The Language of Argument

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C H A P T E R 6 ■ P r o p o s i t i o n a l L o g i c

There is, however, good reason for dealing with simple—trivially simple—

arguments at the start. The analytic approach to a complex issue is first to

break it down into subissues, repeating the process until we reach problems

simple enough to be solved. After these simpler problems are solved, we

can reverse the process and construct solutions to larger and more complex

problems. When done correctly, the result of such an analytic process may

seem dull and obvious—and it often is. The discovery of such a process, in

contrast, often demands the insight of genius.

The methods of analysis to be discussed here are formal in a specific

way. In Chapter 5, we gave the following argument as an example of a

valid argument: “All Senators are paid, and Sam is a Senator, so Sam is

paid.” The point could have been made just as well with many similar

examples: (a) “All Senators are paid, and Sally is a Senator, so Sally is

paid.” (b) “All plumbers are paid, and Sally is a plumber, so Sally is paid.”

(c) “All plumbers are dirty, and Sally is a plumber, so Sally is dirty.” These

arguments are all valid (though not all are sound). Thus, we can change

the person we are talking about, the group that we say the person is in,

and the property that we ascribe to the person and to the group, all with-

out affecting the validity of the argument at all. That flexibility shows that

the validity of this argument does not depend on the particular content of

its premises and conclusion. Instead, the validity of this argument results

solely from its form. Formal validity of this kind is what formal logics try

to capture.

Basic Propositional Connectives

Conjunction

The first system of formal logic that we will examine concerns propositional

(or sentential) connectives. Propositional connectives are terms that allow us to

build new propositions from old ones, usually combining two or more prop-

ositions into a single proposition. For example, given the propositions “John

is tall” and “Harry is short,” we can use the term “and” to conjoin them,

forming a single compound proposition: “John is tall and Harry is short.”

Let us look carefully at the simple word “and” and ask how it func-

tions. “And” is a curious word, for it does not seem to stand for anything,

at least in the way in which a proper name (“Churchill”) and a common

noun (“dog”) seem to stand for things. Instead of asking what this word

stands for, we can ask a different question: What truth conditions gov-

ern this connective? That is, under what conditions are propositions con-

taining this connective true? To answer this question, we imagine every

possible way in which the component propositions can be true or false.

Then, for each combination, we decide what truth value to assign to the

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