Page:Chance, love, and logic - philosophical essays (IA chancelovelogicp00peir 0).pdf/155

 *tive 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, [Greek: a], [Greek: b], [Greek: g], [Greek: d], [Greek: e]. Then it will have a separate character for each of the 31 groups (with non-existence making up 32 or 2^5) shown in the following table:

[Greek: ab]   [Greek: abg]    [Greek: abgd]    [Greek: abgde] [Greek: a]   [Greek: ag]    [Greek: abd]    [Greek: abge] [Greek: b]   [Greek: ad]    [Greek: abe]    [Greek: abde] [Greek: g]   [Greek: ae]    [Greek: agd]    [Greek: agde] [Greek: d]   [Greek: bg]    [Greek: age]    [Greek: bgde] [Greek: e]   [Greek: bd]    [Greek: ade] [Greek: be]   [Greek: bgd] [Greek: gd]   [Greek: bge] [Greek: ge]   [Greek: bde] [Greek: de]   [Greek: gde]