ORDER, NUMBER, QUANTITY 255 



for example, Monday and Tuesday have an actual order 

 which would be different if Tuesday occurred before Mon- 

 day ; similarly father and son may be said to have a certain 

 order by virtue of the fact that the father begets the son 

 but the son does not beget the father. The question then 

 arises as to whether the asymmetrical relation is psycho- 

 logically more primitive than a three- termed relationship, 

 such as Monday-Tuesday-Wednesday, or father-son-grand- 

 son, and whether the greater complexity of this latter type 

 is essential to order. Since the question is largely a verbal 

 matter at this point it may be answered arbitrarily by sug- 

 gesting that order be denied empirical applicability to events 

 unless they exhibit a complexity which is at least threefold. 



What is required, therefore, in the second place, is to 

 specify the character of the relational-structure which must 

 be exhibited by a three-termed complex in order to permit 

 the characterization of this complex as ordered. The re- 

 quired relational-structure may be described as one which 

 possesses betweenness. Betweenness may be defined as a 

 triadic relation in which there is a term such that (a) it has 

 an asymmetrical relation to each of the two other terms, 

 and (b) its relation to one of the terms is the converse of its 

 relation to the other. 1 The element possessing this relation 

 would then be said to be between the other two. For exam- 

 ple, Tuesday is between Monday and Wednesday, since 

 there is a triadic relation characterizing the complex, of 

 such a kind that a certain element (Tuesday) is so situated 

 that it has an asymmetrical relation to Monday (succeeds) 

 and to Wednesday (precedes), and its relation to the one 

 is the converse of its relation to the other. A similar 

 relational-structure is exhibited by any three months in 

 the year, any three points on a line, any three floors of a 

 building, and so on. 



The relational-structure may then be said to determine 

 the serial arrangement. Usually the arrangement which is 



1 This is essentially equivalent to Russell's definition. See his Principles of 

 Mathematics (Cambridge: Cambridge University, 1903), p. 200. 



