100 ROYCE— PRINCIPLES OF THEORETICAL SCIENCE. [April 19, 



of the activities of our own thought. We understand relations, 

 because of our own thinking processes we can at once depict, and in 

 a sense reconstruct or create them. The types of our own con- 

 struction, of our own thoughtful activity, are therefore the rela- 

 tional types. If we are to understand, then, the unity and the 

 system of relational types, we must see how their varieties are re- 

 lated to our own activities as thinkers. 



Now, however, relations are known to us not only as existing in 

 tell world of numbers and of geometry, but as present in the purely 

 logical world, the world of classes and propositions, of syllogisms, 

 and of reasonings in general. I have already mentioned what some 

 of the logical relations are. They are relations such as are expressed 

 by the words '' and," " or" " not" " implies" and so on. These re- 

 lations are as fundamental and as simple as are our thinking proc- 

 esses themselves. We learn about them not through our senses, 

 but through our activity as thinkers. Now what Kempe's re- 

 search suggests, and what my own line of research has tended I 

 hope to bring a very small step further on the road towards defini- 

 tion, and confirmation, is the thought that such geometrical rela- 

 tions as " between,'' such relations as '' greater " and " less," and even 

 such relations as are fundamental in group theory, are capable of 

 being interpreted as instances, as consequences, or as partial views, 

 of the fundamental logical relations themselves. Kempe has shown 

 how a logical class can be viewed as " between " two other classes 

 and how the geometrical " between " can be regarded as a special in- 

 stance of this logical "between" I have shown how the system of Dr. 

 Veblen's principles of geometry could be brought into definite con- 

 nection with the relations which characterize a system of logical 

 classes. The whole research in question is still in a very elementary 

 stage, but enough has been done, I think, to make it at least prob- 

 able that whoever comprehends the most fundamental logical rela- 

 tions, such as a child begins to comprehend when it first says " no," 

 that is, whoever comprehends such relations, as " and " and " or " 

 and " not," and the relation of implication, has already in his hand the 

 means for developing the fundamental concepts of all of the exact 

 sciences, since the relations of these exact sciences are more or less 



