CHAP. XI.] OF SECONDARY PROPOSITIONS. 165 



sense, viz. : Let x represent an act of the mind by which we fix 

 our regard upon that portion of time for which the proposition X 

 18 true ; and let this meaning be understood when it is asserted 

 that x denotes the time for which the proposition X is true. Let 

 us further employ the connecting signs +, -, =, &c., in the fol- 

 lowing sense, viz. : Let x + y denote the aggregate of those por- 

 tions of time for which the propositions X and Tare respectively 

 true, those times being entirely separated from each other. Si- 

 milarly let x - y denote that remainder of time which is left when 

 we take away from the portion of time for which X is true, that 

 (by supposition) included portion for which Fis true. Also, let 

 x = y denote that the time for which the proposition X is true, 

 is identical with the time for which the proposition Yis true. 

 We shall term x the representative symbol of the proposition X, &c. 

 From the above definitions it will follow, that we shall 

 always have 



x + y = y + x -> 



for either member will denote the same aggregate of time. 



Let us further represent by xy the performance in succession 

 of the two operations represented by y and x, i. e. the whole 

 mental operation which consists of the following elements, viz., 

 1st, The mental selection of that portion of time for which the 

 proposition Fis true. 2ndly, The mental selection, out of that 

 portion of time, of such portion as it contains of the time in 

 which the proposition X is true, the result of these successive 

 processes being the fixing of the mental regard upon the whole 

 of that portion of time for which the propositions X and Y are 

 both true. 



From this definition it will follow, that we shall always have 



xy = yx. (1) 



For whether we select mentally, first that portion of time for 

 which the proposition Y is true, then out of the result that con- 

 tained portion for which X is true ; or first, that portion of time 

 for which the proposition X is true, then out of the result that 

 contained portion of it for which the proposition Y is true ; we 

 shall arrive at the same final result, viz., that portion of time for 

 which the propositions X and Y are both true. 



