Subsection 2.2.1 Introduction
An individual proposition has a truth value, which can be either true (T) or false (F). A compound statement derives its truth value from the truth values of its components. We’d like to have a simple tool that lets us consider a compound statement and then see what truth value it would have, given any of the possible combinations of truth values of its components.
Truth tables let us do precisely this. We can use them to define the individual operators that we want to use. And we can simply extend them as a way to compute the truth value of an arbitrarily complex statement.
To simplify the writing out of truth tables (and do a tiny bit of abstraction), we will use variable names, P and q , for statements. Also, we’ll let T stand for “true” and F stand for “false”.
A truth table for a logical statement has one title row. Next, it has one row for each logically possible combination of truth values of the variables in that statement. It has one column for each constituent statement, including the smallest ones (the individual variables), all the intermediate ones, and the complete statement itself.
For example, here’s the outline of a truth table for a statement that contains two variables and uses two operators:
| \(\text {p} \) | \(\text {q} \) | \(\text { p and q }\) | \(\text {not (p and q)}\) |
|---|---|---|---|
| T | T | ||
| T | F | ||
| F | T | ||
| F | F |
Notice that, after the title row, which we won’t count, there are four rows, each corresponding to one of the four logically possible combinations of the two truth values.
Big Idea
Truth tables are a powerful tool for defining the meaning of Boolean logic statements. We’ll see in the next chapter that they’re also a tool for creating Boolean logic proofs.
In the next several sections, we’ll see how to fill in truth tables, starting with definitions of the individual operators.