INDUCTIVE LOGICAL PROBLEM. 127 



sion all the multiples of every number, and to have marked 

 them off, so that at last the prime numbers alone remained, 

 and the factors of every number were exhaustively dis- 

 covered. My problem of 256 series of combinations is the 

 logical analogue, the chief points of difference being that 

 there is a limit to the number of cases, and that prime 

 numbers have no analogue in logic, since every series of 

 combinations corresponds to some law or group of condi- 

 tions. But the analogy is perfect in the point that they 

 are both inverse processes. There is no mode of ascer- 

 taining that a number is prime but by showing that it is 

 not the product of any assignable factors. So there is no 

 mode of ascertaining what laws are embodied in any series 

 of combinations but trying exhaustively the laws which 

 would give them. Just as the results of Eratosthenes' s 

 method have been worked out to a great extent and regis- 

 tered in tables for the convenience of other mathematicians, 

 I have endeavoured to work out the inverse logical pro- 

 blem to the utmost extent which is at present practicable 

 or useful. 



I have thus found that there are altogether fifteen con- 

 ditions or series of conditions which may govern the com- 

 binations of three terms, forming the premises of fifteen 

 essentially different kinds of arguments. The following 

 table contains a statement of these conditions, together 

 with the number of combinations which are contradicted 

 or destroyed by each, and the number of logically distinct 

 variations of which the law is capable. There might be 

 also added, as a sixteenth case, that case in which no 

 special logical condition exists, so that all the eight com- 

 binations remain. 



