Great. We can do course analysis, we can
identify the premises and conclusions, we can put them in standard form.
What's next? Well, the next step is take those parts
and put them in a certain order, and fill in the missing gaps.
We need to learn how to reconstruct arguments.
Are you ready? Well, there are lots of ways to
reconstruct. When you think about constructing a
house, or a building. In order to construct a good building
you've got know, what the goal is, what the standards of a good building are.
The same thing goes for reconstructing arguments.
In order to reconstruct an argument properly, you need to know what the
standards are for reconstruction. We're trying to reconstruct it so as to
meet those standards. Because the goal is not to reconstruct
the argument in order to make it look bad.
The point is going to be reconstruct arguments so as to make them look good.
because by making your opponents look bad or silly, that doesn't do anybody any
good. If you want to learn about their
perspective, and you want to learn from their views, then you need to reconstruct
their argument, so as to make it look as good as possible.
And to do that, you need to know about the standards for arguments.
That is the standards that make arguments good or bad.
So what we're going to do this week is we're going to look first at some
standards for our arguments, validity and soundness in particular, and
they we're going to use those standards to develop a method called reconstruction
or deep analysis, I'll explain those terms later.
And then we are going to apply that methods to a few concrete examples, in
order to be able to take a passage and take those premises and conclusions and
fill them out and get a full fledged argument that if we've done it properly
will be be as good as it can be, and that we can learn from.
That's the goal. Now,
because an argument consists of premises and a conclusion, and the premises are
supposed to be related in the right way to the conclusion, there can be two main
ways an argument can go wrong, two main vices of argument, you might say.
The first is there might be something wrong with the premises. In particular,
they might be false, or at least one of them might be false.
Second, there might be something bad about the relation between the premises
and the conclusion. The premises might fail to give a good
reason for the conclusion. Now each of these problems is something
that we need to avoid and when we do avoid them, we get the corresponding
virtues mainly validity and soundness. And those are the two notions that we
want to discuss in this lecture and the next.
Let's begin with the relation between the premises and the conclusion.
What kind of relation between the premises and the conclusion is good for
an argument or makes an argument good? Well, that depends.
Some arguments are deductive and others are not.
So, let's focus for a moment on deductive arguments.
In deductive arguments, the conclusion is supposed to follow from the premises, but
what does that mean? I mean, what does it mean for a
conclusion to follow from the premises? That's a really hard notion to pin down.
So what logicians usually do and, and what we're going to do, is focus instead
on the notion of validity. And the idea is that a deductive argument
is trying to structure itself so that it's valid.
And we'll explain what validity is, but for now, I want to emphasize that we're
only talking about deductive arguments. There's going to be another class of
arguments called inductive arguments that we'll get to later in this course,
where they don't even pretend to be valid.
They don't even pretend that the conclusion follows from the premises.
But just for simplicity, let's focus on deductive arguments now, and the idea is
that the deductive argument should be structured in such a way that it's valid.
Then the next question is what's validity?
Let's start with a simple example. Suppose that you know Mary but you don't
know her children. However you do know that she has one
child who is pregnant. And you also know that only daughters can
become pregnant. So you have all that you need to know in
order to draw a further conclusion, namely, Mary has at least one daughter.
So here's the argument. Mary has a child who is pregnant.
Only daughters can become pregnant, therefore, Mary has at least one
daughter. Now, if you think about it, there's just
no way, no possibility that both of those premises are true and the conclusion is
false. That is the feature that we're going to
call validity. More generally, we can define validity in
an argument so that an argument is valid if and only if, it's not possible for the
premises to be true and the conclusion false.
That is, it's not possible for there to be a situation where both of those hold,
that is a situation where the premises are true and the conclusion is also
false. Now that might strike you as a pretty
simple notion. But actually that little word possible is
a problem. How do you tell what's possible or what's
not possible? Well, there's no mechanical solution to
that and we'll struggle with that a little bit throughout this course.
But for now, since we're right at the start, let's think of it this way.
Is there any way for you to tell a coherent story?
Where the premises are true and the conclusion is false.
Can you describe a situation with that combination of truth values?
That is, the premises being true and the conclusion false in the same situation.
If you can tell a coherent story with that combination then it's possible and
the argument is not valid. But if there is no way to tell a coherent
story where the premises true and the conclusion is false, then the argument is
valid. Now let's try that test on our example.
Mary has a child who is pregnant. Only daughters can be pregnant.
