Chap 8,9,10 (Ph)

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Stat ist ical General izat ions

STATISTICAL GENERALIZATIONS

One classic example of an inductive argument is an opinion poll. Suppose a candidate wants to know how popular she is with voters. Because it would be practically impossible to survey all voters, she takes a sample of voting opinion and then infers that the opinions of those sampled indicate the over- all opinion of voters. Thus, if 60 percent of the voters sampled say that they will vote for her, she concludes that she will get around 60 percent of the vote in the actual election. As we shall see later, inferences of this kind often

1. The sun is coming out, so the rain will probably stop soon. 2. It’s going to rain tomorrow, so it will either rain or be clear tomorrow. 3. No woman has ever been elected president. Therefore, no woman will

ever be elected president. 4. Diet cola never keeps me awake at night. I know because I drank it just

last night without any problems. 5. The house is a mess, so Jeff must be home from college.

1. The following arguments are not clearly inductive and also not clearly deductive. Explain why. a. All humans are mortal, and Socrates is a human, so Socrates is likely to

be mortal also. b. We checked every continent there is, and every raven in every continent

was observed to be black, so every raven is black. c. If there’s radon in your basement, this monitor will go off. The monitor

is going off, so there must be radon in your basement. (Said by an engi- neer while running the monitor in your basement.)

2. In mathematics, proofs are sometimes employed using the method of mathematical induction. If you are familiar with this procedure, determine whether these proofs are deductive or inductive in character. Explain why.

Discussion Questions

" "

Remember that the difference between induction and deduction is just a matter of what standard we intend to use in evaluating the argument’s support relation (will it be the standard of deductive validity or the standard of inductive strength?). This means that you need to make a judgment call about which standard is the most sensible to apply.

Assuming a standard context, label each of the following arguments as deduc- tive or inductive. Explain what it is about the words or form of argument that indicates whether or not each argument is intended or claimed to be valid. If it is not clear whether the argument is inductive or deductive, say why.

Exercise IGeneral Note:

Remember to not try to do all these in one sitting! It's a lot! My intention is that you'd work on the exercises as we move through the material in class - spacing it out so you can focus on one argument form at a time. Read ahead of where we're at in class, and follow behind with the relevant homework exercises after we've had a chance to discuss them in class.

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Stat ist ical Appl icat ions

STATISTICAL APPLICATIONS

In a statistical generalization, we draw inferences concerning a population from information concerning a sample of that population. If 60 percent of the population sampled said that they would vote for candidate X, we might draw the conclusion that roughly 60 percent of the population will vote for candidate X. With a statistical application (sometimes called a statisti- cal syllogism), we reason in the reverse direction: From information concern- ing a population, we draw a conclusion concerning a member or subset of that population. Here is an example:

Ninety-seven percent of the Republicans in California voted for McCain. Marvin is a Republican from California. Marvin voted for McCain.

Such arguments have the following general form:

X percent of Fs have the feature G. a is an F.

a has the feature G.4

6. I have lots of friends. Most of them think that I would make a great president. So most Americans would probably agree.

7. In exit polls after people had just voted, most people told our candidate that they voted for her, so probably most people did vote for her.

8. Mary told me that all of her older children are geniuses, so her baby will probably be a genius, too.

9. When asked whether they would prefer a tax break or a bloated budget, almost everyone said that they wanted a tax break. So a tax break is over- whelmingly popular with the people.

10. When hundreds of convicted murderers in states without the death penalty were asked whether they would have committed the murder if the state had a death penalty, most of them said that they would not have done it. So most murders can be deterred by the death penalty.

It is often easy to see that a sample is biased, but how can you tell that a sam- ple is not biased? How can you determine whether a sample is big enough?

Discussion Question

Exercise II

In the following passages (A) identify the reference class, (B) identify the sample class, and finally (C) evaluate the generalizations using the criteria of sample size, sample bias, bias in investigation, and bias in interpretation.

Be sure to explain your answers for the evaluation portion!

