CHAPTER VI. 



THE INDIRECT METHOD OF 



THE forms of deductive reasoning as yet considered, are 

 mostly cases of Direct Deduction as distinguished from 

 those which we are now about to treat. The method of 

 Indirect Deduction may be described as that which points 

 out what a thing is, by showing that it cannot be anything 

 else. We can define a certain space upon a map, either by 

 colouring that space, or by colouring all except the space ; 

 the first mode is positive, the second negative. The 

 difference, it will be readily seen, is exactly analogous to 

 that between the direct and indirect modes of proof in 

 geometry. Euclid often shows that two lines are equal, by 

 showing that they cannot be unequal, and the proof rests 

 upon the known number of alternatives, greater, equal or 

 less, which are alone conceivable. In other cases, as for 

 instance in the seventh proposition of the first book, he 

 shows that two lines must meet in a particular point, by 

 showing that they cannot meet elsewhere. 



In logic we can always define with certainty the utmost 

 number of alternatives which are conceivable. The Law 

 of Duality (pp. 6, 74) enables us always to assert that any 

 quality or circumstance whatsoever is either present or 

 absent. Whatever may be the meaning of the terms A 

 and B it is certainly true that 



A = AB.|-AZ> 

 B = AB -I- aB. 



These are universal tacit premises which may be em- 

 ployed in the solution of every problem, and which are 

 such invariable and necessary conditions of all thought, 



G 