Therefore, Mary has a daughter. So is there any way to tell a coherent
story where the two premises are true? That is, where Mary has a child who is
pregnant, and only daughters can be pregnant, but the conclusion is false.
Mary does not have a daughter. Well, just try.
Suppose Mary has only one child and it's a son.
There's the conclusion that's false. Good.
What about that? But then, is that son pregnant?
Well, if the son is not pregnant then the first premise's false.
Mary doesn't have a child who is pregnant.
But if the son is pregnant somehow, don't ask me how, but if the son is pregnant
then the second premise's not true. It can't be true that only daughters can
be pregnant because this child is a son. Okay,
what if Mary has two children? Try that.
Try to tell the story that way. Mary has a daughter and a son.
Now she's got a child who is pregnant, the daughter,
and only daughters can be pregnant, but she has a son.
Wait a minute, she's got a son and a daughter.
So now the conclusion's true, because she does have a daughter even though she also
has a son. Oh, oh, wait.
How about this one? What if Mary has a child who is
biologically female but sees himself as a male?
And so she sees that child as a male, but that child is pregnant, because after
all, they're biologically female. Now are the premises true and the
conclusion false? Does that story make sense?
Wait a minute. Either her child is a daughter or her
child is a son. Now if it's a daughter and its pregnant,
no problem. The conclusion's true.
If it's a son, because that child sees himself as a male, then you've got a
choice. Well, what about the first premise?
The first premise is going to be true. She does have a child, who is pregnant,
but what about the second premise, only daughters can be pregnant.
Wait a minute. If that really is a son, if we're
going to call that a son, then it's not true that only daughters can be pregnant.
So now the second premise is false. So try it again.
Try it with, you know, sex changes, and try it with Hermaphrodites tell the story
any way you want about Mary's children. And there's no way that both premises
come out true when the conclusion is false.
That shows that the argument is valid. It might be just that we can't imagine
the coherent story, which makes it invalid.
But the fact that we've tried hard and looked at all the possibilities we can
think of at least gives us a good reason to think that this argument is valid.
Now some people like to think of it in the reverse direction.
They say, let's imagine that the conclusion is false, and then,
If it has to be the case, that at least one of the premises is false, the
argument is valid. Then you can define the validity as, is necessarily the case
that if the conclusion is false one of the premises is false, or in every
possible situation, if the conclusion's false one of the premises is false.
We can apply this new account of validity to the same old example.
It's got to be the case that if Mary doesn't have a daughter,
then she doesn't have a child who is a pregnant, or else there are at least some
children who are pregnant who are not daughters.
So notice in this case you're reasoning back from the falsehood of the conclusion
to at least one of the premises has to be false.
whereas in the earlier definition you were saying it's not possible in the
situations where the premises are true for the conclusion to be false.
You can look at it either way, either direction.
Just pick the one that works for you and go with that definition, because in the
end, the two definitions are equivalent. It's just a matter of what's going to
help you understand which arguments are valid and which ones are not.
In addition to understanding what validity is, it's also very important to
understand what validity is not. A lot of people get confused by the
notion of validity in this context, because they're thinking that to call an
argument valid must be to call it good, right?
You call a driver's license valid when it's good in the eyes of the law.
But that's not what we're talking about here.
The notion of validity is getting used by logicians here as a technical notion and
it's very, very, very important to remember that to call an argument valid
is not to call it good. For some arguments, like deductive
arguments the invalid might be necessary for them to be good.
But it's not enough and we'll see a lot of examples of that later on.
The second point about what validity is not is that validity does not depend on
whether the premises and the conclusion are actually true or false.
Instead it depends on what's possible whether there is a certain combination,
true premises and a false conclusion, it's even possible.
So, whether the premise is actually true in the actual world is not what's at
issue. And we can see this, by seeing that some
arguments with false premises can still be valid.
And some arguments with true conclusions can be invalid.
So let's look at some examples of that. Indeed there four possibilities.
because remember, the conclusion could be true or false, and the premises could be
all true or at least one false. So we've got four possibilities.
And all of those are possible except for one.
The one combination that's not possible for valid arguments is true premises and
a false conclusion. But if you've got true premises and a
true conclusion, it might be valid, it might not.
If you've got false premises and a true conclusion it might be valid, it might
not. If you got false premises and a false
conclusion, it might be valid, it might not.
So let's look at some examples each of those possibilities in order to better
understand the relation between premises and conclusion that exist when the
argument is valid. It's hard to give examples with true
premises or false conclusion, or any these other combinations when the
truth is controversial. So we're going to have a really simple
example, and we're going to start just by stipulating what the facts are.
