CHAP. V.] PRINCIPLES OF SYMBOLICAL REASONING. 77 



ployed in any stage of the process of analysis intermediate be- 

 tween the change of interpretation of the symbols from the 

 logical to the quantitative system above referred to, and the final 

 restoration of the logical interpretation. 



Secondly. The theorem is perfectly true and intelligible when 

 x and y are logical symbols, provided that the form of the func- 

 tion /(#, y] is such as to represent a class or collection of thing s, 

 in which case the second member is always logically interpretable. 

 For instance, if /(#, y) represent the function 1 - x + xy, we ob- 

 tain on applying the theorem 



1 -x + xy = xy + Ox(\-y) + (\ -x)y + (I -x) (I-?/), 



= xy + (l-x)y+(l-a!)(l-y) 9 

 and this result is intelligible and true. 



Thus we may regard the theorem as true and intelligible for 

 quantitative symbols of the species above described, always ; for 

 logical symbols, always when interpretable. Whensoever there- 

 fore it is employed in this work it must be understood that the 

 symbols 37, y are quantitative and of the particular species referred 

 to, if the expansion obtained is not interpretable. 



But though the expansion is not always immediately inter- 

 pretable, it always conducts us at once to results which are in- 

 terpretable. Thus the expression x - y gives on development 

 the form 



which is not generally interpretable. We cannot take, in thought, 

 from the class of things which are a?'s and not ?/'s, the class of 

 things which are ?/'s and not o?'s, because the latter class is not 

 contained in the former. But if the form x - y presented itself 

 as the first member of an equation, of which the second member 

 was 0, we should have on development 



*(i-y)-(i-*)-o: 



Now it will be shown in the next chapter that the above equa- 

 tion, x and y being regarded as quantitative and of the species 

 described, is resolvable at once into the two equations 

 tf(l-?/) = 0, y(l-#) = 0, 



and these equations are directly interpretable in Logic when lo- 



