Inductive Logic

 

Inductive arguments can be turned into deductive arguments by substituting a universal premise (all, every, always, etc.) For Example:

The following is inductive:

  1. Most animals are furry.
  2. Rover is an animal.
  3. Therefore, Rover is probably furry

The following is deductive:

  1. All animals are furry.
  2. Rover is an animal.
  3. Therefore, Rover is furry.

 

Science is often a combination of inductive and deductive knowledge. String Theory is a theory that says, at the fundamental level, the universe around us is composed of vibrating strings. The problem is that the theory hasn't given us any observable predictions, essentially leaving the inductive aspect out. There have been recent books attacking string theory on these grounds, such as The Trouble With Physics by Lee Smolin and Not Even Wrong by Peter Woit. Of course, Professor Farnsworth from Futurama (below) has also done some work with string theory...

If you are having computer problems, do you ask your grandpa, who can barely use the microwave, or do you ask your friend who works with computers?

So, where are we in the class at this point? We've learned about the structure of arguments (premises and a conclusion), how to recognize arguments, and deductive arguments. Now, we move to inductive arguments. Rememeber, while deductive arguments aim to provide certainty to their conclusions, inductive arguments aim to provide conclusions that are likely or probable based on the best possible evidence or support.

Barrel of Apples Example

One of the best ways to lead into induction is through something called "the barrel of apples example" (this is much easier to get than the prisoners and the hats). Consider the following. There is a barrel full of 100 apples. Without looking inside the barrel itself, we start picking apples out of it. We find that the first few apples we pick are rotten. If we keep taking apples out of the barrel that are rotten, at what point would you bet that every other apple in the barrel that we pull out will be rotten? If you only picked one rotten apple out of the barrel, wouldn’t it be bad reasoning to conclude that all the rest of the apples will be rotten?

Of course, we might have other reasons to believe that all the rest of the apples will be rotten after only seeing one that was rotten. Maybe we know that the apples have been sitting out in the sun for days. Or maybe we know that there are bacteria in the barrel that have likely infected the apples. Scientists (physicists in particular) come to conclusions based on a small amount of experimental data, usually in conjunction with other beliefs about the way the world works.

The more apples we pull, the better conclusion we will be able to reach. If we have 5 rotten apples (as opposed to 1) out of 95, our conclusion about the rest of the apples will be stronger, especially if we have pulled the 5 apples from different sections of the barrel. This is done in elections when a representative sample of voters is taken from different parts of the country, rather than from one area.

What if we pulled 50 rotten apples from the barrel? Would you be sure yet that the other 50 apples are rotten? Would you bet money on it?

What if we pulled 99 rotten apples from the barrel? Would you be sure that the last apple is rotten? Would you bet your life savings on it? Would you bet your life on it?

Most of us would probably not bet our lives on it, because there is always the chance that the last apple is not rotten. It’s possible. And yet, the fact that the other 99 apples are rotten is good inductive evidence to believe that the last apple will be rotten.

Many people aren’t willing to bet anything, even after 99 rotten apples are pulled out. And yet, we make decisions in everyday life all the time based on less conclusive inductive knowledge. Take, for example, when we try to find a good restaurant. How do we know what’s good? We make some calls, look at some reviews, ask our best friend. How many people have we consulted here to reach the conclusion of which restaurant is good?

The point is, we use rather poor inductive arguments all the time in our day to day lives to provide justifications for our actions. We lack certainty that our conclusions are true. The best thing we can do is to make our inductive inferences as strong as possible.

Inductive Arguments

Another way to say it is this: premises of inductive arguments do not prove their conclusions, but rather support them. If someone ate your last microwave burrito, and your roommate Joey loves microwave burritos, then it’s likely that Joey ate your last burrito. But it’s not certain. The fact that Joey likes microwave burritos supports the conclusion that he is the one who ate yours. But it’s really not very strong support for that conclusion. Now, if you later find the burrito wrapper on Joey’s desk, this is even stronger support for the conclusion. But it’s still not as strong as a deductive arguments where the conclusion is proven with certainty.

Strong inductive argument: more support is given for the conclusion.
Weak inductive argument: less support is given for the conclusion.

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With each piece of evidence we find, the conclusion of an inductive argument becomes more likely to be true, just like taking more and more apples out of the barrel.

