130 



ON THE INDUCTIVE LOGICAL PROBLEM. 



second, eleventh, or twelfth type applies, and a certain 

 nnrnber of trials will disclose the proper type and variety ; 

 if there be but two combinations, the law must be of the 

 third type ; and so on. 



The above investigations are complete as regards the 

 possible logical relations of two or three terms. But when 

 we attempt to apply the same kind of method to the rela- 

 tions of four or more terms, the labour becomes impossibly 

 great. Four terms give sixteen combinations compatible 

 with the laws of thought, and the number of possible 

 selections of combinations is no less than 2 16 , or 65,536. 

 The following table shows the extraordinary manner in 

 which the number of possible logical relations increases 

 with the number of terms involved. 



Number 



of 

 terms. 



Number of 



possible 



combinations. 



Number of possible selections of com- 

 binations corresponding to con- 

 sistent or inconsistent logical relations. 



2 



3 

 4 

 5 

 6 



4 

 8 



16 

 32 

 64 



16 



256 



65>536 



4,294,967,296 



18,446,744,073,709,55 1,616 



Some years of continuous labour would be required to 

 ascertain the precise number of types of laws which may 

 govern the combinations of only four things ; and only a 

 small part of such laws would be exemplified or capable of 

 practical application in science. The purely logical inverse 

 problem whereby we pass from combinations to their laws 

 is solved in the preceding pages as far as it is likely to be 

 for a long time to come ; and it is almost impossible that 

 it should ever be carried more than a single step further. 



