## Computing Truth Values

We have seen that a compound statement is truth-functional: Its truth or falsity is a function of the truth or falsity of its component statements. For compound statements with just one connective, the truth table for that connective tells us how to make the computation. But we can construct truth tables for more complex statements with more than one connective.

Consider the conditional statement A(AB). Because the main connective is the conditional, it has the basic logical form pq, and its truth value is a function of p and q. But q in this case is itself a compound statement, AB. So we need to compute this element first.

In general, we work from the inside out, starting with the connectives that apply directly to the components and doing the main connective last.

 A B AB A(AB) T T T T F T F T T F F F F F F T
If there are more components, we need more lines -- with three components, for example, we need eight lines -- but the procedure is the same. Let's construct the truth table for the statement ~A(B v C).

 A B C ~A (B v C) ~A(B v C) T T T F T F T T F F T F T F T F T F T F F F F F F T T T T T F T F T T T F F T T T T F F F T F F
To construct a truth table for a statement with more than one connective:

1. Make a column for each component statement, with enough rows for each possible combination of truth values among the components.

2. Identify the connectives that apply directly to component statements. On each row, determine the truth value of the statement involving just that connective, and enter that truth value in a column under the connective.

3. Repeat step #2 until you reach the main connective (the one outside all parentheses). The truth values in its column are the truth values of the statement as a whole.

Comprehension Questions

1 The column of the following truth table under A>B (from top to bottom) is:
(A>B) v (~A . C)
 A B C A>B ~A ~A . C (A>B) v (~A . C) T T T ? T T F ? T F T ? T F F ? F T T ? F T F ? F F T ? F F F ?
a) F F F F T T T T
b) F F F F T F T F
c) T T F F T T T T
d) T T F F T T T T
2 The column of the following truth table under ~A (from top to bottom) is:
(A>B) v (~A . C)
 A B C A>B ~A ~A . C (A>B) v (~A . C) T T T ? T T F ? T F T ? T F F ? F T T ? F T F ? F F T ? F F F ?
a) T F T F T F T F
b) F F F F T T T T
c) T T F F T T T T
d) T T F F T T F F
3 The column of the following truth table under (~A . C) (from top to bottom) is:
(A>B) v (~A . C)
 A B C A>B ~A ~A . C (A>B) v (~A . C) T T T ? T T F ? T F T ? T F F ? F T T ? F T F ? F F T ? F F F ?
a) F F F F T F T F
b) F F F F T T T T
c) T T F F T T T T
d) T T F F T T F T
4 The column of the following truth table under (A>B) v (~A . C) (from top to bottom) is:
(A>B) v (~A . C)
 A B C A>B ~A ~A . C (A>B) v (~A . C) T T T ? T T F ? T F T ? T F F ? F T T ? F T F ? F F T ? F F F ?
a) F F F F T F T F
b) F F F F T T T T
c) T T F F T F T F
d) T T F F T T T T