CHAP. VI.] OF INTERPRETATION. 83 



bols, x and ?/, and let us represent the development of the given 

 equation by 



axy + bx (1 - y) + c (1 - *) y + d(l - a) (I - y) = ; (1) 

 a, b 9 c, and d being definite numerical constants. 



Now, suppose that any coefficient, as a, does not vanish. 

 Then multiplying each side of the equation by the constituent #y, 

 to which that coefficient is attached, we have 



axy = 0, 

 whence, as a does not vanish, 



ajf-'O, 



and this result is quite independent of the nature of the other co- 

 efficients of the expansion. Its interpretation, on assigning to 

 x and y their logical significance, is " No individuals belonging at 

 once to the class represented by #, and the class represented by y, 

 exist." 



But if the coefficient a does vanish, the term axy does not 

 appear in the development (1), and, therefore, the equation xy = 

 cannot thence be deduced. 



In like manner, if the coefficient b does not vanish, we have 



which admits of the interpretation, "There are no individuals 

 which at the same time belong to the class #, and do not belong 

 to the class y." 



Either of the above interpretations may, however, as will sub- 

 sequently be shown, be exhibited in a different form. 



The sum of the distinct interpretations thus obtained from 

 the several terms of the expansion whose coefficients do not 

 vanish, will constitute the complete interpretation of the equation 

 V = 0. The analysis is essentially independent of the number 

 of logical symbols involved in the function V, and the object of 

 the proposition will, therefore, in all instances, be attained by the 

 following Rule: 



RULE. Develop the function V, and equate to every consti- 

 tuent whose coefficient does not vanish. The interpretation of these 

 results collectively will constitute the interpretation of the aiven 

 equation. 



G 2 



