Let p be the statement "You drink Pepsi." Let q be the statement "You are happy." For each row of the truth table we will consider the promise "If you drink Pepsi, then you will be happy."
| p | q | 
|
| T | T | ? |
In the first row of the truth table, p is true (so you did drink the Pepsi) and q is also true (so you were happy) in that case, the promise "If you drink Pepsi, then you are happy" was clearly not a lie, so the statement
is TRUE:| p | q |
|
| T | T | T |
Now we move on to the second row of the truth table:
| p | q |
|
| T | T | T |
| T | F | ? |
In this case p is true while q is false; this means that you did drink the Pepsi, but you weren't happy. In such a case, it is obvious that the promise "If you drink Pepsi, then you are happy" was a lie, or a broken promise:
| p | q |
|
| T | T | T |
| T | F | F |
Now we go to the third row of the truth table:
| p | q |
|
| T | T | T |
| T | F | F |
| F | T | ? |
In this case p is false while q is true; this means that you didn't drink the Pepsi but you were happy anyway. It is vital that you understand that in this case the promise "If you drink Pepsi, then you are happy" has not been broken, since the promise is contingent on your drinking the Pepsi. If you don't drink the Pepsi, it doesn't matter whether you are happy or not: you can't accuse the promiser of having lied. We say that the statement is "vacuously true" in a situation like this.
| p | q |
|
| T | T | T |
| T | F | F |
| F | T | T |
Finally we move to the last row of table.
| p | q |
|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | ? |
In this case p is false while q is false. This means that you didn't drink the Pepsi, and you weren't happy, either. This situation is similar to the case in the third row: since you didn't drink the Pepsi, the promise "If you drink Pepsi, then you are happy" can't possibly be a lie. The statement is vacuously true:
| p | q |
|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Although we filled in this truth table by referring to a specific "if...then" statement, the result is general. It shows that the only situation in which an "if...then" statement is FALSE is when the antecendant is true while the consequent is false (the second row of this table).
| p | q |
|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |