Generalizations are not much use if you can't apply them back to particular
cases. It rains about 35% of the days in North
Carolina. Great.
But, is it going to rain tomorrow? That's what I want to know.
I don't want to know the general statistic.
I want to to know about tomorrow because I got to decide whether to go on a
picnic. 65% of the snakes in this area are
poisonous. Great.
But I want to know whether the one I'm stepping on right now is poisonous.
I don't want to know the general statistic.
So, we need some way to take these generalizations and apply them down to
particular cases, and that's the job of a form of argument we're going to call
application of a generalization. Not a great name, but they're usually
called statistical syllogisms. And that's an even worse name because
they're not really necessarily statistical in the mathematical sense
because they don't have to have numbers in them. And they're also not syllogisms
like the categorical syllogisms that you studied back in the section on deductive
arguments. So we're going to call them applications
of generalizations, and they just happen all the time.
They really do. You might say, for example, I almost
never like horror movies. And that's a horror movie.
So I don't want to go to it. I'm, because I'm not going to like it.
Or you might say, I don't want to invest in that restaurant because 80% of
restaurants fail during the first two or three years.
And so, this is a restaurant that's new, it's probably going to fail, too.
Or, how about this one. Most people who are on the track team are
pretty thin. Now, you know there's some exceptions to
that, like the people who throw shot put. That's what I used to do.
But still, it's true that most of them do.
And so if you know that Sally is on the track team,
that's all you know about her. Then, you got some reason to believe that
Sally is probably thin. She might throw the shot put, but there
many more people on the track team who run and jump, and they're going to be
thin if they're any good at it. so, although it might be wrong, it's a
pretty good bet that Sally's thin if she's on the track team.
And that's the kind of argument that we're going call an application of a
generalization. Here's my favorite.
Am I wearing shoes right now? You can't see, can you?
Maybe I am, maybe I'm not. But most professors wear shoes when
they're teaching a class. Walter is a professor who's teaching a
class. So, Walter is probably wearing shoes
right now. That's an application of a
generalization. It has the same form as the other
examples that we saw before. To see that form, we can substitute
variables for the terms in the English argument.
So we can substitute the letter F for the reference class,
which is the set of professors who are teaching.
Then, we can substitute the letter G for what's called the attribute class, which
is the people who are wearing shoes. Then, we can substitute the letter a,
usually in lower case, for the individual that we're talking about.
In this case, that individual is me, Walter.
And we can substitute X% for the quantifier if it's most, or almost all,
or something like that, then X% takes its place.
And then, the form of the argument that I just gave about wearing shoes is simply
that X% of F or G, a is F, therefore a is probably G.
Notice that this application of a generalization moves in the opposite
direction from the generalization from a sample.
May I start off saying, well, X% of the f's in the sample are G.
So, X% of f's are G. And that's generalization from a sample.
But then, we take the generalization in that conclusion, and use it as a premise
in a, in an application. So we say, X% of Fs are G, a is F,
therefore, a is probably G. And that's why I said that we're going to
study up to generalizations and then down from generalizations in this part of the
course. And when they work together, we can use
information about the sample to reach a conclusion about the individual.
Image is pretty useful. Notice also that these applications of
generalizations are inductive. They share all the features of inductive
arguments. First of all, they're invalid.
Because it's possible for the premises to be true and the conclusion false.
That is, it's possible that X% of F are G, a is in F,
but it's not true that a is probably G. There might be no chance at all that a is
G. Second,
applications of generalizations are defeasible.
You can add additional information to the premises that make the argument very
weak, and like, completely undermine the argument.
For example, even if you know that Walter's a professor and most professors
wear shoes while they're teaching, a little bit of additional information
might show you that it's just not true that Walter is wearing shoes while he's
teaching. By the way, I'm going to, well, you'll
have to decide for yourself whether I'm really putting my shoes back on now.
But the third feature of inductive arguments is that, They're strong or they
can be strong. They can vary in strength. So even if I don't have shoes on, it's
still might be a fairly strong argument that almost all professors wear shoes
while they're teaching. Walter's a professor and he is teaching,
therefore he probably has shoes on. That can be strong before you get that
additional information. Before you see my feet without shoes on.
and now that you don't way whether I've got shoes on, it can still be a strong
argument. So we have arguments that are not valid,
and they're defeasible, and they vary in strength.
And that makes them inductive arguments. So, how do we tell when an application of
a generalization really is strong. That is, when does it provide strong
reasons for the conclusion. Now, the first standard should be
obvious. It's the same for generalizations, from samples.
The premises have to be true and justified.
If it's not true that I'm a professor, or if it's not true that I'm teaching, or if
you have no reason to believe, you're not justified in believing that I'm a
professor or I'm teaching, then the argument that we've been looking at can't
give you a good reason to believe that I'm wearing shoes because I just don't
fall under the classes that we're talking about.
Secondly, a standard that's specific to these kinds
of arguments is that the strength of the argument varies with how big X is.
If X is 99%,. so 99% of F's are G's,
and a is an F, that's a pretty strong argument that this
a is a G. But if it's 60%,.
then it's not a very strong argument. So, 99 is going to be stronger than 90,
which is stronger than 80, which is stronger than 70, which is stronger than
60. And the strength of the inductive
argument can vary as the percentage X varies.
But notice that if X is 10%,, then it becomes pretty strong argument for the
opposite conclusion. If only 10% of professors wear shoes when
they're teaching, and I am a professor who is teaching.
Then, it's pretty likely that I'm not wearing shoes.
And that's stronger than if it's twenty and 30 and 40.
And when you get close to the middle, the 50%,, if 50% of professors wear shoes
when they're teaching, and I'm a professor, then you can't really reach a
conclusion one way or the other about whether I'm wearing shoes.
