The Relation of the Principles of Logic to the Foundations of Geomety
by Josiah Royce
ISBN 0217499031 (0-217-49903-1)
Find This Book
Softcover, General Books LLC, 2012
› Find signed collectible books: 'The Relation of the Principles of Logic to the Foundations of Geomety'
Book summary: This historic book may have numerous typos, missing text, images, or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1905. Not illustrated. Excerpt: ... is made centrally important. The algebra of logic, so far as I know, has not hitherto been brought into definite relations with the problem of the continuum. This is one of the things that I here accomplish. .This undertaking involves proving all the principles of logic so as to make them applicable to infinite sets of entities at once. This also I have here done. Kempe's "linear triad" of elements is represented, in any logical system of classes, by the classes, or areas a, b, and c, which stand in the relation which is represented in the adjoining diagram by the closed figures so lettered. Any area c which includes the common part of a and 6, and which is included within their logical sum, is, in Kempe's phrase, such that c is between a and 6." I hereafter symbolize this relation, in my own way, as F(cab). The relation in question is called by me the i^-relation, because it is that characteristic of Kempe's "flat collections." The i^-relation, so long as "obverses " or "negatives" exist, follows immediately from, and is equivalent to, an O-relation. For, in the diagram if s is the total surface in which a, b, and c are included, then when "c is between a and b," " a, b, and c (c being the obverse of c) constitute an O-collection," or " are in the O-relation." The outcome of our discussion will show that, while logical relations can be indifferently stated as O-relations, or stated as i^-relations, or (when once addition, multiplication, and negation have been defined) can be stated in terms of equivalence, the i^-relations are the only natural means of expressing the geometrical ordinal relations. This difference, however, between the logical and the geometrical entities, is due to the simple fact that (as Kempe points out), when geometrical sets are ...
More editions of The Relation of the Principles of Logic to the Foundations of Geomety: