Richard L Epstein
Page 17
Manuel: And the coach chose only girls who could jump well and hit
three-pointers.
Lee: She had everything you need to get on the team.
Lee thinks that jumping well and hitting three-pointers are sufficient for getting on
the team. But what Manuel said is that they're necessary. Lee's got it backwards.
This kind of mistake is easy enough to avoid if you translate statements about
necessary or sufficient conditions into conditionals.
Here's another example heard on National Public Radio:
Interviewer: So, will we continue to see home schooling in America?
Interviewee: As long as there are parents who love their kids and are willing
to work hard, yes.
The last person has said that love and willingness to work hard are enough for home
schooling to continue. That may be necessary, but it's certainly not sufficient. Also
needed are laws allowing home schooling, a cultural climate encouraging it, . . . .
Exercises for Section B.2
1. State the contrapositive of:
a. If Flo plays with Spot, then she has to take a bath.
b. If Manuel doesn't get his wheelchair fixed by Wednesday, he can't attend class
Thursday.
c. If Maria goes with Manuel to the dance, then Lee will be home alone on Saturday.
2. We know that the following are equivalent claims:
• If Dick went to the movies, then he got home before 6 p.m.
• If Dick didn't get home before 6 p.m., then he didn't go to the movies.
• For Dick to go to the movies, it's necessary for him to get home before 6 p.m.
Similarly, rewrite each of the following in two ways (using "necessary" or "sufficient"
as appropriate):
a. Suzy will go with Tom to the library if he gets out of practice by 6.
b. For Dick to take Spot for a walk, it's necessary that it not be raining.
c. If Spot got out of the yard, then the gate was unlatched.
3. State which of the following hold:
(i) is necessary for (ii) (i) is both necessary and sufficient for (ii)
(i) is sufficient for (ii) (i) is neither necessary nor sufficient for (ii)
a. (i) Dr. E had his annual physical examination.
(ii) Dr. E had an appointment with his physician.
126 CHAPTER 6 Compound Claims
b. (i) Manuel opened a checking account, (ii) Manuel wrote his first check.
c. (i) Zoe won $47 at blackjack, (ii) Zoe was gambling.
d. (i) Maria is divorced, (ii) Maria has an ex-husband.
e. (i) Suzy is over 21. (ii) Suzy can legally drink in this state.
4. Often we say one condition is necessary or sufficient for another, as in "Being over 16
is necessary for getting a driver's license." That means that the general conditional is
true: "If you can get a driver's license, then you're over 16."
State which of the following hold:
(i) is necessary for (ii) (i) is both necessary and sufficient for (ii)
(i) is sufficient for (ii) (i) is neither necessary nor sufficient for (ii)
a. (i) visiting City Hall (ii) leaving home
b. (i) having the ability to fly (ii) being a bird
c. (i) being a U.S. citizen (ii) being allowed to vote in the U.S.
d. (i) losing at the lottery (ii) buying a lottery ticket
5. What is a necessary condition for there to be a fire?
6. What is a sufficient condition for you to be happy? Is it necessary?
7. Rewrite each of the following as an "if. . . then . . ." claim if that is possible.
If it is not possible, say so.
a. Paying her library fines is required in order for Zoe to get a copy of her transcript.
b. Dick: Since I'm on the way to the store anyway, I'll pick up some dog food.
c. Suzy loves Puff even though he isn't her cat.
d. Of course, Suzy loves Tom despite the coach suspending him for a game.
e. For Tom to get back on the team, he has to do 200 push-ups.
f. Dick apologizing is enough for Zoe to forgive him.
8. The phrase only if does not mean the same as if:
Harry will get into graduate school only if his grades place him in the top 10%
of his graduating class.
Harry will get into graduate school if his grades place him in the top 10%
of his graduating class.
These are not equivalent. The first gives a necessary condition for Harry to get into
graduate school, and it's true. The second gives a sufficient condition, and it's false.
A only if B means the same as if not B, then not A.
Since we know the right-hand side is equivalent to if A, then B, we have:
A only if B is equivalent to if A, then B.
Rewrite each of the following as a conditional and as a statement of a necessary or
sufficient condition.
a. Maria will buy a new dress only if she gets a bonus this month.
b. Flo will go over to play with Spot only if her mother lets her.
c. Lee: Only if Tom is back on the team can we win this weekend.
SECTION B Conditionals 127
9. From Exercise 8 we have that "A only if B" is equivalent to "if A then B".
A if and only if B means if A, then B; and if B, then A.
We use "if and only if" to show that two claims are equivalent: each is necessary and sufficient for the other. For example,
Suzy will marry Tom if and only if he remains faithful to her until graduation.
This means that it is necessary for Tom to stay faithful to Suzy for her to marry him.
But it is also sufficient for Tom to stay faithful to Suzy to ensure that she will marry him.
Give an example of an "if and only if " claim from your own life you know is true.
3. Valid and weak forms of arguments using conditionals
If Spot barks, then Dick will wake up.
Spot barked.
So Dick woke up.
That's valid. It's impossible for the premises to be true and the conclusion false.
If Suzy calls early, then Dick will wake up.
Suzy called early.
So Dick woke up.
This is valid, too.
Notice that these arguments are similar. They have the same form:
If Spot barks, then Dick will wake up. // Suzy calls early, then Dick will wake up.
A B A B
Spot barked. Suzy called early.
A A
So Dick woke up. So Dick woke up.
B B
Any argument of this form is valid (though not necessarily good, since a premise
could be false).
This way of reasoning is sometimes called modus ponens.
We can also reason:
128 CHAPTER 6 Compound Claims
If Spot barks, then Dick will wake up.
Dick didn't wake up.
So Spot didn't bark.
That's valid. After all, if Spot had barked, Dick would have woken up. Similarly:
If Suzy calls early, then Dick will wake up.
A B
Dick didn't wake up.
not B
So Suzy didn't call early,
not A
This way of reasoning is sometimes called modus tollens. Here, again,
"not A" and "not B" are shorthand for "the contradictory of A" and "the contradictory of B . " For example, this argument also uses the indirect way:
If Suzy doesn't call early, then Zoe won't go shopping.
Zoe went shopping.
So Suzy called early.
Recognizing this form can be hard if "not" occurs in the antecedent or
consequent, or if their order is reversed. For example, this uses the indirect way:
Zoe won'
t go shopping if Dick comes home early.
Zoe went shopping.
So Dick didn't come home early.
Zoe won't go shopping if Dick comes home early.
B A
Zoe went shopping.
not B
So Dick didn't come home early.
not A
To help us see how reasoning with conditionals involves possibilities, look at
what Dick has to face every morning:
SECTION B Conditionals 129
There are many ways that Dick could be awakened. And if he doesn't wake
up, then we know that none of those happened.
But it's wrong to reason that if Dick did wake up, then Spot barked. Maybe
Suzy called early. Or maybe Flo came over to play. It's reasoning backwards, over-
looking possibilities, to reason: If A, then B, B, so A. Yet it's easy to get confused
and use this way of reasoning as if it were valid, because it's so similar to the direct
way of reasoning with conditionals.
130 CHAPTER 6 Compound Claims
Just as there's a weak form that's easy to confuse with the direct way, there's a
weak form that's easy to confuse with the indirect way.
If it's the day for the garbageman, then Dick will wake up.
It's not the day for the garbageman. So Dick didn't wake up.
This, too, is overlooking other possibilities. Even though the garbageman didn't
come, maybe Flo came over to play, or Spot barked.
Denying the antecedent
If A, then B
IfA. then B + not A
Usually
not A
Weak
So not B
not B
With this form, too, we have to be alert when "not" shows up in the conditional.
If Dick doesn't wake up, If Dick doesn't wake up, then Dick will miss his class.
then he'll miss his class. A B
Dick woke up. Dick did wake up. So Dick didn't miss his class.
So Dick didn't miss his class. not A not B
But if Dick woke up, can't we at least say that one of those four claims from
the picture are true? No, there could be another possibility:
Here's a chart to summarize the valid and weak forms we've seen.
EXERCISES for Section B. 3 131
These weak forms of arguing with conditionals are clear confusions with valid
forms, mistakes a good reasoner doesn't make. When you see one, don't bother to
repair the argument. For example, suppose you hear:
Maria: If Suzy called early, then Dick woke up.
Lee: So Dick didn't wake up.
The obvious premise to add is "Suzy didn't call early," and probably Lee knows that.
But it makes the argument weak. So Lee's argument is unrepairable.
Exercises for Section B.3
1. Assume that all of the following conditionals are true:
• If Dick and Zoe get another dog, then Spot will be happy.
• If Dick buys Spot a juicy new bone, then Spot will be happy.
• If Dick spends more time with Spot, then Spot will be happy.
• If Spot finally learns how to catch field mice, then Spot will be happy.
Using them:
a. Give two examples of the direct way of reasoning with conditionals.
b. Give two examples of the indirect way of reasoning with conditionals.
c. Give two examples of affirming the consequent. Explain why each is weak
in terms of other possibilities.
d. Give two examples of denying the antecedent. Explain why each is weak in
terms of other possibilities.
2. Give an example (not from the text) of the direct way of reasoning with conditionals.
3. Give an example (not from the text) of the indirect way of reasoning with conditionals.
4. Give an example (not from the text) of affirming the consequent. Show that it is weak.
5. Give an example (not from the text) of denying the antecedent. Show that it is weak.
For Exercises 6-11, if there's a claim you can add to make the argument valid according to
one of the forms we've studied, add it. If the argument is unrepairable, say so.
