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: 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_