I. Introduction

Most logicians would agree that we make use of inductive inference far more frequently than deductive inferences. Still, logicians have traditionally devoted far more attention to deductive inference. After a brief study of inductive inference you may appreciate the logician's preference for deduction; as hard as deductive logic can be, it is a lot less messy than inductive logic.

Let's begin by reminding ourselves of the difference between deductive and inductive arguments. Deductive arguments purport to guarantee the truth of the conclusion from the truth of the premises. A deductive argument which carries this out is valid, others are invalid. Inductive arguments purport only the make the conclusion likely, given the truth of the premises. For deductive arguments we ask: Does the truth of the premises guarantee the truth of the conclusion? For inductive arguments we ask: Does the truth of the premises make it at all likely that the conclusion is true?

It will become obvious once we look closely at particular types of inductive arguments that determining that an inductive argument is a good one is more difficult than determining the validity or invalidity of a deductive argument. There is room for discrimination in the evaluation of inductive inference, whereas evaluating deductive inferences in a formal language involves no judgment calls.

We will isolate a number of different types of inductive inference and individuate types by specifying a form of argument associated with each type. Unlike deductive logic, however, where the determination of the validity or invalidity of an argument type yielded an answer concerning the validity or invalidity of any instance of the form, in inductive logic we cannot evaluate an argument on the basis of its form alone. We can classify inductive arguments on the basis of form; we cannot evaluate them on the basis of form. For each of the argument forms discussed below we will specify fallacies associated with that form, i.e. ways of reasoning incorrectly using the form of inference in question.

II. Inductive Generalization - Arguments from Samples

A. The Form of Arguments from Samples

A common way of expressing the difference between inductive and deductive inference is to say that inductive inferences are ampliative and deductive inferences are not. What does this mean? In inductive inferences we amplify or "go beyond" the evidence contained in our premises. Very often that means that we generalize from particular facts which are reported in the premises. This form of inductive reasoning is quite common. Suppose you spent the day looking at storks. After a day of stork observing, you realize that all the storks you saw had red legs. You conclude, based on your experience, that all storks are red. In drawing this conclusion, you are not making a deductive inference. You would be the first to admit that there is no guarantee that all storks are red legged. Still, you would claim that your experience, that of seeing only red-legged storks after a full day of stork observation, supports, to some degree, the assertion that all storks are red-legged.

Arguments of this type are called inductive generalizations and the most common type of inductive generalization is called an argument from a sample. Our stork example above is an argument from a sample. By observing storks for a day you have a sample of storks. In that sample you found a property, red-leggedness. You infer that the extent to which that property is found in all storks is the same as the extent to which that property is found in the sample of storks, the storks you observed.

We can specify the form of arguments from samples as follows:

X percent of observed Fs are Gs.
X percent of all Fs are Gs.

Notice the crucial difference between the premise and the conclusion. The subject of the premise refers to observed Fs, that is to a sample of Fs. The conclusion refers to all Fs.

Here are a few examples of arguments from samples which we will analyze:

1. A nationwide poll of a random sample of thousands of homeowners revealed that 70 percent of them are opposed to increases in in property taxes. Thus, roughly 70 percent of the adult population opposes such increases.

2. Ignat is told to determine the boiling point of copper. He tests two very pure samples of copper and finds that each sample has a boiling point of 2,567 degrees Celsius. Ignat concludes that this is the boiling point for copper.

3. An investigator studied three thousand cocaine users and discovered that 75 percent of them had used marijuana before trying cocaine. She concludes that 75 percent of all marijuana users will go on to try cocaine.

The above arguments all have the form of arguments from samples. Notice that no explicit reference to percentages need be made in the premises or the conclusion of an argument from a sample. Are any of the above arguments good arguments? To answer that question we need to discuss the standards for good arguments from samples.

b. Evaluating Arguments from Samples

Like deductive arguments, good inductive arguments have premises which are true. So a necessary condition for any inductive argument being good is that it is sound. Since logic is concerned with inference, we don't worry about determining the soundness of arguments. Rather, for inductive as well as deductive arguments, we ask: if the premises are true, does the conclusion follow?

