Unfortunately it was two centuries later before the importance of his findings was recognized and an explanation of their potential begun. In England, Sir William Hamilton began to refine the old syllogisms, and is known for his “quantification of the predicate.” Augustus De Morgan, also an Englishman, moved from the quantification of the predicate to the formation of thirty-two rules or propositions that result. The stage was set now for the man who has come to be known as the father of symbolic logic. His name was George Boole, inventor of Boolean algebra.

In 1854, Boole published “An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities.” In an earlier pamphlet, Boole had said, “The few who think that there is that in analysis which renders it deserving of attention for its own sake, may find it worth while to study it under a form in which every equation can be solved and every solution interpreted.” He was a mild, quiet man, though nonconformist religiously and socially, and his “Investigation” might as well have been dropped down a well for all the immediate splash it made in the scientific world. It was considered only academically interesting, and copies of it gathered dust for more than fifty years.

Only in 1910 was the true importance given to Boole’s logical calculus, or “algebra” as it came to be known. Then Alfred North Whitehead and Bertrand Russell made the belated acknowledgment in their Principia Mathematica, and Russell has said, “Pure mathematics was discovered by Boole, in a work he called ‘The Laws of Thought.’” While his praise is undoubtedly exaggerated, it is interesting to note the way in which mathematics and thought are considered inseparable. In 1928, the first text on the new algebra was published. The work of Hilbert and Ackermann, Mathematical Logic, was printed first in German and then in English.

What was the nature of this new tool for better thinking that Boole had created? Its purpose was to make possible not merely precise, but exact analytical thought. Historically we think in words, and these words have become fraught with semantic ditches, walls, and traps. Boole was thinking of thought and not mathematics or science principally when he developed his logic algebra, and it is indicative that symbolic logic today is often taught by the philosophy department in the university.

Russell had hinted at the direction in which symbolic logic would go, and it was not long before the scientist as well as the mathematician and logician did begin to make use of the new tool. One pioneer was Shannon, mentioned in the chapter on history. In 1938, Claude Shannon was a student at M.I.T. He would later make scientific history with his treatise on and establishment of a new field called information theory; his early work was titled “A Symbolic Analysis of Relay and Switching Circuits.” In it he showed that electrical and electronic circuitry could best be described by means of Boolean logic. Shannon’s work led to great strides in improving telephone switching circuits and it also was of much importance to the designer of digital computers. To see why this is so, we must now look into Boolean algebra itself. As we might guess, it is based on a two-valued logic, a true-false system that exactly parallels the on-off computer switches we are familiar with.

The Biblical promise “Ye shall know the truth, and the truth shall make you free” applies to our present situation. The best way to get our feet wet in the Boolean stream is to learn its so-called “truth tables.”

Conjunctive Boolean Operation

A and B equal CA B C
(A · B = C)———
0 0 0
1 0 0
0 1 0
1 1 1

Disjunctive Boolean Operation

A or B equals CA B C
(Ā ∨ B = C)———
0 0 0
1 0 1
0 1 1
1 1 1