“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 IF AND 1 OR 2 IF 1, THEN 2 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. Table 5 1 | 1 AND 1 = 6 | _T_ | _T_ _F_ | _F_ We know that 6 is true if and only if 1 is true. In other words, the statement “1 AND 1 IF AND ONLY IF 1” is true, no matter what statement 1 may refer to. It is because of this fact that we never use a statement in the form “1 and 1”: it can always be replaced by the plain statement “1.” LOGICAL-TRUTH CALCULATION BY EXAMINING CASES AND REASONING Now you may say that this is all very well, but what good is it? Almost anybody can use these connectives correctly and certainly has had a great deal of practice using them. Why do we need to go into truth values and truth tables?