Truth Value

truth value of the new proposition is always determined by the truth value of the original propositions. For this reason, these connectives are called truth- functional connectives. (Of course, with negation, we start with a single proposi- tion, so there are not really two things to connect.) For example, suppose that “A” and “B” are two true propositions and “G” and “H” are two false propo- sitions. We can then determine the truth values of more complex propositions built from them using conjunction, 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 one expression the negation symbol applies only to the propo- sition “A,” whereas in the other expression it applies to the entire proposition “A & G.” Thus, the first expression above is false, and the second 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 “Nei- ther 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 determine 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 propositions and then, step-by-step, make assignments progressively wider, going from the in- side 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.