The Language of Argument

1 5 3

C a t e g o r i c a l P r o p o s i t i o n s

We then shade out the portions of the circles in which nothing exists, ac-

cording to the proposition we are diagramming. If all squares are rectangles,

there is nothing that is a square that is not a rectangle—that is, there is noth-

ing in region 1. So our diagram looks like this:

Squares 123 Rectangles

Either method of representation seems plausible. Perhaps the first seems more

natural. We shall, however, use the system of overlapping circles, because they

will work better when we get to more complex arguments. They are called Venn

diagrams, after their inventor, John Venn, a nineteenth-century English logician.

Having examined one relationship that can exist between two classes,

it is natural to wonder what other relationships might exist. Going to the

opposite extreme from our first example, two classes may have nothing in

common. This relationship could be expressed by saying, “All triangles

are not squares,” but it is more common and natural to say, “No trian-

gles are squares.” We diagram this claim by indicating that there is nothing

in the overlapping region of things that are both triangles and squares:

Triangles 123 Squares

Aliens 123 Spies

This is one of the relationships that could not be diagrammed by putting one

circle inside another. (Just try it!)

In these first two extreme cases, we have indicated that one class is either

completely included in another (“All squares are rectangles”) or completely

excluded from another (“No triangles are squares”). Sometimes, however,

we claim only that two classes have at least some things in common. We

might say, for example, “Some aliens are spies.” How shall we indicate this

relationship in the following diagram?

In this case, we do not want to cross out any whole region. We do not want to

cross out region 1 because we are not saying that all aliens are spies. Plainly,

we do not want to cross out region 2, for we are actually saying that some per-

sons are both aliens and spies. Finally, we do not want to cross out region 3,

97364_ch07_ptg01_151-176.indd 153 15/11/13 10:29 AM

some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materiallyCopyright 201^3 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights,

affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.