It would not be fair in me to ask you to publish my reply to your reviewer’s criticisms, though if that reply is justified the criticism must have done the prospects of my book some injury, seeing from what a quarter it comes. But I hope you will see your way to publishing the latter part of this letter in _The Monist_, together with your reply to it, if you think it worthy of such a distinction. I have just come across, in this month’s _Nineteenth Century_, another remarkable instance of reasoning which seems to be rendered entirely nugatory by the want of proper definitions. It is asserted that conceptual thought is impossible without language. At first sight this would certainly appear to be a real assertion. It follows from it that since dogs have no language they have no “conceptual thought.” But it may be plainly shown that dogs do entertain “general notions,” which in ordinary English would be included under the head of “conceptual thought.” The apparent contradiction is however explained when it appears that the author distinguishes general notions as “concepts” or “recepts,” according as they _are_ or _are not named_. This being his definition of “conceptual thought” as opposed to other thought, it appears that the assertion is only a truism after all, and conveys no real information whatever. To discuss it further is then mere waste of time. The author of the assertion doubtless _wished_ it to convey some information, but he did not attend to his definitions and so failed to attain his object. EDWARD T. DIXON. FOOTNOTES: [18] The reviewer of Mr. Dixon’s book has read these remarks on his criticism (_The Monist_, Vol. II, No. 1, p. 126) and has given them what seems to him full consideration. He confesses that he misunderstood what Mr. Dixon meant by “a direction.” (See the article “Logic as Relation-Lore” to be published in a subsequent number.) In regard to the requirements for a logical definition he must still abide by his former opinion. The need of a definition arises either from the inaccuracy in the application of a term or from a supposed lack of knowledge as to its signification. Hence to use the term itself in its own definition is to import into the definition the same vagueness or ignorance which it is the very office of a definition to correct. When Mr. Dixon says that it is requisite for a logical definition that the defining assertions “must not be _independent_ of the meaning of the term defined,” what is that but to say that the same must be _dependent_ upon that meaning? which, unless the reviewer again misunderstands the author, is to say that we must understand the meaning of the term before we can understand the definition. ρσλ. MATHEMATICS A DESCRIPTION OF OPERATIONS WITH PURE FORMS. IN REPLY TO MR. EDWARD DIXON.