41 o THE POPULAR SCIENCE MONTHLY 



relations " such as/' " greater than/' " to the right of/' " to the left of/' 

 are transitive. That is, they follow James's axiom of skipped inter- 

 mediaries. Xow all those serial relations that can be expressed in 

 these transitive dyadic relations can also be expressed in terms of the 

 formally triadic relation "between." Thus, let A, B, C, D be four 

 objects in a row. I can say, " B is to the right of A, C is to the right of 

 B." I can conclude that C is to the right of A. And then I can define 

 the relations of order in question. Now it is very easy to see that if B 

 is to the right of A and C is to the right of B, C must be to the right of 

 A so long as one interprets the relations of right and left as we ordi- 

 narily do. But suppose I give you the premises, " B is between A and 

 C, C is between A and D/' and ask you what follows. The conclusion is 

 decidedly hard for most minds to work out. In other words, the triadic 

 relations have a psychological difficulty which we do not feel in the 

 case of the transitive d3^adic relations, although we can express equiva- 

 lent facts in both terms. The difference in question is hardly due to the 

 fact that a set of three objects is more complicated to grasp than a set 

 of two. For a little exercise in attempting to reason in terms of " be- 

 tween," as the geometers often do, will show that the psychological 

 difficult}^ is out of all proportion to the numerical difference between 

 two and three. The grounds for the difference in difficulty are pre- 

 sumably statable only in psycho-physical terms. But the matter is one 

 for psychological research, and should be undertaken. 



Over against these problems of the psychology of deduction which 

 are possibly capable of a more or less direct experimental research, 

 there are vast numbers of problems of deduction which can be 

 attacked more indirectly, some of them by following the records of 

 formation of new habits, some of them by means of more or less exact 

 study of social processes. There exists, for instance, an indefinite range 

 of possibilities for the study of the psychology of the arithmetical 

 processes by a device which, so far as I know, has still been very little 

 used, although I have repeatedly recommended it to students of educa- 

 tional psychology. We hear a good deal of effort to make out the details 

 of the process whereby a child gets control of arithmetical computations. 

 Now it is perfectly easy for any one to put himself near to the beginning 

 of practical arithmetic and into a place where he has to learn very many 

 of bis habits as a computer over again, under conditions that will admit 

 of a pretty careful experimental scrutiny of the way in which the new 

 habits get formed, and which will enable us to make precise records of 

 the growth of the new habits. The device in question consists simply in 

 using, instead of our decadic notation and numeration, a dyadic, triadic, 

 or other such system. Dyadic arithmetic is the simplest of all. In this 

 one uses two digits instead of the digits from to 0. inclusive. That is, 

 one uses only and unity; 1 standing alone will mean unity. If one 



