When we draft a contract or a set of rules, we often have to consider several conditions that give rise to a number of cases. We must avoid: 1. All _conflicts_, in which two statements that disagree apply to the same case. 2. All _loopholes_, in which there is a case not covered by any statement. If we have one statement or condition only, we have to consider 2 possible cases: the condition satisfied or the statement true; the condition not satisfied or the statement false. If we have 2 conditions, we have to consider 4 possible cases: true, true; false, true; true, false; false, false. If we have 3 conditions, we have to consider 8 possible cases one after the other (see Table 6). Table 6 CASE 1ST CONDITION 2ND CONDITION 3RD CONDITION 1 _T_ _T_ _T_ 2 _F_ _T_ _T_ 3 _T_ _F_ _T_ 4 _F_ _F_ _T_ 5 _T_ _T_ _F_ 6 _F_ _T_ _F_ 7 _T_ _F_ _F_ 8 _F_ _F_ _F_ Instead of _T_’s and _F_’s, we would ordinarily use _check-marks_ (✓) and _crosses_ (✕), which, of course, have the same meaning. We may consider and study each case individually. In any event, we must make sure that the proposed contract or set of rules covers all the cases without conflicts or loopholes. The number of possible cases that we have to consider doubles whenever one more condition is added. Clearly, it soon becomes too much work to consider each case individually, and so we must turn to a second method, thoughtful classifying and reasoning about classes of cases. Now suppose that the number of conditions increases: 4 conditions give rise to 16 possible cases; 5, 6, 7, 8, 9, 10, ··· conditions give rise to 32, 64, 128, 256, 512, 1024, ··· cases respectively. Because of the large number of cases, we soon begin to make mistakes while reasoning about classes of cases. We need a more efficient way of knowing whether all cases are covered properly. LOGICAL-TRUTH CALCULATION BY ALGEBRA One of the more efficient ways of reasoning is often called the _algebra of logic_. This algebra is a part of a new science called _mathematical logic_. Mathematical logic is a science that has the following characteristics: It studies chiefly nonnumerical reasoning. It seeks accurate meanings and necessary consequences. Its chief instruments are efficient symbols. Mathematical logic studies especially the logical relations expressed in such words as “or,” “and,” “not,” “else,” “if,” “then,” “only,” “the,” “of,” “is,” “every,” “all,” “none,” “some,” “same,” “different,” etc. The algebra of logic studies especially only the first seven of these words.