When contemplating the properties of this Alphabet I am often inclined to think that Pythagoras perceived the deep logical importance of duality; for while unity was the symbol of identity and harmony, he described the number two as the origin of contrasts, or the symbol of diversity, division and separation. The number four, or the Tetractys, was also regarded by him as one of the chief elements of existence, for it represented the generating virtue whence come all combinations. In one of the golden verses ascribed to Pythagoras, he conjures his pupil to be virtuous:[77]
“By him who stampt The Four upon the Mind,
The Four, the fount of Nature’s endless stream.”
Now four and the higher powers of duality do represent in this logical system the numbers of combinations which can be generated in the absence of logical restrictions. The followers of Pythagoras may have shrouded their master’s doctrines in mysterious and superstitious notions, but in many points these doctrines seem to have some basis in logical philosophy.
The Logical Slate.
To a person who has once comprehended the extreme significance and utility of the Logical Alphabet the indirect process of inference becomes reduced to the repetition of a few uniform operations of classification, selection, and elimination of contradictories. Logical deduction, even in the most complicated questions, becomes a matter of mere routine, and the amount of labour required is the only impediment, when once the meaning of the premises is rendered clear. But the amount of labour is often found to be considerable. The mere writing down of sixty-four combinations of six letters each is no small task, and, if we had a problem of five premises, each of the sixty-four combinations would have to be examined in connection with each premise. The requisite comparison is often of a very tedious character, and considerable chance of error intervenes.
I have given much attention, therefore, to lessening both the manual and mental labour of the process, and I shall describe several devices which may be adopted for saving trouble and risk of mistake.
In the first place, as the same sets of combinations occur over and over again in different problems, we may avoid the labour of writing them out by having the sets of letters ready printed upon small sheets of writing-paper. It has also been suggested by a correspondent that, if any one series of combinations were marked upon the margin of a sheet of paper, and a slit cut between each pair of combinations, it would be easy to fold down any particular combination, and thus strike it out of view. The combinations consistent with the premises would then remain in a broken series. This method answers sufficiently well for occasional use.
A more convenient mode, however, is to have the series of letters shown on p. [94], engraved upon a common school writing slate, of such a size, that the letters may occupy only about a third of the space on the left hand side of the slate. The conditions of the problem can then be written down on the unoccupied part of the slate, and the proper series of combinations being chosen, the contradictory combinations can be struck out with the pencil. I have used a slate of this kind, which I call a Logical Slate, for more than twelve years, and it has saved me much trouble. It is hardly possible to apply this process to problems of more than six terms, owing to the large number of combinations which would require examination.
Abstraction of Indifferent Circumstances.
There is a simple but highly important process of inference which enables us to abstract, eliminate or disregard all circumstances indifferently present and absent. Thus if I were to state that “a triangle is a three-sided rectilinear figure, either large or not large,” these two alternatives would be superfluous, because, by the Law of Duality, I know that everything must be either large or not large. To add the qualification gives no new knowledge, since the existence of the two alternatives will be understood in the absence of any information to the contrary. Accordingly, when two alternatives differ only as regards a single component term which is positive in one and negative in the other, we may reduce them to one term by striking out their indifferent part. It is really a process of substitution which enables us to do this; for having any proposition of the form