The Language of Argument

1 6 0

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

But for the corresponding I proposition to be true, there must be something in

the central region. Thus, they cannot both be true. They also cannot both be

false. The only way for an E proposition to be false is for there to be some-

thing in the central region, but then the corresponding I proposition is not

false but true. The only way for the I proposition to be false is if there is

nothing in the central region, and then the E proposition is not false but

true. Thus, they cannot both be true, and they cannot both be false. In other

words, they always have opposite truth values. This relation is described by

saying that these propositions are contradictories.

More generally, we can produce a diagram for the denial of a proposition

by a simple procedure. The only information given in a Venn diagram is

represented either by shading out some region, thereby indicating that noth-

ing exists in it, or by putting an asterisk in a region, thereby indicating that

something does exist in it. We are given no information about regions that

are unmarked. To represent the denial of a proposition, we simply reverse

the information in the diagram. That is, where there is an asterisk, we put

in shading; where there is shading, we put in an asterisk. Everything else is

left unchanged. Thus, we can see at once that corresponding E and I propo-

sitions are denials of one another, so they must always have opposite truth

values. This makes them contradictories.

The same relation exists between an A proposition and its corresponding

O proposition. Consider their forms:

S P S P

A: All S is P. O: Some S is not P.

The diagram for an A proposition has shading exactly where the correspond-

ing O proposition has an asterisk, and they contain no other information.

Consequently, corresponding A and O propositions cannot both be false and

cannot both be true, so they are contradictories.

1. Is an A proposition a contradictory of its corresponding E proposition?
   Why or why not?


2. Is an I proposition a contradictory of its corresponding O proposition?
   Why or why not?


3. If one proposition is the contradictory of another, is the latter always the
   contradictory of the former? Why or why not?

Exercise III

97364_ch07_ptg01_151-176.indd 160 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.