Inductive Generalizations Induction is the basis for our commonsense beliefs about the world.

1. Inductive Generalizations Induction is the basis for our commonsense beliefs about the world. In the most general sense, inductive reasoning, is that in which we extrapolate from experiences to what we have not yet experiences. In this chapter we are going to focus on: inductive generalizations, statistical syllogisms, and common errors or fallacies that are associated with inductive reasoning in general.

2. Inductive Arguments • Inductive arguments have the following characteristics: • The premises and the conclusion are all empirical propositions. • The conclusion is not deductively entailed by the premises. • The reasoning used to infer the conclusion from the premises is based on the underlying assumption that the regularities described in the premises will persist. • The inference is either that unexamined cases will resemble examined ones or that evidence makes an explanatory hypothesis probable. • Terms such as probably, in all likelihood, and most likely are often used in inductive arguments.

3. Inductive Generalizations (IG) • In an IG, the premises describe a number of observed objects or events as having some particular feature. From this observed set of objects or events a conclusion asserts a claim about all (or most) of the objects and events of the same type have the feature in question. • 85% of polled students at Notre Dame think the football team is great. • So, probably: • 85% of all Notre Dame students think the the football team is great. • The word probably indicates that there is an extrapolation from the polled (observed) group the the entire group.

4. Variations in Inductive Generalizations In inductive generalizations, the premises describe a number of observed objects or events as having some particular feature, and the conclusion asserts, on the basis of these observations, that all or most objects of the same type will have that feature. Enumerative Inductive Generalization (Universal) In these arguments, the premises describe a number of observed objects or events as having some particular feature, and the conclusion asserts, on the basis of this generalization, that all objects or events of the same type will have that feature. This sort of inductive argument can be formally represented as follows: 1. All observed X's are f. Therefore, probably, 2. All X's are f.

5. Restricted Enumerative Inductive Generalization This sort of argument is like the first type with the exception that the conclusion makes a claim about most items in the category rather than about all. The premise may be universal in form, or may be similarly qualified. Formally, this type of argument can be represented as: 1. All (or most) observed X's are f. Therefore, probably, 2. Most X's are f.

6. Inductive Argument to a Singular Conclusion In this type of argument, the conclusion is about a single case. A generalization, about many or most cases, is applied to that single case. These sorts of inductive arguments can be formally represented as follows: 1. All, or most, observed X's are f. 2. Case A is an X. Therefore, probably, 3. Case A is f.

7. Statistical Inductive Argument In a statistical inductive argument, there is a generalization from past experience, but the past experience need not be uniform or nearly uniform as it is in the inductive arguments just mentions. The premises describe a statistical relationship and the conclusion extrapolates that relationship from observed cases to unobserved cases. Formally, these arguments can be represented as follows: 1. N percent of observed X's are f. Therefore, probably, 2. Approximately N percent of unobserved and observed X's are f.

8. Sample and Target Population An inductive generalization generalizes from a sample to an entire class. We use the sample to reach a conclusion about a target class. Sample: the part of a class referred to in the premises of an inductive generalization. Target population: in the conclusion of an inductive generalization, the members of an entire class of things. Property in question: in inductive generalizations, the members of the sample and target population share a property or feature in common. In the process of evaluating an inductive generalization, you must be able to identify a particular sample, target population, and some property in question. Oftentimes, a poll is introduced as evidence for a particular generalization. The sample consists of the actual individuals polled and the target population is typically the broader group we are reasoning about. You may have to make inferences regarding the likely target population.

9. Evaluating Inductive Generalizations The reliability of inductive generalizations depends on two general points: 1. How well the selected sample represents the population. 2. The size of the sample. To evaluate an inductive generalization, we have to know whether the sample is large enough, and representative enough, to be a guide to the population. If your sample is nonrepresentative the inductive generalization likely fails the G condition: the sample does not provide sufficient support for the generalization. The study cannot support the conclusion.

10. Sample Representativeness A representative sample, S, is perfectly representative of a population, P, with respects to a characteristic, x, if the percentage of S that has x is exactly equal to the percentage of P that has x. Representativeness in this strict sense is seldom available. Rather, typically we settle for trying to make samples representative by choosing them in such a way that the variety in the sample will reflect variety in the population. The less confidence we have that the sample of a class or population accurately represents the entire class or population, the less confidence we should have in the inductive generalization based on that sample. Stratified Sample: A sample selected in such a way that significant characteristics within the population are proportionally represented within it. Random Sample: A sample in which every member of the population has an equal chance of being included. Biased Sample: A sample that demonstrably and obviously misrepresents the population.

11. Representativeness • In a strong inductive generalization, the sample must represent the target class. • A representative sample is a sample that possesses all relevant features of a target population and possesses them in proportion that are similar to those of the target population. • The less confidence we have that the sample of a class or population accurately represents the entire class or population, the less confidence we should have in the inductive generalization based on that sample. • A biased sample is a sample that is not representative • If the population is heterogeneous, then the sample should be random or otherwise constituted so as to represent the target population. A random sample of a population is one in which every individual has an equal chance of being selected.

12. Issue of Sample Size Whether a sample is representative is typically more important to evaluating an inductive generalization than the size of the sample. When samples are representative, cogent generalizations can be based on small sample sizes. Generally, the more heterogeneous the population, the larger the sample should be. The more uniform the population, the stronger the inductive generalization is. Except in populations known to be homogeneous, the smaller the sample in an inductive generalization, the more guarded should be the conclusion. When we generalize from the percentage of a random sample that has a certain feature to the percentage of the target class that has that feature, the larger the sample size, the higher the confidence level or the smaller the margin of error.

13. Procedure for Evaluating the Strength of Inductive Generalizations • The first steps ask you to identify the relevant parts of the argument: • First determine that indeed you have an argument and that it is an inductive generalization. • Be able to identify the conclusion, which usually contains the target population. If no explicit target population is mentioned, extrapolate to the most reasonable population. Note the degree of confidence with which the conclusion is stated. • Be able to identify the premises, which usually contain the sample on which the generalization is based. • Note any basic mathematical information (the size of the sample, percentages, margins of error, etc.).

14. Procedure for Evaluating the Strength of Inductive Generalizations The next steps ask you to evaluate the generalization: How variable is the population with regard to the trait or property that the argument is about? Consider any information about how the sample was selected. Try to evaluate the representativeness of the sample, drawing on common sense, background knowledge, etc. Sample Size: Is the sample large enough? Does the sample reflect the diversity of the population? Bias: Is any related factor present in the sample in a frequency different from what we would expect to find in the target population? Note any problems with pseudoprecision or questionable operational definitions. In evaluating the G condition an inductive generalization, you must ask whether the premises provide sufficient grounds to support the conclusion. Usually, your answer will range somewhere between strong and weak support.

15. Example: Thirty percent of American women ages 19 to 39 diet at least once a month, according to a news syndicate poll released last November. These findings are based on telephone interviews with a random sample of women listed in the Los Angeles telephone directory.