## Inductive Arguments

We have seen what inductive arguments are. Inductive arguments are those whose conclusion cannot be conclusive because they state more than what is said in the premises. Here we are going to discuss two kinds of inductive arguments -- arguments by analogy and induction by enumeration

### Argument by Analogy

Consider the following argument:

Somsri and Somporn share a number of characteristics -- (1) they are from the same high school; (2) they love studying mathematics; (3) they got about the same GPA at high school. We know that Somsri has a good GPA in the university, so Somporn will have a good GPA at her university too.

How confident are we that Somporn will have a good GPA as does Somsri? Suppose we do not know Somporn's GPA yet. But since we know something about her, and especially about her similarities with Somsri, then we believe we have some right in concluding that about Somporn too. But since the information about Somporn's GPA is not already included in the premises, we cannot be one hundred percent certain. At most the conclusion is very likely, but we cannot be completely sure.

This argument is an example of arguments from analogy. It can be stated in the following formula:

A and B share the following characteristics -- a, b, c, d
A has the special characteristic e.
Therefore, B has characteristic e too.

We conclude that B has e from the facts that A and B both have a, b, c, and d. You can see that this conclusion can never by validly inferred because we can never be completely certain whether B really has e or not since it is not stated in the premises. (Premises are what we know to be true, and the conclusion is what we want to infer or draw out from them.) Thus arguments by analogy are clearly inductive.

Although we cannot be completely certain that B has e or not, we have some means to decide how much confidence we want to put to the conclusion. So the question is: What are the conditions which will make us more confident that Somporn will have a good GPA too? Firstly and most importantly, there must be some logical connection between the characteristics stated in the premises and the concluded characteristic. If there is no connection whatsoever between Somporn's and Somsri' graduating from the same high school, loving mathematics, having the same GPA in high school, and their having the same GPA in university, then we cannot put much confidence in the conclusion. On the other hand, if there is some connection. That is to say, if we know that anyone, or a large percentage of those who love mathematics usually have good GPA's then we will be more likely to be confident that the conclusion is true.

But how do we know that those who love mathematics are more likely to earn high GPA's? From experience, of course. This is an important in evaluating inductive arguments. Since we cannot rely on what is stated in the premises in evaluating such arguments, we have to rely on our own general knowledge. There are no mechanical means to decide how much confidence we have in the conclusions of inductive arguments. So we have to use experience, background knowledge, common sense, and the like to help us decide how likely the conclusion is to be true. Thus, when you evaluate inductive arguments, what you do is that you have to proceed on a case by case basis. Many logicians have tried to provide general rules, which are supposed to be more or less mechanical, to decide the likeliness of the conclusion, but such attempts mostly fail. And the only recourse left is to consider the arguments one by one, using our own background knowledge and common sense as the case in question requires.

### Induction by Enumeration

This type of arguments is familiar to every student of statistics. Actually it is a very basic kind of statistical thinking, and shows that statistics as a whole is just a branch of inductive thinking. Statistics is a very complex discipline, but in its essence it is a way for us to know more than what our evidence gives us. This very basic statistics is present in argument employing induction by enumeration. It is a projection of a ratio from observed samples to the conclusion of the same ratio in the population we want to know about. For example, consider this argument:

```     Four out of five of the oranges in this basket are sweet. (I tasted them myself.)
Consequently, 80 percent of all the oranges in this basket are sweet.
```

This is an induction by enumeration. As you have studied in statistics, there are two major caveats that you have to realize in order to find out how likely the conclusion is to be true --

1. The sample has to be large enough compared to the population.
2. The sample has to distribute in such a way that they really represent the population.

Thus, if there are about twenty oranges in the basket, tasting five of them can give you some degree of confidence. But if there are two hundred, then, everything being equal, then we are less likely to believe in the conclusion. Furthermore, if the oranges I tasted were not well distributed. For example, if the merchant has put all the few sweet oranges on top of the sour ones, then I will be less likely to be correct in believing the conclusion if I pick only the oranges from the top.