INDUCTIVE LOGICAL PROBLEM. 125 



ordinary modes of calculating numbers of combinations in 

 mathematical science fail to give any aid, and exhaustive 

 examination of the combinations in detail is the only me- 

 thod applicable. 



My mode of solving the problem was as follows : — 

 Having written out the whole of the 256 series of com- 

 binations, I examined them separately, and struck out 

 such as did not fulfil the test of consistency. I then chose 

 some common form of proposition involving two or three 

 terms, and varied it in every possible manner, both by the 

 circular interchange of letters (A, B, C into B, C, A and 

 then into C, A, B) and by the substitution for any one or 

 more of the terms of the corresponding negative terms. 

 For instance, the proposition AB = ABC can be first varied, 

 by circular interchange, so as to give BC=BCA and then 

 CA=CAB. Each of these three can then be thrown into 

 eight varieties by negative change. Thus, AB = ABC gives 

 «B = «BC, Kb = A bC, AB = ABc, ab = abC, and so on. Thus 

 there may possibly exist no less than twenty-four varieties 

 of the law having the general form AB = ABC, meaning that 

 whatever has the properties of A and B has those also of C. 

 It by no means follows that some of the varieties may not 

 be equivalent to others; and trial shows, in fact, that 

 AB = ABC is exactly the same in meaning as Kc = Kcb 

 or Bc = Bc#. Thus the law in question has but eight 

 varieties of distinct logical meaning. I now ascertain, by 

 actual deductive reasoning, which of the 256 series of com- 

 binations result from each of these distinct laws, and mark 

 them olf as soon as found. I now proceed to some other 

 form of law, for instance A=ABC, meaning that whatever 

 has the qualities of A has those also of B and C. I find 

 that it admits of twenty-four variations, all of which are 

 found to be logically distinct ; the combinations being 

 worked out, I am able to mark off twenty-four more of the 

 list of 256 series. I proceed in this way to work out the 



