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 



6 = A6 -I- ab. (2) 



For A in this proposition we substitute its description 

 as given in (i), obtaining 



b = AB6 -I- ab. 



But according to the Law of Contradiction the term 

 ABfe must be excluded from thought or 



AB6 = o. 



Hence it results that b 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 bv 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. 3 Now De Morgan 

 thinks that this proof is entirely needless, because common 



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



