136 THE PRINCIPLES OF SCIENCE. [CHAP. 



table in which each of A, a, B, b appears. These cases 

 are shown in the columns marked with an asterisk. 



We find that seven cases remain for examination, thus 

 characterised 



Four cases exhibiting three combinations, 

 Two cases exhibiting two combinations, 

 One case exhibiting four combinations. 

 It has already been pointed out that a proposition of the 

 form A = AB destroys one combination, Ab, so that this is 

 the form of law applying to the twelfth column. But by 

 changing one or more of the terms in A = AB into its 

 negative, or by interchanging A and B, a and b, we obtain 

 no less than eight different varieties of the one form ; thus 



nth case. 8th case. i$fh case. Mth case. 



A = AB A = A& a = aB a == ab 

 b = ab B = aB b = Ab B = AB 



The reader of the preceding sections will see that each 

 proposition in the lower line is logically equivalent to, and 

 is in fact the contrapositive of, that above it (p. 83). Thus 

 the propositions A = A& and B = aB both give the same 

 combinations, shown in the eighth column of the table, 

 and trial shows that the twelfth, eighth, fifteenth and 

 fourteenth columns are thus accounted for. We come to 

 this conclusion then The general form of proposition 

 A = AB admits of four logically distinct varieties, each 

 capable of expression in two modes. 



In two columns of the table, namely the seventh and 

 tenth, we observe that two combinations are missing. 

 Now a simple identity A = B renders impossible both Ab 

 and aB, accounting for the tenth case ; and if we change 

 B into b the identity A b accounts for the seventh case. 

 There may indeed be two other varieties of the simple 

 identity, namely a = b and a = B ; but it has already 

 been shown repeatedly that these are equivalent respec- 

 tively to A = B and A = b (p. 115). As the sixteenth 

 column has already been accounted for as governed 

 by no special conditions, we come to the following general 

 conclusion : The laws governing the combinations of two 

 terms must be capable of expression either in a partial 

 identity or a simple identity ; the partial identity is capable 

 of only four logically distinct varieties, and the simple 

 identity of two. Every logical relation between two terms 



