CHAP. XI.] OF SECONDARY PROPOSITIONS. 169 



duration of the time to which our discourse refers ; and when dif- 

 ferent assertions of the unconditional truth or falsehood of propo- 

 sitions are jointly made as the premises of a logical demonstration, 

 it is to the same universe of time that those assertions are re- 

 ferred, and not to particular and limited parts of it. In that 

 necessary matter which is the object or field of the exact sciences 

 every assertion of a truth may be the assertion of an " eternal 

 truth." In reasoning upon transient phaenomena (as of some 

 social conjuncture) each assertion may be qualified by an imme- 

 diate reference to the present time, " Now." But in both cases, 

 unless there is a distinct expression to the contrary, it is to the 

 same period of duration that each separate proposition relates. 

 The cases which then arise for our consideration are the fol- 

 lowing : 



1st. To express the Proposition, " The proposition X is true." 



We are here required to express that within those limits of 

 time to which the matter of our discourse is confined the propo- 

 sition X is true. Now the time for which the proposition X is 

 true is denoted by x 9 and the extent of time to which our dis- 

 course refers is represented by 1. Hence we have 



=1 (4) 



as the expression required. 



2nd. To express the Proposition, "The proposition X is 

 falser 



We are here to express that within the limits of time to which 

 our discourse relates, the proposition X is false ; or that within 

 those limits there is no portion of time for which it is true. Now 

 the portion of time for which it is true is x. Hence the required 



equation will be 



x = 0. (5) 



This result might also be obtained by equating to the whole du- 

 ration of time 1, the expression for the time during which the 

 proposition X is false, viz., 1 - x. This gives 



!-*=!, 

 whence x = 0. 



3rd. To express the disjunctive Proposition, "Either the pro- 



