REVIEWS 



LOGIC 



Collected Logical Works. Vol. II. The Laws of Thought. By George Boole. 

 [Pp. xvi + 448.] (London : The Open Court Publishing Company, 1916. 

 Price 15s. net.) 

 This volume is a reprint of Boole s Laws of Thought, originally published in 1854. 

 It is an extension to logical reasoning of the symbolic processes employed in 

 mathematics. 



The earlier chapters deal with Primary propositions which express relations 

 between things, as distinguished from Secondary propositions which express a 

 relation between propositions. In primary propositions the terms are symbolised 

 by letters. Thus if x denote the class " white things " and y the class " sheep," 

 xy is said to denote "white sheep." Since "white sheep" and "sheep that are 

 white " mean the same thing, xy = yx. Hence the logical like the algebraical 

 symbols are commutative. The signs + and - being taken to denote "along 

 with " and " except " respectively, it is easily established that a(x +y) = ax + ay, 

 and that a term may be removed from one side of an equation to the other by 

 changing its sign. Also each side of an equation may be multiplied by the same 

 symbol. If all metals are elements, then obviously all white metals are white 

 elements. Although, however, certain members of one class which have a par- 

 ticular quality may be identical with certain members of another class which have 

 the same quality, we cannot infer that the whole of the former class is identical 

 with the whole of the latter. There is, therefore, in logic no process analogous to 

 the algebraic division of both sides of an equation by the same symbol. 



A contrast still more striking becomes apparent when we try to find if there is 

 anything in logic corresponding to the algebraic x 2 . If to the quality " white " 

 we apply the same quality we get " white white," which expresses no more than 

 " white." Hence in logic xx or x 2 = x. This is a fundamental law, and is called 

 the Law of Duality. Now we know from algebra that the only quantities which 

 satisfy the equation x 2 = x are o and 1. It follows that the logical symbols cannot 

 be regarded as quantities other than o and 1. In fact, throughout the treatment 

 of the subject we find that no numbers occur except o and 1, and it is shown that, 

 while o has its usual signification, 1 must represent the universe, or rather that 

 part of the universe which is under discussion. If then x denote " men " 1 - x 

 will denote all that are " not-men," and since " men " who are also " not-men " do 

 not exist, it follows that x(i - x) = o. Hence the Law of Duality is really the 

 symbolic' expression of the Law of Thought that a thing cannot both be and 



not be. 



We are now in a position to express all the divisions of things with reference to 

 their possessing or not possessing certain qualities. Let x = white, y = sheep, 



then 



xy = white sheep, 

 x(i - v) = all white things which are not sheep, 

 (1 - x)y — all sheep that are not white, 

 (1 - x) (1 - y) = all things that are neither white nor sheep. 

 If we had these expressions together we find that they amount to 1 or the universe. 



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