Sometimes it's not immediately apparent whether an argument is aiming to be inductive or deductive. This is often because arguments have unstated premises (see the textbook for more info). Context and content sometimes make it unclear whether an unstated premise makes an argument deductive or inductive. When it’s unclear whether the argument is deductive or inductive, it’s best to attribute a believable unstated premise to the speaker. In other words, it's best to ask yourself what the speaker is really trying to say.

Three Types of Inductive Arguments

The text identifies three types of inductive arguments: inductive syllogisms, inductive generalizations, and arguments by analogy.

Inductive Syllogism

  1. Most Xs are Ys.
  2. There is an X.
  3. Therefore, this is a Y.

With this inductive argument form, the likelihood of the conclusion depends on how many Xs are actually Ys. (Note that "X" and "Y" are simply variables that change depending on the content of the specific argument.) Most people living in the US are US citizens, but we know there are some who are not.

Inductive arguments usually go from specific claims to general claims: this is because they are trying to give general support for a conclusion. We generalize from samples to establish general statements about a population when we haven’t observed all its members. Generalizing is not as simple a thing as it seems. It requires probabilities and statistics, but we're not going to get into all the complications.

Inductive Generalization

  1. Such and such a percentage of observed Xs are Ys.
  2. Therefore, the same percentage of all Xs are Ys.

How can we evaluate these types of arguments? The observed Xs make up what we call the sample. All the Xs are the target population or just the population. The feature or attribute of interest is the property of having or being Y. And n is equal to the sample size.

The primary question in evaluating such arguments becomes: how likely is it that the same proportion of the target population has that feature/attribute of interest? In making these evaluations, as the text discusses, it helps to understand the sampling frame.

A sample is biased when it contains a disproportionate number of things with a given feature/attribute of interest. If a sample has more New Englanders than anything else, then the sample is biased with respect to New Englanders.

The point is that the sample should be as diverse as possible, and large enough to contain that diversity in the first place.

Random sample: a sample in which every member of the population has a chance of being included. This can be quite complicated to determine, but for our purposes think of the barrel of apples. Is it more likley that you'll get a random sample if you pull every apple from one side of the barrel or if you pull them from all different parts of the barrel? It's pretty clear that pulling apples from as many parts of the barrel as possible will give you the best random sample. If you pull them all from one area, it might be that that is the only area with rotten apples--just as if you take a poll and include only people from Kentucky, your results will be biased.

If you're wondering what the point of this sort of reasoning is, see real-life examples of it in Gallup Polls (click here for their website). Gallup takes random samples from state to state of American opinions, asking many different types of questions. They have a good track record of predicting wins and losses of politcal candidates, among other things. The inductive sampling method works!

Argument from Analogy

This is an argument from a claim that two things have certain characteristics.

  1. X and Y both have property a.
  2. X has feature F.
  3. So Y has feature F.
  1. Paris Hilton and Neil Young are both famous.
  2. Neil Young is a talented musician.
  3. So Paris is a talented musician.

The fact that two things are similar in one respect increases the probability that they are similar in other respects. If two people are famous, then this increases the likelihood that they are talented musicians (since a good amount of famous people are talented musicians), though not by much.

But not every similarity between two things will show that other similarities will be shared. If Paris and Neil young are both Christians, this would not increase the probability that they are both talented musicians (this isn't to say that there aren't some talented Christian musicians out there).

Evaluating arguments by analogy is not an exact science. General information must be known to evaluate arguments by analogy. In the example above, we need to know that many famous people are musicians, and many of those musicians are talented. The argument is made stronger or weaker depending on how well we understand these general claims.

To attack an argument from analogy you must attack the analogy to show that, given the weakness of the analogy, the conclusion isn’t likely to follow from the premises. Here, as the text discusses, it's good to know such things as the error margin and the confidence level.

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Below is the burger joint Crazy Burger in San Diego, CA. Over the last couple years I've made an inductive argument to friends for why it's worth trying. First, I like it and other friends like it. Second, it was featured on the food network. Third, it has won several "best burger" awards. When I put it in perspective, though, is my argument really that strong? If we compare it to the barrel of apples example, how many apples have I really pulled out of the barrel?

