42 THE PRINCIPLES OF SCIENCE. [chap. 



be the whole) of tlie vevtebrata and the mammalia. If it is 

 asked What part ? the propasition affords no answer, except 

 that it is the part which is uianmialian; 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 

 artilicial 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 endlrss 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 

 tlie 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 Quceilam LocpMr., 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^ distinctly says that 

 tlie 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 propositions 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 thi^ 

 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 concerniug things wiiich are outside of tlie class 

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

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



' EnmicJnp(zdia Britannica, Eighth Ed. art. Li;gic, sect. 37, note. 

 8vo. icjuiut,, p. 79. 