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Stat ist ical Appl icat ions

5. Ninety-eight percent of what John says is true. John said that his father is also named John.

John’s father is named John.

6. Ninety-eight percent of what John says is true. John said that the Giants are going to win.

The Giants are going to win.

8. Most people do not understand quantum mechanics. My physics professor is a person.

My physics professor probably does not understand quantum mechanics.

9. Almost all birds can fly. This penguin is a bird.

This penguin can fly.

Although both in science and in daily life, we rely heavily on the methods of inductive reasoning, this kind of reasoning raises a number of perplexing problems. The most famous problem concerning the legitimacy of induction was formulated by the eighteenth-century philosopher David Hume, first in his Treatise of Human Nature and then later in his Enquiry Concerning Human Understanding. A simplified version of Hume’s skeptical argument goes as fol- lows: Our inductive generalizations seem to rest on the assumption that unob- served cases will follow the patterns that we discovered in observed cases. That is, our inductive generalizations seem to presuppose that nature operates uni- formly: The way things are observed to behave here and now are accurate indicators of how things behave anywhere and at any time. But by what right can we assume that nature is uniform? Because this claim itself asserts a con- tingent matter of fact, it could only be established by inductive reasoning. But because all inductive reasoning presupposes the principle that nature is uni- form, any inductive justification of this principle would seem to be circular. It seems, then, that we have no ultimate justification for our inductive reasoning at all. Is this a good argument or a bad one? Why?

Discussion Question

For each of the following statistical applications, (A) identify the reference class, (B) identify the subset class, and finally (C) evaluate the strength of the argument using the percentages or proportions cited and the relevance of the reference class.

4. Three percent of socialists with blue eyes voted for McCain. Maureen is a socialist with blue eyes.

Maureen did not vote for McCain.

Exercise IV

Again, be sure to provide explanations!

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CHAPTER 9 ■ Causal Reason ing

being the element mercury is a sufficient condition for being a metal, but it is not a necessary condition for being a metal, since there are other metals. Similarly, being a metal is a necessary condition for being mercury, but it is not a sufficient condition for being mercury. Of course, some necessary conditions are also sufficient conditions. Being mercury is both necessary and sufficient for being a metallic element that is liquid at twenty degrees Centigrade. Nonetheless, many necessary conditions are not sufficient conditions, and vice versa, so we need to be careful not to confuse the two kinds of conditions.

This distinction becomes complicated when conditions get complex. Our definitions and tests hold for all features, whether positive or negative (such as not having hair) and whether simple or conjunctive (such as having both a beard and a mustache) or disjunctive (such as having either a beard or a mustache). Thus, not having any hair (anywhere) on your head is a suffi- cient condition of not having a beard, so not having a beard is a necessary condition of not having any hair on your head. But not having any hair on your head is not necessary for not having a beard, because you can have some hair on the top of your head without having a beard. Negation can cre- ate confusion, so we need to think carefully about what is being claimed to be necessary or sufficient for what.

Even in simple cases without negation, conjunction, or disjunction, there is a widespread tendency to confuse necessary conditions with sufficient conditions. It is important to keep these concepts straight, for, as we will see, the tests concerning them are fundamentally different.

Which of the following claims are true? Which are false?

1. Being a car is a sufficient condition for being a vehicle. 2. Being a car is a necessary condition for being a vehicle. 3. Being a vehicle is a sufficient condition for being a car. 4. Being a vehicle is a necessary condition for being a car.

Exercise I

14. Driving seventy-five miles per hour (for fun) is a sufficient condition for violating a legal speed limit of sixty-five miles per hour.

15. Driving seventy-five miles per hour (for fun) is a necessary condition for violating a legal speed limit of sixty-five miles per hour.

16. Cutting off Joe’s head is a sufficient condition for killing him. 17. Cutting off Joe’s head is a necessary condition for killing him. 18. Cutting off Joe’s head and then holding his head under water for ten min-

utes is a sufficient condition for killing him.

