DISJUNCTIVE PROPOSITIONS. 87 



qualities is wanting, the combined term will not apply. 

 Hence the negative of the whole term is 



o 



Not-malleable or not-dense or not-metallic. 

 In the above the conjunction or must clearly be inter- 

 preted as unexclusive ; for there may readily be objects 

 which are both not-malleable, and not-dense, and perhaps 

 not-metallic at the same time. If in fact we were required 

 to use or in a strictly exclusive manner, it would be 

 requisite to specify seven distinct alternatives in order to 

 describe the negative of a combination of three single 

 terms. The negatives of four or five terms would consist 

 of fifteen or thirty-one alternatives. This consideration 

 alone is sufficient to prove that the meaning of or can- 

 not be always exclusive in common language. 



Expressed symbolically, we may say that the negative 

 of 



ABC 



is not- A or not-B or not-C ; 



that is, a [ b [ c. 



Reciprocally the negative of 



P I- Q I R 



is pqr. 



Every disjunctive term, then, is the negative of a 

 combined term, and vice versa. 



Apply this result to the combined term AAA, and its 

 negative is 



a -I- a [ a. 



Now since AAA is by the Law of Simplicity equivalent to 

 A, so a -I- a -\- a must be by the Law of Unity equivalent 

 to a. Each law thus necessarily presupposes the other. 



Symbolic expression of the Law of Duality. 



We may now employ our symbol of alternation to express 

 in a clear and formal manner the third Fundamental Law 



