Page:Popular Science Monthly Volume 13.djvu/219

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207
ILLUSTRATIONS OF THE LOGIC OF SCIENCE.

might be, and, for instance, not so much so as a world of pure chance would be.

But we can never get to the bottom of this question until we take account of a highly-important logical principle[1] which I now proceed to enounce. This principle is that any plurality or lot of objects whatever have some character in common (no matter how insignificant) which is peculiar to them and not shared by anything else. The word "character" here is taken in such a sense as to include negative characters, such as incivility, inequality, etc., as well as their positives, civility, equality, etc. To prove the theorem, I will show what character any two things, A and B, have in common, not shared by anything else. The things, A and B, are each distinguished from all other things by the possession of certain characters which may be named A-ness and B-ness. Corresponding to these positive characters, are the negative characters un-A-ness, which is possessed by everything except A, and un-B-ness, which is possessed by everything except B. These two characters are united in everything except A and B; and this union of the characters un-A-ness and un-B-ness makes a compound character which may be termed A-B-lessness. This is not possessed by either A or B, but it is possessed by everything else. This character, like every other, has its corresponding negative un-A-B-lessness, and this last is the character possessed by both A and B, and by nothing else. It is obvious that what has thus been shown true of two things is, mutatis mutandis, true of any number of things. Q. E. D.

In any world whatever, then, there must be a character peculiar to each possible group of objects. If, as a matter of nomenclature, characters peculiar to the same group be regarded as only different aspects of the same character, then we may say that there will be precisely one character for each possible group of objects. Thus, suppose a world to contain five things, α, β, γ, δ, ε. Then it will have a separate character for each of the 31 groups (with non-existence making up 32 or 25) shown in the following table:

Table II.
αβ αβγ αβγδ αβγδε
α αγ αβδ αβγε
β αδ αβε αβδε
γ αε αγδ αγδε
δ βγ αγε βγδε
ε βδ αδε
βε βγδ
γδ βγε
γε βδε
δε γδε
 

This shows that a contradiction is involved in the very idea of a chance-world, for in a world of 32 things, instead of there being only 35

  1. This principle was, I believe, first stated by Mr. De Morgan.