70 PRINCIPLES OF SYMBOLICAL REASONING. [CHAP. V 



x, ?/, 0, as if they were quantitative symbols of the kind above 

 described. We may in fact lay aside the logical interpretation of 

 the symbols in the given equation ; convert them into quantitative sym- 

 bols, susceptible only of the values and 1 ; perform upon them as such 

 all the requisite processes of solution; and finally restore to them their 

 logical interpretation. And this is the mode of procedure which 

 will actually be adopted, though it will be deemed unnecessary 

 to restate in every instance the nature of the transformation em- 

 ployed. The processes to which the symbols x, y, z, regarded 

 as quantitative and of the species above described, are subject, are 

 not limited by those conditions of thought to which they would, 

 if performed upon purely logical symbols, be subject, and a free- 

 dom of operation is given to us in the use of them, without 

 which, the inquiry after a general method in Logic would be a 

 hopeless quest. 



Now the above system of processes would conduct us to no 

 intelligible result, unless the final equations resulting therefrom 

 were in a form which should render their interpretation, after 

 restoring to the symbols their logical significance, possible. 

 There exists, however, a general method of reducing equations 

 to such a form, and the remainder of this chapter will be devoted 

 to its consideration. I shall say little concerning the way in 

 which the method renders interpretation possible, this point 

 being reserved for the next chapter, but shall chiefly confine 

 myself here to the mere process employed, which may be cha- 

 racterized as a process of " development." As introductory to 

 the nature of this process, it may be proper first to make a few 

 observations. 



7 . Suppose that we are considering any class of things with 

 reference to this question, viz., the relation in which its members 

 stand as to the possession or the want of a certain property x. As 

 every individual in the proposed class either possesses or does 

 not possess the property in question, we may divide the class 

 into two portions, the former consisting of those individuals 

 which possess, the latter of those which do not possess, the pro- 

 perty. This possibility of dividing in thought the whole class 

 into two constituent portions, is antecedent to all knowledge of 

 the constitution of the class derived from any other source ; of 



