172 OF SECONDARY PROPOSITIONS. [CHAP. XI. 



3rd. If either X is true or Y is true, then either Z and W 

 are both true, or they are both false. 



It is evident that in the above cases the relation of the ante- 

 cedent to the consequent is not affected by the circumstance that 

 one of those terms or both are of a disjunctive character. Ac- 

 cordingly it is only necessary to obtain, in conformity with the 

 principles already established, the proper expressions for the ante- 

 cedent and the consequent, to affect the latter with the indefinite 

 symbol v, and to equate the results. Thus for the propositions 

 above stated we shall have the respective equations, 



1st. #(1 -y) + (1 -x)y = vz. 



2nd. x = v(y(l-z) + z(l-y)}. 



3rd. x (1 - y) + y (1 - x) = v {zw + (1 - z) (1 - w) } . 



The rule here exemplified is of general application. 



Cases in which the disjunctive and the conditional elements 

 enter in a manner different from the above into the expression of 

 a compound proposition, are conceivable, but I am not aware that 

 they are ever presented to us by the natural exigencies of human 

 reason, and I shall therefore refrain from any discussion of them. 

 No serious difficulty will arise from this omission, as the general 

 principles which have formed the basis of the above applications 

 are perfectly general, and a slight effort of thought will adapt 

 them to any imaginable case. 



13. In the laws of expression above stated those of interpre- 

 tation are implicitly involved. The equation 



x= 1 



must be understood to express that the proposition X is true ; 



the equation 



* = 0, 



that the proposition X is false. The equation 



will express that the propositions -X" and Y are both true toge- 

 ther ; and the equation 



that they are not both together true. 



