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.


Comprehension Questions
1 Identify the following proposition.

No true pianist studies music theory.

a) categorical
b) hypothetical
c) disjunctive
2 Identify the following proposition.

If I were you, I would not graduate without taking logic.

a) categorical
b) hypothetical
c) disjunctive
3 Identify the following proposition.

Either I take Professor Smith for philosophy, or I take Professor Allen for history.

a) categorical
b) hypothetical
c) disjunctive

Disjunctive Syllogisms

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