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.
The great thinkers of ancient Greece first studied the problems of logical reasoning as these problems turned up in philosophy, psychology, and debate. Aristotle originated what was called formal logic. This was devoted mainly to variations of the logical pattern shown above called the syllogism. In the last 150 years, the fine symbolic techniques developed by mathematicians were applied to the problems of the calculation of logical truth, and the result was mathematical logic, much broader and much more powerful than formal logic. A milestone in the development of mathematical logic was The Laws of Thought, written by George Boole, a great English mathematician, and published in 1854. Boole introduced the branch of mathematical logic called the algebra of logic, also called Boolean algebra. In late years, all the branches of mathematical logic have been improved and made easier to use.