84 THE PEINCIPLES OF SCIENCE. [CHAP. 



Hence it results that b is either nothing at all , or it is db ; 

 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 substitution 

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

 doubtful, therefore, whether not-metal is element or not- 

 elemeut. 



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 l : " Erom 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 DC Morgan 

 thinks that this proof is entirely needless, because common 

 logic gives the inference without the use of any geo- 

 metrical reasoning. I conceive however that logic gives 

 the inference only by an indirect process. De Morgan 

 claims " to see identity in Every A is B and every not-B 

 is not- A, by a process of thought prior to syllogism." 

 Whether prior to syllogism or not, I claim that it is not 

 prior to the laws of thought and the process of substitutive 

 inference, by which it may be undoubtedly demonstrated. 



Employment, of the Contrapositive Proposition. 



We can frequently employ the Contrapositive form of a 

 proposition by the method of substitution ; and certain 

 moods of the ancient syllogism, which we have hitherto 

 passed over, may thus be satisfactorily comprehended in 

 our system. Take for instance the following syllogism in 

 the mood Camcstres : 



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



