98 THE PRINCIPLES OF SCIENCE. [chap. 



three-sided rectilinear figure, with or without three equal 

 angles," the last alternatives really express a property of 

 triangles, namely, that some triangles have three equal 

 angles, and some do not have them. If we put P = 

 " Some," meaning by the indefinite adjective " Some," one 

 or more of the undefined properties of triangles with three 

 equal angles, and take 

 A = triangle 



B = three-sided rectilinear figure, 

 C = with three equal angles, 

 then the knowledge implied is expressed in the two 

 propositions 



PA = PEG 

 ^A = j)Bc. 

 These may also be thrown into the form of one pro- 

 position, namely, 



A = PBC -I- pBc ; 

 but these alternatives cannot be reduced, and the propo- 

 sition is quite different from 



A = BC •!• Be. 



Illustrations of the Indirect Method. 



A great variety of arguments and logical problems 

 might be introduced here to show the comprehensive 

 character and powers of the Indirect Method. We can 

 treat either a single premise or a series of premises. 



Take in the first place a simple definition, such as "a 

 triangle is a three-sided rectilinear figure." Let 

 A = triangle 

 B = three-sided 

 C = rectilinear figure; 

 then the definition is of the form 



A = BC. 

 If we take the series of eight combinations of three 

 letters in the Logical Alphabet (p. 94) and strike out 

 those which are inconsistent with the definition, we have 

 the following result : — ABC 



aBc 

 ahC 

 abc. 



