170 OF SECONDARY PROPOSITIONS. [CHAP. XI. 



position X is true or the proposition Y is true ;" it being thereby 

 implied that the said propositions are mutually exclusive, that is to 

 say, that one only of them is true. 



The time for which either the proposition X is true or the 

 proposition Y is true, but not both, is represented by the ex- 

 pression x(\ -y) + y (1 - x). Hence we have 



x(l-y) + u(\-x) = l, (6) 



for the equation required. 



If in the above Proposition the particles either, or, are sup- 

 posed not to possess an absolutely disjunctive power, so that the 

 possibility of the simultaneous truth of the propositions X and Y 

 is not excluded, we must add to the first member of the above 

 equations the term xy. We shall thus have 



xy + x(l-y) + (\-x)y= 1, 

 or x + (1 - x)y = 1. 



4th. To express the conditional Proposition, " If the propo- 

 sition Y is true, the proposition X is true." 



Since whenever the proposition Y is true, the proposition X 

 is true, it is necessary and sufficient here to express, that the time 

 in which the proposition Y is true is time in which the propo- 

 sition X is true ; that is to say, that it is some indefinite portion 

 of the whole time in which the proposition X is true. Now the 

 time in which the proposition Y is true is y, and the whole time 

 in which the proposition X is true is x. Let v be a symbol of 

 time indefinite, then will vx represent an indefinite portion of the 

 whole time x. Accordingly, we shall have 



y = vx 

 as the expression of the proposition given. 



12. When v is thus regarded as a symbol of time indefinite, 

 vx may be understood to represent the whole, or an indefinite 

 part, or no part, of the whole time x ; for any one of these mean- 

 ings may be realized by a particular determination of the arbitrary 

 symbol v. Thus, if v be determined to represent a time in which 

 the whole time x is included, vx will represent the whole time x. 

 If v be determined to represent a time, some part of which is in- 



