Common Inductive Arguments

15 Inductive Applications

Section 1: Introduction

Statistical generalizations tend to be used in two ways in our reasoning: as premises and as conclusions.  As we have seen, in an inductive generalization we start with premises about individuals or small groups and move to a generalization about a whole population.  However, we can also do the reverse, and reason from a conclusion.  That is, we can start with a generalization about a whole population and move to a conclusion about an individual or small group.  In this chapter we will take a close look at the process of applying generalizations to specific cases.  As we will see, these arguments bear a similarity to some of the deductive argument forms discussed in Chapters 10 and 11.

Section 2: Reasoning from a Statistical Generalization

We will refer to an argument which draws a conclusion about an individual or small group on the basis of a statistical generalization an Inductive Application (these arguments are also sometimes called statistical applications or statistical syllogisms).  Let’s take a look at an example.

Ex. 1:

Chandler: I doubt that Tony is a Cubs fan.

Alex: Why?

Chandler: I know Tony loves baseball, but he grew up on the south side of Chicago, and most baseball fans who grew up on the south side of Chicago are White Sox—and not Cubs—fans.

Chandler is reasoning from a generalization.  She starts with a statistical generalization about people who grew up on the south side of Chicago, and uses it to draw a conclusion about an individual—Tony.  We can standardize Chandler’s argument as follows:

  1. Most baseball fans who grew up on the south side of Chicago are White Sox—and not Cubs—fans.
  2. Tony is a baseball fan who grew up on the south side of Chicago.
  3. So, Tony is probably a White Sox—and not a Cubs—fan.

Inductive applications follow a pattern.  Although they may be stated with more or less specificity, all inductive applications include a statistical generalization as a premise.  The statistical claim tells us that many, most, or few members of the group that constitute the subject of the sentence are members of the group that constitute the predicate of the sentence.  Put in terms we introduced in the last chapter, the statistical claim in the argument above claims that most members of the subject class, ‘baseball fans who grew up on the south side of Chicago’, are members of the predicate class, ‘White Sox—and not Cubs—fans’.

Inductive applications also rely on a premise which tells us that an individual is or is not a member of one of the classes.  In this case, we are told that Tony is a member of the subject class—that is, Tony is a baseball fan who grew up on the south side of Chicago.  Given that Tony is a member of the subject class, and that most members of the subject class are also members of the predicate class, Chandler concludes that Tony is also probably a member of the predicate class—that is, that he is probably a White Sox—and not a Cubs—fan.

Alternatively, we could have applied the generalization to draw a different conclusion.  Suppose that we only knew that Tony was a Cubs—and not a White Sox—fan.  We could then reason as follows:

  1. Most baseball fans who grew up on the south side of Chicago are White Sox—and not Cubs—fans.
  2. Tony is a Cubs and not a White Sox fan.
  3. So, Tony probably didn’t grow up on the south side of Chicago.

This is a different way of formulating an inductive application, since although it appeals to the same statistical generalization in premise 1, it adds that Tony is not a member of the predicate class, and concludes he is not a member of the subject class.

Chicago skyline in reflection
“Chicago Skyline reflection in the BEAN” by mikeyexists CC BY-NC-ND 2.0

Perhaps you have already noticed that inductive applications have a lot in common with some other arguments we have looked at—namely Modus Ponens and Modus Tollens.  When we know that the individual in question is a member of the subject class, and thereby infer that the individual is a member of the predicate class, we are drawing an inference that is structurally similar to modus ponens.  Similarly, when we know that the individual in question is not a member of the predicate class, and thereby infer that the individual is not a member of the subject class, we are drawing an inference that is structurally similar to Modus Tollens.  Despite these similarities, there is one fundamental difference, of course, namely that inductive applications are just that—inductive.

Like other kinds of argument, inductive applications are not always explicit.  As noted above, sometimes the quantity in question is ambiguous.  In other cases, the statistical generalization is left implicit.  Consider the following example:

Ex. 2:

Sydney: I am not sure what I am going to do when I get out of college, but I am pretty sure I don’t want to be a teacher.

Gina:   You don’t want to be a teacher?  I thought that you were an English major!

