42 THE PRINCIPLES OF SCIENCE. [CHAP. 



be the whole) of the vertebrata and the mammalia. If it is 

 asked What part ? 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 will think this 

 mode of representing the universal affirmative proposition 

 artificial and complicated. I will not undertake to con- 

 vince them of the opposite at this point of my exposition. 

 Justification for it will be found, not so much in the im- 

 mediate treatment of this proposition, as in the general 

 harmony which it will enable us to disclose between all 

 parts of reasoning. I have no doubt that this is the 

 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 anomaly in every direction. 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 logicians have accepted this 

 view of the universal affirmative proposition. Leibnitz, in 

 his Difficultates Qucedam Logicce, adopts it, saying, " Omne 

 A est B ; id est equivalent AB et A, seu A non B est non- 

 ens." Boole employed the logical equation x = xy con- 

 currently with x = vy ; and Spalding l distinctly says that 

 the proposition " all metals are minerals " might be de- 

 scribed as an assertion of partial identity between the two 

 classes. Hence the name which I have adopted for the 

 proposition. 



Limited Identities. 



An important class of proposition have the form 



AB = AC, 



expressing the identity of the class AB with the class AC. 

 In other words, " Within the sphere of the class A, all the 

 B's are all the C's ; " or again, " The B's and C's, which are 

 A's, are identical." But it will be observed that nothing is 

 asserted concerning things which are outside of the class 

 A ; and thus the identity is of limited extent. It is the 

 proposition B = C limited to the sphere of things called A. 



1 Encyclopedia Britannica, Eighth Ed. art. Logic, sect. 37, note. 

 8vo. reprint, p. 79. 



