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SCIENCE. 



[N. S. Vol. XX. No. 510. 



a number of recent English writers, have 

 so effectively furthered— and secondly, the 

 logic of scientific method, which is now so 

 actively pursued, in France, in Germany 

 and in the English-speaking countries — 

 this whole movement in modern logic, as I 

 hold, is rapidly approaching new solutions 

 of the problem of the fundamental nature 

 and the logic of 7-elations. The problem is 

 one in which we are all equally interested. 

 To De Morgan in England, in an earlier 

 generation, and, in our time, to Charles 

 Peirce in this country, very important 

 stages in the growth of these problems are 

 due. Russell, in his work on the 'Prin- 

 ciples of Mathematics,' has very lately un- 

 dertaken to sum up the results of the logic 

 of relations, as thus far developed, and to 

 add his own interpretations. Tet I think 

 that Russell has failed to get as near to the 

 foundations of the theory of relations as 

 the present state of the discussion permits. 

 For Russell has failed to take account of 

 what I hold to be the most fundamentally 

 important generalization yet reached in the 

 general theory of relations. This is the 

 generalization set forth as early as 1890, 

 by Mr. A. B. Kempe, of London, in a pair 

 of wonderful, but too much neglected, 

 papers, entitled, respectively, 'The Theory 

 of Mathematical Form,' and 'The Analogy 

 between the Logical Theory of Classes and 

 the Geometrical Theory of Points.' A 

 mere hint first as to the more precise form- 

 ulation of the problem at issue, and then 

 later as to Kempe 's special contribution to 

 that problem, may be in order here, despite 

 the impossibility of any adeqiiate state- 

 ment. 



III. 

 The two most obviously and universally 

 important kinds of relations known to the 

 exact sciences, as these sciences at present 

 exist, are : ( 1 ) The relations of the type 

 of equality or equivalence; and (2) the 

 relations of the type of before and after, 



or greater and less. The first of these 

 two classes of relations, viz., the class rep- 

 resented, although by no means exhausted, 

 by the various relations actually called, in 

 diiferent branches of science by the one 

 name equality, this class I say, might well 

 be named, as I myself have proposed, the 

 leveling relations. A collection of objects 

 between any two of which some one rela- 

 tion of this type holds, may be said to be 

 a collection whose members, in some defined 

 sense or other, are on the same level. The 

 second of these two classes of relations, 

 viz., those of the type of before and after, 

 or greater and less— this class of relations, 

 I say, consists of what are nowadays often 

 called the serial relations. And a collec- 

 tion of objects such that, if any pair of 

 these objects be chosen, a determinate one 

 of this pair stands to the other one of the 

 same pair in some determinate relation of 

 this second type, and in a relation which 

 remains constant for all the pairs that can 

 be thus formed out of the members of this 

 collection — any such collection, I say, con- 

 stitutes a one-dimensional open series. 

 Thus, in case of a file of men, if you choose 

 any pair of men belonging to the file, a 

 determinate one of them is, in the file, be- 

 fore the other. In the number series, of 

 any two numbers, a determinate one is 

 greater than the other. Wherever such a 

 state of affairs exists, one has a series. 



Now these two classes of relations, the 

 leveling relations and the serial relations, 

 agree with one another, and differ from 

 one another in very momentous ways. 

 They agree with one another in that both 

 the leveling and the serial relations are 

 what is technically called transitive; that 

 is, both classes conform to what Professor 

 James has called the law of 'skipped inter- 

 mediaries.' Thus, if A is equal to B, and 

 B is equal to C, it follows that A is equal 

 to C. If A is before B, and B is before C, 

 then A is before C. And this property, 



