watched the tryouts and she was great. She hit a couple
three-pointers, and she can really jump.
Richard L Epstein Page 16