In this case, Gina is surprised that Sydney doesn’t want to be a teacher, because she is assuming a connection between students who are English majors and students who are pursuing teaching as a career.  It is hard to know exactly what the assumed connection is in this case without asking Gina, but it is likely that she is assuming something like this: ‘most English majors want to be teachers’.  Given this, the unstated assumption is a statistical generalization, and Gina is reasoning using an inductive application.  This kind of ambiguously stated inductive application is relatively common in our everyday thinking.  We naturally and intuitively generalize on our experiences, and once these generalizations are part of our overall system of belief we can use them to draw conclusions about specific cases.   As we saw in our discussion of missing premises, we often leave out premises that we take to be obvious and generalizations are no exception.  Thus, inductive applications are more common in our reasoning than we might expect.

Section 3: Evaluation: Illusory Forms

Most errors in reasoning with generalizations are found in the process of generalizing, not in the process of applying the general claim to a particular instance.  This is not, perhaps, surprising since the standards for logically strong inductive applications would seem to be clear: in general, the higher the likelihood that members of the subject class are members of the predicate class, the logically stronger the inductive application.  Put otherwise, you have better reason to believe that Sydney wants to be a teacher if 90% of English majors want to be teachers, than you do if only 75% do.  It is important to note, however, that there are two forms of inductive applications.  We will call them (a) and (b):

There are, however, two kinds of mistake we are especially prone to make when dealing with inductive applications.  First, recall from our discussion of deductive arguments that there are illusory argument forms.  As we saw in Chapter 10, arguments with the form Affirming the Consequent and Denying the Antecedent often strike people as deductive forms, even though they are not.  Similarly, we need to be clear that argument forms we will call Affirming the Predicate Class and Denying the Subject Class can seem logically strong, but tend to be logically weak.  We will add ‘probably’ to the conclusions below to remind ourselves that these are inductive arguments.

Affirming the Predicate Class (AP—watch out!)

  1. Most As are Bs.
  2. x is an B.
  3. So, probably x is a A.

Denying the Subject Class (DS—watch out!)

  1. Most As are Bs.
  2. x is not a A.
  3. So, probably x is not an B.

Instances of Affirming the Predicate Class and Denying the Subject Class are often logically weak because in statistical generalizations the predicate class is often much larger than the subject class.  Thus, in most cases of Affirming the Predicate Class knowing that an individual is a member of the predicate class doesn’t tell you anything about whether it is also a member of the subject class.  Similarly, in most cases of Denying the Subject Class, knowing that an individual is not a member of the subject class doesn’t tell you that the individual is not a member of the predicate class.  To illustrate:

Ex. 3: Affirming the Predicate Class (AP)

  1. Most sodas are have a lot of sugar.
  2. Sweet tea has a lot of sugar.
  3. So, probably sweet tea is a kind of soda.

Ex. 4: Denying the Subject Class (DS)

  1. Most sodas have a lot of sugar.
  2. Sweet tea is not a kind of soda.
  3. So sweet tea does not have a lot of sugar.

While we might want to clarify what counts as ‘a lot’ of sugar, these two examples nevertheless illustrate the problem with Affirming the Predicate Class and Denying the Subject Class.  These are both logically weak arguments because, like most statistical generalizations, the predicate class is much larger than the subject class.  In this case, there are many drinks with a lot of sugar (predicate class) that are not sodas (subject class).  That is, sodas are only one of many kinds of drinks with a lot of sugar (energy drinks, juices, lemonades, iced teas, etc.).  For this reason, knowing that sweet tea has a lot of a sugar doesn’t allow us to infer, as in Ex. 3, that sweet tea is a soda.  The same goes for Ex. 4.  Since there are many drinks with a lot of sugar besides sodas, simply knowing that sweet tea is not a kind of soda does not allow us to infer that it doesn’t have a lot of sugar.

