Symbolic Logic
Concurrently with the great strides being made with mechanical computers that could handle mathematics, much work was also being done with the formalizing of the logic. As hinted vaguely in the syllogisms of the early philosophers, thinking did seem amenable to being diagrammed, much like grammar. Augustus De Morgan devised numerical logic systems, and George Boole set up the logic system that has come to be known as Boolean algebra in which reasoning becomes positive or negative terms that can be manipulated algebraically to give valid answers.
John Venn put the idea of logic into pictures, and simple pictures at that. His symbology looks for all the world like the three interlocking rings of a well-known ale. These rings stand for the subject, midterm, and predicate of the older Aristotelian syllogism. By shading the various circles according to the major and minor premises, the user of Venn circles can see the logical result by inspection. Implicit in the scheme is the possibility of a mechanical or electrical analogy to this visual method, and it was not long until mathematicians began at least on the mechanical kind. Among these early logic mechanizers, surprisingly, was Lewis Carroll who of course was mathematician Charles L. Dodgson before he became a writer.
Carroll, who was a far busier man than most of us ever guess, marketed a “Game of Logic,” with a board and colored cardboard counters that handled problems like the following:
All teetotalers like sugar.
No nightingale drinks wine.
By arranging the counters on Carroll’s game board so that: All M are X, and No Y is not-M, we learn that No Y is not-X! This tells the initiate logician that no nightingale dislikes sugar; a handy piece of information for bird-fancier and sugar-broker alike.
Charles, the third Earl Stanhope, was only slightly less controversial than his prime minister, William Pitt. Scientifically he was far out too, writing books on electrical theory, inventing steamboats, microscopes, and printing presses among an odd variety of projects; he also became interested in mechanical logic and designed the “Stanhope Demonstrator,” a contrivance like a checkerboard with sliding panels. By properly manipulating the demonstrator he could solve such problems as:
Eight of ten children are bright.
Four of these children are boys.
What are the minimum and maximum number of bright boys? A simple sliding of scales on the Stanhope Demonstrator shows that two must be boys and as many as four may be. This clever device could also work out probability problems such as how many heads and tails will come up in so many tosses of a coin.
In 1869 William S. Jevons, an English economist and expert logician, built a logic machine. His was not the first, of course, but it had a unique distinction in that it solved problems faster than the human brain could! Using Boolean algebra principles, he built a “logical abacus” and then even a “logical piano.” By simply pressing the keys of this machine, the user could make the answer appear on its face. It is of interest that Jevons thought his machine of no practical use, since complex logical questions seldom arose in everyday life! Life, it seems, was simpler in 1869 than it is today, and we should be grateful that Jevons pursued his work through sheer scientific interest.
More sophisticated than the Jevons piano, the logic machine invented in America by Allan Marquand could handle four terms and do problems like the following:
There are four schoolgirls, Anna, Bertha, Cora, and Dora.
When Anna or Bertha, or both, remain home, Cora is at home.
When Bertha is out, Anna is out.
Whenever Cora is at home, Anna is too.
What can we tell about Dora?
The machine is smart enough to tell us that when Dora is at home the other three girls are all at home or out. The same thing is true when Dora is out.