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. Cuz 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 wron g, 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 daugh ters 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 gonna 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 i t'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, cuz 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 gonna 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 th e
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. Cuz 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 tr ue 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,
cuz 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, cuz 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 not ice
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 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,
cuz 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. Cuz 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 poss ibilities.
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