100 OF ELIMINATION. [CHAP. VII. 



and the number of symbols of quantity which it is possible to 

 eliminate from them. But it is otherwise with the system of 

 Logic. No fixed connexion there prevails between the num- 

 ber of equations given representing propositions or premises, 

 and the number of typical symbols of which the elimination 

 can be effected. From a single equation an indefinite num- 

 ber of such symbols may be eliminated. On the other hand, 

 from an indefinite number of equations, a single class symbol 

 only may be eliminated. We may affirm, that in this peculiar 

 system, the problem of elimination is resolvable under all circum- 

 stances alike. This is a consequence of that remarkable law of 

 duality to which the symbols of Logic are subject. To the equa- 

 tions furnished by the premises given, there is added another 

 equation or system of equations drawn from the fundamental 

 laws of thought itself, and supplying the necessary means for the 

 solution of the problem in question. Of the many consequences 

 which flow from the law of duality, this is perhaps the most 

 deserving of attention. 



3. As in Algebra it often happens, that the elimination of 

 symbols from a given system of equations conducts to a mere 

 identity in the form = 0, no independent relations connecting 

 the symbols which remain ; so in the system of Logic, a like re- 

 sult, admitting of a similar interpretation, may present itself. 

 Such a circumstance does not detract from the generality of 

 the principle before stated. The object of the method upon 

 which we are about to enter is to eliminate any number of sym- 

 bols from any number of logical equations, and to exhibit in the 

 result the actual relations which remain. Now it may be, that 

 no such residual relations exist. In such a case the truth of the 

 method is shown by its leading us to a merely identical propo- 

 sition. 



4. The notation adopted in the following Propositions is 

 similar to that of the last chapter. By / (x) is meant any ex- 

 pression involving the logical symbol x, with or without other 

 logical symbols. By /(I) is meant what/(#) becomes when x 

 is therein changed into 1 ; by /(O) what the same function be- 

 comes when x is changed into 0. 



