140 



THE PRINCIPLES OF SCIENCE. 



[CHAP. 



variations of which the law is capable. There might be 

 also added, as a sixteenth case, that case where no special 

 logical condition exists, so that all the eight combinations 

 remain. 



There are sixty-three series of combinations derived from 

 self-contradictory premises, which with 192, the sum of 

 the numbers of distinct logical variations stated in the 

 third column of the table, and with the one case where 

 there are no conditions or laws at all, make up the whole 

 conceivable number of 256 series. 



We learn from this table, for instance, that two pro- 

 positions of the form A = AB, B = BC, which are such 

 as constitute the premises of the old syllogism Barbara, 

 exclude as impossible four of the eight combinations in 

 which three terms may be united, and that these proposi- 

 tions are capable of taking twenty- four variations by trans- 

 positions of the terms or the introduction of negatives. 

 This table then presents the results of a complete analysis 

 of all the possible logical relations arising in the case of 

 three terms, and the old syllogism forms but one out of 

 fifteen typical forms. Generally speaking, every form can 

 be converted into apparently different propositions ; thus 

 the fourth type A = B, B = BC may appear in the form 

 A = ABC, a = ab, or again in the form of three proposi- 

 tions A = AB, B = BC, aB = aBc ; but all these sets of 

 premises yield identically the same series of combinations, 



