# Inductive Reasoning and Inductive Arguments

First in this tutorial/summary, let's recall the hat problem from Chapter 1 (also see the tutorial).   In that story we noted that the prisoners were not going to be allowed to use inductive reasoning.   Each prisoner had to be sure about what hat he had on his head, otherwise he would answer "no" (he did not know).

For instance, we noted one inductive possibility (probability only) for the first man (x):

If only three hats are used (one on x, one on y, and one on z), then we know x has a higher probability of having on a red hat IF he sees two white hats, one on y and one on z.  If he sees two whites, then there is only one chance for him to have on a white hat but two chances for him to have on a red hat.

x-----------------------------------------y----------------------------z
( probably red )---------------(white hat)---------------(white hat)

Consider the take away message from this example.

1. We can be smart using inductive reasoning even though we do not have certainty for our inference.   We can also be dumb and have less probability for our inference . If x was allowed to choose based on probability, he would be smart to choose red and not very smart choosing white.   Choosing red is wiser because the probability is higher - there are two red hats left but only one white.   So, x would not be certain that he had on a red hat but he would be smart (playing the odds) if he chose red as the most probable.
2. We use inductive reasoning every day and should think hard about how to have less risk, more probability, and stronger inductive inferences . We make decisions that involve risk.   We cannot be certain of what the future will bring.   Something that worked in the past may not work in the future.   We want to make decisions backed by evidence that lesson risk and increase the probability of success.   We want what we will call strong inductive arguments.   We want to avoid weak inductive arguments.
3. Very strong inductive conclusions = reliable beliefs. (See the optional tutorial for C2.) Reliable beliefs are beliefs backed by lots of evidence, even though we cannot be absolutely certain that they are true. We have overwhelming inductive evidence that people will fall off tall buildings if they jump due to gravity, even though we have not and cannot test every human being. Based on the evidence, we have a reliable and practical belief even though we believe in a big generalization.

Categorizing inductive arguments as strong v weak is similar to categorizing arguments as valid or invalid for deductive arguments.   But there will not be a crisp cut off between strong v weak arguments.   See the barrel full of apples example in the textbook (C3).   The point of this example is that there is a sliding scale from weak to strong inductive inferences, but never certainty for any inductive inference, no matter how strong the evidence is for the inference.

Most of our future discussion on inductive reasoning will be on how to make inductive arguments stronger, and avoid weak inductive arguments such as those discussed in Chapter 5.   But let's do the basics first and get some practice just seeing the difference between deductive and inductive reasoning.

## Deductive arguments

Arguments where the goal (to achieve valid and sound arguments) is to provide conclusive evidence for the conclusion; the nature of the inferential claim is such that it is impossible for the premises to be true and the conclusion false.

(Valid or Invalid)

Valid arguments succeed in achieving this goal IF the premises are true (sound argument).   Invalid arguments fail in achieving this goal EVEN IF the premises are true.

## Inductive arguments

Arguments where the goal (to achieve strong and reliable beliefs) is to provide the best available evidence for the conclusion; the nature of the inferential claim is such that it is unlikely that the premises are true and the conclusion false.

Strong inductive arguments achieve this goal - providing the best available evidence.   Weak inductive arguments do not.

Here are some examples:

## Deductive argument Examples

All Internet hackers and spies for the Chinese government are Chinese.
Wen Ho Lee is Chinese.
So, Wen Ho Lee is an Internet hacker and spy for the Chinese government.

All Chinese people are Internet hackers and spies for the Chinese government..
Wen Ho Lee is Chinese.
So, Wen Ho Lee is an Internet hacker and spy for the Chinese government.

Take Away Point: Both arguments are attempting to provide conclusive evidence for the conclusion. They are attempting to deduce a conclusion from a general statement and information about Wen Ho Lee. These arguments are not using any language that would indicate that the conclusions are only probably true. They are both implying a slam dunk conclusion. The first one though fails in this attempt; it is invalid. The second one partially accomplished the goal of conclusive evidence for the conclusion; it is valid. But the premises would have to all be true for the conclusion to be conclusive.

## Inductive argument Examples

After careful observation we have not seen any hummingbirds all day in this forest.
Therefore, probably there are no any hummingbirds in this forest.

After careful observation by trained hummingbird specialists over many weeks, no hummingbirds or signs of hummingbird habitation were found in this forest.
Therefore, probably there are no hummingbirds in this forest.

Take Away Point: Note the important message form these two examples. Both are inductive and both have uncertain conclusions. But clearly the second inductive argument is stronger than the first one. It has more evidence. Many weeks of observation by trained observers v only one day by untrained observers.

Common sense = When something is very important to us, we want the best available evidence for our inductive conclusions. The second argument also has a big generalization as a conclusion, but the conclusion has a higher probability and involves less risk. We want less risk, but we have to use induction every day. So, we should learn how to have less risk for our inductive conclusions.

## Quiz

All inductive arguments involve generalizations to the conclusion and thus involve risk and probability.

All deductive arguments provide conclusive evidence for their conclusions.

All inductive arguments provide the best available evidence for their conclusions.

Some inductive arguments are valid.

All inductive arguments involve risk and have uncertain conclusions, but some are stronger than others.

We can have a sharp cut off between strong and weak inductive arguments just as we can have a sharp cut off between valid and invalid deductive arguments.