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CHAPTER 9 ■ Causal Reason ing

target feature, ~G. It also might be necessary for features A, B, and D. Nothing in Cases 1–4 rules out these possibilities. Thus, even after Case 4, we cannot say simply that C is not a necessary condition. Case 4 shows that candidate feature C is not a necessary conditions for target feature G, but C still might be necessary for something else. The same point applies to sufficient conditions as well. In Table 1, Case 2 ruled out the possibility that candidate feature B is sufficient for target feature G, but none of the cases in Table 1 show that B is not sufficient for target feature C. To avoid confusion, then, it is always important to specify the target feature when talking about what is or is not a necessary or sufficient condition.

THE JOINT TEST

It is also possible to apply these rules simultaneously in the search for possible conditions that are both sufficient and necessary. Any candidate cannot be both sufficient and necessary if it fails either the SCT or the NCT. In Table 2, C is the only possible necessary condition for G, and it is not also a possible sufficient condition for G, since C fails the SCT in Case 1, where C is present and G is absent. In Table 1, however, D is a possible sufficient condition of G, because D is never present when G is absent; and D might also be a necessary condition for G, since G is never present when D is absent. Thus, none of Cases 1–3 in Table 1 eliminates D as a candidate for a condition that is both sufficient and necessary for G. As before, this possibility still might be refuted by Case 4, so any inference to a positive conclusion that some candidate is a necessary and sufficient condition must be defeasible and, hence, inductive.

For each of the following tables determine

a. List the conditions that fail the SCT for feature "G" and provide all the cases that prove that they cannot be sufficient conditions.

b. List the conditions that fail the NCT for feature "G" and provide all the cases that prove that they cannot be sufficient conditions.

EXAMPLE: Case 1: A B �C D �G

Case 2: �A B C D G

Case 3: A �B C D G

a. A, B, and D fail the SCT in Case 1.

b. A fails the NCT in Case 2. B fails the NCT in Case 3.

Exercise III

239

Suff ic ient Cond it ions and Necessary Cond it ions

1. Case 1: A B C D G Case 2: �A B �C D �G Case 3: A �B C �D G

2. Case 1: A B C �D G Case 2: �A B C D G Case 3: A �B C �D G

3. Case 1: A B C D �G Case 2: �A B C D G Case 3: A �B C �D G

4. Case 1: A B �C D G Case 2: �A �B C D G Case 3: A B �C �D �G

5. Case 1: A �B C D �G Case 2: �A B C �D �G Case 3: A �B �C D G

6. Case 1: A B �C D G Case 2: �A �B C D �G Case 3: A �B C �D �G

7. Case 1: A B �C D �G Case 2: �A B �C D �G Case 3: A B �C �D �G

8. Case 1: A B C D �G Case 2: �A �B C D G Case 3: A �B �C �D �G

Imagine that your desktop computer system won’t work, and you want to find out why. After checking to make sure that it is plugged in, you experiment with a new central processing unit (CPU), a new monitor (MON), and new system software (SSW) in the combinations on the table below. The candidates for necessary conditions and sufficient conditions of failure are the plug posi- tion (in or out), the CPU (old or new), the monitor (old or new), and the soft- ware (old or new). For each candidate, say (1) which cases, if any, eliminate it as a sufficient condition of your computer’s failure and (2) which cases, if any, eliminate it as a necessary condition of your computer’s failure. Which candi- dates, if any, are not eliminated as a sufficient condition of failure? As a neces- sary condition of failure? Does it follow that these candidates are necessary conditions or sufficient conditions of failure? Why or why not?

