- 1. Statement 1 true, statement 2 true.
- 2. Statement 1 false, statement 2 true.
- 3. Statement 1 true, statement 2 false.
- 4. Statement 1 false, statement 2 false.
On the right-hand side of this table, we shall put down the truth value of statement 5. Statement 5 is true if both statements 1 and 2 are true; statement 5 is false in the other cases. We know this from our common everyday experience with the meaning of “and” between statements. So we can set up the truth table, and our rule for calculation of logical truth, in the case of and, is shown on [Table 3].
Table 3
| 1 | 2 | 1 and 2 = 5 |
| T | T | T |
| F | T | F |
| T | F | F |
| F | F | F |
“and” and the other words and phrases joining together the original two statements to make new statements are called connectives, or logical connectives. The connectives that we have illustrated in statements 7 to 10 are: or, if ··· then, if and only if, or else.
[Table 4] shows the truth table that applies to statements 7, 8, 9, and 10. This truth table expresses the calculation of the logical truth or falsity of these statements.
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.