The conclusion of an argument from a sample is supported by the evidence only if the sample is representative. A sample is representative when the relevant features of the sample match the relevant features of the population about which we are drawing a conclusion. Suppose we wanted to determine what members of the Occidental community believe about the bombing of Libya, and we ask one student who believes that the bombing was justified. Should we conclude that in general, the Occidental community believes that the bombing was justified? Of course not! The sample of one student is not representative. Suppose we asked many students, but they all happened to be members of the Occidental Republican Student Organization. Can we draw a good inductive conclusion from such evidence? No again. In both cases our samples are not representative.

How do we insure that a sample is representative? There are two questions which we can ask in this connection:

1. Is the sample large enough?

2. Is the sample varied enough?

Asking one Occidental student is not enough to determine the preponderance of opinion on the issue in question. So the first sample failed to be representative because it was not large enough. The second sample, however, could be very large. It still fails to be representative because it yields a negative answer to our second question: it is not varied enough.

A good argument from a sample is one which is sound, and which appeals to a sample which is representative of the target population. A representative sample is one which is large enough and varied enough. This definition of representativeness is very unsatisfying. What is "enough"? When is a sample varied enough and when is it large enough.

There are no mechanical decision procedures for answering these questions. Whether a sample is large enough or varied enough to be representative depends on the inquiry in question. The crucial thing for determining representativeness of a sample is whether the relevant features of the sample match the relevant features of the population. If this match occurs, the sample is representative, otherwise not.

Let's return to the example above. When our sample included only members of the Occidental community who are staunch republicans, our sample, though perhaps quite large, was not varied enough. The sample did not contain those relevant features found in the population at large, e.g. persons who are Democrats, Independents, Marxists, etc. A varied enough sample would contain appropriate numbers of persons of these various political persuasions.

How many democrats should be in our sample? We should study the population at large, determine the percentage of democrats in the population as a whole, and try to match that percentage in our sample. Why make sure that Democrats are in the sample at all? Because the feature of being a Democrat is relevant to the question concerning the bombing of Libya. We have reason to believe that being a Democrat may make a difference in one's views on the subject.

It is very important to appreciate that evaluating and constructing inductive inferences of this sort (of any sort really) depends on background knowledge. Questions concerning the representativeness of a sample cannot be answered without a good deal of knowledge about "how things are in the world."

Not all representative samples are large. Look at the example of the boiling point of copper, above. Ignat uses a sample of only two. Is this a representative sample? It is not large, but there is such a close matching of the features of the sample to the "population" of copper (i.e. to all copper) that a sample of two is enough. Unlike political polling, dealing with copper is dealing with something which is uniform across various instances (under careful experimental conditions, where things like altitude are kept constant.) How large a sample must be depends on the relevant features of the population. The more uniform the population being investigated, the fewer "instances" are needed.

C. Fallacies

There are a number of fallacies associated with arguments from samples. We will consider a few of them. It is usually not hard to spot a fallacious argument from a sample, but it is important to figure out exactly why it is fallacious.

1. Hasty Generalization: This fallacy occurs when the sample is too small. If you inferred that someone you met was a jerk on the basis of one brief encounter with the person, you might be committing the fallacy of hasty generalization. In such cases a larger sample is needed.

2. Biased Statistics: This fallacy occurs when the sample is not varied enough. The homeowner example above (#1) is an example of biased statistics. The sample is quite large, but it is biased toward a certain feature in the population.

D. A Revision in the Form of Arguments from Samples

Our discussion of the form of arguments from samples has involved a simplification which must be corrected. Even in the best arguments from samples, it would be extremely surprising if the percent to which the feature we are testing for in the sample was matched exactly in the population as a whole. If we took a representative sample and found that 57 percent of observed Fs are Gs, it is still unlikely that exactly 57 percent of all Fs are Gs. What is likely is that the percent of Fs which are Gs is somewhere around 57 percent. We need to be explicit about allowing leeway in our conclusion. Taking account of this, we offer the following as the revised form:

X percent of Observed Fs are Gs.
X +/- z percent of all Fs are Gs.

where z represents the degree of departure in either direction from the observed percentage. Notice that the larger we allow z to be, the stronger will be the support of the conclusion by the premises. By weakening the conclusion, however, we weaken what we are establishing. So one has to choose z carefully.