We need to be clear, however, that there are logically strong instances of AP and DS (unlike Affirming the Consequent and Denying the Antecedent).  This is unusual, however, and occurs only when the subject class and the predicate class overlap almost completely.  Here is an example.  Suppose that the employees of a small jewelry store all know the code to the security system that allows them into the store.  The store vigilantly updates its code every three months, and has instructed the employees not to share the information with anybody else.  Assuming that the employees are trustworthy in this respect, the class of ‘people who know the security code’ will almost perfectly overlap with the class of ‘people who work at the store’.  Most, if not all, of the people who know the security code will be employees of the store, and most, if not all, of the employees of the store will know the security code.  In this case, the subject class and predicate class almost perfectly overlap, and we could formulate a logically strong instance of Affirming the Predicate Class as follows:

Ex.5:

  1. Most of the people who know the security code are employees of the store.
  2. Jan is an employee of the store.
  3. So, Jan probably knows the security code.

Again, what makes this kind of case unusual is that the subject and predicate classes almost perfectly overlap.  In the vast majority of our everyday arguments, when we express a statistical generalization we are talking about subject classes that are much smaller than predicate classes.

The fact that instances of AP and DS are commonly weak arguments means that we need to be on the lookout for them.  More specifically, we need to be on the lookout for arguments with premises telling us that an individual is a member of the predicate class or that an individual is not a member of the subject class.  Again, it is not that all such arguments are logically weak, but they often are, and we should take a closer look if we find one.  We will call a logically weak instance of AP or DS a misapplication.

More specifically, a person gives a misapplication when they mistakenly conclude that (i) an individual is a member of the subject class because they are member of the predicate class or (ii) an individual is not a member of the predicate class because they are a member of the subject class.

Section 4: Relevant Subject Classes

When it comes to evaluating inductive applications, the second thing to keep in mind arises from the fact that inductive applications are, like other inductive arguments, defeasible.  That is, the discovery of new information can weaken the logical strength of the argument.  The chief concern, in this context, arises from the fact that individual objects and people are always members of many subject classes.  A person, for example, might be a student, a resident of Pennsylvania, a dog owner, a Steelers fan, a Toyota owner, and so forth, and we need to take this into account in our evaluation of inductive applications.  Consider the following example.  Suppose that we know that Jocelyn is from Nashville, Tennessee.  We might draw the following inference.

Ex. 6:

  1. Most people from Nashville, Tennessee have been to the Country Music Hall of Fame.
  2. Jocelyn is from Nashville, Tennessee.
  3. Jocelyn has probably been to the Country Music Hall of Fame.

However, since Jocelyn is a member of many different subject classes she may be a member of a subject class that would undermine the support the premises lend to the conclusion in Ex. 6.  That is, Jocelyn may be a member of another subject class that is relevant to the logical strength of the argument.  Let us call this a relevant subject class.  Suppose, for example, that she strongly dislikes country music.  Knowing this would significantly undermine our confidence that she’s probably been to the Hall of Fame.

Country Music Hall of Fame at night
“Country Music Hall of Fame & Museum – Nashville, Tennessee” by Timothy Wildey CC BY-NC 2.0

This shows that in evaluating an inductive application we have to look outside or beyond the argument as stated to think about what else we know, or can find out about, the individual in question.  Again, doing so is in accordance with the The Rule of Total Evidence (see Chap 12), and helps us avoid drawing premature conclusions based on partial evidence.  We will call an inductive application that violates this rule a hasty application.  More specifically:

A person gives a hasty application when they mistakenly conclude that an individual is, or is not, a member of some class because they did not adequately consider other relevant subject classes.

It is important to see that a concern about hasty applications is no merely academic matter—it can be absolutely crucial that we avoid the use of hasty applications.  Here are a couple of examples.  For most women, oral contraceptives (birth control pills) are perfectly safe means of family planning; consequently, an individual woman might reasonably conclude on the basis of an inductive application that using oral contraceptives will be perfectly safe.  However, birth control pills are not a safe means of family planning for women with one of a number of different blood clotting disorders.  In these cases, the use of oral contraceptives can raise the chances of pulmonary embolism and other dangerous blood clots.  Thus, for individuals in the subject class ‘women with blood clotting disorders’ taking oral contraceptives is not a perfectly safe means of family planning.

