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. We can give a simple numerical example of Boolean algebra and how it can calculate logical truth. Suppose that we take the truth value of a statement as 1 if it is true and 0 if it is false. Now we have numbers 1 and 0 instead of letters _T_ and _F_. Since they are numbers, we can add them, subtract them, and multiply them. We can also make up simple numerical formulas that will let us calculate logical truth. If _P_ and _Q_ are statements, and if _p_ and _q_ are their truth values, respectively, we have Table 7. Table 7 STATEMENT TRUTH VALUE NOT-_P_ 1 - _p_ _P_ AND _Q_ _pq_ _P_ OR _Q_ _p_ + _q_ - _pq_ IF _P_, THEN _Q_ 1 - _p_ + _pq_ _P_ IF AND ONLY IF _Q_ 1 - _p_ - _q_ + 2_pq_ _P_ OR ELSE _Q_ _p_ + _q_ - 2_pq_ For example, suppose that we have two statements _P_ and _Q_: _P_: John Doe is eligible for insurance. _Q_: John Doe requires a medical examination. To test that the truth value of “_P_ OR _Q_” is _p_ + _q_-_pq_, let us put down the four cases, and calculate the result (see Table 8). Table 8 _p_ _q_ | _p_ + _q_ - _pq_ | 1 1 | 1 + 1 - 1 = 1 0 1 | 0 + 1 - 0 = 1 1 0 | 1 + 0 - 0 = 1 0 0 | 0 + 0 - 0 = 0 Now we know that _P_ or _Q_ is true if and only if either one or both of _P_ and _Q_ are true, and thus we see that the calculation is correct. The algebra of logic (see also Supplement 2) is a more efficient way of calculating logical truth. But it is still a good deal of work to use the algebra. For example, if we have 10 conditions, we shall have 10 letters like _p_, _q_ to handle in calculations. Thus we need a still more efficient way. CALCULATION OF CIRCUITS BY THE ALGEBRA OF LOGIC