74 THE PKINCIPLES OF SCIENCE. [CHAP. 



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- Ab. 



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 -I- b. 



But if he will consider the contents of the last section 

 (p. 73), he will see that the latter expression cannot be 

 correct, otherwise no term could have a corresponding 

 negative term. For the negative of B -|- b is Z>B, or a self- 

 contradictory term ; thus if A were identical with B -j- b 

 its negative a would be non-existent. To say the least, 

 this result would in most cases 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 things, 

 and these things necessarily form the negative notion. 1 

 It follows that any proposition of the form A = B -|- b is 

 just as self-contradictory as one of .the form A = B&. 



It is convenient to recapitulate in this place the three 

 Laws of Thought in their symbolic form, thus 

 Law of Identity A = A. 



Law oi Uuu&fiMiiutioii A.O, o. 



Law of Duality A = AB -I- Ab. 



Various Forms of the Disjunctive Proposition. 



Disjunctive propositions may occur in a great variety of 

 forms, of which the old logicians took insufficient notice. 

 There may be any number of alternatives, each of which 

 may be a combination of any number of simple terms. A 

 proposition, again, may be disjunctive in one or both 

 members. The proposition 



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

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



