The Language of Argument

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B a s i c P r o p o s i t i o n a l C o n n e c t i v e s

determined by the truth value of the original propositions. For this reason,

these connectives are called truth-functional connectives. (Of course, with ne-

gation, we start with a single proposition, so there are not really two things

to connect.) For example, suppose that “A” and “B” are two true proposi-

tions and “G” and “H” are two false propositions. We can then determine the

truth values of more complex propositions built from them using conjunc-

tion, disjunction, and negation. Sometimes the correct assignment is obvious

at a glance:

A & B True

A & G False

~A False

~G True

A ∨ H True

G ∨ H False

~A & G False

As noted earlier, parentheses can be used to distinguish groupings. Some-

times the placement of parentheses can make an important difference, as in

the following two expressions:

~A & G

~(A & G)

Notice that in the first expression the negation symbol applies only to the

proposition “A,” whereas in the other expression it applies to the entire

proposition “A & G.” Thus, the first expression above is false, and the sec-

ond expression is true. Only the second expression can be translated as “Not

both A and G.” Both of these expressions are different from “~A & ~G,”

which means “Neither A nor G.”

As expressions become more complex, we reach a point where it is no

longer obvious how the truth values of the component propositions deter-

mine the truth value of the entire proposition. Here a regular procedure is

helpful. The easiest method is to fill in the truth values of the basic proposi-

tions and then, step-by-step, make assignments progressively wider, going

from the inside out. For example:

~((A^ ∨ G) & ~(~H & B))

~((T ∨ F) & ~(~F & T))

~((T ∨ F) & ~(T & T))

~(T & ~(T))

~(T & F)

~(F)

T

With a little practice, you can master this technique in dealing with other

very complex examples.

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