Lecture 9/29: On Induction
This lecture focused on the following two main points:
1. Distinguish between inductive and deductive arguments.
2. Evaluate the strength of inferences from generalization.
Read Along: Understanding Arguments, Chapter 8 (pages 215–230)
The standard of validity is powerful because, as we’ve discussed in earlier lessons, valid arguments guarantee the truth of the conclusion if you provide true premises. As we have seen, it is easy to get muddled by invalid arguments or arguments that sometimes contain true conclusions and sometimes not, regardless of the truth of the premises. The truth of a conclusion of an invalid argument is more or less accidental, since invalid arguments do not come with a guarantee. Valid arguments provide that guarantee.
Yet, there is something about valid arguments that limit their applicability. Vaguely put, in a deductively valid argument, the conclusion cannot say anything that was not already contained in the premises. Let me explain.
First, you get a sense of what I mean by the phrase in italics by considering a valid argument form. Here’s one:
All As are Bs.
All Bs are Cs.
Therefore, all As are Cs.
Notice that the conclusion is a rearrangement, as it were, of the variables in the premises (not a haphazard rearrangement but one nonetheless). So, there is a sense in which we don’t learn anything new in the conclusion:
the information in the conclusion is already contained in the premises (just rearranged). The premises and the conclusion are all about As, Bs, and Cs. Now, what I’m saying is still vague because there is a sense in which you do learn something new in the conclusion of a valid argument. You learn new ways to understand the relationship between the variables. But, in another sense, valid arguments don’t tell us about anything outside of the content of the variables that are mentioned in the premises. Here’s an example of an argument where the content of the conclusion is beyond that of the premises:
All As are Bs.
All Bs are Cs.
Therefore, all Gs are Qs.
See what I mean? The premises are about As, Bs, and Cs, and the conclusion is about something else entirely different—Gs and Qs. The problem is the argument is invalid. It is certainly possible for the premises to be true and the conclusion to be false. Now, the last argument is just plain silly. But, consider another argument where the conclusion is about something that is not contained in the premises. Suppose I’m interested in finding out what percentage of NFL football fans in Columbia, Missouri, root for the Kansas City Chiefs as opposed to the St. Louis Rams (as you may know, Columbia is located equidistant from St. Louis and Kansas City). I don’t feel like calling all NFL football fans in Columbia (there are probably tens of thousands of them). Instead, I’ll make 1,000 calls. Suppose the result is that 58% of the 1,000 NFL fans I called said they root for the Rams. I think I have enough to formulate an argument with one premise and one conclusion:
1. 58% of the 1,000 NFL fans I called said they root for the Rams.
2. Therefore, 58% of the NFL fans in Columbia, Missouri, root for the Rams.
Now, let’s assign variables to the argument and put it in deductive form to see whether it is valid (which it is not).
A = 58% of the 1,000 NFL fans I called said they root for the Rams.
How should I translate the conclusion into a variable? Well, I certainly cannot assign A again because the conclusion says something other than A. The conclusion says that 58% of all (as opposed to 1,000) Columbia NFL fans root for the Rams. So, I need to assign another variable. This already gives you another sense in which the conclusions of some arguments go beyond what is contained in the premises. Here’s the argument form:
2. Therefore, B.
Clearly, this is an invalid argument.
Before I go on to ask whether the argument is any “good,” I want to make a subtle point. Students often point out that the sample size is small (1,000 out of probably tens of thousands of Columbia NFL football fans). But, I want you to see that the sample size makes no difference to the validity of the argument. Even if I call every NFL football fan in Columbia, the argument would still be invalid. Why? Because the argument form (A; Therefore, B) is invalid. It is possible to imagine an argument of the same form where the premises are true and the conclusion is false. Even in the case where I’ve called every NFL football fan, you can imagine that it is possible that someone is lying; hence, this is a case in which the premises are all true and the conclusion is false. Bottom line: no argument based on an empirical sample is ever valid (now, that’s profound!).
