2o6 Mr. J. J. Murphy on 



as already proposed, the contrapositives of the foregoing 

 four propositions are as follows : — 



'All b is a ; or 

 all that is not B is not A b < a 



a is an includent of b. a = ^b 



o 

 § Some b is not a. ^b< A. 



^ o 



a is a non-includent of b. 



a = ^^ <b. 



'All that is not a is b ; or 

 Everything is either not A or 

 not B a<b 



^ a is an alternative of b, a = iV''<b 



Some things are neither a nor b 



2a <b 



o 



a is a non-alternative of b. 



a = ;«^<b 



It follows from the nature of a relation, that if 



A = ^B, then inversely B = E'^K, 



that is to say, if A is an enclosure of B, B is an includent of 

 A. But we have seen that if 



A = ^B, then 2i = E'b, 



that is to say, not-A is an includent of not-B ; so that in 

 this case the contrapositive relation is of the same form 

 with the inverse relation. Symbolically, 



This is true of the relation E (inclusion) and its derivatives 

 by contradiction and contraposition ; but it is not true of the 

 relation N (exclusion), and its derivatives. 



N and its derivatives are invertible. Symbolically, 



E and its derivatives are not so. Those properties of the E 



§ I have found it difficult to make this contraposition clear to myself 

 without an example, and the following may be serviceable. Animals are 

 either vertebrate or invertebrate, and animals, except those which have nothing 

 analogous to blood, are either red-blooded or white-blooded. Some red- 

 blooded animals are not vertebrate ; and the contrapositive of this proposition 

 is, that some invertebrate animals are not white-blooded. 



