114 OF REDUCTION. [CHAP. VIII. 



CHAPTER VIII. 



ON THE REDUCTION OF SYSTEMS OF PROPOSITIONS. 



1. TN the preceding chapters we have determined sufficiently 

 -*- for the most essential purposes the theory of single pri- 

 mary propositions, or, to speak more accurately, of primary pro- 

 positions expressed by a single equation. And we have estab- 

 lished upon that theory an adequate method. We have shown 

 how any element involved in the given system of equations may 

 be eliminated, and the relation which connects the remaining 

 elements deduced in any proposed form, whether of denial, of af- 

 firmation, or of the more usual relation of subject and predicate. 

 It remains that we proceed to the consideration of systems of 

 propositions, and institute with respect to them a similar series 

 of investigations. We are to inquire whether it is possible from 

 the equations by which a system of propositions is expressed to 

 eliminate, ad libitum, any number of the symbols involved ; to 

 deduce by interpretation of the result the whole of the relations 

 implied among the remaining symbols ; and to determine in par- 

 ticular the expression of any single element, or of any inter- 

 pretable combination of elements, in terms of the other elements, 

 so as to present the conclusion in any admissible form that may 

 be required. These questions will be answered by showing that it 

 is possible to reduce any system of equations, or any of the equa- 

 tions involved in a system, to an equivalent single equation, to 

 which the methods of the previous chapters may be immediately 

 applied. It will be seen also, that in this reduction is involved 

 an important extension of the theory of single propositions, which 

 in the previous discussion of the subject we were compelled to 

 forego. This circumstance is not peculiar in its nature. There 

 are many special departments of science which cannot be com- 

 pletely surveyed from within, but require to be studied also from 

 an external point of view, and to be regarded in connexion with 