Taking Apples Out of a Barrell...

Strong Inductive Argument:

  1. There are 100 apples in a barrel.
  2. I know that 75 of them are not rotten (because I've pulled them out).
  3. Thus, the last 25 will not be rotten either.

 

Weak Inductive Argument:

  1. There are 100 apples in a barrel.
  2. I know that 3 of them are not rotten (because I've pulled them out).
  3. Therefore, the last 97 will not be rotten either.

Below is the Federal Reserve (the Fed) building in Washington D.C. Some say that there is a consipracy to control the world through the Fed, as discussed in the internet sensation movie Zeitgeist. The Libertarian Ron Paul wrote a less conspiracy-oriented book about the Fed and its largely unknown role in our lives. Is either Zeitgeist or Ron Paul making a strong inductive argument? If the matter interests you, I'll let you be the judge...

A few years ago I travelled to the Mayan city of Palenque (above) in the Yucatan. Inside one of the temples is a design like the one below. Some theorists believe that the design depicts an ancient Mayan space man manipulating the controls of a spaceship. Most archaeologists, however, believe that the design depicts a Mayan Lord/God descending into the underworld, which is consistent with other findings and currently existing descendants of the Maya. Which explanation do you think is the stronger inductive argument?

More on Inductive Arguments in General

While it’s true that sample sizes in studies and polls are a small representation of the whole, the generalizations we make in everyday life are usually based on an even smaller representation of the whole—like one or two cases. This is something to be aware of when making generalizations.

When samples start getting large (500 or more) it gets increasingly difficult to lower the error margin. For the error margin to go down even a percent, it takes another 500 added to the sample. This is why most samples generalize from 1000 or 1500 to the whole. The Gallup Poll (mentioned above) for the presidency, for example, takes 1500 people a day from different areas of the country.

Again, there is practically no precision in everyday inductive arguments.

Fallacies in Inductive Reasoning

Because inductive reasoning is not an exact science, it's easy to commit errors in reasoning, or fallacies (we'll discuss more fallacies in more detail later).

A hasty generalization happens when someone overestimates the strength of an argument based on a small sample. The text mentions some other inductive fallacies, but I'll focus on this one in the lecture since it's so common. You often hear members of political parties claiming that members of opposing parties are all a certain way. "All the democrats are morons." "All the republicans are idiots." Such claims are usually only based on a few samples and are therefore hasty generalizations. The reasoning looks like this: some are like this, so all must be like this. This seems also to be a motivation behind a lot of racism and prejudice. Someone has a couple of bad experiences with a particular race, so they commit a hasty generalization and say that everyone of that race is a particular way.

Some Last Words on Inductive Reasoning

Despite the fact that life can be uncertain, we should understand that some beliefs are more reasonable to accept than others. The nature of inductive knowledge always makes it possible that we can be wrong. And still some inductive inferential links are better than others. The belief that the earth is round is much more reasonable to believe than that the earth is flat. The primary reason is that we have satellite images from space showing that the earth is round. It can also be proved using math and geometry by measuring the sun's shadow at different locations, then comparing the distance between the locations (it's more complicated, but that's the basic idea.)

Inductive claims can be shown to be good or bad based on their relation to other alternatives. If we are trying to figure out how to fix our computer, will we listen to our mother who can’t use the DVD player, or someone with a degree in computer science?

From the barrel example we can see that there are some objective standards that help us determine when one inductive inference is better than another: for example, a greater representative sample lends itself to a stronger conclusion. We can have a sliding scale of reasonableness where some inductive arguments are objectively more convincing.

It is important to notice here that very weak ideas can work. If we only take a small sample, we can conclude that what we are trying to prove might be true. Take, for instance, UFO (spacecrafts from another planet, for this example) sightings. Many believers in UFOs will tell you that, given the evidence they have, there is no other conclusion except that UFOs exist. But what sort of evidence are they working with? Usually, they have a few pictures, a personal story, and maybe some quotes from government officials. Is this "undeniable" evidence? The UFO proponent thinks he has created a deductive argument, when in fact he has only created a very weak inductive argument. He might be right, but then my friend Jeff might be right that there are Leprechauns in the forest. After all, he's seen them himself and he has friends who can corroborate his story...

Copyright © Luke Cuddy 2009