AND MULTIPLICATION OF LOGICAL RELATIVES. 213 



excludents, then not-A and not-B are alternatives. Simi- 

 larly, the truth of either of the following implies the truth 

 of the other : — 



A=MB; a=iVb. 



That is to say, if every thing is either A or B, then nothing 

 is both not-A and not-B ; and conversely ■^. 



M, like ISl, is invertible and is not transitive. Every 

 invertible relative has this property, that its second power 

 is equal to its zero power. This follows from the definition 

 that when two terms stand in the same relation to a third, 

 they stand to each other in the zero power of the same 

 relative. Thus 



A=^B, 



B=iVC, 



A=iV"C=iV°C, 



whereof the canonical equation is 



and similarly 



That is to say. 



M^=M°. 



Excludent of excludent is coexcludent. 

 Alternative of alternative is coalternative. 



This combination of properties — equal to its own reci- 

 procal and not equal to its own second power — exists in 

 negative unity and in no other number. 



* The introduction of this relation, which I call alternation, into logic is 

 clue to De Morgan. What I call an alternative he calls a complement. 