At least, not on the basis of that evidence.
Not on the basis of that argument, okay?
So, a second standard for assessing applications of generalizations is to
figure out what the percentage is. But another crucial feature of
applications of generalization is that there can be conflicting reference
classes. So, it might be true that 95% of
professors wear shoes when they're teaching.
But, I'm not just any old professor. I'm an online professor.
This course is online and that's very different.
Because most professors will wear shoes when they teach because students will see
their bare feet. But you can't see my bare feet. So,
maybe, it turns out, that a lot of online professors are really not wearing shoes
when they teach. So, that's a conflicting reference class.
It's a different reference class that conflicts because it points to a
different conclusion than the original reference class.
The original reference class was professors who are teaching.
The conflicting reference class is online professors who are teaching,
or professors who are teaching online. When you run into conflicting reference
classes, as a general rule, what you ought to do is look at the smallest of
those reference classes. because that's usually going to give you
a better estimate of how likely it is that this individual has the general
attribute. That is, how likely it is that a is G.
So, if you want to know whether Walter is wearing shoes, you shouldn't look at the
broad class of all professors. But should instead, look at the narrow
class of online professors, if the course that he's teaching right now is an online
course. But there's a problem that you might run
into also. As you get a narrower, narrower reference
class, you sometimes just don't have enough data to figure out what the exact
percentage is. How do you know what the percentage of
professors who are teaching online is, that are wearing shoes?
How many of them are wearing shoes? I've never seen a survey.
I've seen lots of professors give regular lectures,
but you rarely see the feet of online professors.
So, you might not have enough information to apply that generalization.
So, while the narrower reference class of two reference classes that conflict can
be more accurate, it can also be problematic if you don't have enough
information to support the premise. That says that a certain percentage of that
class, that F, have the general attribute G.
The fallacy of overlooking conflicting reference classes can be a little
confusing, but it's very important. So, let's look at another example, and
this time let's focus on a medical example.
Let's suppose that 90% of the people with a certain condition, a certain medical
condition, a certain illness, die. And Bob has that illness.
So, it looks very likely that Bob will die.
And this is an application of the generalization about people with this
illness. But now, let's suppose we find out that
most people catch this illness when they're old, but Bob is quite young.
So, it turns out that young people with this illness usually survive.
As a matter of fact, only about 20% of the people with this illness, who get it
when they're young, die from the illness. If Bob is young and has this illness, so
now we can rethink, well he probably won't die from the illness.
But, wait a minute. We now have another conflicting reference class because Bob
has a heart condition. And it turns out that even young people,
when they catch this illness, if they have a heart condition, they usually die.
As a matter of fact, 80% of the people with this illness, who have a heart
condition and are young, die from this illness.
So Bob, who is young and has this illness, and also has a heart condition
will probably die from this illness. But wait a minute, it turns out that
there's a new treatment. And of the people who are given this
treatment, only about 30% of them die even if they are young, with this
illness, with a heart condition. And Bob lives in an area where he can get
the treatment. So now, it looks like there's only 30%
chance that Bob will die from this illness.
So what's happening here is as we get more and more information,
the likelihood of Bob dying from this illness starts out really high and then
it goes low. And then it goes high again, and then it
goes low again. And it goes back and forth as we get more
information. And then the question arises,
which of these different generalizations should we use to figure out how likely it
is that Bob will die from the illness? The answer here, as with most cases of
conflicting reference classes, is that we ought to look at the narrowest class that
we can. because if we know that there's a
treatment and Bob is young, and he has a heart condition, and he also has this
illness, and we put all of that together and compare him to other people who are
young with this illness and a heart condition who can get the treatment, and
look at how many of them have died in order to form an estimate of how likely
it is that Bob will die from this illness.
So, we always want to look at the narrowest reference class that we can in
order to get the best estimate of the probability in the conclusion.
But then, there's a problem. There might not be very many people,
remember it's a new treatment. So there are not very many people who are
young, who have this illness to begin with.
Remember, most of the people with the illness are old.
And the treatment's new so it hasn't been tried on very many young people.
And, of course, young people don't often have heart conditions.
So, it might be very difficult to find enough people who are like Bob in all the
essential respects in order to determine whether or not Bob will probably die from
this illness. So there's a kind of a tension here.
More information gives us more accuracy, but only if we have enough information to
be justified in trusting the premises of the application that we're looking at.
And that's one of the tricks in figuring out how to estimate whether or not Bob is
likely to die from the illness. The same points apply to all kinds of
examples. You know, if you want to know the climate
in the area, that's going to be a generalization about days around here in
this time of year. But if you want to know the weather
tomorrow, you need to apply it. And then, you're going to need to look at
specifically what the weather was yesterday.
Specifically, what the humidity is. The more information you can build in,
the better estimate of what the weather's going to be tomorrow.
Or, if you want to bet our sports team. You say, well, they're a very good team,
they won most of their games in the last five years.
But wait a minute, there'll be other players left and now its not likely that
they're going to win. But wait a minute, they got new players
that are even better. That makes it more likely they are going
to win. And again, the information can make the
probability go low and then high, and then low and then high, and you have to
get the most specific information you can to get the most accurate estimate of how
likely it is that this team will win. But, you might not have enough
information about this team with these players under these circumstances because
there just hasn't been enough cases for you to observe.
So, as with the medical example, it's going to be a tension between wanting the
most precise, the most smallest reference class among the different conflicting
reference classes. And yet, you need premises that you have
enough information that you can justify them.
And if you can reach that perfect point. Where you've got enough information to
justify the premises. But that premise is specific enough so
that it has all the relevant information about the case,
that's when we're going to have the best outcome on our applications of
generalizations.