E. Exercises

After distinguishing the premises from the conclusion in the following arguments from samples, evaluate the argument. If the argument is fallacious, state the fallacy committed.

1. It has rained during the last two home football games at our school. Therefore, it will probably rain at all the home games this year.

2. It hasn't rained during the last twenty baseball games. Therefore it will rain during the next game.

3. Pueblo sites are located in many places in Arizona, New Mexico, and Colorado. To determine the number of rooms in pueblo sites that were located by surface surveys but were not excavated, archaeologists examined the relationship between the size of the surface rubble mound prior to excavation and the number of rooms that were ultimately uncovered in each of six sites excavated in the Upper Little Colorado River region. The number of rooms in each site was equal to (0.10 x area of rubble mount in square meters) +4.

4. In 1977 at the University of Pennsylvania, psychiatrists conducted a study to determine the social factors that affect the well-being of coronary patients. There were 93 patients in the study; slightly more than 50 percent of them had pets of some kind (dogs, cats, fish, and one iguana). At the end of a year, one-third of the patients who did not own pets had died but only three animal owners had succumbed. The psychiatrists concluded that pet ownership may have a positive effect on the health of humans.

5. "There is no overestimating the importance of pets to people it seems. Katcher [the psychiatrist in charge of the study mentioned in 4 above] reported that in one questionnaire, on which people were given the opportunity to indicate whether they thought their pet was an animal or a human member of the family, 48 percent responded that the animal was a human family member". (from "Human-Animal Relationship Under Scrutiny", Science, 1981, 214:418.

6. Ignat has to travel to New York from L.A. He wants to take the safest mode of transport, so he compares statistics over the past ten years on accidents involving buses, trains, automobiles, and planes that occurred between L.A. and N.Y.C. Ignat determines that the safety record of buses is far better in terms of lives lost than any of the other forms of travel. As he is about to purchase his ticket, however, he sees a newspaper story about a bus accident in which six people died. Ignat decides to drive to N.Y.C.

7. To test the algae content in a lake, a biologist took a sample of the water at one end. The algae in the sample registered 5 micrograms per liter. Therefore, the algae in the lake at that time registered 5 micrograms per liter.

8. When asked, most Americans say they would not like to live in the Soviet Union. Therefore, most people believe that the Soviet Union is not a desirable place to live.

9. A survey of philosophers attending an annual meeting called "Philosophy and Teaching" revealed that most think that teaching is more important than research. It may safely be concluded, then, that philosophers are first and foremost committed to excellence in teaching.

10. Consumer Reports rated that stereo receiver a "best buy". But my friend Orcutt bought one and had to return it to the company for repairs after just one month. so it cannot be a very good receiver.

III. Analogical Arguments

A. Introduction

Another important type of inductive inference is the argument from analogy. In such arguments we draw attention to the fact that two things or classes of things have some features in common. This information about shared features is the basis for inferring that the two things or classes of things have some further properties in common. Why do we study mice in order to learn about disease in human beings. We reason analogically, by pointing out that there are similarities in the physiology, chemistry, etc. of mice and human beings. We conclude that a further similarity exists, namely, that whatever causes a particular disease in mice is likely to cause the same disease in humans. We reason from certain shared properties to others.

Not all or even most uses of analogy are arguments from analogy. Analogy is important in description, in poetry, and in many other contexts. Boswell, in In Search of a Wife says "I am a weaker man than can well be imagined. My brilliant qualities are like embroidery on gauze. Lawrence Durrell, in Reflections on a Marine Venus compares the close of days in Rhodes to falling fruit: "In Rhodes the days drop softly as fruit from trees." These are not arguments, but they are important uses of analogy outside the scope of logic.

B. The Form of Arguments from Analogy

Arguments from analogy can be represented by the following general schema:

Objects of type X have properties F, G, H, etc.

Objects of type T have properties F, G, H, etc. and also an additional property Z.
Objects of type X have property Z as well.

The premises provide information about features shared in common by the two objects (or sets of objects). The information that one of the objects has an additional feature. The conclusion claims that the additional feature is also had by the other object.

