CHAP. V.] PRINCIPLES OF SYMBOLICAL REASONING. 71 



which knowledge the effect can only be to inform us, more or 

 less precisely, to what further conditions the portions of the class 

 which possess and which do not possess the given property are 

 subject. Suppose, then, such knowledge is to the following effect, 

 viz., that the members of that portion which possess the property 

 x, possess also a certain property w, and that these conditions 

 united are a sufficient definition of them. We may then repre- 

 sent that portion of the original class by the expression ux (II. 6). 

 If, further, we obtain information that the members of the ori- 

 ginal class which do not possess the property #, are subject to a 

 condition v, and are thus defined, it is clear, that those members 

 will be represented by the expression v (1 -x). Hence the class 

 in its totality will be represented by 



ux + v (1 - x)', 



which may be considered as a general developed form for the 

 expression of any class of objects considered with reference to 

 the possession or the want of a given property x. 



The general form thus established upon purely logical 

 grounds may also be deduced from distinct considerations of 

 formal law, applicable to the symbols #, y, 3, equally in their 

 logical and in their quantitative interpretation already referred to 

 (V.6). 



8. Definition. Any algebraic expression involving a sym- 

 bol x is termed a function of #, and may be represented under 

 the abbreviated general form f(x}. Any expression involving 

 two symbols, x and y, is similarly termed a function of x and y, 

 and may be represented under the general form f(x, y\ and so 

 on for any other case. 



Thus the form / (x) would indifferently represent any of the 



following functions, viz., #, 1 -as, , &c. ; and/ (x,y) would 



M I ftl 



equally represent any of the forms x + ?/, x - 2y, - J~, &c. 



x - 2y 



On the same principles of notation, if in any function f(x\ 

 we change x into 1, the result will be expressed by the form 

 /(I) ; if in the same function we change x into 0, the result will 

 be expressed by the form /(O). Thus, if f (x) represent the 



