VI.] THE INDIRECT METHOD OF INFERENCE. 83 



Simple Illustrations. 



In tracing out the powers and results of this metliod, 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 investi<^ate 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 JSfot-element is a Not-metal. 

 While some logicians, as for instance I)e 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 (Jontrapositive Can- 

 version ; the i'ull process is as follows : — 



Firstly, by the Law of Duality we know that 

 Not-iiement is either Metal or Not-metal. 

 If it be metal, we know that it is by the prenn'se 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-Ji is 

 thus described 



l = Ab -I- ab. (2) 



For A in this proposition we substitute its description as 

 given in (i), olitaining 



h = AB/; -I- ab. 



But according to the Law of Contradiction the term 

 ABJ must be excluded from thought, or 



' rhilosopliinil Mafjnzine, D<'ct'iiil)(!r 1852 ; Foiiilli Series, vol. iv. 

 !'• 435. " <Jii liKliioct Deuionstrutioii." 



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