We're going to do something a little different today, something a little out
of the ordinary. I'd like to start off today's lecture by
speaking to possibly the most important issue in philosophy, that is of course,
myself. I have done a lot of incredibly important
things in a very short period of time. One of the most interesting and profound
topics in philosophy is the self, and there's no self worth talking about as
much as myself. So I thought I would talk about my self
during this lecture, to give you an idea of what other selves are like.
Someone that you are now seeing on this screen is a philosophy professor.
And what follows from that? Well, what follows from that, is that
it's not the case. That no one that you're seeing on this
screen is a philosophy professor. Now notice, the Venn diagram for the
premise of that inference consists of two circles, with an x in their intersection.
And, the conclusion of that inference. Is the negation.
Of a statement. Whose Venn diagram consists of those same
two circles, with their intersection shaded.
In general, when you want to, negate. A statement that can be represented using
a Venn diagram, when you want to negate a statement of the form A, E, I, or O, you
can do it by replacing an X everywhere it occurs with shading or with replacing
shading everywhere it occurs with an X. Here's another example of the same point.
Someone that you're seeing right now is not a philosophy professor.
And from that it follows that it's not the case that everyone you're seeing
right now is a philosophy professor. The Venn diagram for the premise of that
inference contains two circles with an x in one of them but not in the other.
The Venn diagram for the conc, conclusion of that inference is the negation of a
statement, the Venn diagram for which contains those same two circles, with
shading in one of them but not in the other.
So once again, you negate one statement. By another statement by replacing shading
in the area where there was originally an x.
We've just discussed how we can use venn diagrams to establish the validity or
invalidity of categorical immediate inferences.
Now, I'd like to show how venn diagrams could be used to establish the validity
or invalidity of a different kind of inference.
A kind of inference that we're going to call a syllogism.
Now syllogisms are different from categorical immediate inferences in a
couple of ways. First, instead of one premise syllogisms
have two premises. Second, the two premises share.
One particular term. A term that we're going to call the
middle term. So, one of the premises is going to be an
a, e, I, or o statement, with two terms. We can call them a subject term and a
middle term. The other premise will be another a, e,
I, or o statement, with two terms. We'll call them the middle term and the
predicate term. The conclusion is going to be an A, E, I,
or O statement, without the middle term. It'll just have the subject term and the
predicate term. Now because categorical inferences
involve three terms, that is to say, they involve three categories, three kinds of
thing. In order to represent them in order to
represent what it is that they say and to figure out what they say is valid or
invalid we're going to need a venn diagram that consists of three circles
rather than just two circles one circle corresponding to the subject term one
circle corresponding to the middle term and one circle corresponding to the
predicate term. Let me give you some examples of how this
works. Here's an example of a syllogism and a
Venn diagram that we can use to represent what the syllogism is saying and why it's
valid. So this syllogism has two premises, both
of which are of the form a. The first premise says all Duke students
are humans. In fact, that's true.
There are no non-human Duke students. So, how do we represent that using this
Venn diagram? Well, we shade out the circle of Duke
students that's outside the circle of humans.
Right? This is the circle representing all the
Duke students. This is the circle representing all the
humans, and the first premise tell us that all Duke students are human.
In other words, whatever Duke students there are, they fall into the category of
human. So, the circle of Duke students outside
the circle of humans gets shaded out. The second premise is also of the form A.
It tells us all humans are animals. Alright so how do we represent that?
Well the way we represent that is by shading out the circle of humans that's
outside the circle of animals. Right?
Because all of the humans are inside the circle of animals.
So, here's what we do, we shade out the part of the, human circle that's outside
the circle of animals. And, what does that tell us?
Well, according to the syllogism all Duke students are animals and looking at the
Venn diagram you can see why that's right.
It's right because if there are no Duke students in this region and there are no
Duke students in this region, the only place where there could be any Duke
students is this region. And that's inside the circle of animals.
So if there are any Duke students they've gotta be animals.
And that's what we at Carolina have been saying for over 80 years.
Here's another example. Here's a syllogism where one premise has
the form A and another premise has the form I, and the conclusion has the form
I. So one premise says all humans are
animals. So to represent that using our venn
diagram, what we do is shade out, the part of the circle of humans that's
outside the circle of animals. And there we go.
Okay so we're showing that if there are any humans they are in that circle of
animals. The other premise says some Duke students
are humans okay so we represent that by putting an X in the circle of Duke
students that's also in the circle of humans.
Well. The only place it could go is right
there. It couldn't go there, because that's
shaded out. So the only place that X could go is
right there in that region. Okay, but if there's an X in that region,
then what does that tell us? That tells us that some Duke students are
animals. Right?
Because there is something that's in the category of Duke students and that's also
in the category of animals. So, using this Venn diagram to represent
the information that we get from these two premises, we can show that this
conclusion follows validly from those two premises.
If these two premises are true, then this conclusion has gotta be true.
And that's what the Venn diagram shows us.
Now let's consider another example. So this syllogism tells us, first, that
some Duke students are humans, and second, that all humans are animals.
But it concludes from those two premises, that all Duke students are animals.