Here is another example: for most people drinking grapefruit juice is perfectly fine and has no adverse health consequences (other than its taste!).  In fact, you might never have heard of anyone for whom grapefruit juice is potentially harmful.  Thus you might reason as follows: for most people, grapefruit juice is perfectly fine, so it is probably perfectly fine for me.  However, if you are taking statins (a class of drug commonly used to lower cholesterol), drinking grapefruit juice can be dangerous.  As it turns out grapefruit juice prevents the body from breaking down some kinds of statins, and this has potentially dangerous consequences.  Thus, if you fall into the subject class ‘person taking statins’ then the consumption of grapefruit juice is not perfectly safe.

It is not difficult to see how important it can be for a physician to have all the relevant available information before him or her prior to making even routine suggestions to patients.  Of course many of our own inductive applications are less important than these; nevertheless, the lesson is the same—in making inductive applications we cannot ignore an individual’s membership in other relevant classes.

Two Questions to Ask of Inductive Applications:

  • Is the individual in question a member of the subject class or not a member of the predicate class? (Failure: Misapplication–AP or DS)
  • Is the individual in question a member of other relevant classes? (Failure: Hasty Application)

Section 5: Evaluating Inductive Arguments—A Summary

This brings us to the end of Unit #5.  We have taken a close look at four of the most common forms of inductive argument, and we have identified the most important questions to ask of these arguments.  Of course in the real world, arguments do not come neatly packaged and categorized, and they are often put together to form long chains of argument.  We have to do the work of breaking down, identifying, and evaluating for ourselves, and the hope is that you are now in a better position to do so.  That is, hopefully you are able to better spot and distinguish Arguments from Analogy, Inferences to the Best Explanation, Inductive Generalizations, and Inductive Applications.  Moreover, you are now in a position to ask the right questions as you evaluate these kinds of argument.  Of course, these questions will not always settle the matter.  Evaluating arguments can be hard, and we don’t always have the information we need.  Nevertheless, knowing the right questions to ask can push conversations and inquiries forward, and can help direct us toward further information that bears on the logical strength of these arguments.

Exercises

Exercise Set 15A:

Directions: Determine whether the following are inductive generalizations or inductive applications.  For inductive applications identify the subject class and predicate class of the appropriate generalization.  For inductive generalizations identify the sample and population.  

#1:

There is no way I am getting one of those little dogs!  Little dogs bark all the time.

#2:

She loves going to the movie theater.  I’ve been with her 3 or 4 times and each time she really enjoyed it.

#3:

I don’t think I am going to call him this afternoon; he just finished paying bills and that typically puts him in a bad mood.

#4:

Math teachers tend to have weird personalities.  I’ve never had a math teacher I would call normal.

#5:

What do you mean you aren’t pre-med?  You are part of the campus emergency medical service (EMS) and everyone there is pre-med!

#6:

Jill is an excellent baker, judging from these macaroons.

#7:

I think Anne will probably enjoy the cake, I mean most people like chocolate.

Exercise Set 15B:

Directions: For each of the following inductive applications identify the subject class and the predicate class.  Then evaluate the argument for logical strength.  If there is a problem with the argument, identify it. 

#1:

Casey is probably in a motorcycle club; I mean he has tattoos and most guys in motorcycle clubs have tattoos.

#2:

Most people from Japan do not speak English, so the newly appointed Japanese ambassador to the United States probably does not speak English.

#3:

I doubt that Casey is in a motorcycle club; I mean he doesn’t have any tattoos, and most guys in motorcycle clubs have tattoos.

#4:

Most college football players never play professionally after college, so the Heisman trophy winner will probably not play professionally after he is out of college.

#5:

Most obese people either suffer from diabetes or are at high risk for developing diabetes.  But since Jack isn’t obese, it is likely that he neither suffers from diabetes nor is at risk for developing it.

#6:

Typically, jobs in retail do not come with health benefits.  So, I doubt these benefits will be part of your new job.

#7:

I am worried I am suffering from heavy metal poisoning.  After all, most victims of heavy metal poisoning suffer from fatigue, and I have been fatigued lately.

 

 

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Arguments in Context by Thaddeus Robinson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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