72 THE PRINCIPLES OF SCIENCE. [CHAP. 



and Commutativeness, must be observed. In uniting 

 terms by the disjunctive symbol we shall find that the 

 same or closely similar laws hold true. The alternatives 

 of either member of a disjunctive proposition are certainly 

 commutative. Just as we cannot properly distinguish 

 between rich and rare gems and rare and rich gems, so we 

 must consider as identical the expression rich or rare gems, 

 and rare or rich gems. In our symbolic language we may 

 say 



A -I- B = B -|. A. 



The order of statement, in short, has no effect upon the 

 meaning of an aggregate of alternatives, so that the 

 Law of Commutativeness holds true of the disjunctive 

 symbol. 



As we have admitted the possibility of joining as alter- 

 natives terms which are not really different, the question 

 arises, How shall we treat two or more alternatives when 

 they are clearly shown to be the same? If we have it 

 asserted that P is Q or E, and it is afterwards proved that 

 Q is but another name for K, the result is that P is either 

 R or E. How shall we interpret such a statement ? What 

 would be the meaning, for instance, of " wreath or anadem " 

 if, on referring to a dictionary, we found anadem described 

 as a wreath ? I take it to be self-evident that the meaning 

 would then become simply "wreath." Accordingly we 

 may affirm the general law 



A -I- A = A. 



Any number of identical alternatives may always be 

 reduced to, and are logically equivalent to, any one of 

 those alternatives. This is a law which distinguishes 

 mathematical terms from logical terms, because it obviously 

 does not apply to the former. I propose to call it the Law 

 of Unity, because it must really "be involved in any 

 definition of a mathematical unit. This law is closely 

 analogous to the Law of Simplicity, AA = A ; and the 

 nature of the connection is worthy of attention. 



Few or no logicians except De Morgan have adequately 

 noticed the close relation between combined and disjunctive 

 terms, namely, that every disjunctive term is the negative 

 of a corresponding combined term, and vice versa. Consider 

 the term 



Malleable dense metal. 



