Introduction to Disjunctive and Hypothetical Syllogisms

Compare these three propositions:

1. Whales are mammals.

2. Either whales are mammals, or they are very large fish.

3. If whales are mammals, then they cannot breathe underwater.

The first is a categorical proposition. The second has
the structure p or q; it is a disjunctive proposition.
The third has the structure, "If p then q;" it is a hypothetical
proposition. Both #2 and #3 contain #1 as a component, but neither
of them asserts that #1 is true. As we saw in Chapter 4, these are
compound statements in which the components are expressed but not
asserted.

What the compound statement asserts is that a certain
relationship exists between the components. The disjunctive
proposition says that whales belong to one or the other of two
wider classes--without saying which one. The hypothetical
proposition tells us what the implication would be if whales are
mammals--without actually saying that they are.

When we use compound propositions in our reasoning, it is the
relationship among the components that is important. We don't
need to break down the component propositions into subject and
predicate terms or to identify their categorical form. So we
won't bother writing out the components. We'll just use single
lower-case letters like p, q, and r to stand for the components
as whole propositions.

As in the case of categorical reasoning, the goal of logical
analysis is to identify the logical forms of compound
propositions and then to identify which propositions are
equivalent, and which arguments are valid, in virtue of their
logical forms.