98 THE PRINCIPLES OF SCIENCE. 



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 



I = A6 -I- ab. (2) 



For A in this proposition we substitute its description 

 as given in (i), obtaining 



1} - AB6 -I- ab. 



But according to the Law of Contradiction the term 

 AB6 must be excluded from thought or 



AB6 = o. 



Hence it results that 6 is either nothing at all, or it is 

 ab ; and the conclusion is 



b = ab. 



As it will often be necessary to refer to a conclusion 

 of this kind I shall call it, as is usual, the Contrapositive 

 Proposition of the original. The reader need hardly be 

 cautioned to observe that from all A s are B s it does not 

 follow that all not-A s are not-B s. For by the Law 

 of Duality we have 



a = aB | ab, 



and it will not be found possible to make any substitu 

 tion in this by our original premise A = AB. It still 

 remains doubtful, therefore, whether not-metal is element 

 or not-element. 



The proof of the Contrapositive Proposition given above 

 is exactly the same as that which Euclid applies in the 

 case of geometrical notions. De Morgan describes Euclid s 

 process as follows d : From every not-B is not-A he pro 

 duces every A is B, thus If it be possible, let this A be 

 not-B, but every not-B is not-A, therefore this A is not-A, 

 which is absurd : whence every A is B. Now De Morgan 

 thinks that this proof is entirely needless, because common 



d Philosophical Magazine, Dec. 1852 ; p. 437. 



