Subsection 2.7.1 Introduction
After we build a truth table for a logical expression, we can read off the values in its last column. It is that column that tells us the truth values for the entire expression as a function of the truth values of its variables. Think of each possible set of input values (represented as a single row) as a “possible world” that might occur. Then the whole column gives a picture of what might happen given any particular world in which we might find ourselves. All sequences of values are possible in that last column. But a few are particularly significant and we give names to expressions that exhibit those values.
Consider the following possible final columns for a truth table with three variables:
| T |
| T |
| T |
| T |
| T |
| T |
| T |
| T |
All values are T.
| T |
| F |
| F |
| T |
| F |
| T |
| F |
| T |
At least one value is T.
| F |
| F |
| F |
| F |
| F |
| F |
| F |
| F |
All values are F.
When all values are T, we’ll say that the corresponding formula is a tautology or that it is valid. In other words, the formula will always be true, no matter what.
When at least one value is T, we’ll say that the corresponding formula is satisfiable. In other words, there’s some circumstance(s) in which it may be true.
When all values are F, we’ll say that the corresponding formula is a contradiction. We’ll also say that it is unsatisfiable. In other words, “no way”.