166 OF SECONDARY PROPOSITIONS. [CHAP. XI. 



By continuing this method of reasoning it may be established, 

 that the laws of combination of the symbols x, y^ z, &c., in the 

 species of interpretation here assigned to them, are identical in 

 expression with the laws of combination of the same symbols, in 

 the interpretation assigned to them in the first part of this 

 treatise. The reason of this final identity is apparent. For in 

 both cases it is the same faculty, or the same combination of fa- 

 culties, of which we study the operations ; operations, the essen- 

 tial character of which is unaffected, whether we suppose them to 

 be engaged upon that universe of things in which all existence 

 is contained, or upon that whole of time in which all events are 

 realized, and to some part, at least, of which all assertions, 

 truths, and propositions, refer. 



Thus, in addition to the laws above stated, we shall have by 

 (4), Chap, ii., the law whose expression is 



x (y + z) = xy + xz ; (2) 



and more particularly the fundamental law of duality (2) Chap, n., 

 whose expression is 



x~ = x, or, x(l -x) = 0; (3) 



a law, which while it serves to distinguish the system of thought 

 in Logic from the system of thought in the science of quantity, 

 gives to the processes of the former a completeness and a gene- 

 rality which they could not otherwise possess. 



8. Again, as this law (3) (as well as the other laws) is satis- 

 fied by the symbols and 1, we are led, as before, to inquire 

 whether those symbols do not admit of interpretation in the pre- 

 sent system of thought. The same course of reasoning which we 

 before pursued shows that they do, and warrants us in the two 

 following positions, viz. : 



1st, That in the expression of secondary propositions, re- 

 presents nothing in reference to the element of time. 



2nd, That in the same system 1 represents the universe, or 

 whole of time, to which the discourse is supposed in any manner 

 to relate. 



As in primary propositions the universe of discourse is some- 

 times limited to a small portion of the actual universe of things, 

 and is sometimes co-extensive with that universe ; so in secon- 



