The fourth dimension

102 THE FOURTH DIMENSION

in the third representative cube is the one in the second wall to the right for the major premiss, the third wall from the front for the minor premiss, and the top layer for the conclusion.

It is easily seen that in the diagram this cube is marked, and so with all the valid conclusions. The regions marked in the total region show which combinations of the four variables, major premiss, minor premiss, figure, and conclusion exist.

That is to say, we objectify all possible conclusions, and build up an ideal manifold, containing all possible combinations of them with the premisses, and then out of this we eliminate all that do not satisfy the laws of logic. The residue is the syllogism, considered as a canon of reasoning.

Looking at the shape which represents the totality of the valid conclusions, it does not present any obvious symmetry, or easily characterisable nature. A striking configuration, however, is obtained, if we project the fourdimensional figure obtained into a three-dimensional one; that is, if we take in the base cube all those cubes which have a marked space anywhere in the series of four regions which start from that cube.

This corresponds to making abstraction of the figures, giving all the conclusions which are valid whatever the figure may be.

Proceeding in this way we obtain the arrangement of marked cubes shown in fig. 57. We see that the valid conclusions are arranged almost symmetrically round one cube—the one on the top of the column starting from AAA. ‘There is one breach of continuity however in this scheme. One cube is unmarked, which if marked would give symmetry. It is the one which would be denoted by the

Fig. 57.