Last week, we talked about propositional logic which is the study of inferences
that are valid, because of the true functional connectives that they use.
This week, we're going to be talking about a different kind of logic, called
categorical logic. Which is also the study of inferences
that are valid. But they're not valid because of the true
functional connectives that they use. They're valid because of something else
that I'd like to describe now. In order to describe what makes these
inferences valid, I'd like you to consider some examples.
So first, consider the inference. All squares are rectangles.
All rectangles have parallel, therefore, all squares have parallel sides.
That inference is obviously valid. There's no possible way for the premises
to be true if the conclusion is false. But notice that even if we replace
squares with another category like, let's say, squirrels, we replace rectangles
with another category like, let's say, mammals, and we replace the category of
having parallel sides with another category like, let's say animals, we
still end up with a valid inference. Let's try that.
All squirrels are mammals, all mammals are animals.
Therefore, all squirrels are animals, still valid.
So, both of these inferences are valid. And they're valid because of some common
feature that they share but what is that common feature?
Is it a truth functional connective that they both use?
Well, no, because neither inference uses any truth functional connective.
In fact, neither inference uses any propositional connective, so it can't be
some because of some truth functional connective that they use that the
inferences are valid. It must be for some other reason that the
inferences are valid. And in the study of categorical logic,
that we'll be doing this week, we'll try to understand what that other reason is.
Let me say something first about the notation that we're going to use.
Last week when we did propositional logic we used lower case letters like p and q
to represent propositions that were being connected, by some truth functional
connective. This week in categorical logic, we're
going to use, upper case letters from a different part of the alphabet, we'll use
upper case F, upper case J and upper case H, to represent categories, that are
being connected together into a single propasition and then as, both of the
inferences that we just discussed, have this form in common.
The first prement says, all of the things that fall into one category, F.
Also fall into a second category, G. For instance, all things that fall into
the category square also fall into the category rectangle, or all things that
fall into the category squirrel also fall into the category mammal.
The second premise, said that all things, that fall into that second category over
here also fall into a third category. So for instance,
all things that fall into the category, rectangle also fall into the category,
having parallel sides or all things that fall into the category, mammal.
Also fall into the category, animal. And then the conclusion said that all
things that fall into that first category also fall into the third category.
For instance all of the things that fall into the category square fall into the
category having parallel sides or all things that fall into the category
squirrel also fall into the category animal.
We can use a certain kind of diagram called a Venn diagram, after it's
inventor John Venn, to represent the relationships that these premises and
this conclusion describe. And using Venn diagrams we can show that
this inference, any particular example of this inference is valid, here's how.
Remember the first premise says, all things that fall into a first category
also fall into a second category. So what we do with a Venn diagram, is we
use a circle. to represent, the first category.
So all the things that fall into that first category, are inside this circle.
And all the things that fall outside that first category, are outside the circle.
So this first category we've called F [SOUND] and it's represented by that
circle. Now if, everything in that first category
is also in the second category, then if we use a circle to represent the second
category, the circle that we use to represent the second category has to be
at least as large as the circle we use to represent the first category because it
has to include everything in the circle that we use to represent the first
category. So,
we could draw a second circle to represent the second category, G.
So if F is the category of being a square then this interior circle represents all
of the squares. If F is the category of being a squirrel
then the interior circle represents all of the squirrels.
If G is the category of being a rectangle, then this exterior circle
represents all the things that are rectangles.
If G is the category of being a mammal, then the exterior circle represents all
the things that are mammals. So the first premise tells us that all
the things in the first category, all the Fs are also Gs, they're also in second
category. But the second premise tells us that all
the things that are in that second category, all the things that are Gs also
fall into a third category, they're Hs. So, we'll represent the category of Hs
using a circle that includes inside it. Everything that the g circle includes
inside it like this. So, the second premise says, all the
things that are in the category of G are also in the category of H.
We can represent that in the Venn diagram, by having the circle of all the
Gs be inside the circle of all the Hs. So, if the category of G is the category
of rectangles, and the category of H is the category of things that have parallel
lines, then the second premise says that all the things that are rectangles also
have parallel lines. If the category of G is the category of
mammal, and the category of H is the category of animal, then the second
premise tells us that all the mammals are also in the category of animals.
So these three circles right here represent the information contained in
premise one and premise two together. The information contained in premise one
is represented by having the F circle inside the G circle.
And the information contained in premise two is represented by having the G circle
inside the H circle. So having these three circles
concentrically arranged as they are right now, represents the information contained
in these two premises. But notice something,
if we put the F circle inside the G circle, and then we put the G circle
inside the H circle. Then, the F circle is going to have to be
inside the H circle. And if the F circle is inside the h
circle, what does that represent? That represents the fact that everything
that is in the F category is also in the h category.
In other words, all F are H. And so, when we construct a Venn diagram,
to represent the information contained in these two premises.
That Venn diagram will have to also represent the information contained in
the conclusion. And so the Venn diagram shows us that
there is no way for the premises to be true and the conclusion to be false.
If the premises are both true, then the conclusion is going to have to be true.
And the venn diagram shows us that. Using venn diagrams, there are a number
of different relationships that we can visually represent between categories.
I just showed you how we can represent the relationship that we're claiming to
hold when we say, all F or G. We're saying all the things that fall
into the F category, whatever that may happen to be, also fall into the G
category, whatever that may happen to be. And we represent that by putting an F
circle inside a G circle to show that all the things that are Fs are also Gs.
Here's how we would represent the claim that some F is a G.
