vii.J INDUCTION. 141 



and are therefore of equivalent logical meaning. The fifth 

 type, or Barbara, can also be thrown into the equivalent 

 forms A = ABC, aB = aBC and A = AC, B = A -|- aBC. 

 In other cases I have obtained the very same logical 

 conditions in four modes of statements. As regards mere 

 appearance and form of statement, the number of possible 

 premises would be very great, and difficult to exhibit 

 exhaustively. 



The most remarkable of all the types of logical condition 

 is the fourteenth, namely, A = BC -|- be. It is that which 

 expresses the division of a genus into two doubly marked 

 species, and might be illustrated by the example " Com- 

 ponent of the physical universe = matter, gravitating, or 

 not-matter (ether), not-gravitating." It is capable of only 

 two distinct logical variations, namely, A = BC -|- be and 

 A = Be -|- Z>C. By transposition or negative change of the 

 letters we can indeed obtain six different expressions of 

 each of these propositions ; but when their meanings are 

 analysed, by working out the combinations, they are found 

 to be logically equivalent to one or other of the above two. 

 Thus the proposition A = BC -|- be can be written in any 

 of the following five other modes, 



a = bC -I- Be, B = CA -|- ca, b = cA -|- Ca, 

 C = AB -I- ab, c = aB -|- Ab. 



I do not think it needful to publish at present the com- 

 plete table of 193 series of combinations and the premises 

 corresponding to each. Such a table enables us by mere 

 inspection to learn the laws obeyed by any set of com- 

 binations of three things, and is to logic what a table of 

 factors and prime numbers is to the theory of numbers, or 

 a table of integrals to the higher mathematics. The table 

 already given (p. 140) would enable a person with but little 

 labour to discover the law of any combinations. If there 

 be seven combinations (one contradicted) the law must be 

 of the eighth type, and the proper variety will be apparent. 

 If there be six combinations (two contradicted), either the 

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

 number 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 



