410 PROCEEDINGS OF THE AMERICAN ACADEMY 



a -\- b = b -f- a 



a+(b + c) = (a + b)+c 



b c = c b 



(a b) c = a (b c) 



a (in -J- ri) == a m -\- a n 



we have a series of identities whose truth or falsity is entirely un- 

 determinable. In order, therefore, fully to define those operations, we 

 will say that all propositions, equations, and identities which are in 

 the general case left by the former definitions undetermined as to truth 

 shall be true, provided they are so in all interpretable cases. 



On Arithmetic. 



Equality is a relation of which identity is a species. 



If we were to leave ecmality without further defining it, then by the 

 last scholium all the formal rules of arithmetic would follow from it. 

 And this completes the central design of this paper, as far as arith- 

 metic is concerned. 



Still it may be well to consider the matter a little further. Imagine, 

 then, a particular case under Boole's calculus, in which the letters are 

 no longer terms of first intention, but terms of second intention, and 

 that of a special kind. Genus, species, difference, property, and acci- 

 dent, are the well-known terms of second intention. These relate par- 

 ticularly to the comprehension of first intentions ; that is, they refer 

 to different sorts of predication. Genus and species, however, have at 

 least a secondary reference to the extension of first intentions. Now- 

 let the letters, in the particular application of Boole's calculus now 

 supposed, be terms of second intention which relate exclusively to the 

 extension of first intentions. Let the differences of the characters of 

 things and events be disregarded, and let the letters signify only the 

 differences of classes as wider or narrower. In other words, the only 

 logical comprehension which the letters considered as terms will have 

 is the greater or less divisibility of the classes. Tims, n in another 

 case of Boole's calculus might, for example, denote " New England 

 State " ; but in the case now supposed, all the characters which make 

 these States what they are being neglected, it would signify only what 



