The Language of Argument

1 5 4

C H A P T E R 7 ■ C a t e g o r i c a l L o g i c

for we are not saying that all spies are aliens. Saying that some aliens are

spies does not rule out the possibility that some spies are homegrown. So we

need some new device to represent claims that two classes have at least some

members in common. We shall do this in the following way:

Aliens Spies

*

Aliens Spies

*

Aliens Spies

*

Here the asterisk indicates that there is at least one person who is both an alien

and a spy. Notice, by the way, that we are departing a bit from an everyday

way of speaking. “Some” is usually taken to mean “more than one”; here we

let it mean “at least one.” This makes things simpler and will cause no trouble,

so long as we remember that this is what we are using “some” to mean.

Given this new method of diagramming class relationships, we can im-

mediately think of other possibilities. The following diagram indicates that

there is someone who is an alien but not a spy. In more natural language, it

represents the claim that some aliens are not spies.

Next we can indicate that there is someone who is a spy but not an alien. More

simply, the claim is that some spies are not aliens, and it is represented like this:

These last three claims are, of course, compatible, because there might be

some aliens who are spies, some aliens who are not spies, and some spies

who are not aliens.

The Four Basic Categorical Forms

Although two classes can be related in a great many different ways, it is pos-

sible to examine many of these relationships in terms of four basic proposi-

tional forms:

A: All S is P. E: No S is P.

I: Some S is P. O: Some S is not P.

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