50 THE PRINCIPLES OF SCIENCE. 



This formula expresses that the class A is identical 

 with the class AB ; and as the latter must be a part at 

 least of the class B, it implies the inclusion of the class A 

 in that of B. Thus we might represent our former ex- 

 ample thus 



Mammalia = Mammalian vertebrata. 



This proposition asserts identity between a part of the 

 vertebrata and the mammalia. If it is asked What part 1 

 the proposition affords no answer except that it is the 

 part which is mammalian ; but the assertion ' mammalia = 

 some vertebrata' tells us no more. 



It is quite likely that some readers may think this 

 mode of representing the universal affirmative proposition 

 of the old logic artificial and complicated. I will not 

 undertake to convince them of the opposite at this point 

 of the system. My justification for it will be found, not 

 in the immediate treatment of this proposition, but in 

 the general harmony which it enables us to discover 

 between all parts of reasoning. I have no doubt that 

 this is the point of critical difficulty in the relation of 

 logical to other forms of reasoning. Grant this mode 

 of denoting that ' all A's are B's/ and I fear no further 

 difficulties ; refuse it, and we find want of analogy and 

 endless complication in every direction. For instance 

 Aristotle, in accepting inclusion of class in class as 

 the fundamental relation of logic, was at once obliged 

 to ignore the existence of the very extensive and all- 

 important class of propositions denoting the similarity 

 of one thing with another. It is on general grounds 

 that I hope to show overwhelming reasons for seeking 

 to reduce every kind of proposition to the form of an 

 identity. 



I may add that not a few previous logicians have 

 accepted this view of the universal affirmative proposition. 

 Boole often employed this mode of expression, and 



