Page:The World and the Individual, Second Series (1901).djvu/103

78 a geometrical system of points. Mr. Kempe shows, in fact, that the system of points possible in a space of any number of dimensions differs from the system of logical classes possible in any “Universe of Discourse,” merely by the addition of a single new property, viz. that which is geometrically expressed by saying that two straight lines have only one point in common. This very striking identification of laws belonging to the kinds of orderly arrangement present in such different realms as a system of ideal logical classes and a system of points in space is associated, in Mr. Kempe’s discussion, with an observation regarding the nature of the generalized relation between, which I here propose to use, although I have no time to state either fully or very exactly the reasoning that I found upon this observation.

If one visible point were between two others on a line, and if all three were (to fix our ideas) luminous points, and if you went just far enough away from the line to be unable longer to observe the place of the point a as diverse from that of the point b, so that the two blended to your eye in one luminous point, then obviously m, the intermediate point, would blend with both of them. Just so, however, if you abstract from the difference between the classes ɑ and b, while still recognizing, in a measure, the possibility of objects that, as a fact, belong to one or to another of them, then, so long as you thus regard the two classes as equivalent, it makes no conscious difference to you whether an object is in the class ɑ, or in the class b, or both at once. So that you do not observe, in that case, Mr. Kempe’s intermediate class m as a class different