10. John Doe is eligible for insurance or else he requires a medical examination.
Now clearly it is troublesome to repeat quantities of words when we are interested only in the way that “and,” “or,” “else,” “not,” “if,” “then,” “only” occur. So, let us use just 1 and 2 for the two original statements, remembering that “1 and 2” means here “statement 1 AND statement 2” and does not mean 1 plus 2. Then we have:
| 3: | not-1 |
| 4: | not-2 |
| 5: | 1 and 2 |
| 6: | 1 and 1 |
| 7: | 1 or 2 |
| 8: | if 1, then 2 |
| 9: | 1 if and only if 2 |
| 10: | 1 or else 2 |
Here then are some simple logical patterns that we can make.
CALCULATION OF LOGICAL TRUTH
Now what can we find out about the logical truth of statements 3 to 10? If we know something about the truth or falsity of statements 1 and 2, what will logically follow about the truth or falsity of statements 3 to 10? In other words, how can we calculate the logical truth of statements 3 to 10, given the truth or falsity of statements 1 and 2?
For example, 3 is not-1; that is, statement 3 is the negative or the denial of statement 1. It follows logically that, if 1 is true, 3 is false; if 1 is false, 3 is true. Suppose that we use T for logically true and F for logically false. Then we can show our calculation of the logical truth of statement 3 in Table 1.
| Table 1 | Table 2 | |||
| 1 | not-1 = 3 | 2 | not-2 = 4 | |
| T | F | T | F | |
| F | T | F | T | |
Our rule for calculation is: For T put F; for F put T. Of course, exactly the same rule applies to statements 2 and 4 ([see Table 2]). The T and F are called truth values. Any meaningful statement can have truth values. This type of table is called a truth table. For any logical pattern, we can make up a truth table.
Let us take another example, “and.” Statement 5 is the same as statement 1 and statement 2. How can we calculate the logical truth of statement 5? We can make up the same sort of a table as before. On the left-hand side of this table, there will be 4 cases: