92 OF INTERPRETATION. [CHAP. VI. 



duced from its arithmetical properties, but must be established 

 experimentally. 



12. We may now collect the results to which we have been 

 led, into the following summary : 



1st. The symbol 1, as the coefficient of a term in a develop- 

 ment, indicates that the whole of the class which that constituent 

 represents, is to be taken. 



2nd. The coefficient indicates that none of the class are to 

 be taken. 



3rd. The symbol - indicates that a perfectly indefinite por- 



tion of the class, i. e. some, none, or all of its members are to be 

 taken. 



4th. Any other symbol as a coefficient indicates that the 

 constituent to which it is prefixed must be equated to 0. 



It follows hence that if the solution of a problem, obtained 

 by development, be of the form 



that solution may be resolved into the two following equations, 

 viz., 



w = A + vC, (3) 



D = 0, (4) 



v being an indefinite class symbol. The interpretation of (3) 

 shows what elements enter, or may enter, into the composition 

 of w, the class of things whose definition is required ; and the 

 interpretation of (4) shows what relations exist among the ele- 

 ments of the original problem, in perfect independence of w. 



Such are the canons of interpretation. It may be added, that 

 they are universal in their application, and that their use is 

 always unembarrassed by exception or failure. 



13. Corollary. If Fbe an independently interpretable logi- 

 cal function, it will satisfy the symbolical law, F(l - F) = 0. 



By an independently interpretable logical function, I mean 

 one which is interpretable, without presupposing any relation 

 among the things represented by the symbols which it involves. 

 Thus x ( 1 - y) is independently interpretable, but x - y is not so. 



