CHAP. VI.] OF INTERPRETATION. 87 



But this equation is not at present in an interpretable form. If 

 we can reduce it to such a form it will furnish the relation 

 required. 



On developing the second member of the above equation, we 

 have 



* - *y + i * ( 1 - y) + 0(1- *) y + J (1 - *) ( 1 - y ), 



and it will be shown hereafter (Prop. 3) that this admits of the 

 following interpretation : 



" Beasts which chew the cud consist of all clean beasts 

 (which also divide the hoof), together with an indefinite re- 

 mainder (some, none, or all) of unclean beasts which do not di- 

 vide the hoof." 



9. Now the above is a particular example of a problem of the 

 utmost generality in Logic, and which may thus be stated : 

 " Given any logical equation connecting the symbols a?, y, z 9 w, 

 required an interpretable expression for the relation of the [class 

 represented by w to the classes represented by the other symbols 

 x, y, z, &c." 



The solution of this problem consists in all cases in deter- 

 mining, from the equation given, the expression of the above 

 symbol w, in terms of the other symbols, and rendering that ex- 

 pression interpretable by development. Now the equation given 

 is always of the first degree with respect to each of the symbols 

 involved. The required expression for w can therefore always 

 be found. In fact, if we develop the given equation, whatever 

 its form may be with respect to iv, we obtain an equation of the 

 form 



w) = Q, (1) 



E and E being functions of the remaining symbols. From the 

 above we have 



E=(E-E)w. 

 Therefore 



and expanding the second member by the rule of development, it- 

 will only remain to interpret the result in logic by the next 

 proposition. 



