v.] DISJUNCTIVE PKOPOSITIONS. 73 



How shall we describe the class of things which are not 

 malleable-dense-metals ? Whatever is included under that 

 term must have all the qualities of malleability, denseness, 

 and metallicity. Wherever any one or more of the qualities 

 is wanting, the combined term will not apply. Hence the 

 negative of the whole term is 



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 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 cannot 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 + 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 -\- a ( a. 



Since AAA is by the Law of Simplicity equivalent to A, 

 so a -|- a } a must be equivalent to a, and the Law of 

 Unity holds true. 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 Funda- 

 mental Law of Thought, which I have called the Law 

 of Duality (p. 6). Taking A to represent any class or 



