inductive arguments

Inductive Arguments

Inductive vs. Deductive Arguments

The difference between a deductive and an inductive argument lies in what it is attempting to show.

A deductive argument is trying to show that its conclusion must follow from its premises.

An argument that successfully does this is deductively valid.

Counterexamples

If a deductive argument is a good one, you won’t be able to think of any counterexamples to it.

Counterexample = a possible situation where the premises of the argument are true but the conclusion is false.

A valid argument will not have any counterexamples.

Inductive Arguments

An inductive argument has a different aim. It is only trying to show that its conclusion is supported by its premises.

In other words, it’s trying to show that the truth of its premises makes it more likely that its conclusion will be true.

Inductive Strength

The quality of an inductive argument is measured by its strength – the degree to which its premises raise the probability of its conclusion.

If they don’t raise the probability very much, the argument is not very strong. If they do, the argument is strong.

Inductive Strength

For inductive arguments, counterexamples don’t show that the argument is bad. This is because an inductive argument is only saying that its conclusion is very likely, not that other possibilities do not exist.

Validity isn’t a feature we look for in good inductive arguments – just strength.

Additional Evidence

Since inductive arguments don’t prove their conclusion, even a strong argument with true premises can be defeated by additional evidence.

Example

For example:

Most cats like to play. Fluffy is a cat. Therefore Fluffy probably likes to play.

This argument is strong, but future evidence might show that its conclusion is still false, even if its premises are true! (Fluffy might be too old to play)

Vs. Valid Argument

Notice how this is different from valid arguments. In a valid argument, so long as the premises are true, no future evidence can show that the conclusion is false.

Of course, future evidence might show that the premises are wrong!

Induction

Today, we will discuss two different types of inductive arguments: •inductive syllogism•inductive generalization

Inductive Syllogism

An inductive syllogism is a method for arguing from a general statement to a more specific one. A general statement is a statement about a group of people:• ‘most teachers are smart’• ‘athletes are strong’• ‘in general, dogs like to play fetch’• ‘many students don’t get enough sleep’• ‘bankers are usually rich’

Inductive Syllogism

A general statement about a certain group can help us make a good guess about a particular member of that group.

For instance, if we know that most bankers are rich, we can make a good guess that Bill the banker is rich, too.

Inductive Syllogism

In standard form:1) Most bankers are rich.2) Bill is a banker.3) Bill is rich.

The general form of the argument I just gave is:1) Most X’s are Y.2) A is an X.C) A is Y.

This type of argument is called an inductive syllogism.

Inductive Syllogism

In standard form:1) Most bankers are rich.2) Bill is a banker.3) Bill is rich.

Notice that this is NOT a valid argument. The premises do not guarantee the conclusion – Bill could be a poor banker. (A counterexample!)But since this is an inductive argument, it doesn’t matter that the argument is not valid.

Inductive Syllogism

In standard form:1) Most bankers are rich.2) Bill is a banker.3) Bill is rich.

The argument does raise the probability that its conclusion is true. If it turns out that most bankers are rich, this makes it more likely that Bill the banker is rich.So this is a strong inductive argument.

More Examples

•Most basketball players are tall. Jane is a basketball player. Therefore Jane is probably tall.•Most university students are smart. Alice is a university student. Therefore Alice is probably smart.•Most Saturdays, the bus runs late. Today is a Saturday. Therefore, the bus will probably run late.

Hidden Premises

An inductive syllogism can have hidden premises.Often, the second premise will be left out if it is common knowledge (i.e., it will become a hidden premise).

1) Most Presidents are very powerful.2) Barack Obama is very powerful.

Similarly, the first premise may be left out if the generalization is common knowledge.

1) Bill is a banker.C) Bill is probably rich.

Different Orders

The premises in an inductive syllogism don’t have to come in the same order, either.The following two arguments are the same:

•Most winter days are cold. Tomorrow is the first day of winter. Therefore, tomorrow will probably be cold.

•Tomorrow is the first day of winter. Most winter days are cold. Therefore, tomorrow will probably be cold.

Hidden Conclusion

Even the conclusion of an inductive syllogism can be ‘hidden’ – particularly if the conclusion is used as a sub-conclusion in a larger argument.

If Bill is rich, then he will buy us lunch. Bill’s a banker, and most bankers are rich – so we’ll probably get a free lunch today!

Strength of Generalization

The strength of an inductive syllogism depends primarily on the strength of the generalization.

If 90% of cats like milk, then ‘most cats like milk, Fifi is a cat, therefore Fifi likes milk’ is a pretty strong argument.

If only 70% of cats like milk, then the argument is much weaker.

Other Evidence

But our assessment of the argument also has to do with the amount of available evidence that has been taken into account.

Consider the example ‘most bankers are rich, Bill is a banker, therefore Bill is probably rich’.

Even though this argument is strong, our assessment of the likelihood of its conclusion may be affected by other available information about Bill.

Other Evidence

For instance, if we find out that Bill invested heavily in the stock market right before it crashed, then we might no longer accept the conclusion that Bill is probably rich.

Or, if we know that Bill is a chronic gambler, we might use this to infer that Bill is bad at managing his money and that he’s therefore less likely to be rich, even though he is a banker.