Exercise IV

(continued)

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Suff ic ient Cond it ions and Necessary Cond it ions

1. Case 1: A B C D G Case 2: �A B �C D �G Case 3: A �B C �D G

2. Case 1: A B C �D G Case 2: �A B C D G Case 3: A �B C �D G

3. Case 1: A B C D �G Case 2: �A B C D G Case 3: A �B C �D G

4. Case 1: A B �C D G Case 2: �A �B C D G Case 3: A B �C �D �G

5. Case 1: A �B C D �G Case 2: �A B C �D �G Case 3: A �B �C D G

6. Case 1: A B �C D G Case 2: �A �B C D �G Case 3: A �B C �D �G

7. Case 1: A B �C D �G Case 2: �A B �C D �G Case 3: A B �C �D �G

8. Case 1: A B C D �G Case 2: �A �B C D G Case 3: A �B �C �D �G

Imagine that your desktop computer system won’t work, and you want to find out why. After checking to make sure that it is plugged in, you experiment with a new central processing unit (CPU), a new monitor (MON), and new system software (SW) in the combinations on the table below. The candidates for necessary conditions and sufficient conditions of failure are the plug position (in or out), the CPU (old or new), the monitor (old or new), and the software (old or new). Make a list of all the features that pass the SCT and those that pass the NCT for the target condition of "Failure".

Exercise IV

(continued)

Another important exercise for the exam! It may look a little more intimidating with the words rather than the letters, but it is really the same game with the same patterns. Stick to your guidelines for the SCT and NCT and you’ll be fine!

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CHAPTER 9 ■ Causal Reason ing

Plug CPU Monitor Software Result

Case 1 ln Old CPU Old MO Old SW Works Case 2 ln Old CPU Old MO New SW Works Case 3 ln Old CPU New MO Old SW Fails Case 4 ln Old CPU New MO New SW Works Case 5 ln Old CPU Old MO Old SW Works Case 6 ln Old CPU Old MO New SW Works Case 7 ln Old CPU New MO Old SW Fails Case 8 ln Old CPU New MO New SW Works Case 9 ln New CPU Old MO Old SW Fails Case 10 ln New CPU Old MO New SW Works Case 11 ln New CPU New MO Old SW Fails Case 12 ln New CPU New MO New SW Works

After a banquet, several diners get sick and die. You suspect that something they ate or drank caused their deaths. The following table records their meals and fates. The target feature is death. The candidates for necessary conditions and suf- ficient conditions of death are the soup, entrée, wine, and dessert. For each candi- date, say (1) which cases, if any, eliminate it as a sufficient condition of death and (2) which cases, if any, eliminate it as a necessary condition of death. Which can- didates, if any, are not eliminated as a sufficient condition of death? Which candi- dates, if any, are not eliminated as a necessary condition of death? Does it follow that these candidates are necessary conditions or sufficient conditions of death? Why or why not?

Diners Soup Entrée Wine Dessert Result

Ann Tomato Chicken White Pie Alive Barney Tomato Fish Red Cake Dead Cathy Tomato Beef Red Ice Cream Alive Doug Tomato Beef Red Cake Alive Emily Tomato Fish Red Pie Dead Fred Tomato Fish Red Cake Dead Gertrude Leek Fish White Pie Alive Harold Tomato Beef White Cake Alive Irma Leek Fish Red Pie Dead Jack Leek Beef Red Ice Cream Alive Ken Leek Chicken Red Ice Cream Alive Leslie Tomato Chicken White Cake Alive

Exercise V

RIGOROUS TESTING

Going back to Table 1, it is easy to see that candidates A, B, C, and D are not eliminated by the NCT as necessary conditions of target G, as G is present in only one case (Case 1) and A, B, C, and D are present there as well. So far, so

Don’t miss how if one of the possible features is present, its alternative is not. So for example, if the cases is one with the New Monitor, then that means it is also at the same time a case where the Old Monitor condition is absent.

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CHAPTER 10 ■ Inference to the Best Explanat ion and from Analogy

Imagine that you offer an explanation, and a critic responds in the following way. Which virtue (explanatoriness, depth, power, falsifiability, modesty, sim- plicity, or conservativeness) is your critic claiming that your explanation lacks?

1. But that won’t explain anything other than this particular case. 2. But that conflicts with everything we know about biology. 3. But you don’t have to claim all of that in order to explain what we see. 4. But that just raises new questions that you need to answer. 5. But that explains only a small part of the story. 6. But that would apply whatever happened.

