88 THE PRINCIPLES OF SCIENCE. 



of Thought, which I have called the Law of Duality. 

 Taking A to represent any class or object or quality, and 

 B any other class, object or quality, we may always 

 assert that A either agrees with B, or does not agree. 



Thus we may say 



A = AB -I- A6. 



This is a formula which will henceforth be constantly 

 employed, and it lies at the basis of reasoning. 



The reader may perhaps wish to know why A is inserted 

 in both alternatives of the second member of the identity, 

 and why the law is not stated in the form 



A = B ! b. 



But if he will consider the contents of the last section 

 (p. 87), he will see that the latter expression cannot be 

 correct, otherwise no term would have any negative. 

 For the negative of B I b is 6B, or a self-contradictory 

 term ; so that if A were identical with B -|- b, its nega- \ 

 tive a would be non-existent. This result would generally 

 be an absurd one, and I see much reason to think that in 

 a strictly logical point of view it would always be absurd. 

 In all probability we ought to assume as a fundamental 

 logical axiom that every term has its negative in thought. 

 We cannot think at all without separating what we think 

 about from other different things, and these things neces- 

 sarily form the negative notion f . If so-, it follows that 1 

 any term of the form B { b is just as self-contradictory 

 as one of the form Bfc. 



It will be convenient to recapitulate in this place the 

 three great Laws of Thought in their symbolic form, thus 



Law of Identity A = A. 



Law of Contradiction Aa = o. 



Law of Duality A = AB [ A6. 



f ' Pure Logic,' p. 65. See also the criticism of this point by De 

 Morgan in the ' Athenaeum,' No. 1892, 3oth January, 1864 ; p. 155. 