We would draw an F circle and a G circle that overlaps it and then mark a spot
that's in the intersection of the F circle and the G circle and that shows
that there is something whatever it is. That falls into the F category, so it's
an F. And it also falls into the G category, so
it's also a G. So we are showing there that some F
whatever it is, is also a G. If we want to represent that no F f is G
that nothing in the F category is in the G category, then we can draw the F circle
and the G circle overlapping and shade out the area where they overlap to
indicate that there is nothing in that intersection.
There is nothing that is both an F and a G and that would represent no F is G.
Finally, if we want to represent the claim that some F is not G.
Then we could draw the F circle, and draw the G circle overlapping it.
But mark a spot inside the F circle, but outside the G circle.
That way we're showing that there is something, whatever exactly it is, that
is in the F category, but it's outside the G category.
So some F, whatever it is, is not a G. And that's how we can use Venn diagrams
to represent different relationships that we could claim to hold between
categories. Now let's see how we can represent the
validity of an inference that uses claims like this by means of Venn diagrams.
Notice that these Venn diagrams already tell us something about the logical
relations between the claims that they represent.
For instance, look at the Venn diagram for some f is g.
It shows us that there is a thing. Don't mind what it is.
But there's a thing that's in the F category and that's also in the G
category. Now consider the Venn diagram for the
claim, no F is G. what it shows us is that there is nothing
that is in the F category and in the G category.
So this claim right here no F is G, is the negation of this claim right here,
some F is G. If this claim is true, if some F is G,
then this claim is not true, it's not true that no F is G.
And similarly if this claim is true, no F is G, then this claim is not true, it's
not true that some F is G. So this claim and this claim are
negations and you can tell that by looking at the Venn diagram that
represent them. This Venn Diagram shows you that there is
something in a particular area and this Venn diagram shows you that there is
nothing in that same area. But, now notice the sum F is not G is
also the negation of the claim all F are G.
If all F are G then it can't possibly be true that some F is not G, but if some F
is not G then it can't possibly be true that all F are G, so this claim and this
claim are negations and yet the way we've done the Venn diagram for this claim, all
F or G, it doesn't show that this claim is the negation of this claim.
So we're going to change the Venn diagram for all F or G so that it indicates that
all F or G is the negation of sum F is not G.
And here's how we're going to do it. We're going to draw an F circle and a G
circle that overlaps it . And we're going to shade out, the area of
the F circle that's outside the G circle to indicate that there's nothing that is
in the F circle that is not also in the G circle.
Now, with this Venn diagram, you can easily see that all F or G, is the
negation, of sum F is not G. The Venn diagram for sum F is not G,
shows you that there's a thing in the F circle, that's outside the G circle.
And the Venn diagram for all F or G, shows you that there's nothing that's in
the F circle that's outside the G circle. And so the Venn diagrams now represent.
The relation between sum F is not G and all F are not G.
The same way that the Venn diagrams represent the relation the sum F is G and
no F is G. Venn diagrams are useful in helping us to
see what the logical relationships are between different claims concerning
what's true about all things of a certain kind, or some things of a certain kind or
no things of a certain kind. Let me give you an example.
Consider a claim of the form all f or g and a claim of a form no f or g.
For instance, all of the sheep are in the field or no sheep are in the field.
Now, those two claims might at first sound inconsistent with each other, like
they can't both be true. But if you look at the Venn Diagram for
all of the sheep are in the field and the Venn Diagram for no sheep are in the
field, you realize actually that both of those claims can be true.
The Venn Diagram for all of the sheep are in the field shows that what that claim
means is that there are no sheep that are not in the field.
. But it could be true that there are no
sheep that are not in the field, even when it's also true that there are no
sheep that are in the field. That can be true, both of those two
claims can be true whenever there are no sheep.
If there are no sheep, then there are no sheep that are not in the field and there
are also no sheep that are in the field. And the Venn Diagram for those two claims
helps us to see that they are consistent with each other.
Consider two claims of the form sum F is G and sum F is not G.
For example, some politician is honest, and some politician is dishonest.
Now. At first, you might hear those two claims
and think. Well wait a second.
Who is this, some politician that you're talking about?
Whoever it is, they can't both be honest and dishonest.
Right? Someone can't be both honest and
dishonest. So whoever this, some politician that
you're talking about is. One Or the other of the two things that
you're saying about them has got to be false.
So you might think that the two claims are just made are inconsistent with each
other. But again the Venn diagram for those
claims helps us to see that they're not inconsistent with each other.
To say that some politician is honest is just to say that there is some
politician. Nevermind who.
Who is honest. And again to say that some politician is
dishonest is just to that there is some politician.
Never mind who. Who's dishonest.
But of course, those two claims could both be true.
There could be one politician who's honest, and another politician who's
dishonest. So once again, the venn diagrams for the
two claims help us to see that those two claims are consistent with each other.
They can both be true under certain circumstances.
It'll be useful for us to label the four kinds of propositions that we've been
talking about. Propositions of the form all f or g,
we're going to label with the uppercase letter A.
Propositions of the form no f is g, we're going to label with the upper case letter
E. Propositions of the form, sum F or G,
we're going to label with an uppercase letter.
I and propositions of the form sum f is not g we're going to label with an
uppercase letter o. Using this labeling scheme, we can now
describe immediate categorical inferences.
As inferences that have a single premise, of the form A, E, I or O.
These inferences occur very commonly in everyday life, and so we're going to
discuss the conditions under which they're valid.
Rahm, Rahm, wake up. .
Oh, wow. I was just having the most boring dream.
You know, some books are so boring. Yeah, but not all books.
Oh. Not all books are boring, but does that
mean that all books are not boring?