The Language of Argument

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Categorical Logic

In Chapter 6, we saw how validity can depend on the external connections among

propositions. This chapter will demonstrate how validity can depend on the internal

structure of propositions. In particular, we will examine two types of categorical

arguments—immediate inferences and syllogisms—whose validity or invalid-

ity depends on relations among the subject and predicate terms in their premises

and conclusions. Our interest in these kinds of arguments is mostly theoretical.

Understanding the theory of the syllogism deepens our understanding of validity,

even if this theory is, in some cases, difficult to apply directly to complex arguments

in daily life.

Beyond Propositional Logic

Armed with the techniques developed in Chapter 6, let’s look at the

following argument:

All squares are rectangles.

All rectangles have parallel sides.

∴ All squares have parallel sides.

It is obvious at a glance that the conclusion follows from the premises, so

this argument is valid. Furthermore, it seems to be valid in virtue of its form.

But it is not yet clear what the form of this argument is. To show the form of

this argument, we might try something of the following kind:

p ⊃ q

q ⊃ r

∴ p ⊃ r

But this is a mistake—and a bad mistake. We have been using the letters “p,”

“q,” and “r” as propositional variables—they stand for arbitrary propositions.

But the proposition “All squares are rectangles” is not itself composed of

two propositions. Nor does it contain “if,” “then” or any other propositional

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