We're going to assume that all Ford cars have four tires,
but some Ford cards do not have four doors.
We're also going to assume that Henry's car is a Ford that has four doors.
And Jane's car is a Chrysler that has only two doors, not four doors.
And we're just going to take those facts for granted and assume that that's the
situation we're talking about, and then we can give examples of all the
combinations that we discussed before. Let's begin with true premises and a true
conclusion. So, here's an example of that sort.
All Fords have four doors. Henry's car is a Ford, therefore, Henry's
car has four doors. Is the first premise true?
Yes, that's what we are assuming, that's one of our assumptions.
Is the second premise true? Yes.
That's another one of our assumptions. Is the conclusion true?
Yes. So they're all true and now is the
argument valid? Is it possible that all Fords have four
doors? Henry's car is a Ford and yet it's not
true that Henry's car has a four doors. I mean, just think about it.
How would that happen? Well, for the conclusion to be false, it
would have to not have four doors. Suppose it has two doors.
Well then, either it's not a Ford or there's some Ford,
namely Henry's Ford, that only has two doors and not four
doors. So, there's just no coherent story you
can tell where the premises of this argument are true and the conclusion's
false. Or in reverse, if you start off with the
assumption that the conclusion's false. You can tell from that, that at least one
of the premises has to be false as well. Nonetheless.
There are other examples, where the premises are true, and the conclusion is
true, but the argument is not valid, instead it's invalid.
Here's an example of that combination. All Ford cars have four tires.
Henry's car, has four tires. Therefore, Henry's car is a Ford.
Now, in this new argument, are all the premises true?
Yes, the first premise says, all Ford cars have four tires.
And that's true by our assumptions. Second premises Henry's car has four
tires and that's also true by our assumptions and is the conclusion true?
Yes our assumptions also tells that Henry's car is a Ford.
But is it possible, is there any way to tell a coherent story where those
premises are true and the conclusion is false?
Yes, absolutely. All that has to happen is that Jane and
Henry switch cars. Then the first premises can be true
because all four cars have four tires, and the second premise is going to be
true, because Henry's car has four times, of course now it's a Chrysler, because he
got it from Jane, but the conclusions can be false.
Henry's car is not a Ford because Ford and Chrysler are different companies.
So, if he switches cars with Jane and he has a Chrysler then he doesn't have a
Ford. His car is not a Ford.
Okay, so now you've got a situation where the premises are true and conclusion
false. It's not the actual situation but its a
possible situation. You can tell a coherent story with the
premises true and conclusions false and that tells you that the argument is
invalid. Next, let's consider an example with
false premises and a true conclusion. Premise one, all Fords have four doors.
Premise two, Henry's car is a Ford. Conclusion, Henry's car has four doors.
Is the first premise true? No, it's not true that all Ford's have
four doors. Our assumptions tell us that.
Second, is Henry's car a Ford? That's true.
So one of the premises is false and the other one's true.
That means they're not all true. And the conclusion, is that true?
Yes, it is true that Henry's car has four doors.
But remember, the fact that that's actually the case doesn't tell us wether
or not is valid. So, is it valid? That depends on wether
it's possible for the premises to be true and a conclusion false.
Premises aren't actually true, but is there a possible story that you could
tell that would be coherent where the premises are true and the conclusions
false? That's the test of validity.
So let's apply it to this case. We'll just imagine, that, the
conclusion's false, that Henry's car does not have four
doors. It's only got two doors. Then, there are really only two
possibilities, either it's a ford or it's not a Ford.
If it is a Ford, then the first premise is false.
It's not true that all Fords have four doors.
But if Henry's car is not a Ford, then, the second premise is false, because it
says that Henry's car is a Ford. So, there's no coherent way in which it
could possibly be true that both of these premises are true and the conclusion is
false so this argument's valid and notice that, that shows that an argument can
valid, even though it's got a false premise.
Now, you might be thinking to yourself this is crazy how can an argument be
valid when one of it's premises are false?
An argument's no good when it's premises are false.
Notice what that does. That confuses the notion of valid.
Like in a valid driver's license where to be vaild is good.
With the technical notion of validity that we're using here.
The technical notion of validity that we're using here has to do with the
relation between the premises and the conclusion.
And in particular, it has to do with possibilities,
and not with the actual falsehood of the premise.
So what we have to ask ourselves is, what would happen if it really were true?
