Lecture 3-1 - Validity

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