On the complete independence of the postulates for betweenness

The purpose of the present paper is to exhibit the "complete existential theory" (in the sense of E. H. Moore+) of each of these sets. This requires the discussion, in the usual way, of 26 = 64 examples for each of the sets (1)-(8), and 27= 128 examples for each of the sets (9)-(11). The results show that sets (1)-(10) are comrpletely independent while set (11) is not. In the case of sets (1), (2), (3), (5), (6), (7), which happen to be the sets which do not contain either postulate 3 or postulate 7, the necessary examples are given in terms of a class K containing only four elements. In the case of sets (4), (8), (9), (10), and (11), some of the examples require the use of a class K containing five elements. These five-element