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!