6. If Flc comes over early to play, then Spot will bark. So Spot barked.
7. Whenever Flo comes over to play, Spot barks. So Flo didn't come over to play.
8. Tom: Suzy will fail Dr. E's class for sure if she doesn't study hard.
Harry: So she'll have to repeat that class, right?
9. Zoe will wash the dishes if Dick cooks. So Dick didn't cook.
10. Suzy: Dr. E won't give an exam today if he doesn't finish grading by this afternoon.
Maria: So Dr. E will give an exam today.
11. If Flo does her homework, then she can watch TV. So Flo did her homework.
132 CHAPTER 6 Compound Claims
12. Here's another valid form of reasoning with conditionals:
Dick: If I study for my math exam this weekend, we won't be able to have a good time
at the beach.
Zoe: But if you don't study for your exam, you'll worry about it like you always do,
and we won't be able to have a good time at the beach. So it looks like this
weekend is shot.
Give another example of a no-matter-what argument.
4. Reasoning in a chain and the slippery slope
Suppose we know that if Dick takes Spot for a walk, then Zoe will cook dinner. And
if Zoe cooks dinner, then Dick will do the dishes. Then we can conclude that if Dick
takes Spot for a walk, he'll do the dishes. We can set up a chain of reasoning, a
chain of conditionals.
Here's another example:
If Manuel's wheelchair isn't fixed tomorrow, then he can't go to classes.
If Manuel can't go to classes, then Lee will have to take notes for him.
If Lee takes notes for Manuel, then Manuel will have to cook dinner.
So if Manuel's wheelchair isn't fixed tomorrow, then Manuel will have to
cook dinner.
The conclusion is another conditional.
Reasoning in a chain is important: We go by little steps. Then if A is true,
we can conclude C.
SECTION B Conditionals 133
But this valid form of argument can be used badly. As Lee said to Maria:
Don't get a credit card! If you do, you'll be tempted to spend money you
don't have. Then you'll max out on your card. Then you'll be in real
debt. And you'll have to drop out of school to pay your bills. You'll
end up a failure in life.
This isn't stated as a series of conditionals, but it's easy to rewrite it that way (that's
Exercise 6 below). Then it will be valid. But it's not a good argument. If you take
the first step (accept the antecedent of the first conditional), then the chain of
conditionals forms a slippery slope for you to slide all the way to the conclusion.
But you can stop the slide: Just point out that one of the conditionals is dubious.
The second one is a good candidate. Or perhaps each one is only a little dubious,
but your reason to believe the conclusion becomes thinner and thinner as the doubt of
each one adds to the doubt of the previous ones.
Slippery slope argument A slippery slope argument is a bad argument
that uses a chain of conditionals, at least one of which is false or dubious.
Zoe: Don't go out with a football player.
Suzy: Why not?
Zoe: You're crazy about football players, and if you go out with one you're sure
to sleep with him.
Suzy: So?
Zoe: Then you'll get pregnant. And you'll marry the guy. But those guys are
su
ch jerks. You'll end up cooking and cleaning for him while he and his
buddies watch football on TV. In twenty years you'll have five kids, no
life, and a lot of regrets.
Suzy: Gosh. I guess you're right. I'll go out with a basketball player instead.
5. Reasoning from hypotheses
Lee: I'm thinking of doing a nursing degree.
Maria: That means you'll have to take summer school.
Lee: Why?
Maria: Look, you're in your second year now. To finish in four years like
you told me you need to, you'll have to take all the upper-division
biology courses your last two years. And you can't take any of those
until you've finished the three-semester calculus course. So you'll
have to take calculus over the summer in order to finish in four years.
Maria has not shown that Lee has to go to summer school. Rather, Maria has
134 CHAPTER 6 Compound Claims
shown on the assumption (hypothesis) that Lee will do a nursing degree, Lee will
have to go to summer school. That is, Maria has proved: If Lee does a nursing
degree, then he'll have to go to summer school.
Reasoning from hypotheses The following are equivalent:
• Start with an hypothesis A and make a good argument for B.
• Make a good argument for If A, then B.
Summary Some claims are made up of other claims. We need to recognize that such
claims must be treated as just one claim.
We looked at two kinds of compound claims in this chapter that involve
possibilities for how things could be: "or" claims and conditionals. There are lots of
confusing issues to master with conditionals: How to say they are false; necessary
and sufficient conditions; valid and weak forms. But we need to do that work,
because conditionals are the way we talk about how things could turn out under
certain conditions.
We found that compound claims are an important way to construct valid
arguments. We can reason with "or" claims by excluding possibilities. We can
reason with conditionals the direct or indirect way, or with a chain of conditionals.
We can reason from hypotheses.
There are typical mistakes people make using these valid forms. Some use
dubious or false premises, like false dilemmas or slippery slope arguments. Others
overlook possibilities by affirming the consequent or by denying the antecedent.
Key Words