372 THE PRINCIPLES OF SCIENCE. 



never occur together, and that they are never both absent ; 

 these propositions are indeed logically equivalent to one, 

 namely AB = Ac. Our classification is then identical with 

 the following bifurcate one : 



AB Ab 



ABC ABc AbC Abe 



= o =o 



If, again, we divide the genus A into three species AB, 

 AC, AD, we are either logically in error, or else we must 

 be understood to imply the existence of three propositions 

 excluding the union within the genus A of the properties 

 of B, C and D, namely AB = ABcc, A.G = A.bCd, and 

 AD = A&cD. It comes to the same thing if we say that 

 our classification is really a bifurcate one, as follows : 



A 



A^ A6 



ABC ABc A6C Abe 



ABCD AECd ABcD ABcd AbCV ~AbCd AbcV Abed 



= O =o=O =0 = O 



The logical necessity of bifurcate classification has been 

 clearly and correctly stated in the 'Outline of a New System 

 of Logic' by George Bentham, a work of which the logical 

 value has been quite overlooked until lately. Mr. Bentham 

 points out, in p. 1 1 3, that every classification must be 

 essentially bifurcate and takes, as an example, the division 

 of vertebrate animals into four subclasses, as follows : 



Mammifera endowed with mammae and lungs. 

 Birds without mammae but with lungs and wings. 



Fish deprived of lungs. 



Reptiles deprived of mammae and wings but with 

 lungs. 



