vi.] THE INDIRECT METHOD OF INFERENCE. 



Simple Illustrations. 



In tracing out the powers and results of this method, we 

 will begin with the simplest possible instance. Let us 

 take a proposition of the common form, A = AB, say, 



A Metal is an Element, 



and let us investigate its full meaning. Any person who 

 has had the least logical training, is aware that we can 

 draw from the above proposition an apparently different 

 one, namely, 



A Not-element is a Not-metal. 



While some logicians, as for instance De Morgan, 1 have 

 considered the relation of these two propositions to be 

 purely self-evident, and neither needing nor allowing 

 analysis, a great many more persons, as I have observed 

 while teaching logic, are at first unable to perceive the 

 close connection between them. I believe that a true and 

 complete system of logic will furnish a clear analysis of 

 this process, which has been called Contrapositive Con- 

 version ; the full process is as follows : 



Firstly, by the Law of Duality we know that 



Not-element is either Metal or Not-metal. 

 If it be metal, we know that it is by the premise an 

 element ; we should thus be supposing that the same thing 

 is an element and a not-element, which is in opposition 

 to the Law of Contradiction. According to the only 

 other alternative, then, the not-element must be a not- 

 metal. 



To represent this process of inference symbolically we 

 take the premise in the form 



A = AB. (i) 



We observe that by the Law of Duality the term not-B is 

 thus described 



b = Ab -I- ab. (2) 



For A in this proposition we substitute its description as 

 given in (i), obtaining 



b = AB& -|. ab. 



But according to the Law of Contradiction the term 

 AB6 must be excluded from thought, or 



1 Philosophical Magazine, December 1852 ; Fourth Series, vol. iv 

 p. 435, "On Indirect Demonstration." 



02 



