178 METHODS IN SECONDARY PROPOSITIONS. [CHAP. XII. 



mentary propositions involved in the terms of the given premises. 

 And as in primary propositions it was seen that the system of 

 denials admitted of conversion into various other forms of propo- 

 sitions (VI. 7), &c., such conversion will be found to be possible 

 here also, the sole difference consisting not in the forms of the 

 equations, but in the nature of their interpretation. 



3. Moreover, as in primary propositions, we can find the ex- 

 pression of any element entering into a system of equations, in 

 terms of the remaining elements (VI. 10), or of any selected 

 number of the remaining elements, and interpret that expression 

 into a logical inference, the same object can be accomplished by 

 the same means, difference of interpretation alone excepted, in 

 the system of secondary propositions. The elimination of those 

 elements which we desire to banish from the final solution, the 

 reduction of the system to a single equation, the algebraic solu- 

 tion and the mode of its development into an interpretable form, 

 differ in no respect from the corresponding steps in the discussion 

 of primary propositions. 



To remove, however, any possible difficulty, it may be de- 

 sirable to collect under a general Rule the different cases which 

 present themselves in the treatment of secondary propositions. 



RULE. Express symbolically the given propositions (XI. 11). 



Eliminate separately from each equation in which it is found the 

 indefinite symbol v (VII. 5). 



Eliminate the remaining symbols which it is desired to banish 

 from the final solution : always before elimination reducing to a 

 single equation those equations in which the symbol or symbols to 

 be eliminated are found (VIII. 7). Collect the resulting equa- 

 tions into a single equation V= 0. 



Then proceed according to the particular form in which it is 

 desired to express the final relation, as 



1st. If in the form of a denial, or system of denials, develop the 

 function V, and equate to all those constituents whose coefficients 

 do not vanish. 



2ndly. If in the form of a disjunctive proposition, equate to 1 

 the sum of those constituents whose coffiedents vanish. 



3rdly. If in the form of a conditional proposition having a sim- 



