THE INDIRECT METHOD OF INFERENCE. 97 



proved indirectly, we may say that the process is a 

 necessary and sufficient one, and the question of its com- 

 parative excellence or usefulness is not worth discussion. 

 As a matter of fact I believe that nearly half our logical 

 conclusions rest upon its employment. 



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 very 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, 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. 

 Now if it be metal, we know that it is by the premise 

 an element ; we should thus be supposing that the very 

 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. 



c 'Philosophical Magazine,' December 1852, Fourth Series, vol. iv. 

 P- 435> 'On Indirect Demonstration,' 



H 



