_Pa. Ma._ Alfred North Whitehead and Bertrand Russell: _Principia Mathematica_, vol. i., Cambridge, 1910. [Other volumes were published in 1912 and 1913.] _P. E._ Bertrand Russell: _Philosophical Essays_, London and New York, 1910. _Ph. L._ Bertrand Russell: _A Critical Exposition of the Philosophy of Leibniz, with an Appendix of Leading Passages_, Cambridge, 1900. _P. M._ Bertrand Russell: _The Principles of Mathematics_, vol. i., Cambridge, 1903. _R. M. M._ _Revue de Métaphysique et de Morale_, Paris. _S. B._ Lewis Carroll: _Sylvie and Bruno_, London, 1889. _S. L._ John Venn: _Symbolic Logic_, London, 1881; 2nd ed., 1894. _S. o. S._ William Stanley Jevons: _The Substitution of Similars, the True Principle of Reasoning derived from a Modification of Aristotle's Dictum_, London, 1869. _T. L. G._ Lewis Carroll: _Through the Looking-Glass, and what Alice found there_, London, 1911. _Z. S._ Gottlob Frege: _Ueber die Zahlen des Herrn H. Schubert_, Jena, 1899. CHAPTER I THE INDEFINABLES OF LOGIC The view that the fundamental principles of logic consist solely of the law of identity was held by Leibniz,[2] Drobisch, Uberweg,[3] and Tweedledee. Tweedledee, it may be remembered,[4] remarked that certain identities "are" logic. Now, there is some doubt as to whether he, like Jevons,[5] understood "are" to mean what mathematicians mean by "=," or, like Schröder[6] and most logicians, to have the same meaning as the relation of subsumption. The first alternative alone would justify our contention; and we may, I think, conclude from an opposition to authority that may have been indicated by Tweedledee's frequent use of the word "contrariwise" that he did not follow the majority of logicians, but held, like Jevons,[7] the mistaken[8] view that the quantification of the predicate is relevant to symbolic logic.