106 THE PRINCIPLES OF SCIENCE. [CHAP. 



mechanical manner that exclusion of self-contradictories 

 which was formerly done upon the slate or upon paper. 

 Accordingly, from the combinations remaining in the upper 

 line we can draw any inference which the premises yield. 

 If we raise the A's we find only one, and that is G, so 

 that A must be C. If we select the c's we again find only 

 one, which is a and also b ; thus we prove that not-C is 

 not- A and not-B. 



When a disjunctive proposition occurs among the 

 premises the requisite movements become rather more 

 complicated. Take the disjunctive argument 



A is either B or C or D, 



A is not C and not D, 



Therefore A is B. 

 The premises are represented accurately as follows : 



A = AB -|. AC -I- AD (i) 



A = Ac (2) 



A = Ad. (3) 



As there are four terms, we choose the series of sixteen 

 combinations and place them on the highest ledge of the 

 board but one. We raise the a's and out of the A's, which 

 remain, we lower the &'s. But we are not to reject all the 

 AJ's as contradictory, because by the first premise A's 

 may be either B's or C's or D's. Accordingly out of the 

 A&'s we must select the c's, and out of these again the d's, 

 so that only Abed will remain to be rejected finally. 

 Joining all the other fifteen combinations together again, 

 and proceeding to premise (2), we raise the a's and lower 

 the AC's, and thus reject the combinations inconsistent 

 with (2) ; similarly we reject the AD's which arje incon- 

 sistent with (3). It will be found that there remain, in 

 addition to all the eight combinations containing a, only 

 one containing A, namely 



ABcd, 



whence it is apparent that A must be B, the ordinary 

 conclusion of the argument. 



In my " Substitution of Similars " (pp. 56 59) I have 

 described the working upon the Abacus of two other 

 logical problems, which it would be tedious to repeat in 

 this place. 



