88 OF INTERPRETATION. [CHAP. VI. 



V 

 If the fraction = = has common factors in its numerator 



JLJ & 



and denominator, we are not permitted to reject them, unless they 

 are mere numerical constants. For the symbols #, ?/, &c., re- 

 garded as quantitative, may admit of such values and 1 as to 

 cause the common factors to become equal to 0, in which case 

 the algebraic rule of reduction fails. This is the case contem- 

 plated in our remarks on the failure of the algebraic axiom of 

 division (II. 14). To express the solution in the form (2), and 

 without attempting to perform any unauthorized reductions, to 

 interpret the result by the theorem of development, is a course 

 strictly in accordance with the general principles of this treatise. 

 If the relation of the class expressed by 1 - w to the other 

 classes, x, y, &c. is required, we deduce from (1), in like manner 

 as above, 



to the interpretation of which also the method of the following 

 Proposition is applicable : 



PROPOSITION III. 



10. To determine the interpretation of any logical equation of 

 the form w=V, in which w is a class symbol, and V a function of 

 other class symbols quite unlimited in its form. 



Let the second member of the above equation be fully ex- 

 panded. Each coefficient of the result will belong to some one 

 of the four classes, which, with their respective interpretations, 

 we proceed to discuss. 



1st. Let the coefficient be 1. As this is the symbol of the 

 universe, and as the product of any two class symbols represents 

 those individuals which are found in both classes, any constituent 

 which has unity for its coefficient must be interpreted without 

 limitation, i. e. the whole of the class which it represents is 

 implied. 



2nd. Let the coefficient be 0. As in Logic, equally with 

 Arithmetic, this is the symbol of Nothing, no part of the class 



