The Language of Argument

1 5 2

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

connective. In fact, if we properly translate the above argument into the

language of propositional logic, we get the following result:

p

q

∴ r

This, of course, is not a valid argument form. But if we look back at the origi-

nal argument, we see that it is obviously valid. This shows that propositional

logic—however adequate it is in its own area—is not capable of explaining the

validity of all valid arguments. There is more to logic than propositional logic.

Categorical Propositions

To broaden our understanding of the notion of validity, we will examine

a modern version of a branch of logic first developed in ancient times—

categorical logic. Categorical logic concerns immediate inferences and

syllogisms that are composed of categorical propositions, so we need to

begin by explaining what a categorical proposition is.

In the argument above, the first premise asserts some kind of relationship be-

tween squares and rectangles; the second premise asserts some kind of relation-

ship between rectangles and things with parallel sides; finally, in virtue of these

asserted relationships, the conclusion asserts a relationship between squares and

things having parallel sides. Our task is to understand these relationships as

clearly as possible so that we can discover the basis for the validity of this argu-

ment. Again, we shall adopt the strategy of starting from simple cases and then

use the insights gained there for dealing with more complicated cases.

A natural way to represent the relationships expressed by the propositions in

an argument is through diagrams. Suppose we draw one circle standing for all

things that are squares and another circle standing for all things that are rectan-

gles. The claim that all squares are rectangles may be represented by placing the

circle representing squares completely inside the circle representing rectangles.

Rectangles Squares

Another way of representing this relationship is to begin with overlapping circles.

Squares 123 Rectangles

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