Page:Mind (New Series) Volume 8.djvu/126

112 CKITICAL NOTICES : also the symbol A B may be used to assert that the concrete and material thing denoted by A belongs to the class of concrete and material things denoted by B. So far there is symbolic coinci- dence. But now take the symbol A BC , which is short for (A B ) C and asserts that the statement A B belongs to the class of statements denoted by C. Here, from the standpoint of Pure Logic, A is a statement, B is a statement, C is a statement, A B is a statement, and A BC is a statement, so that we have homogeneity throughout. But we cannot get this homogeneity in Applied Logic. In Ap- plied Logic A and B may both denote concrete things or classes ; but farther we cannot go. By virtue of our definitions, C cannot denote a concrete thing ; neither can A B nor A BC . These, within the limits of our definitions, must denote statements, and cannot possibly denote anything else. Similarly for A BCD and for state- ments of still higher degrees. Thus, as soon as we pass the limits of jjrimary statements and get into statements of the second- ary, tertiary, etc., degrees, we enter an abstract region of thought (including many important problems in probability) to which none of the Boolian systems can be applied. Any statement, no matter what its degree, can be spoken of as true or false, certain or un- certain, possible or impossible, probable or improbable, within more or less exact limits according to our data ; but none of these epithets can be applied to any portion of space or time, or to any concrete subject whatever; nor can we find any concrete homogeneous substitutes, so far analogous to these abstrac- tions as to be interpretable in the same formulae or symbolic operations. Of course this criticism in no way affects the value of Mr. Whitehead's work, even if that work were restricted to that small part of it on which I may without presumption venture to express an opinion. On the contrary, he has, I think, done even more for exact science than he had contemplated when he embarked upon his arduous undertaking ; he has traced out the line of analogy running through his compared group of algebras so dis- tinctly that it enables one to see not only the very large class of problems to which these algebras can be applied, but also the limits of their application. This is very important, and its im- portance must be accepted as my excuse for drawing attention more to the boundary within which his allied algebras can effec- tively operate than to the undoubted value of their operations within that boundary. Apart from these questions of general principle and of inter- pretation, on which, perhaps, no two logicians could be found in complete agreement, I regard Mr. Whitehead's five chapters on Symbolic Logic as an admirable epitome (with some original improvements) of the results and processes discovered by previous writers ; the whole subject being presented as it appeared to him from his standpoint of comparative algebra. In the matter of notation he adopts Boole's horizontal bar, and in very much the