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If I understand correctly, you define: hole(a,P): neither a nor any bit of a is P atom(a,P): a is P and for any split of a at most one bit of a is P. substance(a,P): a is P, some bit of a is P and no bit of a is P-atom. group(a,P): anything else. You then conclude:
However, unlike a hole, a group does contain a bit of which P holds, since it contains an atom.
But the definition allows for groups that don't contain atoms. For example, something that is not P but has some bit that is P-substance. To be more concrete: Take the property "... is a solid cube". Then a solid sphere would be a group: It is not a hole because some bits of it are solid cubes. It is not an atom, clearly. It is not a substance because it is not a solid cube. Therefore it must be a group. Yet it contains no atoms of solid cube. Probably you don't want substances to necessarily be P, just have some bit that is P and no bit that is P-atom. Then the solid sphere would be a substance with respect to "...is a solid cube". mu'o mi'e xorxes _________________________________________________________________MSN 8 with e-mail virus protection service: 2 months FREE* http://join.msn.com/?page=features/virus