C. Assessing the Strength of Analogical Arguments

Good analogical arguments, like all other arguments, are sound. In this case the similarities claimed in the premises must exist. But this is not sufficient, as the following example makes clear:

Ignat and Fred both have red cars. Fred smashed up his car yesterday. Therefore Ignat smashed up his car as well.

This is an argument from analogy, but a very shabby one. The feature of being a red car, which Ignat's car shares with Fred's, is simply irrelevant to the question of the status of the car's sheet metal. This suggests that in good arguments from analogy the common features mentioned in the premises are relevant to the determination of the further feature. Good arguments from analogy mention only relevant similarities. In addition, for an argument from analogy to be good, there must be few if any relevant dissimilarities between the objects being compared, whether or not those dissimilarities are mentioned in the argument. Naturally, the greater the number of relevant similarities and the smaller the number of relevant dissimilarities, the stronger the argument. An argument which rests on irrelevant similarities commits the fallacy of false analogy.

How do we know when a feature is relevant? One feature is relevant to another if the presence of the first increases or decreases the probability of the presence of the second. To know that a feature is relevant involves background knowledge. In the example above, to identifying the failed inference depends on already knowing that the color of a car is not causally related to its getting into an accident on a particular day.

In order to be confident that the similarities appealed to in an argument from analogy are relevant, we often look for variety in the things we are comparing. Thus when we draw an analogy between the formation of a certain disease in mice and humans on the basis of physiological similarities, our inference is strengthened if the similarity is not just between one type of mice and humans, but between all types. If we can find other animals for which the analogy holds, this can only strengthen our conclusion.

Arguments from analogy are very much like arguments from samples. In good arguments from samples, the sample must be representative, i.e. it must be like the population. It is sometimes hard to determine whether one is confronted with an argument from a sample or an argument from analogy, and there is no mechanical way to decide either way. In such borderline cases one must look at the way the argument is stated. This is another feature of inductive arguments which makes them difficult to deal with. Even the classification of arguments by type is difficult.

D. Exercises - Arguments from Analogy

Evaluate the following arguments.

1. Tar (extracted from cigarette smoke) when smeared on the skin of mice in laboratories causes skin cancers. Therefore, cigarette smoking causes lung cancer in humans.

2. My last pair of Brand X running shoes were comfortable, gave excellent support to my feet and ankles, and lasted a long time. I expect my new pair of Brand X running shoes, which have the same design, to give the same kind of service as my old pair.

3. Wives, be subject tot your husbands as to the Lord, for the husband is head of the wife as Christ also is the head of the church; as the church is subject to Christ, so wives are to be subject to their husbands in every respect. (ST. Paul, Ephesians, 5,22)

4. The force that binds planets to the sun (gravity) obeys the same general form of law as the electrical force that binds electrons to the nucleus of an atom. (Both gravity and electricity decrease in strength with the square of the distance between the bodies or particles.) Therefore, the electron particles, which have negative charges, when attracted by the positive electricity of the nucleus, should move around it in the same way that the planets move around the sun.

IV. Statistical Syllogisms

A. Form

Syllogisms are arguments with two premises and three terms, where one term, the middle term, occurs in both premises, and the two other terms occur once each in the premises and both in the conclusion. The following is a famous example of a deductive syllogism:

All men are mortal.
Socrates is a man.
Socrates is mortal.

The middle term is "men". the other terms are "is a mortal" and "Socrates". In this syllogism the first premise is a universal generalization. It says something about the entire class of men, namely that they are mortal. The second premise claims that an individual is a member of the first class, and the conclusion is that the individual is a member of the second class.

In inductive logic we have syllogisms as well. A difference between a deductive and an inductive syllogism is that the first premise of an inductive syllogism is a statistical generalization rather than a universal generalization. Another difference is that inductive or statistical syllogisms do not purport to guarantee the truth of their conclusions, only to make them probable on the basis of the evidence. The claim that all inductive inferences go from the particular to the general is false. In statistical syllogisms the inference goes the other way.

The following is an example of a statistical syllogism:

Most lovers of Mozart's music hate Falco.
Orcutt is a lover of Mozart's music.
Orcutt hates Falco.
 

