vii. j INDUCTION. 139 



we have the relations, " All A's are B's and all B's are 

 C's," of which the old logical syllogism is the development. 

 We may also have " all A's are all B's, and all B's are C's," 

 or even " all A's are all B's, and all B's are all C's." All 

 such premises admit of variations, greater or less in 

 number, the logical distinctness of which can only be 

 determined by trial in detail. Disjunctive propositions 

 either singly or in pairs were also treated, but were often 

 found to be equivalent to other propositions of a simpler 

 form ; thus A = ABC ( Abe is exactly the same in meaning 

 as AB = AC. 



This mode of exhaustive trial bears some analogy to 

 that ancient mathematical process called the Sieve of 

 Eratosthenes. Having taken a long series of the natural 

 numbers, Eratosthenes is said to have calculated out in 

 succession 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 discovered. 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 a law or group 

 of conditions. But the analogy is perfect in the point that 

 they are both inverse processes. There is no mode of 

 ascertaining 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 Erato- 

 sthenes' method have been worked out to a great extent 

 and registered in tables for the convenience of other 

 mathematicians, I have endeavoured to work out the 

 inverse logical problem 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 numbers of combinations which are contradicted 

 or destroyed by each, and the numbers of logically distinct 



