168 OF SECONDARY PROPOSITIONS. [CHAP. XI 



As 1 denotes the whole duration of time, and x that portion 

 of it for which the proposition X is true, 1 - x will denote that 

 portion of time for which the proposition X is false. 



Again, as xy denotes that portion of time for which the pro- 

 positions X and Y are both true, we shall, by combining this and 

 the previous observation, be led to the following interpretations, 

 viz. : 



The expression x (1 - y) will represent the time during which 

 the proposition X is true, and the proposition Y false. The ex- 

 pression (1 - x) (1 -y} will represent the time during which the 

 propositions X and Y are simultaneously false. 



The expression x(l - y) +y(l - x) will express the time 

 during which either X is true or Y true, but not both ; for that 

 time is the sum of the times in which they are singly and exclu- 

 sively true. The expression xy + (1 - x) (1 - y) will express the 

 time during which X and Y are either both true or both false. 



If another symbol z presents itself, the same principles remain 

 applicable. Thus xyz denotes the time in which the propositions 

 X, Y, and Z are simultaneously true ; (1 - x) (1 -y) (1 - z) the 

 time in which they are simultaneously false; and the sum of 

 these expressions would denote the time in which they are either 

 true or false together. 



The general principles of interpretation involved in the above 

 examples do not need any further illustrations or more explicit 

 statement. 



1 1 . The laws of the expression of propositions may now be 

 exhibited and studied in the distinct cases in which they present 

 themselves. There is, however, one principle of fundamental 

 importance to which I wish in the first place to direct attention. 

 Although the principles of expression which have been laid down 

 are perfectly general, and enable us to limit our assertions of the 

 truth or falsehood of propositions to any particular portions of 

 that whole of time (whether it be an unlimited eternity, or a pe- 

 riod whose beginning and whose end are definitely fixed, or the 

 passing moment) which constitutes the universe of our discourse, 

 yet, in the actual procedure of human reasoning, such limitation 

 is not commonly employed. When we assert that a proposition 

 is true, we generally mean that it is true throughout the whole 



