'O^ TEE LAWS OF THOUtJHT. 163 



.|i3an"titie8 can receive no values distinct from unity and zero, the 

 analogy between the two sciences will still be preserved. 



It is necessary to observe that unity and zero (1 and 0) are virtually 

 included by Professor Boole among his literal symbols. Of course 

 we can give 1 and any meaning we please, provided the meaning 

 once imposed on them be rigidly adhered to. By 0, then, Professor 

 Soole understands Nothing — a class (if the expression may be per- 

 mitted) in which no object whatever is found. On tbe other hand, 

 by 1 he understands tbe universe of conceivable objects. Thus 1 

 and are at two opposite poles ; the former including every thing in 

 its extension; the latter, nothing. The meaning which has been 

 aflSxed to 1 and preserves, in the Logical system as in Algebra, the 

 equations, 



and,0 X ^ =0; S ••• — •••'•• -K ) 



for, the meaning of the former is, that objects which are common to 

 the universe and to the class x are identical with those which con- 

 stitute the class x ; and the latter means, that there are no objects 

 which are common to a class in which nothing is found and to a 

 class X.' both of which propositions are self-evident. Erom the 

 meaning affixed to 1, we see what the meaning of 1 — a: must be. 

 In fact, X and 1 — x are logical contradictories, the latter denoting 

 all conceivable objects except those which belong to the former ; so 



that 



1 — X — not X. (6) 



This value of the symbol 1 being admitted, we can, by the principles 

 of transposition and distribution (see (3)] reduce equation (4) to 

 the form, 



X (l_a;) = ....(7) 



The law here expressed, which is termed the Law of Duality, plays 

 a most important part in the development of logical functions, and 

 in the elimination of symbols. In fact, it may be described as the 

 germ out of which Professor Boole's whole system is made to unfold 

 itself. 



Having shown how concepts, whether taken universally or parti* 

 cularly, are represented, and also how the contradictory of a concept 

 is represented, we have next to notice the manner of expressing 

 judgments. All judgments are regarded by our author as affirm- 

 ative ; the negation, in those which are commonly called negatives 