Exercise I

For each of the following explanations, specify which standard of a good ex- planation, if any, it violates. The standards require that a good explanation be explanatory, deep, powerful, falsifiable, modest, simple, and conservative. A single explanation might violate more than one standard.

1. Although we usually have class at this time in this room, I don’t see any- body in the classroom, because a wicked witch made them all invisible.

2. Although we usually have class at this time in this room, I don’t see any- body in the classroom, because they all decided to skip class today.

3. Although we usually have class at this time in this room, I don’t see any- body in the classroom, because it’s Columbus Day.

4. My house fell down, because it was painted red. 5. My house fell down, because of a powerful earthquake centered on my

property that did not affect anything or anybody else. 6. My house fell down, because its boards were struck by a new kind of sub-

atomic particle. 7. Although I fished here all day, I didn’t catch any fish, because there are

no fish in this whole river. 8. Although I fished here all day, I didn’t catch any fish, because the river

gods don’t like me. 9. Although I fished here all day, I didn’t catch any fish, because I was

unlucky today. 10. That light far up in the night sky is moving quickly, because it is the daily

United Airlines flight from Boston to Los Angeles. 11. That light far up in the night sky is moving quickly, because it is an alien

space ship. 12. That light far up in the night sky looks like it is moving quickly, because

there’s something wrong with my eyes right now.

Exercise II

Don’t just stick to mentioning the standards you think are not being met. Practice addressing each standard and being sure that you explain each answer. This is what you’ll be asked to do on the exam!

Doing this for every single problem is a LOT here, so my encouragement is to do the first few with answers to all 7 criteria, and then after that, when you're feeling good with it, you can just stick to mentioning the criteria that are violated. If a particular standard is giving you trouble, keep answering problems for it until it's feeling more familiar. If that magic isn't happening, contact me!

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b. The drug does not cure this disease in cats. c. The drug has to be injected into the rat’s tail to be effective (that is, it

does not work if it is injected anywhere else in the rat). d. No drug of this general type has been used on humans before. e. The effects of the drug are enhanced by eating cooked foods.

Using the criteria mentioned above, evaluate each of the following arguments as strong or weak. Explain your answers. Be sure to specify the properties on which the analogy is based, as well as any background beliefs on which your evaluation depends.

1. This landscape by Cézanne is beautiful. He did another painting of a similar scene around the same time. So it is probably beautiful, too.

2. My aunt had a Siamese cat that bit me, so this Siamese cat will probably bite me, too.

3. The students I know who took this course last year got grades of A. I am a lot like them, since I am also smart and hardworking; and the course this year covers very similar material. So I will probably get an A.

4. This politician was caught cheating in his marriage, and he will have to face similarly strong temptations in his public duties, so he will probably cheat in political life as well.

5. A very high minimum wage led to increased unemployment in one coun- try. That country’s economy is similar to the economy in a different coun- try. So a very high minimum wage will probably lead to increased unemployment in the other country as well.

6. I feel pain when someone hits me hard on the head with a baseball bat. Your body is a lot like mine. So you would probably feel pain if I hit you hard on the head with a baseball bat. (This is related to the so-called “Problem of Other Minds.”)

7. It is immoral for a doctor to lie to a patient about a test result, even if the doctor thinks that lying is in the patient’s best interest. We know this because even doctors would agree that it would be morally wrong for a financial adviser to lie to them about a potential investment, even if the financial advisor thinks that this lie is in the doctor’s best interests.

8. Chrysler was held legally liable for damages due to defects in the suspension of its Corvair. The defects in the Pinto gas tank caused injuries that were just as serious. Thus, Ford should also be held legally liable for damages due to those defects.

Exercise V

Same story here as with the previous exercise – be sure to address each of the standards at first (don’t leave any out), and try to explain your answers. The exam will grade you mostly on your explanations, so get practice at articulating your thinking.