Obvious Available Evidence

It’s the neglect of obvious available evidence that makes the following inductive syllogisms not very convincing:•Most Americans don’t speak Chinese. The new professor of Chinese literature is American. Therefore, he doesn’t speak Chinese.•Most of the citizens of Zimbabwe are poor. Therefore, the president of Zimbabwe is poor.

Different Argument Form

Inductive syllogisms are easily confused with a different argument form:

1) Most X’s are Y.2) A is a Y.C) A is an X.

An example to show why this is bad:1) Most teachers are adults.2) Bill is an adult.C) Bill is a teacher.

Inductive Generalization

The other form of inductive argument we will discuss today is inductive generalization.

Inductive generalization is a way to argue from specific to general claims. So in a sense, it’s the opposite of the argument style we just looked at.

Inductive Generalization

However, it’s not exactly the opposite:1) Bill is rich.2) Bill is a banker.C) Most bankers are rich.

This would not be a good argument. You can’t go from an observation about one person to a claim about a whole group.Instead, in order to justify the statement that most bankers are rich, we need to look at lots and lots of particular bankers.

Samples

Ideally, when we are trying to find out whether a large percentage of a group has a certain property, we would check every member of the group.

But for a lot of groups, that’s just not possible – there are too many to check. Instead, we look at a sample, or a subset of the group.

Inductive Generalization

Say we look at a sample of bankers and find that 90% of them are rich. We can then use that information to support a conclusion that most bankers are rich.

So the argument form of an inductive generalization is:

1) Most of the observed sample of X’s are Y.C) Most X’s are Y.

Other Examples

We have given our new pet food to 200 cats. All but one liked the food. Therefore, we believe that most cats will love our new pet food.

We polled 1000 people in Hong Kong and asked them if they preferred coffee or tea. 886 people replied tea. Therefore, it seems likely that most Hong Kong people prefer tea.

Representative Samples

The success of an inductive generalization depends on how good the match is between the sample and the entire group.

If our sample of bankers is 90% rich, but bankers on a whole are only 30% rich, our argument will not be a good one.

Samples

Of course, we can’t check this match directly – the whole point of using a sample was that we can’t check the properties of the entire group!

If we knew that bankers were 30% rich, we wouldn’t need to test a sample.

Sample Size

But, there are ways to make educated guesses about how closely the percentages match.

One of the easiest is the size of the sample – if we survey 10% of the bankers, we’ll have a better estimate than if we survey 1% of the bankers.

When all else is equal, a larger sample is better than a smaller sample.

Sample Size

Size of the Group

Sometimes it’s difficult to tell if we’ve sampled a large percentage or a small percentage of the total group – because sometimes we don’t know who belongs in the group!

This is particularly tricky for cases where we have to rely on self-identification – not everyone is going to admit to being a member of the group ‘pornography users’!

Representative Samples

Another factor is whether the sample is representative.

Representative Samples

For instance, say we want to know whether most bankers are rich. But say our sample only contains bankers who went to Harvard.

Our sample is BIASED – not all bankers go to prestigious schools, and whether or not one goes to a prestigious school is likely to influence whether or not one is rich.

Perfectly Representative Samples

A perfectly representative sample does not contain any bias.

In a perfectly representative sample, the percentage of bankers in the sample that went to Harvard would be the same as the percentage of total bankers that went to Harvard.

Relevant Factors

Of course, it is not usually possible to use a perfectly representative sample – the sample will almost always contain a greater or lesser percentage of for example, red-headed bankers, left-handed bankers, etc.

But ideally we’ll make sure that any relevant factors like education match pretty well.

Biased Samples

Other examples of (relevantly) biased samples –

•Using a sample of Americans to make claims about what kinds of movies people (worldwide) will like.•Using a sample of only first-years to make claims

about the work habits of university students.

Sneaky Bias

Sometimes bias can be sneaky:

Phone polls during elections often end up with a sample containing too many old people – old people are more likely to be at home when the phone rings. Also, they are more likely to have a land line!

Random Samples

We’ll also do our best to make sure that even factors that don’t appear relevant will be reasonably represented in the sample.

Often the best way to do this is to use a random sample – a sample where every member of the target group has the same chance at being included in the sample.

Don’t Reverse Terms

Just like with inductive syllogism, it’s important to be careful not to reverse terms with inductive generalization.This is not a good argument:

1) Most of the sample of doctors we surveyed are rich.C) Most rich people are doctors.

The reason this is a bad argument is that, while you might have tested a good sample of doctors, you might not have tested a good sample of rich people.

Anecdotal Evidence

Unfortunately, many people often make inductive generalizations on very bad samples. Sometimes, samples of only one case!

This is called relying on anecdotes, or anecdotal evidence– using a single case, or just a few, to draw conclusions.

It is not a very strong form of argument.

Cherry Picking

You often see this in advertisements – rather than giving a percentage of people who gained a benefit from the product, advertisers will simply show a single person saying that the product worked for them.

Anecdotal Evidence as Argument

Sometimes people try to reject arguments on the basis of anecdotal evidence, too!