Table 4
| 1 or 2 | if 1, then 2 | 1 if and only if 2 | 1 or else 2 | ||
| 1 | 2 | = 7 | = 8 | = 9 | = 10 |
| T | T | T | T | T | F |
| F | T | T | T | F | T |
| T | F | T | F | F | T |
| F | F | F | T | T | F |
The “or” (as in statement 7) that is defined in the truth table is often called the inclusive “or” and means “and/or.” Statement 7, “1 or 2,” is considered to be the same as “1 or 2 or both.” There is another “or” in common use, often called the exclusive “or,” meaning “or else” (as in statement 10). Statement 10, “1 or else 2,” is the same as “1 or 2 but not both” or “either 1 or 2.” In ordinary English, there is some confusion over these two “or’s.” Usually we rely on the context to tell which one is intended. Of course, such reliance is not safe. Sometimes we rely on a necessary conflict between the two statements connected by “or” which prevents the “both” case from being possible. In Latin the two kinds of “or” were distinguished by different words, vel meaning “and/or,” and aut meaning “or else.”
The “if ··· then” that is defined in the truth table agrees with our usual understanding that (1) when the “if clause” is true, the “then clause” must be true; and (2) when the “if clause” is false, the “then clause” may be either true or false. The “if and only if” that is defined in the truth table agrees with our usual understanding that (1) if either clause is true, the other is true; and (2) if either clause is false, the other is false.
In statement 6, there are only two possible cases, and the truth table is shown in Table 5.