§1.  Three Kinds Of Signs (concl.)

In this paper, I purpose to develop an algebra adequate to the treatment of all problems of deductive logic, showing as I proceed what kinds of signs have necessarily to be employed at each stage of the development.  I shall thus attain three objects.  The first is the extension of the power of logical algebra over the whole of its proper realm.  The second is the illustration of principles which underlie all algebraic notation.  The third is the enumeration of the essentially different kinds of necessary inference;  for when the notation which suffices for exhibiting one inference is found inadequate for explaining another, it is clear that the latter involves an inferential element not present to the former.  Accordingly, the procedure contemplated should result in a list of categories of reasoning, the interest of which is not dependent upon the algebraic way of considering the subject.

I shall not be able to perfect the algebra sufficiently to give facile methods of reaching logical conclusions:  I can only give a method by which any legitimate conclusion may be reached and any fallacious one avoided.  But I cannot doubt that others, if they will take up the subject, will succeed in giving the notation a form in which it will be highly useful in mathematical work.  I even hope that what I have done may prove a first step toward the resolution of one of the main problems of logic, that of producing a method for the discovery of methods in mathematics.  (3.364).