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ALFRED N. WHITEHEAD, Universal Algebra with Applications. 109 of any kind but complete statements. By a ' statement ' I mean any sound or symbol, or any combination of sounds or symbols, employed to convey information. A subjectless and predicateless. sound or symbol, like the warning " caw " of a sentinel rook, or the national flag of a passing ship, may be called an elementary statement ; the formal grammatical propositions of ordinary spoken or written speech may be called complex statements. The ulti- mate units of expressed thinking, whether those units be indi- vidually communicated to ear or eye by single symbols or by many, are statements ; and in no sphere or region of investigation can reasoning be expressed without those units. Since, therefore,, statements, and statements alone, constitute the ever indispens- able elements of all expressed reasoning, we should, in my opinion, first investigate the mutual relations of these statements, repre- senting each by its own independent symbol, and call this process of investigation Pure Logic. The moment we begin (as in mathematics and in the traditional logic) to represent things things which are not statements by separate symbols, we are no- longer in the domain of Pure (or Abstract) Logic, but in that of Applied Logic. A system of Symbolic Logic thus built up wholly of statements has one great advantage which no other system can possibly possess, namely, the advantage of homogeneity of matter. Mr. Whitehead, who was acquainted with my earlier, but not with my recent, papers when he wrote, admits (p. 112) that the existing systems of logic the traditional as well as the modern symbolic can be thus entirely constructed on a basis of pure statements ; but he has preferred to follow the method of Boole as simplified by Venn, Schroder, Peirce and others. In this he was quite right. For one thing, the principle which underlies my method v however important, did not come within the scope and purpose of his work ; and, for another, it did not appear (in my earlier papers) to lead to any essential difference in the symbolic pro- cesses. That this is no longer the case my recent papers in MIND and in the Proceedings of the Mathematical Society will show; but the new development is still further removed than the old from the allied algebras which it has been the great aim of Mr. Whitehead to unite into one general comprehensive system. This task, judging of the whole from my knowledge of a part, I con- sider him to have accomplished with rare ability. His opening chapter, "On the Nature of a Calculus," is very interesting, and may be understood by any one of ordinary educa- tion and intelligence. If the reader knows something of common algebra he will grasp the author's meaning more easily ; but, for much of this chapter, even this modicum of preliminary knowledge is not absolutely indispensable. When, however, we enter upon the second chapter, which treats of Manifolds, we find ourselves on very different territory. A reader previously unacquainted with the subject cannot read this straight through, as he would a novel or a paragraph in a newspaper; he wUl have to make