Instead of "most" we could specify some percentage. The general form of statistical syllogisms is:
 

X percent of all Fs are Gs.
a is an F.
a is a G.

where "X" is a variable for some numerical value, "F" and "G" represent classes of things, and "a" is a variable for an individual. We call the class represented by "F" the reference class, the class represented by "G" the attribute class.
 

B. Evaluating Statistical Syllogisms
 

The closer X is to 100, the stronger the argument. This is reflects what we said earlier about the strength of an inductive argument depending on the relative strength of the premises and the conclusion. Just as we could strengthen an argument by weakening the conclusion, an argument can be strengthened by strengthening the premises. Of course, one can only strengthen the premises when one has the evidence to back up the strengthened claim being made.

Statistical syllogisms can also be used to reach negative conclusions. Consider the following version of the Orcutt argument:

Few lovers of Mozart's music like Falco
Orcutt is a lover of Mozart's music.
Orcutt is not a person who likes Falco.

Appropriate changes must be made in the claim that the closer X is to 100, the better the statistical syllogism, in light of the above. As an exercise, state this full condition in your own words.

C. The Fallacy of Incomplete Evidence

Consider the following argument:
 

97 percent of all nurses are female.
Fred is a nurse.
Fred is a female.

The fallaciousness of this argument should hit you like a brick. It fits the form of a statistical syllogism. Where does it go wrong. The problem is that the evidence provided is incomplete. There is other evidence which is relevant to the question of whether the individual in question is a female. That evidence is that the individual belongs to another reference class, the reference class of persons whose name is "Fred". We know that almost all (perhaps all) human members of this reference class are males. By leaving that crucial bit of evidence out of our argument, we have committed the fallacy of incomplete evidence. In light of this we can say that statistical syllogisms face the requirement of total evidence. We must consider all the relevant reference classes to which the individual mentioned in the argument belongs before we can draw the conclusion.

What constitutes a relevant reference class? The reference class A is relevant to the reference class B if membership in A effects the likelihood that the individual is a member of B.

D. Exercises

1. Come up with at least five pairs of classes which are relevant to one another.

2. The following arguments commit the fallacy of incomplete evidence. In each case, discuss what relevant evidence is ignored.

a. Most Russians don't speak English, so the newly appointed Russian ambassador to the United Nations probably doesn't speak any English.

b. Most movie actors aren't politicians, so Ronald Reagan, a former movie actor, is not a politician.

c. Most Americans earn less than $100,000 a year, so the President of General Motors earns less than $100,000 a year.

E. Special types of Statistical Syllogism

We can locate special forms of statistical syllogisms by concentrating on the statistical generalization which serves as the first premise of such arguments. We briefly mention the various types and leave for an exercise the discussion of the standards by which such arguments are assessed. (Hint: Apply the standards for statistical syllogisms.)

a. Arguments from authority:

Most of what authority a has t say on subject matter S is correct.

a says p about S.
p is correct.
 

b. Arguments against the Person [Argumentum Ad Hominem]

Most of what authority a has t say on subject matter S is false.

a says p about S.
p is false.

c. Arguments from Consensus

When most people agree on a cl aim about subject matter S, the claim is usually true.
p is a claim most people make about S.
p is true.

F. More Exercises - Classify and Evaluate the following arguments:

1. Charles Colson, a former White house Aide and a convicted perjurer, charged that the CIA knew about the Watergate break-in in advance. Referring to Colson's charge, William Colby, former CIA Director, said, "His lack of credibility should cause the charge to fall of its own weight." (Reconstruct Colby's argument.)

2. My favorite fashion model, Cheryl Teigs, says that Cover Girl cosmetics are the safest for delicate skins. Therefore, Cover girl is the best brand for sensitive skins.

3. Most people believe that smoking marijuana is dangerous to one's health. Therefore, smoking marijuana is dangerous to one's health.

4. During both the Ford and the Carter administrations, economic advisors to the President agreed that the best way to encourage the United States to conserve oil would be to impose a tariff of $3 a barrel on imported oil. Therefore, such a tariff would be the best conservation measure.

5. Many prisoners have complained that conditions in the county jail are unsanitary. However, these persons are outlaws, so we can safely deny the truth of their charges.