Now is that inference valid? Let's use the venn diagram to try to find
out. Well, this premise says all humans are
animals. So how do we represent that on the Venn
diagram? By shading out the portion of the circle
of humans that's outside the circle of animals.
Alright? So we shade out that portion to show that
there's nothing in the category of humans that's outside the category of animals.
All humans, if they're already, are animals.
Some Duke students are humans, the other premise tells us.
Now how do we represent that information? Well, by drawing an axe that is inside
the circle of Duke students, and also inside the circle of humans.
Now where could that go? Only one place for it to go and that's
right here. Now, from that information, can we
conclude that all Duke students are animals?
No. We can't and here's why not.
We have not shaded out the region of Duke students that's outside the circle of
animals. We don't know if there's something in
that region or not. There could very well be something in
that region for all that we've been told by the premises.
The premises leave it open that maybe most Duke students are not animals at
all, but rather plants or microbes or inorganic materials.
So, this inference is not valid, because there is a possible way for the premises
to be true, as is represented by this diagram, but the conclusion to be false,
because maybe there are things that are out here.
Now, let's look at another syllogism. Where both of the premises and the
conclusion are all of the I form. Some Duke students are humans.
Some humans are animals. Therefore, the syllogisms concludes, some
Duke students are animals. Now, it that syllogism valid?
Let's see if we can use our Venn Diagram to find out.
So, one premise says some Duke students are humans.
Let's try to represent that information. In order to represent that information we
need to make a little mark a little X that's inside the circle of Duke students
but also inside the circle of humans. So, we make that mark.
Wait a second. It looks like there are two different
places where we could make that mark. We could make it in this region or in
this region. Because both of these regions are in the
circle of Duke students, and also in the circle of humans.
So where should we make the mark? Let's hedge our bets and make it between
those two regions, right here. Okay.
The second premise is some humans are animals.
So we need to make a mark that's inside the circle of humans but also inside the
circle of animals. But wait.
Do we make that mark right here or right here?
The premise doesn't give us enough information to decide.
So once again, let's hedge our bets and make that mark, right here on the
boundary between those two regions. So.
Finally, the conclusion says, some Duke students are animals.
Is that right? Well, you might think it is, because this
x and this x both look like they're in the circle of Duke students and also in
the circle of animals. But that's not right.
See this X is on the border of the circle of animals.
Maybe it's inside that circle. Maybe it's outside that circle.
We don't know. So this X right here might not be inside
the circle of animals. This X right here is on the border of the
circle of the Duke's students. So maybe it's inside the circle.
Maybe it's outside the circle. We don't know.
So this x, while it's certainly in the circle of Duke students might not be in
the circle of animals. And this x, while it's certainly in the
circle of animals might not be in the circle of Duke students.
So, this Venn diagram doesn't tell us whether or not there is anything that is
both a Duke student and an animal. It doesn't tell us that there is anything
in this region right here. And if it doesn't tell us whether there's
anything in this region right here, that means that the information provided by
the premises is not enough to guarantee that the conclusion is true.
So using this venn diagram we can figure out that this syllogism right here is not
valid. The premises don't guarantee the truth of
the conclusion. Now notice since every syllogism has a
subject term, a middle term and a predicate term, it's going to be about
three different categories, a subject category, a middle category and a
predicate category. And so you're going to need a Venn
diagram with three different circles to represent the information conveyed by
that inference. Now that Venn diagram is going to have
eight different regions to it. There's going to be a region that's
inside the S circle, inside the P circle and inside the M circle, inside all three
circles. There's going to be a region that's
inside both the S and P circle, but outside the M circle.
There's going to be a region inside the S and the M circles but outside the P
circle, a region inside the P and the M circles but outside the S circle.
Then there's going to be a region that's in the S circle, but outside the M and
the P circles. A region that's in the, in the M circle,
but outside the S and the P circles. And a region that's in the P circle, but
outside the S and the M circles. And finally, there'll be an eighth region
that's outside all three circles. Now.
Thinking about the fact that every Venn diagram representing the information
conveyed in the syllogism is going to have those eight regions.
I want you to think about why it is that there would be 256 possible different
kinds of syllogism. That concludes our lecture on syllogism.
I want to point something out about the logic that we've learned over the last
two weeks. Both the propositional logic and the
categorical logic. While both of them are useful tools in
understanding why certain kinds of arguments are valid in virtue of their
form, neither of them provides a complete treatment of all valid arguments.
Not even of all valid arguments that are valid in virtue of their form.
For instance, consider the argument, for every number, there is another number
larger than it. Therefore, there is no number that is the
largest. That argument is valid and it is valid in
virtue of its form. If we replace the word number with some
other word the argument is still going to be valid.
. But nothing that we've studied so far in
propositional logic or in syllogistic logic and categorical logic can help us
understand why that argument is valid in virtue of its form.
In order to understand why that argument is valid in virtue of its form we'd need
to get into more advanced kinds of logic. There are lots more advanced kinds of
logic, but we're not going to be discussing them in this course.
If you're interested, I recommend another course in logic to you.