OF ARTS AND SCIENCES : SEPTEMBER 10, 1867. 411 



essentially belongs to a class which has the same relations to higher 

 and lower classes which the class of New England States has, — that 

 is, a collection of six. 



In this case, the sign of identity will receive a special meaning. 

 For, if m denotes what essentially belongs to a class of the rank of 

 "sides of a cube," then m = n will imply, not that every New Eng- 

 land State is a side of a cube, and conversely, but that whatever 

 essentially belongs to a class of the numerical rank of " New England 

 States " essentially belongs to a class of the rank of " sides of a cube, 

 and conversely. Identity of this particular sort may be termed equality, 

 and be denoted by the sign = .* Moreover, since the numerical rank 

 of a logical sum depends on the identity or diversity (in first intention) 

 of the integrant parts, and since the numerical rank of a logical 

 product depends on the identity or diversity (in first intention) of parts 

 of the factors, logical addition and multiplication can have no place in 

 this system. Arithmetical addition and multiplication, however, will not 

 be destroyed, a b = c will imply that whatever essentially belongs at 

 once to a class of the rank of a, and to another independent class of 

 the rank of b belongs essentially to a class of the rank of c, and con- 

 versely, a -j- b = c implies that whatever belongs essentially to a 

 class which is the logical sum of two mutually exclusive classes of the 

 ranks of a and b belongs es.-entially to a class of the rank of c, and 

 conversely. It is plain that from these definitions the same theorems 

 follow as from those given above. Zero and unity will, as before, de- 

 note the classes which have respectively no extension and no compre- 

 hension ; only the comprehension here spoken of is, of course, that 

 comprehension which alone belongs to letters in the system now con- 

 sidered, that is, this or that degree of divisibility ; and therefore unity 

 will be what belongs essentially to a class of any rank independent of 

 its divisibility. These two classes alone are common to the two sys- 

 tems, because the first intentions of these alone determine, and are de- 

 termined b^r, their second intentions. Finally, the laws of the Boolian 



* Thus, in one point of view, identity is a species of equality, and, in another, 

 the reverse is the case. This is because the Being of the copula may be consid- 

 ered on the one hand (with De Morgan) as a special description of " inconvertible, 

 transitive relation," while, on the other hand, all relation may be considered as a 

 special determination of being. If a Hegelian should be disposed to see a contra- 

 diction here, an accurate analysis of the matter will show him that it is only a 

 verbal one. 