That all Fords have four doors is not true in the actual world, but we're
concerned with possibility. And if all Fords did have four doors, and
if Henry's car was a Ford, then it would have to have four doors.
So, that possibility of the premise being true, even though it's not, is what's
crucial for determining validity. Because it's not possible for the
premises to be true, and the conclusion false.
That makes it valid in our technical sense.
Even if it's not valid, in the common sense notion of validity as goodness,
we're not saying that the argument's a good argument.
We're saying that it meets this technical definition of validity.
That logicians use. Now the only combination of truth values
in premise and conclusion, that you cannot get with a valid argument
is to have true premises, an a false conclusion.
So here's an example of that. Premise one,
some Ford cars do not have four doors. Premise two, Henry's car is a Ford.
Conclusion, Henry's car does not have four doors.
The premises by our assumptions are both true and the conclusion is false and it's
not valid because it's easy to see how it might be possible for the premises to be
true and the conclusion false. It's simple.
Even if some Ford's don't have four doors, Henry's car is one of the Ford's
that does have four doors, and then both the premises can be true and the
conclusions false. So that's how you can get an invalid
argument with true premises and a false conclusion.
But you don't [LAUGH] really even need that.
Look. Every argument that has, true premises
and a false conclusion, has to be invalid.
Because if it does in fact actually have true premises and a false conclusion,
then it's possible, for it to have true premises and a false conclusion.
So you can know right off the bat that every argument with true premises and a
false conclusion is invalid. What you can't know is for the other
combinations. Then you have to think of what is
possible instead of simply what is actual.
So far we've only looked at arguments with all and some and we've looked at
Henry and Ford and Chrysler and so on. But the same points are going to apply to
lots of different arguments with very different forms.
So lets look at one example of that. Premise one, David, is either a swimmer
or a golfer. Premise two, David, is a swimmer,
therefore, conclusion, David is not a golfer.
Okay, is it possible for the premises to be true and the conclusion false?
How could you tell a coherent story where both premises are true and the conclusion
is false? Just think about it.
How could that happen? Oh I've got it!
He could be both a swimmer and a golfer, like me.
I play golf, and I also swim, and lots of people do.
Now of course, if you have or, and you say something like he's either male or
female, maybe you can't have both. But with swimming and golfing you can
just be both a swimmer and also a golfer. And then the premises can be true when
the conclusion is false, which shows that this argument is not valid.
Now let's try this other example which is a lot like the last one, but it's
different in an important way. Premise one, David is either a swimmer or
a golfer. Premise two, David is not a swimmer,
therefore conclusion, David is a golfer. Is there any way?
Is it possible? Is there anyway to tell a coherent story
where, the premises are true and the conclusion is false?
We know. Well just think about it, the four
possibilities. Suppose that David is both a swimmer and
also a golfer. Well then the conclusion's true, right?
So you can't have two premises and a false conclusion because then in that
case then the conclusion is true. Now, suppose that David is a golfer, but
he's not a swimmer. Well again, the conclusion's true.
So that's not a case where the premise's is true and the conclusion's false.
but suppose he's not a golfer but he is a swimmer Well wait a minute.
In that case the second premise is wrong, because it says, he's not a swimmer and
we're, in this story, imagining that he is a swimmer.
Now suppose that he's neither a swimmer nor a golfer.
Well then the conclusion is false, and that second premise is true.
But wait a minute, now the first premise is false, because
the first premise says, he either a swimmer or a golfer.
In this story it's saying that he's neither.
So, those are the four possibilities and there's none of them where the premises
are true and the conclusions false. So it's not possible for the premises to be
true and conclusions to be false in this case, so this argument is valid.
And did you notice something? I didn't make assumptions like in Henry,
and the Ford, and the Chrysler, because we don't need to know whether David
really is a swimmer or a golfer. We don't need to know the actual facts of
the world at all. We could tell that this argument is valid
without knowing what kinds of sports David does.
And that shows you that whether an argument is valid or not depends on
what's possible, not on what's actual. because you can know that the argument's
valid, even if you don't know whether in the actual world he's a swimmer or golfer
or neither or both or one but not the other.
We haven't been through all of the possibilities,
but we have seen that you can have invalid arguments with true premises and
true conclusions, and you can have valid arguments with
false premises and true conclusions, and we've got a little table that shows
us the other possibilities. Instead of going through all of those
other possibilities myself, I think it'd be better, if.
You did a few exercises, and that'll, make sure that you
understand this notion of validity before we go on and try to show how validity is
related to soundness.