Here’s the next question: is the argument any good? More generally, are arguments from empirical samples
“good”? We’re going to need to be subtle about this and think this through very carefully. If we define “good arguments” in terms of a guarantee for truth, then, by definition the argument is no good. I think this is a common reason for students to reject arguments from empirical sampling (especially after they have gone through all the lessons of validity unscathed). But, suppose we expand our definition of a “good argument” to include good reasons for believing a conclusion.
Now, is the argument any good? That is, do we have good reasons for believing that 58% of Columbia NFL fans root for the Rams, given the poll I conducted? Does it make a difference to you if I significantly increase the poll size from 1,000 to 60,000 (assuming there are that many NFL fans in Columbia)? If you are a holdout and refuse to believe that the argument is good, even if I increase the poll size, then I’ll ask you to consider what would make a good reason to believe in the conclusion that 58% of the NFL fans in Columbia root for the Rams. What kind of argument would convince you of the truth of the conclusion? As I said, no argument from empirical sampling is ever valid; hence, there is no way to make a valid deductive argument to support the conclusion. The desire of a population of people is not something that you can know by definition or by logical reasoning. You have to ask people what they like and dislike. So, it seems that we can never guarantee the truth of the conclusion about people’s NFL team allegiances. All such conclusions would be supported by invalid arguments. But, still, it would be unreasonable to suggest that no poll data or sampling data could ever give
us some good reason for believing the conclusion inferred. Here’s what I’m trying to convince you about arguments from sample data: while in our argument about who roots for the Rams, the premises do not provide a guarantee that our conclusion is true, they do provide us with something weaker but still worthwhile—a good reason for thinking the conclusion is true, albeit believing the conclusion is a little risky.
The argument from sample data is called an inductive argument (http://en.wikipedia.org/wiki/Inductive_reasoning). Distilling everything I just told you about the difference between inductive and deductive arguments, the bottom line is that inductive arguments are never valid and are not intended to be valid. So, we don’t evaluate them in the same way. In particular, the arguments about NFL fans are what the authors call “statistical generalizations” (pages 219–220). The example the authors use concerns Canadian quarters, but the lesson learned is the same as I provided in the commentary. How do we evaluate inductive arguments? We say they are to some degree strong or weak. In general, there are two ways to evaluate the strength of an inductive argument.
First is sample size (pages 220–222). The greater the percentage of Columbians we poll, the more confident we are that our poll results reflect the truth about Columbian NFL allegiances. We encountered this before. Some students are not confident of the conclusion that 58% of Columbians root for the Rams because I haven’t called enough people (1,000 is too small of a percentage). Their comfort level with the conclusion is increased the greater the sample size.
Though beware: no matter how many people are sampled, there remains a possibility that further information can undercut the argument (e.g., that some of the respondents to my poll were lying). This constitutes a difference between inductive and deductive arguments. The validity of a deductive argument is never lost by adding more premises. Philosophers say that for this reason, valid arguments are indefeasible, while inductive arguments are defeasible.
A second way to evaluate the strength of an inductive argument is by determining whether the sample is biased (pages 222–224). Suppose I were to call 10,000 as opposed to 1,000 Columbians, with the result that 78% said they are Rams fans. At that point, you might say, “See, I told you—the sample size makes a big difference!” The change of percentage of Rams fans increased dramatically as I increased the sample size tenfold. But, suppose
you were to inquire a little further about the source of my sample (perhaps you are a little skeptical that the percentage would change so drastically). Suppose my answer is that I found all the Columbians who have signed up for more information from the St. Louis Rams Web site (http://www.stlouisrams.com/splash/). Aha! What does this tell you about the reliability of my sample data? Since the list comes from a Rams fan Web site, it isn’t a fair sample of all Columbians. That is, the data are skewed. So, the inference from the poll data (where 78% say they are Rams fans) to the conclusion “78% of Columbians are Rams fans” is very weak.
You should notice by now a further difference between inductive and deductive arguments. An argument is either valid or not. There is no in-between. But, the strength of an inductive argument is a matter of degree depending (in sum) on the sample size and biasness of the sample.