58 THE PRINCIPLES OF SCIENCE. [chap. 



more full and accurate result from the former. We may 

 show this most simply as follows : — 



By the first Law of Thought it is evident that 



AA = AA ; 

 and if we have given the proposition A = ABC, we may 

 substitute for both the A's in the second side of the above, 

 obtaining 



AA = ABC . ABC. 

 But from the property of logical symbols expressed in the 

 Law of Simplicity (p. 33) some of the repeated letters may 

 be made to coalesce, and we have 



A = ABC . C. 



Substituting again for ABC its equivalent A, we obtain 



A=AC, 

 the desired result. 



By a similar process of reasoning it may be shown that 

 we can always drop out any term appearing in one member 

 of a proposition, provided that we substitute for it the 

 whole of the other member. This process was described in 

 my first logical Essay,^ as Intrinsic Elimination, but it 

 might perhaps be better entitled the Ellipsis of Terms. 

 It enables us to get rid of needless terms by strict 

 substitutive reasoning. 



Inference of a Simple from, Two Partial Identities. 



Two terms may be connected together by two partial 

 identities in yet another manner, and a case of inference 

 then arises which is of the highest importance. In the 

 two premises 



A = AB (i) 



B = AB (2) 



the second member of each is the same ; so that we can by 

 obvious substitution obtain 



A = B. 

 Thus, in plain geometry we readily prove that " Every 

 equilateral triangle is also an equiangular triangle," and we 

 can with equal ease prove that " Every equiangular triangle 

 is an equilateral triangle.' Thence by substitution, as 

 explained above, we pass to the simple identity, 



Equilateral triangle = equiangular triangle. 

 * Pure Logic, p. 19. 



